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# Forward-Looking Sonar Patch Matching: Modern CNNs, Ensembling, and
Uncertainty
Arka Mallick1 and Paul Plöger1 and Matias Valdenegro-Toro2 1Department of
Computer Science, Hochschule Bonn-Rhein-Sieg, 53757 Sankt Augustin, Germany
<EMAIL_ADDRESS>Research Center for Artificial Intelligence,
28359 Bremen, Germany<EMAIL_ADDRESS>
###### Abstract
Application of underwater robots are on the rise, most of them are dependent
on sonar for underwater vision, but the lack of strong perception capabilities
limits them in this task. An important issue in sonar perception is matching
image patches, which can enable other techniques like localization, change
detection, and mapping. There is a rich literature for this problem in color
images, but for acoustic images, it is lacking, due to the physics that
produce these images. In this paper we improve on our previous results for
this problem (Valdenegro-Toro et al, 2017), instead of modeling features
manually, a Convolutional Neural Network (CNN) learns a similarity function
and predicts if two input sonar images are similar or not. With the objective
of improving the sonar image matching problem further, three state of the art
CNN architectures are evaluated on the Marine Debris dataset, namely DenseNet,
and VGG, with a siamese or two-channel architecture, and contrastive loss. To
ensure a fair evaluation of each network, thorough hyper-parameter
optimization is executed. We find that the best performing models are DenseNet
Two-Channel network with 0.955 AUC, VGG-Siamese with contrastive loss at 0.949
AUC and DenseNet Siamese with 0.921 AUC. By ensembling the top performing
DenseNet two-channel and DenseNet-Siamese models overall highest prediction
accuracy obtained is 0.978 AUC, showing a large improvement over the 0.91 AUC
in the state of the art.
## I Introduction
More than two-thirds of our planet’s surface is covered by oceans and other
water bodies. For a human, it is often impossible to explore it extensively.
The need for venturing into potentially dangerous underwater scenarios appear
regularly, for example, finding new energy sources, monitoring tsunamis,
global warming, wreckage search, or maybe just to learn about deep sea
ecosystems. This motivates design and deployment of robots in underwater
scenarios, and much research goes in this direction. Some exploration or
monitoring tasks require the robot to ”see” underwater, to make intelligent
decisions. But the underwater environment is very difficult for optical
cameras, as light is attenuated and absorbed by the water particles. And a lot
of real-life monitoring and mapping tasks take place in a cluttered and turbid
underwater scenario. The limited visibility range of an optical sensor is a
big challenge. Hence, sonar is a more practical choice for underwater sensing,
as acoustic waves can travel long distances with comparatively little
attenuation.
An underwater robot, equipped with sonar image sensors, regularly needs to
perform basic tasks such as object detection and recognition, navigation,
manipulation etc. In underwater scenarios, sonar patch matching functionality
is very useful in several applications such as data association in
simultaneous localization and mapping (SLAM), object tracking, sonar image
mosaicing [1] etc. Patch matching, in general, is heavily used in computer
vision and image processing applications for low-level tasks like image
stitching [2], deriving structure from motion [3], also in high-level tasks
such as object instance recognition [4], object classification [5], multi-view
reconstruction [6], image-retrieval etc.
Typical challenges in patch matching tasks are different viewing points,
variations in scene insonification, occlusion, and different sensor settings.
For sonar patch matching the common challenges with acoustic vision adds to
the overall complexity. For example, low signal-to-noise ratio, lower
resolution, unwanted reflections, less visibility etc. Because of these
challenges, the underlying object features might not be so prominent as in a
normal optical image. It has also been found that it is very challenging to
manually design features for sonar images, and popular hand designed features
such as SIFT [7] are not always very effective in sonar images [8]. For these
reasons, patch matching for sonar images remains a topic of research interest.
Figure 1: Use of convolutional network for learning general similarity
function for image patches. The patches in the image are samples taken from
the data used in this work. Inspired from Zagoruyko et al. [9]
## II State of the Art
Sonar image patch matching is more difficult than normal optical matching
problem. This is because sonar images have additional challenges such as non-
uniform insonification, low signal-to-noise ratio, poor contrast [10], low
resolution, low feature repeatability [11] etc. But sonar image matching has
important applications like in sonar registration, mosaicing [12], [1] and
mapping of seabed surface [13] etc. While Kim et al. [12] used Harris corner
detection and matched key-points to register sonar images, Hurtos et al. [1]
incorporated Fourier-based features for registration of FLS images.
Negahdaripour et al. [13] estimated mathematical models from the dynamics of
object movements and it’s shadows. Vandrish et al. [14] used SIFT [7] for
sidescan sonar image registration. Even though these approaches achieve
considerable success in respective goals, were found to be most effective when
the rotation/translation between the frames of sonar images are comparatively
smaller. Block-matching was performed on segmented sonar images by Pham et al.
[15], using Self-Organizing Map for the registration and mosaicing task.
Recently CNNs have been applied for this problem, Zbontar et al[16] for stereo
matching in color images, and Valdenegro-Toro et al [8] for sonar images,
which is based on Zagoryuko et al [9], and is the state of the art for sonar
image patch matching at 0.91 AUC on the Marine Debris dataset. CNNs are
increasingly being used for sonar image processing [17]. The main reason
behind such a rise of CNN usage is that it can learn sonar-specific
information from the data directly. No complex manual feature design or
rigorous data pre-processing steps are needed, which makes the task less
complex and good prediction accuracy can be achieved.
## III Matching as Binary Classification
We formulate the matching problem as learning a classifier. A classification
model is given two images, and it decides if the images match or not. This
decision can be modeled as a score in $y\in[0,1]$, or a binary output decision
$y=0,1$.
For this formulation, we use AUC, the area under the ROC curve (Receiver
Operating Characteristic) as the primary metric to assess performance, as we
are interested in how separable are the score distributions between matches
and non-matches.
TABLE I: Best hyper-parameter values for DenseNet Two-Channel (DTC).
Name | Value | Name | Value
---|---|---|---
Layers | 2-2-2 | Pooling | avg
Growth rate (gr) | 12 | Number of filter | 32
DenseNet dropout | 0.2 | Compression | 0.5
Bottleneck | False | Batch size | 128
Optimizer | Adadelta | Learning rate | 0.03
## IV Matching Architectures
In this section we describe the neural network architectures we selected as
trunk for the meta-architectures like two-channel and siamese networks, which
are used for matching.
### IV-A Hyper-Parameter Tuning
For each architecture, we tuned their hyper-parameters using a validation set,
in order to maximize accuracy. Each range of hyper-parameters was set
individually for each architecture, considering width, filter values at each
layer, drop probabilities, dense layer widths, etc. Overall, we performed 10
runs of different hyper-parameter combinations for each architecture. Details
of the hyper-parameter tuning are available at [18].
### IV-B DenseNet Two-Channel Network
In DenseNet [19] each layer connects to every layer in a feed-forward fashion.
With the basic idea to enhance the feature propagation, each layer of DenseNet
blocks takes the feature-maps of the previous stages as input.
TABLE II: Best hyperparameter values for DenseNet Siamese (DS).
Name | Value | Name | Value
---|---|---|---
Number of filter | 16 | Layers | 2-2
Growth rate | 30 | DenseNet dropout | 0.4
Compression | 0.3 | Bottleneck | False
FC output | 512 | FC dropout | 0.7
Pooling | flatten | Batch size | 64
Optimizer | Adadelta | Learning rate | 0.07
In DenseNet two channel the the sonar patches are supplied as inputs in two
channels format, the network by itself divides each patch into one channel and
learn the features from the patches and then finally compare them using the
Sigmoid activation function at the end with FC layer of single output.
Hyper-parameters for this architecture are shown in Table I.
### IV-C DenseNet Siamese Network
In this architecture the branches of the Siamese network are DenseNet.
Following the classic Siamese model each branch of the Siamese network shares
weights between them and gets trained simultaneously on two input patches and
then learns the features from the inputs. Through the shared neurons the
Siamese network is able to learn the similarity function and be able to
discriminate between the two input patches. The role of the DenseNet branches
are feature extraction, the decision making or prediction part is taken care
of by the Siamese network.
Figure 2: DenseNet Siamese architecture.
In Figure 2 the basic architecture is displayed for the DenseNet-Siamese
network. The two DenseNet branches are designed to share weights between them.
The extracted features are concatenated and connected through a FC layer,
followed by ReLU activation and where applicable Batch Normalization and
Dropout layers. The output is then connected to another FC layer with single
output, for binary prediction score of matching (1) or non-matching (0).
Sigmoid activation function and binary cross entropy loss function is used for
this final FC layer. As mentioned in Figure 2 the size of the output of the FC
layer and value of dropout probability etc. hyper-parameters are shown in
Table II.
### IV-D Contrastive Loss
Using Contrastive loss [20] higher dimensional input data (e.g. a pair of
images) can be mapped in a much lower dimensional output manifold, where
similar pairs are placed closer to each other and the dissimilar pairs have
larger distances between them depending on their dissimilarity. Using this
loss function the distance between two input patches projected in the output
manifold can be predicted and if the distance is closer to 0 then the input
pairs are matching, otherwise its dissimilar (above threshold). The formulas
for this loss are shown in Equations 1 and 2.
$D_{W}(\vec{X}_{1},\vec{X}_{2})=\left\lVert
G_{W}(\vec{X}_{1})-G_{W}(\vec{X}_{2})\right\rVert$ (1)
$L=(1-Y)\frac{1}{2}(D_{W})^{2}+\frac{Y}{2}\\{\max(0,m-D_{W})\\}^{2}$ (2)
Here L is the loss term, the formula presented here is the most generalized
form of the loss function, suitable for batch training. $\vec{X_{1}}$,
$\vec{X_{2}}$ represents a pair of input image vectors. Y are the labels, 0
for similar pair and 1 for dissimilar pair. $D_{w}$ is the parameterized
distance function to be learned by the neural network. $m>0$ is the margin
that defines a radius around $G_{w}$. The dissimilar pairs only contribute to
the loss function if their distance is within the radius. We use $m=1$ for our
experiments. One of the ideas for evaluating this loss function is to use it
with a Siamese network, as the loss function takes a pair of images as input,
indicating their similarity, matching pairs having closer distances in the
learned embedding than non-matching ones, and the distance between pairs can
be used as a score with a threshold.
### IV-E VGG Siamese Network
The VGG network [21] is a CNN which was conceptualized by K. Simonyan and A.
Zisserman from the University of Oxford (Visual Geometry Group). This network
performed very well in ImageNet challenge 2014. The architecture/s has very
small 3x3 Conv filters and depth varying from 16 to 19 weight layers. This
network generalizes very well with different kinds of data. VGG network has
been chosen as the branches of the Siamese (Figure 3) network It’s role is to
extract features, similar to the DenseNet-Siamese, the final decision making
and prediction is done by the Siamese network. The network is trained with
Contrastive loss. The output of this network is euclidean distance between the
two input sonar patches, projected into lower dimension using Contrastive
loss. The hyper-parameters of this network are shown in Table III.
Figure 3: VGG Siamese network with contrastive loss.
Since contrastive loss returns projected distance, close to zero means
similarity and higher values means dissimilarity. Although, in our original
data and matching formulation, labels close to one represents similarity
between patches. Hence the labels for train, validation and test data here are
all flipped:
$Y_{\text{new}}=1-Y_{\text{old}}$ (3)
Equation 3 is applied to all ground truth labels, meaning that for this
evaluation input label zero means similarity (match) between patches.
TABLE III: Best hyper-parameter values for VGG Siamese network with Contrastive loss (CL). Name | Value | Name | Value
---|---|---|---
Conv filters | 16 | Kernel size | 3
FC Layers | 1 | FC output | 2048
Batch normalization | False | Dropout | 0.6
Batch size | 256 | Optimizer | Nadam
Conv Initializer | random normal | FC Initializer | glorot normal
Learning rates | 0.0002 | |
## V Experimental Evaluation
### V-A Dataset
We use the Marine Debris dataset, matching task, 111Available at
https://github.com/mvaldenegro/marine-debris-fls-datasets/releases/ to
evaluate our models. This dataset contains 47K labeled sonar image patch
pairs, captured using a ARIS Explorer 3000 Forward-Looking sonar, generated
from the original 2627 labeled object instances. We exclusively use the D
dataset, on which the training and testing sets were generated using different
sets of objects, with the purpose of testing a truly generic image matching
algorithm that is not object specific. The training set contains 39840 patch
pairs, while the test set contains 7440 patch pairs.
### V-B Comparative Analysis of AUC
Our main results are presented in Table IV and Figure 4, where we present the
AUC and the ROC curves on the test set, correspondingly.
DenseNet two-channel has highest mean AUC (10 trials) of $0.955\pm 0.009$ with
max AUC of 0.966. With total parameters of only 51,430. DenseNet-Siamese has
highest mean AUC (10 trials) of $0.921\pm 0.016$, Max AUC 0.95 with total
parameters of 16,725,485. VGG-Siamese network with Contrastive loss have mean
AUC (10 trials) of $0.949\pm 0.005$ and highest AUC value in a single run as
0.956. With total number of parameters of 3,281,840. These AUC values are
considerably better than Valdenegro-Toro [8], with improvements from $0.910$
to $0.966$ (almost 5 AUC points).
It is notable that our best performing model is a two-channel network,
indicating that this meta-architecture is better suited for the matching
problem than a siamese one, and that there is a considerable reduction in the
number of parameters, from $1.8$M to $51$K, which hints at increased
generalization.
A comparison of predictions between all our three architectures is provided in
Figure 5.
TABLE IV: Comparative analysis on the AUC and total number of parameters in the best performing networks. Network | AUC | Best AUC | # of Params
---|---|---|---
Two-Channel DenseNet | $0.955\pm 0.009$ | 0.966 | 51K
Siamese DenseNet | $0.921\pm 0.016$ | 0.95 | 16.7M
Siamese VGG | $0.949\pm 0.005$ | 0.956 | 3.3M
Two-Channel CNN [8] | 0.910 | 0.910 | 1.8M
Siamese CNN [8] | 0.855 | 0.855 | 1.8M
Figure 4: Comparison of ROC curves for best hyper-parameter architecture
configurations and top AUC. Figure 5: Comparison of predictions across
multiple models, DenseNet Siamese (DS), DenseNet Two-Channel (DTC), and VGG
Siamese Contrastive Loss (CL). Note that Siamese VGG produce distances which
are not in the range $[0,1]$, while the other architectures give scores in the
$[0,1]$ range.
### V-C Monte Carlo Dropout Analysis
Normally Dropout is only applied in the training phase, where it provides
regularization to avoid overfitting. In test time all the connections/nodes
remain present and dropout is not applied, though the weights are adjusted
according to the dropout ratio during training. So every time a prediction on
test data is obtained, they are deterministic. For Monte Carlo dropout the
dropout is also applied in the inference/test time, which introduces
randomness, as the connections are dropped randomly according to the dropout
probability. This prediction process is stochastic i.e the model could predict
different predictions for same test data. The main goal of Monte Carlo Dropout
[22] is to generate samples of the predictive posterior distribution of an
equivalent Bayesian Neural Wetwork, which quantifies epistemic uncertainty.
We would like to evaluate uncertainty for our best performing model, the
DenseNet two-channels (AUC 0.966). This model is trained with Dropout with
$p=0.2$. For this evaluation the MC-Dropout during inference time is enabled
explicitly. 20 forward passes for each of the test images are made and the
mean score and standard deviation is computed. The standard deviation is a
measure of uncertainty, with increasing value indicating more uncertainty.
Figures 6 and 7 present these results in terms of the most uncertain patch
pairs in Figure 6, and the most certain (least uncertain) images in Figure 7.
These results give insights on what the model thinks are its most difficult
samples (high uncertainty), and in particular, the most uncertain examples
(highest standard deviation) are the ones close to being out of distribution,
where the patches are positioned near the border of the FLS polar field of
view, which probably confuses the model.
The lowest uncertainty results in Figure 7 indicate the easiest patch pairs to
discriminate, either the same object in relatively similar poses, or radically
different objects or background in each pair. In both cases the model is quite
confident of these predictions.
Figure 8 shows a large selection of patch pairs and their uncertainty
estimates, showing that the model is not always confident, particularly for
predictions with scores in between zero and one, even for pairs that a human
would consider to be easy to match or reject.
Figure 6: MC-Dropout predictions of DTC with highest standard deviation over
20 forward passes. Ground truth label 1 indicated matching. It is clear that
the low signal-to-noise for sonar is affecting the predictions, and unwanted
reflections and occlusions are also challenging.
Figure 7: MC-Dropout predictions of DTC with lowest standard deviation over 20
forward passes. These results show that the network learned some of the
similarity functions with great confidence. For object-object non-matching
pairs usual std values are much higher than other categories. Figure 8: MC-
Dropout predictions of DTC with highest standard deviation over 20 forward
passes, presenting 20 sonar image patch pairs from the test dataset with index
1000 to 1019, and corresponding mean prediction and standard deviation.
### V-D Ensemble
The performance of the DenseNet-Siamese(DS) is good for non-matching pair
predictions. DenseNet two-channel(DTC) is overall very good, but most
uncertain in object-object non matching pairs.
This observation led to the hypothesis that making an ensemble of these two
classifiers might improve overall predictive capability. For this experiment a
few of the previously trained models of DTC and DS are loaded, and their
predictions on the test data are averaged,i.e. same weights for DS and DTC
both. These evaluation results are displayed in Table V. The ROC AUC
calculated on the average prediction is found to be higher than the individual
scores each time.
TABLE V: After combining DS and DTC models, with AUC presented in first two
columns, the Ensemble is encoded and its prediction accuracy (AUC) gets much
improved, presented in the third column.
DS model AUC | DTC model AUC | Ensemble AUC
---|---|---
0.95 | 0.959 | 0.97
0.952 | 0.959 | 0.97
0.952 | 0.963 | 0.973
0.952 | 0.966 | 0.971
0.952 | 0.972 | 0.978
Ensemble accuracies (AUC) are consistently better than each model
individually. If the underlying models, which encode the ensemble, has low
AUC, the ensemble AUC is found to be much-improved. For example the first
result presented in Table V where the ensemble accuracy is much higher (0.97
AUC) than the underlying model predictions (0.95 and 0.959 AUCs). By encoding
an ensemble of the DenseNet-Siamese model with AUC 0.952 and the DenseNet two-
channel model with 0.972 AUC, the resulting ensemble AUC is found to be 0.978,
which is the highest AUC on test data obtained in any other experiment during
the scope of this work. This indicates that both DS and DTC models are
complementary and could be used together if higher AUC is required in an
application.
## VI Conclusions and Future Work
In this work we present new neural network architectures for matching of sonar
image patches, including extensive hyper-parameter tuning, and explore their
performance in terms of area under the ROC curve, uncertainty as modeled by
MC-Dropout, and performance as multiple models are ensembles. The results in
this work are proven to be improvements over the state of the art on the same
dataset. Using DenseNet two-Channel network, average prediction accuracy
obtained is 0.955 area under ROC curve (AUC). VGG-Siamese (with Contrastive
loss function) and DenseNet-Siamese perform the prediction with an average AUC
of 0.949 and 0.921 respectively. All these results are an improvement over the
result of 0.910 AUC from Valdenegro-Toro [8]. Furthermore, by encoding an
ensemble of DenseNet two-channel and DenseNet-Siamese models with respective
highest AUC scores, prediction accuracy for the Ensemble obtained is 0.978
AUC, which is overall highest accuracy obtained in the Marine Debris Dataset
for the matching task.
We expect that our results motivate other researchers to build applications on
top of our matching networks.
## References
* [1] N. Hurtós, Y. Petillot, J. Salvi _et al._ , “Fourier-based registrations for two-dimensional forward-looking sonar image mosaicing,” in _Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on_. Ieee, 2012, pp. 5298–5305.
* [2] M. Brown and D. G. Lowe, “Automatic panoramic image stitching using invariant features,” _International journal of computer vision_ , vol. 74, no. 1, pp. 59–73, 2007.
* [3] N. Molton, A. J. Davison, and I. D. Reid, “Locally planar patch features for real-time structure from motion.” in _Bmvc_. Citeseer, 2004, pp. 1–10.
* [4] D. G. Lowe, “Object recognition from local scale-invariant features,” in _Computer vision, 1999. The proceedings of the seventh IEEE international conference on_ , vol. 2. Ieee, 1999, pp. 1150–1157.
* [5] B. Yao, G. Bradski, and L. Fei-Fei, “A codebook-free and annotation-free approach for fine-grained image categorization,” in _Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on_. IEEE, 2012, pp. 3466–3473.
* [6] S. M. Seitz, B. Curless, J. Diebel, D. Scharstein, and R. Szeliski, “A comparison and evaluation of multi-view stereo reconstruction algorithms,” in _null_. IEEE, 2006, pp. 519–528.
* [7] D. G. Lowe, “Distinctive image features from scale-invariant keypoints,” _International journal of computer vision_ , vol. 60, no. 2, pp. 91–110, 2004\.
* [8] M. Valdenegro-Toro, “Improving sonar image patch matching via deep learning,” in _Mobile Robots (ECMR), 2017 European Conference on_. IEEE, 2017, pp. 1–6.
* [9] S. Zagoruyko and N. Komodakis, “Learning to compare image patches via convolutional neural networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2015, pp. 4353–4361.
* [10] S. Emberton, L. Chittka, and A. Cavallaro, “Underwater image and video dehazing with pure haze region segmentation,” _Computer Vision and Image Understanding_ , vol. 168, pp. 145–156, 2018.
* [11] N. Hurtós, N. Palomeras, S. Nagappa, and J. Salvi, “Automatic detection of underwater chain links using a forward-looking sonar,” in _OCEANS-Bergen, 2013 MTS/IEEE_. IEEE, 2013, pp. 1–7.
* [12] K. Kim, N. Neretti, and N. Intrator, “Mosaicing of acoustic camera images,” _IEE Proceedings-Radar, Sonar and Navigation_ , vol. 152, no. 4, pp. 263–270, 2005.
* [13] S. Negahdaripour, M. Aykin, and S. Sinnarajah, “Dynamic scene analysis and mosaicing of benthic habitats by fs sonar imaging-issues and complexities,” in _Proc. OCEANS_ , vol. 2011, 2011, pp. 1–7.
* [14] P. Vandrish, A. Vardy, D. Walker, and O. Dobre, “Side-scan sonar image registration for auv navigation,” in _Underwater Technology (UT), 2011 IEEE Symposium on and 2011 Workshop on Scientific Use of Submarine Cables and Related Technologies (SSC)_. IEEE, 2011, pp. 1–7.
* [15] M. T. Pham and D. Guériot, “Guided block-matching for sonar image registration using unsupervised kohonen neural networks,” in _Oceans-San Diego, 2013_. IEEE, 2013, pp. 1–5.
* [16] J. Zbontar and Y. LeCun, “Stereo matching by training a convolutional neural network to compare image patches,” _Journal of Machine Learning Research_ , vol. 17, no. 1-32, p. 2, 2016.
* [17] M. Valdenegro-Toro, “Objectness scoring and detection proposals in forward-looking sonar images with convolutional neural networks,” in _IAPR Workshop on Artificial Neural Networks in Pattern Recognition_. Springer, 2016, pp. 209–219.
* [18] A. Mallick, “Sonar patch matching via deep learning,” Master’s thesis, Bonn-Rhein-Sieg University of Applied Sciences, 2019.
* [19] G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, “Densely connected convolutional networks.” in _CVPR_ , vol. 1, no. 2, 2017, p. 3.
* [20] R. Hadsell, S. Chopra, and Y. LeCun, “Dimensionality reduction by learning an invariant mapping,” in _2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06)_ , vol. 2. IEEE, 2006, pp. 1735–1742.
* [21] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _arXiv preprint arXiv:1409.1556_ , 2014.
* [22] Y. Gal and Z. Ghahramani, “Dropout as a Bayesian approximation: Representing model uncertainty in deep learning,” _arXiv:1506.02142_ , 2015.
|
the predicted reheating temperature is
\begin{equation}\label{eqn:ad122}
T_{R} = 2.11 \times 10^{13}\GeV,
\end{equation}
and using (<ref>) the predicted baryon-to-entropy ratio is
\begin{equation}\label{eqn:ad123}
\frac{n_{B}}{s} = 5.05 \times 10^{-5}.
\end{equation}
The numerical computation gives a baryon-to-entropy ratio for $\tau_{\Phi} = 5T_{asy}$ with $m_{\Phi} = 10^{16}\GeV$, $A^{\frac{1}{2}} = 10^{13}\GeV$, and $\sin 2\theta =1$ of
\begin{equation}\label{eqn:ad124}
\frac{n_{B}}{s} = 4.84 \times 10^{-5},
\end{equation}
which is in very good agreement with the analytical prediction.
Figures <ref> - <ref> show the evolution of the baryon-to-entropy ratio, the comoving condensate asymmetry, and the transferred asymmetry over time in the $\tau_{\Phi} = 5T_{asy}$ scenario.
Numerically calculated comoving condensate asymmetry, $n_{c}$, vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.
Numerically calculated comoving transferred asymmetry, $\hat{n}_{c}$, vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.
Numerical baryon-to-entropy ratio vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.
Comparison of numerically calculated comoving condensate asymmetry with the analytical estimate from (<ref>), $n_{c}$, vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.
Comparison of numerically calculated comoving transferred asymmetry with the analytical estimate using (<ref>), $\hat{n}_{c}$, vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.
Figure <ref> shows the condensate asymmetry oscillating about zero with decreasing amplitude, until the condensate has completely decayed and the condensate asymmetry is zero. Since in this case we are considering $\tau_{\Phi} > T_{asy}$, this occurs over a large number of phase oscillations of the field, and therefore a large number of oscillations of the asymmetry. Similarly, Figure <ref> shows the transferred asymmetry increasing as an asymmetry is initially generated in the condensate and transferred through the initial inflaton decays. The transferred asymmetry then continues oscillating while the inflaton scalars continue decaying, transferring the asymmetry to the Standard Model until the inflaton condensate has decayed completely. The transferred asymmetry then settles at a small, positive, finite value, corresponding to the total asymmetry transferred to the Standard Model, and thus the baryon-to-entropy ratio shown in Figure <ref>. As in the case of $\tau_{\Phi} = T_{asy}$, we can see from Figures <ref> and <ref>, that the numerically calculated comoving condensate and transferred asymmetries are in perfect agreement with the comoving condensate and transferred asymmetries calculated analytically using the threshold approximation for $\tau_{\Phi} = 5T_{asy}$, demonstrating that the threshold approximation accurately models the behaviour of the condensate and transferred asymmetries, as well as accurately predicting the baryon-to-entropy ratio, both in the case of rapid decay of the inflaton condensate and decay over many oscillations of the asymmetry.
§.§ Validity of the Analytical Approximation of the Decay of the Condensate Asymmetry
In this section we examine the validity of the analytical approximation of the decay of the condensate asymmetry, and thus the validity of the analytical treatment of the threshold approximation. In our treatment thus far, we have assumed that the condensate asymmetry decays as
\begin{equation}\label{eqn:ad125}
n_{c}\left(t \right) = n_{0}\left(t \right)e^{-\Gamma_{\Phi}\left(t - t_{\ast}\right)},
\end{equation}
where $n_{0}\left(t \right)$ is the condensate asymmetry without decay. We will show that this is true if
\begin{equation}\label{eqn:ad126}
\Gamma_{\Phi} << m_{\Phi}^{2},
\end{equation}
\begin{equation}\label{eqn:ad127}
H\Gamma_{\Phi} << m_{\Phi}^{2}.
\end{equation}
We are considering rapid oscillations in the amplitude of the inflaton field, $\phi$, in this model, meaning that $m_{\Phi}^{2} >> H^{2}$, and up to the complete decay of the inflaton condensate we have that $\Gamma_{\Phi} \lesssim H$. From these two conditions, it follows that (<ref>), (<ref>) are satisfied.
We can prove (<ref>) is true under these conditions. As a starting point, we assume that the field solutions in the case of condensate decay are
\begin{equation}\label{eqn:ad128}
\phi_{i}\left(t \right) = \phi_{i0}\left(t \right)e^{-\frac{\Gamma_{\Phi}t}{2}}, \; \; \; i = 1,2
\end{equation}
where $\phi_{i0}\left(t \right)$ is the solution to the field equations without the decay terms. The full field equations with the decay terms are
\begin{equation}\label{eqn:ad129}
\ddot{\phi}_{1} + 3H\dot{\phi}_{1} + \Gamma_{\Phi}\dot{\phi}_{1} = -m_{1}^{2}\phi_{1},
\end{equation}
\begin{equation}\label{eqn:ad130}
\ddot{\phi}_{2} + 3H\dot{\phi}_{2} + \Gamma_{\Phi}\dot{\phi}_{2} = -m_{2}^{2}\phi_{2},
\end{equation}
and in order to verify the analytical approximation of the decay of the condensate asymmetry we need to substitute (<ref>) into the field equations with the decay terms. Using (<ref>) we have that
\begin{equation}\label{eqn:ad131}
\dot{\phi}_{i}\left(t \right) = \left[ \dot{\phi}_{i0}\left(t \right) - \frac{\Gamma_{\Phi}}{2}\phi_{i0}\left(t \right) \right]e^{-\frac{\Gamma_{\Phi}t}{2}},
\end{equation}
\begin{equation}\label{eqn:ad132}
\ddot{\phi}_{i}\left(t \right) = \left[\ddot{\phi}_{i0}\left(t\right) - \Gamma_{\Phi}\dot{\phi}_{i0}\left(t \right) + \frac{\Gamma_{\Phi}^{2}}{4} \right]e^{-\frac{\Gamma_{\Phi}t}{2}}.
\end{equation}
Substituting (<ref>), (<ref>) into the field equations (<ref>), (<ref>) we find that both sides are proportional to $e^{-\frac{\Gamma_{\Phi}t}{2}}$, and can therefore cancel the exponentials. The field equations with decay terms in terms of the field solution (<ref>) are then
\begin{equation}\label{eqn:ad133}
\ddot{\phi}_{i0}\left(t \right) + 3H\dot{\phi}_{i0}\left(t \right) = - \left[m_{i}^{2} - \frac{\Gamma_{\Phi}^{2}}{4} - \frac{3H\Gamma_{\Phi}}{2} \right]\phi_{i0}\left(t \right).
\end{equation}
These are the same as the field equations with no decay term if
\begin{equation}\label{eqn:ad134}
\frac{\Gamma_{\Phi}^{2}}{4} << m_{i}^{2}, \; \; \frac{3H\Gamma_{\Phi}}{2} << m_{i}^{2},
\end{equation}
and if these conditions are satisfied then we can infer that (<ref>) and (<ref>) are also satisfied, and are true for $\Gamma_{\Phi}^{2} << m_{i}^{2}$ and $H^{2} << m_{i}^{2}$. These conditions are generally satisfied for a rapidly oscillating field, which is a basic assumption of the dynamics of this model.
The condensate asymmetry is given by
\begin{equation}\label{eqn:ad135}
n\left(t \right) = \dot{\phi}_{1}\phi_{2} - \dot{\phi}_{2}\phi_{1}.
\end{equation}
Substituting (<ref>) and (<ref>) into the condensate asymmetry (<ref>) we find that
\begin{equation}\label{eqn:ad136}
n\left(t \right) = \left(\dot{\phi}_{10}\phi_{20} - \dot{\phi}_{20}\phi_{10}\right)e^{-\Gamma_{\Phi}t},
\end{equation}
which is simply
\begin{equation}\label{eqn:ad137}
n\left(t \right) = n_{0}\left(t \right)e^{-\Gamma_{\Phi}t}.
\end{equation}
The condensate asymmetry for a decaying condensate is therefore equal to the condensate asymmetry with no decay multiplied by $e^{-\Gamma_{\Phi}t}$, as put forward in (<ref>) in Section <ref>, and the analytical approximation of the decay of the condensate asymmetry is therefore valid in the limit of rapid oscillations, as we assume throughout this model.
§ BARYON ASYMMETRY WASHOUT BY INFLATON EXCHANGE
So far we have assumed that the condensate decays via $B$-conserving processes, and that the mean asymmetry formed in the inflaton condensate is transferred equally to the Standard Model particles. This means that we assume there are no additional processes happening which could remove the asymmetry from the particle plasma. In this section we will consider a $B$-violating process which follows the annihilation of two anti-fermions $\bar{\psi}$ to a virtual inflaton, which then decays to a pair of fermions $\psi$, which can subtract baryon number from the Universe and potentially wipe out the asymmetry. Such a process would arise from inflaton exchange combined with the interaction $\lambda_{\psi}\bar{\psi}^{c}\psi \Phi$, responsible for inflaton condensate decay. The rate of this process is given by
\begin{equation}\label{eqn:ad138}
\Gamma_{\Delta B} = n_{\Phi} \langle \sigma v \rangle,
\end{equation}
where we assume this process is relativistic, $v=1$. At reheating $t = t_{R}$, the number density of the inflaton particles is $n_{\Phi} \sim T_{R}^{3}$, and dimensionally the rate is given by
\begin{equation}\label{eqn:ad139}
\Gamma_{\Delta B} \sim \frac{\lambda_{\psi}^{4}A^{2}T_{R}^{5}}{m_{\Phi}^{8}},
\end{equation}
where inflaton exchange implies an amplitude proportional to $\lambda_{\psi}^{2}A$. This process will not be significant - and therefore washout due to inflaton exchange will not pose a problem for Affleck-Dine baryogenesis with quadratic symmetry-breaking terms - provided that $\Gamma_{\Delta B} < H\left(T_{R}\right)$. Using $H\left(T_{R}\right) \sim T_{R}^{2}/M_{pl}$, this requires that
\begin{equation}\label{eqn:ad140}
\frac{\lambda_{\psi}^{4}A^{2}T_{R}^{5}}{m_{\Phi}^{8}} < H\left(T_{R}\right) \sim \frac{T_{R}^{2}}{M_{pl}}.
\end{equation}
Rearranging (<ref>) gives the following constraint on the coupling $\lambda_{\psi}$
\begin{equation}\label{eqn:ad141}
\lambda_{\psi}^{4} \lesssim \frac{m_{\Phi}^{8}}{M_{pl}A^{2} T_{R}^{3}} \Rightarrow \lambda_{\psi}^{2} \lesssim \frac{m_{\Phi}^{4}}{M_{pl}^{\frac{1}{2}} A T_{R}^{\frac{3}{2}}}.
\end{equation}
Normalising the inflaton mass and the reheating temperature using $m_{\phi} = 10^{13}\GeV$ and $T_{R} = 10^{8}\GeV$, this can be expressed as
\begin{equation}\label{eqn:ad142}
\lambda_{\psi}^{2} \lesssim \left(6.5 \times 10^{4}\right)\left(\frac{m_{\Phi}^{2}}{A}\right)\left(\frac{m_{\Phi}}{10^{13}\GeV}\right)^{2}\left(\frac{10^{8} \GeV}{T_{R}}\right)^{\frac{3}{2}}.
\end{equation}
Provided that this condition is satisfied, the asymmetry will not be washed out by inflaton exchange. This can be rewritten as a constraint on the reheating temperature if we consider the fact that the inflaton decay rate is given by $\Gamma_{\Phi} = \lambda_{\psi}^{2}m_{\Phi}/4\pi $ [132], and $\Gamma_{\Phi} = H_{R}$. The reheating temperature is then
\begin{equation}\label{eqn:ad143}
T_{R} \approx \lambda_{\psi} \sqrt{m_{\Phi} M_{pl}},
\end{equation}
which allows the constraint (<ref>) to be expressed as
\begin{equation}\label{eqn:ad144}
T_{R} \lesssim \left(\frac{m_{\Phi}^{2}}{A}\right)^{\frac{2}{7}}\left(\frac{M_{pl}}{m_{\Phi}}\right)^{\frac{1}{7}}m_{\Phi}.
\end{equation}
Provided this is satisfied, there is little risk of the asymmetry being washed out by inflaton exchange. Since $A << m_{\Phi}^{2}$ and $m_{\Phi} < M_{pl}$, then this is generally satisfied for Affleck-Dine baryogenesis with quadratic symmetry-breaking terms if $T_{R} < m_{\Phi}$. A more in-depth examination of washout would require a specific model of the inflaton decay processes and the transfer of the baryon asymmetry.
§ CONSISTENCY WITH NON-MINIMALLY COUPLED INFLATION
In this section we consider the case of non-minimally coupled inflation with a quartic potential, which is naturally compatible with the potential of the model (<ref>). We explore the conditions needed for the analytic expressions of the baryon asymmetry we have derived in this work to be consistent with non-minimally coupled inflation. To achieve this we need that the inflaton dynamics stop being dominated by the non-minimal coupling before the Affleck-Dine dynamics become significant. This can be stated as requiring that the point of non-minimal coupling domination, $\phi_{c}$, must be larger than the threshold point $\phi_{\ast}$, $\phi_{\ast} < \phi_{c}$. From Section <ref>, we have that once $H^{2} = 4A$, at a point we denote as $\phi_{AD}$, the phase field $\theta$ becomes dynamical, and we need $\phi_{AD} < \phi_{\ast}$ in order for no asymmetry to be produced while the potential is in its $\left|\Phi\right|^{4}$ dominated regime and for the threshold approximation of the asymmetry to hold. We therefore require that $\phi_{AD} < \phi_{\ast} < \phi_{c}$ (see Figure <ref>) in order for this model of Affleck-Dine baryogenesis via quadratic $B$-violating terms as described by the threshold approximation to be compatible with the AD field being a non-minimally coupled inflaton.
Schematic of the inflaton potential illustrating the relationship between the different regimes of the potential as thresholds in the dominant dynamics of the inflaton field, assuming $\phi$ is canonically normalised, producing the characteristic plateau potential of non-minimally coupled models.
In the Palatini formalism, the point at which the non-minimal coupling ceases to be the dominant contribution to the dynamics of the inflaton field is when $\Omega^{2} = 1$. This represents the point at which the canonical inflaton in the Einstein frame becomes equivalent to the Jordan frame inflaton, and the potential is approximately given by the Jordan frame potential. We have that
\begin{equation}\label{eqn:ad145}
\Omega^{2} = 1 + \frac{\xi \phi_{c}^{2}}{M_{pl}^{2}} \approx \mathcal{O}\left(1\right).
\end{equation}
Since this corresponds to the edge of the plateau, we have
\begin{equation}\label{eqn:ad146}
\frac{\xi \phi_{c}^{2}}{M_{pl}^{2}} \sim 1 \Rightarrow \phi_{c} = \frac{M_{pl}}{\sqrt{\xi}},
\end{equation}
in the Palatini formalism.
In the metric formalism, the condition for equivalence of the inflaton between frames is (from the definition of the canonical field transformation in Section <ref>)
\begin{equation}\label{eqn:ad147}
\frac{\Omega^{2} + 6\xi^{2}\phi_{c}^{2}/M_{pl}^{2}}{\Omega^{4}} =1,
\end{equation}
for $\Omega^{2} = 1$. Using the slow-roll inflation condition on the non-minimal coupling, $\xi >> 1$, we have
\begin{equation}\label{eqn:ad148}
\frac{6\xi^{2}\phi_{c}^{2}}{M_{pl}^{2}} = 1 \Rightarrow \phi_{c} = \frac{M_{pl}}{\sqrt{6}\xi}.
\end{equation}
Using the condition $\phi_{\ast} = m_{\Phi}/\sqrt{\lambda_{\Phi}} < \phi_{c}$, in the Palatini formalism (<ref>) we therefore require
\begin{equation}\label{eqn:ad149}
\frac{m_{\Phi}}{\sqrt{\lambda_{\Phi}}} < \frac{M_{pl}}{\sqrt{\xi}},
\end{equation}
which gives the condition on the inflaton mass for the model to be consistent with non-minimally coupled dynamics as
\begin{equation}\label{eqn:ad150}
m_{\Phi} < \frac{\sqrt{\lambda_{\Phi}}M_{pl}}{\sqrt{\xi}}.
\end{equation}
The comoving curvature power spectrum for a quartic potential in the Palatini formalism is (from (<ref>)[In Chapter 6 we derive the primordial curvature power spectrum and the inflation observables in the Palatini formalism generalised to include an inflaton mass term. We find that the power spectrum and observables are the same as those for conventional Palatini inflation with a quartic inflaton potential. See e.g. [62] for a recent review of the calculation of the observables in quartic Palatini inflation.])
\begin{equation}\label{eqn:ad151}
\mathcal{P}_{\mathcal{R}} = \frac{\lambda_{\Phi}N^{2}}{12\pi^{2}\xi },
\end{equation}
and using $N = 55$ as an estimate of the pivot scale and taking the amplitude of $\mathcal{P}_{\mathcal{R}}$ to be $A_{s} = 2.1 \times 10^{-9}$ [12], $\xi$ is
\begin{equation}\label{eqn:ad152}
\xi = 1.22\times 10^{10}\lambda_{\Phi}.
\end{equation}
Using $\xi$ normalised to the Palatini power spectrum $\mathcal{P}_{\mathcal{R}}$ given in (<ref>), the constraint on the inflaton mass (<ref>) becomes
\begin{equation}\label{eqn:ad153}
m_{\Phi} < 2.2 \times 10^{13}\GeV.
\end{equation}
For the case of the metric formalism (<ref>), for $\phi_{\ast} < \phi_{c}$, we have that
\begin{equation}\label{eqn:ad154}
\frac{m_{\Phi}}{\sqrt{\lambda_{\Phi}}} < \frac{M_{pl}}{\sqrt{6}\xi},
\end{equation}
which gives
\begin{equation}\label{eqn:ad155}
m_{\Phi} < \frac{\sqrt{\lambda_{\Phi}}M_{pl}}{\sqrt{6}\xi}.
\end{equation}
The comoving curvature power spectrum in the metric formalism is given by (from (<ref>))
\begin{equation}\label{eqn:ad156}
\mathcal{P}_{\mathcal{R}} = \frac{\lambda_{\Phi}}{72\pi^{2}\xi^{2} }N^{2},
\end{equation}
and using $N =55$ and $A_{s} = 2.1 \times 10^{-9}$, this gives the value of the non-minimal coupling to be
\begin{equation}\label{eqn:ad157}
\xi = 4.5 \times 10^{4}\sqrt{\lambda_{\Phi}}.
\end{equation}
Using $\xi$ normalised to the metric power spectrum $\mathcal{P}_{\mathcal{R}}$ given in (<ref>) this gives
\begin{equation}\label{eqn:ad158}
m_{\Phi} < 2.2 \times 10^{13}\GeV,
\end{equation}
in order for the model to be consistent with non-minimally coupled inflaton dynamics in the metric formalism, which is the coincidentally the same numerical constraint for the inflaton mass for the Palatini formalism. This means that this model is compatible with an embedding into a non-minimally coupled inflation model within a well defined upper bound on the inflaton mass.
§ BARYON ISOCURVATURE PERTURBATIONS IN NON-MINIMALLY COUPLED INFLATION
In this model, it is possible that the angular component of the inflaton field may undergo quantum fluctuations during inflation which result in density perturbations uncorrelated with the perturbations in the photon energy density. These are isocurvature fluctuations, as defined in Section <ref>. In Affleck-Dine baryogenesis models, the phase of the AD field can produce isocurvature fluctuations in the local baryon density, and therefore in baryon number density [133] - [137]. These isocurvature fluctuations in baryon number density can be observable, and this can lead to constraints on inflation models used in conjunction with the Affleck-Dine model in order for the model to be compatible with observations. In this section, we explore the baryon isocurvature perturbations generated in this model of Affleck-Dine inflation with quadratic $B$-violating terms, and examine whether this model is compatible with the bounds on the isocurvature fraction from the Planck experiment (2018).
We define the isocurvature perturbations as
\begin{equation}\label{eqn:ad159}
I = \left(\frac{\delta \rho_{i}}{\rho_{i}}\right)_{iso},
\end{equation}
for a given species $i$. In order to compare the isocurvature perturbations in baryon energy density in this model to the isocurvature bounds defined using Planck data [36], which are normalised to cold dark matter isocurvature perturbations, we begin with the isocurvature perturbations in cold dark matter density
\begin{equation}\label{eqn:ad160}
I = \left(\frac{\delta \rho_{CDM}}{\rho_{CDM}}\right)_{iso},
\end{equation}
and then use the fact that $\delta \rho_{CDM} \rightarrow \delta \rho_{B}$ to find the effective cold dark matter isocurvature perturbations due to baryon number perturbations
\begin{equation}\label{eqn:ad161}
I = \frac{\rho_{B}}{\rho_{CDM}}\left(\frac{\delta \rho_{B}}{\rho_{B}}\right)_{iso} = \frac{\Omega_{B}}{\Omega_{CDM}}\left(\frac{\delta \rho_{B}}{\rho_{B}}\right)_{iso},
\end{equation}
where the $\Omega$ here are the total energy densities as defined in Section <ref>. The isocurvature fraction $\beta_{iso}$ defined in [36] is
\begin{equation}\label{eqn:ad162}
\beta_{iso} = \frac{\mathcal{P}_{I}}{\mathcal{P}_{\mathcal{R}} + \mathcal{P}_{I}},
\end{equation}
assuming that the adiabatic and isocurvature fluctuations are uncorrelated. $\mathcal{P}_{\mathcal{R}} = A_{s}$ is the primordial curvature power spectrum, and to calculate $\mathcal{P}_{I}$ we need to calculate the power spectrum of the baryon isocurvature fluctuations $\delta \rho_{B}/\rho_{B}$ due to the fluctuations of the inflaton field, given by $\delta \theta$.
For non-minimally coupled inflation with a complex field, we have that
\begin{equation}\label{eqn:ad163}
\Phi = \frac{\phi}{\sqrt{2}}e^{i\theta} = \frac{1}{\sqrt{2}}\left(\phi_{1} + i\phi_{2}\right),
\end{equation}
where $\theta$ is an effectively massless field during inflation and the evolution of the inflaton field is purely along the radial direction, corresponding to $\theta = 0$ along the $\phi_{1}$ direction. Writing the inflaton as an effectively constant background plus a fluctuation in the field, we have that
\begin{equation}\label{eqn:ad164}
\Phi = \frac{\bar{\phi}}{\sqrt{2}}e^{i\delta\theta} \simeq \frac{\bar{\phi}}{\sqrt{2}}\left( 1 + i\delta \theta \right) = \frac{1}{\sqrt{2}}\left(\bar{\phi}_{1} + i\delta \phi_{2}\right),
\end{equation}
for small $\delta \theta$, where
\begin{equation}\label{eqn:ad165}
\bar{\phi}_{1} = \bar{\phi}, \; \; \; \delta \phi_{2} = \bar{\phi}\delta \theta.
\end{equation}
We have that the inflaton kinetic term in both the metric and Palatini formalisms for a complex inflaton field is given by
\begin{equation}\label{eqn:ad166}
\frac{1}{\Omega^{2}}\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi = \frac{1}{2\left(1 + \frac{\xi \phi_{1}^{2}}{M_{pl}^{2}} + \frac{\xi \phi_{2}^{2}}{M_{pl}^{2}}\right)}\left( \partial_{\mu}\phi_{1}\partial^{\mu}\phi_{1} + \partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}\right),
\end{equation}
and during slow-roll inflation we have that
\begin{equation}\label{eqn:ad167}
\frac{\xi \bar{\phi}_{1}^{2}}{M_{pl}^{2}} >> 1, \frac{\xi \phi_{2}^{2}}{M_{pl}^{2}},
\end{equation}
and the kinetic term for the angular field can therefore be written
\begin{equation}\label{eqn:ad168}
\frac{M_{pl}^{2}}{2\xi \bar{\phi}_{1}^{2}}\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}.
\end{equation}
We define a canonical field $\chi_{2}$ such that
\begin{equation}\label{eqn:ad169}
\frac{d\chi_{2}}{d\phi_{2}} = \frac{M_{pl}}{\sqrt{\xi}\bar{\phi}_{1}} \Rightarrow \chi_{2} = \frac{M_{pl}}{\sqrt{\xi}\bar{\phi}_{1}} \phi_{2},
\end{equation}
and substituting this into (<ref>) we find the canonical kinetic term to be
\begin{equation}\label{eqn:ad170}
\frac{M_{pl}^{2}}{2\xi \bar{\phi}_{1}^{2}}\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2} \rightarrow \frac{1}{2}\partial_{\mu}\chi_{2}\partial^{\mu}\chi_{2}.
\end{equation}
In terms of fluctuations of the inflaton field we therefore have
\begin{equation}\label{eqn:ad171}
\delta \chi_{2} = \frac{M_{pl}}{\sqrt{\xi}\bar{\phi}_{1}} \delta \phi_{2} = \frac{M_{pl}}{\sqrt{\xi}\bar{\phi}}\bar{\phi}\delta \theta,
\end{equation}
\begin{equation}\label{eqn:ad172}
\Rightarrow \delta \theta = \frac{\sqrt{\xi}}{M_{pl}}\delta \chi_{2}.
\end{equation}
Since $\chi_{2}$ is a canonically normalised field, its power spectrum is the standard expression given in Section <ref>
\begin{equation}\label{eqn:ad173}
\mathcal{P}_{\delta \chi_{2}} = \left(\frac{H}{2\pi}\right)^{2},
\end{equation}
and from (<ref>) we have that the corresponding power spectrum in $\delta \theta$ is
\begin{equation}\label{eqn:ad174}
\mathcal{P}_{\delta \theta} = \frac{\xi}{M_{pl}^{2}}\mathcal{P}_{\delta \chi_{2}} \Rightarrow \mathcal{P}_{\delta \theta} = \frac{\xi H^{2}}{4\pi^{2}M_{pl}^{2}}.
\end{equation}
The fluctuations in the baryon energy density relate to fluctuations in baryon number density through $\rho_{B} = m_{B}n_{B}$ for a fixed mass of the baryonic species, and we have
\begin{equation}\label{eqn:ad175}
\frac{\delta \rho_{B}}{\rho_{B}} = \frac{\delta n_{B}}{n_{B}}.
\end{equation}
In this model we have that
\begin{equation}\label{eqn:ad176}
n_{B} \propto \sin \left(2\theta \right),
\end{equation}
from (<ref>), (<ref>), (<ref>), for an initial phase angle $\theta$, and we can write the fluctuation in baryon number density as
\begin{equation}\label{eqn:ad177}
\frac{\delta n_{B}}{n_{B}} = \frac{d n_{B}}{d\theta}\frac{\delta \theta}{n_{B}}.
\end{equation}
Using (<ref>), we have that
\begin{equation}\label{eqn:ad178}
\frac{d n_{B}}{d\theta} = 2\cos \left(2\theta \right),
\end{equation}
and using (<ref>) and (<ref>) substituted into (<ref>), the fluctuation in baryon number density in terms of the fluctuation in $\theta$ is
\begin{equation}\label{eqn:ad179}
\frac{\delta n_{B}}{n_{B}} \propto \frac{2\delta \theta}{\tan \left(2\theta \right)}.
\end{equation}
The isocurvature fluctuation in baryon number density from (<ref>) is then
\begin{equation}\label{eqn:ad180}
I = \frac{\Omega_{B}}{\Omega_{CDM}}\frac{\delta n_{B}}{n_{B}} = \frac{\Omega_{B}}{\Omega_{CDM}}\frac{2\delta \theta}{\tan \left(2\theta \right)},
\end{equation}
the power spectrum of the baryon isocurvature fluctuations is thus
\begin{equation}\label{eqn:ad181}
\mathcal{P}_{I} = \left(\frac{\Omega_{B}}{\Omega_{CDM}}\right)^{2}\frac{4}{\tan^{2} \left(2\theta \right)}\mathcal{P}_{\delta \theta},
\end{equation}
and using (<ref>), we can write (<ref>) as
\begin{equation}\label{eqn:ad182}
\mathcal{P}_{I} = \left(\frac{\Omega_{B}}{\Omega_{CDM}}\right)^{2}\frac{\xi H^{2}}{\tan^{2} \left(2\theta \right)\pi^{2}M_{pl}^{2}}.
\end{equation}
The ratio $\mathcal{P}_{I}/\mathcal{P}_{\mathcal{R}}$ from (<ref>) can be written in terms of the isocurvature fraction
\begin{equation}\label{eqn:ad183}
\frac{\mathcal{P}_{I}}{\mathcal{P}_{\mathcal{R}}} = \frac{\beta_{iso}}{1 - \beta_{iso}},
\end{equation}
and using the observational limit on the isocurvature fraction, $\beta_{iso, lim}$, [36] we can constrain this ratio
\begin{equation}\label{eqn:ad184}
\frac{\mathcal{P}_{I}}{\mathcal{P}_{\mathcal{R}}} = \left(\frac{\Omega_{B}}{\Omega_{CDM}}\right)^{2}\frac{\xi H^{2}}{\tan^{2} \left(2\theta \right)\pi^{2}M_{pl}^{2}\mathcal{P}_{\mathcal{R}}} < \frac{\beta_{iso, lim}}{1 - \beta_{iso, lim}}.
\end{equation}
With some rearrangement this can be recast as a constraint on the Hubble parameter during inflation
\begin{equation}\label{eqn:ad185}
H_{inf} < \frac{\Omega_{CDM}}{\Omega_{B}}\frac{M_{pl}\pi A_{s}^{\frac{1}{2}}}{\sqrt{\xi}}\tan \left(2\theta \right) \left(\frac{\beta_{iso, lim}}{1 - \beta_{iso, lim}} \right)^{\frac{1}{2}}.
\end{equation}
Using $A_{s} = 2.1 \times 10^{-9}$, $\Omega_{CDM}/\Omega_{B} = 5.3$ and $\beta_{iso, lim} = 0.038$ from [36] this is
\begin{equation}\label{eqn:ad186}
H_{inf} < 3.6\times 10^{14}\left(\frac{\tan\left(2\theta \right)}{\sqrt{\xi}}\right)\GeV,
\end{equation}
which is the requirement in order to produce an isocurvature fraction consistent with observations.
The Einstein frame potential on the plateau in both the metric and Palatini formalisms for a quartic inflaton potential (see e.g. [62] for a review of the derivation of (<ref>) in both formalisms) is
\begin{equation}\label{eqn:ad187}
V_{E} = \frac{\lambda_{\Phi}M_{pl}^{4}}{4\xi^{2}},
\end{equation}
and from (<ref>) we have that
\begin{equation}\label{eqn:ad188}
H_{inf} = \left(\frac{\lambda_{\Phi}}{12}\right)^{\frac{1}{2}}\frac{M_{pl}}{\xi}.
\end{equation}
Using (<ref>) and (<ref>), we can write a constraint on the non-minimal coupling $\xi$ needed in order to produce an acceptable isocurvature fraction in this model,
\begin{equation}\label{eqn:ad189}
\left(\frac{\lambda_{\Phi}}{12}\right)^{\frac{1}{2}}\frac{M_{pl}}{\xi} < 3.6\times 10^{14}\left(\frac{\tan\left(2\theta \right)}{\sqrt{\xi}}\right)\GeV,
\end{equation}
\begin{equation}\label{eqn:ad190}
\xi > 3.7 \times 10^{6}\frac{\lambda_{\Phi}}{\tan^{2}\left(2\theta \right)}.
\end{equation}
We can now examine $\xi$ in the metric and Palatini formalisms and test whether the non-minimal coupling in each formalism satisfies the constraint (<ref>). The non-minimal coupling $\xi$ normalised from the comoving curvature power spectrum in the Palatini formalism is $\xi = 1.22\times 10^{10}\lambda_{\Phi}$ (<ref>) which easily satisfies the constraint (<ref>) provided that $\tan \left(2\theta \right)$ isn't exceptionally small, and also shows that the constraint on $\xi$ ultimately does not depend on the size of the inflaton self-coupling $\lambda_{\Phi}$. This shows that this model in the Palatini formalism produces an isocurvature perturbation in the baryon number density compatible with observations.
In the metric formalism we find $\xi = 4.5 \times 10^{4}\sqrt{\lambda_{\Phi}}$ (<ref>), and therefore
\begin{equation}\label{eqn:ad191}
\lambda_{\Phi} < 1.5\times 10^{-4}\tan^{4}\left(2\theta \right),
\end{equation}
is required in order for the constraint (<ref>) to be satisfied and isocurvature perturbations to be compatible with observations in the metric formalism.
Therefore in metric non-minimally coupled inflation, isocurvature perturbations place a significant constraint on the inflaton self-coupling. This also means that the baryon isocurvature fraction can be close to the present CMB bound if $\lambda_{\Phi} \sim 10^{-4}$, and so potentially observable as the bound improves, in contrast to the case of Palatini inflation where it will generally be much smaller than the CMB bound.
§ SEMI-CLASSICAL TREATMENT OF THE COHERENTLY OSCILLATING CONDENSATE
Throughout our derivation and analysis of the Affleck-Dine baryogenesis model with quadratic symmetry-breaking terms, we have assumed that $\phi_{1}$ and $\phi_{2}$ can be treated as classical fields. However, we must consider whether the classical treatment of the fields is valid in this context.
The inflaton field after inflation forms an oscillating coherent quantum condensate. In a second quantised theory, a coherent condensate can be treated as a classical field because the occupation number of the field states is much larger than one [138] [139]. In order for this to be satisfied, we require in the general sense that $X > m_{X}$ for the field $X$ which we are considering. However, in our model the condensate decays when $\phi < \phi_{\ast} = m_{\Phi}/\sqrt{\lambda_{\Phi}}$, and so $\phi < m_{\Phi}$ generally, meaning that the condensate can no longer be considered classically. Nevertheless, we will show that the classical calculation of the asymmetry is still correct.
The oscillating classical field corresponds to a coherent state of the field $| \phi_{i}\left(t \right) \rangle$. In general, the expectation value of the field operator in the coherent state $| \phi_{i}\left(t \right) \rangle$ is given by its classical value,
\begin{equation}\label{eqn:ad192}
\langle \phi_{i}\left(t \right) | \hat{\phi}_{i}| \phi_{i}\left(t \right) \rangle = \phi_{i,cl}\left(t \right),
\end{equation}
where $\phi_{i, cl}$ is the classical value of the field $\phi_{i}$ which satisfies the classical equations of motion. Since we are treating the components of the fields $\phi_{1}$ and $\phi_{2}$ as independent scalar fields, the coherent state of the complex field is the product of the coherent states of $\phi_{1}$ and $\phi_{2}$
\begin{equation}\label{eqn:ad193}
| \Phi \left( t \right) \rangle = | \phi_{1}\left(t \right) \rangle | \phi_{2}\left(t \right) \rangle.
\end{equation}
The baryon number density operator is given by
\begin{equation}\label{eqn:ad194}
\hat{n} = \hat{\dot{\phi}}_{1}\hat{\phi}_{2} - \hat{\dot{\phi}}_{2}\hat{\phi}_{1},
\end{equation}
and the expectation value of the baryon number density in the coherent state $| \phi_{1}\left(t \right) \rangle | \phi_{2}\left(t \right) \rangle$ is given by
\begin{multline}\label{eqn:ad195}
\langle \Phi \left( t \right) | \hat{n} | \Phi \left( t \right) \rangle = \langle \Phi \left( t \right) | \hat{\dot{\phi}}_{1}\hat{\phi}_{2} - \hat{\dot{\phi}}_{2}\hat{\phi}_{1} | \Phi \left( t \right) \rangle = \langle \phi_{1}\left(t \right) | \langle \phi_{2}\left(t \right) | \hat{\dot{\phi}}_{1}\hat{\phi}_{2} - \hat{\dot{\phi}}_{2}\hat{\phi}_{1}| \phi_{1}\left(t \right) \rangle | \phi_{2}\left(t \right) \rangle \\
= \langle \phi_{1} \left| \hat{\dot{\phi}}_{1} \right| \phi_{1} \rangle \langle \phi_{2} \left| \hat{\phi}_{2} \right| \phi_{2} \rangle - \langle \phi_{1} \left| \hat{\phi}_{1} \right| \phi_{1} \rangle \langle \phi_{2} \left| \hat{\dot{\phi}}_{2} \right| \phi_{2} \rangle.
\end{multline}
Thus the expectation value of the baryon asymmetry operator in the coherent state is given by
\begin{equation}\label{eqn:ad196}
\langle \Phi \left( t \right) | \hat{n} | \Phi \left( t \right) \rangle = \dot{\phi}_{1}\phi_{2} - \dot{\phi}_{2}\phi_{1} \equiv n_{cl}.
\end{equation}
Therefore the expectation value of the baryon asymmetry operator in the coherent state $i$ is equal to the baryon number density, $n_{cl}$, calculated using the classical fields $\phi_{1}$ and $\phi_{2}$. This is true even if the classical oscillating field is no longer a good approximation to the coherent state.
Since $\phi_{i} < m_{\Phi}$ in this model, the variance of the expectation value of the field becomes large compared to the squared classical field, which means that the quantum fluctuations of the fields around their expectation values will be large, and the fields cannot be considered classical. However, the length scale of the field fluctuations at the onset of inflaton decay must be smaller than the horizon. The present observed Universe evolves from a spacetime volume much larger than the size of the horizon at inflaton decay, therefore the observed baryon asymmetry today will be given by its spatial average, which corresponds to the classical value of the asymmetry. This means that the observed baryon-to-entropy ratio is equivalent to the baryon asymmetry calculated using the classical fields, $n_{cl}$, in this case, despite the fact that the fields themselves cannot be considered classical. The classical calculation of the baryon-to-entropy ratio is therefore still correct in the Affleck-Dine baryogenesis model with quadratic symmetry-breaking terms.
§ SUMMARY
In this chapter we have considered an Affleck-Dine baryogenesis scenario resulting from a model of non-minimally coupled inflation with a complex inflaton charged under a global $U(1)$ symmetry, with the complex inflaton taking the role of the AD field. We considered inflation from a renormalisable $U(1)$-symmetric $\left| \Phi \right|^{2} + \left| \Phi \right|^{4}$-type potential, with a quadratic $U(1)$-violating term $A\left( \Phi^{2} + \Phi^{\dagger^{2}}\right)$. This term introduces a phase dependence, altering the field's trajectory from being purely radial (along the $\phi_{1}$ direction) into a varying elliptical orbit in the complex plane, corresponding to an oscillating asymmetry.
At the end of slow-roll inflation, the inflaton is coherently oscillating. Initially these are $\left|\Phi\right|^{4}$ dominated oscillations, and then later becomes $\Phi^{2}$ oscillations as the oscillations become damped and the inflaton field oscillates about its minimum deep in the $\Phi^{2}$ region of the potential. At this stage the coherent condensate decays in a $B$-conserving process and subsequently reheats the universe. While the field undergoes coherent oscillations, it undergoes a periodic phase oscillation between its eigenstates $\Phi \leftrightarrow \Phi^{\dagger}$. This means that while the condensate is decaying, it is also periodically rotating between its $\Phi$ and $\Phi^{\dagger}$ states. So for each half-cycle in the phase oscillation, either $\Phi$ or $\Phi^{\dagger}$ scalars will be dominantly decaying, and will therefore produce either a baryon or anti-baryon number contribution to the particle plasma. Due to the fact that the condensate is decaying away while this is happening, it means that the produced baryon number and anti-baryon number of each oscillation doesn't exactly cancel with the charge produced in the previous half-cycle. This results in an overall asymmetry in baryon number being transferred to the particle content of the Standard Model, and this is how the observed small but finite baryon asymmetry is generated.
We were interested in studying the baryon asymmetry generated in this Affleck-Dine baryogenesis model from a quadratic symmetry-breaking term. In order to do this we derived the $U(1)$ asymmetry using a threshold approximation, wherein the asymmetry is generated when the potential is strongly dominated by its quadratic terms. We modelled the asymmetry of the condensate, and its evolution into the asymmetry which is transferred to the Standard Model, both in the case where the lifetime of the condensate is much larger than the oscillation period of the inflaton phase, $\tau_{\Phi} > T_{asy}$, and in the case where the condensate decays away before a phase rotation can occur, $\tau_{\Phi} < T_{asy}$. We find that in both cases the asymmetry transferred to the Standard Model derived using the threshold approximation could account for the observed baryon-to-entropy ratio. For $\tau_{\Phi} < T_{asy}$ it is typically much greater, and for $\tau_{\Phi} > T_{asy}$ it can be much less, due to the effect of suppression of the asymmetry due to averaging over a large number of oscillations. Given that the typical baryon asymmetry generated is much larger than that observed, the suppression due to oscillations is advantageous, requiring a smaller suppression of the quadratic symmetry-breaking terms.
We examine the dynamics of the angular component of the inflaton field $\theta$, and determine that the perturbations of the $\theta$ field do not significantly evolve before the potential becomes $\Phi^{2}$ dominated, provided that the constraint (<ref>) is satisfied, which is easily compatible with the constraints on $A^{\frac{1}{2}}/m_{\Phi}^{2}$, (<ref>) and (<ref>), derived in order to generate the observed baryon-to-entropy ratio from the transferred asymmetry. We numerically verified the constraint (<ref>) and showed that when this is violated there is an additional suppression of the asymmetry due to the damping of the phase during $\Phi^{4}$ oscillations.
We verify the validity of the threshold approximation in the derivation of the asymmetry, and the calculation of the baryon-to-entropy ratio using a numerical computation. In the numerical computation, we include decay terms in the field equations to account for the decay of the condensate, and we also model the dilution of the radiation energy density due to the expansion of the Universe. In the analytical calculation we assume that at time $t = t_{R}$, the condensate instantly decays to radiation and use this to calculate the reheating temperature. In the numerical computation of Section <ref>, we model the condensate as decaying away over a time period $\sim \Gamma_{\Phi}^{-1}$, and then calculate the reheating temperature directly from the energy density of radiation. We verify that the threshold approximation is reasonable for the analytical calculation for cases of the asymmetry generated with $\tau_{\Phi} = T_{asy}$ and $\tau_{\Phi} > T_{asy}$, with the predictions of the baryon-to-entropy ratio obtained numerically being within $10\%$ of the analytical predictions and the behaviour of the comoving condensate and transferred asymmetries in each case when modelled numerically is in perfect agreement with the behaviour predicted analytically using the threshold approximation. We also demonstrated that washout of the asymmetry through $B$-violating inflaton exchange is unlikely to pose a problem in this model for the range of reheating temperatures we would consider. This shows that this model provides a robust general framework for studying Affleck-Dine baryogenesis due to quadratic terms in models with a complex inflaton as the Affleck-Dine field.
In Section <ref> we discuss the compatibility of the Affleck-Dine baryogenesis via quadratic $B$-violating potential terms model with the dynamics of non-minimally coupled inflation. We find that in order for this to be realised, we require the non-minimally coupled dynamics of the inflaton to become insignificant before the threshold of $\Phi^{2}$ domination, and before the Affleck-Dine dynamics become significant. In both the metric and Palatini formalisms of non-minimally coupled inflation, we find that the model presented in this chapter can fulfill this requirement if the mass of the inflaton is smaller than $2.2 \times 10^{13}\GeV$.
We discuss baryon isocurvature perturbations in Section <ref> and we derive the constraints needed to ensure that the isocurvature fraction generated by a non-minimally coupled inflaton in this framework is not too large, and therefore consistent with the isocurvature bounds given in the Planck results [36]. We find that the isocurvature fraction predicted in this model can be consistent with the observational limit in both the metric and Palatini formalisms subject to constraints on the non-minimal coupling, $\xi$, (<ref>), (<ref>) and the inflaton self-coupling in the metric case (<ref>). In particular, in the metric case there is a significant upper bound on the size of the inflaton self-coupling, $\lambda_{\Phi} \sim 10^{-4}$. This also means that in the metric case, the baryon isocurvature perturbations could potentially become observable with an improved CMB bound on the isocurvature fraction in the future.
We also examine the validity of the classical treatment of the transfer of the asymmetry to the Standard Model. Since the inflaton field initially forms a coherently oscillating condensate with a large occupation number, the inflaton field can be treated as a classical field. However, when the inflaton decays, its amplitude $\phi$ is less than $m_{\Phi}$ and the coherent state is no longer in the classical limit. This raises the question of whether the classical treatment of the asymmetry in order to generate the observed baryon-to-entropy ratio is necessarily correct. We find that due to the spatial averaging of the baryon asymmetry, it is equal to the expectation value of the transferred asymmetry operator, which is equal to its classical value. Therefore, while it may not be true that the asymmetry can be considered strictly classical, the baryon asymmetry calculated using the classical fields still gives the correct value of the asymmetry.
While in this work we have considered a general Affleck-Dine scenario without studying a specific decay path of the inflaton or resulting transfer route of the $U(1)$-asymmetry to baryon number, it is possible that this work could be applied to a leptogenesis scenario from the decay of the inflaton to right-handed neutrinos, and then a subsequent transfer of the asymmetry to baryon number via sphaleron processes. This would allow a more in-depth analysis of the possibility of baryon asymmetry washout, and could also provide a framework for a model of inflation, baryogenesis and dark matter. This study is in progress.
CHAPTER: Q-BALLS FROM NON-MINIMALLY COUPLED INFLATION IN PALATINI GRAVITY
In this chapter we discuss the formulation of Q-balls in the framework of a non-minimally coupled Palatini inflation model. We present a study of the Palatini inflation model for a $\left|\Phi\right|^{2} + \left|\Phi\right|^{4}$-type potential in the Jordan frame (the first time an inflaton mass term has been included in Palatini inflation) and present the values for the inflationary observables. We derive the Q-ball field equation in a non-minimally coupled Palatini framework, discuss the existence conditions of these Q-balls and derive an inflaton mass range for which the model can both support inflation and produce Q-balls. In doing so, we will, for the first time, derive Q-ball solutions for the case of a complex scalar with a non-canonical kinetic term, corresponding to a new class of Q-ball. We derive the zeroes of the Q-ball equation and use these to solve the equation numerically. We verify the existence of Q-ball solutions over a range of field values and present some important properties of these Q-balls, including energy, charge and radius. We also present an analytical approximation of the Q-ball solution and obtain expressions for the energy and charge using this calculation. We show analytically that these Q-balls are stable and derive the energy-charge relation, both of which are then confirmed numerically. We consider the effects of curvature on these Q-balls and derive approximate relations between the inflaton self-coupling, $\lambda$, and the size and global energy of the Q-balls in order to predict, as a first approximation, the radius needed for one of these Q-balls to collapse to a black hole following formation in the presence of curvature. We discuss reheating in the case of Q-balls and Q-ball derived Black Holes, and speculate on the post-inflation cosmology which could result from an early matter-dominated era of Q-balls or their associated Black Holes. Finally the observability of the model is discussed, including the possibility of observable gravitational waves from Q-ball formation and/or decay and Q-ball dark matter. We also comment on the implications for the case of a real inflaton and oscillons.
§ Q-BALLS IN THE CONTEXT OF NON-MINIMALLY COUPLED PALATINI INFLATION
Q-balls are a subset of the class of field theory solutions known as non-topological solitons. Solitons are extended objects which can be visualised as droplets composed of many particles, held together by an attractive interaction between the particles. Their stability arises due to a conserved quantity of the theory, if the solitons are non-topological in nature.
In the case of Q-balls, the objects are composed of complex scalar particles charged under a conserved Noether symmetry of the theory, and the objects themselves carry a large charge. Charged scalar field configurations stable against small perturbations were originally theorised to exist by Rosen [140] in 1968. These ideas were then explored further and non-topological solitons were formally described by Friedberg, Lee et al [141, 142] later in the 20th century. The formulation of Q-balls as we understand them today was developed by Coleman in 1985 [50].
Since then, Q-balls have been studied extensively in the context of supersymmetry (SUSY) [105] - [113],[143, 144], generally forming from scalar condensates in the flat directions of supersymmetric potentials. They are often incorporated into supersymmetric theories in conjunction with Affleck-Dine baryogenesis [100] - [102] as a dark matter candidate, either as complete Q-balls or as a source of particle dark matter when the Q-balls decay [103] - [108]. The globally charged nature of Q-balls makes them a good dark matter candidate [109] - [113], [143] - [145], [146] - [152] both as compact objects or as the seeds of particle dark matter from decays. The possibility of dark matter being accounted for by Primordial Black Holes seeded from the collapse of supersymmetric Q-balls has also recently been considered [153].
Many numerical simulations have been performed to model the formation of Q-balls from scalar condensates [154] - [160]. In [160], Hiramatsu et al showed that the fragmentation of a neutral condensate in a model based on a complex scalar first forms oscillons which then fragment to $\pm$ Q-ball pairs. Using Hiramatsu et al's results [160] as a benchmark, we will consider the possibility that the Q-balls in this model may also form from the fragmentation of the inflaton condensate into neutral lumps of scalars, which then decay into pairs of $\pm$ Q-balls.
The fragmentation of a scalar condensate is a complex process which depends heavily on the underlying particle physics and requires detailed numerical simulations to model. As such, a quantitative description of the process is beyond the scope of the research presented in this thesis. However, it is possible that the fragmentation of the inflaton condensate in this case could arise as a result of tachyonic preheating [161] - [165].
Tachyonic preheating is the process by which an inflaton condensate becomes unstable due to the amplified growth of specific wavelengths of perturbations arising due to a tachyonic instability of the inflaton potential. This can lead to the formation of overdensities within a previously homogeneous condensate, which - if soliton solutions are admissible in the underlying field theory - can lead to the condensate breaking apart to form solitons or other related objects [85, 86], the precise nature of which depends on the nature of the inflaton field itself.
In the work this chapter is based on we were interested in exploring Q-balls in a non-SUSY context. More specifically, we were interested in whether Q-balls composed of inflaton scalars could be produced in a model of non-minimally coupled Palatini inflation, possibly as a result of tachyonic preheating, whether the model could inflate and reheat successfully in conjunction with the production of Q-balls, what the properties of these Q-balls would be, how they would compare to the Q-balls of Coleman's solution, and what the broader implications for cosmology could be from the presence of these Q-balls.
§ THE MODEL
In this section we introduce the non-minimally coupled Palatini inflation model within which we want to produce inflatonic Q-balls. This model describes inflation driven by a complex scalar inflaton $\Phi$ charged under a global $U(1)$ symmetry and non-minimally coupled to gravity. In particular, it differs from existing analyses of Palatini inflation by the inclusion of an inflaton mass term, which has not been previously considered. We use the $\left( +, -, -, - \right)$ convention for the metric and $M_{pl}$ should be taken to be the reduced Planck mass.
The Jordan frame inflaton action is given by
\begin{equation}\label{eqn:q1}
S_{J} = \int d^{4}x \sqrt{-g} \left[ -\frac{1}{2}M_{pl}^{2}\left( 1 + \frac{2\xi \mid \Phi \mid^{2}}{M_{pl}^{2}}\right)R + \partial_{\mu}\Phi^{ \dagger} \partial^{\mu}\Phi - V\left(\mid \Phi \mid \right) \right],
\end{equation}
where the Jordan frame potential is
\begin{equation}\label{eqn:q2}
V\left( \mid \Phi \mid \right) = m^{2} \mid \Phi \mid^{2} + \lambda \mid \Phi \mid^{4},
\end{equation}
where $m$ is the inflaton mass and $\lambda$ is the inflaton self-coupling. In order to recast the model in terms of conventional General Relativity, we perform a conformal transformation on the spacetime metric, given by
\begin{equation}\label{eqn:q3}
g_{\mu \nu} \longrightarrow \tilde{g}_{\mu \nu} = \Omega^{2}g_{\mu \nu},
\end{equation}
where the conformal factor $\Omega^{2}\left(\Phi \right)$ is
\begin{equation}\label{eqn:q4}
\Omega^{2} = 1 + \frac{2\xi \left| \Phi \right|^{2}}{M_{pl}^{2}}.
\end{equation}
In the Palatini formulation, the Ricci scalar transforms as (from (<ref>))
\begin{equation}\label{eqn:q5}
R \longrightarrow \tilde{R} = \frac{R}{\Omega^{2}},
\end{equation}
and the integration measure, $\sqrt{-g}$ in (<ref>), transforms as
\begin{equation}\label{eqn:q6}
\sqrt{-det \left(\frac{\tilde{g}_{\mu\nu}}{\Omega^{2}}\right)} = \sqrt{-\frac{1}{\Omega^{8}}det\left(\tilde{g}_{\mu\nu}\right)} = \frac{1}{\Omega^{4}}\sqrt{-det \tilde{g}_{\mu\nu}} = \frac{\sqrt{-\tilde{g}}}{\Omega^{4}}.
\end{equation}
The conformal transformation (<ref>) acts purely on the metric of the theory. This means that the transformation does not act on coordinates and the transformation on the derivative term in (<ref>) is due to the implicit factor of the metric used in the contraction of the indices
\begin{equation}\label{eqn:q7}
\partial_{\mu}\Phi^{\dagger} \partial^{\mu}\Phi = g_{\mu \nu}\partial^{\nu}\Phi^{\dagger} \partial^{\mu}\Phi,
\end{equation}
\begin{equation}\label{eqn:q8}
\Rightarrow \tilde{g}_{\mu \nu}\partial^{\nu}\Phi^{\dagger} \partial^{\mu}\Phi = g_{\mu \nu}\Omega^{2}\partial^{\nu}\Phi^{\dagger} \partial^{\mu}\Phi = \Omega^{2}\partial_{\mu}\Phi^{\dagger} \partial^{\mu}\Phi.
\end{equation}
The transformed action is then
\begin{equation}\label{eqn:q9}
S_{E} = \int d^{4}x \sqrt{-\tilde{g}} \left[ -\frac{1}{2}M_{pl}^{2}\tilde{R} + \frac{1}{\Omega^{2}}\partial_{\mu}\Phi^{ \dagger} \partial^{\mu}\Phi - V_{E}\left(\mid \Phi \mid \right) \right],
\end{equation}
where we are henceforth working in the Einstein frame (denoted by a subscript 'E' for the action or potential, or a tilde for geometric or other quantities). Our Einstein frame potential is then defined as
\begin{equation}\label{eqn:q10}
V_{E}\left(\mid \Phi \mid \right) = \frac{V\left(\mid \Phi \mid \right)}{\Omega^{4}}.
\end{equation}
As in Chapter 4, all calculations are performed in the Einstein frame in this chapter unless explicitly stated otherwise.
§ SLOW ROLL PARAMETERS AND INFLATIONARY OBSERVABLES
In this section we calculate the observables for non-minimally coupled Palatini inflation for the potential (<ref>) and check the compatibility of this model with the observations from the Planck satellite experiment. For ease of calculation we write the complex inflaton field in the form
\begin{equation}\label{eqn:q11}
\Phi = \frac{\phi}{\sqrt{2}}e^{i\theta},
\end{equation}
which means that the conformal factor (<ref>) becomes
\begin{equation}\label{eqn:q12}
\Omega^{2} = 1 + \frac{\xi \phi^{2}}{M_{pl}^{2}},
\end{equation}
and the Jordan frame potential is
\begin{equation} \label{eqn:q13}
V\left( \phi \right) = \frac{1}{2}m^{2}\phi^{2} + \frac{\lambda}{4}\phi^{4}.
\end{equation}
The Einstein frame potential in terms of (<ref>) is thus
\begin{equation}\label{eqn:q14}
V_{E}\left( \phi \right) = \frac{1}{2\Omega^{4}}m^{2}\phi^{2} + \frac{\lambda}{4\Omega^{4}}\phi^{4} = \frac{m^{2}\phi^{2}}{2\left( 1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}\right)^{2}} + \frac{\lambda \phi^{4}}{4\left( 1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}\right)^{2}},
\end{equation}
\begin{equation}\label{eqn:q15}
\Rightarrow V_{E} = \frac{\lambda M_{pl}^{4}}{4 \xi^{2}\left( 1 + \frac{M_{pl}^{2}}{\xi \phi^{2}}\right)^{2}}\left[ 1 + \frac{2 m^{2}}{\lambda \phi^{2}}\right].
\end{equation}
Since we are assuming that inflation takes place upon the plateau of the potential, and we expect the field to be very large in this regime, we define what we will refer to as "the plateau limit", where
\begin{equation}\label{eqn:q16}
\frac{\xi \phi^{2}}{M_{pl}^{2}} >> 1.
\end{equation}
It is assumed throughout this chapter that this approximation is a reflection of the dominant dynamics during and at the instantaneous end of slow-roll inflation, and more generally while the inflaton is otherwise on its plateau.
From (<ref>), we have that $M_{pl}^{2}/\xi \phi^{2} << 1$, and can therefore approximate
\begin{equation}\label{eqn:q17}
\left( 1 + \frac{M_{pl}^{2}}{\xi \phi^{2}}\right)^{-2} \thickapprox 1 - 2\frac{M_{pl}^{2}}{\xi \phi^{2}},
\end{equation}
\begin{equation}\label{eqn:q18}
\Rightarrow V_{E} = \frac{\lambda M_{pl}^{4}}{4 \xi^{2}}\left[ 1 + \frac{2 m^{2}}{\lambda \phi^{2}}\right] \left( 1 - \frac{2M_{pl}^{2}}{\xi \phi^{2}} \right).
\end{equation}
To leading order in small quantities, the Einstein frame potential is,
\begin{equation}\label{eqn:q19}
V_{E} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left[ 1 + \frac{2m^{2}}{\lambda \phi^{2}} - \frac{2M_{pl}^{2}}{\xi \phi^{2}} \right],
\end{equation}
which can be rewritten as
\begin{equation}\label{eqn:q20}
V_{E} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left[ 1 - \frac{2M_{pl}^{2}}{\xi \phi^{2}}\beta \right],
\end{equation}
where $\beta$ is defined as
\begin{equation}\label{eqn:q21}
\beta = 1 - \frac{\xi m^{2}}{\lambda M_{pl}^{2}}.
\end{equation}
To canonically normalise the inflaton field, we perform the field rescaling
\begin{equation}\label{eqn:q22}
\frac{d\sigma}{d\phi} = \frac{1}{\sqrt{1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}}},
\end{equation}
\begin{equation}\label{eqn:q23}
\Rightarrow \int d\sigma = \int \frac{d\phi}{\sqrt{1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}}}.
\end{equation}
Writing $a = \xi/M_{pl}^{2}$ we can use
\begin{equation}\label{eqn:q24}
\int \frac{dx}{\sqrt{1 + ax^{2}}} = \frac{1}{\sqrt{a}}\sinh^{-1} \left(\sqrt{a}x\right) + C,
\end{equation}
and the rescaled scalar field is therefore
\begin{equation}\label{eqn:q25}
\sigma \left(\phi \right) = \frac{M_{pl}}{\sqrt{\xi}}\sinh^{-1} \left(\frac{\sqrt{\xi}}{M_{pl}}\phi\right),
\end{equation}
where $\sigma \rightarrow \phi$ as $\phi \rightarrow 0$. It is important to note that this is a field rescaling and not a truly canonically normalised scalar field, due to the complex nature of the field $\Phi$. When discussing inflation in this model specifically, it can be regarded as a "canonical inflaton" since the inflation dynamics are determined by the radial component, $\phi$, of the inflaton field. Elsewhere in this chapter, $\sigma$ should be taken as a field redefinition, and the rescaling (<ref>) will be used multiple times in calculations throughout this chapter.
It follows from (<ref>) that
\begin{equation}\label{eqn:q26}
\phi \left(\sigma \right) = \frac{M_{pl}}{\sqrt{\xi}}\sinh \left(\frac{\sqrt{\xi}}{M_{pl}}\sigma \right).
\end{equation}
The hyperbolic sine function is
\begin{equation}\label{eqn:q27}
\sinh\left(x \right) = \frac{e^{x} - e^{-x}}{2},
\end{equation}
and we can therefore rewrite (<ref>) as
\begin{equation}\label{eqn:q28}
\phi \left(\sigma \right) = \frac{M_{pl}}{2\sqrt{\xi}}\left(e^{\frac{\sqrt{\xi}}{M_{pl}}\sigma} - e^{-\frac{\sqrt{\xi}}{M_{pl}}\sigma} \right).
\end{equation}
The rescaled field will be large in the same regime as the physical inflaton field. From (<ref>), we can say that the exponential arguments of (<ref>) will be large during inflation. This means that the decreasing exponential will be significantly smaller than the increasing exponential while on the plateau. During inflation, we can therefore say that the inflaton field in terms of the canonically normalised scalar is
\begin{equation}\label{eqn:q29}
\phi = \frac{M_{pl}}{2\sqrt{\xi}} e^{\frac{\sqrt{\xi}}{M_{pl}}\sigma},
\end{equation}
to a good approximation. Substituting (<ref>) into the Einstein frame potential (<ref>) gives
\begin{equation}\label{eqn:q30}
V_{E}\left(\sigma \right) = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left[ 1 - 8\beta e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma} \right].
\end{equation}
This is the form of the inflaton potential in the Einstein frame which we will use in deriving the expressions for the inflationary observables (see Figure <ref>). In existing analyses of Palatini inflation, $m = 0$ and $\beta = 1$. Here we are generalising to $m \neq 0$ and $\beta \neq 1$.
Schematic of the inflaton potential in terms of the quasi-canonical field $\sigma$ indicating the edge of the plateau, as used in defining the plateau limit used in this research.
The slow roll parameters are given by
\begin{equation}\label{eqn:q31}
\epsilon = \frac{M_{pl}^{2}}{2}\left(\frac{V_{E}'}{V_{E}} \right)^{2},
\end{equation}
\begin{equation}\label{eqn:q32}
\eta = M_{pl}\frac{V_{E}''}{V_{E}},
\end{equation}
where a prime $'$ denotes a derivative with respect to a field, $\phi$, in this case. Working with the assumption that $\phi$ is large we approximate
\begin{equation}\label{eqn:q33}
V_{E} \thickapprox \frac{\lambda M_{pl}^{4}}{4\xi^{2}},
\end{equation}
while on the plateau. This is another approximation which will be used in this work when the plateau limit is invoked. The first and second derivatives of (<ref>) are
\begin{equation}\label{eqn:q34}
\frac{\partial V_{E}}{\partial \sigma} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\cdot \left(-8\beta \right) \cdot \left(-\frac{2\sqrt{\xi}}{M_{pl}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left(\frac{16\beta \sqrt{\xi}}{M_{pl}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma},
\end{equation}
\begin{equation}\label{eqn:q35}
\frac{\partial^{2}V_{E}}{\partial \sigma^{2}} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left(-\frac{32\beta \xi}{M_{pl}^{2}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma}.
\end{equation}
Using (<ref>), (<ref>) the slow roll parameters are thus
\begin{equation}\label{eqn:q36}
\epsilon = \frac{M_{pl}^{2}}{2}\left( \frac{4\xi^{2}}{\lambda M_{pl}^{4}} \cdot \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left(\frac{16\beta \sqrt{\xi}}{M_{pl}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma}\right)^{2} = 128 \beta^{2} \xi e^{-4\frac{\sqrt{\xi}}{M_{pl}}\sigma},
\end{equation}
\begin{equation}\label{eqn:q37}
\eta = M_{pl}^{2} \frac{4\xi^{2}}{\lambda M_{pl}^{4}}\cdot \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left(-\frac{32\beta \xi}{M_{pl}^{2}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma} = -32\beta \xi e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma}.
\end{equation}
The number of e-folds of inflation is calculated using
\begin{equation}
\begin{split}\label{eqn:q38}
N = -\frac{1}{M_{pl}^{2}}\int^{\sigma_{end}}_{\sigma} & \frac{V_{E}}{V_{E}'} d\sigma = -\frac{1}{M_{pl}^{2}}\int^{\sigma_{end}}_{\sigma} \frac{M_{pl}}{12\beta \sqrt{\xi}}\exp \left(\frac{2\sqrt{\xi}}{M_{pl}}\sigma \right) d\sigma \\
& = \frac{1}{32\xi \beta} \left[ \exp \left(\frac{2\sqrt{\xi}}{M_{pl}}\sigma \right) - \exp \left(\frac{2\sqrt{\xi}}{M_{pl}}\sigma_{end} \right)\right].
\end{split}
\end{equation}
Working with the assumption that $\sigma_{end} << \sigma $, the number of e-folds is given by
\begin{equation}\label{eqn:q39}
N = \frac{1}{32\xi \beta}\exp \left(\frac{2\sqrt{\xi}}{M_{pl}}\sigma \right).
\end{equation}
This can be rewritten to give an expression for $\sigma \left(N \right)$
\begin{equation}\label{eqn:q40}
\sigma \left(N \right) = \frac{M_{pl}}{2\sqrt{\xi}} \ln \left(32\xi \beta N \right).
\end{equation}
Substituting (<ref>) into (<ref>) and (<ref>) to find $\eta$ and $\epsilon$ in terms of the number of e-folds of inflation, we find
\begin{equation}\label{eqn:q41}
\epsilon = 128 \beta^{2} \xi \exp \left[-4\frac{\sqrt{\xi}}{M_{pl}} \left(\frac{M_{pl}}{2\sqrt{\xi}}\ln \left(32\xi \beta N \right)\right)\right] = \frac{1}{8\xi N^{2}},
\end{equation}
\begin{equation}\label{eqn:q42}
\eta = -32\beta \xi \exp \left[-2\frac{\sqrt{\xi}}{M_{pl}} \left(\frac{M_{pl}}{2\sqrt{\xi}} \ln \left(32\xi \beta N \right)\right)\right] = -\frac{1}{N}.
\end{equation}
These are the standard expressions for $\epsilon$ and $\eta$ in Palatini inflation, and are independent of the $\beta$ parameter. This means that $\eta$ and $\epsilon$ are not affected by the presence of the mass term during inflation, provided that $\beta > 0$.
It is instructive to examine the size of the $\epsilon$ parameter relative to the $\eta$ parameter in this model to check the consistency of the slow-roll approximation in this inflation model. From Section <ref> we have that slow-roll inflation ends when $\left| \eta \right| = 1$, and from (<ref>) we can see that the end of slow-roll corresponds to $N = 1$. Using (<ref>), we find that the value at the end of slow-roll inflation is $\epsilon = 1.03 \times 10^{-10}$, which is significantly smaller than the $\eta$ parameter. This shows that this model is consistent with slow-roll inflation, and that $\epsilon << \eta$ throughout.
The scalar spectral index, $n_{s}$ in terms of $\eta$ and $\epsilon$ is
\begin{equation}\label{eqn:q43}
n_{s} = 1 + 2\eta - 6\epsilon.
\end{equation}
Since $\epsilon << \eta$ in this case we approximate
\begin{equation}\label{eqn:q44}
n_{s} \approx 1 - \frac{2}{N}.
\end{equation}
The tensor-to-scalar ratio $r$ is
\begin{equation}\label{eqn:q45}
r \approx 16\epsilon = 16 \cdot \frac{1}{8\xi N^{2}} = \frac{2}{\xi N^{2}}.
\end{equation}
and the primordial curvature power spectrum is
\begin{equation}\label{eqn:q46}
\mathcal{P}_{\mathcal{R}} = \frac{V_{E}}{24\pi^{2} \epsilon M_{pl}^{4}} = \frac{\lambda}{12\xi \pi^{2}} N^{2}.
\end{equation}
From the results of the Planck satellite experiment [12] (2018), the observed amplitude of the power spectrum is $A_{s} = 2.1 \times 10^{-9}$. Using $\lambda = 0.1$ and $N = 55$ as an estimate of the self-coupling and the pivot scale, (<ref>) gives a non-minimal coupling of $\xi = 1.2163 \times 10^{9}$. Using the same estimate, the inflationary observables are $n_{s} = 0.9636$ and $r = 6.01 \times 10^{-13}$. The scalar spectral index is within the bounds of the 2018 results from the Planck satellite (assuming $\Lambda$CDM and no running of the spectral index), $n_{s} = 0.9649 \pm 0.0042$ (1-$\sigma$) [12], whilst the tensor-to-scalar ratio is heavily suppressed, as is typically the case in Palatini inflation models. These values of the observables are only an estimate of the predictions of the model since, as we will discuss further in Section <ref> - <ref>, the post-inflationary cosmology of this model may have an effect on the number of e-folds of inflation needed for the model to inflate successfully, and consequently the location of the pivot scale, leading to an adjustment in $n_{s}$ and $r$.
The inflaton field can be expressed in terms of the number of e-folds as
\begin{equation}\label{eqn:q47}
\phi \left(N \right) = \frac{M_{pl}}{2\sqrt{\xi}} \exp \left(\frac{\sqrt{\xi}}{M_{pl}} \left( \frac{M_{pl}}{2\sqrt{\xi}} \ln \left(32\xi \beta N \right)\right)\right) = 2\sqrt{2}M_{pl} \sqrt{\beta N},
\end{equation}
so the mass term directly influences the amplitude of the field throughout inflation. Defining the end of slow-roll inflation to be when $\left| \eta \right| \approx 1$, giving $N_{end} \approx 1$, the value of the field at the end of slow-roll inflation is then
\begin{equation}\label{eqn:q48}
\phi_{end} = 2\sqrt{2}M_{pl} \sqrt{\beta}.
\end{equation}
$\beta$ will be in the range $0.1-1$ for the values of inflaton mass relevant here. In the following section we consider limits on the inflaton mass in this model needed to produce successful inflation, and we consider how this governs the effect of the mass term on the value of the field at the end of slow-roll inflation.
§.§ The Upper Bound on the Inflaton Mass
For the purposes of the research presented here, we are primarily interested in Q-ball solutions which are compatible with non-minimally coupled Palatini inflation. Since the Q-balls will be formed from inflaton scalars, the effects of the inflaton mass from the presence of the $\beta$ parameter on the Einstein frame potential, and subsequently on the existence of Q-ball solutions, must be established, including the range of inflaton masses for which inflation is possible. It is important to ascertain the limits placed on the inflaton mass by the necessity that the potential be compatible with inflation first, in order to later establish the range of inflaton masses for which Q-ball solutions are compatible with Palatini inflation.
We start from the Einstein frame potential
\begin{equation}\label{eqn:q49}
V_{E}\left(\phi \right) = \frac{m^{2} \phi^{2}}{2\Omega^{4}} + \frac{\lambda \phi^{4}}{4\Omega^{4}},
\end{equation}
and differentiate to give
\begin{equation}\label{eqn:q50}
\frac{\partial V_{E}}{\partial \phi} = \frac{1}{\Omega^{6}}\left[m^{2}\phi + \left(\lambda - \frac{\xi m^{2}}{M_{pl}^{2}}\right) \phi^{3} \right].
\end{equation}
The inflaton potential must have a positive gradient with respect to the inflaton for inflation to occur, we therefore require
\begin{equation}\label{eqn:q51}
\frac{\partial V_{E}}{\partial \phi} > 0 \Rightarrow \lambda - \frac{\xi m^{2}}{M_{pl}^{2}} > 0.
\end{equation}
This means that, in order for inflation to occur in this non-minimally coupled Palatini model, the inflaton mass must obey the upper bound
\begin{equation} \label{eqn:q52}
m^{2} < \frac{\lambda M_{pl}^{2}}{\xi},
\end{equation}
where the upper limit on $m^{2}$ corresponds to $\beta =0$. For the purposes of studying these Q-balls numerically, we will consider masses which give a value for $\beta$ in the range $\sim 0.1 - 1$, which correspond to inflaton field values at the end of inflation (approximated as the end of slow-roll) in the range $\phi_{end} \sim \left(1 - 3 \right) M_{pl}$. The size of this mass term directly affects the shape of the potential, but will only have a significant effect on the value of the field at the end of inflation for inflation masses squared close to the upper limit (<ref>).
§ DERIVATION OF THE Q-BALL EQUATION
In this section we derive the Q-ball equation in non-minimally coupled Palatini gravity. We work in the Einstein frame in flat space with the action
\begin{equation} \label{eqn:q53}
S = \int d^{4}x \; \; \frac{1}{\Omega^{2}}\partial_{\mu}\Phi^{\dagger} \partial^{\mu}\Phi - \frac{1}{\Omega^{4}}V\left(\left| \Phi \right| \right).
\end{equation}
We will show that the flat space calculation is a valid approximation for the Q-ball solutions that we will obtain. While it is true that gravitational effects can affect the stability of Q-balls within certain limits, or alter the size of the Q-balls (see e.g. [166], [167]), the attractive interaction between scalars dominates over any gravitational effects for Q-balls of the size we are working with in this model. The effects of gravity, and indeed the presence of a non-minimal coupling of gravity to the scalar field in the Jordan frame are factors to consider in the post-inflationary cosmology of this model and may affect the survival of any relic Q-balls to the present day. We we will later consider a first-pass approximation of the effects of curvature on these Q-ball solutions in Section <ref> of this chapter, and proceed with the understanding that the purpose of this work is to ascertain the flat space properties of these Q-balls. Expansion could affect the formation of these Q-balls and their subsequent evolution, and this is something we touch upon in Sections <ref> - <ref>. For the purposes of establishing the existence of Q-balls made from scalars with a non-minimal coupling to gravity without the cosmological considerations of their existence we derive the Q-ball equation using the energy-momentum tensor for a scalar field in a non-expanding spacetime, and compare these Q-balls to the conventional flat space Q-balls as derived in [50].
In order to derive the Q-ball equation we begin by using the method of Lagrange multipliers in order to minimise the energy of the inflaton field with respect to the conserved $U(1)$ charge of the field
\begin{equation} \label{eqn:q54}
E_{Q} = E + \omega \left( Q - \int d^{3}x \rho_{Q} \right),
\end{equation}
which gives a "Q-ball action", $E_{Q}$, which we will refer to as the Q-ball energy functional. This can be extremised to obtain the field equations of the theory, the solutions of which - subject to a number of conditions which we discuss in Sections <ref> - <ref> - correspond to Q-balls. The global energy, $E$, and charge, $Q$, are
\begin{equation} \label{eqn:q55}
Q = \int d^{3}x \; j^{0} = \int d^{3}x \rho_{Q},
\end{equation}
\begin{equation} \label{eqn:q56}
E = \int d^{3}x \; T^{00} = \int d^{3}x \rho_{E},
\end{equation}
where $j^{0}$ is the temporal component of the conserved $U(1)$ Noether current, $j^{\mu}$, and $T^{00}$ is the temporal component of the energy-momentum tensor $T^{\mu \nu}$ for a complex scalar field, given by
\begin{equation} \label{eqn:q57}
T^{\mu \nu} = \frac{\partial \mathcal{L}}{\partial \left(\partial_{\mu}\phi_{a}\right)}\eta^{\nu \rho}\partial_{\rho}\phi_{a} - \delta^{\mu}_{\rho} \eta^{\nu \rho}\mathcal{L},
\end{equation}
where $\eta^{\nu \rho}$ is the Minkowski metric. The energy density of the $\Phi$ field is therefore
\begin{equation} \label{eqn:q58}
\rho_{E} = T^{00} = \frac{1}{\Omega^{2}}\partial_{t}\Phi^{\dagger} \partial_{t}\Phi + \frac{1}{\Omega^{2}}\partial_{i}\Phi^{\dagger} \partial^{i}\Phi + \frac{V(\mid \Phi \mid)}{\Omega^{4}}.
\end{equation}
This includes the effect of the non-canonical kinetic term of $\Phi$, which has not been previously studied in the context of Q-ball solutions.
The conserved Noether current $j^{\mu}$ of the $U(1)$ symmetry of the model is defined by
\begin{equation} \label{eqn:q59}
\partial_{\mu} j^{\mu} = 0, j^{\mu} = \frac{\partial \mathcal{L}}{\partial \left(\partial_{\mu}\phi_{a}\right)}\delta \phi_{a}.
\end{equation}
The temporal components of (<ref>) are
\begin{equation} \label{eqn:q60}
j^{0} = \frac{\partial \mathcal{L}}{\partial \left(\partial_{t}\Phi \right)} i\Phi - \frac{\partial \mathcal{L}}{\partial \left(\partial_{t} \Phi^{\dagger}\right)} i\Phi^{\dagger},
\end{equation}
which from (<ref>) gives the charge density of the inflaton field to be
\begin{equation} \label{eqn:q61}
\rho_{Q} = \frac{i}{\Omega^{2}}\left( \Phi \partial_{t}\Phi^{\dagger} - \Phi^{\dagger}\partial_{t}\Phi \right).
\end{equation}
Substituting (<ref>) and (<ref>) into the Q-ball energy functional (<ref>) gives
\begin{equation} \label{eqn:q62}
E_{Q} = \int d^{3}x \left[ \frac{1}{\Omega^{2}}\partial_{t}\Phi^{\dagger} \partial_{t}\Phi + \frac{1}{\Omega^{2}}\partial_{i}\Phi^{\dagger} \partial^{i}\Phi + \frac{V(\left|\Phi \right|)}{\Omega^{4}} - \frac{\omega i}{\Omega^{2}}\left( \Phi \partial_{t}\Phi^{\dagger} - \Phi^{\dagger}\partial_{t}\Phi \right) \right] + \omega Q.
\end{equation}
The temporal derivative terms in (<ref>) can be rewritten as
\begin{equation}
\begin{split} \label{eqn:q63}
\left(\partial_{t}\Phi - i\omega \Phi \right) \left(\partial_{t}\Phi^{\dagger} + i\omega \Phi^{\dagger} \right)
& = \partial_{t}\Phi \partial_{t}\Phi^{\dagger} + i\omega \Phi^{\dagger} \partial_{t}\Phi - i\omega \Phi \partial_{t} \Phi^{\dagger} - \omega^{2}\Phi^{\dagger}\Phi \\
& = \mid \partial_{t}\Phi - i\omega \Phi \mid^{2},
\end{split}
\end{equation}
in order to rewrite the Q-ball energy functional in a more insightful way. If we define the time derivatives in the integrand of (<ref>) as
\begin{equation} \label{eqn:q64}
I = \frac{1}{\Omega^{2}}\left[ \partial_{t}\Phi^{\dagger}\partial_{t}\Phi + i\omega \Phi^{\dagger}\partial_{t}\Phi - i\omega \Phi \partial_{t}\Phi^{\dagger} \right],
\end{equation}
these can be rewritten as in (<ref>)
\begin{equation} \label{eqn:q65}
I = \frac{1}{\Omega^{2}}\left[ \mid \partial_{t}\Phi - i\omega \Phi \mid^{2} - \omega^{2} \mid \Phi \mid^{2} \right].
\end{equation}
Substituting (<ref>) back into the Q-ball energy functional (<ref>) we have that
\begin{equation} \label{eqn:q66}
E_{Q} = \int d^{3}x \left[ \frac{1}{\Omega^{2}} \mid \partial_{t}\Phi - i \omega \Phi \mid^{2} - \frac{1}{\Omega^{2}} \omega^{2} \mid \Phi \mid^{2} + \frac{1}{\Omega^{2}}\partial_{i}\Phi ^{\dagger} \partial^{i} \Phi + \frac{1}{\Omega^{4}}V\left(\mid \Phi \mid \right) \right] + \omega Q,
\end{equation}
whereupon it is clear that a solution which extremises $E_{Q}$ is
\begin{equation} \label{eqn:q67}
\Phi \left(x,t \right) = \Phi \left( x \right) e^{i \omega t} \longrightarrow \partial_{t} \Phi = i\omega \Phi.
\end{equation}
Applying (<ref>), the Q-ball energy functional (<ref>) becomes
\begin{equation} \label{eqn:q68}
E_{Q} = \int d^{3}x \left[ \frac{1}{\Omega^{2}} \mid \overrightarrow{\nabla} \Phi \mid^{2} + \frac{1}{\Omega^{4}} V\left(\mid \Phi \mid \right) - \frac{1}{\Omega^{2}}\omega^{2} \mid \Phi \mid^{2} \right].
\end{equation}
\begin{equation} \label{eqn:q69}
V_{\omega} \left( \mid \Phi \mid \right) = \frac{1}{\Omega^{4}}V \left( \mid \Phi \mid \right) - \frac{1}{\Omega^{2}}\omega^{2}\mid \Phi \mid^{2},
\end{equation}
be defined as the Q-ball potential. Assuming that the Q-balls are spherically symmetric, we assume that the coordinate dependence of the inflaton field, when extremised with respect to its charge, is purely radial, which gives
\begin{equation} \label{eqn:q70}
\Phi \left(\textbf{x}\right) = \frac{\phi \left(\textbf{r}\right)}{\sqrt{2}} e^{i\omega t} = \frac{\phi\left(r\right) \hat{r}}{\sqrt{2}}e^{i\omega t},
\end{equation}
as the field solution corresponding to Q-balls.
Since the coordinate dependence of the field is purely radial, the gradient operator is reduced to
\begin{equation} \label{eqn:q71}
\overrightarrow{\nabla}\Phi = \frac{\partial \Phi}{\partial r}\hat{r} \\
\longrightarrow \frac{1}{\Omega^{2}} \left| \overrightarrow{\nabla}\Phi \right| = \frac{1}{\Omega^{2}}\left| \frac{1}{\sqrt{2}}\frac{\partial \phi}{\partial r}\hat{r} \right|^{2} = \frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2}.
\end{equation}
Rewriting in spherical polar coordinates, we have that the Q-ball energy functional is
\begin{equation} \label{eqn:q72}
E_{Q} = \int dr \; 4\pi r^{2} \left[ \frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2} + V_{\omega} \left(\phi \right) \right] + \omega Q.
\end{equation}
We define the function $\mathcal{L}_{Q}$ (which we refer to as the "Q-ball effective Lagrangian") to be extremised as
\begin{equation} \label{eqn:q73}
\mathcal{L}_{Q}= 4\pi r^{2} \left[ \frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2} + V_{\omega} \left(\phi \right) \right],
\end{equation}
and apply the Euler-Lagrange equations (derived for a complex scalar field in flat space in (<ref>))
\begin{equation} \label{eqn:q74}
\frac{\partial \mathcal{L}_{Q}}{\partial \phi} - \frac{d}{dr}\left(\frac{\partial \mathcal{L}_{Q}}{\partial \left(\partial_{r}\phi\right)}\right) = 0,
\end{equation}
in order to extremise (<ref>).
Using (<ref>), the second term of (<ref>) is
\begin{multline} \label{eqn:q75}
\frac{d}{dr}\left(\frac{\partial \mathcal{L}_{Q}}{\partial \left(\partial_{r}\phi\right)}\right) = \frac{4\pi r^{2}}{\Omega^{2}} \frac{\partial \phi}{\partial r}\frac{d}{dr}\left(\frac{\partial \mathcal{L}_{Q}}{\partial \left(\partial_{r}\phi\right)}\right) \\
= \frac{8\pi r}{\Omega^{2}}\frac{\partial \phi}{\partial r} + \frac{4 \pi r^{2}}{\Omega^{2}}\frac{\partial^{2}\phi}{\partial r^{2}} - \frac{8 \pi r^{2}}{M_{pl}^{2}}\frac{\xi \phi}{\Omega^{4}}\left(\frac{\partial \phi}{\partial r}\right)^{2},
\end{multline}
and the first term of (<ref>) is
\begin{equation} \label{eqn:q76}
\frac{\partial \mathcal{L}_{Q}}{\partial \phi} = 4\pi r^{2} \frac{\partial V_{\omega}}{\partial \phi} - \frac{4 \pi r^{2}}{M_{pl}^{2}}\frac{\xi \phi}{\Omega^{4}}\left(\frac{\partial \phi}{\partial r}\right)^{2}.
\end{equation}
Substituting (<ref>) and (<ref>) into (<ref>), dividing through by a factor of $4\pi r^{2}$ and multiplying by $\Omega^{2}$ gives the Q-ball equation
\begin{equation} \label{eqn:q77}
\frac{\partial^{2}\phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} - K\left(\phi \right) \left(\frac{\partial \phi}{\partial r}\right)^{2} = \Omega^{2}\frac{\partial V_{\omega}}{\partial \phi},
\end{equation}
\begin{equation} \label{eqn:q78}
K(\phi) = \frac{\xi \phi}{M_{pl}^{2} \Omega^{2}}.
\end{equation}
Equation (<ref>) corresponds to an entirely new and different form of the Q-ball equation than that for a scalar theory with canonical kinetic terms, and therefore to a new class of Q-ball. The result of the Q-ball equation for the minimally coupled case [50] can be recovered by setting $\Omega^{2} = 1 \left(\xi = 0 \right)$
\begin{equation}\label{eqn:q79}
\frac{\partial^{2} \phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} = \frac{\partial V_{\omega}}{\partial \phi},
\end{equation}
where in this instance
\begin{equation}\label{eqn:q80}
V_{\omega} \left( \phi \right) = \frac{1}{2}m^{2}\phi^{2} + \frac{\lambda}{4}\phi^{4} - \frac{1}{2}\omega^{2}\phi^{2}.
\end{equation}
Where we see that the difference the presence of the non-minimal coupling for (<ref>) makes as compared to (<ref>) is in the additional gradient squared term, and the dependence of the conformal factor in the Q-ball potential and on the right-hand side. This shows that, although the non-minimal coupling of the scalar field to gravity is recast in the Einstein frame using the conformal transformation, its effects are still manifest in the construction of the Q-balls in the mathematical sense, in that the attractive interaction binding the scalars into Q-balls seems to contain an additional component due to the non-minimal coupling. Physically, in the Jordan frame, this would be gravitational in nature, whereas as the Q-balls are studied here in the Einstein frame, it is merely a component of the regular attractive interaction between the inflaton scalars.
§.§ Existence Conditions of Q-balls
The existence of Q-ball solutions in a given theory is heavily dependent on the form of the potential, and the realisation of physical Q-balls relies on the $\omega$ parameter being within an acceptable range. The physical reasons for this will be discussed in later sections. In this section we explore the form of the potential in this model of non-minimally coupled Palatini Q-balls, derive the constraints on the existence of Q-balls in this model from the potential and compare the results to those derived in the case of conventional Q-balls.
§.§.§ Rescaling the Non-Canonical Q-ball Equation in Terms of a Quasi-Canonical Scalar
The Palatini Q-ball equation in terms of the rescaled field $\sigma$ has a similar (but not the same) form as the general form of the Q-ball equation for the conventional case (<ref>), and the dynamics can therefore be understood as being similar in this form.
To illustrate this, we begin with the non-canonical Q-ball equation derived at the beginning of the section
\begin{equation}\label{eqn:q81}
\frac{\partial^{2} \phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} - \frac{\xi \phi}{M_{pl}^{2} \Omega^{2}} \left(\frac{\partial \phi}{\partial r}\right)^{2} = \Omega^{2}\frac{\partial V_{\omega}}{\partial \phi},
\end{equation}
and apply the rescaling
\begin{equation}\label{eqn:q82}
\frac{d\phi}{d r} \longrightarrow \Omega \frac{d\sigma}{d r},
\end{equation}
such that
\begin{equation}\label{eqn:q83}
\frac{d^{2}\phi}{d r^{2}} = \frac{d}{d r}\left(\frac{d\phi}{d r}\right) = \frac{d}{d r}\left(\Omega \frac{d\sigma}{d r}\right) = \Omega \frac{d^{2}\sigma}{d r^{2}} + \frac{d\Omega}{d r }\frac{d\sigma}{d r},
\end{equation}
\begin{equation}\label{eqn:q84}
\frac{d V_{\omega}}{d\phi} = \frac{d \sigma}{d \phi} \frac{d V_{\omega}}{d \phi} = \frac{1}{\Omega}\frac{d V_{\omega}}{d \sigma}.
\end{equation}
Substituting (<ref>) - (<ref>) into (<ref>) gives the rescaled Q-ball equation
\begin{equation}\label{eqn:q85}
\Omega \frac{d^{2}\sigma}{d r^{2}} + \frac{2\Omega}{r} \frac{d\sigma}{d r} + \frac{d\Omega}{d r} \frac{d \sigma}{d r} - \frac{\xi \phi \Omega^{2}}{M_{pl}^{2} \Omega^{2}}\left(\frac{d \sigma}{d r}\right)^{2} = \frac{\Omega^{2}}{\Omega}\frac{d V_{\omega}}{d \sigma}.
\end{equation}
We can write
\begin{equation}\label{eqn:q86}
\frac{d\Omega}{d r } = \frac{d\phi}{d r} \frac{d\Omega}{d\phi} = \frac{d\sigma}{d r} \frac{d\phi}{d\sigma} \frac{d\Omega}{d\phi},
\end{equation}
\begin{equation}\label{eqn:q87}
\frac{d\Omega}{d\phi} = \frac{d}{d\phi}\sqrt{1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}} = \frac{1}{2} \frac{2\xi \phi}{M_{pl}^{2}}\left(1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}\right)^{-\frac{1}{2}} = \frac{\xi \phi}{M_{pl}^{2} \Omega}.
\end{equation}
\begin{equation}\label{eqn:q88}
\frac{d\Omega}{d r} = \frac{d\sigma}{d r} \frac{d\phi}{d\sigma} \frac{d\Omega}{d\phi} = \frac{d\sigma}{d r} \Omega \frac{\xi \phi}{M_{pl}^{2} \Omega} = \frac{d\sigma}{d r} \frac{\xi \phi}{M_{pl}^{2}}.
\end{equation}
Using (<ref>) on the third term of (<ref>), the rescaled Q-ball equation becomes
\begin{equation}\label{eqn:q89}
\Omega \frac{d^{2}\sigma}{d r^{2}} + \frac{2\Omega}{r} \frac{d\sigma}{d r} + \frac{\xi \phi}{M_{pl}^{2}}\left(\frac{d \sigma}{d r}\right)^{2} - \frac{\xi \phi}{M_{pl}^{2}}\left(\frac{d \sigma}{d r}\right)^{2} = \Omega\frac{d V_{\omega}}{d \sigma}.
\end{equation}
We can then cancel the gradient squared terms and divide through by $\Omega$ to obtain the Q-ball equation in terms of the $\sigma$ field
\begin{equation}\label{eqn:q90}
\frac{d^{2}\sigma}{d r^{2}} + \frac{2}{r}\frac{d\sigma}{d r} = \frac{d V_{\omega}}{d\sigma},
\end{equation}
which is exactly the same form as the Q-ball equation for $\phi$ in the $\Omega \rightarrow 1$ limit (<ref>), but with $V_{\omega}$ given by (<ref>). This shows that this is not the same as the equation for a canonically normalised field, as in that case we would have $\sigma^{2}$ rather than $\phi^{2}(\sigma)/\Omega^{2}(\sigma)$ in (<ref>). Therefore (<ref>) will produce significantly different Q-ball solutions compared to the conventional Q-ball solution, although it does suggest that there are some broad similarities between the two solutions.
It is at this point that we highlight the fact that it is not in general possible to transform a complex field such as $\Phi$ to a canonically normalised scalar in the strict sense, and that the transformation used in this case to compare the dynamics of the non-minimally coupled case to conventional Q-balls is quasi-canonical. The rescaled field $\sigma$ is therefore not interpreted as a physically meaningful field, but a function of the radial component $\phi$ of the physical complex inflaton used to illustrate some of the properties of the non-canonical Q-ball solution.
§.§.§ Q-ball Existence and Coleman's Mechanical Analogy in the Case of Non-Minimally Coupled Palatini Q-balls
In this section we discuss the application of Coleman's mechanical analogy [50], originally used as a tool for understanding the precise nature of Q-ball solutions and the dependence of their existence on the form of the scalar potential, to Q-balls in the non-minimally coupled Palatini case. This analogy was originally derived for Q-balls in the canonical scalar field case, and the underlying dynamics in this model therefore look a little different, as we will discuss later.
We first introduce the analogy in the context of the conventional case of a canonically normalised scalar, for which $\sigma = \phi$. If we interpret the field $\sigma$ $(\phi)$ as a position coordinate, and $r$ as a time coordinate, then (<ref>) ((<ref>)) looks like it is describing the damped motion of a particle moving in the potential $-V_{\omega}$. Figure <ref> schematically shows the potential $-V_{\omega}$ and the motion of the "particle" within it .
Schematic potential and particle motion as described by (<ref>) and (<ref>).
The "particle" begins its motion at the black circle on the right, at a starting point we will refer to as $\phi_{0}$. The local maximum at the origin of the potential $\phi = 0$ corresponds to the field in the vacuum at $r \rightarrow \infty$, and the idea of this analogy is that in order to obtain a Q-ball solution in a given potential $V_{\omega}$, the conditions must be precisely right such that the particle rolling down from $\phi_{0}$ on the right comes to rest exactly at the origin as illustrated in Figure <ref>. One such condition is the initial placement, or starting location of the particle, $\phi_{0}$. If the particle is initially placed too close to $\phi = 0$ - $\phi_{0}$ is too small - then the particle will not gain enough momentum to reach the top of the local maximum at $\phi = 0$, and it will roll backwards to oscillate about the local minimum to the right in Figure <ref> (clear circle on the right hand side). This is known as undershoot, and can also occur if $\phi_{0}$ is sufficiently high but the gradient of the potential is too shallow. Conversely, if $\phi_{0}$ is too large (starting placement is too far from $\phi = 0$, or the potential itself is too steep on the approach to the positive $\phi$ local minimum, the field will gather too much momentum and will roll clear of the local maximum at $\phi = 0$ and into the negative $\phi$ regime, where it will oscillate about the negative $\phi$ local minimum on the left in Figure <ref> (clear circle on the left hand side). This is known as overshoot.
Plots of $-V_{\omega}\left(\phi \right)$ for $\phi > 0$ are shown in Figure <ref> and Figure <ref>. Interestingly, the inverted potential before rescaling has the same form as the inverted potential in the conventional case (comparing to the $\phi > 0$ region of the schematic in Figure <ref>) corresponding to (<ref>). This demonstrates that the idea of Coleman's mechanical analogy can still be applied to explore the dependence of the existence of non-canonical Q-balls on the form of the scalar field potential in the underlying theory, although the underlying dynamics are different. As shown at the beginning of this section, in the non-minimally coupled Palatini case, the Q-ball equation in terms of $\phi$, (<ref>), has an additional gradient squared term with a negative sign and a dependence on the conformal factor on the right-hand side, both explicitly and in the form of the Q-ball potential, $V_{\omega}$, in the non-minimally coupled case. The gradient squared term can be interpreted as some external energy input, which makes the motion of the particle dynamically different from the simple Newtonian motion described in (<ref>), although the inverted potential looks very similar and the general idea as an analogy for the precise conditions on the potential and the field itself to produce Q-balls is still applicable. When considered in terms of the rescaled field $\sigma$, the Q-ball equation, has the same form as in the conventional case but with a modified potential $V_{\omega}$, (<ref>). Equation (<ref>) can therefore also be used as a means for comparison to the conventional case when considering the mechanical analogy with the understanding that, in our definition of the quasi-canonical field, $\phi$ is a function of $\sigma$ such that $V_{\omega}\left(\phi \right) = V_{\omega}\left(\phi \left(\sigma \right) \right)$.
$-V_{\omega}(\phi)$ for a non-minimally coupled Palatini Q-ball solution with large $\omega$ .
$-V_{\omega}(\phi)$ for a non-minimally coupled Palatini Q-ball solution with small $\omega$.
We can now use the rules derived from this analogy [50] for conventional minimally-coupled Q-balls to derive an analogous set of rules for the case of non-minimally coupled Q-balls to ensure that the Q-ball potential is compatible with producing Q-ball solutions. To avoid undershoot, the initial placement of the particle ($\phi_{0}, -V_{\omega}\left(\phi_{0}\right)$), must sit at least at, or higher than zero in order for the particle to reach the local maximum of the origin, otherwise the particle will not gather sufficient momentum. This condition can be expressed as
\begin{equation}\label{eqn:q91}
max\left(-V_{\omega}\right) \geq 0,
\end{equation}
as it is in the conventional model of Q-balls [50], and is significant because it implies that there is a lower bound on $\omega$ for which Q-balls can exist in a given potential. For the $V_{\omega}$ used in this model (<ref>), this result does not appear to produce anything meaningful analytically, which is not an unexpected result as the inflaton potential in the Einstein frame corresponds to a plateau. It is possible that there is another way to attain a lower bound on the value of $\omega$ for non-minimally coupled Palatini Q-balls, and we will return to this possibility in Section <ref> when numerical solutions of the Q-balls are explored, although a definitive lower bound on $\omega$ in this model is at the point of submitting this thesis unconfirmed.
To avoid overshoot we require that there is an extremum at $\phi = 0$ which is a local maximum, so that the particle can come to rest in the vacuum and not continue rolling into the $\phi < 0$ region
\begin{equation}\label{eqn:q92}
\left.\frac{d^{2}\left(-V_{\omega} \right)}{d\phi^{2}}\right|_{\phi = 0} < 0,
\end{equation}
this condition may be satisfied if and only if
\begin{equation}\label{eqn:q93}
\left.\omega^{2} < \frac{d^{2}V}{d\phi^{2}} \right|_{\phi = 0}.
\end{equation}
Calculating the left-hand side of (<ref>) we have
\begin{equation}\label{eqn:q94}
\frac{d^{2}\left(-V_{\omega} \right)}{d\phi^{2}} = \frac{\omega^{2}}{\Omega^{2}} - \frac{m^{2}}{\Omega^{4}} - \frac{3\lambda \phi^{2}}{\Omega^{4}}.
\end{equation}
At $\phi = 0$, $\Omega =1$ and (<ref>) therefore gives the result
\begin{equation}\label{eqn:q95}
\left.\frac{d^{2}\left(-V_{\omega} \right)}{d\phi^{2}}\right|_{\phi = 0} = \omega^{2} - m^{2} < 0 \Rightarrow \omega^{2} < m^{2},
\end{equation}
and from (<ref>) we find
\begin{equation}\label{eqn:q96}
\left.\omega^{2} < \frac{d^{2}V}{d\phi^{2}} \right|_{\phi = 0} = m^{2}.
\end{equation}
This means that
\begin{equation}\label{eqn:q97}
\omega < m,
\end{equation}
is a hard constraint on the existence of non-minimally coupled Palatini Q-balls, and is the same result as that on the existence of conventional minimally-coupled Q-balls derived from the mechanical analogy in [50], as the condition to avoid overshoot.
§.§ Q-ball Stability
Stability of Q-balls is an important property, particularly when discussing Q-balls as cosmological objects. A Q-ball is said to be absolutely stable if [50]
\begin{equation}\label{eqn:q98}
E < mQ,
\end{equation}
holds. $E$ is the energy of the Q-ball and $mQ$ can be interpreted as the energy of $Q$ free quanta of scalar particles of mass $m$ in the vacuum. This essentially means that in order to be absolutely stable the energy of a Q-ball must be less than the sum of the energies of the individual component scalars it is composed of when treated as free quanta. This difference in energy corresponds to the binding energy of the Q-ball. Classical stability implies a solution that is stable against small perturbations. As an approximation, the classical stability of a Q-ball can be gauged by the sign of the derivative of the charge with respect to $\omega$, i.e. if $\partial Q/\partial \omega <0$ then a Q-ball is classically stable [141] [142]. The absolute stability of the numerically generated non-minimally coupled Palatini Q-balls is discussed in Section <ref>.
§.§ Mass Range of the Inflaton from Inflation and the Existence of Q-balls
In Section <ref>, we demonstrated that the form of the inflaton potential - and the state of the inflaton field at the end of inflation - depends explicitly on the size of the mass term (<ref>), which means that the size of this mass term determines the initial conditions for Q-balls in non-minimally coupled Palatini inflation. It is therefore important that we establish the range of inflaton masses which are compatible with the existence condition (<ref>) of these Q-balls. In Section <ref>, we derived the constraint (<ref>) which gives an upper bound on the inflaton mass squared, from the condition that the inflaton potential must be of positive gradient in order for inflation to proceed. In this section, we use this, and the existence condition of non-minimally coupled Palatini Q-balls derived in the previous section, (<ref>), to derive a range of inflaton masses for which both inflation can occur and inflatonic Q-balls can exist. We will refer to this range of compatible inflaton masses which produce both inflation and Q-balls as the "Q-ball window".
We begin with the non-canonical Q-ball potential
\begin{equation}\label{eqn:q99}
V_{\omega} \left(\phi \right) = \frac{1}{\Omega^{4}}\left(\frac{1}{2} m^{2} \phi^{2} + \frac{\lambda}{4} \phi^{4} \right) - \frac{\omega^{2} \phi^{2}}{2\Omega^{2}},
\end{equation}
and expand to leading order in $M_{pl}^{2}/\xi \phi^{2}$ to give
\begin{equation}\label{eqn:q100}
V_{\omega} \left(\phi \right) = \frac{M_{pl}^{4}}{2\xi^{2}\phi^{2}}\left( m^{2} + \omega^{2} - \frac{2m^{2}M_{pl}^{2}}{\xi \phi^{2}} \right) - \frac{\lambda M_{pl}^{6}}{2 \xi^{3}\phi^{2}} + \frac{\lambda M_{pl}^{4}}{4\xi^{2}} - \frac{\omega^{2}M_{pl}^{2}}{2\xi}.
\end{equation}
In other words,
\begin{equation} \label{eqn:q101}
V_{\omega}\left(\phi \right) = \frac{M_{pl}^{4}}{2\xi^{2}\phi^{2}}\left[ m^{2} + \omega^{2} - \frac{\lambda M_{pl}^{2}}{\xi} \right] + \hspace{1mm}higher\hspace{1.0mm} order\hspace{1.0mm} and \hspace{1.0mm} constant ~\hspace{0.8mm} terms.
\end{equation}
For (<ref>) to yield a Q-ball solution, the solution $\phi \left(r \right)$ must decrease as $r$ increases from zero. This is true if the $\phi$ dependent term on the right hand side of (<ref>) is positive. The existence condition for a Q-ball from the Q-ball potential (<ref>) is therefore
\begin{equation}\label{eqn:q102}
m^{2} + \omega^{2} > \frac{\lambda M_{pl}^{2}}{\xi}.
\end{equation}
In Section <ref> we will analytically demonstrate that if $\omega < m$, as required in order for Q-balls to exist, then the condition (<ref>) must also be satisfied in order for the right-hand side of the Q-ball equation to have zeroes for some $\phi \neq 0$. Both interpretations of (<ref>) correspond to a necessary condition on the Q-ball potential for Q-ball solutions to exist in the model.
We define
\begin{equation}\label{eqn:q103}
\omega_{c}^{2} = \frac{\lambda M_{pl}^{2}}{\xi},
\end{equation}
and use this to rewrite (<ref>)
\begin{equation} \label{eqn:q104}
m^{2} + \omega^{2} > \omega_{c}^{2}.
\end{equation}
If we combine (<ref>) with the condition for the inflaton potential to be compatible with inflation (<ref>), which we can write as $m^{2} < \omega_{c}^{2}$, we find the condition for inflation and the existence of Q-balls in terms of the $\omega$ parameter and the inflaton mass to be
\begin{equation}\label{eqn:q105}
m^{2} < \omega_{c}^{2} < m^{2} + \omega^{2}.
\end{equation}
The existence of Q-balls also requires that $\omega < m$ (<ref>), and the right-hand side of the inequality (<ref>) can be constrained
\begin{equation}\label{eqn:q106}
\omega^{2} + m^{2} < 2m^{2}.
\end{equation}
(<ref>) can therefore be stated as
\begin{equation}\label{eqn:q107}
m^{2} < \omega_{c}^{2} < 2m^{2},
\end{equation}
which may be equivalently written as an range of inflaton masses squared
\begin{equation} \label{eqn:q108}
\frac{\omega_{c}^{2}}{2} < m^{2} < \omega_{c}^{2}.
\end{equation}
This gives the range in inflaton mass squared (the Q-ball window) for which the inflaton potential is compatible with both inflation and with the existence of Q-balls. It also provides a mass window within which we can numerically search for Q-ball solutions which are compatible with inflation, the topic of the next section. From (<ref>), the Q-ball window seems to favour inflaton masses close to the upper bound derived from inflation (<ref>), and while Q-balls may form for inflaton masses larger than the upper bound (<ref>) in this model, inflation would not be possible in conjunction with these Q-balls.
§ NUMERICAL SOLUTIONS OF THE Q-BALL EQUATION
In this section we discuss and present the results of the numerical analysis of non-minimally coupled Palatini Q-balls. In this project we solved the non-canonical Q-ball equation (<ref>) numerically for a fixed mass within the range specified in (<ref>), over a range of $\omega$ for the following boundary conditions at $r=0$
\begin{equation} \label{eqn:q109}
\phi \left(r =0\right) = \phi_{0} \;\;\;\; ; \;\;\;
\frac{\partial \phi}{\partial r} \left(r=0\right) = 0.
\end{equation}
For each $\omega$ we ran the code for trial values of the inflaton field, corresponding to $\phi_{0}$, and scanned for Q-ball solutions. The details of this methodology are outlined in the following subsection. When discussing numerical results, the inflaton mass $m$ and the parameter $\omega$ are listed in units of $\omega_{c}$.
§.§ Zeroes of the Q-ball Equation
In this section we discuss the zeroes of the right-hand side Q-ball equation (<ref>), and how they relate to the existence of Q-ball solutions. These zeroes are shown in Figure <ref>, and correspond to fixed points of the Q-ball equation, which in turn correspond to the extrema of the Q-ball effective potential $V_{\omega}\left(\phi\right)$, shown schematically in Figure <ref>. The fixed points of the Q-ball equation can be related to the mechanical analogy outlined in Section <ref> and more importantly, used to constrain $\phi_{0}$ when searching for Q-ball solutions numerically.
The non-minimally coupled Q-ball equation derived in this work (<ref>) has three zeroes (shown in Figure <ref>). These correspond to two stable fixed points, which are symmetric attractors about $\Omega^{2}\partial V_{\omega}/\partial \phi = 0$, and one unstable fixed point.
Plot illustrating the zeroes of the Q-ball equation.
Figure <ref> shows an example of the right hand side of the Q-ball equation as a function of $\phi$. The points where the function crosses zero are the zeroes or fixed points of the Q-ball equation. The attractor fixed points are the furthest to the left (denoted $\phi_{-}$) in the negative region of $\phi$ and the furthest to the right (denoted $\phi_{+}$) in the positive regime of $\phi$. These provide a reference for determining the value of $\phi$ at $r = 0$, $\phi_{0}$, for which a Q-ball solution exists. If the field gets caught by - and starts to oscillate around - the positive fixed point, $\phi_{+}$, as $r$ increases, then $\phi_{0}$ is too low to produce a Q-ball. The lowest $\phi_{0}$ which produces this result can then be used as a lower bound for the range of $\phi_{0}$ which could generate a Q-ball solution for a given $\omega$. In the opposite limit, if $\phi_{0}$ is too high then the field can drop below zero as $r$ increases and begin to oscillate around the negative fixed point, $\phi_{-}$. The lowest $\phi_{0}$ which produces this result can then act as an upper bound on the range of $\phi_{0}$ for which there could exist a Q-ball solution for the chosen $\omega$. The fixed points of the Q-ball equation therefore produce behaviour in the function of the field over distance, $\phi \left(r \right)$, which can be used to restrict the parameter space of $\phi_{0}$ used when searching for Q-ball solutions numerically.
This can be understood in terms of the mechanical analogy of Q-balls outlined in Section <ref> by noting that the fixed points of the Q-ball equation correspond to the extrema of the inverted Q-ball effective potential $-V_{\omega}$, in the analogy. The local minimum in the $\phi >0$ region of Figure <ref> corresponds to the positive attractor fixed point $\phi_{+}$ (first zero from the right in Figure <ref>). The unstable fixed point corresponds to the local maximum at $\phi =0$ in Figure <ref>, and is the central fixed point in Figure <ref>. The local minimum in the $\phi < 0$ region of the potential in Figure <ref> corresponds to the symmetric negative fixed point $\phi_{-}$ (furthest zero from the right in Figure <ref>).
This illustrates how precise the values of the parameters need to be to obtain a Q-ball solution numerically, as well as how strict the conditions on the potential in order to produce Q-balls are. The field profile for a Q-ball solution is represented by $\phi\left(r\right)$ asymptoting to zero as $r \rightarrow \infty$, corresponding to the field coming to rest exactly at the top of the local maximum at $\phi = 0$ in Figure <ref> from its starting point $\phi_{0}$ in the language of Coleman's analogy. It is also important to emphasise that the existence of the zeroes of the Q-ball equation determine the existence of the Q-ball solutions themselves, without the fixed points there cannot be Q-ball solutions.
We now derive the condition for the existence of zeroes of the Q-ball equation. The requirement for extrema to exist is that $\partial V_{\omega}/\partial \phi = 0$ for some $\phi \neq 0$. Using (<ref>) and setting the first derivative to zero gives the location of the symmetric $\phi \neq 0$ zeroes to be
\begin{equation} \label{eqn:q110}
\phi = \pm \frac{M_{Pl}}{\sqrt{\xi}} \frac{\sqrt{m^{2} - \omega^{2}}}{\sqrt{m^{2} + \omega^{2} - \omega_{c}^{2} } }.
\end{equation}
In order for the Q-ball solution to be stable we require that $\omega^{2} < m^{2}$. This will be verified both numerically later in this section and analytically in Section <ref>. The range of values of $m^{2}$ for which relevant zeros of the Q-ball equation exist is therefore
\begin{equation}\label{eqn:q111}
m^{2} + \omega^{2} > \omega_{c}^{2},
\end{equation}
which is exactly the condition (<ref>) derived from the Q-ball equation. We confirm numerically in Section <ref> that zeroes with $\phi \neq 0$ exist over the range of $\omega$ satisfying (<ref>) and (<ref>).
§.§ Numerical Results
In this section we present the results of solving the non-minimally coupled Palatini Q-ball equation numerically, and discuss the properties of these Q-balls. We present ten Q-ball solutions for the inflaton field of mass $m = 0.9\omega_{c}$, which is within the Q-ball window (<ref>), close to the upper bound of inflaton masses which inflate the model (<ref>), for a range of $\omega$ from $0.89\omega_{c}$ to $0.707155\omega_{c}$. This produces Q-balls with initial field values in the range $\phi_{0} \sim 10^{13} - 10^{17} \GeV$, illustrating that Q-balls could be produced from the kind of field values predicted at the end of non-minimally coupled Palatini inflation in this model. We present the radii, charge and energy of the Q-balls, in addition to some other parameters in relation to stability of the Q-balls which we define later in this section. The bounds on $\omega$ are chosen because they lie close to the bounds for which a Q-ball solution can be obtained for $m = 0.9\omega_{c}$. The upper bound, $\omega = 0.89\omega_{c}$, lies close to the stability limit of $\omega = m$ for Q-balls of the chosen mass, whereas the lower limit was discovered numerically while searching for Q-ball solutions for $m = 0.9\omega_{c}$. It may be expected that there are Q-balls with $\phi_{0}$ larger than this - of particular interest being a $\phi_{0} \sim 10^{18}\GeV$ Q-ball, discussed in depth in Section <ref> - but the precision on $\omega$ would need to be even greater to find it. It is interesting to note that the lower limit is close to being $\omega \sim \omega_{c}/\sqrt{2}$, which may have some kind of theoretical relevance to the existence of Q-balls in this framework.
Since there is no fixed definition of the radius of the Q-ball, two different possibilities are considered in this work for the purposes of evaluating the numerical Q-ball solutions. One possibility considered is a definition we will call $r_{X}$ - referred to as "the $X$ point" when discussing the numerical procedure - and a second definition, $r_{Z}$ - similarly "the $Z$ point" - as two different possibilities for understanding the concept of the Q-ball's "edge" when evaluating the properties numerically. The $X$ point is defined as being at a distance $r = r_{X}$ from the centre of the Q-ball $r = 0$ at which the field has decreased to $1 \%$ of its initial value $\phi_{0}$. This definition allows the properties of the Q-ball solutions to be examined using a fixed definition of what the radius is for all $\phi_{0}$. The $Z$ point is defined as being the point at which the code used to find a Q-ball solution for a given $\phi_{0}$ and $\omega$ cuts off having found a Q-ball solution. This distance from the centre of the Q-ball appears in the Q-ball profile, $\phi \left(r \right)$, as the point at which the field stops asymptoting towards zero and begins to increase again, moving away from the expected behaviour of a Q-ball solution. This occurs at $r = r_{Z}$ and this point corresponds to the maximum value of $r$ for which the Q-ball solution is valid for a given $\phi_{0}$ and $\omega$. Although less well defined than the $X$ point, since it will be different for every Q-ball solution, the $Z$ point cutoff is the more useful definition for calculating the energy $E$ and charge $Q$ of the Q-balls since the $Z$ point corresponds to the upper value of $r$ to which the Q-ball equation is integrated.
Two tables of values for the Q-ball properties are therefore presented, corresponding to those calculated at the $X$ point and the $Z$ point, as an exploration into how the radius of the Q-ball should be interpreted numerically and whether it makes any significant difference to the predicted properties of the Q-balls. Quantities with a subscript $X$ are calculated at the $X$ point, and those with a $Z$ are calculated at the $Z$ point.
Figures <ref> - <ref> show two Q-ball solutions, calculated numerically and using an analytical approximation derived in Section <ref>, (<ref>). Figures <ref> and <ref> show a Q-ball with $\phi_{0} \sim 10^{16}\GeV$, and the Figures <ref> and <ref> show a Q-ball with $\phi_{0} \sim 10^{13}\GeV$, with $\omega$ close to the upper bound of compatibility with the existence of Q-balls, $\omega = m$. The inflaton self-coupling is $\lambda = 0.1$ throughout and the corresponding non-minimal coupling is $\xi = 1.2163 \times 10^{9}$, using the estimate from (<ref>) derived in Section <ref>.
Q-ball solution for $\omega = 0.709\omega_{c} $, $m=0.9\omega_{c}$ and $\phi_{0} = 1.3464098 \times 10^{16} \GeV$ obtained by solving (<ref>) numerically.
Q-ball solution for $\omega = 0.709\omega_{c} $, $m=0.9\omega_{c}$ and $\phi_{0} = 1.3464098 \times 10^{16} \GeV$ obtained using the analytical solution (<ref>).
Q-ball solution for $\omega = 0.89\omega_{c}$, $m=0.9\omega_{c}$ and $\phi_{0} = 4.7918\times 10^{13} \GeV$ obtained by solving (<ref>) numerically.
Q-ball solution for $\omega = 0.89\omega_{c}$, $m=0.9\omega_{c}$ and $\phi_{0} = 4.7918\times 10^{13} \GeV$ obtained using the analytical approximation (<ref>).
In Tables <ref> and <ref> we present the numerical properties of the Q-ball solutions using the $r_{Z}$ and $r_{X}$ definitions of the Q-ball radius, respectively. Quantities recorded are the $\omega$ parameter; the value of the field at $r = 0$, $\phi_{0}$; the radius of the Q-ball, $r_{Z}$ or $r_{X}$; the value of the field at the defined edge of the Q-ball, and the total energy, $E$, charge, $Q$, and the energy-charge ratio, $E/Q$ of the Q-ball.
In order to assess whether the Q-balls found are stable, we introduce the quantities $\Delta_{\omega}$ and $\Delta_{m}$. $\Delta_{\omega}$ is defined by
\begin{equation}\label{eqn:q112}
\Delta_{\omega} = \frac{1}{\omega}\left(\frac{E}{Q} - \omega \right).
\end{equation}
This quantity measures the extent to which the $\omega$ parameter for each Q-ball solution is equal to the energy-charge ratio. The relationship between $\omega$ and the energy-charge ratio is established analytically for this model in Section <ref>.
$\Delta_{m}$ is defined by
\begin{equation}\label{eqn:q113}
\Delta_{m} = \frac{1}{m}\left(\frac{E}{Q} - m \right).
\end{equation}
This quantity provides a measure of absolute stability. As discussed in Section <ref>, absolute stability is defined by the energy of a Q-ball in relation to the energy of its component scalars if they were free particles, $E < mQ$. $\Delta_{m} < 0$ is therefore the condition for a Q-ball is to be absolutely stable.
max width=
$\omega / \omega_{c}$ $\phi_{0} / \GeV$ $\phi_{Z}/\GeV$ $r_{Z}/\GeV^{-1}$ $Q_{Z}$ $E_{Z}/\GeV$ $(E/Q)_{Z}/\GeV$ $\Delta_{\omega, Z}$ $ \Delta_{m, Z}$
$0.707155$ $3.2217991 \times 10^{17} $ $1.61 \times 10^{10}$ $8.94 \times 10^{-10}$ $2.17 \times 10^{14}$ $3.34 \times 10^{27}$ $1.54 \times 10^{13}$ $-2.79 \times 10^{-4}$ $-0.21$
$0.709$ $1.3464098 \times 10^{16}$ $1.63 \times 10^{10}$ $3.79 \times 10^{-11}$ $1.56 \times 10^{10}$ $2.40 \times 10^{23}$ $1.54 \times 10^{13}$ $9.71 \times 10^{-4}$ $-0.21$
$0.71$ $8.855792 \times 10^{15}$ $2.05 \times 10^{10}$ $2.50 \times 10^{-11}$ $4.37 \times 10^{9}$ $6.76 \times 10^{22}$ $1.55 \times 10^{13}$ $1.64 \times 10^{-3}$ $-0.21$
$0.72$ $1.960795 \times 10^{15}$ $9.76 \times 10^{9}$ $6.09 \times 10^{-12}$ $4.43 \times 10^{7}$ $6.99 \times 10^{20}$ $1.58 \times 10^{13}$ $7.71 \times 10^{-3}$ $-0.19$
$0.73$ $1.090258 \times 10^{15}$ $6.60 \times 10^{9}$ $3.73 \times 10^{-12}$ $7.09 \times 10^{6}$ $1.14 \times 10^{20}$ $1.61 \times 10^{13}$ $1.30 \times 10^{-2}$ $-0.18$
$0.74$ $7.45339 \times 10^{14}$ $1.61 \times 10^{10}$ $2.72 \times 10^{-12}$ $2.16 \times 10^{6}$ $3.54 \times 10^{19}$ $1.64 \times 10^{13}$ $1.76 \times 10^{-2}$ $-0.16$
$0.75$ $5.61953 \times 10^{14}$ $1.14 \times 10^{10}$ $2.28 \times 10^{-12}$ $8.85 \times 10^{5}$ $1.48 \times 10^{19}$ $1.67 \times 10^{13}$ $2.17 \times 10^{-2}$ $-0.15$
$0.80$ $2.29632 \times 10^{14}$ $1.96 \times 10^{10}$ $1.47 \times 10^{-12}$ $5.60 \times 10^{4}$ $1.01 \times 10^{18}$ $1.80 \times 10^{13}$ $3.51 \times 10^{-2}$ $-8.00 \times 10^{-2}$
$0.85$ $1.16877 \times 10^{14}$ $1.57 \times 10^{10}$ $1.49 \times 10^{-12}$ $1.01 \times 10^{4}$ $1.93 \times 10^{17}$ $1.91 \times 10^{13}$ $3.51 \times 10^{-2}$ $-2.24 \times 10^{-2}$
$0.89$ $4.7918 \times 10^{13}$ $8.82 \times 10^{9}$ $2.70 \times 10^{-12}$ $3.98 \times 10^{3}$ $7.82 \times 10^{16}$ $1.97 \times 10^{13}$ $1.50 \times 10^{-2}$ $3.74 \times 10^{-3}$
Table listing important properties of the $m=0.9\omega_{c}$ Q-balls, calculated using the $Z$ point definition of the Q-ball radius. $\phi_{0}$ denotes the value of the field at $r = 0$. $\phi_{Z}$, $E_{Z}$, $Q_{Z}$, $(E/Q)_{Z}$, $\Delta_{\omega, Z}$ and $\Delta_{m, Z}$ are calculated numerically at the point $r_{Z}$.
max width=
$\omega / \omega_{c}$ $\phi_{0}/\GeV$ $\phi_{X}/\GeV$ $r_{X}/\GeV^{-1}$ $Q_{X}$ $E_{X}/\GeV$ $(E/Q)_{X}/\GeV$ $\Delta_{\omega, X}$ $\Delta_{m, X}$
$0.707155$ $3.2217991 \times 10^{17} $ $3.22 \times 10^{15}$ $8.89 \times 10^{-10}$ $2.15 \times 10^{14}$ $3.30 \times 10^{27}$ $1.54 \times 10^{13}$ $-4.47 \times 10^{-4}$ $-0.22$
$0.709$ $1.3464098 \times 10^{16}$ $1.35 \times 10^{14}$ $3.70 \times 10^{-11}$ $1.54 \times 10^{10}$ $2.38 \times 10^{23}$ $1.54 \times 10^{13}$ $ -1.27 \times 10^{-3}$ $-0.21$
$0.71$ $8.855792 \times 10^{15}$ $8.86 \times 10^{13}$ $2.43 \times 10^{-11}$ $4.34 \times 10^{9}$ $6.70 \times 10^{22}$ $1.54 \times 10^{13}$ $-7.78 \times 10^{-4}$ $-0.21$
$0.72$ $1.960795 \times 10^{15}$ $1.96 \times 10^{13}$ $5.38 \times 10^{-12}$ $4.42 \times 10^{7}$ $6.97 \times 10^{20}$ $1.58 \times 10^{13}$ $6.58 \times 10^{-3}$ $-0.19$
$0.73$ $1.090258 \times 10^{15}$ $1.09 \times 10^{13}$ $3.03 \times 10^{-12}$ $7.08 \times 10^{6}$ $1.14 \times 10^{20}$ $1.61 \times 10^{13}$ $1.23 \times 10^{-2}$ $-0.18$
$0.74$ $7.45339 \times 10^{14}$ $7.45 \times 10^{12}$ $2.13 \times 10^{-12}$ $2.16 \times 10^{6}$ $3.54 \times 10^{19}$ $1.64 \times 10^{13}$ $1.71 \times 10^{-2}$ $-0.16$
$0.75$ $5.61953 \times 10^{14}$ $5.62 \times 10^{12}$ $1.66 \times 10^{-12}$ $8.85 \times 10^{5}$ $1.48 \times 10^{19}$ $1.67 \times 10^{13}$ $2.13 \times 10^{-2}$ $-0.15$
$0.80$ $2.29632 \times 10^{14}$ $2.30 \times 10^{12}$ $8.97 \times 10^{-13}$ $5.60 \times 10^{4}$ $1.01 \times 10^{18}$ $1.80 \times 10^{13}$ $3.48 \times 10^{-2}$ $-8.02 \times 10^{-2}$
$0.85$ $1.16877 \times 10^{14}$ $1.17 \times 10^{12}$ $7.84 \times 10^{-13}$ $1.01 \times 10^{4}$ $1.93 \times 10^{17}$ $1.91 \times 10^{13}$ $3.49 \times 10^{-2}$ $-2.26 \times 10^{-2}$
$0.89$ $4.7918 \times 10^{13}$ $4.79 \times 10^{11}$ $1.21 \times 10^{-12}$ $3.96 \times 10^{3}$ $7.79 \times 10^{16}$ $1.97 \times 10^{13}$ $1.50 \times 10^{-2}$ $3.69 \times 10^{-3}$
Table listing important properties for the $m=0.9\omega_{c}$ Q-balls calculated using the $X$ point definition of the Q-ball radius. $\phi_{0}$ is the value of the scalar field at $r = 0$. $\phi_{X}$, $E_{X}$, $Q_{X}$, $(E/Q)_{X}$, $\Delta_{\omega, X}$ and $\Delta_{m, X}$ are calculated numerically at the point $r_{X}$.
The energy-charge ratio increases by less than an order of magnitude from the largest $\phi_{0}$ Q-ball to the smallest at $\omega = 0.89\omega_{c}$. Comparing the data in Tables <ref> and <ref> also shows that the calculated energy, $E$, charge $Q$, and energy-charge ratio, $E/Q$, using the $X$ point definition of radius are comparable to the values obtained for these quantities calculated using the $Z$ point definition for each Q-ball. The radii for each Q-ball are similarly close in magnitude for each definition, with the difference between $r\left(X\right)$ and $r\left(Z\right)$ becoming slightly more pronounced in the smaller $\phi_{0}$ Q-balls.
The larger $\phi_{0}$ Q-balls have generally larger radii and the increase in radius is linear with $\phi_{0}$ to a good approximation for $\phi_{0} > M_{Pl}/\sqrt{\xi} \; ( \sim 6.9 \times 10^{13} \GeV)$. Increasing $\phi_{0}$ also increases the Q-ball energy and charge, as expected since a physically larger Q-ball (larger radius) will be composed of a larger number of scalars and will therefore carry a greater charge and energy. Figures <ref> and <ref> show the relationships between the logarithm of the energy of the Q-balls and the logarithm of $\phi_{0}$, and the logarithm of the charge of the Q-balls and the logarithm of $\phi_{0}$ respectively. We can see from these plots that $E$ and $Q$ are proportional to $\phi_{0}^{3}$ to a good approximation for the larger $\phi_{0}$ Q-balls, and that the approximate proportionality becomes less representative for the smaller $\phi_{0}$ Q-balls. This proportionality is predicted by the analytical approximation derived in Section <ref>, and is therefore a useful test of the validity of the numerical Q-ball solutions.
Log-log plot of $\ln Q_{z}$, vs. $\ln \phi_{0}$ for the $m=0.9\omega_{c}$ Q-balls.
Log-log plot of $\ln E_{z}$, vs. $\ln \phi_{0}$ for the $m=0.9\omega_{c}$ Q-balls.
As we can see from Tables <ref> and <ref>, $\left| \Delta_{\omega} \right| \sim 10^{-4} - 10^{-2}$ in general. This shows that $\omega$ is close to the value of the energy-charge ratio, and therefore to a good approximation can be equated to the chemical potential for the Q-ball solutions considered in this work. $ \left| \Delta_{\omega} \right| $ increases as $\phi_{0}$ decreases, and from Figure <ref> it is clear that that $ \Delta_{\omega} $ increases as Q-ball radius decreases in general. This is consistent with the patterns seen in the other quantities considered, in that the behaviour patterns of the properties for the smaller $\phi_{0}$ Q-balls are less well-defined than for the larger $\phi_{0}$ Q-balls. It is possible that this is a reflection of the Q-balls becoming unstable as the potential begins to deviate from the plateau ($\phi_{0} \rightarrow M_{Pl}/\sqrt{\xi}$).
$\Delta_{\omega,X}$ vs $r_{X}$ for the $m=0.9\omega_{c}$ Q-balls.
We can see from Tables <ref> and <ref>, and clearly in Figure <ref>, that $\Delta_{m}$ is negative for all Q-ball solutions considered except the $\omega = 0.89\omega_{c}$ Q-ball, meaning that all of the Q-balls considered are absolutely stable except for the $\omega = 0.89\omega_{c}$ case. Since the magnitude of $\Delta_{m}$ for $\omega = 0.89\omega_{c}$ is still small although it is positive, it is possible that $\Delta_{m}$ is actually negative but smaller than the level of numerical errors in the Q-ball calculation.
$\Delta_{m,X}$ vs $r_{X}$ for the $m=0.9\omega_{c}$ Q-balls.
The most significant result of the numerical analysis is that absolutely stable Q-balls of $\phi_{0} \geq 10^{17} \GeV$ can be generated in this model. This shows that the Q-ball mass window allows for the formation of Q-balls from field values we expect at the end of non-minimally coupled Palatini inflation (see Section <ref>). It is therefore possible that these Q-balls could form from the fragmentation of the inflaton condensate following tachyonic preheating at the end of non-minimally coupled Palatini inflation [168]. The presence of these Q-balls could significantly affect the post-inflation cosmology of the model, with consequences such as faster reheating and the production of gravitational waves from the decay of these Q-balls, leading to measurable effects. This is discussed further in Sections <ref> - <ref>.
§ ANALYTICAL APPROXIMATION OF THE Q-BALL SOLUTION AND ESTIMATE OF Q-BALL PROPERTIES
In this section we derive an analytical approximation of the Q-ball solution, and calculate the energy and charge in this approximation as an estimate of the Q-ball properties. We compare these to the results obtained from the numerical solutions in Section <ref> and derive some important relations between energy and charge.
In this model we are assuming that inflation ends while the inflaton field is on the inflationary plateau. Using the plateau limit (<ref>), we can approximate the right hand side of the Q-ball equation (<ref>) to be
\begin{equation}\label{eqn:q114}
\Omega^{2}\frac{\partial V_{\omega}}{\partial \phi} = - \frac{\gamma}{\phi},
\end{equation}
to leading order in $1/\phi^{2}$, where $\gamma$ is defined as
\begin{equation}\label{eqn:q115}
\gamma = \frac{M_{pl}^{2}}{\xi}\left( m^{2} + \omega^{2} - \frac{\lambda M_{pl}^{2}}{\xi}\right).
\end{equation}
In the plateau limit we have that $\phi >> M_{pl}/\sqrt{\xi}$, and we also assume that
\begin{equation}\label{eqn:q116}
\left| \frac{\partial^{2}\phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} \right| >> K\left(\phi \right) \left(\frac{\partial \phi}{\partial r}\right)^{2},
\end{equation}
where $K\left(\phi \right) \sim 1/\phi$ on the plateau (we will confirm that (<ref>) is consistent below). Using (<ref>) and (<ref>), the Q-ball equation can be approximated as
\begin{equation}\label{eqn:q117}
\frac{\partial^{2}\phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} = - \frac{\gamma}{\phi},
\end{equation}
where $\gamma > 0$ for Q-ball solutions to exist.
We assume that $\phi$ does not deviate from its initial value $\phi_{0}$ too much, and also that
\begin{equation}\label{eqn:q118}
\frac{\partial \phi}{\partial r} \rightarrow 0,
\end{equation}
as $r \rightarrow 0$.
Setting $\phi = \phi_{0}$ on the right hand side of (<ref>), we make the following ansatz
\begin{equation}\label{eqn:q119}
\phi \left( r \right) = \phi_{0} - A r^{2},
\end{equation}
where $A$ is a coefficient to be determined. In order for the field profile to have the typical Q-ball shape, we require that the right hand side of the equation is negative. This means that
\begin{equation}\label{eqn:q120}
\gamma > 0 \longrightarrow m^{2} + \omega^{2} > \frac{\lambda M_{pl}^{2}}{\xi}.
\end{equation}
Substituting (<ref>) into (<ref>), we find that the left hand side becomes a constant, $-6A$, and that the approximation (<ref>) holds because
\begin{equation}\label{eqn:q121}
K\left(\phi \right) \left(\frac{\partial \phi}{\partial r}\right)^{2} \approx \frac{4A^{2}r^{2}}{\phi_{0}},
\end{equation}
is a very small number for $\phi >> M_{pl}/\sqrt{\xi}$, and goes to zero as $r \rightarrow 0$. This leaves
\begin{equation}\label{eqn:q122}
-6A = -\frac{\gamma}{\phi_{0}} \Rightarrow A = \frac{\gamma}{6 \phi_{0}},
\end{equation}
and approximation of the field profile (<ref>) is therefore
\begin{equation}\label{eqn:q123}
\phi \left( r \right) = \phi_{0} - \frac{\gamma}{6 \phi_{0}}r^{2}.
\end{equation}
The assumption that $\phi \approx \phi_{0}$ holds to a good approximation if
\begin{equation}\label{eqn:q124}
r^{2} < \frac{r_{Q}^{2}}{4} = \frac{6 \phi_{0}^{2}}{4\gamma},
\end{equation}
since applying (<ref>) at $r_{Q}/2$ gives $\phi \left(r_{Q}/2\right) = \frac{3}{4}\phi_{0}$.
To a good approximation, we can therefore say that the Q-ball solution follows
\begin{equation}\label{eqn:q125}
\phi \left( r \right) = \phi_{0}\left[ 1 - \left(\frac{r}{r_{Q}}\right)^{2} \right],
\end{equation}
for values of $r < r_{Q}/2$, where the assumption (<ref>) is consistent with this solution. We can use this to estimate the Q-ball properties, such as energy and charge, which can be used as a check on the numerical solution and also to determine the parameter dependence of the numerical Q-ball solutions.
§.§ Analytical Estimate of Q-ball Energy and Charge
In this section we derive analytical expressions for the energy, $E$, and the global charge, $Q$, of the non-minimally coupled Palatini Q-balls. Assuming $\phi \approx \phi_{0}$, and working in the plateau limit, we have that the extremised Q-ball energy $E$ is approximately (from (<ref>))
\begin{equation}\label{eqn:q126}
E = \int^{r_{Q}/2}_{0} 4\pi r^{2} \; dr \left[\frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2} + \frac{V\left(\phi \right)}{\Omega^{4}}\right].
\end{equation}
For $r < r_{Q}/2$, and assuming $\phi \approx \phi_{0}$ we approximate the Jordan frame potential to be
\begin{equation}\label{eqn:q127}
V\left(\phi \right) \approx \frac{\lambda}{4}\phi_{0}^{4},
\end{equation}
since $\lambda\phi_{0}^{4} >> m^{2}\phi_{0}^{2}$ in the Jordan frame in this case, and the conformal factor is $\Omega^{2} \approx \xi \phi_{0}^{2}/M_{pl}^{2}$. Using (<ref>), we have that
\begin{equation}\label{eqn:q128}
\frac{\partial \phi}{\partial r} = -\frac{2r \phi_{0}}{r_{Q}^{2}}.
\end{equation}
Using (<ref>) and (<ref>), we find that the Q-ball energy (<ref>) becomes
\begin{equation}\label{eqn:q129}
E = \left( \frac{\omega^{2}\phi_{0}^{2}}{2\Omega^{2}} + \frac{\lambda \phi_{0}^{4}}{4\Omega^{4}}\right) \int^{r_{Q}/2}_{0} 4\pi r^{2} \; dr + \frac{2\phi_{0}^{2}}{r_{Q}^{4}}\int^{r_{Q}/2}_{0} 4\pi r^{4} \; dr.
\end{equation}
Performing the integration gives
\begin{equation}\label{eqn:q130}
E = \left( \frac{\omega^{2}\phi_{0}^{2}}{2\Omega^{2}} + \frac{\lambda \phi_{0}^{4}}{4\Omega^{4}}\right)\frac{4\pi}{3}\left(\frac{r_{Q}}{2}\right)^{3} + \frac{2\phi_{0}^{2}}{r_{Q}^{4}}\frac{4\pi}{5}\left(\frac{r_{Q}}{2}\right)^{5},
\end{equation}
and with some rearrangement this is
\begin{equation}\label{eqn:q131}
E = \frac{\pi M_{pl}^{2}}{2 \xi}\left[ \frac{1}{6}\left(\omega^{2} + \frac{\lambda M_{pl}^{2}}{2\xi}\right) + \frac{1}{10 r_{Q}^{2}}\right]r_{Q}^{3}.
\end{equation}
Using (<ref>) and (<ref>), we have that
\begin{equation}\label{eqn:q132}
\frac{1}{r_{Q}^{2}} = \frac{\gamma}{6\phi_{0}^{2}} = \frac{M_{pl}^{2}}{6\xi \phi_{0}^{2}}\left(m^{2} + \omega^{2} - \omega_{c}^{2} \right),
\end{equation}
and since we are working in the plateau limit, $M_{pl}^{2}/\xi \phi_{0}^{2} << 1$, we can say that
\begin{equation}\label{eqn:q133}
\frac{M_{pl}^{2}}{6\xi \phi_{0}^{2}}\left(m^{2} + \omega^{2} - \omega_{c}^{2} \right) << \frac{1}{6}\left(m^{2} + \omega^{2} - \omega_{c}^{2} \right).
\end{equation}
This means that the $1/r_{Q}^{2}$ term is small compared to the $1/6$ term in (<ref>), and the Q-ball energy can be approximated as
\begin{equation}\label{eqn:q134}
E = \frac{\pi M_{pl}^{2}}{12 \xi}\left[ \omega^{2} + \frac{\lambda M_{pl}^{2}}{2\xi}\right] r_{Q}^{3}.
\end{equation}
Note that this expression holds for $r < r_{Q}/2$ and is therefore a lower bound estimate on the Q-ball energies calculated numerically. Similarly, the Q-ball charge can be approximated analytically. The charge density of the Q-balls from (<ref>) using the Q-ball ansatz (<ref>) is
\begin{equation}\label{eqn:q135}
\rho_{Q} = \frac{\omega \phi^{2}}{\Omega^{2}},
\end{equation}
where for $\phi \approx \phi_{0}$ and $\Omega^{2} \approx \xi \phi_{0}^{2}/M_{pl}^{2}$ this is
\begin{equation}\label{eqn:q136}
\rho_{Q} = \frac{\omega M_{pl}^{2}}{\xi}.
\end{equation}
The global charge from (<ref>) using (<ref>) is thus
\begin{equation}\label{eqn:q137}
Q = \int^{r_{Q}/2}_{0} 4\pi r^{2} \; dr \frac{\omega M_{pl}^{2}}{\xi} = \frac{4\pi \omega M_{pl}^{2}}{\xi}\left( \frac{r_{Q}}{2}\right)^{3},
\end{equation}
and integrating for $r < r_{Q}/2$ gives the Q-ball charge to be
\begin{equation}\label{eqn:q138}
Q = \frac{\pi \omega M_{pl}^{2}}{6 \xi} r_{Q}^{3}.
\end{equation} |
# Super narrow peaks in excitation spectrum of alkali spin polarization:
non-adiabatic case of spin dynamics
E. N. Popov<EMAIL_ADDRESS>Laboratory of Quantum Processes and
Measurements, ITMO University,
199034, 3b Kadetskaya Line, Saint-Petersburg, Russia A. A. Gaidash A. V.
Kozubov Laboratory of Quantum Processes and Measurements, ITMO University,
199034, 3b Kadetskaya Line, Saint-Petersburg, Russia Department of
Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute
of Russian Academy of Sciences, 119991, 8 Gubkina St, Moscow, Russia S. P.
Voskoboynikov Higher School of Software Engineering, Peter the Great
St.Petersburg Polytechnic University,
195251, 29 Polytechnicheskaya, Saint-Petersburg, Russia
###### Abstract
We theoretically describe the phenomenon of non-adiabatic spin dynamics, which
occurs in a gas cell filled by alkali vapor in presence of a strong
alternating magnetic field and pump light. Steep increase of the spin
polarization occurs if frequency of the magnetic field is equal to the certain
value. Although, the observable effect relies on the periodic field that
consists of two perpendicular components defined by harmonics with the same
amplitudes and different frequencies. Considered spin effect cannot be
explained by a resonance, because the own Larmor frequency of spin precession
is absent without a constant component of magnetic field. Moreover, there are
some clearly visible peaks in the excitation spectrum of spin polarization,
and they are super narrow in comparison to relaxation rate. Detailed analysis
according to proposed quantum model results in the reasoning of the effect via
qualitative properties of non-adiabatic dynamics of atomic spin.
††preprint: APS/123-QED
## I INTRODUCTION
The atomic spin dynamics in a gas cell has been studied for over half of the
century. The main advantage of that physical model is the large lifetime of
non-excited spin states in comparison to crystals and liquids, since any atom
moves as a free particle between collisions [1, 2, 3]. Moreover, in a mixture
consisting of an alkali vapor and an inert buffer gas, only a small percentage
of collisions results in spin reversal. These factors make it possible to
observe some evident magnetic effects in an alkali vapor, which are
implemented in precise measurements for metrology and navigation [4, 5], i.e.
magnetometers [6, 7, 8, 9, 10, 11, 12, 13], gyroscopes [14, 15, 16, 17, 18,
19, 20, 21] and frequency standards [22, 23]. A rapid development in the study
of a gas cell with spin dynamics is induced by the implementation of optical
approaches to scanning and excitation of a spin polarization [24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35].
In an alkali vapor, one of the most explored and utilized in the practice
magnetic effects is the electron paramagnetic resonance (EPR) [36, 37, 38]. A
necessary condition for the EPR observation is the presence of a constant
magnetic field, which defines the own Larmor frequency of the alkali spin
precession. If the frequency of periodic perturbation of the spin system is
close to or equal to the own Larmor frequency, the EPR is arising as
oscillations of spin polarization. The width of the EPR spectral line is
determined by a rate of population mixing among alkali Zeeman sublevels. The
latter is the most important parameter of the system, because the precision of
different gas cell sensors depends on lifetime of spin states [39, 40, 41, 42,
43, 44, 45, 46, 47, 48].
In this work, we theoretically explore a phenomenon in alkali vapor: its spin
polarization steeply increases near a certain frequency of an external
magnetic field in the presence of a circularly-polarized pump light. To
observe the effect, the following properties of the magnetic field should be
satisfied:
* •
The external magnetic field is periodic with the period that is much less than
the relaxation time of a spin polarization;
* •
The averaged over a period external magnetic field is equal to zero, thus the
constant component of the magnetic field in a gas cell with alkali vapor is
absent;
* •
Both the magnitude and direction of the magnetic field vary;
* •
Zeeman shift of sublevels with non-zero angular momentum projection is much
larger than broadening of these sublevels due to population mixing to
equilibrium state.
Listed above properties regard to non-adiabatic dynamics, defined by rapid
variation of an alkali spin polarization in comparison to its relaxation.
Withal, magnetic field should be periodic to analyze the steady dynamics
through its constant characteristics. It is essential, that the origin of the
effect differs from EPR induced by an externally applied periodic force.
Though, to exclude an EPR we need to disable a trend of Larmor spin
precession, therefore a constant component of magnetic field should be absent.
Hereafter in the paper, we use a term the spin effect to refer to the
phenomenon under consideration.
The listed above properties may belong to a magnetic field defined by various
temporal dependencies. Here, we consider the special case that is sufficient
for exploring the spin effect:
$\displaystyle\mathbf{B}(t)=B_{0}\mathbf{l_{z}}\cos\left(\Omega
t\right)+B_{0}\mathbf{l_{x}}\cos\left(2\Omega t\right),$ (1)
$\displaystyle\gamma B_{0}\gg\Gamma,\qquad\Omega\gg\Gamma,$ (2)
where $B_{0}$ is an amplitude of the magnetic field, $\mathbf{l_{x}}$ and
$\mathbf{l_{z}}$ are orthonormal spatial vectors, $\Omega$ is a frequency of
the magnetic field, $\gamma$ is gyromagnetic ratio of the oriented alkali
atoms, $\Gamma$ is the relaxation rate of spin polarization.
Previously, various magnetic effects has been studied in condition of the non-
adiabatic dynamics of oriented spins. There are some experimental and
theoretical works, in which non-adiabatic dynamics is essential [49, 50, 51,
52, 53, 54, 55, 56, 57, 58]. However, the cited papers do not focus on the
non-adiabatic dynamics and it’s properties. Moreover, to the best of our
knowledge, the synchronization of alkali spin motion due to periodic non-
adiabatic dynamics in presence of a strong alternating magnetic field was not
analyzed in detail. We suggest exploring a spin effect induced by properties
of non-adiabatic dynamics using a quantum model of alkali vapor.
Figure 1: The physical system for studying alkali spin dynamics in a gas cell.
To create a magnetic field, the double Helmholtz coils located around the
cell. A pump light propagates along the Z-axis and covers the cell
homogeneously. A scanning light may be directed along any axis. The atomic
spin projection onto the path of scanning light can be measured by the angle
of scanning light polarization rotation $\Delta\psi$; they are proportional to
each other
The spin effect cannot be explained in terms of spin resonance, since a
constant magnetic field is absent. Parametric resonance [59] is not a suitable
candidate for explanation of the spin effect as well, since there are no
internal parameters of the atom-field system, which determine the own
frequency of the resonance and make periodic oscillations. Though, we explain
the spin effect by the qualitative properties of non-adiabatic dynamics of
atomic spins in alkali vapor. Mathematically, the non-adiabatic case of atomic
spin motion in the presence of a strong alternating magnetic field corresponds
to a dynamic system with unpredictable behavior. Since the dynamics cannot be
described as an explicit temporal dependence, the search for physical effects
becomes a non-trivial task. Only via a numerical solution, we can explore the
phase-space trajectory of a non-adiabatic dynamic system, that is considered
in the paper.
The paper is organized as follows. In Sec. II we describe an optical method of
spin driving inside a gas cell with vapor of alkali. Also, the simplified
scheme for an experiment is suggested. In Sec. III we propose a general
quantum model of alkali spin dynamics in presence of a strong alternating
magnetic field and relaxation. Calculations according to the model and
resulting excitation spectrum of the spin effect are demonstrated in Sec. IV.
In Sec. V the explanation of the spin effect is provided. Sec. VI concludes
the article.
## II Physical system under consideration
In this section, we propose a principal optical scheme for the spin effect
observation, see Fig. 1. Also, a structure of Rb-87 energy levels with excited
transitions is shown in the Fig. 2. The scheme consists of a gas cell with
vapor of 87Rb and buffer gas, Helmholtz coils for generating the magnetic
field Eq. (1), a source of circularly polarized pump light (linearly polarized
laser and quarter-wave plate) and source of scanning light (low-power linearly
polarized laser). The pump light orients rubidium spins along the Z-axis, and
the scanning light detects an expectation value of the atomic spin projection
onto the path of the scanning light propagation. Note, the similar scheme of
spin perturbation was considered in the works [57] and [58] in partially-
adiabatic case.
### II.1 Gas cell
A gas cell contains a vapor of $\mathrm{{}^{87}Rb}$ and a mixture of inert
gases. The gas mixture composes of diatomic nitrogen in order to reduce alkali
fluorescence and some noble monatomic gases as a buffer. It is important that
spin polarization can be formed by alkali atoms populating different hyperfine
ground levels. Therefore, to define a certain group of non-excited alkali
atoms by $\mathrm{Rb_{F=1}}$ and $\mathrm{Rb_{F=2}}$.
Temperature within the cell is about $80^{\circ}\,\mathrm{C}$. Under the
conditions, the concentration of the alkali is 5–6 orders of magnitude less
than the concentration of the buffer. Since an electron cloud of alkali has an
enlarged radius and anisotropic form, multiple collisions between an excited
$\mathrm{{}^{87}Rb}$ atom and $\mathrm{N}_{2}$ molecules lead to rapid non-
radiative transitions from the upper to one of the two ground levels in the
$\mathrm{D1}$ line (see Fig. 2). Moreover, since the buffer environment
freezes free motion of alkali, relaxation due to collisions with a cell’s
surface is significantly small (up to be considered absent). The latter
properties of the gas cell are necessary for the inducing of non-adiabatic
dynamics of atomic spins in an external magnetic field.
Figure 2: D1 line of Rubidium 87. Rectangles denote Zeeman sublevels in the
hyperfine structure. Degree of black filling shows its population: the darker,
the higher population. Red arrows indicate atomic transitions
$F=1\leftrightarrow F^{\prime}=2$ induced by circularly polarized pump light.
Orange arrows indicate weak perturbation on transitions $F=2\leftrightarrow
F^{\prime}=1$ and $F=2\leftrightarrow F^{\prime}=2$ excited by linearly
polarized scanning light. Symbols $F$ and $F^{\prime}$ denote total angular
momentum of ground and excited levels respectively, $m$ is a projection of
total angular momentum onto the path of the pump light propagation
### II.2 Spin polarization pumping
As shown in the Fig. 1, a circularly polarized pump light propagates along the
Z-axis. It orients alkali spins along the path of propagation. The frequency
of a pump light is equal to the one of a transition between the levels with a
total angular momentum $F\\!=\\!1$ and $F^{\prime}\\!=\\!2$. In the presence
of an optical pump, alkali atoms populate the upper levels with one-sided
change of the angular momentum, which is projected along the path of the light
propagation. Therefore, alkali atoms accumulate non-zero angular momentum
directed along the Z-axis. Recent known data about the D1 line of
$\mathrm{{}^{87}Rb}$ can be found in [60].
According to the optical scheme shown in the Fig. 1, spins of
$\mathrm{Rb_{F=1}}$ and $\mathrm{Rb_{F=2}}$ are oriented through depopulation
and repopulation effects, respectively. Both effects create the non-
equilibrium population among Zeeman sublevels belonging to the ground levels
of the D1 line. The nature of depopulation is a selective depletion of certain
sublevels by a circularly polarized light. It should be noted, that
repopulation process is more efficient than depopulation. As presented in
[61], repopulation is based on the feature of collisional decay, that is a
nucleus spin state is not destroyed during transition of an excited alkali
atom from upper to ground levels. Therefore, spin polarization of
$\mathrm{Rb_{F=2}}$ is produced by preserved alkali nucleus spin.
### II.3 Spin polarization scanning
Atomic ensemble with a spin polarization becomes circular birefringent due to
inducing of optical anisotropy. The magnitude of the circular birefringence is
determined by the expectation value of the atomic spin projection onto the
path of light propagation. Since refraction indices for orthogonal circular
light components are different, the linear polarization plane of the passing
light rotates around the path of propagation. Therefore, alkali spin
polarization dynamics can be measured via deviation of scanning light
polarization as follows:
$\langle\hat{F}_{\alpha}\rangle\propto\Delta\psi(t),$ (3)
where $\Delta\psi(t)$ is the angular rotation of the polarization plane shown
in the Fig. 1, $\hat{F}_{\alpha}$ is the operator of the total angular
momentum projection onto the $\alpha$-axis. By selection of the scanning light
direction, one may measure any spin polarization component of the alkali vapor
in the gas cell.
The spectral linewidth of the scanning light should be narrow enough for
resolution of the alkali hyperfine structure. It is necessary for measurement
of the spin polarization of alkali vapor only from $\mathrm{Rb_{F=2}}$. As
shown in Fig. 2, the scanning light frequency should be close to
$F=2\leftrightarrow F^{\prime}=2$ transition frequency. At the same time, it
should be far from the frequency of $F=1\leftrightarrow F^{\prime}=2$
transition. If the latter conditions are satisfied, then $\mathrm{Rb_{F=2}}$
atoms affect the polarization of the scanning light significantly stronger
than $\mathrm{Rb_{F=1}}$ atoms do. Withal, the light frequency should be
detuned from the optical resonance in order to exclude redundant depletion of
the ground level with $F=2$. Since the ground level with $F=1$ is broadened by
a pump light, the spin polarization measurements from $\mathrm{Rb_{F=1}}$ are
not advisable.
## III Master equation
Here, we would like to propose the math model of alkali dynamics affected by a
monochromatic light and an alternating magnetic field. To correctly describe
the spin effect, we take into account common relaxation processes listed in
the Table 1. The model is based on the semi-classical theory of atom-field
interaction: alkali vapor is described by a density operator, and the fields
are described by classical vectors dependent on time. The density operator is
a $16\times 16$ matrix in the basis of non-perturbed Zeeman sublevels that
belong to D1 line of $\mathrm{{}^{87}Rb}$.
Table 1: Relaxation processes in the gas cell, their origins and typical rates with corresponding mapping Process | Origin of the | Rate,
---|---|---
description | process | Mapping
Mixing of ground | Alkali spin exchange | $\begin{array}[]{c}\Gamma\sim 10^{2}\>\text{Hz},\vspace{0.1cm}\\\ \hat{\rho}-\hat{\rho}_{0}\end{array}$
Zeeman sublevels | and atomic collision
population | with cell’s walls
Decay from excited | Non-ellastic collisions | $\begin{array}[]{c}\delta_{dcy}\sim 10^{8}\>\text{Hz},\vspace{0.1cm}\\\ \mathcal{R}\left\\{\hat{\rho}\right\\}\end{array}$
to ground levels | between excited alkali
without fluorescence | and buffer atoms
Inhomogeneous | Velocities mixing to | $\begin{array}[]{c}\delta_{mix}\sim 10^{9}\>\text{Hz},\vspace{0.1cm}\\\ \mathcal{M}\left\\{\hat{\rho}\right\\}\end{array}$
Doppler broadening | Maxwell-Boltzmann
of optical resonance | distribution
Decoherence of dipole | Alkali-buffer | $\begin{array}[]{c}\delta_{dec}\sim 10^{10}\>\text{Hz},\vspace{0.1cm}\\\ \mathcal{D}\left\\{\hat{\rho}\right\\}\end{array}$
oscillations on | elastic collisions
optical transitions | without decay
The master equation for alkali density matrix, that describes the certain
velocity group inside the Maxwell distribution, is as follows:
$\begin{array}[]{l}\displaystyle
i\hbar\left(\frac{\partial\hat{\rho}}{\partial
t}+kv_{z}\hat{\rho}\right)=\left[\hat{H},\hat{\rho}\right]-\Gamma\left(\hat{\rho}-\hat{\rho}_{0}\right)-\vspace{0.2
cm}\\\
\displaystyle-\delta_{mix}\mathcal{M}\left\\{\hat{\rho}\right\\}-\delta_{dcy}\mathcal{R}\left\\{\hat{\rho}\right\\}-\delta_{dec}\mathcal{D}\left\\{\hat{\rho}\right\\},\end{array}$
(4)
where $\hat{H}$ is the Hamiltonian without relaxation, $k$ is the wavenumber,
$v_{z}$ is the alkali velocity projection onto the path of the pump light,
operator $\hat{\rho}_{0}$ corresponds to the state of thermodynamic
equilibrium with mixed population among Zeeman sublevels. Operator $\hat{H}$
comprises the interaction of alkali with the pump light $\hat{V}_{E}$, the
interaction of alkali with magnetic field $\hat{V}_{B}$ and unperturbed
Hamiltonian $\hat{H}_{0}$:
$\displaystyle\hat{H}=\hat{H}_{0}+\hat{V}_{E}+\hat{V}_{B},$ (5)
$\displaystyle\hat{V}_{E}=-\left(\mathbf{\hat{d}\cdot
E}\right),\qquad\hat{V}_{B}=\sum\limits_{n=1}^{2}g_{n}\gamma_{e}\left(\mathbf{\hat{F}_{n}\cdot
B}\right),$ (6)
$\displaystyle\mathbf{E}=\frac{\mathcal{E}}{2}\,\mathbf{l}_{+}e^{-i\omega
t}+c.c.,\qquad k=\frac{\omega}{c}.$ (7)
Here $\mathcal{E}$ is a constant amplitude of the pump light, $\mathbf{l}_{+}$
is the unit vector of the circular polarization, $\omega$ is the frequency of
the pump light, the dipole operator $\mathbf{\hat{d}}$ describes all optical
transitions between Zeeman sublevels in D1 line of $\mathrm{{}^{87}Rb}$ [60],
$\gamma_{e}$ is the electron gyromagnetic ratio, $g_{n}$ is the g-factor of
the ground hyperfine levels, $\mathbf{\hat{F}}_{n}$ is the total angular
momentum operator corresponding to atoms $\mathrm{Rb_{F=n}}$, and magnetic
field $\mathbf{B}$ is defined by Eq. (1).
Below we define mappings $\mathcal{M}$, $\mathcal{D}$ and $\mathcal{R}$ from
the master equation (4):
1. 1.
The mapping $\mathcal{M}$ determines the transition of the dependence of
density operator on alkali velocities to the uniform distribution:
$\mathcal{M}\left\\{\hat{\rho}\right\\}=\hat{\rho}-\int\limits_{-\infty}^{\infty}\hat{\rho}\>\mu(v_{z})\,\mathrm{d}v_{z},$
(8)
where $\mu(v_{z})$ is the Maxwell–Boltzmann distribution.
2. 2.
The mapping $\mathcal{D}$ nullifies non-diagonal elements with an optical
frequency of phase rotation:
$\mathcal{D}\left\\{\hat{\rho}\right\\}=-\hat{P}_{e}\hat{\rho}\hat{P}_{g}-\hat{P}_{g}\hat{\rho}\hat{P}_{e},$
(9)
where $\hat{P}_{g}$ and $\hat{P}_{e}$ are projection operators to the space of
ground hyperfine levels $F$ and excited hyperfine levels $F^{\prime}$ in D1
line respectively, see Fig. 2.
3. 3.
The mapping $\mathcal{R}$ decomposes the full density matrix $\hat{\rho}$ to a
tensor product of an electron $\hat{\rho}_{e}^{(0)}$ and nuclear
$\hat{\rho}^{\prime}_{n}$ density matrices:
$\displaystyle\mathcal{R}\left\\{\hat{\rho}\right\\}=-\hat{P}_{e}\hat{\rho}\hat{P}_{e}+\hat{\rho}^{\prime}_{n}\otimes\hat{\rho}_{e}^{(0)},$
(10)
$\displaystyle\hat{\rho}^{\prime}_{n}=\sum\limits_{m=-\nicefrac{{1}}{{2}}}^{\nicefrac{{1}}{{2}}}\langle{m;\,\scriptstyle
5^{2}P_{1/2}}\,|\,\hat{\rho}\,|\,m;\,{\scriptstyle 5^{2}P_{1/2}}\rangle,$ (11)
$\displaystyle\hat{\rho}_{e}^{(0)}=\sum\limits_{m=-\nicefrac{{1}}{{2}}}^{\nicefrac{{1}}{{2}}}\frac{1}{2}\,|\,m;\,{\scriptstyle
5^{2}S_{1/2}}\rangle\langle m;\,{\scriptstyle 5^{2}S_{1/2}}\,|,$ (12)
where $|m;level\rangle$ is a spin-orbital state of an external electron with a
projection $m$ of total angular momentum.
Note, the electron density matrix reduces to the one of equilibrium state,
however the nuclear density matrix does not. The partial trace in Eq. (11)
leads to a non-equilibrium distribution of population among Zeeman sublevels
of $\mathrm{Rb_{F=2}}$. Due to this feature, alkali spin polarization during
collision decay is preserved [45, 61].
Figure 3: Peaks in the $Rb_{F=2}$ spin polarization dependence on the
frequency of periodic magnetic field determined by Eq. (1); curves are defined
by Eqs. (16) and (17), correspondingly. Parameters for the evaluation are as
follows: magnetic field amplitude $B_{0}=27\ \mu T$, the relaxation rate
$\Gamma=1\ kHz$, the pump light amplitude $\mathcal{E}=100\ V/m$, the
temperature is $80^{\circ}\mathrm{C}$
## IV Non-adiabatic spin dynamics evaluation
In order to proceed with the numerical solution of the master equation (4),
let us prior define the vectors $\mathbf{S_{[1]}}$ and $\mathbf{S_{[2]}}$ for
describing a spin polarization formed by alkali atoms $\mathrm{Rb_{F=1}}$ and
$\mathrm{Rb_{F=2}}$ respectively:
$\displaystyle\mathbf{S_{[n]}}=S_{[n],x}\mathbf{l_{x}}+S_{[n],y}\mathbf{l_{y}}+S_{[n],z}\mathbf{l_{z}},$
(13) $\displaystyle
S_{[n],\alpha}=\mathrm{Tr}\left\\{\hat{\rho}_{n}\hat{\Sigma}_{n,\alpha}\right\\},\qquad\alpha\in\\{x,y,z\\},$
(14)
$\displaystyle\hat{\rho}_{n}=\int\limits_{-\infty}^{\infty}\hat{P}_{n}\hat{\rho}\hat{P}_{n}\mu(v_{z})\,\mathrm{d}v_{z},$
(15)
where $\mathbf{l_{\alpha}}$ are orthonormal spatial vectors; $\hat{P}_{n}$ is
the operator of projection to the ground hyperfine level with the total
angular momentum $F=n$; symbol $\hat{\Sigma}_{n,\alpha}$ denotes a Pauli
matrix equivalent to a spin-n particle; value $S_{[n],\alpha}$ is a component
of spin polarization formed by $\mathrm{Rb}_{F=n}$ atoms; cropped density
matrix $\hat{\rho}_{n}$ belongs to a subspace of $F=n$ hyperfine level, its
rank is equal to $2n+1$ according to the number of Zeeman sublevels. Note, the
cropped density matrix is averaged over alkali velocities.
As the vectors above have been defined, we come to a solution of the master
equation (4) with satisfied conditions of non-adiabatic dynamics. The solution
is density matrix dependent on time for any frequency $\Omega$ of the magnetic
field defined by Eq. (1). For the last period of the solution, we calculate
spin polarization components via Eqs. (13) – (15) for a wide range of
frequencies $\Omega$. Finally, frequency dependence of two convolutions are
presented in the Fig. 3, the latter demonstrates some peaks of spin
polarization:
$\displaystyle\mathcal{C}_{1}(\Omega)=\max\limits_{t\in
T_{last}}\,\left|\mathbf{S_{[2]}}(t)\right|=\mathrm{sup}\,\left|\>\mathbf{S_{[2]}}\>\right|,$
(16)
$\displaystyle\mathcal{C}_{2}(\Omega)=\left|\>\frac{1}{T}\\!\\!\\!\int\limits_{t\in
T_{last}}\\!\\!\\!\\!\\!\mathbf{S_{[2]}}(t)\,\mathrm{d}t\>\right|=\left|\>\overline{\mathbf{S_{[2]}}}\>\right|.$
(17)
Regarding the selective scanning of alkali spin polarization (see Fig. 2), we
observe only $\mathbf{S_{[2]}}$ vector.
These convolutions give general information about spin dynamics. The first one
describes the radius of the sphere that contains a trajectory of the vector
$\mathbf{S_{[2]}}$ moving in the three-dimensional space of spin polarization
components. The larger the $\mathcal{C}_{1}$, the more oriented alkali spins
are. The second one describes alkali vapor magnetization, determined by the
mean spin polarization over time. It can be measured by rotation of the
linearly polarized scanning light, i.e. angle $\Delta\psi$ in the Fig. 1 and
Sec. II.3.
According to Fig. 3, the highest peak is the second from the right, it appears
near the frequency $\Omega\approx 33.2\ kHz$. There are no additional peaks in
the region above $50\ kHz$. Low frequencies are not considered in the work,
since corresponding dynamics are close to adiabatic case there.
### IV.1 Width of the highest peak
Relaxation of a spin polarization occurs due to population mixing among the
ground Zeeman sublevels, see the Table 1. This process is defined by the
relaxation rate $\Gamma$, which determines the form of peaks.
In the Fig. 3, the second from the right (at $\approx 33.2\ kHz$) peak of the
convolution $\mathcal{C}_{2}$ is of interest. The high altitude of the peak is
more important than its narrow width from the practical point of view: the
less altitude, the lower signal-to-noise ratio. It is so since approximately
only one alkali atom in a hundred creates spin polarization, hence detection
of the effect may be challenging. Also, the $\mathcal{C}_{2}$ is more contrast
compare to the convolution $\mathcal{C}_{1}$.
Figure 4: The second from the right peak in the Fig. 3 calculated for the
different relaxation rates $\Gamma$. There is only convolution Eq. (17) in the
picture
In Fig. 4 one may observe dependence of its shape on different rates of
$\Gamma$. Note, that the peak altitude and relaxation rate $\Gamma$ are not in
inverse proportion. Therefore, a the spin effect behavior differs from a
driven damped oscillator in condition of the resonance, when an amplitude of
oscillations and the damping ratio are inverse proportional. Furthermore, the
results are not well approximated by the Lorentz function as for a damped
oscillator. Regarding the latter, we consider steep increase of spin
polarization as non-adiabatic effect of spin dynamics, though its physical
nature is different from the EPR.
Another observable parameter is a width of the peak dependent on relaxation
rate $\Gamma$, we specify it by the half width at half maximum (HWHM). We
would like to compare the width of the peak dependence on relaxation rates
$\Gamma$ with the corresponding dependence in case of the EPR under the same
conditions of pumping and relaxation. The data for EPR is obtained by solving
the master equation (4) implying magnetic field is defined as follows (instead
of Eq. (1) for the spin effect):
$\mathbf{B}=B_{dc}\mathbf{l_{z}}+B_{ac}\mathbf{l_{x}}\cos\left(\Omega
t\right),\qquad B_{dc}\gg B_{ac},$ (18)
where the constant magnetic field $B_{dc}$ determines the resonance frequency,
and the alternating magnetic field with the amplitude $B_{ac}$ induces the
resonance dynamics of alkali spin. In the Fig. 5 one may observe calculated
HWHMs for a given relaxation rates $\Gamma$ as in Fig. 4. As a result, we
estimate that the HWHM of the observed peak for the spin effect is $3.5$ times
less than the one of the EPR. Note, that HWHM and relaxation rate are
proportional in both cases.
Figure 5: Comparison of the peak’s width in the Fig. 4 with the width of the
EPR calculated under the same conditions of pumping and relaxation. When the
EPR was modeling, a magnetic field was defined by Eq. (18) with $B_{dc}=10\
\mu T$ and $B_{ac}=10\ nT$. These parameters are chosen to match the frequency
of the peak ($\approx 33.2\ kHz$)
### IV.2 Spin dynamics near peak polarization
Further we explore the steady non-adiabatic dynamics of the spin polarization
created by atoms $\mathrm{Rb}_{F=2}$, when the frequency of the magnetic
field, defined by Eq. (1), maximizes
$\left|\overline{\mathbf{S_{[2]}}}\right|$, i.e. $\Omega\approx 33.2\ kHz$,
see the Fig. 3. We consider the three components of the spin polarization,
which can be measured by scanning light: $S_{[2],x}(t)$, $S_{[2],y}(t)$, and
$S_{[2],z}(t)$. The trajectory of the vector $\mathbf{S_{[2]}}(t)$ in three-
dimension space of spin polarization components and its parametric
representation are plotted in the Fig. 6. It is mirror symmetric about two
planes: $S_{x}S_{z}$ and $S_{y}S_{z}$. A little deviation from the mirror
symmetry about $S_{x}S_{z}$ the plane occurs due to location of the magnetic
field in the $XZ$ plane, see Fig. 1. At the same time, the trajectory is not
mirror symmetric about the $S_{x}S_{y}$ plane. We assume that the axis of the
general shift of the spin polarization is correlated to the direction of pump
light.
The average radius and the Z-shift of the trajectory is described by two
convolutions $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$, defined in the Eqs. (16)
and (17). So alteration of these trajectory characteristics may be estimated
by the observation of the latter convolutions dependence on frequency
$\Omega$, shown in Fig. 3. If the frequency deviates from the value at the
point of maximum, the trajectory collapses to the axis-origin and reshapes to
isotropic tangle clew. Reshaping of the trajectory explains the higher rate of
the collapse of $\mathcal{C}_{2}$ convolution compare to the one of
$\mathcal{C}_{1}$ convolution. Note, the spin polarization trajectories are
different for each of four peaks shown in the Fig. 3. Despite we provide the
figure for only one, all other trajectories exhibit similar properties: two
mirror symmetries and “protuberant” along the path of pump light propagation.
To observe correlations, time dependencies $S_{\alpha}(t)$ in the Fig. 6 are
plotted together with the magnetic field dynamics. Surprisingly, non-adiabatic
dynamics of spin polarization involves strong higher harmonics, which are
absent in the dynamics of the magnetic field.
Figure 6: The purple parametric 3d-curve drawn by three temporal dependencies
of the spin polarization components $S_{[2],\alpha}$, where
$\alpha\in\\{x,y,z\\}$. There is a steady dynamics of spin polarization shown
in the picture. Blue curves demonstrate two components of the periodic
magnetic field, defined by Eq. (1). Note, the index $[2]$ in $S_{[2],\alpha}$
is dismissed for the sake of visual simplicity. The purple curves are
calculated for the frequency $\Omega\approx 33.2\ kHz$, which matches the
maximum of the peak in the Fig. 3
## V Analysis
We investigate the nature of steep increase of an alkali spin polarization
under non-adiabatic dynamics in presence of the strong alternating magnetic
field. As discussed in the previous section, the spin effect appears to be not
a resonance. We claim that the spin effect is a phenomenon of unpredictable
periodic spin dynamics, that lasts while spin state decays. It occurs only in
the non-adiabatic case. Then, the main condition of the spin effect to be
observed is a synchronous spin motion inside an alkali vapor. If atomic spins
are not synchronized, the ensemble of atoms tends to a mixed spin state with
equilibrium distribution, and the spin polarization is vanished. Note, that
observed effect does not require the same spin dynamics for all atoms.
In case of EPR, one component of the vapor magnetization approaches to a
saturated state and then does not vary; this component is often called
longitudinal. At the same time, both transverse components rapidly oscillate,
but in most cases its magnitudes are small in comparison to the longitudinal
component. Thus, EPR cannot be interpreted as a non-adiabatic process.
In order to explain the synchronous spin motion, let us describe an alkali
atom $\mathrm{Rb}_{L=2}$, that is moving as free particle between two time
points: the moment of initial spin state creation due to pump photon
absorption, and the moment of spin state decay due to collision with the cell
walls or with another alkali atom. Time interval of the free path of motion is
estimated by inverse of relaxation rate $\Gamma$. As the first order
approximation, we can use the Pauli equation to model a free alkali atom in an
alternating magnetic field:
$\displaystyle
i\hbar\dot{\varphi}=\gamma\hbar\left(\hat{\sigma}\cdot\mathbf{B}(t)\right)\varphi,$
(19)
$\displaystyle\hat{\sigma}=\frac{1}{2}\left(\hat{\sigma}_{x}\mathbf{l_{x}}+\hat{\sigma}_{y}\mathbf{l_{y}}+\hat{\sigma}_{z}\mathbf{l_{z}}\right),$
(20) $\displaystyle\varphi(t)=\hat{U}(t,t_{0})\varphi_{0},$ (21)
where $\varphi$ is the spinor, $\gamma$ is the gyromagnetic ratio,
$\hat{\sigma}_{\alpha}$ are Pauli matrices, where $\alpha\in\\{x,y,z\\}$,
$\varphi_{0}$ is the spinor state at the initial time $t_{0}$,
$\hat{U}(t,t_{0})$ is the unitary time-evolution operator.
Table 2: The highest four frequencies, that lead to periodic solution of the Pauli equation (19), and corresponding solution properties Frequency | Initial spin | Spin components
---|---|---
from the set | direction | averaged over a period ($\times 10^{3}$)
$\Omega_{\mathcal{A}}/\gamma B_{0}$ | $\langle\varphi_{0}|\hat{\sigma}|\varphi_{0}\rangle$ | $\overline{\langle\varphi|\hat{\sigma}_{x}|\varphi\rangle}$ | $\overline{\langle\varphi|\hat{\sigma}_{y}|\varphi\rangle}$ | $\overline{\langle\varphi|\hat{\sigma}_{z}|\varphi\rangle}$
$0.099$ | $=\mathbf{l_{x}}/2$ | -48.4 | 0 | -26.4
$=\mathbf{l_{y}}/2$ | 0 | -5.8 | 0
$=\mathbf{l_{z}}/2$ | -22.6 | 0 | 56.3
$0.126$ | $=\mathbf{l_{x}}/2$ | 1.5 | 0 | -56.9
$=\mathbf{l_{y}}/2$ | 0 | 31.7 | 0
$=\mathbf{l_{z}}/2$ | 1.6 | 0 | 57.5
$0.175$ | $=\mathbf{l_{x}}/2$ | -4.7 | 0 | 196.8
$=\mathbf{l_{y}}/2$ | 0 | -35.6 | 0
$=\mathbf{l_{z}}/2$ | -52.5 | 0 | -17.6
$0.259$ | $=\mathbf{l_{x}}/2$ | -44.3 | 0 | 5.5
$=\mathbf{l_{y}}/2$ | 0 | -82.2 | 0
$=\mathbf{l_{z}}/2$ | 4.2 | 0 | 58.4
The Pauli equation (19) with the alternating magnetic field $\mathbf{B}(t)$,
defined by Eq. (1), generates a set $\mathcal{A}$ of frequencies, which lead
to a periodic solution independent of the spinor state at the initial time:
$\forall\varphi(t):\quad\varphi(t+2\pi/\Omega_{\mathcal{A}})=\varphi(t),$ (22)
where $\Omega_{\mathcal{A}}\in\mathcal{A}$. The set $\mathcal{A}$ is discrete
and upper-bounded by a maximal frequency. A sequence of the highest
frequencies in the set $\mathcal{A}$ corresponds to coordinates of the peaks
in the Fig. 3. Note, that if a frequency from the absolute complement of
$\mathcal{A}$ is substituted into the Pauli equation, the solution can be
periodic only for the certain initial state.
In the Table 2 we present the four highest frequencies from the set
$\mathcal{A}$ divided by $\gamma B_{0}$ and corresponding properties of the
Pauli equation solutions. Although, periodicity does not depend on an initial
state for the frequency $\Omega_{\mathcal{A}}$, the solution of the Pauli
equation (19) does. To describe a behavior of the spin dynamics, we define
spin components averaged over a period. The latter corresponds to the second
convolution expressed by Eq. (17). It is evaluated for a given three
(orthogonal) initial states and four different frequencies
$\Omega_{\mathcal{A}}$.
Data in the Table 2 shows us, that maximal magnetization occurs when frequency
$\Omega$ is equal to the second highest $\Omega_{\mathcal{A}}$ ($\approx 33.2\
kHz$ and corresponds to $\Omega_{\mathcal{A}}/\gamma B_{0}\approx 0.175$). The
averaged spin polarization is quasi-collinear with the Z-axis, which matches
with the path of the pump light propagation.
Based on the arguments above, we propose the following statements that explain
the studied spin effect:
1. 1.
Permanent relaxation and pump of the alkali spin polarization leads to jumps
of atomic spin between different quantum states;
2. 2.
If the dynamics is non-adiabatic and periodic, alkali spins can synchronously
move starting from a random the initial spin state;
3. 3.
Condition of the synchronous moving is the certain frequency/amplitude rate of
an alternating magnetic field, which is determined by corresponding temporal
profile;
4. 4.
The effect of synchronized and repetitive motion of the spins appears as steep
increase of an averaged spin polarization over a period.
## VI Conclusion
We have considered a non-adiabatic spin polarization dynamics, which occurs in
alkali vapor under optical spin orientation and in the presence of a strong
alternating magnetic field. By solving the master equation for a density
matrix, we have revealed the effect that resembles a resonance. If the
frequency of a periodic magnetic field without a constant component is equal
to a certain value, an alkali spin polarization steeply increases. However,
with a certainty we claim, that it is not an electron spin resonance, as it is
revealed by the analysis of the peaks that may be observed in the dependence
of the spin polarization on the magnetic field frequency. We interpret the
effect as a fundamental property of non-adiabatic spin polarization dynamics
in presence of the external magnetic field, when spins that belong to an
alkali ensemble move synchronously only for a certain frequency.
Mathematically, it can be explained by the existence of Pauli equation
solutions with periodic behavior, if periodic magnetic field has the certain
profile.
The important result from the practical point of view is that the width of one
of the found peaks is significantly narrower than the width of EPR induced in
the same medium with similar conditions, see Fig. 5. Therefore, the effect may
find its place in the field of precise sensing.
## Funding
This work was financially supported by Russian Ministry of Education (Grant
No. 2019-0903)
###### Acknowledgements.
We thank our colleagues A.D. Kiselev and G.P. Miroshnichenko for fruitful
discussions during the research.
## References
* Balabas _et al._ [2010a] M. V. Balabas, T. Karaulanov, M. P. Ledbetter, and D. Budker, Polarized alkali-metal vapor with minute-long transverse spin-relaxation time, Phys. Rev. Lett. 105, 070801 (2010a).
* Balabas _et al._ [2010b] M. V. Balabas, K. Jensen, W. Wasilewski, H. Krauter, L. S. Madsen, J. H. Müller, T. Fernholz, and E. S. Polzik, High quality anti-relaxation coating material for alkali atom vapor cells, Opt. Express 18, 5825 (2010b).
* Chi _et al._ [2020] H. Chi, W. Quan, J. Zhang, L. Zhao, and J. Fang, Advances in anti-relaxation coatings of alkali-metal vapor cells, Applied Surface Science 501, 143897 (2020).
* Goldfarb and Stoll [2018] D. Goldfarb and S. Stoll, _EPR Spectroscopy: Fundamentals and Methods_ (John Wiley and Sons, 2018).
* Weil and Bolton [2007] J. A. Weil and J. R. Bolton, _Electron Paramagnetic Resonance: Elementary Theory and Practical Applications_ (John Wiley and Sons, 2007).
* Pomerantsev _et al._ [1972] N. M. Pomerantsev, V. M. Ryzhkov, and G. V. Skrotskii, _Physical Principles of Quantum Magnetometry_ (Nauka, Moscow, 1972) p. 448.
* Farr and Otten [1974] W. Farr and E. W. Otten, A rb-magnetometer for a wide range and high sensitivity, Applied physics 3, 367 (1974).
* Aleksandrov _et al._ [1999] E. B. Aleksandrov, M. V. Balabas, A. K. Vershovskii, and A. S. Pazgalev, Multiple-quantum radio-frequency spectroscopy of atoms: Application to the metrology of geomagnetic fields, Technical Physics 44, 1025 (1999).
* Budker and Romalis [2007] D. Budker and M. Romalis, Optical magnetometry, Nature Physics 3, 227 (2007).
* Bison _et al._ [2003] G. Bison, R. Wynands, and A. Weis, A laser-pumped magnetometer for the mapping of human cardiomagnetic fields, Applied Physics B 76, 325 (2003).
* Budker and Kimball [2013] D. Budker and D. F. J. Kimball, _Optical Magnetometry_ (Cambridge University Press, 2013).
* Aleksandrov and Vershovskii [2009] E. B. Aleksandrov and A. K. Vershovskii, Modern radio-optical methods in quantum magnetometry, Phys.-Usp. 52, 573 (2009).
* Zhivun _et al._ [2014a] E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields, Applied Physics Letters 105, 10.1063/1.4902028 (2014a).
* Woodman _et al._ [1987] K. F. Woodman, P. W. Franks, and M. D. Richards, The nuclear magnetic resonance gyroscope: a review, Journal of Navigation 40, 366–384 (1987).
* Donley [2010] E. A. Donley, Nuclear magnetic resonance gyroscopes, 2010 IEEE Sensors , 17 (2010).
* Vershovskii _et al._ [2018] A. K. Vershovskii, Y. A. Litmanovich, A. S. Pazgalev, and V. G. Peshekhonov, Nuclear magnetic resonance gyro: Ultimate parameters, Gyroscopy and Navigation 9, 162 (2018).
* Kornack _et al._ [2005] T. W. Kornack, R. K. Ghosh, and M. V. Romalis, Nuclear spin gyroscope based on an atomic comagnetometer, Phys. Rev. Lett. 95, 230801 (2005).
* Walker and Larsen [2016] T. G. Walker and M. S. Larsen, Chapter eight - spin-exchange-pumped nmr gyros (Academic Press, 2016) pp. 373–401.
* Popov _et al._ [2018a] E. N. Popov, K. A. Barantsev, N. A. Ushakov, A. N. Litvinov, L. B. Liokumovich, A. N. Shevchenko, F. V. Sklyarov, and A. V. Medvedev, Behavior of signal from optical circuit of quantum rotation sensor based on nuclear magnetic resonance, Gyroscopy and Navigation 9, 183 (2018a).
* Zhang _et al._ [2020] K. Zhang, N. Zhao, and Y.-H. Wang, Closed-loop nuclear magnetic resonance gyroscope based on rb-xe, Scientific Reports 10, 2258 (2020).
* Gao _et al._ [2024] G. Gao, J. Hu, F. Tang, W. Liu, X. Zhang, B. Wang, D. Deng, M. Zhu, and N. Zhao, Stability improvement of nuclear magnetic resonance gyroscope with self-calibrating parametric magnetometer, Phys. Rev. Appl. 21, 014042 (2024).
* Camparo and Frueholz [1987] J. Camparo and R. Frueholz, A comparison of various alkali gas-cell atomic-frequency standards, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 34, 607 (1987).
* Jau and Carter [2020] Y.-Y. Jau and T. Carter, Vapor-cell-based atomic electrometry for detection frequencies below 1 khz, Phys. Rev. Appl. 13, 054034 (2020).
* Happer [1972] W. Happer, Optical pumping, Rev. Mod. Phys. 44, 169 (1972).
* Happer _et al._ [2010] W. Happer, Y.-Y. Jau, and T. Walker, _Optically Pumped Atoms_ (John Wiley and Sons, 2010).
* Kubo [1972] T. Kubo, Optical detection of magnetic resonance in alkali-metal, vapor using d(1) and d(2) radiation simultaneously, Appl Opt. 11, 1521 (1972).
* Aleksandrov and Zapasskii [1981] E. B. Aleksandrov and V. S. Zapasskii, Magnetic resonance in the faraday-rotation noise spectrum, JETP 54, 64 (1981).
* Happer and Van Wijngaarden [1987] W. Happer and W. A. Van Wijngaarden, An optical pumping primer, Hyperfine Interactions 38, 435 (1987).
* Suter [2020] D. Suter, Optical detection of magnetic resonance, Magnetic Resonance 1, 115 (2020).
* Školnik _et al._ [2009] G. Školnik, N. Vujičić, and T. Ban, Optical pumping of the zeeman components in the rubidium vapor, Optics Communications 282, 1326 (2009).
* Kozlov _et al._ [2016] A. N. Kozlov, S. A. Zibrov, A. A. Zibrov, A. V. Yudin, V. I. Taichenachev, V. P. Yakovlev, A. S. Tsygankov, E. A. Zibrov, V. V. Vassiliev, and V. L. Velichansky, Structure of magnetic resonance in 87rb atoms, JETP 122, 823 (2016).
* Talker _et al._ [2019] E. Talker, P. Arora, Y. Barash, D. Wilkowski, and U. Levy, Efficient optical pumping of alkaline atoms for evanescent fields at dielectric-vapor interfaces, Opt. Express 27, 33445 (2019).
* Franzen and Emslie [1957] W. Franzen and A. G. Emslie, Atomic orientation by optical pumping, Phys. Rev. 108, 1453 (1957).
* Kastler [1963] A. Kastler, Displacement of energy levels of atoms by light, J. Opt. Soc. Am. 53, 902 (1963).
* Kastler [1966] A. Kastler, Les methodes optiques pour l’etude des resonances hertziennes, in _Preprint les Prix Nobel en 1966_ (Conference Nobel, Stockholm, 1966).
* Zavoisky [1945] E. Zavoisky, Spin-magnetic resonance in paramagnetics, J. Phys. USSR , 211 (1945).
* Dehmelt [1957] H. G. Dehmelt, Modulation of a light beam by precessing absorbing atoms, Phys. Rev. 105, 1924 (1957).
* Bell and Bloom [1957] W. E. Bell and A. L. Bloom, Optical detection of magnetic resonance in alkali metal vapor, Phys. Rev. 107, 1559 (1957).
* Hawkins [1961] W. B. Hawkins, Cesium transition probabilities for optical pumping, Phys. Rev. 123, 544 (1961).
* Anderson and Ramsey [1963] L. W. Anderson and A. T. Ramsey, Study of the spin-relaxation times and the effects of spin-exchange collisions in an optically oriented sodium vapor, Phys. Rev. 132, 712 (1963).
* Gamblin and Carver [1965] R. L. Gamblin and T. R. Carver, Polarization and relaxation processes in ${\mathrm{he}}^{3}$ gas, Phys. Rev. 138, A946 (1965).
* Aymar _et al._ [1967] M. Aymar, M. Bouchiat, and J. Brossel, Relaxation of optically pumped rb in collisions against kr atoms, Physics Letters A 24, 753 (1967).
* Bouchiat _et al._ [2003] M. A. Bouchiat, J. Brossel, and L. C. Pottier, Evidence for Rb‐Rare‐Gas Molecules from the Relaxation of Polarized Rb Atoms in a Rare Gas. Experimental Results, The Journal of Chemical Physics 56, 3703 (2003).
* Bernabeu and Tornos [1985] E. Bernabeu and J. Tornos, Relaxation of quadrupole orientation in an optically pumped alkali vapour, Il Nuovo Cimento D 5, 315 (1985).
* Franz and Franz [1966] F. A. Franz and J. R. Franz, Excited-state mixing in the optical pumping of alkali-metal vapors, Phys. Rev. 148, 82 (1966).
* Bouchiat and Brossel [1966] M. A. Bouchiat and J. Brossel, Relaxation of optically pumped rb atoms on paraffin-coated walls, Phys. Rev. 147, 41 (1966).
* Breault [1964] R. O. Breault, _A Measurement of the Spin Relaxation Time of Optically Pumped Rubidium Vapor_ , Master’s thesis, Western Michigan University (1964).
* Cutler _et al._ [2020] T. Cutler, W. Hamlyn, J. Renger, K. Whittaker, D. Pizzey, I. Hughes, V. Sandoghdar, and C. Adams, Nanostructured alkali-metal vapor cells, Phys. Rev. Appl. 14, 034054 (2020).
* Appelt _et al._ [1995] S. Appelt, G. Wäckerle, and M. Mehring, A magnetic resonance study of non-adiabatic evolution of spin quantum states, Zeitschrift für Physik D 34, 75 (1995).
* Gaitan [1999] F. Gaitan, Berry’s phase in the presence of a non-adiabatic environment with an application to magnetic resonance, Journal of Magnetic Resonance 139, 152 (1999).
* Zhivun _et al._ [2014b] E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields, Applied Physics Letters 105, 192406 (2014b), https://pubs.aip.org/aip/apl/article-pdf/doi/10.1063/1.4902028/13802950/192406_1_online.pdf .
* Liu _et al._ [2021] G. Liu, V. Guarrera, S. Gu, and S. Zhang, Response dynamics of an alkali-metal–noble-gas hybrid trispin system, Phys. Rev. A 104, 032827 (2021).
* Titimbo _et al._ [2023] K. Titimbo, D. C. Garrett, S. S. Kahraman, Z. He, and L. V. Wang, Numerical modeling of the multi-stage stern–gerlach experiment by frisch and segrè using co-quantum dynamics via the bloch equation, Journal of Physics B: Atomic, Molecular and Optical Physics 56, 205004 (2023).
* Giuseppe _et al._ [2021] B. Giuseppe, B. Valerio, D. Yordanka, F. Alessandro, and V. Antonio, Spin dynamic response to a time dependent field, Applied Physics B 127, 128 (2021).
* Zhang _et al._ [2017] R. Zhang, Z.-G. Wang, X. Peng, W.-H. Li, S.-J. Li, and H. Guo, Spin dynamics of magnetic resonance with parametric modulation in a potassium vapor cell*, Chinese Physics B 26, 030701 (2017).
* Gong _et al._ [2007] F. Gong, Y.-Y. Jau, and W. Happer, Magnetic resonance reversals in optically pumped alkali-metal vapor, Phys. Rev. A 75, 053415 (2007).
* Su _et al._ [2023] S. Su, Z. Xu, X. He, C. Yin, M. Kong, X. Zhang, Y. Ruan, K. Li, and Q. Lin, An integrated single-beam three-axis high-sensitivity magnetometer, Sensors 23, 10.3390/s23063148 (2023).
* Chen _et al._ [2022] L. Chen, M. Yao, Y. Shi, J. Duan, S. Kang, P. Wang, Z. Yu, S. Guo, and X. Liu, Microwave magnetic field strength imaging based on Rabi resonance with alkali-atom vapor cell, Applied Physics Letters 120, 234001 (2022), https://pubs.aip.org/aip/apl/article-pdf/doi/10.1063/5.0098071/16449633/234001_1_online.pdf .
* Weigert [2002] S. Weigert, Quantum parametric resonance, Journal of Physics A: Mathematical and General 35, 4169 (2002).
* Steck _et al._ [2001] D. A. Steck, M. Gehm, and T. Tiecke, Alkali d-line data, https://steck.us/alkalidata/ (2001).
* Popov _et al._ [2018b] E. N. Popov, V. A. Bobrikova, S. P. Voskoboinikov, K. A. Barantsev, S. M. Ustinov, A. N. Litvinov, A. K. Vershovskii, S. P. Dmitriev, V. A. Kartoshkin, A. S. Pazgalev, and M. V. Petrenko, Features of the formation of the spin polarization of an alkali metal at the resolution of hyperfine sublevels in the 2s1/2 state, JETP Letters 108, 513 (2018b).
* Preston [1996] D. W. Preston, Doppler‐free saturated absorption: Laser spectroscopy, American Journal of Physics 64, 1432 (1996).
* Happer _et al._ [1984] W. Happer, E. Miron, S. Schaefer, D. Schreiber, W. A. van Wijngaarden, and X. Zeng, Polarization of the nuclear spins of noble-gas atoms by spin exchange with optically pumped alkali-metal atoms, Phys. Rev. A 29, 3092 (1984).
* Zeng _et al._ [1985] X. Zeng, Z. Wu, T. Call, E. Miron, D. Schreiber, and W. Happer, Experimental determination of the rate constants for spin exchange between optically pumped k, rb, and cs atoms and ${}^{129}\mathrm{Xe}$ nuclei in alkali-metal–noble-gas van der waals molecules, Phys. Rev. A 31, 260 (1985).
* Knize _et al._ [1988] R. Knize, Z. Wu, and W. Happer, Optical pumping and spin exchange in gas cells (Academic Press, 1988) pp. 223–267.
* Walker [1989] T. G. Walker, Estimates of spin-exchange parameters for alkali-metal–noble-gas pairs, Phys. Rev. A 40, 4959 (1989).
* Walker and Happer [1997] T. G. Walker and W. Happer, Spin-exchange optical pumping of noble-gas nuclei, Rev. Mod. Phys. 69, 629 (1997).
* Happer, W. [1985] Happer, W., Spin exchange, past, present and future, Ann. Phys. Fr. 10, 645 (1985).
* Sinatra [2022] A. Sinatra, Noble-gas and alkali spins exchange excitations, Nature Physics 18, 487 (2022).
|
# On the subgroup structure of the full Brauer group of Sweedler Hopf algebra
Giovanna Carnovale Juan Cuadra Dipartimento di Matematica Pura Universidad de
Almería ed Applicata Dpto. Álgebra y Análisis Matemático via Trieste 63
E-04120 Almería, Spain I-35121 Padua, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
We introduce a family of three parameters $2$-dimensional algebras
representing elements in the Brauer group $BQ(k,H_{4})$ of Sweedler Hopf
algebra $H_{4}$ over a field $k$. They allow us to describe the mutual
intersection of the subgroups arising from a quasitriangular or
coquasitriangular structure. We also define a new subgroup of $BQ(k,H_{4})$
and construct an exact sequence relating it to the Brauer group of Nichols
$8$-dimensional Hopf algebra with respect to the quasitriangular structure
attached to the $2\times 2$-matrix with $1$ in the $(1,2)$-entry and zero
elsewhere.
MSC:16W30, 16K50
## Introduction
The Brauer group of a Hopf algebra is an extremely complicated invariant that
reflects many aspects of the Hopf algebra: its automorphisms group, its Hopf-
Galois theory, its second lazy cohomology group, (co)quasitriangularity, etc.
It is very difficult to describe all its elements and to find their
multiplication rules. For the most studied case, that of a commutative and
cocommutative Hopf algebra, these are the results known so far: the first
explicit computation was done by Long in [14] for the group algebra
$k{\mathbb{Z}}_{n},$ where $n$ is square-free and $k$ algebraically closed
with $char(k)\nmid n$; DeMeyer and Ford [12] computed it for
$k{\mathbb{Z}}_{2}$ with $k$ a commutative ring containing $2^{-1}$. Their
result was extended by Beattie and Caenepeel in [2] for $k{\mathbb{Z}}_{n},$
where $n$ is a power of an odd prime number and some mild assumptions on $k$.
In [4] Caenepeel achieved to compute the multiplication rules for a subgroup,
the so-called split part, of the Brauer group for a faithfully projective
commutative and cocommutative Hopf algebra $H$ over any commutative ring $k$.
These results were improved in [6] and allowed him to compute the Brauer group
of Tate-Oort algebras of prime rank. For a unified exposition of these results
the profuse monograph [5] is recommended.
Since the Brauer group was defined for any Hopf algebra with bijective
antipode ([7], [8]), it was a main goal to compute it for the smallest
noncommutative noncocommutative Hopf algebra: Sweedler’s four dimensional Hopf
algebra $H_{4}$, which is generated over the field $k$ ($char(k)\neq 2$) by
the group-like $g$, the $(g,1)$-primitive element $h$ and relations
$g^{2}=1,h^{2}=0,gh=-hg$. A first step was the calculation in [20] of the
subgroup $BM(k,H_{4},R_{0})$ induced by the quasitriangular structure
$R_{0}=2^{-1}(1\otimes 1+g\otimes 1+1\otimes g-g\otimes g).$ It was shown to
be isomorphic to the direct product of $(k,+)$, the additive group of $k$, and
$BW(k)$, the Brauer-Wall group of $k$. It was later proved in [9] that the
subgroups $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ arising from all the
quasitriangular structures $R_{t}$ and the coquasitriangular structures
$r_{s}$ of $H_{4}$ respectively, with $s,t\in k$, are all isomorphic.
In this paper we introduce a family of three parameters $2$-dimensional
algebras $C(a;t,s)$, for $a,t,s\in k,$ that represent elements in
$BQ(k,H_{4})$. They will allow us to shed a ray of light on the subgroup
structure of $BQ(k,H_{4})$ and will provide some evidences about the
difficulty of the computation of this group. The algebra $C(a;t,s)$ is
generated by $x$ with relation $x^{2}=a$ and has a $H_{4}$-Yetter-Drinfeld
module algebra structure with action and coaction:
$g\cdot x=-x,\quad h\cdot x=t,\quad\rho(x)=x\otimes g+s\otimes h.$
We list the main properties of these algebras in Section 2 (Lemma 2.1) and we
show that $C(a;t,s)$ is $H_{4}$-Azumaya if and only if $2a\neq st$. When
$s=lt$ they represent elements in $BM(k,H_{4},R_{l})$ and this subgroup is
indeed generated by the classes of $C(a;1,t)$ with $2a\neq t$ together with
$BW(k)$, Proposition 2.6. The same statement holds true for
$BC(k,H_{4},r_{l})$ when $t=sl$ replacing $C(a;1,t)$ by $C(a;s,1)$,
Proposition 2.5.
Using the description of $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ in terms
of these algebras, Section 3 is devoted to analyze the intersection of these
subgroups inside $BQ(k,H_{4})$. Let $i_{t}$ and $\iota_{s}$ denote the
inclusion map of the former and the latter respectively. It is known that
$BW(k)$ is contained in any of the above subgroups. Theorem 3.5 states that:
1. (1)
$Im(i_{t})\cap Im(\iota_{s})\neq BW(k)$ iff $ts=1$. If this is the case,
$Im(i_{t})=Im(\iota_{s})$;
2. (2)
$Im(i_{t})\cap Im(i_{s})\neq BW(k)$ if and only if $t=s$;
3. (3)
$Im(\iota_{t})\cap Im(\iota_{s})\neq BW(k)$ if and only if $t=s$.
A remarkable property of our algebras is that they represent the same class in
$BQ(k,H_{4})$ if and only if they are isomorphic, Corollary 3.4.
A morphism from the automorphism group of $H_{4}$ to $BQ(k,H_{4})$ was
constructed in [19], allowing to consider $k^{\cdot 2}$ as a subgroup of
$BQ(k,H_{4})$. In Section 4 we show that the subgroup $BM(k,H_{4},R_{l})$ is
conjugated to $BM(k,H_{4},R_{l\alpha^{2}})$ inside $BQ(k,H_{4})$, for
$\alpha\in k^{\cdot}$, by a suitable representative of $k^{\cdot 2}$, Lemma
4.1.
Any $H_{4}$-Azumaya algebra possesses two natural ${\mathbb{Z}}_{2}$-gradings:
one stemming from the action of $g$ and one from the coaction (after
projection) of $g$. In Section 6 we introduce the subgroup
$BQ_{grad}(k,H_{4})$ consisting of those classes of $BQ(k,H_{4})$ that can be
represented by $H_{4}$-Azumaya algebras for which the two
${\mathbb{Z}}_{2}$-gradings coincide. On the other hand, the Drinfeld double
of $H_{4}$ admits a Hopf algebra map $T$ onto Nichols $8$-dimensional Hopf
algebra $E(2)$. This map is quasitriangular as $E(2)$ is equipped with the
quasitriangular structure $R_{N}$ corresponding to the $2\times 2$-matrix $N$
with $1$ in the $(1,2)$-entry and zero elsewhere, see (5.1). If we consider
the associated Brauer group $BM(k,E(2),R_{N})$, then Theorem 5.2 claims that
$T$ induces a group homomorphism $T^{*}$ fitting in the following exact
sequence
$\begin{array}[]{l}\begin{CD}1\longrightarrow{\mathbb{Z}}_{2}@>{}>{}>BM(k,E(2),R_{N})@>{T^{*}}>{}>BQ_{grad}(k,H_{4})\longrightarrow
1.\end{CD}\end{array}$
So in order to compute $BQ(k,H_{4})$ one should first understand
$BM(k,E(2),R_{N})$. This new problem cannot be attacked with the available
techniques for computations of groups of type BM, [20], [10], [11]. Those
computations were achieved by finding suitable invariants for a class by means
of a Skolem-Noether-like theory. In the Appendix we underline some obstacles
to the application of these techniques to the computation of
$BM(k,E(2),R_{N})$: the set of elements represented by algebras for which the
action of one of the standard nilpotent generators of $E(2)$ is inner
coincides with the set of classes represented by ${\mathbb{Z}}_{2}$-graded
central simple algebras and this is not a subgroup of $BM(k,E(2),R_{N}),$
Theorems 6.1, 6.3. Moreover, $BM(k,E(2),R_{N})$ seems to be much more complex
than the groups of type BM treated until now since, according to Proposition
5.3, each group $BM(k,H_{4},R_{t})$ may be viewed as a subgroup of it.
## 1 Preliminaries
In this paper $k$ is a field, $H$ will denote a Hopf algebra over $k$ with
bijective antipode $S$, coproduct $\Delta$ and counit $\varepsilon$. Tensor
products $\otimes$ will be over $k$ and, for vector spaces $V$ and $W$, the
usual flip map is denoted by $\tau:V\otimes W\to W\otimes V$. We shall adopt
the Sweedler-like notations $\Delta(h)=h_{(1)}\otimes h_{(2)}$ and
$\rho(m)=m_{(0)}\otimes m_{(1)}$ for coproducts and right comodule structures
respectively. For $H$ coquasitriangular (resp. quasitriangular), the set of
all coquasitriangular (resp. quasitriangular) structures will be denoted by
$\cal U$ (resp. $\cal T$).
Yetter-Drinfeld modules. Let us recall that if $A$ is a left $H$-module with
action $\cdot$ and a right $H$-comodule with coaction $\rho$ the two
structures combine to a left module structure for the Drinfeld double
$D(H)=H^{*,cop}\bowtie H$ of $H$ (cfr. [15]) if and only if they satisfy the
so-called Yetter-Drinfeld compatibility condition:
$\rho(l\cdot b)=l_{(2)}\cdot b_{(0)}\otimes
l_{(3)}b_{(1)}S^{-1}(l_{(1)}),\quad\forall l\in H,b\in A.$ (1.1)
Modules satisfying this condition are usually called Yetter-Drinfeld modules.
If $A$ is a left $H$-module algebra and a right $H^{op}$-comodule algebra
satisfying (1.1) we shall call it a Yetter-Drinfeld $H$-module algebra.
The Brauer group (see [7], [8]). Suppose that $A$ is a Yetter-Drinfeld
$H$-module algebra. The $H$-opposite algebra of $A$, denoted by
$\overline{A}$, is the underlying vector space of $A$ endowed with product
$a\circ c=c_{(0)}(c_{(1)}\cdot a)$ for every $a,c\in A$. The same action and
coaction of $H$ on $A$ turn $\overline{A}$ into a Yetter-Drinfeld $H$-module
algebra. Given two Yetter-Drinfeld $H$-module algebras $A$ and $B$ we can
construct a new Yetter-Drinfeld module $A\\#B$ whose underlying vector space
is $A\otimes B$, with action $h\cdot(a\otimes b)=h_{(1)}\cdot a\otimes
h_{(2)}\cdot b$ and with coaction $a\otimes b\mapsto a_{(0)}b_{(0)}\otimes
b_{(1)}a_{(1)}$. This object becomes a Yetter-Drinfeld module algebra if we
provide it with the multiplication
$(a\\#b)(c\\#d)=ac_{(0)}\\#(c_{(1)}\cdot b)d.$
For every finite dimensional Yetter-Drinfeld module $M$ the algebras ${\rm
End}(M)$ and ${\rm End}(M)^{op}$ can be naturally provided of a Yetter-
Drinfeld module algebra structure through (1.2) and (1.3) below respectively:
$\begin{array}[]{l}(h\cdot f)(m)=h_{(1)}\cdot f(S(h_{(2)})\cdot
m),\vspace{2pt}\\\ \rho(f)(m)=f(m_{(0)})_{(0)}\otimes
S^{-1}(m_{(1)})f(m_{(0)})_{(1)},\end{array}$ (1.2) $\begin{array}[]{l}(h\cdot
f)(m)=h_{(2)}\cdot f(S^{-1}(h_{(1)})\cdot m),\vspace{2pt}\\\
\rho(f)(m)=f(m_{(0)})_{(0)}\otimes f(m_{(0)})_{(1)}S(m_{(1)}),\end{array}$
(1.3)
where $h\in H,f\in End(M),m\in M.$ A finite dimensional Yetter-Drinfeld module
algebra $A$ is called $H$-Azumaya if the following module algebra maps are
isomorphisms:
$\begin{array}[]{ll}F\colon A\\#{\overline{A}}\rightarrow{\rm
End}(A),&F(a\\#b)(c)=ac_{(0)}(c_{(1)}\cdot b),\vspace{2pt}\\\
G\colon\overline{A}\\#{{A}}\rightarrow{\rm
End}(A)^{op},&G(a\\#b)(c)=a_{(0)}(a_{(1)}\cdot c)b.\end{array}$ (1.4)
The algebras ${\rm End}(M)$ and ${\rm End}(M)^{op}$, for a finite dimensional
Yetter-Drinfeld module $M$, provided with the preceding structures are
$H$-Azumaya.
The following relation $\sim$ established on the set of isomorphism classes of
$H$-Azumaya algebras is an equivalence relation: $A\sim B$ if there exist
finite dimensional Yetter-Drinfeld modules $M$ and $N$ such that $A\\#{\rm
End}(M)\cong B\\#{\rm End}(N)$ as Yetter-Drinfeld module algebras. The set of
equivalence classes of $H$-Azumaya algebras, denoted by $BQ(k,H)$, is a group
with product $[A][B]=[A\\#B]$, inverse element $[\overline{A}]$ and identity
element $[End(M)]$ for finite dimensional Yetter-Drinfeld modules $M$. This
group is called the full Brauer group of $H$. The adjective full is used to
distinguish it from the subgroups presented next, that receive the same name
in the literature.
Given a left $H$-module algebra $A$ with action $\cdot$ and a quasitriangular
structure $R=R^{(1)}\otimes R^{(2)}$ on $H$, a right $H^{op}$-comodule algebra
structure $\rho$ on $A$ is determined by
$\rho(a)=(R^{(2)}\cdot a)\otimes R^{(1)},\quad\forall a\in A.$
We will call this coaction the coaction induced by $\cdot$ and $R$. It is
well-known that $(A,\cdot,\rho)$ satisfies the Yetter-Drinfeld condition. This
allows the definition of the subgroup $BM(k,H,R)$ of $BQ(k,H)$ whose elements
are equivalence classes of $H$-Azumaya algebras with coaction induced by $R$
([8, §1.5]). To underline that a representative $A$ of a given class in
$BQ(k,H)$ represents a class in $BM(k,H,R)$ we shall say that $A$ is an
$(H,R)$-Azumaya algebra. The inclusion map will be denoted by $i\colon
BM(k,H,R)\to BQ(k,H)$. For $H$ finite dimensional $BQ(k,H)=BM(k,D(H),{\cal
R}),$ where ${\cal R}$ is the natural quasitriangular structure on the
Drinfeld double $D(H)$.
Dually, given a right $H^{op}$-comodule algebra $A$ with coaction $\varrho$
and a coquasitriangular structure $r$ on $H$, a $H$-module algebra structure
$\cdot$ on $A$ is determined by
$h\cdot a=a_{(0)}r(h\otimes a_{(1)}),\quad\forall a\in A,h\in H,$
and $(A,\cdot,\varrho)$ becomes a Yetter-Drinfeld module algebra. We will call
this action the action induced by $\chi$ and $r$. The subset $BC(k,H,r)$ of
$BQ(k,H)$ consisting of those classes admitting a representative whose action
is induced by $r$ is a subgroup ([8, §1.5]). To stress that a representative
$A$ of a class in $BQ(k,H)$ represents a class in $BC(k,H,r)$ we shall say
that $A$ is an $(H,r)$-Azumaya algebra. The inclusion of $BC(k,H,r)$ in
$BQ(k,H)$ will be denoted by $\iota\colon BC(k,H,r)\to BQ(k,H)$.
On Sweedler Hopf algebra. In the sequel we will assume that $char(k)\neq 2.$
Let $H_{4}$ be Sweedler Hopf algebra, that is, the Hopf algebra over $k$
generated by a grouplike element $g$ and an element $h$ with relations,
coproduct and antipode:
$g^{2}=1,\quad h^{2}=gh+hg=0,\quad\Delta(h)=1\otimes h+h\otimes g,\quad
S(g)=g,\quad S(h)=gh.$
The Hopf algebra $H_{4}$ has a family of quasitriangular (indeed triangular)
structures. They were classified in [18] and are given by:
$R_{t}=\frac{1}{2}(1\otimes 1+1\otimes g+g\otimes 1-g\otimes
g)+\frac{t}{2}(h\otimes h+h\otimes gh+gh\otimes gh-gh\otimes h),$
where $t\in k$. It is well-known that $H_{4}$ is self-dual so that $H_{4}$ is
also cotriangular. Let $\\{1^{*},g^{*},h^{*},(gh)^{*}\\}$ be the basis of
$H^{*}_{4}$ dual to $\\{1,g,h,gh\\}$. We will often make use of the Hopf
algebra isomorphism
$\begin{array}[]{rl}\phi\colon H_{4}&\to H_{4}^{*}\\\ 1&\mapsto
1^{*}+g^{*}=\varepsilon\\\ h&\mapsto h^{*}+(gh)^{*}\\\ g&\mapsto
1^{*}-g^{*}\\\ gh&\mapsto h^{*}-(gh)^{*}.\end{array}$
So, the cotriangular structures of $H_{4}$ can be obtained applying the
isomorphism $\phi\otimes\phi$ to the $R_{t}$’s. They are:
$\begin{array}[]{c|rrrr}r_{t}&1&g&h&gh\\\ \hline\cr 1&1&1&0&0\\\ g&1&-1&0&0\\\
h&0&0&t&-t\\\ gh&0&0&t&t\\\ \end{array}$
The Drinfeld double $D(H_{4})=H_{4}^{*,cop}\bowtie H_{4}$ of $H_{4}$ is
isomorphic to the Hopf algebra generated by $\phi(h)\bowtie 1$,
$\phi(g)\bowtie 1$, $\varepsilon\bowtie g$ and $\varepsilon\bowtie h$ with
relations:
$\begin{array}[]{l}(\phi(h)\bowtie 1)^{2}=0;\\\ (\phi(g)\bowtie
1)^{2}=\varepsilon\bowtie 1;\\\ (\phi(h)\bowtie 1)(\phi(g)\bowtie
1)+(\phi(g)\bowtie 1)(\phi(h)\bowtie 1)=0;\\\ (\varepsilon\bowtie h)^{2}=0;\\\
(\varepsilon\bowtie h)(\varepsilon\bowtie g)+(\varepsilon\bowtie
g)(\varepsilon\bowtie h)=0;\\\ (\varepsilon\bowtie g)^{2}=\varepsilon\bowtie
1;\\\ (\phi(h)\bowtie 1)(\varepsilon\bowtie g)+(\varepsilon\bowtie
g)(\phi(h)\bowtie 1)=0;\\\ (\phi(g)\bowtie 1)(\varepsilon\bowtie
h)+(\varepsilon\bowtie h)(\phi(g)\bowtie 1)=0;\\\ (\varepsilon\bowtie
g)(\phi(g)\bowtie 1)=(\phi(g)\bowtie 1)(\varepsilon\bowtie g);\\\
(\phi(h)\bowtie 1)(\varepsilon\bowtie h)-(\varepsilon\bowtie h)(\phi(h)\bowtie
1)=(\phi(g)\bowtie 1)-(\varepsilon\bowtie g)\end{array}$
and with coproduct induced by the coproducts in $H_{4}$ and $H_{4}^{*,cop}$.
For $l\in H_{4}$ we will sometimes write $\phi(l)$ instead of $\phi(l)\bowtie
1$ and $l$ instead of $1\bowtie l$ for simplicity.
Let us recall that a Yetter-Drinfeld $H_{4}$-module $M$ with action $\cdot$
and coaction $\rho$ becomes a $D(H_{4})$-module by letting $1\bowtie l$ act as
$l$ for every $l\in H_{4}$ and $(\phi(l)\bowtie
1).m=m_{(0)}(\phi(l)(m_{(1)}))$ for $m\in M$. Conversely, a $D(H_{4})$-module
$M$ becomes naturally a Yetter-Drinfeld module with $H_{4}$-action obtained by
restriction and $H_{4}$-coaction given by
$\rho(m)=\frac{1}{2}(\phi(1+g).m\otimes 1+\phi(1-g).m\otimes
g+\phi(h+gh).m\otimes h+\phi(h-gh)\otimes gh).$
We will often switch from one notation to the other according to convenience.
Centers and centralizers. If $A$ is a Yetter-Drinfeld $H$-module algebra, and
$B$ is a Yetter-Drinfeld submodule algebra of $A$, the left and the right
centralizer of $B$ in $A$ are defined to be:
$C^{l}_{A}(B):=\\{a\in A~{}|~{}ba=a_{(0)}(a_{(1)}\cdot b)\ \forall b\in B\\},$
$C^{r}_{A}(B):=\\{a\in A~{}|~{}ab=b_{(0)}(b_{(1)}\cdot a)\ \forall b\in B\\}.$
For the particular case $B=A$ we have the right center $Z^{r}(A)$ and the left
center $Z^{l}(A)$ of $A$. Both are trivial when $A$ is $H$-Azumaya, [8,
Proposition 2.12].
## 2 Some low dimensional representatives in $BQ(k,H_{4})$
In this section we shall introduce a family of 2-dimensional representatives
of classes in $BQ(k,H_{4})$ that will turn out to be easy to compute with.
They appeared for the first time in [16] and a particular case of them is
treated in [1, Section 1.5].
Let $a,\,t,\,s\in k$. The algebra $C(a)$ generated by $x$ with relation
$x^{2}=a$ is acted upon by $H_{4}$ by
$g\cdot 1=1,\quad g\cdot x=-x,\quad h\cdot 1=0,\qquad h\cdot x=t,$
and it is a right $H_{4}$-comodule via
$\rho_{s}(1)=1\otimes 1,\quad\quad\rho_{s}(x)=x\otimes g+s\otimes h.$
It is not hard to check that $C(a)$ with this action and coaction is a left
$H_{4}$-module algebra and a right $H^{op}$-comodule algebra. We shall denote
it by $C(a;t,s)$.
###### Lemma 2.1
Let notation be as above.
1. (1)
$C(a;t,s)$ is a Yetter-Drinfeld module algebra with the preceding structures.
2. (2)
As a module algebra $C(a;t,s)\cong C(a^{\prime};t^{\prime},s^{\prime})$ if and
only if there is $\alpha\in k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$ and
$t=\alpha t^{\prime}$.
3. (3)
As a comodule algebra $C(a;t,s)\cong C(a^{\prime};t^{\prime},s^{\prime})$ if
and only if there is $\alpha\in k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$
and $s=\alpha s^{\prime}$.
4. (4)
As a Yetter-Drinfeld module algebra $C(a;t,s)\cong
C(a^{\prime};t^{\prime},s^{\prime})$ if and only if there exists $\alpha\in
k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$, $t=\alpha t^{\prime}$ and
$s=\alpha s^{\prime}$.
5. (5)
The module structure on $C(a;t,s)$ is induced by its comodule structure and a
cotriangular structure $r_{l}$ if and only if $t=sl$.
6. (6)
The comodule structure on $C(a;t,s)$ is induced by its module structure and a
triangular structure $R_{l}$ if and only if $s=lt$.
7. (7)
The $H_{4}$-opposite algebra of $C(a;t,s)$ is $C(st-a;t,s)$.
8. (8)
$C(a;t,s)$ is an $H_{4}$-Azumaya algebra if and only if $2a\neq st$.
Proof: Let $x$ and $y$ be algebra generators in $C(a;t,s)$ and
$C(a^{\prime};t^{\prime},s^{\prime})$ respectively with $x^{2}=a$ and
$y^{2}=a^{\prime}$.
(1) We verify condition (1.1) for $b=x$ and $l=h$. The other cases are easier
to check.
$\begin{array}[]{l}h_{(2)}\cdot x_{(0)}\otimes
h_{(3)}x_{(1)}S^{-1}(h_{(1)})\\\ \hskip 48.36958pt=g\cdot x\otimes(-gh)+g\cdot
s\otimes(gh)(-gh)+h\cdot x\otimes g^{2}\\\ \hskip 56.9055pt+h\cdot s\otimes
gh+x\otimes hg+s\otimes h^{2}\\\ \hskip 48.36958pt=x\otimes gh+t\otimes
1-x\otimes gh\\\ \hskip 48.36958pt=\rho_{s}(h\cdot x).\end{array}$
(2) An algebra isomorphism $f\colon C(a;t,s)\to
C(a^{\prime};t^{\prime},s^{\prime})$ must map $x$ to $\alpha y$ for some
$\alpha\in k^{\cdot}$. Then $a=x^{2}=(\alpha y)^{2}=\alpha^{2}a^{\prime}$.
Besides, $h.f(x)=f(h.x)$ implies $t^{\prime}\alpha=t$. It is easy to verify
that the condition is also sufficient.
(3) In the above setup $\rho_{s^{\prime}}(f(x))=(f\otimes{\rm id})\rho_{s}(x)$
implies $s^{\prime}\alpha=s$. It is not hard to check that this condition is
also sufficient.
(4) It follows from the preceding statements.
(5) If the module structure on $C(a;t,s)$ is induced by its comodule structure
$\rho_{s}$ and some $r_{l}\in{\cal U},$ then $t=h\cdot x=xr_{l}(h\otimes
g)+sr_{l}(h\otimes h)=sl.$ Conversely, if $t=sl$, then
$\begin{array}[]{l}g\cdot 1=1=1r_{l}(g\otimes 1);\qquad h\cdot
1=0=1r_{l}(h\otimes 1);\vspace{2pt}\\\ g\cdot x=-x=xr_{l}(g\otimes
g)+sr_{l}(g\otimes h)=x_{(0)}r_{l}(g\otimes x_{(1)});\vspace{2pt}\\\ h\cdot
x=t=xr_{l}(h\otimes g)+sr_{l}(h\otimes h)=x_{(0)}r_{l}(h\otimes
x_{(1)}).\end{array}$
Therefore the action is induced by the coaction and $r_{l}$.
(6) If the comodule structure on $C(a;t,s)$ is induced by the action and some
$R_{l}\in{\cal T},$ then
$x\otimes g+s\otimes h=\rho_{s}(x)=(R_{l}^{(2)}\cdot x)\otimes
R_{l}^{(1)}=\frac{1}{2}(2x\otimes g)+\frac{l}{2}(2t\otimes h)=x\otimes
g+lt\otimes h$
hence $s=lt$. Conversely, if $s=lt$ then
$\begin{array}[]{l}\rho_{s}(1)=1\otimes 1=(R_{l}^{(2)}\cdot 1)\otimes
R_{l}^{(1)},\\\ \rho_{s}(x)=x\otimes g+s\otimes h=(R_{l}^{(2)}\cdot x)\otimes
R_{l}^{(1)},\end{array}$
so the comodule structure is induced by the action and $R_{l}$.
(7) $\overline{C(a;t,s)}$ has $1,\,x$ as a basis and $1$ is the unit. The
action and coaction on $1$ and $x$ are as for $C(a;t,s)$. By direct
computation, $x\circ x=x(g\cdot x)+s(h\cdot x)=-a+st,$ so
$\overline{C(a;t,s)}=C(st-a;t,s)$.
(8) The algebra $C(a;t,s)$ is $H_{4}$-Azumaya if and only if the maps $F$ and
$G$ defined in (1.4) are isomorphisms. The space $C(a;t,s)\\#C(a;t,s)$ has
ordered basis $1\\#1,\,1\\#x,\,x\\#1,\,x\\#x$ while ${\rm End}(C(a;t,s))$ has
basis $1^{*}\otimes 1,\,1^{*}\otimes x,\,x^{*}\otimes 1,\,x^{*}\otimes x$ with
the usual identification $C(a;t,s)^{*}\otimes C(a;t,s)\cong{\rm
End}(C(a;t,s))$. Then for every $b,c\in C(a;t,s)$ we have
$\begin{array}[]{l}F(b\\#c)(1)=bc,\quad F(b\\#c)(x)=bx(g\cdot c)+sb(h\cdot
c),\vspace{2pt}\\\ G(1\\#b)(c)=cb,\quad G(x\\#b)(c)=x(g\cdot c)b+s(h\cdot
c)b.\end{array}$
The matrices associated with $F$ and $G$ with respect to the given bases are
respectively
$\left(\begin{array}[]{cccc}1&0&0&a\\\ 0&1&1&0\\\ 0&st-a&a&0\\\
1&0&0&st-a\end{array}\right)\qquad\left(\begin{array}[]{cccc}1&0&0&a\\\
0&1&1&0\\\ 0&a&st-a&0\\\ 1&0&0&st-a\end{array}\right)$
whose determinants $-(st-2a)^{2}$ and $(st-2a)^{2}$ are nonzero if and only if
$2a\neq st$. $\Box$
We have seen so far that the algebras $C(a;s,t)$ can be viewed as
representatives of classes in $BM(k,H_{4},R_{l})$ or in $BC(k,H_{4},r_{l})$
for suitable $l\in k$. It is known that these groups are all isomorphic to
$(k,+)\times BW(k),$ where $BW(k)$ is the Brauer-Wall group of $k$. We aim to
find to which pair $(\beta,[A])\in(k,+)\times BW(k)$ do the class of
$C(a;t,s)$ correspond. The group $BM(k,H_{4},R_{0})$ was computed in [20]. The
computation of $BC(k,H_{4},r_{0})$ follows from self-duality of $H_{4}$. It
was shown in [9] that all groups $BC(k,H_{4},r_{t})$ (hence, dually, all
$BM(k,H_{4},R_{t})$) are isomorphic. We shall use the description of
$BM(k,E(1),R_{t})$ given in [11] beause this might allow generalizations. In
the mentioned paper the Brauer group $BM(k,E(n),R_{0})$ is computed for the
family of Hopf algebras $E(n)$, where $E(1)=H_{4}$. We shall recall first
where do the isomorphism of the different Brauer groups $BC$ and $BM$ stem
from. The notion of lazy cocycle plays a key role here.
We recall from [3] that a lazy cocycle on $H$ is a left 2-cocycle $\sigma$
such that twisting $H$ by $\sigma$ does not modify the product in $H$. In
other words: for every $h,l,m\in H,$
$\sigma(h_{(1)}\otimes l_{(1)})\sigma(h_{(2)}l_{(2)}\otimes
m)=\sigma(l_{(1)}\otimes m_{(1)})\sigma(h\otimes l_{(2)}m_{(2)})$ (2.1)
$\sigma(h_{(1)}\otimes
l_{(1)})h_{(2)}l_{(2)}=h_{(1)}l_{(1)}\sigma(h_{(2)}\otimes l_{(2)})$ (2.2)
It turns out that a lazy left cocycle is also a right cocycle. Given a lazy
cocycle $\sigma$ for $H$ and a $H^{op}$-comodule algebra $A$, we may construct
a new $H^{op}$-comodule algebra $A_{\sigma}$, which is equal to $A$ as a
$H^{op}$-comodule, but with product defined by:
$a\bullet b=a_{(0)}b_{(0)}\sigma(a_{(1)}\otimes b_{(1)}).$
The group of lazy cocycles for $H_{4}$ is computed in [3]. Lazy cocycles are
parametrized by elements $t\in k$ as follows:
$\begin{array}[]{c|rrrr}\sigma_{t}&1&g&h&gh\\\ \hline\cr 1&1&1&0&0\\\
g&1&1&0&0\\\ h&0&0&\frac{t}{2}&\frac{t}{2}\\\
gh&0&0&\frac{t}{2}&-\frac{t}{2}\\\ \end{array}$
We have the following group isomorphisms:
1. (2.3)
$\Psi_{t}:BC(k,H_{4},r_{0})\rightarrow
BC(k,H_{4},r_{t}),[A]\mapsto[A_{\sigma_{t}}],$ constructed in [9, Proposition
3.1].
2. (2.4)
$\Phi_{t}:BM(k,H_{4},R_{t})\rightarrow BC(k,H_{4},r_{t}),[A]\mapsto[A^{op}].$
We explain how $A^{op}$ is equipped with the corresponding structure. The left
$H_{4}$-module algebra $A$ becomes a right $H_{4}^{*}$-comodule algebra. Then
$A^{op}$ is a right $H_{4}^{*,op}$-comodule algebra. The quasitriangular
structure $R_{t}$ is a coquasitriangular structure in $H_{4}^{*}$. Then
$A^{op}$ may be endowed with the left $H_{4}^{*}$-action stemming from the
comodule structure and $R_{t}$. On the other hand, $A^{op}$ may be viewed as
an $H_{4}^{op}$-comodule algebra through the isomorphism
$\phi:H_{4}\rightarrow H_{4}^{*}$. The coquasitriangular structure $R_{t}$ on
$H_{4}^{*}$ corresponds to the coquasitriangular structure $r_{t}$ on $H_{4}$
via $\phi.$
An isomorphism between $BM(k,H_{4},R_{0})$ and $BM(k,H_{4},R_{t})$ can be
constructed combining the above ones. Thus, the crucial step is to analyze the
sought correspondence for $BM(k,H_{4},R_{0})$.
The Brauer group $BM(k,H_{4},R_{0})$ is computed in [20] through the split
exact sequence (see also [1, Theorem 3.8] for an alternative approach):
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k,+)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BM(k,H_{4},R_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{BW(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{1.}$
The map $j^{*}:BM(k,H_{4},R_{0})\to BW(k),[A]\mapsto[A]$ is obtained by
restricting the $H_{4}$-action of $A$ to a $k{\mathbb{Z}}_{2}$-action via the
inclusion map $j:k{\mathbb{Z}}_{2}\rightarrow H_{4}$. This map is split by
$\pi^{*}:BW(k)\rightarrow BM(k,H_{4},R_{0}),[B]\mapsto[B]$, where $B$ is
considered as an $H_{4}$-module by restriction of scalars via the algebra
projection $\pi:H_{4}\rightarrow k{\mathbb{Z}}_{2},g\mapsto g,h\mapsto 0$. A
class $[A]$ lying in the kernel of $j^{*}$ is a matrix algebra with an inner
action of $H_{4}$ such that the restriction to $k{\mathbb{Z}}_{2}$ is strongly
inner. Thus there exist uniquely determined $u,w\in A$ such that
$g\cdot a=uau^{-1},\quad h\cdot a=w(g\cdot a)-aw\quad\forall a\in A,$ (2.5)
$u^{2}=1,\quad wu+uw=0,\quad w^{2}=\beta,$ (2.6)
for certain $\beta\in k$. Mapping $[A]\mapsto\beta$ defines a group
isomorphism $\chi\colon Ker(j^{*})\\\ \cong(k,+)$. We will determine
$j^{*}([C(a;t,s)])$ and $\chi([C(a;t,s)]\pi^{*}j^{*}([C(a;t,s)]^{-1}))$
whenever this is well-defined. To this purpose, we will first describe all
products of two algebras of type $C(a;t,s)$.
###### Lemma 2.2
Let $x,y$ be generators for $C(a;t,s)$ and
$C(a^{\prime};t^{\prime},s^{\prime})$ respectively, with relations,
$H_{4}$-actions and coactions as above. The product
$C(a;t,s)\\#C(a^{\prime};t^{\prime},s^{\prime})$ is isomorphic to the
generalized quaternion algebra with generators $X=x\\#1$ and $Y=1\\#y$,
relations, $H_{4}$-action and and $H_{4}$-coaction:
$X^{2}=a,\quad Y^{2}=a^{\prime},\quad XY+YX=st^{\prime},$ $g\cdot X=-X,\quad
g\cdot Y=-Y,\quad h\cdot X=t,\quad h\cdot Y=t^{\prime},$ $\rho(X)=X\otimes
g+s\otimes h,\quad\rho(Y)=Y\otimes g+s^{\prime}\otimes h.$
Proof: By direct computation:
$X^{2}=(x\\#1)(x\\#1)=a\\#1,\quad Y^{2}=(1\\#y)(1\\#y)=a^{\prime}\\#1,\quad
XY=x\\#y,$ $YX=(1\\#y)(x\\#1)=x\\#(g\cdot y)+s\\#(h\cdot
y)=-XY+st^{\prime}\\#1.$
The formulas for the action and the coaction follow immediately from the
definition of action and coaction on a $\\#$-product. $\Box$
Elements in $BW(k)$ are represented by graded tensor products of the following
three type of algebras: $C(1)$ generated by the odd element $x$ with
$x^{2}=1$; classically Azumaya algebras having trivial
${\mathbb{Z}}_{2}$-action; and $C(a)\\#C(1),$ where $C(a)$ is generated by the
odd element $y$ with $y^{2}=a\in k^{\cdot}$ ([13, Theorem IV.4.4]).
###### Proposition 2.3
For $a\neq 0$ let $[C(a;t,0)]\in BM(k,H_{4},R_{0})$ denote the class of
$C(a;t,0).$ Then
$[C(a;t,0)]=(t^{2}(4a)^{-1},[C(a)])\in(k,+)\times BW(k),$
so the group $BM(k,H_{4},R_{0})$ is generated by $BW(k)$ and the classes
$[C(a;1,0)]$.
Proof: It is clear that if $a\neq 0$ then $j^{*}([C(a;t,0)])=[C(a)]$ and that
$\pi^{*}([C(a)])=[C(a;0,0)]$. Thus, $[C(a;t,0)\\#C(-a;0,0)]\in{\rm
Ker}(j^{*})$. We shall compute its image through $\chi$. By Lemma 2.2,
$C(a;t,0)\\#C(-a;0,0)$ is generated by $X$ and $Y$ with relations,
$H_{4}$-action and $H_{4}$-coaction:
$X^{2}=a,\quad Y^{2}=-a,\quad XY+YX=0,$ $g\cdot X=-X,\quad g\cdot Y=-Y,\quad
h\cdot X=t,\quad h\cdot Y=0,$ $\rho(X)=X\otimes g,\quad\rho(Y)=Y\otimes g.$
We look for the element $w$ satisfying (2.5) and (2.6). This element must be
odd with respect to the ${\mathbb{Z}}_{2}$-grading induced by the $g$-action,
hence $w=\lambda X+\mu Y$ for some $\lambda,\mu\in k$. Condition $h\cdot
X=-wX-Xw$ implies $t=-2\lambda a$ and condition $h\cdot Y=-wY-Yw$ implies
$0=-2\mu a$ so $w^{2}=a\lambda^{2}=t^{2}(4a)^{-1}$. Thus
$[C(a;t,0)]=(t^{2}(4a)^{-1},[C(a)])$ and we have the first statement. For the
second one, let $(\beta,[A])\in(k,+)\times BW(k)$. If $\beta=0$ there is
nothing to prove. If $\beta\neq 0$, the class
$[C((4\beta)^{-1}t^{2};t,0)]=[C((4\beta)^{-1};1,0)]=(\beta,[C((4\beta)^{-1})])$,
so $BM(k,H_{4},R_{0})\cong(k,+)\times BW(k)$ is generated by $BW(k)$ and the
$[C(a;1,0)]$ for $a\neq 0.$ $\Box$
###### Lemma 2.4
Let $A$ be a $D(H_{4})$-module algebra.
1. (1)
If the $h$-action on $A$ is trivial, then $A$ is $(H_{4},R_{0})$-Azumaya if
and only if it is $(H_{4},R_{t})$-Azumaya for every $t\in k$.
2. (2)
If the $\phi(h)$-action on $A$ is trivial, then $A$ is $(H_{4},r_{0})$-Azumaya
if and only if it is $(H_{4},r_{t})$-Azumaya for every $t\in k$.
3. (3)
The representatives of $BW(k)$ in $BC(k,H_{4},r_{t})$ and $BM(k,H_{4},R_{s})$
all coincide when viewed inside $BQ(k,H_{4})$.
Proof: (1) It follows from the form of the elements in ${\cal T}$ that if $A$
is $(H_{4},R_{0})$-Azumaya and the action of $h$ on $A$ is trivial (i.e., if
it lies in $BW(k)$), then its comodule structure $\rho_{t}$ induced by $R_{t}$
coincides with the comodule structure $\rho_{0}$ induced by $R_{0}$. Hence,
the maps $F$ and $G$ with respect to the action and $\rho_{t}$ are the same as
the maps $F$ and $G$ with respect to the action and $\rho_{0}$, so $A$ is
$(H_{4},R_{t})$-Azumaya for every $t\in k$.
(2) It is proved as (1).
(3) The first statement shows that the representatives of $BW(k)$ inside the
different $BM(k,H_{4},R_{t})$ coincide. The second statement shows the same
for $BC(k,H_{4},r_{t})$. Therefore we may assume $s=t=0$. The elements of this
copy of $BW(k)$ consist of ${\mathbb{Z}}_{2}$-graded Azumaya algebras $A$
where the grading is induced by the action of $g$. The $h$-action is trivial.
If the coaction $\rho$ is induced by $R_{0}$, then $a\in A$ is odd if and only
if $\rho(a)=a\otimes g$. The action $\rightharpoonup$ induced on $A$ by
$r_{0}$ and $\rho$ is as follows: $h\rightharpoonup a=0$ for every $a\in A$
and $g\rightharpoonup a=-a$ if and only if $\rho(a)=a\otimes g$, that is, the
original action on $A$ and $\rightharpoonup$ coincide. Thus, the maps $F$ and
$G$ coincide in all cases and $A$ represents an element in $BW(k)\subset
BM(k,H_{4},R_{0})$ if and only if it represents an element in $BW(k)\subset
BC(k,H_{4},r_{0})$. $\Box$
###### Proposition 2.5
The group $BC(k,H_{4},r_{s})$ is generated by the Brauer-Wall group and the
classes $[C(a;s,1)]$ for $2a\neq s$.
Proof: We will first deal with the case $s=0$. We will show that the
isomorphism $\Phi_{0}:BM(k,H_{4},R_{0})\rightarrow BC(k,H_{4},r_{0}),$
$[A]\mapsto[A^{op}]$ in (2.4) maps $[C(a;1,0)]$ to $[C(a;0,1)]$ and
$BW(k)\subset BM(k,H_{4},R_{0})$ to $BW(k)\subset BC(k,H_{4},r_{0})$. The
class $[C(a;1,0)]$ is mapped to the class of the algebra $C(a)^{op}$ with
comodule structure
$\rho(x)=x\otimes(1^{*}-g^{*})+1\otimes(h^{*}+(gh)^{*})=x\otimes\phi(g)+1\otimes\phi(h)$
and $H_{4}$-action induced by the cotriangular structure $r_{0}$, that is,
$g\cdot x=-x$ and $h\cdot x=0$. The algebra $C(a)^{op}$ with these structures
is just $C(a;0,1)$.
Let $A$ be a representative of a class in $BW(k)\subset BM(k,H_{4},R_{0})$
with action $\cdot$ for which $h\cdot a=0$ for all $a\in A$. The class $[A]$
is mapped by $\Phi_{0}$ to the class of $A^{op}$ with coaction
$\rho(a)=a\otimes 1^{*}+(g\cdot a)\otimes g^{*}+(h\cdot a)\otimes
h^{*}+(gh\cdot a)\otimes(gh)^{*}\in A\otimes\phi(k{\mathbb{Z}}_{2}).$
Therefore $[A^{op}]\in BW(k)\subset BC(k,H_{4},r_{0})$.
We now take $s\in k$ arbitrary and use the isomorphism
$\Psi_{s}:BC(k,H_{4},r_{0})\rightarrow BC(k,H_{4},r_{s})$ in (2.3) to prove
the statement. We will show that $[C(a;0,1)]$ is mapped to
$[C(a+2^{-1}s;s,1)]$ through $\Psi_{s}$. Recall that $\Psi_{s}$ maps the class
of $C(a;0,1)$ to the class of the algebra $C(a;0,1)_{\sigma_{s}}$. It is
generated by $x$ with relation
$x\bullet x=x^{2}\sigma_{s}(g\otimes g)+x\sigma_{s}(h\otimes
g)+x\sigma_{s}(g\otimes h)+\sigma_{s}(h\otimes h)=a+\frac{s}{2},$
with (same) coaction $\rho(x)=x\otimes g+1\otimes h$ and action induced by
$\rho$ and $r_{s}$, that is:
$g\cdot x=r_{s}(g\otimes g)x+r_{s}(g\otimes h)=-x,\quad h\cdot
x=r_{s}(h\otimes g)x+r_{s}(h\otimes h)=s.$
Then $\Psi_{s}([C(a;0,1)])=[C(a+\frac{s}{2};s,1)]$.
Since the coaction is not changed by $\Psi_{s}$ the class of an element $A$
for which the image of the coaction is in $A\otimes k{\mathbb{Z}}_{2}$ is
again of this form. Hence the classes in $BW(k)\subset BC(k,H_{4},r_{0})$
correspond to the classes in $BW(k)\subset BC(k,H_{4},r_{s})$. $\Box$
###### Proposition 2.6
The group $BM(k,H_{4},R_{t})$ is generated by the Brauer-Wall group and the
classes $[C(a;1,t)]$ for $2a\neq t$.
Proof: Through the isomorphism $\Phi_{t}:BM(k,H_{4},R_{t})\rightarrow
BC(k,H_{4},r_{t})$ in (2.4), the class $[C(a;1,t)]$ is mapped to $[C(a;t,1)]$
and the classes in $BW(k)\subset BM(k,H_{4},R_{t})$ correspond to the classes
in $BW(k)\subset BC(k,H_{4},r_{t})$. The $H_{4}$-comodule structure on the
algebra $C(a)^{op}$ is:
$\rho(x)=x\otimes(1^{*}-g^{*})+1\otimes(h^{*}+(gh)^{*})=x\otimes\phi(g)+1\otimes\phi(h)$
The $H_{4}$-action induced by the cotriangular structure $r_{t}$ on $H_{4}$
gives $h\cdot x=t$. Therefore this algebra is $C(a;t,1)$. Finally, the
statement concerning $BW(k)$ is proved as in the preceding theorem. $\Box$
###### Remark 2.7
That $BM(k,H_{4},R_{t})$ is generated by $BW(k)$ and the classes $[C(a;1,t)]$
for $2a\neq t$ was first discovered in [1, Theorem 3.8 and Page 392] as a
consequence of the Structure Theorems for $(H_{4},R_{t})$-Azumaya algebras.
Since we will strongly use Proposition 2.6 later, for the reader’s convenience
we offered this alternative and self-contained approach. Notice that it mainly
relies on Lemma 2.2 that will be another key result for us in the sequel.
## 3 Fitting $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ into $BQ(k,H_{4})$
As groups $BM(k,H_{4},R_{t})\cong BC(k,H_{4},r_{s})$ for every $s,t\in k$.
However, their images in $BQ(k,H_{4})$ through the natural embeddings
$i_{t}\colon BM(k,H_{4},R_{t})\to
BQ(k,H_{4})\quad\textrm{and}\quad\iota_{s}\colon BC(k,H_{4},r_{s})\to
BQ(k,H_{4})$
do not coincide in general. In this section we will describe the mutual
intersections of these images.
###### Proposition 3.1
Let $0\neq t\in k$ then $Im(i_{t})=Im(\iota_{t^{-1}})$
Proof: Given $t\neq 0$, by Lemma 2.1, $[C(a;1,t)]\in Im(i_{t})\cap
Im(\iota_{t^{-1}})$ for every $a\neq 2t.$ Besides, by Lemma 2.4,
$i_{t}(BW(k))=\iota_{s}(BW(k))$ for any $s\in k$. Since the elements of
$BW(k)$ and the $[C(a;1,t)]$’s generate $BM(k,H_{4},R_{t})$ and
$BC(k,H_{4},r_{t^{-1}})$ we are done. $\Box$
Given $[A]$ in $BQ(k,H_{4})$, there are two natural
${\mathbb{Z}}_{2}$-gradings on $A$, the one coming from the $g$-action, for
which $|a|=1$ iff $g\cdot a=-a$ for $0\neq a\in A$ and the one arising from
the coaction, for which $\deg(a)=1$ if and only if $({\rm
id}\otimes\pi)\rho(a)=a\otimes g$ where $\pi$ is the projection onto
$k{\mathbb{Z}}_{2}$. If we view $A$ as a $D(H_{4})$-module, the grading
$|\cdot|$ is associated with the $1\bowtie g$-action whereas the grading
$\deg$ is associated with the $\phi(g)\bowtie 1$-action. Let us observe that
for the classes $C(a;t,s)$ the two natural gradings coincide, for every
$a,\,t,\,s\in k$.
###### Lemma 3.2
Let $[A]\in BQ(k,H_{4})$ and $[B]$ in $i_{0}(BW(k))$. As a $H_{4}$-module
algebra,
* (1)
$A\\#B\cong A\widehat{\otimes}B$, the ${\mathbb{Z}}_{2}$-graded tensor product
with respect to the $\deg$-grading on $A$ and the natural $|\cdot|$-grading on
$B$.
* (2)
$B\\#A\cong B\widehat{\otimes}A$, the ${\mathbb{Z}}_{2}$-graded tensor product
with respect to the $|\cdot|$-grading on $A$ and the natural $|\cdot|$-grading
on $B$.
Proof: The two gradings on $B$ coincide and we have, for homogeneous $b\in B$
and $c\in A$ (for the $\deg$-grading):
$(a\\#b)(c\\#d)=ac_{(0)}\\#(c_{(1)}\cdot b)d=ac\\#(g^{\deg(c)}\cdot
b)d=(-1)^{\deg(c)|b|}ac\\#bd.$
For homogeneous $b\in B$ and $c\in A$ (for the $|\cdot|$-grading):
$(d\\#c)(b\\#a)=db_{(0)}\\#(b_{(1)}\cdot c)a=db\\#(g^{|b|}\cdot
c)a=(-1)^{|c||b|}db\\#ca.$
$\Box$
It follows from Propositions 2.5, 2.6 and Lemma 3.2 that all elements in
$Im(i_{t})$ and $Im(\iota_{t})$ can be represented by algebras for which the
two ${\mathbb{Z}}_{2}$-gradings coincide, since this property is respected by
the $\\#$-product. Indeed, this kind of representatives give rise to a
subgroup that we will study in Section 5.
We will show now that groups of type $BC$ or $BM$ either intersect only in
$BW(k)$ or coincide and that the latter happens only in the situation of
Proposition 3.1.
###### Theorem 3.3
Consider the class of $C(a;t,s)$ in $BQ(k,H_{4})$. Then:
1. (1)
$[C(a;t,s)]\in Im(i_{l})$ if and only if $s=lt$;
2. (2)
$[C(a;t,s)]\in Im(\iota_{l})$ if and only if $sl=t$.
Proof: (1) We know from Lemma 2.1 that if the action (resp. coaction) of
$C(a;t,s)$ comes from the cotriangular (resp. triangular) structure, then the
indicated relations among the parameters hold. We only need to show that the
condition is still necessary if we change representative in the class.
Let us assume that $[C(a;t,s)]\in Im(i_{l})$ for some $l\in k$. Then
$[C(a;t,s)]=[C(b;1,l)][A]=[A][C(b;1,l)]$ for some $[A]\in i_{l}(BW(k))$ and
$b\in k$ with $2b\neq l$. Hence $[C(a;t,s)\\#C(l-b;1,l)]=[A]\in i_{l}(BW(k))$.
We may choose $A$ so that the $h$-action and the $\phi(h)$-action on $A$ are
trivial.
Since $[C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}]$ is trivial in $BQ(k,H_{4})$,
there is a $D(H_{4})$-module $P$ such that
$C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}\cong{\rm End(P)}$ as $D(H_{4})$-module
algebras. Then ${\rm End(P)}$ has a strongly inner $D(H_{4})$-action. In other
words, there is a convolution invertible algebra map $\nu\colon
D(H_{4})\to{\rm End}(P)$ such that
$(m\bowtie n)\cdot f=\nu(m_{(2)}\bowtie n_{(1)})f\nu^{-1}(m_{(1)}\bowtie
n_{(2)})$
for every $m\bowtie n\in D(H_{4}),f\in{\rm End(P)}$, where $\nu^{-1}$ denotes
the convolution inverse of $\nu$. In particular, for $u=\nu(\varepsilon\bowtie
g)$ and $w=\nu(\varepsilon\bowtie h)u$ we have
$\begin{array}[]{l}g\cdot f=ufu^{-1},\quad h\cdot f=w(g\cdot
f)-fw,\vspace{5pt}\\\ u^{2}=1,\quad w^{2}=0,\quad uw+wu=0.\end{array}$
We should be able to find $U,\,W\in C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ such
that
$\begin{array}[]{l}U^{2}=1,\quad g\cdot Z=UZU^{-1},\vspace{5pt}\\\ g\cdot
W=-W,\quad W^{2}=0,\quad h\cdot Z=W(g\cdot Z)-ZW\end{array}$
for all $Z$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}.$
Using the presentation of $C(a;t,s)\\#C(l-b;1,l)$ in Lemma 2.2 we may write
$W=\sum_{0\leq i,j\leq 1}X^{i}Y^{j}\\#\alpha_{ij}$ with
$\alpha_{ij}\in\overline{A}$ homogeneous of degree $i+j+1\;{\rm mod}\ 2$ with
respect to the $g$-grading. Since the action of $h$ on $1\\#\overline{A}$ is
trivial we have, for homogeneous $\gamma\in\overline{A}$:
$\begin{array}[]{ll}0&=h\cdot(1\\#\gamma)\vspace{2pt}\\\
&=W(g\cdot(1\\#\gamma))-(1\\#\gamma)W\vspace{2pt}\\\
&=(-1)^{|\gamma|}\sum_{0\leq i,j\leq
1}X^{i}Y^{j}\\#\alpha_{ij}\gamma-\sum_{0\leq i,j\leq
1}(X^{i}Y^{j})_{(0)}\\#((X^{i}Y^{j})_{(1)}\cdot\gamma)\alpha_{ij}\vspace{2pt}\\\
&=(-1)^{|\gamma|}[1\\#\alpha_{00}\gamma+Y\\#\alpha_{01}\gamma+X\\#\alpha_{10}\gamma+XY\\#\alpha_{11}\gamma]\vspace{2pt}\\\
&\phantom{=}-1\\#\gamma\alpha_{00}-Y\\#(-1)^{|\gamma|}\gamma\alpha_{01}-X\\#(-1)^{|\gamma|}\gamma\alpha_{10}-XY\\#\gamma\alpha_{11}.\end{array}$
From here we deduce that the odd elements $\alpha_{00},\alpha_{11}$ and the
even elements $\alpha_{10},\alpha_{01}$ belong to the
${\mathbb{Z}}_{2}$-center of $\overline{A}.$ Hence $\alpha_{00},\alpha_{11}$
are zero and $\alpha_{10},\alpha_{01}$ are scalars. So, we can write $W=\alpha
X\\#1+\beta Y\\#1$ for some $\alpha,\beta\in k$ and we will get:
$\begin{array}[]{rl}\alpha t+\beta&=h\cdot W=-2W^{2}=0,\\\
t&=h\cdot(X\\#1)=\alpha(-2a+ts),\\\ 1&=h\cdot(Y\\#1)=-\alpha
s-2\beta(l-b)=\alpha(-s+2t(l-b)).\end{array}$
Combining the second equation with the third one multiplied by $t$ and using
$\alpha\neq 0$ we obtain
$a=ts-t^{2}(l-b).$ (3.1)
The $|\cdot|$-grading and the $\deg$-grading on
$C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ coincide. Therefore:
$\nu(\phi(g)\bowtie 1)f\nu(\phi(g)\bowtie 1)^{-1}=\phi(g)\cdot f=g\cdot
f=ufu^{-1}\qquad\forall f\in{\rm End}(P).$
Since ${\rm End}(P)$ is central and $\nu$ is an algebra morphism,
$u^{\prime}:=\nu(\phi(g)\bowtie 1)=\lambda u$ with $\lambda=\pm 1$ (both
possibilities will be analyzed later). The element
$w^{\prime}:=\nu(\phi(h)\bowtie 1)$ satisfies
$\phi(h)\cdot f=w^{\prime}f-(\phi(g)\cdot f)w^{\prime}\qquad\forall f\in{\rm
End}(P).$
Thus, we can take $W^{\prime}$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ such
that
$W^{\prime}U+UW^{\prime}=0,\quad(W^{\prime})^{2}=0\quad\phi(h)\cdot
Z=W^{\prime}Z-(g\cdot Z)W^{\prime}$
for all $Z$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}.$ Arguing as for $W$
before, we see that $W^{\prime}=\gamma X\\#1+\delta Y\\#1$ for some
$\gamma,\delta\in k$. It follows from the last relation of $D(H_{4})$ in §1
that
$\nu(\varepsilon\bowtie hg)\nu(\phi(h)\bowtie 1)+\nu(\phi(h)\bowtie
1)\nu(\varepsilon\bowtie hg)=\nu(\phi(g)\bowtie 1)\nu(\varepsilon\bowtie
g)-\nu(\varepsilon\bowtie g)^{2}.$
This implies $WW^{\prime}+W^{\prime}W=\lambda-1.$ Besides,
$0=\phi(h)\cdot W^{\prime}=2(W^{\prime})^{2}=s\gamma+\delta l.$
Now, by direct computation:
$\begin{array}[]{rl}\lambda-1&=WW^{\prime}+W^{\prime}W\vspace{2pt}\\\
&=\alpha((X-tY)(\gamma X+\delta Y)+(\gamma X+\delta Y)(X-tY))\vspace{2pt}\\\
&=\alpha\gamma(2a-ts)+\alpha\delta(s-2t(l-b))\\\
&=-t\gamma-\delta.\end{array}$
Let us first assume $\lambda=1$. Then, $\gamma(s-tl)=0$. If $\gamma=0,$ then
$\delta=0$ and so $W^{\prime}=0$. This means that the $\phi(h)$-action is
identically zero, yielding $s=l=0$. Otherwise, $s=tl$ and we are done.
We finally show that the possibility $\lambda=-1$ can not occur. If
$\lambda=-1$, then $\delta=2-t\gamma$ and $s\gamma=-(2-t\gamma)l$. On the
other hand,
$l=\phi(h)\cdot(Y\\#1)=W^{\prime}(Y\\#1)+(Y\\#1)W^{\prime}=s\gamma+2(2-t\gamma)(l-b)$
(3.2)
Moreover,
$\begin{array}[]{rl}0&=(W^{\prime})^{2}\vspace{2pt}\\\
&=\gamma^{2}a+\delta^{2}(l-b)+\gamma\delta s\vspace{2pt}\\\
&\stackrel{{\scriptstyle(\ref{a=})}}{{=}}\gamma^{2}(ts-t^{2}(l-b))+(2-t\gamma)^{2}(l-b)+\gamma(2-t\gamma)s\vspace{2pt}\\\
&=2(l-b)(2-2t\gamma)+2\gamma s\end{array}$
From here, $s\gamma=(2t\gamma-2)(l-b).$ Substituting this in (3.2) we get
$l=2b$, contradicting the fact that $C(b;1,l)$ is $(H_{4},R_{l})$-Azumaya.
(2) If $l\not=0$, then $Im(\iota_{l})=Im(i_{l^{-1}})$ by Proposition 3.1 and
the statement follows from (1). It remains to show that $[C(a;t,s)]\in
Im(\iota_{0})$ implies $t=0$. If $[C(a;t,s)]\in Im(\iota_{0}),$ there exists
$b\in k^{\cdot}$ and an $H_{4}$-Azumaya algebra $A$ with trivial $h$-action
and trivial $\phi(h)$-action such that $[C(a;t,s)]=[A\\#C(b;0,1)].$ Then
$C(a;t,s)\\#C(-b;0,1)\\#\overline{A}\cong{\rm End}(P)$ for some
$D(H_{4})$-module $P$. Arguing as in (1) we see that there is $W=\alpha
X\\#1+\beta Y\\#1\in(C(a;t,s)\\#C(-b;0,1))\\#\overline{A}$ for some
$\alpha,\beta\in k$ such that
$\begin{array}[]{ll}&\hskip 11.0pth\cdot Z=W(g\cdot Z)-ZW,\\\ 0&=h\cdot
W=-2W^{2}=\alpha t+\beta,\\\ t&=h\cdot(X\\#1)=-2a\alpha,\\\
0&=h\cdot(Y\\#1)=2b\beta.\end{array}$
From here if follows that $t=0$. $\Box$
###### Corollary 3.4
Let $[C(a;t,s)]$, $[C(b;p,q)]$ be in $BQ(k,H_{4})$. Then
$[C(a;t,s)]=[C(b;p,q)]$ if and only if $C(a;t,s)\cong C(b;p,q)$.
Proof: We analyze the case $t\neq 0$, the other cases are treated similarly.
If $[C(a;t,s)]=[C(b;p,q)]$ and $p=0$ then $[C(a;t,s)]\in Im(\iota_{0}),$
contradicting Theorem 3.3. Then $tp\neq 0$ and we may reduce to the case
$[C(a;1,s)]=[C(b;1,q)]\in Im(i_{q})$. Applying again Theorem 3.3 we see that
$s=q$ and the equality of classes is an equality in $BM(k,H_{4},R_{q})$.
Applying $\Phi_{0}^{-1}\Psi_{q}^{-1}\Phi_{q}$ we obtain the equality
$[C(a-2^{-1}q;1,0)]=[C(b-2^{-1}q;1,0)]$ in $BM(k,H_{4},R_{0})$. From
Proposition 2.3, we obtain $(4a-2q)^{-1}=(4b-2q)^{-1}$ and we have the
statement. $\Box$
###### Theorem 3.5
Let $i_{t}:BM(k,H_{4},R_{t})\to BQ(k,H_{4})$ and
$\iota_{s}:BC(k,H_{4},r_{s})\to BQ(k,H_{4})$ be the natural embeddings in
$BQ(k,H_{4})$. Then:
1. (1)
$Im(i_{t})\cap Im(\iota_{s})\neq i_{0}(BW(k))$ if and only if $ts=1$. If this
is the case, then $Im(i_{t})=Im(\iota_{s})$;
2. (2)
$Im(i_{t})\cap Im(i_{s})\neq i_{0}(BW(k))$ if and only if $t=s$;
3. (3)
$Im(\iota_{t})\cap Im(\iota_{s})\neq i_{0}(BW(k))$ if and only if $t=s$.
Proof: This is a consequence of Propositions 2.3, 2.5, 2.6, 3.1 and Theorem
3.3. $\Box$
## 4 The action of $Aut(H_{4})$ on $Im(i_{t})$ and $Im(\iota_{s})$
For a Hopf algebra $H$, a group morphism from ${\rm Aut}_{\rm Hopf}(H)$ to
$BQ(k,H_{4})$ has been constructed in [8], where the case of $H_{4}$ was also
analized. The image of an automorphism $\alpha$ can be represented as follows.
Let us denote by $H_{\alpha}$ the right $H$-comodule $H$ with left $H$-action
$l\cdot m=\alpha(l_{(2)})mS^{-1}(l_{(1)})$. Then $A_{\alpha}={\rm
End}(H_{\alpha})$ can be endowed of the $H$-Azumaya algebra structure:
$\begin{array}[]{l}(l\cdot f)(m)=l_{(1)}\cdot f(S(l_{(2)})\cdot
m),\vspace{2pt}\\\ \rho(f)(m)=\sum f(m_{(0)})_{(0)}\otimes
S^{-1}(m_{(1)})f(m_{(0)})_{(1)}.\end{array}$
The assignment $\alpha\mapsto[A_{\alpha^{-1}}]$ defines a group morphism ${\rm
Aut}_{\rm Hopf}(H)\to BQ(k,H)$. The image of ${\rm Aut}_{\rm Hopf}(H)$ acts on
$BQ(k,H)$ by conjugation. An easy description of
$[B(\alpha)]:=[A_{\alpha}][B][A_{\alpha}]^{-1}$ for any representative $B$ has
been given in [8, Theorem 4.11]. As an algebra $B(\alpha)$ coincides with $B$,
while the $H$-action and $H$-coaction are:
$h\cdot_{\alpha}b=\alpha(h)\cdot
b,\quad\rho_{\alpha}(b)=b_{(0)}\otimes\alpha^{-1}(b_{(1)}).$ (4.1)
When $H=H_{4}$ the Hopf automorphism group is ${\rm Aut}_{\rm
Hopf}(H_{4})\cong k^{\cdot}$ and consists of the morphisms that are the
identity on $g$ and multiply $h$ by a nonzero scalar $\alpha$. The module
$H_{\alpha}$ has action
$\begin{array}[]{l}g\cdot g=g,\quad g\cdot h=-h,\vspace{2pt}\\\ h\cdot
g=\alpha hg+g^{2}S^{-1}(h)=-(1+\alpha)gh,\quad h\cdot h=0,\end{array}$
and the kernel of the group morphism consists of $\\{\pm 1\\}$. We may thus
embed $(k^{\cdot})^{2}\cong k^{\cdot}/\\{\pm 1\\}$ into $BQ(k,H_{4})$ (cf.
[19]). We shall denote by $K$ the image of this group morphism.
We analyze this action on the classes and subgroups described in the previous
sections.
###### Lemma 4.1
Let $\alpha\in k^{\cdot}$. Then:
1. (1)
$[A_{\alpha}][C(a;t,s)][A_{\alpha}]^{-1}=[C(a;\alpha t,s\alpha^{-1})]$.
2. (2)
$K$ acts trivially on $i_{0}(BW(k))$.
In particular, $BM(k,H_{4},R_{l\alpha^{2}})$ is conjugate to
$BM(k,H_{4},R_{l})$ in $BQ(k,H_{4})$ while $BM(k,H_{4},R_{0})$ and
$BC(k,H_{4},r_{0})$ are normalized by $K$.
Proof: (1) It follows from direct computation that
$h\cdot_{\alpha}x=\alpha t,\quad g\cdot_{\alpha}x=-x,\quad\rho(x)=x\otimes
g+s\alpha^{-1}\otimes h.$
(2) Since: the action of an automorphism of $H_{4}$ is trivial on $g$; the
action of $h$ is trivial on a representative of a class in $BW(k)$; and the
comodule map on a representative $A$ of a class in $BW(k)$ has image in
$A\otimes k{\mathbb{Z}}_{2}$, the formulas in (4.1) do not modify the action
and coaction on $A$ therefore $[A]=[A_{\alpha}][A][A_{\alpha}]^{-1}$ for every
$[A]\in i_{0}(BW(k))$.
Since $Im(i_{l})$ is generated by $i_{0}(BW(k))$ and the classes $[C(a;1,l)]$,
we see that $Im(i_{l})$ is conjugate to $Im(i_{\alpha^{2}l})$ in
$BQ(k,H_{4})$. If $l=0$ we get the statement concerning $Im(i_{0})$. The
statement concerning $BC(k,H_{4},r_{0})$ follows because this group is
generated by $i_{0}(BW(k))$ and the classes $[C(a;0,1)]$. $\Box$
###### Remark 4.2
The observation that $Im(i_{0})$ is normalized by $K$ has already been proved
in [21, §4]. Lemma 4.1 should be seen as a generalization of that result.
It is shown in [18] that $(H_{4},R_{t})$ is equivalent to $(H_{4},R_{s})$ if
and only if $t=\alpha^{2}s$ for some $\alpha\in k^{\cdot}$. The above lemma
shows that the Brauer groups of type $BM$ are conjugate in $BQ(k,H_{4})$ if
the corresponding triangular structures are equivalent. This is a general
fact:
###### Proposition 4.3
Let $R$ and $R^{\prime}$ be two equivalent quasitriangular structures on $H$
and let $\alpha\in{\rm Aut}_{\rm Hopf}(H)$ be such that
$(\alpha\otimes\alpha)(R^{\prime})=R$. Then the images of $BM(k,H,R)$ and
$BM(k,H,R^{\prime})$ are conjugate by the image of $\alpha$ in $BQ(k,H)$.
Proof: If $B$ represents an element in $BM(k,H,R)$ then there will be an
action $\cdot$ on $B$ such that the coaction $\rho$ is given by
$\rho(b)=(R^{(2)}\cdot b)\otimes R^{(1)}$ for all $b\in B$. The image of
$\alpha$ in $BQ(k,H)$ is represented by $A_{\alpha^{-1}}$. A representative of
$[A_{\alpha}]^{-1}[B][A_{\alpha}]$ is given by the algebra $B$ with action
$h\cdot_{\alpha^{-1}}b=\alpha^{-1}(h)\cdot b$. The coaction is given by
$\rho_{\alpha}(b)=(R^{(2)}\cdot
b)\otimes\alpha(R^{(1)})=(\alpha(R^{(2)})\cdot_{\alpha}b)\otimes\alpha(R^{(1)})=R^{\prime(2)}\cdot_{\alpha}b\otimes
R^{\prime(1)},$
so the coaction on $[A_{\alpha}]^{-1}[B][A_{\alpha}]$ is induced by
$R^{\prime}$ and $\cdot_{\alpha}$. $\Box$
For the dual statement, the proof is left to the reader.
###### Proposition 4.4
Let $r$ and $r^{\prime}$ be two equivalent coquasitriangular structures on $H$
and let $\alpha\in{\rm Aut}_{\rm Hopf}(H)$ be such that
$r^{\prime}(\alpha\otimes\alpha)=r$. Then the images of $BC(k,H,r)$ and
$BM(k,H,r^{\prime})$ are conjugate by the image of $\alpha$ in $BQ(k,H)$.
## 5 The subgroup $BQ_{grad}(k,H_{4})$
In this section we shall analyze the classes that can be represented by
$H_{4}$-Azumaya algebras for which the gradings coming from the $g$-action and
the comodule structure coincide. They form a subgroup that will be related to
the Brauer group $BM(k,E(2),R_{N})$ of Nichols $8$-dimensional Hopf algebra
$E(2)$ with respect to the quasitriangular structure $R_{N}$ attached to the
$2\times 2$-matrix $N$ with $1$ in the $(1,2)$-entry and zero elsewhere.
Let $BQ_{grad}(k,H_{4})$ be the set of classes that can be represented by a
$H_{4}$-Azumaya algebra $A$ for which the $|\cdot|$-grading and the
$\deg$-grading coincide. In other words, the classes in $BQ_{grad}(k,H_{4})$
can be represented by $D(H_{4})$-module algebras on which the actions of $g$
and $\phi(g)$ coincide. The last defining relation of $D(H_{4})$ in Section 1
implies that the action of $h$ and $\phi(h)$ on such representatives commute.
Clearly, $BQ_{grad}(k,H_{4})$ is a subgroup of $BQ(k,H_{4})$.
###### Proposition 5.1
$BQ_{grad}(k,H_{4})$ is normalized by $K$.
Proof: Let $[A]\in BQ_{grad}(k,H_{4})$ with $|a|=\deg(a)$ for every $a\in A$
and let $[A_{\alpha}]\in K$. Then $[A_{\alpha}\\#A\\#\overline{A_{\alpha}}]$
is represented by $A$ with action and coaction determined by (4.1). Since $g$
is fixed by all Hopf automorphisms of $H_{4}$ we have
$g\cdot_{\alpha}a=g\cdot a,\quad({\rm id}\otimes\pi)\rho_{\alpha}(a)=({\rm
id}\otimes\pi)\rho(a),$
so the two gradings are not modified by conjugation by $[A_{\alpha}]$. $\Box$
The subgroup $BQ_{grad}(k,H_{4})$ consists of those classes that can be
represented by module algebras for the quotient of $D(H_{4})$ by the Hopf
ideal $I$ generated by $\phi(g)\bowtie 1-\varepsilon\bowtie g$. Let us denote
by $\pi_{I}$ the canonical projection onto $D(H_{4})/I$.
Let $E(2)$ be the Hopf algebra with generators $c,\,x_{1},\,x_{2},$ with
relations
$c^{2}=1,\quad x_{i}^{2}=0,\quad cx_{i}+x_{i}c=0,\ i=1,2,\quad
x_{1}x_{2}+x_{2}x_{1}=0,$
coproduct
$\Delta(c)=c\otimes c,\quad\Delta(x_{i})=1\otimes x_{i}+x_{i}\otimes c,$
and antipode
$S(c)=c,\quad S(x_{i})=cx_{i}.$
The Hopf algebra morphism
$\begin{array}[]{rl}T\colon D(H_{4})&\longrightarrow E(2)\\\ \phi(g)\bowtie
1&\mapsto c\\\ \varepsilon\bowtie g&\mapsto c\\\ \varepsilon\bowtie h&\mapsto
x_{1}\\\ \phi(h)\bowtie 1&\mapsto cx_{2}\end{array}$
determines a Hopf algebra isomorphism $D(H_{4})/I\cong E(2)$. The canonical
quasitriangular structure ${\cal R}$ on $D(H_{4})$ is
$\begin{array}[]{rl}{\cal R}&=\frac{1}{2}[\varepsilon\bowtie(1\otimes
1^{*}+g\otimes g^{*}+h\otimes h^{*}+gh\otimes(gh)^{*})\bowtie
1]\vspace{3pt}\\\ &\hskip
3.0pt+\frac{1}{2}[\varepsilon\bowtie(1\otimes\varepsilon+g\otimes\varepsilon+1\otimes\phi(g)-g\otimes\phi(g)\vspace{3pt}\\\
&\hskip
20.0pt+h\otimes\phi(h)+h\otimes\phi(gh)+gh\otimes\phi(h)-gh\otimes\phi(gh))\bowtie
1]\end{array}$
so $(\pi_{I}\otimes\pi_{I})({\cal R})$ is a quasitriangular structure for
$D(H_{4})/I\cong E(2)$. Applying $T\otimes T$ to ${\cal R}$ we have:
$\begin{array}[]{rl}(T\otimes T)({\cal R})&=\frac{1}{2}(1\otimes 1+1\otimes
c+c\otimes 1-c\otimes c\\\ &\hskip 20.0pt+x_{1}\otimes cx_{2}+x_{1}\otimes
x_{2}+cx_{1}\otimes cx_{2}-cx_{1}\otimes x_{2})\end{array}$ (5.1)
The quasitriangular structures on $E(n)$ were computed in [17]. They are in
bijection with $n\times n$-matrices with entries in $k$. For a given matrix
$M$ the corresponding quasitriangular structure is denoted by $R_{M}$. The map
$T$ induces a quasitriangular morphism from $(D(H_{4}),{\cal R})$ onto
$(E(2),R_{N}),$ where $N$ is the $2\times 2$-matrix with $1$ in the
$(1,2)$-entry and zero elsewhere. If $A$ is a representative of a class in
$BQ_{grad}(k,H_{4})$ on which the ideal $I$ acts trivially, then $A$ is an
$E(2)$-module algebra and the maps $F$ and $G$ on $A\otimes A$ are the same as
those induced by $R_{N}$, so $A$ is $(E(2),R_{N})$-Azumaya.
###### Theorem 5.2
The group $BM(k,E(2),R_{N})$ fits into the following exact sequence
$\begin{array}[]{l}\begin{CD}1\longrightarrow{\mathbb{Z}}_{2}@>{}>{}>BM(k,E(2),R_{N})@>{T^{*}}>{}>BQ_{grad}(k,H_{4})\longrightarrow
1.\end{CD}\end{array}$
Proof: Restriction of scalars through $T$ provides a group morphism $T^{*}$
from $BM(k,E(2),R_{N})$ to $BQ(k,H)$ whose image is $BQ_{grad}(k,H_{4})$. The
kernel of $T^{*}$ consists of those classes $[A]$ such that $A\cong{\rm
End}(P)$ as $D(H_{4})$-module algebras, for some $D(H_{4})$-module $P$. The
class $[A]$ may be non-trivial only if $g$ and $\phi(g)$ act differently on
$P$ even though they act equally on ${\rm End}(P)$. The $\phi(g)$\- and
$g$-action on ${\rm End}(P)$ are strongly inner, hence there are elements $U$
and $u$ in ${\rm End}(P)$ such that $\phi(g)\cdot f=UfU^{-1}=ufu^{-1}=g\cdot
f$ for every $f\in{\rm End}(P).$ Since ${\rm End}(P)$ is a central algebra,
$U^{2}=u^{2}=1$, $uU=Uu$. From here, $U=\pm u,$ and if $[{\rm End}(P)]\neq 1$
in $BM(k,E(2),R_{N})$ we necessarily have $U=-u$. The actions of $g$ and
$\phi(g)$ on $P$ are given by the element $u$ and $U$ respectively, so for
every non-trivial $[A]$ in ${\rm Ker}(T^{*})$ we have $A\cong{\rm End}(P)$ for
some $D(H_{4})$-module $P$ for which $g$ acts as $-\phi(g)$. We claim that
there is at most one non-trivial element in ${\rm Ker}(T^{*})$.
Given any pair of such elements ${\rm End}(P)$ and ${\rm End}(Q)$ representing
classes in ${\rm Ker}(T^{*})$ we have ${\rm End}(P)\\#{\rm End}(Q)\cong{\rm
End}(P\otimes Q)$ as $D(H_{4})$-module algebras by [7, Proposition 4.3], where
$P\otimes Q$ is a $D(H_{4})$-module. Then, the actions of $g$ and $\phi(g)$ on
$P\otimes Q$ coincide, so $P\otimes Q$ is an $E(2)$-module. Thus, $[{\rm
End}(P)][{\rm End}(Q)]$ is trivial in $BM(k,E(2),R_{N})$ for every choice of
$P$ and $Q$. Therefore, ${\rm Ker}(T^{*})$ is either trivial or isomorphic to
${\mathbb{Z}}_{2}$. The proof is completed once we provide a non-trivial
element. Let us consider $P=k^{2}$ on which $g,\,h,\,\phi(g)$ and $\phi(h)$
act via the following matrices $u,\,w,\,U,\,W$, respectively:
$u=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\quad
w=\left(\begin{array}[]{cc}0&0\\\ -2&0\end{array}\right),\quad U=-u,\,\quad
W=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right).$
Then $P$ is a $D(H_{4})$-module but not an $E(2)$-module. On the other hand,
the $D(H_{4})$-module algebra structure on ${\rm End}(P)$ is in fact an
$E(2)$-module algebra structure:
$g\cdot f=ufu^{-1}=UfU^{-1}=\phi(g)\cdot f;$ (5.2) $h\cdot
f=wfu^{-1}+fuw,\quad\phi(h)\cdot f=Wf-UfU^{-1}W.$ (5.3)
Moreover, ${\rm End}(P)$ is $(E(2),R_{N})$-Azumaya because it is
$H_{4}$-Azumaya. We claim that the class of ${\rm End}(P)$ is not trivial in
$BM(k,E(2),R_{N})$. Indeed, if it were trivial, then the $E(2)$-action on
${\rm End}(P)$ given by $c.f=g.f$, $x_{1}.f=h.f$ and $(cx_{2}).f=\phi(h).f$
would be strongly inner. In other words, there would exist a convolution
invertible algebra morphism $p\colon E(2)\to{\rm End}(P)$ for which $l\cdot
f=\sum p(l_{(1)})fp^{-1}(l_{(2)})$ for every $l\in E(2)$. Putting
$u^{\prime}=p(c)$ we have $c.f=u^{\prime}f(u^{\prime})^{-1}=ufu^{-1}$. Since
${\rm End}(P)$ is a central simple algebra, we necessarily have
$u^{\prime}=\lambda u$ and since $(u^{\prime})^{2}=1$ we have $\lambda=\pm 1$.
Putting $w^{\prime}=p(x_{1})$ we have
$x_{1}.f=w^{\prime}fu^{\prime}-fw^{\prime}u^{\prime}$ and since
$u^{\prime}w^{\prime}=-w^{\prime}u^{\prime}$, we have $\lambda
w^{\prime}fu+\lambda fuw^{\prime}=x_{1}.f=h.f=wfu+fuw$ for every $f\in{\rm
End}(P)$. Using $uw=-wu$ we see that $(\lambda w^{\prime}-w)f=f(\lambda
w^{\prime}-w)$ so $w=\lambda w^{\prime}+\mu$ for some $\mu\in k$. Using once
more skew-commutativity of $u$ with $w$ and $w^{\prime}$ we see that $\mu=0$.
Putting $W^{\prime}=p(cx_{2})$ and using that
$u^{\prime}W^{\prime}=-W^{\prime}u^{\prime}$ we see that
$W^{\prime}f-ufuW^{\prime}=(cx_{2}).f=\phi(h).f=Wf-ufuW$ for every $f\in{\rm
End}(P)$. From here, we deduce that $u(W^{\prime}-W)=\nu\in k$. Using skew-
commutativity of $u$ with $W$ and $W^{\prime}$ we conclude that $\nu=0$ so
$W^{\prime}=W$. Then $W^{\prime}w^{\prime}-w^{\prime}W^{\prime}=\lambda(Ww-
wW)\neq 0$ so that relation $(cx_{2})x_{1}-x_{1}(cx_{2})=0$ in $E(2)$ cannot
be respected. Hence, $[{\rm End}(P)]\neq 1$ in $BM(k,E(2),R_{N})$ and ${\rm
Ker}(T^{*})\cong{\mathbb{Z}}_{2}$. $\Box$
The following proposition shows that the groups $BM(k,H_{4},R_{l})$ may be
viewed inside $BM(k,E(2),R_{N})$ and it also describes the image through
$T^{*}$ of them.
###### Proposition 5.3
For every $(\lambda,\mu)\in k\times k$ there is a group homomorphism
$\Theta_{\lambda,\mu}\colon BM(k,H_{4},R_{\lambda\mu})\to BM(k,E(2),R_{N})$
satisfying:
1. (1)
The image of $\Theta_{0,0}$ is the subgroup isomorphic to $BW(k)$ represented
by elements with trivial $x_{1}$\- and $x_{2}$-action and
$Ker(\Theta_{0,0})\cong(k,+)$.
2. (2)
$\Theta_{\lambda,\mu}$ is injective if and only if $(\lambda,\mu)\neq(0,0)$.
3. (3)
For $(\lambda,\mu)\neq(0,0),$ the image of $T^{*}\Theta_{\lambda,\mu}$ is
$Im(i_{\mu\lambda^{-1}})$ if $\lambda\neq 0$ and $Im(\iota_{\mu^{-1}\lambda})$
if $\mu\neq 0$.
Proof: For every $(\lambda,\mu)\in k\times k$ the map
$\theta_{\lambda,\mu}\colon E(2)\to H_{4}$ mapping $c\to g$, $x_{1}\to\lambda
h$ and $x_{2}\to\mu h$ is a Hopf algebra projection. A direct computation
shows that
$(\theta_{\lambda,\mu}\otimes\theta_{\lambda,\mu})(R_{N})=R_{\lambda\mu}$ so
the pull-back of $\theta_{\lambda,\mu}$ induces the desired homomorphism
$\Theta_{\lambda,\mu}$.
(1) Let $(\lambda,\mu)=(0,0)$. Then any element in $BM(k,H_{4},R_{0})$ can be
written as a pair of the form $([C(a;t,0)],[B])$ for $[B]\in BW(k)$. The image
through $\Theta_{0,0}$ of such an element is $[C(a)][B]\in BW(k)$ with trivial
$x_{i}$-action on $C(a)$. Clearly, $BW(k)=Im(\Theta_{0,0})$. That
$Ker(\Theta_{0,0})$ is isomorphic to $(k,+)$ follows from the isomorphism
$BM(k,H_{4},R_{0})\cong(k,+)\times BW(k)$ and the fact that $(k,+)$ is
realized as classes admitting a representative that is trivial when viewed as
a $k{\mathbb{Z}}_{2}$-module algebra.
(2) Let $(\lambda,\mu)\neq(0,0)$. If $\Theta_{\lambda,\mu}([A])=1$ then $A$ is
isomorphic to an endomorphism algebra with strongly inner $E(2)$-action. In
other words, $A\cong{\rm End}(P)$ and there is a convolution invertible
algebra map $p\colon E(2)\to A$ such that $l\cdot a=\sum
p(l_{(1)})ap^{-1}(l_{(2)})$ for every $l\in E(2),a\in A$. There are elements
$u,v,w\in A$ with $u$ invertible such that $c\cdot a=g\cdot a=uau^{-1}$,
$x_{1}\cdot a=(va-av)u=\lambda h\cdot a$ and $x_{2}\cdot a=(wa-aw)u=\mu h\cdot
a$. Then
$0=\mu x_{1}\cdot a-\lambda x_{2}\cdot a=((\mu v-\lambda w)a-a(\mu v-\lambda
w))u\quad\forall a\in A,$
and since $u$ is invertible and $A$ is central we have $\mu v-\lambda w=\eta$
for some $\eta\in k$. The relation between $v$ and $w$ gives $\eta=0$ and so
$\mu v=\lambda w.$ Thus, the same elements $u,v$ and $w$ ensure that the
$H_{4}$-action on $A$ is strongly inner. Therefore $[A]=1$ in
$BM(k,H_{4},R_{\lambda\mu})$. The converse follows from (1).
(3) Let us now assume that $(\lambda,\mu)\neq(0,0).$ It is immediate to see
that if $[A]\in BW(k)\subset BM(k,H_{4},R_{\lambda\mu})$ is represented by an
algebra with trivial $h$-action, then $\Theta_{\lambda,\mu}([A])$ is
represented by an algebra with trivial $x_{1}$\- and $x_{2}$-action. Hence
$T^{*}\Theta_{\lambda,\mu}(BM(k,H_{4},R_{\lambda\mu}))\subset i_{0}(BW(k))$
and the restriction of $T^{*}\Theta_{\lambda,\mu}$ to $BW(k)$ is an
isomorphism onto $i_{0}(BW(k))$. Let us now consider the class
$[C(a;1,\lambda\mu)]\in BM(k,H_{4},R_{\lambda\mu})$. Its image through
$\Theta_{\lambda,\mu}$ is the algebra generated by $x$ with $x^{2}=a$, with
$c\cdot x=-x$, $x_{1}\cdot x=\lambda$ and $x_{2}\cdot x=\mu$. A direct
verification shows that
$T^{*}\Theta_{\lambda,\mu}([C(a;1,\lambda\mu)])=[C(a;\lambda,\mu)]$. Then the
image of $T^{*}\Theta_{\lambda,\mu}$ is $Im(i_{\mu\lambda^{-1}})$ if
$\lambda\neq 0$ and $Im(\iota_{\mu^{-1}\lambda})$ if $\mu\neq 0$. $\Box$
Theorem 5.2 shows that one should understand $BM(k,E(2),R_{N})$ in order to
compute $BQ(k,H_{4})$. In view of Proposition 5.3, $BM(k,E(2),R_{N})$ seems to
be much more complex that the groups of type BM treated in [10, 11, 20].
## 6 Appendix
This last section is devoted to the analysis of some difficulties occurring in
the study of the structure of $(E(2),R_{N})$-Azumaya algebras. We show that
the set of classes represented by ${\mathbb{Z}}_{2}$-graded central simple
algebras (with respect to the grading induced by the $c$-action) is not a
subgroup of $BM(k,E(2),R_{N})$.
Let us consider the braiding $\psi_{VW}$ determined by $R_{N}$ between two
left $E(2)$-modules $V$ and $W$. Let $v\in V$ and $w\in W$ be homogeneous
elements with respect to the ${\mathbb{Z}}_{2}$-grading induced by the
$c$-action. By direct computation it is:
$\begin{array}[]{l}\psi_{VW}(v\otimes w)=\sum R_{N}^{(2)}\cdot w\otimes
R_{N}^{(1)}\cdot v\vspace{2pt}\\\ \hskip 48.36958pt=(-1)^{|v||w|}w\otimes
v+(-1)^{|w|+1}(-1)^{(|v|+1)(|w|+1)}(x_{2}\cdot w)\otimes(x_{1}\cdot
v).\end{array}$
If we denote by $\psi_{0}$ the braiding associated with the
${\mathbb{Z}}_{2}$-grading we have
$\psi_{VW}(v\otimes w)=\psi_{0}(v\otimes w)+(-1)^{|w|+1}\psi_{0}(x_{1}\cdot
v\otimes x_{2}\cdot w).$ (6.1)
Let $F$ and $G$ be the maps in (1.4) defining an $(E(2),R_{N})$-Azumaya
algebra $A$ and let $F_{0}$ and $G_{0}$ be the maps defining an
$(E(2),R_{0})$-Azumaya algebra, that is, the maps determining when an
$E(2)$-module algebra is ${\mathbb{Z}}_{2}$-graded central simple. It is not
hard to verify by direct computation that, for homogeneous $a,\,b,\,d\in A$
with respect to the $c$-action we have:
$F(a\\#b)(d)=F_{0}(a\\#b)(d)+(-1)^{|d|+1}F_{0}(a\\#x_{1}\cdot b)(x_{2}\cdot
d)$ (6.2) $G(a\\#b)(d)=G_{0}(a\\#b)(d)+(-1)^{|a|+1}F_{0}(x_{2}\cdot
a\\#b)(x_{1}\cdot d)$ (6.3)
Notice that if either $x_{1}$ or $x_{2}$ acts trivially, then $F=F_{0}$ and
$G=G_{0}$. So in this case, $A$ is $(E(2),R_{N})$-Azumaya if and only if it is
${\mathbb{Z}}_{2}$-graded central simple (i.e. $A$ is $(E(2),R_{0})$-Azumaya).
We will say that the $x_{i}$-action on an $E(2)$-module algebra $A$ is inner
if there exists an odd element $v\in A$ such that $x_{i}\cdot a=v(c\cdot
a)-av$ for every $a\in A$.
###### Theorem 6.1
Let $A$ be an $(E(2),R_{N})$-Azumaya algebra. The following assertions are
equivalent:
1. (1)
The $x_{1}$-action on $A$ is inner;
2. (2)
The $x_{2}$-action on $A$ is inner;
3. (3)
$A$ is a ${\mathbb{Z}}_{2}$-graded central simple algebra.
In addition, the $E(2)$-action on $A$ is inner if and only if $A$ is a central
simple algebra.
Proof: (1) $\Rightarrow$ (3) Let $v_{1}\in A$ be an odd element such that
$x_{1}\cdot a=v_{1}(c\cdot a)-av_{1}$ for all $a\in A$. Applying equality
(6.2) to any homogeneous $b$ and $d$ in $A$ gives:
$\begin{array}[]{ll}F(a\\#b)(d)&=F_{0}(a\\#b)(d)+F_{0}(a\\#b)((x_{2}\cdot
d)v_{1})\vspace{2pt}\\\ &\hskip 10.0pt+(-1)^{|d|}F_{0}(a\\#bv_{1})(x_{2}\cdot
d)\end{array}$ (6.4)
This equality extends to all elements $a$ and $b$ in $A$. If $A$ were not
${\mathbb{Z}}_{2}$-graded central simple, there would exist an element
$0\neq\sum_{i}a_{i}\\#b_{i}$ in $Ker(F_{0})$. Then
$(\sum_{i}a_{i}\\#b_{i})(1\\#v_{1})=\sum_{i}a_{i}\\#b_{i}v_{1}\in Ker(F_{0})$
and for every $f$ in $A$ we would have
$F_{0}(\sum_{i}a_{i}\\#b_{i})(f)=F_{0}(\sum_{i}a_{i}\\#b_{i}v_{1})(f)=0$. It
follows from (6.4) that $\sum_{i}a_{i}\\#b_{i}\in Ker(F),$ contradicting the
injectivity of $F$.
(2) $\Rightarrow$ (3) Similarly to (1) $\Rightarrow$ (3) replacing $F$ by $G$.
(3) $\Rightarrow$ (1), (2) Suppose that $A$ is a ${\mathbb{Z}}_{2}$-graded
central simple algebra. If $A$ is a central simple algebra then the
$E(2)$-action on $A$ is inner by the Skolem-Noether theorem. If $A$ is not
central simple then it is of odd type ([13, Theorem 3.4, Definition 3.5]) and
it is $(H_{4},R_{0})$-Azumaya for the subalgebra of $E(2)$, isomorphic to
$H_{4}$ generated by $c$ and $x_{i}$. By [1, Theorem 3.4] the $x_{i}$-action
is inner.
Let us finally assume that the $E(2)$-action on $A$ is inner. Then $A$ is a
${\mathbb{Z}}_{2}$-graded central simple algebra. Since $E(2)$ acts innerly on
$A$ then it acts trivially on its center $Z(A)$. Besides it is immediately
seen that $Z(A)$ is contained in the right and left $E(2)$-center, that are
trivial because $A$ is assumed to be $E(2)$-Azumaya. Hence $Z(A)$ must be
trivial and so $A$ is also a central algebra. By the structure theorems of
${\mathbb{Z}}_{2}$-graded central simple algebras ([13, Theorem IV.3.4]), $A$
is central simple. $\Box$
###### Proposition 6.2
Let $A$ and $B$ be two equivalent $(E(2),R_{N})$-Azumaya algebras. Then the
$x_{i}$-action on $A$ is inner if and only if it is so on $B$.
Proof: Let $P$ and $Q$ be finite dimensional $E(2)$-modules for which
$A\\#{\rm End}(P)\cong B\\#{\rm End}(Q)$. If the $x_{i}$-action on $A$ is
inner then it is so on $A\\#{\rm End}(P)$ by [11, Proposition 4.6], hence it
is so on $B\\#{\rm End}(Q)$, which is a ${\mathbb{Z}}_{2}$-graded central
simple algebra by Theorem 6.1. For $i=1,2$, let $W_{i},v_{i}$ be odd elements
in $B\\#{\rm End}(Q)$ and ${\rm End}(Q)$ respectively inducing the
$x_{i}$-action. We recall that $x_{j}\cdot v_{i}=0$ because the action on
${\rm End}(Q)$ is strongly inner, while $x_{j}\cdot W_{i}$ is a scalar for
every pair $i,j$ because $x_{j}\cdot W_{i}$ belongs to the graded center of
$B\\#{\rm End}(Q)$. The odd elements $T_{i}=W_{i}-1\\#v_{i}-(x_{2}\cdot
W_{i})(1\\#v_{1})\in B\\#{\rm End}(Q)$ for $i=1,2$ are such that $x_{j}\cdot
T_{i}=x_{j}\cdot W_{i}$ for every $i$ and $j$. Moreover, for every homogeneous
$f\in{\rm End}(Q)$ with respect to the $c$-action we have:
$\begin{array}[]{l}(-1)^{|f|}T_{i}(1\\#f)=W_{i}(c\cdot 1\\#c\cdot
f)-1\\#v_{i}(c\cdot f)-(x_{2}\cdot W_{i})(1\\#v_{1}(c\cdot f))\vspace{2pt}\\\
\hskip 36.98866pt=(1\\#f)W_{i}-(1\\#fv_{i})-(x_{2}\cdot
W_{i})(1\\#fv_{1})-(x_{2}\cdot W_{i})(x_{1}\cdot(1\\#f))\vspace{2pt}\\\ \hskip
36.98866pt=(1\\#f)[W_{i}-1\\#v_{i}-(x_{2}\cdot W_{i})(1\\#v_{1})]-(x_{2}\cdot
W_{i})(x_{1}\cdot(1\\#f))\vspace{2pt}\\\ \hskip
36.98866pt=(1\\#f)T_{i}-(x_{2}\cdot W_{i})(x_{1}\cdot(1\\#f)).\end{array}$
In other words,
$(1\\#f)T_{i}=(-1)^{|f||T_{i}|}T_{i}(1\\#f)+(x_{2}\cdot
T_{i})(x_{1}\cdot(1\\#f)),$
so by (6.1) the element $T_{i}\in C^{l}_{B\\#{\rm End}(Q)}({\rm End}(Q)),$ the
left centralizer of ${\rm End}(Q)$ in $B\\#{\rm End}(Q)$, that is, $T_{i}\in
B\\#1$ by the double centralizer theorem [1, Theorem 2.3]. Besides, for every
homogeneous $b\in B$ we have:
$\begin{array}[]{rl}T_{i}(c\cdot
b\\#1)-(b\\#1)T_{i}&=(-1)^{|b|}W_{i}(b\\#1)-(b\\#v_{i})-(x_{2}\cdot
W_{i})(b\\#v_{1})\vspace{2pt}\\\ &\hskip
10.0pt-(b\\#1)W_{i}+(b\\#v_{i})+(x_{2}\cdot W_{i})(b\\#v_{1})\vspace{2pt}\\\
&=x_{i}\cdot(b\\#1).\end{array}$
Hence the $x_{i}$-action on $B$ is inner. $\Box$
We conclude by showing that, contrarily to the cases treated in the literature
([10, 11, 20]), a Skolem-Noether-like approach is probably not appropriate for
the computation of $BM(k,E(2),R_{N})$ because the set of classes admitting a
representative with inner action is not a subgroup.
###### Theorem 6.3
The classes in $BM(k,E(2),R_{N})$ that are represented by
${\mathbb{Z}}_{2}$-graded central simple algebras do not form a subgroup.
Proof: Let $t\neq 0,1$ and $q\neq 2$ be in $k$. We consider the representative
$C(1;t,2)$ generated by $x$ with $x^{2}=1$, $c\cdot x=-x$, $x_{1}\cdot x=t$
and $x_{2}\cdot x=2$ and the representative $C(1;1,q)$ generated by $y$ with
$y^{2}=1$, $c\cdot y=-y$, $x_{1}\cdot y=1$ and $x_{2}\cdot y=q$. Both are
$(E(2),R_{N})$-Azumaya because $C(1;1,2t)$ is $(H_{4},R_{2t})$-Azumaya,
$C(1;1,q)$ is $(H_{4},R_{q})$-Azumaya and $C(1;t,2),C(1;1,q)$ are obtained
from these ones respectively by pulling back through $\theta_{\lambda,\mu}$.
They are also ${\mathbb{Z}}_{2}$-graded central simple algebras. Their product
$C(1;t,2)\\#C(1;1,q)$ is generated by the odd elements $X$ and $Y$ with
$X^{2}=1$, $Y^{2}=1$ and $XY+YX=2$. The element $X-Y$ is easily seen to lie in
the ${\mathbb{Z}}_{2}$-graded center, so $C(1;t,2)\\#C(1;1,q)$ is not a
${\mathbb{Z}}_{2}$-graded central simple algebra. If $B$ were another
representative of $[C(1;t,2)\\#C(1;1,q)]$ that is a ${\mathbb{Z}}_{2}$-graded
central simple algebra, then by Theorem 6.1, the $x_{1}$-action on it would be
inner. By Proposition 6.2, $x_{1}$ would act innerly on $C(1;t,2)\\#C(1;1,q)$.
Applying again Theorem 6.1, $C(1;t,2)\\#C(1;1,q)$ would be
${\mathbb{Z}}_{2}$-graded central simple. $\Box$
Acknowledgements
This research was partially supported by the Azioni Integrate Italia-España
AIIS05E34A Algebre, coalgebre, algebre di Hopf e loro rappresentazioni. The
second named author is also supported by projects MTM2008-03339 from MCI and
FEDER and P07-FQM-03128 from Junta de Andalucía.
## References
* [1] Armour A.; Chen H.-X.; Zhang Y. Structure theorems of $H_{4}$-Azumaya algebras. J. Algebra 305 (2006), 360-393.
* [2] Beattie, M.; Caenepeel, S. The Brauer-Long group of ${\mathbb{Z}}/p^{t}{\mathbb{Z}}$-dimodule algebras. J. Pure Appl. Algebra 60 (1989), 219-236.
* [3] Bichon, J.; Carnovale G. Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras. J. Pure Appl. Algebra 204 no. 3 (2006), 627-665.
* [4] Caenepeel, S. Computing the Brauer-Long group of a Hopf algebra I: the cohomological theory. Israel J. Math. 72 Nos. 1-2 (1990), 38-83.
* [5] Caenepeel, S. Brauer groups, Hopf algebras and Galois Theory. K-Monographs in Mathematics 4. Kluwer Academic Publishers, Dordrecht, 1998.
* [6] Caenepeel, S. The Brauer-Long group revisited: the multiplication rules. Algebra and Number Theory (Fez), 61-86. Lecture Notes in Pure Appl. Math 208. Marcel-Dekker, New York, 2000.
* [7] Caenepeel, S.; Van Oystaeyen, F.; Zhang, Y. Quantum Yang-Baxter Module Algebras. K-theory 8 no. 3 (1994), 231-255.
* [8] Caenepeel, S.; Van Oystaeyen, F.; Zhang, Y. The Brauer group of Yetter-Drinfeld module algebras. Trans. Amer. Math. Soc. 349 no. 9 (1997), 3737-3771.
* [9] Carnovale, G. Some isomorphisms for the Brauer groups of a Hopf algebra. Comm. Algebra 29 no. 11 (2001), 5291-5305.
* [10] Carnovale, G.; Cuadra, J. The Brauer group of some quasitriangular Hopf algebras. J. Algebra 259 no. 2 (2003), 512-532.
* [11] Carnovale, G.; Cuadra, J. Cocycle twisting of $E(n)$-module algebras and applications to the Brauer group. K-Theory 33 (2004), 251–276.
* [12] DeMeyer, F.; Ford, T. Computing the Brauer-Long group of ${\mathbb{Z}}_{2}$-dimodule algebras. J. Pure Appl. Algebra 54 (1988), 197-208.
* [13] Lam, T. Y. Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67. American Mathematical Society, Providence, RI, 2005.
* [14] Long, F.W. A generalization of the Brauer group of graded algebras. Proc. London Math. Soc. 29 no. 3 (1974), 237-256.
* [15] Majid, S. Doubles of quasitriangular Hopf algebras. Comm. Algebra 19 (1991), 3061-3073.
* [16] Montgomery, S.; Schneider, H.-J. Skew derivations of finite-dimensional algebras and actions of the Taft Hopf algebra. Tsukuba J. Math. 25 no. 2 (2001), 337-358.
* [17] Panaite, F; Van Oystaeyen, F. Quasitriangular structures for some pointed Hopf algebras of dimension $2^{n}$. Comm. Algebra 27 no. 10 (1999), 4929-4942.
* [18] Radford, D.E. Minimal quasitriangular Hopf algebras. J. Algebra 157 no. 2 (1993), 285-315.
* [19] Van Oystaeyen, F.; Zhang, Y. Embedding the Hopf automorphism group into the Brauer group. Can. Math. Bull. 41 (1998), 359-367.
* [20] Van Oystaeyen, F.; Zhang, Y. The Brauer group of Sweedler’s Hopf algebra $H_{4}$. Proc. Amer. Math. Soc. 129 no. 2 (2001), 371-380.
* [21] Van Oystaeyen, F.; Zhang, Y. Computing subgroups of the Brauer group of $H_{4}$. Comm. Algebra 30 no. 10 (2002), 4699-4709.
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# High order conservative LDG-IMEX methods for the degenerate nonlinear non-
equilibrium radiation diffusion problems***The first and the fourth authors
are supported by National Key R&D Program of China No. 2022YFA1004500, NSFC
No. 11971025 and No. 92270112, NSF of Fujian Province No. 2023J02003. The
second author is partially supported by NSFC No. 12031013, Shanghai pilot
innovation project No. 21JC1403500 and the Strategic Priority Research Program
of Chinese Academy of Sciences Grant No. XDA25010401. The third author is
supported by NSFC No. 12071214.
Shaoqin Zheng<EMAIL_ADDRESS>Min Tang<EMAIL_ADDRESS>Qiang
Zhang<EMAIL_ADDRESS>Tao Xiong<EMAIL_ADDRESS>
###### Abstract
In this paper, we develop a class of high-order conservative methods for
simulating non-equilibrium radiation diffusion problems. Numerically, this
system poses significant challenges due to strong nonlinearity within the
stiff source terms and the degeneracy of nonlinear diffusion terms. Explicit
methods require impractically small time steps, while implicit methods, which
offer stability, come with the challenge to guarantee the convergence of
nonlinear iterative solvers. To overcome these challenges, we propose a
predictor-corrector approach and design proper implicit-explicit time
discretizations. In the predictor step, the system is reformulated into a
nonconservative form and linear diffusion terms are introduced as a
penalization to mitigate strong nonlinearities. We then employ a Picard
iteration to secure convergence in handling the nonlinear aspects. The
corrector step guarantees the conservation of total energy, which is vital for
accurately simulating the speeds of propagating sharp fronts in this system.
For spatial approximations, we utilize local discontinuous Galerkin finite
element methods, coupled with positive-preserving and TVB limiters. We
validate the orders of accuracy, conservation properties, and suitability of
using large time steps for our proposed methods, through numerical experiments
conducted on one- and two-dimensional spatial problems. In both homogeneous
and heterogeneous non-equilibrium radiation diffusion problems, we attain a
time stability condition comparable to that of a fully implicit time
discretization. Such an approach is also applicable to many other reaction-
diffusion systems.
###### keywords:
non-equilibrium radiation diffusion , predictor-corrector procedure ,
conservative , high order , local discontinuous Galerkin method , IMEX
[1]organization=School of Mathematical Sciences, addressline=Xiamen
University, city=Xiamen, postcode=Fujian 361005, country=PR China
[2]organization=Institute of Natural Sciences, addressline=Shanghai Jiao Tong
University, city=Shanghai, postcode=200240, country=PR China
[3]organization=Department of Mathematics, addressline=Nanjing University,
city=Nanjing, postcode=210093, country=PR China [4]organization=School of
Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical
Modeling and High-Performance Scientific Computing, addressline=Xiamen
University, city=Xiamen, postcode=Fujian 361005, country=PR China
## 1 Introduction
In scenarios where radiation interacts dynamically with a material, but does
not fully reach thermodynamic equilibrium, a commonly employed description
involves a system of coupled time-dependent nonlinear diffusion equations.
This system is known for its strong nonlinearity and close coupling, and it
finds extensive applications across various fields, including inertial
confinement fusion [47], astrophysics [8], and Z-pinch experiments [38]. In
this work, we specifically explore a two-temperature (2T) model. This model
comprises a radiation diffusion equation within the framework of a gray
approximation and a material energy balance equation [8, 36, 50]. The 2T model
writes:
$\begin{cases}\dfrac{\partial E}{\partial t}-\nabla\cdot\left(D_{r}\nabla
E\right)=\sigma\left(T^{4}-E\right),\\\\[5.0pt] \dfrac{\partial T}{\partial
t}-\nabla\cdot\left(D_{t}\nabla
T\right)=\sigma\left(E-T^{4}\right),\end{cases}$ (1.1)
where $E({\bf x},t)$ is the radiation energy and $T({\bf x},t)$ is the
material temperature. The energy exchange between materials and photons is
controlled by the photon absorption cross-section $\sigma$, which is modeled
by
$\sigma(T)=\frac{z({\bf x})^{3}}{T^{3}},$ (1.2)
where $z({\bf x})$ is a spatial dependent material coefficient that represents
an atomic mass number. The following flux-limited energy radiation diffusion
coefficient $D_{r}$ is widely used [8, 23]:
$D_{r}=\frac{1}{3\sigma+\frac{|\nabla E|}{E}}.$ (1.3)
Moreover, the material conduction coefficient has the following form [43]
$D_{t}=\kappa T^{\frac{5}{2}},$ (1.4)
where $\kappa$ is a constant.
Adding the two equations of (1.1) together, one gets
$\dfrac{\partial(E+T)}{\partial t}=\nabla\cdot\left(D_{r}\nabla
E\right)+\nabla\cdot\left(D_{t}\nabla T\right),$ (1.5)
which indicates that the system (1.1) conserves the energy $E+T$ over the
space. Energy conservation is not only physically important, but also crucial
to capture a useful simplification of the 2T model. As $\sigma$ approaches
$\infty$, the radiation energy tends to approximate the thermal equilibrium,
namely $E\approx T^{4}$, (1.5) would lead to the gray radiative diffusion
equation [26]
$\dfrac{\partial(T^{4}+T)}{\partial t}=\nabla\cdot\left(D_{r}\nabla
T^{4}\right)+\nabla\cdot\left(D_{t}\nabla T\right).$ (1.6)
The gray radiation equation is an important approximation widely employed for
studying diverse radiative heat transfer phenomena, including radiative
transfer in stellar atmospheres.
Non-equilibrium radiation diffusion equations have attracted extensive
research efforts. Marshak, for instance, developed a time-dependent radiative
transfer model to investigate the impact of radiation on shock wave behavior
[30]. Analytical solutions have also been provided for specific Marshak wave
problems in previous studies [37, 6]. However, solving the system (1.1)
numerically presents substantial challenges, mainly due to the following
several reasons:
* 1.
The source terms might be very stiff. Specifically, the coefficient $\sigma$,
as defined in (1.2), tends to be very large for large values of $z$ or small
values of $T$. This leads to pronounced energy exchanges and a close coupling
between $E$ and $T^{4}$. Dealing with such stiff source terms when $\sigma$ is
large requires implicit time discretizations, resulting in a highly nonlinear
system with multiscale coefficients.
* 2.
The diffusion terms $\nabla\cdot\left(D_{r}\nabla E\right)$ and
$\nabla\cdot\left(D_{t}\nabla T\right)$ are nonlinear and degenerate. The
coefficients $D_{r}$ and $D_{t}$ defined in (1.3)-(1.4) nonlinearly depend on
$T$, and they may vary by several orders of magnitude when the temperature $T$
varies over a wide range. Moreover, when $T$ is small, both $D_{r}$ and
$D_{t}$ are close to $0$, which result in degenerate diffusion. Consequently,
solutions exhibit sharp fronts that propagate at finite speeds, akin to shock
wave solutions in hyperbolic systems [46].
* 3.
A challenging condition number for a nonlinear iteration. When the temperature
$T$ is low, energy $E$ and temperature $T$ operate at significantly different
scales, with $E$ roughly proportionate to $T^{4}$. Consequently, using a fully
implicit time discretization for both $E$ and $T$, combined with Newton
iteration, can lead to a coefficient matrix with a challenging condition
number. It is essential to devise suitable iterative solvers and efficient
preconditioners to tackle this challenge effectively.
When solving (1.1) with explicit schemes, it is relatively straightforward to
maintain the conservation of total energy $E+T$, however a significant
drawback is the requirement of very small time steps. A parabolic time step
$\Delta t=\mathcal{O}(h^{2})$ with $\Delta t$ being the time step and $h$
being the mesh size is needed due to the nonlinear diffusion terms, and the
stiff source terms lead to a severe time step constraint as well [55]. Fully
implicit schemes offer the advantage of using large time steps. However, the
presence of strong nonlinearity and multiscale coefficients, when employing a
large time step, the solutions from the previous time step may not serve as a
suitable initial guess. As a result, iterative solvers used in fully implicit
methods may struggle to converge effectively [1, 20]. In past years,
significant research efforts have been dedicated to developing efficient
iterative solvers with effective preconditioners. One notable approach is the
Jacobian-free Newton-Krylov method [23, 24], which combines Newtonian external
iterations and conjugate gradient-like (Krylov) internal iterations, resulting
in superlinear convergence without the need to form Jacobian matrices.
Additionally, a physical-based preconditioning Newton-Krylov method was
explored in [33], and an operator-split preconditioner was investigated in
[31]. Various other efficient time discretization methods have also been
examined in [21, 29, 9, 32, 35, 22] and references therein. Two semi-implicit
schemes allowing for large time steps for the gray radiation diffusion
equation (1.6) have been developed in [46]. Furthermore, radiation diffusion
models are frequently coupled with hydrodynamic equations, and large
deformations of complex fluid flows lead to mesh distortions. Consequently,
for spatial discretization, there have been some works built upon distorted
meshes [19, 40, 56, 62], as well as moving meshes [54, 55].
Schemes that efficiently handle non-equilibrium radiation diffusion equations,
while being robust and capable of accommodating large time steps for accurate
sharp front capturing, remain relatively rare in the existing literature. To
design such a scheme, several requirements have to be met:
* I
Preservation of the energy equation (1.5) at the discrete level. This
indicates that the source terms in the two equations in (1.1) are in balance
with each other. As far as (1.5) is preserved at the discrete level, when
$\sigma$ is large and the system reaches the thermodynamic equilibrium
$E=T^{4}$, the scheme’s accuracy can be controlled by the gray radiation
equation (1.6);
* II
Conservation of energy. Because of the presence of the nonlinear degenerate
diffusion terms, the solution exhibits hyperbolic properties at the front. As
is well-established in the study of hyperbolic simulations, preserving the
conservation properties of the scheme’s discretization is of paramount
importance to accurately capture finite-speed front propagations.
* III
Positivity Preserving. The system is only physically meaningful for positive
radiation energy $E$ and positive material temperature $T$. The radiation
temperature $T_{r}=E^{1/4}$ and all coefficients in (1.1) depend on $T$.
Hence, it would be crucial to maintain positivity of $E$ and $T$ in order to
ensure the robustness of the scheme and obtain physically meaningful
solutions.
* IV
Efficient iterative solvers which can guarantee convergence and the providing
of a good initial guess. Fully implicit time discretizations coupled with
proper spatial discretizaitons may possibly meet the above mentioned
requirements, but due to strong nonlinearities and multiscale variations of
coefficients, one has to provide a good initial guess and a suitable
preconditioner in order to make an iterative solver converging properly.
We note that radiation diffusion equations belongs to reaction-diffusion
equations which appear in many other fields, such as in material sciences
[34], chemical reactions [14], ecology and biology systems [18], and many
numerical approaches have been developed for reaction-diffusion equations,
e.g. [63, 61, 27] and reference therein. However, for the radiation diffusion
equations (1.1), the main challenges are the degeneracy in the nonlinear
diffusion operators and stiffness in the source terms. Both are caused by the
low material temperature $T$, making degeneracy, stiffness and nonlinearity
strongly coupled, while standard reaction diffusion equations usually involve
only linear diffusion but various nonlinear reactions. The design of an
efficient iterative solver for radiation diffusion equations is very
challenging. In this paper, to address such difficulties, we present a novel
and efficient method for (1.1) utilizing implicit-explicit (IMEX) time
discretizations, designed to satisfy all four of the aforementioned criteria.
Some techniques we develop can also be applied to other reaction diffusion
systems. We introduce a predictor-corrector procedure as our proposed
approach. In the predictor step, we multiply both sides of the second equation
in (1.1) with $4T^{3}$, introduce $B=T^{4}$, and obtain a simpler system for
$B$ and $E$ [16]. This new system does not meet requirements I, II, and III.
However, we can design an efficient iterative solver capable of ensuring
convergence even for large time steps. The predictor step can provide a good
initial guess. Then, in the corrector step we solve the energy conservative
equation (1.5), and substitute $E+T$ into the first equation of (1.1) to
further update $E$. Requirements I and II are satisfied in the corrector step
and the bad condition number for solving $E$ and $T$ together is avoided.
Finally, requirement III can be attained through the utilization of suitable
spatial discretizations.
For requirement IV, several strategies are employed to alleviate the
challenges associated with solving a highly nonlinear system. First of all,
implicit treatments of nonlinear diffusion terms are avoided by using the idea
in [51, 49] from adding the same linear diffusion terms on both sides of those
equations, which however are discretized explicitly and implicitly
respectively. For both the predictor and corrector steps, only a mildly
nonlinear system needs to be solved, and a simple Picard iteration is adopted.
Secondly, a local discontinuous Galerkin (LDG) finite element method is
employed for spatial discretiztions, due to a shock wave-like behavior in the
solutions of radiation [30]. The LDG method was introduced by Cockburn and Shu
in [12, 13] for convection-diffusion problems, which is well-suited for h-p
adaptivity and is also very good at shock or sharp gradient capturing. For a
first order in space LDG method, the iterative matrix for $E$ and $B$ can be
shown to be an M-matrix [16]. This can guanrantee the convergence of the
Picard iteration in the predictor step. Finally, for second and third orders
of discretizations, appropriate spatial limiters are applied to control
numerical oscillations for sharp gradient solutions and preserve the
positivity of radiation energy $E$ and material temperature $T$. After spatial
discretizations, the nonlinearity iteration can be written in the form of a
linear system, with nonlinearities mainly appear in the diagonal part of the
resulting system, a fast convergence can be obtained.
The rest of this paper is organized as follows. In section 2, an efficient
IMEX time discretization is introduced via a predictor-corrector procedure,
followed by an LDG space discretization in section 3. In section 4, one and
two spatial dimensional numerical examples are performed to verify the high
order accuracy, conservation and large time step conditions, and good
performances for capturing sharp fronts of the radiation energy in both
homogeneous and heterogeneous media. Finally, a conclusion is drawn in section
5.
## 2 Time discretization
In this section, we present an efficient IMEX time discretization for (1.1),
employing a predictor-corrector procedure. The predictor step, though non-
conservative, ensures fast and robust nonlinear convergence. In contrast, the
corrector step maintains conservation, which is crucial for accurately
propagating sharp fronts. This novel approach significantly mitigates
nonlinearity as compared to a fully implicit time discretization, while
maintaining a generous time step stability condition and ensuring fast and
robust convergence. We begin with a first-order IMEX scheme, which serves as a
general framework that can be readily extended to higher orders by integrating
a multistage Runge-Kutta (RK) or multistep IMEX time-marching approach. Here,
we adopt an IMEX RK scheme. The IMEX time discretization can be combined with
any front-capturing spatial discretization. In this section, we keep space
continuous, with a detailed description of the spatial discretization provided
in the next section.
### 2.1 First order IMEX scheme
Let the discrete time steps be $t_{n}\,(n=0,1,\ldots)$ and the time step size
be $\Delta t=t_{n+1}-t_{n}$. Instead of solving (1.1) directly, we propose a
predictor-corrector procedure:
* 1.
The predictor step:
To mitigate the stiffness of the source terms caused by $\sigma=z^{3}/T^{3}$,
as in [16], we multiply both sides of the second equation in (1.1) by
$4T^{3}$. (1.1) can be rewritten as:
$\begin{cases}\dfrac{\partial E}{\partial t}-\nabla\cdot\left(D_{r}\nabla
E\right)=\sigma\left(B-E\right),\\\\[10.0pt] \dfrac{\partial B}{\partial
t}-4\kappa\left[\nabla\cdot\left(T^{\frac{11}{2}}\nabla
T\right)-3T^{\frac{9}{2}}|\nabla
T|^{2}\right]=4z^{3}\left(E-B\right),\\\\[10.0pt] T=B^{1/4}.\end{cases}$ (2.1)
As discussed in the introduction, $T^{4}$ in the source term of (1.1) needs to
be treated implicitly to ensure thermal dynamic equilibrium ($E=T^{4}$) in the
stiff regime when $T$ is small. To achieve this without resorting to nonlinear
iterations for a given $\sigma$, we introduce a new variable, $B=T^{4}$, and
update a system for both $E$ and $B$ in this predictor step. Utilizing the
equation for $B$ not only avoids the stiffness associated with $\sigma$ in the
second equation but also maintains balance between $E$ and $B$, ensuring they
remain at the same scale when $T$ is small, thus preventing unfavorable
condition numbers. Furthermore, in this equation, we continue to use the
material temperature $T$ rather than $B$ for the diffusion terms, thereby
preventing negative powers of $B$ which can lead to stiffness when $B=T^{4}$
is small. Once we have determined $B$, we can calculate $T$ using the
relationship $B=T^{4}$. It is noteworthy that throughout this process and in
the following, we consistently use $E$ and $T$ as our input and output
variables.
To update (2.1) for $E$ and $B$, the diffusion terms remain nonlinear. An
implicit treatment would lead to a complex nonlinear system. Following the
approach presented in [51, 49], we introduce two linear diffusion terms
$\alpha_{0}\Delta E$ and $\beta_{0}\Delta B$ with constant coefficients
$\alpha_{0}$, $\beta_{0}$ on both sides of (2.1) for $E$ and $B$,
respectively. Starting from $E^{n}$ and $T^{n}$, a first-order IMEX scheme for
updating predicted solutions $\tilde{E}^{n+1}$ and $\tilde{T}^{n+1}$ is
defined as follows:
$\begin{cases}\frac{\tilde{E}^{n+1}-E^{n}}{\Delta
t}-\nabla\cdot\left(D^{n}_{r}\nabla E^{n}\right)+\alpha_{0}\,\Delta
E^{n}=\alpha_{0}\,\Delta\tilde{E}^{n+1}+\tilde{\sigma}^{n+1}\left(\tilde{B}^{n+1}-\tilde{E}^{n+1}\right),\\\\[10.0pt]
\frac{\tilde{B}^{n+1}-B^{n}}{\Delta t}-\mathcal{H}^{n}+\beta_{0}\,\Delta
B^{n}=\beta_{0}\,\Delta\tilde{B}^{n+1}+4z^{3}\left(\tilde{E}^{n+1}-\tilde{B}^{n+1}\right),\\\\[10.0pt]
\tilde{T}^{n+1}=(\tilde{B}^{n+1})^{1/4},\end{cases}$ (2.2)
where
$\mathcal{H}=4\kappa\left[\nabla\cdot\left(T^{\frac{11}{2}}\nabla
T\right)-3T^{\frac{9}{2}}|\nabla T|^{2}\right].$ (2.3)
Here, the superscript $n$ or $n+1$ denotes the corresponding values at time
step $t_{n}$ or $t_{n+1}$ respectively, similarly in the following. As we can
see, in each equation, two added identical terms are discretized differently,
one explicitly and the other implicitly. This approach allows us to achieve
time stability close to that of an implicit scheme while only needing to solve
linearly implicit diffusion operators [49, 51]. From the above, we observe
that by introducing $B$ and adding the linear diffusion terms, when we solve
(2.2) with an IMEX method, the only nonlinearity arises from the source term
$\sigma(E-B)$ due to the coefficient $\sigma$ appearing in the first equation.
If $\sigma$ is constant or a predefined function, (2.2) becomes a simple
linear system. Additionally, if $\kappa=0$ for $\mathcal{H}$ in (2.3), (2.2)
is in a conservative form, and such a 2T model can be efficiently solved.
* 2.
The corrector step:
However, if $\kappa$ is not zero, the term $\mathcal{H}$ in (2.3) within the
equation of $B$ is in a non-conservative form. This non-conservative form can
result in incorrect sharp front propagation, as will be demonstrated in our
numerical examples. To address this issue, a corrector step is required. To
ensure the conservation of total energy, as shown in (1.5), we begin by
replacing the second equation in the original equation (1.1) with (1.5),
resulting in the following:
$\begin{cases}\dfrac{\partial E}{\partial t}-\nabla\cdot\left(D_{r}\nabla
E\right)=\sigma\left(T^{4}-E\right),\\\\[10.0pt] \dfrac{\partial Q}{\partial
t}=\nabla\cdot\left(D_{r}\nabla E\right)+\nabla\cdot\left(D_{t}\nabla
T\right),\\\\[10.0pt] T=Q-E.\end{cases}$ (2.4)
The equation for $Q$ is in a conservative form, allowing for easy conservation
of total energy across space. Following (2.2), we also introduce two linear
diffusion terms, $\alpha_{0}\Delta E$ and $\gamma_{0}\Delta Q$, on both sides
of (2.4). Using a first-order IMEX time discretization, we obtain:
$\begin{cases}\dfrac{E^{n+1}-E^{n}}{\Delta t}-\nabla\cdot\left(D^{n}_{r}\nabla
E^{n}\right)+\alpha_{0}\,\Delta E^{n}=\alpha_{0}\,\Delta
E^{n+1}+\tilde{\sigma}^{n+1}\left((T^{n+1})^{4}-E^{n+1}\right),\\\\[10.0pt]
\dfrac{Q^{n+1}-Q^{n}}{\Delta t}-\nabla\cdot\left(D^{n}_{r}\nabla
E^{n}\right)-\nabla\cdot\left(D^{n}_{t}\nabla T^{n}\right)+\gamma_{0}\,\Delta
Q^{n}=\gamma_{0}\,\Delta Q^{n+1},\\\\[10.0pt]
T^{n+1}=Q^{n+1}-E^{n+1}.\end{cases}$ (2.5)
Solving for $E$ and $Q$ ensures good convergence and the conservation of $Q$.
As we can see, in (2.5), adding these linear diffusion terms is crucial to
simplify the nonlinear system. With this approach, we can first solve a linear
system for $Q$ from the second equation. Then, we can use $T=Q-E$ to replace
the $T^{4}$ term in the first equation, allowing us to solve this mildly
nonlinear equation to further update $E$. The nonlinearity is simply
$(Q-E)^{4}$ for $E$ and only appears in the diagonal part of the mass matrix
after spatial discretization.
###### Remark 1.
In the corrector step, we utilize the values obtained from the predictor step
to determine the coefficients $\sigma$ in the source term and to provide an
initial guess for the corrector step. One could contemplate simplifying the
scheme by eliminating the predictor step and directly employing a Picard
iteration in the corrector step to handle these coefficients. However, we
would mention that $\sigma=z^{3}/T^{3}$ and $T^{4}=(Q-E)^{4}$ appears as a
nonlinear term. In case of $T$ being small, with a stiff coefficient, the
nonlinear iteration for solving $E$ in (2.5) may not converge well, for
example, the stand Marshak wave problem in Example 4.4. Instead, in the
predictor step, except $\sigma$, others linearly depend on $E$ and $B$. Even
with a stiff $\sigma$, it appears in the diagonal part of the mass matrix, so
that a fast and robust convergence can be obtained. Starting from a good
initial guess provided by the predictor step, the corrector step can also
converge well.
### 2.2 High order IMEX scheme
The first-order IMEX scheme with a predictor-corrector procedure has offered a
highly efficient and versatile framework for solving the 2T model (1.1). To
attain high-order accuracy in time while preserving these desirable
properties, we employ a globally stiffly accurate IMEX RK time discretization
[7]. However, a multistep IMEX method can also be utilized [2].
We consider a system of additive ordinary differential equations:
$\frac{\mathrm{d}\bm{y}}{\mathrm{d}t}=L(t,\bm{y})+N(t,\bm{y}),\quad\bm{y}\left(t_{0}\right)=\bm{y}_{0},$
(2.6)
where $\bm{y}=(y_{1},y_{2},\ldots,y_{d})^{T}$. $L(t,\bm{y})$ and $N(t,\bm{y})$
are linear and nonlinear operators, respectively. $L(t,\bm{y})$ will be
discretized implicitly, while $N(t,\bm{y})$ will be discretized explicitly. An
$s$-stage IMEX RK time discretization can be represented by a double Butcher
tableau
$\begin{array}[]{c|c}\hat{c}&\hat{A}\\\ \hline\cr\vspace{-0.25cm}\hfil\\\
&\hat{b}^{T}\end{array},\ \ \ \qquad\begin{array}[]{c|c}{c}&{A}\\\
\hline\cr\vspace{-0.25cm}\hfil\\\ &{b^{T}}\end{array}.$
Here $A=\left(a_{ij}\right)$ and
$\hat{A}=\left(\hat{a}_{ij}\right)\in\mathbf{R}^{s\times s}$. $\hat{A}$ is a
strictly lower triangular matrix for explicit parts. For the implicit part,
$A$ can be taken as a lower triangular matrix with a nonzero diagonal to get
an efficient implementation, which is usually referred to as a diagonally
implicit RK (DIRK) scheme. The vectors are $b^{T}=(b_{1},b_{2},\ldots,b_{s})$,
$\hat{b}^{T}=(\hat{b}_{1},\hat{b}_{2},\ldots,\hat{b}_{s})$,
$c^{T}=(c_{1},c_{2},\ldots,c_{s})$, and
$\hat{c}^{T}=(\hat{c}_{1},\hat{c}_{2},\ldots,\hat{c}_{s})$, where
$c_{i}=\sum\limits_{j=1}^{i}a_{ij}$ and
$\hat{c}_{i}=\sum\limits_{j=1}^{i-1}\hat{a}_{ij}$. Denoting $t_{n}^{(j)}=$
$t_{n}+c_{j}\Delta t$, $\hat{t}_{n}^{(j)}=$ $t_{n}+\hat{c}_{j}\Delta t$, the
solution of (2.6) can be updated from time level $t^{n}$ to $t^{n+1}$ in the
following way:
$\begin{cases}\bm{Y}^{(1)}=\bm{y}_{n},\\\\[5.0pt]
\bm{Y}^{(i)}=\bm{y}_{n}+\Delta
t\sum\limits_{j=1}^{i-1}\hat{a}_{ij}N\left(\hat{t}_{n}^{(j)},\bm{Y}^{(j)}\right)+\Delta
t\sum\limits_{j=1}^{i}a_{ij}L\left(t_{n}^{(j)},\bm{Y}^{(j)}\right),\quad 2\leq
i\leq s,\\\\[5.0pt] \bm{y}_{n+1}=\bm{y}_{n}+\Delta
t\sum\limits_{i=1}^{s}\hat{b}_{i}N\left(\hat{t}_{n}^{(i)},\bm{Y}^{(i)}\right)+\Delta
t\sum\limits_{i=1}^{s}b_{i}L\left(t_{n}^{(i)},\bm{Y}^{(i)}\right).\end{cases}$
(2.7)
The IMEX RK scheme is called to be globally stiffly accurate, if the
coefficients satisfy [7] are required to satisfy:
$\hat{c}_{s}=c_{s}=1,\text{ and }a_{sj}=b_{j},\,\hat{a}_{sj}=\hat{b}_{j},\quad
j=1,2,\ldots,s.$
With such an IMEX scheme, the final updating of $\bm{y}_{n+1}$ in (2.7)
coincides with the last stage of updating $\bm{Y}^{(s)}$, so that we can take
$\bm{y}_{n+1}=\bm{Y}^{(s)}$ and avoid the last cumulative step. In A, Butcher
tableaux from first order to third order, which are adopted in this work, are
provided.
If we choose $L$ and $N$ in (2.6) based on the first order scheme (2.2) and
(2.5), with (2.7), the updating of the solutions at $t^{n+1}$ from $t^{n}$ can
be presented as follows:
$\displaystyle\frac{E^{n+1}-E^{n}}{\Delta t}$
$\displaystyle=\sum_{i=1}^{s-1}\hat{b}_{i}\left[\nabla\cdot(D^{(i)}_{r}\nabla
E^{(i)})-\alpha_{0}\Delta
E^{(i)}\right]+\sum_{i=1}^{s}b_{i}\left[\alpha_{0}\Delta
E^{(i)}+{\sigma}^{(i)}(B^{(i)}-E^{(i)})\right],$ (2.8a)
$\displaystyle\frac{Q^{n+1}-Q^{n}}{\Delta t}$
$\displaystyle=\sum_{i=1}^{s-1}\hat{b}_{i}\left[\nabla\cdot\left(D^{(i)}_{r}\nabla
E^{(i)}\right)+\nabla\cdot\left(D^{(i)}_{t}\nabla
T^{(i)}\right)-\gamma_{0}\Delta
Q^{(i)}\right]+\gamma_{0}\sum_{i=1}^{s}b_{i}\Delta Q^{(i)},$ (2.8b)
$\displaystyle T^{n+1}$ $\displaystyle=Q^{n+1}-E^{n+1}.$ (2.8c)
Here similarly the superscript $(i)$ denotes variables at the time stage
$t_{n}^{(i)}$ or $\hat{t}_{n}^{(i)}$, and the intermediate stage values for
$2\leq i\leq s$ are obtained from:
* 1.
the predictor step:
$\displaystyle\frac{\tilde{E}^{(i)}-E^{n}}{\Delta t}$
$\displaystyle=\text{RHS}_{E}^{(i)}+a_{ii}\left[\alpha_{0}\Delta\tilde{E}^{(i)}+\tilde{\sigma}^{(i)}(\tilde{B}^{(i)}-\tilde{E}^{(i)})\right],$
(2.9a) $\displaystyle\frac{\tilde{B}^{(i)}-B^{n}}{\Delta t}$
$\displaystyle=\text{RHS}_{B}^{(i)}+a_{ii}\left[\beta_{0}\Delta\tilde{B}^{(i)}+z^{3}(\tilde{E}^{(i)}-\tilde{B}^{(i)})\right],$
(2.9b) $\displaystyle\tilde{T}^{(i)}$ $\displaystyle=(\tilde{B}^{(i)})^{1/4},$
(2.9c)
* 2.
the corrector step:
$\displaystyle\frac{E^{(i)}-E^{n}}{\Delta t}$
$\displaystyle=\text{RHS}_{E}^{(i)}+a_{ii}\left[\alpha_{0}\Delta
E^{(i)}+\tilde{\sigma}^{(i)}(B^{(i)}-E^{(i)})\right],$ (2.10a)
$\displaystyle\frac{Q^{(i)}-Q^{n}}{\Delta t}$
$\displaystyle=\text{RHS}_{Q}^{(i)}+a_{ii}\gamma_{0}\Delta Q^{(i)},$ (2.10b)
$\displaystyle T^{(i)}$ $\displaystyle=Q^{(i)}-E^{(i)}.$ (2.10c)
The shorthand notations in (2.9)-(2.10) are defined as:
$\begin{cases}\text{RHS}_{E}^{(i)}=\sum\limits_{j=1}^{i-1}\left[\hat{a}_{ij}\left(\nabla\cdot(D^{(j)}_{r}\nabla
E^{(j)})-\alpha_{0}\Delta E^{(j)}\right)+a_{ij}\left(\alpha_{0}\Delta
E^{(j)}+\sigma^{(j)}(B^{(j)}-E^{(j)})\right)\right],\\\\[5.0pt]
\text{RHS}_{B}^{(i)}=\sum\limits_{j=1}^{i-1}\left[\hat{a}_{ij}\left(\mathcal{H}^{(j)}-\beta_{0}\Delta
B^{(j)}\right)+a_{ij}\left(\beta_{0}\Delta
B^{(j)}+z^{3}(E^{(j)}-B^{(j)})\right)\right],\\\\[5.0pt]
\text{RHS}_{Q}^{(i)}=\sum\limits_{j=1}^{i-1}\left[\hat{a}_{ij}\left(\nabla\cdot\left(D^{(j)}_{r}\nabla
E^{(j)}\right)+\nabla\cdot\left(D^{(j)}_{t}\nabla
T^{(j)}\right)-\gamma_{0}\Delta Q^{(j)}\right)+a_{ij}\gamma_{0}\Delta
Q^{(j)}\right].\end{cases}$
For the first stage, we take
$\tilde{E}^{(1)}=E^{n},\,\tilde{B}^{(1)}=(T^{n})^{4},\,E^{(1)}=E^{n},\,Q^{(1)}=E^{n}+T^{n}$.
### 2.3 Picard iteration
For the first-order IMEX scheme (2.2) and (2.5), or the high-order IMEX scheme
in the intermediate stages (2.9)-(2.10), each system is mildly nonlinear. Here
we will describe how to solve those mildly nonlinear systems with a simple
Picard iteration. The iteration does not rely on any specific spatial
discretization, so we keep space continuous first.
Taking the high-order IMEX scheme for the predictor step (2.9) as an example,
the updating $\tilde{E}$ and $\tilde{B}$ can be rewritten as:
$\displaystyle\left(\frac{1}{\Delta
t}-a_{ii}\alpha_{0}\Delta+a_{ii}\tilde{\sigma}^{(i)}\right)\tilde{E}^{(i)}-a_{ii}\tilde{\sigma}^{(i)}\tilde{B}^{(i)}=\text{RHS}_{E}^{(i)}+\frac{1}{\Delta
t}E^{n},$ (2.11a) $\displaystyle\left(\frac{1}{\Delta
t}-a_{ii}\beta_{0}\Delta+a_{ii}z^{3}\right)\tilde{B}^{(i)}-a_{ii}z^{3}\tilde{E}^{(i)}=\text{RHS}_{B}^{(i)}+\frac{1}{\Delta
t}B^{n}.$ (2.11b)
For a Picard iteration, starting from the iterative number $l=0$, we set
$\tilde{E}^{i,0}=E^{(i-1)}$ and $\tilde{B}^{i,0}=B^{(i-1)}$, and update
$\tilde{E}^{i,l+1}$ and $\tilde{B}^{i,l+1}$ from $\tilde{E}^{i,l}$ and
$\tilde{B}^{i,l}$ iteratively as:
$\displaystyle\left(\frac{1}{\Delta
t}-a_{ii}\alpha_{0}\Delta+a_{ii}\tilde{\sigma}^{i,l}\right)\tilde{E}^{i,l+1}-a_{ii}\tilde{\sigma}^{i,l}\tilde{B}^{i,l+1}=\text{RHS}_{E}^{(i)}+\frac{1}{\Delta
t}E^{n},$ (2.12a) $\displaystyle\left(\frac{1}{\Delta
t}-a_{ii}\beta_{0}\Delta+a_{ii}z^{3}\right)\tilde{B}^{i,l+1}-a_{ii}z^{3}\tilde{E}^{i,l+1}=\text{RHS}_{B}^{(i)}+\frac{1}{\Delta
t}B^{n}.$ (2.12b)
As observed, when $\tilde{\sigma}^{i,l}$ is set based on the previous
iterative step, (2.12) gives rise to a linear system for $\tilde{E}^{i,l+1}$
and $\tilde{B}^{i,l+1}$ that exhibits diagonal dominance. This property arises
from the positivity of $\sigma$ and $z({\bf x})$, resulting in rapid
convergence when employing iterative methods to solve it.
Similar to (2.11), the corrector step (2.10) for $E$ and $Q$ can be rewritten
as:
$\displaystyle\left(\frac{1}{\Delta
t}-a_{ii}\alpha_{0}\Delta+a_{ii}\tilde{\sigma}^{(i)}\right)E^{(i)}-a_{ii}\tilde{\sigma}^{(i)}B^{(i)}=\text{RHS}_{E}^{(i)}+\frac{1}{\Delta
t}E^{n},$ (2.13a) $\displaystyle\left(\frac{1}{\Delta
t}-a_{ii}\gamma_{0}\Delta\right)Q^{(i)}=\text{RHS}_{Q}^{(i)}+\frac{1}{\Delta
t}Q^{n}.$ (2.13b)
Here, $Q^{(i)}$ can be readily obtained by solving the linear system (2.13b),
which also possesses diagonal dominance. Subsequently, we solve (2.13a)
through a Picard iteration, with initial values
$B^{i,0}=\tilde{B}^{(i)}=(\tilde{T}^{(i)})^{4}$, as follows:
$\displaystyle\left(\frac{1}{\Delta
t}-a_{ii}\alpha_{0}\Delta+a_{ii}\tilde{\sigma}^{(i)}\right){E}^{i,l+1}=a_{ii}\tilde{\sigma}^{(i)}{B}^{i,l}+\text{RHS}_{E}^{(i)}+\frac{1}{\Delta
t}E^{n},$ (2.14a) $\displaystyle B^{i,l+1}=(Q^{(i)}-E^{i,l+1})^{4}.$ (2.14b)
We solve the linear system (2.14a) to get $E^{i,l+1}$, and then update
$B^{i,l+1}$ from (2.14b). Since the initial values are set as
$E^{i,0}=\tilde{E}^{(i)}$ and $B^{i,0}=(Q^{(i)}-E^{i,0})^{4}$ from the
predictor step, a fast convergence can be obtained for the corrector step.
The above procedures are similar for the first order IMEX scheme (2.2) and
(2.5), we omit them to save space.
## 3 LDG spatial discretization
For the first-order IMEX scheme (2.2)-(2.5), or the high-order IMEX scheme
(2.8)-(2.10), we can couple them with any front capturing spatial
discretizations [28, 5, 1, 49, 58]. In this work, we utilize an LDG finite
element method. The LDG method offers great flexibility for h-p adaptivity and
excels at capturing sharp gradient propagations.
### 3.1 Some notations
For a computational domain $\Omega\subseteq\mathbb{R}^{2}$, we consider a
partition $\mathcal{T}_{h}$ of $\Omega$ with a set of non-overlapping
rectangular elements $\\{I_{i,j}\\}$, which can cover the whole domain
$\Omega$. Here $I_{ij}=I_{i}\times I_{j}$,
$I_{i}=\left[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}\right]$ and
$I_{j}=\left[y_{j-\frac{1}{2}},y_{j+\frac{1}{2}}\right]$ for
$i=1,2,\cdots,N_{x},j=1,2,\cdots,N_{y}$. We denote the element length and
width as $h_{i}^{x}=x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}}$,
$h_{j}^{y}=y_{j+\frac{1}{2}}-y_{j-\frac{1}{2}}$, respectively.
$h=\max\limits_{i,j}\\{h_{i}^{x},h_{j}^{y}\\}$ is the maximum edge size of
these elements. The center of the element $I_{i,j}$ is $(x_{i},y_{j})$. We
also assume that $\mathcal{T}_{h}$ is quasi-uniform, namely,
$\max\limits_{i}\\{h/h_{i}^{x}\\}$ and $\max\limits_{j}\\{h/h_{j}^{y}\\}$ are
upper bounded by a given positive constant.
With the above partition, we follow [51] to give some notations which will be
used in the following. Given any non-negative integer vector
$\mathbf{k}=\left(k_{1},k_{2}\right)$, we define a finite-dimensional discrete
piecewise polynomial space as follows
$W_{h}^{\mathbf{k}}=\\{u\in
L^{2}(\Omega):u|_{K}\in\mathcal{Q}^{\mathbf{k}}(K),\forall
K\in\mathcal{T}_{h}\\},$
where $\mathcal{Q}^{\mathbf{k}}(K)$ consists of tensor product polynomials of
degree not exceeding $k_{\ell}$ along the $\ell$-th direction on each element
$K$, for $\ell=1,2$. Besides, we denote
${\mathbf{W}}_{h}^{\mathbf{k}}=W_{h}^{\mathbf{k}}\times W_{h}^{\mathbf{k}}$ as
a vector space, where each component belongs to $W_{h}^{\mathbf{k}}$. We
define a unit normal vector $\mathbf{n}^{e}$ on each edge $e$ of
$\mathcal{T}_{h}$ as follows: if $e\in\partial\Omega$, $\mathbf{n}^{e}$ is
defined as the unit normal vector pointing outside of $\Omega$; for an
interior edge $e=\partial K^{+}\cap\partial K^{-}$, the outward unit normal
vectors of $e$ taken from the elements $K^{+}$ and $K^{-}$ are denoted by
$\mathbf{n}^{+}$ and $\mathbf{n}^{-}$, respectively. Here we fix
$\mathbf{n}^{e}$ as one of $\mathbf{n}^{\pm}$. If we denote $u^{+}$ and
$u^{-}$ as the values of a function $u$ on $e$, taken from $K^{+}$ and $K^{-}$
respectively, then the jump $[[u]]$ over an edge $e$ for a scalar-valued
function $u$ is defined as
$[[u]]|_{e}=-\left(u^{+}\mathbf{n}^{+}+u^{-}\mathbf{n}^{-}\right)\cdot\mathbf{n}^{e}.$
For a vector-valued function $\mathbf{v}$, the jump
$[[\mathbf{v}\cdot\mathbf{n}]]$ is defined as
$[[\mathbf{v}\cdot\mathbf{n}]]|_{e}=-\left(\mathbf{v}^{+}\cdot\mathbf{n}^{+}+\mathbf{v}^{-}\cdot\mathbf{n}^{-}\right)\mathbf{n}^{e}\cdot\mathbf{n}^{+}.$
Accordingly, we express the averages of $u$ and $\mathbf{v}\cdot\mathbf{n}$ as
$\\{\\{u\\}\\}|_{e}=-\frac{1}{2}\left(u^{+}\mathbf{n}^{+}-u^{-}\mathbf{n}^{-}\right)\cdot\mathbf{n}^{e},\quad\\{\\{\mathbf{v}\cdot\mathbf{n}\\}\\}|_{e}=-\frac{1}{2}\left(\mathbf{v}^{+}\cdot\mathbf{n}^{+}-\mathbf{v}^{-}\cdot\mathbf{n}^{-}\right)\mathbf{n}^{e}\cdot\mathbf{n}^{+}.$
In this work, we take $\mathbf{n}^{e}=\mathbf{n}^{-}$, then
$[[u]]|_{e}=u^{+}-u^{-},\quad\\{\\{u\\}\\}|_{e}=\frac{1}{2}\left(u^{+}+u^{-}\right),$
and
$[[\mathbf{v}\cdot\mathbf{n}]]|_{e}=-(\mathbf{v}^{+}-\mathbf{v}^{-})\cdot\mathbf{n}^{-},\quad\\{\\{\mathbf{v}\cdot\mathbf{n}\\}\\}|_{e}=-\frac{1}{2}\left(\mathbf{v}^{+}+\mathbf{v}^{-}\right)\cdot\mathbf{n}^{-}.$
In our implementation, we use an orthogonal basis of $W_{h}^{\bm{k}}$ with a
uniform rectangular partition $h_{x}=h_{x}^{i}$ for $1\leq i\leq N_{x}$ and
$h_{y}=h_{y}^{j}$ for $1\leq j\leq N_{y}$. In this case, the numerical
solution can be expressed as
$u_{h}(x,y)=\sum\limits_{i=1}^{N_{x}}\sum\limits_{j=1}^{N_{y}}\sum\limits_{m=1}^{k_{1}}\sum\limits_{n=1}^{k_{2}}u_{i,j}^{m,n}H_{i}^{m}(x)H_{j}^{n}(y),\quad(x,y)\in\Omega,$
$u_{h}$ is $E_{h}$, $T_{h}$, or $B_{h}$ respectively. The local basis of
$\mathcal{Q}^{\mathbf{k}}(K)$ on each element $K$ is denoted as
$H_{i}^{m}(x)H_{j}^{n}(y)$ for $K=I_{i,j}\in\mathcal{T}_{h}$, and
$\mathbf{k}=(k_{1},k_{2})$. For example, up to third order, the local bases
are chosen as follows
$\displaystyle
H_{i}^{1}(x)=1,\,H_{i}^{2}(x)=\frac{x-x_{i}}{h_{x}},\,H_{i}^{3}(x)=\left(\frac{x-x_{i}}{h_{x}}\right)^{2}-\frac{1}{12},\quad
x\in I_{i},$ $\displaystyle
H_{j}^{1}(y)=1,\,H_{j}^{2}(y)=\frac{y-y_{j}}{h_{y}},\,H_{j}^{3}(y)=\left(\frac{y-y_{j}}{h_{y}}\right)^{2}-\frac{1}{12},\quad
y\in I_{j},$
with zero extension outside the cell $I_{i}$ or $I_{j}$ respectively.
### 3.2 First order IMEX-LDG scheme
With the above notations, a fully-discrete LDG scheme utilizing a first-order
IMEX scheme (2.2) and (2.5) is defined as follows. First, for the predictor
step (2.2), the scheme reads: given $E_{h}^{n},T_{h}^{n},B_{h}^{n}\in
W_{h}^{\mathbf{k}}$ and $\mathbf{p}^{n}_{h}$, $\mathbf{q}^{n}_{h}$,
$\mathbf{r}^{n}_{h}\in{\mathbf{W}}_{h}^{\mathbf{k}}$, we find
$\tilde{E}_{h}^{n+1},\tilde{B}_{h}^{n+1}\in W_{h}^{\mathbf{k}}$ and
$\tilde{\mathbf{p}}^{n+1}_{h}$,
$\tilde{\mathbf{q}}^{n+1}_{h}\in{\mathbf{W}}_{h}^{\mathbf{k}}$, such that for
any $\mu$, $\nu$, $\zeta\in W_{h}^{\mathbf{k}}$, and $\boldsymbol{\upsilon}$,
$\boldsymbol{\xi}\in{\mathbf{W}}_{h}^{\mathbf{k}}$, such that
$\displaystyle\frac{1}{\Delta
t}\left(\tilde{E}_{h}^{n+1}-E_{h}^{n},\mu\right)$
$\displaystyle=\mathcal{G}_{h}^{n}(\mu)-\alpha_{0}\mathcal{L}^{n}_{h,\mathbf{p}}(\mu)+\alpha_{0}\mathcal{L}^{n+1}_{h,\tilde{\mathbf{p}}}(\mu)+\left(\tilde{\sigma}_{h}^{n+1}(\tilde{B}^{n+1}_{h}-\tilde{E}^{n+1}_{h}),\mu\right),$
(3.1a) $\displaystyle\frac{1}{\Delta
t}\left(\tilde{B}_{h}^{n+1}-B_{h}^{n},\nu\right)$
$\displaystyle=\mathcal{H}_{h}^{n}(\nu)-\beta_{0}\mathcal{L}^{n}_{h,\mathbf{q}}(\nu)+\beta_{0}\mathcal{L}^{n+1}_{h,\tilde{\mathbf{q}}}(\nu)+\left(z^{3}(\tilde{B}^{n+1}_{h}-\tilde{E}^{n+1}_{h}),\nu\right),$
(3.1b) $\displaystyle(\tilde{T}_{h}^{n+1},\zeta)$
$\displaystyle=((\tilde{B}_{h}^{n+1})^{1/4},\zeta),$ (3.1c)
$\displaystyle(\tilde{\mathbf{p}}_{h}^{n+1},\boldsymbol{\upsilon})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(\tilde{E}_{h}^{n+1},\boldsymbol{\upsilon}),$
(3.1d) $\displaystyle(\tilde{\mathbf{q}}_{h}^{n+1},\boldsymbol{\xi})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(\tilde{B}_{h}^{n+1},\boldsymbol{\xi}),$
(3.1e)
with
$\mathcal{G}_{h}^{n}(\mu)=\boldsymbol{\mathcal{C}}_{h}((D_{r}\mathbf{p}_{h})^{n},\mu),\quad\mathcal{H}_{h}^{n}(\nu)=\boldsymbol{\mathcal{D}}_{h}(T_{h}^{n},\mathbf{r}_{h}^{n},\nu),\quad$
where
$\mathcal{L}^{n+1}_{h,\tilde{\mathbf{p}}}(\mu)=\boldsymbol{\mathcal{L}}_{h}(\tilde{\mathbf{p}}_{h}^{n+1},\mu),\quad\mathcal{L}^{n+1}_{h,\tilde{\mathbf{q}}}(\nu)=\boldsymbol{\mathcal{L}}_{h}(\tilde{\mathbf{q}}_{h}^{n+1},\nu),\quad\mathcal{L}^{n}_{h,\mathbf{p}}(\mu)=\boldsymbol{\mathcal{L}}_{h}(\mathbf{p}_{h}^{n},\mu),\quad\mathcal{L}^{n}_{h,\mathbf{q}}(\nu)=\boldsymbol{\mathcal{L}}_{h}(\mathbf{q}_{h}^{n},\nu).$
Correspondingly, for the corrector step (2.5), the scheme is defined as: we
look for $E_{h}^{n+1},Q_{h}^{n+1}\in W_{h}^{\mathbf{k}}$ and
$\mathbf{p}^{n+1}_{h}$,
$\mathbf{w}^{n+1}_{h}\in{\mathbf{W}}_{h}^{\mathbf{k}}$, for any $\mu$,
$\phi\in W_{h}^{\mathbf{k}},$ and $\boldsymbol{\upsilon}$,
$\boldsymbol{\eta}\in{\mathbf{W}}_{h}^{\mathbf{k}}$, such that
$\displaystyle\frac{1}{\Delta t}\left(E_{h}^{n+1}-E_{h}^{n},\mu\right)$
$\displaystyle=\mathcal{G}_{h}^{n}(\mu)-\alpha_{0}\mathcal{L}^{n}_{h,\mathbf{p}}(\mu)+\alpha_{0}\mathcal{L}^{n+1}_{h,\mathbf{p}}(\mu)+\left(\tilde{\sigma}_{h}^{n+1}(B^{n+1}_{h}-E^{n+1}_{h}),\mu\right),$
(3.2a) $\displaystyle\frac{1}{\Delta
t}\left(Q_{h}^{n+1}-Q_{h}^{n},\phi\right)$
$\displaystyle=\mathcal{E}_{h}^{n}(\phi)-\gamma_{0}\mathcal{L}^{n}_{h,\mathbf{w}}(\phi)+\gamma_{0}\mathcal{L}^{n+1}_{h,\mathbf{w}}(\phi),$
(3.2b) $\displaystyle T^{n+1}_{h}$ $\displaystyle=Q^{n+1}_{h}-E^{n+1}_{h},$
(3.2c) $\displaystyle(\mathbf{p}_{h}^{n+1},\boldsymbol{\upsilon})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(E_{h}^{n+1},\boldsymbol{\upsilon}),$
(3.2d) $\displaystyle(\mathbf{w}_{h}^{n+1},\boldsymbol{\eta})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(Q_{h}^{n+1},\boldsymbol{\eta}),$
(3.2e)
with
$\mathcal{E}_{h}^{n}(\phi)=\boldsymbol{\mathcal{F}}_{h}((D_{t}\mathbf{r})_{h}^{n},\phi)+\boldsymbol{\mathcal{C}}_{h}((D_{r}\mathbf{p}_{h})^{n},\phi),$
where
$\mathcal{L}^{n}_{h,\mathbf{p}}(\mu)=\boldsymbol{\mathcal{L}}_{h}(\mathbf{p}_{h}^{n},\mu),\quad\mathcal{L}^{n}_{h,\mathbf{w}}(\phi)=\boldsymbol{\mathcal{L}}_{h}(\mathbf{w}_{h}^{n},\phi).$
Those operators in (3.1)-(3.2) are defined as:
$\displaystyle\boldsymbol{\mathcal{L}}_{h}\left(\mathbf{p}_{h},\mu\right)=$
$\displaystyle-\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}\mathbf{p}_{h}\cdot\nabla\mu
d{\bf x}+\sum\limits_{K\in\mathcal{T}_{h}}\int_{\partial
K}\mathbf{n}^{e}\cdot\hat{\mathbf{p}}_{h}\mu ds,$
$\displaystyle\boldsymbol{\mathcal{K}}_{h}\left(E_{h},\boldsymbol{\upsilon}\right)=$
$\displaystyle-\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}E_{h}\nabla\cdot\boldsymbol{\upsilon}d{\bf
x}+\sum\limits_{K\in\mathcal{T}_{h}}\int_{\partial
K}\mathbf{n}^{e}\cdot\boldsymbol{\upsilon}\hat{E}_{h}ds,$
$\displaystyle\boldsymbol{\mathcal{C}}_{h}(D_{r}\mathbf{p}_{h},\mu)=$
$\displaystyle-\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}D_{r,h}\mathbf{p}_{h}\cdot\nabla\mu
d{\bf x}+\sum\limits_{K\in\mathcal{T}_{h}}\int_{\partial
K}\hat{D}_{r,h}\mathbf{n}^{e}\cdot\hat{\mathbf{p}}_{h}\mu ds,$
$\displaystyle\boldsymbol{\mathcal{F}}_{h}(D_{t}\mathbf{r}_{h},\phi)=$
$\displaystyle-\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}D_{t,h}\mathbf{r}_{h}\cdot\nabla\phi
d{\bf x}+\sum\limits_{K\in\mathcal{T}_{h}}\int_{\partial
K}\hat{D}_{t,h}\mathbf{n}^{e}\cdot\hat{\mathbf{r}}_{h}\phi ds.$
$\displaystyle\boldsymbol{\mathcal{D}}_{h}(T_{h},\mathbf{r}_{h},\nu)=$
$\displaystyle-4\kappa\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}\left(T^{\frac{11}{2}}_{h}\mathbf{r}_{h}\cdot\nabla\nu+3T^{\frac{9}{2}}_{h}|\mathbf{r}_{h}|^{2}\nu\right)d{\bf
x}$ $\displaystyle+4\kappa\sum\limits_{K\in\mathcal{T}_{h}}\int_{\partial
K}\hat{d}_{h}(T_{h};\mathbf{n}^{e})\mathbf{n}^{e}\cdot\hat{\mathbf{r}}_{h}\nu
ds.$
In the above and below, $(\cdot,\cdot)$ denotes the usual $L^{2}$ inner
product on $\Omega$. We can find that $\mathbf{p}_{h}$, $\mathbf{q}_{h}$,
$\mathbf{r}_{h}$, and $\mathbf{w}_{h}$ approximate $\nabla E$, $\nabla B$,
$\nabla T$, and $\nabla Q$ respectively.
The variables with $\hat{\cdot}$ are numerical fluxes. For such a diffusion
system, we adopt an alternating left-right flux, namely
$\displaystyle\hat{u}_{h}=u_{h}^{-},\quad\hskip 39.83385pt\text{for a scalar-
valued function }\,u=E,B,T,Q,$
$\displaystyle\mathbf{n}^{e}\cdot\hat{\mathbf{v}}_{h}=\mathbf{n}^{-}\cdot\mathbf{v}_{h}^{+},\quad\text{for
a vector-valued function
}\,\mathbf{v}=\mathbf{p},\mathbf{q},\mathbf{r},\mathbf{w}.$
The alternating right-left flux or central fluxes as in [10, 17, 48, 51] can
also be used. As used in [12, 52, 53],
$\hat{D}_{r,h}=\\{\\{D_{r,h}\\}\\}|_{e}$ is defined as a central numerical
flux in approximating of $D_{r,h}$, and $\hat{d}_{h}(T_{h};\mathbf{n}^{e})$ in
$\boldsymbol{\mathcal{D}}_{h}(T_{h},\mathbf{r}_{h},\nu)$ is an approximation
to $T^{\frac{11}{2}}$,
$\hat{d}_{h}(T_{h};\mathbf{n}^{e})=\left\\{\begin{array}[]{ll}[[\frac{2}{13}T_{h}^{\frac{13}{2}}]]|_{e}/[[T_{h}]]|_{e},&\text{
if }[[T_{h}]]|_{e}\neq 0,\\\\[10.0pt] (T_{h}^{\frac{11}{2}})^{-},&\text{
otherwise. }\end{array}\right.$
$\hat{D}_{t,h}$ is chosen to approximate $D_{t}$ in the same manner as
$\hat{d}_{h}(T_{h};\mathbf{n}^{e})$.
### 3.3 High order IMEX-LDG scheme
By applying a high-order IMEX RK scheme (2.7) to (3.1)-(3.2), similar to the
first-order scheme, the updating of the solutions at $t^{n+1}$ from $t^{n}$ is
given as follows:
$\displaystyle\frac{1}{\Delta t}\left(E_{h}^{n+1}-E_{h}^{n},\mu\right)$
$\displaystyle=\sum\limits_{i=1}^{s}\left[\hat{b}_{i}\left(\mathcal{G}_{h}^{(i)}(\mu)-\alpha_{0}\mathcal{L}^{(i)}_{h,\mathbf{p}}(\mu)\right)+b_{i}\left(\alpha_{0}\mathcal{L}^{(i)}_{h,\mathbf{p}}(\mu)+\left({\sigma}_{h}^{(i)}(B^{(i)}_{h}-E^{(i)}_{h}),\mu\right)\right)\right],$
(3.3a) $\displaystyle\frac{1}{\Delta
t}\left(Q_{h}^{n+1}-Q_{h}^{n},\phi\right)$
$\displaystyle=\sum\limits_{i=1}^{s}\left[\hat{b}_{i}\left(\mathcal{E}_{h}^{(i)}(\phi)-\gamma_{0}\mathcal{L}^{(i)}_{h,\mathbf{w}}(\phi)\right)+b_{i}\gamma_{0}\mathcal{L}^{(i)}_{h,\mathbf{w}}(\phi)\right],$
(3.3b) $\displaystyle T^{n+1}_{h}$ $\displaystyle=Q^{n+1}_{h}-E^{n+1}_{h},$
(3.3c) $\displaystyle(\mathbf{p}_{h}^{n+1},\boldsymbol{\upsilon})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(E_{h}^{n+1},\boldsymbol{\upsilon}),$
(3.3d) $\displaystyle(\mathbf{w}_{h}^{n+1},\boldsymbol{\eta})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(Q_{h}^{n+1},\boldsymbol{\eta}).$
(3.3e)
The intermediate stage values for $2\leq i\leq s$ are obtained from:
* 1.
the predictor step:
$\displaystyle\frac{1}{\Delta
t}\left(\tilde{E}_{h}^{(i)}-E_{h}^{n},\mu\right)$
$\displaystyle=\text{RHS}_{E}^{(i)}(\mu)+a_{ii}\left[\alpha_{0}\mathcal{L}^{(i)}_{h,\tilde{\mathbf{p}}}(\mu)+\left(\tilde{\sigma}_{h}^{(i)}(\tilde{B}^{(i)}_{h}-\tilde{E}^{(i)}_{h}),\mu\right)\right],$
(3.4a) $\displaystyle\frac{1}{\Delta
t}\left(\tilde{B}_{h}^{(i)}-B_{h}^{n},\nu\right)$
$\displaystyle=\text{RHS}_{B}^{(i)}(\nu)+a_{ii}\left[\beta_{0}\mathcal{L}^{(i)}_{h,\tilde{\mathbf{q}}}(\nu)+\left(z^{3}(\tilde{B}^{(i)}_{h}-\tilde{E}^{(i)}_{h}),\nu\right)\right],$
(3.4b) $\displaystyle(\tilde{T}_{h}^{(i)},\zeta)$
$\displaystyle=((\tilde{B}_{h}^{(i)})^{1/4},\zeta),$ (3.4c)
$\displaystyle(\tilde{\mathbf{p}}_{h}^{(i)},\boldsymbol{\upsilon})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(\tilde{E}_{h}^{(i)},\boldsymbol{\upsilon}),$
(3.4d) $\displaystyle(\tilde{\mathbf{q}}_{h}^{(i)},\boldsymbol{\xi})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(\tilde{B}_{h}^{(i)},\boldsymbol{\xi});$
(3.4e)
* 2.
the corrector step:
$\displaystyle\frac{1}{\Delta t}\left(E_{h}^{(i)}-E_{h}^{n},\mu\right)$
$\displaystyle=\text{RHS}_{E}^{(i)}(\mu)+a_{ii}\left[\alpha_{0}\mathcal{L}^{(i)}_{h,\tilde{\mathbf{p}}}(\mu)+\left(\tilde{\sigma}_{h}^{(i)}(B^{(i)}_{h}-E^{(i)}_{h}),\mu\right)\right],$
(3.5a) $\displaystyle\frac{1}{\Delta
t}\left(Q_{h}^{(i)}-Q_{h}^{n},\phi\right)$
$\displaystyle=\text{RHS}_{Q}^{(i)}(\phi)+a_{ii}\gamma_{0}\mathcal{L}^{(i)}_{h,\mathbf{w}}(\phi),$
(3.5b) $\displaystyle T^{(i)}_{h}$ $\displaystyle=Q^{(i)}_{h}-E^{(i)}_{h},$
(3.5c) $\displaystyle(\mathbf{p}_{h}^{(i)},\boldsymbol{\upsilon})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(E_{h}^{(i)},\boldsymbol{\upsilon}),$
(3.5d) $\displaystyle(\mathbf{w}_{h}^{(i)},\boldsymbol{\eta})$
$\displaystyle=\boldsymbol{\mathcal{K}}_{h}(Q_{h}^{(i)},\boldsymbol{\eta}).$
(3.5e)
The shorthand notations in (3.4)-(3.5) are defined as:
$\begin{cases}\text{RHS}_{E}^{(i)}(\mu)=\sum\limits_{j=1}^{i-1}\left[\hat{a}_{ij}\left(\mathcal{G}_{h}^{(j)}(\mu)-\alpha_{0}\mathcal{L}^{(j)}_{h,\mathbf{p}}(\mu)\right)+a_{ij}\left(\alpha_{0}\mathcal{L}^{(j)}_{h,\mathbf{p}}(\mu)+\left(\tilde{\sigma}_{h}^{(j)}(B^{(j)}_{h}-E^{(j)}_{h}),\mu\right)\right)\right],\\\\[5.0pt]
\text{RHS}_{B}^{(i)}(\nu)=\sum\limits_{j=1}^{i-1}\left[\hat{a}_{ij}\left(\mathcal{H}_{h}^{(j)}(\nu)-\beta_{0}\mathcal{L}^{(j)}_{h,\mathbf{q}}(\nu)\right)+a_{ij}\left(\beta_{0}\mathcal{L}^{(j)}_{h,\mathbf{q}}(\nu)+\left(z^{3}(\tilde{B}^{(j)}_{h}-\tilde{E}^{(j)}_{h}),\nu\right)\right)\right],\\\\[5.0pt]
\text{RHS}_{Q}^{(i)}(\phi)=\sum\limits_{j=1}^{i-1}\left[\hat{a}_{ij}\left(\mathcal{E}_{h}^{(j)}(\phi)-\gamma_{0}\mathcal{L}^{(j)}_{h,\mathbf{w}}(\phi)\right)+a_{ij}\gamma_{0}\mathcal{L}^{(j)}_{h,\mathbf{w}}(\phi)\right].\end{cases}$
As we can observe, each of the intermediate steps (3.4)-(3.5) in a high-order
scheme shares the same structure as the first-order scheme (3.1)-(3.2). Thus,
the first-order scheme provides a general framework for coupling with a multi-
stage IMEX RK method, making it convenient to extend to higher orders.
Furthermore, the choice of operators and numerical fluxes remains consistent
with the first-order scheme.
### 3.4 Picard iteration for a full scheme
To represent our scheme as a mildly nonlinear system more effectively, we
introduce notations for matrices and vectors. We define a long vector
$\bm{u_{c}}$ to represent the coefficients of a two-dimensional numerical
solution $u_{h}$, as follows:
$\bm{u_{c}}=\left(u_{1,1}^{1,1},\cdots,u_{1,1}^{k_{1},k_{2}},u_{2,1}^{1,1},\cdots,u_{2,1}^{k_{1},k_{2}},\cdots,u_{N_{x},1}^{1,1},\cdots,u_{N_{x},1}^{k_{1},k_{2}},u_{1,2}^{1,1},\cdots,u_{N_{x},N_{y}}^{k_{1},k_{2}}\right)^{T}.$
(3.6)
Such a vector $\bm{u_{c}}$ is the value to be updated by a numerical method in
the LDG framework. For instance, when taking the nonlinear integrals
$\left(\sigma_{h}E_{h},\mu\right)$ and $\left(\sigma_{h}B_{h},\mu\right)$ in
(3.1) and (3.2), practical approximations are employed. These approximations
involve Gaussian quadrature integration along each direction, facilitated by
the use of an interpolation operator $\mathcal{I}_{h}$ within each cell
$K=I_{i}\times I_{j}$, i.e.
$\left(\sigma_{h}E_{h},\mu\right)_{K}\approx\int_{K}\mathcal{I}_{h}\left(\sigma_{h}E_{h}\mu\right)d{\bf
x}=\dfrac{h_{x}h_{y}}{4}\sum\limits_{G=1}^{k_{1}+1}\sum\limits_{G^{\prime}=1}^{k_{2}+1}\hat{\omega}_{G}\hat{\omega}_{G^{\prime}}(\sigma_{h}E_{h}\mu)\left(\frac{h_{x}}{2}\eta_{G}+x_{i},\frac{h_{y}}{2}\eta_{G^{\prime}}+y_{j}\right),$
where $\eta_{G}$ are Gaussian quadrature points on the interval $[-1,1]$ with
corresponding weights $\hat{\omega}_{G}$. We define
$k_{1,2}=(k_{1}+1)(k_{2}+1)$, therefore, by taking $\mu$ over the local bases
$H^{m}_{i}(x)H^{n}_{j}(y)$ in each cell, $(\sigma_{h}E_{h},\mu)$ can be
formatted as a matrix $\Lambda_{h,T}$ multiplied by a long vector
$\bm{E_{c}}$, that is $\Lambda_{h,T}\bm{E_{c}}$, where $\Lambda_{h,T}$ is a
$k_{1,2}$-block at most tri-diagonal matrix in the size of $N_{x}N_{y}$. Here
$\sigma$ is $T$-dependent, so that $\Lambda_{h,T}$ depends on $T$, and
$\Lambda_{h,T}\bm{E_{c}}$ is nonlinear.
With similar matrix-by-vector notations, the high order fully-discrete IMEX-
LDG scheme (3.4), corresponding to (2.11), can be written in the following
form
$\displaystyle\left(\frac{1}{\Delta
t}M_{h}-a_{ii}\alpha_{0}S_{h}+a_{ii}\Lambda_{h,\tilde{T}}^{(i)}\right)\tilde{\bm{E}}_{\bm{c}}^{(i)}-a_{ii}\Lambda_{h,\tilde{T}}^{(i)}\tilde{\bm{B}}_{\bm{c}}^{(i)}=F_{E}^{(i)},$
(3.7a) $\displaystyle\left(\frac{1}{\Delta
t}M_{h}+a_{ii}Z_{h}-a_{ii}\beta_{0}S_{h}\right)\tilde{\bm{B}}_{\bm{c}}^{(i)}-a_{ii}Z_{h}\tilde{\bm{E}}_{\bm{c}}^{(i)}=F_{B}^{(i)},$
(3.7b)
where $M_{h}\bm{E_{c}}$, $M_{h}\bm{B_{c}}$, $S_{h}\bm{E_{c}}$,
$S_{h}\bm{B_{c}}$, $Z_{h}\bm{E_{c}}$ and $Z_{h}\bm{B_{c}}$ are matrix-by-
vector forms of $(E_{h},\mu)$, $(B_{h},\nu)$,
$(\mathcal{L}_{h,\mathbf{p}},\mu)$, $(\mathcal{L}_{h,\mathbf{q}},\nu)$,
$(z^{3}E_{h},\nu)$ and $(z^{3}B_{h},\nu)$, respectively. The right-hand-side
long vectors $F_{E}^{(i)}$ and $F_{B}^{(i)}$ are corresponding to
$(\text{RHS}_{E}^{(i)},\mu)+(E_{h},\mu)/\Delta t$ and
$(\text{RHS}_{B}^{(i)},\nu)+(B_{h},\nu)/\Delta t$ respectively.
Similar to (3.7), the corrector step (3.5), corresponding to (2.13), can be
written as
$\displaystyle\left(\frac{1}{\Delta
t}M_{h}-a_{ii}\alpha_{0}S_{h}+a_{ii}\Lambda_{h,\tilde{T}}^{(i)}\right)\bm{E_{c}}^{(i)}-a_{ii}\Lambda_{h,\tilde{T}}^{(i)}\bm{B_{c}}^{(i)}=F_{E}^{(i)},$
(3.8a) $\displaystyle\left(\frac{1}{\Delta
t}M_{h}-a_{ii}\gamma_{0}S_{h}\right)\bm{Q_{c}}^{(i)}=F^{(i)}_{Q}+\frac{1}{\Delta
t}M_{h}\bm{Q_{c}}^{n}.$ (3.8b)
For the predictor step, (3.7) can be expressed as a nonlinear algebraic system
$A(\tilde{U}^{(i)})\tilde{U}^{(i)}=F(U^{n},U^{(1)},U^{(2)},\ldots,U^{(i-1)}),$
(3.9)
with
$U=\left[\begin{array}[]{c}\bm{E_{c}}\\\ \bm{B_{c}}\end{array}\right],\quad
F=\left[\begin{array}[]{c}F_{E}^{(i)}\\\ F_{B}^{(i)}\end{array}\right],\quad
A(U)=\left[\begin{array}[]{cc}M_{1}(U)&M_{2}(U)\\\
M_{3}&M_{4}\end{array}\right].$
The corresponding submatrices in $A(U)$ are given by
$\displaystyle M_{1}(U)=\frac{1}{\Delta
t}M_{h}-a_{ii}\alpha_{0}S_{h}+a_{ii}\Lambda_{h,T},\quad
M_{2}(U)=-a_{ii}\Lambda_{h,T},$ $\displaystyle M_{3}=-a_{ii}Z_{h},\quad
M_{4}=\frac{1}{\Delta t}M_{h}+a_{ii}Z_{h}-a_{ii}\beta_{0}S_{h}.$
As we can see, only $M_{1}(U)$ and $M_{2}(U)$ mildly depend on $U$ due to
$\Lambda_{h,T}$. Hence an iteration procedure, corresponding to (2.12), can be
written as
$A(U^{i,l})U^{i,l+1}=F(U^{n},U^{(1)},U^{(2)},\ldots,U^{(i-1)}),$ (3.10)
with the initial value taking to be $U^{i,0}=U^{(i-1)}$.
A detailed procedure of the Picard iteration for solving (3.10) is as follows:
starting from an iterative number $l=0$, we set $E_{h}^{i,0}=E_{h}^{(i-1)}$,
$B_{h}^{i,0}=B_{h}^{(i-1)}$, and update the unknowns $B_{h}^{i,l+1}$ and
$E_{h}^{i,l+1}$ iteratively through the following three steps:
* 1.
Step 1: compute $A(U^{i,l})$
$A(U^{i,l})=\left[\begin{array}[]{cc}M_{1}(U^{i,l})&M_{2}(U^{i,l})\\\
M_{3}&M_{4}\end{array}\right];$
* 2.
Step 2: if $z({\bf x})$ is piecewise constant, $M_{3}$ is an easy invertible
diagonal matrix. From (3.10), by Gauss elimination, we update $E_{h}^{i,l+1}$
and $B_{h}^{i,l+1}$ with their long vectors by
$\begin{cases}\left(M_{1}{(M_{3})}^{-1}M_{4}-M_{2}\right)\bm{B_{c}}^{i,l+1}=M_{1}{(M_{3})}^{-1}F_{B}^{(i)}-F_{E}^{(i)},\\\
M_{3}\bm{E_{c}}^{i,l+1}=F_{B}^{(i)}-M_{4}\bm{B_{c}}^{i,l+1}.\end{cases}$
(3.11)
Otherwise, we solve (3.10) with the GMRES method [39].
* 3.
Step 3: we take $\tilde{E}_{h}^{(i)}=E_{h}^{i,l+1}$ and
$\tilde{B}_{h}^{(i)}=B_{h}^{i,l+1}$ when iteration stops. The stop criteria is
$\|\bm{E_{c}}^{i,l+1}-\bm{E_{c}}^{i,l}\|<\delta.$
In our numerical tests, we take an $L^{2}$ norm and a threshold
$\delta=10^{-8}$.
For the corrector step (3.8), $\bm{Q_{c}}^{(i)}$ is solved directly from
(3.8b), and then obtain $\bm{E_{c}}^{(i)}$ and $\bm{B_{c}}^{(i)}$ iteratively
by
$\displaystyle\left(\frac{1}{\Delta
t}M_{h}-a_{ii}\alpha_{0}S_{h}+a_{ii}\Lambda_{h,\tilde{T}}^{(i)}\right)\bm{E_{c}}^{i,l+1}=a_{ii}\Lambda_{h,\tilde{T}}^{(i)}\bm{B_{c}}^{i,l}+F_{E}^{(i)},$
(3.12a)
$\displaystyle\bm{B_{c}}^{i,l+1}=(\bm{Q_{c}}^{(i)}-\bm{E_{c}}^{i,l+1})^{4}.$
(3.12b)
with the initial values $B_{h}^{i,0}=\tilde{B}_{h}^{(i)}$. The stop criteria
is the same as Step 3 in the predictor step.
For the Picard iteration in the predictor step, we can show that, if $E_{h}$
and $B_{h}\in W_{h}^{(0,0)}$, and the implicit part of the IMEX RK time
discretizations has a nonnegative diagonal, the resulting matrix
$A(\tilde{U}^{(i)})$ in (3.9) is an $\mathbf{M}$ matrix if
$\tilde{U}^{(i)}\geq 0$:
###### Theorem 1.
The matrix $A(\tilde{U}^{(i)})$ in (3.9) is an $\mathbf{M}$ matrix for
piecewise constant $\mathcal{Q}^{0}$ finite elements if $\tilde{U}^{(i)}\geq
0$.
We can easily show that the mass matrix $A(\tilde{U}^{(i)})$ has a nonnegative
diagonal, while off-diagonal entries are all non-positive. Besides,
$A(U^{(i)})$ is diagonally dominant, so $A(U^{(i)})$ is an $\mathbf{M}$
matrix, namely its inverse has all nonnegative arguments [4]. Thanks to this
property, rapid convergence can be assured. A similar fast convergence is also
observed in the iteration for the correction step (3.12), with
$\Lambda_{h,\tilde{T}}$ obtained from the predictor step and initial values
taken from $\tilde{B}$.
### 3.5 Positivity preserving and TVB limiters
For the solutions of (1.1), both the radiation energy $E({\bf x},t)$ and the
material temperature $T({\bf x},t)$ remain positive at all times [57].
Unfortunately, the schemes described above cannot preserve the positivity of
the solutions when they are close to zero. For solutions in $W_{h}^{(0,0)}$,
if the right-hand side terms $F_{E}^{(i)}$ and $F_{B}^{(i)}$ in (3.7) are non-
negative, the solutions without correction preserve positivity. However, this
requires a time step size on the order of $\mathcal{O}(h^{2})$. Furthermore,
an M-matrix cannot be guaranteed for higher-order schemes. In this work, to
develop an efficient scheme with larger time steps, we rely on the following
procedure to preserve positivity and control numerical oscillations.
Here we employ a positivity preserving limiter denoted as $P\Pi_{h}$ from
[59]. The $P\Pi_{h}$ limiter has also been applied to porous medium equations
in [49]. Additionally, TVB limiters [11] are utilized to control numerical
oscillations in the case of non-smooth solutions for second and third-order
schemes. As mentioned earlier, even for a first-order scheme with solutions in
$W^{0,0}_{h}$, the cell averages may not necessarily remain positive when
using large time step sizes. For diffusion systems, a cut-off limiter is
commonly employed to ensure a lower bound on the solutions [41, 51, 55].
For a one-dimensional case, we denote $\bar{u}_{j}$ as the cell average of the
numerical solution $u_{h}$ in the cell $I_{j}$ and $u_{\text{cold}}$ as the
minimum value of $u({\bf x},0)$. First, for the limiter $P\Pi_{h}$, it is
applied as follows:
1. 1.
Check the cell average $\bar{u}_{j}$ in each cell $I_{j}$, if it is less than
$u_{\text{cold}}$, we set
$P\Pi_{h}u_{h}=u_{\text{cold }};$
2. 2.
Then if $\bar{u}_{j}\geq u_{\text{cold}}$, but at least one value is less than
$u_{\text{cold}}$ appearing at two endpoints of the cell $I_{j}$ or at any
Gaussian point inside the cell $I_{j}$, we set
$P\Pi_{h}u_{h}=P\Pi^{1}_{h}u_{h}^{1}$, where $u_{h}^{1}$ is a local
$L^{2}$-projection of the solution $u_{h}$ to a linear polynomial within this
cell, where
$P\Pi^{1}_{h}u_{h}^{1}=\begin{cases}\left[1-2h_{j}^{-1}\left(x-x_{j}\right)\right]\bar{u}_{j}+2u_{\text{cold
}}h_{j}^{-1}\left(x-x_{j}\right),&\text{ if
}u_{h,j+\frac{1}{2}}^{1,-}<u_{\text{cold}},\\\ \,\\\
\left[1+2h_{j}^{-1}\left(x-x_{j}\right)\right]\bar{u}_{j}-2u_{\text{cold
}}h_{j}^{-1}\left(x-x_{j}\right),&\text{ if
}u_{h,j-\frac{1}{2}}^{1,+}<u_{\text{cold}};\end{cases}$
3. 3.
Otherwise, we set $P\Pi_{h}u_{h}|_{I_{j}}=u_{h}|_{I_{j}}$.
After applying the PP limiter $P\Pi_{h}$, a generalized TVB limiter is further
applied to control numerical oscillations for second and third order schemes:
$\displaystyle\begin{cases}u_{j-\frac{1}{2}}^{+,(\bmod)}=\text{TVB minmod
}\left(\bar{u}_{j}-u_{j-\frac{1}{2}}^{+},\frac{\theta}{2}\left(\bar{u}_{j}-\bar{u}_{j-1}\right),\frac{\theta}{2}\left(\bar{u}_{j+1}-\bar{u}_{j}\right),\frac{1}{4}\left(\bar{u}_{j+1}-\bar{u}_{j-1}\right)\right),\\\
u_{j+\frac{1}{2}}^{-,(\bmod)}=\text{TVB minmod
}\left(u_{j+\frac{1}{2}}^{-}-\bar{u}_{j},\frac{\theta}{2}\left(\bar{u}_{j}-\bar{u}_{j-1}\right),\frac{\theta}{2}\left(\bar{u}_{j+1}-\bar{u}_{j}\right),\frac{1}{4}\left(\bar{u}_{j+1}-\bar{u}_{j-1}\right)\right),\end{cases}$
where
$\displaystyle\text{TVB minmod }(a,b,c,d)=\begin{cases}a,&\text{ if }|a|\leq
Mh^{2},\\\ \operatorname{minmod}(a,b,c,d),&\text{ otherwise, }\end{cases}$
and
$\displaystyle\operatorname{minmod}(a,b,c,d)=\begin{cases}\operatorname{sign}(a)\min(|a|,|b|,|c|),&\text{
if }ab>0,ac>0,\\\ 0,&\text{ otherwise. }\end{cases}$
If $u_{j-\frac{1}{2}}^{+,(\bmod)}$ is different from $u_{j-\frac{1}{2}}^{+}$,
or $u_{j+\frac{1}{2}}^{-,(\bmod)}$ is different from $u_{j+\frac{1}{2}}^{-}$,
the cell $I_{j}$ is identified as a troubled cell, and the polynomial in this
cell is modified to be
$u_{h}(x)=\bar{u}_{j}+\left(u_{j+\frac{1}{2}}^{-,(\bmod)}+u_{j-\frac{1}{2}}^{+,(\bmod)}\right)h_{j}^{-1}\left(x-x_{j}\right).$
In the generalized TVB limiter, we take the parameters $M=0.1$ and
$\theta=1.5$ in our numerical tests.
For the two-dimensional case, we denote $\bar{u}_{i,j}$ as the cell average of
the numerical solution $u_{h}$ in the cell $I_{ij}$ and $u_{\text{cold}}$ as
the minimum value of $u_{h}({\bf x},0)$.
$u_{i\pm\frac{1}{2},j\pm\frac{1}{2}}=u_{h}(x_{i\pm\frac{1}{2}},y_{j\pm\frac{1}{2}})$
are denoted the values of the endpoints within the cell $I_{i,j}$. The PP
limiter $P\Pi_{h}$ is applied as follows:
1. 1.
Check the cell average $\bar{u}_{i,j}$ in each cell $I_{i,j}$, if it is less
than $u_{\text{cold}}$, we set
$P\Pi_{h}u_{h}=u_{\text{cold }};$
2. 2.
Then if $\bar{u}_{i,j}\geq u_{\text{cold}}$, but at least one value is less
than $u_{\text{cold}}$ appearing at those endpoints of the cell $I_{ij}$ or at
any Gaussian point inside the cell $I_{ij}$, we set
$P\Pi_{h}u_{h}=P\Pi_{h}^{1}u_{h}^{1}$, where
$u_{h}^{1}=\bar{u}_{i,j}+\alpha_{1}\psi_{x}^{i}+\alpha_{2}\psi_{y}^{j},\quad\psi_{x}^{i}=\frac{x-x_{i}}{h_{x}^{i}},\,\psi_{y}^{j}=\frac{y-y_{j}}{h_{y}^{j}}.$
$u_{h}^{1}$ is a local $L^{2}$-projection of the solution $u_{h}$ to
$\mathcal{P}^{1}$ within this cell $I_{i,j}$. For $u^{1}_{h}$, the extremum
occurs only at four endpoints of the cell $I_{i,j}$, and due to
$\bar{u}_{i,j}\geq u_{\text{cold}}$ at most two adjacent points are less than
$u_{\text{cold}}$. We set those end-point values to be $u_{\text{cold}}$ if
they are less than $u_{\text{cold}}$. Next we only need to determine the
parameters $\alpha_{1}$, $\alpha_{2}$ by using two end-point values, and we
take two minimum point values at those end points. They are determined as
follows:
* (a)
if only one end-point value at $(x_{i-\frac{1}{2}},y_{j-\frac{1}{2}})$ is
modified, we have
$\displaystyle P\Pi_{h}^{1}u^{1}_{h}=$
$\displaystyle\bar{u}_{i,j}+\left(\bar{u}_{i,j}-u_{\text{cold}}+\frac{u^{1}_{i+\frac{1}{2},j-\frac{1}{2}}-u^{1}_{i-\frac{1}{2},j+\frac{1}{2}}}{2}\right)\psi_{x}^{i}$
$\displaystyle+\left(\bar{u}_{i,j}-u_{\text{cold}}-\frac{u^{1}_{i+\frac{1}{2},j-\frac{1}{2}}-u^{1}_{i-\frac{1}{2},j+\frac{1}{2}}}{2}\right)\psi_{y}^{j},$
* (b)
otherwise if two end-point values at $(x_{i-\frac{1}{2}},y_{j-\frac{1}{2}})$
and $(x_{i-\frac{1}{2}},y_{j+\frac{1}{2}})$ are changed, we take
$P\Pi_{h}^{1}u^{1}_{h}=\bar{u}_{i,j}+2(\bar{u}_{i,j}-u_{\text{cold}})\psi_{x}^{i}.$
Other cases can be determined similarly, we omit them to save space;
3. 3.
Otherwise, we set $P\Pi_{h}u_{h}|_{I_{ij}}=u_{h}|_{I_{ij}}$.
For the two-dimensional generalized TVB limiter, we define
$w^{1}(y)=u_{h}(x_{i},y)$ in the cell $I_{i,j}$ and then modify $w^{1}(y)$
along $x=x_{i}$ as in the one-dimensional case, that is,
$\displaystyle\begin{cases}u_{j-\frac{1}{2}}^{+,(\bmod)}=\text{TVB
minmod}\left(\bar{u}_{i,j}-w^{1}(y_{j-\frac{1}{2}}^{+}),\frac{\theta}{2}\left(\bar{u}_{i,j}-\bar{u}_{i,j-1}\right),\frac{\theta}{2}\left(\bar{u}_{i,j+1}-\bar{u}_{i,j}\right),\frac{1}{4}\left(\bar{u}_{i,j+1}-\bar{u}_{i,j-1}\right)\right),\\\
u_{j+\frac{1}{2}}^{-,(\bmod)}=\text{TVB
minmod}\left(w^{1}(y_{j+\frac{1}{2}}^{-})-\bar{u}_{i,j},\frac{\theta}{2}\left(\bar{u}_{i,j}-\bar{u}_{i,j-1}\right),\frac{\theta}{2}\left(\bar{u}_{i,j+1}-\bar{u}_{i,j}\right),\frac{1}{4}\left(\bar{u}_{i,j+1}-\bar{u}_{i,j-1}\right)\right).\end{cases}$
If $u_{j-\frac{1}{2}}^{+,(\bmod)}$ is different from
$w^{1}(y_{j-\frac{1}{2}}^{+})$, or $u_{j+\frac{1}{2}}^{-,(\bmod)}$ is
different from $w^{1}(y_{j+\frac{1}{2}}^{-})$, the cell $I_{i,j}$ is
identified as a troubled cell along $x=x_{i}$ and $w^{1}(y)$ is modified to be
$w^{1}(y)=\bar{u}_{i,j}+\left(u_{j+\frac{1}{2}}^{-,(\bmod)}+u_{j-\frac{1}{2}}^{+,(\bmod)}\right)h_{j}^{-1}\left(y-y_{j}\right).$
Similarly, along $y=y_{j}$ we can also define $w^{2}(x)=u_{h}(x,y_{j})$, and
$\displaystyle\begin{cases}u_{i-\frac{1}{2}}^{+,(\bmod)}=\text{TVB
minmod}\left(\bar{u}_{i,j}-w^{2}(x_{i-\frac{1}{2}}^{+}),\frac{\theta}{2}\left(\bar{u}_{i,j}-\bar{u}_{i-1,j}\right),\frac{\theta}{2}\left(\bar{u}_{i+1,j}-\bar{u}_{i,j}\right),\frac{1}{4}\left(\bar{u}_{i+1,j}-\bar{u}_{i-1,j}\right)\right),\\\
u_{i+\frac{1}{2}}^{-,(\bmod)}=\text{TVB
minmod}\left(w^{2}(x_{i+\frac{1}{2}}^{-})-\bar{u}_{i,j},\frac{\theta}{2}\left(\bar{u}_{i,j}-\bar{u}_{i-1,j}\right),\frac{\theta}{2}\left(\bar{u}_{i+1,j}-\bar{u}_{i,j}\right),\frac{1}{4}\left(\bar{u}_{i+1,j}-\bar{u}_{i-1,j}\right)\right).\end{cases}$
If $u_{i-\frac{1}{2}}^{+,(\bmod)}\neq w^{2}(x_{i-\frac{1}{2}}^{+})$, or
$u_{i+\frac{1}{2}}^{-,(\bmod)}\neq w^{2}(x_{i+\frac{1}{2}}^{-})$, the cell
$I_{i,j}$ is identified as a troubled cell along $y=y_{j}$ and $w^{2}(x)$ is
modified to be
$w^{2}(x)=\bar{u}_{i,j}+\left(u_{i+\frac{1}{2}}^{-,(\bmod)}+u_{i-\frac{1}{2}}^{+,(\bmod)}\right)h_{i}^{-1}\left(x-x_{i}\right).$
If $I_{i,j}$ is a troubled cell along either the $x$ or the $y$ direction, we
set $u_{h}(x,y)$ in the cell $I_{i,j}$ as
$u_{h}(x,y)=w^{1}(y)+w^{2}(x)-\bar{u}_{i,j}$.
### 3.6 Algorithm flowchart
Finally we present our high order IMEX-LDG scheme updating procedure from time
level $t_{n}$ to $t_{n+1}$ in Algorithm 1.
Input: $E^{n}$, $T^{n}$
Output: $E^{n+1}$, $T^{n+1}$
1 $E^{(1)}=E^{n}$, $B^{(1)}=(T^{n})^{4}$;
2 for _$i=2$ to $s$_ do
3 $E^{i,0}=E^{(i-1)}$, $B^{i,0}=B^{(i-1)}$, $l=0$;
4 $M_{3}=-a_{ii}Z_{h},\quad M_{4}=\frac{1}{\Delta
t}M_{h}+a_{ii}Z_{h}-a_{ii}\beta_{0}S_{h}$;
5 while _$\|E^{i,l+1}-E^{i,l}\|\geq\delta$ or $l=0$_ do
6 $M_{1}=\frac{1}{\Delta
t}M_{h}-a_{ii}\alpha_{0}S_{h}+a_{ii}\Lambda_{h,T},\quad
M_{2}=-a_{ii}\Lambda_{h,T}$;
7 update $E^{i,l+1}$ and $B^{i,l+1}$ by (3.11);
8 apply the limiters to $E^{i,l+1}$ and $B^{i,l+1}$;
9 $l=l+1$;
10 end while
11 $\tilde{E}^{(i)}=E^{i,l+1}$, $\tilde{B}^{(i)}=B^{i,l+1}$,
$\tilde{T}^{(i)}=(\tilde{B}^{(i)})^{1/4}$ ;
12 obtain $Q^{(i)}$ by (3.8b);
13 $E^{i,0}=\tilde{E}^{(i)}$, $l=0$;
14 while _$\|E^{i,l+1}-E^{i,l}\|\geq\delta$ or $l=0$_ do
15 update $E^{i,l+1}$ by (3.12a);
16 apply the limiters to $E^{i,l+1}$;
17 $B^{i,l+1}=(Q^{(i)}-E^{i,l+1})^{4}$;
18 apply the limiters to $B^{i,l+1}$;
19 $l=l+1$;
20
21 end while
22 $E^{(i)}=E^{i,l+1}$, $T^{(i)}=Q^{(i)}-E^{(i)}$;
23
24 end for
25$E^{n+1}=E^{(s)}$, $T^{n+1}=T^{(s)}$;
26
Algorithm 1 The updating from $t^{n}$ to $t^{n+1}$ with an $s$-stage
$(k+1)$-th order IMEX method
###### Remark 2.
(Numerical boundary treatment) There are mainly three types of boundary
conditions in our numerical experiments: periodic, Neumann, and mixed boundary
conditions. The numerical fluxes at the boundary under periodic or Neumann
boundary conditions are chosen as in [12].
Taking 1D with $\Omega=[x_{L},x_{R}]$ as an example, for a periodic boundary
condition, we take $\phi_{L}^{-}=\phi_{R}^{-}$, $\phi_{R}^{+}=\phi_{L}^{+}$
where $\phi=u$ or $u_{x}$. For a Neumann boundary condition, we set
$\hat{u}_{L}=u_{L}^{+}$, $\hat{u}_{R}=u_{R}^{-}$, while
$\hat{v}_{L}=\hat{v}_{R}=0$ where $v=u_{x}$. Here $\phi^{\pm}_{L}$,
$\phi^{\pm}_{R}$, are the left and right limits at $x_{L}$ and $x_{R}$
respectively. Similarly for $u^{+}_{L}$ and $u^{-}_{R}$.
For a mixed boundary condition, e.g. in the Marshak wave problem in section 4,
the physical boundary and initial conditions on the domain
$x\in\Omega=[x_{L},x_{R}]=[0,1]$ are given as
$\begin{cases}\dfrac{E}{4}-\dfrac{1}{6\sigma}\dfrac{\partial E}{\partial
x}=1,\quad\dfrac{\partial T}{\partial x}=0,\quad x=0,\\\\[5.0pt]
\dfrac{E}{4}+\dfrac{1}{6\sigma}\dfrac{\partial E}{\partial
x}=0,\quad\dfrac{\partial T}{\partial x}=0,\quad x=1,\\\\[5.0pt]
E|_{t=0}=1.0\times 10^{-5},\quad T|_{t=0}=E^{1/4}|_{t=0}.\end{cases}$ (3.13)
Taking the left boundary $x_{L}=x_{\frac{1}{2}}=0$ as an example, we omit the
artificial term $\frac{|\nabla E|}{E}$ in $D_{r}$ and the boundary condition
is simplified to [60, 19, 16]
$\dfrac{E}{4}-\dfrac{1}{2}D_{r}\frac{\partial E}{\partial x}=1.$
Hence, in both the predictor and corrector steps, we set the numerical fluxes
as
$\hat{E}_{h}=E_{h}^{+},\quad\hat{D}_{r,h}\hat{p}_{h}=\dfrac{\hat{E}_{h}}{2}-2.$
Similarly at the right boundary $x_{N+\frac{1}{2}}=1$, we take
$\hat{E}_{h}=E_{h}^{-},\quad\hat{D}_{r,h}\hat{p}_{h}=-\dfrac{\hat{E}_{h}}{2}.$
In the corrector step, we set
$\hat{Q}_{h}=\hat{E}_{h}+\hat{T}_{h},\quad\hat{w}_{h}=3\sigma(\hat{T}_{h})\hat{D}_{r,h}\hat{p}_{h}.$
## 4 Numerical Examples
In this section, we conduct numerical experiments to validate the high-order
accuracy, conservation properties, suitability for large time steps, and
effectiveness in capturing sharp fronts in both homogeneous and heterogeneous
media using our proposed schemes. For the added diffusion terms, we take the
coefficients as $\alpha_{0}=\tau\max\\{\frac{1}{3\sigma}\\}$,
$\beta_{0}=\tau\max\\{D_{t}\\}$ and $\gamma_{0}=\alpha_{0}+\beta_{0}$ with
$\tau=0.6$ [49]. In space, the $\mathcal{Q}^{\bm{k}}$ basis with $k$-th
piecewise polynomial in each direction for $k=0,1,2$ is taken.
Correspondingly, an $s$-stage $(k+1)$-th order globally stiffly accurate IMEX
RK time discretization is employed in time with the double Butcher tableau
given in A. In the following, our schemes are denoted as $(k+1)$-th order
methods for $k=0,1,2$ with $s=2,3,5$, respectively.
###### Example 4.1.
(Accuracy test in 1D) First we consider a 1D example with smooth initial
values and periodic boundary conditions at the equilibrium, which are given by
$\begin{cases}T(x,0)=0.8+0.1\sin(x),\\\\[5.0pt] E(x,0)=T(x,0)^{4},\end{cases}$
(4.1)
on the computational domain $[-\pi,\pi]$. We take $\kappa=0.1$ and a
homogeneous medium $z(x)=1$. The problem is run to time $t=5$ using the 1st,
2nd, and 3rd order methods, respectively. Since the exact solution is not
available, we compute the numerical errors by comparing numerical solutions
with a reference solution. In this case, the source term is not stiff as $T$
is away from $0$, so we compute the reference solution by a 3rd order LDG
method with a 3rd order explicit strong-stability-preserving RK time
discretization [15], on a much refined mesh $N=1024$. In Tables 4.1-4.3, we
show the numerical $L^{2}$ errors and orders of accuracy for the 1st, 2nd, and
3rd order schemes with different time steps, respectively. From these tables,
we can see that our methods achieve the corresponding orders of accuracy when
the time step is $\Delta t=\mathcal{O}(h).$ From the numerical results, we
find that larger $\Delta t$ lead to larger errors. In Table 4.3, for the 3rd
order method, an order reduction can be observed for a large ratio of $\Delta
t/h$. The order increases with further mesh refinements. Such a phenomeno may
be due to a high order IMEX time discretization.
Table 4.1: The numerical $L^{2}$ errors and orders of accuracy for the 1st order scheme with different time steps for Example 4.1. $t=5$. N | $\Delta t$ | $L^{2}$ error | order | $L^{2}$ error | order | $L^{2}$ error | order
---|---|---|---|---|---|---|---
E | T | B
4 | $\frac{h}{2}$ | 3.46e-2 | - | 1.74e-2 | - | 3.51e-2 | -
8 | 1.73e-2 | 1.00 | 8.65e-3 | 1.01 | 1.76e-2 | 0.99
16 | 8.68e-3 | 1.00 | 4.34e-3 | 0.99 | 8.84e-3 | 1.00
32 | 4.34e-3 | 1.00 | 2.17e-3 | 1.00 | 4.42e-3 | 1.00
64 | 2.17e-3 | 1.00 | 1.09e-3 | 1.00 | 2.21e-3 | 1.00
128 | 1.08e-3 | 1.00 | 5.43e-4 | 1.00 | 1.10e-3 | 1.00
4 | $h$ | 3.47e-2 | - | 1.75e-2 | - | 3.52e-2 | -
8 | 1.74e-2 | 1.00 | 8.68e-3 | 1.01 | 1.77e-2 | 0.99
16 | 8.70e-3 | 1.00 | 4.36e-3 | 1.00 | 8.86e-3 | 1.00
32 | 4.35e-3 | 1.00 | 2.18e-3 | 1.00 | 4.43e-3 | 1.00
64 | 2.17e-3 | 1.00 | 1.09e-3 | 1.00 | 2.21e-3 | 1.00
128 | 1.09e-3 | 1.00 | 5.44e-4 | 1.00 | 1.11e-3 | 1.00
4 | $3h$ | 3.52e-2 | - | 1.77e-2 | - | 3.57e-2 | -
8 | 1.77e-2 | 0.99 | 8.85e-3 | 1.00 | 1.80e-2 | 0.99
16 | 8.86e-3 | 1.00 | 4.44e-3 | 1.00 | 9.02e-3 | 1.00
32 | 4.43e-3 | 1.00 | 2.22e-3 | 1.00 | 4.51e-3 | 1.00
64 | 2.21e-3 | 1.00 | 1.11e-3 | 1.00 | 2.25e-3 | 1.00
128 | 1.11e-3 | 1.00 | 5.55e-4 | 1.00 | 1.13e-3 | 1.00
4 | $5h$ | 3.53e-2 | - | 1.78e-2 | - | 3.58e-2 | -
8 | 1.79e-2 | 0.98 | 9.00e-3 | 0.98 | 1.82e-2 | 0.97
16 | 9.04e-3 | 0.99 | 4.55e-3 | 0.98 | 9.21e-3 | 0.99
32 | 4.56e-3 | 0.99 | 2.30e-3 | 0.99 | 4.64e-3 | 0.99
64 | 2.28e-3 | 1.00 | 1.15e-3 | 1.00 | 2.32e-3 | 1.00
128 | 1.14e-3 | 1.00 | 5.74e-4 | 1.00 | 1.16e-3 | 1.00
Table 4.2: The numerical $L^{2}$ errors and orders of accuracy for the 2nd order scheme with different time steps for Example 4.1. $t=5.$ N | $\Delta t$ | $L^{2}$ error | order | $L^{2}$ error | order | $L^{2}$ error | order
---|---|---|---|---|---|---|---
E | T | B
4 | $\frac{h}{2}$ | 1.25e-2 | - | 6.18e-3 | - | 1.27e-2 | -
8 | 3.00e-3 | 2.06 | 1.57e-3 | 1.98 | 3.04e-3 | 2.06
16 | 7.36e-4 | 2.02 | 3.86e-4 | 2.02 | 7.48e-4 | 2.02
32 | 1.83e-4 | 2.01 | 9.60e-5 | 2.01 | 1.86e-4 | 2.01
64 | 4.58e-5 | 2.00 | 2.40e-5 | 2.00 | 4.66e-5 | 2.00
128 | 1.14e-5 | 2.00 | 5.99e-6 | 2.00 | 1.16e-5 | 2.00
4 | $h$ | 1.25e-2 | - | 6.19e-3 | - | 1.27e-2 | -
8 | 3.00e-3 | 2.06 | 1.57e-3 | 1.98 | 3.04e-3 | 2.06
16 | 7.36e-4 | 2.02 | 3.86e-4 | 2.02 | 7.48e-4 | 2.02
32 | 1.83e-4 | 2.01 | 9.60e-5 | 2.01 | 1.86e-4 | 2.01
64 | 4.58e-5 | 2.00 | 2.40e-5 | 2.00 | 4.66e-5 | 2.00
128 | 1.14e-5 | 2.00 | 5.99e-6 | 2.00 | 1.16e-5 | 2.00
4 | $3h$ | 1.25e-2 | - | 6.29e-3 | - | 1.27e-2 | -
8 | 3.03e-3 | 2.05 | 1.60e-3 | 1.97 | 3.10e-3 | 2.04
16 | 7.41e-4 | 2.03 | 3.90e-4 | 2.04 | 7.58e-4 | 2.03
32 | 1.85e-4 | 2.01 | 9.63e-5 | 2.02 | 1.87e-4 | 2.02
64 | 4.61e-5 | 2.00 | 2.41e-5 | 2.00 | 4.68e-5 | 2.00
128 | 1.15e-5 | 2.00 | 6.03e-6 | 2.00 | 1.17e-5 | 2.00
4 | $5h$ | 1.30e-2 | - | 6.29e-3 | - | 1.27e-2 | -
8 | 3.18e-3 | 2.03 | 1.93e-3 | 1.71 | 3.81e-3 | 1.74
16 | 9.11e-4 | 1.80 | 4.37e-4 | 2.14 | 8.77e-4 | 2.12
32 | 2.00e-4 | 2.19 | 9.99e-5 | 2.13 | 1.93e-4 | 2.18
64 | 4.81e-5 | 2.05 | 2.52e-5 | 1.99 | 4.92e-5 | 1.97
128 | 1.20e-5 | 2.01 | 6.28e-6 | 2.00 | 1.22e-5 | 2.00
Table 4.3: The numerical $L^{2}$ errors and orders of accuracy for the 3rd order scheme with different time steps for Example 4.1. $t=5.$ N | $\Delta t$ | $L^{2}$ error | order | $L^{2}$ error | order | $L^{2}$ error | order
---|---|---|---|---|---|---|---
E | T | B
4 | $\frac{h}{2}$ | 1.71e-3 | - | 1.03e-3 | - | 1.78e-3 | -
8 | 1.97e-4 | 3.12 | 1.14e-4 | 3.18 | 2.04e-4 | 3.12
16 | 2.49e-5 | 2.99 | 1.47e-5 | 2.96 | 2.61e-5 | 2.97
32 | 3.11e-6 | 3.00 | 1.86e-6 | 2.98 | 3.31e-6 | 2.98
64 | 3.89e-7 | 3.00 | 2.34e-7 | 2.99 | 4.15e-7 | 2.99
128 | 4.86e-8 | 3.00 | 2.93e-8 | 3.00 | 5.20e-8 | 3.00
4 | $h$ | 1.72e-3 | - | 1.04e-3 | - | 1.78e-3 | -
8 | 1.97e-4 | 3.12 | 1.14e-4 | 3.18 | 2.04e-4 | 3.12
16 | 2.49e-5 | 2.99 | 1.47e-5 | 2.96 | 2.62e-5 | 2.96
32 | 3.12e-6 | 3.00 | 1.87e-6 | 2.97 | 3.33e-6 | 2.98
64 | 3.91e-7 | 3.00 | 2.36e-7 | 2.98 | 4.21e-7 | 2.98
128 | 4.90e-8 | 3.00 | 2.97e-8 | 2.99 | 5.29e-8 | 2.99
4 | 3$h$ | 1.98e-3 | - | 1.15e-3 | - | 2.06e-3 | -
8 | 2.21e-4 | 3.17 | 1.28e-4 | 3.17 | 2.36e-4 | 3.12
16 | 2.99e-5 | 2.89 | 1.78e-5 | 2.85 | 3.30e-5 | 2.84
32 | 4.28e-6 | 2.80 | 2.81e-6 | 2.66 | 5.30e-6 | 2.64
64 | 7.37e-7 | 2.54 | 6.50e-7 | 2.11 | 1.31e-6 | 2.01
128 | 1.25e-7 | 2.56 | 1.07e-7 | 2.61 | 2.16e-7 | 2.60
256 | 1.97e-8 | 2.66 | 1.60e-8 | 2.74 | 3.23e-8 | 2.74
512 | 2.85e-9 | 2.79 | 2.21e-9 | 2.85 | 4.48e-9 | 2.85
4 | 5$h$ | 2.13e-3 | - | 1.19e-3 | - | 2.20e-3 | -
8 | 4.29e-4 | 2.31 | 2.37e-4 | 2.32 | 4.82e-4 | 2.19
16 | 6.43e-5 | 2.74 | 3.65e-5 | 2.70 | 7.39e-5 | 2.71
32 | 1.06e-5 | 2.61 | 8.27e-6 | 2.14 | 1.68e-5 | 2.13
64 | 1.94e-6 | 2.44 | 1.64e-6 | 2.33 | 3.31e-6 | 2.35
128 | 3.70e-7 | 2.39 | 3.09e-7 | 2.41 | 6.18e-7 | 2.42
256 | 6.82e-8 | 2.44 | 5.69e-8 | 2.44 | 1.15e-7 | 2.43
512 | 1.11e-8 | 2.61 | 8.99e-9 | 2.66 | 1.82e-8 | 2.66
###### Example 4.2.
Next we consider the following initial boundary data [29], with a sharp
transition in the initial values
$\begin{cases}E(x,0)=E_{L}+(E_{R}-E_{L})\dfrac{1+\tanh[50(x-0.25)]}{2},\\\\[5.0pt]
T(x,0)=E(x,0)^{1/4},\\\\[5.0pt] \dfrac{E}{4}-\dfrac{1}{6\sigma}\dfrac{\partial
E}{\partial x}=1,\,\,\,\,\dfrac{\partial T}{\partial x}=0,x=0,\\\\[5.0pt]
\dfrac{E}{4}+\dfrac{1}{6\sigma}\dfrac{\partial E}{\partial
x}=V_{R},\dfrac{\partial T}{\partial x}=0,x=1,\end{cases}$ (4.2)
where $V_{R}=1\times 10^{-3}$ , $E_{L}=4$ and $E_{R}=4\times 10^{-3}.$ We
consider two cases $\kappa=0$ and $\kappa=0.1$ in $D_{t}$. For the case of
$\kappa=0$, the system of $E$ and $B$ itself is in a conservative form. We
take $z(x)=1$ and a time step size $\Delta t=\frac{1}{5}h$. The results at
$t=1$ for $\kappa=0$ and $t=0.5$ for $\kappa=0.1$ are shown in Fig. 4.1 and
4.2, respectively. Reference solutions are obtained by a 1st order explicit RK
LDG method on $1024$ elements with a small enough time step, which is denoted
as “ref”. The left column is the radiation temperature $T_{r}$ and the right
column is the material temperature $T$. Two different mesh sizes are
considered, $N=64$ and $N=128$. “WL” refers to numerical solutions obtained
with limiters, and correspondingly “NL” refers to numerical solutions without
limiters. It can be observed that the higher the order, the closer the
numerical solutions are, as compared to the reference solutions. In addition,
the 1st order solutions perform well without limiters, but the 2nd order and
3rd order solutions have small oscillations in front of the sharp gradient.
After applying limiters, these oscillations can be well controlled. As we can
see from Fig. 4.1 and Fig. 4.2, refining the mesh from $N=64$ to $N=128$, the
numerical solutions match the reference solutions better, no matter with or
without limiters. Especially, deviations due to the application of the
limiters are also reduced with the mesh refinement.
Figure 4.1: The numerical results for Example 4.2 with $\kappa=0$. 1st, 2nd,
and 3rd order schemes are used. “WL”: with limiters, “NL”: without limiters.
“ref” represents the reference solutions. Top row: $N=64$; Bottom row:
$N=128$. Left column: radiation temperature $T_{r}$; Right column: material
temperature $T$. $t=1$, $\Delta t=\frac{h}{5}.$
Figure 4.2: The numerical results for Example 4.2 with $\kappa=0.1,$ and
$N=128$. 1st, 2nd, and 3rd order schemes are used. “WL” : with limiters, “NL”:
without limiters. “ref” represents the reference solutions. Top row: $N=64$;
Bottom row: $N=128$. Left column: radiation temperature $T_{r}$; Right column:
material temperature $T$. $t=0.5$, $\Delta t=\frac{h}{5}.$
###### Example 4.3.
We now consider an example with a periodic boundary condition to verify the
conservation errors. The initial values are taken to be
$\begin{cases}E(x,0)=E_{L}+(E_{R}-E_{L})\dfrac{1+\tanh[200(x-0.5)^{2}]}{2},\\\\[5.0pt]
T(x,0)=E(x,0)^{1/4},\end{cases}$ (4.3)
with $E_{L}=1,E_{R}=0.0001$, which are shown in the left column of Fig. 4.3.
We run the solution with $\kappa=0.5$ up to time $t=1$. We take $z(x)=1$ and
mesh numbers $N=128$. 1st, 2nd, and 3rd order schemes are used. “c” denotes
numerical results obtained with the conservation corrector step, while “nc” is
without the corrector step. As we have observed numerically, without the
corrector step, a smaller time step is needed for the convergence of
iteration. We take $\Delta t=\frac{1}{30}h$ for the 2nd method, and $\Delta
t=\frac{1}{50}h$ for the 3rd order method, without a corrector step, and
$\Delta t=\frac{1}{5}h$ for all others. In Fig. 4.3, on the right column, we
show the time evolution of conservation errors for the total energy $Q=E+T$.
We can observe that with a corrector step, the errors are much smaller than
those without a corrector step. In Fig. 4.4, we show the numerical solutions
for different orders with or without a corrector step. As we can see, with a
corrector step, all results match each other well. Without a corrector step,
we can clearly observe a deviation, especially for first and second order
methods. However, we would note that with a corrector step, the method itself
is conservative for the total energy, although limiters for second and third
order methods would slightly destroy such a conservation.
Figure 4.3: The numerical results for Example 4.3 with $\kappa=0.5$ and
$N=128$. 1st, 2nd, and 3rd order schemes are used. “c”: with a conservation
corrector step; “nc”: without a corrector step. Left: initial values; Right:
time evolution of conservation errors. $\Delta t=\frac{1}{30}h$ for “2nd, nc”,
$\Delta t=\frac{1}{50}h$ for “3rd, nc” and $\Delta t=\frac{1}{5}h$ for all
others.
Figure 4.4: The numerical results for Example 4.3 with $\kappa=0.5$ and
$N=128$. 1st, 2nd, and 3rd order schemes are used. “c”: with a conservation
corrector step; “nc”: without a corrector step. Left: radiation energy $E$;
Right: material temperature $T$. $\Delta t=\frac{1}{30}h$ for “2nd, nc”,
$\Delta t=\frac{1}{50}h$ for “3rd, nc” and $\Delta t=\frac{1}{5}h$ for all
others.
###### Example 4.4.
Here we consider the standard homogeneous Marshak wave problem [30, 45, 59]
with the initial and boundary conditions (3.13), where the atomic mass number
$z(x)=1$. This is a benchmark problem for radiation diffusion problems. It is
very challenging since the left inflow boundary value is not consistent with
the initial datum. Similarly, we consider both $\kappa=0$ and $\kappa=0.1$ in
$D_{t}$, respectively. The reference solutions are computed by a 1st order
explicit RK LDG method on $1024$ elements with a small enough time step. In
Fig. 4.5 and Fig. 4.6, we show the results at different times $t=1,2$ and
$t=1,1.5$, respectively. The left column is for the radiation temperature
$T_{r}$ and the right column is for the material temperature $T$. A mesh
refinement from $N=64$ to $N=128$ is also considered. As we can see that the
1st order numerical results are very deviated from the reference solutions,
second and third order methods capture the sharp fronts more accurately. With
mesh refinement, all results are getting closer to the reference solutions,
indicating that our methods are convergent with mesh refinement. The results
are also consistent with those in [33, 22]. As compared to [60, 16], a
relatively larger time step size $\Delta t=\frac{1}{5}h$ can be used, where
$\Delta t=\frac{4}{125}h$ with $h=\frac{1}{80}$ in [60] and $\Delta
t=\frac{1}{8}h^{2}$ with $h=\frac{1}{16}$ in [16]. In Fig 4.7, we compare the
results with and without a corrector step. In the case of $\kappa=0$, the
system $E$ and $B$ in (2.1) itself is in a conservative form, although it only
conserves the total energy $Q=E+T$ up to an error of numerical precision, the
results with or without a corrector step are almost the same. For
$\kappa=0.1$, the results with a corrector step clearly match the reference
solutions better, especially for first and second order methods. This has
demonstrated that the corrector step is very necessary.
###### Remark 3.
As discussed in Remark 1, if we eliminate the predictor step and directly
employ a Picard iteration in the corrector step, it may not work well for some
challenge problems. For this standard Marshak wave problem, if we consider
$E(x,0)=10^{-7}$, $N=64$ and $\Delta t=\frac{h}{5}$, even a first order scheme
does not converge well. Such an approach with higher orders work even worse.
However, our predictor-corrector procedure works well for these test cases.
With the results in the previous Example 4.3, we have shown that our methods
can ensure both conservation and robustness.
Figure 4.5: The numerical results of 1st, 2nd, and 3rd order schemes for
Example 4.4 with $\kappa=0,$ $\Delta t=\frac{h}{5},$ time $t=1$ and $t=2.$
“ref” represents the reference solutions. Top: $N=64$; Bottom: $N=128$. Left
column: radiation temperature $T_{r}$; Right column: material temperature $T$.
Figure 4.6: The numerical results of 1st, 2nd, and 3rd order schemes for
Example 4.4 with $\kappa=0.1,$ $\Delta t=\frac{h}{5},$ time $t=1$ and $t=1.5.$
“ref” represents the reference solutions. Top: $N=64$; Bottom: $N=128$. Left
column: radiation temperature $T_{r}$; Right column: material temperature $T$.
Figure 4.7: The numerical results of 1st, 2nd and 3rd order schemes for
Example 4.4 with $t=1,$ $\Delta t=\frac{h}{5},$ $N=64$. “ref” means the
reference solutions. Left column: radiation temperature $T_{r}$; Right column:
material temperature $T$. Top: $\kappa=0$; Bottom: $\kappa=0.1$. “c” and “nc”
denote numerical results with or without a corrector step, respectively.
###### Example 4.5.
In this example, we consider a heterogeneous Marshak problem [30, 45, 59] with
the initial and boundary conditions (3.13) with $z=3$ inside the interval
$[\frac{1}{3},\frac{2}{3}]$ and $z=1$ elsewhere. We take $\kappa=0.1$ and two
different times $t=1,3$. In Fig. 4.8, we show the results of 2nd and 3rd order
methods, on a mesh $N=180$ and the time step is taken to be $\Delta
t=\frac{1}{15}h$. The 1st order results are not presented since they deviate a
lot as shown above. From the numerical results, we can observe that the
radiation temperature $T_{r}$ and the material temperature propagate much
slower with a large value of $z$ in the middle region. Besides, the radiation
temperature $T_{r}$ is approaching the material temperature $T$ in the middle
region, reaching a thermodynamic equilibrium. In Fig. 4.9, we compare the
numerical results between the 2nd and 3rd order methods. “ref” represents the
reference solution obtained by the 1st order explicit RK LDG method on $1024$
elements with a small enough time step. We can find that the solutions of the
3rd order method match the reference better than the 2nd order ones,
especially for the radiation temperature. From this example, we can see that
high order methods can capture fine structures than corresponding lower order
methods.
Figure 4.8: The numerical results of 2nd and 3rd order schemes for Example 4.5
at time $t=1$ and $t=3$. $N=180$ and $\Delta t=\frac{h}{15}$. Left: 2nd order;
Right: 3rd order.
Figure 4.9: The comparison between the results of 2nd and 3rd order schemes
for Example 4.5 at time $t=1$ and $t=3$. Left: radiation temperature $T_{r}$;
Right: material temperature $T$. $N=180$ and $\Delta t=\frac{h}{15}$.
###### Example 4.6.
(Accuracy test in 2D) In this example we test the errors and convergence
orders of accuracy at the equilibrium in the 2D case. We consider a 2D
radiation diffusion problem with two given source terms $f_{1}(x,y,t)$ and
$f_{2}(x,y,t)$, which are chosen properly so that exact solutions are
available for the following system:
$\begin{cases}\dfrac{\partial E}{\partial t}-\nabla\cdot\left(D_{r}\nabla
E\right)=\sigma\left(T^{4}-E\right)+f_{1}(x,y,t),\\\\[5.0pt] \dfrac{\partial
T}{\partial t}-\nabla\cdot\left(D_{t}\nabla
T\right)=\sigma\left(E-T^{4}\right)+f_{2}(x,y,t),\end{cases}$ (4.4)
where
$T(x,y,t)=\left(0.8+0.1\sin\left(2\pi\left(x-t\right)\right)\right)\left(0.8+0.1\sin\left(2\pi\left(y-t\right)\right)\right)$
and $E(x,y,t)=T(x,y,t)^{4}$, $\Omega=[0,1]^{2}$. For convenience we remove the
artificially added term in the energy radiation coefficient, that is
$D_{r}=\frac{1}{3\sigma}$.
We take $\kappa=0.01$ and $z(x)=1$. The problem is run to time $t=0.5$ using
the 1st, 2nd, and 3rd order methods, respectively. We list the numerical
$L^{2}$ errors and orders of accuracy with different time step sizes in Table
4.4. Similar results as the 1D case are obtained.
Table 4.4: The numerical $L^{2}$ errors and orders of accuracy for the 1st, 2nd and 3rd order schemes with different time steps for Example 4.6. $t=0.5.$ | N | $\Delta t$ | $L^{2}$ error | order | $L^{2}$ error | order | $L^{2}$ error | order
---|---|---|---|---|---|---|---|---
| E | T | B
1st | $16\times 16$ | $\frac{h}{2}$ | 1.22e-2 | - | 1.02e-2 | - | 1.23e-2 | -
$32\times 32$ | 6.21e-3 | 0.98 | 5.17e-3 | 0.98 | 6.23e-3 | 0.98
$64\times 64$ | 3.13e-3 | 0.99 | 2.60e-3 | 0.99 | 3.14e-3 | 0.99
$16\times 16$ | $h$ | 1.85e-2 | - | 1.80e-2 | - | 2.10e-2 | -
$32\times 32$ | 9.54e-3 | 0.96 | 9.35e-3 | 0.95 | 1.08e-2 | 0.95
$64\times 64$ | 4.84e-3 | 0.98 | 4.77e-3 | 0.97 | 5.50e-3 | 0.98
$16\times 16$ | $2h$ | 3.67e-2 | - | 4.07e-2 | - | 4.48e-2 | -
$32\times 32$ | 1.89e-2 | 0.96 | 2.14e-2 | 0.93 | 2.33e-2 | 0.94
$64\times 64$ | 9.65e-3 | 0.97 | 1.10e-2 | 0.96 | 1.19e-2 | 0.97
2nd | $16\times 16$ | $\frac{h}{2}$ | 1.11e-3 | - | 1.04e-3 | - | 1.14e-3 | -
$32\times 32$ | 2.78e-4 | 2.00 | 2.63e-4 | 1.99 | 2.90e-4 | 1.98
$64\times 64$ | 6.93e-5 | 2.00 | 6.57e-5 | 2.00 | 7.27e-5 | 2.00
$16\times 16$ | $h$ | 2.56e-3 | - | 2.89e-3 | - | 2.51e-3 | -
$32\times 32$ | 6.33e-4 | 2.01 | 7.45e-4 | 1.95 | 6.40e-4 | 1.97
$64\times 64$ | 1.57e-4 | 2.01 | 1.88e-4 | 1.99 | 1.61e-4 | 1.99
$16\times 16$ | $2h$ | 9.68e-3 | - | 1.01e-2 | - | 8.91e-3 | -
$32\times 32$ | 2.34e-3 | 2.05 | 2.79e-3 | 1.86 | 2.33e-3 | 1.94
$64\times 64$ | 5.81e-4 | 2.01 | 7.22e-4 | 1.95 | 5.93e-4 | 1.97
3rd | $16\times 16$ | $\frac{h}{2}$ | 7.63e-5 | - | 2.40e-5 | - | 2.59e-5 | -
$32\times 32$ | 1.13e-5 | 2.75 | 3.26e-6 | 2.88 | 3.31e-6 | 2.97
$64\times 64$ | 1.59e-6 | 2.83 | 4.32e-7 | 2.91 | 4.22e-7 | 2.97
$16\times 16$ | $h$ | 4.56e-4 | - | 1.11e-4 | - | 1.23e-4 | -
$32\times 32$ | 7.17e-5 | 2.67 | 1.62e-5 | 2.78 | 1.50e-5 | 3.04
$64\times 64$ | 1.08e-5 | 2.73 | 2.37e-6 | 2.78 | 1.95e-6 | 2.94
$16\times 16$ | $2h$ | 2.80e-3 | - | 8.52e-4 | - | 1.11e-3 | -
$32\times 32$ | 4.55e-4 | 2.62 | 1.10e-4 | 2.95 | 1.21e-4 | 3.20
$64\times 64$ | 7.16e-5 | 2.67 | 1.61e-5 | 2.77 | 1.47e-5 | 3.04
###### Example 4.7.
Now we consider a 2D blast wave problem (1.1) with initial and boundary
conditions given by
$\begin{cases}E(x,y,0)=10^{-3}+100e^{-100((x-1)^{2}+(y-1)^{2})},\\\
T(x,y,0)=E(x,y,0)^{\frac{1}{4}},\\\ \frac{\partial
E}{\partial\bm{n}}|_{\partial\Omega}=0,\quad\frac{\partial
T}{\partial\bm{n}}|_{\partial\Omega}=0,\end{cases}$ (4.5)
where $\Omega=[0,1]\times[0,1]$, $\kappa=0.01$.
First we consider a homogeneous case with $z(x,y)=1$. In Fig. 4.10 and Fig.
4.11, we show the numerical solutions $T_{r}$ and $T$ at time $t=0.5,1.5,2$ on
a mesh of $64\times 64$, respectively. We take a time step $\Delta
t=\frac{1}{5}h$. From top to bottom, numerical results obtained by the 1st,
2nd, and 3rd order methods are presented, respectively. We can observe that
our methods can capture the sharp fronts of the blast wave well. Moreover, in
Fig. 4.12, we show the cutting plots for the numerical solutions along $y=x+1$
at time $t=2$ on a mesh of $N\times N$ for $N=64,128,512$. We can see that the
three methods converge and match each other. 2nd and 3rd order methods are
clearly better than the 1st order method. From zoom-in figures, we can see the
results of 3rd order method are slightly better than the 2nd order ones.
###### Example 4.8.
Finally we consider the problems (1.1)-(4.5) in a heterogeneous medium. The
atomic mass number $z(x,y)$ is $1$ everywhere except in two inner square
regions $\frac{3}{16}<x<\frac{7}{16}$, $\frac{9}{16}<y<\frac{13}{16}$ and
$\frac{9}{16}<x<\frac{13}{16}$, $\frac{3}{16}<y<\frac{7}{16}$, where the value
of $z(x,y)$ is $3$, as shown in Fig. 4.13. In Fig. 4.14 and Fig. 4.15, we
present the numerical solutions $T_{r}$ and $T$ at time $t=1,2,2.5$ on a mesh
of $64\times 64$, respectively. As compared to $\Delta t=\frac{1}{32}h$ in
[16, 62], we use a lager time step $\Delta t=\frac{1}{10}h$. From top to
bottom, numerical results obtained by the 1st, 2nd, and 3rd order methods are
presented, respectively. From the results, we find that the results of the 1st
order method are very smeared due to numerical viscosities. 2nd and 3rd order
methods perform better. In Fig. 4.16, we show the cutting plots for the
numerical solutions along $y=x+1$ at $t=2$ on a mesh of $N\times N$ for
$N=64,128,512$. We can observe that by a mesh refinement, the results of 2nd
and 3rd methods match each other. In addition, we observe that when the front
of the radiation temperature propagates to the interface of two different
materials, the process is hindered by a region with dense atoms. Higher energy
exchange $\sigma=z({\bf x})^{3}/T^{3}$ occurs in these regions, resulting in a
well balance between the material and radiation temperatures. The temperature
propagates faster in the lower $z({\bf x})$ region. Our results agree with
those in [57, 55].
###### Remark 4.
We are not able to prove the convergence of the Picard iteration for (3.9)
theoretically. For the 2nd and 3rd order methods in space, the matrix
$A(U^{(i)})$ may not be diagonally dominant, so that $A(U^{(i)})$ cannot be
guaranteed to be an $\mathbf{M}$ matrix. However, in our numerical examples,
all results converge quickly under a tolerance $\delta=10^{-8}$. In Table 4.5,
we show an average number of iterations per stage for the nonlinear algebraic
system (3.9) in the heterogeneous medium. We can observe that in such
challenging cases, our methods with large time steps converge around four
iterations, which show the robustness of our proposed methods.
Table 4.5: Average iteration numbers per stage for some examples in the heterogeneous medium. | | mesh | time step | an average number of iteration per stage
---|---|---|---|---
1D heterogeneous Marshak wave(Example 4.5) | 2nd | 180 | $\frac{1}{15}h$ | 4.02
3rd | 5.05
2D heterogeneous Blast wave(Example 4.8) | 1st | $64\times 64$ | $\frac{1}{10}h$ | 4.12
2nd | 3.99
3rd | 4.00
Figure 4.10: The numerical results for Example 4.7. From top to bottom: the
radiation temperature $T_{r}$ for 1st, 2nd, and 3rd order schemes. From left
to right: time $t=0.5,1.5,2.$ $\Delta t=\frac{1}{5}h$.
Figure 4.11: The numerical results for Example 4.7. From top to bottom: the
material temperature $T$ for 1st, 2nd, and 3rd order schemes. From left to
right: time $t=0.5,1.5,2.$ $\Delta t=\frac{1}{5}h$.
Figure 4.12: The numerical results for Example 4.7 at time $t=2$. Left: the
radiation temperature $T_{r}$; Right: the material temperature $T$. $\Delta
t=\frac{1}{5}h$.
$y$$x$01$\frac{3}{16}$$\frac{7}{16}$$\frac{9}{16}$$\frac{13}{16}$$\frac{3}{16}$$\frac{7}{16}$$\frac{9}{16}$$\frac{13}{16}$1313
Figure 4.13: The atomic mass number $z$ in the heterogeneous case for Example
4.8.
Figure 4.14: The numerical results for Example 4.8. From top to bottom: the
radiation temperature $T_{r}$ for 1st, 2nd, and 3rd order schemes. From left
to right: time $t=1.0,2.0,2.5.$ $\Delta t=\frac{1}{10}h$.
Figure 4.15: The numerical results for Example 4.8. From top to bottom: the
material temperature $T$ for 1st, 2nd, and 3rd order schemes. From left to
right: time $t=1.0,2.0,2.5.$ $\Delta t=\frac{1}{10}h$.
Figure 4.16: The numerical results for Example 4.8 at time $t=2$. Left: the
radiation temperature $T_{r}$; Right: the material temperature $T$. $\Delta
t=\frac{1}{10}h$.
## 5 Conclusion
In this paper, we have developed a class of high order conservative LDG-IMEX
methods for non-equilibrium radiation diffusion problems. The proposed scheme
is based on a predictor-corrector approach. During the predictor step, we
solve a reformulated system to improve convergence and approach thermodynamic
equilibrium. Subsequently, in the corrector step, we solve the original system
using nonlinear coefficients and initial estimates obtained from the predictor
step. This dual-step process ensures the conservation of total energy and
robust convergence. We incorporate linear diffusion terms to circumvent
implicit discretization for nonlinear diffusion terms. This approach permits
the use of larger time step sizes, in contrast to the restrictive parabolic
time step conditions $\Delta t=\mathcal{O}(h^{2})$ associated with purely
explicit discretizations. Local discontinuous Galerkin finite element
approximations in space are employed. Numerical examples in both 1D and 2D
illustrate the benefits of high-order conservative methods in accurately
capturing steep solution fronts within both homogeneous and heterogeneous
media. Our proposed methods exhibit robustness, as confirmed by numerical
experiments. However, theoretical analysis to ensure the convergence of such a
nonlinear iteration is a challenging task, which we plan to explore in our
future work. The extension to a 3D radiation diffusion model [25] or three-
temperature (3T) model [44, 56], and other reaction-diffusion equations [42]
will also be investigated.
## Appendix A IMEX Butcher tableau
In this paper, the double Butcher tableaux we used from [3] are listed below:
###### Tableau A.1.
2 stage 1st order:
0 | 0 | 0 | | 0 | 0 | 0
---|---|---|---|---|---|---
1 | 0 | 1 | | 1 | 1 | 0
| 0 | 1 | | | 1 | 0
.
###### Tableau A.2.
3 stage 2nd order:
0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0
---|---|---|---|---|---|---|---|---
$\gamma$ | 0 | $\gamma$ | 0 | | $\gamma$ | $\gamma$ | 0 | 0
1 | $\gamma$ | 0 | $\gamma$ | | 1 | 0 | 1 | 0
| $\gamma$ | 0 | $\gamma$ | | | 0 | 1 | 0
,
where $\gamma=\frac{1}{2}$.
###### Tableau A.3.
5 stage 3rd order:
0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 0
---|---|---|---|---|---|---|---|---|---|---|---|---
1/2 | 0 | 1/2 | 0 | 0 | 0 | | 1/2 | 1/2 | 0 | 0 | 0 | 0
2/3 | 0 | 1/6 | 1/2 | 0 | 0 | | 2/3 | 11/18 | 1/18 | 0 | 0 | 0
1/2 | 0 | -1/2 | 1/2 | 1/2 | 0 | | 1/2 | 5/6 | -5/6 | 1/2 | 0 | 0
1 | 0 | 3/2 | -3/2 | 1/2 | 1/2 | | 1 | 1/4 | 7/4 | 3/4 | -7/4 | 0
| 0 | 3/2 | -3/2 | 1/2 | 1/2 | | | 1/4 | 7/4 | 3/4 | -7/4 | 0
.
## References
* [1] T. Arbogast, C.-S. Huang, and X. Zhao. Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes. Journal of Computational Physics, 399:108921, 2019.
* [2] U. M. Ascher, S. J. Ruuth, and B. T. Wetton. Implicit-explicit methods for time-dependent partial differential equations. SIAM Journal on Numerical Analysis, 32:797–823, 1995.
* [3] U.M. Ascher, S.J. Ruuth, and R.J. Spiteri. Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25(2-3):151–167, 1997.
* [4] A. Berman and R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. SIAM, 1994.
* [5] Marianne Bessemoulin-Chatard and Francis Filbet. A finite volume scheme for nonlinear degenerate parabolic equations. SIAM Journal on Scientific Computing, 34(5):B559–B583, 2012.
* [6] S. Bingjing and G.L. Olson. Benchmark results for the non-equilibrium Marshak diffusion problem. Journal of Quantitative Spectroscopy and Radiative Transfer, 56(3):337–351, 1996.
* [7] S. Boscarino, L. Pareschi, and G. Russo. Implicit-explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit. SIAM Journal on Scientific Computing, 35(1):A22–A51, 2013.
* [8] R.L. Bowers and J.R. Wilson. Numerical Modeling in Applied Physics and Astrophysics. Jones and Bartlett Publishers, Inc., 1991.
* [9] P.N. Brown, D.E. Shumaker, and C.S. Woodward. Fully implicit solution of large-scale non-equilibrium radiation diffusion with high order time integration. Journal of Computational Physics, 204(2):760–783, 2005.
* [10] P. Castillo, B. Cockburn, D. Schötzau, and C. Schwab. Optimal a priori error estimates for the $hp$-version of the local discontinuous Galerkin method for convection-diffusion problems. Mathematics of Computation, 71(238):455–478, 2002.
* [11] B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Mathematics of Computation, 52(186):411–435, 1989.
* [12] B. Cockburn and C.-W. Shu. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis, 35(6):2440–2463, 1998.
* [13] B. Cockburn and C.-W. Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 16(3):173–261, 2001.
* [14] Irving R. Epstein and John A. Pojman. An introduction to nonlinear chemical dynamics. Oxford Academic, 1998.
* [15] S. Gottlieb and C.-W. Shu. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation, 67(221):73–85, 1998.
* [16] Z. Huang and Y. Li. Monotone finite point method for non-equilibrium radiation diffusion equations. BIT Numerical Mathematics, 56(2):659–679, 2016.
* [17] J. Jang, F. Li, J.-M. Qiu, and T. Xiong. Analysis of asymptotic preserving DG-IMEX schemes for linear kinetic transport equations in a diffusive scaling. SIAM Journal on Numerical Analysis, 52(4):2048–2072, 2014.
* [18] A.M. Jokisaari, P.W. Voorhees, J.E. Guyer, J. Warren, and O.G. Heinonen. Benchmark problems for numerical implementations of phase field models. Computational Materials Science, 126:139–151, 2017.
* [19] K.S. Kang. ${P}_{1}$ Nonconforming Finite Element Multigrid Method for Radiation Transport. SIAM Journal on Scientific Computing, 25(2):369–384, 2003.
* [20] D.I. Ketcheson, C.B. Macdonald, and S. Gottlieb. Optimal implicit strong stability preserving Runge–Kutta methods. Applied Numerical Mathematics, 59(2):373–392, 2009.
* [21] D.A. Knoll, L. Chacon, L.G. Margolin, and V.A. Mousseau. On balanced approximations for time integration of multiple time scale systems. Journal of Computational Physics, 185(2):583–611, 2003.
* [22] D.A. Knoll, R.B. Lowrie, and J.E. Morel. Numerical analysis of time integration errors for nonequilibrium radiation diffusion. Journal of Computational Physics, 226(2):1332–1347, 2007.
* [23] D.A. Knoll, W.J. Rider, and G.L. Olson. An efficient nonlinear solution method for non-equilibrium radiation diffusion. Journal of Quantitative Spectroscopy and Radiative Transfer, 63(1):15–29, 1999.
* [24] D.A. Knoll, W.J. Rider, and G.L. Olson. Nonlinear convergence, accuracy, and time step control in nonequilibrium radiation diffusion. Journal of Quantitative Spectroscopy and Radiative Transfer, 70(1):25–36, 2001.
* [25] X. Lai, Z. Sheng, and G. Yuan. Monotone finite volume scheme for three dimensional diffusion equation on tetrahedral meshes. Communications in Computational Physics, 21(1):162–181, 2017.
* [26] E.W. Larsen, G.C. Pomraning, and V.C. Badham. Asymptotic analysis of radiative transfer problems. Journal of Quantitative Spectroscopy and Radiative Transfer, 29:285–310, 1983.
* [27] Shu Liu, Siting Liu, Stanley Osher, and Wuchen Li. A first-order computational algorithm for reaction-diffusion type equations via primal-dual hybrid gradient method. Journal of Computational Physics, 500:112753, 2024.
* [28] Yuanyuan Liu, Chi-Wang Shu, and Mengping Zhang. High order finite difference WENO schemes for nonlinear degenerate parabolic equations. SIAM Journal on Scientific Computing, 33(2):939–965, 2011.
* [29] R.B. Lowrie. A comparison of implicit time integration methods for nonlinear relaxation and diffusion. Journal of Computational Physics, 196(2):566–590, 2004.
* [30] R.E. Marshak. Effect of radiation on shock wave behavior. The Physics of Fluids, 1(1):24–29, 1958.
* [31] V.A. Mousseau and D.A. Knoll. New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion. Journal of Computational Physics, 190(1):42–51, 2003.
* [32] V.A. Mousseau and D.A. Knoll. Temporal accuracy of the nonequilibrium radiation diffusion equations applied to two-dimensional multimaterial simulations. Nuclear Science and Engineering, 154(2):174–189, 2006.
* [33] V.A. Mousseau, D.A. Knoll, and W.J. Rider. Physics-based preconditioning and the Newton–Krylov method for non-equilibrium radiation diffusion. Journal of Computational Physics, 160(2):743–765, 2000.
* [34] J. D. Murray. Mathematical Biology II: Spatial Models and Biomedical Applications, volume 3rd Edition. Springer New York, 2003.
* [35] C.C. Ober and J.N. Shadid. Studies on the accuracy of time-integration methods for the radiation–diffusion equations. Journal of Computational Physics, 195(2):743–772, 2004.
* [36] D.L. Peterson, R.L. Bowers, J.H. Brownell, A.E. Greene, K.D. McLenithan, T.A. Oliphant, N.F. Roderick, and A.J. Scannapieco. Two-dimensional modeling of magnetically driven Rayleigh-Taylor instabilities in cylindrical Z pinches. Physics of Plasmas, 3(1):368–381, 1996.
* [37] G.C. Pomraning. The non-equilibrium Marshak wave problem. Journal of Quantitative Spectroscopy and Radiative Transfer, 21(3):249–261, 1979.
* [38] A.C. Robinson and C.J. Garasi. Three-dimensional z-pinch wire array modeling with ALEGRA-HEDP. Computer Physics Communications, 164(1-3):408–413, 2004.
* [39] Y. Saad and M.H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on scientific and Statistical Computing, 7(3):856–869, 1986.
* [40] Z. Sheng, J. Yue, and G. Yuan. Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes. SIAM Journal on Scientific Computing, 31(4):2915–2934, 2009.
* [41] Y. Shi, H. Yong, C. Zhai, J. Qi, and P. Song. A functional expansion tally method for gray radiative transfer equations in implicit Monte Carlo. Journal of Computational and Theoretical Transport, 47(7):581–598, 2018.
* [42] J. Smoller. Shock Waves and Reaction—Diffusion Equations, volume 258. Springer Science & Business Media, 2012.
* [43] L. Spitzer Jr and R. Härm. Transport phenomena in a completely ionized gas. Physical Review, 89(5):977, 1953.
* [44] S. Su and J. Wu. A vertex-centered and positivity-preserving finite volume scheme for two-dimensional three-temperature radiation diffusion equations on general polygonal meshes. Numerical Mathematics: Theory, Methods and Applications, 13(1):220–252, 2020.
* [45] M. Tang, L. Wang, and X. Zhang. Accurate front capturing asymptotic preserving scheme for nonlinear gray radiative transfer equation. SIAM Journal on Scientific Computing, 43(3):B759–B783, 2021.
* [46] M. Tang and X. Zhang. Semi-implicit front capturing schemes for the degenerate nonlinear radiative diffusion equation. Journal of Computational Physics, 436:110290, 2021.
* [47] N.J. Turner and J.M. Stone. A module for radiation hydrodynamic calculations with ZEUS-2D using flux-limited diffusion. The Astrophysical Journal Supplement Series, 135(1):95, 2001.
* [48] H. Wang, C.-W. Shu, and Q. Zhang. Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems. SIAM Journal on Numerical Analysis, 53(1):206–227, 2015.
* [49] H. Wang, Q. Zhang, S. Wang, and C.-W. Shu. Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems. Science China Mathematics, 63(1):183–204, 2020.
* [50] A.M. Winslow. Multifrequency-gray method for radiation diffusion with Compton scattering. Journal of Computational Physics, 117(2):262–273, 1995.
* [51] T. Xiong, W. Sun, Y. Shi, and P. Song. High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations. Journal of Computational Physics, page 111308, 2022.
* [52] Y. Xu and C.-W. Shu. Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Communications in Computational Physics, 7(1):1, 2010.
* [53] J. Yan and C.-W. Shu. Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. Journal of Scientific Computing, 17(1):27–47, 2002.
* [54] X. Yang, W. Huang, and J. Qiu. A moving mesh finite difference method for equilibrium radiation diffusion equations. Journal of Computational Physics, 298:661–677, 2015.
* [55] X. Yang, W. Huang, and J. Qiu. Moving mesh finite difference solution of non-equilibrium radiation diffusion equations. Numerical Algorithms, 82(4):1409–1440, 2019.
* [56] Y. Yu, X. Chen, and G. Yuan. A finite volume scheme preserving maximum principle for the system of radiation diffusion equations with three-temperature. SIAM Journal on Scientific Computing, 41(1):B93–B113, 2019.
* [57] G. Yuan, X. Hang, Z. Sheng, and J. Yue. Progress in numerical methods for radiation diffusion equations. Chinese Journal of Computational Physics, 26(4):475, 2009.
* [58] Peng Zhang and Tao Xiong. High order implicit finite difference schemes with a semi-implicit WENO reconstruction for nonlinear degenerate parabolic equations. Journal of Computational Physics, 467:111442, 2022.
* [59] Q. Zhang and Z.-L. Wu. Numerical simulation for porous medium equation by local discontinuous Galerkin finite element method. Journal of Scientific Computing, 38(2):127–148, 2009.
* [60] R. Zhang, X. Yu, X. Cui, and T. Feng. Discontinuous finite element method for 1D non-equilibrium radiation diffusion equations. Chinese Journal of Computational Physics, 29(5):641, 2012.
* [61] Rongpei Zhang, Xijun Yu, Jiang Zhu, and Abimael F.D. Loula. Direct discontinuous galerkin method for nonlinear reaction–diffusion systems in pattern formation. Applied Mathematical Modelling, 38:1612–1621, 2014.
* [62] X. Zhao, Y. Chen, Y. Gao, C. Yu, and Y. Li. Finite volume element methods for nonequilibrium radiation diffusion equations. International Journal for Numerical Methods in Fluids, 73(12):1059–1080, 2013.
* [63] Jianfeng Zhu, Yong-Tao Zhang, Stuart A. Newman, and Mark Alber. Application of discontinuous galerkin methods for reaction-diffusion systems in developmental biology. Journal of Scientific Computing, 40:391–418, 2009.
|
This paper studies the performative prediction problem which optimizes a stochastic loss function with data distribution that depends on the decision variable. We consider a setting where the agent(s) provides samples adapted to the learner's and agent's previous states. The said samples are used by the learner to optimize a loss function. This closed loop algorithm is studied as a state-dependent stochastic approximation (SA) algorithm, where we show that it finds a fixed point known as the performative stable solution. Our setting models the unforgetful nature and the reliance on past experiences of agent(s). Our contributions are three-fold. First, we demonstrate that the SA algorithm can be modeled with biased stochastic gradients driven by a controlled Markov chain (MC) whose transition probability is adapted to the learner's state. Second, we present a novel finite-time performance analysis of the state-dependent SA algorithm. We show that the expected squared distance to the performative stable solution decreases as ${\cal O}(1/k)$, where $k$ is the iteration number. Third, numerical experiments are conducted to verify our findings.
§ INTRODUCTION
Many supervised learning algorithms are built around the assumption that learners can obtain samples from a static distribution independent of the state of the learner and/or the agent who provides the sample. This assumption is reasonable for static tasks such as image classification. Oftentimes, it simplifies the design and analysis of algorithms such as stochastic gradient methods.
On the other hand, in certain applications agents can be performative where the samples are drawn from a decision-dependent distribution. This is relevant to the framework of strategic classification [Hardt et al., 2016, Cai et al., 2015, Kleinberg and Raghavan, 2020]. For instance, when training a classifier for loan applications, given the classifier published by the learner (bank), as the best response strategy, the agent(s) (loan applicants) may manipulate their profile prior to the submission, e.g., by spending more with credit cards, making unnatural transactions, etc., in order to increase their chance of successful application. The latter manipulation effectively shifts the data distribution and may affect the convergence properties, or even the stability of learning algorithms.
Earlier works [Bartlett, 1992, Quiñonero-Candela et al., 2009] studied the effects of exogenous changes with shifts in data distribution.
Recently, [Perdomo et al., 2020] considered the convergence properties of learning algorithms when they are agnostic to the shifted distributions. Specifically, the learner is interested in the following performative prediction problem:
min_ θ∈^d V(θ) = _ z ∼D(θ) [ ℓ( θ; z ) ],
$\ell( \theta; z )$ is the loss function given the sample $z \in {\sf Z}$. The loss function is strongly-convex with respect to (w.r.t.) the parameter $\theta \in \RR^d$, and the gradient map $\grd_\theta \ell( \theta; z )$ is Lipschitz continuous w.r.t. $z$, $\theta$. In addition, the distribution ${\cal D}(\theta)$ on ${\sf Z}$ is parameterized by the decision vector $\theta$, which captures the distribution shift due to the learner's state. In other words, problem (<ref>) finds a parameter $\theta$ which minimizes the expected loss that takes care of the decision-dependent distribution.
Despite the strong convexity of $\ell(\theta ;z)$, problem (<ref>) is non-convex in general due to the coupling with $\theta$ in the data distribution ${\cal D}(\theta)$. As a remedy, [Perdomo et al., 2020] studied population-based algorithms that converge to a performative stable point, $\theta_{PS}$, which is a fixed point to the system $\theta = \argmin_{ \theta' \in \RR^d } \EE_{ z \sim {\cal D}(\theta) } \big[ \ell( \theta'; z ) \big]$.
Along the same line, [Mendler-Dünner et al., 2020] analyzed stochastic algorithms which deploy minibatches of i.i.d. samples from the shifted distribution at each iteration, [Izzo et al., 2021, Miller et al., 2021] studied gradient estimation techniques and developed algorithms that converge to an optimal solution of (<ref>) through introducing a gradient correction term (also see [Munro, 2020] which has considered a related setting), [Drusvyatskiy and Xiao, 2020] studied the stability of proximal stochastic gradient algorithms (and their variants) in the performative prediction setting, [Brown et al., 2020] studied population-based algorithms where the state-dependent distribution is updated iteratively.
This paper studies the convergence of stochastic algorithms where only one sample (or a minibatch of samples) is required at each iteration. Specifically, we consider a greedy deployment scheme similar to [Mendler-Dünner et al., 2020] where the learner deploys the most recent model after each update round. Moreover, the agent(s) is modeled with a memory property such that the update of his or her state depends on the past state. The closed-loop algorithm can be studied as a state-dependent stochastic approximation (SA) algorithm. In contrast to the setting analyzed in [Mendler-Dünner et al., 2020, Drusvyatskiy and Xiao, 2020] where only the learner's state is incrementally updated and the agent draws i.i.d. samples from the distribution shifted by the learner's state, the agent's state evolves according to a controlled Markov chain (MC) whose stationary distribution is the shifted distribution.
Our study is motivated by the stateful (or unforgetful) nature of the agents who depend on past experiences when adapting to a shifted target data distribution. For example, a loan applicant may take months to build up his/her credit history to adapt to changes in the published classifier. Several questions naturally arise from such dynamical performative prediction problems: will the stochastic algorithm converge to a performative stable point similar to [Mendler-Dünner et al., 2020, Drusvyatskiy and Xiao, 2020]? what is the sample complexity? This paper addresses these questions as we make the following contributions:
* We develop a fully state-dependent performative prediction framework which extends the analysis in [Mendler-Dünner et al., 2020, Drusvyatskiy and Xiao, 2020]. The proposed extension relies on a state-dependent stochastic approximation (SA) algorithm with noise originating from a controlled Markov chain [cf. Algorithm 1].
* Our main result consists of a finite-time convergence analysis of the state-dependent SA algorithm under a setting which does not assume the iterates to be bounded a-priori. Previous works either assumed the latter condition a-priori (e.g., [Benveniste et al., 2012]), or they require a compact constraint set (e.g., [Atchadé et al., 2017]). Using a novel analysis, we show that the mean squared error between the SA iterates and the unique performative stable solution [cf. (<ref>)] converges at a rate of ${\cal O}(1/k)$ in expectation. Additionally, we discuss about the convergence to an approximate stationary point of (<ref>) when the loss function $\ell(\theta;z)$ is not strongly convex (possibly non-convex).
* We demonstrate the efficacy of the SA algorithm with several experiments. We show that it has a comparable performance as in [Mendler-Dünner et al., 2020] which assumes an ideal setting with i.i.d. samples taken from the shifted distribution.
The rest of this paper is organized as follows. <ref> formally describes the performative prediction problem and a state-dependent SA algorithm for tackling the problem, <ref> presents the main theoretical results for the convergence of the state dependent SA algorithm, <ref> gives an overview of the proof strategy, and <ref> presents the numerical experiments.
Related Works Analysis for state-dependent stochastic approximation (SA) algorithms with controlled MC, which extend over the classical SA [Robbins and Monro, 1951], has been considered in a number of works. Benveniste et al., 2012, Kushner and Yin, 2003 studied the asymptotic convergence of such algorithms, also see [Tadić et al., 2017] which analyzed the case of biased SA.
Recent works have analyzed the finite-time performance of state-dependent SA algorithms that are related to ours. Atchadé et al., 2017 considered a proximal SA algorithm where the proximal function has a compact domain; Karimi et al., 2019 analyzed the plain SA algorithm without projection but assumed that the updates are bounded; Sun et al., 2018, Doan et al., 2020 studied SA algorithms with a static MC not suitable for performative prediction.
Our analysis relaxes the restrictions of these prior works and focuses on convergence to a performative stable solution unique to (<ref>) [cf. see (<ref>)].
Lastly, our analysis technique is related to the recent endeavors on obtaining finite time bounds for reinforcement learning (RL) algorithms. Notice that (<ref>) can in fact be regarded as a special case of policy optimization [Sutton and Barto, 2018]. To this end, recent works [Wu et al., 2020, Xu et al., 2020, Zhang et al., 2020] studied the sample complexity of actor critic algorithms in finding a stationary point of an average reward function, where controlled MCs are considered. In comparison to our analysis, the latter works have only studied the convergence to a stationary point of a simple cost function.
Notations We denote ${\sf Z}$ as the state space of samples provided by the agent(s) and ${\cal Z}$ is a $\sigma$-algebra on ${\sf Z}$. A Markov transition kernel is a map given by $\MK : {\sf Z} \times {\cal Z} \rightarrow \RR_+$. At the state $z \in {\sf Z}$, the next state is drawn as $z' \sim \MK( z, \cdot )$. It holds for any measurable function $f : {\sf Z} \rightarrow \RR$ that $\EE[ f(z') | z ] = \int_{\sf Z} f(z') \MK( z, {\rm d}\!~z') =: \MK f(z)$. Unless otherwise specified, the operator $\grd$ takes the gradient of a function w.r.t. the first argument for a multivariate function, e.g., $\grd \ell( \theta; z )$ denotes the gradient of $\ell(\theta; z)$ taken w.r.t. $\theta$. For $x, y \in \RR^d$, we denote the inner product as $\pscal{x}{y} = x^\top y$.
§ STATE-DEPENDENT PERFORMATIVE PREDICTION
Due to the intractability of Problem (<ref>), we are interested in evaluating the performative stable (PS) solution:
θ_PS = _ θ∈^d _ z ∼D(θ_PS) [ ℓ( θ; z ) ].
Note that the expectation is taken with respect to $z \sim {\cal D}( \theta_{PS})$. It is known that $\theta_{PS}$ is in general different from an optimal or stationary solution to (<ref>); see <cit.>. Instead, $\theta_{PS}$ is a fixed point solution to the procedure when the learner repeatedly update the parameter $\theta$ with the drifted data distribution provided by the agent.
We consider a state dependent stochastic approximation (SA) algorithm motivated by the stateful nature of agents.
The latter is modeled using a controlled Markov chain.
For any $\theta$, we define a Markov transition kernel $\MK_{\theta}$ which induces a Markov chain with a unique stationary distribution $\pi_{\theta}(\cdot)$ such that $\EE_{z \sim \pi_{\theta}(\cdot)} [ \grd \ell(\theta; z) ] = \EE_{z' \sim {\cal D}(\theta)} [ \grd \ell( \theta ; z' )]$.
The learner and agent interact through the following rules:
Algorithm 1: State-dependent SA
Input: initialization $\theta_0$, step sizes $\{ \gamma_k \}_{k \geq 0}$.
For $k=0,1,2, \ldots$
Agent draws $z_{k+1} = {\sf SAMPLE}( \theta_k, z_k )$.
Learner updates
θ_k+1=θ_k-γ_k+1 ℓ(θ_k; z_k+1),
and deploys $\theta_{k+1}$.
${\sf SAMPLE}(\theta , z )$:
Draw the next sample as
z' ∼_ θ ( z, ·),
where $\MK_{\theta} : {\sf Z} \times {\cal Z} \rightarrow \RR_+$ is a Markov transition kernel.
Output: $z' \in {\sf Z}$.
We observe that (<ref>) is a standard SA recursion based on $\grd \ell(\theta_{k}; z_{k+1})$, where the learner deploys the most recent model $\theta_k$ and takes the sample $z_{k+1}$ directly from the agent.
Specifically, we consider a state-dependent setting where the sampling of $z_{k+1}$ can be affected by both the learner's and agent's states. Formally, the state-dependency is captured by modeling the samples sequence $\{z_k\}_{k \geq 1}$ as a controlled MC in (<ref>).
Notice that the stationary distribution $\pi_{\theta}(\cdot)$ does not need to be the same as ${\cal D}(\theta)$ as long as the former yields an unbiased gradient. However, for simplicity, we assume $\pi_{\theta}(\cdot) \equiv {\cal D}(\theta)$ for any $\theta \in \RR^d$.
The greedy deployment scheme studied by [Mendler-Dünner et al., 2020] assumed that the agents draw $z_{k+1} {\sim} {\cal D}( \theta_k )$ as independent samples. The latter implies that $\grd \ell( \theta_k; z_{k+1})$ is an unbiased estimator of $\EE_{z' \sim {\cal D}(\theta_k)} [ \grd \ell( \theta_k; z' )]$. As a significant departure, in Algorithm 1, the stochastic gradient $\grd \ell( \theta_k; z_{k+1})$ is a biased estimator for $\EE_{z' \sim {\cal D}(\theta_k)} [ \grd \ell( \theta_k; z' )]$.
For any $\theta \in \RR^d$, we observe
[ ℓ( θ; z_k+1 ) | z_k ] = _θ_k ℓ( θ; z_k ) = ∫_Z ℓ( θ; z ) _θ_k ( z_k, d z ) .
Since $\MK_{\theta_k} ( z_k, \cdot ) \neq \pi_{\theta_k}(\cdot)$, we have $\EE[ \grd \ell ( \theta; z_{k+1} ) \!~|\!~ z_k ] \neq \EE_{z' \sim {\cal D}(\theta_k)} [ \grd \ell( \theta_k; z' )]$.
Under the setting (<ref>) with restricted access to the shifted data distribution ${\cal D}(\theta_k)$, one possibility to obtain an unbiased gradient estimate
is to hold $\theta_k$ as fixed and repeat the sampling process $z' = {\sf SAMPLE}(\theta_k , z)$ indefinitely. In this case, we have the unbiased estimate $\lim_{ n \rightarrow \infty } \MK_{\theta_k}^n {\grd \ell}( \theta ; z ) = \EE_{z' \sim {\cal D}(\theta_k)} [ \grd \ell(\theta; z') ]$ for any initial agent state $z \in {\sf Z}$.
Instead of searching for an unbiased gradient estimator through repeated MC updates, the state-dependent SA algorithm uses the instantaneous samples from (<ref>) that are co-evolving with the learner's iterate in (<ref>).
§.§ Example of Controlled Markov Chain (<ref>)
The model (<ref>) captures the stateful and stochastic nature of the agent as the sample $z_{k+1}$ depends on the previous one $z_k$. Concretely, our study is motivated by the following application example which satisfies $\pi_{\theta}(\cdot) \equiv {\cal D}(\theta)$ (a case where $\pi_{\theta}(\cdot) \neq {\cal D}(\theta)$ will be discussed in Appendix <ref>):
[Strategic Classification with Adapted Best Response] We consider the problem of strategic classification [Hardt et al., 2016] involving some agents and a learner. In a typical scenario, the agent provides the best-response (i.e., optimized) samples upon knowing the current learner's state $\theta_k$. Ideally, the sample $z_{k+1} \sim {\cal D}(\theta_k)$ shall be drawn as
z_k+1 ∈_ z' ∈Z U( z'; z̃_k+1 , θ_k ), z̃_k+1 ∼D_0,
where ${\cal D}_0$ is the base distribution and $U( z'; z, \theta )$ is strongly concave in $z'$ for any $(z,\theta)$. The best response $\max_{ z' \in {\sf Z} }U( z'; z , \theta )$ perturbs the base sample $z$ in favor of the agent.
In practice, the exact maximization in (<ref>) is not obtained unless the agent(s) are given sufficient time to respond to the learner's state. Instead, we consider a setting where the agent(s) improve their responses via a gradient ascent dynamics evolving simultaneously with the learner.
Concretely, consider a setting where the learning problem (<ref>) utilizes data provided by $m$ agents.
Let ${\cal D}_0$ be the empirical distribution of $m$ data points $\{ \bar{d}_1,..., \bar{d}_m \}$, where $\bar{d}_i \in {\sf Z}$ is the initial data held by agent $i$.
At iteration $k$, a subset of agents ${\cal I}_k \subset \{1,...,m\}$ (selected uniformly with $|{\cal I}_k| = pm$, $p \in (0,1]$) becomes aware of the learner's state and they search for the best response through a gradient descent update. Then, the learner selects uniformly an agent $i_k \in \{1,...,m\}$ and requests the data sample $z_{k+1}$ from him/her.
In summary, the inexact best response dynamics follows:
Step 1: d_i^k+1 =
d_i^k + αU( d_i^k; d̅_i, θ_k ), i ∈I_k,
d_i^k, i ∉I_k,
Step 2: z_k+1 = d_i_k^k+1 ,
where the gradient is taken w.r.t. the first argument of $U( \cdot )$, $\alpha > 0$ is the agents' response rate (stepsize).
For the initialization, we set $d_i^0 = \bar{d}_i$ for all $i=1,...,m$.
The above dynamics highlights the stateful nature of the agents as they improve the responses based on their past experiences.
The best response dynamics executed by the agent(s) in (<ref>) leads naturally to a controlled MC (<ref>). Specifically, the MC's state is given by the tuple $\hat{z}_k = \{ d_1^k, ..., d_m^k , z_k \}$ and the application of the Markov kernel $\MK_{\theta_k}$ to $\hat{z}_k$ yields the inexact best response dynamics (<ref>). Furthermore, the latter admits a stationary distribution where
$\lim_{n \rightarrow \infty} \MK_{\theta_k}^n \grd \ell( \theta; \hat{z} ) = \EE_{ z \sim {\cal D}(\theta_k)} [ \grd \ell( \theta; z ) ]$.
We provide detailed properties about the controlled MC in Appendix <ref>. $\square$
The analysis of our state-dependent SA algorithm (<ref>), (<ref>) entails unique challenges not found in the literature. First, the agent's states $\{ z_k \}_{k \geq 1}$ form a controlled MC whose transition probabilities are changing according to the learner's states $\{ \theta_k \}_{k \geq 0}$. Second, a plain SA update is used in (<ref>) which does not require a compact constraint set as in the prior works such as [Atchadé et al., 2017]. In fact, $\theta_k$ can be unbounded when the step size is not carefully selected.
§ MAIN RESULTS
Let us define the following shorthand notations
f( θ_1 ; θ_2 ) = _z ∼D(θ_2) [ ℓ( θ_1; z) ], f(θ_1;θ_2) = _ z ∼D(θ_2) [ ℓ( θ_1 ; z) ],
where the first argument $\theta_1$ controls the loss function value and the second argument $\theta_2$ controls the distribution shift.
Notice that $\grd f(\theta_{PS}; \theta_{PS}) = 0$. We consider the following assumptions. First, the learner's loss is strongly convex in $\theta$, and its gradient map is Lipschitz continuous in $(\theta,z)$, i.e.,
For each $z \in {\sf Z}$, there exists $\mu>0$ such that
ℓ( θ; z ) ≥ℓ( θ'; z ) + ℓ( θ'; z ) θ- θ' + (μ/2) θ' - θ^2, ∀ θ, θ' ∈^d.
There exists $L \geq 0$ such that
ℓ( θ; z ) - ℓ( θ' ; z' ) ≤L { θ- θ' + z - z' }, ∀ θ, θ' ∈^d, z,z' ∈Z.
Notice that as a consequence, the expected objective function $f(\theta_1, \theta_2)$ and gradient $\grd f(\theta_1; \theta_2)$ are $\mu$-strongly convex in $\theta_1$, and $L$-Lipschitz in $\theta_1$, respectively. These are standard assumptions in the optimization literature. As indicated by [Drusvyatskiy and Xiao, 2020], these conditions are necessary for finding a performative stable solution in (<ref>).
Second, we have the following assumption on the oscillation of the stochastic gradient $\grd \ell(\theta;z)$:
There exists $\sigma \geq 0$ such that
sup_ z ∈Z ℓ( θ; z ) - f( θ; θ_PS ) ≤σ( 1 + θ- θ_PS ), ∀ θ∈^d.
The above is slightly stronger than the assumptions on second order moments typically found in the stochastic gradient literature, e.g., [Bottou et al., 2018], as we require a uniform bound on the gradient noise. This condition is common for the algorithms using Markovian samples [Sun et al., 2018, Srikant and Ying, 2019, Karimi et al., 2019], which requires that the oscillation of stochastic gradient is controlled. For strategic classicaition problems, it is satisfied for the finite dataset setting in <Ref>.
Moreover, similar to [Doan et al., 2020], this bound is adapted to the growth of $\| \theta - \theta_{PS} \|$ which is compatible with the strong convexity of the loss function $\ell(\theta; z)$.
Our next set of assumptions pertain to the Markov kernels $\MK_{\theta} $ that generate $\{ z_k \}_{k \geq 1}$:
There exists a solution $\widehat{\grd \ell}: \RR^d \times {\sf Z} \rightarrow \RR^d$ to the Poisson equation:
ℓ( θ'; z) - f( θ'; θ) = ℓ (θ'; z) - _θ ℓ ( θ'; z), ∀ θ, θ' ∈^d, z ∈Z.
Consider the Poisson equation's solution $\widehat{\grd \ell} (\cdot; \cdot )$ defined in <Ref>. There exists $\Lph \geq 0$ such that
sup_ z ∈Z _θ ℓ( θ; z) - _θ' ℓ (θ'; z) ≤ θ- θ' , ∀ θ, θ' ∈^d.
For Assumption <ref>, the existence of $\widehat{\grd \ell}$ in (<ref>) holds under mild assumptions on the Markov chains (MCs). For instance, it holds when the MC is irreducible and aperiodic, and satisfying a Lyapunov drift condition, or in the simpler case, when the MC is uniform geometrically ergodic, see <cit.>.
In the case for performative prediction with stateful agents, the above condition holds when repeated applications of the iterative map (<ref>) where agents adopt their data to the current learner's model $\theta$ converges linearly to the best response.
Assumption <ref> is a smoothness condition on the kernel $\MK_\theta$ with respect to $\theta$. The condition can be satisfied when the Markov kernel is only slightly modified when $\theta$ is perturbed.
<Ref> is also linked to our next assumption which is central to the study of performative prediction. Particularly, we require the distribution map $\cal D(\theta)$ to be $\epsilon$-sensitive w.r.t. $\theta$:
There exists $\epsilon \geq 0$ such that
W_1( D(θ), D(θ') ) =
inf_ J ∈J( D(θ), D(θ') )
_ (z,z') ∼J [ z - z' _1 ]
≤ϵ θ- θ' ,
for any $\theta, \theta' \in \RR^d$. Notice that $W_1(\cdot)$ denotes the Wasserstein-1 distance and ${\cal J}( {\cal D}(\theta), {\cal D}(\theta') )$ is the set of all joint distributions on ${\sf Z} \times {\sf Z}$ with ${\cal D}(\theta), {\cal D}(\theta')$ as its marginal distribution.
The above is a common condition for performative prediction, e.g., [Perdomo et al., 2020]. Intuitively, it allows for performative prediction algorithm to behave stably as the perturbation to ${\cal D}(\theta)$ is under control. In the subsequent analysis, we demonstrate that carefully controlling the step size in relation to $\mu, \epsilon, L$ is crucial to the convergence of Algorithm 1.
Before presenting our main result, we notice that Assumptions <ref>, <ref>, <ref> imply that there exists constants $\overline{L}$, $\widehat{L} > 0$ such that for any $z \in {\sf Z}$, $\theta \in \RR^d$,
ℓ( θ; z) ≤L (1 + θ- θ_PS ), max{ ℓ( θ; z) , _θ ℓ( θ; z ) } ≤(1 + θ- θ_PS ),
In other words, $\grd \ell(\theta; z)$, $\widehat{\grd \ell}(\theta; z)$, $\MK_\theta \widehat{\grd\ell}( \theta; z )$ are all locally bounded functions.
Notice that $\overline{L}$ is proportional to $\sigma$ in Assumption <ref>, while $\widehat{L}$ is proportional to the maximum mixing time of the Markov chain induced by the kernel $\MK_\theta$ over all $\theta \in \RR^d$.
Our main result for the state-dependent SA algorithm is summarized as follows:
Under Assumptions <ref>–<ref>. Suppose that the problem parameters satisfy $\epsilon < \frac{\mu}{L}$, the step sizes $\{\gamma_{k}\}_{k\geq 1}$ are non-increasing and satisfy for any $k \geq 1$,
γ_k-1/γ_k≤1+ γ_k ( μ - L ϵ)/ 4 , γ_k ≤min{ μ- L ϵ/2L^2, μ- L ϵ/2 , min{ (μ- L ϵ)/3, 3 } / + 3 ( μ- L ϵ) , 1/6 }.
Then for any $k \geq 1$, the expected distance between $\theta_{k}$ and the performative stable solution $\theta_{PS}$ satisfies
[ θ_k - θ_PS ^2 ] ≤∏_i=1^k ( 1 - γ_i μ- L ϵ/2 ) θ_0 - θ_PS ^2 + C γ_k ,
where $\EE[\cdot]$ is the expectation taken over all the randomness in (<ref>), (<ref>), and we have defined:
C :=
3 Δ + 4 ς/ μ- L ϵ ( 2 (2 σ^2 + ) + (μ- L ϵ) + ( + 3 (μ- L ϵ) ) Δ ) ,
with $\varsigma := 1 + \gamma_1 ( \mu - L \epsilon ) / 4$, and $\Cone, \Ctwo, \Cthree$, $\overline{\Delta}$ are constants defined in (<ref>), (<ref>), respectively.
The above establishes the finite-time convergence of the studied state-dependent SA algorithm (<ref>), (<ref>). To understand this result, we observe that the step size conditions in (<ref>) can be satisfied by a variety of step size schedules. For instance, it can be satisfied by the constant step size $\gamma_k \equiv \gamma$; and the diminishing step size $\gamma_{k}=a_{0}/(a_{1}+k) = {\cal O}(1/k)$ with appropriate $a_0, a_1 > 0$.
Moreover, we require the SA algorithm to work in the regime when $\epsilon < \mu / L$. This is similar to [Perdomo et al., 2020] which ensures that the solution $\theta_{PS}$ is a stable fixed point to (<ref>).
The main result is stated on the expected squared distance of $\| \theta_k - \theta_{PS} \|^2$ in (<ref>). Here, the bound consists of a transient term and a fluctuation term. The transient term decays sub-exponentially as ${\cal O}( \exp( - \frac{\mu-L\epsilon}{2} \sum_{i=1}^k \gamma_i ) )$ and is scaled by the initial gap $\| \theta_0 - \theta_{PS} \|^2$. The fluctuation term is in the order of ${\cal O}(\gamma_k)$ and is scaled by $\mathds{C}$ which depends on the oscillation of stochastic gradient (via $\sigma$) and the mixing time of the controlled Markov chain (via $\LZ$).
With a diminishing step size schedule such as $\gamma_{k}= c_{0}/( c_{1}+k)$, <Ref> shows that the state dependent SA algorithm finds the performative stable solution $\theta_{PS}$ at the rate of ${\cal O}(1/k)$ in expectation.
On Non-strongly-convex Loss Function An obvious shortcoming with <Ref> is the requirement of strongly convex loss functions [cf. <Ref>]. Below, we comment on the convergence of the state-dependent SA algorithm (<ref>), (<ref>) when the loss function is not strongly convex (possibly non-convex). Notice that in the absence of <Ref>, the performative stable solution $\theta_{PS}$ may not be well defined. We resort to finding a stationary point to the performative prediction problem (<ref>).
Our idea is to view (<ref>), (<ref>) as a biased SA algorithm with mean field $h(\theta) = \EE_{z \sim {\cal D}(\theta)} [ \grd \ell( \theta; z )] = \grd f(\theta; \theta)$. This mean field is correlated with the gradient for the performative loss in (<ref>). Under additional assumptions on $\ell(\theta;z)$, ${\cal D}(\theta)$, in Appendix <ref> we show
h(θ) _ z ∼D(θ) [ ℓ( θ; z ) ] ≥ h(θ) ^2 / 2 - c_0 , ∀ θ∈^d,
where ${\rm c}_0 > 0$ is a bias dependent on the sensitivity of distribution shift [cf. <Ref>].
We define the constant ${\rm c}_0$ in (<ref>) of Appendix <ref>, which is shown to be dependent on $\ell(\theta; z)$, $\grd_{\theta} \log(p_{\mathcal{D}(\theta)}(z))$.
It should be pointed that the algorithm (<ref>) may not provide a `good' solution to the non-convex performative learning problem (<ref>). However, if the state-dependent distribution is not too sensitive to the change of state, the following corollary shows that (<ref>) would still converge to a ${\rm c}_{0}$-neighborhood of a stationary solution. Before we discuss the main statement, we need two additional assumptions:
The function $V(\theta) = \EE_{ z \sim {\cal D}(\theta) } [ \ell( \theta; z ) ]$ is continuously differentiable and there exists $L_V \geq 0$ such that $\| \grd V(\theta) - \grd V(\theta') \| \leq L_V \| \theta - \theta' \|$ for any $\theta, \theta' \in \RR^d$.
There exists $\sigma \geq 0$ such that $\| \grd \ell(\theta; z) - \grd f(\theta; \theta) \| \leq \sigma$ for any $\theta \in \RR^d, z \in {\sf Z}$.
The above are stronger conditions than Assumptions <ref>, <ref>, yet are reasonable settings for certain non-convex loss functions.
For instance, as we will show in Appendix <ref>, Assumption <ref> holds if $\ell(\theta ; z), \nabla_{\theta} \log \left(p_{D(\theta)}(z)\right)$ are bounded, and $\nabla_{\theta} \log \left(p_{D(\theta)}(z)\right)$ to be Lipschitz w.r.t. $\theta$, e.g., when $\mathcal{D}(\theta)$ is `smooth', e.g., it is Gaussian or softmax distribution.
Assumption <ref> holds under similar condition as Assumption <ref>, e.g., ${\sf Z}$ is compact.
We obtain the following as a corollary of <cit.>:
Under Assumptions <ref>, <ref>, <ref>, <ref>, and let (<ref>) holds. With a step size sequence that decays in the order of $\gamma_k = {\cal O}(1/\sqrt{k})$, it holds for any $K \geq 1$ that
[ V( θ_K ) ^2 ] = O ( logK / √(K) + c_0 ), ( K = k ) = γ_k / ∑_j=1^K γ_j, ∀ k ∈{1,...,K}.
Note that ${\sf K} \in \{1,...,K\}$ is a discrete r.v. independent of the randomness in the SA algorithm and $\EE[\cdot]$ denotes the total expectation.
See the details in Appendix <ref>.
The above corollary shows that even without the strong convexity assumption, the state dependent SA algorithm finds an ${\cal O}(\log K / \sqrt{K} + {\rm c}_0)$-stationary solution to the performative prediction problem (<ref>) in at most $K$ iterations.
§ PROOF OUTLINE OF <REF>
We outline the main steps in proving <Ref>. Our proof strategy consists in tracking the progress of the mean squared error $\Delta_k \eqdef \EE[ \| \theta_k - \theta_{PS} \|^2 ]$.
To simplify notations, we define $\tilde{\mu} \eqdef \mu - L \epsilon$, the scalar product $G_{m:n} = \prod_{i=m}^n (1 - \gamma_i \tilde{\mu})$, for $n > m \geq 1$, and $G_{m:n} = 1$ if $n \leq m$.
The following lemma describes the one-step progress of the SA algorithm.
Under Assumptions <ref>, <ref>, <ref>, <ref>. For any $k \geq 0$, it holds
θ_k+1 - θ_PS ^2 ≤( 1 - 2 γ_k+1 μ̃ + 2L^2 γ_k+1^2 ) θ_k - θ_PS ^2
+ 2 σ^2 γ_k+1^2 - 2 γ_k+1 θ_k - θ_PS ℓ( θ_k; z_k+1 ) - f( θ_k; θ_k ) .
The proof can be found in Appendix <ref>, which involves a simple expansion of the squared error.
The above lemma suggests that the sensitivity parameter shall satisfy $\epsilon < {\mu} / {L}$ to ensure $\tilde{\mu} > 0$. Furthermore, the step size condition
$\sup_{k \geq 1} \gamma_{k} \leq {\tilde{\mu}} / (2 L^2)$ in (<ref>) leads to $1 - 2 \gamma_{k+1} \tilde{\mu} + 2L^2 \gamma_{k+1}^2 \leq 1 - \gamma_{k+1} \tilde{\mu}$ such that the first term in the r.h.s. of (<ref>) is a contraction.
Under the above premises and suppose $z_{k+1} \overset{\text{i.i.d.}}{\sim} {\cal D}( \theta_k )$ as in the case of [Mendler-Dünner et al., 2020, Drusvyatskiy and Xiao, 2020], the stochastic gradient in (<ref>) is conditionally unbiased. As such, <Ref> leads to the recursion $\Delta_{k+1} \leq ( 1 - \gamma_{k+1} \tilde{\mu} ) \Delta_k + 2 \sigma^2 \gamma_{k+1}^2$, implying $\Delta_k = {\cal O}(\gamma_k)$.
However, for the state-dependent SA algorithm (<ref>), (<ref>), the stochastic gradient $\grd \ell( \theta_k; z_{k+1} )$ is conditionally biased and is driven by a controlled MC. Under the stepsize condition $\sup_{k \geq 1} \gamma_{k} \leq {\tilde{\mu}} / (2 L^2)$, taking the total expectation and solving the recursion in (<ref>) yield
Δ_k ≤G_1:k Δ_0 + 2 σ^2 ∑_s=0^k-1 G_s+2:k γ_s+1^2
+ 2 ∑_s=0^k-1 G_s+2:k γ_s+1 [ θ_PS - θ_s ℓ( θ_s; z_s+1) - f(θ_s; θ_s) ] .
It can be shown that the first two terms are bounded by ${\cal O}(\gamma_k)$. We are interested in analyzing the last term when the samples $\{ z_{k} \}_{k \geq 1}$ are drawn according to (<ref>). Observe that:
Under Assumptions <ref>–<ref> and the stepsize conditions in (<ref>). For any $k \geq 1$, it holds
2 ∑_s=0^k-1 G_s+2:k γ_s+1 [ θ_PS - θ_s ℓ( θ_s; z_s+1) - f(θ_s; θ_s) ] ≤
∑_s=2^k γ_s^2 G_s+1:k ( + Δ_s-1 + Δ_s-2 ) + γ_1 G_2:k { (1 + 3 Δ_0 ) + γ_1 } + γ_k { 1 + 3 Δ_k-1 },
where we have defined the constants:
:= ςL + 4 ςL + ( 1 + μ̃ ) ς, := 2 ςL,
:= + 2 ( 1 + μ̃ )ς.
The analysis is inspired by [Benveniste et al., 2012] and has been adopted in recent works such as [Atchadé et al., 2017, Karimi et al., 2019]; see the details in Appendix <ref>. To handle the controlled MC, our technique involves applying the Poisson equation in <Ref> and decomposing the gradient error $\grd \ell( \theta_s; z_{s+1}) - \grd f(\theta_s; \theta_s)$ into Martingale and finite difference terms.
A key difference between <Ref> and analysis in the previous works is that the latter assumed that the stochastic gradients $\grd \ell(\theta_k; z_{k+1} )$ are bounded which greatly simplifies the proof. Our assumptions are significantly weaker as the latter actually grows as ${\cal O}( 1 + \| \theta_k - \theta_{PS} \|)$ [cf. <Ref>].
From the analysis standpoint, this demands a new proof technique as we present next.
To proceed, observe that substituting <Ref> into (<ref>) yields:
Δ_k ≤G_1:k Δ_0 + ∑_s=1^k-1 γ_s+1^2 G_s+2:k ( 2 σ^2 + + Δ_s + Δ_s-1 )
+ γ_1 G_2:k { (1 + 3 Δ_0 ) + γ_1 ( 2 σ^2 + ) } + γ_k { 1 + 3 Δ_k-1 }.
With the first step size condition in (<ref>), we can apply the auxiliary result in <Ref> from the appendix, which simplifies the upper bound as
Δ_k ≤G_1:k Δ_0 + ( 2/ μ̃ (2 σ^2 + ) + ) γ_k + ∑_s=1^k-1 γ_s+1^2 G_s+2:k ( Δ_s + Δ_s-1 )
+ γ_1 G_2:k { (1 + 3 Δ_0 ) + γ_1 ( 2 σ^2 + ) } + 3 γ_k Δ_k-1 .
Observe that the first row in (<ref>) is already in a similar form to the bound presented in the theorem. The key issue lies with the last term $3 \gamma_k \LZ \Delta_{k-1}$ which may be unbounded. We show that our choice of step sizes in (<ref>) ensures the convergence of $\Delta_k$ to ${\cal O}(\gamma_k)$:
Suppose that $\{ \Delta_k \}_{k \geq 0}$ satisfy (<ref>) and the step sizes $\{ \gamma_k \}_{k \geq 1}$ satisfy (<ref>). It holds (i)
sup_k ≥0 Δ_k ≤Δ := 3 Δ_0 + ς/ 9 ^2 ( 2 (2 σ^2 + ) + (μ- L ϵ) ),
and (ii) the following inequality holds for any $k \geq 1$:
Δ_k ≤∏_i=1^k ( 1 - γ_i μ̃/2) Δ_0 + { 3 Δ + 4 ς/ μ̃ ( 2 (2 σ^2 + ) + μ̃ + ( + 3 μ̃ ) Δ ) } γ_k.
Proving the above lemma requires one to establish the stability of the system (<ref>), which demands a sufficiently small $\gamma_k$ to control the remainder term $3 \LZ \gamma_k \Delta_{k-1}$. Our analysis relies on the special structure of this inequality system; see the proof details in Appendix <ref>.
The convergence bound (<ref>) follows from the boundedness of $\Delta_k$.
Finally, we obtain <Ref> by applying <Ref>.
§ NUMERICAL EXPERIMENTS
This section considers two performative prediction problems to corroborate with our theories. All the experiments are performed with Python on a server using a single thread of an Intel Xeon 6138 CPU. Further details about the experiments below can be found in Appendix <ref>.
Gaussian Mean Estimation
The first problem is concerned with Gaussian mean estimation using synthetic data. Our aim is to validate Theorem <ref> using a simple experiment. Here, (<ref>) is specified as $\min_{ \theta \in \RR } \EE_{ z \sim {\cal D}(\theta)} [ (z-\theta)^2/2 ]$ with ${\cal D}(\theta) \equiv {\cal N}( \bar{z} + \epsilon \theta ; \sigma^2 )$. For $0<\epsilon<1$, the performative stable solution has a closed form $\theta_{PS} = \frac{ \bar{z} }{ 1 - \epsilon }$. For the state-dependent SA, the agent follows an autoregressive (AR) model $z_{k+1} = (1 - \rho) z_k + \rho \tilde{z}_{k+1}$ with independent $\tilde{z}_{k+1} \sim {\cal N}( \bar{z} + \epsilon \theta_k; \sigma^2 )$ and regression parameter $\rho \in (0,1)$. This AR recursion is a controlled MC with a stationary distribution that yields the unbiased gradient of (<ref>), details about the MC are in Appendix <ref>.
We consider a large variance setting with $\bar{z} = 10$, $\sigma = 50$, $\epsilon = 0.1$. The step size is $\gamma_k = \frac{c_0}{c_1+k}$, $c_0 = \frac{500}{\tilde{\mu}}, c_1 = \frac{800}{\tilde{\mu}^2}$. In Fig. <ref> (left), we compare $|\theta_k - \theta_{PS}|^2$ against the iteration number $k$ for the Gaussian estimation problem using our state-dependent SA and greedy deploy [Mendler-Dünner et al., 2020] algorithms. As observed, both algorithms have an asymptotic convergence rate of ${\cal O}(1/k)$ towards $\theta_{PS}$ which is predicted by <Ref>. As $\rho \downarrow 0$, the state-dependent SA algorithm delivers a smaller error as the AR model has a stationary distribution with lower variance.
Gaussian mean estimation – (Left) Under different regression parameter $\rho$; Strategic Classification – (Middle) Under Linear BR $U_{\sf q}(\cdot)$ and different agent response rate $\alpha$ [cf. (<ref>)]; (Right) Under Logistics BR $U_{\sf lg}(\cdot)$. The shaded region shows the 90% confidence interval over 20 trials.
Strategic Classification
The second problem is a strategic classification (SC) problem similar to [Perdomo et al., 2020]
for a credit scoring classifier with dataset[Available at <https://www.kaggle.com/c/GiveMeSomeCredit/data>.].
Our aim is to showcase the effects when the agent adapts slowly to the shifted distribution. In particular, our theory implies that while the algorithm will still converge to $\theta_{PS}$, a slower convergence rate will be observed. To specify (<ref>), let $z \equiv (x,y)$ where $x \in \RR^d$ is feature vector, $y \in \{0,1\}$ is label. The learner finds $\theta \in \RR^d$ that minimizes:
_z ∼D(θ)[ ℓ( θ; z) ], where ℓ( θ; z ) = β/2 θ^2 + log( 1 + exp( θx ) ) - y θx.
Observe that when $\beta > 0$, $\ell(\theta;z)$ is a $\beta$-strongly convex function w.r.t. $\theta$ satisfying <Ref>.
For any $\theta \in \RR^d$, the shifted data distribution ${\cal D}(\theta)$ is obtained through evaluating the best response (BR) in (<ref>) of Example <ref>. We consider two types of utility functions adopted by the agents:
U_q(z^';z, θ) = θx'- x'-x^2/2ϵ, U_lg(z^';z, θ) = y θx' - log(1+exp(θx'))- x'-x^2/2ϵ,
where $z \equiv (x,y)$ is the original unshifted data. The label $y \in \{0,1\}$ is unchanged in the BR. Notice that $U_{\sf q}(\cdot)$, $U_{\sf lg}(\cdot)$ have respectively linear and logistics costs. Both utility functions include a quadratic regularizer where $\epsilon$ controls the sensitivity of the distribution shift [cf. <Ref>].
With a published $\theta_k$, the agent(s) maximize the utility function prior to giving data to the learner for the next round.
For both $U_{\sf q}(\cdot)$ and $U_{\sf lg}(\cdot)$, the BR obtained steers the classifier in favor of the agent(s).
Furthermore, $U_{\sf lg}(\cdot)$ is motivated by logistic regression which favors towards samples with label `1'.
For details of the numerical experiments, we set $\beta = 1000/m$ in (<ref>), $\epsilon = 0.01$ in the utility functions, and in (<ref>), we set number of selected agents as $|{\cal I}_k| = 5$, agents' response rate (stepsize) as $\alpha = 0.5 \epsilon$ unless otherwise specified.
The step size for (<ref>) is $\gamma_k = {c_0} / {(c_1+k)}$, $c_0 = {100} / {\tilde{\mu}}, c_1 = { 8 L^2 } / {\tilde{\mu}^2}$.
We first consider when $\theta_{PS}$ is computed with ${\cal D}(\theta)$ defined by the linear BR function $U_{\sf q}(\cdot)$ and compare our state-dependent SA with greedy deploy scheme [Mendler-Dünner et al., 2020] and repeated risk minimization [Perdomo et al., 2020]. As shown in Fig. <ref> (middle), all algorithms converge to $\theta_{PS}$. As $\alpha \downarrow 0$, the state-dependent SA converges at slightly slower rates as the agents adapt to the distribution shift with increased mixing time of the MC.
The result corroborates with <Ref> which established the ${\cal O}(1/k)$ convergence rate with state-dependent SA.
Our last experiment pertains to the same SC problem as before, but we consider a different setting where $\theta_{PS}$ is computed with ${\cal D}(\theta)$ defined by the logistics BR function $U_{\sf lg}(\cdot)$.
The agents follow a more complicated dynamics since the BR does not admit a closed form solution. Again, we aim to confirm Theorem <ref> that Algorithm 1 converges under the stateful agent setting. We compare the distance $\| \theta_k - \theta_{PS}\|^2$ versus iteration $k$.
In addition to showing the convergence result as predicted in Theorem <ref>, we aim to observe the effects when the learner adopts a lazy deployment scheme [cf. [Mendler-Dünner et al., 2020]], by varying the number of $\theta$-update by the learner per the agents' update.
Fig. <ref> (right) shows the error $\| \theta_k - \theta_{PS} \|^2$ against the number of adaptation steps performed at the agents via (<ref>) as we illustrate the convergence rate from the perspectives of the agents.
We observe that the error decreases at a faster rate when the number of learner's iteration increases. More details about our numerical experiments can be found in Appendix <ref>.
Conclusions & Limitations We consider a state-dependent SA algorithm for performative prediction. We showed a convergence rate of ${\cal O}(1/k)$ in mean-squared error towards the performative stable solution when the agents provide data drawn from a controlled MC. Our study paved the first step towards understanding and applying performative prediction in a dynamical setting.
There are several limitations. First, despite being a reasonable fixed point, $\theta_{PS}$ can be different from the optimal solution to (<ref>). An open problem is whether the state dependent SA converges when gradient correction, e.g., [Izzo et al., 2021], is used. Second, our convergence analysis of state-dependent SA algorithms relies on smoothness conditions on the controlled MC and distributions [cf. <Ref>, <ref>]. An open problem is to verify if these are necessary for our results to hold.
[Atchadé et al., 2017]
Y. F. Atchadé, G. Fort, and E. Moulines.
On perturbed proximal gradient algorithms.
The Journal of Machine Learning Research, 180
(1):0 310–342, 2017.
[Bartlett, 1992]
P. L. Bartlett.
Learning with a slowly changing distribution.
In Proceedings of the fifth annual workshop on Computational
learning theory, pages 243–252, 1992.
[Benveniste et al., 2012]
A. Benveniste, M. Métivier, and P. Priouret.
Adaptive algorithms and stochastic approximations, volume 22.
Springer Science & Business Media, 2012.
[Borkar, 2021]
V. S. Borkar.
A concentration bound for contractive stochastic approximation.
Systems & Control Letters, 153:0 104947, 2021.
[Bottou et al., 2018]
L. Bottou, F. E. Curtis, and J. Nocedal.
Optimization methods for large-scale machine learning.
Siam Review, 600 (2):0 223–311, 2018.
[Brown et al., 2020]
G. Brown, S. Hod, and I. Kalemaj.
Performative prediction in a stateful world.
arXiv preprint arXiv:2011.03885, 2020.
[Cai et al., 2015]
Y. Cai, C. Daskalakis, and C. Papadimitriou.
Optimum statistical estimation with strategic data sources.
In Conference on Learning Theory, pages 280–296. PMLR, 2015.
[Doan et al., 2020]
T. T. Doan, L. M. Nguyen, N. H. Pham, and J. Romberg.
Finite-time analysis of stochastic gradient descent under markov
arXiv preprint arXiv:2003.10973, 2020.
[Douc et al., 2018]
R. Douc, E. Moulines, P. Priouret, and P. Soulier.
Markov chains.
Springer, 2018.
[Drusvyatskiy and Xiao, 2020]
D. Drusvyatskiy and L. Xiao.
Stochastic optimization with decision-dependent distributions.
arXiv preprint arXiv:2011.11173, 2020.
[Ginart et al., 2021]
T. Ginart, E. Zhang, Y. Kwon, and J. Zou.
Competing ai: How does competition feedback affect machine learning?
In International Conference on Artificial Intelligence and
Statistics, pages 1693–1701. PMLR, 2021.
[Hardt et al., 2016]
M. Hardt, N. Megiddo, C. Papadimitriou, and M. Wootters.
Strategic classification.
In Proceedings of the 2016 ACM conference on innovations in
theoretical computer science, pages 111–122, 2016.
[Izzo et al., 2021]
Z. Izzo, L. Ying, and J. Zou.
How to learn when data reacts to your model: Performative gradient
arXiv preprint arXiv:2102.07698, 2021.
[Kaggle, 2012]
Give me some credit, 2012.
URL <https://www.kaggle.com/c/GiveMeSomeCrdit/data>.
[Kaledin et al., 2020]
M. Kaledin, E. Moulines, A. Naumov, V. Tadic, and H.-T. Wai.
Finite time analysis of linear two-timescale stochastic approximation
with markovian noise.
In Conference on Learning Theory, 2020.
[Karimi et al., 2019]
B. Karimi, B. Miasojedow, E. Moulines, and H.-T. Wai.
Non-asymptotic analysis of biased stochastic approximation scheme.
In Conference on Learning Theory, pages 1944–1974. PMLR,
[Kleinberg and Raghavan, 2020]
J. Kleinberg and M. Raghavan.
How do classifiers induce agents to invest effort strategically?
ACM Transactions on Economics and Computation (TEAC),
80 (4):0 1–23, 2020.
[Kushner and Yin, 2003]
H. Kushner and G. G. Yin.
Stochastic approximation and recursive algorithms and
applications, volume 35.
Springer Science & Business Media, 2003.
[Mendler-Dünner et al., 2020]
C. Mendler-Dünner, J. Perdomo, T. Zrnic, and M. Hardt.
Stochastic optimization for performative prediction.
Advances in Neural Information Processing Systems, 33, 2020.
[Miller et al., 2021]
J. Miller, J. C. Perdomo, and T. Zrnic.
Outside the echo chamber: Optimizing the performative risk.
arXiv preprint arXiv:2102.08570, 2021.
[Munro, 2020]
E. Munro.
Learning to personalize treatments when agents are strategic.
arXiv preprint arXiv:2011.06528, 2020.
[Nesterov, 2012]
Y. Nesterov.
Efficiency of coordinate descent methods on huge-scale optimization
SIAM Journal on Optimization, 220 (2):0
341–362, 2012.
[Patrascu and Necoara, 2015]
A. Patrascu and I. Necoara.
Efficient random coordinate descent algorithms for large-scale
structured nonconvex optimization.
Journal of Global Optimization, 610 (1):0
19–46, 2015.
[Perdomo et al., 2020]
J. C. Perdomo, C. Zrnic, Tijana aCnd Mendler-Dünner, and M. Hardt.
Performative prediction.
In ICML, 2020.
[Qu and Wierman, 2020]
G. Qu and A. Wierman.
Finite-time analysis of asynchronous stochastic approximation and $ q
In Conference on Learning Theory, pages 3185–3205. PMLR,
[Quiñonero-Candela et al., 2009]
J. Quiñonero-Candela, M. Sugiyama, N. D. Lawrence, and A. Schwaighofer.
Dataset shift in machine learning.
Mit Press, 2009.
[Richtárik and Takáč, 2014]
P. Richtárik and M. Takáč.
Iteration complexity of randomized block-coordinate descent methods
for minimizing a composite function.
Mathematical Programming, 1440 (1):0 1–38,
[Robbins and Monro, 1951]
H. Robbins and S. Monro.
A stochastic approximation method.
The annals of mathematical statistics, pages 400–407, 1951.
[Robert and Casella, 2013]
C. Robert and G. Casella.
Monte Carlo statistical methods.
Springer Science & Business Media, 2013.
[Srikant and Ying, 2019]
R. Srikant and L. Ying.
Finite-time error bounds for linear stochastic approximation and TD
In Conference on Learning Theory, pages 2803–2830. PMLR,
[Sun et al., 2018]
T. Sun, Y. Sun, and W. Yin.
On markov chain gradient descent.
In NeurIPS, 2018.
[Sutton and Barto, 2018]
R. S. Sutton and A. G. Barto.
Reinforcement learning: An introduction.
MIT press, 2018.
[Tadić et al., 2017]
V. B. Tadić, A. Doucet, et al.
Asymptotic bias of stochastic gradient search.
Annals of Applied Probability, 270 (6):0
3255–3304, 2017.
[Wu et al., 2020]
Y. Wu, W. Zhang, P. Xu, and Q. Gu.
A finite time analysis of two time-scale actor critic methods.
In NeurIPS, 2020.
[Xu et al., 2020]
T. Xu, Z. Wang, and Y. Liang.
Non-asymptotic convergence analysis of two time-scale (natural)
actor-critic algorithms.
arXiv preprint arXiv:2005.03557, 2020.
[Zhang et al., 2020]
S. Zhang, B. Liu, H. Yao, and S. Whiteson.
Provably convergent two-timescale off-policy actor-critic with
function approximation.
In International Conference on Machine Learning, pages
11204–11213. PMLR, 2020.
§ SUPPLEMENTARY INFORMATION FOR SECTION <REF>
§.§ Example on Gaussian Estimation (A Case where $\pi_{\theta}(\cdot) \neq {\cal D}(\theta)$)
Consider the following instance of (<ref>) with:
min_ θ∈ _ z ∼D(θ) [ ( z - θ)^2 / 2 ] where D(θ) ≡N( z̅ + ϵθ; σ^2 ).
Following (<ref>), the state-dependent SA algorithm reads
θ_k+1 = θ_k - γ_k+1 ℓ(θ_k; z_k+1) = θ_k - γ_k+1 ( θ_k - z_k+1 ).
where the sequence $\{ z_k \}_{k \geq 1}$ is generated by an autoregressive (AR) model, with $\rho \in (0,1]$,
z_k+1 = (1 - ρ) z_k + ρz̃_k+1 where z̃_k+1 ∼D( θ_k ) = N( z̅ + ϵθ_k ; σ^2 ),
such that the draw of $\tilde{z}_{k+1}$ are independent.
We show that the algorithm (<ref>), (<ref>) can be analyzed as a state-dependent SA (<ref>), (<ref>) considered in our framework. Precisely, we show that the controlled MC in (<ref>) admits a stationary distribution $\pi_{\theta_k}(\cdot)$ such that $\EE_{z \sim \pi_{\theta_k}(\cdot)} [ \grd \ell(\theta; z) ] = \EE_{z' \sim {\cal D}(\theta_k)} [ \grd \ell( \theta ; z' )]$.
Observe that (<ref>) defines a controlled MC with a transition kernel denoted by $\MK_{\theta_k}: \RR \times \RR \rightarrow \RR_+$ at the $k$th iteration, as in (<ref>). For every $\theta \in \RR$, $z \in \RR$, the kernel $\MK_{\theta}$ has a unique stationary distribution given by
lim_ n →∞ _θ^n (z, ·) = π_θ(·) ≡N ( z̅ + ϵθ ; ρ/2-ρ σ^2 ).
Notice that the above is different from the distribution ${\cal D}(\theta)$ desired in (<ref>) unless $\rho = 1$. In the latter case, the AR model (<ref>) reduces to drawing i.i.d. samples from ${\cal D}(\theta)$. For general $\rho < 1$, it still satisfies the asymptotically unbiasedness of the stochastic gradient estimate. In particular,
_ z ∼π_θ(·) [ ℓ( θ; z ) ] = _ z ∼π_θ(·) [ θ- z ] = _ z ∼D(θ) [ ℓ( θ; z ) ] .
The key observation is that for this particular performative prediction problem (<ref>), the gradient of the loss function is linear in the sample $z$. As such, with (<ref>) yielding a stationary distribution that has the same mean as ${\cal D}(\theta)$, the asymptotic unbiasedness property is unaffected.
In fact, the stationary distribution in (<ref>) has a reduced variance compared to ${\cal D}(\theta)$. Therefore, we expect the estimation error of $\theta_{PS}$ to be more stable using (<ref>), (<ref>) than [Mendler-Dünner et al., 2020].
§.§ Details on Example <ref> for Adapted Best Response
We continue the discussions in the paper with the procedure (<ref>).
When $\theta \in \RR^d$ is fixed, the procedure in (<ref>) is modelled as a Markov Chain (MC) with unique stationary distribution that corresponds to the best response distribution ${\cal D}(\theta)$ described in (<ref>).
To this end, we model the state of the MC by the tuple $\hat{z} \equiv ( d_1, ..., d_m, z )$. Consider the state space given by ${\sf Z}^{m+1}$ and
denote $\MK_{\theta} : {\sf Z}^{m+1} \times {\cal Z}^{m+1} \rightarrow \RR_+$ as the Markov transition kernel.
We remark that there is a slight abuse of notation here as the stochastic gradient $\grd \ell( \theta; \hat{z} )$ used by the learner depends only on the last term, $z$, in the agents' state variable $\hat{z}$. We have decided to use the current notation in the main paper to avoid introducing complicated notation for the implementation focused readers. Nevertheless, the SC example fits our proposed model.
Turning back on the MC. Observe when the current state is $\hat{z}$, under the action of kernel $\MK_{\theta}$, by following the description in (<ref>), we obtain the next state $\hat{z}' = ( d_1', ..., d_m', z' )$ as
d_j' = d_j + α1_ j ∈I U( d_j; d̅_j, θ), j=1,...,m, z' = d_i',
with probability
\[
\frac{1}{ { m \choose pm } } \times \frac{1}{m} ,
\]
for any ${\cal I} \subseteq \{1,...,m\}$, $|{\cal I}| = pm$ and $i \in \{1,...,m \}$.
At each transition, the data points $\{ d_1,..., d_m \}$ are updated by the first equation in (<ref>). The latter can be treated as a special case of the random block coordinate gradient descent (RBCD) algorithm to the separable problem:
max_ d_i , i =1,...,m ∑_i=1^m U( d_i ; d̅_i, θ).
Note that the optimal solution to the above, $\{ d_1^\star, ..., d_m^\star \}$, is a set of data points that forms the empirical distribution ${\cal D}(\theta)$.
Furthermore, it is known that the RBCD algorithm converges linearly with high probability and almost surely to the optimal solution for strongly concave maximization; see [Richtárik and Takáč, 2014, Patrascu and Necoara, 2015].
With the above observations, the MC induced by $\MK_\theta$ has a stationary distribution $\pi_\theta (\cdot)$ where for any measurable function $f : {\sf Z}^{m+1} \rightarrow \RR^n$, it holds
lim_k →∞ _θ^k f( d_1,..., d_m, z ) = 1/m ∑_i=1^m f( d_1^⋆, ..., d_m^⋆, d_i^⋆) = _z' ∼D(θ) [ f( d_1^⋆, ..., d_m^⋆, z') ],
for any initial state $\hat{z} = ( d_1, ..., d_m, z )$. The above identity can be derived from the fact that the RBCD algorithm converges almost surely to the optimal solution to (<ref>) and the random variable $z^k$ is uniformly drawn from $\{ d_1^k, ..., d_m^k \}$.
Furthermore, for any $L$-Lipschitz continuous $f$, it holds
_θ^k f( d_1,..., d_m, z ) - _z' ∼D(θ) [ f( d_1^⋆, ..., d_m^⋆, z') ]
(a)≤ 1/m ∑_i=1^m [ f( d_1^k, ..., d_m^k, d_i^k )] - f( d_1^⋆, ..., d_m^⋆, d_i^⋆)
(b)≤ L (1 + 1/m ) ∑_i=1^m [ d_i^k - d_i^⋆ ] ≤C ρ^k,
where (a) uses $\MK_\theta^k f( d_1,..., d_m, z ) = \sum_{i=1}^m \EE[ f( d_1^k , ..., d_m^k , d_i^k )] / m$ and the expectation is taken with respect to the random subset selection of ${\cal I}_k$ in (<ref>). In the expression that follows (b), the constants $\overline{\rm C}$, $\rho \in [0,1)$ depend on the initial value $\hat{z}$ and the strong concavity property of $U(\cdot)$. The above property is important for establishing the existence of the solution to Poisson equation in <Ref>.
§ CONVERGENCE ANALYSIS WITH NON-CONVEX LOSS FUNCTION
We first verify the inequality (<ref>) by observing the following expression for the gradient of performative loss:
V(θ) = ∫_Z ℓ( θ; z) p_ D(θ) (z) d z = _ z ∼D(θ) [ ℓ(θ; z) ] + _z ∼D(θ) [ ℓ(θ;z) _θlog( p_ D(θ) ( z) ) ],
where we have denoted $p_{ {\cal D}(\theta) }(z)$ as the probability distribution function for ${\cal D}(\theta)$. The above identity is derived using chain rule and the property $\grd_\theta \log p_{ {\cal D}(\theta) } (z) = \frac{ \grd_\theta p_{ {\cal D}(\theta) } (z) }{ p_{ {\cal D}(\theta) } (z) }$ similar to the policy gradient theorem; see <cit.>.
Observe that
V( θ) h (θ) = h(θ) ^2 + _z ∼D(θ) [ ℓ(θ;z) _θlog( p_ D(θ) ( z) ) ] h(θ)
We note
| _z ∼D(θ) [ ℓ(θ;z) _θlog( p_ D(θ) ( z) ) ] h(θ) | ≤1/2 h(θ)^2 + 1/2 _z ∼D(θ) [ ℓ(θ;z) _θlog( p_ D(θ) ( z) ) ] ^2
≤1/2 h(θ)^2 + 1/2 _z ∼D(θ) [ |ℓ(θ;z)|^2 _θlog( p_ D(θ) ( z) ) ^2 ] ≤1/2 h(θ) ^2 + c_0
where we have used the Jensen's inequality and set
c_0 := sup_ θ∈^d 1/2 _z ∼D(θ) [ |ℓ(θ;z)|^2 _θlog( p_ D(θ) ( z) ) ^2 ].
The above can be shown to be bounded when the loss function is bounded (e.g., a sigmoid loss), and the state-dependent distribution has bounded gradient w.r.t. $\theta$ (e.g., a soft-max distribution).
Together, we obtain the desired inequality:
V( θ) h (θ) ≥1/2 h(θ) ^2 - c_0, ∀ θ∈^d.
Notice that (<ref>) also implies
V( θ) ≤ h( θ) + _z ∼D(θ) [ ℓ(θ;z) _θlog( p_ D(θ) ( z) ) ]
≤ h( θ) + _z ∼D(θ) [ |ℓ(θ;z)| _θlog( p_ D(θ) ( z) ) ] ≤ h( θ) + √(2 c_0).
Proof of Corollary <ref> Notice that (<ref>), (<ref>) imply A1, A2 of [Karimi et al., 2019], respectively. Moreover, the stated assumptions in the corollary imply A3, A5-A7 of [Karimi et al., 2019]. Applying Theorem 2 from [Karimi et al., 2019] shows that
[ V( θ_ K ) ^2 ] ≲[ h( θ_ K ) ^2 ] + c_0 ≲1 + ∑_k=1^K γ_k^2 / ∑_k=1^K γ_k + c_0,
where we have omitted the constants from [Karimi et al., 2019]. Note that ${\sf K} \in \{1,...,K\}$ is a discrete r.v. selected independently with the probability $\PP ( {\sf K} = k ) = \gamma_k / \sum_{j=1}^K \gamma_j$.
Setting the step sizes as $\gamma_k = {\cal O}(1/\sqrt{k})$ shows the desired bound in the corollary.
§ MISSING PROOFS IN SECTION <REF>
Below, we present the detailed proof for the lemmas presented in <ref>.
For <Ref>, if $b=0$, then the smoothness parameter of the function $\MK_{\theta} \widehat{\grd \ell} ( \theta ; z)$ is invariant with the optimality gap $\| \theta - \theta_{PS} \|$.
If $b = 1$, then the smoothness of the composite function $\MK_{\theta} \widehat{\grd \ell} ( \theta ; z)$ grows with the optimality gap $\| \theta - \theta_{PS} \|$, which can be justified provided that the Markov kernel is smooth w.r.t. $\theta$, e.g., $\| \MK_{\theta} - \MK_{\theta'} \|_{TV} \leq {\rm C}_P \| \theta - \theta' \|$ for some constant ${\rm C}_P$. However, as we shall reveal later in the analysis, the convergence of the SA scheme with $b=1$ would require the following assumption on the fourth moment of $\| \theta_k - \theta_{PS}\|$, i.e.,
For any $t \geq 1$, there exists a constant ${\rm C}_m$ such that
√( [ θ_t - θ_PS ^4 ] ) ≤C_m [ θ_t - θ_PS ^2 ].
The above depends on the tail of the distribution of $\| \theta_t - \theta_{PS} \|$ which may be difficult to verify. Our following analysis will be derived on two cases: (a) <Ref> with $b = 0$, or (b) <Ref> with $b = 1$ and <Ref>.
§.§ Proof of Lemma <ref>
We begin our analysis by observing that as $\grd f(\theta_{PS}; \theta_{PS} ) = 0$, we have:
\begin{align*}
& \left\|\theta_{k+1}-\theta_{PS}\right\|^{2} = \left\|\theta_{k}-\gamma_{k+1} \nabla \ell( \theta_{k}; z_{k+1} )-\theta_{PS}\right\|^{2} \\
% & = \left\|\theta_{k}-\theta_{PS}+\nabla f(\theta_{PS}; \theta_{PS})-\gamma_{k+1} \nabla \ell ( \theta_{k}; z_{k+1} ) \right\|^{2} \\
& = \underbrace{ \left\|\theta_{k}-\theta_{PS}\right\|^{2}}_{ =: B_1 } - \, 2 \gamma_{k+1} \underbrace{ \pscal{ \theta_k -\theta_{PS} }{\nabla \ell( \theta_{k}; z_{k+1} ) - \nabla f(\theta_{PS}; \theta_{PS}) } }_{=:B_2} \\
& \qquad + \gamma_{k+1}^2 \underbrace{ \left\| \nabla f(\theta_{PS}; \theta_{PS})- \nabla \ell ( \theta_{k}; z_{k+1} ) \right\|^{2} }_{ =: B_3 }
\end{align*}
The inner product can be lower bounded as
\begin{eqnarray}
\begin{aligned}
B_{2}&= \pscal{ \theta_k -\theta_{PS}}{ \nabla \ell ( \theta_{k}; z_{k+1} ) -\nabla f(\theta_{PS}; \theta_{PS}) }\\
% &=\EE\left<\theta_k -\theta_{PS}|\nabla \ell(z_{k+1};\theta_{k})-\nabla f(\theta_{PS}; \theta_{PS})-\nabla f(\theta_{k}; \theta_{k})+\nabla f(\theta_{k}; \theta_{k})-\nabla f(\theta_{k}; \theta_{PS})+\nabla f(\theta_{k}; \theta_{PS})\right>\\
&= \pscal{ \theta_k -\theta_{PS} } { \nabla \ell ( \theta_{k}; z_{k+1} ) -\nabla f(\theta_{k}; \theta_{k}) } + \pscal { \theta_k -\theta_{PS} }{\nabla f(\theta_{k};\theta_{k})-\nabla f(\theta_{k}; \theta_{PS}) } \\
&\quad + \pscal{ \theta_k -\theta_{PS} }{ \nabla f(\theta_{k};\theta_{PS})-\nabla f(\theta_{PS}; \theta_{PS}) }\\
&\overset{(a)}{\geq} \pscal{ \theta_k -\theta_{PS}} {\nabla \ell( \theta_{k}; z_{k+1} )-\nabla f(\theta_{k}; \theta_{k}) } \\
& \quad - \left\|\theta_{k}-\theta_{PS} \right\| \left\|\nabla f(\theta_{k}; \theta_{k})- \nabla f(\theta_{k}; \theta_{PS}) \right\|+ \mu \left\|\theta_{k}-\theta_{PS} \right\|^{2}\\
&\overset{(b)}{\geq} \left<\theta_k -\theta_{PS}|\nabla \ell( \theta_{k}; z_{k+1} ) -\nabla f(\theta_{k}; \theta_{k})\right> +(\mu-L\varepsilon) \left\|\theta_{k}-\theta_{PS} \right\|^{2}
\end{aligned}
\end{eqnarray}
where (a) is due to the Cauchy-schwarz inequality and the $\mu$-strong convexity of $\nabla f(\cdot; \cdot)$; (b) is due to the $L$-smoothness of $f$ and the $\epsilon$-sensitivity of the distribution [c.f Assumption <ref>]; also see [Perdomo et al., 2020]. Furthermore,
\begin{eqnarray}
\begin{aligned}
B_{3}&= \left\| \nabla \ell ( \theta_{k}; z_{k+1} )-\nabla f(\theta_{PS}; \theta_{PS})+\nabla \ell ( \theta_{PS}; z_{k+1} ) -\nabla \ell ( \theta_{PS}; z_{k+1} ) \right\|^{2} \\
&\leq 2 \left( \left\| \nabla\ell ( \theta_{PS}; z_{k+1} )-\nabla\ell ( \theta_{k}; z_{k+1} ) \right\|^2+\left\|\nabla f(\theta_{PS}; \theta_{PS})-\nabla\ell ( \theta_{PS}; z_{k+1} ) \right\|^2 \right)\\
&\leq 2L^2 \left\|\theta_{k}-\theta_{PS} \right\|^2 + 2 \sigma^2
\end{aligned}
\end{eqnarray}
where the third inequality is due to Assumptions <ref>, <ref>.
Combing the bounds for $B_1$, $B_{2}$ and $B_3$, we can get the desired inequality.
\begin{eqnarray}\label{eq:recursion}
\begin{aligned}
& \left\|\theta_{k+1}-\theta_{PS}\right\|^2 \\
&\leq \left\|\theta_{k}-\theta_{PS}\right\|^2+ 2 \gamma_{k+1}^2\cdot \left(\sigma^2+L^2 \left\|\theta_{k}-\theta_{PS}\right\|^2\right)\\
&\quad -2\gamma_{k+1}\left(\left<\theta_k -\theta_{PS}|\nabla \ell( \theta_{k}; z_{k+1} )-\nabla f(\theta_{k}; \theta_{k})\right> +(\mu-L\varepsilon) \left\|\theta_{k}-\theta_{PS} \right\|^{2}\right) \\
& = \big( 1 - 2 \gamma_{k+1} ( \mu - L \epsilon ) + 2 \gamma_{k+1}^2 L^2 \big) \| \theta_k - \theta_{PS} \|^2 \\
& \quad + 2 \gamma_{k+1}^2 \sigma^2 - 2\gamma_{k+1} \pscal{ \theta_k - \theta_{PS} }{ \grd \ell( \theta_k; z_{k+1} ) - \grd f( \theta_k; \theta_k ) }.
\end{aligned}
\end{eqnarray}
It is noted that if we consider a case when the SA scheme (<ref>) is non-state-dependent, e.g., $z_{k+1}$ is drawn from $\mathcal{D}(\theta_{k})$ independently, then proving Lemma <ref> suffices to show our desired Theorem <ref> since the last term in equation (<ref>) is zero mean when conditioned on the previous iterates [cf. (<ref>)].
§.§ Proof of Lemma <ref>
Applying Assumption <ref> shows that the sum of inner product can be evaluated as
\begin{eqnarray*}
\begin{aligned}
& \sum_{s=1}^{k}\gamma_{s}G_{s+1:k}\EE\pscal{ \theta_{PS} - \theta_{s-1} } {\nabla \ell( \theta_{s-1}; z_s )-\nabla f(\theta_{s-1}; \theta_{s-1}) } \\
&= \sum_{s=1}^{k}\gamma_{s}G_{s+1:k}\EE\pscal{ \theta_{PS} - \theta_{s-1} } {\widehat{\nabla\ell}( \theta_{s-1}; z_s )-\MK_{\theta_{s-1}} \widehat{\nabla\ell}( \theta_{s-1}; z_s )} \equiv \EE\left(A_1+A_2+A_3+A_4+A_5\right),
\end{aligned}
\end{eqnarray*}
where we decomposed the sum of inner product into five sub-terms $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$ such that
\begin{align*}
&A_{1}:=-\sum_{s=2}^{k}\gamma_{s}G_{s+1:k}\pscal{ \theta_{s-1}-\theta_{PS}} { \widehat{\nabla\ell}( \theta_{s-1}; z_s )-\MK_{\theta_{s-1}}\widehat{\nabla \ell}( \theta_{s-1}; z_{s-1} ) } \\
&A_{2}:=-\sum_{s=2}^{k}\gamma_{s}G_{s+1:k} \pscal{ \theta_{s-1}-\theta_{PS} }{ \MK_{\theta_{s-1}} \widehat{\nabla\ell}( \theta_{s-1}; z_{s-1} )-\MK_{\theta_{s-2}}\widehat{\nabla\ell}( \theta_{s-2}; z_{s-1} ) }\\
&A_{3}:=-\sum_{s=2}^{k}\gamma_{s}G_{s+1:k} \pscal{ \theta_{s-1}-\theta_{s-2} } { \MK_{\theta_{s-2}}\widehat{\nabla\ell}( \theta_{s-2}; z_{s-1} ) }\\
&A_{4}:=-\sum_{s=2}^{k}(\gamma_{s}G_{s+1:k}-\gamma_{s-1}G_{s:k})\pscal{ \theta_{s-2}-\theta_{PS} }{ \MK_{\theta_{s-2}}\widehat{\nabla\ell}(\theta_{s-2}; z_{s-1}) }\\
&A_{5}:=-\gamma_{1}G_{2:k}\pscal{ \theta_{0}-\theta_{PS} } {\widehat{\nabla\ell}( \theta_{0}; z_1 ) } +
\gamma_{k} \pscal{ \theta_{k-1}-\theta_{PS} }{ \MK_{\theta_{k-1}}\widehat{\nabla\ell}( \theta_{k-1}; z_k ) } .
\end{align*}
We remark that a similar decomposition can be found in [Benveniste et al., 2012]. However, [Benveniste et al., 2012] proceeded with the analysis by assuming that $\theta_k$ stays in the compact set for all $k \geq 0$. We do not make such assumption in this work.
For $A_{1},$ we note that $\widehat{\nabla\ell}( \theta_{s-1}; z_s )-\MK_{\theta_{s-1}}\widehat{\nabla\ell}( \theta_{s-1}; z_{s-1} )$ is a martingale difference sequence and therefore we have $\EE\left[A_{1}\right]=0$ by taking the total expectation.
For $A_{2}$, as $\theta_{k+1}=\theta_{k}-\gamma_{k+1} \nabla {\ell}\left( \theta_{k}; z_{k+1} \right) $, we get $\theta_{s-1}-\theta_{s-2}=-\gamma_{s-1} \nabla {\ell} ( \theta_{s-2}; z_{s-1})$. Applying the smoothness condition <Ref> shows that
A_2 =-∑_s=2^kγ_sG_s+1:kθ_s-1-θ_PS _θ_s-1 ∇ℓ( θ_s-1; z_s-1 )-_θ_s-2∇ℓ( θ_s-2; z_s-1)
≤∑_s=2^k γ_s G_s+1:k θ_s-1-θ_PS θ_s-1-θ_s-2
≤∑_s=2^k γ_s-1 γ_s G_s+1:k θ_s-1-θ_PS ℓ( θ_s-2; z_s-1 ).
Combining with the implied bound (<ref>) from the assumptions as well as (<ref>) yield
\begin{align*}
&%\leq \varsigma \Lph \overline{L} \, \sum_{s=2}^k \gamma_{s}^2 G_{s+1:k} \|\theta_{s-1}-\theta_{PS} \| \left( 1 + b \norm{\theta_{s-1}-\theta_{PS}} \right) \left( 1 + \| \theta_{s-2} - \theta_{PS} \| \right) \\
\leq \varsigma \Lph \overline{L} \, \sum_{s=2}^k \gamma_{s}^2 G_{s+1:k} \|\theta_{s-1}-\theta_{PS} \| \left( 1 + \| \theta_{s-2} - \theta_{PS} \| \right)
\\
&%\leq \varsigma \Lph \overline{L} \, \sum_{s=2}^k \gamma_{s}^2 G_{s+1:k} \left\{ \frac{1}{2} + \frac{1}{2} \| \theta_{s-2} - \theta_{PS} \|^2 + (1+b) \| \theta_{s-1} - \theta_{PS} \|^2 + b \| \theta_{s-1} - \theta_{PS} \|^2 \| \theta_{s-2} - \theta_{PS} \| \right\}
\leq \varsigma \Lph \overline{L} \, \sum_{s=2}^k \gamma_{s}^2 G_{s+1:k} \left\{ \frac{1}{2} + \frac{1}{2} \| \theta_{s-2} - \theta_{PS} \|^2 + \| \theta_{s-1} - \theta_{PS} \|^2
%+ \| \theta_{s-2} - \theta_{PS} \|
\right\}
\\
& %\leq \varsigma \Lph \overline{L} \, \Big\{ \frac{1}{2} \sum_{s=2}^k \gamma_s^2 G_{s+1:k} + \frac{1}{2} \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \| \theta_{s-2} - \theta_{PS} \|^2 + (1+b)\sum_{s=2}^k \gamma_s^2 G_{s+1:k} \| \theta_{s-1} - \theta_{PS} \|^2 \Big\} \\
\leq \varsigma \Lph \overline{L} \, \Big\{ \frac{1}{2} \sum_{s=2}^k \gamma_s^2 G_{s+1:k} + \frac{1}{2} \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \| \theta_{s-2} - \theta_{PS} \|^2 + \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \| \theta_{s-1} - \theta_{PS} \|^2 \Big\} ,
%& \qquad + b \, \varsigma \Lph \overline{L} \, \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \| \theta_{s-1} - \theta_{PS} \|^2 \| \theta_{s-2} - \theta_{PS} \|
% \\
% &\leq \sum_{s=2}^k \Lph (\sigma+1)\gamma_{s}\gamma_{s-1} G_{s+1:k} \left( 1+2\norm{\theta_{s-1}-\theta_{PS}}^2\right)\cdot \left( 1 + \norm{\grd f(\theta_{s-2};\theta_{PS})-\grd f(\theta_{PS};\theta_{PS})}\right)
% \\
% &\leq \sum_{s=2}^k \Lph (\sigma+1)\gamma_{s}\gamma_{s-1} G_{s+1:k} \left( 1+2\norm{\theta_{s-1}-\theta_{PS}}^2\right)\cdot \left( 1 + L\norm{\theta_{s-2}-\theta_{PS}}\right)
% \\
% &\leq \sum_{s=2}^k \Lph (\sigma+1)\gamma_{s}\gamma_{s-1} G_{s+1:k} \left( 1+L\norm{\theta_{s-2}-\theta_{PS}} +2\norm{\theta_{s-1}-\theta_{PS}}^{2}+2L \norm{\theta_{s-1}-\theta_{PS}}^{2}\norm{\theta_{s-2}-\theta_{PS}}\right)
% \\
% &\leq \sum_{s=2}^k \Lph (\sigma+1)\gamma_{s}\gamma_{s-1} G_{s+1:k} \left( 1+L+L\norm{\theta_{s-2}-\theta_{PS}}^{2} +2\norm{\theta_{s-1}-\theta_{PS}}^{2}+2L \norm{\theta_{s-1}-\theta_{PS}}^{2}\norm{\theta_{s-2}-\theta_{PS}}\right)
% \\
% &\leq \Lph (\sigma+1)\left( 1+L\right)\sum_{s=2}^k \gamma_{s}\gamma_{s-1} G_{s+1:k}
% + \Lph L(\sigma+1)\sum_{s=2}^k \gamma_{s}\gamma_{s-1} G_{s+1:k}\Delta_{s-2}\\
% &\quad +2\Lph (\sigma+1)\sum_{s=2}^k \gamma_{s}\gamma_{s-1} G_{s+1:k}\Delta_{s-1}
% +2L\Lph (\sigma+1)\sum_{s=2}^k \gamma_{s}\gamma_{s-1} G_{s+1:k}\Delta_{s-1}\sqrt{\Delta_{s-2}}
\end{align*}
where the second inequality applies $a(1 + c) \leq \frac{1}{2} + \frac{1}{2} c^2 + a^2 $ for any $a,c\in \RR$.
For $A_{3}$, again using (<ref>), we observe that
A_3 =-∑_s=2^kγ_sG_s+1:k θ_s-1-θ_s-2 _θ_s-2∇ℓ( θ_s-2; z_s-1 )
≤∑_s=2^kγ_s G_s+1:kθ_s-1-θ_s-2·_θ_s-2∇ℓ( θ_s-2; z_s-1 )
≤∑_s=2^kγ_sγ_s-1 G_s+1:k ℓ( θ_s-2; z_s-1 )·( 1 + θ_s-2 - θ_PS )
≤ςL ∑_s=2^kγ_s^2 G_s+1:k (1 + θ_s-2 - θ_PS )^2
≤2 ςL ∑_s=2^kγ_s^2 G_s+1:k { 1 + θ_s-2 - θ_PS ^2 } .
For $A_{4}$, we notice that
A_4 =-∑_s=2^k(γ_sG_s+1:k-γ_s-1G_s:k) θ_s-2-θ_PS _θ_s-2∇ℓ( θ_s-2; z_s-1)
≤∑_s=2^k | γ_sG_s+1:k-γ_s-1G_s:k | θ_s-2-θ_PS ·_θ_s-2ℓ(θ_s-2; z_s-1 ) .
It can be shown that $| \gamma_{s}G_{s+1:k}-\gamma_{s-1}G_{s:k} | \leq ( 1 + \tilde{\mu} ) \varsigma \gamma_s^2 G_{s+1:k} $, therefore
A_4 ≤( 1 + μ̃ ) ς∑_s=2^k γ_s^2 G_s+1:k θ_s-2-θ_PS ( 1 + θ_s-2 - θ_PS )
≤( 1 + μ̃ ) ς∑_s=2^k γ_s^2 G_s+1:k { 1/2 + 3/2 θ_s-2 - θ_PS ^2 }
≤( 1 + μ̃ ) ς{ 1/2 ∑_s=2^k γ_s^2 G_s+1:k + 3/2 ∑_s=2^k γ_s^2 G_s+1:k θ_s-2 - θ_PS ^2 }.
Finally, for $A_{5}$, we have
\begin{align*}
A_{5}&=-\gamma_{1}G_{2:k}\pscal{ \theta_{0}-\theta_{PS}} { \widehat{\nabla\ell}(\theta_{0}; z_{1} ) } +
\gamma_{k} \pscal{ \theta_{k-1}-\theta_{PS}} {\MK_{\theta_{k-1}}\widehat{\nabla\ell} ( \theta_{k-1}; z_{k} ) }\\
&\leq \gamma_{1} G_{2:k} \left\|\theta_{0}-\theta_{PS} \right\| \left\|\widehat{\grd\ell}( \theta_{0}; z_{1} )\right\|+
\gamma_{k} \left\|\theta_{k-1}-\theta_{PS}\right\|\left\|\MK_{\theta_{k-1}}\widehat{\grd\ell}( \theta_{k-1}; z_{k}) \right\| \\
& \leq \gamma_1 \widehat{L} \, G_{2:k} \| \theta_0 - \theta_{PS} \| \big( 1 + \| \theta_0 - \theta_{PS} \| \big) + \gamma_k \LZ \, \| \theta_{k-1} - \theta_{PS} \| \big( 1 + \| \theta_{k-1} - \theta_{PS} \| \big) \\
& \leq \frac{\gamma_1 \widehat{L} \, G_{2:k}}{2} + \frac{ \gamma_k \LZ }{2} + \frac{3 \gamma_1 \widehat{L}}{2} \, G_{2:k} \| \theta_0 - \theta_{PS} \|^2 + \frac{3 \gamma_k \LZ}{2} \, \| \theta_{k-1} - \theta_{PS} \|^2
% \\
% &\leq \gamma_{1} G_{2:k} \norm{\theta_{0}-\theta_{PS}}\cdot \hat{\sigma}\left(1+\norm{\grd f(\theta_{0}; \theta_{PS}}\right) + \gamma_{k}\left\|\theta_{k-1}-\theta_{PS}\right\| \cdot {\LZ}\left( 1+ \norm{\grd f(\theta_{k-1}; \theta_{PS})} \right)\\
% & {\color{red}\leq \gamma_{1} G_{2:k} \norm{\theta_{0}-\theta_{PS}}\cdot \hat{\sigma}\left(1+\norm{\grd f(\theta_{0}; \theta_{PS})-\grd f(\theta_{PS}; \theta_{PS})}\right)} \\
% &\quad +\gamma_{k}\left\|\theta_{k-1}-\theta_{PS}\right\| \cdot {\LZ}\left( 1+ \norm{\grd f(\theta_{k-1}; \theta_{PS})-\grd f(\theta_{PS}; \theta_{PS})} \right)\\
% & \leq \gamma_{1} G_{2:k} \norm{\theta_{0}-\theta_{PS}}\cdot \hat{\sigma}\left(1+L\norm{\theta_{0}-\theta_{PS}}\right)
% +\gamma_{k}\left\|\theta_{k-1}-\theta_{PS}\right\| \cdot {\LZ}\left( 1+ L\norm{\theta_{k-1}- \theta_{PS}} \right)\\
% & \leq \gamma_{1} G_{2:k} \cdot \hat{\sigma}\left(\norm{\theta_{0}-\theta_{PS}}+L\norm{\theta_{0}-\theta_{PS}}^{2}\right)
% +\gamma_{k} {\LZ}\left( \left\|\theta_{k-1}-\theta_{PS}\right\|+ L\norm{\theta_{k-1}- \theta_{PS}}^{2} \right)\\
% & \leq \gamma_{1} G_{2:k} \cdot \hat{\sigma}\left(1+\norm{\theta_{0}-\theta_{PS}}^{2}+L\norm{\theta-\theta_{PS}}^{2}\right) +\gamma_{k} {\LZ}\left( 1+\left\|\theta_{k-1}-\theta_{PS}\right\|^{2}+ L\norm{\theta_{k-1}- \theta_{PS}}^{2} \right)\\
% & \leq \gamma_{1}\hat{\sigma}G_{2:k}+\gamma_{k}\LZ+ \gamma_{1} \hat{\sigma}(1+L)G_{2:k}\norm{\theta_{0}-\theta_{PS}}^{2}+\gamma_{k} {\LZ}(1+L)\norm{\theta_{k-1}-\theta_{PS}}^{2}\\
% & \leq \gamma_{1}\hat{\sigma}G_{2:k}+\gamma_{k}\LZ+ \gamma_{1} \hat{\sigma}(1+L)G_{2:k}\Delta_{0}+\gamma_{k}{\LZ}(1+L)\Delta_{k-1} \\
% & \leq \gamma_{1}\hat{\sigma}G_{2:k}+\gamma_{k}\LZ+ \gamma_{1} \hat{\sigma}(1+L)G_{2:k}\Delta_{0}+\gamma_{k} {\LZ}(1+L)\Delta_{k-1}
\end{align*}
Summing up $A_{1}$ to $A_{5}$ and taking the full expectation yield:
\begin{align*}
& 2 \big| \EE\left[A_1+A_2+A_3+A_4+A_5\right] \big| \\
%& \leq \varsigma \Lph \overline{L} \, \Big\{ \sum_{s=2}^k \gamma_s^2 G_{s+1:k} + \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \Delta_{s-2} + 2(1+b) \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \Delta_{s-1} + 2 b {\rm C}_m \, \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \Delta_{s-1} \sqrt{\Delta_{s-2}} \Big\} \\
%& \quad + 4 \varsigma \overline{L} \LZ \sum_{s=2}^{k}\gamma_{s}^2 G_{s+1:k} \big\{ 1 + \Delta_{s-2} \big\} + \{ 1 + \mu - L \epsilon \} \varsigma \LZ \Big\{ \sum_{s=2}^{k} \gamma_{s}^{2} G_{s+1:k} + 3 \sum_{s=2}^{k} \gamma_{s}^{2} G_{s+1:k} \Delta_{s-2} \Big\} \\
%& \quad + \gamma_1 \widehat{L} \, G_{2:k} + \gamma_k \LZ + 3 \gamma_1 \widehat{L} \, G_{2:k} \Delta_0 + 3 \gamma_k \LZ \, \Delta_{k-1}
& \leq \varsigma \Lph \overline{L} \, \Big\{ \sum_{s=2}^k \gamma_s^2 G_{s+1:k} + \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \Delta_{s-2} + 2\sum_{s=2}^k \gamma_s^2 G_{s+1:k} \Delta_{s-1} \Big\} \\
& \quad + 4 \varsigma \overline{L} \LZ \sum_{s=2}^{k}\gamma_{s}^2 G_{s+1:k} \big\{ 1 + \Delta_{s-2} \big\} + ( 1 + \tilde{\mu} ) \varsigma \LZ \Big\{ \sum_{s=2}^{k} \gamma_{s}^{2} G_{s+1:k} + 3 \sum_{s=2}^{k} \gamma_{s}^{2} G_{s+1:k} \Delta_{s-2} \Big\} \\
&\quad + \gamma_1 \widehat{L} \, G_{2:k} + \gamma_k \LZ + 3 \gamma_1 \widehat{L} \, G_{2:k} \Delta_0 + 3 \gamma_k \LZ \, \Delta_{k-1} .
\end{align*}
Recall the following constants:
:= ςL + 4 ςL + ( 1 + μ̃ ) ς, := 2 ςL, := ςL + 4 ςL + 3 ( 1 + μ̃ )ς.
We obtain the desirable bound for the lemma:
\begin{align*}
& 2 \big| \EE\left[A_1+A_2+A_3+A_4+A_5\right] \big| \\
%& \leq \varsigma \Lph \overline{L} \, \Big\{ \sum_{s=2}^k \gamma_s^2 G_{s+1:k} + \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \Delta_{s-2} + 2(1+b) \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \Delta_{s-1} + 2 b {\rm C}_m \, \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \Delta_{s-1} \sqrt{\Delta_{s-2}} \Big\} \\
%& \quad + 4 \varsigma \overline{L} \LZ \sum_{s=2}^{k}\gamma_{s}^2 G_{s+1:k} \big\{ 1 + \Delta_{s-2} \big\} + \{ 1 + \mu - L \epsilon \} \varsigma \LZ \Big\{ \sum_{s=2}^{k} \gamma_{s}^{2} G_{s+1:k} + 3 \sum_{s=2}^{k} \gamma_{s}^{2} G_{s+1:k} \Delta_{s-2} \Big\} \\
%& \quad + \gamma_1 \widehat{L} \, G_{2:k} + \gamma_k \LZ + 3 \gamma_1 \widehat{L} \, G_{2:k} \Delta_0 + 3 \gamma_k \LZ \, \Delta_{k-1}
& \leq \sum_{s=2}^k \gamma_s^2 G_{s+1:k} \big( \Cone + \Ctwo \Delta_{s-1} + \Cthree \Delta_{s-2} \big) + \LZ \gamma_k \big\{ 1 + 3 \Delta_{k-1} \big\} + \gamma_1 G_{2:k} \big( \LZ (1 + 3 \Delta_0 ) + \gamma_1 \Cone \big).
\end{align*}
This concludes the proof.
§.§ Proof of Lemma <ref>
Consider the inequality in (<ref>). We consider a non-negative upper bound sequence $\{ {\rm U}_k \}_{k \geq 0}$ defined by the recursion:
U_k = G_1:k U_0 + ( 2/ μ̃ (2 σ^2 + ) + ) γ_k + ∑_s=1^k-1 γ_s+1^2 G_s+2:k ( U_s + U_s-1 )
+ γ_1 G_2:k { (1 + 3 U_0 ) + γ_1 ( 2 σ^2 + ) } + 3 γ_k U_k-1 ,
for any $k \geq 1$, and we have defined ${\rm U}_0 = \Delta_0$.
Notice that by construction, we have $\Delta_k \leq {\rm U}_k$ for any $k \geq 0$.
Using the convention that ${\rm U}_{-1} = 0$, we observe that for any $k \geq 1$,
U_k = ( 1 - γ_k μ̃ ) U_k-1 + ( 2/ μ̃ (2 σ^2 + ) + ) ( γ_k - (1 - γ_k μ̃ ) γ_k-1 )
+ γ_k^2 ( U_k-1 + U_k-2 ) + 3 ( γ_k U_k-1 - (1 - γ_k μ̃) γ_k-1 U_k-2 )
≤( 1 - γ_k μ̃ + γ_k^2 ) U_k-1 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ςγ_k^2 + ( + 3 μ̃ )γ_k γ_k-1 U_k-2
+ 3 ( γ_k U_k-1 - γ_k-1 U_k-2 )
≤( 1 - γ_k μ̃ / 2 ) U_k-1 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ςγ_k^2 + ( + 3 μ̃ )γ_k γ_k-1 U_k-2
+ 3 ( γ_k U_k-1 - γ_k-1 U_k-2 ),
where the last inequality is due to $\gamma_k \leq \tilde{\mu}/2 \Ctwo$.
We prove part (i) of the lemma.
From (<ref>), we consider an upper bound sequence $\{ \oU_k \}_{k \geq -1}$ defined by the recursion:
_k = ( 1 - γ_k μ̃ / 2 ) _k-1 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ςγ_k^2 + ( + 3 μ̃ )γ_k γ_k-1 _k-2
+ 3 ( γ_k _k-1 - γ_k-1 _k-2 ), ∀k ≥1.
We have also defined $\oU_0 = {\rm U}_0$, $\oU_{-1} = 0$.
For any $t \geq 1$, summing up the equation (<ref>) from $k=1$ to $k=t$ yields
∑_k=1^t _k = ∑_k=1^t { ( 1 - γ_k μ̃ / 2 ) _k-1 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ςγ_k^2 + ( + 3 μ̃ )γ_k γ_k-1 _k-2 }
+ 3 ∑_k=1^t ( γ_k _k-1 - γ_k-1 _k-2 ),
Rearranging terms leads to
_t = _0 + ∑_k=1^t { ( + 3 μ̃ ) γ_k γ_k-1 _k-2 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ςγ_k^2 - μ̃/2 γ_k _k-1 } + 3 γ_t _t-1
Using the step size conditions $\gamma_k \leq \gamma_{k-1}$, $\gamma_k \leq \big( \Cthree + 3 \LZ \tilde{\mu} \big)^{-1} \min\{ \tilde{\mu} / 2 , 3 \LZ \}$, $\gamma_k \leq (6 \LZ)^{-1}$,
_t ≤_0 + 3 γ_t _t-1
+ ∑_k=1^t { [ ( + 3 μ̃ ) γ_k^2 - μ̃/2 γ_k ] _k-1 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ςγ_k^2 }
≤3 γ_t _t-1 + _0 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ς∑_k=1^t γ_k^2
≤1/2 _t-1 + { _0 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ς∑_k=1^t γ_k^2 },
where we obtain the first inequality after shifting the summation's index and it is noted that $\oU_{-1}=0.$ Rearranging terms and solving the recursion lead to
_t ≤( 1/2 )^t _0 + ∑_s=1^t ( 1/2 )^t-s { _0 + ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ς∑_k=1^s γ_k^2 }
≤3 _0 + 2 μ̃ ς( 2/ μ̃ (2 σ^2 + ) + ) ∑_ℓ=1^t γ_ℓ^2 ( 1/2 )^t-ℓ
≤3 _0 + μ̃ ς/ 9 ^2 ( 2/ μ̃ (2 σ^2 + ) + )
Recall that $\overline{\Delta} \eqdef 3 \oU_0 + \frac{ \varsigma}{ 9 \LZ^2 } \Big( 2 (2 \sigma^2 + \Cone) + \tilde{\mu} \LZ \Big)$, the above shows $\Delta_t \leq {\rm U}_t \leq \oU_t \leq \overline{\Delta}$ for any $t \geq 1$, thus establishing part (i).
We now proceed to proving part (ii) of the lemma. We define $\oG_{m:n} = \prod_{\ell=m}^n (1 - \gamma_\ell \tilde{\mu}/2)$ and observe from (<ref>) that
U_k ≤_1:k U_0 + ∑_s=1^k _s+1:k { ( 2/ μ̃ (2 σ^2 + ) + ) μ̃ ςγ_s^2 + ( + 3 μ̃ )γ_s γ_s-1 U_s-2 }
+ 3 ∑_s=1^k _s+1:k { ( γ_s U_s-1 - γ_s-1 U_s-2 ) }.
Notice that
∑_s=1^k _s+1:k ( γ_s U_s-1 - γ_s-1 U_s-2 )
= ∑_s=1^k _s+1:k ( γ_s U_s-1 + (1 - γ_s μ̃ / 2) ( γ_s-1 U_s-2 - γ_s-1 U_s-2 ) - γ_s-1 U_s-2 )
= ∑_s=1^k { ( _s+1:k γ_s U_s-1 - _s:k γ_s-1 U_s-2 ) - γ_s γ_s-1 μ̃ U_s-2 / 2 } ≤γ_k U_k-1 ≤γ_k Δ.
By Lemma <ref>, we have $\sum_{s=1}^k \oG_{s+1:k} \gamma_s^2 \leq 4 \gamma_k / \tilde{\mu}$ and the following is obtained
U_k ≤_1:k U_0 + { 4 ς/μ̃ ( 2 (2 σ^2 + ) + μ̃ ) + 3 Δ + 4 ς/μ̃ ( + 3 μ̃ ) Δ } γ_k .
The proof is completed.
§.§ Auxiliary Lemmas
Let $a>0$ and $\left(\gamma_{k}\right)_{k \geq 1}$ be a non-increasing sequence such that $\gamma_{1}< 2 / a$. If $\gamma_{k-1} / \gamma_k \leq 1 + (a/2)\gamma_k$ for any $k \geq 1$, then for any $k \geq 2$,
∑_j=1^k γ_j^2 ∏_ℓ=j+1^k(1-γ_ℓ a) ≤2/a γ_k.
The proof is elementary. Observe that:
∑_j=1^k γ_j^2 ∏_ℓ=j+1^k(1-γ_ℓ a) = γ_k ∑_j=1^k γ_j ∏_ℓ=j+1^k γ_ℓ-1 / γ_ℓ (1-γ_ℓ a)
≤γ_k ∑_j=1^k γ_j ∏_ℓ=j+1^k (1 + (a/2) γ_ℓ) (1-γ_ℓ a)
≤γ_k ∑_j=1^k γ_j ∏_ℓ=j+1^k (1-γ_ℓ (a/2) )
= 2 γ_k/a ∑_j=1^k ( ∏_ℓ=j+1^k (1 - γ_ℓa/2 ) - ∏_ℓ'=j^k (1 - γ_ℓ' a/2 ) )
= 2 γ_k/a ( 1 - ∏_ℓ'=1^k (1 - γ_ℓ' a/2 ) ) ≤2 γ_k/a.
The proof is concluded.
§ DETAILS OF THE NUMERICAL EXPERIMENTS
This section provides details about the numerical experiments on the second problem of strategic classification (SC) in <ref>. Moreover, we provide additional experiment results to better illustrate the performance of the state dependent SA algorithm for this problem.
The experiments conducted in this section are based on the Credit simulator provided at <https://github.com/zykls/performative-prediction>.
Our experiments are conducted on a server with Intel Xeon Gold 6138 CPU. The Python codes are executed in a single-thread environment.
There are two roles in the SC problem – learner and agents. The learner utilizes agents' information to obtain a classifier $f_{\theta}$. Meanwhile, individual agents hope to be assigned to a favorable class. To do so, they modify their features and thereby shifting the data distribution towards the target ${\cal D}(\theta)$.
Specifically, our experiments are done on the dataset with $m=18357$ samples as we select $d=3$ features to build the classifier. Each (original) data sample is given by $\bar{z}_i = (\bar{x}_i, \bar{y}_i)$ with the label $\bar{y}_i \in \{0,1\}$ and selected feature $\bar{x}_i \in \RR^3$. We associate each data sample to an agent. The task for the learner (bank) is to design a classifier that distinguishes whether the application of an individual (agent) who want to default a loan should be granted or not.
We simulate the adapted best response presented in Example 1 of the main paper. In this setting, the agents rely on their past experience to present data to the learner that is favorable to to agents. The latter is achieved by a gradient descent step that depends on the current learner's state ($\theta_k$), past agent's state ($z_k$) and the original data (${\cal D}_0$).
As the dynamics is coupled between the agents' and learner's update, we present the overall algorithm based on (<ref>), (<ref>) as follows:
Algorithm 2: State-dependent SA with Adapted Best Response.
Input: initial iterate $\theta_0 \in \RR^d$, agents' state $x_i^0 = \bar{x}_i$, $i \in \{1,...,m\}$ such that $\bar{x}_i$ is the $i$th original feature vector, step sizes $\{ \gamma_k \}_{k \geq 0}$, agents' response rate $\alpha > 0$, update parameter ${\sf b}$.
For $k=0,1,2, \ldots$
* A subset of agents, ${\cal I}_k$ with $|{\cal I}_k| = {\sf b}$, is selected uniformly from $\{1,...,m\}$. They adapt their feature vectors based on past experience and $\theta_k$ as:
x_i^k+1 = x_i^k + αU( x_i^k ; z̅_i, θ_k ), ∀ i ∈I_k, x_i^k+1 = x_i^k, ∀ i ∉I_k.
* An agent $i_k \in \{1,...,m\}$ is drawn uniformly to present data. Set $z_{k+1} = ( x_{i_k}^{k+1} , y_{i_k})$.
* The learner computes the $k+1$th iterate by:
θ_k+1=θ_k-γ_k+1 ℓ(θ_k; z_k+1).
The most recent iterate $\theta_{k+1}$ is deployed and made available to the agent(s).
Steps 1 & 2 in Algorithm 2 resemble the adaptive best response update in (<ref>). We emphasize that these two steps are agnostic to the learner as the latter only sees $z_{k+1}$ at iteration $k$, similarly, the last step is not known to the agents as the latter only sees the classifier given as $\theta_{k+1}$.
Furthermore, we recall that the following two types of utility functions are considered as $U(\cdot)$:
U_q(x^';z, θ) = θx'- x'-x^2/2ϵ,
U_lg(x^';z, θ) = y θx' - log(1+exp(θx'))- x'-x^2/2ϵ.
In step 1, the agents' response rate $\alpha$ and parameter ${\sf b}$ control the speed of adaptation among the group of $m$ agents. These parameters will affect the mixing time of the MC which determines the bounds in <Ref>.
Overall, we observe that the agents' states and learner's iterates are evolving simultaneously, highlighting the coupled nature in the analysis of the state-dependent SA algorithm.
In cases such as $U_{\sf lg}(\cdot)$ where the ideal best response $\argmax_{x'} U(x'; z, \theta)$ must be obtained via an iterative algorithm. From an algorithmic standpoint, the stateful nature for the agent is necessary for the performative prediction algorithm to converge to $\theta_{PS}$.
Additional Experiments Next, we provide additional experiments to illustrate the performance of the state-dependent SA algorithm from a few additional perspectives. Unless otherwise specified, we adopt the same parameters set in the experiments presented in the main paper.
In particular, we set $\beta = 1000/m$ in (<ref>), $\epsilon = 0.01$ in the utility functions, and in (<ref>), we set number of selected agents $|{\cal I}_k| = 5$, agents' response rate $\alpha = 0.5 \epsilon$.
The step size for (<ref>) is $\gamma_k = {c_0} / {(c_1+k)}$, $c_0 = {100} / {\tilde{\mu}}, c_1 = { 8 L^2 } / {\tilde{\mu}^2}$, where $L, \tilde{\mu}$ are estimated as $\sqrt{2 \beta m + \| X \|_F^2 / 2}$, $(1-\epsilon) \beta - \epsilon \| X \|_F^2 / 4m$, respectively. By default, the SA algorithm is executed as presented in Algorithm 2 with a batch size of ${\sf batch} = 1$ and the agents perform only ${\sf BR}=1$ best response update per SA update in step 3 of Algorithm 2.
Besides, we compare the convergence rates of the algorithms from the perspective of the agents – measured by the number of BR updates performed by the agents. This is the setting used in the plot of Fig. <ref> (right) and is denoted with the $x$-axis label of `no. of agent update'. We also compare the convergence from the perspective of the learner – measured by the number of samples requested from the agents by the learner. This setting is denoted with the $x$-axis label of `no. of samples drawn by learner'.
Convergence of SA algorithm with varying number of learner's updates per iteration.
Effects of Stateful Updates at Agents Notice that the comparison has been made in Fig. <ref> (right). Here, we again plot the convergence of the SA algorithm
to illustrate the convergence rates from the learner's perspective as well. We observe that the SA algorithms with stateful update converges as $k$ increases.
We vary the `learner's iteration' parameter to observe the effects on convergence when the learner is adapting at faster rate than the agents. This is achieved by repeating steps 2 and 3 in Algorithm 2 for multiple times. Notice that this setting is similar to the lazy deploy scheme in [Mendler-Dünner et al., 2020]. From the figure, we observe that doing so improves the convergence from the agents' perspective, while the sample efficiency (from the learner's perspective) is unaffected.
Convergence of SA algorithm with varying number of minibatch size.
Effects of Minibatch Size In this experiment, we consider the setting of $U_{\sf lg}(\cdot)$ and we draw different batch size of samples ($\hat{\sf b} \in \{1,5,10\}$) per iteration. To implement this, at step 2 of Algorithm 2, the learner draws $\hat{\sf b}$ agents uniformly as $\hat{\cal I}_k$, and at step 3, we update the iterate through:
\[
\theta_{k+1} = \theta_k - \gamma_{k+1} \frac{1}{\hat{\sf b}} \sum_{ j \in \hat{\cal I}_k } \grd \ell( \theta_k ; z_{k+1, j}).
\]
In Fig. <ref>, we compare the error $\| \theta_k - \theta_{PS} \|^2$ in terms of the number of agents' best response update[Since the agents only perform one best response update per iteration, the $x$-axis here is equivalent to the iteration number $k$.] and the number of samples drawn by the learner.
We find that increasing the minibatch reduces the variance of the gradient estimate, yet it can be less sample efficient from the learner's perspective.
Convergence of SA algorithm with varying number of best responses.
Effects of Number of Adaptive Best Responses In this experiment, we consider the setting of $U_{\sf lg}(\cdot)$ and at each iteration, the agents execute multiple rounds of adapted best response (${\sf BR} \in \{1,2,4\}$) to simulate the scenario when the agents are allowed with more time to respond to the published classifier $\theta_k$. To implement this, we repeat the update in (<ref>) of step 1 in Algorithm 2 for ${\sf BR}$ times.
Notice that this is reverse of Fig. <ref> (right) where the learner performs multiple iterations per agents' best response update.
In Fig. <ref>, we compare the error $\| \theta_k - \theta_{PS} \|^2$ in terms of the number of agent update and the number of samples drawn by the learner. We observe that increasing the number of best responses improves the performance slightly. However, as a drawback, it requires more computations/updates at the agents to reach the same performance.
|
theoremTheorem propositionProposition lemmaLemma corollaryCorollary
remarkRemark definitionDefinition assumptionAssumption
# Revisiting Tree Isomorphism:
AHU Algorithm with Primes Numbers
Florian Ingels
(Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France)
###### Abstract
The AHU algorithm has been the state of the art since the 1970s for
determining in linear time whether two unordered rooted trees are isomorphic
or not. However, it has been criticized (by Campbell and Radford) for the way
it is written, which requires several (re)readings to be understood, and does
not facilitate its analysis. In this paper, we propose an alternative version
of the AHU algorithm, which addresses this issue by being designed to be
clearer to understand and implement, with the same theoretical complexity and
equally fast in practice.. Whereas the key to the linearity of the original
algorithm lay on the careful sorting of lists of integers, we replace this
step by the multiplication of lists of prime numbers, and prove that this
substitution causes no loss in the final complexity of the new algorithm.
Keywords: tree isomorphism, AHU algorithm, prime numbers multiplication
## 1 Introduction
### 1.1 Context
The Aho, Hopcroft and Ullman (AHU) algorithm, introduced in the 1970s [1,
Example 3.2], establishes that the tree isomorphism problem can be solved in
linear time, whereas the more general graph isomorphism problem is still an
open problem today, where no proof of NP-completeness nor polynomial algorithm
is known [34], even though very efficient algorithms exist [27, 4].
As far as we know, AHU remains the only state-of-the-art algorithm for
practically determining whether two trees are isomorphic. Recently, Liu [25]
proposed to represent a tree by a polynomial of two variables, computable in
linear time, and where two trees have the same polynomial if and only if they
are isomorphic. Unfortunately, the existence of an algorithm to determine the
equality of two polynomials in polynomial time is still an open question [32].
We should also mention [10], which proposes an alternating logarithmic time
algorithm for tree isomorphism – under NC complexity class framework, that is,
problems efficiently solvable on a parallel computer [5].
One criticism made of the AHU algorithm concerns the way the algorithm is
presented in the original article, which is claimed to be
> utterly opaque. Even on second or third reading. When an algorithm is
> written it should be clear, it should persuade, and it should lend itself to
> analysis.
To the best of our knowledge, this remark seems to have remained a dead letter
in the community, and no alternative, clearer version of the algorithm seems
ever to have been published – with the exception of Campbell and Radford
themselves, but with quasilinear complexity instead of linear.
In this article, we propose to revisit the AHU algorithm by giving an
alternative version that is intended to be easier to understand and implement,
with the same theoretical complexity and equally fast in practice. This
variant is based on replacing multiset hashing, originally carried out in the
form of sorting lists of integers, by the elementary principle of multiplying
lists of prime numbers.
The rest of this section is devoted to introducing the notations and
definitions useful for the rest of the paper. Section 2 presents the AHU
algorithm, while Section 3 presents our variant of the same algorithm and its
complexity analysis; both versions are then compared numerically in Section 4.
### 1.2 Tree isomorphisms
A rooted tree $T$ is a connected directed graph without any undirected cycle
such that (i) there exists a special node called the root and (ii) any node
but the root has exactly one parent. The parent of a node $u$ is denoted by
$\operatorname{\mathcal{P}}(u)$, whereas its children are denoted by
$\operatorname{\mathcal{C}}(u)$. The leaves $\operatorname{\mathcal{L}}(T)$ of
$T$ are the nodes without any children. Rooted trees are said to be unordered
if the order among siblings is not significant. In this paper, we use _trees_
to refer to unordered rooted trees.
The degree of a node $u$ is defined as
$\deg(u)=\\#\operatorname{\mathcal{C}}(u)$ and the degree of a tree $T$ as
$\deg(T)=\max_{u\in T}\deg(u)$. The depth $\operatorname{\mathcal{D}}(u)$ of a
node $u$ is the length of the path between $u$ and the root. The depth
$\operatorname{\mathcal{D}}(T)$ of $T$ is the maximal depth among all nodes.
The level of a node $u$ is defined as
$\operatorname{\mathcal{D}}(T)-\operatorname{\mathcal{D}}(u)$. The sets of
nodes of level $d$ in a tree $T$ is denoted by $T^{d}$, and the mapping
$d\mapsto T^{d}$ can be constructed in linear time by a simple traversal of
$T$.
Two trees $T_{1}$ and $T_{2}$ are said to be _isomorphic_ if there exists a
bijective mapping $\varphi:T_{1}\to T_{2}$ so that (i) the roots are mapped
together and (ii) for any $u,v\in T_{1}$,
$v\in\operatorname{\mathcal{C}}(u)\iff\varphi(v)\in\operatorname{\mathcal{C}}(\varphi(u))$.
Such a mapping $\varphi$ is called a _tree isomorphism_. In other words, two
trees are isomorphic if one can be obtained from the other by simply swapping
the children of each node. An example of isomorphic trees is provided in
Figure 1.
Whenever two trees $T_{1}$ and $T_{2}$ are isomorphic, we note $T_{1}\simeq
T_{2}$. It is well known that $\simeq$ is an equivalence relation on the set
of trees [37]. The _tree isomorphism problem_ consists in deciding whether two
trees are isomorphic or not.
Figure 1: Two isomorphic trees.
For the broader graph isomorphism problem, it is not usual to explicitly
construct the isomorphism $\varphi$ – let us mention nonetheless [14, Section
3.3] and [19] – but rather to compute a certificate of non-isomorphism. For
instance, Weisfeiler-Lehman algorithms, also known as colour refinement
algorithms [18, 21], colour the nodes of each graph according to certain
rules, and the histograms of the colour distributions are then compared: if
they diverge, the graphs are not isomorphic. This test is not complete in the
sense that there are non-isomorphic graphs with the same colour histogram –
even though the distinguishing power of these algorithms is constantly being
improved [16].
While the graph isomorphism problem is not solved in the general case, it is
solved for trees by virtue of the AHU algorithm, which is built on a colouring
principle similar to that of Weisfeiler-Lehman.
## 2 The Aho, Hopcroft and Ullman algorithm
In this section we introduce AHU algorithm, that solves the tree isomorphism
problem. First, we present the general principle of the algorithm in Section
2.1, before reproducing and commenting in Section 2.2 the original
presentation of the algorithm as it can be found in [1].
### 2.1 Principle
In [11], Campbell and Radford provide a very clear, step-by-step exposition of
the intuitions that lead to the AHU algorithm, and we invite the interested
reader to consult it. For the sake of self-containment, we offer here another
intuition of how the AHU algorithm works, presented as a colouring process,
thus making the connection with Weisfeiler-Lehman algorithms for graph
isomorphism.
The core idea behind AHU algorithm is to provide each node in trees $T_{1}$
and $T_{2}$ a canonical representative of its equivalence class for $\simeq$,
thus containing all the information about its descendants. The trees are
isomorphic if and only if the canonical representatives of the roots are
identical.
The nodes of both trees are simultaneously browsed in ascending levels.
Suppose that each node $u$ of level $d-1$ has been assigned a colour $c(u)$,
supposed to represent its equivalence class for the relation $\simeq$. Each
node $u$ of level $d$ is associated with a multiset
$\operatorname{\operatorname{\mathcal{C}}_{c}}(u)=\\{\\!\\{c(v):v\in\operatorname{\mathcal{C}}(u)\\}\\!\\}$
– if $u$ is a leaf, this multiset is denoted $\emptyset$. Each distinct
multiset is given a colour, which is assigned to the corresponding nodes. An
illustration is provided in Figure 2.
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Figure 2: Assigning colours to nodes in AHU algorithm.
In the end, the trees are isomorphic if and only if their roots receive the
same colour. Moreover, after processing level $d$, if the multiset of colours
assigned to the nodes of level $d$ differs from one tree to the other, we can
immediately conclude that the trees are not isomorphic.
In practice, colours are represented by integers. The pseudocode for this
ideal version of the AHU algorithm is given in Algorithm 1. We say ideal
because it ignores an important implementation problem. Indeed, multisets must
be treated carefully: in line 9 to find out whether a colour has already been
assigned to multiset $\operatorname{\operatorname{\mathcal{C}}_{c}}(u)$, and
in line 13 to check whether the colours assigned to level $d$ coincide between
the trees.
Input: $T_{1},T_{2}$
Output: $\top$ if and only if $T_{1}\simeq T_{2}$
1 if
_$\operatorname{\mathcal{D}}(T_{1})\neq\operatorname{\mathcal{D}}(T_{2})$_
then
2 return $\bot$
3else
4 for _$d$ from $0$ to $\operatorname{\mathcal{D}}(T_{1})$_ do
5 $k\leftarrow 0$
6 Let $f:\emptyset\mapsto 0$
7 for _$i\in\\{1,2\\}$ and $u\in T_{i}^{d}$_ do
8
$\operatorname{\operatorname{\mathcal{C}}_{c}}(u)\leftarrow\\{\\!\\{c(v):v\in\operatorname{\mathcal{C}}(u)\\}\\!\\}$
9 if _$f(\operatorname{\operatorname{\mathcal{C}}_{c}}(u))$ _is not defined__
then
10 $k\leftarrow k+1$
11 Define $f(\operatorname{\operatorname{\mathcal{C}}_{c}}(u))=k$
12
13 $c(u)\leftarrow f(\operatorname{\operatorname{\mathcal{C}}_{c}}(u))$
14
15 if _$\\{\\!\\{c(u):u\in T_{1}^{d}\\}\\!\\}\neq\\{\\!\\{c(u):u\in
T_{2}^{d}\\}\\!\\}$_ then
16 return $\bot$
17
18 return $\top$
Algorithm 1 IdealAHU
The latter can be addressed using pigeonhole sort [8] in linear time; whereas
for the former, there are two main options: either by using hash functions
specifically designed for multisets [12, 26], or by treating these multisets
as lists that are sorted before being hashed or compared. The second approach
is commonly used, both by AHU and by Weisfeiler-Lehman algorithms.
Before examining the actual AHU algorithm in detail, we investigate the
complexity of Algorithm 1. If we assume that determining whether
$f(\operatorname{\operatorname{\mathcal{C}}_{c}}(u))$ is defined or not, at
line 9, can be accomplished in constant time (e.g. assuming a perfect hash
function [23] working with multisets), then we have the following result.
Algorithm 1 runs in $O(n)$, where $n=\\#T_{1}=\\#T_{2}$. Proof. Fix a level
$d$ and a node $u\in T_{i}^{d}$. Building
$\operatorname{\operatorname{\mathcal{C}}_{c}}(u)$ requires $O(\deg(u))$;
noticing that $\sum_{u\in T_{i}^{d}}\deg(u)=\\#T_{i}^{d-1}$ and that the
comparison in line 13 can be done in $O(\\#T_{i}^{d})$ – e.g. with pigeonhole
sort [8]; summing over $d$ leads to the result.
Notably, this establishes that the tree isomorphism problem is solvable in
linear time, provided that the assumption made above is valid. Note that there
also exists a naive algorithm for tree isomorphism in $O(n^{2})$ [11] which
makes extensive use of Knuth tuples [22].
### 2.2 Original algorithm
The description of the AHU algorithm in the original article [1, Example 3.2]
is quite different from what has been presented previously. For the sake of
self-containedness, we reproduce it here, where only minor changes have been
made to fit the notations used in this paper:
1. 1.
GreenBlackFirst, assign to all leaves in $T_{1}$ and $T_{2}$ the integer $0$.
2. 2.
Assume by induction that all nodes at level $d-1$ of $T_{1}$ and $T_{2}$ have
been assigned an integer. Let $L_{1}$ (respectively $L_{2}$) be the list of
nodes in $T_{1}$ (respectively $T_{2}$) at level $d-1$ sorted by non-
decreasing value of the assigned integers.
3. 3.
Assign to the nonleaves of $T_{1}$ at level $d$ a tuple of integers by
scanning the list $L_{1}$ from left to right and performing the following
actions:
* •
For each vertex on list $L_{1}$ take the integer assigned to $u$ to be the
next component of the tuple associated with $\operatorname{\mathcal{P}}(u)$.
* •
On completion of this step, each nonleaf $w$ of $T_{1}$ at level $d$ will have
a tuple $(i_{1},i_{2},\dots,i_{k})$ associated with it, where
$i_{1},\dots,i_{k}$ are the integers, in non-decreasing order, associated with
the children of $w$.
* •
Let $S_{1}$ be the sequence of tuples created for the vertices of $T_{1}$ on
level $d$.
4. 4.
Repeat Step 3 for $T_{2}$ and let $S_{2}$ be the sequence of tuples created
for the vertices of $T_{2}$ on level $d$.
5. 5.
Sort $S_{1}$ and $S_{2}$ lexicographically. Let $S_{1}^{\prime}$ and
$S_{2}^{\prime}$, respectively, be the sorted sequence of tuples.
6. 6.
If $S_{1}^{\prime}$ and $S_{2}^{\prime}$ are not identical, then halt: the
trees are not isomorphic. Otherwise, assign the integer $1$ to those vertices
of $T_{1}$ on level $d$ represented by the first distinct tuple on
$S_{1}^{\prime}$, assign the integer $2$ to the vertices represented by the
second distinct tuple, and so on. As these integers are assigned to the
vertices of $T_{1}$ on level $d$, replace $L_{1}$ by the list of the vertices
so assigned. Append the leaves of $T_{1}$ on level $d$ to the front of
$L_{1}$. Do the same for $L_{2}$. $L_{1}$ and $L_{2}$ can now be used for the
assignment of tuples to nodes at level $d+1$ by returning to Step 3.
7. 7.
If the roots of $T_{1}$ and $T_{2}$ are assigned the same integer, $T_{1}$ and
$T_{2}$ are isomorphic.
Note that, in Step 5, the authors resort to a variant of radix sort [1,
Algorithm 3.2]. Actually, the tree isomorphism problem and AHU algorithm are
only introduced in the book as an application example of this sorting
algorithm.
To analyse the complexity of AHU algorithm, the authors make the assumption
that trees are sufficiently small so that they can be described by a $k$ bit
word (i.e. with a 64-bit machine, $\\#T<2^{64}$). For the purpose of this
paper, we reframe this assumption as follows.
For any considered tree $T$, $\log\\#T=O(1)$.
With this assumption, they show that tree isomorphism can be solved in linear
time.
AHU algorithm runs in $O(n)$ where $n=\\#T_{1}=\\#T_{2}$. Proof. See the
proofs in [1, Example 3.2] for the whole algorithm and especially [1,
Algorithm 3.2] for sorting lists $S_{1}$ and $S_{2}$ in Step 5.
Assumption 2.2 is made to ensure that the largest integer manipulated in the
various lists is not too large, and therefore that the (linear) sorting
algorithm for these lists can effectively consider these numbers as integers
and not as sequences of 0s and 1s.
If Assumption 2.2 is relaxed, there are (large) trees for which the algorithm
runs in $O(n\log n)$; see [11].
As already mentioned in the introduction, Campbell and Radford describe this
formulation of the algorithm as “utterly opaque. Even on second or third
reading.” (sic) [11]. In their view, this is detrimental to understanding the
algorithm and being able to analyse and implement it. Based on this
observation, the natural question that arises is whether it is possible to
find a version of AHU algorithm that is easier to understand and analyze,
while remaining linear under Assumption 2.2, and manipulates only elementary
concepts, just as original AHU does by sorting lists of integers.
Hash functions designed for multisets have already been mentioned [12, 26],
but they involve advanced concepts, which would make implementation difficult
for non-specialists. For this reason, they are beyond the scope of this paper.
Instead, we propose an algorithm that uses only elementary concepts, replacing
hash of multisets by multiplications of primes numbers.
## 3 Revisiting AHU algorithm
In Algorithm 1, we need to associate a unique integer
$f(\operatorname{\operatorname{\mathcal{C}}_{c}}(u))$ to each distinct
multiset $\operatorname{\operatorname{\mathcal{C}}_{c}}(u)$ of integers
encountered. There is a particularly simple and fundamental example where
integers are associated with multisets: prime factorization. Indeed, through
the fundamental theorem of arithmetic, there is a bijection between integers
and multisets of primes. For example, $360=2^{3}\cdot 3^{2}\cdot 5$ is
associated to the multiset $\\{\\!\\{2,2,2,3,3,5\\}\\!\\}$. Note that this
bijection is well known [9], and has already been successfully exploited in
the literature for prime decomposition, but also usual operations such as
product, division, gcd and lcd of numbers [36].
To the best of our knowledge, this link has never been exploited to replace
multiset hashing, a fortiori in the context of graph isomorphism algorithms –
such as Weisfeiler-Lehman, or AHU for trees. Note, however, that this approach
has been used in the context of evaluating poker hands [35], where prime
multiplication has been preferred to sorting cards by value in order to get a
unique identifier for each distinct possible hand.
Since the original AHU sorts lists of integers, the main difficulty in making
this substitution is to ensure that the complexity of multiplying lists of
primes does not exceed that of sorting lists of integers.
In Section 3.1, we present our version of AHU algorithm which uses
multiplication of primes; while Section 3.2 presents its complexity analysis
and shows that, under the same Assumption 2.2 as original AHU, our algorithm
is still linear.
### 3.1 AHU algorithm with primes
Suppose that each node $u$ at level $d$ has received a prime number $c(u)$,
assuming that all nodes at that level and of the same class of equivalence
have received the same number. Then, to a node $u$ at level $d$, instead of
associating the multiset
$\operatorname{\operatorname{\mathcal{C}}_{c}}(u)=\\{\\!\\{c(v):v\in\operatorname{\mathcal{C}}(u)\\}\\!\\}$,
we associate the number $N(u)=\prod_{v\in\operatorname{\mathcal{C}}(u)}c(v)$.
The nodes of level $d$ are then renumbered with prime numbers – where each
distinct number $N(u)$ gets a distinct prime. The fundamental theorem of
arithmetic ensures that two identical multisets
$\operatorname{\operatorname{\mathcal{C}}_{c}}(\cdot)$ receive the same number
$N(\cdot)$. The pseudocode for this new version of AHU is presented in
Algorithm 2.
This version of AHU differs from the ideal version presented in Algorithm 1 by
only few lines: line 4, which defines additional variables required for this
version; line 9 (respectively line 8 in Algorithm 1), which replaces the
multiset $\operatorname{\operatorname{\mathcal{C}}_{c}}(u)$ with the number
$N(u)$; and line 11 (respectively line 10), that replaces the increment of $k$
by providing a new prime number $p$.
Input: $T_{1},T_{2}$
Output: $\top$ if and only if $T_{1}\simeq T_{2}$
1 if
_$\operatorname{\mathcal{D}}(T_{1})\neq\operatorname{\mathcal{D}}(T_{2})$_
then
2 return $\bot$
3else
4 $P\leftarrow[2,3,5,7,11,13]$ and $N_{\text{sieve}}\leftarrow 16$
5 for _$d$ from $0$ to $\operatorname{\mathcal{D}}(T_{1})$_ do
6 Let $f:1\mapsto 2$
7 $p\leftarrow 2$
8 for _$i\in\\{1,2\\}$ and $u\in T_{i}^{d}$_ do
9 $N(u)\leftarrow\displaystyle\prod_{v\in\operatorname{\mathcal{C}}(u)}c(v)$
10 if _$f(N(u))$ _is not defined__ then
11 $N_{\text{sieve}},P,p\leftarrow\textsc{NextPrime}(N_{\text{sieve}},P,p)$
12 Define $f(N(u))=p$
13
14 $c(u)\leftarrow f(N(u))$
15
16 if _$\\{\\!\\{c(u):u\in T_{1}^{d}\\}\\!\\}\neq\\{\\!\\{c(u):u\in
T_{2}^{d}\\}\\!\\}$_ then
17 return $\bot$
18
19 return $\top$
Algorithm 2 PrimesAHU
The subroutine NextPrime, introduced in Algorithm 3, returns the next prime
not already used at the current level; if there is no unassigned prime in the
current list $P$, then new primes are generated using a segmented version of
the sieve of Eratosthenes.
Input: $N_{\text{sieve}}$, the largest number for which the sieve of
Eratosthenes has already been done, the list $P$ of primes $\leq
N_{\text{sieve}}$ in ascending order, with $\operatorname{length}(P)\geq 6$,
and a prime $p\in P$
1 if _$p$ _is the last of element of_ $P$_ then
2
$N_{\text{sieve}},P\leftarrow\textsc{SieveOfEratosthenes}(N_{\text{sieve}},P)$
/* At least one new prime has been added to $P$ */
3
4Let $p^{\prime}$ be the next prime after $p$ in $P$
return $N_{\text{sieve}},P,p^{\prime}$
Algorithm 3 NextPrime
Let us denote $p_{n}$ the $n$-th prime number. There are well known bounds on
the value of $p_{n}$ [13, 31] – with $\ln$ denoting the natural logarithm and
$n\geq 6$:
$n(\ln n+\ln\ln n-1)<p_{n}<n(\ln n+\ln\ln n).$ (1)
Suppose we have the list of all primes $P\leq N_{\text{sieve}}$, where
$N_{\text{sieve}}$ is the largest integer sieved so far. With $\\#P=n-1$, to
generate $p_{n}$, we simply resume the sieve up to the integer $\lceil n(\ln
n+\ln\ln n)\rceil$, starting from $\lfloor n(\ln n+\ln\ln n-1)\rfloor$ or
$N_{\text{sieve}}+1$, whichever is greater – to make sure there is no overlap
between two consecutive segments of the sieve. With this precaution in mind,
the total complexity of the segmented sieve is the same as if we had directly
performed the sieve in one go [6]; i.e., $O(N\log\log N)$ for a sieve
performed up to integer $N$. Therefore, to generate the first $n$ prime
numbers, according to (1), the complexity of the sieve can be evaluated as
$O\left(n\cdot(\log n+\log\log n)\cdot\log\log\big{(}n\cdot(\log n+\log\log
n)\big{)}\right).$ We refer the reader to [29] for practical considerations on
the implementation of the segmented sieve of Eratosthenes.
Note that other sieve algorithms exist, with better complexities – such as
Atkin sieve [2] or the wheel sieve [30]; the sieve of Eratosthenes has the
merit of being the simplest to implement and sufficient for our needs. Also, a
better asymptotic complexity but with a worse constant can be
counterproductive for producing small primes – which is rather our case since
we generate the primes in order.
### 3.2 Complexity analysis
There are several points to adress when analysing Algorithm 2: (i) the
complexity for testing whether or not $f(N)$ is defined in line 10; (ii) the
number of primes required by the algorithm, and the complexity for generating
them; and (iii) the complexity of multiplication in line 9. To simplify the
notations, let us assume that we run the algorithm with $T_{1}=T_{2}=T$ – this
is the worst case, since, if $T_{1}\not\simeq T_{2}$, we do not visit all the
levels.
#### (i)
The question of determining in $O(1)$ whether $f(N)$ is defined or not is not
trivial in Algorithm 1 where $f(\cdot)$ hashes multisets; however, for
integers it is possible by virtue of [15].
Note also that [15] also provides a way to create a table, which associates
with integer $i$ the $i$-th prime number, searchable in $O(1)$. An application
of this table is found in line 14: one can use pigeonhole sort to compare the
two lists, with complexity $O(\\#T^{d}+p_{n})$, where $p_{n}$ is the biggest
prime in the list; but many holes will be unnecessary (as $c(u)$ is
necessarily prime). Using the table, one can use only $n$ holes, one for each
prime number, which reduces the complexity to $O(\\#T^{d}+n)$. Since the
primes are reallocated at each level, at level $d$ we need as many primes as
there are different equivalence classes at that level – i.e. $\\#\\{c(u):u\in
T^{d}\\}$. This number is $\leq\\#T^{d}$, therefore the complexity of the sort
collapses to $O(\\#T^{d})$.
#### (ii)
As already discussed, to generate the first $n$ primes, the sieve must be
carried out up to the integer $N=n\cdot(\ln n+\ln\ln n)$, for total complexity
$O(N\log\log N)$.
The number of primes required by Algorithm 2 at level $d$ is $\\#\\{c(u):u\in
T^{d}\\}$. Thus, in total, the number of primes needed is exactly
$\displaystyle\max_{d\in[\\![0,\operatorname{\mathcal{D}}(T)]\\!]}\\#\\{c(u):u\in
T^{d}\\}$. We call this number the _width_ of $T$ and denote it by
$\operatorname{\mathcal{W}}(T)$. We have the following key result.
For any tree $T$,
$\operatorname{\mathcal{W}}(T)\cdot\left(\ln\operatorname{\mathcal{W}}(T)+\ln\ln\operatorname{\mathcal{W}}(T)\right)=O(\\#T)$.
Proof. The proof can be found in Appendix A.
It follows that generating the primes required for Algorithm 2 is done in
$O(\\#T\log\log\\#T)$.
#### (iii)
Let us denote $M(n)$ the complexity for multiplying two $n$-bits numbers,
where $M(n)$ depends on the algorithm used: $O(n^{2})$ for usual schoolbook
algorithm, $O(n^{1.585})$ with Karatsuba algorithm [20], and $O(n\cdot\log
n\cdot\log\log n)$ with Schönhage–Strassen algorithm [33]. The fastest known
algorithm was recently introduced [17], with complexity $O(n\cdot\log n)$ –
even if this result is, by the authors’ own admission, primarily theoretical.
For our complexity proof, we assume that the algorithm used is
Schönhage–Strassen; note that, in practice, for small integers, algorithms
with a worse complexity but a better constant are used. Nonetheless, we assume
in the sequel that $M(n)=n\cdot\log n\cdot\log\log n$, for the purpose of the
proof.
Multiplying two $n$-bits numbers together yields a $2n$-bits number.
Therefore, if we sequentially multiply $m$ numbers of $n$ bits together, the
total complexity can be evaluated as $M(n)+M(2n)+\dots+M((m-1)n)$, which is
$O(m\cdot M(mn))$. A better way is to adopt a divide and conquer approach and
multiply two numbers which themselves are the recursive product of $m/2$
numbers. This strategy leads to a complexity of $O(M(mn))$ by virtue of the
Master Theorem [7].
#### Conclusion
Combining all the above discussions, we get the following result.
Algorithm 2 runs in
$O\left(\\#T\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\cdot\log\left(\deg(T)\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\right)\cdot\log\log\left(\deg(T)\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\right)+\\#T\log\log\\#T\right)$
where $p_{\operatorname{\mathcal{W}}(T)}$ is the largest prime needed by the
algorithm. Proof. By the previous discussion in (ii), we consider separately
the generation of primes, whose total complexity is $O(\\#T\log\log\\#T)$.
Now, fix $d\in[\\![0,\operatorname{\mathcal{D}}(T)]\\!]$ and $u\in T^{d}$.
Computing $N(u)$ implies multiplying $\deg(u)$ primes with at most $\log
p_{\operatorname{\mathcal{W}}(T)}$ bits, which is, following (iii),
$O\left(M\left(\deg(u)\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\right)\right)$ – with $M(n)=n\cdot\log
n\cdot\log\log n$ as stated earlier. Lines 10 to 13 are $O(1)$ from the
discussion in (i). Sorting the lists in line 14 is $O(\\#T^{d})$ – also from
(i).
Processing level $d$ thus requires
$O\left(\\#T^{d-1}\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\cdot\log\left(\deg(T)\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\right)\cdot\log\log\left(\deg(T)\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\right)+\\#T^{d}\right),$
noticing that $\sum_{u\in T^{d}}\deg(u)=\\#T^{d-1}$ and bounding other
occurrences of $\deg(u)$ by $\deg(T)$. Summing over $d$ leads to the claim.
As already stated, the original AHU algorithm is linear only under the
assumption that trees are not too large – recall Assumption 2.2. The term
$\log\log\\#T$ coming from the generation of primes immediately vanishes. We
now analyse the term
$\log p_{\operatorname{\mathcal{W}}(T)}\cdot\log\left(\deg(T)\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\right)\cdot\log\log\left(\deg(T)\cdot\log
p_{\operatorname{\mathcal{W}}(T)}\right).$
First, since $\deg(T)<\\#T$, we have $\log\deg(T)=O(1)$. Using (1) we have
$p_{\operatorname{\mathcal{W}}(T)}<\operatorname{\mathcal{W}}(T)\left(\ln\operatorname{\mathcal{W}}(T)+\ln\ln\operatorname{\mathcal{W}}(T)\right),$
where $\ln$ is the natural logarithm. Using Proposition 3.2, we have
$p_{\operatorname{\mathcal{W}}(T)}=O(\\#T)$. It follows immediately that $\log
p_{\operatorname{\mathcal{W}}(T)}=O(1)$. The nested logarithms follows without
difficulty.
Finally, we have proven the following result.
Algorithm 2 runs in $O(n)$ with $n=\\#T_{1}=\\#T_{2}$.
## 4 Numerical experiments
We established that PrimesAHU is equivalent in complexity (under the same
assumption) to the original version. However, this theoretical result would be
of little interest, especially with regard to the “ease of implementation”
argument, if the constant were much larger, resulting in disproportionately
long calculation times compared with the original algorithm. We show here that
this is not the case, by comparing on random trees two Python implementations
of the algorithm, where (i) PrimesAHU turns out to be faster in the case when
the trees are isomorphic, and (ii) of comparable time in the case when they
are not.
Section 4.1 provides some insights about our implementation of the two
algorithms, while Section 4.2 presents the results obtained.
### 4.1 Comments on the implementation
We implemented in Python the two algorithms discussed in this paper, the
original AHU (oAHU for short) and PrimesAHU (pAHU for short), making extensive
use of the `treex` library [3], designed to manipulate trees.
We tried, as far as possible, to re-implement all the auxiliary functions used
by the algorithms, to avoid an unfair advantage, linked, for example, to the
use of highly optimised functions in Python. Typically, to multiply lists of
prime numbers, we have implemented and used the recursive procedure described
in Section 3.2-(ii), even though the `numpy.prod` function is faster when
lists become sufficiently large. Note, however, that for multiplication
operations `*`, we let Python choose the appropriate algorithm (schoolbook for
small numbers and Karatsuba for large numbers), whereas we chose
Schönhage–Strassen for the purposes of our proof of complexity in Section 3.2
– despite being slower in practice for small numbers.
Concerning perfect hash function for integers, we used Python dictionaries.
Note also our implementation of the segmented sieve of Eratosthenes ignores
multiples of $2$ and $3$, thus making the sieve $6$ times faster.
Experiments have been conducted on a HP Elite Notebook with 32 Go of RAM and
Intel Core i7-1365U processor.
### 4.2 Results
Provided an integer $n$, to treat the case $T_{1}\simeq T_{2}$, we generate a
random recursive tree $T$ of size $n$ [38], generate a copy $T^{\prime}$ of
$T$, and measure the computation time taken for both algorithms – oAHU and
pAHU – on the couple $(T,T^{\prime})$. To treat the case $T_{1}\not\simeq
T_{2}$, we directly generate a pair $(T_{1},T_{2})$ of random recursive trees
of size $n$ and measure the computation time for both algorithm.
We generated $100$ couples for both cases and with $n=10^{i}$ for each
$i\in[\\![1,6]\\!]$. The results are depicted in Figure 3. As expected, both
algorithms behave linearly; pAHU is faster in the case $T_{1}\simeq T_{2}$, as
can be seen in Figure 3(a), whereas it achieves comparable time to oAHU in the
case $T_{1}\not\simeq T_{2}$, as soon as the trees are not too small – see
Figure 3(b). Note also that it is about 10 times faster to conclude
$\not\simeq$ than $\simeq$ on the examples considered.
It is not our intention here to explain the differences in performance between
the two algorithms – that is a topic requiring more detailed analysis – but
suffice it to say that our algorithm, pAHU, is capable of performing just as
well as oAHU in practice, which was our objective in this section.
(a) $T_{1}\simeq T_{2}$ case
(b) $T_{1}\not\simeq T_{2}$ case
Figure 3: Computation time (in seconds, log scale) for testing isomorphism of
random recursive trees of sizes $10^{i}$, $i\in[\\![1,6]\\!]$, using either
oAHU or pAHU algorithm, with $100$ couples of trees tested for each size and
each case.
## Conclusion and perspectives
Following a remark by Campbell & Radford, who deplored the lack of clarity of
the AHU algorithm (fundamental to understanding the tree isomorphism problem),
we proposed a variant of this same algorithm: (i) with the same theoretical
complexity; (ii) just as fast in practice, and (iii) intended to be simple to
understand and implement. AHU works by sorting lists of integers, with the aim
of computing a unique hash of multisets. We proposed instead to use an equally
elementary concept, the multiplication of prime numbers, to also compute an
invariant – thanks to the fundamental theorem of arithmetic.
We mentioned previously that the Weisfeiler-Lehman algorithms, used for graph
isomorphism, use integer list sorts just like AHU to determine the next colour
to assign to each node. This raises the question of whether these sorts can
also be replaced by our idea of multiplying lists of prime numbers. While this
issue is outside the scope of this paper, and remains to be investigated, let
us nonetheless mention two points that may prove challenging.
First, the way Weisfeiler-Lehman operates can lead to processing as many
colours as there are nodes in the graph, and therefore having to generate as
many prime numbers – requiring the sieve of Eratosthenes to be run up to an
integer supra-linear in the size of the graph.
Next, we would multiply lists whose size depends on the degree of the current
node; in a dense or complete graph, this means lists whose size is comparable
to the number of nodes in the graph. The complexity of performing these
multiplications could prove far more expensive than for trees.
Since (1-dimensional) Weisfeiler-Lehman can be implemented in
$O((\\#V+\\#E)\log\\#V)$ for a graph $G=(V,E)$, it remains to be investigated
whether or not the additional complexities mentioned above exceeds that of the
original algorithm. See [21, Section 3.1] and references therein for a more
precise description of the Weisfeiler-Lehman algorithms.
### Acknowledgements
The author would like to thank Dr. Romain Azaïs and Dr. Jean Dupuy for their
helpful suggestions on the first draft of the article.
## References
* [1] Alfred V. Aho, John E. Hopcroft and Jeffrey D. Ullman “The Design and Analysis of Computer Algorithms” Reading, Mass.: Addison-Wesley, 1974
* [2] Arthur Atkin and Daniel Bernstein “Prime sieves using binary quadratic forms” In _Mathematics of Computation_ 73.246, 2004, pp. 1023–1030
* [3] Romain Azaïs, Guillaume Cerutti, Didier Gemmerlé and Florian Ingels “Treex: a Python package for manipulating rooted trees” In _Journal of Open Source Software_ 4.38, 2019, pp. 1351
* [4] László Babai “Graph isomorphism in quasipolynomial time” In _Proceedings of the forty-eighth annual ACM symposium on Theory of Computing_ , 2016, pp. 684–697
* [5] David A Barrington “Bounded-width polynomial-size branching programs recognize exactly those languages in NC” In _Proceedings of the eighteenth annual ACM symposium on Theory of computing_ , 1986, pp. 1–5
* [6] Carter Bays and Richard H Hudson “The segmented sieve of Eratosthenes and primes in arithmetic progressions to $10^{12}$” In _BIT Numerical Mathematics_ 17.2 Springer, 1977, pp. 121–127
* [7] Jon Louis Bentley, Dorothea Haken and James B Saxe “A general method for solving divide-and-conquer recurrences” In _ACM SIGACT News_ 12.3 ACM New York, NY, USA, 1980, pp. 36–44
* [8] Paul E Black “Dictionary of algorithms and data structures”, 1998
* [9] Wayne D. Blizard “Multiset theory” In _Notre Dame Journal of formal logic_ 30.1, 1989, pp. 36–66
* [10] Samuel R. Buss “Alogtime algorithms for tree isomorphism, comparison, and canonization” In _Kurt Gödel Colloquium on Computational Logic and Proof Theory_ , 1997, pp. 18–33 Springer
* [11] Douglas M Campbell and David Radford “Tree isomorphism algorithms: Speed vs. clarity” In _Mathematics Magazine_ 64.4 Taylor & Francis, 1991, pp. 252–261
* [12] Dwaine Clarke et al. “Incremental multiset hash functions and their application to memory integrity checking” In _International conference on the theory and application of cryptology and information security_ , 2003, pp. 188–207 Springer
* [13] Pierre Dusart “The $k$-th prime is greater than $k(\ln k+\ln\ln k-1)$ for $k\geq 2$” In _Mathematics of computation_ JSTOR, 1999, pp. 411–415
* [14] Scott Fortin “The graph isomorphism problem”, 1996
* [15] Michael L Fredman, János Komlós and Endre Szemerédi “Storing a sparse table with 0 (1) worst case access time” In _Journal of the ACM (JACM)_ 31.3 ACM New York, NY, USA, 1984, pp. 538–544
* [16] Martin Grohe, Pascal Schweitzer and Daniel Wiebking “Deep weisfeiler leman” In _Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)_ , 2021, pp. 2600–2614 SIAM
* [17] David Harvey and Joris Van Der Hoeven “Integer multiplication in time O (n log n)” In _Annals of Mathematics_ 193.2 JSTOR, 2021, pp. 563–617
* [18] Ningyuan Teresa Huang and Soledad Villar “A Short Tutorial on The Weisfeiler-Lehman Test And Its Variants” In _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2021, pp. 8533–8537 IEEE
* [19] Florian Ingels and Romain Azaïs “Isomorphic unordered labeled trees up to substitution ciphering” In _International Workshop on Combinatorial Algorithms_ , 2021, pp. 385–399 Springer
* [20] Anatolii Alekseevich Karatsuba and Yu P Ofman “Multiplication of many-digital numbers by automatic computers” In _Doklady Akademii Nauk_ 145.2, 1962, pp. 293–294 Russian Academy of Sciences
* [21] Sandra Kiefer “Power and limits of the Weisfeiler-Leman algorithm”, 2020
* [22] Donald E Knuth “Art of Computer Programming, Volume 4, Fascicle 4, The: Generating All Trees–History of Combinatorial Generation” Addison-Wesley Professional, 2013
* [23] Donald E Knuth “The Art of Computer Programming, Vol. 3: Sorting and Searching,” 1973” In _This book describes the binary search on_ , pp. 409–426
* [24] Donald E Knuth “The Art of Computer Programming: Fundamental Algorithms, volume 1” Addison-Wesley Professional, 1997
* [25] Pengyu Liu “A tree distinguishing polynomial” In _Discrete Applied Mathematics_ 288 Elsevier, 2021, pp. 1–8
* [26] Jeremy Maitin-Shepard, Mehdi Tibouchi and Diego F Aranha “Elliptic curve multiset hash” In _The Computer Journal_ 60.4 Oxford University Press, 2017, pp. 476–490
* [27] Brendan D. McKay and Adolfo Piperno “Practical graph isomorphism, II” In _Journal of symbolic computation_ 60 Elsevier, 2014, pp. 94–112
* [28] Shin-ichi Nakano and Takeaki Uno “Efficient generation of rooted trees” In _National Institute for Informatics (Japan), Tech. Rep. NII-2003-005E_ 8, 2003, pp. 4–63
* [29] Melissa e O’neill “The genuine sieve of Eratosthenes” In _Journal of Functional Programming_ 19.1 Cambridge University Press, 2009, pp. 95–106
* [30] Paul Pritchard “Linear prime-number sieves: A family tree” In _Science of computer programming_ 9.1 Elsevier, 1987, pp. 17–35
* [31] Barkley Rosser “Explicit bounds for some functions of prime numbers” In _American Journal of Mathematics_ 63.1 JSTOR, 1941, pp. 211–232
* [32] Nitin Saxena “Progress on Polynomial Identity Testing.” In _Bull. EATCS_ 99, 2009, pp. 49–79
* [33] Arnold Schönhage and Volker Strassen “Schnelle multiplikation grosser zahlen” In _Computing_ 7.3 Springer, 1971, pp. 281–292
* [34] Uwe Schöning “Graph isomorphism is in the low hierarchy” In _Journal of Computer and System Sciences_ 37.3 Elsevier, 1988, pp. 312–323
* [35] Kevin Suffecool “Cactus kev’s poker hand evaluator”, 2005
* [36] Paul Tarau “Towards a generic view of primality through multiset decompositions of natural numbers” In _Theoretical Computer Science_ 537 Elsevier, 2014, pp. 105–124
* [37] Gabriel Valiente “Algorithms on trees and graphs” Springer, 2002
* [38] Yazhe Zhang “On the number of leaves in a random recursive tree”, 2015
## Appendix A Proof of Proposition 3.2
Let $k\geq 1$ be a fixed integer. A tree $T$ such that
$\operatorname{\mathcal{W}}(T)=k$ can be obtained by placing $k$ trees
$T_{i}$, $i\in[\\![1,k]\\!]$, under a common root, so that $T_{i}\not\simeq
T_{j}$ for $i\neq j$. Note that this construction by no means encompasses all
types of trees $T$ with $\operatorname{\mathcal{W}}(T)=k$. On the other hand,
by cleverly choosing the $T_{i}$’s, we can build a tree with the minimum
number of nodes among all trees verifying $\operatorname{\mathcal{W}}(T)=k$.
First, $T_{1}$ would be the tree with a unique node. Then, $T_{2}$ the only
tree with two nodes. Then, $T_{3}$ and $T_{4}$ would be the two non-isomorphic
trees with three nodes; $T_{5}$ to $T_{8}$ the four non-isomorphic trees with
four nodes, and so on until we reach $T_{k}$. See Figure 4 for an example with
$k=5$. It should be clear that this construction ensures that
$\operatorname{\mathcal{W}}(T)=k$ and $\\#T$ is minimal. The explicit
generation of $T_{1},\dots,T_{k}$ can be carried following [28].
$a_{1}$$a_{2}$$a_{3}$$k-b_{3}$$b_{2}$$b_{3}$ Figure 4: The smallest tree so
that $\operatorname{\mathcal{W}}(T)=5$. We have $b_{3}<5\leq b_{4}$ and
$\\#T=1+1\cdot a_{1}+2\cdot a_{2}+3\cdot a_{3}+4\cdot(5-b_{3})=14$.
Following this construction, the total number of nodes in $T$, that we denote
by $t_{k}$, is therefore closely related to the number of non-isomorphic trees
and their cumulative sum. Let us denote $a_{n}$ the number of non-isomorphic
trees of size $n$, and $b_{n}$ the number of non-isomorphic trees of size at
most $n$ – so that $b_{n}=\sum_{i=1}^{n}a_{i}$. Let $n$ be the integer so that
$b_{n}<k\leq b_{n+1}$. All trees with size up to $n-1$ are used in the
construction, as well as $k-b_{n}$ trees of size $n+1$ (no matter which ones).
Therefore,
$t_{k}=1+\sum_{i=1}^{n}i\cdot a_{i}+(n+1)(k-b_{n}).$
Table 1 provides the first values for $a_{n}$, $b_{n}$ and $t_{k}$.
$n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$
---|---|---|---|---|---|---|---|---
$a_{n}$ | $1$ | $1$ | $2$ | $4$ | $9$ | $20$ | $48$ | $115$
$b_{n}$ | $1$ | $2$ | $4$ | $8$ | $17$ | $37$ | $85$ | $200$
$t_{n}$ | $2$ | $4$ | $7$ | $10$ | $14$ | $18$ | $22$ | $26$
Table 1: First values of $a_{n}$, $b_{n}$ and $t_{n}$. $a_{n}$ is sequence
A000081 in OEIS, and $b_{n}$ sequence A087803. See OEIS Foundation Inc.
(2023), The On-Line Encyclopedia of Integer Sequences, Published
electronically at https://oeis.org.
Following the previous discussion, we have the following result. For any tree
$T$, $\\#T\geq t_{\operatorname{\mathcal{W}}(T)}$.
Our goal here is to prove the following proposition. It should be clear that
combining upcoming Proposition A and Lemma A yields Proposition 3.2. For any
$k\geq 1$, $k\cdot(\ln k+\ln\ln k)=O(t_{k})$.
##### A few asymptotic equivalents
Before starting the proof of Proposition A, we require some preliminary
results. We start by the following technical lemma.
Let $(u_{n})_{n\in\mathbb{n}}$ be a sequence so that
$u_{n}\underset{+\infty}{\sim}c\cdot d^{n}\cdot n^{-\alpha}$, with
$c,\alpha\geq 0$ and $d>1$. Then
$\sum_{k=1}^{n}u_{k}\underset{+\infty}{\sim}c\cdot\frac{d}{d-1}\cdot
d^{n}\cdot n^{-\alpha}$. Proof. Obviously the sequence $\sum d^{n}\cdot
n^{-\alpha}$ diverges, and therefore we have
$\sum_{k=1}^{n}u_{k}\underset{+\infty}{\sim}c\sum_{k=1}^{n}d^{k}\cdot
k^{-\alpha}$. Then,
$\frac{c\sum_{k=1}^{n}d^{k}\cdot k^{-\alpha}}{c\cdot\frac{d}{d-1}\cdot
d^{n}\cdot
n^{-\alpha}}=\frac{d-1}{d}\sum_{k=1}^{n}d^{k-n}\left(\frac{k}{n}\right)^{-\alpha}\underset{j=n-k}{=}\frac{d-1}{d}\sum_{j=1}^{n}\left(1-\frac{j}{n}\right)^{-\alpha}d^{-j}.$
With bounds
$1\leq\left(1-\frac{j}{n}\right)^{-\alpha}\leq\left(1-\frac{1}{n}\right)^{-\alpha}$,
it is easy to see that the right-hand term goes to $1$ as $n\to\infty$.
From [24, Section 2.3.4.4], we have $a_{n}\underset{+\infty}{\sim}c\cdot
d^{n}\cdot n^{-3/2}$ with $c\approx 0.439924$ and $d\approx 2.955765$. From
Lemma A, we immediately derive
$b_{n}\underset{+\infty}{\sim}c\cdot\frac{d}{d-1}\cdot d^{n}\cdot n^{-3/2}$.
Finally, noticing that $i\cdot a_{i}\underset{+\infty}{\sim}c\cdot d^{i}\cdot
i^{-1/2}$, we derive from Lemma A that $\sum_{i=1}^{n}i\cdot
a_{i}\underset{+\infty}{\sim}c\cdot\frac{d}{d-1}\cdot d^{n}\cdot n^{-1/2}$.
##### Proof of Proposition A
Note that, by definition $k\leq b_{n+1}$, and $t_{k}\geq\sum_{i=1}^{n}i\cdot
a_{i}$; so it suffices to prove that $b_{n+1}\cdot(\ln b_{n+1}+\ln\ln
b_{n+1})=O\left(\sum_{i=1}^{n}i\cdot a_{i}\right)$. Having established this,
some straightforward calculations allow to derive
$b_{n+1}\cdot(\ln b_{n+1}+\ln\ln
b_{n+1})\underset{+\infty}{\sim}c\cdot\frac{d}{d-1}\cdot
d^{n+1}\cdot(n+1)^{-1/2}\cdot\ln d.$
Since $\displaystyle\sum_{i=1}^{n}i\cdot
a_{i}\underset{+\infty}{\sim}c\cdot\frac{d}{d-1}\cdot d^{n}\cdot n^{-1/2}$, we
can immediately conclude that
$\frac{b_{n+1}\cdot(\ln b_{n+1}+\ln\ln b_{n+1})}{\sum_{i=1}^{n}i\cdot
a_{i}}\xrightarrow[n\to\infty]{}d\ln d,$
which completes the proof.
|
# Memory Complexity of Entropy Estimation
Tomer Berg, Or Ordentlich and Ofer Shayevitz
###### Abstract
We observe an infinite sequence of independent identically distributed random
variables $X_{1},X_{2},\ldots$ drawn from an unknown distribution $p$ over
$[n]$, and our goal is to estimate the entropy
$H(p)=-\operatorname{\mathbb{E}}[\log p(X)]$ within an $\varepsilon$-additive
error. To that end, at each time point we are allowed to update a finite-state
machine with $S$ states, using a possibly randomized but time-invariant rule,
where each state of the machine is assigned an entropy estimate. Our goal is
to characterize the minimax memory complexity $S^{*}$ of this problem, which
is the minimal number of states for which the estimation task is feasible with
probability at least $1-\delta$ asymptotically, uniformly in $p$.
Specifically, we show that there exist universal constants $C_{1}$ and $C_{2}$
such that $S^{*}\leq C_{1}\cdot\frac{n(\log n)^{4}}{\varepsilon^{2}\delta}$
for $\varepsilon$ not too small, and $S^{*}\geq C_{2}\cdot\max\\{n,\frac{\log
n}{\varepsilon}\\}$ for $\varepsilon$ not too large. The upper bound is proved
using approximate counting to estimate the logarithm of $p$, and a finite
memory bias estimation machine to estimate the expectation operation. The
lower bound is proved via a reduction of entropy estimation to uniformity
testing. We also apply these results to derive bounds on the memory complexity
of mutual information estimation.
## I Introduction
The problem of inferring properties of an underlying distribution given sample
access is called statistical property estimation. A typical setup is as
follows: given independent samples $X_{1},\ldots,X_{n}$ from an unknown
distribution $p$, the objective is to estimate a property $g(p)$ (e.g.,
entropy, support size, $L_{p}$ norm, etc.) under some resource limitation. A
prominent example of such a limitation is the amount of available samples, and
this limitation gives rise to the notion of sample complexity, namely the
minimal number of samples one needs to see in order to estimate $g(p)$ with
some given accuracy. Many real–world machine learning and data analysis tasks
are limited by insufficient samples, and the challenge of inferring properties
of a distribution given a small sample size is encountered in a variety of
settings, including text data, customer data, and the study of genetic
mutations across a population. The sample complexity of property estimation
and, specifically, of entropy estimation, have therefore received much
attention in the literature (see Section II for details).
However, in many contemporary settings, collecting enough samples for accurate
estimation is less of a problem, and the bottleneck shifts to the
computational resources available for the task and, in particular, the
available memory size. In this work, we therefore focus on the problem of
estimation under memory constraints, and, in particular, entropy estimation.
In order to isolate the effect that finite memory has on the fundamental
limits of the problem, we let the number of samples we process be arbitrarily
large.
Formally, the problem is defined as follows. Let $\Delta_{n}$ be the
collection of all distributions over $[n]$. The Shannon entropy of
$p\in\Delta_{n}$ is $H(p)=-\sum_{x\in[n]}p(x)\log p(x)$. Given independent
samples $X_{1},X_{2},\ldots$ from an unknown $p\in\Delta_{n}$, we would like
to accurately estimate $H(p)$ using limited memory. To that end, an $S$-state
entropy estimator is a finite-state machine with $S$ states, defined by two
functions: The (possibly randomized) memory update function
$f:[S]\times[n]\rightarrow[S]$, describing the transition between states as a
function of an input sample, and the entropy estimate function
$\hat{H}:[S]\rightarrow[0,\log n]$, assigning an entropy estimate to each
state. Letting $M_{t}$ denote the state of the memory at time $t$, this
finite-state machine evolves according to the rule:
$\displaystyle M_{0}$ $\displaystyle=s_{\text{init}},$ (1) $\displaystyle
M_{t}$ $\displaystyle=f(M_{t-1},X_{t})\in[S],$ (2)
for some predetermined initial state $s_{\text{init}}\in[S]$. If the machine
is stopped at time $t$, it outputs the estimation $\hat{H}(M_{t})$. We
restrict the discussion to time-invariant memory update function $f$, since
storing the time index necessarily incurs a memory cost, and, furthermore,
since the number of samples is unbounded, simply storing the code generating a
time-varying algorithm may require unbounded memory. We say that an
$\epsilon$-error occurred at time $t$ if our estimate $\hat{H}(M_{t})$ is
$\varepsilon$-far from the correct entropy. Our figure of merit for the
estimator is taken to be its worst-case asymptotic $\epsilon$-error
probability:
$\displaystyle\operatorname{\mathsf{P_{e}}}(f,\hat{H},\epsilon)$
$\displaystyle=\underset{p\in\Delta_{n}}{\sup}\limsup_{t\rightarrow\infty}\Pr\left(|\hat{H}(M_{t})-H(p)|>\varepsilon\right).$
(3)
We are interested in the _minimax memory complexity_
$S^{*}(n,\varepsilon,\delta)$, defined as the smallest integer $s$ for which
there exist $(f,\hat{H})$ such that
$\operatorname{\mathsf{P_{e}}}(f,\hat{H},\epsilon)\leq\delta$.
Our main result is an upper bound on $S^{*}(n,\varepsilon,\delta)$, which
shows that $\log\frac{n}{\varepsilon^{2}}+o\left(\log n\right)$ bits suffice
for entropy estimation when $\varepsilon>10^{-5}$, thus improving upon the
best known upper bounds thus far ([1, 2]). While our focus here is on
minimizing the memory complexity of the problem in the limit of infinite
number of available samples, we further show that the estimation algorithm
attaining this memory complexity upper bound only requires $\tilde{O}(n^{c})$
samples, for any $c>1$.111The $\tilde{O}$ suppresses poly-logarithmic terms.
Thus, in entropy estimation one can achieve almost optimal sample complexity
and memory complexity, simultaneously. Our proposed algorithm approximates the
logarithm of $p(x)$, for a given $x\in[n]$, using a Morris counter [3]. The
inherent structure of the Morris counter is particularly suited for
constructing a nearly-unbiased estimator for $\log p(x)$, making it a natural
choice for memory efficient entropy estimation. In order to compute the mean
of these estimators, $\operatorname{\mathbb{E}}[\widehat{\log p(X)}]$, in a
memory efficient manner, a finite-memory bias estimation machine (e.g., [4,
5]) is leveraged for simulating the expectation operator. The performance of a
scheme based on this high-level idea is analyzed, and yields the following
upper bound on the memory complexity:
###### Theorem 1.
For any $c>1$, $\beta>0$, $0<\delta<1$ and
$\varepsilon=10^{-5}+\beta+\psi_{c}(n)$, we have
$\displaystyle S^{*}(n,\varepsilon,\delta)\leq\frac{(c+1)^{4}n\cdot(\log
n)^{4}}{\beta^{2}\delta},$ (4)
where
$\displaystyle\psi_{c}(n)=\min\left\\{1+n^{-(c-1)+\sqrt{\frac{c}{\log
n}}},2\cdot 10^{8}\cdot n^{-\frac{1}{2}\cdot(c-1)+\sqrt{\frac{c}{8\log
n}}}\right\\}=O\left(n^{-\frac{1}{2}\cdot(c-1)+\sqrt{\frac{c}{8\log
n}}}\right).$ (5)
Moreover, there is an algorithm that attains (4) whenever the number of
samples is $\Omega\left(\frac{n^{c}\cdot\mathop{\mathrm{poly}}(\log
n)}{\delta}\cdot\mathop{\mathrm{poly}}(\log(1/\delta))\right)$, and returns an
estimation of $H(p)$ within an $\varepsilon$-additive error with probability
at least $1-3\delta$.
Note that while $\psi_{c}(n)$ vanishes for large $n$, our bound is always
limited to $\varepsilon>10^{-5}$. This small bias is due to inherent
properties of the Morris counter, on which we elaborate in Section III. As in
this work we are more interested in the large entropy regime (in which the
entropy grows with the alphabet size $n$), the limitation of the attainable
additive error to values above $10^{-5}$ is typically a moderate one. We also
note that if $n$ is large and $\varepsilon$ not too small, one can choose $c$
arbitrarily close to $1$, resulting in an algorithm whose sample complexity
has similar dependence on $n$ as those of the limited-memory entropy
estimation algorithms proposed in [1] and [2], while requiring far less memory
states. This result might be of practical interest for applications in which
memory is a scarcer resource than samples, e.g., a limited memory high-speed
router that leverages entropy estimation to monitor IP network traffic [6].
Furthermore, we derive two lower bounds on the memory complexity. The first
lower bound shows that when $H(p)$ is close to $\log{n}$, the memory
complexity cannot be too small. This bound is obtained via a reduction of
entropy estimation to uniformity testing, by noting that thresholding the
output of a good entropy estimation machine around $\log{n}$ can be used to
decide whether $p$ is close to the uniform distribution or not. The bound then
follows from the $\Omega(n)$ lower bound of [7] on uniformity testing. The
second lower bound follows from the observation that, if the number of states
is too small, there must be some value of the entropy at distance greater than
$\varepsilon$ from all estimate values, hence for this value of the entropy we
err with probability $1$. Combining these lower bounds yields the following.
###### Theorem 2.
For any $\varepsilon>0$, we have
$\displaystyle S^{*}(n,\varepsilon,\delta)\geq\frac{\log n}{2\varepsilon}.$
(6)
Furthermore, if $\varepsilon<\frac{1}{4\ln 2}$, then
$\displaystyle S^{*}(n,\varepsilon,\delta)\geq n(1-2\sqrt{\varepsilon\ln 2}).$
(7)
One of several open problems posed by the authors of [1] is to prove a lower
bound on the space requirement of a sample optimal algorithm for entropy
estimation. Theorem 2 answers this question by giving a lower bound on the
memory size needed when the number of samples is infinite, which clearly also
holds for any finite number of samples. In the concluding section of the
paper, we extend our results to the mutual information estimation problem. Let
$(X,Y)\sim p_{XY}$, where $p_{XY}$ is an unknown distribution over
$[n]\times[m]$ such that the marginal distribution of $X$ is $p_{X}$ and the
marginal distribution of $Y$ is $p_{Y}$. The mutual information between $X$
and $Y$ is given as $I(X;Y)=H(X)+H(Y)-H(X,Y)$. We derive the following bounds
on the memory complexity of mutual information estimation, namely the minimal
number of states needed to estimate $I(X;Y)$ with additive error at most
$\varepsilon$ with probability of at least $1-\delta$, which we denote as
$S_{\text{MI}}^{*}(n,m,\varepsilon,\delta)$.
###### Theorem 3.
For any $c>1$, $\beta>0$ and $\varepsilon=3\cdot
10^{-5}+\beta+O\left(n^{-\frac{1}{2}\cdot(c-1)}\vee
m^{-\frac{1}{2}\cdot(c-1)}\right)$,
$\displaystyle
S_{\text{MI}}^{*}(n,m,\varepsilon,\delta)\leq\frac{(c+1)^{6}nm\cdot(\log
nm)^{6}}{\beta^{2}\delta}.$ (8)
For $\varepsilon<\frac{1}{12\ln 2}$, and if
$\frac{n}{\log^{3}n}=\Omega(\log^{7}m)$ and
$\frac{m}{\log^{3}m}=\Omega(\log^{7}n)$ both hold, then
$\displaystyle
S_{\text{MI}}^{*}(n,m,\varepsilon,\delta)=\Omega\left(\frac{n\cdot
m}{\log^{3}n\cdot\log^{3}m}\right).$ (9)
## II Related work
The study of estimation under memory constraints has received far less
attention then the sample complexity of statistical estimation. References
[8], [9] studied this setting for hypothesis testing with finite memory, and
[10], [4] have studied estimating the bias of a coin using a finite state
machine. It has then been largely abandoned, but recently there has been a
revived interest in space-sample trade-offs in statistical estimation, and
many works have addressed different aspects of the learning under memory
constraints problem over the last few years. See, e.g., [11, 12, 13, 14, 15,
16, 17, 18, 19] for a non exhaustive list of recent works.
The problem of estimating the entropy with limited independent samples from
the distribution has a long history. It was originally addressed by [20], who
suggested the simple and natural empirical plug-in estimator. This estimator
outputs the entropy of the empirical distribution of the samples, and its
sample complexity [21] is
$\Theta\left(\frac{n}{\varepsilon}+\frac{\log^{2}n}{\varepsilon^{2}}\right)$.
[21] showed that the plug-in estimator is always consistent, and the resulting
sample complexity was shown to be linear in $n$. In the last two decades, many
efforts were made to improve the bounds on the sample complexity. Paninski
[22, 23]] was the first to prove that it is possible to consistently estimate
the entropy using sublinear sample size. While the scaling of the minimal
sample size of consistent estimation was shown to be $\frac{n}{\log n}$ in the
seminal results of [24, 25], the optimal dependence of the sample size on both
$n$ and $\varepsilon$ was not completely resolved until recently. In
particular, $\Omega\left(\frac{n}{\varepsilon\log n}\right)$ samples were
shown to be necessary, and the best upper bound on the sample complexity was
relied on an estimator based on linear programming that can achieve an
additive error $\varepsilon$ using $O\left(\frac{n}{\varepsilon^{2}\log
n}\right)$ samples [26]. This gap was partially amended in [27] by a different
estimator, which requires $O\left(\frac{n}{\varepsilon\log n}\right)$ samples
but is only valid when $\varepsilon$ is not too small. The sharp sample
complexity was shown by [28, 29, 29] to indeed be
$\displaystyle\Theta\left(\frac{n}{\varepsilon\log
n}+\frac{\log^{2}n}{\varepsilon^{2}}\right).$ (10)
The space-complexity (which is the minimal memory in bits needed for the
algorithm) of estimating the entropy of the empirical distribution of the data
stream is well-studied for worst-case data streams of a given length, see
[30], [6], [31]. Reference [32] addressed the problem of deciding if the
entropy of a distribution is above or beyond than some predefined threshold,
using algorithms with limited memory. The trade-off between sample complexity
and space/communication complexity for the entropy estimation of a
distribution is the subject of a more recent line of work. The earliest work
on the subject is [1], where the authors constructed an algorithm which is
guaranteed to work with
$O(k/\varepsilon^{3}\cdot\operatorname{polylog}(1/\varepsilon))$ samples and
any memory size $b\geq 20\log\left(\frac{k}{\varepsilon}\right)$ bits (which
corresponds to $O(n^{20}/\varepsilon^{20})$ memory states in our setup). Their
upper bound on the sample complexity was later improved by [2] to
$O(k/\varepsilon^{2}\cdot\operatorname{polylog}(1/\varepsilon))$ with space
complexity of $O\left(\log\left(\frac{k}{\varepsilon}\right)\right)$ bits. In
both the above works, the constant in the space complexity upper bound can be
shown to actually be smaller than $20$ by a careful analysis, but it cannot be
made smaller than $2$.
## III Preliminaries
In this section, we introduce mathematical notations and some relevant
background for the paper.
### III-A Notations
We write $[n]$ to denote the set $\\{1,\ldots,n\\}$, and consider discrete
distributions over $[n]$. We use the notation $p_{i}$ to denote the
probability of element $i$ in distribution $p$. When $X$ is a random variable
on $[n]$, $p_{X}$ denotes the random variable obtained by evaluating $p$ in
location $X$. The entropy of $p$ is defined as $H(p)=-\sum_{x\in[n]}p_{x}\log
p_{x}=\operatorname{\mathbb{E}}_{X\sim p}(-\log p_{X})$, where $H(p)=0$ for a
single mass distribution and $H(p)=\log n$ a uniform distribution over $[n]$.
The total variation distance between distributions $p$ and $q$ is defined as
half their $\ell^{1}$ distance, i.e.,
$d_{\mathsf{TV}}(p,q)=\frac{1}{2}||p-q||_{1}=\frac{1}{2}\sum_{i=1}^{n}|p_{i}-q_{i}|$,
and their KL (Kullback–Leibler) divergence is defined as
$D_{\text{KL}}(p||q)=\sum_{i=1}^{n}p_{i}\log\frac{p_{i}}{q_{i}}$. Logarithms
are taken to base 2.
### III-B Morris Counter
Suppose one wishes to implement a counter that counts up to $m$. Maintaining
this counter exactly can be accomplished using $\log m$ bits. In the first
example of a non-trivial streaming algorithm, Morris gave a randomized
“approximate counter”, which allows one to retrieve a constant multiplicative
approximation to $m$ with high probability using only $O(\log\log m)$ bits
(see [3]). The Morris Counter was later analyzed in more detail by Flajolet
[33], who showed that $O(\log\log m+\log(1/\varepsilon)+\log(1/\delta))$ bits
of memory are sufficient to return a $(1\pm\varepsilon)$ approximation with
success probability $1-\delta$. A recent result of [34] shows that $O(\log\log
m+\log(1/\varepsilon)+\log\log(1/\delta))$ bits suffice for the same task.
The original Morris counter is a random state machine with the following
simple structure: At each state $l=1,2,3,\ldots,$ an increment causes the
counter to transition to state $l+1$ with probability $2^{-l}$, and to remain
in state $l$ with probability $1-2^{-l}$. This is formally a discrete time
pure birth process, which can be seen in Figure 1.
$1$$2$$3$$\cdots$$l$$\cdots$$2^{-1}$$2^{-2}$$2^{-3}$$2^{-l}$ Figure 1: The
original Morris counter
The performance of the above counter was characterized by Flajolet, who proved
the following theorem.
###### Theorem 4 ([33]).
Let $C_{m}$ be the value of the Morris counter after $m$ increments. It holds
that
$\displaystyle\operatorname{\mathbb{E}}(C_{m})=\log m+\mu+g(\log m)+\phi(m),$
(11)
where $\mu\approx-0.273$ is a known constant, $g(\cdot)$ is a periodic
function of amplitude less than $10^{-5}$,
$|\phi(m)|\leq\min\left\\{1,\frac{2^{\sqrt{16\log m}}\cdot(\log
m)^{4.5}}{2m}\right\\}$ and the expectation is over the randomness of the
counter.222In [33], Flajolet bounded $\phi(m)$ with $O(m^{-0.98})$. Here, we
carefully follow the constants in his derivation and provide an explicit upper
bound on the the error terms, since we are interested in bounds that can be
applied for finite $m$.
Thus, if we are interested in approximating $\log m$ using the counter, then
taking our estimator to be $C_{m}-\mu$ guarantees that on average our additive
error will not be more than $10^{-5}+\phi(m)$, a property that we leverage in
our entropy estimation algorithm.
### III-C Finite-State Bias Estimation Machine
In the bias estimation problem, we are given access to i.i.d samples drawn
from the $\mathsf{Bern}(p)$ distribution, and we wish to estimate the value of
$p$ under the expected quadratic loss (also known as mean squared error
distortion measure). The $S$-state randomized machine with the state diagram
depicted in Figure 2, was purposed by [10] and later carefully analyzed by
[4], where it was shown to asymptotically induce a
$\mathrm{Binomial}(S-1,\theta)$ stationary distribution on the memory state
space. Thus, when this machine is initiated with a $\mathop{\mathrm{Bern}}(p)$
distribution and is run for a long enough time, it outputs an estimate
$\hat{p}$ that has
$\operatorname{\mathbb{E}}(\hat{p}-p)^{2}\leq\frac{1}{S-1}$. [4] further
showed that the machine is order-optimal, by proving a lower bound of
$\operatorname{\mathbb{E}}(\hat{p}-p)^{2}\geq\Omega(1/S)$ for any finite-state
estimator.
$1$$2$$\cdots$$S-1$$S$$q$$p$$\frac{1}{S-1}\cdot q$$\frac{S-2}{S-1}\cdot
q$$\frac{S-2}{S-1}\cdot p$$\frac{1}{S-1}\cdot p$$\frac{2}{S-1}\cdot
q$$\frac{2}{S-1}\cdot p$$\frac{S-2}{S-1}\cdot q$$\frac{1}{S-1}\cdot
p$$\frac{S-2}{S-1}\cdot p$$q$$\frac{1}{S-1}\cdot q$$p$ Figure 2: Randomized
bias estimation machine with $q=1-p$
## IV Upper Bound - Entropy Estimation Algorithm
In this section we prove Theorem 1, that is, we show the existence of an
$S$-states randomized entropy estimation machine with
$S\leq\frac{(c+1)^{4}n\cdot(\log n)^{4}}{\beta^{2}\delta}$ states, for
$\varepsilon=10^{-5}+\beta$ and $n$ large enough. The basic idea is to let
nature draw some $X$ from $p$ and use a Morris counter to approximate $-\log
p_{X}$, then, since we are looking for $H(p)=\operatorname{\mathbb{E}}(-\log
p_{X})$, use a bias estimation machine to simulate the averaging operation, by
randomly generating coin tosses with bias that is proportionate to our
estimate of $-\log p_{X}$. The bias estimation machine is incremented whenever
a count is concluded in second Morris counter that simulates a clock, thus
essentially averaging our estimates over $x$ values. For a sufficiently large
number of samples, this averaging converges (approximately) to the mean of
$-\log p_{X}$, and thus outputs an approximation to the true underlying
entropy. We divide our presentation to four parts: in the first part we
describe the algorithm; in the second part we analyze the total number of
states used by the algorithm; in the third part we assume the bias estimation
machine is fed with an infinite number of i.i.d. samples and analyze the
performance of the algorithm; and in the fourth part we relax this assumption
by studying the mixing time of the Markovian process induced by our bias
estimation machine. This allows us to prove an upper bound on the number of
samples the developed algorithm requires.
### IV-A Description of the algorithm
1. 1.
In each iteration of the algorithm we collect a fresh sample $X\sim p$, and
store its value, which requires $n$ states. Denote the realization of this
random variable by $x$. We proceed to estimate $\log p_{x}$ based on more
fresh samples, using Morris counters.
2. 2.
We use two Morris counters - one that approximates a clock, and one that
approximates a count for $x$ values:
* •
The first counter has $M=\log T+1$ states, where $T$ is the smallest integral
power of $2$ larger than $\lceil n^{c}\rceil$ for some $c>1$. The counter
stops when we arrive at state $M$, which corresponds to some _random_ time
$N$. This counter essentially approximates a clock that counts until $\lceil
n^{c}\rceil$ samples from $p$ are observed and it uses at most $c\log n+2$
states.
* •
The second counter runs in parallel to the first one and approximates the
logarithm of the number of observed $x$ values. It stops when the first
counter arrived at state $M$ and output its value, which we denote as
$C_{N_{x}}$. This counter also has $M$ states. The randomness used for the two
counters is statistically independent. In the event that the second counter
reached state $M$ before the first one, it also stops and outputs the value
$M$.333We ignore this case in the analysis since its contribution to the
estimation error can be shown to have a negligible effect. This follows since
if $p_{x}\leq 1-n^{-(c-1)}$ the probability of this event is extremely small,
otherwise, the contribution of $p_{x}$ to the entropy is negligible.
3. 3.
Denoting the number of observed $x$ values in the previous stage as $N_{x}$,
we define $C_{N_{x}}^{\text{centered}}\triangleq
C_{N_{x}}-\mu-\operatorname{\mathbb{E}}\log N$ to be the centralized output of
the second counter. As we argue below, this is an almost unbiased estimator
for $-\log p_{x}$.
We now use the bias estimation machine with
$S_{\text{bias}}=\frac{M^{2}}{\beta^{2}\delta}+1$ states to simulate the
expectation operation, where $\beta>0$ is such that
$\varepsilon=10^{-5}+\beta$. Specifically, each time the first Morris counter
concludes a count, we generate a $\mathop{\mathrm{Ber}}(\theta_{N_{x}})$
random variable, with
$\theta_{N_{x}}=-\frac{C_{N_{x}}^{\text{centered}}}{M}+a$, and use it as the
input to our bias estimation machine. The offset $a\triangleq
1-\frac{\operatorname{\mathbb{E}}(\log N)+\mu}{M}$ guarantees that
$\theta_{N_{x}}\in[0,1)$ with probability $1$, as
$\theta_{N_{x}}=1-\frac{C_{N_{x}}}{M}$ and $1\leq C_{N_{x}}\leq M$ as it is
the output of a Morris counter with $M$ states. Our estimator for the entropy
$\hat{H}$ is the bias estimate of the machine, after subtraction of the known
offset $a$ and multiplication by $M$, that is, $\hat{H}=M(\hat{\theta}-a)$.
### IV-B Number of states in our machine
In each time point, our algorithm keeps the value of $x$, the state of the
Morris counter approximating the clock, the state of the Morris counter
approximating the logarithm of the $x$ counter, and the state of the bias
estimation machine. Thus, calculating the product of the number of states
needed at each step, and recalling that $M=\log T+1\leq(c+1)\log n$, the total
number of states is
$\displaystyle S=n\cdot M\cdot M\cdot S_{\text{bias}}=n\cdot
M^{2}\cdot\left(\frac{M^{2}}{\beta^{2}\delta}+1\right)\leq\frac{(c+1)^{4}n\cdot(\log
n)^{4}}{\beta^{2}\delta}.$ (12)
### IV-C Analysis of the algorithm
We first show that the random run time of the first Morris counter $N$ is
concentrated around $\lceil n^{c}\rceil$, by upper bounding $\Pr(N<m)$ in
Lemma 1, and then using it to upper bound
$\operatorname{\mathbb{E}}\left(N^{-\alpha}\right)$ in Lemma 2. We then show
that $\operatorname{\mathbb{E}}(C_{N_{x}}^{\text{centered}})$ is close to
$H(p)$ in expectation in Lemma 3. We conclude by showing in Lemma 6 that the
output of the bias estimation machine converges to the expectation of
$\theta_{N_{X}}$, which then implies that $\hat{H}=M(\hat{\theta}-a)$ is an
$\varepsilon$-additive estimator of $H(p)$ with probability at least
$1-\delta$. Expectations are taken with respect to the randomness at different
stages of the algorithm, and we implicitly state the random variables with
respect to which the expectations are taken, apart for the case where the
expectation is taken over all the randomness of the algorithm.
###### Lemma 1.
For $m=2^{\ell},1\leq\ell\leq M-1$, it holds that
$\displaystyle\Pr(N<m)\leq e\cdot 2^{-\frac{1}{2}\cdot(M-\ell-1)^{2}}$ (13)
###### Proof.
let $\tau_{k}$ be the time it takes to move from state $k$ to state $k+1$ in
the first Morris counter and note that $\tau_{k}\sim\text{Geo}(2^{-k})$.
Clearly, $N=\sum_{k=1}^{M-1}\tau_{k}$. The moment generating function of
$\tau_{k}$ is
$\displaystyle\mathbb{M}_{\tau_{k}}(s)\triangleq\operatorname{\mathbb{E}}\left(e^{s\tau_{k}}\right)=\frac{1}{1+2^{k}(e^{-s}-1)},$
(14)
and it is defined for all $s<-\ln(1-2^{-k})$. The moment generating function
of $N$ is therefore
$\displaystyle\mathbb{M}_{N}(s)=\prod_{k=1}^{M-1}\mathbb{M}_{\tau_{k}}(s)=\prod_{k=1}^{M-1}\frac{1}{1+2^{k}(e^{-s}-1)}.$
(15)
From Chernoff bound, it holds that $\Pr(N<m)\leq
e^{-sm}\cdot\mathbb{M}_{N}(s)$ for any $s<0$. Setting $s=-\ln(1+1/m)$, we get
$\displaystyle\Pr(N<m)$
$\displaystyle\leq\left(1+\frac{1}{m}\right)^{m}\cdot\prod_{k=1}^{M-1}\frac{1}{1+\frac{2^{k}}{m}}\leq
e\cdot 2^{-\sum_{k=1}^{M-1}\log\left(1+\frac{2^{k}}{m}\right)}.$ (16)
Now let $m=2^{\ell}$ for $1\leq\ell\leq M-1$. We conclude by lower bounding
the exponent:
$\displaystyle\sum_{k=1}^{M-1}\log(1+2^{k-\ell})$
$\displaystyle=\sum_{k=0}^{\ell-1}\log(1+2^{-k})+\sum_{k=1}^{M-\ell-1}\log(1+2^{k})$
(17) $\displaystyle\geq\sum_{k=1}^{M-\ell-1}k$ (18)
$\displaystyle\geq\frac{1}{2}\cdot(M-\ell-1)^{2}.$ (19)
∎
###### Lemma 2.
For any $0<\alpha\leq 1$, we have that
$\displaystyle\operatorname{\mathbb{E}}\left(N^{-\alpha}\right)\leq(e+1)n^{-c\cdot\alpha+v_{n}(\alpha)},$
(20)
where $v_{n}(\alpha)\triangleq\sqrt{\frac{\alpha^{3}c}{\log n}}$.
###### Proof.
Appealing to Lemma 1, for any $1\leq\ell\leq M-1,m=2^{\ell}$, we have
$\displaystyle\operatorname{\mathbb{E}}(N^{-\alpha})$
$\displaystyle\leq\Pr(N<m)+m^{-\alpha}\cdot\Pr(N\geq m)$ (21)
$\displaystyle\leq e\cdot
2^{-\frac{1}{2}\cdot(M-\ell-1)^{2}}+2^{-\ell\cdot\alpha}.$ (22)
Setting $\ell=M-1-\sqrt{2\alpha\cdot(M-1)}$ and recalling that $n^{c}\leq
2^{M-1}$, we get
$\displaystyle\operatorname{\mathbb{E}}(N^{-\alpha})\leq e\cdot
2^{-\alpha(M-1)}+2^{-\alpha(M-1)\cdot\left(1-\sqrt{\frac{2\alpha}{M-1}}\right)}\leq
e\cdot n^{-c\cdot\alpha}+n^{-c\cdot\alpha\left(1-\sqrt{\frac{\alpha}{c\log
n}}\right)}.$ (23)
∎
###### Lemma 3.
Let $\psi_{c}(n)=\min\left\\{1+(e+1)n^{-(c-1)+v_{n}(1)},C\cdot
n^{-\frac{1}{2}\cdot(c-1)+v_{n}(1/2)}\right\\}$, where $C=2(e+1)\cdot 10^{8}$.
Then
$\displaystyle|\operatorname{\mathbb{E}}(C_{N_{X}}^{\text{centered}})+H(p)|\leq
10^{-5}+\psi_{c}(n).$ (24)
###### Proof.
According to Theorem 4, the value of the Morris counter after $m$ updates is
close to $\log m$ in expectation, up to some small bias. Using this fact we
show that, given $N$ and $X=x$,
$\operatorname{\mathbb{E}}(C_{N_{X}}^{\text{centered}})$ is close to the
expected logarithm of a normalized Binomial($N,p_{x}$) random variable. Taking
the expectation over $X$, this gives us $-H(p)$ plus some bias. Write
$\displaystyle\operatorname{\mathbb{E}}(C_{N_{X}}^{\text{centered}})$
$\displaystyle=\operatorname{\mathbb{E}}(\operatorname{\mathbb{E}}_{C_{N_{X}}|_{X=x,N}}(C_{N_{x}}-\mu-\operatorname{\mathbb{E}}(\log
N)))$ (25) $\displaystyle=\operatorname{\mathbb{E}}(\log
N_{X}-\operatorname{\mathbb{E}}(\log N)+g(\log N_{X})+\phi(N_{X}))$ (26)
$\displaystyle=\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N}\right)+\operatorname{\mathbb{E}}(\gamma_{N_{X}})$
(27) $\displaystyle=\operatorname{\mathbb{E}}(\log
p_{X})+\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N\cdot
p_{X}}\right)+\operatorname{\mathbb{E}}(\gamma_{N_{X}})$ (28)
$\displaystyle=-H(p)+\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N\cdot
p_{X}}\right)+\operatorname{\mathbb{E}}(\gamma_{N_{X}}),$ (29)
where $\gamma_{N_{x}}=g(\log N_{x})+\phi(N_{x})$. We conclude the proof by
showing that $\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N\cdot
p_{X}}\right)$ is small in Lemma 4, and showing that
$\operatorname{\mathbb{E}}(\gamma_{N_{X}})$ is small in Lemma 5. ∎
###### Lemma 4.
It holds that
$\displaystyle 0\leq\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N\cdot
p_{X}}\right)\leq(e+1)n^{-(c-1)+v_{n}(1)}.$ (30)
###### Proof.
We first show that $\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N\cdot
p_{X}}\right)\geq 0$. By Jensen’s inequality and convexity of
$t\mapsto-\log(t)$
$\displaystyle\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N\cdot
p_{X}}\right)$
$\displaystyle=\operatorname{\mathbb{E}}_{X,N}\left[\operatorname{\mathbb{E}}_{N_{X}|N,X}\left(-\log\frac{N\cdot
p_{X}}{N_{X}}\right)\right]$ (31)
$\displaystyle\geq-\operatorname{\mathbb{E}}_{X,N}\left[\log\left(N\cdot
p_{X}\cdot\operatorname{\mathbb{E}}_{N_{X}|N,X}\left[\frac{1}{N_{X}}\right]\right)\right].$
(32)
To establish non-negativity of
$\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N\cdot p_{X}}\right)$, it
therefore suffices to show that
$\operatorname{\mathbb{E}}_{N_{X}|N,X=x}\left[\frac{1}{N_{X}}\right]\leq\frac{1}{p_{x}\cdot
N}$. To that end, recall that given $X=x$ and $N$, we have
$N_{X}\sim\mathop{\mathrm{Bin}}(N-1,p_{x})+1$. Thus, we indeed have
$\displaystyle\operatorname{\mathbb{E}}_{N_{X}|N,X=x}\left[\frac{1}{N_{X}}\right]$
$\displaystyle=\sum_{m=0}^{N-1}\frac{1}{m+1}{N-1\choose
m}p_{x}^{m}(1-p_{x})^{N-m-1}$
$\displaystyle=\sum_{m=0}^{N-1}\frac{1}{p_{x}\cdot N}{N\choose
m+1}p_{x}^{m+1}(1-p_{x})^{N-m-1}$
$\displaystyle=\frac{1-(1-p_{x})^{N}}{p_{x}\cdot N}$
$\displaystyle\leq\frac{1}{p_{x}\cdot N}.$ (33)
To upper bound $\operatorname{\mathbb{E}}\left(\log\frac{N_{X}}{N\cdot
p_{X}}\right)$, we use Jensen’s inequality and the concavity of $t\mapsto\log
t$, to obtain
$\displaystyle\operatorname{\mathbb{E}}_{N_{X}|N,X=x}\left(\log\frac{N_{X}}{N\cdot
p_{X}}\right)$
$\displaystyle\leq\log\left(\frac{\operatorname{\mathbb{E}}_{N_{X}|N,X=x}[N_{X}]}{N\cdot
p_{x}}\right)$ (34) $\displaystyle=\log\left(1+\frac{1-p_{x}}{N\cdot
p_{x}}\right)$ (35) $\displaystyle\leq\frac{1}{N\cdot p_{x}}.$ (36)
Thus, overall,
$\displaystyle\operatorname{\mathbb{E}}\left[\log\frac{N_{X}}{N\cdot
p_{X}}\right]$
$\displaystyle\leq\operatorname{\mathbb{E}}_{N,X}\left[\frac{1}{N\cdot
p_{X}}\right]$
$\displaystyle=\operatorname{\mathbb{E}}_{X}\left[\frac{1}{p_{X}}\right]\operatorname{\mathbb{E}}_{N}\left[\frac{1}{N}\right]$
$\displaystyle=n\cdot\operatorname{\mathbb{E}}_{N}\left[\frac{1}{N}\right]$
and appealing to Lemma 2 with $\alpha=1$, we have
$\operatorname{\mathbb{E}}\left[\log\frac{N_{X}}{N\cdot
p_{X}}\right]\leq(e+1)n^{-(c-1)+v_{n}(1)}$. ∎
###### Lemma 5.
It holds that
$\displaystyle\operatorname{\mathbb{E}}(\gamma_{N_{X}})\leq
10^{-5}+\min\\{1,C\cdot n^{-\frac{1}{2}\cdot(c-1)+v_{n}(1/2)}\\}.$ (37)
###### Proof.
Note that $\operatorname{\mathbb{E}}(g(\log N_{x}))\leq 10^{-5}$ is explicit
in Theorem 4 for any $x\in[n]$, and in particular,
$\operatorname{\mathbb{E}}(g(\log N_{X}))\leq 10^{-5}$. Thus, it remains to
upper bound $\operatorname{\mathbb{E}}(\phi(N_{X}))$. It is straightforward to
verify that $\phi(x)\leq\min\left\\{1,\frac{2\cdot 10^{8}}{\sqrt{x}}\right\\}$
for all $x\geq 1$, and consequently,
$\displaystyle\operatorname{\mathbb{E}}(\phi(N_{X}))\leq\operatorname{\mathbb{E}}\left[\min\left\\{1,\frac{2\cdot
10^{8}}{\sqrt{N_{X}}}\right\\}\right]\leq\min\left\\{1,2\cdot
10^{8}\operatorname{\mathbb{E}}\left[\sqrt{\frac{1}{N_{x}}}\right]\right\\}.$
(38)
From Jensen’s inequality, concavity of $t\mapsto\sqrt{t}$, and equation (33),
$\displaystyle\operatorname{\mathbb{E}}\left[\sqrt{\frac{1}{N_{X}}}\right]$
$\displaystyle=\operatorname{\mathbb{E}}_{N,X}\left[\operatorname{\mathbb{E}}_{N_{X}|N,X}\left[\sqrt{\frac{1}{N_{X}}}\right]\right]$
(39)
$\displaystyle\leq\operatorname{\mathbb{E}}_{N,X}\left[\sqrt{\operatorname{\mathbb{E}}_{N_{X}|N,X}\left[\frac{1}{N_{X}}\right]}\right]$
(40)
$\displaystyle\leq\operatorname{\mathbb{E}}_{N,X}\left[\sqrt{\frac{1}{p_{X}\cdot
N}}\right]$
$\displaystyle=\operatorname{\mathbb{E}}_{N}\left[\sqrt{\frac{1}{N}}\right]\operatorname{\mathbb{E}}_{X}\left[\sqrt{\frac{1}{p_{X}}}\right].$
(41)
Note that, again using Jensen’s inequality and concavity of
$t\mapsto\sqrt{t}$, we have
$\displaystyle\operatorname{\mathbb{E}}_{X}\left[\sqrt{\frac{1}{p_{X}}}\right]=\sum_{x=1}^{n}\sqrt{p_{x}}\leq
n\sqrt{\frac{1}{n}\sum_{x=1}^{n}p_{x}}=\sqrt{n}.$ (42)
Appealing to Lemma 2 with $\alpha=0.5$, we have
$\displaystyle\operatorname{\mathbb{E}}(N^{-0.5})\leq(e+1)n^{-\frac{c}{2}+v_{n}(1/2)}.$
(43)
Thus, substituting (42) and (43) into (41) and then into (38), and recalling
that $C=2(e+1)10^{8}$, we obtain the claimed result. ∎
###### Lemma 6.
Define the estimator $\hat{H}=M(\hat{\theta}-a)$, where $\hat{\theta}$ is the
output of the bias estimation machine with
$S_{\text{bias}}=\frac{M^{2}}{\beta^{2}\delta}+1$ states, to which we feed a
$\mathsf{Bern}(\theta_{N_{X}})$ sample at each iteration of our approximate
counter. Then,
$\displaystyle\Pr(|\hat{H}-(H(p)+b)|>\beta)\leq\delta,$ (44)
where $|b|\leq 10^{-5}+\psi_{c}(n)$.
###### Proof.
Averaging over $X$, we have that the overall bias of the binary random
variable we feed to the bias estimation machine is
$\displaystyle\theta\triangleq\operatorname{\mathbb{E}}(\theta_{N_{X}})=\operatorname{\mathbb{E}}\left(-\frac{C_{N_{X}}^{\text{centered}}}{M}+a\right)=\frac{H(p)+b}{M}+a,$
(45)
where $|b|\leq 10^{-5}+\psi_{c}(n)$ is an unknown offset that arises from
Lemma 3. Recalling that
$\operatorname{\mathbb{E}}(\hat{\theta}-\theta)^{2}\leq\frac{1}{S_{\text{bias}}-1}$,
$\displaystyle\operatorname{\mathbb{E}}(\hat{H}-(H(p)+b))^{2}=M^{2}\cdot\operatorname{\mathbb{E}}(\hat{\theta}-\theta)^{2}\leq\beta^{2}\delta,$
(46)
thus we have from Chebyshev’s inequality,
$\displaystyle\Pr(|\hat{H}-(H(p)+b)|>\beta)\leq\frac{\operatorname{\mathbb{E}}(\hat{H}-(H(p)+b))^{2}}{\beta^{2}}\leq\delta.$
(47)
Note that our upper bound on the additive error in estimation of $H(p)$ is
$\beta+|b|\leq\beta+10^{-5}+\psi_{c}(n)$, which limits our results to
estimation error $\varepsilon>10^{-5}+\psi_{c}(n)$. ∎
### IV-D Sample complexity of our algorithm
In our algorithm, the number of observed samples is unbounded. In practice we
only need to observe $O(t_{\mathsf{mix}}(p))$ samples, where
$t_{\mathsf{mix}}(p)$ is the mixing time of our machine whenever the input is
$\mathop{\mathrm{Bern}}(p)$ samples, i.e., the minimal time it takes for the
total variation distance between the marginal distribution and the limiting
distribution to be small. In our case the bias estimation machine is only
incremented after an iteration of the first Morris counter is completed, and
the run time of each iteration is a random variable that is only bounded in
expectation. However, we note that this in fact implies the existence of a
good algorithm that has a bounded sample complexity; namely, running our
entropy estimation algorithm on $L$ samples is equivalent to running the bias
estimation machine from [10] on a random number of samples $k=k(L)$ times with
$p=\theta=\operatorname{\mathbb{E}}(\theta_{N_{X}})$. The randomness in $k(L)$
follows since the runtime $N_{i}$ of each iteration of the Morris counter
procedure is a random variable. Below, we use Chernoff’s bound to upper bound
the probability that $k(L)$ is small. This event is considered as an error in
our analysis. We now upper bound the mixing time of the bias estimation
machine from [10]. Whenever $k(L)$ is greater than this mixing time, the error
of our algorithms with $L$ samples is close to its asymptotic value.
To upper bound the mixing time, we use the coupling method. Recall that the
transition matrix $P$ of a Markov process $\\{X_{t}\\}_{t=0}^{\infty}$
supported on $\mathcal{X}$ is a matrix whose elements are
$\Pr(X_{t+1}=x^{\prime}|X_{t}=x)=P(x,x^{\prime})$, for any
$x,x^{\prime}\in\mathcal{X}\times\mathcal{X}$. We define a coupling of Markov
chains with transition matrix $P$ to be a process
$\\{X_{t},Y_{t}\\}_{t=0}^{\infty}$ with the property that both
$\\{X_{t}\\}_{t=0}^{\infty}$ and $\\{Y_{t}\\}_{t=0}^{\infty}$ are Markov
chains with transition matrix $P$, although the two chains may be correlated
and have different initial distributions. Given a Markov chain on
$\mathcal{X}$ with transition matrix $P$, a Markovian coupling of two
$P$-chains is a Markov chain $\\{X_{t},Y_{t}\\}_{t=0}^{\infty}$ with state
space $\mathcal{X}\times\mathcal{X}$, which satisfies, for all
$x,y,x^{\prime},y^{\prime}$,
$\displaystyle\Pr(X_{t+1}=x^{\prime}|X_{t}=x,Y_{t}=y)$
$\displaystyle=P(x,x^{\prime})$ (48)
$\displaystyle\Pr(Y_{t+1}=y^{\prime}|X_{t}=x,Y_{t}=y)$
$\displaystyle=P(y,y^{\prime}).$ (49)
Let $P^{t}(x_{0})$ be the marginal distribution of the chain at time $t$ when
initiated at $x_{0}$, and let $\pi$ be the unique stationary distribution.
Define the $\delta$-mixing time as
$\displaystyle
t_{\delta}^{*}\triangleq\min\\{t:d_{\mathsf{TV}}(P^{t}(x_{0}),\pi)\leq\delta\\},$
(50)
and $t_{\mathsf{mix}}\triangleq t_{1/4}^{*}$. We now show that the bias
estimation machine with $S$ states mixes in $\Theta(S\log S)$ time, uniformly
for all $p\in(0,1]$.
###### Theorem 5.
Let $t_{\mathsf{mix}}(p)$ denote the mixing time of the bias estimation
machine with $S$ states when the input is i.i.d $\mathsf{Bern}(p)$, and define
the worst-case mixing time to be $t^{*}=\max_{p\in(0,1]}t_{\mathsf{mix}}(p)$.
Then
$\displaystyle\ln(2)\cdot(S-1)\log(S-1)\leq t^{*}\leq 4S\log S.$ (51)
###### Proof.
The transition probabilities of the bias estimation machine of Figure 2 are
given, for $1<k<S$, as
$\displaystyle X_{t+1}|_{X_{t}=k}=\begin{cases}k+1,&\textit{w.p.
}\frac{S-k}{S-1}\cdot p,\\\ k,&\textit{w.p. }\frac{k-1}{S-1}\cdot
p+\frac{S-k}{S-1}\cdot q,\\\ k-1,&\textit{w.p. }\frac{k-1}{S-1}\cdot
q,\end{cases}$ (52)
and for the extreme states $\\{1,S\\}$ as
$\displaystyle X_{t+1}|_{X_{t}=1}=\begin{cases}2,&\textit{w.p. }p,\\\
1,&\textit{w.p. }q,\end{cases}\hskip
11.38109ptX_{t+1}|_{X_{t}=S}=\begin{cases}S,&\textit{w.p. }p,\\\
S-1,&\textit{w.p. }q.\end{cases}$ (53)
We construct a Markovian coupling in which the two chains stay together at all
times after their first simultaneous visit to a single state, that is
$\displaystyle\text{if }X_{s}=Y_{s}\text{ then }X_{t}=Y_{t}\text{ for all
}t\geq s.$ (54)
The following lemma, due to [35](Theorem 5.4), will give us an upper bound on
the mixing time using this coupling.
###### Lemma 7.
Let $\\{(X_{t},Y_{t})\\}$ be a Markovian coupling satisfying (54), for which
$X_{0}=x_{0}$ and $Y_{0}=y_{0}$. Let $\tau_{\mathsf{couple}}$ be the
coalescence time of the chains, that is,
$\displaystyle\tau_{\mathsf{couple}}\triangleq\min\\{t:X_{t}=Y_{t}\\}.$ (55)
Then
$\displaystyle t_{\mathsf{mix}}\leq
4\max_{x_{0},y_{0}\in\mathcal{X}}\operatorname{\mathbb{E}}(\tau_{\mathsf{couple}}).$
(56)
Assume w.l.o.g. that $x_{0}<y_{0}$ and let $U_{t}$ be an i.i.d sequence drawn
according to the $\mathop{\mathrm{Unif}}(0,1)$ distribution. We construct a
coupling on $(X_{t},Y_{t})$ such that, at each time point
$t<\tau_{\mathsf{couple}}$, $X_{t}$ and $Y_{t}$ are incremented in the
following manner:
$\displaystyle X_{t+1}|_{X_{t}=i}=\begin{cases}i+1,&\textit{if
}U_{t}\leq\frac{S-i}{S-1}\cdot p,\\\ i,&\textit{if }\frac{S-i}{S-1}\cdot p\leq
U_{t}\leq 1-\frac{i-1}{S-1}\cdot q,\\\ i-1,&\textit{if }1-\frac{i-1}{S-1}\cdot
q\leq U_{t}\leq 1,\end{cases}$ (57)
and
$\displaystyle Y_{t+1}|_{Y_{t}=j}=\begin{cases}j+1,&\textit{if
}U_{t}\leq\frac{S-j}{S-1}\cdot p,\\\ j,&\textit{if }\frac{S-j}{S-1}\cdot p\leq
U_{t}\leq 1-\frac{j-1}{S-1}\cdot q,\\\ j-1,&\textit{if }1-\frac{j-1}{S-1}\cdot
q\leq U_{t}\leq 1.\end{cases}$ (58)
One can validate that the transition probabilities are the correct ones, for
example
$\displaystyle\Pr(X_{t+1}=i|X_{t}=i)$
$\displaystyle=\Pr\left(\frac{S-i}{S-1}\cdot p\leq U_{t}\leq
1-\frac{i-1}{S-1}\cdot q\right)$ (59) $\displaystyle=1-\frac{i-1}{S-1}\cdot
q-\frac{S-i}{S-1}\cdot p$ (60) $\displaystyle=\frac{i-1}{S-1}\cdot
p+\frac{S-i}{S-1}\cdot q,$ (61)
and, similarly, $\Pr(Y_{t+1}=j|Y_{t}=j)=\frac{j-1}{S-1}\cdot
p+\frac{S-j}{S-1}\cdot q$. The other transition probabilities are easily
calculated. Note that $i<j$ implies $\frac{S-j}{S-1}<\frac{S-i}{S-1}$, thus
$Y_{t}$ cannot move right unless $X_{t}$ moves right and $X_{t}$ cannot move
left unless $Y_{t}$ moves left. Moreover, since $x_{0}<y_{0}$, we have $i<j$
for all $t<\tau_{\mathsf{couple}}$. This follows from construction, since
$\frac{S-i}{S-1}\cdot p$ is always smaller than $1-\frac{j-1}{S-1}\cdot q$,
implying that $X_{t}$ cannot jump over $Y_{t}$ when they are one-state apart.
Thus, the distance process $D_{t}\triangleq Y_{t}-X_{t}$, is a non-increasing
function of $t$, with initial state $D_{0}=y_{0}-x_{0}$, that can only
decrease by one unit at a time or stay unchanged. We have
$\displaystyle\Pr(D_{t+1}=D_{t}-1)$
$\displaystyle=\Pr(X_{t+1}=X_{t}+1,Y_{t=1}=Y_{t})+\Pr(Y_{t+1}=Y_{t}-1,X_{t=1}=X_{t})$
(62) $\displaystyle=\Pr\left(\frac{S-Y_{t}}{S-1}\cdot p\leq
U_{t}\leq\frac{S-X_{t}}{S-1}\cdot p\right)+\Pr\left(1-\frac{Y_{t}-1}{S-1}\cdot
q\leq U_{t}\leq 1-\frac{X_{t}-1}{S-1}\cdot q\right)$ (63)
$\displaystyle=\frac{Y_{t}-X_{t}}{S-1}\cdot p+\frac{Y_{t}-X_{t}}{S-1}\cdot q$
(64) $\displaystyle=\frac{D_{t}}{S-1}.$ (65)
The expected coupling time is now the expected time it takes for $D_{t}$ to
decrease from $D_{0}$ to $D_{t}$, thus in order to maximize it under the given
coupling, we need to maximize $D_{0}$, which corresponds to setting
$X_{0}=1,y_{0}=S$. For $D_{0}=S-1$, consider the process $M_{t}\triangleq
D_{0}-D_{t}$, which is a non-decreasing function of $t$ that goes from $0$ to
$S-1$ and has
$\Pr(M_{t+1}=M_{t}+1)=\Pr(D_{t+1}=D_{t}-1)=\frac{D_{t}}{S-1}=1-\frac{M_{t}}{S-1}$.
Then this process is no other than the Coupon Collector process with $S-1$
coupons, and the expected coupling time in our chain in identical to the
expected number of coupons collected until the set contains all $S-1$ types,
which according to [35], Proposition 2.3., is
$\displaystyle\operatorname{\mathbb{E}}(\tau_{\mathsf{couple}})=(S-1)\cdot\sum_{k=1}^{S-1}\frac{1}{k}\leq(S-1)(\ln(S-1)+1)\leq
S\log(S).$ (66)
To show that this upper bound is indeed tight, consider the case of $p=1$. In
this case, the chain of Figure 2 is simply the Coupon Collector process with
$S-1$ coupons, thus, letting $\tau$ be the (random) time it takes to collect
all coupons, we have
$\displaystyle\operatorname{\mathbb{E}}(\tau)=(S-1)\cdot\sum_{k=1}^{S-1}\frac{1}{k}\geq\ln(2)\cdot(S-1)\log(S-1).$
(67)
∎
From [35], Eq. (4.34), we have that the $\delta$-mixing time $t_{\delta}^{*}$
can be upper bounded in terms on the mixing time by
$\displaystyle
t_{\delta}^{*}\leq\left\lceil\log\left(\frac{1}{\delta}\right)\right\rceil\cdot
t_{\mathsf{mix}}.$ (68)
Let
$\displaystyle k\triangleq 8\log\left(\frac{1}{\delta}\right)\frac{(\log
4n^{c})^{2}}{\beta^{2}\delta}\log\left(\frac{(\log
4n^{c})^{2}}{\beta^{2}\delta}\right),$ (69)
and note that from equation (68), Theorem 5, and substituting
$S_{\text{bias}}=\frac{M^{2}}{\beta^{2}\delta}+1$, we have that the
$\delta$-mixing time of the bias estimation machine is at most $k$. Let
$N_{1},N_{2},\ldots,N_{k}$ be the first $k$ i.i.d. Morris counter running
times, which are all distributed as $N$ in the analysis from Section IV. Lemma
8 uses the concentration of $N$ to show that, with probability $1-\delta$, the
number of samples we need to observe until the bias machine mixes is not
large.
###### Lemma 8.
Let $m=4n^{c}\cdot\ln\left(\frac{5k}{\delta}\right)$. Then
$\displaystyle\Pr\left(\sum_{i=1}^{k}N_{i}>k\cdot m\right)\leq\delta.$ (70)
###### Proof.
Recall that for all $i\in\mathbb{N}$, we have
$\displaystyle\mathbb{M}_{N_{i}}(s)=\prod_{j=1}^{M-1}\frac{1}{1+2^{j}(e^{-s}-1)},$
(71)
which is defined for all $s<-\ln(1-2^{-(M-1)})=-\ln(1-1/T)$. As $T\leq
2n^{c}$, $\mathbb{M}_{N_{i}}(t)$ is well defined for $s=-\ln(1-1/4n^{c})$,
thus we have from Chernoff bound,
$\displaystyle\Pr(N_{i}>m)$ $\displaystyle\leq
e^{-sm}\cdot\mathbb{M}_{N_{i}}(s)$ (72)
$\displaystyle=\left(1-\frac{1}{4n^{c}}\right)^{m}\cdot\prod_{j=1}^{M-1}\frac{1}{1-\frac{2^{j}}{4n^{c}}}$
(73)
$\displaystyle\leq\left(1-\frac{1}{4n^{c}}\right)^{m}\cdot\prod_{j=1}^{M-1}\frac{1}{1-2^{-j}}$
(74) $\displaystyle\leq
5\exp\left\\{-\frac{m}{4n^{c}}\right\\}=\frac{\delta}{k}$ (75)
where in (74) we used the fact that
$\frac{2^{j}}{4n^{c}}\leq\frac{2^{j}}{2T}=2^{j-M}$, and in (75) we used the
bound $\prod_{j=1}^{M}(1-2^{-j})\geq\frac{1}{4}+\frac{1}{2^{M+1}}$, which can
be proved via induction. Consequently, the probability that at least one of
the random variables $N_{1},\ldots,N_{k}$ is greater than $m$ is at most
$1-\left(1-\frac{\delta}{k}\right)^{k}\leq\delta$, which implies the statement
of the lemma. ∎
We conclude with the following lemma, which connects Theorem 5 and Lemma 8 to
show that our entropy estimator performs well even if the number of input
samples is $\tilde{O}(n^{c}/\delta)$.
###### Lemma 9.
Let the algorithm of Theorem 1 run on $L=k\cdot m$ samples, and output the
estimate $\hat{H}_{M_{L}}$. Then with probability at least $1-3\delta$,
$\hat{H}_{M_{L}}$ is within $\varepsilon$-additive error from $H(p)$.
###### Proof.
Lemma 8 implies that, with probability at least $1-\delta$, after observing
$k\cdot m$ samples, the bias estimation machine has been incremented at least
$k$ times. Recall that, by definition, after $t\geq t_{\delta}^{*}$ increments
of the bias estimation machine, we have that
$d_{\mathsf{TV}}(P^{t}(x_{0}),\pi)\leq\delta$, and that our $S$-states entropy
estimator has $\sum_{i\in\hat{H}_{\bar{\varepsilon}}}\pi_{i}<\delta$, where
$\hat{H}_{\bar{\varepsilon}}=\\{i\in[S]:|\hat{H}_{i}-H(p)|>\varepsilon\\}$.
Thus, from a union bound, a fraction of $2\delta$ of the distribution
$P^{t}(x_{0})$ (at most) is supported on $\hat{H}_{\bar{\varepsilon}}$.
Putting it all together, we have that a finite-time algorithm that outputs an
estimate $\hat{H}(M_{L})$ after
$\displaystyle L=k\cdot
m=\Omega\left(\frac{n^{c}\cdot\mathop{\mathrm{poly}}(\log
n)}{\delta}\cdot\mathop{\mathrm{poly}}(\log(1/\delta))\right)$ (76)
will be $\varepsilon$-far from the correct entropy with probability at most
$3\delta$. ∎
## V Lower Bounds
In this section we prove Theorem 2. The $\Omega(n)$ bound is proved via
reduction to uniformity testing. For the $\frac{\log n}{2\varepsilon}$ bound,
we use a simple quantization argument. Assume that $S<\frac{\log
n}{2\varepsilon}$. Then there must be two consecutive estimate values
$\hat{H}_{1},\hat{H}_{2}\in[0,\log n]$ such that
$\hat{H}_{2}-\hat{H}_{1}>2\varepsilon$. This implies that
$H=(\hat{H}_{1}+\hat{H}_{2})/2$ has
$|H-\hat{H}_{1}|=|H-\hat{H}_{2}|>\varepsilon$. Thus, for this value of the
entropy, we have $\Pr(|\hat{H}(M_{t})-H|>\varepsilon)=1$ for all
$t\in\mathbb{N}$.
### V-A Proof of the $(1-2\sqrt{\varepsilon\ln 2})n$ bound
An ($\varepsilon,\delta$) uniformity tester can distinguish (with probability
$0<\delta<1/2$) between the case where $p$ is uniform and the case where $p$
is $\varepsilon$-far from uniform in total variation. Assume we have an
($\varepsilon,\delta$) entropy estimator. Then we can obtain an
($\tilde{\varepsilon}=\sqrt{\varepsilon\ln 2},\delta$) uniformity tester using
the following protocol: the tester declares that $p$ is uniform if
$\hat{H}>\log n-\varepsilon$, and that $p$ is $\tilde{\varepsilon}$-far from
uniform if $\hat{H}<\log n-\varepsilon$. We now argue that this is indeed an
($\tilde{\varepsilon},\delta$) uniformity tester, in which case the
$(1-2\tilde{\varepsilon})n$ lower bound will follow immediately from the lower
bound on uniformity testing of [7]. If $p=u$, where $u$ is the uniform
distribution over $[n]$, then $H(p)=\log n$ and $\hat{H}>\log n-\varepsilon$
with probability at least $1-\delta$, so our tester will correctly declare
“uniform” with probability at least $1-\delta$. If
$d_{\mathsf{TV}}(p,u)>\sqrt{\varepsilon\ln 2}$, then from Pinsker’s inequality
( [36], Lemma $11.6.1$),
$\displaystyle 2\varepsilon<\frac{2}{\ln 2}d_{\mathsf{TV}}(p,u)^{2}\leq
D(p||u)=\log n-H(p),$ (77)
which implies $H(p)<\log n-2\varepsilon$ and $\hat{H}<\log n-\varepsilon$ with
probability at least $1-\delta$. Thus, our tester will correctly declare “far
from uniform” with probability at least $1-\delta$.
## VI Memory Complexity of Mutual Information Estimation
We extend our results to the problem of mutual information estimation. The
upper bound follows by a slight tweaking of our entropy estimation machine,
and the lower bound follows by noting the close relation between mutual
information and joint entropy, and lower bounding the memory complexity of the
latter.
### VI-A Sketch of Upper Bound achieving algorithm
1. 1.
Let nature randomly draw some $(X,Y)=(x,y)\in[n]\times[m]$ according to
$p_{XY}$, and keep that value of $(x,y)$ for the following stages. This
requires $n\cdot m$ states.
2. 2.
We use four Morris counters - one that approximates a clock, one that
approximates a count for $x$ values, one that approximates a count for $y$
values, and one that approximates a count for the pair $(x,y)$:
* •
The first counter has $M=\log T+1$ states, where $T$ is the next power of $2$
after $\lceil(n\cdot m)^{c}\rceil$. The random stopping time of the counter is
$N$. This counter approximates a clock that counts until $\lceil(n\cdot
m)^{c}\rceil$ samples from the distribution are observed. This takes at most
$c\log(n\cdot m)+2$ states.
* •
The second, third and fourth counters run in parallel to the first one and
approximate a counter for $x$, a counter for $y$, and a counter for the pair
$(x,y)$, and we denote their outputs as $C_{N_{x}}$, $C_{N_{y}}$ and
$C_{N_{xy}}$, respectively. Each of these counters contains $M$ states.
3. 3.
Let $C_{\text{MI}}=C_{N_{x}}+C_{N_{y}}-C_{N_{xy}}$, and define
$C_{\text{MI}}^{\text{centered}}=C_{\text{MI}}-\mu-\operatorname{\mathbb{E}}\log
N$. For the reasons outlined in Section IV, the expectation of
$C_{\text{MI}}^{\text{centered}}$ is equal to $-\log p_{x}-\log p_{y}+\log
p_{xy}$ plus some bias
$\displaystyle|b_{\text{MI}}|\leq 3\cdot
10^{-5}+O\left(n^{-\frac{1}{2}\cdot(c-1)}\vee
m^{-\frac{1}{2}\cdot(c-1)}\right).$ (78)
4. 4.
We simulate the expectation operation using a bias estimation machine with
$S_{\text{bias}}=\frac{9M^{2}}{\beta^{2}\delta}+1$ states. Let
$\theta_{N_{xy}}=-\frac{C_{\text{MI}}^{\text{centered}}}{3M}+a$, where
$a\triangleq\frac{2M-\operatorname{\mathbb{E}}(\log N)-\mu}{3M}$ is a known
offset that we add to guarantee that $\theta_{N_{xy}}\in[0,1)$ with
probability $1$. Averaging over $(X,Y)=(x,y)$, our bias is
$\theta_{\text{MI}}=\operatorname{\mathbb{E}}(\theta_{N_{XY}})=\frac{I(X;Y)+b_{\text{MI}}}{3M}+a$.
After subtraction of $a$ and multiplication by $3M$, that is, setting
$\hat{I}=3M(\hat{\theta}_{\text{MI}}-a)$ we get an (almost) unbiased estimator
for $I(X;Y)$. As we have
$\displaystyle\operatorname{\mathbb{E}}(\hat{I}-(I(X;Y)+b_{\text{MI}}))^{2}=9M^{2}\cdot\operatorname{\mathbb{E}}(\hat{\theta}_{\text{MI}}-\theta_{\text{MI}})^{2}\leq\beta^{2}\delta,$
(79)
Chebyshev’s inequality implies that
$\Pr(|\hat{I}-(I(X;Y)+b_{\text{MI}})|>\beta)\leq\delta$. The total number of
states is thus the product of the number of states at each step,
$\displaystyle S=nm\cdot
M^{4}\cdot\left(\frac{9M^{2}}{\beta^{2}\delta}+1\right)\leq\frac{9nm\cdot(c+1)^{6}(\log
nm)^{6}}{\beta^{2}\delta},$ (80)
for $\beta>0$, $\varepsilon=3\cdot
10^{-5}+\beta+O\left(n^{-\frac{1}{2}\cdot(c-1)}\vee
m^{-\frac{1}{2}\cdot(c-1)}\right)$.
### VI-B Lower Bound
For simplicity of proof, let $\varepsilon,\delta\geq\frac{1}{100}$, and recall
that $\varepsilon<\frac{1}{12\ln 2}$. Our lower bound from Theorem 2 implies
that for joint entropy estimation of $H(X,Y)$ where $(X,Y)\in[n]\times[m]$,
the memory complexity is $\Omega(n\cdot m)$. Assume that we have a mutual
information estimation machine that returns an estimate of $I(X;Y)$ with
additive error at most $\varepsilon$ with probability at least $1-\delta$
using $S_{\text{MI}}^{*}(n,m,\varepsilon,\delta)$ states. We show below an
algorithm that uses this machine as a black box and estimates
$H(X,Y)=H(X)+H(Y)-I(X;Y)$ with additive error of at most $3\varepsilon$ with
probability at least $1-3\delta$ using $S_{\text{MI}}^{*}\cdot
O(\log^{3}n\cdot\log^{3}m)$ states. Since estimation of $H(X,Y)$ requires
$S^{*}(n\cdot m,3\varepsilon,3\delta)=\Omega(n\cdot m)$, this must imply that
$\displaystyle
S_{\text{MI}}^{*}(n,m,\varepsilon,\delta)>\Omega\left(\frac{n\cdot
m}{\log^{3}n\cdot\log^{3}m}\right).$ (81)
We now describe such an algorithm. The algorithm has $3$ modes. It starts in
mode $1$, in which $H(X)$ is estimated. It then moves to mode $2$, in which
$H(Y)$ is estimated, and finally it moves to mode $3$ in which $I(X;Y)$ is
estimated. The estimation of these $3$ quantities is done using
$\displaystyle\tilde{S}=\max\\{S^{*}(n,\varepsilon,\delta),S^{*}(m,\varepsilon,\delta),S_{\text{MI}}^{*}(n,m,\varepsilon,\delta)\\}$
(82)
states. Those states are “reused” once the algorithm switches its mode of
operation. The current mode is stored using $S_{1}=3$ states.
In mode $1$, the algorithm estimates $H(X)$ using the Morris-counter based
machine we introduced in Section IV, which uses $S^{*}(n,\varepsilon,\delta)$
states. After a long enough time, the estimate will be accurate enough, and we
will then stop the machine and store that value. In order to decide if enough
time has passed, we must ensure that the bias estimation machine with
$S_{\text{Bias}}=O(\log^{2}n)$ number of states, to which we feed
$\mathrm{Ber}(\theta_{N_{X}})$ samples, is sufficiently mixed. From Theorem 5,
we have that the mixing time of the bias estimation machine is at most
$4S_{\text{Bias}}\log S_{\text{Bias}}\leq O(\log^{3}n)$. Thus, we will let the
machine run for $\log^{k}n$ samples of independent
$\mathrm{Ber}(\theta_{N_{X}})$ random variables for $k>>3$, and then stop it,
which would guarantee it is sufficiently mixed. To determine when to stop the
machine without using a memory consuming clock, we will use another Morris
counter with $S_{2}=O(\log\log^{k}n)=O(\log\log n)$ states. We will also store
the state of the bias estimation machine, which corresponds to our estimate
$\hat{H}(X)$ of $H(X)$, using $S_{3}=S_{\text{Bias}}=O(\log^{2}n)$ states. At
this point, the algorithm switches to mode $2$, and estimates $H(Y)$ with the
algorithm from Section IV, which uses $S^{*}(m,\varepsilon,\delta)$ states. As
in mode $1$, we use a Morris counter of $S_{4}=O(\log\log m)$ states to
determine when the machine is sufficiently mixed and can be stopped, and store
the state of the bias estimation machine, which corresponds to the estimate
$\hat{H}(Y)$ of $H(Y)$, using $S_{5}=O(\log^{2}m)$ states. The process then
moves to state $3$ where $I(X;Y)$ is estimated using the black-box machine
with $S_{\text{MI}}^{*}(n,m,\varepsilon,\delta)$ states. From this time
onward, the machine estimates $H(X,Y)$ as
$\hat{H}(X)+\hat{H}(Y)-\hat{I}(X;Y)$, where $\hat{I}(X;Y)$ is the current
estimate of the black box machine. All in all, this algorithm produces a
$(3\varepsilon,3\delta)$ (recall that we assumed $\delta,\varepsilon\geq
1/100$) estimate of $H(X,Y)$ using
$\displaystyle S\leq\tilde{S}\prod_{i=1}^{5}S_{i}=\tilde{S}\cdot
O(\log^{3}n\cdot\log^{3}m),$ (83)
which implies that
$\displaystyle\tilde{S}=\Omega\left(\frac{S}{\log^{3}n\log^{3}m}\right)=\Omega\left(\frac{S^{*}(n,m,3\varepsilon,3\delta)}{\log^{3}n\log^{3}m}\right)=\Omega\left(\frac{n\cdot
m}{\log^{3}n\log^{3}m}\right).$ (84)
Finally, since Theorem 1 states that
$S^{*}(n,\varepsilon,\delta)=O(n\cdot\log^{4}n)$ and
$S^{*}(m,\varepsilon,\delta)=O(m\cdot\log^{4}m)$, and we assumed that
$\frac{n}{\log^{3}n}=\Omega(\log^{7}m)$ and
$\frac{m}{\log^{3}m}=\Omega(\log^{7}n)$, we must therefore have that
$\displaystyle
S_{\text{MI}}^{*}(n,m,\varepsilon,\delta)=\Omega\left(\frac{n\cdot
m}{\log^{3}n\cdot\log^{3}m}\right).$ (85)
## Acknowledgements
This work was supported by the ISF under Grants 1641/21 and 1766/22.
## References
* [1] J. Acharya, S. Bhadane, P. Indyk, and Z. Sun, “Estimating entropy of distributions in constant space,” arXiv preprint arXiv:1911.07976, 2019\.
* [2] M. Aliakbarpour, A. McGregor, J. Nelson, and E. Waingarten, “Estimation of entropy in constant space with improved sample complexity,” arXiv preprint arXiv:2205.09804, 2022.
* [3] R. Morris, “Counting large numbers of events in small registers,” Communications of the ACM, vol. 21, no. 10, pp. 840–842, 1978.
* [4] F. Leighton and R. Rivest, “Estimating a probability using finite memory,” IEEE Transactions on Information Theory, vol. 32, no. 6, pp. 733–742, 1986\.
* [5] T. Berg, O. Ordentlich, and O. Shayevitz, “Deterministic finite-memory bias estimation,” in Conference on Learning Theory, pp. 566–585, PMLR, 2021\.
* [6] A. Chakrabarti, K. Do Ba, and S. Muthukrishnan, “Estimating entropy and entropy norm on data streams,” Internet Mathematics, vol. 3, no. 1, pp. 63–78, 2006.
* [7] T. Berg, O. Ordentlich, and O. Shayevitz, “On the memory complexity of uniformity testing,” in Conference on Learning Theory, pp. 3506–3523, PMLR, 2022.
* [8] T. M. Cover et al., “Hypothesis testing with finite statistics,” The Annals of Mathematical Statistics, vol. 40, no. 3, pp. 828–835, 1969.
* [9] M. E. Hellman and T. M. Cover, “Learning with finite memory,” The Annals of Mathematical Statistics, pp. 765–782, 1970.
* [10] F. J. Samaniego, “Estimating a binomial parameter with finite memory,” IEEE Transactions on Information Theory, vol. 19, no. 5, pp. 636–643, 1973.
* [11] J. Steinhardt and J. Duchi, “Minimax rates for memory-bounded sparse linear regression,” in Conference on Learning Theory, pp. 1564–1587, 2015.
* [12] J. Steinhardt, G. Valiant, and S. Wager, “Memory, communication, and statistical queries,” in Conference on Learning Theory, pp. 1490–1516, 2016.
* [13] R. Raz, “Fast learning requires good memory: A time-space lower bound for parity learning,” Journal of the ACM (JACM), vol. 66, no. 1, p. 3, 2018\.
* [14] Y. Dagan and O. Shamir, “Detecting correlations with little memory and communication,” in Conference On Learning Theory, pp. 1145–1198, 2018\.
* [15] A. Jain and H. Tyagi, “Effective memory shrinkage in estimation,” in 2018 IEEE International Symposium on Information Theory (ISIT), pp. 1071–1075, IEEE, 2018.
* [16] Y. Dagan, G. Kur, and O. Shamir, “Space lower bounds for linear prediction in the streaming model,” in Conference on Learning Theory, pp. 929–954, 2019\.
* [17] V. Sharan, A. Sidford, and G. Valiant, “Memory-sample tradeoffs for linear regression with small error,” in Symposium on Theory of Computing (STOC), 2019.
* [18] S. Garg, P. K. Kothari, P. Liu, and R. Raz, “Memory-sample lower bounds for learning parity with noise,” in 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021, p. 60, Schloss Dagstuhl-Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2021.
* [19] A. Pensia, V. Jog, and P.-L. Loh, “Communication-constrained hypothesis testing: Optimality, robustness, and reverse data processing inequalities,” arXiv preprint arXiv:2206.02765, 2022.
* [20] G. P. Basharin, “On a statistical estimate for the entropy of a sequence of independent random variables,” Theory of Probability & Its Applications, vol. 4, no. 3, pp. 333–336, 1959.
* [21] A. Antos and I. Kontoyiannis, “Convergence properties of functional estimates for discrete distributions,” Random Structures & Algorithms, vol. 19, no. 3-4, pp. 163–193, 2001.
* [22] L. Paninski, “Estimation of entropy and mutual information,” Neural computation, vol. 15, no. 6, pp. 1191–1253, 2003.
* [23] L. Paninski, “Estimating entropy on m bins given fewer than m samples,” IEEE Transactions on Information Theory, vol. 50, no. 9, pp. 2200–2203, 2004\.
* [24] G. Valiant and P. Valiant, “A clt and tight lower bounds for estimating entropy.,” in Electron. Colloquium Comput. Complex., vol. 17, p. 179, 2010\.
* [25] G. Valiant and P. Valiant, “Estimating the unseen: an n/log (n)-sample estimator for entropy and support size, shown optimal via new clts,” in Proceedings of the forty-third annual ACM symposium on Theory of computing, pp. 685–694, 2011.
* [26] P. Valiant and G. Valiant, “Estimating the unseen: Improved estimators for entropy and other properties.,” in NIPS, pp. 2157–2165, 2013.
* [27] G. Valiant and P. Valiant, “The power of linear estimators,” in 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 403–412, IEEE, 2011.
* [28] J. Jiao, K. Venkat, Y. Han, and T. Weissman, “Minimax estimation of functionals of discrete distributions,” IEEE Transactions on Information Theory, vol. 61, no. 5, pp. 2835–2885, 2015.
* [29] Y. Wu and P. Yang, “Minimax rates of entropy estimation on large alphabets via best polynomial approximation,” IEEE Transactions on Information Theory, vol. 62, no. 6, pp. 3702–3720, 2016.
* [30] A. Lall, V. Sekar, M. Ogihara, J. Xu, and H. Zhang, “Data streaming algorithms for estimating entropy of network traffic,” ACM SIGMETRICS Performance Evaluation Review, vol. 34, no. 1, pp. 145–156, 2006.
* [31] S. Guha, A. McGregor, and S. Venkatasubramanian, “Sublinear estimation of entropy and information distances,” ACM Transactions on Algorithms (TALG), vol. 5, no. 4, pp. 1–16, 2009.
* [32] S. Chien, K. Ligett, and A. McGregor, “Space-efficient estimation of robust statistics and distribution testing.,” in ICS, pp. 251–265, Citeseer, 2010\.
* [33] P. Flajolet, “Approximate counting: a detailed analysis,” BIT Numerical Mathematics, vol. 25, no. 1, pp. 113–134, 1985.
* [34] J. Nelson and H. Yu, “Optimal bounds for approximate counting,” arXiv preprint arXiv:2010.02116, 2020.
* [35] D. A. Levin and Y. Peres, Markov chains and mixing times, vol. 107. American Mathematical Soc., 2017.
* [36] T. M. Cover and J. A. Thomas, Elements of information theory. John Wiley & Sons, 2012.
|
fixmeTotalCounter [500] Human-centered computing HCI theory, concepts and
models rightsretained 978-1-4503-6819-3/20/04
# Conceptualizing the Relationship between AI Explanations and User Agency
Iyadunni Adenuga
Jonathan Dodge
Penn State University University Park, PA 16802, USA<EMAIL_ADDRESS>Penn
State University University Park, PA 16802, USA<EMAIL_ADDRESS>
(2023)
###### Abstract
We grapple with the question: _How, for whom and why should explainable
artificial intelligence (XAI) aim to support the user goal of agency?_ In
particular, we analyze the relationship between agency and explanations
through a user-centric lens through case studies and thought experiments. We
find that explanation serves as one of several possible first steps for agency
by allowing the user convert forethought to outcome in a more effective manner
in future interactions. Also, we observe that XAI systems might better cater
to laypersons, particularly “tinkerers,” when combining explanations and user
control, so they can make meaningful changes.
###### keywords:
Explainability; Agency; AI systems
###### doi:
https://doi.org/10.1145/3334480.XXXXXXX
††conference: CHI Workshops ’23 April 23–28, 2023, Hamburg, Germany
## 1 Introduction
Complex technologies are commonplace in today’s society, with examples
including reinforcement learning, deep neural networks, or other forms of
artificial intelligence (AI). Criticisms have plagued the acceptance of these
technologies due to the opaque nature of the algorithms and the erasure of
user influence (i.e., creating an automated experience). For example, high-
stakes scenarios (e.g. law enforcement, medicine, etc.) traditionally require
human experts that go through rigorous training, who are then accountable to
human stakeholders. Thus, it is unsurprising that such decision makers prefer
worse-performing, interpretable models over opaque models [45].
Beyond experts, laypeople also desire a level of control and understanding of
the complex AI systems that affect them [48, 40]. Legal regimes (e.g.,
European Union General Data Protection Regulation [18] and White House
Executive Order [23]) align with such observations by highlighting the
importance of human agency over these systems and the need for these systems
to explain and justify their results.
However, making AI systems more _agentic_ is not as widely studied as making
them _explainable_. This paper attempts to describe how designing for agency
fits with XAI, namely: 1) the relationship between agency and explanations and
2) agency in scenarios with two and three user groups.
## 2 What is an Explanation?
Explanation is a human phenomenon that strongly relates with peoples’ mental
models, understanding and knowledge of “why an outcome happened” [26]. Its
social interactive characteristic [26] means there’s some level of
communication (which may be continuous) occurring between the explain-_er_ and
explain-_ee_.
Relationship between Agency and Explanations in an XAI system
Today, AI systems are a major part of our environment. If the target users do
not understand the model, they usually cannot assess or appropriately rely
[38] on it. To address this, post-hoc methods aim to make opaque AI methods
(e.g. neural networks, ensemble models, etc.) more “understandable” without
compromising accuracy [4, 14]. There are two main approaches employed by post-
hoc techniques: _opaque box_ (operates on the input/output boundary; e.g.,
LIME [36], LORE [21]) and _transparent box_ (operates on the internal
structures; e.g., deconvnet model method [50] and network dissection [8, 9]).
[90pt] From Zhang and Lim paper [51], user interface of Counterfactual
Explanation for Voice-Emotion Recognition system (best viewed digitally).
Existing XAI systems that utilize the opaque- and transparent-box approaches
described above do not fit the requirements laid out in prior work for
“everyday” explanations understandable to the layperson [33]. AI explanations
created based on human characteristics (e.g. preferences, reasoning and
perception methods) are more relatable and effective [46, 51, 29]. In their
work about connecting existing XAI techniques to user expectations for
explanations, Liao et al. [30] propose a “question-driven framework” that
encourages an interactive explanation experience [30] via meaningful
interrogative dialogue [33].
## 3 What is Agency?
People have an innate need to control the course of their lives and predict
the outcomes of situations, no matter the difficulty [6]. Humans feel a sense
of agency when we believe that our _“conscious intention caused a voluntary
action”_ [47]. Agency is an internal “human” feeling that is outwardly
expressed by intentional actions. If people do not feel in control, they might
abandon the on-going task or distrust their actions, especially in hard
situations [5].
A technology that affords agency is “flexible” to the user’s interactions
inputs and interests such that they can modify their experience [49, 43]. The
control a person has in a typical environment (such as while utilizing
technology) can be weighted by: 1) the presence of relevant actions; 2) the
relationship between the actions of a user and the outcome in the environment;
3) the ability of a user to predict the outcome of their actions, and; 4) the
ability of the user to trace the cause of an outcome [42].
Researchers have shown agentic interactions have positive effects such as
improved user experience and satisfaction and more appropriate trust [19, 24,
44]. The many AI systems stakeholders with low technical knowledge should also
experience these benefits, as per the ACM Code of Ethics: _“…all people are
stakeholders in computing”_ [2].
## 4 How are Agency and Explanation Related?
The answer to this question is not straightforward, but we will attempt an
answer for AI systems. Existing human-centered XAI systems prioritize
providing explanations in an understandable, visually appealing format with an
assumption of improved agency in the represented artifact. There is no
_direct_ measure for the “actual” agency a user experienced while interacting
with such a system. _Self-reporting_ only measures agency _perceived_ by
users, which is a proxy for “actual” agency. Teasing out the modalities of the
relationship between explanation and agency is the first step in deducing the
“actual” agency in XAI systems.
The agency process starts from a person’s _forethought_ to their performance
of an action and then, observation of the action’s outcome. The aim of
explanations is to improve the consumers’ understanding of their environment.
Providing explanations can contribute to consumers’ sense of agency by
informing their initial _forethought_ so they perform the appropriate actions
to successfully complete their task. People with higher need for control are
more likely to seek more information and clarifications in a work environment
[20]. This shows that even before the introduction of explanations to a
scenario, an individual has an inherent agency requirement—and that such
requirements will vary among users.
Studies on designing agency in AI systems, such as interactive machine
learning, have primarily focused on users with technical know-how, and in its
absence, requires additional technical training for end-users for them to
understand and use the provided agentic functions [41, 15]. End-users with no
access to technical training can still benefit from an agentic experience with
explanations.
[20pt] The original Rube Goldberg machine, as depicted in the cartoon
“Professor Butts and the Self-Operating Napkin.” The machine functions as
follows: _“Soup spoon (A) is raised to mouth, pulling string (B) and thereby
jerking ladle (C), which throws cracker (D) past toucan (E). Toucan jumps
after cracker and perch (F) tilts, upsetting seeds (G) into pail (H). Extra
weight in pail pulls cord (I), which opens and ignites lighter (J), setting
off skyrocket (K), which causes sickle (L) to cut string (M), allowing
pendulum with attached napkin to swing back and forth, thereby wiping chin.”_
https://en.wikipedia.org/wiki/Rube_Goldberg_machine Users can take an active
role in their absorption of an explanation. Zhang and Lim [51] studied
providing relatable explanations for a vocal-emotion recognition system, the
participants preferred and utilized more effectively the “Counterfactual
Sample + Cues” explanation (Figure 2). The user interface for this explanation
required active play-through and listening to alternative voices to detect
vocal differences. Another method for involving people in the explanation
process is to obtain input from them to create “selective” explanations [29].
Here, the user customizes the types of received explanations to their taste.
Tastes vary, for example GenderMag [10] identified facets describing people’s
cognitive styles. One important axis is _learning style_ , with people who
gain understanding by “tinkering” with the technology on one end. To cater to
tinkerers, XAI system designs should have control functions. These functions
would accept different kinds of user inputs and then provide appropriate
visible outcomes, allowing the system to _“be actionable”_ [28]. On the other
end of the learning style axis are people who gain understanding _by process_.
Process-oriented learners may benefit from assessment _processes_ , such as
After-Action Review for AI (AAR/AI) [13]. Later, Khanna et al. [27] found
participants helped participants examine and effectively use explanations to
identify AI faults, observing a moderate-sized practical effect.
There are situations when XAI systems cannot honor user inputs [41]. How
should the system react? For low-stakes scenarios, illusory agency may be a
useful tool. Game designers use this as a complementary mechanism to preserve
their rigid game-story narrative [12, 32]. To allow for continued user agency,
the user is able to observe the effect of their input but the input has no
effect on the underlying algorithms. Vaccaro et al. [44] showed in a social
media setting that users _“felt more satisfied with the presence of controls”_
regardless of their effectiveness. Some everyday systems that already utilize
illusory agency include crosswalk buttons and elevator close-door buttons.
Illusory agency should only be designed to supplement the already present
“real” agentic experience in low-stakes scenarios so as to avoid user
deception and minimize the effects of ethical issues. Example of such
scenarios that might benefit from illusory agency include XAI systems in
training environments [17].
## 5 How Does One Increase or Decrease Agency?
We will use two examples to illustrate adjusting agency. The first example is
to consider wiping your mouth with a napkin using direct manipulation vs with
a Rube Goldberg machine, depicted in Figure 4. Rube Goldberg machines are
famous for having a simple input, which then initiates a complicated chain
reaction generating a simple output. In this case, the simple input is lifting
a spoon, the long chain reaction is via the crackers, toucan, string, etc.,
and the simple output is wiping one’s mouth.
[10pt] Notional curve depicting agency as a function of number of controls.
When moving from no controls to few controls, agency gain enjoys a direct
relationship. But at some point, the extra controls will overwhelm the user,
either taking the form of a plateau (red curve) or even a downturn (blue
curve). [-35pt] 1) Add/remove source documents 2) Add/remove sections, where
sections are subtopics of the document title 3) Add/remove words and/or
sentences 4) View the order relation of the summary sentences in a concept
graph 5) View actual sentences contributed by each document to the overall
summary output Key functionality found in Living Documents.
As Figure 4 shows, the machine automates the functioning of the napkin to the
point that its use is involuntary. Suppose we changed the simple input to be
pushing a button, which is more typical of modern technology. Now, consider
how much agency the user has in each case. It seems fairly obvious that agency
would be highest with direct manipulation, then with button-interface Rube
Goldberg machine, and finally the unmodified Rube Goldberg machine. The
reasoning is that with direct manipulation, one could manifest whatever wiping
approach they desire: arbitrary direction, length of time, and so on. Note
that all of the previously compared interactions lead to same _outcome_ , but
are different in terms of controllability [37]. According to Shneiderman [39],
high levels of automation and human control can co-exist in a technical
artifact. They illustrate this in their description of the digital camera and
elevator where agency is afforded by the inclusion of a button and settings
page respectively. This is similar to the surface-level agency button
introduced above, to the Rube Goldberg machine. Would increasing agency
require addition of extra buttons/settings or introducing a more manual and
influential process (i.e., less automation)? If we assume adding a feature and
accompanying button increases perceived agency, what is the amount of UI
complexity at which agency gains diminish or turn negative (See Figure 5 for
an illustration)?
Thus, we have illustrated a tension between manual processes (which have the
highest agency) vs automation (which reduces agency). The open question is,
how much agency does one lose when a process undergoes automation? To answer
this question, we turn to our second example: Living Documents [3]. Living
Documents is an interactive multi-document text summarization system,
providing the control functions found in Table 5.
What would agency treatment levels look like in Living Documents? The highest-
level of agency is full access to all functionality in the system (full-
agency, see Table 5) while the automation level has no user controls (no-
agency). This means it would work like a typical text summarization system
such that the only input-output operation is the user providing the source
documents and receiving the summary result. The interaction designer can
decide on intermediate level(s) based on specified criteria. Our criterion is
“magnitude of impact” (document $\rightarrow$ sections $\rightarrow$
sentences/words), so our some-agency treatment has functionality 3 to 5 in
Table 5.
## 6 Agency and/or Explanations, for whom?
Now, we would like to broaden the previously discussed system-user agency
cases to where there are _multiple types_ of users, leading to a more complex
tradeoff relationship. From an explanation perspective, we know that
explanations may need to account for different domain expertise, cognitive
abilities, context of use, and audience. Users have varied needs for an XAI
system [30] and do not have a homogeneous process for interacting with models
[41]. The agency perspective is less well studied.
Consider the case of a rideshare application called Co-opRide, which is an
algorithmic manager for _two_ user groups: drivers and riders. What is the
right agency balance to strike between these three parties (the third is Co-
opRide)? Suppose Co-opRide offers a design feature that drivers can veto
riders. This would increase the agency of the drivers at the expense of the
riders’ agency, as well as that of the system provider. Imagine being a rider
receiving vetoes from several drivers, based on your low population density
location or even worse, cultural markers present in your name. This might lead
to long wait times and negative customer sentiment. Should the XAI system
alert riders that a driver vetoed them? Each time? How should the XAI system
provide notification of the veto and/or explain the decision? If there are no
satisfying agentic actions available, why should the system provide
explanation at all? In the case where the rider’s waiting time increases over
time as a result of receiving multiple driver vetoes, provision of
explanations by the algorithmic XAI platform becomes even more imperative.
The example of the driver veto feature suggests “The Agency Tradeoff Game”
might be zero-sum, though this it is not totally clear that it cannot be
positive-sum or negative-sum. It is also an open question whether or not a
stable solution exists. As an example, “Hotelling’s game” (see margin) has a
stable solution with two players, but the three-player version has no stable
solution ([35], Chapter 3).
††margin: Hotelling’s Game [22] Suppose two competing shops are located along
the length of street, with customers spread equally along the street. Each
customer will always choose the nearer shop. With two shops, the consumer
ideal has shops at $\frac{1}{4}$ and $\frac{3}{4}$. However, this is unstable,
since both shops can claim more customers by moving toward the middle. The
stable solution has both at $\frac{1}{2}$. (E.g., this is why Lowe’s and Home
Depot are often co-located.)
XAI platforms interact with groups of humans; as a result, agency occurring on
a _collective_ basis becomes relevant. When individuals perform a joint
action, they can feel individual and/or collective (joint) agency [31]. The
individual perceived self-efficacy of multiple members of a group forms
collective efficacy which can lead to meaningful social change [5, 7]. People
are usually interested in the experiences of their fellow people and this has
led to calls for social explanations [30, 16]. Enabling social explanations
means there can be a joint platform for “knowledge sharing” and “social
learning” [16]. People can confidently contest the decisions by an AI system
and if some form of collective agency has been designed in the system, they
can effect popular meaningful change.
For an example of collective agency as a result of the conditions in an AI
system, consider work by Calacci and Pentland [11]. Shipt, a grocery delivery
service with an AI algorithmic manager, was initially explainable and
transparent about its wage calculation process. The introduction of a wage-
processing opaque-box algorithm to Shipt led to the implementation of social
explanations, albeit on a platform (called Shipt Calculator [11]) external to
the AI system. Workers anonymously provided screenshots of their payment
history and the Shipt Calculator aggregated the payment information and
provided the observed wage difference to the workers. These authors discovered
a paycut for 41 percent of the workers in their study. Similar occurrences
with Doordash led to change in the pay and tipping model as well as a class-
action lawsuit [34, 25]. XAI systems that allow social explanations and
collective agency would ensure a cooperative approach so that only beneficial
improvements are implemented on the platform.
## 7 Concluding Remarks
We do not claim that our statement of the relationship between agency and
explanation is complete. As an example, perhaps agency and explanations might
share multiple simultaneous relationships. We believe that recognizing and
formalizing these relationship(s) would ensure that XAI designers take the
closely-related extra step of designing for _agency_ while working on
explainability.
## References
* [1]
* [2] ACM. 2018. ACM Code of Ethics and Professional Conduct. (2018). https://www.acm.org/code-of-ethics
* [3] Iyadunni J Adenuga, Benjamin V Hanrahan, Chen Wu, and Prasenjit Mitra. 2022. Living Documents: Designing for User Agency over Automated Text Summarization. In CHI Conference on Human Factors in Computing Systems Extended Abstracts. 1–6.
* [4] Alejandro Barredo Arrieta, Natalia Díaz-Rodríguez, Javier Del Ser, Adrien Bennetot, Siham Tabik, Alberto Barbado, Salvador García, Sergio Gil-López, Daniel Molina, Richard Benjamins, and others. 2020. Explainable Artificial Intelligence (XAI): Concepts, taxonomies, opportunities and challenges toward responsible AI. Information fusion 58 (2020), 82–115.
* [5] Albert Bandura. 1982. Self-efficacy mechanism in human agency. American psychologist 37, 2 (1982), 122.
* [6] Albert Bandura. 1996a. Reflections on human agency: Part I. Constructivism in the Human Sciences 1, 2 (1996), 3.
* [7] Albert Bandura. 1996b. Reflections on human agency: Part II. Constructivism in the Human Sciences 1, 3/4 (1996), 5.
* [8] David Bau, Bolei Zhou, Aditya Khosla, Aude Oliva, and Antonio Torralba. 2017. Network dissection: Quantifying interpretability of deep visual representations. In Proceedings of the IEEE conference on computer vision and pattern recognition. 6541–6549.
* [9] David Bau, Jun-Yan Zhu, Hendrik Strobelt, Agata Lapedriza, Bolei Zhou, and Antonio Torralba. 2020. Understanding the role of individual units in a deep neural network. Proceedings of the National Academy of Sciences 117, 48 (2020), 30071–30078.
* [10] Margaret Burnett, Simone Stumpf, Jamie Macbeth, Stephann Makri, Laura Beckwith, Irwin Kwan, Anicia Peters, and William Jernigan. 2016. GenderMag: A method for evaluating software’s gender inclusiveness. Interacting with Computers 28, 6 (2016), 760–787.
* [11] Dan Calacci and Alex Pentland. 2022. Bargaining with the Black-Box: Designing and Deploying Worker-Centric Tools to Audit Algorithmic Management. Proc. ACM Hum.-Comput. Interact. 6, CSCW2, Article 428 (nov 2022), 24 pages. DOI:http://dx.doi.org/10.1145/3570601
* [12] Timothy Day and Jichen Zhu. 2017. Agency informing techniques: Communicating player agency in interactive narratives. In Proceedings of the 12th International Conference on the Foundations of Digital Games. 1–4.
* [13] Jonathan Dodge, Roli Khanna, Jed Irvine, Kin-ho Lam, Theresa Mai, Zhengxian Lin, Nicholas Kiddle, Evan Newman, Andrew Anderson, Sai Raja, Caleb Matthews, Christopher Perdriau, Margaret Burnett, and Alan Fern. 2021. After-Action Review for AI (AAR/AI). ACM Trans. Interact. Intell. Syst. 11, 3–4, Article 29 (sep 2021), 35 pages. DOI:http://dx.doi.org/10.1145/3453173
* [14] Filip Karlo Došilović, Mario Brčić, and Nikica Hlupić. 2018. Explainable artificial intelligence: A survey. In 2018 41st International convention on information and communication technology, electronics and microelectronics (MIPRO). IEEE, 0210–0215.
* [15] John J Dudley and Per Ola Kristensson. 2018. A review of user interface design for interactive machine learning. ACM Transactions on Interactive Intelligent Systems (TiiS) 8, 2 (2018), 1–37.
* [16] Upol Ehsan, Q Vera Liao, Michael Muller, Mark O Riedl, and Justin D Weisz. 2021. Expanding explainability: Towards social transparency in ai systems. In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems. 1–19.
* [17] Krzysztof Fiok, Farzad V Farahani, Waldemar Karwowski, and Tareq Ahram. 2022. Explainable artificial intelligence for education and training. The Journal of Defense Modeling and Simulation 19, 2 (2022), 133–144.
* [18] GDPR. 2018. European Union General Data Protection Regulation, Article 15 - “Right of access by the data subject”. (2018). http://www.privacy-regulation.eu/en/article-15-right-of-access-by-the-data-subject-GDPR.htm Accessed: 1/16/2019.
* [19] Andreas Girgensohn, Sara A Bly, Frank Shipman, John S Boreczky, and Lynn Wilcox. 2001. Home Video Editing Made Easy-Balancing Automation and User Control.. In INTERACT, Vol. 1. 464–471.
* [20] Michael Goller and Christian Harteis. 2017. Human agency at work: Towards a clarification and operationalisation of the concept. In Agency at work. Springer, 85–103.
* [21] Riccardo Guidotti, Anna Monreale, Fosca Giannotti, Dino Pedreschi, Salvatore Ruggieri, and Franco Turini. 2019. Factual and counterfactual explanations for black box decision making. IEEE Intelligent Systems 34, 6 (2019), 14–23.
* [22] Harold Hotelling. 1929. Stability in Competition. The Economic Journal 39, 153 (1929), 41–57. http://www.jstor.org/stable/2224214
* [23] White House. 2022. Blueprint for an AI Bill of Rights. (2022). https://www.whitehouse.gov/ostp/ai-bill-of-rights/ Last accessed: 10/13/22.
* [24] Jina Huh, Martha Pollack, Hadi Katebi, Karem Sakallah, and Ned Kirsch. 2010. Incorporating user control in automated interactive scheduling systems. In Proceedings of the 8th ACM Conference on Designing Interactive Systems. 306–309.
* [25] Catie Keck. 2019. DoorDash tip-skimming scheme prompts class action lawsuit seeking all those tips that didn’t go to drivers. (Jul 2019). https://gizmodo.com/doordash-tip-skimming-scheme-prompts-clash-action-lawsu-1836820630
* [26] Frank C Keil. 2006. Explanation and understanding. Annual review of psychology 57 (2006), 227.
* [27] Roli Khanna, Jonathan Dodge, Andrew Anderson, Rupika Dikkala, Jed Irvine, Zeyad Shureih, Kin-ho Lam, Caleb R Matthews, Zhengxian Lin, Minsuk Kahng, and others. 2022. Finding AI’s faults with AAR/AI: An empirical study. ACM Transactions on Interactive Intelligent Systems (TiiS) 12, 1 (2022), 1–33.
* [28] Todd Kulesza, Margaret Burnett, Weng-Keen Wong, and Simone Stumpf. 2015\. Principles of explanatory debugging to personalize interactive machine learning. In Proceedings of the 20th international conference on intelligent user interfaces. 126–137.
* [29] Vivian Lai, Yiming Zhang, Chacha Chen, Q Vera Liao, and Chenhao Tan. 2023. Selective Explanations: Leveraging Human Input to Align Explainable AI. arXiv preprint arXiv:2301.09656 (2023).
* [30] Q Vera Liao, Daniel Gruen, and Sarah Miller. 2020. Questioning the AI: informing design practices for explainable AI user experiences. In Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems. 1–15.
* [31] Janeen D Loehr. 2022. The sense of agency in joint action: An integrative review. Psychonomic Bulletin & Review (2022), 1–29.
* [32] Esther MacCallum-Stewart and Justin Parsler. 2007. Illusory agency in vampire: The masquerade–Bloodlines. Dichtung Digital. Journal für Kunst und Kultur digitaler Medien 9, 1 (2007), 1–17.
* [33] Brent Mittelstadt, Chris Russell, and Sandra Wachter. 2019. Explaining explanations in AI. In Proceedings of the conference on fairness, accountability, and transparency. 279–288.
* [34] Andy Newman. 2019. Doordash changes tipping model after uproar from customers. (Jul 2019). https://www.nytimes.com/2019/07/24/nyregion/doordash-tip-policy.html?action=click& module=Intentional& pgtype=Article
* [35] Martin J. Osborne. 2004. An introduction to game theory. Oxford Univ. Press, New York, NY [u.a.]. http://gso.gbv.de/DB=2.1/CMD?ACT=SRCHA&SRT=YOP&IKT=1016&TRM=ppn+369342747&sourceid=fbw_bibsonomy
* [36] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. 2016. " Why should i trust you?" Explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining. 1135–1144.
* [37] Quentin Roy, Futian Zhang, and Daniel Vogel. 2019. Automation Accuracy Is Good, but High Controllability May Be Better. In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems (CHI ’19). Association for Computing Machinery, New York, NY, USA, 1–8. DOI:http://dx.doi.org/10.1145/3290605.3300750
* [38] Max Schemmer, Niklas Kuehl, Carina Benz, Andrea Bartos, and Gerhard Satzger. 2023. Appropriate Reliance on AI Advice: Conceptualization and the Effect of Explanations. In Proceedings of the 28th International Conference on Intelligent User Interfaces (IUI ’23). Association for Computing Machinery, New York, NY, USA, 410–422. DOI:http://dx.doi.org/10.1145/3581641.3584066
* [39] Ben Shneiderman. 2020. Human-centered artificial intelligence: Reliable, safe & trustworthy. International Journal of Human–Computer Interaction 36, 6 (2020), 495–504.
* [40] Ben Shneiderman, Catherine Plaisant, Maxine Cohen, Steven Jacobs, Niklas Elmqvist, and Nicholoas Diakopoulos. 2016. Grand Challenges for HCI Researchers. Interactions 23, 5 (aug 2016), 24–25. DOI:http://dx.doi.org/10.1145/2977645
* [41] Alison Smith-Renner, Varun Kumar, Jordan Boyd-Graber, Kevin Seppi, and Leah Findlater. 2020. Digging into User Control: Perceptions of Adherence and Instability in Transparent Models. In Proceedings of the 25th International Conference on Intelligent User Interfaces (IUI ’20). Association for Computing Machinery, New York, NY, USA, 519–530. DOI:http://dx.doi.org/10.1145/3377325.3377491
* [42] Suzanne C Thompson, Wade Armstrong, and Craig Thomas. 1998. Illusions of control, underestimations, and accuracy: a control heuristic explanation. Psychological bulletin 123, 2 (1998), 143.
* [43] David Thue, Vadim Bulitko, Marcia Spetch, and Trevon Romanuik. 2010. Player agency and the relevance of decisions. In Joint International Conference on Interactive Digital Storytelling. Springer, 210–215.
* [44] Kristen Vaccaro, Dylan Huang, Motahhare Eslami, Christian Sandvig, Kevin Hamilton, and Karrie Karahalios. 2018. The Illusion of Control: Placebo Effects of Control Settings. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems (CHI ’18). Association for Computing Machinery, New York, NY, USA, 1–13. DOI:http://dx.doi.org/10.1145/3173574.3173590
* [45] Michael Veale, Max Van Kleek, and Reuben Binns. 2018. Fairness and Accountability Design Needs for Algorithmic Support in High-Stakes Public Sector Decision-Making. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems (CHI ’18). ACM, New York, NY, USA, Article 440, 14 pages. DOI:http://dx.doi.org/10.1145/3173574.3174014
* [46] Danding Wang, Qian Yang, Ashraf Abdul, and Brian Y Lim. 2019. Designing theory-driven user-centric explainable AI. In Proceedings of the 2019 CHI conference on human factors in computing systems. 1–15.
* [47] Daniel M Wegner and Thalia Wheatley. 1999. Apparent mental causation: Sources of the experience of will. American psychologist 54, 7 (1999), 480.
* [48] Allison Woodruff, Sarah E. Fox, Steven Rousso-Schindler, and Jeffrey Warshaw. 2018. A Qualitative Exploration of Perceptions of Algorithmic Fairness. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems (CHI ’18). Association for Computing Machinery, New York, NY, USA, 1–14. DOI:http://dx.doi.org/10.1145/3173574.3174230
* [49] Peta Wyeth. 2007. Agency, tangible technology and young children. In Proceedings of the 6th international conference on Interaction design and children. 101–104.
* [50] Matthew D Zeiler and Rob Fergus. 2014. Visualizing and understanding convolutional networks. In European conference on computer vision. Springer, 818–833.
* [51] Wencan Zhang and Brian Y Lim. 2022. Towards relatable explainable AI with the perceptual process. In Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems. 1–24.
|
# Rigidity of Totally Geodesic Hypersurfaces in Negative Curvature
Ben Lowe
###### Abstract.
Let $M$ be a closed hyperbolic manifold containing a totally geodesic
hypersurface $S$, and let $N$ be a closed Riemannian manifold homotopy
equivalent to $M$ with sectional curvature bounded above by $-1$. Then it
follows from the work of Besson-Courtois-Gallot that $\pi_{1}(S)$ can be
represented by a hypersurface $S^{\prime}$ in $N$ with volume less than or
equal to that of $S$. We study the equality case: if $\pi_{1}(S)$ cannot be
represented by a hypersurface $S^{\prime}$ in $N$ with volume strictly smaller
than that of $S$, then must $N$ be isometric to $M$? We show that many such
$S$ are rigid in the sense that the answer to this question is positive. On
the other hand, we construct examples of $S$ for which the answer is negative.
## 1\. Introduction
There is a sharp upper bound for the area of a minimal surface in a Riemannian
3-manifold with a negative upper bound on its sectional curvature. Namely, if
$(M,g)$ is a Riemannian 3-manifold containing a minimal surface $\Sigma$, and
if the sectional curvature $K_{g}$ satisfies $K_{g}\leq-1$, then
(1.1) $\text{Area}_{g}(\Sigma)\leq 2\pi|\chi(\Sigma)|.$
It is simple to check this inequality using the Gauss equation and the Gauss
Bonnet formula, but the question of what happens in the case of equality is
more interesting. Equality in (1.1) implies that $\Sigma$ is totally geodesic
and hyperbolic (constant curvature $-1$) in its induced metric. The
straightforward way for this to happen is if $g$ itself is hyperbolic, but is
this the only way?
V. Lima showed that the answer to this question is negative by constructing
non-hyperbolic examples of metrics on $\Sigma\times\mathbb{R}$ with sectional
curvature at most $-1$ for which equality in (1.1) is attained [Lim19]. In
section 6 we construct closed non-hyperbolic Riemannian 3-manifolds with
sectional curvature at most $-1$ that contain totally geodesic hyperbolic
surfaces. These give examples of metrics for which rigidity in the case of
equality in the area inequality (1.1) fails, even in the case when the ambient
manifold is closed. The main goal of this paper is to give conditions under
which equality in (1.1) forces the ambient metric to be hyperbolic.
In fact, our arguments will work in any dimension. For a closed hyperbolic
$n$-manifold $\Sigma$ and a closed $n+1$-manifold $N$ with sectional curvature
less than or equal to $-1$ the following holds. Suppose we have a map
$F:\Sigma\rightarrow N$ so that the induced map
$\pi_{1}(\Sigma)\rightarrow\pi_{1}(N)$ is injective. Assume also that there is
a closed hyperbolic manifold homotopy equivalent to $N$ via a homotopy
equivalence that sends $F(\Sigma)$ to the homotopy class of a totally geodesic
hypersurface. Then as we explain in Section 3.4, the next theorem is a
consequence of the work of Besson-Courtois-Gallot.
###### Theorem 1.1.
There exists a smooth map $\overline{F}$ homotopic to $F$ so that the
$n$-volume of the image $\overline{F}(\Sigma)$ of $\Sigma$ in $N$ is smaller
than the volume of $\Sigma$ in its hyperbolic metric. The map $\overline{F}$
has the property that in the case that the volume of $\overline{F}(\Sigma)$ is
equal to the volume of $\Sigma$ in its hyperbolic metric, then:
1. (1)
The image $\overline{F}(\Sigma)$ of $\Sigma$ under $\overline{F}$ is a totally
geodesic hypersurface in $N$ with induced metric the hyperbolic metric on
$\Sigma$.
2. (2)
The volume of any immersed hypersurface in $N$ homotopic to and distinct from
$\overline{F}(\Sigma)$ up to reparametrization is greater than that of
$\overline{F}(\Sigma)$.
###### Remark 1.2.
If $N$ is three dimensional, the previous theorem is essentially a consequence
of the fact that $\overline{\Sigma}$ is homotopic to a least area minimal
immersion [SY79] and (1.1.)
Theorem 1.1 leaves open the question of whether equality forces not just the
image of some map homotopic to $F$, but also the ambient metric $N$, to be
hyperbolic. We will show that the answer to this question depends on how the
image of $\Sigma$ sits inside $N$.
We say that a hypersurface $\Sigma$ in a Riemannian n-manifold $N$ is well-
distributed in $N$ if every point in the universal cover $\tilde{N}$ of $N$ is
contained in an embedded solid hypercube all of whose hyperfaces lie in lifts
of $\Sigma$ to $\tilde{N}$. The following is our first main theorem.
###### Theorem 1.3.
Let $N$ be a Riemannian manifold with sectional curvature at most $-1$ that
contains a totally geodesic hyperbolic hypersurface $\Sigma$. Then if $\Sigma$
is well-distributed in $N$, $N$ must be hyperbolic.
Let $M$ be a closed hyperbolic manifold that contains infinitely many totally
geodesic hypersurfaces $\Sigma_{k}$ (see [MR03], [Mil76] for examples.) Then
using Theorem 1.3, we can prove the following:
###### Theorem 1.4.
There exists $K\in\mathbb{N}$ so that the following holds. Fix $\Sigma_{k}$
for $k>K$. Let $N$ be a Riemannian manifold homotopy equivalent to $M$ with
sectional curvature at most $-1$. Assume that there is a totally geodesic
hyperbolic hypersurface in $N$ whose fundamental group injectively includes to
a subgroup conjugate to $\pi_{1}(\Sigma_{k})$ in $\pi_{1}(N)$, where we have
used the homotopy equivalence to identify $\pi_{1}(M)$ and $\pi_{1}(N)$. Then
$N$ is isometric to $M$ in its hyperbolic metric.
###### Remark 1.5.
Rather than assuming that $\Sigma$ (resp. $\Sigma_{k}$) was totally geodesic
and hyperbolic in the previous two theorems, we could instead have assumed, in
view of Theorem 1.1, that $\Sigma$ (resp. $\Sigma_{k}$) was not homotopic in
$N$ to an immersed hypersurface with volume smaller than the volume of
$\Sigma$ (resp. $\Sigma_{k}$) in its hyperbolic metric.
Although it is unclear how sharp Theorem 1.4 is, certainly some assumption on
the hypersurface in the model case is needed by the constuction in Section 6.
To the author’s knowledge, this is the first example of a rigidity phenomenon
for minimal hypersurfaces that is sensitive to how the minimal hypersurface is
distributed in the ambient space. We comment that Farrell-Jones proved that if
$N$ is a closed manifold homotopy equivalent to a closed hyperbolic manifold
$M$ of dimension above four, then $N$ and $M$ must be homeomorphic [FJ89] but
not necessarily diffeomorphic [FJ90]. The sectional curvature of such an $N$
can moreover be taken to be pinched between $-1-\epsilon$ and $-1$ for any
$\epsilon>0$. By geometrization, all closed negatively curved 3-manifolds have
hyperbolic metrics.
The results of this paper are evidence for the following conjecture.
###### Conjecture 1.6.
For every $D>0$ and $k\in\mathbb{N}$ , there exists $A=A(D,k)$ so that the
following holds. Let $N$ be a closed Riemannian manifold of dimension $k$,
sectional curvature bounded above by $-1$, and diameter at most $D$. Then if
$N$ contains a totally geodesic hyperbolic hypersurface with volume greater
than $A$, $N$ must be isometric to a hyperbolic manifold.
One could formulate a similar conjecture for totally geodesic submanifolds
$\Sigma$ of higher codimension. In this case, one would need to modify the
statement by requiring that $\Sigma$ is not contained in some higher
dimensional totally geodesic submanifold with small volume, and one would need
to allow for the possibility that the universal cover of $N$ is some other
rank one symmetric space. It might also be interesting to drop the compactness
assumption on $N$.
### 1.1. Related Work
The study of the kinds of rigidity questions considered in this paper goes
back to Calabi, who showed the following. For a Riemannian 2-sphere $S$ with
sectional curvature between 0 and 1, every simple closed geodesic must have
length at least $2\pi$ [Pog46]. When such a geodesic has length exactly
$2\pi$, then $S$ must be isometric to the unit sphere. (See [AH98], where a
proof of this result is given and attributed to Calabi.)
For $\Sigma$ a minimal sphere in a Riemannian manifold $M$ of sectional
curvature at most 1, inequality (1.1) is true with the direction of the
inequality reversed. In contrast to the negative curvature case discussed
above and the examples of V. Lima, there is a general rigidity statement in
the positive curvature three dimensional setting, due to Mazet-Rosenberg
[MR14]. Mazet recently proved a version in higher codimension [Maz20]. We also
mention that Espinar-Rosenberg obtained rigidity statements for surfaces of
constant mean curvature $H\in(0,1)$ that have the largest possible area given
that the ambient space has sectional curvature at most $-1$.
There are many examples of closed hyperbolic manifolds that contain closed
totally geodesic hypersurfaces (see [MR03] for the three dimensional case, and
[Mil76] for examples coming from so-called arithmetic hyperbolic manifolds of
simplest type.) An arithmetic hyperbolic manifold that contains one totally
geodesic hypersurface must contain infinitely many. In the other direction,
Bader-Fisher-Miller-Stover [BFMS21] and Margulis-Mohamaddi (the latter in
dimension 3) [MM22] proved that a finite volume hyperbolic manifold with
infinitely many totally geodesic hypersurfaces must be arithmetic. There are
examples of closed hyperbolic 3-manifolds that contain no totally geodesic
surfaces [MR03]. It is an open problem whether there are closed hyperbolic
manifolds in dimension above three that contain no totally geodesic
hypersurfaces.
Let $M$ be a closed hyperbolizable 3-manifold. Calegari-Marques-Neves
introduced an entropy functional on metrics $g$ defined by asymptotic counts
of minimal surfaces in $(M,g)$, and proved it to be uniquely minimized at the
hyperbolic metric over all metrics with sectional curvature at most $-1$
[CMN22]. The way we move information obtained via homogenous dynamics from
constant curvature to variable curvature in the proof of Theorem 1.4 is
inspired by their ideas.
Results that assume a negative sectional curvature upper bound often have
analogues that instead assume a negative Ricci or scalar curvature lower
bound. For example, in our joint work with Neves [LN21] we prove the following
analogue of inequality (1.1). Let $(M,g_{hyp})$ be a closed hyperbolic
3-manifold containing a closed totally geodesic surface $\Sigma$. Then for any
metric $g$ on $M$ with scalar curvature at least $-6$, the area of any surface
homotopic to $\Sigma$ must be at least
(1.2) $Area_{g_{hyp}}(\Sigma)=-2\pi\chi(\Sigma).$
For the latter theorem, rigidity in the case of equality holds, in that
equality in (1.2) implies that $g$ must be hyperbolic. In the setting of our
paper, however, whether equality in (1.1) implies that $M$ must be hyperbolic
depends on the homotopy class of $\Sigma$. Thus, perhaps surprisingly, a
sectional curvature upper bound is in this context a weaker constraint than a
scalar curvature lower bound.
### 1.2. Outline
This paper has three main parts. The reader is encouraged to assume that $M$
and $N$ are three dimensional on a first reading.
In the first part, we prove Theorem 1.3. Let $N$ be a negatively curved
manifold containing a totally geodesic hyperbolic hypersurface $\Sigma$ that
satisfies the well-distribution condition. Every point in the universal cover
of $N$ is contained in some solid embedded cube $\Box$ whose hyperfaces lie in
lifts $\tilde{\Sigma}$ of $\Sigma$ to the universal cover. The crucial point
for our argument is that there is an isometric embedding $\Phi$ of the
boundary of $\Box$ in $\mathbb{H}^{n}$. After constructing $\Phi$, we define a
new metric on $\mathbb{H}^{n}$ by gluing the unbounded connected component of
$\mathbb{H}^{n}-\Phi(\partial\Box)$ to $\Box$. Using the Rauch comparison
theorem, we are able to argue that this metric must have been isometric to
$\mathbb{H}^{n}$. We thus conclude that the metric on the interior of $\Box$
has constant curvature $-1$. Since the point we chose was arbitrary, this
shows that $N$ has constant curvature $-1$.
In the second part, we find examples of totally geodesic hypersurfaces
$\Sigma$ in closed hyperbolic manifolds $M$ that satisfy a slightly stronger
version of the well distribution condition. Here we rely on work by Mozes-Shah
that implies uniform distribution statements for totally geodesic
hypersurfaces in closed hyperbolic manifolds. The strong well distribution
condition will imply that for any Riemannian manifold $N$ homotopy equivalent
to $M$ with sectional curvature at most $-1$, any totally geodesic
hypersurface in $N$ corresponding to $\Sigma$ will satisfy the well
distribution condition. This allows us to apply Theorem 1.3 to prove Theorem
1.4.
In the third part, we construct examples of closed Riemannian manifolds $M$
with sectional curvature at most $-1$ that contain totally geodesic hyperbolic
hypersurfaces without themselves being hyperbolic.
## 2\. Acknowledgements
We are grateful to Laurent Mazet for explaining to us the construction in
Proposition 6.2. We also thank Antoine Song and Matthew Stover. The author was
supported by NSF grant DMS-2202830.
## 3\. Background
In this section we collect some facts that we will need in the paper.
### 3.1.
Informally speaking, the Rauch comparison theorem states that if one
Riemannian manifold is more negatively curved than the other, then its Jacobi
fields will grow faster than those of the other. We will need the following
corollary of that theorem (see [CE08][Cor. 1.35, Remark 1.37].)
###### Corollary 3.1 (Corollary of Rauch Comparison Theorem).
Let $(M,g)$ be a Riemannian manifold of dimension $n$ and let $m\in M$. Assume
the sectional curvature $K_{g}$ of $(M,g)$ satisfies $K_{g}\leq-1$. Choose a
point $m_{0}\in\mathbb{H}^{n}$, and let $I:TM_{m}\rightarrow
T\mathbb{H}^{n}_{m_{0}}$ be a linear isometry between the tangent spaces. Let
$c:[0,1]\rightarrow M$ be a geodesic segment in $(M,g)$ so that $exp_{m}$ is a
nonsingular embedding on $s\cdot exp_{m}^{-1}(c(t))$ for $0\leq s\leq 1$ and
$0\leq t\leq 1$. Then we have
(3.1) $L[c]\geq L[exp_{m_{0}}\circ I\circ exp_{m}^{-1}(c)].$
Moreover, in the case of equality the image of the map
$(s,t)\mapsto exp_{m}(s\cdot exp_{m}^{-1}(c(t))),\hskip 5.69054pt0\leq s\leq
1,\hskip 2.84526pt0\leq t\leq 1$
is a solid totally geodesic triangle, and every 2-plane tangent to the image
has constant sectional curvature $-1$.
### 3.2.
Let $M$ be a closed hyperbolic n-manifold. Then $M$ is the quotient of
$\mathbb{H}^{n}$ by a discrete group isomorphic to $\pi_{1}(M)$ acting
properly discontinuously and by isometries. If $N$ is closed negatively curved
Riemannian manifold, then $M$ is homotopy equivalent to $N$ if and only if
$\pi_{1}(M)$ is isomorphic to $\pi_{1}(N)$. Assume this is the case, and fix a
homotopy equivalence $F:M\rightarrow N$. We can lift $F$ to a map
$\tilde{F}:\tilde{M}\rightarrow\tilde{N}$ that commutes with the actions of
$\pi_{1}(M)$ and $\pi_{1}(N)$ by deck transformations.
The boundary at infinity $\partial_{\infty}(X)$ of a simply connected
negatively curved manifold $X$ is the set of geodesic rays in $X$ up to the
equivalence relation of remaining at finite distance for all time when
parametrized by arc-length. For any $x\in X$, the exponential map defines a
bijection between the unit sphere in the tangent space to $X$ and
$\partial_{\infty}(X)$. $\partial_{\infty}(X)$ is topologized so that this map
is a homeomorphism for all $x$. Every geodesic in $X$ is uniquely determined
by its two endpoints in $\partial_{\infty}(X)$, and conversely any two
distinct points in $\partial_{\infty}(X)$ determine a geodesic.
The fundamental groups $\pi_{1}(M)$ and $\pi_{1}(N)$ act by homeomorphisms on
$\partial_{\infty}(\tilde{M})$ and $\partial_{\infty}(\tilde{N})$. The map
$\tilde{F}$ induces a homeomorphism
$\tilde{F}_{\infty}:\partial_{\infty}\tilde{M}\rightarrow\partial_{\infty}\tilde{N}$
between the boundaries at infinity of $\tilde{M}$ and $\tilde{N}$, that
commutes with the actions of $\pi_{1}(M)$ and $\pi_{1}(N)$.
Suppose that $\Sigma$ and $\Sigma_{N}$ are closed totally geodesic hyperbolic
hypersurfaces of $M$ and $N$ so that $F_{*}\pi_{1}(\Sigma)$ is conjugate to
$\pi_{1}(\Sigma_{N})$ in $\pi_{1}(N)$. Let $\tilde{\Sigma}$ be a lift of
$\Sigma$ to $\tilde{M}$, and suppose that the intersection
$G=\cap_{i=1}^{n-k}\gamma_{i}\tilde{\Sigma}$
is transverse for some $\gamma_{1},..,\gamma_{n-k}\in\pi_{1}(M)$. Then $G$ is
a $k$-dimensional totally geodesic subspace in $\tilde{M}\cong\mathbb{H}^{n}$.
The equatorial $n-2$-sphere $\tilde{F}_{\infty}(\gamma_{i}\tilde{\Sigma})$ in
$\partial_{\infty}(N)$ bounds a unique totally geodesic hyperplane that lifts
$\Sigma_{N}$, which by an abuse of notation we call
$\gamma_{i}\tilde{\Sigma}_{n}$. It will be important for us that
$\cap_{i=1}^{n-k}\gamma_{i}\tilde{\Sigma}_{N}$
is a totally geodesic k-plane with boundary at infinity
$\tilde{F}_{\infty}(\partial_{\infty}G)$. This implies, for example, that if
$\Box$ is an embedded hypercube in $\tilde{M}$ whose hyperfaces are totally
geodesic and extend to totally geodesic hyperplanes containing lifts
$\gamma\tilde{\Sigma}$ of $\Sigma$ to $\tilde{M}$, then the region $\Box_{N}$
bounded by the corresponding lifts of $\Sigma_{N}$ in $\tilde{N}$ will also be
a hypercube.
### 3.3.
We state the result from homogeneous dynamics that we will need. Let
$\Sigma_{k}$ be a sequence of distinct totally geodesic hypersurfaces in a
closed hyperbolic n-manifold $M$. Denote by $\hat{\Sigma}_{k}$ their lifts to
the Grassmann bundle $Gr_{n-1}(M)$ of unoriented tangent $n-1$ planes to $M$.
The $\hat{\Sigma}_{k}$ define probability measures $\mu_{k}$ on $Gr_{n-1}(M)$
by
$\mu_{k}(f):=\frac{1}{Vol(\hat{\Sigma}_{k})}\int_{\hat{\Sigma}_{k}}fd\hat{\Sigma}_{k}.$
for $f$ a continuous function on $Gr_{n-1}(M)$, and where $d\hat{\Sigma}_{k}$
is the volume form for the hyperbolic metric on $\Sigma_{k}$. The hyperbolic
metric induces a metric on $Gr_{n-1}(M)$ on which we denote the volume
measure, normalized to have unit volume, by $\mu_{Leb}$. It follows from
Ratner’s measure classification theorem that any weak-$*$ limit of the
$\mu_{k}$ is equal to a convex combination of measures supported on totally
geodesic hypersurfaces and $\mu_{Leb}$. That we can rule out ergodic
components that are supported on totally geodesic hypersurfaces is a
consequence of work by Mozes-Shah [MS95][Theorem 1.1]:
###### Theorem 3.2.
The $\mu_{k}$ weak-$*$ converge to $\mu_{Leb}$.
In fact, we will only require a weaker statement. We say that a surface
$\Sigma$ is $\epsilon$-dense if every tangent plane in $Gr_{n-1}(M)$ is at a
distance of at most $\epsilon$ from some tangent plane to $\Sigma$, where the
distance is measured in the natural metric on $Gr_{n-1}(M)$. The corollary
below follows from Theorem (3.2).
###### Corollary 3.3.
For every $\epsilon>0$ there is $K$ so that $\Sigma_{k}$ is $\epsilon$-dense
if $k>K$.
### 3.4.
Finally, we explain how Theorem 1.1 follows from the work of Besson-Courtois-
Gallot. That $\overline{F}$ is homotopic to a smooth map with Jacobian
pointwise smaller than 1 is a direct consequence of [BCG99][Theorem 1.10],
taking the manifold $X$ in the statement of that theorem to be the universal
cover $\tilde{N}$ of $N$, and the representation to correspond to the action
of of $\pi_{1}(\Sigma)\subset\pi_{1}(N)$ on $\tilde{N}$ by deck
transformations. That the action of $\pi_{1}(\Sigma)$ is convex cocompact
follows from the fact that $N$ is homotopy equivalent to a closed hyperbolic
manifold via a homotopy equivalence that sends $F(\Sigma)$ to a totally
geodesic hypersurface. For the equality case, if no immersion homotopic to
$\overline{F}$ has smaller volume than the volume of $\Sigma$ in its
hyperbolic metric, then the volume of $\overline{F}(\Sigma)$ is equal to the
volume of $\Sigma$ in the hyperbolic metric, and the Jacobian of
$\overline{F}(\Sigma)$ must be everywhere equal to 1. In that case
[BCG99][Theorem 1.2] implies that the differential of $\overline{F}$ is at
every point an isometry onto its image. The image $\overline{F}(\Sigma)$ is
therefore isometric to the hyperbolic metric on $\Sigma$ in its induced
metric.
Since $\overline{F}(\Sigma)$ minimizes volume over all hypersurfaces homotopic
to it, we know that $\overline{F}(\Sigma)$ is a minimal hypersurface. Thus at
every point the principal curvatures $\lambda_{1},..,\lambda_{n}$ sum to zero.
If any were nonzero, then we could find $\lambda_{i}>0$ and $\lambda_{j}<0$.
The tangent plane to $\overline{F}(\Sigma)$ spanned by the corresponding
principal directions would then have sectional curvature strictly less than
one by the Gauss equation and the fact that the sectional curvature of $N$ is
less than or equal to $-1$, which contradicts the fact that
$\overline{F}(\Sigma)$ is hyperbolic in its induced metric. If there were some
other immersed hypersurface in the homotopy class of $\overline{F}(\Sigma)$
with the same volume, then one could argue as in [BCG08][Proof of Theorem 1.2]
to show that it is equal to $\overline{F}(\Sigma)$.
## 4\. Proof of Theorem 1.4
In this section we give the proof of Theorem 1.3. Again we recommend that the
reader assume $N$ is three dimensional on a first reading.
Assume that $\Sigma$ is a closed totally geodesic hypersurface in $N$ that
satisfies the well-distribution condition. This means that every point
$p\in\tilde{N}$ is contained in an embedded solid hypercube $\Box$ whose
hyperfaces are contained in lifts of $\Sigma$ to $\tilde{N}$. We claim that
there is an isometric embedding $\Phi$ of the boundary $\partial\Box$ of
$\Box$ in $\mathbb{H}^{n}$.
### 4.1.
To illustrate some of the ideas, we first give a proof of a similar statement
with the boundary of the cube $\partial\Box$ replaced by the boundary $\Delta$
of a solid embedded tetrahedron $\Delta\subset\tilde{N}$ whose faces are
contained in lifts of $\Sigma$. We assume also that $N$ is three dimensional.
We originally tried to work with tetrahedra rather than cubes, but it does not
seem possible to follow the approach of Section 5 using tetrahedra.
We claim that $\Delta$ isometrically embeds in $\mathbb{H}^{3}$. Label the
vertices of $\Delta$ as $A,B,C,D$. Note that the faces of $\Delta$ meet at
constant angles, since the faces are totally geodesic– i.e., for any edge of
$\Delta$, the angle that the two faces meeting at that edge make at any point
on the edge is the same.
Then since the face $ABC$ is contained in a totally geodesic plane isometric
to $\mathbb{H}^{2}$, we can map $ABC$ isometrically into some copy of
$\mathbb{H}^{2}$ contained in $\mathbb{H}^{3}$. Call this map $\Phi$.
Note also that the choice of $\Phi$ together with a choice of orientation for
$\mathbb{H}^{3}$ and for the triangle $ABC$ uniquely determines linear
isometries between the tangent spaces to $A$, $B$, and $C$ and respectively
$\Phi(A)$, $\Phi(B)$, and $\Phi(C)$– fix such a linear isometry
$d\Phi_{A}:T_{A}(M)\rightarrow T_{\Phi(A)}(\mathbb{H}^{3}),$
and similarly for $B$ and $C$. We can then extend $\Phi$ to the line segment
$AD$ by sending it to the geodesic ray beginning at $\Phi(A)$ with tangent
vector at $\Phi(A)$ equal to the image under $d\Phi_{A}$ of the tangent vector
to $AD$ at $A$, and similarly for $B$ and $C$.
By the fact that the faces of $\Delta$ meet along constant angles, we know
that $\Phi(AD)$ and $\Phi(BD)$ intersect, and that triangle $\Phi(ABD)$ is
isometric to triangle $ABD$, and similarly for triangles $ACD$ and $BCD$. We
can thus extend $\Phi$ to an isometry on all of $\Delta$ as desired.
### 4.2.
In the cube case, it is possible by a similar but more complicated argument to
define an isometric embedding $\Phi$ of $\partial\Box$ in $\mathbb{H}^{n}$ “by
hand.” Instead, we give a short proof that this is possible using a developing
map argument.
Note that every point $p$ on a hyperface of $\Box$ has a neighborhood in
$\partial\Box$ that isometrically embeds in $\mathbb{H}^{n}$. If $p$ is
contained in the interior of a hyperface then this is immediate. Otherwise we
can first define an isometric embedding $\Phi^{\prime}$ on the intersection of
a neighborhood of $p$ with a hyperface $H$ containing $p$. Then the fact that
hyperfaces of $\Box$ meet along constant angles, since they are contained in
totally geodesic hypersurfaces, allows us to extend $\Phi^{\prime}$ to a
neighborhood of $p$ in $\partial\Box$.
To define $\Phi$, we fix a point $p$ in $\partial\Box$, and define $\Phi$ on a
neighborhood of $p$ in $\partial\Box$ to be some isometric embedding of that
neighborhood in $\mathbb{H}^{n}$. Then for any $p^{\prime}\in\partial\Box$ and
any path $\gamma$ in $\partial\Box$ joining $p$ to $p^{\prime}$, we can use
the fact that each point has a neighborhood in $\partial\Box$ that
isometrically embeds in $\mathbb{H}^{n}$ to extend $\Phi$ along the path
$\gamma$ and define $\Phi(p^{\prime})$. Since $\partial\Box$ is simply
connected, $\Phi(p^{\prime})$ is well-defined independent of the choice of
$\gamma$. Therefore the map $\Phi$ is a well-defined local isometry onto its
image from $\partial\Box$ into $\mathbb{H}^{n}$. Note that $\Phi$ is an
isometry restricted to each hyperface of $\partial\Box$ and that the images of
opposite hyperfaces under $\Phi$ are disjoint. Therefore $\Phi$ is an
isometric embedding onto its image.
### 4.3.
We define a new Riemannian manifold $H$ as follows. First, we take the closure
of the non-compact connected component of the complement of $\Phi(\Box)$ in
$\mathbb{H}^{n}$ equipped with the hyperbolic metric. We then “fill in” the
boundary $\Phi(\Box)$ by gluing in the region $R_{\Box}$ bounded by $\Box$ in
$\tilde{N}$ to obtain $H$. Since $\Phi$ is an isometry, this defines a
Riemannian manifold which we denote by $H$. Note that apriori the metric
tensor of $H$ is only continuous.
We claim that $H$ must be isometric to $\mathbb{H}^{n}$. Our proof is similar
to some of the arguments in [SZ90] (see also the recent paper [GS22] and the
references therein.)
Every totally geodesic two-dimensional plane $S$ in $\mathbb{H}^{n}$ that
intersects $\Phi(\Box)$ transversely corresponds to a totally geodesic surface
with boundary $S^{\prime}$ in $H$, whose polygonal boundary is contained in
$\Phi(\Box)$. The surface $S^{\prime}$ is isometric to $S$ with the compact
region bounded by some $m$-sided polygon $P^{\prime}$ removed. Note that
$P^{\prime}$ is isometric to a polygon $P$ in $S\cong\mathbb{H}^{2}$, and
label the vertices of $P$ (resp. $P^{\prime}$) as $v_{1},..,v_{\ell}$ (resp.
$v_{1}^{\prime},..,v_{\ell}^{\prime}$.)
First consider the case that $P$ is a triangle. Note that $P$ has the same
angles and side-lengths as a geodesic triangle in $\mathbb{H}^{2}$. Then since
$R_{\Box}$ is geodesically convex and has curvature bounded above by $-1$, the
equality case of Corollary 3.1 implies that $P$ bounds a totally geodesic
hyperbolic solid triangle in $R_{\Box}$, and $S^{\prime}$ thus extends to an
embedded hyperbolic plane in $H$.
To prove the general case, we will show that for any $i,j$ with $1\leq
i,j\leq\ell$ and $\ell-|i-j|$ or $|i-j|$ less than $k$, the length of
$v_{i}^{\prime}v_{j}^{\prime}$ is greater than or equal to that of
$v_{i}v_{j}$. Take this as the inductive hypothesis, assume that it holds for
$k$, and choose $i$ and $j$ with $|i-j|=k-1$. Relabeling if necessary we can
assume $i=1$ and $j=k$.
First we claim that for any $m$ such that $1<m<k$, the angle
$<v_{m-1}^{\prime}v_{k}^{\prime}v_{m}^{\prime}$ is less than or equal to
$<v_{m-1}v_{k}v_{m}$. To see this, by the inductive hypothesis
$v_{m-1}^{\prime}v_{k}^{\prime}$ and $v_{m}^{\prime}v_{k}^{\prime}$ have
length greater than or equal to respectively $v_{m-1}v_{k}$ and $v_{m}v_{k}$,
and the length of $v_{m-1}^{\prime}v_{m}^{\prime}$ is the same as that of
$v_{m-1}v_{m}$. The claim then follows from Corollary 3.1.
Figure 1.
Second, we claim that angle $>v_{1}^{\prime}v_{k}^{\prime}v_{k+1}^{\prime}$ is
greater than or equal to angle $>v_{1}v_{k}v_{k+1}$. Note that the sum of
$>v_{1}^{\prime}v_{k}^{\prime}v_{k+1}^{\prime}$ and
(4.1) $\sum_{p=2}^{k-1}>v_{p-1}^{\prime}v_{k}^{\prime}v_{p}^{\prime}$
is equal to $>v_{k-1}^{\prime}v_{k}^{\prime}v_{k+1}^{\prime}$, which is equal
to $>v_{k-1}v_{k}v_{k+1}$. Therefore since (4.1) is smaller than or equal to
the same sum with the $v_{i}^{\prime}$ replaced by $v_{i}$, this implies that
$>v_{1}^{\prime}v_{k}^{\prime}v_{k+1}^{\prime}$ is greater than or equal to
$>v_{1}v_{k}v_{k+1}$. Using Corollary 3.1 we can then conclude that the length
of $v_{1}^{\prime}v_{k+1}^{\prime}$ is greater than or equal to the length of
$v_{1}v_{k+1}$. In the case of equality, we could then use the equality case
of Corollary 3.1 to conclude that $v_{1}^{\prime}v_{k}^{\prime}$ has length
equal to $v_{1}v_{k}$ and $v_{1}^{\prime}v_{k+1}^{\prime}$ has length equal to
$v_{1}v_{k+1}$, and that $\Delta v_{1}^{\prime}v_{k+1}^{\prime}v_{k}^{\prime}$
can be filled in by a totally geodesic hyperbolic triangle. Continuing in this
way we see that we must have had equality at every previous stage of the
induction, and we finally get that the length of
$v_{1}^{\prime}v_{3}^{\prime}$ equals the length of $v_{1}v_{3}$, and so
$\Delta v_{1}^{\prime}v_{2}^{\prime}v_{3}^{\prime}$ can be filled in by a
hyperbolic triangle. Removing $\Delta
v_{1}^{\prime}v_{2}^{\prime}v_{3}^{\prime}$ we obtain a new polygon with one
fewer side on which we can repeat the same argument. If we never have
equality, then we can conclude the finite induction to get that
$v_{1}^{\prime}v_{\ell}^{\prime}$ has length strictly greater than
$v_{1}v_{\ell}$, which is a contradiction since the two have the same length.
We have thus shown that $S^{\prime}$ extends to an embedded totally geodesic
hyperbolic plane.
### 4.4.
We explain why this implies that $H$ has constant curvature $-1$. Take a large
metric sphere $B$ in $\mathbb{H}^{n}$ that contains $\Phi(\Box)$, and let
$\mathcal{C}$ be the set of round (but not necessarily great) circles
contained in the metric sphere $\partial B$. There is a natural disk bundle
$\mathcal{D}$ over $\mathcal{C}$ whose fiber over $C\in\mathcal{C}$ is equal
to the totally geodesic disc in $B$ bounded by $C$. We define a map $f$ from
$\mathcal{D}$ to the Grassmann bundle $Gr_{2}(\overline{R_{\Box}})$ of
unoriented tangent 2-planes to the closure of $R_{\Box}$ as follows. Here
$Gr_{2}(\overline{R_{\Box}})$ is defined by viewing $\overline{R_{\Box}}$ as a
subset of $N$ and pulling back $Gr_{2}(N)$ to $\overline{R_{\Box}}$.
Suppose $d\in\mathcal{D}$ corresponds to a circle $C\in\mathcal{C}$ that
bounds a totally geodesic disc $D$ containing $d$. We take the unique totally
geodesic tangent plane $S$ that intersects $\partial B$ in $D$, which by what
we have shown above corresponds to a totally geodesic plane $S^{\prime}$ in
$H$. This $S^{\prime}$ contains an isometric copy $D^{\prime}$ of the
hyperbolic disc $S\cap B$. We define $f(d)$ to be the tangent plane
$\Pi^{\prime}$ to the point $d^{\prime}$ on $D^{\prime}$ corresponding to $d$
if it is contained in $\overline{R_{\Box}}$, or else the parallel transport of
$\Pi^{\prime}$ along the geodesic joining $d^{\prime}$ to its nearest point
projection to $\partial\overline{R_{\Box}}$.
Note that $f$ is continuous, and $f$ is injective near every point that maps
into the interior of $R_{\Box}$. Since $\mathcal{D}$ and the interior of
$Gr_{2}(\overline{R_{\Box}})$ both have dimension $3n-4$, the invariance of
domain theorem implies that $f$ is a local homeomorphism onto its image near
any such point $d$. Since the image of $f$ is both open and closed, the fact
that $Gr_{2}(\overline{R_{\Box}})$ is connected implies that $f$ is
surjective. It follows that $R_{\Box}$ has constant curvature $-1$, which
implies that $H$ is isometric to $\mathbb{H}^{n}$.
## 5\. Proof of Theorem 1.5
Let $M$ be a closed hyperbolic manifold, and let $N$ be a Riemannian manifold
homotopy equivalent to $N$ with sectional curvature at most $-1$. Fix a
homotopy equivalence $F:M\rightarrow N$. Let $\Sigma$ be a totally geodesic
hypersurface in $M$, and assume that $N$ contains a totally geodesic
hypersurface $\Sigma_{N}$ whose fundamental group includes to
$\pi_{1}(\Sigma)$ up to conjugacy, where we’ve used $F$ to identify
$\pi_{1}(M)$ and $\pi_{1}(N)$. To show that $\Sigma_{N}$ satisfies the well-
distribution property, we will actually need $\Sigma$ to satisfy a stronger
condition, which we define now.
###### Definition 5.1.
We say that $\Sigma$ satisfies the strong well-distribution property if the
following holds. Let $\mathcal{B}_{\Sigma}$ be the set of solid embedded
hypercubes in $\mathbb{H}^{n}=\tilde{M}$ each of whose boundary hyperfaces is
contained in a lift of $\Sigma$ to the universal cover $\mathbb{H}^{n}$. Then
for each geodesic $\gamma$ in the universal cover
$\tilde{M}\cong\mathbb{H}^{n}$ of $M$, we can find hypercubes
$\Box_{i}\in\mathcal{B}_{\Sigma}$, $i\in\mathbb{Z}$, so that the following
holds. This sequence of hypercubes is then said to enclose $\gamma.$
1. (1)
Exactly two hyperfaces $H_{i,top}$ and $H_{i,bottom}$ of $\Box_{i}$ intersect
$\gamma$
2. (2)
The totally geodesic hyperplanes that contain the hyperfaces of $\Box_{i}$,
excluding $H_{i,top}$ and $H_{i,bottom}$, do not intersect $\gamma$.
3. (3)
$\Box_{i-1}$ and $\Box_{i}$ satisfy the interlocking property: the totally
geodesic hyperplane that contains $H_{i-1,top}$ is contained in the region
between those of $H_{i,top}$ and $H_{i,bottom}$.
4. (4)
Each point on $\gamma$ is contained in some $\Box_{i}$.
Now let $\Sigma_{k}$ be an infinite sequence of distinct closed totally
geodesic hypersurfaces in $M$. We claim that for $k$ large enough,
$\Sigma_{k}$ satisfies the strong well-distribution property. By Corollary 3.3
we know that for every $\epsilon>0$ the $\Sigma_{k}$ are $\epsilon$-dense in
$Gr_{n-1}(M)$ for $k$ large enough.
For $\epsilon>0$ sufficiently small, we define the following hypercube
$\Box_{model}(\epsilon)$ with totally geodesic faces in $\mathbb{H}^{n}$. Take
some point $P$ in $\mathbb{H}^{n}$, and let $E_{1},..,E_{n}$ be an orthonormal
basis for the tangent space at that point. For each $i$, let $H_{i}^{+}$ and
$H_{i}^{-}$ be the totally geodesic hyperplanes orthogonal to the geodesic
through $P$ in the direction of $E_{i}$, and at signed distance of
respectively $\epsilon$ and $-\epsilon$ from $P$. Let $U_{i}$ be the connected
component of the complement of $H_{i}^{+}\cup H_{i}^{-}$ in $\mathbb{H}^{n}$
containing $P$. We define $\Box_{model}(\epsilon)$ to be the closure of the
intersection $\cap_{i=1}^{n}U_{i}$, where we have taken $\epsilon$ small
enough that this intersection is a solid embedded hypercube.
We say that two hypercubes $\Box$ and $\Box^{\prime}$ with totally geodesic
hyperbolic faces are $\delta$-close if there is a labeling of the vertices
$V_{1},..,V_{\ell}$ of $\Box$ and the vertices
$V_{1}^{\prime},..,V_{\ell}^{\prime}$ of $\Box^{\prime}$ such that
1. (1)
For all $i,j$ such that $V_{i}$ and $V_{j}$ are adjacent the length of
$V_{i}V_{j}$ and the length of $V_{i}^{\prime}V_{j}^{\prime}$ differ by less
than $\delta$.
2. (2)
For all $i,j,p$ such that $V_{i}$ and $V_{p}$ are adjacent to $V_{j}$ the
angle $>V_{i}V_{j}V_{p}$ differs from the angle
$>V_{i}^{\prime}V_{j}^{\prime}V_{p}^{\prime}$ by less than $\delta$.
Denote by $\mathcal{B}_{\Sigma_{k}}^{\epsilon}$ the set of hypercubes $\Box$
in $\mathcal{B}_{\Sigma_{k}}$ that are $\delta$-close to
$\Box_{model}(\epsilon)$, for some small $\delta$ be be specified later. The
reason for defining $\mathcal{B}_{\Sigma_{k}}^{\epsilon}$ is that having lots
of elements of $\mathcal{B}_{\Sigma_{k}}$ of controlled shape will be useful
for enclosing $\gamma$.
Fix $\epsilon>0$ small enough that $\Box_{model}(\epsilon)$ is defined. For
every geodesic $\gamma$ in the universal cover, we claim that we can find a
collection of solid hypercubes $\Box$ contained in
$\mathcal{B}_{\Sigma_{k}}^{\epsilon}$ that enclose $\gamma$, provided that $k$
was chosen sufficiently large and $\delta$ was chosen sufficiently small at
the start. Let $\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$ be the set of
hypercubes $\Box$ in the universal cover so that the following holds:
1. (1)
$\Box$ is the lift of some element of $\mathcal{B}_{\Sigma_{k}}^{\epsilon}$ to
the universal cover
2. (2)
$\gamma$ intersects the boundary of $\Box$ in exactly two points $p_{1}$ and
$p_{2}$ on opposite hyperfaces $H_{top}$ and $H_{bottom}$ of $\Box$
3. (3)
$p_{1}$ and $p_{2}$ are at a distance of less than $\delta$ from the centers
of $H_{top}$ and $H_{bottom}$ respectively. Here the centers of $H_{top}$ and
$H_{bottom}$ are the endpoints of the geodesic segment of shortest length
beginning at $H_{bottom}$ and ending at $H_{top}$.
4. (4)
$\gamma$ makes an angle between $\pi/2-\delta$ and $\pi/2+\delta$ with
$H_{top}$ and $H_{bottom}$ at respectively $p_{1}$ and $p_{2}$
Here $\delta$ is chosen small enough that for every $\gamma$ and $\Box$
satisfying the above four conditions, $\gamma$ will be disjoint from all of
the totally geodesic hyperplanes that contain hyperfaces of $\Box$ different
from $H_{top}$ and $H_{bottom}$. We emphasize that all of the requirements we
impose on $\delta$ and $k$ will be independent of $\gamma$. That they can be
satisfied follows from the fact that given any $\delta^{\prime}$, any embedded
copy of $\Box_{model}(\epsilon)$ in $\mathbb{H}^{n}$ is
$\delta^{\prime}$-close to some element of
$\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$, provided $\delta$ was chosen
small enough and $k$ was chosen large enough at the start.
If $k$ was chosen large enough, then every point on $\gamma$ will be contained
in some element of $\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$. Suppose that
$p\in\gamma$ is contained in some
$\Box_{p}\in\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$, whose boundary
intersects $\gamma$ at $p_{1}$ and $p_{2}$. Then provided $\delta$ was chosen
small enough and $k$ was chosen large enough, we can find some
$\Box_{p_{1}}\in\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$ that contains
$p_{1}$, and whose boundary intersects $\gamma$ at points $p^{\prime}$ and
$p^{\prime\prime}$ so that
$\min(d(p^{\prime},p_{1}),d(p^{\prime\prime},p_{1})>\epsilon/4.$
In this case, again provided $\delta$ was chosen small enough, $\Box_{p_{1}}$
and $\Box_{p}$ satisfy the interlocking property, and in a similar way we can
find a $\Box_{p_{2}}$ containing $p_{2}$ in its interior so that
$\Box_{p_{2}}$ and $\Box_{p}$ satisfy the interlocking property. Continuing in
this way, we can find a collection of
$\Box\in\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$ that enclose $\gamma$.
This shows that $\Sigma_{k}$ satisfies the strong well-distribution property
for large enough $k$.
### 5.1.
Suppose that $\Sigma$ satisfies the strong well-distribution property. Then we
claim that $\Sigma_{N}$ is well-distributed in $N$. First, note that each
hypercube $\Box$ bounded by lifts of $\Sigma$ to the universal cover
corresponds to a hypercube $\Box^{\prime}$ in $N$ bounded by lifts of
$\Sigma_{N}$. This can be seen as follows. Suppose that $\Box$ is bounded by
$2n+2$ totally geodesic hyperplanes $H_{1},..,H_{2n+2}$ that are lifts of
$\Sigma$.
Recall the discussion in 3.2. The map $F$ defines a homeomorphism between the
boundary at infinity of $\mathbb{H}^{n}$ and the boundary at infinity of the
universal cover $\tilde{N}$ of $N$. Taking the images of the boundaries at
infinity of the $H_{i}$ we obtain $2n+2$ $n-2$-discs in
$\partial_{\infty}\tilde{N}$. These bound totally geodesic hyperplanes that
project down to $\Sigma_{N}$. We thus obtain a hypercube $\Box^{\prime}$ in
$\tilde{N}$ whose sides are contained in lifts of $\Sigma_{N}$.
We claim that $\Sigma_{N}$ satisfies the well distribution property in $N$.
Let $\gamma_{N}$ be a geodesic in the universal cover $\tilde{N}$. Let
$\gamma$ be a geodesic in $\tilde{M}=\mathbb{H}^{n}$ with the same endpoints
at infinity as $\gamma_{N}$, identifying the two boundaries at infinity as
above. Since $\Sigma$ satisfies the strong well-distribution property, we can
find a sequence of hypercubes $\Box_{i}:i\in\mathbb{Z}$ in $\mathbb{H}^{n}$
whose hyperfaces are contained in lifts of $\Sigma$ so that the following are
true:
1. (1)
Exactly two hyperfaces $H_{i,top}$ and $H_{i,bottom}$ of $\Box_{i}$ intersect
$\gamma$
2. (2)
$H_{i-1,top}$ is contained in the region between $H_{i,top}$ and
$H_{i,bottom}$
3. (3)
Each point on $\gamma$ is contained in some $\Box_{i}$.
The hyperplanes corresponding to the faces of $\Box_{i}$ excluding $H_{i,top}$
and $H_{i,bottom}$ bound an infinite solid rectangular “pillar” that contains
$\gamma$.
Figure 2. Hypercubes enclosing $\gamma$
The hypercubes $\Box_{i}^{\prime}$ corresponding to the $\Box_{i}$ satisfy the
same three properties in $\tilde{N}$, but with $\gamma$ replaced by
$\gamma_{N}$. Item (1) follows from the fact that a totally geodesic
hyperplane $H$ in $\tilde{N}$ will intersect a given geodesic $\gamma$ exactly
if the two endpoints of $\gamma$ in $\partial_{\infty}\tilde{N}$ are in
separate components of the complement of $\partial_{\infty}H$ in
$\partial_{\infty}\tilde{N}$. This implies that whether $\gamma$ intersects
$H$ is determined by what happens in $\mathbb{H}^{n}$ for the corresponding
geodesic and lift of $\Sigma$. The second property follows in a similar way.
The first two properties imply that the subset of $\gamma$ of points contained
in some $\Box_{i}^{\prime}$ is both open and closed, and so must be all of
$\gamma$.
Therefore, since every point in $N$ is contained in some geodesic, every point
$p$ of $\tilde{N}$ is contained in an embedded hypercube $\Box_{p}^{\prime}$
as above. By Theorem 1.3, this implies that $N$ has constant curvature $-1$.
Since $N$ is homotopy equivalent to $M$, $N$ must be isometric to $M$ by
Mostow’s rigidity theorem.
## 6\. Examples Where Rigidity Fails
In this section we prove the following theorem. We expect that a similar
construction works in higher dimensions, but for simplicity we focus on the
three dimensional case.
###### Theorem 6.1.
There exists a closed hyperbolic 3-manifold $(M,g_{hyp})$ and a metric $g$ on
$M$ that is not isometric to $g_{hyp}$ such that the following hold
1. (1)
$(M,g_{hyp})$ contains a closed totally geodesic surface $S$.
2. (2)
$S$ is a totally geodesic hyperbolic surface in the metric $(M,g)$
3. (3)
The sectional curvature of $g$ is less than or equal to $-1$.
Our construction will be based on the following proposition.
###### Proposition 6.2.
Let $\ell>1$ and let $g_{\ell}$ be a metric on $S\times\mathbb{R}$ with
constant curvature $-\ell$ so that $S\times\\{0\\}$ is a totally geodesic
surface in the metric $g_{\ell}$. There exists a smooth metric $g$ on
$S\times\mathbb{R}$ satisfying the following conditions:
1. (1)
$g=g_{\ell}$ outside of a compact set
2. (2)
$g$ has sectional curvature $K_{g}\leq-1$
3. (3)
$S\times\\{0\\}$ is totally geodesic in $g$ with constant sectional curvature
$-1$.
V. Lima constructed non-hyperbolic examples as in the proposition, but with
point (1) excluded. The examples we give were explained to us by Laurent
Mazet, and we are grateful to him for allowing us to include them. Our
verifications closely follow [Lim19][section 3].
###### Proof.
Let $(S,g_{hyp})$ be a hyperbolic surface, and let $f$ be a function defined
by
$f(t):=\frac{1}{\ell}\cosh(\ell t)+\chi(t)(1-\frac{1}{\ell})$
for $\chi(t)$ a smooth non-negative function satisfying, for $\epsilon>0$ and
$M$ respectively sufficiently small and large to be specified later:
1. (1)
$\chi(t)=1$ near $t=0$, $\chi(t)=0$ for $|t|>M$
2. (2)
$\max(|\chi^{\prime}(t)|,|\chi^{\prime\prime}(t)|)<\epsilon$
We take $\ell>1$. Define $g$ to be the warped product metric
$g:=f^{2}(t)g_{hyp}^{2}+dt^{2}.$
Writing $g_{\ell}$ as
$g_{\ell}=\frac{1}{\ell^{2}}\cosh^{2}(\ell t)g_{hyp}^{2}+dt^{2},$
Only the second item in the statement of the proposition requires an argument.
In computing the curvature of $g$, we follow [Lim19]. For a vector field $V$
on $(S\times\mathbb{R},g)$, define the projection to $S$ by
$V^{S}:=V-g(V,\partial_{t})\partial_{t}$
The curvature tensor $R$ of $g$ can be written in terms of the curvature
tensor $R_{S}$ of $(S,g_{hyp})$ as
(6.1) $\displaystyle R(X,Y,Z)$
$\displaystyle=R_{S}(X^{S},Y^{S})Z^{S}-\left(\frac{f^{\prime}}{f}\right)^{2}(g(X,Z)Y-g(Y,Z)X)$
(6.2) $\displaystyle+(\log
f)^{\prime\prime}g(Z,\partial_{t})(g(Y,\partial_{t})X-g(X,\partial_{t})Y)$
(6.3) $\displaystyle-(\log
f)^{\prime\prime}(g(X,Z)g(Y,\partial_{t})-g(Y,Z)g(X,\partial_{t})\partial_{t},$
for $X$, $Y$, and $Z$ smooth vector fields. Given a plane $\Pi$ spanned by
unit length orthogonal vector fields $X,Y$, we have that the sectional
curvature of $\Pi$ is given by
(6.4) $\displaystyle K_{\Pi}$
$\displaystyle=\frac{K_{S}-(f^{\prime})^{2}}{f^{2}}+\left(\frac{-K_{S}+(f^{\prime})^{2}-f^{\prime\prime}f}{f^{2}}\right)(g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2})$
For $|t|$ smaller than some $t_{0}$ we have that $\chi(t)$ is equal to $1$.
The numerator of the coefficent of
$(g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2})$ is then equal to
(6.5) $\displaystyle=1+\sinh^{2}(\ell t)-\ell\cosh\ell
t(\frac{1}{\ell}\cosh(\ell t)+\chi(t)(1-\frac{1}{\ell}))$ (6.6)
$\displaystyle=(1-\ell)\cosh\ell t,$
which is less than zero, so we have that (6.4) is maximized when
$(g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2})=0$. In this case, we just need
to show that the initial term
(6.7) $\frac{K_{S}-(f^{\prime})^{2}}{f^{2}}=\frac{-1-(\sinh(\ell
t)+\chi^{\prime}(t)(1-\frac{1}{\ell}))^{2}}{\left(\frac{1}{\ell}\cosh(\ell
t)+\chi(t)(1-\frac{1}{\ell})\right)^{2}}$
is at most $-1$. This will actually be true for all $t$, not just $|t|<t_{0}$,
which we verify now. Note that this holds at $t=0$. It is then enough to show
that the quantity (6.7) is decreasing in $t$ for $t>0$ and increasing in $t$
for $t<0$. For the derivative of the quantity (6.7) to be negative, we need
(6.8)
$\displaystyle 0>$ $\displaystyle-2(\sinh(\ell
t)+\chi^{\prime}(t)(1-\frac{1}{\ell}))(\ell\cosh(\ell
t)+\chi^{\prime\prime}(t)(1-\frac{1}{\ell}))(\frac{1}{\ell}\cosh(\ell
t)+\chi(t)(1-\frac{1}{\ell}))+$ $\displaystyle(1+(\sinh(\ell
t)+\chi^{\prime}(t)(1-\frac{1}{\ell}))^{2})(\sinh(\ell
t)+\chi^{\prime}(t)(1-\frac{1}{\ell})).$
Choose $t_{0}$ so that $\chi(t)=1$ for $|t|<t_{0}$. We first do the case
$|t|<t_{0}$. Assume that $t>0$. To verify (6.8), it is enough to show that
(6.9) $0\geq\sinh(\ell t)(-2\cosh^{2}(\ell t)-2\ell\cosh(\ell
t)\chi(t)(1-\frac{1}{\ell})+1+\sinh^{2}(\ell t)),$
where the RHS of (6.9) is all of the terms of the RHS of (6.8) that don’t
contain $\chi^{\prime}(t)$ or $\chi^{\prime\prime}(t)$. Since $\sinh(\ell
t)(-2\ell\cosh(\ell t)\chi(t)(1-\frac{1}{\ell}))$ is always non-positive
provided $t>0$, for (6.9) to hold it is enough that
(6.10) $0\geq-2\cosh^{2}(\ell t)+1+\sinh^{2}(\ell t),$
which is true since the RHS equals $-\cosh^{2}(\ell t)$. Thus the quantity
(6.7) is negative if $t_{0}>t>0$, and the same reasoning shows it is positive
if $-t_{0}<t<0$.
Now assume that $t\geq t_{0}$ (the case $t\leq-t_{0}$ is similar.) Then by the
previous computations, we will be able to conclude that (6.7) is smaller than
$-1$ for all $t$ as long as $-\sinh(\ell t)\cosh^{2}(\ell t)$ is larger in
absolute value than all of the terms of the RHS of (6.8) that, when expanded
out, contain $\chi^{\prime}(t)$ or $\chi^{\prime\prime}(t)$. This will be true
provided $\epsilon$ was taken small enough at the start, since
$\max(|\chi^{\prime}(t)|,|\chi^{\prime\prime}(t)|)<\epsilon$ and no term in
(6.8) has more than three factors equal to $\sinh(\ell t)$ or $\cosh(\ell t)$.
We have thus shown that (6.7) is smaller than $-1$ for all $t$.
We now check the case that $|t|>t_{0}$. Because the initial term in (6.4) is
always less than or equal to $-1$, if the second term in (6.4) is negative
then we are done. If not, it is enough to check (6.4) in the case that
$g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2}$ assumes its maximum possible
value:
$g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2}=1.$
The quantity (6.4) is then equal to $-f^{\prime\prime}/f$. Therefore we need
to show that
$-1\geq-f^{\prime\prime}/f=-\frac{\ell^{2}\cosh(\ell
t)+\ell\chi^{\prime\prime}(t)(1-\frac{1}{\ell})}{\cosh(\ell
t)+\ell\chi(t)(1-\frac{1}{\ell})})).$
Since $\chi\leq 1$, it is enough to show that
(6.11) $1\leq\frac{\ell^{2}\cosh(\ell t)}{\cosh(\ell
t)+\ell-1}+\frac{\chi^{\prime\prime}(t)(\ell-1)}{\cosh(\ell t)+\ell-1}.$
Note that
$\ell^{2}\cosh(\ell t)\geq\ell\cosh(\ell t)+\ell(\ell-1),$
so the first term on the RHS of (6.11) is greater than or equal to
$\frac{\ell\cosh(\ell t)+\ell(\ell-1)}{\cosh(\ell t)+\ell-1}=\ell.$
We can make the second term on the RHS of (6.11) as small as desired by
choosing $\epsilon$ small enough because $|\chi^{\prime\prime}|<\epsilon$.
Therefore inequality (6.11) holds, which completes the proof of Proposition
6.2.
∎
We also need the following lemma.
###### Lemma 6.3.
Let $M$ be a closed hyperbolic 3-manifold containing an embedded two-sided
totally geodesic surface $S$. Then for any $L>0$, there exists a finite cover
$M_{0}$ of $M$ which contains a totally geodesic surface $S_{0}$ isometric to
$S$ and with normal injectivity radius greater than $L$.
Recall that $S_{0}$ has normal injectivity radius greater than $L$ in $M_{0}$
if no two distinct geodesic segments of length less than $L$ beginning normal
to $S_{0}$ intersect. This means that the normal exponential map
$S_{0}\times(-L,L)\rightarrow M_{0}$ is a diffeomorphism onto its image, for
$S_{0}\times(-L,L)$ identified with a tubular neighborhood of the zero section
of the normal bundle of $S_{0}$ in $M_{0}$ in the natural way.
###### Proof.
This is a consequence of subgroup separability for
$\pi_{1}(S,p)\subset\pi_{1}(M,p)$, for some choice of basepoint $p$ on $S$
[MR03][Lemma 5.3.6]. This means that for any finite subset
$\\{g_{1},..g_{n}\\}\subset\pi_{1}(M,p)$ disjoint from $\pi_{1}(S,p)$, we can
choose a finite index subgroup of $\pi_{1}(M,p)$ containing $\pi_{1}(S,p)$ but
none of $g_{1},..g_{n}$.
Let $F$ be the Fuchsian cover of $M$ corresponding to $\pi_{1}(S)$. Fix a
connected polyhedral fundamental domain $P$ for the action of $\pi_{1}(M)$ on
the universal cover $\mathbb{H}^{3}$ of $M$. Then a fundamental domain in
$\mathbb{H}^{3}$ for the cover $F$ is tiled by copies of $P$, and any fixed
normal neighborhood of $S$ in $F$ is contained in the projections of finitely
many such copies. The images of the projections to $F$ of the copies of $P$
correspond to cosets of $\pi_{1}(S)$ in $\pi_{1}(M)$. Choose finitely many
coset representatives $g_{1},..,g_{n}$ so that the corresponding projections
of copies of $P$ contain the $L$-neighborhood of $S$ in $F$.
Now use subgroup separability to choose a finite index subgroup of
$\pi_{1}(M)$ containing $\pi_{1}(S)$ but none of $g_{1},..,g_{n}$. The finite
cover of $M$ corresponding to this finite index subgroup is then also covered
by $F$, and the restriction of the covering map to the $L$-neighborhood of $S$
in $F$ is a diffeomorphism onto its image. This shows that we can make the
normal injectivity radius of $S$ as large as desired by passing to finite
covers. ∎
We can now prove Theorem 6.1. (Compare [Low21][Section 6], where a similar
construction appeared.) Take a closed hyperbolic 3-manifold $(M,g_{hyp})$
containing an embedded two-sided totally geodesic surface $S$. See [MR03] for
examples.
Let $F\cong S\times\mathbb{R}$ be the Fuchsian cover of $(M,g_{hyp})$
corresponding to $S$. Let $g_{\ell}$ be the homothetic scaling of the
hyperbolic metric on $F$ that has constant curvature $-\ell$ for some
$\ell>1$. We apply Proposition 6.2 to obtain a metric $g_{K}$ on $F$ that
agrees with $g_{\ell}$ outside a compact set $K$, and so that $S\times\\{0\\}$
is a totally geodesic hyperbolic surface.
Next we apply Lemma 6.3 to find a finite cover of $M^{\prime}$ so that $F$
also covers $M^{\prime}$, and the restriction of the covering map
$\rho:F\rightarrow M^{\prime}$ to $K$ is injective. We can then define a
metric on $M^{\prime}$ to be the scaled hyperbolic metric with constant
curvature $-\ell$ outside of $\rho(K)$, and to be the pushforward of $g_{K}$
under $\rho$ on $\rho(K)$. Since $g_{\ell}$ and $g_{K}$ agree outside of the
compact set $K$, this gives a smooth well-defined metric on $M^{\prime}$,
which completes the proof.
## References
* [AH98] Lars Andersson and Ralph Howard, _Comparison and rigidity theorems in semi-Riemannian geometry_ , Comm. Anal. Geom. 6 (1998), no. 4, 819–877. MR 1664893
* [BCG99] Gérard Besson, Gilles Courtois, and Sylvestre Gallot, _Lemme de Schwarz réel et applications géométriques_ , Acta Math. 183 (1999), no. 2, 145–169. MR 1738042
* [BCG08] Gérard Besson, Gilles Courtois, and Sylvain Gallot, _Rigidity of amalgamated products in negative curvature_ , J. Differential Geom. 79 (2008), no. 3, 335–387. MR 2433927
* [BFMS21] Uri Bader, David Fisher, Nicholas Miller, and Matthew Stover, _Arithmeticity, superrigidity, and totally geodesic submanifolds_ , Ann. of Math. (2) 193 (2021), no. 3, 837–861. MR 4250391
* [CE08] Jeff Cheeger and David G. Ebin, _Comparison theorems in Riemannian geometry_ , AMS Chelsea Publishing, Providence, RI, 2008, Revised reprint of the 1975 original. MR 2394158
* [CMN22] Danny Calegari, Fernando C. Marques, and André Neves, _Counting minimal surfaces in negatively curved 3-manifolds_ , Duke Math. J. 171 (2022), no. 8, 1615–1648. MR 4432012
* [FJ89] F. T. Farrell and L. E. Jones, _Negatively curved manifolds with exotic smooth structures_ , J. Amer. Math. Soc. 2 (1989), no. 4, 899–908. MR 1002632
* [FJ90] by same author, _Rigidity and other topological aspects of compact nonpositively curved manifolds_ , Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 59–64. MR 1001606
* [GS22] M. Ghomi and J. Spruck, _Rigidity of nonpositively curved manifolds with convex boundary_ , https://arxiv.org/abs/2210.05588 (2022).
* [Lim19] Vanderson Lima, _Area estimates and rigidity of non-compact $H$-surfaces in 3-manifolds_, Proc. Amer. Math. Soc. 147 (2019), no. 10, 4499–4512. MR 4002559
* [LN21] B. Lowe and A. Neves, _Minimal surface entropy and average area ratio_ , https://arxiv.org/abs/2110.09451 (2021).
* [Low21] Ben Lowe, _Deformations of totally geodesic foliations and minimal surfaces in negatively curved 3-manifolds_ , Geom. Funct. Anal. 31 (2021), no. 4, 895–929. MR 4317506
* [Maz20] L. Mazet, _Rigidity of riemannian manifolds containing an equator_ , https://arxiv.org/abs/2010.01994 (2020).
* [Mil76] John J. Millson, _On the first Betti number of a constant negatively curved manifold_ , Ann. of Math. (2) 104 (1976), no. 2, 235–247. MR 422501
* [MM22] Amir Mohammadi and Gregorii Margulis, _Arithmeticity of hyperbolic 3-manifolds containing infinitely many totally geodesic surfaces_ , Ergodic Theory Dynam. Systems 42 (2022), no. 3, 1188–1219. MR 4374970
* [MR03] Colin Maclachlan and Alan W. Reid, _The arithmetic of hyperbolic 3-manifolds_ , Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957
* [MR14] Laurent Mazet and Harold Rosenberg, _On minimal spheres of area $4\pi$ and rigidity_, Comment. Math. Helv. 89 (2014), no. 4, 921–928. MR 3284299
* [MS95] Shahar Mozes and Nimish Shah, _On the space of ergodic invariant measures of unipotent flows_ , Ergodic Theory Dynam. Systems 15 (1995), no. 1, 149–159. MR 1314973
* [Pog46] A. Pogorelov, _A theorem regarding geodesics on closed convex surfaces_ , Rec. Math. [Mat. Sbornik] N.S. 18(60) (1946), 181–183. MR 0017557
* [SY79] R. Schoen and Shing Tung Yau, _Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature_ , Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332
* [SZ90] V. Schroeder and W. Ziller, _Local rigidity of symmetric spaces_ , Trans. Amer. Math. Soc. 320 (1990), no. 1, 145–160. MR 958901
|
# Choosing the Right Weights:
Balancing Value, Strategy, and Noise in Recommender Systems
Smitha Milli Cornell Tech
<EMAIL_ADDRESS>Emma Pierson Cornell Tech
<EMAIL_ADDRESS>Nikhil Garg Cornell Tech
<EMAIL_ADDRESS>
###### Abstract
Many recommender systems are based on optimizing a linear weighting of
different user behaviors, such as clicks, likes, shares, etc. Though the
choice of weights can have a significant impact, there is little formal study
or guidance on how to choose them. We analyze the optimal choice of weights
from the perspectives of both users and content producers who strategically
respond to the weights. We consider three aspects of user behavior: value-
faithfulness (how well a behavior indicates whether the user values the
content), strategy-robustness (how hard it is for producers to manipulate the
behavior), and noisiness (how much estimation error there is in predicting the
behavior). Our theoretical results show that for users, upweighting more
value-faithful and less noisy behaviors leads to higher utility, while for
producers, upweighting more value-faithful and strategy-robust behaviors leads
to higher welfare (and the impact of noise is non-monotonic). Finally, we
discuss how our results can help system designers select weights in practice.
## 1 Introduction
Most widely-used recommender systems are based on prediction and optimization
of multiple behavioral signals. For example, a video platform may predict
whether a user will click on a video, how long they will watch it, and whether
or not they will give it a thumbs-up. These predictions need to then be
aggregated into a final score that items for a user will be ranked by.
Typically, the aggregation is done through a linear combination of the
different signals. For example, leaked documents from TikTok (Smith, 2021)
described the objective for ranking as $\mathbb{P}(\text{like)}\cdot
w_{\text{like}}+\mathbb{P}(\text{comment})\cdot
w_{\text{comment}}+\mathbb{E}[\text{playtime}]\cdot
w_{\text{playtime}}+\mathbb{P}(\text{play})\cdot w_{\text{play}}$. Twitter
also recently open-sourced the exact weights on the ten behaviors they use for
ranking (Twitter, 2023).
Unfortunately, the chosen weights can often lead to unintended consequences.
For example, when Facebook introduced emoji reactions, they gave all emoji
reactions a weight five times that of the standard thumbs-up. However, after
evidence that the high weight on the angry reaction led to more
misinformation, toxicity, and low-quality content, its weight was eventually
reduced from five to four, to one and a half, and then finally, to zero times
that of a thumbs up (Merrill and Oremus, 2021). Furthermore, the weights can
have a significant impact on content producers who strategically adapt in
response to them. For example, leaked documents from Facebook stated that,
_“Research conducted in the EU reveals that political parties feel strongly
that the change to the algorithm has forced them to skew negative in their
communications on Facebook, with the downstream effect of leading them into
more extreme policy positions,”_ (Hagey and Horwitz, 2021; Morris, 2021).
Even though they can have a major effect on the emergent dynamics of the
platform, these weights are rarely the topic of formal research and there
exist few guidelines for system designers on how to choose them. In this
paper, we study how to optimally choose weights (for users and producers) when
behaviors can vary along three dimensions that designers consider in practice:
_value-faithfulness_ , _strategy-robustness_ , and _noisiness_. Firstly,
value-faithfulness is how indicative a behavior is of whether the user values
the content or not. This concept is referenced, for example, by TikTok, which
has stated that behaviors are “weighted based on their value to a user”
(TikTok, 2020). Secondly, strategy-robustness refers to how hard it is for
producers to manipulate the behavior. A prime example of this is YouTube’s
shift from focusing less on views and more on explicit user behaviors such as
likes and dislikes, in an effort to curb the rise of clickbait video titles
(Youtube, 2019). Lastly, noisiness, refers to the variance in machine learning
predictions of the behavior. Variance is a common consideration in machine
learning and depends, among other factors, on training set size. Netflix
increasingly relied on implicit behaviors (e.g. views) over explicit behaviors
(e.g. ratings) due to their greater prevalence (Gomez-Uribe and Hunt, 2015).
To study the optimal weight design problem, we posit a model in which two
producers compete for the attention of one user. The recommender system ranks
producers based on a linear combination of predictions of $k$ behaviors.
However, producers can strategically adapt their items to increase the
probability of different user behaviors. User utility depends on being shown a
high value producer, and a producer’s utility is the probability they are
ranked highly minus their costs of strategic manipulation. We find that, for
the user, upweighting behaviors that are more value-faithful and less noisy
leads to higher utility (and strategy-robustness has no impact), while for
producers, upweighting behaviors that are more value-faithful and strategy-
robust leads to higher welfare, i.e., higher average utility (and the impact
of noise is non-monotonic).
In practice, the weight vector that platforms use is chosen by employees,
typically product and engineering teams, based on both performance in A/B
tests and qualitative human judgment (Twitter, 2023; TikTok, 2020; Merrill and
Oremus, 2021; Hagey and Horwitz, 2021). Understanding how different behaviors
compare on the three aspects studied—value-faithfulness, noisiness, and
strategy-robustness—can help system designers narrow down the most relevant
weight vectors to test. This is particularly important as a full grid search
is often too expensive to run as the number of behaviors grows, and some
aspects (such as long-run strategic behavior) are difficult to measure. We
provide examples in three settings—e-commerce, TikTok, and Twitter—to
illustrate how this analysis may be applied. Considering these aspects can
help anticipate potential negative side effects, and in certain cases, our
theoretical results imply constraints on the user-optimal weight vector that
can be used to narrow the search space of weights.
## 2 Related Work
##### Designing weights in recommender systems.
Many recommender systems use a weighted combination of various behavior
predictions to rank items (Smith, 2021; Twitter, 2023). The choice of weights
is typically not automated and is chosen by employees based on performance in
A/B tests, insights from user surveys, and qualtitative judgement (TikTok,
2020; Twitter, 2023; Cameron et al., 2022). The weights can have a large
impact on the emergent dynamics of the platform. For instance, when Facebook
modified their weights in 2018 as part of their transition to the “Meaningful
Social Interactions Metric,” Buzzfeed CEO Jonah Peretti warned that the
changes were promoting the virality of divisive content, thereby incentivizing
its production (Hagey and Horwitz, 2021).
To bypass the manual weight creation process, some prior work tries to rank
content by directly optimizing for users’ latent _value_ for content (Milli et
al., 2021; Maghakian et al., 2022). Latent value is unobserved and must be
inferred from observed behaviors, but importantly, the relationship between
value and the behaviors is empirically learned rather than manually specified
through weights. While more elaborate, automated approaches are intriguing,
the use of simple linear weights is widespread as it provides system designers
with a more interpretable design lever that can be used to shape the platform.
Moreover, the models used in these automated approaches are typically not
capable of accounting for complex effects like strategic behavior, which
humans may be more adept at factoring into their selection.
Despite the importance of the weights, there exists little formal study of
them or guidelines on how to select them. In this paper, we study how to
choose weights when behaviors can vary along three dimensions: _value-
faithfulness_ , _strategy-robustness_ , and _noisiness_. Value-faithfulness
indicates the degree to which a behavior reveals a user’s genuine preference
for an item. Although defining true value can be challenging (Lyngs et al.,
2018), we focus on users’ reflective preferences. Recent research has shown
that an overreliance on learning from implicit, less value-faithful signals
can cause recommender systems to be misaligned with users’ stated, reflective
preferences (Lu et al., 2018; Kleinberg et al., 2022; Agan et al., 2023). Our
work aims to offer guidelines for weight selection when behaviors exhibit
variability in not just value-faithfulness, but also noisiness and strategy-
robustness. For instance, in Section 4, we demonstrate that, for users, there
is a trade-off between choosing value-faithful behaviors and behaviors with
lower estimation noise.
##### Strategic classification, ranking, and recommendation.
Our model considers how strategic producer behavior (alongside estimation
noise and behavior value-faithfulness) should affect the design of recommender
weights. Strategic behavior by content producers, particularly motivated
organizations like news outlets and political parties, has been well-
documented (Morris, 2021; Christin, 2020; Hagey and Horwitz, 2021; Smith,
2023; Meyerson, 2012). In our theoretical model, as in practice, producers
compete against each other (and so, for example, effort by multiple producers
may cancel each other out in equilibria).
In machine learning, strategic adaptation has been primarily been studied in
the field of strategic _classification_ (Brückner et al., 2012; Hardt et al.,
2016). Kleinberg and Raghavan (2020) study how to set linear weights on
observed features that can be strategically changed; Braverman and Garg (2020)
show that noisier signals can lead to better equilibrium outcomes when there
is heterogeneity in producer’s cost functions. Relatively less work in machine
learning has focused on the problem of strategic _contests_ in which
participants must compete to receive desired outcomes, with Liu et al. (2022)
being a notable exception – they consider a rank competition setting in a
single dimension. There exists a rich theory of contests in economics (Hillman
and Riley, 1989; Baye et al., 1996; Lazear and Rosen, 1981; Che and Gale,
2000; Tullock, 1980), and our model is most similar to the classic model of
rank-order tournaments by Lazear and Rosen (1981). To this literature, our
work contributes an analysis of how strategic behavior interacts with value-
faithfulness and noisiness in order to influence the recommender system weight
design problem.
Prior work has also specifically analyzed strategy in recommender systems
(Ben-Porat and Tennenholtz, 2018; Ben-Porat et al., 2020; Jagadeesan et al.,
2022; Hron et al., 2022), studying properties such as genre formation and
producer profit at equilibria (Jagadeesan et al., 2022) or the algorithmic
factors that lead to existence of equilibria (Hron et al., 2022). However, the
prior work does not model the fact that the ranking objective on a recommender
system is typically a linear weighting of _multiple_ behaviors with some
behaviors being easier to game than others. In our work, we focus on the
design of these weights.
## 3 Model
We model a system with one user and two producers. Each producer
$i\in\\{-1,+1\\}$ creates an item whose true value111Here, we consider an
item’s true value to the user to be how the user would value the item upon
reflection (as opposed to their immediate, automatic preference for the item).
to the user is $v(i)$, where $v(1)>v(-1)$. Users can interact with a
producer’s item through $k$ different behaviors. For example, a user may
_click_ , _like_ , and/or _watch_ a video. To rank the producers, the
recommender system creates predictions $\mathbf{y}\in\mathbb{R}^{k}$ of
whether the user will engage with the item using each of these $k$ behaviors
(using historical data from the user and the producers). It then combines the
predictions into a final score $\mathbf{w}^{T}\mathbf{y}$, using a weight
vector $\mathbf{w}\in\mathbb{R}^{k}_{\geq 0}$. User utility depends on being
shown a high value producer, and producer utility depends on being ranked
first.
The platform’s design challenge is to choose weights
$\mathbf{w}\in\mathbb{R}_{\geq 0}^{k}$, to maximize user utility or producer
welfare, i.e., the average utility of both producers. We assume
$\|\mathbf{w}\|_{p}=1$ for some $p$-norm $\|\cdot\|_{p}$, examples of which
include the $\ell^{1}$ or $\ell^{2}$ norm.
We postulate that the predictions $\mathbf{y}(i)\in\mathbb{R}^{k}$
corresponding to each producer $i\in\\{-1,+1\\}$ are:
$\displaystyle\mathbf{y}(i)=v(i)+\mathbf{b}(i)+\bm{\xi}(i)+\mathbf{e}(i),$ (1)
where $v$ reflects the item’s true value; $\mathbf{b}_{j}(i)$ corresponds, for
each behavior $j$ and item $i$, to the user’s bias for engaging with that
item; $\bm{\xi}(i)$ is a noise vector, reflecting variance in the predictions
due to finite sample sizes (Domingos, 2000); $\mathbf{e}_{j}(i)$ corresponds
to producer $i$’s effort in strategically manipulating users to engage in
behavior $j$ on their item.
#### Behavior characteristics
We now detail each component of the prediction $\mathbf{y}(i)$ introduced
above, which represent the three primary behavior characteristics we study:
value-faithfulness, noisiness, and strategy-robustness.
Value-faithfulness. Some behaviors are more indicative of whether the user
values the item than others. For example, explicitly liking an item is more
indicative of value than simply clicking on the item. To model this, each item
$i$ has a behavior-specific bias: $\mathbf{b}(i)\in\mathbb{R}^{k}$. The sum
$v(i)+\mathbf{b}(i)\in\mathbb{R}^{k}$ captures how likely a user is to engage
in a behavior on producer $i$’s item (in the absence of any strategic effort
from the producer). Then, the _value-faithfulness_ of a behavior $j\in[k]$ is
$\displaystyle\mathbf{VF}_{j}=\mathbb{E}[\mathbf{y}_{j}(1)-\mathbf{y}_{j}(-1)]=(v(1)+\mathbf{b}_{j}(1))-(v(-1)+\mathbf{b}_{j}(-1))\,.$
(2)
The higher a behavior’s value-faithfulness, the more likely the user is to
engage in that behavior on the higher-valued item compared to the lower-valued
item. For example, a like has higher value-faithfulness than a click because a
user is more likely to only like a high-valued item while they may click on
both high and low-valued items. Without loss of generality, we assume that
$\mathbf{VF}>\mathbf{0}$ (if a behavior does not have positive value-
faithfulness, we can always consider the opposite of the behavior instead,
e.g. not clicking instead of clicking).
Noisiness. The predictions for the behaviors are made by a machine learning
model trained on a finite dataset. Some behavior predictions may be noisier
than others, for example, especially if that behavior is much sparser (e.g.
likes) and the system has less data on it than other behaviors (e.g. clicks)
(Chen et al., 2018). We model this estimation error as random behavior-
specific noise $\bm{\xi}(i),\bm{\xi}(-i)\sim\mathcal{N}(0,\Sigma)$ where
$\Sigma\in\mathbb{R}^{k\times k}$ is a diagonal matrix. We say that a behavior
$j$ is _noisier_ than a behavior $k$ if its variance $\Sigma_{jj}$ is higher
than $\Sigma_{kk}$.
Strategy-robustness. Given a weight vector $\mathbf{w}$, producers will
strategically adapt their items to get a higher score under $\mathbf{w}$.
Though we consider an item’s true value to be fixed, a producer can put effort
$\mathbf{e}\in\mathbb{R}^{k}_{\geq 0}$ into increasing the probability that
the user interacts with their item with each of the $k$ behaviors. For
example, without increasing the quality of their content, the producer may
craft a clickbait title to entice the user into clicking on it. The producer
incurs a cost $c(\mathbf{e})$ for their effort. The cost is quadratic with
some behaviors being higher cost to manipulate than others:
$c(\mathbf{e})=\frac{1}{2}\mathbf{e}^{\intercal}A\mathbf{e}$ where $A$ is a
diagonal matrix and all entries on the diagonal are unique and positive. We
say that a behavior $i$ is more _strategy-robust_ than behavior $j$ if
$A_{ii}>A_{jj}$.
#### Ranking, utility, and equilibria
Producer one is ranked first if
$\mathbf{w}^{\intercal}\mathbf{y}(1)-\mathbf{w}^{\intercal}\mathbf{y}(-1)>0$.
Or equivalently, producer one is ranked first if
$\epsilon(\mathbf{w})<\mathbb{E}[\mathbf{w}^{\intercal}\mathbf{y}(1)-\mathbf{w}^{\intercal}\mathbf{y}(-1)]$,
where
$\epsilon(\mathbf{w})=\mathbf{w}^{\intercal}\bm{\xi}(-1)-\mathbf{w}^{\intercal}\bm{\xi}(1)$
is the difference in noise terms. Letting $F_{\epsilon}$ be the distribution
of $\epsilon(\mathbf{w})$, the probability that producer $i$ is ranked first
is
$\displaystyle\mathbb{P}_{\mathbf{w}}(R(i)=1\mid\mathbf{e})=\begin{cases}F_{\epsilon}(\mathbb{E}[\mathbf{w}^{\intercal}\mathbf{y}(1)-\mathbf{w}^{\intercal}\mathbf{y}(-1)])&i=1\\\
1-F_{\epsilon}(\mathbb{E}[\mathbf{w}^{\intercal}\mathbf{y}(1)-\mathbf{w}^{\intercal}\mathbf{y}(-1)])&i=-1\end{cases}$
(3)
where $R(i)$ is a random variable indicating producer $i$’s rank.
An individual producer’s expected utility is the probability they are ranked
first minus the incurred cost of manipulation:
$\displaystyle\mathcal{U}_{\text{prod}}^{i}(\mathbf{e}(i),\mathbf{e}(-i);\mathbf{w})=\mathbb{P}_{\mathbf{w}}(R(i)=1\mid\mathbf{e})-c(\mathbf{e}(i))\,.$
(4)
The producer welfare is defined as the average utility of the producers,
$\displaystyle\mathcal{W}_{\text{prod}}(\mathbf{e}(1),\mathbf{e}(-1);\mathbf{w})$
$\displaystyle=\frac{1}{2}\sum_{i}\mathcal{U}_{\text{prod}}^{i}(\mathbf{e}(i),\mathbf{e}(-i);\mathbf{w})=\frac{1}{2}-\frac{\sum_{i}c(\mathbf{e}(i))}{2}\,.$
(5)
The user’s utility is the probability the higher-valued producer is ranked
first:
$\displaystyle\mathcal{U}_{\text{user}}(\mathbf{e}(1),\mathbf{e}(-1);\mathbf{w})=\mathbb{P}_{\mathbf{w}}(R(1)=1\mid\mathbf{e})\,.$
(6)
Producer $i$’s best response given fixed features for the other producer is
$\displaystyle\mathbf{BR}^{i}(\mathbf{e}(-i);\mathbf{w})=\operatorname*{arg\,max}_{\mathbf{q}\in\mathbb{R}^{k}_{\geq
0}}\,\mathcal{U}_{\text{prod}}^{i}(\mathbf{q},\mathbf{e}(-i);\mathbf{w})\,.$
(7)
At a (pure Nash) equilibrium, the best responses of both producers are at a
fixed point, defined formally below.
###### Definition 3.1 (Equilibrium).
Given a fixed weight vector $\mathbf{w}$, an equilibrium consists of a pair of
efforts $(\mathbf{e}^{*}(1),\mathbf{e}^{*}(-1))$ that satisfy
$\mathbf{BR}^{i}(\mathbf{e}^{*}(-i);\mathbf{w})=\mathbf{e}^{*}(i)$ for
$i\in\\{-1,+1\\}$.
## 4 User utility without strategic adaptation
First, we analyze user utility in the absence of strategic adaptation from
producers, i.e., when $\mathbf{e}\triangleq\mathbf{0}$. Even without any
strategic adaptation from producers, it is not obvious what the weights a
practitioner should choose are. (We do not analyze producer welfare in the
non-strategic setting because producer welfare is constant when producers do
not manipulate, i.e.,
$\mathcal{W}_{\text{prod}}(0,0;\mathbf{w})=\frac{1}{2}$.)
One might assume that for the user it would be best to give the most weight to
the most value-faithful behaviors. However, the value-faithful behaviors may
not be the easiest to predict, and thus, may introduce more noise into the
rankings. Indeed, our analysis suggests that the optimal weight vector depends
on a trade-off between choosing behaviors that are more value-faithful and
choosing behaviors with lower estimation noise. This conclusion is formalized
in Theorem 4.1 and illustrated in Figure 1. Omitted proofs for this section
can be found in Appendix A.
Figure 1: This example demonstrates how the user’s optimal weight vector
$\mathbf{w}$ is affected by two factors: value-faithfulness and noise. There
are two behaviors being considered, and the second one is both more value-
faithful and noisier than the first. Here, the weight vector is normalized so
that $||\mathbf{w}||_{1}=1$, and thus, a weight of $0.5$ corresponds to equal
weight on both behaviors. The optimal weight vector assigns more weight
$\mathbf{w}_{2}$ to the second behavior as its relative value-faithfulness
increases and less weight as its relative noisiness increases. The exact
parameters used for the figure can be found in Appendix C.
###### Theorem 4.1.
Without any strategic adaptation, the weight vector that maximizes user
utility is
$\displaystyle\mathbf{w}^{*}=(\Sigma^{-1}\mathbf{VF})/\|\Sigma^{-1}\mathbf{VF}\|_{p}\,.$
(8)
Consequently, the optimal weight on a behavior increases as its value-
faithfulness increases or its variance decreases, stated formally below.
###### Corollary 4.1.
For any behavior, $j\in[k]$, the user-optimal weight on the behavior
$\mathbf{w}^{*}_{j}$ monotonically increases in the behavior’s value-
faithfulness $\mathbf{VF}_{j}$ and monotonically decreases in the behavior’s
variance $\Sigma_{jj}$.
User utility under the optimal weight vector also increases in value-
faithfulness and decreases in variance. Under any weight vector, increasing
the value-faithfulness or decreasing the variance of a behavior will increase
the probability that the higher-valued producer is ranked first, i.e., user
utility. Since this is true for _any_ weight vector, it is also true under the
optimal weight vector.
###### Theorem 4.2.
Let
$\mathcal{U}_{\text{user}}(\mathbf{0},\mathbf{0};\mathbf{w}^{*}(\mathbf{VF},\Sigma))$
be user utility under the user-optimal weight vector $\mathbf{w}^{*}$. For any
behavior, $j\in[k]$, optimal user utility
$\mathcal{U}_{\text{user}}(\mathbf{0},\mathbf{0};\mathbf{w}^{*}(\mathbf{VF},\Sigma))$
monotonically increases in the behavior’s value-faithfulness $\mathbf{VF}_{j}$
and monotonically decreases in the behavior’s variance $\Sigma_{jj}$.
###### Proof.
We can write user utility $\mathcal{U}_{\text{user}}$ as
$\displaystyle\mathcal{U}_{\text{user}}(\mathbf{0},\mathbf{0};\mathbf{w})=\mathbb{P}_{\mathbf{w}}(R(1)=1)=F_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\mathbf{w}^{\intercal}\mathbf{VF}}{2\sqrt{\mathbf{w}^{\intercal}\Sigma\mathbf{w}}}\right)\right]\,.$
(9)
For _any_ weight vector $\mathbf{w}\in\mathbb{R}^{k}_{\geq 0}$, user utility
$\mathcal{U}_{\text{user}}(\mathbf{0},\mathbf{0};\mathbf{w})$ is monotonically
increasing in $\mathbf{VF}_{j}$ and monotonically decreasing in $\Sigma_{jj}$
for any behavior $j\in[k]$. Thus, user utility under the _optimal_ weight
vector must also be monotonically increasing in $\mathbf{VF}_{j}$ and
monotonically decreasing in $\Sigma_{jj}$. ∎
Aspect of behavior $j$ | Optimal user utility $\mathcal{U}_{\text{user}}^{*}$ | Optimal producer welfare $\mathcal{W}_{\text{prod}}^{*}$
---|---|---
(5.1) | (Theorem 5.1)
Value-faithfulness $\mathbf{VF}_{j}$ | Increases | Increases
Noise $\Sigma_{jj}$ | Decreases | Non-monotonic
Strategy-robustness $A_{jj}$ | Constant | Increases
Table 1: A summary of our results on the impact of value-faithfulness, noise,
and strategy-robustness on user utility under the user-optimal weight vector
and producer welfare under the producer-optimal weight vector. Figure 2: This
example demonstrates how the user-optimal and producer-optimal weight vector
is affected by three aspects of behavior: value-faithfulness, variance, and
strategy-robustness. There are two behaviors being considered, and the second
one scores higher on all three aspects. The exact parameters used for the
figure can be found in Appendix C.
## 5 User and producer welfare under strategic adaptation
In this section, we analyze user utility and producer welfare under the full
model described in Section 3. Omitted proofs can be found in Appendix B. Given
the weights $\mathbf{w}$ chosen by the designer, producers strategically exert
effort $\mathbf{e}\in\mathbb{R}^{k}_{\geq 0}$ into increasing the probability
that the user engages with their item through each behavior. The choice of
weights must balance between three aspects of behavior at once: value-
faithfulness, strategy-robustness, and noisiness.
In 5.1, we derive the unique equilibrium strategy for producers and find that
the strategy is symmetric, i.e., both producers exert the same effort at
equilibrium.
###### Proposition 5.1 (Equilibrium).
The unique equilibrium strategy for both producers is
$\mathbf{e}^{*}(1)=\mathbf{e}^{*}(-1)=f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})A^{-1}\mathbf{w}$
where $f_{\epsilon}$ is the density of the difference in noise terms
$\epsilon(\mathbf{w})\sim\mathcal{N}(0,2\mathbf{w}^{\intercal}\Sigma\mathbf{w})$.
Symmetric equilibria are commonly seen in the literature on _contests_ in
which agents exert effort towards attaining outcomes that are allocated based
on relative rank (Olszewski and Siegel, 2016; Bodoh-Creed and Hickman, 2018;
Liu et al., 2022). The intuition for the symmetry in our setting is that
producer utility is linear in the probability of being ranked first (which is
zero-sum between producers) and the cost of manipulation (where the cost
function is the same between producers). Thus, if one producer had found it
valuable to expend effort to improve their ranking probability (at the expense
of the other), then the other producer would equally have found it valuable to
do the same. Hence, they have to have the same effort at equilibrium.
Since the equilibrium is symmetric, the probability that producer one is
ranked first is the same as in the non-strategic setting, as producer effort
cancels out. Consequently, for users, both the optimal weight vector and user
utility under the optimal weight vector remain the same in the strategic
setting. Thus, for users, it is better to up-weigh value-faithful behaviors
and down-weigh noisy behaviors (and strategy-robustness has no impact on the
optimal weight vector). (This result would not hold in a model in which effort
is productive, improving producer quality for the user.)
###### Corollary 5.1.
Even with strategic adaptation, the user-optimal weight vector is the same as
in Theorem 4.1. As in the non-strategic setting (Theorem 4.2), for any
behavior $j$, user utility under the user-optimal weight vector is
monotonically increasing in value-faithfulness $\mathbf{VF}_{j}$,
monotonically decreasing in noisiness $\Sigma_{jj}$, and is constant in
strategy-robustness $A_{jj}$.
###### Proof.
From 5.1, the equilibrium strategy for producers is symmetric:
$\mathbf{e}^{*}(1)=\mathbf{e}^{*}(-1)$. When the strategies are symmetric,
then user utility
$\mathcal{U}_{\text{user}}(\mathbf{e}^{*}(1),\mathbf{e}^{*}(-1);\mathbf{w})$
is equal to user utility without strategic adaptation
$\mathcal{U}_{\text{user}}(\mathbf{0},\mathbf{0};\mathbf{w})$. Thus, for
users, the results from the non-strategic setting still hold in the strategic
setting. ∎
At equilibrium, the probability that each producer is ranked first is the same
as it would be if neither producer exerted any effort. Thus, a producer’s
effort is essentially wasted effort but is required in order to “keep up” with
their competitor. Hence, the weight vector that maximizes producer welfare is
the one that most disincentives manipulation among producers. Unlike the user-
optimal weight vector, it is difficult to find a closed-form solution for the
producer-optimal weight vector. The following proposition shows that the
optimal weight vector for producers is the solution to a non-convex
optimization problem. Solving for the weight vector requires minimizing the
product of a convex quadratic form ($\mathbf{w}^{\intercal}A^{-1}\mathbf{w}$)
and a non-convex term (the Gaussian density
$f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})$).
###### Proposition 5.2.
The weight vector that maximizes producer welfare at equilibrium is
$\displaystyle\mathbf{w}^{*}\in\operatorname*{arg\,min}_{\mathbf{w}:||\mathbf{w}||=1}f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})^{2}\mathbf{w}^{\intercal}A^{-1}\mathbf{w}$
(10)
where $f_{\epsilon}$ is the density of the difference in noise terms
$\epsilon(\mathbf{w})\sim\mathcal{N}(0,2\mathbf{w}^{\intercal}\Sigma\mathbf{w})$.
###### Proof.
By 5.1, the unique and symmetric equilibrium strategy for producers is
$\mathbf{e}^{*}(1)=\mathbf{e}^{*}(-1)=f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})A^{-1}\mathbf{w}$.
Thus, producer welfare at equilibrium is equal to
$\displaystyle\mathcal{W}_{\text{prod}}(\mathbf{e}^{*}(1),\mathbf{e}^{*}(-1);\mathbf{w})=\frac{1}{2}-c(\mathbf{e}^{*}(1))=\frac{1}{2}-\frac{f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})^{2}}{2}\mathbf{w}^{\intercal}A^{-1}\mathbf{w}\,,$
(11)
and the optimal weight vector minimizes
$f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})^{2}\mathbf{w}^{\intercal}A^{-1}\mathbf{w}$.
∎
We can, however, characterize how producer welfare at equilibrium under the
optimal weight vector changes as the three aspects of behavior—value-
faithfulness, strategy-robustness, and noisiness—change (Theorem 5.1) We find
that producer welfare under the optimal weight vector increases as strategy-
robustness and value-faithfulness increase: both strategy-robustness and
value-faithfulness disincentivize producer manipulation; strategy-robustness
does so directly, and value-faithfulness does so by making the gap between the
producers’ pre-manipulation scores larger. On the other hand, the relationship
between noisiness and producer welfare is non-monotonic. Intuitively, when
noise is very high, producer welfare is high because manipulation is
disincentivized since the ranking outcome is primarily determined by
randomness rather than their scores. Conversely, when noise is very low,
producer welfare is also high because producers gain little from any
incremental increase in their score, so manipulation is also disincentivized.
###### Theorem 5.1.
Let $\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)$ be the optimal
producer welfare given the exogenous parameters for value-faithfulness
$\mathbf{VF}$, noise $\Sigma$, and strategy-robustness $A$, i.e.,
$\displaystyle\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)=\max_{\mathbf{w}}\mathcal{W}_{\text{prod}}(\mathbf{e}_{\mathbf{w}}^{*}(1),\mathbf{e}_{\mathbf{w}}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})$
(12)
where $\mathbf{e}_{\mathbf{w}}^{*}(1)$ and $\mathbf{e}_{\mathbf{w}}^{*}(-1)$
are the unique equilibrium strategies in response to $\mathbf{w}$.
For any behavior, $j\in[k]$, the optimal producer welfare
$\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)$ is monotonically
increasing in the behavior’s strategy robustness $A_{jj}$, monotonically
increasing in its value-faithfulness $\mathbf{VF}_{j}$, and is not necessarily
monotonic in its variance $\Sigma_{jj}$.
Furthermore, at the limit, as any of strategy robustness $A_{jj}$, value-
faithfulness $\mathbf{VF}_{j}$, or variance $\Sigma_{jj}$ go to infinity,
producer welfare under the optimal weight vector
$\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)$ reaches the maximum
possible value, i.e.,
$\displaystyle\lim_{z\rightarrow\infty}\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)=1/2~{}~{}~{}\text{for
}~{}z\in\\{A_{jj},\mathbf{VF}_{j},\Sigma_{jj}\\}\,.$ (13)
Figure 2 shows an example of how the user-optimal and producer-optimal weight
on a behavior changes as its value-faithfulness, variance, and strategy-
robustness increase. For producers, the optimal weight is calculated through
numerical approximation. The producer-optimal weight changes in the same way,
as described in Theorem 5.1, that the optimal producer welfare changes as a
function of these three aspects. Though the user and producer-optimal weight
vector remain similar as value-faithfulness changes, they diverge drastically
as noisiness or strategic-robustness change.
## 6 Discussion
We analyzed how three aspects of behavior — value-faithfulness, noisiness, and
strategy-robustness — affect the optimal weight vector and welfare for users
and producers. In practice, the weight vector is chosen based on both
performance in A/B tests and qualitative human judgment (Twitter, 2023;
Merrill and Oremus, 2021). Understanding how these three aspects of behavior
affect user and producer welfare could help platform designers (typically the
product and engineering teams) narrow the search space of weights that are
most relevant to test, since a full grid search may become prohibitively
expensive as the number of behaviors grows.
### 6.1 Example: the three aspects in e-commerce, TikTok, and Twitter
Through an example, we illustrate how system designers could think through
these three aspects to help choose the weights for their platform. First, they
could start by listing out the behaviors and theorizing rankings for each of
the three aspects. Table 2 shows one hypothesized ranking of three behaviors
for three different settings: an e-commerce setting, TikTok, and Twitter. Note
that for certain platforms, such as TikTok or Twitter, users are typically
both consumers and producers. Our hypothesized rankings for value-faithfulness
and strategy-robustness coincide: value-faithful behaviors tend to be more
explicit, reflective behaviors that are harder to game. For the noisiness
rankings, we assume that less frequent behaviors (i.e., smaller training sets)
result in greater variance in estimation, as often seen in theory and practice
(Chen et al., 2018).
For example, for Twitter, we consider three behaviors: _like_ , _retweet_ ,
and _reply_. From the recent open-sourcing of the Twitter algorithm, we know
that these behaviors are used in practice by Twitter. Since Twitter does not
use a separate behavior specifically for quote retweets222On Twitter, users
can “retweet” tweets posted by others to share them with their followers. A
regular retweet simply shares the original tweet. In a quote tweet, the user
shares the original tweet with their own message added to it., we assume that
_retweet_ includes both regular retweets and quote retweets (Twitter, 2023).
We hypothesize that the value-faithfulness and strategy-robustness ordering is
$\mathtt{like}\succ\mathtt{retweet}\succ\mathtt{reply}$ (meaning that likes
are more value-faithful and strategy-robust than the others). Retweets are
ranked lower than likes because quote retweets often express disagreement with
the original poster (Minot et al., 2021). Based on the prevalence of the
behaviors (McClain et al., 2021), we hypothesize the reverse ranking for
noisiness: $\mathtt{reply}\succ\mathtt{retweet}\succ\mathtt{like}$.
Ultimately, the rankings provided here are just for illustration. Platforms
can use a combination of domain expertise and measurement studies to determine
how behaviors rank on these three aspects. Indeed, Facebook conducted surveys
to measure how much users value different kinds of interactions (i.e.,
measuring value-faithfulness) and used them in choosing the weights: _“the
base weight of all the interactions are derived based on producer-side
experiments which measure value to the originator/producer (of the content)”_
(Cameron et al., 2022).333The quote is from a leaked document from the
Facebook Files titled “The Meaningful Social Interactions Metric Revisited:
Part Two”.
The exercise of thinking through these aspects may help foresee certain
negative side effects, especially when it comes to strategic adaptation, which
often takes longer to measure than the typical duration of an A/B test. For
instance, on Twitter, producers sometimes post offensive or sensationalized
messages that receive few likes but many replies and quote tweets expressing
disagreement (a phenomenon called “ratioing” (Troy, 2022; Minot et al.,
2021)). By theorizing why some behaviors may be less strategy-robust,
designers may recognize that overweighing replies or retweets could
inadvertently incentivize producers to game the rankings by intentionally
posting offensive messages that are widely disliked but still gain engagement
in the form of replies or retweets.
Hypothesized behavior rankings
---
| e-commerce | TikTok | Twitter
Value-faithfulness | $\mathtt{order}\succ\mathtt{cart}\succ\mathtt{click}$ | $\mathtt{like}\succ\mathtt{comment}\succ\mathtt{play}$ | $\mathtt{like}\succ\mathtt{RT}\succ\mathtt{reply}$
Strategy-robustness | $\mathtt{order}\succ\mathtt{cart}\succ\mathtt{click}$ | $\mathtt{like}\succ\mathtt{comment}\succ\mathtt{play}$ | $\mathtt{like}\succ\mathtt{RT}\succ\mathtt{reply}$
Noisiness | $\mathtt{order}\succ\mathtt{cart}\succ\mathtt{click}$ | $\mathtt{comment}\succ\mathtt{like}\succ\mathtt{play}$ | $\mathtt{reply}\succ\mathtt{RT}\succ\mathtt{like}$
Theory-implied constraints on user-optimal $\mathbf{w}^{*}$
| e-commerce | TikTok | Twitter
Implied constraints | None | $\mathbf{w}^{*}_{\mathtt{like}}>\mathbf{w}^{*}_{\mathtt{comment}}$ | $\mathbf{w}^{*}_{\mathtt{like}}>\mathbf{w}^{*}_{\mathtt{RT}}>\mathbf{w}^{*}_{\mathtt{reply}}$
Table 2: Three types of platforms and hypothesized rankings of their behaviors
according to the three aspects: value-faithfulness, strategy-robustness, and
noisiness.
Further, our results suggest certain constraints on the ordering of the user-
optimal weights based on the rankings of value-faithfulness and noisiness.
Such constraints can help reduce the range of weights requiring testing, which
is particularly beneficial as the total number of behaviors increases (e.g.,
Twitter uses 10 different behaviors for ranking (Twitter, 2023)). In our
hypothesized ranking for TikTok, likes are both more value-faithful and less
noisy than comments, and thus, in the user-optimal weight vector, we must have
$\mathbf{w}^{*}_{\mathtt{like}}>\mathbf{w}^{*}_{\mathtt{comment}}$. Similarly,
for Twitter, the dominance relations would imply that
$\mathbf{w}^{*}_{\mathtt{like}}>\mathbf{w}^{*}_{\mathtt{retweet}}>\mathbf{w}^{*}_{\mathtt{reply}}$.
### 6.2 Limitations and open questions
Our insights into the three aspects of behavior were derived from a simple
theoretical model that examined one user and two items or producers. In
reality, however, there are numerous users and items—an important direction
for extension of our model. Furthermore, the three aspects of each behavior
might differ depending on the item and user in question. For instance, some
users may be more likely to click on items that they don’t value. In the
simulations provided in Appendix C, we demonstrate how heterogeneity may
affect the optimal weight vector.
Moreover, addressing the issue of producers’ strategic adaptation remains a
challenge in practical applications. Theoretical models of strategic response
are often simplistic, and further research is needed to connect theory and
practice more effectively. One prevalent method for accounting for strategic
effects involves periodically retraining (in this case, periodically changing
the weights), but this may not always be optimal (Perdomo et al., 2020). In
the context of recommender systems, conducting producer-side A/B tests to
gauge strategic effects can be complex due to the need to avoid interference
with ongoing user-side A/B tests and to minimize violations of the Stable Unit
Treatment Value Assumption (SUTVA) (Nandy et al., 2021).
Broader impacts. The weights chosen can have a large effect on the emergent
dynamics of the platform, in some cases even increasing toxicity and
misinformation (Merrill and Oremus, 2021). Our paper describes guidelines for
choosing weights, but it is essential to always be evaluating properties of
the system, e.g., tracking measures of toxicity or misinformation, to mitigate
any possible unintended side effects. Ultimately, we hope that our framework
and its three aspects—value-faithfulness, noisiness, and strategy-
robustness—can be a helpful one for recommender system designers to use when
grappling with the difficult question of what weights to choose.
## Acknowledgements
We thank Gabriel Agostini, Sidhika Balachandar, Ben Laufer, Raj Movva, Kenny
Peng, and Luke Thorburn for feedback on a draft version of the paper.
## References
* Smith [2021] Ben Smith. How tiktok reads your mind, Dec 2021. URL https://www.nytimes.com/2021/12/05/business/media/tiktok-algorithm.html.
* Twitter [2023] Twitter. Twitter’s recommendation algorithm - heavy ranker and twhin embeddings, Mar 2023. URL https://github.com/twitter/the-algorithm-ml/tree/main/projects/home/recap.
* Merrill and Oremus [2021] Jeremy B Merrill and Will Oremus. Five points for anger, one for a ‘like’: How facebook’s formula fostered rage and misinformation. _The Washington Post_ , 26, 2021.
* Hagey and Horwitz [2021] Keach Hagey and Jeff Horwitz. Facebook tried to make its platform a healthier place. it got angrier instead. _The Wall Street Journal_ , 16, 2021.
* Morris [2021] Loveday Morris. In poland’s politics, a “social civil war” brewed as facebook rewarded online anger, Oct 2021. URL https://www.washingtonpost.com/world/2021/10/27/poland-facebook-algorithm/.
* TikTok [2020] TikTok. How tiktok recommends videos #foryou, Jun 2020. URL https://newsroom.tiktok.com/en-us/how-tiktok-recommends-videos-for-you.
* Youtube [2019] Youtube. Continuing our work to improve recommendations on youtube, Jan 2019. URL https://blog.youtube/news-and-events/continuing-our-work-to-improve/.
* Gomez-Uribe and Hunt [2015] Carlos A Gomez-Uribe and Neil Hunt. The netflix recommender system: Algorithms, business value, and innovation. _ACM Transactions on Management Information Systems (TMIS)_ , 6(4):1–19, 2015.
* Cameron et al. [2022] Dell Cameron, Shoshana Wodinsky, Mack DeGeurin, and Thomas Germain. Read the facebook papers for yourself, Apr 2022. URL https://gizmodo.com/facebook-papers-how-to-read-1848702919.
* Milli et al. [2021] Smitha Milli, Luca Belli, and Moritz Hardt. From optimizing engagement to measuring value. In _Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency_ , pages 714–722, 2021.
* Maghakian et al. [2022] Jessica Maghakian, Paul Mineiro, Kishan Panaganti, Mark Rucker, Akanksha Saran, and Cheng Tan. Personalized reward learning with interaction-grounded learning (igl). _arXiv preprint arXiv:2211.15823_ , 2022.
* Lyngs et al. [2018] Ulrik Lyngs, Reuben Binns, Max Van Kleek, and Nigel Shadbolt. So, tell me what users want, what they really, really want! In _Extended Abstracts of the 2018 CHI Conference on Human Factors in Computing Systems_ , pages 1–10, 2018.
* Lu et al. [2018] Hongyu Lu, Min Zhang, and Shaoping Ma. Between clicks and satisfaction: Study on multi-phase user preferences and satisfaction for online news reading. In _The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval_, pages 435–444, 2018.
* Kleinberg et al. [2022] Jon Kleinberg, Sendhil Mullainathan, and Manish Raghavan. The challenge of understanding what users want: Inconsistent preferences and engagement optimization. In _Proceedings of the 23rd ACM Conference on Economics and Computation_ , pages 29–29, 2022.
* Agan et al. [2023] Amanda Y Agan, Diag Davenport, Jens Ludwig, and Sendhil Mullainathan. Automating automaticity: How the context of human choice affects the extent of algorithmic bias. Technical report, National Bureau of Economic Research, 2023.
* Christin [2020] Angèle Christin. _Metrics at work: journalism and the contested meaning of algorithms_. Princeton University Press, 2020.
* Smith [2023] Ben Smith. _Traffic: Genius, rivalry, and delusion in the billion-dollar race to go viral_. Penguin Press, 2023.
* Meyerson [2012] Eric Meyerson. Youtube now: Why we focus on watch time. _YouTube Creator Blog. August_ , 10:2012, 2012.
* Brückner et al. [2012] Michael Brückner, Christian Kanzow, and Tobias Scheffer. Static prediction games for adversarial learning problems. _The Journal of Machine Learning Research_ , 13(1):2617–2654, 2012.
* Hardt et al. [2016] Moritz Hardt, Nimrod Megiddo, Christos Papadimitriou, and Mary Wootters. Strategic classification. In _Proceedings of the 2016 ACM conference on innovations in theoretical computer science_ , pages 111–122, 2016.
* Kleinberg and Raghavan [2020] Jon Kleinberg and Manish Raghavan. How do classifiers induce agents to invest effort strategically? _ACM Transactions on Economics and Computation (TEAC)_ , 8(4):1–23, 2020.
* Braverman and Garg [2020] Mark Braverman and Sumegha Garg. The Role of Randomness and Noise in Strategic Classification. In Aaron Roth, editor, _1st Symposium on Foundations of Responsible Computing (FORC 2020)_ , volume 156 of _Leibniz International Proceedings in Informatics (LIPIcs)_ , pages 9:1–9:20, Dagstuhl, Germany, 2020\. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. ISBN 978-3-95977-142-9. doi: 10.4230/LIPIcs.FORC.2020.9. URL https://drops.dagstuhl.de/opus/volltexte/2020/12025.
* Liu et al. [2022] Lydia T Liu, Nikhil Garg, and Christian Borgs. Strategic ranking. In _International Conference on Artificial Intelligence and Statistics_ , pages 2489–2518. PMLR, 2022.
* Hillman and Riley [1989] Arye L Hillman and John G Riley. Politically contestable rents and transfers. _Economics & Politics_, 1(1):17–39, 1989.
* Baye et al. [1996] Michael R Baye, Dan Kovenock, and Casper G De Vries. The all-pay auction with complete information. _Economic Theory_ , 8:291–305, 1996.
* Lazear and Rosen [1981] Edward P Lazear and Sherwin Rosen. Rank-order tournaments as optimum labor contracts. _Journal of political Economy_ , 89(5):841–864, 1981.
* Che and Gale [2000] Yeon-Koo Che and Ian Gale. Difference-form contests and the robustness of all-pay auctions. _Games and Economic Behavior_ , 30(1):22–43, 2000\.
* Tullock [1980] Gordon Tullock. Efficient rent seeking. _Toward a Theory of the Rent-Seeking Society_ , 1980.
* Ben-Porat and Tennenholtz [2018] Omer Ben-Porat and Moshe Tennenholtz. A game-theoretic approach to recommendation systems with strategic content providers. _Advances in Neural Information Processing Systems_ , 31, 2018.
* Ben-Porat et al. [2020] Omer Ben-Porat, Itay Rosenberg, and Moshe Tennenholtz. Content provider dynamics and coordination in recommendation ecosystems. _Advances in Neural Information Processing Systems_ , 33:18931–18941, 2020.
* Jagadeesan et al. [2022] Meena Jagadeesan, Nikhil Garg, and Jacob Steinhardt. Supply-side equilibria in recommender systems. _arXiv preprint arXiv:2206.13489_ , 2022.
* Hron et al. [2022] Jiri Hron, Karl Krauth, Michael I Jordan, Niki Kilbertus, and Sarah Dean. Modeling content creator incentives on algorithm-curated platforms. _arXiv preprint arXiv:2206.13102_ , 2022.
* Domingos [2000] Pedro Domingos. A unified bias-variance decomposition. In _Proceedings of 17th international conference on machine learning_ , pages 231–238. Morgan Kaufmann Stanford, 2000.
* Chen et al. [2018] Irene Chen, Fredrik D Johansson, and David Sontag. Why is my classifier discriminatory? _Advances in neural information processing systems_ , 31, 2018.
* Olszewski and Siegel [2016] Wojciech Olszewski and Ron Siegel. Large contests. _Econometrica_ , 84(2):835–854, 2016.
* Bodoh-Creed and Hickman [2018] Aaron L Bodoh-Creed and Brent R Hickman. College assignment as a large contest. _Journal of Economic Theory_ , 175:88–126, 2018.
* Minot et al. [2021] Joshua R Minot, Michael V Arnold, Thayer Alshaabi, Christopher M Danforth, and Peter Sheridan Dodds. Ratioing the president: An exploration of public engagement with obama and trump on twitter. _PloS one_ , 16(4):e0248880, 2021.
* McClain et al. [2021] Colleen McClain, Regina Widjaya, Gonzalo Rivero, and Aaron Smith. The behaviors and attitudes of us adults on twitter. 2021\.
* Troy [2022] Cassandra Lynn-Collins Troy. Get ratioed: Questioning the fossil fuel industry’s social license to operate on twitter. _The Journal of Public Interest Communications_ , 6(1):4–4, 2022.
* Perdomo et al. [2020] Juan Perdomo, Tijana Zrnic, Celestine Mendler-Dünner, and Moritz Hardt. Performative prediction. In _International Conference on Machine Learning_ , pages 7599–7609. PMLR, 2020.
* Nandy et al. [2021] Preetam Nandy, Divya Venugopalan, Chun Lo, and Shaunak Chatterjee. A/b testing for recommender systems in a two-sided marketplace. _Advances in Neural Information Processing Systems_ , 34:6466–6477, 2021.
## Appendix A Proofs for Section 4
###### Lemma A.1.
Let $\|\cdot\|_{p}$ be a $p$-norm. The partial derivative of
$\|\mathbf{w}\|_{p}$ is
$\displaystyle\frac{\partial}{\partial\mathbf{w}_{i}}\|\mathbf{w}\|_{p}=\left(\frac{|\mathbf{w}_{i}|}{\|\mathbf{w}\|_{p}}\right)^{p-1}\operatorname{sgn}(\mathbf{w}_{i})\,.$
(14)
###### Proof.
The $p$-norm of a vector $\mathbf{w}$ is equal to
$\|\mathbf{w}\|_{p}=\left(\sum_{j}|\mathbf{w}_{j}|^{p}\right)^{(1/p)}$ for
some $p\geq 1$. Taking the derivative with respect to a component
$\mathbf{w}_{i}$ yields,
$\displaystyle\frac{\partial}{\partial\mathbf{w}_{i}}\|\mathbf{w}\|_{p}=\frac{1}{p}\left(\sum_{j}|\mathbf{w}_{j}|^{p}\right)^{\frac{1}{p}-1}\cdot
p|\mathbf{w}_{i}|^{p-1}\operatorname{sgn}(\mathbf{w}_{i})=\left(\frac{|\mathbf{w}_{i}|}{\|\mathbf{w}\|_{p}}\right)^{p-1}\operatorname{sgn}(\mathbf{w}_{i})\,.$
(15)
∎
### Proof of Theorem 4.1
###### Proof.
User utility in the non-strategic setting is equal to
$\displaystyle\mathcal{U}_{\text{user}}(\mathbf{0},\mathbf{0};\mathbf{w})=\mathbb{P}_{\mathbf{w}}(R(1)=1)=F_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\mathbf{w}^{\intercal}\mathbf{VF}}{2\sqrt{\mathbf{w}^{\intercal}\Sigma\mathbf{w}}}\right)\right]\,.$
(16)
Since the error function $\mathrm{erf}$ is monotonically increasing on
$[0,\infty)$, the optimal weight vector is simply one which solves the
following optimization problem (for now, let us ignore the constraint that the
weight vector satisfy $\|\mathbf{w}\|_{p}=1$):
$\displaystyle\max_{\mathbf{w}\geq 0}g(\mathbf{w})\text{ where
}g(\mathbf{w})=(\mathbf{w}^{\intercal}\mathbf{VF})/\sqrt{\mathbf{w}^{\intercal}\Sigma\mathbf{w}}\,.$
(17)
Since the objective function is scale-invariant, i.e.,
$g(\mathbf{w})=g(a\mathbf{w})$ for any scalar $a\geq 0$, one can rewrite the
problem as
$\displaystyle\max_{\mathbf{w}\geq 0}\mathbf{w}^{\intercal}\mathbf{VF}\text{
such that }\mathbf{w}^{\intercal}\Sigma\mathbf{w}=1$ (18)
because one can always scale any optimal weight vector for the original
problem in Equation 17 so that $\mathbf{w}^{\intercal}\Sigma\mathbf{w}=1$ is
satisfied. Let $\tilde{\mathbf{w}}=\Sigma^{1/2}\mathbf{w}$ and
$\mathbf{z}=\Sigma^{-1/2}\mathbf{VF}$. Then, the constrained optimization
problem in Equation 18 can be rewritten as
$\displaystyle\max_{\tilde{\mathbf{w}}\geq
0}\tilde{\mathbf{w}}^{\intercal}\mathbf{z}\text{ such that
}\tilde{\mathbf{w}}^{\intercal}\tilde{\mathbf{w}}=1\,.$ (19)
The unique optimal solution to Equation 19 is
$\tilde{\mathbf{w}}=\mathbf{z}/\|\mathbf{z}\|_{2}$. Thus, the unique optimal
weight vector that solves Equation 18 is
$\mathbf{w}=(\Sigma^{-1}\mathbf{VF})/\|\Sigma^{-1}\mathbf{VF}\|_{2}$.
Therefore, the solution set to the original problem in Equation 17 consists of
all vectors $\\{\alpha\Sigma^{-1}\mathbf{VF}\mid\alpha>0\\}$. Thus, the
optimal weight vector that is unit-norm with respect to the $p$-norm
$\|\cdot\|_{p}$ is
$\mathbf{w}=(\Sigma^{-1}\mathbf{VF})/\|\Sigma^{-1}\mathbf{VF}\|_{p}$. ∎
### Proof of 5.1
###### Proof.
Define the vector
$\mathbf{z}=\Sigma^{-1}\mathbf{VF}=(\Sigma_{11}^{-1}\mathbf{VF}_{1},\dots,\Sigma_{kk}^{-1}\mathbf{VF}_{k})$.
By Theorem 4.1, the user-optimal weight vector is
$\mathbf{w}^{*}=\mathbf{z}/\|\mathbf{z}\|_{p}$. To prove that the optimal
weight vector $\mathbf{w}^{*}$ is increasing in the value-faithfulness
$\mathbf{VF}_{i}$ and decreasing in the variance $\Sigma_{i}$ of a behavior
$j\in[k]$, it suffices to prove that for $\mathbf{z}\geq 0$, the optimal
weight vector $\mathbf{w}^{*}$ is increasing in $\mathbf{z}_{j}$. To do so, we
can show that the partial derivative is non-negative:
$\displaystyle\frac{\partial}{\partial\mathbf{z}_{j}}\frac{\mathbf{z}_{j}}{\|\mathbf{z}\|_{p}}$
$\displaystyle=\frac{1}{\|\mathbf{z}\|_{p}^{2}}\left(\|\mathbf{z}\|_{p}-\mathbf{z}_{j}\frac{\partial}{\partial\mathbf{z}_{j}}\|\mathbf{z}\|_{p}\right)$
(20)
$\displaystyle=\frac{1}{\|\mathbf{z}\|_{p}^{2}}\left(\|\mathbf{z}\|_{p}-\mathbf{z}_{j}\left(\frac{\mathbf{z}_{j}}{\|\mathbf{z}\|_{p}}\right)^{p-1}\right)$
(21)
$\displaystyle=\frac{\|\mathbf{z}\|_{p}^{p}-\mathbf{z}_{j}^{p}}{\|\mathbf{z}\|_{p}^{p+1}}\geq
0\,,$ (22)
∎
where Equation 21 uses Lemma A.1, the partial derivative of the $p$-norm, and
the fact that $\mathbf{z}\geq 0$.
## Appendix B Proofs for Section 5
### Proof of 5.1
###### Proof.
The utility for producer $i$ is equal to
$\displaystyle\mathcal{U}_{\text{prod}}^{i}(\mathbf{e}(i),\mathbf{e}(-i);\mathbf{w})=\begin{cases}F_{\epsilon}(\mathbb{E}[\mathbf{w}^{\intercal}\mathbf{y}(1)-\mathbf{w}^{\intercal}\mathbf{y}(-1)])-c(\mathbf{e}(1))&i=1\\\
1-F_{\epsilon}(\mathbb{E}[\mathbf{w}^{\intercal}\mathbf{y}(1)-\mathbf{w}^{\intercal}\mathbf{y}(-1)])-c(\mathbf{e}(-1))&i=-1\end{cases}\,,$
(23)
where $F_{\epsilon}$ is the CDF of the difference in noise terms
$\epsilon(\mathbf{w})=\mathbf{w}^{\intercal}\bm{\xi}(-1)-\mathbf{w}^{\intercal}\bm{\xi}(1)\sim\mathcal{N}(0,2\mathbf{w}^{\intercal}\Sigma\mathbf{w})$
and the mean difference in producer scores is equal to
$\mathbb{E}[\mathbf{w}^{\intercal}\mathbf{y}(1)-\mathbf{w}^{\intercal}\mathbf{y}(-1)]=\mathbf{w}^{\intercal}\mathbf{VF}+\mathbf{w}^{\intercal}\mathbf{e}(1)-\mathbf{w}^{\intercal}\mathbf{e}(-1)$.
At the equilibrium, the first-order conditions for both producers must be
satisfied:
$\displaystyle\nabla_{\mathbf{e}(1)}\mathcal{U}_{\text{prod}}^{1}(\mathbf{e}(1),\mathbf{e}(-1);\mathbf{w})=f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF}+\mathbf{w}^{\intercal}\mathbf{e}(1)-\mathbf{w}^{\intercal}\mathbf{e}(-1))\mathbf{w}-A\mathbf{e}(1)=0\,,$
(24)
$\displaystyle\nabla_{\mathbf{e}(-1)}\mathcal{U}_{\text{prod}}^{-1}(\mathbf{e}(-1),\mathbf{e}(1);\mathbf{w})=f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF}+\mathbf{w}^{\intercal}\mathbf{e}(1)-\mathbf{w}^{\intercal}\mathbf{e}(-1))\mathbf{w}-A\mathbf{e}(-1)=0\,,$
(25)
where $f_{\epsilon}$ is the density of $\epsilon(\mathbf{w})$. Subtracting
Equations 24 and 25 shows that the equilibrium strategy is symmetric, i.e,
$\mathbf{e}(1)=\mathbf{e}(-1)$ at equilibrium. Substituting
$\mathbf{e}(1)=\mathbf{e}(-1)$ into either equation yields
$\mathbf{e}(i)=f_{\mathbf{w}}(\mathbf{w}^{\intercal}\mathbf{VF})A^{-1}\mathbf{w}$
for $i\in\\{-1,+1\\}$.
To prove sufficiency, we need to consider the second-order conditions and show
that the Hessian of each producer’s utility is negative-definite at the
equilibrium efforts. The Hessian is given by
$\displaystyle\nabla^{2}_{\mathbf{e}(i)}\mathcal{U}_{\text{prod}}^{i}(\mathbf{e}(i),\mathbf{e}(-i);\mathbf{w})$
$\displaystyle=\nabla_{\mathbf{e}(i)}f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF}+\mathbf{w}^{\intercal}\mathbf{e}(i)-\mathbf{w}^{\intercal}\mathbf{e}(-i))\mathbf{w}-A\mathbf{e}(i)$
(26)
$\displaystyle=f^{\prime}_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF}+\mathbf{w}^{\intercal}\mathbf{e}(i)-\mathbf{w}^{\intercal}\mathbf{e}(-i))\mathbf{w}\mathbf{w}^{\intercal}-A\,.$
(27)
When both producers exert equal effort, the Hessian simplifies to
$f^{\prime}_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})\mathbf{w}\mathbf{w}^{\intercal}-A$.
By assumption $\mathbf{VF}>0$, $\mathbf{w}\geq 0$, and $||\mathbf{w}||=1$,
which ensures that the dot product $\mathbf{w}^{\intercal}\mathbf{VF}$ is
positive. When $\mathbf{w}^{\intercal}\mathbf{VF}>0$, the derivative of the
zero-mean Gaussian density
$f^{\prime}_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})$ is negative, making
the Hessian negative-definite. Consequently,
$\mathbf{e}(1)=\mathbf{e}(-1)=f_{\mathbf{w}}(\mathbf{w}^{\intercal}\mathbf{VF})A^{-1}\mathbf{w}$
represents the unique equilibrium. ∎
### Proof of Theorem 5.1
###### Proof.
By 5.1, the unique and symmetric equilibrium strategy for producers is
$\mathbf{e}_{\mathbf{w}}^{*}(1)=\mathbf{e}_{\mathbf{w}}^{*}(-1)=f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})A^{-1}\mathbf{w}$
where $f_{\epsilon}$ is the density of the difference in noise terms
$\epsilon(\mathbf{w})=\mathbf{w}^{\intercal}\bm{\xi}(-1)-\mathbf{w}^{\intercal}\bm{\xi}(1)\sim\mathcal{N}(0,2\mathbf{w}^{\intercal}\mathbf{w})$.
Thus, producer welfare at equilibrium is equal to
$\displaystyle\mathcal{W}_{\text{prod}}(\mathbf{e}_{\mathbf{w}}^{*}(1),\mathbf{e}_{\mathbf{w}}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})$
$\displaystyle=\frac{1}{2}-\frac{1}{2}c(\mathbf{e}^{*}(1))-\frac{1}{2}c(\mathbf{e}^{*}(-1))$
(28)
$\displaystyle=\frac{1}{2}-\frac{f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})^{2}}{2}\mathbf{w}^{\intercal}A^{-1}\mathbf{w}$
(29)
$\displaystyle=\frac{1}{2}-\frac{1}{4\sqrt{\pi}\mathbf{w}^{\intercal}\Sigma\mathbf{w}}\exp\left(-\frac{(\mathbf{w}^{\intercal}\mathbf{VF})^{2}}{\mathbf{w}^{\intercal}\Sigma\mathbf{w}}\right)\mathbf{w}^{\intercal}A^{-1}\mathbf{w}\,.$
(30)
From the above expression, it is clear that for any fixed weight vector
$\mathbf{w}$, producer welfare
$\mathcal{W}_{\text{prod}}(\mathbf{e}^{*}(1),\mathbf{e}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})$
is monotonically increasing as the strategy-robustness $A_{jj}$ or
value-faithfulness $\mathbf{VF}_{j}$ of any behavior $j\in[k]$ increases.
Thus, producer welfare under the optimal weight vector
$\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)$ must also be
monotonically increasing in strategy-robustness and value-faithfulness.
However, the optimal producer welfare
$\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)$ is not necessarily
monotonic in the behavior’s variance $\Sigma_{jj}$. The variance only affects
producer welfare by changing
$f_{\epsilon}(\mathbf{w}^{\intercal}\mathbf{VF})$, the density of the
difference in noise terms
$\epsilon(\mathbf{w})\sim\mathcal{N}(0,2\mathbf{w}^{\intercal}\mathbf{w})$.
Let $\sigma^{2}=2\mathbf{w}^{\intercal}\Sigma\mathbf{w}$ be the variance of
$\epsilon(\mathbf{w})$. For any fixed weight vector
$\mathbf{w}\in\mathbb{R}_{\geq 0}^{k}$, as $\sigma^{2}$ approaches $0^{+}$ or
$+\infty$, producer welfare approaches its maximum possible value:
$\displaystyle\lim_{\sigma^{2}\rightarrow
0^{+}}\mathcal{W}_{\text{prod}}(\mathbf{e}_{\mathbf{w}}^{*}(1),\mathbf{e}_{\mathbf{w}}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})=1/2\,,$
(31)
$\displaystyle\lim_{\sigma^{2}\rightarrow\infty}\mathcal{W}_{\text{prod}}(\mathbf{e}_{\mathbf{w}}^{*}(1),\mathbf{e}_{\mathbf{w}}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})=1/2\,.$
(32)
Thus, the optimal producer welfare also approaches the maximum possible value
as $\sigma^{2}$ approaches either $0^{+}$ or $+\infty$:
$\displaystyle 1/2\geq\lim_{\sigma^{2}\rightarrow
0^{+}}\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)\geq\lim_{\sigma^{2}\rightarrow
0^{+}}\mathcal{W}_{\text{prod}}(\mathbf{e}_{\mathbf{w}}^{*}(1),\mathbf{e}_{\mathbf{w}}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})=1/2\,,$
(33) $\displaystyle
1/2\geq\lim_{\sigma^{2}\rightarrow\infty}\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)\geq\lim_{\sigma^{2}\rightarrow\infty}\mathcal{W}_{\text{prod}}(\mathbf{e}_{\mathbf{w}}^{*}(1),\mathbf{e}_{\mathbf{w}}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})=1/2\,,$
(34)
Therefore, the only way for optimal producer welfare to be monotonic in a
behavior’s variance is if it is constant over $\sigma^{2}\in(0,\infty)$, i.e.,
is always equal to $1/2$, for any given value-faithfulness vector
$\mathbf{VF}$ or cost matrix $A$. This is clearly untrue in general, and thus,
optimal producer welfare is not necessarily monotonic in a behavior’s variance
$\Sigma_{jj}$.
Finally, for any fixed weight vector $\mathbf{w}$, we have that producer
welfare approaches the maximal possible value as any of the three aspects of
behavior go to $+\infty$:
$\displaystyle\lim_{z\rightarrow\infty}\mathcal{W}_{\text{prod}}(\mathbf{e}_{\mathbf{w}}^{*}(1),\mathbf{e}_{\mathbf{w}}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})=1/2$
(35)
for any $z\in\\{A_{jj},\mathbf{VF}_{j},\Sigma_{jj}\mid j\in[k]\\}$.
Furthermore,
$\displaystyle
1/2\geq\lim_{z\rightarrow\infty}\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)\geq\lim_{z\rightarrow\infty}\mathcal{W}_{\text{prod}}(\mathbf{e}_{\mathbf{w}}^{*}(1),\mathbf{e}_{\mathbf{w}}^{*}(-1);\mathbf{VF},\Sigma,A,\mathbf{w})=1/2\,,$
(36)
and thus, the optimal producer welfare also approaches the maximum possible
value:
$\lim_{z\rightarrow\infty}\mathcal{W}_{\text{prod}}^{*}(\mathbf{VF},\Sigma,A)=1/2$.
∎
## Appendix C Simulations
### Parameters for Figure 1
The parameters used for the simulation are
$\displaystyle\text{value: }v(1)=1,v(-1)=0\,,$ (37)
$\displaystyle\text{behavioral biases:
}\mathbf{b}(-1)=\mathbf{0},\mathbf{b}(1)_{1}=0.75\,,$ (38)
$\displaystyle\text{variance: }\Sigma_{11}=1\,,$ (39) $\displaystyle\text{cost
of manipulation: }A=I\,.$ (40)
The bias $\mathbf{b}(1)_{2}$ and variance $\Sigma_{22}$ of the second behavior
is adjusted so that the second behavior has the relative value-faithfulness
and variance given by the $x$ and $y$-axes.
### Parameters for Figure 2
The default parameters used for each of the subplots are
$\displaystyle\text{value: }v(1)=1,v(-1)=0\,,$ (41)
$\displaystyle\text{behavioral biases:
}\mathbf{b}(-1)=\mathbf{0},\mathbf{b}(1)_{1}=\mathbf{b}(1)_{2}=0.75\,,$ (42)
$\displaystyle\text{variance: }\Sigma_{11}=1,\Sigma_{22}=3\,,$ (43)
$\displaystyle\text{cost of manipulation: }A=I\,.$ (44)
The three subplots are generated by adjusting the bias $\mathbf{b}(1)_{2}$,
variance $\Sigma_{22}$, or cost $A_{22}$ of the second behavior so that it has
the relative value-faithfulness, variance, or strategy-robustness given by the
$x$-axis.
Figure 3: How the optimal weight vector changes as value-faithfulness and
variance change in the homogeneous setting. The default parameters for the
simulations are $\mu_{\texttt{click}}=\mu_{\texttt{rec}}=1$ and
$\Sigma_{11}=\Sigma_{22}=2$. The left figure is generated by plotting the
optimal weight vector as $\mu_{\texttt{rec}}$ increases (and consequently,
when value-faithfulness increases), and the right figure is generated by
increasing $\Sigma_{22}$.
Figure 4: A comparison of the empirical and theoretical optimal weight vector
in the homogeneous and heterogeneous setting. In both settings, the variance
on click is $\Sigma_{11}=0.1$ and the variance on recommend is
$\Sigma_{22}=2$. In the homogeneous setting, $\mu_{\texttt{click}}=1$ and
$\mu_{\texttt{rec}}=3$. In the heterogeneous setting,
$\alpha_{\texttt{click}}=0$, $\beta_{\texttt{click}}=2$,
$\alpha_{\texttt{rec}}=2$, and $\beta_{\texttt{click}}=4$. Thus, the mean
predictions across items are the same in both the homogeneous and
heterogeneous setting: $\mu_{\texttt{click}}=(\mu_{\texttt{i, click}})/n_{+}$
and $\mu_{\texttt{rec}}=(\mu_{\texttt{i, rec}})/n_{+}$. However, while the
theory optimal weight vector and empirical optimal weight vector closely match
in the homogeneous setting, they have a distinct gap in the heterogeneous
setting.
### C.1 Additional simulations with $n>2$ items
In simulation, we consider the non-strategic setting, i.e., when
$\mathbf{e}\triangleq\mathbf{0}$, and extend it to a setting with $n$
producers or items. Here, the user values $n_{+}$ items and doesn’t value
$n_{-}$ items. All valued items are assumed to have the same positive value
$v_{+}$ while unvalued items have a value of zero. In the two-item setting, we
defined a user’s utility as the probability that the higher-valued item is
ranked first (Equation 6). Then, a natural metric to optimize for in the $n$
item setting is the probability that a randomly-picked valued item is ranked
above a randomly-picked unvalued item, i.e., the AUC.
The user can interact with the items using two different behaviors: (1)
_click_ and (2) _recommend_. In our simulations, we assume that recommend is
more value-faithful than click but also higher variance. To extend value-
faithfulness to the setting with $n$ items, we define the mean value-
faithfulness $\overline{\mathbf{VF}}$ as the mean difference in behavior
scores between valued items and unvalued items:
$\displaystyle\overline{\mathbf{VF}}=\frac{1}{n_{+}}\sum_{i:v(i)>0}\mathbf{y}(i)-\frac{1}{n_{-}}\sum_{i:v(i)=0}\mathbf{y}(i)\,.$
(45)
We investigate two settings: one in which, given their value, each item has
the same mean behavior predictions, and the other, in which items are
heterogeneous. In both, we compare (a) the weight vector that maximizes the
empirical AUC and (b) the user-optimal weight vector given by Theorem 4.1 (in
which we substitute $\overline{\mathbf{VF}}$ for $\mathbf{VF}$).
In the homogeneous setting, the predictions are simulated as
$\displaystyle\mathbf{y}(i)\sim\begin{cases}\mathcal{N}(\mathbf{0},\Sigma)&v(i)=0\\\
\mathcal{N}\left(\begin{bmatrix}\mu_{\texttt{click}}\\\
\mu_{\texttt{rec}}\end{bmatrix},\Sigma\right)&v(i)=v_{+}\end{cases}\,,$ (46)
i.e., all unvalued items have the same mean prediction of $0$ for both clicks
and recommend while all valued items have the same mean prediction
$\mu_{\texttt{click}}>0$ and $\mu_{\texttt{rec}}>0$ for clicks and recommend.
In the heterogeneous setting, the behavior predictions for unvalued items are
simulated the same way, i.e. $\mathbf{y}(i)\sim(\mathbf{0},\Sigma)$ for $i$
such that $v(i)=0$. But for valued items, the mean of the behavior predictions
is heterogeneous, i.e.,
$\displaystyle\mu_{i,\texttt{click}}$
$\displaystyle\sim\mathrm{Unif}[\alpha_{\texttt{click}},\beta_{\texttt{click}}]\,,$
(47) $\displaystyle\mu_{i,\texttt{rec}}$
$\displaystyle\sim\mathrm{Unif}[\alpha_{\texttt{rec}},\beta_{\texttt{rec}}]\,,$
(48) $\displaystyle\mathbf{y}(i)$
$\displaystyle\sim\mathcal{N}\left(\begin{bmatrix}\mu_{i,\texttt{click}}\\\
\mu_{i,\texttt{rec}}\end{bmatrix},\Sigma\right)\,.$ (49)
Figure 3 shows that in the homogeneous setting, the theory-optimal weight
vector and the empirical-optimal weight vector match closely even as value-
faithfulness and variance change. However, Figure 4 demonstrates that in the
heterogeneous setting, the theory-optimal weight vector and the empirical-
optimal weight vector may not necessarily coincide.
|
# Nodal d-wave pairing from spin fluctuations in a thermally disordered anti-
ferromagnet
Nick Bultinck Department of Physics, Ghent University, Krijgslaan 281, 9000
Gent, Belgium
###### Abstract
We consider electron pairing in a two-dimensional thermally disordered
itinerant anti-ferromagnet. It is shown that transverse spin fluctuations in
such a state can give rise to superconductivity with a sizeable critical
temperature $T_{c}$. Below $T_{c}$ there is quasi-long-range spin-singlet and
$d_{x^{2}-y^{2}}$ superconducting order, together with fluctuating triplet
order at momentum $(\pi,\pi)$. The singlet pairs we find are tightly bound
together, and the pair wavefunction has a purely inter-sublattice structure
due to the U$(1)$ spin rotation symmetry of the anti-ferromagnet.
_Introduction –_ It is well-known that anti-ferromagnetic (AFM) spin
fluctuations generate an attractive interaction between electrons which favors
pairing in the spin-singlet $d_{x^{2}-y^{2}}$-channel [1, 2, 3, 4]. In the
original approach for connecting AFM fluctuations and $d$-wave
superconductivity [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], the normal state is a
conventional Fermi liquid. In the underdoped cuprates, however, the normal
state does not seem to fit the standard Fermi liquid mould. For example, one
of the intriguing properties of the underdoped cuprate normal state is the
pseudogap phenomenon, manifested in the form of a suppressed density of states
at the Fermi energy and Fermi arcs in photoemission spectra. Motivated by
these experimental observations, we consider a magnetic pseudogap state in the
form of a thermally disordered AFM –i.e. an AFM above its critical temperature
(which is $T=0$), but well below the mean-field transition temperature– as the
parent state for superconductivity. This magnetic pseudogap state has a
suppressed density of states at the Fermi energy due to the fluctuating anti-
ferromagnetism. It also produces a spectral weight consisting of $4$ small
Fermi pockets centered at $(\pm\pi/2,\pm\pi/2)$ with faint backsides,
resembling the Fermi arcs seen in photoemission. Inspired by the well-
established connection between AFM spin fluctuations and $d$-wave pairing in
Fermi liquids, our goal here is to investigate the role of spin fluctuations
for mediating superconductivity in such a thermally disordered AFM. In
particular, we focus on spin fluctuations which are the finite-temperature
remnants of the AFM Goldstone modes.
To set up our calculations we start from the effective theory for a thermally
disordered AFM presented in Ref. [11]. The effective action is derived from a
simple mean-field and random phase approximation (RPA). However, one of its
main advantages is that the effective theory contains scattering vertices
between the electrons and Goldstone modes which are not entirely
phenomenological. Instead these vertices are calculated starting from the
microscopic Hamiltonian. And as the tendency of an interaction to promote
$d$-wave pairing crucially relies on its structure in momentum space, having
an explicit expression for the scattering vertices is key for our analysis.
Within the effective theory we find that spin fluctuations indeed give rise to
superconductivity with a critical temperature which is high compared to
conventional phonon-driven superconductors. Crucially, due to the symmetry
properties of the AFM, the gap function we find cannot be a featureless
$s$-wave state but instead needs to have a non-trivial momentum dependence. A
further important property of the effective theory is that the fermion fields
are defined in a `rotating frame' [12, 13, 14, 15], in order to ensure that
long-wavelength Goldstone modes decouple from the electrons [11, 16]. It is
also this rotating frame which ultimately ensures that the superconducting
order parameter we find is spin-singlet. At the end of the manuscript we
discuss a few predictions of our theory which can be used to test whether
superconductivity in a particular model or material indeed originates from
spin fluctuations in a thermally disordered AFM.
Our approach is closely related to both the spin-bag [17, 18] and spin-fermion
[7, 8, 16, 19, 20, 21, 22, 23, 24] models studied in the early days of
high-$T_{c}$ superconductivity. However, it also differs from these pre-
existing theories in some crucial ways. In particular, unlike in the spin-
fermion model, the bare coupling between electrons and spin fluctuations in
our effective theory cannot be described by a purely local and instantaneous
interaction. Our approach also differs from the original spin-bag theory
because we focus on transverse spin fluctuations, previously studied in [25,
26, 27], and use a rotating frame which removes strong inter-band scattering
terms 111See Refs. [52, 53, 54] for a complementary perspective. In these
works, the strong inter-band scattering terms from transverse spin
fluctuations are not eliminated via a rotating frame, but their effect on the
electron Green’s function is explicitly taken into account via an infinite sum
of diagrams in an eikonal approximation. and leads to a spin-singlet
superconductor with nodes on the Fermi surface. Moreover, we consider spin
fluctuations with a non-zero thermal mass to generate the effective
attraction, and use the complete dynamical interaction to calculate $T_{c}$.
Figure 1: (a) Mean-field valence band energy of the AFM insulator. The mean-
field hybridization strength between spin up and down electrons is $\sim
1.93$. The Fermi surface obtained by $\sim 12\%$ hole doping is shown in red.
(b) Goldstone mode dispersion relation
$\tilde{\omega}_{\tilde{{\mathbf{q}}}}$. (c)-(d) Averaged absolute values of
the vertices in Eq. (6) for electrons in the valence band ($\alpha=\beta=0)$
as a function of the pseudo-momentum transfer $\tilde{{\mathbf{q}}}$, where
the averaging is over the incoming pseudo-momentum. Results are obtained on a
$34\times 34$ pseudo-momentum grid.
_Model and results –_ As a microscopic model we consider the Hubbard model on
the square lattice:
$H=-t\sum_{\langle
ij\rangle}\sum_{\sigma}c^{\dagger}_{i,\sigma}c_{j,\sigma}-t^{\prime}\sum_{\langle\langle
ij\rangle\rangle}c^{\dagger}_{i,\sigma}c_{j,\sigma}+h.c.+U\sum_{i}n_{i,\uparrow}n_{i,\downarrow}\,,$
(1)
where the first (second) sum is over nearest (next nearest) neighbors. In the
third term $n_{i,\sigma}$ is the number of electrons with spin $\sigma$ on
site $i$. We take $t\equiv 1$, $t^{\prime}=-0.35$, and $U=5$. At half filling,
the system is an insulating antiferromagnet. We assume that the AFM moments
are ordered in the $x$-direction, and work with following imaginary-time
effective action to describe the interacting electrons and AFM Goldstone modes
at mean-field+RPA level:
$S_{\rm eff}=S_{el}+S_{V}+S_{B}+S_{el-B}$ (2)
The first term is the kinetic energy term for the fermions
$S_{el}=\int\mathrm{d}\tau\sum_{\tilde{{\mathbf{k}}}}\sum_{\alpha}\bar{\psi}_{\tilde{{\mathbf{k}}},\alpha}(\partial_{\tau}+E_{\tilde{{\mathbf{k}}},\alpha}-\mu)\psi_{\tilde{{\mathbf{k}}},\alpha}\,,$
(3)
where $\alpha=0,1$ labels the mean-field bands of the AFM insulator – $0$
($1$) is the valence (conduction) band. The mean-field valence band energy
$E_{\tilde{{\mathbf{k}}},0}$ is shown in Fig. 1 (a). The conduction band is
separated by a gap of order $U$. The AFM breaks translation over one site, but
is invariant under the action of
$T^{\prime}_{{\mathbf{r}}}=e^{i{\mathbf{r}}\cdot{\mathbf{Q}}\,\sigma^{z}/2}T_{{\mathbf{r}}}$,
where ${\mathbf{Q}}=(\pi,\pi)$, $\sigma^{i}$ are the Pauli spin matrices, and
$T_{\mathbf{r}}$ implements a translation by lattice vector ${\mathbf{r}}$.
The pseudo-momenta $\tilde{{\mathbf{k}}}$ are the conserved quantum numbers
(modulo reciprocal lattice vectors) associated with the
$T^{\prime}_{\mathbf{r}}$ symmetry. The relation between states with pseudo-
momentum $\tilde{{\mathbf{k}}}$ and crystal momentum ${\mathbf{k}}$ is given
by:
$f^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}:=c^{\dagger}_{\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}\hskip
2.84544pt\,,\hskip
14.22636ptf^{\dagger}_{\tilde{{\mathbf{k}}},\downarrow}:=c^{\dagger}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}\,,$
(4)
where $f^{\dagger}_{\tilde{{\mathbf{k}}},\sigma}$ creates electrons with
pseudo-momentum $\tilde{{\mathbf{k}}}$.
Invariance under spin rotations around the $x$-axis implies that the effective
theory is symmetric under
$f^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}\leftrightarrow
f^{\dagger}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}},\downarrow}$. This symmetry
acts on the fermion fields
$\bar{\psi}_{\tilde{{\mathbf{k}}},\alpha}=\sum_{\sigma=\uparrow,\downarrow}u(\tilde{{\mathbf{k}}})_{\alpha,\sigma}\bar{\psi}_{\tilde{{\mathbf{k}}},\sigma}$,
where $u(\tilde{{\mathbf{k}}})_{\alpha,\sigma}$ are the coefficients of the
mean-field single-particle states [29], as
$\bar{\psi}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}},\alpha}\leftrightarrow\pm\bar{\psi}_{\tilde{{\mathbf{k}}},\alpha}$.
The signs depend on a choice of real gauge for
$u(\tilde{{\mathbf{k}}})_{\alpha,\sigma}$. Here we use a gauge where all signs
are positive.
The second term $S_{V}$ in Eq. (2) contains the instantaneous two-body
interaction for the electrons, which consists of the microscopic Hubbard
interaction, and an interaction which is generated by integrating out the
field conjugate to the Goldstone field $\phi({\mathbf{r}})$ [11]. The
instantaneous interaction is written out in detail in the supplementary
material [29].
The dynamics of the Goldstone field is described by the third term in the
effective action:
$S_{B}=\frac{1}{2}\int\mathrm{d}\tau\,\sum_{\tilde{{\mathbf{q}}}}\left(-\phi_{-\tilde{{\mathbf{q}}}}\partial^{2}_{\tau}\phi_{\tilde{{\mathbf{q}}}}+\omega_{\tilde{{\mathbf{q}}}}^{2}\phi_{\tilde{{\mathbf{q}}}}\phi_{-\tilde{{\mathbf{q}}}}\right)\,,$
(5)
where
$\omega^{2}_{\tilde{{\mathbf{q}}}}=\tilde{\omega}_{\tilde{{\mathbf{q}}}}^{2}+m^{2}$,
with $\tilde{\omega}^{2}_{\tilde{{\mathbf{q}}}}\sim
c^{2}\tilde{{\mathbf{q}}}^{2}$ near $\tilde{{\mathbf{q}}}=0$, and
$\tilde{\omega}^{2}_{\tilde{{\mathbf{q}}}}\sim
c^{2}({\mathbf{Q}}-\tilde{{\mathbf{q}}})^{2}$ near
$\tilde{{\mathbf{q}}}={\mathbf{Q}}$ ($c\approx 1)$. The complete Goldstone
dispersion $\tilde{\omega}_{\tilde{{\mathbf{q}}}}$ is shown in Fig. 1 (b). $m$
is a thermal mass for the Goldstone modes which takes into account that 2D AFM
are disordered at non-zero temperatures. Below we will treat $m$ as a
phenomenological parameter. The final term in Eq. (2) contains the electron-
boson interaction
$\begin{split}S_{el-B}=&\int\mathrm{d}\tau\,\frac{1}{\sqrt{N}}\sum_{\tilde{{\mathbf{q}}},\tilde{{\mathbf{k}}}}\bar{\psi}_{\tilde{{\mathbf{k}}},\alpha}\psi_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}},\beta}\times\\\
&\left(g_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})\,\phi_{-\tilde{{\mathbf{q}}}}+f_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})\,i\partial_{\tau}\phi_{-\tilde{{\mathbf{q}}}}\right)\,,\end{split}$
(6)
where $N$ is the number of lattice sites. Following the procedure explained in
Ref. [11], the vertex functions $g$ and $f$ are constructed from solutions to
the Bethe-Salpeter equation for the Hubbard model (which also determines the
Goldstone mode spectrum $\tilde{\omega}_{\tilde{{\mathbf{q}}}}$). In Figs. 1
(c-d) these vertices are shown for electron scattering within the valence band
($\alpha=\beta=0$). In particular, we show the absolute value of the vertices,
averaged over the incoming pseudo-momentum, as a function of the pseudo-
momentum transfer $\tilde{{\mathbf{q}}}$.
A final crucial property of the effective action is that the fermion fields
are defined in a `rotating frame', which means that the bare/microscopic
fermions $\bar{\psi}^{b}_{{\mathbf{r}},\sigma}$ are related to the fields in
Eq. (3) by
$\bar{\psi}_{{\mathbf{r}},\sigma}=\sum_{\sigma^{\prime}}R_{\sigma\sigma^{\prime}}(\tau,{\mathbf{r}})\bar{\psi}^{b}_{{\mathbf{r}},\sigma^{\prime}}$.
The $2\times 2$ matrix field $R(\tau,{\mathbf{r}})$ is defined as
$R(\tau,{\mathbf{r}})=\exp\left(-i\left[\phi_{z}(\tau,{\mathbf{r}})\sigma^{z}+\phi_{y}(\tau,{\mathbf{r}})\sigma^{y}\right]\right)\,,$
(7)
where $\phi_{z}(\tau,{\mathbf{r}})$ and $\phi_{y}(\tau,{\mathbf{r}})$ contain
momenta $\tilde{{\mathbf{k}}}$ in the first magnetic Brillouin zone, i.e. the
Brillouin zone with reciprocal vectors $(\pi,\pm\pi)$, and are related to the
Goldstone field in Eq. (5) by
$\phi=\phi_{z}+(-1)^{{\mathbf{Q}}\cdot{\mathbf{r}}}\phi_{y}$. The reason for
defining the fermion fields in a rotating frame is that these fermions
decouple from the low-frequency and long-wavelength Goldstone modes [11, 16],
i.e. in this basis we have $\lim_{\tilde{{\mathbf{q}}}\rightarrow
0,{\mathbf{Q}}}g_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})=0$.
We now proceed by changing the chemical potential $\mu$ to hole-dope the AFM.
Upon changing the electron density one should actually redetermine the optimal
mean-field state and the RPA collective modes that go into the construction of
our effective action. However, for metallic systems calculating the scattering
vertices $g$ and $f$ is challenging, as the Goldstone modes are partly hidden
inside the particle-hole continuum. We will therefore keep the effective
action as is, and simply change $\mu$. For our purposes we expect this to be a
reasonable approximation at small doping – in particular, we expect that the
gross features in the momentum dependence of the vertex functions $g$ and $f$
will not change.
To study electron pairing we ignore the empty conduction band, and work
exclusively with the valence band which crosses the Fermi energy. We emphasize
that this step can only be justified by using the rotating frame, which
eliminates inter-band scattering terms of order $U$ [11, 30]. We work at a
hole doping of $\sim 12\%$. The resulting Fermi surface is shown in Fig. 1
(a). To study superconductivity we sum the usual Cooper ladder diagrams, where
each rung consists of both the instantaneous interaction contained in $S_{V}$,
and the interaction generated by tree-level Goldstone mode exchange. The
Cooper ladder sum diverges when following equation has a solution with
$\lambda=-1$:
$\lambda\hat{\Delta}(i\omega^{\prime},\tilde{{\mathbf{k}}}^{\prime})=T\sum_{i\omega}\frac{1}{N}\sum_{\tilde{{\mathbf{k}}}}\frac{V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})}{\omega^{2}+(E_{\tilde{{\mathbf{k}}}}-\mu)^{2}}\hat{\Delta}(i\omega,\tilde{{\mathbf{k}}})\,,$
(8)
where
$V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})$
scatters a pair of electrons with pseudo-momenta
$(\tilde{{\mathbf{k}}},-\tilde{{\mathbf{k}}})$ and frequencies
$(i\omega,-i\omega)$ to a pair of electrons with
$(\tilde{{\mathbf{k}}}^{\prime},-\tilde{{\mathbf{k}}}^{\prime})$ and
$(i\omega^{\prime},-i\omega^{\prime})$, due to both the instantaneous
interaction and Goldstone mode exchange. The explicit expression for $V$ is
given in the supplementary material [29], but let us note here that the
frequency dependence of $V$ comes from both the Goldstone mode propagator and
the direct coupling of the electrons to $\partial_{\tau}\phi$ [see Eq. (6)].
In Eq. (8) we have also introduced the notation
$E_{\tilde{{\mathbf{k}}}}:=E_{\tilde{{\mathbf{k}}},0}$, and used that
$E_{\tilde{{\mathbf{k}}}}=E_{-\tilde{{\mathbf{k}}}}$. To obtain $\lambda$, we
use following ansatz
$\hat{\Delta}(i\omega,\tilde{{\mathbf{k}}})=\frac{\Delta(\tilde{{\mathbf{k}}})}{\omega^{2}+\Omega^{2}}\,,$
(9)
where $\Delta(-\tilde{{\mathbf{k}}})=-\Delta(\tilde{{\mathbf{k}}})$ as
required by fermion antisymmetry, and $\Omega$ is a variational parameter
corresponding to an inverse retardation timescale. In Fig. 2 (a) we show
$\lambda+1$ obtained from a variational calculation with this ansatz as a
function of $T$ and $\Omega$, where we have used a thermal Goldstone mass of
$m=0.01$ (for details see [29]). We see that $\lambda$ reaches $-1$ at a
highest temperature of $\sim 0.028$ when $\Omega\approx 0.5$, which is roughly
half the Goldstone bandwidth. Due to the variational nature of our calculation
we have thus obtained a lower bound for $T_{c}$ (within the ladder sum
approximation) which, when using a representative value of $t=0.3$ eV,
corresponds to $\sim 100$ K. In Fig. 2 (b) we show the corresponding gap
function $\Delta(\tilde{{\mathbf{k}}})$. It has the important property
$\Delta(\tilde{{\mathbf{k}}}+{\mathbf{Q}})=-\Delta(\tilde{{\mathbf{k}}})$
(recall that the shift symmetry over momentum ${\mathbf{Q}}$ is a result of
the U$(1)$ spin rotation symmetry). We have also calculated $\lambda$ and
$\Delta(\tilde{{\mathbf{k}}})$ using a thermal mass $m=0.1$ and found that in
this case $T_{c}\gtrsim 0.023$, and the gap function remains essentially
unchanged [29].
Figure 2: (a) $\lambda+1$, with $\lambda$ defined in Eq. (8), as obtained from
a variational calculation with the ansatz in Eq. (9). (b) Corresponding gap
function $\Delta(\tilde{{\mathbf{k}}})$. Results are obtained on a $34\times
34$ pseudo-momentum grid using $m=0.01$.
In the spin basis, the gap function is given by
$\Delta_{\sigma\sigma^{\prime}}(\tilde{{\mathbf{k}}})=u_{0,\sigma}(\tilde{{\mathbf{k}}})u_{0,\sigma^{\prime}}(-\tilde{{\mathbf{k}}})\Delta(\tilde{{\mathbf{k}}})$.
As the AFM breaks spin rotation symmetry, the gap is generically an admixture
of a singlet and a triplet component. To see this, we rewrite the gap function
in the crystal momentum basis, where it takes the form
$\Delta_{\sigma\sigma^{\prime}}({\mathbf{k}})=\Delta^{S}({\mathbf{k}})\sigma^{y}_{\sigma\sigma^{\prime}}+\Delta^{T}_{\mathbf{Q}}({\mathbf{k}})\sigma^{z}_{\sigma\sigma^{\prime}}$
[29]. $\Delta^{S}({\mathbf{k}})$ is a zero-momentum singlet gap, and
$\Delta^{T}_{\mathbf{Q}}({\mathbf{k}})$ is a momentum-${\mathbf{Q}}$ triplet
gap [31, 32, 33, 34, 35]. Written out explicitly, the singlet part is given by
$\Delta^{S}({\mathbf{k}})=u(-{\mathbf{k}}-{\mathbf{Q}}/2)_{0\downarrow}u({\mathbf{k}}+{\mathbf{Q}}/2)_{0\uparrow}\Delta({\mathbf{k}}+{\mathbf{Q}}/2)\,.$
(10)
Note that from $\Delta(-{\mathbf{k}})=-\Delta({\mathbf{k}})$,
$\Delta({\mathbf{k}}+{\mathbf{Q}})=-\Delta({\mathbf{k}})$ and
$u({\mathbf{k}})_{0\downarrow}=u({\mathbf{k}}+{\mathbf{Q}})_{0\uparrow}$ it
follows that $\Delta^{S}(-{\mathbf{k}})=\Delta^{S}({\mathbf{k}})$ as required.
Finally, we calculate the superconducting order parameter in terms of the
microscopic fermions. We start from the anomalous Green's function
$\displaystyle\langle\mathcal{T}c^{\dagger}_{{\mathbf{r}},\sigma}(\tau)c^{\dagger}_{0,\sigma^{\prime}}(0)\rangle$
$\displaystyle=$ $\displaystyle\langle
R^{*}_{\sigma\tilde{\sigma}}(\tau,{\mathbf{r}})R^{*}_{\sigma^{\prime}\tilde{\sigma}^{\prime}}(0,0)\bar{\psi}_{{\mathbf{r}},\tilde{\sigma}}(\tau)\bar{\psi}_{0,\tilde{\sigma}^{\prime}}(0)\rangle$
(11) $\displaystyle\approx$ $\displaystyle\langle
R^{*}_{\sigma\tilde{\sigma}}(\tau,{\mathbf{r}})R^{*}_{\sigma^{\prime}\tilde{\sigma}^{\prime}}(0,0)\rangle\times$
$\displaystyle\langle\bar{\psi}_{{\mathbf{r}},\tilde{\sigma}}(\tau)\bar{\psi}_{0,\tilde{\sigma}^{\prime}}(0)\rangle\,,$
where summation over repeated indices is implied, $\mathcal{T}$ is the time-
ordering operator, and we have used that the bare electrons are related to the
fermion fields $\bar{\psi}_{{\mathbf{r}},\sigma}$ via a rotation with
$R(\tau,{\mathbf{r}})$ as explained above. In the second line we have
approximated the correlation function by a product of the correlation
functions of the matrix field $R$ and the fermions in the rotating frame.
Using $R^{*}=\sigma^{y}R\sigma^{y}$, it immediately follows from the results
of Ref. [36] that a saddle-point approximation of the non-linear sigma model
describing the dynamics of $R$ leads to following expression for the $R$ two-
point function at non-zero temperature:
$\langle
R^{*}_{ss^{\prime}}(\tau,{\mathbf{r}})R^{*}_{\tilde{s}\tilde{s}^{\prime}}(0,0)\rangle=\sigma^{y}_{\tilde{s}s}\sigma^{y}_{s^{\prime}\tilde{s}^{\prime}}D(\tau,{\mathbf{r}})\,,$
(12)
where $D(\tau,{\mathbf{r}})$ is the Fourier transform of
$((i\nu)^{2}-\omega_{\mathbf{q}}^{2})^{-1}$. As
$\langle\bar{\psi}_{{\mathbf{r}},\tilde{\sigma}}(\tau)\bar{\psi}_{0,\tilde{\sigma}^{\prime}}(0)\rangle$
is the Fourier transform of
$\Delta_{\tilde{\sigma}\tilde{\sigma}^{\prime}}({\mathbf{k}})/(\omega^{2}+\Omega^{2})$,
we find that the (equal-time) superconducting order parameter is given by
$\langle
c^{\dagger}_{-{\mathbf{k}},\sigma}c^{\dagger}_{{\mathbf{k}},\sigma^{\prime}}\rangle=\sigma^{y}_{\sigma\sigma^{\prime}}\frac{1}{N}\sum_{\mathbf{q}}\frac{n(\omega_{\mathbf{q}})-n(-\omega_{\mathbf{q}})}{\omega_{\mathbf{q}}}\Delta^{S}({\mathbf{k}}-{\mathbf{q}})\,,$
(13)
where $n(\omega)=1/(\exp(\omega/T)-1)$ is the Bose-Einstein distribution
function, and a momentum-independent prefactor has been dropped. Note that the
contraction of $\Delta_{\sigma\sigma^{\prime}}({\mathbf{k}})$ with the $R$
two-point function annihilates the momentum-${\mathbf{Q}}$ triplet component,
such that the superconducting order parameter is translation invariant and
pure singlet. This is similar to how a spin-rotation invariant Green's
function was obtained in Refs. [36, 11, 37, 38, 39]. In Fig. 3 we plot the
order parameter (13) both in momentum and real space. It clearly has a nodal
$d$-wave structure, whose origin can be traced back to the single-band nature
of the pairing function, which imposes
$\Delta(-\tilde{{\mathbf{k}}})=-\Delta(\tilde{{\mathbf{k}}})$, together with
the $C_{4}$ and time-reversal symmetries. From Fig. 3 (b) we also see that the
electron pairs are tightly bound together: the pair wavefunction becomes
negligibly small when the electrons are separated by more than $\sim 5$
lattice sites. From the same figure we also see that the pair wavefunction is
only non-zero when the electrons occupy different sublattices. This is a
direct consequence of the U$(1)$ spin rotation symmetry of the AFM, which
imprints on the gap function the property
$\Delta({\mathbf{k}}+{\mathbf{Q}})=-\Delta({\mathbf{k}})$. From Eq. (11) it is
also clear that the pair wavefunction will decay faster if the spin moments
become more disordered. If we use a thermal mass $m=0.1$ such that the spin
correlations become more short-ranged, we find that the pair wavefunction is
essentially zero if the electrons are separated by more than one lattice site
[29].
Figure 3: (a) Plot of the superconducting order parameter $\langle
c^{\dagger}_{-{\mathbf{k}}}\sigma^{y}c^{\dagger}_{{\mathbf{k}}}\rangle$ as a
function of momentum ${\mathbf{k}}$. (b) Real-space order parameter/pair
wavefunction $\langle
c^{\dagger}_{\mathbf{r}}\sigma^{y}c^{\dagger}_{0}\rangle$. Results are
obtained for a $34\times 34$ system using $m=0.01$.
_Discussion –_ We have shown that with microscopically calculated scattering
vertices, finite-temperature remnants of AFM Goldstone modes can mediate a
sizeable attractive interaction between electrons which leads to spin-singlet
$d_{x^{2}-y^{2}}$ superconducting order. The fact that $T_{c}$ drops only
moderately upon increasing the thermal mass of the Goldstone modes by an order
of magnitude is consistent with the expectation that pairing is predominantly
mediated by short-wavelength and high-energy transverse spin fluctuations, as
these couple most strongly to the electrons. The effective theory used in this
work is obtained from a simple mean-field+RPA analysis, which makes it unclear
how reliable our estimate for $T_{c}$ is. An interesting route for future
research would therefore be to improve our results by using the two-particle
self-consistent approach [40] applied to broken-symmetry systems [41], or by
combining the functional renormalization group with mean-field theory [34, 35,
42, 43].
In this work we have ignored Landau damping of the Goldstone modes. We expect
that this will only quantitatively change our results, and here we only aim to
get an order of magnitude estimate for $T_{c}$. Our analysis does reveal a few
distinct universal features of electron pairing in fluctuating itinerant AFM
which can be used to test the theory in more controlled numerical studies, or
perhaps even in experiment. In particular, our key predictions are: (1) the
presence of fluctuating triplet order at momentum ${\mathbf{Q}}$, (2) a pair
wavefunction which decays faster than the spin correlation function, and (3) a
purely inter-sublattice pair wavefunction. The latter was also found in the
spin-bag approach [17, 18], which is not surprising since it is a consequence
of the symmetries of the AFM.
The theory presented here is undoubtedly too simplified to capture the rich
physics of the underdoped cuprates. For example, it does not take charge
stripes, nematicity and pair density-wave orders into account, all of which
can intertwine with each other, the AFM, and the uniform superconductor in
intricate ways [44, 45]. Nevertheless, we believe that our approach can shed
light on the role of the higher-temperature pseudogap state as a parent state
for superconductivity. Of course, this requires one to assume that the
pseudogap state observed in experiment is indeed a fluctuating AFM, which is
still a matter of current debate. But at least for the Hubbard model the
magnetic pseudogap scenario has received considerable numerical support [46,
47, 48, 49, 37, 50, 51].
_Ackowledgements –_ The author would like to thank Andrey Chubukov, Walter
Metzner and Pietro Bonetti for helpful discussions, and Rafael Fernandes for
pointing out Ref. [31]. This research is supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant agreement No. 101076597 - SIESS).
## References
* Hirsch [1985] J. E. Hirsch, Attractive interaction and pairing in fermion systems with strong on-site repulsion, Phys. Rev. Lett. 54, 1317 (1985).
* Emery [1986] V. Emery, The mechanisms of organic superconductivity, Synthetic Metals 13, 21 (1986), proceedings of the Workshop "Synthetic Metals III".
* Scalapino _et al._ [1986] D. J. Scalapino, E. Loh, and J. E. Hirsch, d-wave pairing near a spin-density-wave instability, Phys. Rev. B 34, 8190 (1986).
* Miyake _et al._ [1986] K. Miyake, S. Schmitt-Rink, and C. M. Varma, Spin-fluctuation-mediated even-parity pairing in heavy-fermion superconductors, Phys. Rev. B 34, 6554 (1986).
* Scalapino _et al._ [1987] D. J. Scalapino, E. Loh, and J. E. Hirsch, Fermi-surface instabilities and superconducting d-wave pairing, Phys. Rev. B 35, 6694 (1987).
* Lee and Read [1987] P. A. Lee and N. Read, Why is ${\mathrm{t}}_{\mathrm{c}}$ of the oxide superconductors so low?, Phys. Rev. Lett. 58, 2691 (1987).
* Monthoux _et al._ [1991] P. Monthoux, A. V. Balatsky, and D. Pines, Toward a theory of high-temperature superconductivity in the antiferromagnetically correlated cuprate oxides, Phys. Rev. Lett. 67, 3448 (1991).
* Monthoux _et al._ [1992] P. Monthoux, A. V. Balatsky, and D. Pines, Weak-coupling theory of high-temperature superconductivity in the antiferromagnetically correlated copper oxides, Phys. Rev. B 46, 14803 (1992).
* Scalapino [1994] D. Scalapino, Antiferromagnetic fluctuations and dx2-y2 pairing in the cuprates, Physica C: Superconductivity 235-240, 107 (1994).
* Scalapino [2012] D. J. Scalapino, A common thread: The pairing interaction for unconventional superconductors, Rev. Mod. Phys. 84, 1383 (2012).
* Vasiliou _et al._ [2024] K. Vasiliou, Y. He, and N. Bultinck, Electrons interacting with goldstone modes and the rotating frame, Phys. Rev. B 109, 045155 (2024).
* Shraiman and Siggia [1988] B. I. Shraiman and E. D. Siggia, Mobile Vacancies in a Quantum Heisenberg Antiferromagnet, Phys. Rev. Lett. 61, 467 (1988).
* Shraiman and Siggia [1990] B. I. Shraiman and E. D. Siggia, Mobile vacancy in a quantum antiferromagnet: Effective hamiltonian, Phys. Rev. B 42, 2485 (1990).
* Schulz [1990] H. J. Schulz, Effective action for strongly correlated fermions from functional integrals, Phys. Rev. Lett. 65, 2462 (1990).
* Schulz [1994] H. J. Schulz, Functional Integrals for Correlated Electrons, arXiv e-prints , cond-mat/9402103 (1994), arXiv:cond-mat/9402103 [cond-mat] .
* Schrieffer [1995] J. R. Schrieffer, Ward's identity and the suppression of spin fluctuation superconductivity, Journal of Low Temperature Physics 99, 397 (1995).
* Schrieffer _et al._ [1988] J. R. Schrieffer, X.-G. Wen, and S.-C. Zhang, Spin-bag mechanism of high-temperature superconductivity, Phys. Rev. Lett. 60, 944 (1988).
* Schrieffer _et al._ [1989] J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Dynamic spin fluctuations and the bag mechanism of high-tc superconductivity, Phys. Rev. B 39, 11663 (1989).
* Chubukov [1995] A. V. Chubukov, Quasiparticle spectrum in a nearly antiferromagnetic fermi liquid: Shadow and flat bands, Phys. Rev. B 52, R3840 (1995).
* Altshuler _et al._ [1995] B. L. Altshuler, L. B. Ioffe, and A. J. Millis, Critical behavior of the t=0 2${\mathit{k}}_{\mathit{f}}$ density-wave phase transition in a two-dimensional fermi liquid, Phys. Rev. B 52, 5563 (1995).
* Chubukov _et al._ [1997] A. V. Chubukov, P. Monthoux, and D. K. Morr, Vertex corrections in antiferromagnetic spin-fluctuation theories, Phys. Rev. B 56, 7789 (1997).
* Schmalian _et al._ [1998] J. Schmalian, D. Pines, and B. Stojković, Weak pseudogap behavior in the underdoped cuprate superconductors, Phys. Rev. Lett. 80, 3839 (1998).
* Abanov _et al._ [2003] A. Abanov, A. V. Chubukov, and J. Schmalian, Quantum-critical theory of the spin-fermion model and its application to cuprates: Normal state analysis, Advances in Physics 52, 119 (2003).
* Chubukov _et al._ [2008] A. V. Chubukov, D. Pines, and J. Schmalian, A spin fluctuation model for d-wave superconductivity, in _Superconductivity: Conventional and Unconventional Superconductors_, edited by K. H. Bennemann and J. B. Ketterson (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) pp. 1349–1413.
* Frenkel and Hanke [1990] D. M. Frenkel and W. Hanke, Spirals and spin bags: A link between the weak- and the strong-coupling limits of the hubbard model, Phys. Rev. B 42, 6711 (1990).
* Rowe _et al._ [2015] W. Rowe, I. Eremin, A. T. Rømer, B. M. Andersen, and P. J. Hirschfeld, Doping asymmetry of superconductivity coexisting with antiferromagnetism in spin fluctuation theory, New Journal of Physics 17, 023022 (2015).
* Rømer _et al._ [2016] A. T. Rømer, I. Eremin, P. J. Hirschfeld, and B. M. Andersen, Superconducting phase diagram of itinerant antiferromagnets, Phys. Rev. B 93, 174519 (2016).
* Note [1] See Refs. [52, 53, 54] for a complementary perspective. In these works, the strong inter-band scattering terms from transverse spin fluctuations are not eliminated via a rotating frame, but their effect on the electron Green's function is explicitly taken into account via an infinite sum of diagrams in an eikonal approximation.
* [29] See Supplemental Material at XXX.
* Chubukov and Morr [1997] A. V. Chubukov and D. K. Morr, Electronic structure of underdoped cuprates, Physics Reports 288, 355 (1997), i.M. Lifshitz and Condensed Matter Theory.
* Psaltakis and Fenton [1983] G. C. Psaltakis and E. W. Fenton, Superconductivity and spin-density waves: organic superconductors, Journal of Physics C: Solid State Physics 16, 3913 (1983).
* Murakami and Fukuyama [1998] M. Murakami and H. Fukuyama, Backward scattering and coexistent state in two-dimensional electron system, Journal of the Physical Society of Japan 67, 2784 (1998).
* Kyung [2000] B. Kyung, Mean-field study of the interplay between antiferromagnetism and d-wave superconductivity, Phys. Rev. B 62, 9083 (2000).
* Reiss _et al._ [2007] J. Reiss, D. Rohe, and W. Metzner, Renormalized mean-field analysis of antiferromagnetism and $d$-wave superconductivity in the two-dimensional hubbard model, Phys. Rev. B 75, 075110 (2007).
* Wang _et al._ [2014] J. Wang, A. Eberlein, and W. Metzner, Competing order in correlated electron systems made simple: Consistent fusion of functional renormalization and mean-field theory, Phys. Rev. B 89, 121116 (2014).
* Borejsza and Dupuis [2004] K. Borejsza and N. Dupuis, Antiferromagnetism and single-particle properties in the two-dimensional half-filled Hubbard model: A nonlinear sigma model approach, Phys. Rev. B 69, 085119 (2004).
* Wu _et al._ [2018] W. Wu, M. S. Scheurer, S. Chatterjee, S. Sachdev, A. Georges, and M. Ferrero, Pseudogap and fermi-surface topology in the two-dimensional hubbard model, Phys. Rev. X 8, 021048 (2018).
* Scheurer _et al._ [2018] M. S. Scheurer, S. Chatterjee, W. Wu, M. Ferrero, A. Georges, and S. Sachdev, Topological order in the pseudogap metal, Proceedings of the National Academy of Sciences 115, E3665 (2018).
* Bonetti and Metzner [2022] P. M. Bonetti and W. Metzner, Su(2) gauge theory of the pseudogap phase in the two-dimensional hubbard model, Phys. Rev. B 106, 205152 (2022).
* Y.M. Vilk and A.-M.S. Tremblay [1997] Y.M. Vilk and A.-M.S. Tremblay, Non-Perturbative Many-Body Approach to the Hubbard Model and Single-Particle Pseudogap, J. Phys. I France 7, 1309 (1997).
* Del Re [2023] L. Del Re, Two-particle self-consistent approach for broken symmetry phases, arXiv e-prints , arXiv:2312.16280 (2023).
* Yamase _et al._ [2016] H. Yamase, A. Eberlein, and W. Metzner, Coexistence of incommensurate magnetism and superconductivity in the two-dimensional hubbard model, Phys. Rev. Lett. 116, 096402 (2016).
* Vilardi _et al._ [2020] D. Vilardi, P. M. Bonetti, and W. Metzner, Dynamical functional renormalization group computation of order parameters and critical temperatures in the two-dimensional hubbard model, Phys. Rev. B 102, 245128 (2020).
* Fradkin _et al._ [2015] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015).
* Tranquada [2020] J. M. Tranquada, Cuprate superconductors as viewed through a striped lens, Advances in Physics 69, 437 (2020).
* Macridin _et al._ [2006] A. Macridin, M. Jarrell, T. Maier, P. R. C. Kent, and E. D'Azevedo, Pseudogap and antiferromagnetic correlations in the hubbard model, Phys. Rev. Lett. 97, 036401 (2006).
* Kyung _et al._ [2006] B. Kyung, S. S. Kancharla, D. Sénéchal, A.-M. S. Tremblay, M. Civelli, and G. Kotliar, Pseudogap induced by short-range spin correlations in a doped mott insulator, Phys. Rev. B 73, 165114 (2006).
* Gunnarsson _et al._ [2015] O. Gunnarsson, T. Schäfer, J. P. F. LeBlanc, E. Gull, J. Merino, G. Sangiovanni, G. Rohringer, and A. Toschi, Fluctuation diagnostics of the electron self-energy: Origin of the pseudogap physics, Phys. Rev. Lett. 114, 236402 (2015).
* Wu _et al._ [2017] W. Wu, M. Ferrero, A. Georges, and E. Kozik, Controlling feynman diagrammatic expansions: Physical nature of the pseudogap in the two-dimensional hubbard model, Phys. Rev. B 96, 041105 (2017).
* Wietek _et al._ [2021] A. Wietek, Y.-Y. He, S. R. White, A. Georges, and E. M. Stoudenmire, Stripes, antiferromagnetism, and the pseudogap in the doped hubbard model at finite temperature, Phys. Rev. X 11, 031007 (2021).
* Simkovic _et al._ [2022] F. Simkovic, R. Rossi, A. Georges, and M. Ferrero, Origin and fate of the pseudogap in the doped Hubbard model, arXiv e-prints , arXiv:2209.09237 (2022).
* Sedrakyan and Chubukov [2010] T. A. Sedrakyan and A. V. Chubukov, Pseudogap in underdoped cuprates and spin-density-wave fluctuations, Phys. Rev. B 81, 174536 (2010).
* Ye and Chubukov [2019] M. Ye and A. V. Chubukov, Hubbard model on a triangular lattice: Pseudogap due to spin density wave fluctuations, Phys. Rev. B 100, 035135 (2019).
* Ye _et al._ [2023] M. Ye, Z. Wang, R. M. Fernandes, and A. V. Chubukov, Location and thermal evolution of the pseudogap due to spin fluctuations, Phys. Rev. B 108, 115156 (2023).
— Supplementary Material —
## Appendix A Appendix A: Definition of
$u(\tilde{{\mathbf{k}}})_{\alpha,\sigma}$ and $S_{V}$
Underlying the effective action in the main text is the mean-field Hamiltonian
in the pseudo-momentum basis:
$H_{MF}=\sum_{\tilde{{\mathbf{k}}}}f_{\tilde{{\mathbf{k}}}}^{\dagger}h(\tilde{{\mathbf{k}}})f_{\tilde{{\mathbf{k}}}}\,,$
(14)
where
$f_{\tilde{{\mathbf{k}}}}=(f_{\tilde{{\mathbf{k}}},\uparrow},f_{\tilde{{\mathbf{k}}},\downarrow})^{T}$,
with $f^{\dagger}_{\tilde{{\mathbf{k}}},\sigma}$ defined in Eq. (4) in the
main text. The precise form of the mean-field Hamiltonian is
$h(\tilde{{\mathbf{k}}})=\left(\begin{matrix}\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}&M\\\
M&\varepsilon_{\tilde{{\mathbf{k}}},\downarrow}\end{matrix}\right)\,,$ (15)
where $M\sim 1.93$ is the self-consistently determined (using the parameters
for the Hubbard model given in the main text) hybridization between the spin
up and down electrons which produces the AFM order, and
$\displaystyle\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}$ $\displaystyle=$
$\displaystyle\varepsilon(\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2)\,,$ (16)
$\displaystyle\varepsilon_{\tilde{{\mathbf{k}}},\downarrow}$ $\displaystyle=$
$\displaystyle\varepsilon(\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2)\,.$ (17)
Here, $\varepsilon({\mathbf{k}})=-2t(\cos k_{x}+\cos
k_{y})-2t^{\prime}(\cos(k_{x}+k_{y})+\cos(k_{x}-k_{y})$ is the dispersion of
the Hubbard model in Eq. (1). The mean-field energies are defined as the
eigenvalues of $h(\tilde{{\mathbf{k}}})$, and are given by
$E_{\tilde{{\mathbf{k}}},\alpha}=\frac{1}{2}\left(\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}+\varepsilon_{\tilde{{\mathbf{k}}},\downarrow}\pm\sqrt{(\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}-\varepsilon_{\tilde{{\mathbf{k}}},\downarrow})^{2}+4M^{2}}\right)\,.$
(18)
The states $|u(\tilde{{\mathbf{k}}})_{\alpha}\rangle$ are the corresponding
eigenstates of $h(\tilde{{\mathbf{k}}})$. Note that the mean-field Hamiltonian
satisfies
$h(\tilde{{\mathbf{k}}}+{\mathbf{Q}})=\sigma^{x}h(\tilde{{\mathbf{k}}})\sigma^{x}$,
which is a manifestation of the U$(1)$ spin rotation symmetry around the
$x$-axis. The eigenstates of $h(\tilde{{\mathbf{k}}})$ (in a real gauge) thus
satisfy
$|u(\tilde{{\mathbf{k}}}+{\mathbf{Q}})_{\alpha}\rangle=\pm\sigma^{x}|u(\tilde{{\mathbf{k}}})_{\alpha}\rangle$.
We partially fix the gauge by requiring these signs to all be positive.
Written out explicitly, the instantaneous interaction in the effective theory
used in the main text is given by
$S_{V}=\int\mathrm{d}\tau\frac{1}{N}\sum_{\tilde{{\mathbf{q}}},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime}}\left[\frac{U}{2}\left(\bar{\psi}_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}}}\Lambda_{\tilde{{\mathbf{q}}}}(\tilde{{\mathbf{k}}})\psi_{\tilde{{\mathbf{k}}}}\right)\left(\bar{\psi}_{\tilde{{\mathbf{k}}}^{\prime}+\tilde{{\mathbf{q}}}}\Lambda_{-\tilde{{\mathbf{q}}}}(\tilde{{\mathbf{k}}}^{\prime})\psi_{\tilde{{\mathbf{k}}}^{\prime}}\right)-\frac{c}{2aw}\big{(}\bar{\psi}_{\tilde{{\mathbf{k}}}}f_{\tilde{{\mathbf{q}}}}(\tilde{{\mathbf{k}}})\psi_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}}}\big{)}\left(\bar{\psi}_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{q}}}}f_{\tilde{{\mathbf{q}}}}^{\dagger}(\tilde{{\mathbf{k}}}^{\prime})\psi_{\tilde{{\mathbf{k}}}^{\prime}}\right)\right]\,.$
(19)
The first term in Eq. (19) is simply the microscopic Hubbard interaction
$\frac{U}{2}\sum_{\mathbf{r}}:n_{\mathbf{r}}^{2}:$ rewritten in the mean-field
basis. The unitary basis transformation from the orbital basis to the mean-
field basis gives rise to the form factors:
$\left[\Lambda_{{\mathbf{q}}}({\mathbf{k}})\right]_{\alpha\beta}=\langle
u(\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}})_{\alpha}|u(\tilde{{\mathbf{k}}})_{\beta}\rangle\,.$
(20)
The second term in Eq. (19) is obtained by integrating out the field conjugate
to the Goldstone field $\phi({\mathbf{r}})$ [11]. As in the main text
$c\approx 1$ is the Goldstone mode velocity, $a$ is the lattice constant, and
$w\approx 1$ is a dimensionless number which is fixed by the requirement that
in the rotating frame it should hold that
$\lim_{\tilde{{\mathbf{q}}}\rightarrow
0,{\mathbf{Q}}}g_{\tilde{{\mathbf{q}}}}=0$ [11]. The matrices
$\left[f_{\tilde{{\mathbf{q}}}}(\tilde{{\mathbf{k}}})\right]_{\alpha\beta}$
contain the scattering vertices
$f_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})$ used in Eq. (6).
## Appendix B Appendix B: Definition of
$V(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})$
The interaction
$V(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})$
which scatters a pair of electrons with frequencies $(i\omega,-i\omega)$ and
pseudo-momenta $(\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}})$ to a pair of
electrons with $(i\omega^{\prime},-i\omega^{\prime})$ and pseudo-momenta
$(\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}}^{\prime})$ can be written
as
$V(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})=V_{I}(\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})+V_{G}(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})\,.$
(21)
The first term comes from the instantaneous interaction contained in $S_{V}$
and is given by
$V_{I}(\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})=U\Lambda_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{k}}}^{\prime},00}(\tilde{{\mathbf{k}}})\Lambda_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}},00}(-\tilde{{\mathbf{k}}})-\frac{c}{aw}f_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}},00}(\tilde{{\mathbf{k}}}^{\prime})f_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{k}}}^{\prime},00}(-\tilde{{\mathbf{k}}}^{\prime})\,,$
(22)
where we have put the band indices equal to zero because we only consider
scattering within the valence band which crosses the Fermi energy. The second
term in Eq. (21) is generated by tree-level Goldstone mode exchange and is
given by
$V_{G}(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})=\frac{\left[g_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}},00}(\tilde{{\mathbf{k}}}^{\prime})+(\omega-\omega^{\prime})f_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}},00}(\tilde{{\mathbf{k}}}^{\prime})\right]\left[g_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{k}}}^{\prime},00}(-\tilde{{\mathbf{k}}}^{\prime})-(\omega-\omega^{\prime})f_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{k}}}^{\prime},00}(-\tilde{{\mathbf{k}}}^{\prime})\right]}{(i\omega-i\omega^{\prime})^{2}-\omega_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}}}^{2}}\,.$
(23)
Figure 4: Results for $\lambda$ on a $34\times 34$ system using a thermal mass
$m=0.01$ and keeping the $N_{\omega}=1000$ (left) and $N_{\omega}=2000$
(right) smallest Matsubara frequencies.
## Appendix C Appendix C: Variational solution of the gap equation
In this appendix we explain in more detail how we solve the gap equation
$\lambda\hat{\Delta}(i\omega^{\prime},\tilde{{\mathbf{k}}}^{\prime})=T\sum_{i\omega}\frac{1}{N}\sum_{\tilde{{\mathbf{k}}}}\frac{V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})}{\omega^{2}+(E_{\tilde{{\mathbf{k}}}}-\mu)^{2}}\hat{\Delta}(i\omega,\tilde{{\mathbf{k}}})$
(24)
variationally with the ansatz
$\hat{\Delta}(i\omega,\tilde{{\mathbf{k}}})=\frac{\Delta(\tilde{{\mathbf{k}}})}{\omega^{2}+\Omega^{2}}\,.$
(25)
As a first step we multiply both sides of Eq. (24) with
$\hat{\Delta}^{*}(i\omega^{\prime},\tilde{{\mathbf{k}}}^{\prime})/(\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2})$, and sum over both
$i\omega^{\prime}$ and $\tilde{{\mathbf{k}}}^{\prime}$. We obtain
$\displaystyle\lambda\sum_{\tilde{{\mathbf{k}}}}\Delta^{*}(\tilde{{\mathbf{k}}}^{\prime})\Delta(\tilde{{\mathbf{k}}}^{\prime})\sum_{i\omega^{\prime}}\frac{1}{(\omega^{\prime
2}+\Omega^{2})^{2}}\frac{1}{\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2}}=$ (26)
$\displaystyle\sum_{\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime}}\Delta^{*}(\tilde{{\mathbf{k}}}^{\prime})\Delta(\tilde{{\mathbf{k}}})\frac{T}{N}\sum_{i\omega,i\omega^{\prime}}\frac{1}{\omega^{\prime
2}+\Omega^{2}}\frac{1}{\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2}}V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})\frac{1}{\omega^{2}+(E_{\tilde{{\mathbf{k}}}}-\mu)^{2}}\frac{1}{\omega^{2}+\Omega^{2}}\,,$
where we have filled in the ansatz for the gap function (25). By defining the
matrices
$\displaystyle D(\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})$
$\displaystyle=$
$\displaystyle\delta_{\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime}}\sum_{i\omega^{\prime}}\frac{1}{(\omega^{\prime
2}+\Omega^{2})^{2}}\frac{1}{\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2}}$ (27) $\displaystyle
M(\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})$ $\displaystyle=$
$\displaystyle\frac{T}{N}\sum_{i\omega,i\omega^{\prime}}\frac{1}{\omega^{\prime
2}+\Omega^{2}}\frac{1}{\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2}}V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})\frac{1}{\omega^{2}+(E_{\tilde{{\mathbf{k}}}}-\mu)^{2}}\frac{1}{\omega^{2}+\Omega^{2}}$
(28)
we can bring Eq. (26) in the form
$\lambda\langle\Delta|D|\Delta\rangle=\langle\Delta|M|\Delta\rangle\,,$ (29)
where $|\Delta\rangle$ is the vector with components
$\Delta(\tilde{{\mathbf{k}}})$. Via following trivial rewriting of this
equation
$\lambda\left(\langle\Delta|D^{1/2}\right)\left(D^{1/2}|\Delta\rangle\right)=\left(\langle\Delta|D^{1/2}\right)D^{-1/2}MD^{-1/2}\left(D^{1/2}|\Delta\rangle\right)$
(30)
it becomes clear that the optimal $\lambda$ corresponds to the smallest
eigenvalue of $D^{-1/2}MD^{-1/2}$, and the gap function is obtained from the
corresponding eigenvector. In practice we evaluate the sums over Matsubara
frequencies in Eqs. (27) and (28) with a frequency cutoff. For the results
presented in the main text we have kept the 1000 smallest Matsubara
frequencies. We have checked that our results are converged upon changing the
cutoff. For example, in Fig. 4 we compare results obtained by using 1000 and
2000 Matsubara frequencies. We find that the change in $\lambda$ is
negligible.
## Appendix D Appendix D: Singlet and triplet components of the gap function
By solving the gap equation we find a gap function which is defined in pseudo-
momentum space and in the mean-field band basis. To interpret the gap function
physically it is easier to go back to the crystal momentum and spin basis. In
this appendix we work out this transformation for the spin-singlet and triplet
components of the gap function.
### D.1 Singlet part
To obtain the singlet component we start from the equations
$\displaystyle\langle
f^{\dagger}_{-\tilde{{\mathbf{k}}},\downarrow}f^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}\rangle$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}c^{\dagger}_{\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}\rangle$
(31) $\displaystyle=$ $\displaystyle
u(-\tilde{{\mathbf{k}}})_{0\downarrow}u(\tilde{{\mathbf{k}}})_{0\uparrow}\,\Delta(\tilde{{\mathbf{k}}})\,,$
(32)
where the first line follows from the definition of
$f^{\dagger}_{\tilde{{\mathbf{k}}},\sigma}$, and the second line follows from
the definition of $\Delta(\tilde{{\mathbf{k}}})$ as the gap function in the
mean-field basis. Similarly we also have the equalities
$\displaystyle\langle
f^{\dagger}_{-\tilde{{\mathbf{k}}},\uparrow}f^{\dagger}_{\tilde{{\mathbf{k}}},\downarrow}\rangle$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}c^{\dagger}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}\rangle$
(33) $\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2+{\mathbf{Q}},\uparrow}c^{\dagger}_{\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2+{\mathbf{Q}},\downarrow}\rangle$
(34) $\displaystyle=$ $\displaystyle
u(-\tilde{{\mathbf{k}}})_{0\uparrow}u(\tilde{{\mathbf{k}}})_{0\downarrow}\,\Delta(\tilde{{\mathbf{k}}})\,.$
(35)
Combining Eqs. (31), (32), (33), (34) and (35) we find
$\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}+{\mathbf{Q}}/2,\downarrow}c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\uparrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}})_{0\downarrow}u({\mathbf{k}})_{0\uparrow}\,\Delta({\mathbf{k}})$
(36) $\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}+{\mathbf{Q}}/2,\uparrow}c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}+{\mathbf{Q}})_{0\uparrow}u({\mathbf{k}}+{\mathbf{Q}})_{0\downarrow}\,\Delta({\mathbf{k}}+{\mathbf{Q}})$
(37)
A trivial shift in momentum then gives
$\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}},\downarrow}c^{\dagger}_{{\mathbf{k}},\uparrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}-{\mathbf{Q}}/2)_{0\downarrow}u({\mathbf{k}}+{\mathbf{Q}}/2)_{0\uparrow}\,\Delta({\mathbf{k}}+{\mathbf{Q}}/2)$
(38) $\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}},\uparrow}c^{\dagger}_{{\mathbf{k}},\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}+{\mathbf{Q}}/2)_{0\uparrow}u({\mathbf{k}}-{\mathbf{Q}}/2)_{0\downarrow}\,\Delta({\mathbf{k}}-{\mathbf{Q}}/2)\,.$
(39)
As $\Delta({\mathbf{k}}+{\mathbf{Q}})=-\Delta({\mathbf{k}})$ and
$u({\mathbf{k}}+{\mathbf{Q}})_{0,\uparrow}=u({\mathbf{k}})_{0,\downarrow}$ the
right-hand side Eq. (38) is equal to minus the right-hand side of Eq. (39).
Eqs. (38) and (39) thus constitute the singlet-component of the gap.
### D.2 Triplet part
To obtain the triplet component we follow essentially the same steps as to
obtain the singlet component. Starting from the equalities
$\displaystyle\langle
f^{\dagger}_{-\tilde{{\mathbf{k}}},\uparrow}f^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}\rangle$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}c^{\dagger}_{\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}\rangle$
(40) $\displaystyle=$ $\displaystyle
u(-\tilde{{\mathbf{k}}})_{0\uparrow}u(\tilde{{\mathbf{k}}})_{0\uparrow}\,\Delta(\tilde{{\mathbf{k}}})$
(41)
and
$\displaystyle\langle
f^{\dagger}_{-\tilde{{\mathbf{k}}},\downarrow}f^{\dagger}_{\tilde{{\mathbf{k}}},\downarrow}\rangle$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}c^{\dagger}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}\rangle$
(42) $\displaystyle=$ $\displaystyle
u(-\tilde{{\mathbf{k}}})_{0\downarrow}u(\tilde{{\mathbf{k}}})_{0\downarrow}\,\Delta(\tilde{{\mathbf{k}}})$
(43)
we find
$\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}-{\mathbf{Q}}/2,\uparrow}c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\uparrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}})_{0\uparrow}u({\mathbf{k}})_{0\uparrow}\,\Delta({\mathbf{k}})$
(44) $\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}-{\mathbf{Q}}/2,\downarrow}c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}+{\mathbf{Q}})_{0\downarrow}u({\mathbf{k}}+{\mathbf{Q}})_{0\downarrow}\,\Delta({\mathbf{k}}+{\mathbf{Q}})$
(45)
Shifting the momentum then gives
$\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}+{\mathbf{Q}},\uparrow}c^{\dagger}_{{\mathbf{k}},\uparrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}-{\mathbf{Q}}/2)_{0\uparrow}u({\mathbf{k}}+{\mathbf{Q}}/2)_{0\uparrow}\Delta({\mathbf{k}}+{\mathbf{Q}}/2)$
(46) $\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}+{\mathbf{Q}},\downarrow}c^{\dagger}_{{\mathbf{k}},\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}+{\mathbf{Q}}/2)_{0\downarrow}u({\mathbf{k}}-{\mathbf{Q}}/2)_{0\downarrow}\,\Delta({\mathbf{k}}-{\mathbf{Q}}/2)\,,$
(47)
which corresponds to the triplet component with crystal momentum
${\mathbf{Q}}$.
## Appendix E Appendix E: Additional numerical results
In this final appendix we present additional numerical results obtained using
a thermal mass $m=0.1$. Other parameters are identical to the ones used in the
main text. The system size is $34\times 34$.
In Fig. 5 (a) we plot $\lambda+1$ as a function of $T$ and $\Omega$. We see
that the highest $T_{c}\approx 0.023$ occurs for $\Omega\approx 0.5$. The
corresponding gap function shown in Fig. 5 (b) is identical to the one
obtained in the main text using a smaller thermal mass $m=0.01$.
In Fig. 6 we show the superconducting order parameter defined in Eqs. (11) and
(13) in the main text. Compared to the results in the main text obtained with
a smaller thermal mass, we see that the order parameter in Fig. 6 is much
smoother in momentum space, and hence more short-ranged in real-space. This
illustrates the statement made in the main text that our theory predicts a
system with shorter-range AFM fluctuations to also have a shorter-range pair
wavefunction.
Figure 5: (a) $\lambda+1$ as a function of $T$ and $\Omega$ using a thermal
mass $m=0.1$. (b) Gap function giving rise to the highest $T_{c}$ at
$\Omega=0.5$. Results are obtained on a $34\times 34$ system. Figure 6:
Superconducing order parameter and corresponding pair wavefunction obtained
with $m=0.1$. (a) Superconducting order parameter as a function of crystal
momentum. (b) Pair wavefunction in real-space. Results are obtained on a
$34\times 34$ system.
|
1–LABEL:LastPageNov. 23, 2016Jan. 16, 2018 [Theory of computation]: AAA—BBB;
[Mathematics of computing]: CCC—DDD
* A preliminary version of this paper has appeared at FORTE 2016, held as Part of the 11th International Federated Conference on Distributed Computing Techniques, DisCoTec 2016
# Fencing Programs with Self-Invalidations and Self-Downgrades
Parosh Aziz Abdullaa aUppsala University, Sweden<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, Mohamed Faouzi Atiga ,
Stefanos Kaxirasa , Carl Leonardssona , Alberto Rosb bUniversidad de
Murcia, Spain<EMAIL_ADDRESS>and Yunyun Zhua
# Mending Fences with Self-Invalidation and Self-Downgrade*
Parosh Aziz Abdullaa aUppsala University, Sweden<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, Mohamed Faouzi Atiga ,
Stefanos Kaxirasa , Carl Leonardssona , Alberto Rosb bUniversidad de
Murcia, Spain<EMAIL_ADDRESS>and Yunyun Zhua
###### Abstract.
Cache coherence protocols based on self-invalidation and self-downgrade have
recently seen increased popularity due to their simplicity, potential
performance efficiency, and low energy consumption. However, such protocols
result in memory instruction reordering, thus causing extra program behaviors
that are often not intended by the programmers. We propose a novel formal
model that captures the semantics of programs running under such protocols,
and features a set of fences that interact with the coherence layer. Using the
model, we design an algorithm to analyze the reachability and check whether a
program satisfies a given safety property with the current set of fences. We
describe a method for insertion of optimal sets of fences that ensure
correctness of the program under such protocols. The method relies on a
counter-example guided fence insertion procedure. One feature of our method is
that it can handle a variety of fences (with different costs). This diversity
makes optimization more difficult since one has to optimize the total cost of
the inserted fences, rather than just their number. To demonstrate the
strength of our approach, we have implemented a prototype and run it on a wide
range of examples and benchmarks. We have also, using simulation, evaluated
the performance of the resulting fenced programs.
###### Key words and phrases:
automatic fence insertion, cache coherence protocol, self-invalidation, self-
downgrade
## 1\. Introduction
### Background.
Traditional cache coherence protocols, either directory-based or snooping-
based, are transparent to the programmer in the sense that they respect the
memory consistency model of the system, and hence there is no effect on memory
ordering due to the coherence protocol. On the other hand, there is an ever
larger demand on hardware designers to increase efficiency both in performance
and power consumption. The quest to increase performance while maintaining
transparency has led to complex coherence protocols with many states and
relying on directories, invalidations, broadcasts, etc, often at the price of
high verification cost, area (hardware cost) and increased energy consumption.
Therefore, many researchers have recently proposed ways to simplify coherence
without compromising performance but at the price of relaxing the memory
consistency model [LW95, CKS+11, KK11, RK12, SKA13, KR13, HHB+14, RDK15, SA15,
RK15a, DRHK15, KRHK16, RK16]. Principal techniques among these proposals are
Self-Invalidation (Si) and Self-Downgrade (Sd).
A protocol with Self-Invalidation (Si) allows old copies of the data to be
kept, without invalidation on each store operation by another core. This
eliminates the need for tracking readers [LW95]. In an Si protocol,
invalidation of data from a cache is caused by synchronization instructions
executed by the core local to the cache.
Correspondingly, in a protocol with Self-Downgrade (Sd), downgrades are not
caused by read operations in other cores, but again by synchronization
instructions. Sd eliminates the need to track the last writer of a cache line
[RK12].
A protocol with both self-invalidation and self-downgrade (SiSd) does not need
a directory, thus removing a main source of complexity and scalability
constraints in traditional cache coherence protocols [RK12]. But this comes at
a price: SiSd protocols induce weak memory semantics that allow reordering or
memory instructions. The behavior of a program may now deviate from its
behavior under the standard Sequentially Consistent (SC) semantics [Lam79],
sometimes leading to subtle errors that are hard to detect and correct.
In the context of weak memory, hardware designers provide memory fence
instructions to help the programmer to eliminate the undesired behaviors. A
fence instruction, executed by a thread, limits the allowed reorderings
between instructions issued before and after the fence instruction. To enforce
consistency under SiSd, fences should also be made visible to caches, such
that necessary invalidations or downgrades may be performed. In this paper, we
consider different types of fences. Each type eliminates a different kind of
non-SC behavior, and may have different impact on the program performance. In
fact, unnecessary fences may significantly jeopardize program performance.
This is particularly true for the fences considered in this work, since they
both incur latency, and affect the performance of the cache coherence
subsystem as a whole. These fences cause the invalidation of the contents of
the cache. Hence the more fences the less caching and the higher traffic we
have. Thus, it is desirable to find the _optimal_ set of fences, which
guarantee correctness at minimal performance cost. There are multiple ways of
defining optimality. The one we adopt is to calculate the number of
occurrences of fences in the source program, with the costs of different
fences taken into account.
### Challenge.
One possibility to make SiSd transparent to the program is to require the
programmer to ensure that the program does not contain any data races. In
fact, data race freedom is often required by designers of SiSd protocols in
order to guarantee correct program behavior [CKS+11, KK11]. However, this
approach would unnecessarily disqualify large sets of programs, since many
data races are in reality not harmful. Examples of correct programs with races
include lock-free data structures (e.g., the Chase-Lev Work-stealing queue
algorithm [CL05]), transactional memories (e.g., the TL2 algorithm [DSS06]),
and synchronization library primitives (e.g. pthread_spin_lock in glibc). In
this paper, we consider a different approach where fences are inserted to
restore correctness. This means that we may insert sufficiently many fences to
achieve program correctness without needing to eliminate all its races or non-
SC behaviors. The challenge then is to find sets of fences that guarantee
program correctness without compromising efficiency. Manual fence placement is
time-consuming and error-prone due to the complex behaviors of multithreaded
programs [HS08]. Thus, we would like to provide the programmer with a tool for
automatic fence placement. There are several requirements to be met in the
design of fence insertion algorithms. First, a set of fences should be sound,
i.e., it should have enough fences to enforce a sufficiently ordered behavior
for the program to be correct. Second, the set should be optimal, in the sense
that it has a lowest total cost among all sound sets of fences. We define the
cost to be the number of occurrences of fences in the source program, with the
costs of different fences taken into account. In general, there may exist
several different optimal sets of fences for the same program. Our experiments
(Section 7) show that different choices of sound fence sets may impact
performance and network traffic.
To carry out fence insertion we need to be able to perform program
verification, i.e., to check correctness of the program with a given set of
fences. This is necessary in order to be able to decide whether the set of
fences is sound, or whether additional fences are needed to ensure
correctness. A critical task in the design of formal verification algorithms
is to define the program semantics under the given memory model.
### Our Approach.
We present a method for automatic fence insertion in programs running in the
presence of SiSd. The method is applicable to a large class of self-
invalidation and self-downgrade protocols such as the ones in [LW95, CKS+11,
KK11, RK12, SKA13, KR13, HHB+14, RDK15, SA15, RK15a, DRHK15, KRHK16]. Our goal
is to eliminate incorrect behaviors that occur due to the memory model induced
by SiSd. We will not concern ourselves with other sources of consistency
relaxation, such as compiler optimizations. We formulate the correctness of
programs as safety properties. A safety property is an assertion that some
specified “erroneous”, or “bad”, program states can never occur during
execution. Such bad states may include e.g., states where a programmer-
specified assert statement fails, or where uninitialized data is read. To
check a safety property, we check the reachability of the set of “bad” states.
We provide an algorithm for checking the reachability of a set of bad states
for a given program running under SiSd. In the case that such states are
reachable, our algorithm provides a counter-example (i.e., an execution of the
program that leads to one of the bad states). This counter-example is used by
our fence insertion procedure to add fences in order to remove the counter-
examples introduced by SiSd semantics. Thus, we get a counter-example guided
procedure for inferring the optimal sets of fences. The termination of the
obtained procedure is guaranteed under the assumption that each call to the
reachability algorithm terminates. As a special case, our tool detects when a
program behaves incorrectly already under SC. Notice that in such a case, the
program cannot be corrected by inserting any set of fences.
### Contributions.
We make the following main contributions:
* •
We define a novel formal model that captures the semantics of programs running
under SiSd, and employs a set of fences that interact with the coherence
layer. The semantics support the essential features of typical assembly code.
* •
We develop a tool, Memorax, available at https://github.com/memorax/memorax,
that we have run successfully on a wide range of examples under SiSd and under
Si. Notably, our tool detects for the first time four bugs in programs in the
Splash-2 benchmark suite [WOT+95], which have been fixed in a recent Splash-3
release [SLKR16]. Two of these are present even under SC, while the other two
arise under SiSd. We employ the tool to infer fences of different kinds and
evaluate the relative performance of the fence-augmented programs by
simulation in GEMS.
We augment the semantics with a reachability analysis algorithm that can check
whether a program satisfies a given safety property with the current set of
fences. Inspired by an algorithm in [LNP+12] (which uses dynamic analysis
instead of verification as backend), we describe a counter-example guided
fence insertion procedure that automatically infers the optimal sets of fences
necessary for the correctness of the program. The procedure relies on the
counter-examples provided by the reachability algorithm in order to refine the
set of fences. One feature of our method is that it can handle different types
of fences with different costs. This diversity makes optimization more
difficult since one has to optimize the total cost of the inserted fences,
rather than just their number. Upon termination, the procedure will return all
optimal sets of fences.
### Related Work.
Adve and Hill proposed SC-for-DRF as a contract between software and hardware:
If the software is data race free, the hardware behaves as sequentially
consistent [AH90]. Dynamic self-invalidation (for DRF programs) was first
proposed by Lebeck and Wood [LW95]. Several recent works employ self-
invalidation to simplify coherence, including SARC coherence [KK11], DeNovo
[CKS+11, SKA13, SA15], and VIPS-M [RK12, KR13, RDK15, RK15a, KRHK16].
A number of techniques for automatic fence insertion have been proposed, for
different memory models and with different approaches. However, to our
knowledge, we propose the first counter-example guided fence insertion
procedure in the presence of a variety of fences (with different costs). In
our previous work [AAC+12], we propose counter-example guided fence insertion
for programs under TSO with respect to safety properties (also implemented in
Memorax). Considering the SiSd model makes the problem significantly more
difficult. TSO offers only one fence, whereas the SiSd model offers a variety
of fences with different costs. This diversity makes the optimization more
difficult since one has to minimize the total cost of the fences rather than
just their number.
The work presented in [KVY10] proposes an insertion procedure for different
memory models w.r.t. safety properties. This procedure computes the set of
needed fences in order to not reach each state in the transition graph.
Furthermore, this procedure assigns a unique cost for all fences. The
procedure is not counter-example based, and requires some modification to the
reachability procedure.
In [BDM13], the tool Trencher is introduced, which inserts fences under TSO to
enforce robustness (formalised by Shasha and Snir in [SS88]), also using an
exact, model-checking based technique. Musketeer [AKNP14] uses static analysis
to efficiently overapproximate the fences necessary to enforce robustness
under several different memory models. In contrast to our work, the fence
insertion procedures in [BDM13] and [AKNP14] first enumerate all solutions and
then use linear programming to find the optimal set of fences.
In [GHS11] and [GHS09], the authors discuss fence insertion and verification
of satisfiability modulo theory (SMT) of relaxed memory models under
transactional memories. Our work, in comparison, addresses the reachability
problem.
The program semantics under SiSd is different from those under other weak
memory models (e.g. TSO and POWER). Hence existing techniques cannot be
directly applied. To our knowledge, ours is the first work that defines the
SiSd model, proposes a reachability analysis and describes a fence insertion
procedure under SiSd.
There exist works on the verification of cache coherence protocols. This paper
is orthogonal to these works since we are concerned with verification of
programs running on such architectures and not the protocols themselves.
## 2\. Self-Invalidation, Self-Downgrade, and their Fences
In this section, we recall the notions of self-invalidation and self-
downgrade, and describe the main features of the system architecture and the
protocol we consider. We also introduce two fences that are defined under the
protocol.
### 2.1. Self-Invalidation and Self-Downgrade
Self-invalidation eliminates the need to track sharers of a cache line in a
directory structure [LW95]. We consider that invalidation of shared data in
caches is caused by fences inserted in the programs and not as a result of
writes from other cores.
Correspondingly, self-downgrade eliminates the need to track the last writer
(i.e., the owner, in a Moesi-like protocol) of a cache line [RK12]. This is
because downgrades are also not performed as a consequence of read operations,
but by means of fence instructions inserted in the programs.
A protocol that implements self-invalidation together with self-downgrade does
not need a directory, thus removing one of the main sources of complexity and
scalability constraints in traditional cache coherence protocols [RK12].
We first set the stage for the architecture and the coherence protocol we
study in this work, by discussing some of their details: i.e., how memory
accesses are resolved, and how the self-invalidation and the self-downgrade
are performed when a fence is encountered.
##### System architecture
We assume a standard multicore architecture with a number of cores, each with
a private L1 cache. The proposals and algorithms described in this paper are
more widely applicable to systems with several levels of private caches. The
last level cache (LLC) of the system is logically shared among all the cores.
### 2.2. Cache coherence protocol
We also assume a very simple version of a self-invalidation/self-downgrade
protocol with only three stable states in the L1 cache (invalid –I–, clean
–C–, and dirty –D–) and only two stable states in the LLC (invalid –I– and
valid –V–). There are no invalidations or downgrades, which means that there
are no transient states to account for the arrival of such coherence actions.
There are no requests other than from the L1s to the LLC (and from LLC to
memory). There is no distinction of data into private or shared as in [RK12],
as this would distract from our discussion. Such optimizations are
straightforward extensions in our approach.
##### Basic actions:
To connect with the formal specification of the system behavior that follows
in Section 4 we present here some necessary —if somewhat mundane— details of
the basic actions in our assumed system.
* •
A read request that misses in the L1 cache issues a request to the LLC. If it
hits in the LLC, a reply containing the data is sent. In case of a miss, main
memory is accessed to get the data block. When the data arrives to the L1
cache, the miss is resolved and the data can be accessed. The block is stored
in an L1 cache line in clean state.
* •
A write request is always resolved immediately, even if the block is not
present in the L1 (in this case, the miss status handling register –MSHR– can
temporarily hold the new data). This is because writes are assumed to be data-
race-free, i.e, they are always ordered with respect to conflicting reads
[AH90]. In this case, writes do not require “write permission”.
* •
After writing in an L1 cache line (e.g., one word), if the data block is
missing it is fetched from the LLC. The block is merged with the written word.
Before merging the data, the cache line is in a transient state and once
merged transitions to dirty.
* •
An atomic read-modify-write (RMW) request (e.g., test-and-set –TAS–) needs to
reach the LLC, get the data block and send it back to the L1 cache. During
this operation, the corresponding cache line in the LLC is blocked, so no
other RMW request can proceed. When the data arrives at the L1 cache it is
read and possibly modified. If modified, the data is written-back in the LLC,
unblocking the LLC line at the same time. Otherwise the LLC line will be
simply unblocked. This blocking operation —common in other protocols for
directory operations that generate new messages (indirection, invalidations,
etc.)— is only necessary in this protocol for RMW requests. Once the
transaction finishes, the data block remains in the L1 in clean state.
* •
Evictions of clean cache lines only require a change of state to invalid.
However, evictions of dirty cache lines need to write back to the LLC the data
that have been modified locally. This is necessary to avoid overwriting
unrelated data in the LLC cache line (a different part of the LLC line may
have been modified independently without a conflict). When modified data are
written-back an acknowledgement message is sent to the L1 to signal the
completion of the corresponding writes.
* •
To keep track of the locally modified data in an L1 cache line, it is
necessary to keep information in the form of a dirty bit per word (byte),
either with the L1 cache lines [CKS+11], or in the write-buffer or MSHRs
[RK12].
### 2.3. Self-Invalidation and Self-Downgrade fences
Since the described protocol has neither invalidation on writes nor downgrades
on reads, we need to ensure that a read operation sees the latest value
written, when this is intended by the program. Typically, the program contains
synchronization to enforce an order between conflicting writes and reads.
In this context, to ensure that a read gets the latest value of a
corresponding write, two things need to happen: first, the data in the
writer’s cache must be self-downgraded and put back in the LLC sometime after
the write but before the read; second, if there is a (stale) copy in the
reader’s cache, it must be self-invalidated sometime before the read. The
self-downgrade and self-invalidation also need to be ordered the same as the
write and read are ordered by synchronization.
Prior proposals [LW95, KK11, CKS+11, SKA13, ADC11, RK12, KR13] invariably
offer SC for data-race-free (DRF) programs [AH90]. In general, such proposals
can be thought of as employing a single fence causing the self-invalidation
and self-downgrade of cached data, _on every synchronization in the DRF
program_ (e.g., [RK12].)
Our approach is fundamentally different. We make no assumption as to what
constitutes synchronization (perhaps ordinary accesses relying on SC
semantics, or algorithms involving atomic RMW operations). We insert _fences_
in a program to cause self invalidation and self downgrade in such a way as to
produce the desired behavior.
With only a single fence, ensuring that a read sees the latest value of a
write causes the self-invalidation _and_ self-downgrade of _both_ the reader’s
and the writer’s cache. In many cases, this is unnecessary.
One of the contributions of our work is to propose two separate fences, which
we call “load-load fence” (llfence) and “store-store fence” (ssfence) to
address the above problem. An llfence self-invalidates only _clean_ data in
the cache (at word level), while an ssfence writes back only _dirty_ data to
the LLC (again at word level), so all the data in the L1 cache of the process
are clean.
The separation of self-invalidation and self-downgrade into two fences affects
performance in two ways: first, we reduce the fence latency when we do not
have to self-downgrade; second, we eliminate extraneous misses (that cost in
performance) when we only need to self-downgrade.
### 2.4. Improving self-invalidation of partially dirty cache lines: the DoI
state
The llfence defined above operates efficiently for cache lines that are
entirely clean, or entirely dirty. In particular, for clean data they take a
single cycle, while for dirty data they do not perform any action. However,
cache lines that contain both clean and dirty words are not self-invalidated
efficiently. Consider, for example, a cache line with one clean word and one
dirty word (its dirty bit is set). The llfence must invalidate the clean word
(if it were to be accessed afterwards), without affecting the dirty word. If
we invalidate the whole cache line we also have to write the dirty data to the
LLC. This would have the same impact (for this cache line) as a single full
fence, and would offer no advantage from using the llfence instead.
In order to improve the efficiency of llfence operations we propose that they
operate at word granularity, being able to self-invalidate the clean words in
a single cycle and leaving untouched the dirty words. Thus, we introduce a new
state for L1 cache lines, called DoI (dirty or invalid), for exactly this
purpose. A cache line in this state contains words that are either dirty (with
the dirty bit set) or invalid (with the dirt bit unset). An llfence
transitions any partially dirty cache line to the DoI state. No write back is
performed for its dirty words. This allows an efficient, one-cycle
implementation of llfence, since now the only necessary action for a llfence
is to change the cache-line state from dirty to DoI.
## 3\. Overview
In this section, we give an informal overview of the main concepts in our
framework. We describe the semantics (configurations and runs) of programs
running under SiSd, the notion of safety properties, the weak memory model
induced by SiSd, the roles of fences, and optimal sets of fences. This will be
formalized in later sections.
data x=0 y=0 z=0process P0registers $r0beginL1: y$x := 1;L2: y$y := 1;L3:
y$$r0 := z;endprocess P1registers $r1$r2 $r3beginL4: z := 1;L5: $r1 := x;L6:
$r2 := y;L7: $r3 := x;end Figure 1. A simple program ${\mathcal{P}}$.
##### Example.
We will use the toy program ${\mathcal{P}}$ in Figure 1 as a running example.
The program is written in a simple assembly-like programming language. The
syntax and semantics of the language are formally defined in Section 4.
${\mathcal{P}}$ consists of two processes P0 and P1 that share three variables
x, y, and z. Process P0 has one register $r0, and process P1 has three
registers $r1, $r2, and $r3. Process P0 has three instructions labeled with
L1, L2, L3, and process P1 has four instructions labeled with L4, L5, L6, L7.
To simplify the presentation, we assume that each cache line holds only one
variable. We also assume that the underlying protocol contains both Si and Sd.
It is straightforward to extend our framework to the case where a cache line
may hold several variables, and to the case where the protocol only contains
one of Si and Sd. In ${\mathcal{P}}$, all the instructions have unique labels.
Therefore, to simplify the presentation, we identify each label with the
corresponding instruction, e.g., L1 and x := 1 in P0.
##### Configurations.
ptx=0y=0z=0ptL1$r0=0ptL4 $r1=0$r2=0 $r3=0$c_{0}$LLCP0P1ptx=0y=0z=0ptL3$r0=0x=1
y=1ptL5 $r1=0$r2=0 $r3=0z=1$c_{1}$ptx=1y=1z=0ptL3$r0=0ptL6 $r1=0$r2=0 $r3=0x=0
z=1$c_{2}$ptx=1y=1z=0ptL3$r0=0ptend $r1=0$r2=1 $r3=0x=0 y=1 z=1 $c_{3}$
ptx=0y=1z=0ptL3$r0=0x=1ptend $r1=0$r2=1 $r3=0x=0 z=1
$c_{4}$LLCP0P1ptx=1y=0z=0ptend $r0=0 y=1 z=0ptend $r1=0$r2=0 $r3=0x=0 z=1
$c_{5}$
Figure 2. Configurations
A configuration is a snapshot of the global state of the system, and consists
of two parts, namely the local and shared parts. The local part defines the
local states of the processes, i.e., it defines for each process: (i) its next
instruction to be executed, (ii) the values stored in its registers, and (iii)
the variables (memory locations) that are currently cached in its L1, together
with their status: invalid, clean, or dirty; and the current value of the
variable in case it is valid. The shared part defines, for each variable, its
value in the LLC. Figure 2 shows different configurations of ${\mathcal{P}}$.
Each configuration is depicted as three fields, representing the LLC, P0, and
P1 respectively. ${\mathcal{P}}$ starts its execution from the initial
configuration $c_{0}$, where the values of all variables are $0$ in the LLC.
P0 and P1 are about to execute the instructions labeled L1 and L4,
respectively. The values of all registers are $0$. None of the variables is
valid in the L1 of the processes. In contrast, in $c_{4}$, the value of y is
$1$ in the LLC. P0 is about to execute the instruction L3, while P1 has ended
its execution. The value of the register $r2 is $1$. The variables x, and z
are valid in the L1 of P1, with values $0$ and $1$ respectively. The variable
z is dirty in the L1 of P1 (marked by underlining z=1), while x is clean (not
underlined). Finally, there is a dirty copy of x with value $1$ in P0.
##### Safety Properties.
Suppose that, together with the program ${\mathcal{P}}$, we are given a safety
property $\phi$ which states that a certain set ${\it Bad}$ of configurations
will not occur during any execution of ${\mathcal{P}}$. For the example, we
assume that ${\it Bad}$ is the set of configurations where (i) P1 has ended
its execution, and (ii) the registers $r2 and $r3 have values $1$ and $0$
respectively. For instance, $c_{3}$ and $c_{4}$ are members of ${\it Bad}$. We
are interested in checking whether ${\mathcal{P}}$ satisfies $\phi$ under the
SiSd semantics. Note that the set ${\it Bad}$ is not reachable in
${\mathcal{P}}$ under SC semantics, which also means that ${\mathcal{P}}$
satisfies $\phi$ under SC.
##### Runs.
The semantics of a program boils down to defining a transition relation on the
set of configurations. The execution of the program can be viewed as a run,
consisting of a sequence of transitions, i.e., events that take the program
from one configuration to another by changing the local states of the
processes and the shared parts. Such a transition will either be performed by
a given process when it executes an instruction, or it occurs due to a system
event. We consider three kinds of system events: ${\tt fetch}$, ${\tt evict}$,
and ${\tt wrllc}$. They model respectively fetching a value from LLC to L1,
invalidating an L1 entry, and writing a dirty L1 entry through to the LLC. The
system events are decoupled from program instructions and execute
independently.
$c_{0}$$\rho_{1}$$c_{1}$L4*L1*L2*$\rho_{2}$$c_{2}$L5*evict*(P0,x) evict*(P0,y)
$\rho_{3}$$c_{3}$L6*L7 Figure 3. The run $\pi_{1}$.
In Figure 3 we show one example run $\pi_{1}$ of ${\mathcal{P}}$. It consists
of three sequences $\rho_{1}$, $\rho_{2}$, $\rho_{3}$ of transitions, and
takes us from $c_{0}$ through $c_{1}$ and $c_{2}$ to $c_{3}$. $\rho_{1}$:
Starting from $c_{0}$, P1 executes L4. Since z is invalid in the L1 of P1, it
is fetched from the LLC. In Figure 3, the star in L4* is to simplify the
notation, and it indicates that the instruction L4 is preceded by fetch
event111In the examples of this section, fetch events always precede read or
write events. In general, fetch events may occur anywhere along the run. of
the process (here P1) on the relevant variable (here z). Consequently, a dirty
copy of z with value $1$ is stored in the L1 of P1. Next, P0 executes L1 and
L2, putting dirty copies (with values $1$) of x and y in its L1, reaching the
configuration $c_{1}$. $\rho_{2}$: P1 executes L5, fetching x from the LLC,
and storing a clean copy with value $0$. P0 evicts the variable x. An evict
event may only be performed on clean variables. To simplify the notation, we
augment the evict event in Figure 3 by a star. This indicates that it is
preceded by an wrllc event on x. The latter updates the value of x to $1$ in
the LLC, and makes x clean in the L1 of P0. Next, P0 evicts y in a similar
manner, thus reaching $c_{2}$. $\rho_{3}$: P1 executes L6. Since y is invalid
in the L1 of P1, it is fetched from the LLC, and stored with value $1$ as
clean. The register $r2 will be assigned the value $1$. Finally, P1 executes
L7. Since x is valid, it need not be fetched from the memory (L7 is therefore
not starred in Figure 3), and hence $r3 is assigned the value $0$. Thus we
reach $c_{3}$, which is in the set ${\it Bad}$. And so ${\mathcal{P}}$
violates the safety property $\phi$ under the SiSd semantics.
##### Weak Memory Model.
Although the configuration $c_{3}$ is not reachable from $c_{0}$ under SC
semantics, we demonstrated above that it is reachable under SiSd semantics.
The reason is that SiSd introduces a weak memory semantics in the form of
reorderings of events. In the case of $\pi_{1}$, we have a read-read
reordering. More precisely, the read event L7 overtakes the read event L6, in
the sense that L6 is issued before L7, but the value assigned to $r2 in L6
(coming from the write on y in L2) is more recent than the value assigned to
$r3 in L7 (which is the initial value of x). To prevent event reorderings, we
use fences. In this paper, we use four types of fences, namely ${\tt
llfence}$, ${\tt ssfence}$, ${\tt fence}$, and ${\tt syncwr}$. In this
section, we only describe the first three types.
data x=0 y=0 z=0process P0registers $r0beginL1: y$x := 1;L2: y$y := 1;L3:
y$$r0 := z;endprocess P1registers $r1$r2 $r3beginL4: z := 1;L5: $r1 := x;L6:
$r2 := y;L8: llfence;L7: $r3 := x;end Figure 4. The program
${\mathcal{P}}_{1}$.
$c_{0}$L1*L2*evict*(P0,y)L4*L5*L6*evict(P1,x)evict(P1,y)L8$c_{4}$L7 Figure 5.
The run $\pi_{2}$.
##### LL Fences.
To forbid the run $\pi_{1}$, we insert an ${\tt llfence}$ between L6 and L7,
obtaining a new program ${\mathcal{P}}_{1}$ (Figure 4). Intuitively, an ${\tt
llfence}$ (load-load-fence) blocks when there are clean entries in the L1 of
the process, and hence forbids the reordering of two read (load) operations.
For instance, in the above example, L8 cannot be executed before x has become
invalid in the L1 of P1, and hence the new value $1$ of x will be assigned to
$r3 in L7. Therefore, ${\mathcal{P}}_{1}$ does not contain the run $\pi_{1}$
any more.
##### SS Fences.
Despite the fact that the insertion of L8 eliminates the run $\pi_{1}$, the
program ${\mathcal{P}}_{1}$ still does not satisfy the safety property $\phi$:
the set ${\it Bad}$ is still reachable from $c_{0}$ in ${\mathcal{P}}_{1}$,
this time through a run $\pi_{2}$ (Figure 5) that leads to the configuration
$c_{4}$ (which is also a member of ${\it Bad}$). In $\pi_{2}$, P0 performs L1
and L2 and then evicts only y, which means that the values of x and y in the
LLC will be $0$ resp. $1$. Now, P1 will perform the instructions L4, L5, L6.
Next P1 evicts x and then y which means that now P1 does not contain any clean
variables, and hence L8 is enabled. Notice that these evict events are not
followed by stars (since they concern clean copies of the variables). Finally,
P1 executes L7. Since x is invalid in the L1 of P0, it is fetched from the LLC
(where its value is $0$ since it was never evicted by P0), and hence $r3 will
be assigned the value $0$. Thus, we are now in $c_{4}$.
data x=0 y=0 z=0process P0registers $r0beginL1: y$x := 1;L9: ssfence;L2: y$y
:= 1;L3: y$$r0 := z;endprocess P1registers $r1$r2 $r3beginL4: z := 1;L5: $r1
:= x;L6: $r2 := y;L7: $r3 := x;end Figure 6. The program ${\mathcal{P}}_{2}$.
Notice again that $\pi_{2}$ is not possible under SC (in the SC semantics,
fences have no effect, so they are equivalent to empty statements). The reason
why $\pi_{2}$ is possible under SiSd is due to a write-write reordering. More
precisely, the write event L1 is issued before the write event L2, but L2
takes effect (updates the LLC) before L1. To forbid the run $\pi_{2}$, we
insert an ${\tt ssfence}$ between L1 and L2, obtaining the program
${\mathcal{P}}_{2}$ (Figure 6). An ${\tt ssfence}$ (store-store-fence) is only
enabled when there are no dirty entries in the L1 of the process. Hence it
forbids the reordering of two write operations. For instance, in the above
example, L9 cannot be executed before x has been evicted by P0, and hence the
value of x in the LLC will be updated to $1$.
$c_{0}$L4*L5*L6*evict(P1,x)evict(P1,y)L8L7*L1*evict*(P0,x)L9L2*$c_{5}$L3*
Figure 7. The run $\pi_{3}$.
In fact, no configuration in ${\it Bad}$ is reachable from $c_{0}$ in
${\mathcal{P}}_{2}$, which means that ${\mathcal{P}}_{2}$ indeed satisfies the
property $\phi$. Thus, we have found a sound set of fences for ${\mathcal{P}}$
w.r.t. $\phi$. It is interesting to observe that, although ${\mathcal{P}}_{2}$
is correct w.r.t. $\phi$, the program still contains runs that are impossible
under SC, e.g., the run $\pi_{3}$ given in Figure 7.
##### Full Fences.
Consider a safety property $\phi^{\prime}$ defined by (unreachability of) a
new set of configurations ${\it Bad}^{\prime}$. The set ${\it Bad}^{\prime}$
contains all configurations in ${\it Bad}$, and also all configurations where
(i) the processes P0 and P1 have both terminated, and (ii) both $r0 and $r3
have values $0$. We show that ${\mathcal{P}}_{2}$ violates $\phi^{\prime}$,
i.e., the set ${\it Bad}^{\prime}$ is reachable from $c_{0}$ in
${\mathcal{P}}_{2}$. To that end, we construct the run $\pi_{3}$, depicted in
Figure 7. (The run can be explained similarly to $\pi_{1}$ and $\pi_{2}$.) At
the end of $\pi_{3}$, we reach the configuration $c_{5}$ which is in ${\it
Bad}^{\prime}$.
Notice that the run $\pi_{3}$ is not possible under SC, while it is feasible
under the SiSd semantics even in the presence of the two fences at L8 and L9.
The reason $\pi_{3}$ is possible under SiSd is write-read reordering. More
precisely, read events may overtake write events (although not the other way
round). In $\pi_{3}$, the write event L4 is issued before the read events L5,
L6, and L7, but L4 does not take effect (does not update the LLC) before the
read events. There are several ways to prevent the reachability of the set
${\it Bad}^{\prime}$. One is to replace the ${\tt llfence}$ at L8 and the
${\tt ssfence}$ at L9 by the full fence ${\tt fence}$, thus obtaining the
program ${\mathcal{P}}_{3}$ (Figure 8). A full fence ${\tt fence}$ is only
enabled when the L1 of the process is empty, and hence it forbids all
reorderings of events of the process. In ${\mathcal{P}}_{3}$, no configuration
in ${\it Bad}^{\prime}$ is reachable from $c_{0}$. Thus we have inserted a
sound set of fences in ${\mathcal{P}}$ w.r.t. the set ${\it Bad}^{\prime}$.
data x=0 y=0 z=0process P0registers $r0beginL1: y$x := 1;L9: fence;L2: y$y :=
1;L3: y$$r0 := z;endprocess P1registers $r1$r2 $r3beginL4: z := 1;L5: $r1 :=
x;L6: $r2 := y;L8: fence;L7: $r3 := x;end Figure 8. The program
${\mathcal{P}}_{3}$.
##### Optimal Sets of Fences.
We will describe some optimal sets of fences for the program ${\mathcal{P}}$.
As we will notice, this task is not trivial even for ${\mathcal{P}}$. Our
framework allows to make use of different kinds of fences. We saw above three
examples of fences (and we introduce another one in Section 4). The motivation
is that different kinds of fences may have different costs. Using a more
“light-weight” fence may both increase program performance and reduce network
traffic (see Section 7). In that respect, the cost of a full fence is higher
than that of an ${\tt llfence}$ or an ${\tt ssfence}$. The cost assignment is
provided by the user of our tool. Let us assume that an ${\tt llfence}$ or an
${\tt ssfence}$ costs 1 unit, and that a full fence costs 2 units. Let $F_{1}$
be the set of fences where there is an ${\tt ssfence}$ after L1, and an ${\tt
llfence}$ after L6. Then, $F_{1}$ is optimal for the program ${\mathcal{P}}$
w.r.t. the property $\phi$. First, $F_{1}$ is sound since ${\mathcal{P}}_{2}$
(which is the result of inserting the two fences in ${\mathcal{P}}$) satisfies
$\phi$, i.e., it does not reach ${\it Bad}$ from $c_{0}$. Second, $F_{1}$ has
the minimal cost that guarantees unreachability of ${\it Bad}$. The set
$F_{2}$ which we get by replacing both the ${\tt llfence}$ and ${\tt ssfence}$
by full fences is also sound. It is also minimal w.r.t. the number of fences
(which is $2$). However, it is not optimal w.r.t. $\phi$ since it has a larger
cost than $F_{1}$. On the other hand, $F_{2}$ is optimal w.r.t. the set
$\phi^{\prime}$. In fact, there are several optimal sets of fences w.r.t.
$\phi^{\prime}$ (12 sets to be exact, as reported by our tool). One such a set
is $F_{3}$ which we get by inserting an ${\tt ssfence}$ after L1, an ${\tt
llfence}$ after L2, and an ${\tt ssfence}$ followed by an ${\tt llfence}$
after L6. The set $F_{3}$ is not minimal w.r.t. the number of fences, but
optimal w.r.t. the property $\phi^{\prime}$. Notice that there are at least
$2^{15}$ ways to insert three types of fences in the simple program of Figure
1. (Each type may or may not be inserted in any particular position.)
## 4\. Programs – Syntax and Semantics
In this section, we formalize SiSd and Si protocols, by introducing a simple
assembly-like programming language, and defining its syntax and semantics.
### 4.1. Syntax
The syntax of programs is given by the grammar in Figure 9. A program has a
finite set of processes which share a number of variables (memory locations)
$\mathcal{M}$. A variable $x\in\mathcal{M}$ should be interpreted as one
machine word at a particular memory address. For simplicity, we assume that
all the variables and process registers take their values from a common finite
domain ${\mathcal{V}}$ of values. Each process contains a sequence of
instructions, each consisting of a program label and a statement. To simplify
the presentation, we assume that all instructions (in all processes) have
unique labels. For a label $\lambda$, we apply three functions: ${\sf
Proc}\left(\lambda\right)$ returns the process $p$ in which the label occurs.
${\sf Stmt}\left(\lambda\right)$ returns the statement whose label id is
$\lambda$. ${\sf Next}\left(\lambda\right)$ returns the label of the next
statement in the process code, or ${\tt end}$ if there is no next statement.
$\begin{array}[]{@{}r@{\;}c@{\;}l@{}}\langle{\it{pgm}}\rangle&::=&\tt{data}\;{\langle{\it{vdecl}}\rangle}^{+}{\langle{\it{proc}}\rangle}^{+}\\\
\\\
\langle{\it{vdecl}}\rangle&::=&\langle{\it{var}}\rangle\;\textrm{\textquotesingle}{=}\textrm{\textquotesingle}\;\left(\textrm{\textquotesingle}{*}\textrm{\textquotesingle}\;|\;\langle{\it{val}}\rangle\right)\\\
\\\
\langle{\it{proc}}\rangle&::=&\tt{process}\;\langle{\it{pid}}\rangle\;\par\tt{registers}\;{\langle{\it{reg}}\rangle}^{*}\;\langle{\it{stmts}}\rangle\\\
\\\
\langle{\it{stmts}}\rangle&::=&\tt{begin}\;{(\langle{\it{label}}\rangle\;\textrm{\textquotesingle}{:}\textrm{\textquotesingle}\;\langle{\it{stmt}}\rangle\;\textrm{\textquotesingle}{;}\textrm{\textquotesingle})}^{+}\;\tt{end}\\\
\\\
\langle{\it{stmt}}\rangle&::=&\langle{\it{var}}\rangle\;\textrm{\textquotesingle}{:=}\textrm{\textquotesingle}\;\langle{\it{expr}}\rangle\;\;\;|\;\;\;\langle{\it{reg}}\rangle\;\textrm{\textquotesingle}{:=}\textrm{\textquotesingle}\;\langle{\it{var}}\rangle\;\;\;|\\\
&&\langle{\it{reg}}\rangle\;\textrm{\textquotesingle}{:=}\textrm{\textquotesingle}\;\langle{\it{expr}}\rangle\;\;\;\;|\;\;\;\tt{llfence}\;\;\;|\;\;\;\tt{fence}\;\;\;|\\\
&&\tt{cas}\;\textrm{\textquotesingle}{(}\textrm{\textquotesingle}\;\langle{\it{var}}\rangle\;\textrm{\textquotesingle}{,}\textrm{\textquotesingle}\;\langle{\it{expr}}\rangle\;\textrm{\textquotesingle}{,}\textrm{\textquotesingle}\;\langle{\it{expr}}\rangle\;\textrm{\textquotesingle}{)}\textrm{\textquotesingle}\;\;\;|\\\
&&\tt{syncwr}\;\textrm{\textquotesingle}{:}\textrm{\textquotesingle}\;\langle{\it{var}}\rangle\;\textrm{\textquotesingle}{:=}\textrm{\textquotesingle}\;\langle{\it{expr}}\rangle\;\;\;|\;\;\;\tt{ssfence}\;\;\;|\\\
&&\tt{cbranch}\;\textrm{\textquotesingle}{(}\textrm{\textquotesingle}\;\langle{\it{bexpr}}\rangle\;\textrm{\textquotesingle}{)}\textrm{\textquotesingle}\;\langle{\it{label}}\rangle\\\
\end{array}$
Figure 9. The grammar of concurrent programs.
$\begin{array}[]{@{}r@{\;}c@{\;}l@{}}$c$&:&\mbox{(${\sf LConf},{\sf LLC})$}\\\
\\\ ${\sf LConf}$&:&\mbox{for each process $p$, returns the local
configuration of $p$}\\\ \\\ \mbox{$(\lambda,{\sf RVal},{\sf
L1})$}&:&\mbox{Local configuration of $p$}\\\ \\\ \mbox{$\lambda$}&:&\mbox{the
label of the next statement to execute in $p$}\\\ \\\ \mbox{${\sf
RVal}$}&:&\mbox{the values of the local registers in $p$}\\\ \\\ \mbox{${\sf
L1}$}&:&\mbox{$({\sf Valid},{\sf LStatus},{\sf LVal})$, the state of the L1
cache of $p$}\\\ \\\ \mbox{${\sf Valid}\subseteq\mathcal{M}$}&:&\mbox{the set
of shared variables currently in the valid state}\\\ \\\ \mbox{${\sf
LStatus}:{\sf Valid}\to\\{{\tt dirty},{\tt clean}\\}$}&:&\mbox{for each
$x\in{\sf Valid}$, whether $x$ is dirty or clean}\\\ \\\ \mbox{${\sf
LVal}:{\sf Valid}\to{\mathcal{V}}$}&:&\mbox{for each $x\in{\sf Valid}$, its
current value in the L1 cache of $p$}\\\ \\\
\mbox{LLC$:\mathcal{M}\to{\mathcal{V}}$}&:&\mbox{shared part of $c$, defines
for each $x\in\mathcal{M}$ its value in the LLC}\par\\\ \end{array}$ Figure
10. The definition of a configuration $c$
### 4.2. Configurations
As illustrated in Figure 10, a local configuration of a process $p$ is a
triple $(\lambda,{\sf RVal},{\sf L1})$, where $\lambda$ is the label of the
next statement to execute in $p$, ${\sf RVal}$ defines the values of the local
registers, and ${\sf L1}$ defines the state of the L1 cache of $p$. In turn,
${\sf L1}$ is a triple $({\sf Valid},{\sf LStatus},{\sf LVal})$. Here ${\sf
Valid}\subseteq\mathcal{M}$ defines the set of shared variables that are
currently in the valid state, and ${\sf LStatus}$ is a function from ${\sf
Valid}$ to the set $\\{{\tt dirty},{\tt clean}\\}$ that defines, for each
$x\in{\sf Valid}$, whether $x$ is dirty or clean, and ${\sf LVal}$ is a
function from ${\sf Valid}$ to ${\mathcal{V}}$ that defines for each $x\in{\sf
Valid}$ its current value in the L1 cache of $p$. The shared part of a
configuration is given by a function ${\sf LLC}$ that defines for each
variable $x\in\mathcal{M}$ its value ${\sf LLC}(x)$ in the LLC. A
configuration $c$ then is a pair $({\sf LConf},{\sf LLC})$ where ${\sf LConf}$
is a function that returns, for each process $p$, the local configuration of
$p$. In the formal definition below, our semantics allows system events to
occur non-deterministically. This means that we model not only instructions
from the program code itself, but also events that are caused by unpredictable
things as hardware prefetching, software prefetching, program preemption,
false sharing, multiple threads of the same program being scheduled on the
same core, etc.
A transition $t$ is either performed by a given process when it executes an
instruction, or is a system event. In the former case, $t$ will be of the form
$\lambda$, i.e., $t$ models the effect of a process $p$ performing the
statement labeled with $\lambda$. In the latter case, $t$ will be equal to
$\omega$ for some system event $\omega$. For a function $f$, we use
$f\left[{a}\leftarrow{b}\right]$, to denote the function $f^{\prime}$ such
that $f^{\prime}(a)=b$ and $f^{\prime}(a^{\prime})=f(a^{\prime})$ if
$a^{\prime}\neq a$. We write $f(a)=\bot$ to denote that $f$ is undefined for
$a$.
Below, we give an intuitive explanation of each transition. The formal
definition can be found in Figure 11 where we assume $c=\left({\sf LConf},{\sf
LLC}\right)$, and ${\sf LConf}(p)=(\lambda,{\sf RVal},{\sf L1})$, and ${\sf
L1}=({\sf Valid},{\sf LStatus},{\sf LVal})$, ${\sf
Proc}\left(\lambda\right)=p$, and ${\sf Stmt}\left(\lambda\right)=\sigma$. We
leave out the definitions for local instructions, since they have standard
semantics.
### 4.3. Semantics
#### 4.3.1. Instruction Semantics.
Let $p$ be one of the processes in the program, and let $\lambda$ be the label
of an instruction in $p$ whose statement is $\sigma$. We will define a
transition relation $\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}$,
induced by $\lambda$, on the set of configurations. The relation is defined in
terms of the type of operation performed by the given statement $\sigma$. In
all the cases only the local state of $p$ and LLC will be changed. The local
states of the rest of the processes will not be affected. This mirrors the
principle in SiSd that L1 cache controllers will communicate with the LLC, but
never directly with other L1 caches.
Read $(\$r:=x)$: Process $p$ reads the value of $x$ from L1 into the register
$\$r$. The L1 and the LLC will not change. The transition is only enabled if
$x$ is valid in the L1 cache of $p$. This means that if $x$ is not in L1, then
a system event ${\tt fetch}$ must occur before $p$ is able to execute the read
operation.
Write $(x:=e)$: An expression $e$ contains only registers and constants. The
value of $x$ in L1 is updated with the evaluation of $e$ where registers have
values as indicated by ${\sf RVal}$, and $x$ becomes dirty. The write is only
enabled if $x$ is valid for $p$.
Fence $({\tt fence})$: A full fence transition is only enabled when the L1 of
$p$ is empty. This means that before the fence can be executed, all entries in
its L1 must be evicted (and written to the LLC if dirty). So $p$ must stall
until the necessary system events (${\tt wrllc}$ and ${\tt evict}$) have
occurred. Executing the fence has no further effect on the caches.
$\begin{array}[]{@{}c@{}}\textbf{Instruction Semantics}\\\ \\\
\textrm{\footnotesize$c=\left({\sf LConf},{\sf LLC}\right)$, ${\sf
LConf}(p)=(\lambda,{\sf RVal},{\sf L1})$,}\\\ \textrm{\footnotesize${\sf
L1}=({\sf Valid},{\sf LStatus},{\sf LVal})$, ${\sf
Proc}\left(\lambda\right)=p$, ${\sf Stmt}\left(\lambda\right)=\sigma$}\\\ \\\
{\displaystyle\frac{\sigma=(\$r:=x)\;,\;x\in{\sf
Valid}}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left({\sf Next}\left(\lambda\right),{\sf
RVal}\left[{\$r}\leftarrow{{\sf LVal}(x)}\right],{\sf L1}\right)}\right],{\sf
LLC}\right)}}\\\ \\\ \frac{\sigma=(x:=e)\;,\;x\in{\sf
Valid}\;}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left({\sf Next}\left(\lambda\right),{\sf
RVal},\left({\sf Valid},{\sf LStatus}\left[{x}\leftarrow{{\tt
dirty}}\right],{\sf LVal}\left[{x}\leftarrow{{\sf
RVal}(e)}\right]\right)\right)}\right],{\sf LLC}\right)}\\\ \\\
{\displaystyle\frac{\sigma={\tt fence}\;,\;{\sf
Valid}=\varnothing}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left({\sf Next}\left(\lambda\right),{\sf RVal},{\sf
L1}\right)}\right],{\sf LLC}\right)}}\\\ \\\ {\displaystyle\frac{\sigma={\tt
ssfence}\;,\;\forall x\in\mathcal{M}.\;\left(x\in{\sf Valid}\Rightarrow{\sf
LStatus}(x)={\tt
clean}\right)}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left({\sf Next}\left(\lambda\right),{\sf RVal},{\sf
L1}\right)}\right],{\sf LLC}\right)}}\\\ \\\ {\displaystyle\frac{\sigma={\tt
llfence}\;,\;\forall x\in\mathcal{M}.\;\left(x\in{\sf Valid}\Rightarrow{\sf
LStatus}(x)={\tt
dirty}\right)}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left({\sf Next}\left(\lambda\right),{\sf RVal},{\sf
L1}\right)}\right],{\sf LLC}\right)}}\\\ \\\ {\displaystyle\frac{\sigma=({\tt
syncwr}:x\hskip 2.0pt{\tt:=}\hskip 2.0pte)\;,\;x\not\in{\sf
Valid}}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left({\sf Next}\left(\lambda\right),{\sf RVal},{\sf
L1}\right)}\right],{\sf LLC}\left[{x}\leftarrow{{\sf
RVal}(e)}\right]\right)}}\\\ \\\ {\displaystyle\frac{\sigma={\tt
cas}(x,e_{0},e_{1})\;,\;x\not\in{\sf Valid}\;,\;{\sf LLC}(x)={\sf
RVal}(e_{0})}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left({\sf Next}\left(\lambda\right),{\sf RVal},{\sf
L1}\right)}\right],{\sf LLC}\left[{x}\leftarrow{{\sf
RVal}(e_{1})}\right]\right)}}\\\ \\\ {\displaystyle\frac{\sigma=({\tt
cbranch}(e)\;\lambda^{\prime}),\;{\sf RVal}(e)={\tt
true}}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left(\lambda^{\prime},{\sf RVal},{\sf
L1}\right)}\right],{\sf LLC}\right)}}\\\ \\\ {\displaystyle\frac{\sigma=({\tt
cbranch}(e)\;\lambda^{\prime}),\;{\sf RVal}(e)={\tt
false}}{c\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left({\sf Next}\left(\lambda\right),{\sf RVal},{\sf
L1}\right)}\right],{\sf LLC}\right)}}\\\ \\\ \textbf{System Event
Semantics}\\\ \\\ {\displaystyle\frac{\omega=({\tt
fetch}({p,x}))\;,\;x\not\in{\sf
Valid}\;}{c\stackrel{{\scriptstyle\omega}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left(\lambda,{\sf RVal},\left({\sf
Valid}\cup\\{x\\},{\sf LStatus}\left[{x}\leftarrow{{\tt clean}}\right],{\sf
LVal}\left[{x}\leftarrow{{\sf LLC}(x)}\right]\right)\right)}\right],{\sf
LLC}\right)}}\\\ \\\ {\displaystyle\frac{\omega=({\tt
wrllc}({p,x}))\;,\;x\in{\sf Valid}\;,\;{\sf LStatus}(x)={\tt
dirty}\;}{c\stackrel{{\scriptstyle\omega}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left(\lambda,{\sf RVal},\left({\sf Valid},{\sf
LStatus}\left[{x}\leftarrow{{\tt clean}}\right],{\sf
LVal}\right)\right)}\right],{\sf LLC}\left[{x}\leftarrow{{\sf
LVal}(x)}\right]\right)}}\\\ \\\ {\displaystyle\frac{\omega=({\tt
evict}({p,x}))\;,\;x\in{\sf Valid}\;,\;{\sf LStatus}(x)={\tt
clean}}{c\stackrel{{\scriptstyle\omega}}{{\longrightarrow}}\left({\sf
LConf}\left[{p}\leftarrow{\left(\lambda,{\sf RVal},\left({\sf
Valid}\setminus\\{x\\},{\sf LStatus}\left[{x}\leftarrow{\bot}\right],{\sf
LVal}\left[{x}\leftarrow{\bot}\right]\right)\right)}\right],{\sf
LLC}\right)}\par}\end{array}$
Figure 11. Semantics of programs running under SiSd.
SS-Fence $({\tt ssfence})$: Similarly, an ${\tt ssfence}$ transition is only
enabled when there are no dirty entries in the L1 cache of $p$. So $p$ must
stall until all dirty entries have been written to the LLC by ${\tt wrllc}$
system events. In contrast to a full fence, an ${\tt ssfence}$ permits clean
entries to remain in the L1.
LL-Fence $({\tt llfence})$: This is the dual of an SS-Fence. An ${\tt
llfence}$ transition is only enabled when there are no clean entries in the L1
cache of $p$. In other words, the read instructions before and after an ${\tt
llfence}$ cannot be reordered.
Synchronized write $({\tt syncwr}:x\hskip 2.0pt{\tt:=}\hskip 2.0pte)$: A
synchronized write is like an ordinary write, but acts directly on the LLC
instead of the L1 cache. For a ${\tt syncwr}$ transition to be enabled, $x$
may not be in the L1. (I.e., the cache must invalidate $x$ before executing
the ${\tt syncwr}$.) When it is executed, the value of $x$ in the LLC is
updated with the evaluation of the expression $e$ under the register valuation
${\sf RVal}$ of $p$. The L1 cache is not changed.
CAS $({\tt cas}(x,e_{0},e_{1}))$: A compare and swap transition acts directly
on the LLC. The ${\tt cas}$ is only enabled when $x$ is not in the L1 cache of
$p$, and the value of $x$ in the LLC equals $e_{0}$ (under ${\sf RVal}$). When
the instruction is executed, it atomically writes the value of $e_{1}$
directly to the LLC in the same way as a synchronized write would.
#### 4.3.2. System Event Semantics.
The system may non-deterministically (i.e., at any time) perform a system
event. A system event is not a program instruction, and so will not change the
program counter (label) of a process. We will define a transition relation
$\stackrel{{\scriptstyle\omega}}{{\longrightarrow}}$, induced by the system
event $\omega$. There are three types of system events as follows.
Eviction $({\tt evict}({p,x}))$: An ${\tt evict}({p,x})$ system event may
occur when $x$ is valid and clean in the L1 of process $p$. When the event
occurs, $x$ is removed from the L1 of $p$.
Write-LLC $({\tt wrllc}({p,x}))$: If the entry of $x$ is dirty in the L1 of
$p$, then a ${\tt wrllc}({p,x})$ event may occur. The value of $x$ in the LLC
is then updated with the value of $x$ in the L1 of $p$. The entry of $x$ in
the L1 of $p$ becomes clean.
Fetch $({\tt fetch}({p,x}))$: If $x$ does not have an entry in the L1 of $p$,
then $p$ may fetch the value of $x$ from the LLC, and create a new, clean
entry with that value for $x$ in its L1.
### 4.4. Program Semantics under an SI Protocol
In a self-invalidation protocol without self-downgrade, a writing process will
be downgraded and forced to communicate its dirty data when another process
accesses that location in the LLC. This behavior can be modelled by a
semantics where writes take effect atomically with respect to the LLC. Hence,
to modify the semantics given in Section 4.3 such that it models a program
under an Si protocol, it suffices to interpret all write instructions as the
corresponding ${\tt syncwr}$ instructions.
### 4.5. Transition Graph and the Reachability Algorithm
Our semantics allows to construct, for a given program ${\mathcal{P}}$, a
finite transition graph, where each node in the graph is a configuration in
${\mathcal{P}}$, and each edge is a transition. Figure 12 shows four nodes in
the transition graph of the program in Figure 1. The configurations $c_{2}$
and $c_{3}$ are those depicted in Figure 2, while $c_{6}$ is the configuration
we get from $c_{2}$ by adding a clean copy of y with value $1$ to the L1 of
P1; and $c_{7}$ is the configuration we get from $c_{6}$ by updating the label
of P1 to L7, and the value of $r2 to $1$. A run is a sequence
$c_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}c_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}c_{2}\cdots\stackrel{{\scriptstyle
t_{n}}}{{\longrightarrow}}c_{n}$, which is a path in the transition graph,
where $t_{i}(0\leq{}i\leq{}n)$ is either a label $\lambda$ or a system event
$\omega$. Figure 12 shows the path of the run $\rho_{3}$.
$c_{2}$$c_{6}$$c_{7}$$c_{3}$fetch(P1,y)L6L7 Figure 12. Part of the transition
graph of the program in Figure 1.
Together with the program, the user provides a safety property $\phi$ that
describes a set ${\it Bad}$ of configurations that are considered to be
errors. Checking $\phi$ for a program ${\mathcal{P}}$ amounts to checking
whether there is a run leading from the initial configuration to a
configuration in ${\it Bad}$. To do that, the input program under SiSd is
translated to the code recognized by the reachability analysis tool chosen by
the user. The translated code simulates all the behaviors which are allowed in
the SiSd semantics. Also, there is instrumentation added to simulate the
caches. Verifying the input program amounts to verifying the translated code
which is analyzed under SC. If a bad configuration is encountered, a witness
run is returned by the tool. Otherwise, the program is declared to be correct.
## 5\. Litmus Tests and Comparison with Other Memory Models
In this section, we will first compare the behavior of SiSd with other weak
memory models. We do this by presenting a sequence of litmus tests that
differentiate the SiSd semantics from the other models. Each program consists
of a number of threads that share a number of variables. Later, we will
describe a number of additional litmus tests to clarify the bahavior of the
SiSd model.
### 5.1. Comparing with Other Memory Models
##### Sequential Consistency (SC)
The SiSd model is weaker than SC. Consider the classic SB (Dekker’s) algorithm
shown in Figure 13. Under the SC semantics, the program does not have any runs
satisfying the assertion, since the operations performed by the process $P0$
are not reordered. If $\$r_{1}=0$, then $x:=1$ must have been executed and the
value 1 of $x$ is updated to the memory. At this time, $P0$ sees the value of
y as 0 in the memory, which means that $y:=1$ is not executed by $P1$ yet.
When $P1$ executes $y:=1$ and $\$r_{2}:=x$, the value of $x$ in the memory is
already $1$ and thus $1$ is assigned to $\$r_{2}$.
Any run of a program ${\mathcal{P}}$ under SC can be simulated by a run of
${\mathcal{P}}$ under SiSd as the following: 1) right after each write
operation, the assigned value of the variable is updated to LLC and the
variable is invalidated from the local cache immediately; 2) right before each
read operation, the value of the variable is fetched from LLC and after
reading the value, the variable is immediately invalidated from the local
cache.
Initially: x=0, y=0process P0registers $r1beginL1: y$x := 1;L2: y$$r1 :=
y;endprocess P1registers $r2beginL3: y := 1;L4: $r2 := x;endAssertion: $r1=0,
$r2=0 Figure 13. SB (Dekker’s) algorithm.
##### Total Store Order (TSO)
The SiSd model and TSO are not comparable. As illustrated in Figure 14, the
TSO model inserts a store buffer between each process and the shared memory.
When a process executes a write instruction, the instruction is appended to
the end of the buffer of the process. At any point of the execution, the
instruction at the head of the buffer may nondeterministically be removed and
applied to update the memory. When a process reads the value of a variable, it
fetches the value from the most recent write operation on the variable in its
buffer. If such a write operation is missing, then the value is fetched from
the memory.
Figure 14. TSO Initially: x=0, y=0process P0registersbeginL1: y$x := 1;L2: y$y
:= 1;endprocess P1registers $r1 $r2beginL3: $r1 := x;L4: $r2 :=
y;endAssertion: $r1=0, $r2=1 Figure 15. Program MP.
First, we show that TSO is not weaker than SiSd. Consider the program MP in
Figure 15. Under the TSO semantics, the program MP does not have any runs that
satisfy the assertion. The reason is that the two write operations performed
by $P0$ will reach the memory in the same order as they are performed, i.e,
$x:=1$ and then $y:=1$. Furthermore, the two read operations performed by $P1$
are not re-ordered according to the TSO semantics. Therefore, if $\$r_{2}=1$
then $P1$ has already seen that the value of $x$ is equal to $1$ when it
performs the assignment $\$r_{1}:=x$. In contrast, the program MP has the
following run under SiSd that satisfies the assertion. First, the process $P0$
assigns $1$ to $x$ and $y$ respectively, but only updates the value of $y$ to
the LLC. Next, $P1$ fetches the values of $x$ and $y$ ($0$ and $1$
respectively) from the LLC, and assigns them to the registers using the
instructions $\$r_{1}:=x$ and $\$r_{2}:=y$, which means the assertion will be
satisfied.
Initially: x=0, y=0process P0registers $r1$r2 $r3 $r4beginL1: y$x := 1;L2: y$x
:= 2;L3: y$x := 3;L4: y$x := 4;L5: y$$r1 := y;L6: y$$r2 := y;L7: y$$r3 :=
y;L8: y$$r4 := y;endprocess P1registers $r5$r6 $r7 $r8beginL9: y := 1;L10: y
:= 2;L11: y := 3;L12: y := 4;L13: $r5 := x;L14: $r6 := x;L15: $r7 := x;L16:
$r8 := x;endAssertion:$r1=1, $r2=2, $r3=3, $r4=4,$r5=1, $r6=2, $r7=3, $r8=4
Figure 16. Program ReadSeq.
The program ReadSeq in Figure 16 shows that SiSd is not weaker than TSO. Under
TSO, the program exhibits a run that satisfies the assertion as follows.
First, $P0$ executes the instructions $x:=1$, $x:=2$, $x:=3$, and $x:=4$, one
by one, and the corresponding operations are put in its buffer. Then, $P1$
executes the instructions $y:=1$, $y:=2$, $y:=3$, and $y:=4$, one by one,
again putting the corresponding operations in its buffer. Now, the memory is
updated with $x:=1$ after which $P1$ executes $\$r_{5}:=x$, thus assigning $1$
to $\$r_{5}$. Following this, the memory is updated with $x:=2$ after which
$P1$ executes $\$r_{6}:=x$, thus assigning $2$ to $\$r_{6}$. Finally $3$ and
$4$ are assigned to $\$r_{7}$ and $\$r_{8}$ respectively in similar manners. A
similar sequence of operations is performed assigning $1$, $2$, $3$ and $4$ to
$\$r_{1}$, $\$r_{2}$, $\$r_{3}$, and $\$r_{4}$ respectively.
However, under the SiSd semantics, the ReadSeq program does not have any runs
that can satisfy the assertion. More specifically, since the processes do not
have any store buffers, at most three different values of a variable can be
kept in this example. I.e., one in the local cache of the process which
executes the write operation of the variable, one in the local cache of the
process which executes the read operation of the variable, and one in the LLC.
When there are four or more values assigned to a variable in a similar manner
as in ReadSeq, the assertion that each process reads all the values of the
variable in the same order as it is written by the other process cannot be
satisfied.
Figure 17. PSO
##### Partial Store Order (PSO)
The SiSd model and PSO are not comparable. In PSO, a store buffer is inserted
for each variable, between each process and the shared memory as illustrated
in Figure 17 .
Initially: x=0, y=0process P0registersbeginL1: y$x := 1;endprocess P1registers
$r1beginL2: $r1 := x;L3: y := 1;endprocess P2registers $r2 $r3beginL4: $r2 :=
y;L5: $r3 := x;endAssertion: $r1=1, $r2=1, $r3=0 Figure 18. Program WRC.
To illustrate the difference between SiSd and PSO, we consider the program WRC
in Figure 18. Under the PSO semantics, the program WRC does not have any runs
that satisfy the assertion. More precisely, if $\$r_{1}=1$ then the write
operation $x:=1$ by $P0$ must have reached the memory before the write
operation $y:=1$ has been performed by $P2$. Furthermore, if $\$r_{2}=1$ holds
then the write operation $y:=1$ must have reached the memory before the
instruction $\$r_{3}:=x$ has been performed by $P2$. Since read instructions
are not re-ordered in PSO and since $x:=1$ reaches the memory before $y:=1$ it
follows that the value of $x$ is equal to $1$ in the memory when $\$r_{3}:=x$
is performed by $P2$, and hence $\$r_{3}=1$.
The program WRC has the following run under SiSd that satisfies the assertion.
The process $P2$ fetches the initial value $0$ of the variable $x$. The
process $P0$ assigns $1$ to $x$ and updates the value to the LLC. The process
$P1$ fetches the value $1$ of $x$ from the LLC and then executes the
instruction $\$r_{1}:=x$ which means that the value of $\$r_{1}$ is equal to
$1$. Then, $P1$ assigns the value $1$ to the variable $y$ and updates the
value to the LLC. The process $P2$ fetches the value of $y$ from the LLC and
then executes the instruction $\$r_{2}:=y$ which means that $\$r_{2}=1$.
Finally, $P2$ executes the instruction $\$r_{3}:=x$ and we get $\$r_{3}=0$.
As PSO is weaker than TSO, we can use the same example in Figure 16 to show
that the ReadSeq program also has a run under PSO satisfying the assertion,
which means that SiSd is not weaker than PSO.
##### POWER/ARM
The SiSd model and POWER are not comparable. We show that the POWER memory
model is not weaker than SiSd with the program PwrEg in Figure 19. The program
does not have any runs that satisfy the assertion under POWER, since cycles of
the form write-(sync)$\rightarrow$write-(read from)$\rightarrow$ read-(address
dependency)$\rightarrow$read-(from read)$\rightarrow$ are not allowed by
POWER. The sync instruction maintains the order between the two write
instructions and the address dependency maintains the order between the two
read instructions, which make the cycle impossible.
Initially: x=0, y=0process P0registers beginL1: y$x := 1;L2: sync;L3: y$y :=
1;endprocess Pregisters $r1 $r2beginL4: $r1 := y;L5: $r2 := (&x +
0*$r1);endAssertion: $r1=1, $r2=0 Figure 19. PwrEg Initially: x=0, y=0process
P0registers beginL1: y$x := 1;L2: fence;L3: y$y := 1;endprocess P1registers
$r1 $r2beginL4: $r1 := y;L5: $r2 := x;endAssertion: $r1=1, $r2=0 Figure 20.
SisdEg
However, under SiSd the program SisdEg in Figure 20 has the run that satisfies
the assertion. The process $P1$ fetches the initial value 0 of the variable
$x$. The process $P0$ assigns 1 to $x$, updates the value to the LLC, assigns
1 to $y$, and updates the value to the LLC again. The process $P1$ fetches the
value 1 of $y$ from the LLC and then executes the instruction $\$r_{1}:=y$.
Therefore, the value of $\$r_{1}$ is equal to 1. Finally, $P1$ executes the
instruction $\$r_{2}:=x$ and we get $\$r_{2}=0$.
We can show that SiSd is not weaker than POWER with the program ReadSeq in
Figure 16.
We can also show that the SiSd model and ARM are not comparable with the
programs in Figure 16, Figure 19 and Figure 20.
##### Relaxed Memory Order (RMO)
We can show that SiSd is not weaker than RMO again with the program ReadSeq in
Figure 16, since RMO allows the reorder between both 1) write and write
operations and 2) read and read/write operations.
We keep the other direction open since we have not found any examples as the
proof.
### 5.2. Further Litmus Tests
We describe the behavior of the SiSd model for three more litmus tests.
##### Load-Buffering (LB)
The LB program in Figure 21 does not have any runs that can satisfy the
assertion under SiSd. If $\$r_{1}=1$, then process $P1$ must have executed the
write instruction $x:=1$ and updated the value 1 of $x$ to the LLC before $P0$
has executed the instruction $\$r_{1}:=x$. This means when $P1$ executes the
read instruction $\$r_{2}:=y$, the value of $y$ has not been updated by $P0$,
and thus $\$r_{2}=0$.
Initially: x=0, y=0process P0registers $r1beginL1: y$$r1 := x;L2: y$y :=
1;endprocess P1registers $r2beginL3: $r2 := y;L4: x := 1;endAssertion: $r1=1,
$r2=1 Figure 21. Program LB.
##### ISA2
The program ISA2 in Figure 22 has the following run under SiSd that satisfies
the assertion. The process $P2$ fetches the initial value $0$ of the variable
$x$. The process $P0$ assigns $1$ to $x$ and $y$, and updates the value to the
LLC. The process $P1$ fetches the value $1$ of $y$ from the LLC and then
executes the instruction $\$r_{1}:=y$, which means that the value of $\$r_{1}$
is equal to $1$. Then, $P1$ assigns the value $1$ to the variable $z$ and
updates the value to the LLC. The process $P2$ fetches the value of $z$ from
the LLC and then executes the instruction $\$r_{2}:=z$, which means that
$\$r_{2}=1$. Finally, $P2$ executes the instruction $\$r_{3}:=x$ and we get
$\$r_{3}=0$.
Initially: x=0, y=0, z=0process P0registersbeginL1: y$x := 1;L2: y$y :=
1;endprocess P1registers $r1beginL3: $r1 := y;L4: z := 1;endprocess
P2registers $r2 $r3beginL5: $r2 := z;L6: $r3 := x;endAssertion: $r1=1, $r2=1,
$r3=0 Figure 22. Program ISA2.
##### IRIW
The program IRIW in Figure 23 has the following run under SiSd that satisfies
the assertion. The processes $P1$ and $P3$ fetch the initial values $0$ of the
variables $y$ and $x$ respectively. The process $P0$ assigns $1$ to $x$ and
updates the values to the LLC. The process $P1$ fetches the value $1$ of $x$
from the LLC and executes the instruction $\$r_{1}:=x$, which means that the
value of $\$r_{1}$ is equal to $1$. Then $P1$ executes the instruction
$\$r_{2}:=y$, which means that $\$r_{2}=0$. After that, the process $P2$
assigns the value $1$ to the variable $y$ and updates the value to the LLC.
The process $P3$ fetches the value of $y$ from the LLC and then executes the
instruction $\$r_{3}:=y$, which means that $\$r_{3}=1$. Finally, $P3$ executes
the instruction $\$r_{4}:=x$ and we get $\$r_{4}=0$.
Initially: x=0, y=0process P0registersbeginL1: y$x := 1;endprocess P1registers
$r1 $r2beginL2: $r1 := x;L3: $r2 := y;endprocess P2registersbeginL4: y :=
1;endprocess P3registers $r3 $r4beginL5: $r3 := y;L6: $r4 := x;endAssertion:
$r1=1, $r2=0, $r3=1, $r4=0 Figure 23. Program IRIW.
## 6\. Fence Insertion
In this section we describe our fence insertion procedure, which is closely
related to the algorithm described in [LNP+12]. Given a program
${\mathcal{P}}$, a cost function $\kappa$ and a safety property $\phi$, the
procedure finds all the sets of fences that are optimal for ${\mathcal{P}}$
w.r.t. $\phi$ and $\kappa$.
In this section we take _fence constraint_ (or _fence_ for short) to mean a
pair $(\lambda,f)$ where $\lambda$ is a statement label and $f$ is a fence
instruction. A fence constraint $(\lambda,f)$ should be interpreted as the
notion of inserting the fence instruction $f$ into a program, between the
statement labeled $\lambda$ and the next statement (labeled by ${\sf
Next}\left(\lambda\right)$)222This definition can be generalized. Our
prototype tool does indeed support a more general definition of fence
positions, which is left out of the article for simplicity.. For a program
${\mathcal{P}}$ and a set $F$ of fence constraints, we define
${{\mathcal{P}}}\oplus{F}$ to mean the program ${\mathcal{P}}$ where all fence
constraints in $F$ have been inserted. To avoid ambiguities in the case when
$F$ contains multiple fence constraints with the same statement label (e.g
$(\lambda,{\tt llfence})$ and $(\lambda,{\tt ssfence})$), we assume that
fences are always inserted in some fixed order.
[Soundness of Fence Sets] For a program ${\mathcal{P}}$, safety property
$\phi$, and set $F$ of fence constraints, the set $F$ is sound for
${\mathcal{P}}$ w.r.t. $\phi$ if ${{\mathcal{P}}}\oplus{F}$ satisfies $\phi$
under SiSd.
A _cost function_ $\kappa$ is a function from fence constraints to positive
integer costs. We extend the notion of a cost function to sets of fence
constraints in the natural way: For a cost function $\kappa$ and a set $F$ of
fence constraints, we define $\kappa(F)=\sum_{c\in{}F}\kappa(c)$.
[Optimality of Fence Sets] For a program ${\mathcal{P}}$, safety property
$\phi$, cost function $\kappa$, and set $F$ of fence constraints, $F$ is
optimal for ${\mathcal{P}}$ w.r.t. $\phi$ and $\kappa$ if $F$ is sound for
${\mathcal{P}}$ w.r.t. $\phi$, and there is no sound fence set $G$ for
${\mathcal{P}}$ w.r.t. $\phi$ where $\kappa(G)<\kappa(F)$.
Observe that the optimality is evaluated with the number of occurrences of
fences in the source program and the costs of different fences.
In order to introduce our algorithm, we define the notion of a _hitting set_.
[Hitting Set] For a set $S=\\{S_{0},\cdots,S_{n}\\}$ of sets
$S_{0},\cdots,S_{n}$, and a set $T$, we say that $T$ is a hitting set of $S$
if $T\cap{}S_{i}\neq\varnothing$ for all $0\leq{}i\leq{}n$.
For example $\\{a,d\\}$ is a hitting set of
$\\{\\{a,b,c\\},\\{d\\},\\{a,e\\}\\}$. For a set $S$ of sets, hitting sets of
$S$ can be computed using various search techniques, such as constraint
programming. We will assume that we are given a function
$\textrm{{hits}}(S,\kappa)$ which computes all hitting sets for $S$ which are
cheapest w.r.t. $\kappa$. I.e., for a set $S$ of finite sets, and a cost
function $\kappa$, the call $\textrm{{hits}}(S,\kappa)$ returns the set of all
sets $T$ with $T\subseteq\bigcup_{S_{i}\in{}S}S_{i}$ such that
* •
$T$ is a hitting set of $S$, and
* •
there is no hitting set $T^{\prime}$ of $S$ such that
$\kappa(T^{\prime})<\kappa(T)$.
Fencins(${\mathcal{P}}$,$\phi$,$\kappa$)
---
MM1: opt := $\varnothing$; // Optimal fence sets
MM2: req := $\varnothing$; // Known requirements
MM3:
while($\exists\textrm{{F}}\in\textrm{{hits}}(\textrm{{req}},\kappa)\setminus\textrm{{opt}}$){
MM4: $\pi$ :=
$\textrm{{reachable}}(\textrm{${{\mathcal{P}}}\oplus{\textrm{{F}}}$},\phi)$;
MM5: if($\pi=\perp$){
MM6: opt := $\textrm{{opt}}\cup\\{\textrm{{F}}\\}$;
MM7: }else{ // $\pi$ is a witness run.
MM8: C :=
$\textrm{{analyze\\_witness}}(\textrm{${{\mathcal{P}}}\oplus{\textrm{{F}}}$},\pi)$;
MM9: if($\textrm{{C}}=\varnothing$){ // error under SC!
MM10: return $\varnothing$;
MM11: }
MM12: req := $\textrm{{req}}\cup\\{\textrm{{C}}\\}$;
MM13: }
MM14: }
MM15: return opt;
Figure 24. The fence insertion algorithm.
We present our fence insertion algorithm in Figure 24. The algorithm keeps two
variables opt and req. Both are sets of fence constraint sets, but are
intuitively interpreted in different ways. The set opt contains all the
optimal fence constraint sets for ${\mathcal{P}}$ w.r.t. $\phi$ and $\kappa$
that have been found thus far. The set req is used to keep track of the
requirements that have been discovered for which fences are necessary for
soundness of ${\mathcal{P}}$. We maintain the following invariant for req: Any
fence constraint set $F$ which is sound for ${\mathcal{P}}$ w.r.t. $\phi$ is a
hitting set of req. As the algorithm learns more about ${\mathcal{P}}$, the
requirements in req will grow, and hence give more information about what a
sound fence set may look like. Notice that the invariant holds trivially in
the beginning, when $\textrm{{req}}=\varnothing$.
In the loop from lines 24-24 we repeatedly compute a candidate fence set F
(line 24), insert it into ${\mathcal{P}}$, and call the reachability analysis
to check if F is sound (line 24). We assume that the call
$\textrm{{reachable}}(\textrm{${{\mathcal{P}}}\oplus{\textrm{{F}}}$},\phi)$
returns $\perp$ if $\phi$ is unreachable in
${{\mathcal{P}}}\oplus{\textrm{{F}}}$, and a witness run otherwise. If
${{\mathcal{P}}}\oplus{\textrm{{F}}}$ satisfies the safety property $\phi$,
then F is sound. Furthermore, since F is chosen as one of the cheapest (w.r.t.
$\kappa$) hitting sets of req, and all sound fence sets are hitting sets of
req, it must also be the case that F is optimal. Therefore, we add F to opt on
line 24.
If ${{\mathcal{P}}}\oplus{\textrm{{F}}}$ does not satisfy the safety property
$\phi$, then we proceed to analyze the witness run $\pi$. The witness analysis
procedure is outlined in Section 6.1. The analysis will return a set C of
fence constraints such that any fence set which is restrictive enough to
prevent the erroneous run $\pi$ must contain at least one fence constraint
from C. Since every sound fence set must prevent $\pi$, this means that every
sound fence set must have a non-empty intersection with C. Therefore we add C
to req on line 24, so that req will better guide our choice of fence set
candidates in the future.
Note that in the beginning, $\textrm{{hits}}(\textrm{{req}},\kappa)$ will
return a singleton set of the empty set, namely $\\{\varnothing\\}$. Then F is
chosen as the empty set $\varnothing$ and the algorithm continues. A special
case occurs when the run $\pi$ contains no memory access reorderings. This
means that ${\mathcal{P}}$ can reach the bad states even under the SC memory
model. Hence it is impossible to correct ${\mathcal{P}}$ by only inserting
fences. The call
$\textrm{{analyze\\_witness}}(\textrm{${{\mathcal{P}}}\oplus{\textrm{{F}}}$},\pi)$
will in this case return the empty set. The main algorithm then terminates,
also returning the empty set, indicating that there are no optimal fence sets
for the given problem.
### 6.1. Witness Analysis
The analyze_witness function takes as input a program ${\mathcal{P}}$ (which
may already contain some fences inserted by the fence insertion algorithm),
and a counter-example run $\pi$ generated by the reachability analysis. The
goal is to find a set $G$ of fences such that
* •
all sound fence sets have at least one fence in common with $G$ and
* •
$G$ contains no fence which is already in ${\mathcal{P}}$.
It is desirable to keep $G$ as small as possible, in order to quickly converge
on sound fence sets.
There are several ways to implement analyze_witness to satisfy the above
requirements. One simple way builds on the following insight: Any sound fence
set must prevent the current witness run. The only way to do that, is to have
fences preventing some access reordering that occurs in the witness. So a set
$G$ which contains all fences preventing some reordering in the current
witness satisfies both requirements listed above.
As an example, consider Figure 25. On the left, we show part of a program
${\mathcal{P}}$ where the thread P0 performs three memory accesses L0, L1 and
L2. On the right, we show the corresponding part of a counter-example run
$\pi$. We see that the store L0 becomes globally visible at line 7, while the
loads L1 and L2 access the LLC at respectively lines 3 and 5. Hence the order
between the instructions L0 and L1 and the order between L0 and L2 in the
program code, is opposite to the order in which they take effect w.r.t. the
LLC in $\pi$. We say that L0 is _reordered_ with L1 and L2. The loads are not
reordered with each other. Let us assume that $\pi$ does not contain any other
memory access reordering.
Program fragment | Witness run
---|---
| process P0
---
...
L0: x := 1;
L1: $\$r_{0}$ := y;
L2: $\$r_{1}$ := z;
...
| | ...
---|---
1. | fetch(P0,x)
2. | L0: x := 1
3. | fetch(P0,y)
4. | L1: $\$r_{0}$ := y
5. | fetch(P0,z)
6. | L2: $\$r_{1}$ := z
| ...
7. | wrllc(P0,x)
| ...
Figure 25. Left: Part of a program ${\mathcal{P}}$, containing three
instructions of the thread P0. Right: A part of a counter-example run $\pi$ of
${\mathcal{P}}$.
The reordering is caused by the late ${\tt wrllc}$ on line 7\. Hence, this
particular error run can be prevented by the following four fence constraints:
$c_{0}=(\texttt{L0},{\tt ssfence})$, $c_{1}=(\texttt{L1},{\tt ssfence})$,
$c_{2}=(\texttt{L0},{\tt fence})$, and $c_{3}=(\texttt{L1},{\tt fence})$. The
fence set returned by $\textrm{{analyze\\_witness}}({\mathcal{P}},\pi)$ is
$G=\\{c_{0},c_{1},c_{2},c_{3}\\}$. Notice that $G$ satisfies both of the
requirements for analyze_witness.
## 7\. Experimental Results
We have implemented our fence insertion algorithm together with a reachability
analysis for SiSd in the tool Memorax. It is publicly available at
https://github.com/memorax/memorax. We apply the tool to a number of
benchmarks (Section 7.1). Using simulation, we show the positive impact of
using different types of fences, compared to using only the full fence, on
performance and network traffic (Section 7.2).
### 7.1. Fence Insertion Results
We evaluate the automatic fence insertion procedure by running our tool on a
number of different benchmarks containing racy code. For each example, the
tool gives us all optimal sets of fences. We run our tool on the same
benchmarks both for SiSd and for the Si protocol.333Our methods could also run
under a plain Sd protocol. However, to our knowledge, no cache coherence
protocol employs only Sd without Si. The results for SiSd are given in Table
1. We give the benchmark sizes in lines of code. All benchmarks have 2 or 3
processes. The fence insertion procedure was run single-threadedly on a 3.07
GHz Intel i7 CPU with 6 GB RAM.
| Only full fence | Mixed fences
---|---|---
Benchmark | Size (LOC) | Time | #solutions | #fences | Time | #solutions | Fences / proc
bakery | 45 | 17.3 s | 4 | 5 | 108.1 s | 16 | 2 sw, 4 ll, 1 ss
cas | 32 | $<$0.1 s | 1 | 2 | $<$0.1 s | 1 | 1 ll, 1 ss
clh | 37 | 4.4 s | 4 | 4 | 3.7 s | 1 | 3 sw, 2 ll, 1 ss
dekker | 48 | 2.0 s | 16 | 3 | 2.9 s | 16 | 1 sw,2 ll, 1 ss
mcslock | 67 | 15.6 s | 4 | 2 | 33.0 s | 4 | 1 ll, 1 ss
testtas | 38 | $<$0.1 s | 1 | 2 | $<$0.1 s | 1 | 1 ll, 1 ss
srbarrier | 60 | 0.3 s | 9 | 3 | 0.4 s | 4 | 2 ll, 1 ss
treebarrier | 56 | 33.2 s | 12 | 1 / 2 | 769.9 s | 132 | 1 ll, 1 ss
dclocking | 44 | 0.8 s | 16 | 4 | 0.9 s | 16 | 1 sw, 2 ll, 1 ss
postgresql | 32 | $<$0.1 s | 4 | 2 | 0.1 s | 4 | 1 ll, 1 ss
barnes 1 | 30 | 0.2 s | 1 | 1 | 0.5 s | 1 | 1 ll / 1 ss
barnes 2 | 96 | 16.3 s | 16 | 1 | 16.1 s | 16 | 1 ss
cholesky | 98 | 1.6 s | 1 | 0 | 1.6 s | 1 | 0
radiosity | 196 | 25.1 s | 1 | 0 | 24.6 s | 1 | 0
raytrace | 101 | 69.3 s | 1 | 0 | 70.1 s | 1 | 0
volrend | 87 | 376.2 s | 1 | 0 | 376.9 s | 1 | 0
Table 1. Automatic fence insertion for SiSd.
The first set of benchmarks are classical examples from the context of lock-
free synchronization. They contain mutual exclusion algorithms: a simple CAS
lock –cas–, a test & TAS lock – _tatas_ – [Sco13], Lamport’s bakery algorithm
–bakery– [Lam74], the MCS queue lock –mcsqueue– [MCS91], the CLH queue lock
–clh– [MLH94], and Dekker’s algorithm –dekker– [Dij02]. They also contain a
work scheduling algorithm –postgresql–444http://archives.postgresql.org/pgsql-
hackers/2011-08/msg00330.php, and an idiom for double-checked locking
–dclocking– [SH96], as well as two process barriers –srbarrier– [Sco13] and
–treebarrier– [MCS91]. The second set of benchmarks are based on the Splash-2
benchmark suite [WOT+95]. We use the race detection tool Fast&Furious [RK15b]
to detect racy parts in the Splash-2 code. We then manually extract models
capturing the core of those parts.
In four cases the tool detects bugs in the original Splash-2 code. The
_barnes_ benchmark is an n-body simulation, where the bodies are kept in a
shared tree structure. We detect two bugs under SiSd: When bodies are inserted
(_barnes 2_), some bodies may be lost. When the center of mass is computed for
each node (_barnes 1_), some nodes may neglect entirely the weight of some of
their children. Our tool inserts fences that prevent these bugs. The
_radiosity_ model describes a work-stealing queue that appears in the Splash-2
_radiosity_ benchmark. Our tool detects data race from it. After careful code
inspection we find that it is possible for all workers but one to terminate
prematurely, leaving one worker to do all remaining work, which is caused by
data race. [SLKR16] The _volrend_ model is also a work-stealing queue. Our
tool detects that it is possible for some tasks to be performed twice. The
bugs in _radiosity_ and _volrend_ can occur even under SC. Hence the code
cannot be fixed only by adding fences. Instead we manually correct it.
For each benchmark, we apply the fence insertion procedure in two different
modes. In the first one (“Only full fence”), we use only full fences. In the
table, we give the total time for computing all optimal sets, the number of
such sets, and the number of fences to insert into each process. For
_treebarrier_ , one process (the root process) requires only one fence, while
the others require two. Notice also that if a benchmark has one solution with
zero fence, that means that the benchmark is correct without the need to
insert any fences.
In the second set of experiments (“Mixed fences”), we allow all four types of
fences, using a cost function assigning a cost of ten units for a full fence,
five units for an ${\tt ssfence}$ or an ${\tt llfence}$, and one unit for a
synchronized write. These cost assignments are reasonable in light of our
empirical evaluation of synchronization cost in Section 7.2. We list the
number of inserted fences of each kind. In _barnes 1_ , the processes in the
model run different codes. One process requires an ${\tt llfence}$, the other
an ${\tt ssfence}$.
In addition to running our tool for SiSd, we have also run the same benchmarks
for Si. As expected, ${\tt ssfence}$ and ${\tt syncwr}$ are no longer
necessary, and ${\tt fence}$ may be downgraded to ${\tt llfence}$. Otherwise,
the inferred fence sets are the same as for SiSd. Since Si allows fewer
behaviors than SiSd, the inference for Si is mostly faster. Each benchmark is
fenced under Si within 71 seconds.
### 7.2. Simulation Results
Here we show the impact of different choices of fences when executing
programs. In particular we show that an optimal fence set using the “Mixed
fences” cost function yields a better performance and network traffic compared
to an optimal fence set using the “Only full fence” cost function. Here
network traffic refers to the traffic in both the on-chip interconnection
network and the memory bus. We account for all the traffic due to coherence
messages. We evaluate the micro-benchmarks analyzed in the previous section
and the whole Splash-2 benchmark suite [WOT+95], running the applications from
beginnig to end, but presenting results only for the parallel phase of the
applications. All programs are fenced according to the optimal fence sets
produced by our tool as described above.
##### Simulation Environment:
We use the Wisconsin GEMS simulator [MSB+05]. We model an in-order processor
that with the Ruby cycle-accurate memory simulator (provided by GEMS) offers a
detailed timing model. The simulated system is a 64-core chip multiprocessor
implementing the SiSd protocol described in Section 2 and 32KB, 4-way, private
L1 caches and a logically shared but physically distributed L2 with 64 banks
of 256KB, 16-way each.
##### Cost of Fences:
Our automatic fence insertion tool employs different weights in order to
insert the optimal amount of fences given the cost of each fence. Here, we
calculate the weights based on an approximate cost of fences obtained by our
simulations.
The effect of fences on performance is twofold. First, there is a cost to
execute the fence instructions (fence latency); the more fences and the more
dirty blocks to self-downgrade, the higher the penalty. Second, fences affect
cache miss ratio (due to self-invalidation) and network traffic (due to extra
fetches caused by self-invalidations and write-throughs caused by self-
downgrades). The combined effect on cache misses and network traffic also
affects performance.
(a) Cycles
(b) Traffic
Figure 26. Percentage of cycles and traffic that each type of fence cost
We calculate the cost of fences in terms of execution as indicated in equation
1, where $\mathit{latency}_{\mathit{fence}}$ is the time in cycles required by
the fence, $\mathit{misses}_{\mathit{si}}$ is the number of misses caused by
self-invalidation, and $\mathit{latency}_{\mathit{miss}}$ is the average
latency of such misses. According to this equation, and considering a protocol
implementing the DoI state described in Section 2, the average percentage of
cycles (execution time) employed by each type of fence when running the
Splash2 benchmarks is the following: the cost of an llfence is 64.4%, the cost
of an ssfence is 22.4%, and the cost of a syncwr is 13.2%, as shown in Figure
26a.
$\mathit{time}_{\mathit{fence}}=\mathit{latency}_{\mathit{fence}}+\mathit{misses}_{\mathit{si}}\times\mathit{latency}_{\mathit{miss}}$
(1)
The cost of the fences in traffic is calculated as indicated in equation 2,
where $\mathit{sd}$ is the number of self-downgrades,
$\mathit{traffic}_{\mathit{wt}}$ is the traffic caused by a write-through, and
$\mathit{traffic}_{\mathit{miss}}$ is the traffic caused by a cache miss. In
percentage, the cost of the each type of fence on average in terms of traffic
is 40.4% for an llfence, 50.3% for an ssfence, and 9.3% for a syncwr, as shown
in Figure 26b. Thus, the weights assigned to fences in our tool seem
reasonable.
$\mathit{traffic}_{\mathit{fence}}=\mathit{sd}*\mathit{traffic}_{\mathit{wt}}+\mathit{misses}_{\mathit{si}}\times\mathit{traffic}_{\mathit{miss}}$
(2)
##### Cache Misses:
As mentioned, the fences affect the cache miss rate. Figure 27 shows clearly
the effect of self-invalidation and self-downgrade on misses. First we show
the misses due to cold capacity and conflict misses (_Cold-cap-conf_), which,
in general, are not affected by the type of fences. However, in some cases
reducing the self-invalidation can give the appearance of extra capacity
misses because of having a more occupied cache. The graph does not plot
(a) Micro-benchmarks
(b) Splash2 benchmarks
Figure 27. Normalized cache misses under different fence sets and protocol
states
coherence misses since fenced programs on Sisd coherence do not induce cache
misses due to coherence transactions. The second kind of miss is named as
_Bypass_. These misses are due to atomic operations which cannot use the data
in the private cache, but need to access it from the shared cache. They are
very frequent in the micro-benchmarks (Figure 27a), which are synchronization
intensive, but almost unnoticeable for the Splash2 benchmarks (Figure 27b).
Finally, we show the misses caused by self-invalidation _Self-inv_. These are
the ones which number is reduced, when applying the mixed fences, but also
when employing the DoI state, since dirty words are not invalidated.
##### Traffic:
As already mentioned, traffic is also affected by the type of fences employed.
Figure 28 shows the traffic in the on-chip network generated by these
applications. The use of llfence, ssfence, syncwr is able to reduce the
traffic requirements by 11.1% for the micro-benchmarks and 1.6% for the
Splash2 applications, on average, compared to using full fences. Additionally,
when employing the DoI state, this reduction reaches 21.3% and 1.9%, on
average, for the micro-benchmarks and the Splash2, respectively. Again, the
more synchronization is required by the applications, the more traffic can be
saved by employing mixed fences.
(a) Micro-benchmarks
(b) Splash2 benchmarks
Figure 28. Normalized network traffic under different fence sets and protocol
states
##### Execution Time:
Finally, we show the impact on execution time, which is affected by the
reductions in cache misses and traffic. Figure 29 shows simulated execution
time for both the micro-benchmarks (Figure 29a) and the Splash2 benchmarks
(Figure 29b). The use of mixed fences improves the execution time compared to
using full fences by 10.4% for the micro-benchmarks and by 1.0% for the
Splash2 benchmarks. The DoI-mixed column shows the execution time results for
the same mixed fence sets as the mixed column. But in DoI case, llfences are
implemented in GEMS using an extra L1 cache line state (the Dirty-or-Invalid
state). This feature is an architectural optimization of the SiSd protocol.
Implementing the DoI state further improves the performance of the mixed
fences, by 20.0% for the micro-benchmarks and 2.1% for the Splash2, on
average, compared to using of full fences. Mixed fences are useful for
applications with more synchronization. Applications using more
synchronization would benefit to a large extent from the use of mixed fences.
(a) Micro-benchmarks
(b) Splash2 benchmarks
Figure 29. Execution time under different fence sets and protocol states
## 8\. Conclusions and Future Work
We have presented a uniform framework for automatic fence insertion in
programs that run on architectures that provide self-invalidation and self-
downgrade. We have implemented a tool and applied it on a wide range of
benchmarks. There are several interesting directions for future work. One is
to instantiate our framework in the context of abstract interpretation and
stateless model checking. While this will compromise the optimality criterion,
it will allow more scalability and application to real program code. Another
direction is to consider robustness properties [BMM11]. In our framework, this
would mean that we consider program traces (in the sense of Shasha and Snir
[SS88]), and show that the program will not exhibit more behaviors under SiSd
than under SC. While this may cause over-fencing, it frees the user from
providing correctness specifications such as safety properties. Also, the
optimality of fence insertion can be evaluated with the number of the times
that each fence is executed. This measurement will provide more accuracy when,
for instance, fences with different weights are inserted in a loop computation
in a branching program.
##### Acknowledgment
This work was supported by the Uppsala Programming for Multicore Architectures
Research Center (UPMARC), the Swedish Board of Science project, “Rethinking
the Memory System”, the “Fundación Seneca-Agencia de Ciencia y Tecnología de
la Región de Murcia” under the project “Jóvenes Líderes en Investigación” and
European Commission FEDER funds.
## References
* [AAC+12] Parosh Aziz Abdulla, Mohamed Faouzi Atig, Yu-Fang Chen, Carl Leonardsson, and Ahmed Rezine. Counter-example guided fence insertion under TSO. In TACAS, pages 204–219. Springer, 2012.
* [ADC11] Thomas J. Ashby, Pedro Díaz, and Marcelo Cintra. Software-based cache coherence with hardware-assisted selective self-invalidations using bloom filters. IEEE Transactions on Computers (TC), 60(4):472–483, April 2011\.
* [AH90] Sarita V. Adve and Mark D. Hill. Weak ordering – a new definition. In ISCA, pages 2–14, 1990.
* [AKNP14] Jade Alglave, Daniel Kroening, Vincent Nimal, and Daniel Poetzl. Don’t sit on the fence - A static analysis approach to automatic fence insertion. In CAV, pages 508–524, 2014.
* [BDM13] Ahmed Bouajjani, Egor Derevenetc, and Roland Meyer. Checking and enforcing robustness against TSO. In Programming Languages and Systems, pages 533–553. Springer, 2013\.
* [BMM11] Ahmed Bouajjani, Roland Meyer, and Eike Möhlmann. Deciding robustness against total store ordering. In ICALP (2), volume 6756 of LNCS, pages 428–440. Springer, 2011.
* [CKS+11] Byn Choi, Rakesh Komuravelli, Hyojin Sung, Robert Smolinski, Nima Honarmand, Sarita V. Adve, Vikram S. Adve, Nicholas P. Carter, and Ching-Tsun Chou. DeNovo: Rethinking the memory hierarchy for disciplined parallelism. In PACT, pages 155–166, 2011.
* [CL05] David Chase and Yossi Lev. Dynamic circular work-stealing deque. In SPAA, pages 21–28, 2005.
* [Dij02] E. W. Dijkstra. Cooperating sequential processes. 2002\.
* [DRHK15] Mahdad Davari, Alberto Ros, Erik Hagersten, and Stefanos Kaxiras. An efficient, self-contained, on-chip, directory: DIR1-SISD. In PACT, pages 317–330, 2015.
* [DSS06] David Dice, Ori Shalev, and Nir Shavit. Transactional locking II. In DISC, volume 4167 of Lecture Notes in Computer Science, pages 194–208, 2006.
* [GHS09] Rachid Guerraoui, Thomas A. Henzinger, and Vasu Singh. Software transactional memory on relaxed memory models. In Computer Aided Verification, 21st International Conference, CAV 2009, Grenoble, France, June 26 - July 2, 2009. Proceedings, pages 321–336, 2009.
* [GHS11] Rachid Guerraoui, Thomas A. Henzinger, and Vasu Singh. Verification of STM on relaxed memory models. Formal Methods in System Design, 39(3):297–331, 2011.
* [HHB+14] Derek R. Hower, Blake A. Hechtman, Bradford M. Beckmann, Benedict R. Gaster, Mark D. Hill, Steven K. Reinhardt, and David A. Wood. Heterogeneous-race-free memory models. In ASPLOS, pages 427–440, 2014.
* [HS08] Maurice Herlihy and Nir Shavit. The Art of Multiprocessor Programming. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 2008.
* [KK11] Stefanos Kaxiras and Georgios Keramidas. SARC coherence: Scaling directory cache coherence in performance and power. IEEE Micro, 30(5):54–65, September 2011.
* [KR13] Stefanos Kaxiras and Alberto Ros. A new perspective for efficient virtual-cache coherence. In ISCA, pages 535–547, 2013.
* [KRHK16] Konstantinos Koukos, Alberto Ros, Erik Hagersten, and Stefanos Kaxiras. Building heterogeneous unified virtual memories UVMS without the overhead. ACM TACO, 13(1):1:1–1:22, 2016.
* [KVY10] Michael Kuperstein, Martin Vechev, and Eran Yahav. Automatic inference of memory fences. In FMCAD, pages 111–119. IEEE, 2010.
* [Lam74] Leslie Lamport. A new solution of Dijkstra’s concurrent programming problem. Communications of the ACM, 17, August 1974.
* [Lam79] Leslie Lamport. How to make a multiprocessor computer that correctly executes multiprocess programs. IEEE Transactions on Computers (TC), 28(9):690–691, September 1979\.
* [LNP+12] Feng Liu, Nayden Nedev, Nedyalko Prisadnikov, Martin T. Vechev, and Eran Yahav. Dynamic synthesis for relaxed memory models. In PLDI, pages 429–440, 2012.
* [LW95] Alvin R. Lebeck and David A. Wood. Dynamic self-invalidation: Reducing coherence overhead in shared-memory multiprocessors. In ISCA, pages 48–59, 1995.
* [MCS91] J. M. Mellor-Crummey and M. L. Scott. Algorithms for scalable synchronization on shared-memory multiprocessors. ACM Transactions on Computer Systems (TOCS), 9, February 1991.
* [MLH94] Peter Magnusson, Anders Landin, and Erik Hagersten. Queue locks on cache coherent multiprocessors. In Proceedings of the Eighth International Parallel Processing Symposium, pages 165–171. IEEE, 1994.
* [MSB+05] Milo M.K. Martin, Daniel J. Sorin, Bradford M. Beckmann, Michael R. Marty, Min Xu, Alaa R. Alameldeen, Kevin E. Moore, Mark D. Hill, and David A. Wood. Multifacet’s general execution-driven multiprocessor simulator (GEMS) toolset. Computer Architecture News, 33(4):92–99, September 2005.
* [RDK15] Alberto Ros, Mahdad Davari, and Stefanos Kaxiras. Hierarchical private/shared classification: the key to simple and efficient coherence for clustered cache hierarchies. In HPCA, pages 186–197, 2015.
* [RK12] Alberto Ros and Stefanos Kaxiras. Complexity-effective multicore coherence. In PACT, pages 241–252, 2012.
* [RK15a] Alberto Ros and Stefanos Kaxiras. Callback: Efficient synchronization without invalidation with a directory just for spin-waiting. In ISCA, pages 427–438, 2015.
* [RK15b] Alberto Ros and Stefanos Kaxiras. Fast&furious: A tool for detecting covert racing. In PARMA and DITAM, pages 1–6, 2015.
* [RK16] Alberto Ros and Stefanos Kaxiras. Racer: TSO consistency via race detection. In 49th IEEE/ACM Int’l Symp. on Microarchitecture (MICRO), 2016\.
* [SA15] Hyojin Sung and Sarita V. Adve. DeNovoSync: Efficient support for arbitrary synchronization without writer-initiated invalidations. In ASPLOS, pages 545–559, 2015.
* [Sco13] Michael L. Scott. Shared-Memory Synchronization. Morgan & Claypool, 2013.
* [SH96] Douglas C. Schmidt and Tim Harrison. Double-checked locking - an optimization pattern for efficiently initializing and accessing thread-safe objects. In PLoP, 1996.
* [SKA13] Hyojin Sung, Rakesh Komuravelli, and Sarita V. Adve. DeNovoND: Efficient hardware support for disciplined non-determinism. In ASPLOS, pages 13–26, 2013.
* [SLKR16] Christos Sakalis, Carl Leonardsson, Stefanos Kaxiras, and Alberto Ros. Splash-3: A properly synchronized benchmark suite for contemporary research. In ISPASS, 2016.
* [SS88] Dennis Shasha and Marc Snir. Efficient and correct execution of parallel programs that share memory. ACM Transactions on Programming Languages and Systems (TOPLAS), 10(2):282–312, 1988.
* [WOT+95] Steven Cameron Woo, Moriyoshi Ohara, Evan Torrie, Jaswinder Pal Singh, and Anoop Gupta. The SPLASH-2 programs: Characterization and methodological considerations. In ISCA, pages 24–36, 1995.
|
# Time delays in anisotropic systems
Ulf Saalmann Jan M. Rost Max-Planck-Institut für Physik komplexer Systeme,
Nöthnitzer Str. 38, 01187 Dresden, Germany
###### Abstract
Scattering properties and time delays for general (non-symmetric) potentials
in terms of the respective S-matrices are discussed paradigmatically in one
dimension and in comparison to symmetric potentials. Only for the latter the
Wigner and Smith time delays coincide. Considering asymmetric potentials also
reveals that only one version of S-matrices used in the literature (the one
with reflection coefficients on the diagonal) generalizes to the asymmetric
case. Finally, we give a criterion how to identify a potential with intrinsic
symmetry which behaves like an asymmetric one if it is merely offset from the
scattering center.
## I Introduction
Time delays related to scattering phases [1, 2] have been discussed for a long
time in transport problems [3, 4]. More recently, they have been addressed in
acoustics [5], electromagnetics [6], and from a fundamental perspective of
quantum trajectories [7], and since about a decade in the context of photo-
ionization by ultra-short laser pulses. Experimentally, photo-ionization time
delays have been extracted from streaking the momenta of electrons released by
a short XUV pulse with a moderate IR field [8] or so called RABBIT
measurements aiming at the same time-delay information of the released
electron wave-packet by using IR sidebands of the XUV photo-ionizing pulse
train [9, 10]. The link of the photo-ionization time delay to the Wigner-Smith
time delay from scattering theory, as well as the delays in general emerging
from these setups have been a source of ongoing debate [11, 12, 13]. This is
not surprising since the setups are quite intricate and become even more
cumbersome, if the long-range Coulomb interaction comes into play, which is
the case for almost all experiments performed.
Recent experimental advances have made it possible to measure time delays
originating in photo-ionizing molecules [14, 15, 16, 17, 18, 19], that is from
anisotropic potentials. This success motivates to ask for the theoretical
foundation of time delays and their formulation for general interactions,
since almost always time delay and S-matrices are discussed in the context of
single-centered, often spherically-symmetric potentials [20].
In the following we elucidate basic properties of time delays in the simplest
setting which is general enough to be sensitive to the properties of
anisotropic (and isotropic, parity-respecting) potentials. Since
characteristic features (such as the difference between proper and partial
time delays) can only be uncovered in a system with at least two independent
scattering channels, we do not investigate photo-ionization but scattering in
one dimension from a generic short-range potential, a scenario which provides
two scattering channels. Such type of scattering is relevant in complex media,
in wave-guides, or generally, for transport problems.
Additional motivation is provided by the fact, that symmetric potentials in 1D
hide subtleties of scattering and related time delays in at least two aspects:
(i) Two different versions of the S-matrix $\boldsymbol{S}$ are pursued in the
literature [21, 22, 23, 24], which have different eigenvalues. Yet, both
fulfill the criteria for S-matrices, derived from overarching principles of
flux conservation and time-reversal invariance (for a real potential), namely
that $\boldsymbol{S}$ is unitary and symmetric. However, only the version
which is a symmetric matrix with respect to incoming and outgoing channels
[21] remains symmetric in case of anisotropic potentials. (ii) Furthermore,
without symmetric interaction, the two commonly used formulations of time
delay, namely partial time delays and proper time delays do not agree,
prompting the question what their respective meaning is.
Scattering in one dimension was mostly theoretically investigated [22, 23, 24,
25] long before time delays have become popular, however, to the best of our
knowledge never with a discussion or even a focus on situations where the
scattering potential is not symmetric.
## II Scattering in 1D
For our context a potential $V(x)$ is short range if at large distances $x$
the solutions of the time-independent Schrödinger equation
$[-\mathrm{d}^{2}/\mathrm{d}x^{2}+2V(x)-2E]\psi(x)=0$ are free waves,
$\psi(|x|\gg 1)\propto e^{\pm\mathrm{i}kx}$ with $k\,{=}\,\sqrt{2E}>0$, see
also App. 1. We will use atomic units $e\,{=}\,\hbar\,{=}\,m_{\rm
e}\,{\equiv}\,1$ and consider for convenience a particle of mass $m_{\rm e}$,
unless stated otherwise.
### II.1 The S-matrix and its parameterization
There are two channels in an 1D scattering scenario. Most easily [26] they are
described by reflection ($r$) and transmission ($t$) amplitudes for incoming
waves from the left or the right side. Asymptotically those wave function
read, with $k\,{=}\,\sqrt{2E}$ and
$\\{\lim_{x\to{-}\infty}\psi(x,E),\;\lim_{x\to{+}\infty}\psi(x,E)\\}$,
$\displaystyle\psi_{\rm l}(x,E)$
$\displaystyle=\\{\mathrm{e}^{{+\mathrm{i}kx}}{+}r_{\rm
l}(E)\,\mathrm{e}^{{-\mathrm{i}kx}},\;t_{\rm
l}(E)\mathrm{e}^{{+\mathrm{i}kx}}\\}$ (1a) $\displaystyle\psi_{\rm r}(x,E)$
$\displaystyle=\\{t_{\rm
l}(E)\,\mathrm{e}^{{-\mathrm{i}kx}},\;\mathrm{e}^{{-\mathrm{i}kx}}{+}r_{\rm
r}(E)\mathrm{e}^{{+\mathrm{i}kx}}\\}.$ (1b)
By means of the four reflection and transmission amplitudes in Eq. (1) one
gets immediately the scattering matrix
$\boldsymbol{S}(E)=\left(\\!\\!\begin{array}[]{cccc}r_{\rm l}(E)&t_{l}(E)\\\
t_{\rm r}(E)&r_{\rm r}(E)\end{array}\\!\\!\right)\,.$ (2)
This S-matrix connects the amplitudes for incoming ($a_{\rm l,r}$) and
outgoing ($b_{\rm l,r}$) waves [27]
$\left(\\!\\!\begin{array}[]{cccc}b_{\rm l}\\\ b_{\rm
r}\end{array}\\!\\!\right)=\boldsymbol{S}\left(\\!\\!\begin{array}[]{cccc}a_{\rm
l}\\\ a_{\rm r}\end{array}\\!\\!\right).$ (3)
For both channels (l, r) particle-flux conservation guarantees
$|r(E)|^{2}+|t(E)|^{2}=1$. Furthermore, for real potentials $V(x)$ the
complex-conjugated channels $\psi_{\rm l,r}^{*}$ are valid solutions as well.
Comparing $[\psi_{\rm l}^{*}{-}r_{\rm l}^{*}\psi_{\rm l}]/t_{\rm l}^{*}$ with
$\psi_{\rm r}$ implies $t_{\rm l}=t_{\rm r}\equiv t$ and $r_{\rm l}/t_{\rm
l}={-}r_{\rm r}^{*}/t_{\rm r}^{*}$. Those four conditions render the
scattering matrix unitary
($\boldsymbol{S}^{\dagger}\boldsymbol{S}\,{=}\,\boldsymbol{1}$) and symmetric
($\boldsymbol{S}^{{\scriptscriptstyle\mathsf{T}}}{=}\,\boldsymbol{S}$).
Note that the diagonal elements of $\boldsymbol{S}$ connect the in- and
outgoing channels on the same side (l$\to$l, r$\to$r). There is another
version in the literature [22, 23, 24] that mixes the channels on the diagonal
(l$\to$r, r$\to$l). If the scattering potential is symmetric, the channel-
mixing version fulfills all symmetries required for an S-matrix, namely that
$\boldsymbol{S}$ is unitary and symmetric. However, if the potential does not
have parity, this channel-mixing version is no longer symmetric and therefore
is not a proper S-matrix. Since the overwhelming majority of published work
discusses only symmetric potentials, this shortcoming of the channel-mixing
S-matrix has not been pointed out.
For the unitary and symmetric S-matrix, three real parameters are sufficient
to define reflection and transmission amplitudes in the general form
$\displaystyle t(E)$
$\displaystyle=\cos(\alpha(E))\,\mathrm{e}^{{\mathrm{i}\beta(E)}}$ (4a)
$\displaystyle r_{\rm l,r}(E)$
$\displaystyle=\mathrm{i}\,\sin(\alpha(E))\,\mathrm{e}^{{\mathrm{i}[\beta(E)\pm\gamma(E)]}}.$
(4b)
The basis $\psi_{\rm l,r}$ of incoming waves from the left and the right is
only one of infinitely many choices. The one which diagonalizes the S-matrix
with a suitable unitary transformation $\boldsymbol{U}_{s}$ stands out and has
the physical meaning, that the ratio of incoming waves from the right and left
is not changed for the outgoing waves upon scattering. Due to the properties
of the S-matrix, its diagonal representation
$\boldsymbol{D}[\mathrm{e}^{{\mathrm{i}\boldsymbol{s}(E)}}]=\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\\!(E)\,\boldsymbol{S}(E)\,\boldsymbol{U}_{\\!s}(E)$
contains the eigenvalues given by pure phases
$\mathbfit{s}(E)=(s_{1}\;s_{2})^{\scriptscriptstyle\mathsf{T}}$. They read in
terms of reflection and transmission amplitudes or in their parameterization
(4)
$\displaystyle\mathrm{e}^{{\mathrm{i}\,s_{1,2}(E)}}$
$\displaystyle=\mbox{{\Large$\frac{r_{\rm l}+r_{\rm
r}}{2}$}}\pm\sqrt{\Big{[}\mbox{{\Large$\frac{r_{\rm l}-r_{\rm
r}}{2}$}}\Big{]}^{2}+t^{2}}$ (5a)
$\displaystyle=\mathrm{e}^{{\mathrm{i}[\beta+\chi_{\pm}]}}$ (5b)
$\displaystyle\chi_{\pm}$
$\displaystyle\equiv\mbox{atan}(\pm\sqrt{1{-}\sin^{2}\\!\alpha\cos^{2}\\!\gamma},\sin\alpha\cos\gamma).$
(5c)
If the potential is symmetric, i.e., $V(-x)=V(x)$, then $r_{\rm l}=r_{\rm r}$
and therefore $\gamma(E)=0$.
We close this section with a note on the channel-mixing S-matrix [22, 23, 24]:
Even if it is a valid S-matrix (i. e. for the case of a symmetric potential
where it is symmetric) its eigenvalues differ from those of the proper
S-matrix (2) while the time delays, to be discussed below, agree. Only the
time delays must agree since they are observables based on Hermitian
operators. The S-matrix itself is not Hermitian and therefore not an
observable. It provides a description of scattering, whose parameterization
can be done in different ways, as long as they are consistent with the
fundamental properties of a collision process.
### II.2 Partial time delays
As it has become clear from its definition, the S-matrix connects the two
input with the two output channels. This means that a wave-packet sent from
one side splits upon scattering and leaves the interaction region towards both
sides. Any change in the incoming configuration will lead to a different
partition of outgoing waves to the left and the right. Time delays, however,
are meaningful for the channels which diagonalize the S-matrix and thereby
keep the ratio of waves entering and leaving the scattering region from the
left and the right the same. Consequently, partial time delays are defined as
$\widetilde{\tau}_{\\!j}(E)=\mbox{{\Large$\frac{\mathrm{d}}{\mathrm{d}E}$}}s_{\\!j}(E)$
(6)
with $s_{\\!j}$ given in Eqs. (5).
### II.3 Proper time delays
Another, potentially more intuitive way of defining a time delay is to
consider the dwell time $\tau$, i. e. the reduced or increased time it takes a
wave-packet to traverse the region of interaction $|x|\leq X$ compared to the
traversal time $\tau_{0}$ of a free wave-packet,
$\displaystyle\tau$
$\displaystyle=\mbox{{\Large$\int$}}\\!\mathrm{d}t\mbox{{\Large$\int\limits_{\scriptscriptstyle|x|\leq
X}$}}\\!\\!\\!\mathrm{d}x\,\rho(x,t)-\tau_{0}$ (7a)
$\displaystyle=\mbox{{\Large$\int$}}\\!\mathrm{d}t\,t\;\big{[}J({+}X,t)-J({-}X,t)\big{]}-\tau_{0}\,,$
(7b)
where $\tau_{0}$ are the corresponding integrals for the free wave-packet
involving $\rho_{0}$ and $J_{0}$, respectively. The expression (7b) follows
from (7a) by means of the continuity equation and contains the time-dependent
current $J$ (or $J_{0}$) of the wave-packet at the left and right boundary of
the scattering region. Note that in 1D the current and the current density are
the same. If $X$ is chosen sufficiently large for the asymptotic description
(1) to be valid, it is obvious that $\tau$, defined spectrally by
$\boldsymbol{\psi}(E)$, can be calculated by means of the reflection and
transmission coefficients or the S-matrix for any given wave-packet. Without
details, which can be found elsewhere [28, 24], the result is
$\tau=\mbox{{\Large$\int$}}\\!\mathrm{d}E\,\boldsymbol{\psi}^{*}(E)\boldsymbol{Q}(E)\boldsymbol{\psi}(E),$
(8)
with Smith’s life-time matrix [2]
$\boldsymbol{Q}(E)=-\mathrm{i}\,\boldsymbol{S}^{\dagger}(E)\mbox{{\Large$\frac{\mathrm{d}}{\mathrm{d}E}$}}\boldsymbol{S}(E),$
(9)
where the time delay $\tau$ appears now as an expectation value of
$\boldsymbol{Q}(E)$ with the state $\boldsymbol{\psi}(E)$. Indeed,
$\boldsymbol{Q}$ is a Hermitian matrix since $\boldsymbol{S}$ is unitary.
Diagonalizing the life-time matrix $\boldsymbol{Q}$
$\boldsymbol{D}[\boldsymbol{q}(E)]=\boldsymbol{U}_{\\!q}{\\!\\!}^{\dagger}\\!(E)\,\boldsymbol{Q}(E)\,\boldsymbol{U}_{\\!q}(E),$
(10)
yieds real eigenvalues which are called proper time delays
$\overline{\tau}_{\\!j}(E)=q_{\\!j}(E).$ (11)
Since we consider a 2$\times$2 problem, they correspond to the minimal and
maximal dwell time $\tau$ in Eq. (8) with the respective eigenvectors
corresponding to the combination of incoming and outgoing waves that minimize
or maximize the time delay.
We have already introduced three different bases for the scattering channels,
waves coming in from the left and the right to define the S-matrix in the
first place, and the two bases which diagonalize the S-matrix giving partial
time delays and the one which diagonalizes Smith’s life-time matrix whose
eigenvalues are the proper time delays. A fourth basis is often introduced if
one wants to use a basis of real functions, namely linear combinations
$\\{\cos(kx),\sin(kx)\\}$ of left- and right-traveling waves which have even
and odd parity, respectively. For completeness, we also give the S-matrix in
the parity basis
$\boldsymbol{S}_{\rm
p}(E)=\mathrm{e}^{{\mathrm{i}\beta}}\sin\\!\alpha\left(\\!\\!\begin{array}[]{cccc}\mathrm{i}\,\cos\gamma{+}\cot\\!\alpha&\sin\gamma\\\
\sin\gamma&\mathrm{i}\,\cos\gamma{-}\cot\\!\alpha\end{array}\\!\\!\right)$
(12)
with details of the derivation given in App. 2.
Note that the S-matrix is always defined with respect to the center of
incoming and outgoing waves. A potential with $V(-x)=V(x)$ we have called
symmetric. There can be, however, the case, that the potential is symmetric
about about a point $x_{\rm cen}\neq 0$, which we call intrinsically
symmetric. Scattering from such a potential will have formally a full S-matrix
in any generic basis including the parity basis (12). This prompts the
question, if and how one could tell from experimental time delays, if the
potential has intrinsic symmetry or not. We will come back to this question
later.
## III Symmetry and time delays
### III.1 Symmetric potentials and the Wigner-Smith time delay
For a symmetric potential, $V(-x)=V(x)$, the S-matrix is diagonal in the
parity basis (12) since $\gamma=0$. The eigenphases reduce to
$\boldsymbol{s}_{\rm
sym}(E)=\left(\\!\\!\begin{array}[]{cccc}\alpha(E)+\beta(E)\\\
\pi-\alpha(E)+\beta(E)\end{array}\\!\\!\right).$ (13)
Also the life-time matrix becomes diagonal in this basis with the eigenvalues
$\boldsymbol{q}_{\rm
sym}(E)=\left(\\!\\!\begin{array}[]{cccc}\alpha^{\prime}(E)+\beta^{\prime}(E)\\\
-\alpha^{\prime}(E)+\beta^{\prime}(E)\end{array}\\!\\!\right),$ (14)
where there prime denotes derivation with respect to $E$. Hence, partial and
proper time delays ($j{=}1,2$) agree for symmetric potentials
$\widetilde{\tau}_{\\!j}(E)=\overline{\tau}_{\\!j}(E).$ (15)
The equivalence holds similarly for spherical potentials in 3D, with the
eigen-basis given by spherical harmonics. It is this equivalence which has led
to the notion of “Wigner-Smith time delays”.
### III.2 Asymmetric potentials
For arbitrary potentials the equivalence (15) does not hold any longer. With
the eigenvalues from Eqs. (5) we obtain
$\widetilde{\tau}_{1,2}=\mbox{{\Large$\frac{\mathrm{d}}{\mathrm{d}E}$}}s_{1,2}=\beta^{\prime}\mp\frac{\cos\\!\alpha\,\cos\gamma\,\alpha^{\prime}-\sin\\!\alpha\,\sin\gamma\,\gamma^{\prime}}{\sqrt{1{-}\cos^{2}\\!\gamma\,\sin^{2}\\!\alpha}}$
(16a) and for the proper time delays from diagonalizing $\boldsymbol{Q}$
$\overline{\tau}_{1,2}=q_{1,2}=\beta^{\prime}\pm\sqrt{\alpha^{\prime}{}^{2}+\sin^{2}\\!\\!\alpha\,\gamma^{\prime}{}^{2}},$
(16b)
where again we have dropped the energy dependence and indicate energy
derivatives with a prime. Obviously, these expression simplify to Eqs. (13)
and (14) for symmetric potentials ($\gamma{=}0$, $\gamma^{\prime}{=}0$).
Although proper and partial time delays differ, the time delay averaged over
the channels is the same for both
$\mbox{{\Large$\frac{1}{2}$}}\mbox{{$\sum$}}_{j=1,2}\widetilde{\tau}_{\\!j}(E)=\mbox{{\Large$\frac{1}{2}$}}\mbox{{$\sum$}}_{j=1,2}\overline{\tau}_{\\!j}(E)=\mbox{{\Large$\frac{\mathrm{d}}{\mathrm{d}E}$}}\beta(E)\,.$
(17)
This result generalizes to higher dimensions (2D, 3D) where the sum runs
formally over infinitely many channels, cf. App. 3.
Since the eigenchannels of $\boldsymbol{S}$ differ from those of
$\boldsymbol{Q}$, the natural question arises: What is the dwell time
$\tau_{\\!j}$ of a (shape-conserving) eigenchannel $j$ of $\boldsymbol{S}$?
Indeed the expectation value of $\boldsymbol{Q}$ in an eigenfunction
$\boldsymbol{\psi}_{\\!j}$ of the partial time delay produces this partial
time delay,
$\displaystyle\tau_{\\!j}\equiv\boldsymbol{\psi}_{\\!j}^{*}\boldsymbol{Q}\boldsymbol{\psi}_{\\!j}=-\mathrm{i}\boldsymbol{\psi}_{\\!j}^{*}\boldsymbol{S}^{\dagger}\boldsymbol{S}^{\prime}\boldsymbol{\psi}_{\\!j}=\widetilde{\tau}_{\\!j}$
(18)
as expected and shown in App. 4.
Figure 1: Upper raw: Time delays for a symmetric ($V_{1}$, a) and an
asymmetric ($V_{2}$, b) potential as a function of energy $E$, see Eq. (19)
and insets. Partial time delays $\widetilde{\tau}_{\\!j}$ (dashed lines)
according to Eq. (6) are compared to proper time delays
$\overline{\tau}_{\\!j}$ (solid lines) given in Eq. (11). The averages of both
$[\widetilde{\tau}_{1}{+}\widetilde{\tau}_{2}]/2$ and
$[\overline{\tau}_{1}{+}\overline{\tau}_{2}]/2$ are identical (thin gray
lines). Blue and red refer to the first and second channel, respectively.
Bottom raw: The corresponding parameters $\alpha$ (solid-blue line), $\beta$
(green-dotted), $\gamma$ (red-dashed) as a function of energy $E$, cf. Eqs.
(4).
### III.3 Generic examples
We illustrate the time-delay behavior for generic symmetric and asymmetric
potentials with two examples in Fig. 1. The potentials are defined by
$V_{1,2}(x)=-V_{0}\sum_{j=-2}^{+2}f_{\\!j}^{[1,2]}\,\mathrm{e}^{{-[x/d-2j]^{2}}}$
(19)
with $V_{0}=2$ eV and $d=1$ Å, to be specific. The prefactors are
$f_{\\!j}{}^{[1]}\,{=}\,1$ and $f_{\\!j}{}^{[2]}\,{=}\,1{+}j/3$ for the
symmetric and asymmetric potential, respectively.
One can see in Fig. 1a that for the symmetric potential partial and proper
time delays are identical for all energies $E$, as stated in Sect. III.1
above.
However, time delays differ for the asymmetric potential (Fig. 1b), where the
proper time delays form an envelope for the partial time delays. This is to be
expected as the eigenvalues of $\boldsymbol{Q}$ are the minimal and maximal
dwell times. No scattering states (and therefore not even a shape-conserving
one) can fall below or exceed those values. Nevertheless, both time delays can
agree at certain energies $E_{=}$, i. e.,
$\widetilde{\tau}_{\\!j}(E_{=})=\overline{\tau}_{\\!j}(E_{=})$, if
$\alpha^{\prime}=-\sin\\!\alpha\,\cos\\!\alpha\,\cot\\!\gamma\,\gamma^{\prime},$
(20)
as can be easily derived from Eqs. (16). Similarly one can find those energies
$E_{\times}$ where the partial time delays cross, i. e.
$\widetilde{\tau}_{\\!1}(E_{\times})=\widetilde{\tau}_{\\!2}(E_{\times})$
which requires
$\alpha^{\prime}=\tan\\!\alpha\,\tan\\!\gamma\,\gamma^{\prime}.$ (21)
For completeness we also present the energy dependence of the S-matrix
parameters ($\alpha,\beta,\gamma$) in Figs. 1c,d, which confirm that
$\gamma(E)=0$ for a symmetric potential, but finite for an asymmetric one.
## IV The dependence of time delays on spatial properties
Time delay is not immune to shifting the potential in a coordinate system,
which may be surprising given its relative character, that is a delay relative
to free motion at a given energy. Yet, scattering and subsequently time delay
define a coordinate system, in particular a scattering center through incoming
and outgoing waves and the S-matrix. The location of the potential relative to
the scattering center will have an influence on the time delay. A very loose
analogy is the echo of an object placed at some distance in front of a
reflecting wall, which plays the role of the scattering center: The echo one
receives will depend on the object as well as on its distance to the wall.
Similarly, for angular differential cross sections parameterized with partial
waves, the amplitudes of the partial waves depend on the location of the
origin of the coordinate system relative to the target. We will elucidate the
dependence of the time delays on spatial properties of the scattering scenario
below with examples.
### IV.1 Position of the potential with respect
to the scattering state
The results so far render time delay a useful observable, if carefully
assessed in a specific physical situation. What makes time delay, however,
quite cumbersome is the fact that it depends also on the location of the
potential with respect to the incoming wave-packet [5]. While it is natural
(albeit not necessary) to place an intrinsically symmetric potential at the
origin rendering it symmetric, no obvious choice exists a priori for an
asymmetric potential. In a realistic situation time delays are extracted from
asymptotic electron wave-packets. Those wave-packets have a clear origin, but
it is the location of the potential (e. g. the molecule) with respect to that
origin which matters for the time delays. This location is not uniquely
“defined” and could be difficult to determine.
One should note, however, that general time delays in the absence of
particular spectral features have only become of interest with the advent of
ultra-short laser pulses. Before, time delays were mostly discussed in
relation to a resonance. At the resonance energy, the time delays are
drastically enhanced for all channels sensitive to the resonance, see Fig. 2b,
where time delays are shown for the potential
$V_{4}(x)=\mathrm{e}^{{-[x/d-1/2]^{2}}}\mbox{atan}\big{(}2\sin(2x/d)\big{)}.$
(22)
In such a situation the location of the potential plays a subdominant role.
This is probably the reason, that the difference between various time delay
definitions and the dependence of those time delays on the location of the
potential has seen little attention to date.
Figure 2: Upper row: Time delays for a displaced potential ($V_{3}$, a) and a
potential with a resonance ($V_{4}$, b) as a function of energy $E$, see Eqs.
(23), (22) and insets. See caption of Fig. 1 for line styles.
Bottom row: Corresponding parameters $\gamma$ (red-dashed) and
$\gamma{-}2E\gamma^{\prime}$ (gray-dot-dashed) as a function $E$. Note that
for the symmetric potential (panel c) $\gamma\sim\sqrt{E}$ and thus
$\gamma{-}2E\gamma^{\prime}\sim\mbox{const}$.
However, the consequence of shifting the potential can be significant and is
particularly dramatic for a symmetric potential. We shift the one defined in
Eq. (19) according to
$V_{3}(x)=V_{1}(x-\delta x)$ (23)
which turns it from a symmetric one into one with only intrinsic symmetry. The
result is shown for a particular displacement of $\delta x=2d$ in Fig. 2a. Not
only do proper and partial time delays no longer agree, also the partial time
delays differ substantially from the ones where the location of the potential
is chosen such that it becomes symmetric as in Fig. 1a. Yet, the intrinsic
symmetry of the potential is still reflected in a more subtle relation as one
can see in Fig. 2c, namely that $\gamma(E)\,{-}\,2E\gamma^{\prime}(E)=0$ holds
for all energies $E$.
Figure 3: Time delays for a symmetric ($V_{1}$, a) and an asymmetric ($V_{2}$,
b) potential as a function a displacement $\delta x$ with respect to the
scattering center, see Eq. (19) and insets. The energy is $E=2$ eV in both
cases. See caption of Fig. 1 for line styles.
This relation, which is not fulfilled for a generically asymmetric potential,
see Fig. 2d, can be understood from (4) and the following consideration
regarding the effect on the time delays when shifting the potential by $\delta
x$, see Fig. 3. A little thought reveals that shifting the potential’s
position by $\delta x$ will only change $\gamma$ according to
$\displaystyle\gamma_{\\!\delta x}(E)$ $\displaystyle=\gamma_{0}(E)+2k\delta
x,$ (24a) such that $\displaystyle\gamma^{\prime}_{\\!\delta x}(E)$
$\displaystyle=\gamma^{\prime}_{0}(E)+2\delta x/k,$ (24b)
where $\gamma_{0}(E)$ refers to some reference position of the potential.
Partial and proper time delays can still be obtained from Eqs. (16) with
$\gamma$ and $\gamma^{\prime}$ replaced by $\gamma_{\\!\delta x}$ and
$\gamma^{\prime}_{\\!\delta x}$, respectively. The difference
$\Delta\overline{\tau}=\overline{\tau}_{1}-\overline{\tau}_{2}$ grows for
large displacement $\delta x$ linearly with a slope of $2/k$. Taking into
account $\hbar$, the slope defines a velocity, which (multiplied with $\delta
x$) represents the increased/reduced time a particle needs to reach/leave a
displaced potential.
The minimal “gap” $\Delta\overline{\tau}_{\rm min}=2\alpha^{\prime}$ of the
proper time delays occurs according to Eqs. (16b) and (24b) for
$\gamma^{\prime}_{\delta x_{\rm min}}=0$, realized with $\delta x_{\rm
min}=-k\gamma^{\prime}_{0}/2$. The resulting
$\gamma_{\\!\delta x_{\rm min}}=\gamma_{0}-k^{2}\gamma^{\prime}_{0}$ (25)
varies for generically asymmetric potentials with energy in the expression for
the partial-time-delay difference
$\Delta\widetilde{\tau}_{\rm
min}=2\frac{\cos\\!\alpha\,\cos(\gamma_{0}{-}k^{2}\gamma^{\prime}_{0})\,\alpha^{\prime}}{\sqrt{1{-}\sin^{2}\\!\alpha\,\cos^{2}(\gamma_{0}{-}k^{2}\gamma^{\prime}_{0})}}$
(26)
at the minimal gap of the proper time delays. Figure 3 illustrates this for a
particalur energy.
For potentials with intrinsic symmetry the reference in Eqs. (24) can be
chosen such that $\gamma_{0}\,{=}\,0$ (by making $x_{\rm cen}\,{=}\,0$), which
entails $\gamma^{\prime}_{0}\,{=}\,0$. Once this potential is offset from the
center, $\gamma$ becomes finite and energy-dependent. Yet, as follows directly
from (24), $\gamma_{\delta x}{-}\,2E\gamma^{\prime}_{\delta
x}\,{=}\,\mbox{const}$ for all energies $E$. For a generically asymmetric
potential (not possessing an $x_{\rm cen}$) this is not possible.
We may conclude that the interplay of partial and proper time delays reveal
the symmetry of the potential, despite their sensitivity to its location: If
the minimal gap of the proper time delays coincides at all energies with the
maximal gap of the partial time delays, the underlying potential has intrinsic
symmetry.
### IV.2 Reflection-less potentials
Apparently, time delays are quite sensitive to the quantum-mechanical
interference of transmitted and reflected waves. Hence, one would expect a
radically different behavior, if reflection is suppressed. This can be double-
checked by investigating the time delays of a reflection-less potential. It is
well known [29] that certain potentials show perfect transmission
$|t(E)|^{2}=1$ for all energies $E$, e.g.,
$V_{5}(x)=\mbox{{\Large$\frac{1}{d^{2}\cosh^{2}(x/d)}$}}\,.$ (27)
In Fig. 4 we show partial and proper time delays for $V_{5}(x)$ and its
shifted version $V_{6}(x)=V_{5}(x-\delta x)$ with $\delta x=3$ Å. As can be
seen, all time delays agree even for the displaced potential. This reveals
that subtle interference effects due to the position of the potential as well
as the difference of proper and partial time delays are of pure quantum nature
and vanish in a (semi-)classical setting, as provided by the potential free of
reflections which behaves as a classical system would do (full transmission
and zero reflection). Of course, in such a situation only a single channel is
left and therefore even in full quantum mechanics, no interference can occur.
This lets the proper and partial time delays collapse to a single time delay
which is identical to the average one. All of this follows directly from the
parameters (4) in this case. It is $\alpha\,{=}\,0$, thus
$\alpha^{\prime}\,{=}\,0$ and $\gamma$ being irrelevant.
Figure 4: Time delays for a reflection-less potential ($V_{5}$, a) and the
same one displaced ($V_{6}$, b) , cf. Eq. (27) and see insets, as a function
of energy $E$. See caption of Fig. 1 for line styles.
## V Conclusion
We have discussed scattering and ensuing time delays for generic potentials
(without symmetry), paradigmatically in one dimension. For potentials
symmetric to the origin, the standard case almost exclusively discussed also
in 3D with centrally-symmetric potentials, proper and partial time delays
agree which has given the notion of Wigner-Smith time delays. This is not the
case for generic potentials whose location relative to the collision origin
have significant influence on the various time delays. With asymmetric
potentials, one also notices a widespread use of a definition of the S-matrix
which is only symmetric (as it should be from general considerations) for
symmetric potentials and therefore not suitable to describe scattering for
generic potentials. Finally, we have provided a criterion which allows one to
identify from the time delays an intrinsically symmetric potential (symmetric
with respect to some position $x_{\rm cen}$) located at an arbitrary position.
## Acknowledgment
We dedicate this analysis to Ravi Rau, who has always been an inspiration and
role model to view a problem from different perspectives and to articulate it
in a clear and transparent manner. We wish him many productive years to come.
## A Appendix
### 1 Coulombic systems
Long-range Coulomb potentials lead to infinite time delays if referenced with
free motion [2]. (Note, that finite values, reported in experiments [8, 9],
are the consequence of the measurement [11].) Therefore, free motion must be
replaced by motion in a pure Coulomb potential as reference [11]. Then time
delays are finite and measure the (short-range) deviation from a pure Coulomb
interaction. In 3D, many of the ideas and techniques presented here carry over
to Coulomb systems with asymptotic wavefunctions
$\exp(\pm[\mathrm{i}kr+\frac{1}{k}\ln(2kr)])$, i. e., amended by the
logarithmic Coulomb phase. However, dynamics with Coulomb interaction differs
substantially in 1D and 3D and disguises the general analogy between 1D and 3D
scattering.
### 2 Analogy to 3D scattering in a basis
of spherical harmonics
The treatment described so far is specific to 1D, but has a clear relation to
the 3D situation: The continuous solid angle $\\{\vartheta,\varphi\\}$ in 3D
gets replaced in 1D by two discrete directions $d\,{=}\,{-}1$ (left) and
$d\,{=}\,{+}1$ (right), respectively. The relation between 1D and 3D becomes
transparent with the commonly-used spherical-harmonics basis $Y_{\ell
m}(\vartheta,\varphi)$ with $\ell\,{=}\,0\ldots\infty$ and
$m\,{=}\,{-}\ell\ldots{+}\ell$ for the angular degrees of freedom. In 1D one
can use two ($m\,{=}\,0,1$) “angular” functions
$y_{m}(d)=d^{m}\\!/\\!\sqrt{2}$, which are orthonormal $\sum_{d=\pm
1}y_{m}(d)y_{m^{\prime}}(d)=\delta_{mm^{\prime}}$. Whereas the description
with a finite $\ell_{\rm max}$ is approximate but numerically accurate in 3D,
the description in 1D in terms of the $y_{\\!m}$ is exact.
Instead of the traditional form (1) in the Cartesian coordinate $x$, one can
write the two continuum states ($j{=}1,2$) asymptotically in terms of the
$y_{m}(d)$ and real radial functions $\phi_{mj}(r)$ in the discrete “angle”
$d\,{=}\,\mbox{sgn}(x)$ and the radial distance $r\,{=}\,|x|$, respectively,
as
$\displaystyle\psi_{j}(r,d,E)$
$\displaystyle=\sum_{m=0,1}y_{m}(d)\phi_{mj}(r,E)$ (28)
$\displaystyle\left.{\phi^{\infty}_{01}(r,E)\atop\phi^{\infty}_{12}(r,E)}\right\\}$
$\displaystyle=\cos(kr)\pm\cos\alpha\cos(kr{+}\beta)$
$\displaystyle\qquad\pm\cos\gamma\sin\alpha\sin(kr{+}\beta)$ (28a)
$\displaystyle\left.{\phi^{\infty}_{11}(r,E)\atop\phi^{\infty}_{02}(r,E)}\right\\}$
$\displaystyle=-\sin\gamma\sin\alpha\cos(kr{+}\beta),$ (28b)
where we have omitted the dependence of $\alpha$, $\beta$ and $\gamma$ on the
energy $E{=}k^{2}\\!/2$ and used the notation
$\phi^{\infty}(r,E)\equiv\lim_{r\to\infty}\phi(r,E)$. In general, in both
functions ($j\,{=}\,1,2$) the two “angular” channels ($m\,{=}\,0,1$) couple,
as can be seen in Eqs. (28). For symmetric potentials, where $\gamma=0$ [cf.
Eqs. (4)], this is not the case leading to the simplification
$\displaystyle\left.{\phi^{\infty}_{01}(r,E)\atop\phi^{\infty}_{12}(r,E)}\right\\}$
$\displaystyle=\cos(kr)\pm\cos(kr{\pm}\alpha{+}\beta)$ (29a)
$\displaystyle\left.{\phi^{\infty}_{11}(r,E)\atop\phi^{\infty}_{02}(r,E)}\right\\}$
$\displaystyle=0,$ (29b)
where one can directly read off the eigenphases (13). Note, that radial wave-
packets built from either symmetric ($m{=}0$) or anti-symmetric ($m{=}1$)
states, given in Eq. (29a), will keep their symmetry throughout the scattering
process.
The asymptotic expressions of the radial functions (28) have the form
$\phi^{\infty}_{mj}(r,E)=a_{mj}\mathrm{e}^{{-\mathrm{i}kr}}+b_{mj}\mathrm{e}^{{+\mathrm{i}kr}},$
(30)
from which the S-matrix is directly obtained by means of Eq. (3). Since we
have used the parity basis $y_{m=0,1}$ we get $\boldsymbol{S}$ given in Eq.
(12).
In 3D the analogous form of Eqs. (28) is the most efficient way to calculate
the continuum functions at energy $E$, since an equivalent form of Eqs. (1) is
not available. The calculations can be done by means of the coupled-channel
renormalized Numerov method [30].
### 3 Sum of time delays
It is shown that the sum of proper ($\overline{\tau}_{\\!j}$) and partial
($\widetilde{\tau}_{\\!j}$) time delays are equal [3]. We will use repeatedly
the unitarity of $\boldsymbol{U}_{\\!s}$ as well as $\boldsymbol{S}$ and the
possibility to change the order of matrices under the trace. Therewith
$\displaystyle\sum_{j}\overline{\tau}_{\\!j}$
$\displaystyle=\sum_{j}q_{j}=\mathrm{tr}(\boldsymbol{U}_{\\!q}{\\!\\!}^{\dagger}\\!\boldsymbol{Q}\boldsymbol{U}_{\\!q})=\mathrm{tr}(\boldsymbol{Q})=-\mathrm{i}\,\mathrm{tr}\big{(}\boldsymbol{S}^{\dagger}\boldsymbol{S}^{\prime}\big{)}$
$\displaystyle=-\mathrm{i}\,\mathrm{tr}\Big{(}\boldsymbol{U}_{\\!s}\boldsymbol{D}[\mathrm{e}^{{-\mathrm{i}\boldsymbol{s}}}]\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\big{(}\boldsymbol{U}_{\\!s}\boldsymbol{D}[\mathrm{e}^{{\mathrm{i}\boldsymbol{s}}}]\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\big{)}^{\prime}\Big{)}$
$\displaystyle=-\mathrm{i}\,\mathrm{tr}\big{(}\boldsymbol{D}[\mathrm{e}^{{-\mathrm{i}\boldsymbol{s}}}]\boldsymbol{D}^{\prime}[\mathrm{e}^{{\mathrm{i}\boldsymbol{s}}}]\big{)}-\mathrm{i}\,\mathrm{tr}\big{(}\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\boldsymbol{U}_{\\!s}^{\prime}+(\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger})^{\prime}\boldsymbol{U}_{\\!s}\big{)}$
$\displaystyle=\mathrm{tr}\big{(}\boldsymbol{D}[\mathrm{e}^{{-\mathrm{i}\boldsymbol{s}}}]\boldsymbol{D}[\boldsymbol{s}^{\prime}\mathrm{e}^{{\mathrm{i}\boldsymbol{s}}}]\big{)}-\mathrm{i}\,\mathrm{tr}\big{(}(\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\boldsymbol{U}_{\\!s})^{\prime}\big{)}$
$\displaystyle=\sum_{\\!j}s^{\prime}_{\\!j}=\sum_{j}\widetilde{\tau}_{\\!j}.$
(31)
As above, $\boldsymbol{D}[]$ denotes a diagonal matrix.
Note that we have nowhere used the fact that we treat a 1D system with
2$\times$2 matrices. Thus Eq. (3) holds for any dimension.
### 4 Dwell time for a scattering eigenchannel
We assume that $\boldsymbol{v}_{\\!j}$ is an eigenvector of the scattering
matrix $\boldsymbol{S}\boldsymbol{v}_{\\!j}=\boldsymbol{v}_{\\!j}s_{\\!j}$.
Therefrom follows
$\displaystyle s^{\prime}_{\\!j}$
$\displaystyle=\boldsymbol{v}_{\\!j}{}^{\prime}\boldsymbol{S}\boldsymbol{v}_{\\!j}+\boldsymbol{v}_{\\!j}\boldsymbol{S}\boldsymbol{v}^{\prime}_{\\!j}+\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}$
$\displaystyle=s_{j}[\boldsymbol{v}_{\\!j}{}^{\prime}\boldsymbol{v}_{\\!j}+\boldsymbol{v}_{\\!j}\boldsymbol{v}^{\prime}_{\\!j}]+\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}=\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}$
(32)
since $\boldsymbol{v}_{\\!j}\boldsymbol{v}_{\\!j}\,{=}\,1$ and thus the term
in brackets vanishes. Therewith we can calculate the expectation value of
$\boldsymbol{Q}$ for the scattering eigenstate
$\displaystyle\tau_{\\!j}$
$\displaystyle\equiv\boldsymbol{v}_{\\!j}\boldsymbol{Q}\boldsymbol{v}_{\\!j}=-\mathrm{i}\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\dagger}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}$
$\displaystyle=-\mathrm{i}\;s_{\\!j}\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}=-\mathrm{i}\;s_{\\!j}s^{\prime}_{\\!j}=\widetilde{\tau}_{\\!j},$
(33)
where in the 2nd line we have used Eq. (4).
### 5 Matrices, eigenvalues and eigenvectors
in paramterized form
For completeness, the matrices $\boldsymbol{S}$ and $\boldsymbol{Q}$ and their
eigenforms are given in terms of the parametrization (4)
$\displaystyle\boldsymbol{S}$
$\displaystyle=\left(\\!\\!\begin{array}[]{cccc}\mathrm{i}\mathrm{e}^{{\mathrm{i}[\beta+\gamma]}}\sin\alpha&\mathrm{e}^{{\mathrm{i}\beta}}\cos\alpha\\\
\mathrm{e}^{{\mathrm{i}\beta}}\cos\alpha&\mathrm{i}\mathrm{e}^{{\mathrm{i}[\beta-\gamma]}}\sin\alpha\end{array}\\!\\!\right)$
(34) $\displaystyle s_{1,2}$
$\displaystyle=\mathrm{e}^{{\mathrm{i}[\beta+\mbox{atan}(\pm\sqrt{1-\cos^{2}\\!\gamma\sin^{2}\\!\alpha},\cos\gamma\sin\alpha)]}}$
(34c) $\displaystyle\boldsymbol{v}_{\\!1,2}$
$\displaystyle=\frac{1}{N_{s}}\left(\\!\\!\begin{array}[]{cccc}-\sin\alpha\sin\gamma\pm\sqrt{1-\cos^{2}\\!\gamma\sin^{2}\\!\alpha}\\\
\cos\alpha\end{array}\\!\\!\right)$ (34f)
with $N_{s}$ ensuring normalization. Note that the eigenvectors of
$\boldsymbol{S}$ can be chosen real (which is generally not the case for
unitary matrices). And further
$\displaystyle\boldsymbol{Q}$
$\displaystyle=\left(\\!\\!\begin{array}[]{cccc}+\sin^{2}\\!\alpha\,\gamma^{\prime}&\mathrm{e}^{{-\mathrm{i}\gamma}}[\alpha^{\prime}-\mathrm{i}\,\eta]\\\
\mathrm{e}^{{+\mathrm{i}\gamma}}[\alpha^{\prime}+\mathrm{i}\,\eta]&-\sin^{2}\\!\alpha\,\gamma^{\prime}\end{array}\\!\\!\right)$
(35) $\displaystyle\qquad\eta\equiv\cos\alpha\sin\alpha\,\gamma^{\prime}$
$\displaystyle q_{1,2}$
$\displaystyle=\beta^{\prime}-\sqrt{\alpha^{\prime}{}^{2}+\sin^{2}\\!\alpha\gamma^{\prime}{}^{2}}$
(35c) $\displaystyle\boldsymbol{w}_{\\!1,2}$
$\displaystyle=\frac{1}{N_{q}}\left(\\!\\!\begin{array}[]{cccc}\pm\sqrt{\alpha^{\prime}{}^{2}+\sin^{2}\\!\alpha\,\gamma^{\prime}{}^{2}}+\sin^{2}\\!\alpha\,\gamma^{\prime}\\\
\mathrm{e}^{{\mathrm{i}\gamma}}[\alpha^{\prime}+\mathrm{i}\,\cos\alpha\sin\alpha\,\gamma^{\prime}]\end{array}\\!\\!\right)$
(35f)
again with the normalization $N_{q}$ not explicitly given. The unitary
matrices referred to in the text are
$\boldsymbol{U}_{\\!s}\,{=}\,(\boldsymbol{v}_{\\!1}\;\boldsymbol{v}_{\\!2})$
and
$\boldsymbol{U}_{\\!q}\,{=}\,(\boldsymbol{w}_{\\!1}\;\boldsymbol{w}_{\\!2})$.
## References
* [1] E. P. Wigner, _Lower limit for the energy derivative of the scattering phase shift._ Phys. Rev. 98, 145 (1955).
* [2] F. T. Smith, _Lifetime matrix in collision theory._ Phys. Rev. 118, 349 (1960).
* [3] C. Texier, _Wigner time delay and related concepts: Application to transport in coherent conductors._ Physica E 82, 16 (2016).
* [4] S. Rotter, P. Ambichl, and F. Libisch, _Generating particle-like scattering states in wave transport._ Phys. Rev. Lett. 106, 120602 (2011).
* [5] U. R. Patel, Y. Mao, and E. Michielssen, _Wigner-Smith time delay matrix for acoustic scattering: Theory and phenomenology._ J. Acoust. Soc. Am. 153, 2769 (2023).
* [6] U. R. Patel and E. Michielssen, _Wigner-Smith time-delay matrix for electromagnetics: Theory and phenomenology._ IEEE Trans. Antennas Propag. 69, 902 (2021).
* [7] L. Dupuy, G. Parlant, B. Poirier, and Y. Scribano, _Direct and accurate calculation of dwell times and time delays using quantum trajectories._ Phys. Lett. A 456, 128548 (2022).
* [8] M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, _Delay in photo-emission._ Science 328, 1658 (2010).
* [9] K. Klünder, J. M. Dahlström, M. Gisselbrecht, T. Fordell, M. Swoboda, D. Guénot, P. Johnsson, J. Caillat, J. Mauritsson, A. Maquet, R. Ta\̈mathrm{i}eb, and A. L’Huillier, _Probing single-photon ionization on the attosecond time scale._ Phys. Rev. Lett. 106, 143002 (2011).
* [10] M. Isinger, R. J. Squibb, D. Busto, S. Zhong, A. Harth, D. Kroon, S. Nandi, C. L. Arnold, M. Miranda, J. M. Dahlström, E. Lindroth, R. Feifel, M. Gisselbrecht, and A. L’Huillier, _Photo-ionization in the time and frequency domain._ Science 358, 893 (2017).
* [11] U. Saalmann and J. M. Rost, _Proper time delays measured by optical streaking._ Phys. Rev. Lett. 125, 113202 (2020).
* [12] B. Fetić, W. Becker, and D. B. Milošević, _Can we measure the Wigner time delay in a photo-ionization experiment?_ arXiv:2210.05219 [quant-ph] (2022).
* [13] K. I. Elghazawy and C. H. Greene, _Wigner time delay in photo-ionization: a 1D model study._ J. Phys. B 56, 175201 (2023).
* [14] M. Huppert, I. Jordan, D. Baykusheva, A. von Conta, and H. J. Wörner, _Attosecond delays in molecular photo-ionization._ Phys. Rev. Lett. 117, 093001 (2016).
* [15] J. Fuchs, N. Douguet, S. Donsa, F. Martin, J. Burgdörfer, L. Argenti, L. Cattaneo, and U. Keller, _Time delays from one-photon transitions in the continuum._ Optica 7, 154 (2020).
* [16] J. Rist, K. Klyssek, N. M. Novikovskiy, M. Kircher, I. Vela-Pérez, D. Trabert, S. Grundmann, D. Tsitsonis, J. Siebert, A. Geyer, N. Melzer, C. Schwarz, N. Anders, L. Kaiser, K. Fehre, A. Hartung, S. Eckart, L. P. H. Schmidt, M. S. Schöffler, V. T. Davis, J. B. Williams, F. Trinter, R. Dörner, P. V. Demekhin, and T. Jahnke, _Measuring the photo-electron emission delay in the molecular frame._ Nat. Commun. 12, 6657 (2021).
* [17] F. Holzmeier, J. Joseph, J. C. Houver, M. Lebech, D. Dowek, and R. R. Lucchese, _Influence of shape resonances on the angular dependence of molecular photo-ionization delays._ Nat. Commun. 12, 7343 (2021).
* [18] S. Heck, M. Han, D. Jelovina, J.-B. Ji, C. Perry, X. Gong, R. Lucchese, K. Ueda, and H. J. Wörner, _Two-center interference in the photo-ionization delays of Kr 2._ Phys. Rev. Lett. 129, 133002 (2022).
* [19] X. Gong, W. Jiang, J. Tong, J. Qiang, P. Lu, H. Ni, R. Lucchese, K. Ueda, and J. Wu, _Asymmetric attosecond photo-ionization in molecular shape resonance._ Phys. Rev. X 12, 011002 (2022).
* [20] A. S. Kheifets, _Wigner time delay in atomic photo-ionization._ J. Phys. B 56, 022001 (2023).
* [21] E. Merzbacher, Quantum mechanics (John Wiley & Sons, Inc., New York, 1998).
* [22] M. S. de Bianchi, _Levinson’s theorem, zero-energy resonances, and time delay in one-dimensional scattering systems._ J. Math. Phys. 35, 2719 (1994).
* [23] Y. Nogami and C. K. Ross, _Scattering from a non-symmetric potential in one dimension as a coupled-channel problem._ Am. J. Phys. 64, 923 (1996).
* [24] H. M. Nussenzveig, _Average dwell time and tunneling._ Phys. Rev. A 62, 042107 (2000).
* [25] L. J. Boya, _Quantum-mechanical scattering in one dimension._ Riv. del Nuovo Cim. 31, 75 (2008).
* [26] There are other possibilities, we outline one in App. 2.
* [27] The asymptotic forms (1) imply $a_{\rm l}{=}1$, $a_{\rm r}{=}0$, $b_{\rm l}{=}r_{\rm l}$, $b_{\rm r}{=}t_{\rm l}$ for (1a) and $a_{\rm l}{=}0$, $a_{\rm r}{=}1$, $b_{\rm l}{=}t_{\rm r}$, $b_{\rm r}{=}r_{\rm r}$ for (1b), respectively.
* [28] W. Jaworski and D. M. Wardlaw, _Time delay in tunneling: Transmission and reflection time delays._ Phys. Rev. A 37, 2843 (1988).
* [29] B. Y. Tong, _Scattering states of the Sech-squared potential._ Solid State Commun. 104, 679 (1997).
* [30] B. R. Johnson, _The renormalized Numerov method applied to calculating bound states of the coupled-channel Schrödinger equation._ J. Chem. Phys. 69, 4678 (1978).
|
# Physics guided machine learning using simplified theories
Suraj Pawar School of Mechanical & Aerospace Engineering, Oklahoma State
University, Stillwater, OK 74078, USA. Omer San<EMAIL_ADDRESS>School of
Mechanical & Aerospace Engineering, Oklahoma State University, Stillwater, OK
74078, USA. Burak Aksoylu Texas A&M University-San Antonio, Department of
Mathematical, Physical, and Engineering Sciences, San Antonio, TX 78224, USA.
Adil Rasheed Department of Engineering Cybernetics, Norwegian University of
Science and Technology, 7465 Trondheim, Norway. Department of Mathematics and
Cybernetics, SINTEF Digital, 7034 Trondheim, Norway. Trond Kvamsdal
Department of Mathematical Sciences, Norwegian University of Science and
Technology, 7491 Trondheim, Norway. Department of Mathematics and
Cybernetics, SINTEF Digital, 7034 Trondheim, Norway.
###### Abstract
Recent applications of machine learning, in particular deep learning, motivate
the need to address the generalizability of the statistical inference
approaches in physical sciences. In this letter, we introduce a modular
physics guided machine learning framework to improve the accuracy of such
data-driven predictive engines. The chief idea in our approach is to augment
the knowledge of the simplified theories with the underlying learning process.
To emphasise on their physical importance, our architecture consists of adding
certain features at intermediate layers rather than in the input layer. To
demonstrate our approach, we select a canonical airfoil aerodynamic problem
with the enhancement of the potential flow theory. We include features
obtained by a panel method that can be computed efficiently for an unseen
configuration in our training procedure. By addressing the generalizability
concerns, our results suggest that the proposed feature enhancement approach
can be effectively used in many scientific machine learning applications,
especially for the systems where we can use a theoretical, empirical, or
simplified model to guide the learning module.
Physics guided machine learning, feature engineering, trustworthy artificial
intelligence, generalizibility, hybrid neurophysical modeling
Advanced machine learning (ML) models like deep neural networks are powerful
tools for finding patterns in complicated datasets. However, as these
_universal approximators_ Hornik, Stinchcombe, and White (1989); Hagan and
Menhaj (1994); Hagan, Demuth, and Beale (1997); Vidal _et al._ (2017) become
complex, the number of trainable parameters (weights) quickly explodes,
adversely affecting their interpretability, and hence their trustworthiness.
Using these models in combination with other traditional models compromises
the trustworthiness of the overall system. While such techniques have been
emerging in both scientific computing and ML fields Krasnopolsky and Fox-
Rabinovitz (2006); Lee and Oh (1997); Tang and Hsieh (2001); Marsland,
Shapiro, and Nehmzow (2002); Raissi, Perdikaris, and Karniadakis (2019); Zhu
_et al._ (2019); Swischuk _et al._ (2019); Sekar _et al._ (2019); Beck,
Flad, and Munz (2019); Rahman, Rasheed, and San (2018); Maulik _et al._
(2019a, b); Pan and Duraisamy (2020); Mohan _et al._ (2020); Muralidhar _et
al._ (2020); Qian _et al._ (2020); Pawar _et al._ (2020a); Ahmed _et al._
(2020a, b); Pawar _et al._ (2020b); Pawar, Ahmed, and San (2020); Pawar and
San (2020); Ahmed _et al._ (2020c), they offer many opportunities to fuse
topics in domain-specific knowledge, numerical linear algebra, and theoretical
computer science Golub _et al._ (2006) towards improving the generalizability
of these models. In this letter, we focus on a modular approach for improving
generalizability by augmenting input features with simplified physics-based
models or theories. By the term of _generalizability_ , we refer to the
performance of the learning engine for unseen configurations. Such new
situations are pervasive in scientific applications. In many cases, there is a
grand challenge in making accurate predictions using data-driven methods based
strictly on statistical inference. For example, in fluid mechanics, potential
flow and boundary layer theories often provide simplistic models pertaining to
a certain level of approximations. A remaining task is how to combine these
models for making more generalizable learning architectures. Therefore, our
approach provides intermediate features based on approximate physics to
statistical models that can relate or fulfill the gap between these simplified
theories and more accurate models (i.e., experimental or high fidelity
computational fluid dynamics simulation data sets).
Figure 1: Physics guided machine learning (PGML) framework to train a learning
engine between processes $A$ and $B$: (a) a conceptual PGML framework which
shows different ways of incorporating physics into machine learning models.
The physics can be incorporated using feature enhancement of the ML model
based on the domain knowledge, embedding simplified theories directly into ML
models, and corrector approach in which the output of the ML model is
constrained using the governing equations of the system, and (b) an overview
of the typical neural network architecture for the PGML framework.
A wide variety of engineering tasks such as optimal control, design
optimization, uncertainty quantification requires the prediction of the
quantity of interest in real-time. In such scenarios, the partial differential
equation (PDE) based discretization methods are not feasible as they can take
days or weeks to simulate different configurations. Reduced order models
(ROMs) are the state-of-the-art models that construct the basis from the past
data, and then solve the governing equations after projecting them on the low
dimensional manifold Lucia, Beran, and Silva (2004); Taira _et al._ (2020);
Fonn _et al._ (2019). The main limitation of such projection based intrusive
ROMs is that it requires a complete description of the dynamics of the system,
and often this information is unavailable or inadequate for the desired
application. Recently model-free prediction using machine learning has proven
successful for many physical systems Pathak _et al._ (2018); White, Ushizima,
and Farhat (2019); Geneva and Zabaras (2020). One of the main challenges with
these non-intrusive models is the prediction for unseen data and its
interpretation. Even though probabilistic machine learning methods like
Bayesian neural networks can give the uncertainty bound on the predicted state
Zhu _et al._ (2019); Maulik _et al._ (2020), the generalizability of non-
intrusive models is not on par with physics-based models. On the other hand,
the simplified models like the lumped-capacitance model for heat transfer
analysis, and Blasius boundary layer model in fluid mechanics are highly
generalizable to different conditions. Therefore, it is important to leverage
the knowledge of physical systems into learning, and to this end, we make
progress towards physics guided machine learning by embedding features from
simplified physical models into the neural network architecture. The proposed
framework is flexible enough to be applied to many physical systems and it has
great potential in scientific machine learning.
We now introduce different components of the physics guided machine learning
(PGML) framework as depicted in Figure 1(a). In supervised machine learning,
the input vector $\mathbf{x}\in\mathbb{R}^{m}$ is fed to the machine learning
model (for example, the neural network in our case), and the mapping from the
input vector to output vector $\mathbf{y}\in\mathbb{R}^{n}$ is learned through
training. The neural network is trained to learn the function $F_{\theta}$,
parameterized by $\theta$, that includes the weights and biases of each
neuron. The function $F_{\theta}$ should be such that it approximates the
known labels and the cost function is minimized. Usually, for the regression
problems, the cost function is the mean squared error between true and
predicted output, i.e.,
$C(\mathbf{x},\theta)=||\mathbf{y}-F_{\theta}(\mathbf{x})||_{2}$. The weights
and biases of the neural network are optimized using any gradient-descent
algorithm like stochastic gradient descent. In the PGML framework, the neural
network is augmented with the output of the simplified physics-based model.
The features extracted from simplified physics-based models can be either
combined with input features, or they can be merged into hidden layers along
with latent variables. During the training, the PGML framework is provided
with $(\mathbf{x},G(\mathbf{x}))$, where $G$ is the model based on simplified
theories for the problem at hand, and the parameters $\theta$ are optimized
based on the true output $\mathbf{y}$. The features from the simplified model
$G$ can also be fed at an intermediate hidden layer along with learned latent
representation.
The PGML framework allows us to extract the knowledge from the simplified
physics-based model and combine it with the latent variables of the system
discovered by the neural network at intermediate hidden layers. This
information from the physics-based model assists the neural network in
constraining its output to a manifold of physically realizable models. It
encourages the neural network to learn the physically consistent
representation of the quantity of interest drawn from complex distributions
such as pressure and velocity field of fluid simulations. Another advantage of
the PGML framework is that it brings interpretability to otherwise black-box
models. We highlight here that the PGML framework allows us to incorporate the
physics of the problem even during the prediction stage, and not just the
training as in the case of approaches like regularization based on governing
equations.
In a nutshell, we highlight that there might be a handful of simplified models
(e.g., similarity solutions, lumped parameter models, zone models, zero or
one-dimensional models, etc) to approximate or describe the underlying
physical processes in many disciples. For example, in aerodynamics, the use of
simplistic panel methods is a proven approach for analysis of inviscid flow
over bodies, especially for a smaller angle of attacks where the potential
flow theory becomes valid. The execution time for these simplified models is
significantly smaller compared to the run time needed for full-fledged CFD
simulations. Therefore, in the PGML method, we propose to fuse the knowledge
coming from such simplified theories in a statistical inference architecture.
This is accomplished by a feature enhancement procedure as described in Figure
1(b). Specifically, it constitutes a predictor-corrector philosophy where we
first run a simplified theory (computationally less demanding) to estimate an
intermediate prediction as an input for the overall ML architecture. We then
estimate the quantity of interest based on the enhancement procedure
abstracted by $G(\mathbf{x})$. We hypothesize that the predictive performance
and generalizability of statistical inference will be significantly improved
by the involvement of $G(\mathbf{x})$ as illustrated with the aerodynamic
performance prediction task.
_Results and Discussion_ — We demonstrate the PGML framework using the
aerodynamic performance prediction task. This is a problem relevant to many
applications such as predictive controlZha _et al._ (2007), and design
optimizationLeGresley and Alonso (2000). The prediction of flow around an
airfoil is a high-dimensional, multiscale, and nonlinear problem that can be
solved using high-fidelity methods like computational fluid dynamics (CFD).
However, these methods are computationally intractable as the design space
exploration increases. In certain flow regimes, the simplified methods like
panel codes come with a non-negligible difference between the actual dynamics
and approximate models for real-world problems. The full-order CFD simulations
typically take extensive computational run time, thus limiting their use in
many inverse modelling methodologies that require a model run to be performed
in each iteration. To overcome these challenges, combining CFD models with
machine learning to build a data-driven surrogate model is gaining widespread
popularityZhang, Sung, and Mavris (2018); Bhatnagar _et al._ (2019); Rajaram
_et al._ (2020). In this example, we exploit the relevant physics-based
features from panel methods through the PGML framework to enhance the
generalizability of data-driven surrogate models.
The training data for the neural network is generated using a series of
numerical simulations. The main goal of this work is to demonstrate how a
simplified model can be used in the PGML framework and therefore, to reduce
the computational complexity, we utilize XFOILDrela (1989) for the aerodynamic
analysis of different airfoils instead of full CFD simulation to get the most
accurate results for the forces on the airfoil. The lift coefficient data were
obtained for different Reynolds numbers between $1\times 10^{6}$ to $4\times
10^{6}$ and several angles of attacks in the range of $-20$ to $+20$. A total
of 168 sets of two-dimensional airfoil geometry were generated for training
the neural network. This includes NACA 4-digit, NACA210, NACA220, and NACA250
series, and each airfoil is represented by 201 points. The maximum thickness
of all airfoils in the training dataset was between 6% to 18% of the chord
length. We use the NACA23012 and NACA23024 airfoil geometry as the test
dataset to evaluate the predictive capability of the trained neural network.
The simplified model used to generate the physics-based feature corresponds to
the Hess-Smith panel methodHess (1990) based on potential flow theory. We
highlight that our testing airfoils are selected not only from a different
NACA230 series (i.e., not used in the training dataset) but also the maximum
thickness of 24% is well beyond the thickness ratio limit included in our
training data set.
Figure 2: The representative neural network architecture of the PGML framework
used in this study for aerodynamic forces prediction task. The latent
variables at the third hidden layers are augmented with the physical
parameters of the flow (i.e., the Reynolds number and angle of attack) and the
prediction from the Hess-Smith panel method (i.e., lift coefficient and
pressure drag coefficient denoted as $\bar{C_{L}}$ and $\bar{C_{DP}}$,
respectively).
The neural network architecture used in this study has four hidden layers with
20 neurons in each hidden layer. The physical parameters, i.e., the Reynolds
number and the angle of attack are concatenated at the third hidden layer
along with the latent variables at that layer. In the PGML model, we augment
the latent variables at the third layer with the lift coefficient and the
pressure drag coefficient predicted by the panel method along with physical
parameters of the flow (i.e., the Reynolds number and angle of attack).
Therefore, the third layer of the PGML neural network has 24 latent variables.
The representative neural network for the PGML framework to predict the
aerodynamic forces on an airfoil is displayed in Figure 2. We utilize ensemble
of neural networks to quantify the predictive uncertaintyTibshirani (1996);
Heskes (1997); Lakshminarayanan, Pritzel, and Blundell (2017). In this method,
a small number of neural networks are trained from different initialization
and the variance of the ensemble’s prediction is interpreted as the epistemic
uncertainty. This method is appealing due to its simplicity, scalability, and
strong empirical results of the uncertainty estimate that are as good as the
Bayesian neural networksLakshminarayanan, Pritzel, and Blundell (2017). We
train multiple neural network models using different values of the initial
weights and biases. The weights and biases of each model are initialized using
the Glorot uniform initializer and different random seed numbers are used to
ensure that the different values of weights and biases are assigned for each
model. The ensemble of all these models indicates the model uncertainty
estimate of the predicted lift coefficient. Figure 3 shows the actual and
predicted lift coefficient for the NACA23012 and NACA23024 airfoil geometry.
The reference _True_ performance is obtained by XFOIL. The ML corresponds to a
simple feed-forward neural network that uses the airfoil $x$ and $y$
coordinates as the input features, and the physical parameters of the flow are
concatenated at the third hidden layer along with the latent variables at that
layer.
Figure 3: Actual versus predicted lift coefficient ($C_{L}$) for NACA23012 and
NACA23024 airfoils at $Re=3\times 10^{6}$ using ML and PGML framework. The
dashed red curve represent the average of the predicted lift coefficient by
all ML models (i.e., testing runs with different initialization seeds). Both
airfoil geometries were not included in the training. Here we note that there
is a significant reduction of uncertainty in performance when we use the
proposed PGML framework, especially for the smaller angle of attacks where the
embedded simplistic physics-based model is valid. The physics embedding in
these tests are based on the utilization of the Hess-Smith panel method, which
is limited mostly for the angle of attacks between -10 and +10 degrees, which
further verifies the success of the proposed PGML framework.
As shown in Figure 3, we can see that the uncertainty in the prediction of the
lift coefficient by the PGML model is less than the ML model for both
NACA23012 and NACA23024 airfoils. The proposed PGML framework provides
significantly more accurate predictions with reduced uncertainty for the angle
of attacks between -10 and +10 degrees. This further illustrates the viability
of the proposed PGML framework, since the physics embedding considered here
employs constant source panels and a single vortex to approximate the
potential flow around the airfoil. We can also notice that the uncertainty is
higher for the angle of attacks outside the range of -10 to +10 degrees. This
finding is not surprising as the Hess-Smith panel method is a proven method
for analysis of inviscid flow over airfoil for the smaller angle of attacks
regime. We highlight that the maximum thickness of an airfoil included in the
training dataset is 18% of the chord length. Therefore, the uncertainty in the
prediction of the lift coefficient by the ML model is higher for the NACA23024
airfoil compared to the NACA23012 airfoil. The inclusion of physics-based
features from the panel method in the PGML model leads to a reduction in this
uncertainty estimate. These results clearly show the advantage of augmenting
the neural network model with simplified theories and the potential of the
PGML framework for physical systems. One of the important caveats with any
neural network is its design and the hyperparameters. The neural architecture
search and hyperparameter optimization are important processes for the success
of the PGML framework. If the network is shallow, then it has less expressive
capabilities and that deteriorates the prediction. On the other hand, if the
neural network architecture is very deep and there is no sufficient training
data, then the network fits the training data very well. However, its
generalizability is reduced and this is usually indicated by an increase in
the loss of the validation dataset after a few iterations of the training.
Some of the strategies to mitigate overfitting issues in the deep neural
network are using $L_{1}/L_{2}$ regularization, applying dropout, early
stopping, data augmentation, and $k-$fold cross-validation. Overall, our
findings suggest that the absorption of physical information into ML methods
improves the modeling uncertainty of the ML architectures.
_Concluding Remarks_ — The data-driven models derived from the data collected
from satellite measurements, internet of things (IoT) devices, and numerical
simulation are increasingly being applied for scientific applications in fluid
dynamics. While these data-driven models are successful, it might be vital to
complement them with physical laws that have been studied for many decades. To
this end, physics-informed machine learning approaches such as embedding soft
and hard constraints designed based on governing laws of the physical system,
have been proposed Raissi, Perdikaris, and Karniadakis (2019); Zhu _et al._
(2019); Pan and Duraisamy (2020); Mohan _et al._ (2020). In this work, we
illustrated how physics-based models derived from simplified approximations of
the physical system can be coupled within data-driven models like neural
networks. The PGML framework introduced in this study allows us to take
advantage of the generalizability of physics-based simplistic models and the
robustness of data-driven models. We demonstrated the proof-of-concept for the
aerodynamic performance analysis of airfoil geometry. The physics-based
features computed from the simplistic panel method are coupled with the latent
representation learned at the intermediate layers of the neural network. The
inclusion of these physics-based features assists the neural network model in
reducing the uncertainty of the lift coefficient prediction. Therefore, the
PGML framework is successful in improving the generalizability of data-driven
models.
Additionally, the PGML framework will also be useful for physical systems
where the data is scarce. For example, the generation of training data is
computationally expensive when dealing with larger problems such as the
optimization of a wind farm layout Samorani (2013) or the shape optimization
of a three-dimensional wing Epstein _et al._ (2009). The simplified models
like the actuator disk theory can be used along with a velocity field
generated from few high-fidelity numerical simulations to build a physics-
guided neural network-based surrogate model of the wind farm that can be
coupled with any optimization algorithm. Another area where the PGML framework
may bear huge potential is the digital twin technologies for physical
systemsRasheed, San, and Kvamsdal (2020), where the digital replica of the
physical system is build for tasks such as real-time control, efficient
operation, and accurate forecast of maintenance schedules. In the present
study, we started with XFOIL to assess the feasibility of the proposed
framework. Despite its simplicity, this is the first step towards
demonstrating the PGML framework for problems in fluid dynamics. In our future
work we plan to extend the PGML approach to complex and high-dimensional
problems like prediction of terrain induced atmospheric boundary layer flows
or flows around bluff bodies immersed in fluid to illustrate the true
capability of the PGML framework.
This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Advanced Scientific Computing Research under
Award Number DE-SC0019290. O.S. gratefully acknowledges their support.
The data that supports the findings of this study are available within the
article. Implementation details and Python scripts can be accessed from the
Github repositoryPawar (2020).
## References
* Hornik, Stinchcombe, and White (1989) K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks 2, 359–366 (1989).
* Hagan and Menhaj (1994) M. T. Hagan and M. B. Menhaj, “Training feedforward networks with the marquardt algorithm,” IEEE Transactions on Neural Networks 5, 989–993 (1994).
* Hagan, Demuth, and Beale (1997) M. T. Hagan, H. B. Demuth, and M. Beale, _Neural network design_ (PWS Publishing Co., Boston, 1997).
* Vidal _et al._ (2017) R. Vidal, J. Bruna, R. Giryes, and S. Soatto, “Mathematics of deep learning,” arXiv preprint arXiv:1712.04741 (2017).
* Krasnopolsky and Fox-Rabinovitz (2006) V. M. Krasnopolsky and M. S. Fox-Rabinovitz, “Complex hybrid models combining deterministic and machine learning components for numerical climate modeling and weather prediction,” Neural Networks 19, 122–134 (2006).
* Lee and Oh (1997) J.-W. Lee and J.-H. Oh, “Hybrid learning of mapping and its Jacobian in multilayer neural networks,” Neural Computation 9, 937–958 (1997).
* Tang and Hsieh (2001) Y. Tang and W. W. Hsieh, “Coupling neural networks to incomplete dynamical systems via variational data assimilation,” Monthly Weather Review 129, 818–834 (2001).
* Marsland, Shapiro, and Nehmzow (2002) S. Marsland, J. Shapiro, and U. Nehmzow, “A self-organising network that grows when required,” Neural Networks 15, 1041–1058 (2002).
* Raissi, Perdikaris, and Karniadakis (2019) M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics 378, 686–707 (2019).
* Zhu _et al._ (2019) Y. Zhu, N. Zabaras, P.-S. Koutsourelakis, and P. Perdikaris, “Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data,” Journal of Computational Physics 394, 56–81 (2019).
* Swischuk _et al._ (2019) R. Swischuk, L. Mainini, B. Peherstorfer, and K. Willcox, “Projection-based model reduction: Formulations for physics-based machine learning,” Computers & Fluids 179, 704–717 (2019).
* Sekar _et al._ (2019) V. Sekar, Q. Jiang, C. Shu, and B. C. Khoo, “Fast flow field prediction over airfoils using deep learning approach,” Physics of Fluids 31, 057103 (2019).
* Beck, Flad, and Munz (2019) A. Beck, D. Flad, and C.-D. Munz, “Deep neural networks for data-driven LES closure models,” Journal of Computational Physics 398, 108910 (2019).
* Rahman, Rasheed, and San (2018) S. Rahman, A. Rasheed, and O. San, “A hybrid analytics paradigm combining physics-based modeling and data-driven modeling to accelerate incompressible flow solvers,” Fluids 3, 50 (2018).
* Maulik _et al._ (2019a) R. Maulik, O. San, J. D. Jacob, and C. Crick, “Sub-grid scale model classification and blending through deep learning,” Journal of Fluid Mechanics 870, 784–812 (2019a).
* Maulik _et al._ (2019b) R. Maulik, O. San, A. Rasheed, and P. Vedula, “Subgrid modelling for two-dimensional turbulence using neural networks,” Journal of Fluid Mechanics 858, 122–144 (2019b).
* Pan and Duraisamy (2020) S. Pan and K. Duraisamy, “Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability,” SIAM Journal on Applied Dynamical Systems 19, 480–509 (2020).
* Mohan _et al._ (2020) A. T. Mohan, N. Lubbers, D. Livescu, and M. Chertkov, “Embedding hard physical constraints in neural network coarse-graining of 3D turbulence,” arXiv preprint arXiv:2002.00021 (2020).
* Muralidhar _et al._ (2020) N. Muralidhar, J. Bu, Z. Cao, L. He, N. Ramakrishnan, D. Tafti, and A. Karpatne, “Physics-guided deep learning for drag force prediction in dense fluid-particulate systems,” Big Data 8, 431–449 (2020).
* Qian _et al._ (2020) E. Qian, B. Kramer, B. Peherstorfer, and K. Willcox, “Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems,” Physica D: Nonlinear Phenomena 406, 132401 (2020).
* Pawar _et al._ (2020a) S. Pawar, S. E. Ahmed, O. San, and A. Rasheed, “Data-driven recovery of hidden physics in reduced order modeling of fluid flows,” Physics of Fluids 32, 036602 (2020a).
* Ahmed _et al._ (2020a) S. E. Ahmed, O. San, A. Rasheed, and T. Iliescu, “A long short-term memory embedding for hybrid uplifted reduced order models,” Physica D: Nonlinear Phenomena , 132471 (2020a).
* Ahmed _et al._ (2020b) S. Ahmed, S. Pawar, O. San, A. Rasheed, and M. Tabib, “A nudged hybrid analysis and modeling approach for realtime wake-vortex transport and decay prediction,” arXiv preprint arXiv:2008.03157 (2020b).
* Pawar _et al._ (2020b) S. Pawar, S. E. Ahmed, O. San, A. Rasheed, and I. M. Navon, “Long short-term memory embedded nudging schemes for nonlinear data assimilation of geophysical flows,” Physics of Fluids 32, 076606 (2020b).
* Pawar, Ahmed, and San (2020) S. Pawar, S. E. Ahmed, and O. San, “Interface learning in fluid dynamics: Statistical inference of closures within micro–macro-coupling models,” Physics of Fluids 32, 091704 (2020).
* Pawar and San (2020) S. Pawar and O. San, “Data assimilation empowered neural network parameterizations for subgrid processes in geophysical flows,” arXiv preprint arXiv:2006.08901 (2020).
* Ahmed _et al._ (2020c) S. E. Ahmed, O. San, K. Kara, R. Younis, and A. Rasheed, “Interface learning of multiphysics and multiscale systems,” Physical Review E 102, 053304 (2020c).
* Golub _et al._ (2006) G. H. Golub, M. W. Mahoney, P. Drineas, and L.-H. Lim, “Bridging the gap between numerical linear algebra, theoretical computer science, and data applications,” SIAM News 39, 1–3 (2006).
* Lucia, Beran, and Silva (2004) D. J. Lucia, P. S. Beran, and W. A. Silva, “Reduced-order modeling: new approaches for computational physics,” Progress in Aerospace Sciences 40, 51–117 (2004).
* Taira _et al._ (2020) K. Taira, M. S. Hemati, S. L. Brunton, Y. Sun, K. Duraisamy, S. Bagheri, S. T. Dawson, and C.-A. Yeh, “Modal analysis of fluid flows: Applications and outlook,” AIAA Journal 58, 998–1022 (2020).
* Fonn _et al._ (2019) E. Fonn, H. van Brummelen, T. Kvamsdal, and A. Rasheed, “Fast divergence-conforming reduced basis methods for steady Navier–Stokes flow,” Computer Methods in Applied Mechanics and Engineering 346, 486–512 (2019).
* Pathak _et al._ (2018) J. Pathak, B. Hunt, M. Girvan, Z. Lu, and E. Ott, “Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach,” Physical Review Letters 120, 024102 (2018).
* White, Ushizima, and Farhat (2019) C. White, D. Ushizima, and C. Farhat, “Neural networks predict fluid dynamics solutions from tiny datasets,” arXiv preprint arXiv:1902.00091 (2019).
* Geneva and Zabaras (2020) N. Geneva and N. Zabaras, “Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks,” Journal of Computational Physics 403, 109056 (2020).
* Maulik _et al._ (2020) R. Maulik, K. Fukami, N. Ramachandra, K. Fukagata, and K. Taira, “Probabilistic neural networks for fluid flow surrogate modeling and data recovery,” Physical Review Fluids 5, 104401 (2020).
* Zha _et al._ (2007) G.-C. Zha, B. F. Carroll, C. D. Paxton, C. A. Conley, and A. Wells, “High-performance airfoil using coflow jet flow control,” AIAA Journal 45, 2087–2090 (2007).
* LeGresley and Alonso (2000) P. LeGresley and J. Alonso, “Airfoil design optimization using reduced order models based on proper orthogonal decomposition,” in _Fluids 2000 conference and exhibit_ (2000) p. 2545.
* Zhang, Sung, and Mavris (2018) Y. Zhang, W. J. Sung, and D. N. Mavris, “Application of convolutional neural network to predict airfoil lift coefficient,” in _2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference_ (2018) p. 1903.
* Bhatnagar _et al._ (2019) S. Bhatnagar, Y. Afshar, S. Pan, K. Duraisamy, and S. Kaushik, “Prediction of aerodynamic flow fields using convolutional neural networks,” Computational Mechanics 64, 525–545 (2019).
* Rajaram _et al._ (2020) D. Rajaram, T. G. Puranik, A. Renganathan, W. J. Sung, O. J. Pinon-Fischer, D. N. Mavris, and A. Ramamurthy, “Deep Gaussian process enabled surrogate models for aerodynamic flows,” in _AIAA Scitech 2020 Forum_ (2020) p. 1640.
* Drela (1989) M. Drela, “XFOIL: An analysis and design system for low Reynolds number airfoils,” in _Low Reynolds number aerodynamics_ (Springer, 1989) pp. 1–12.
* Hess (1990) J. L. Hess, “Panel methods in computational fluid dynamics,” Annual Review of Fluid Mechanics 22, 255–274 (1990).
* Tibshirani (1996) R. Tibshirani, “A comparison of some error estimates for neural network models,” Neural Computation 8, 152–163 (1996).
* Heskes (1997) T. Heskes, “Practical confidence and prediction intervals,” in _Advances in neural information processing systems_ (1997) pp. 176–182.
* Lakshminarayanan, Pritzel, and Blundell (2017) B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simple and scalable predictive uncertainty estimation using deep ensembles,” in _Advances in neural information processing systems_ (2017) pp. 6402–6413.
* Samorani (2013) M. Samorani, “The wind farm layout optimization problem,” in _Handbook of wind power systems_ (Springer, 2013) pp. 21–38.
* Epstein _et al._ (2009) B. Epstein, A. Jameson, S. Peigin, D. Roman, N. Harrison, and J. Vassberg, “Comparative study of three-dimensional wing drag minimization by different optimization techniques,” Journal of Aircraft 46, 526–541 (2009).
* Rasheed, San, and Kvamsdal (2020) A. Rasheed, O. San, and T. Kvamsdal, “Digital twin: Values, challenges and enablers from a modeling perspective,” IEEE Access 8, 21980–22012 (2020).
* Pawar (2020) S. Pawar, “PGML,” https://github.com/surajp92/PGML (2020).
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# On assessing excess mortality in Germany during the COVID-19 pandemic
Giacomo De Nicola 111Corresponding author<EMAIL_ADDRESS>muenchen.de Department of Statistics, LMU Munich, Germany Göran Kauermann
Department of Statistics, LMU Munich, Germany Michael Höhle Department of
Mathematics, University of Stockholm, Sweden
###### Abstract
Coronavirus disease 2019 (COVID-19) is associated with a very high number of
casualties in the general population. Assessing the exact magnitude of this
number is a non-trivial problem, as relying only on officially reported
COVID-19 associated fatalities runs the risk of incurring in several kinds of
biases. One of the ways to approach the issue is to compare overall mortality
during the pandemic with expected mortality computed using the observed
mortality figures of previous years. In this paper, we build on existing
methodology and propose two ways to compute expected as well as excess
mortality, namely at the weekly and at the yearly level. Particular focus is
put on the role of age, which plays a central part in both COVID-19-associated
and overall mortality. We illustrate our methods by making use of age-
stratified mortality data from the years 2016 to 2020 in Germany to compute
age group-specific excess mortality during the COVID-19 pandemic in 2020.
## 1 Introduction
First identified in Wuhan, China, in December 2019, the Coronavirus disease
2019 (COVID-19) caused by the SARS-CoV-2 virus developed into a worldwide
pandemic during the spring of 2020 (Velavan and Meyer, 2020). One of the
challenges for scientists has been to evaluate its impact in terms of life
loss across different countries and regions of the world. A possible way to do
this is through directly looking at the number of people who died while they
were confirmed to be infected. This measure, often defined as
COVID-19-associated mortality, is certainly more robust than other pandemic-
related quantities such as e.g. the number of reported COVID-19 cases, for
which it has become clear that there is a non-negligible discrepancy between
cases detected through tests and the number of individuals who were infected
(Lau et al., 2021; Schneble et al., 2021). Nonetheless, the raw number of
COVID-related fatalities can be subject to biases and interpretative issues as
well. In particular, this number might also be biased downwards, as COVID-19
cases can still remain unreported until and after the point of death.
Moreover, it is not always straightforward to identify if COVID-19 was the
primary cause of death: Some patients might have a SARS-CoV-2 infection, but
the actual contribution of the virus to the death might be minimal (Vincent
and Taccone, 2020). To deal with these issues, comparing all-cause mortality
is generally considered a more robust alternative for assessing the damage
done by the pandemic, and to compare its impact between regions or countries.
A first look at this matter for Germany was provided by Stang et al. (2020),
who looked at data from the first wave ranging from calendar weeks 10 to 26 in
2020. The authors came to the conclusion that a moderate excess mortality was
observable for this period in Germany, in particular for the elderly. Morfeld
et al. (2021) consider regional variation in mortality in Germany during the
first wave (see also Morfeld et al., 2020). A calculation of the years of life
lost over the course of the pandemic in Germany in 2020 was pursued by Rommel
et al. (2021). International analyses on excess mortality due to COVID-19
include e.g. Krieger et al. (2020), looking at data from Massachusetts,
Vandoros (2020) who focuses on England and Wales, and Michelozzi et al. (2020)
investigating mortality in Italian cities. Global analyses in this direction
were pursued by Karlinsky and Kobak (2021) and Aburto et al. (2021).
Monitoring excess mortality has a long tradition as part of analysing the
impact of pandemics (Johnson and Mueller, 2002; Simonsen et al., 2013). With
the EuroMOMO project, Europe also runs an early-warning system specifically
dedicated to mortality monitoring (Mazick et al., 2007). However, no unified
methodological definition exists for deciding if the currently observed death
counts are higher than what would be expected. A very simple approach is to
compare the currently observed deaths for a selected time-period with the
average of death counts for a similar period in previous
years222https://www.destatis.de/DE/Themen/Gesellschaft-
Umwelt/Bevoelkerung/Sterbefaelle-Lebenserwartung/sterbefallzahlen.html.
Alternatively, the expected value can be computed by an underlying time-series
model based on past values, e.g. including seasonality and excluding past
phases of excess, as done in the EuroMOMO project (see e.g. Vestergaard et
al., 2020; Nørgaard et al., 2021). These approaches, however, do not come
without problems, as the age structure within a population can change
significantly over time. Given that both general and COVID-related mortality
are heavily dependent on age (Dowd et al., 2020; Levin et al., 2020), raw
comparisons not taking age into account will often lead to biased estimates.
More sophisticated approaches thus need to adjust for different or changing
age structures in the population. The latter point is of particular relevance
when looking at aging populations (Kanasi et al., 2016) and the infectious
risks for the elderly (Kline and Bowdish, 2016). Such age-adjustments have a
long tradition in demography when comparing mortality across different regions
with different age-structure (Keiding and Clayton, 2014; Kitagawa, 1964). A
general discussion on aging populations and mortality can be found in Crimmins
and Zhang (2019).
In this paper, we build on existing methodology to propose two ways of
calculating expected mortality taking age into account, respectively at the
weekly and at the yearly level. These methods are compared to the existing
benchmarks on data from Germany over the years 2016-2019, for which age-
stratified information is available. We furthermore apply those methods to
assess age group-specific excess mortality in Germany during the COVID-19
pandemic in 2020. The remainder of the manuscript is structured as follows. In
Section 2 we look at yearly expected mortality, while the weekly view is
pursued in Section 3. Section 4 ends the paper with some interpretative
caveats and concluding remarks.
## 2 Yearly Excess Mortality
We first look at yearly data and tackle the question of whether there was
excess mortality in Germany in 2020. In order to obtain an age adjustment for
mortality data we calculate expected deaths based on official life tables.
Life tables give the probability $q_{x}$ of a person who has completed $x$
years of age to die before completing their next life-year, i.e. before their
$x+1^{th}$ birthday. In our analysis we consider the death table provided for
the year 2017/2019 from the Federal Statistical Office of Germany (Destatis,
2020). The calculation of a life table, as simple as it sounds, is not
straightforward and is an age-old actuarial problem. First references date far
back, to Price (1771) and Dale (1772). A historical digest of the topic is
provided by Keiding (1987). Over the last decades, the calculation of the
German life-tables made use of different methods proposed in Becker (1874),
Raths (1909) and Farr (1859). We will come back to this point and demonstrate
that further adjustments are recommendable to relate the expected number of
deaths to recently observed ones. In particular, with increasing life
expectancy, the average age of the German population has been steadily
increasing (see e.g. Buttler, 2003), and this has some effect on the validity
of life tables, as discussed in Dinkel (2002). Generally, an aging population
leads to increasingly high yearly death tolls (see e.g. Klenk et al., 2007).
To quantify excess mortality one therefore needs to account for age effects,
e.g., leading to the standardized mortality ratio (SMR, see e.g. Rothman et
al., 2008). The SMR is defined as the ratio of observed death counts over
expected deaths and thus allows for an age adjusted view, meaning that instead
of pure death counts one takes the (dynamic) age structure into account.
Figure 1: Lexis Diagram
Calculating excess mortality on a yearly basis requires to calculate expected
fatalities using life tables provided by the relevant statistical bureau. We
make use of data provided by the Federal Statistical Office of Germany
(Destatis, 2020). A straightforward way of obtaining the expected number of
deaths for age group $A$ in year $y$ is to calculate
$\displaystyle e_{A,y}=\sum_{x\in A}q_{x}P_{x,y}$ (1)
where $P_{x,y}$ is the population size of individuals aged $x$ years at the
beginning of year $y$, and $q_{x}$ are, e.g., the age-specific death
probabilities in the most recent German life table from the years 2017/19,
calculated following Raths (1909). More specifically, let $D_{x}$ be the
cumulated number of individuals that died aged $x$ year old, i.e. before their
$x+1$-th birthday in the considered years 2017 to 2019. Let $P_{x,y}$ denote
the population size of $x$ year old individuals on December 31st in year
$y\in\\{2016,2017,2018,2019\\}$. Then $q_{x}$ provided in the German life-
tables is defined as
$\displaystyle
q_{x}=\frac{D_{x}}{\displaystyle\sum_{y=2016}^{2018}\frac{P_{x,y}+P_{x,y+1}}{2}+\frac{D_{x}}{2}}$
(2)
We label (1) in combination with (2) as Method 1 below. We will see that this
quantity is biased for estimating the expected number of deaths of $x$ year
old people in year $y$. To motivate this we look at the Lexis diagram in
Figure 1, and for simplicity we replace the calculation in (2) by looking at a
single year only, i.e from $y=t$ to $y=t+1$. This leads to $D_{x}=I+II$, where
$I$ and $II$ refer to the observed deaths in the two triangles in Figure 1.
Note that following the calculation principle (2) of the Statistisches
Bundesamt we would obtain $q_{x}$ as
$\displaystyle
q_{x}=\frac{D_{x}}{\displaystyle\frac{P_{x,t}+P_{x,t+1}}{2}+\frac{D_{x}}{2}}$
(3)
where $P_{x,t}$ and $P_{x,t+1}$ are the population sizes of $x$ year old
indicated in Figure 1. That is, $q_{x}$ is the probability of dying in
triangles $I$ and $II$. Let us define with $\tilde{q}_{x}$ the probability of
an individual aged $x$ years old at the beginning of year $t$ (i.e. on
December 31st in year $t-1$) to die before year $t+1$ starts. In other words
$\tilde{q}_{x}$ is the probability of dying in triangles $II$ and $III$. In
fact, this is the probability we are interested in. It is easy to see that
$\tilde{q}_{x}\neq q_{x}$. Assuming that the probability of dying in triangle
$I$ is roughly equal to the probability of dying in triangle $II$, and
assuming the same relationship for triangles $III$ and $IV$ holds, we can
conclude the approximate equivalence
$\displaystyle\tilde{q}_{x}=\frac{1}{2}q_{x}+\frac{1}{2}q_{x+1}$ (4)
which leads to the expected number of deaths
$\displaystyle\tilde{e}_{A,y}=\sum_{x\in A}\tilde{q}_{x}P_{x,y}.$ (5)
We label (5) as Methods 2 below. The adjustment is still not complete, and in
fact it can be shown that (5) is biased (see Hartz et al., 1983). Note that
individuals dying in triangle $III$ count as $x+1$ years old, so that part of
the deaths contributes to a different age group. We may now assume for
simplicity that the probability of dying in triangles $II$ and $III$ is
roughly the same, which leads to the following calculation. Let
$A=[a_{l},a_{r}]$
$\displaystyle\hat{e}_{A,y}=0.5\cdot\tilde{q}_{a_{l-1}}P_{a_{l-1},y}+\sum_{x=a_{l}}^{a_{r-1}}\tilde{q}_{x}p_{x,y}+0.5\cdot\tilde{q}_{a_{r}}P_{a_{r},y}$
(6)
where $\tilde{q}_{-1}=\tilde{q}_{0}$ and $P_{-1,y}=P_{0,y}$ gives the
approximation for the youngest age group. Accordingly, for $a_{r}=\max(x)$ we
take the full fraction of the last year, that is we add an additional
$0.5\cdot\tilde{q}_{a_{r}}p_{a_{r},y}$ to the formula above. We label (6) as
Method 3 below.
Based on these three methods we can now compare expected and observed
fatalities over the last years using the same 2017/2019 life-table as basis.
Note that, when looking at different years, one may more accurately also take
different life tables to account for changing life expectancy. We omit this
point for simplicity since we only look at five years, and changes in life
expectancy over this short period were moderate (Wenau et al., 2019). Figure 2
gives a first overview of the results for all age groups combined. We plot the
observed death counts (black dots), and the expected counts based on the
different methods are visualised as dashed lines in different colours. We can
see that Method 1, which uses (1), clearly underestimates the expected death
counts. Method 2 and Methods 3 perform equally well, which is not surprising,
since we do not take an age-specific view. The latter is carried out in Figure
3 for all different age groups available from the data. This age-specific view
shows how Methods 2 and 3 differ, and that overall Method 3 shows the better
fit. We can quantify this discrepancy by calculating the root mean squared
error for the different age groups, where we explicitly exclude year 2020 due
to the COVID-19 pandemic. The results of this can be found in Table 1.
Having established that Method 3 performs better than the other two over past
years, we can use the expected number of fatalities computed with this method
for 2020 to quantify the excess mortality during the first calendar year of
the corona pandemic in Germany. Table 2 contains expected and observed
mortality for all age groups in 2020, as well as the absolute and percentage
variations between the two. We see from the table that, for the entire
population, the age-adjusted excess mortality was in the order of 1% in 2020.
We stress that these results in terms of COVID-19 impact need to be
interpreted with utmost care: We here focus on the methodological aspects, and
defer the subject-matter discussion of the results to Section 4.
Table 1: Age-specific root mean squared error for expected yearly fatalities calculated with different methods over the years 2016 to 2019. Year 2020 is excluded due to the COVID-19 pandemic. The smallest value for each age group is highlighted in bold. | 0-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90+ | Overall
---|---|---|---|---|---|---|---|---|---
Method 1 | 302.4 | 121.9 | 413.8 | 2221.8 | 2801.3 | 2112.7 | 24244.9 | 18374.0 | 47942.7
Method 2 | 273.0 | 189.2 | 1052.8 | 2648.0 | 2969.6 | 10362.0 | 7038.7 | 18374.0 | 13676.7
Method 3 | 358.2 | 97.7 | 471.6 | 1775.9 | 1207.1 | 1760.1 | 7570.3 | 3413.2 | 13670.8
Age group | Expected 2020 | Observed 2020 | Absolute diff. | Relative diff.
---|---|---|---|---
$[00,30)$ | 7471 | 7298 | -173 | -2%
$[30,40)$ | 6663 | 6832 | 169 | +3%
$[40,50)$ | 15420 | 15704 | 284 | +2%
$[50,60)$ | 58929 | 57606 | -1323 | -2%
$[60,70)$ | 118047 | 118547 | 500 | +0%
$[70,80)$ | 199569 | 201844 | 2275 | +1%
$[80,90)$ | 379917 | 378404 | -1513 | +0%
$[90,\infty)$ | 193238 | 199761 | 6523 | +3%
Total | 979255 | 985996 | 6771 | +1%
Table 2: Expected and observed yearly mortality in 2020 for each of the six
age-groups, computed with Method 3. Figure 2: Expected deaths computed by year
with the three different methods described, for all age groups combined.
Realized fatalities are shown as black dots. Methods 2 and 3 are visually
indistinguishable, as age groups are pooled together.
Figure 3: Expected deaths by calendar year and age group computed with the
three different methods described. Realized fatalities are shown as black
dots.
## 3 Weekly Excess Mortality
The yearly view presented in the previous section does not allow to take
within-year seasonality into account for the expected deaths. We therefore now
look at weekly excess mortality statements. Classical standardization
approaches such as direct and indirect standardization can be used to adjust
the observed values for age effects, see e.g. Kitagawa (1964). We will focus
on indirect standardization, but given an appropriate choice of reference
population, direct standardization approaches are straightforward adaptations.
Figure 4: Weekly mortality probability estimates $\hat{q}_{t,a}$ as well as
the range (min-max) of the corresponding mortality probablities of the past
four years and their mean $\overline{q}_{t,x}$.
Let $q_{t,x}$ be the mortality probability specific to age $x$ and time period
$t$. In what follows, the considered time period will be one International
Organization for Standardization (ISO) week, but other intervals (e.g. months)
are also imaginable. We estimate $q_{t,x}$ by dividing the number of observed
deaths at age $x$ during time period $t$, defined as $D_{t,x}$, by the
corresponding population at the beginning of the time period, i.e. $P_{t,x}$.
To be specific, we define
$\displaystyle\hat{q}_{t,x}=\frac{D_{t,x}}{P_{t,x}}.$ (7)
Since the age-stratified population is only available as a point estimate for
the 31st of December of each year, we use linear interpolation to estimate
$P_{t,x}$. Furthermore, the exact population of the current year, i.e. on
December 31st, 2020, is not known at the time of analysis. We thus use a
corresponding population projection: Similarly to Ragnitz (2021), we use the
Destatis variant G2-L2-W2333https://www.destatis.de/DE/Themen/Gesellschaft-
Umwelt/Bevoelkerung/Bevoelkerungsvorausberechnung/Publikationen/Downloads-
Vorausberechnung/bevoelkerung-bundeslaender-2060-5124205199024.html. The
corresponding estimates of weekly mortality probabilities (7) are shown in
Figure 4. We see that in age groups $\geq 50$ years a substantial weekly
excess mortality is observable from week 45 on, with more pronounced excess
mortality for the elderly.
A weekly SMR-based excess mortality measure for the entire year 2020 can now
be computed as follows. Let $t$ denote a specific ISO week in 2020, i.e. this
will serve as notational shorthand for ISO week 2020-W$t$, where
$t=1,\ldots,53$. We form the expected age-time mortality probability for this
week by computing the average of the mortality of the same week over the last
4 years, i.e.
$\overline{q}_{t,x}=\frac{1}{4}\sum_{y=2016}^{2019}\hat{q}_{y\text{-W}t,x},\quad
t=1,\ldots,53.$
Because the years 2016-2019 do not have an ISO week 53, we define
$y\text{-W}53$ for $y=2016,\ldots,2019$ as
$\frac{1}{2}(q_{y\text{-W}52}+q_{(y+1)\text{-W}01})$. The indirect
standardization now computes the expected number of deaths for week $t$ as
$\overline{e}_{t,x}=\overline{q}_{t,x}\cdot P_{t,x}$
This corresponds to the expected number of deaths in week $t$ at age $x$, if
the current population would have been subject to the average death
probability over the past 4 years. Since fatalities are not given with exact
ages but rather by age group, we indicate this by using $q_{t,A}$, $P_{t,A}$
and $e_{t,A}$, where $A$ denotes the age classes. For the available Destatis
mortality data the six groups are
$[00-30),[30-40),[40-50),[50-60),[60-70),[70-80),[80,\infty)$. Fig. 4 shows
$\hat{q}_{t,A}$ as well as $\overline{q}_{t,x}$ for Germany. Note that the
comparison for week 53 with the past year is done using the imputation scheme
described above. Also note that this computation is equivalent to computing,
for each reference year $y$, the expected number of deaths for the relevant
week in 2020, and then taking the average of the expected deaths. In other
words: by applying the mortality probabilities for the same week of the
reference year $y$ to our study population (i.e. 2020-W$t$) and then averaging
the four expected fatalities, we get:
$\overline{e}_{t,x}=\frac{1}{4}\sum_{y=2016}^{2019}q_{y\text{-W}t,x}\cdot
P_{t,x}.$
One can now define the absolute excess mortality in week $t$ and age-group $A$
as $D_{t,A}-e_{t,A}$. Instead of focusing on absolute differences, it is
better in terms of interpretation to look at relative estimates of excess
mortality given by the standardized mortality ratio (SMR)
$\displaystyle SMR_{t,A}=\frac{D_{t,A}}{\overline{e}_{t,A}}.$ (8)
Figure 5: Weekly SMR estimates for the eight different age groups.
We plot the corresponding weekly estimate resulting from (8) for all age
groups in Figure 5. As already seen in the incidence plots, we note that in
the older age groups the first approx. 10 weeks of the year had a rather low
SMR, followed by a small increase consistent with the first COVID-19 wave.
Furthermore, substantial increases are then seen in in the $\geq 50$ year old
age groups starting from week 45, coinciding with the 2nd wave, and reaching
up to 40% more deaths than expected in certain weeks.
If we instead aggregate observed and expected counts per year, we could also
generate yearly excess-mortality statements similar to Tab. 2 (see e.g. Höhle,
2021 for comparison).
### Direct standardization
Whereas the indirect standardization strategy, pursued above, extrapolates the
average death probability from the past to the current population, an
alternative is to apply the mortality probabilities from each reference year
to a common standard population and then compare these numbers. This approach
is, e.g., used by Statistics
Austria444https://www.statistik.at/web_de/presse/125475.html and uses the
Eurostat 2013 population as common
reference555https://ec.europa.eu/eurostat/documents/3859598/5926869/KS-
RA-13-028-EN.PDF/e713fa79-1add-44e8-b23d-5e8fa09b3f8f:
$\overline{e}_{y\text{-W}t,A}^{\text{s}}=q_{y\text{-W}t,A}\cdot
P_{a}^{\text{s}},$
where $P_{A}^{\text{s}}$ denotes the size of the standard population in age-
group $A$ and the expected number of deaths for the week $t$ in 2020 is given
by
$e_{t,A}^{s}=\sum_{y=2016}^{2019}\overline{e}_{y\text{-W}t,A}^{\text{s}}/4$.
## 4 Discussion
The COVID-19 pandemic posed numerous challenges to scientists. One of those
challenges lies in estimating the number of casualties brought upon by the
pandemic. To tackle this issue, we pursued an approach based on comparing
observed all-cause mortality in 2020 with the number of fatalities that would
have been expected in the same year without the advent of COVID-19. Building
on existing methodology, we proposed two simple ways of computing expected
mortality at the yearly and at the weekly level. We then put those method to
work to obtain estimates for excess mortality in 2020 in Germany. The two
approaches yield similar results at the aggregate level, and highlight how
2020 was characterized by an overall excess mortality of approximately 1%. The
light excess mortality was apparently driven by a spike in fatalities related
to COVID-19 at the end of the year in older age groups.
Interpreting COVID-19 mortality has become a politically sensitive issue,
where the same underlying data are used to either enhance or downplay the
consequences of COVID-19 infections. We therefore stress that our interests
are methodological, and that the presented results are restricted to the
calendar year 2020 for Germany as a whole. Altogether, the mild mortality in
the older age groups during the first weeks (e.g. due to a mild influenza
season) balanced the excess in the higher age groups which came later in the
year. Clearly noticeable is the second wave during Nov-Dec 2020, which also
continued in the early months of 2021. To better account for such seasonality,
excess mortality computations for influenza are often pursued by season
instead of calendar year, i.e. in the northern hemisphere for the period from
July in Year $X$ to June in Year $X+1$ (Nielsen et al., 2011). Similarly, the
impact of COVID-19 cases and fatalities was not only temporally, but also
spatially heterogeneous, with strong peaks in Dec 2020 in the federal states
of Saxony, Brandenburg and Thuringia (Höhle, 2021). Hence, using mortality
aggregates over periods and regions only provides a partial picture of the
impact of COVID-19. Furthermore, the mortality figures observed in 2020
naturally incorporate the effects of all types of pandemic management
consequences, which include changes in the behavior of the population
(voluntary or due to interventions). Disentangling the complex effects of all-
cause mortality and the COVID-19 pandemic is a delicate matter, which takes
experts in several disciplines (demographers, statisticians, epidemiologists)
to solve. Timely analysis of all-cause mortality data is just one building
block of this process; Nevertheless, the pandemic has shown the need to do
this in near real-time based on sound data while adjusting for age structure.
Our analysis was motivated by the fact that many of the methods that have been
applied to tackle this issue so far fail to take the changing age structure of
the population into account. This can lead to biased results, and especially
so for the rapidly aging developed countries. In the case of Germany, for
example, the absolute number of people aged 80 or more increased by
approximately 20% from 2016 to 2020. Such a remarkable increase will naturally
have an effect on overall mortality, and as such direct comparisons in the
number of casualties across different years will lead to significant
overestimation of the excess mortality. Our approaches are instead robust to
such changes in population structure, and can be used regardless of the
demographic context. Note that, for both of our approaches, it would also be
possible to obtain confidence intervals through imposing simple distributional
assumptions. The same methodologies could be used to pursue a similar analysis
for any country in which mortality data and a mortality table are available,
for any given year. A natural use for the proposed methodology would also be
to assess the overall damages caused by the pandemic when it will be finally
considered a thing of the past. All in all, we hope the proposed methods will
help shedding light on the issue of computing the expected number of
fatalities, and in the assessment of potential excess mortality.
## References
* Aburto et al. (2021) Aburto, J. M., J. Schöley, L. Zhang, I. Kashnitsky, C. Rahal, T. I. Missov, M. C. Mills, J. B. Dowd, and R. Kashyap (2021). Recent Gains in Life Expectancy Reversed by the COVID-19 Pandemic. medRxiv.
* Becker (1874) Becker, K. (1874). Zur Berechnung von Sterbetafeln an die Bevölkerungsstatistik zu stellende Anforderungen: Gutachten über die Frage: Welche Unterlagen hat die Statistik zu beschaffen, um richtige Mortalitätstafeln zu gewinnen? Verlag des Königlichen statistischen Bureaus.
* Buttler (2003) Buttler, G. (2003). Steigende Lebenserwartung – was verspricht die Demographie? Zeitschrift für Gerontologie und Geriatrie 36, 90 – 94\.
* Crimmins and Zhang (2019) Crimmins, E. M. and Y. S. Zhang (2019). Aging Populations, Mortality, and Life Expectancy. Annual Review of Sociology 45(1), 69–89.
* Dale (1772) Dale, W. (1772). Calculations Deduced from First Principles, in the Most Familiar Manner, by Plain Arithmetic, for the Use of the Societies Instituted for the Benefit of Old Age: Intended as an Introduction to the Study of the Doctrine of Annuities. By a Member of One of the Societies. London: J. Ridley.
* Destatis (2020) Destatis (2020). Sterbetafel 2017/2019. Technical report, Statistisches Bundesamt.
* Dinkel (2002) Dinkel, R. H. (2002). Die langfristige Entwicklung der Sterblichkeit in Deutschland. Zeitschrift für Gerontologie und Geriatrie 35, 400 – 405\.
* Dowd et al. (2020) Dowd, J. B., L. Andriano, D. M. Brazel, V. Rotondi, P. Block, X. Ding, Y. Liu, and M. C. Mills (2020). Demographic science aids in understanding the spread and fatality rates of COVID-19. Proceedings of the National Academy of Sciences 117(18), 9696–9698.
* Farr (1859) Farr, W. (1859). On the Construction of Life-Tables, Illustrated by a New Life-Table of the Healthy Districts of England. Philosophical Transactions of the Royal Society of London 149, 837 – 878.
* Hartz et al. (1983) Hartz, A. J., E. E. Giefer, and R. G. Hoffmann (1983). A comparison of two methods for calculating expected mortality. Statistics in Medicine 2(3), 381–386.
* Höhle (2021) Höhle, M. (2021). Age-Structure Adjusted All-Cause Mortality. https://staff.math.su.se/hoehle/blog/2021/03/01/mortadj.html.
* Johnson and Mueller (2002) Johnson, N. P. and J. Mueller (2002). Updating the accounts: global mortality of the 1918-1920 ”Spanish” influenza pandemic. Bulletin of the History of Medicine, 105–115.
* Kanasi et al. (2016) Kanasi, E., S. Ayilavarapu, and J. Jones (2016). The aging population: demographics and the biology of aging. Periodontology 2000 72(1), 13–18.
* Karlinsky and Kobak (2021) Karlinsky, A. and D. Kobak (2021). The World Mortality Dataset: Tracking excess mortality across countries during the COVID-19 pandemic. medRxiv.
* Keiding (1987) Keiding, N. (1987). The Method of Expected Number of Deaths, 1786-1886-1986. International Statistical Review 55(1), 1–20.
* Keiding and Clayton (2014) Keiding, N. and D. Clayton (2014). Standardization and control for confounding in observational studies: a historical perspective. Statistical Science, 529–558.
* Kitagawa (1964) Kitagawa, E. M. (1964). Standardized comparisons in population research. Demography 1(1), 296–315.
* Klenk et al. (2007) Klenk, J., K. Rapp, G. Büchele, U. Keil, and S. K. Weiland (2007, 04). Increasing life expectancy in Germany: quantitative contributions from changes in age- and disease-specific mortality. European Journal of Public Health 17(6), 587–592.
* Kline and Bowdish (2016) Kline, K. A. and D. M. Bowdish (2016). Infection in an aging population. Current Opinion in Microbiology 29, 63–67.
* Krieger et al. (2020) Krieger, N., J. T. Chen, and P. D. Waterman (2020). Excess mortality in men and women in Massachusetts during the COVID-19 pandemic. Lancet 395(10240), 1829.
* Lau et al. (2021) Lau, H., T. Khosrawipour, P. Kocbach, H. Ichii, J. Bania, and V. Khosrawipour (2021). Evaluating the massive underreporting and undertesting of COVID-19 cases in multiple global epicenters. Pulmonology 27(2), 110–115.
* Levin et al. (2020) Levin, A. T., W. P. Hanage, N. Owusu-Boaitey, K. B. Cochran, S. P. Walsh, and G. Meyerowitz-Katz (2020). Assessing the age specificity of infection fatality rates for COVID-19: systematic review, meta-analysis, and public policy implications. European journal of epidemiology, 1–16.
* Mazick et al. (2007) Mazick, A. et al. (2007). Monitoring excess mortality for public health action: potential for a future European network. Weekly releases (1997–2007) 12(1), 3107.
* Michelozzi et al. (2020) Michelozzi, P., F. de’Donato, M. Scortichini, P. Pezzotti, M. Stafoggia, M. D. Sario, G. Costa, F. Noccioli, F. Riccardo, A. Bella, M. Demaria, P. Rossi, S. Brusaferro, G. Rezza, and M. Davoli (2020). Temporal dynamics in total excess mortality and COVID-19 deaths in Italian cities. BMC Public Health 20, 1238.
* Morfeld et al. (2020) Morfeld, P., B. Timmermann, J. Groß, S. DeMattheis, P. Lewis, P. Cocco, and T. Erren (2020). COVID-19: Spatial resolution of excess mortality in Germany and Italy. Journal of Infection.
* Morfeld et al. (2021) Morfeld, P., B. Timmermann, V. J. Groß, P. Lewis, and T. C. Erren (2021). COVID-19: Wie änderte sich die Sterblichkeit?–Mortalität von Frauen und Männern in Deutschland und seinen Bundesländern bis Oktober 2020\. Deutsche Medizinische Wochenschrift 146(02), 129–131.
* Nielsen et al. (2011) Nielsen, J., A. Mazick, S. Glismann, and K. Mølbak (2011). Excess mortality related to seasonal influenza and extreme temperatures in Denmark, 1994-2010. BMC Infectious Diseases 11(1), 350.
* Nørgaard et al. (2021) Nørgaard, S. K., L. S. Vestergaard, J. Nielsen, L. Richter, D. Schmid, N. Bustos, T. Braye, M. Athanasiadou, T. Lytras, G. Denissov, et al. (2021). Real-time monitoring shows substantial excess all-cause mortality during second wave of COVID-19 in Europe, October to December 2020. Eurosurveillance 26(2), 2002023.
* Price (1771) Price, R. (1771). Observations on Reversionary Payments; On Schemes for Providing Annuities for Widows, and for Persons in Old Age; on the Method of Calculating the Values of Assurances on Lives, and on the National Debt. London: Cadell.
* Ragnitz (2021) Ragnitz, J. (2021, Jan). Hat die Corona-Pandemie zu einer Übersterblichkeit in Deutschland geführt? - Aktualisierung 15.1.2021. Technical report, ifo Institute.
* Raths (1909) Raths, J. (1909). Die Sterblichkeitsmessung in der allgemeinen Bevölkerung. In Denkschriften und Verhandlungen des 6. Internationalen Kongressess für Versicherungswissenschaften, pp. 115–129. Wien.
* Rommel et al. (2021) Rommel, A., E. von der Lippe, D. Plaß, T. Ziese, M. Diercke, S. Haller, A. Wengler, et al. (2021). COVID-19-Krankheitslast in Deutschland im Jahr 2020. Technical report, Robert Koch-Institut.
* Rothman et al. (2008) Rothman, K. J., S. Greenland, and T. L. Lash (2008). Modern epidemiology. Lippincott Williams & Wilkins.
* Schneble et al. (2021) Schneble, M., G. De Nicola, G. Kauermann, and U. Berger (2021). Spotlight on the dark figure: Exhibiting dynamics in the case detection ratio of COVID-19 infections in Germany. medRxiv.
* Simonsen et al. (2013) Simonsen, L., P. Spreeuwenberg, R. Lustig, R. J. Taylor, D. M. Fleming, M. Kroneman, M. D. Van Kerkhove, A. W. Mounts, W. J. Paget, et al. (2013). Global mortality estimates for the 2009 Influenza Pandemic from the GLaMOR project: a modeling study. PLoS Med 10(11), e1001558.
* Stang et al. (2020) Stang, A., F. Standl, B. Kowall, B. Brune, J. Böttcher, M. Brinkmann, U. Dittmer, and K.-H. Jöckel (2020). Excess mortality due to COVID-19 in Germany. Journal of Infection 81(5), 797–801.
* Vandoros (2020) Vandoros, S. (2020). Excess mortality during the Covid-19 pandemic: Early evidence from England and Wales. Social Science & Medicine 258, 113101.
* Velavan and Meyer (2020) Velavan, T. P. and C. G. Meyer (2020). The COVID-19 epidemic. Tropical medicine & international health 25(3), 278.
* Vestergaard et al. (2020) Vestergaard, L. S., J. Nielsen, L. Richter, D. Schmid, N. Bustos, T. Braeye, G. Denissov, T. Veideman, O. Luomala, T. Möttönen, et al. (2020). Excess all-cause mortality during the COVID-19 pandemic in Europe–preliminary pooled estimates from the EuroMOMO network, March to April 2020. Eurosurveillance 25(26), 2001214.
* Vincent and Taccone (2020) Vincent, J.-L. and F. S. Taccone (2020). Understanding pathways to death in patients with COVID-19. The Lancet Respiratory Medicine 8(5), 430–432.
* Wenau et al. (2019) Wenau, G., P. Grigoriev, and V. Shkolnikov (2019). Socioeconomic disparities in life expectancy gains among retired German men, 1997–2016. Journal of Epidemiology & Community Health 73(7), 605–611.
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# Quantum Mathematics in Artificial Intelligence
Dominic Widdows<EMAIL_ADDRESS>
IonQ, Inc., 4505 Campus Drive, College Park, MD 20740, USA Kirsty Kitto
<EMAIL_ADDRESS>
University of Technology Sydney, PO Box 123, Broadway, NSW 2007, Australia
Trevor Cohen<EMAIL_ADDRESS>
University of Washington, Box 358047, 850 Republican St, Seattle, WA 98109,
USA
###### Abstract
In the decade since 2010, successes in artificial intelligence have been at
the forefront of computer science and technology, and vector space models have
solidified a position at the forefront of artificial intelligence. At the same
time, quantum computers have become much more powerful, and announcements of
major advances are frequently in the news.
The mathematical techniques underlying both these areas have more in common
than is sometimes realized. Vector spaces took a position at the axiomatic
heart of quantum mechanics in the 1930s, and this adoption was a key
motivation for the derivation of logic and probability from the linear
geometry of vector spaces. Quantum interactions between particles are modelled
using the tensor product, which is also used to express objects and operations
in artificial neural networks.
This paper describes some of these common mathematical areas, including
examples of how they are used in artificial intelligence (AI), particularly in
automated reasoning and natural language processing (NLP). Techniques
discussed include vector spaces, scalar products, subspaces and implication,
orthogonal projection and negation, dual vectors, density matrices, positive
operators, and tensor products. Application areas include information
retrieval, categorization and implication, modelling word-senses and
disambiguation, inference in knowledge bases, decision making, and semantic
composition.
Some of these approaches can potentially be implemented on quantum hardware.
Many of the practical steps in this implementation are in early stages, and
some are already realized. Explaining some of the common mathematical tools
can help researchers in both AI and quantum computing further exploit these
overlaps, recognizing and exploring new directions along the way.
## 1 Introduction
Vector space models have solidified a position at the forefront of artificial
intelligence (AI), and are a common building block for many kinds of systems.
From early uses in information retrieval and image processing, they have
spread to semantic modeling in many systems, reaching even greater prominence
during the past decade due to their use in deep learning. However, vector
spaces have a long and illustrious history, and as an analytical technique,
vector spaces were established in the mid-1800s. They also provide the
foundation for the underlying mathematical model for quantum theory.
This paper aims to support readers who are familiar with the use of vectors in
AI, and interested in mathematical concepts from quantum theory, to make use
of this promising set of results and novel techniques in the field of AI.
Section 2 summarizes the use of vector spaces in AI and quantum mechanics,
including an explanation of how the representation of wavefunctions leads to
the study of vector spaces and linear operators in formal quantum mechanics.
Section 3 explains the quantum model for information retrieval, including the
use of the quantum logic of subspaces for modelling concepts and query
operations. Section 4 looks in more detail at models for categories and
hypernyms, motivating the use of density matrices and positive operators from
quantum mechanics. Section 5 surveys vector models for semantic composition,
in particular the use of tensor products and related operators. Section 6
presents some of the early implementations of these methods on physical
quantum hardware, adding fresh vitality to this area of research.
Our hope is that AI practitioners with a grounding in linear algebra will find
that they already know much of the mathematics they need to understand large
parts of quantum theory, and to begin exploring ways in which they might apply
methods inspired by quantum mathematics to the generation of new results in
AI.
## 2 Vector Space Models in AI and Quantum Physics
Vectors have their roots in the beginning of Euclidean geometry, with the
notion that a straight line can be drawn between any two points and
indefinitely extended (Elements of Geometry, Book I, Axioms 1 and 2). By the
mid-1840s, 3-dimensional vectors and 1-dimensional scalars were thoroughly
explored in Hamilton’s work on quaternions; Cauchy and Hamilton collaborated
on the development of matrices and linear algebra; and the algebra of vectors
or ‘extended quantities’ (German Ausdehnungslehre) and their products in any
number of dimensions is fully-formed in the work of ? (?), a summary of which
is made available by ? (?).
Since then vectors have played a ubiquitous role in scientific advances across
a great range of fields, including differential geometry, electromagnetism,
relativity, machine learning: and of particular interest in this paper,
artificial intelligence and quantum theory.
### 2.1 Vector Models in AI
The use of vector spaces in AI goes back at least to information retrieval in
the 1960’s (?), the vector model of the SMART retrieval system being a
particularly influential example dating from this pioneering period (?).
Introductions to the vector model for information retrieval are given by ? (?,
Ch 5) and ? (?, Ch 14). In such a system, a document collection is processed
into a large table or matrix $M$ whose rows represent terms, whose columns
represent documents, and where the entry $M_{ij}$ contains a number measuring
the extent to which term $i$ appears in document $j$ (as is depicted in Table
1).
Table 1: A term–document matrix | Doc1 | Doc2 | Doc3 | … | Docm
---|---|---|---|---|---
Term1 | $M_{11}$ | $M_{12}$ | $M_{13}$ | … | $M_{1m}$
Term2 | $M_{21}$ | $M_{22}$ | $M_{23}$ | … | $M_{2m}$
Term3 | $M_{31}$ | $M_{32}$ | $M_{33}$ | … | $M_{3m}$
… | … | … | … | … | …
Termn | $M_{n1}$ | $M_{n2}$ | $M_{n3}$ | … | $M_{nm}$
The rows (or columns) in any such matrix can be added together by adding the
corresponding coordinates, and multiplied by a scalar by multiplying each
coordinate, so it is immediately apparent that the rows (or columns) of any
such matrix can be thought of as vectors in a vector space, whose dimension is
the number of columns (or rows).
Such a general description is naturally available as a model for many other
situations that are relevant to AI. Common examples include:
* •
The adjacency matrix of any weighted graph (?, Chapter VIII).
* •
A collection of grayscale images, each of the same width and height, where a
row represents an image, columns represent individual pixels, and each
coordinate represents the shade for that pixel in that image (?, Ch 3).
* •
A dataset from a medical study where each row corresponds to an individual
patient and columns correspond to a measurable property or “vital sign” such
as height, weight, age, or blood pressure values. Many statistical learning
techniques begin by assuming their inputs can be modelled in such a fashion
(?).
* •
Any collection of data projected into a lower-dimensional space, for example,
the ‘best fit’ subspace produced by a decomposition technique such as Singular
Value Decomposition. In the case of textual data, the technique of projecting
a term-document matrix onto such a subspace is referred to as Latent Semantic
Analysis (?), and today it has many variants, surveyed from an AI point of
view by ? (?) and from a machine learning point of view by ? (?, Ch. 8).
* •
The activity states of a connectionist network (?, ?, Definition 2.1). Work on
connectionist representations in AI contributed greatly to the development of
today’s powerful neural network models (?, Ch 10).
* •
The output of a nonlinear training / learning algorithm, for example, the
feature-weights learned by a neural network. Such models have become
increasingly prominent in the last few years: techniques that use several
intervening layers in the network are often referred to as deep learning, and
whether or not a network is deep in this sense, the inputs, outputs, and
learned parameters for intervening layers are typically represented as vectors
(?, Ch. 10). When applied to textual data the resulting vector models are
often referred to as word embeddings in more recent papers since ? (?).
Thus there are many related techniques for deriving collections of vectors
from empirical observations, and these techniques have become a crucial
cornerstone of AI. Key mathematical benefits include expressing addition of
vectors as a pairwise-sum of the corresponding coordinates (for vectors
$x=(x_{1},\ldots,x_{n})$ and $y=(y_{1},\ldots,y_{n})$, their vector sum
$x+y=(x_{1}+y_{1},\ldots,x_{n}+y_{n})$), and measuring similarity (for
example, using a scalar product where $x\cdot y=\sum_{i=1}^{n}x_{i}y_{i}$).
For a beginner’s introduction to measuring similarity and distance, and the
relationship between scalar products, cosine similarity, and Euclidean
distance, see ? (?, Ch 4, 5).
The graded way similarity can be expressed in vector models contrasts with
classical Boolean logic: for example, the introduction of cosine-similarity
between query vectors and document vectors as a continuous (hence graded)
measure of relevance in information retrieval enabled systems to return the
most relevant documents ranked in order. This property became vital once
document collections became large enough that simply marking documents as
‘relevant’ or ‘not relevant’ left too many ‘relevant’ documents for a user to
read.
The use of vector sums for generating graded probabilities in this manner has
been termed ‘superposition’ in physics. It enables a combination that is ‘some
of each’, which contrasts with both Boolean set intersection which is
sometimes too specific, and Boolean set union which is sometimes too general.
Historically, this modeling requirement was recognized very early in the
development of quantum theory, where the same vector sum technique is used to
effectively represent particles that can be measured in more than one way
(this usage is explained in more detail in Section 2.3).
The adoption of similar methods in AI is more recent, but stems from a similar
motivation, as expressed for example by ? (?):
> In the subsymbolic paradigm, the semantically interpretable entities are
> activation patterns, and these combine by superposition: activation patterns
> superimpose upon each other, the way that wave-like structures do in
> physical systems. This difference is another manifestation of moving the
> formalization from the discrete to the continuous …
The core motivation for using vector spaces in AI still has its roots in the
benefits of using continuous representations to complement and sometimes
improve upon discrete Boolean models. These representations can then be put to
good use in problems involving classification (?, Ch 14), (?, Ch 3-5) and
logical reasoning (?). Machine learning techniques for building such
representations have improved considerably, with the use of deep neural
networks, sequence modelling, transformers and attention-based techniques (for
a survey, see ? (?, Ch 16)). Vector models are used throughout AI and related
fields, all the way from pioneering work in information retrieval to today’s
neural network language models. So it makes sense to ask whether vector
methods derived from other areas of science are useful for learning and
reasoning with such models.
### 2.2 Why Quantum Theory?
Quantum physics appears often and prominently in the science and technology
news. Two popular quantum physics topics are immediately relevant to AI:
firstly, the suggestion that natural intelligence may involve quantum physics
directly, put forth most famously by ? (?); secondly, the increasing
technological progress being made in quantum computing, described by e.g., ?
(?, ?). The popular appeal of quantum physics and computing is easy to
appreciate: it challenges established norms; it is at forefront of innovation;
it is real yet mysterious (?).
It is commonly said that quantum theory applies only to particularly small
physical systems, because many of the discrete quantization and interference
effects explained by quantum mechanics are not significantly observable at
large scale. However, the mathematics developed as part of vector models in
quantum theory can also be used to describe larger human-scale phenomena, such
as individuals or groups making decisions or searching for information. One of
the pioneers and leaders in this area has been the physicist Diederik Aerts
(see ? (?) and onwards). The techniques and models proposed by Aerts and
others are sometimes called quantum-inspired, quantum-like, or generalized
quantum models (?). This makes a case that quantum mathematics is worth
investigating in situations outside the traditional microscopic domain of
quantum physics, without answering the challenging ontological questions that
arise when asking how quantum physics might affect larger systems. In this
context, the main argument in favor of quantum mathematical models is
utilitarian: many solutions to known problems have been improved in a number
of different ways. Books and papers discussing the application of quantum
mathematics are available in several fields including: information retrieval
(?); machine learning (?); cognitive science (?); and economics (?).
Within AI itself, the book by ? (?) provides a graduate-level text on quantum
computing and AI, grounded in computational theory, physics, and quantum
algorithms. In contrast, this paper provides a much more thorough introduction
to tensors and their uses. Potential quantum approaches in AI are well-
surveyed in the article by ? (?), which gives a thorough introduction to
qubits and quantum gates. In comparison, our work here focusses upon two
decade’s hindsight and progress in the use of vector models and related
mathematics including subspaces, projections, and tensor products which has
flourished in AI.
Simultaneously, the increasing power of quantum computers has turned quantum
mechanics from a fascinating mystery to an increasingly practical opportunity,
and quantum advantages such as the promise of exponential memory savings are
bound to attract the attention of researchers in AI. Some recent advances in
AI and quantum computing are presented in Section 6.
A question remains: _Why_ does the mathematics of quantum theory appear to
support these types of scenarios? Many different explanations are possible
depending upon the philosophical stance that one adopts, but for now we will
remain agnostic. We will return to this important question in Section 7.6,
with more insights from the mathematics of quantum theory to support our
proposed answer to this question. For now we will continue with a brief
introduction to the topic of quantum theory itself.
### 2.3 Formal Quantum Mechanics
To appreciate the appeal of quantum-inspired techniques in vector space models
we must work to bridge a gap between the physical results of quantum theory
(as predicted by quantum mechanics), and its mathematical formulation (in
terms of state vectors and operators on Hilbert space). In the process, we
hope to show that topics considered to be philosophically challenging in
quantum physics, such as non-locality, quantum disjunction, and entanglement,
are mathematically quite straightforward. Many of them rely upon simple
operations such as projecting lines onto planes, finding the plane spanned by
two lines, and multiplying vectors to generate square matrices which can be
added together. The mathematical formulation of quantum mechanics mainly uses
standard linear algebra, which means that many of its techniques are already
familiar to those working in AI. An excellent introduction is provided by ?
(?) and of course ? (?). Of particular note, the formalism of quantum theory
requires us to move from a model of reality that assumes localised particles,
to one that assumes waves to be fundamental, and these can be represented as
vectors using Fourier series.
Quantum theory was invented to describe the behavior of subatomic particles,
and has successfully explained and predicted many behaviors that could not be
adequately accounted for by late 19th century classical conceptualizations.
These classical models understood the universe as made up of particles,
imagined to consist of Euclidean ‘points’ (that is, things that have no
spatial extent of their own, and can be located anywhere in a continuous
space), and vibrating waves such as light.111 This is an accepted if a
somewhat simplified view of physics before 1900. For example, Sir Isaac Newton
postulated the existence of distinct corpuscles of light, and both Democritus
and Plato postulated the existence of particles with some sort of shape or
spatial extent. In the early decades of the 20th century, particle-like
behavior was observed in light using experiments that increased our
understanding of the polarization and interference of photons (?, Ch 1). More
experiments revealed that matter exhibited wave-like as well as particle-like
behaviour. A new theory was needed, one in which both ‘matter’ such as atoms,
electrons, and protons, and ‘radiation’ such as light exhibited both wave-like
properties (such as interference) and particle-like properties (such as coming
in discrete chunks).
### 2.4 Waves as Vectors Using Fourier Series
Mathematically, any wave can be represented as a function. For example, a
standing wave on a string of length $L$ is represented as a function
$f:x\in[0,L]\rightarrow\mathbb{R}$, and a dynamic (moving) wave is represented
as a function of two variables
$f:(x\in[0,L],t\in\mathbb{R)}\rightarrow\mathbb{R}$. Real functions can be
multiplied by any real number, and added together pairwise, and with these
natural definitions, the standard vector space axioms are satisfied: thus the
set of all such real functions forms a vector space. Functions to the reals
are vectors.
Moreover, piecewise smooth functions of one variable can be represented as
weighted sums of their basic constituents. Such representations were
introduced by Joseph Fourier (1768–1830), who demonstrated that under certain
continuity conditions a real function $f$ over the interval $[0,2\pi]$ can be
uniquely represented as a sum of harmonic functions,
$f(x)=\sum_{0}^{\infty}a_{k}\sin(kx)+b_{k}\cos(kx).$
This sum is called the Fourier series of $f$, and the $a_{k}$ and $b_{k}$ are
called its Fourier coefficients. The example in Figure 1 shows the first few
harmonic functions that sum up to form closer and closer approximations to a
square wave, the expansion in this case being
$\sum_{n=1,3,5,\ldots}\sin(nx)/n$.
Figure 1: Fourier series approximations for a square wave, showing
contributions from the first two and a the first four nonzero harmonics.
In vector space terms,222Introductions to vector spaces — basis vectors,
coordinates, dimensions, and linear dependence — can be found in texts on
linear algebra (?), quantum computing (?, Ch 2), natural language processing
(?, Ch 5), and in many online tutorials. the sine and cosine functions form a
basis for the vector space of real functions, and the coefficients $a_{k}$,
$b_{k}$ are the coordinates of the vector $f$ in this basis. For example, in
the case of the square wave of Figure 1, the $a_{k}$ coordinates are
$(0,1,0,\frac{1}{3},0,\frac{1}{5},0,\frac{1}{7},\ldots)$ and the $b_{k}$
coordinates are all zero. To represent an arbitrary function with perfect
precision, an infinite number of such coordinates may be needed, so the vector
space is infinite dimensional in principle. Infinite dimensions are needed in
quantum mechanics to represent potential values of the position and momentum
operators (?, Ch 2), but in quantum computing, the most typical vector space
building block is the 2-dimensional complex vector space $\mathbb{C}^{2}$
which is used to represent the state of a single qubit (?, §1.2) (?, §2).
The vector space of piecewise smooth functions has a natural scalar product:
for two real functions $f$ and $g$ defined over the interval $[a,b]$, we
define their scalar product to be $\int_{a}^{b}f(x)g(x)\,dx$. (This scalar
product is often called the inner product and is typically written $\langle
f,g\rangle$ in functional analysis.) Crucially for the Fourier theory, for all
natural numbers $n$, $m$, the inner product
$\int_{0}^{2\pi}\sin(nx)\sin(mx)\,dx$ evaluates to 0, unless $n=m$ in which
case it evaluates to 1. The same result holds for products of cosine
functions, and all products of a sine and a cosine function. Thus these sine
and cosine functions form not only a basis, but an orthonormal basis for the
vector space of smooth real functions. (An orthonormal basis is one where each
basic vector has unit length and each pair of distinct basis vectors are
orthogonal to one another. Note that in machine learning, ‘one-hot encodings’
form a kind of orthonormal basis (?, Ch 2).
Representing functions using coordinates via Fourier series expansion is only
one option. Many other such expansions were later found, for single-variable
functions over the real numbers, and also for functions over other spaces such
as the sphere, the so-called spherical harmonic functions. The basis of
spherical harmonic functions is crucial to the solution of the Schrödinger
equation for the hydrogen atom, and in fact each possible energy level for an
electron corresponds to a particular spherical harmonic function. Because of
this, the number and distribution of spherical harmonic functions accounts
directly for the structure of the periodic table of elements in chemistry (?,
§39), but they are not just limited to physical systems. Interestingly, this
mathematical formalism has been fruitfully used in the rendering of images, a
point that was first recognised by ? (?). Spherical harmonics can be seen as a
way to encode information over a sphere, in general, beyond their historical
origins in solving partial differential equations in mathematical physics.
By the end of the 19th century this rich mathematical theory was fully mature,
before quantum physics was developed. Functions, series, limits, conditions
for convergence, and the uses of different bases of functions to represent
different physical systems (typically arising from solutions to partial
differential equations) were well-understood. Around this time David Hilbert
added to the earlier vector spaces of Hermann ? (?), working specifically with
vector spaces with a well-defined scalar product and where every convergent
sequence has a limit. Such a space is today called a Hilbert space.
Later, Paul ? (?) introduced the bra-ket notation, designed especially for
calculations involving wavefunctions of quantum systems represented as vectors
in such a Hilbert space. In Dirac notation, the wavefunction $\psi$ is written
as a ‘ket’ vector $\ket{\psi}$, and then the scalar product of $\ket{\phi}$
and $\ket{\psi}$ is written as $\braket{\phi}{\psi}$. The object $\bra{\phi}$
in this expression has its own interpretation as a covector or dual vector,
called a ‘bra’ vector in Dirac notation: for any vector space $V$ over the
field $F$, the dual space $V^{*}$ is the space of $F$-linear mappings from $V$
to $F$, whereupon it is easy to show that $V^{*}$ is isomorphic to $V$, and a
particular scalar product (written as a bilinear form $\langle u,v\rangle$)
induces one such isomorphism by the identity $u\rightarrow\langle
u,\\_\rangle$ (?, §2.1.4). One of the key reasons for using Hilbert space as
the setting for quantum mechanics is that the existence of such a scalar
product makes it possible to switch between the ket vector $\ket{\psi}$ and
the bra covector $\bra{\psi}$ whenever this is convenient, and the similarity
between the angle bracket notation for bilinear forms and the bra-ket notation
makes this elegantly obvious. Introductions to Dirac notation are given by ?
(?, Ch 2), ? (?, p. 104), ? (?, Ch 2), and the original presentation in Dirac
(?) is still very readable.
Figure 2: Projection using the outer product of the x axis with itself,
$\ket{x}\bra{x}$.
Figure 2 shows the projection of a ket vector $\ket{z}$ onto the $x$-axis
whose ket vector is $\ket{x}$. The projection operator is constructed as
$\ket{x}\bra{x}$, which represents the outer product of the vector $\ket{x}$
with itself. For $\ket{x}$, this outer product is a diagonal matrix with zero
values except for a 1 in the top leftmost quadrant. Multiplying $\ket{z}$ by
this matrix results in its projection onto the x axis, $P_{x}(\ket{z})$. Of
note, the length of this projected vector is $\sqrt{2}/2$, which is also the
scalar product between $\ket{z}$ and $\ket{x}$, or $\braket{z}{x}$ (as well as
the cosine of the angle between these unit length vectors). This projection
can be interpreted probabilistically — the length of the projection on
$\ket{x}$ gives the probability amplitude of observing the superposition $z$
in state ${x}$. Squaring this probability amplitude gives the probability
itself, in this case 0.5.
It should also be noted that elements of the dual space $V^{*}$ are linear
operators on $V$ and vectors in their own right: and elements of $V$ are also
linear operators on $V^{*}$. Matrices present another such example. An
$m\times n$ real matrix can be used to represent a linear mapping from
$\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ using matrix multiplication. At the same
time, it is easy to see that that matrix addition adds the coordinates of two
$m\times n$ matrices just like vector addition, and that the set of all
$m\times n$ is also a vector space — matrices are both vectors and operators
on vectors.
Another crucial definition related to dual vectors and linear operators is
that of the adjoint operator. The adjoint of a linear operator $A$ is the
operator $A^{\dagger}$ such that
$\braket{A^{\dagger}(\phi)}{\psi}=\braket{\phi}{A(\psi)}$ for all
$\ket{\phi}$, $\ket{\psi}$. The definition of a Hilbert space guarantees that
adjoints exist and are unique, and that $(A^{\dagger})^{\dagger}=A$ (?,
§2.1.6).
### 2.5 Observation and Uncertainty
Experimental measurements on quantum systems are represented in quantum theory
using an _observable_ , which is a linear operator $A$ that acts on the vector
$\ket{\phi}$ to give a vector $\ket{A\phi}$. If this observation is performed
twice in quick succession, the same state is always observed, so $\ket{A\phi}$
is invariant under the action of $A$. Such a situation occurs mathematically
for states represented by eigenvectors of the operator $A$, which are those
vectors for which $A\ket{\psi_{k}}=\lambda_{k}\ket{\psi_{k}}$ (i.e. vectors
that do not change their orientation when the operator is applied), where
$\lambda_{k}$ is the $k^{\mathrm{th}}$ eigenvalue of $A$. The probability of
observing the system $\ket{\phi}$ in the eigenstate $\ket{\psi_{k}}$ is given
empirically by $|\braket{\phi}{\psi_{k}}|^{2}$. As well as being probability
amplitudes, in geometric terms the scalar products
$\\{\braket{\phi}{\psi_{i}}\\}$ are also the coordinates for the vector
$\ket{\phi}$ when it is expanded in the basis given by the eigenvectors
$\\{\ket{\psi_{i}}\\}$. So to understand the way $A$ operates on $\ket{\phi}$,
it is enough to write the vector $\ket{\phi}$ as an expansion
$\sum\lambda_{i}\ket{\psi_{i}}$, where $\lambda_{i}=\braket{\phi}{\psi_{i}}$.
Such an expansion works only if the operator $A$ has a suitable spectrum of
eigenvectors with real eigenvalues (they must be real if their squares are to
be interpreted as probabilities). It follows from linear algebra that $A$ must
be self-adjoint, that is, $A=A^{\dagger}$ or more explicitly,
$\braket{A\psi}{\phi}=\braket{\psi}{A\phi}$ for all $\ket{\phi}$,
$\ket{\psi}$. In terms of coordinates and matrices, an operator is self-
adjoint if and only if it is represented by a matrix that is equal to its
conjugate transpose: that is, $a_{ji}=\bar{a}_{ij}$ for all $i,j$. Such a
matrix is called Hermitian.
Other results in quantum mechanics can now be derived relatively easily. For
example, a _quantum superposition_ arises whenever two (or more) state vectors
are added together, a scenario that commonly occurs in many vector space AI
models. Similarly, the Heisenberg Uncertainty Principle follows from the fact
that linear operators do not usually commute with one another (as is well-
known from matrix multiplication, $AB$ and $BA$ are not usually the same), and
the size of the uncertainty is bounded by the commutator $AB-BA$. This follows
directly from the mathematical representation, which as we have seen, follows
from the basic requirement that the system be represented in a way that is
amenable to linear superposition (hence the use of vectors), and that
observations be represented in such a way that performing the same observation
twice in succession gives the same answer.
Understanding the relationship between such mathematical implications and
behaviors in the physical world is one of the key areas of research in quantum
foundations. For example, on the Uncertainty Principle itself, ? (?, p. 114)
pointed out “It is surprising that such a famous principle in physics is
implied by the choice of mathematical representation for state and observable
in Hilbert space.” While the philosophical status of this implication may
still be debated, the claim that a change of basis leads to different and
unpredictable measurements has gone from a paradox in the 1930s, to the design
of encryption protocols such as BB84 in the 1980s (?, Ch 3), to commercial
offerings of quantum key distribution today. Other properties such as
nonlocality and entanglement will be introduced and discussed mathematically
in this paper: as described by authors including ? (?), although some of these
mathematical consequences were first perceived as awkward difficulties for
realistic physics, they have now become cornerstones of how quantum computers
work in practice.
## 3 The Quantum Formulation of Information Retrieval
Information retrieval was pioneering in the use of vector spaces for
representing language and information, and the first area related to AI and
computational linguistics to be described thoroughly from a quantum
theoretical point of view (by ? (?)). Because of this head start, information
retrieval and related word vector representations are a prominent source of
examples in this paper. This section introduces these applications of vectors
and quantum theory to information retrieval, explaining the way that cosine
similarity can be seen as a projection operator, and how projection operators
have been used to model other logical operations including conditionals,
negation and disjunction in the logic of subspaces and projections which is
still called ‘quantum logic’ following ? (?).
The vector space model for information retrieval (summarized by ? (?) and ?
(?, Ch 6)) stands out as one of the first AI models that was given a
distinctly quantum theoretical formulation, developed by ? (?) and ? (?, Ch
7). Van Rijsbergen realized that the Hilbert-space formulation of quantum
mechanics and the vector space model for information retrieval have much in
common, and are in many ways identical. This paved the way for much work in
understanding and exploiting commonalities between quantum mechanics and
information retrieval. A thorough summary of progress in the subsequent decade
is given by ? (?).
Van Rijsbergen (?) began to join these areas using the simple observation that
if we use the Euclidean norm to measure distance, then the vector spaces used
in information retrieval are all trivially Hilbert spaces (since they are
finite-dimensional, they are also complete). Typically, vector-based
information retrieval systems use the cosine measure to give similarities
between queries and documents, which gives the same ranking as using Euclidean
distance with normalized vectors (as explained in detail by ? (?, §5.5)), so
while the strictly metric properties of Euclidean space are often not
emphasized when calculating similarity, they are present mathematically. Given
a query vector $q$ and a document vector $d$ (both of unit length), the cosine
similarity $q\cdot d$ is also the length of the projection of $d$ onto $q$.
Using Dirac’s bra-ket notation, this can easily be written as $\braket{q}{d}$.
This theory was then developed using the non-commutative behavior of
projection operators to model the interaction of a document being relevant to
a particular query, and about a particular topic. It can be exploited to model
conditional logic and implication for IR in vector spaces (?, Ch 5). Thus
there are ways to adapt conditional operators to vector space models.
Importantly, much of the work involved can proceed (and historically has
proceeded) quite independently of quantum mechanics. Powerful logical
operations in Hilbert spaces are available, and whether or not they are
inherently ‘quantum’ is partly a historical question. Here, we will argue that
they are useful either way, but we will return to this this question in
Section 7.6.
### 3.1 An Intuitive Primer: Quantum Logic as Vector Subspaces and
Projections
Recognizing that similarity is implemented as a kind of projection operator
takes the parallel between QM and IR further, because the observables in a
quantum system are often represented as projection operators in a Hilbert
space. The logical structure of such projections was discovered and analyzed
in a seminal paper called The Logic of Quantum Mechanics (?), and has since
been called just ‘quantum logic’, or sometimes the ‘standard logic’ on a
Hilbert space (?, Ch 1). Geometrically speaking, the key to understanding the
contrast between quantum logic and Boolean logic is to see that Boolean logic
is modelled by subsets, whereas quantum logic is modelled by subspaces, that
is, subsets that are themselves vector spaces. For example, in 3-dimensional
space, any arbitrary collection of points can be considered a subset, but only
the lines and planes are subspaces. Note also that with a scalar product, each
subspace $P$ can also be used to define the operation of orthogonal projection
onto that subspace, and if $\ket{p_{1}},\ldots,\ket{p_{n}}$ is an orthonormal
basis for $P$, this projection takes the particularly simple form
$\pi_{P}=\sum_{i=1}^{n}\ket{p_{i}}\bra{p_{i}}$ (?, §2.1.6). The ‘similarity’
between a vector $\ket{a}$ and the subspace $P$ can then be defined as the
magnitude of $\pi_{P}(\ket{v})$, and when $P$ is one-dimensional and $\ket{a}$
has length one, this recovers the familiar definition of cosine similarity
between two vectors.
Figure 3: Vector Subspaces contrasted with Venn Diagrams of Sets. On the left,
a Euclidean picture of the quantum logic of subspaces, and on the right, a
Venn diagram picture of Boolean logic of sets.
A sketch highlighting some of the differences between Boolean and quantum
logic is given in Figure 3. In the realm of quantum logic (depicted left), we
can see that if a point $p$ can be written as $\lambda A+\mu B$ for some
$\lambda$ and $\mu$, it is considered part of the disjunction $A\vee B$ even
if $p$ is itself in neither $A$ or $B$. The negation $C$ is not the entire
rest of the space, but is strictly the orthogonal complement $C^{\perp}$. This
is a big contrast with Boolean logic — in the familiar Venn diagram on the
right hand side of Figure 3, if the point $p$ is in neither $A$ nor $B$, then
by definition it is part of $C=\neg(A\vee B)$. In quantum logic, if $p=\lambda
A+\mu B+\nu C$ and none of these coefficients are zero, then we cannot say
either that $p\in A\vee B$ or that $p\in C=\neg(A\vee B)$, but instead $p$ has
some of each. The conjunction operator behaves the same way in both quantum
and classical logic (because the intersection of two subspaces is itself a
subspace, for example, the intersection of two planes is typically a line),
but disjunction and negation behave very differently. Finally, in the quantum
scenario we find that if the set ${A,B,C}$ forms an orthonormal basis, then
the coefficients $\lambda$, $\mu$, and $\nu$ correspond precisely to the
probability amplitudes for observing $p$ in one of these basis states in
quantum mechanics (?, Ch 2).
#### 3.1.1 Subspace Disjunction
In quantum logic, the disjunction of two one-dimensional subspaces (lines
through the origin) is the plane that spans them both. In Figure 3 (left), the
example is that the disjunction of the lines $OA$ and $OB$ is the plane $OAB$,
and the notation is usually simplified by leaving out the origin $O$ and
saying that the disjunction of the lines $A$ and $B$ is the plane $A\vee B$.
Note that this includes points that are not in either of the initial lines
(these points correspond to a superposition in quantum mechanics.) In logical
terms, we say that quantum logic is non-distributive, because it fails to
satisfy the distributive law $P\wedge(Q\vee R)=(P\vee Q)\wedge(P\vee R)$.
More intuitively, the quantum disjunction interpolates (and sometimes
extrapolates) from its inputs. A conversational example that contrasts such
inclusion might be:
> It’s 70 or 80 miles from Oxford to Birmingham and buses leave at 9am or
> 11am.
Without even thinking about it, an adult reader will assume that the actual
distance (about 74 miles) is expressed perfectly well by the phrase “70 or 80
miles”, but will not assume that there is also a 10am bus departure. In the
first case, the combination is continuous, whereas in the second case, it is
discrete, and while Boolean logic gives a better model for the latter, the
former is more effectively modelled by quantum disjunction. (This is just an
illustrative example, and several other approaches such as fuzzy logic can be
motivated in a similar way.) The notion that an object can be modelled in a
space that combines $A$ and $B$ without it being part of either $A$ or part of
$B$ is a feature of quantum systems and their non-separability that we will
encounter again later (and more strikingly) with tensor product
representations and quantum entanglement. With subspaces, this affects the way
that categories or natural kinds might be modelled in an IR system, a point
discussed more fully by ? (?, Ch. 2) and ? (?, §3.3).
#### 3.1.2 Subspace Negation
The negation of a subspace in quantum logic is not just its set complement,
but its orthogonal complement: for example, for a plane in 3-dimensions, the
orthogonal complement is the linear subspace normal to the plane. For example,
in Figure 3, the orthogonal complement to the plane $A\vee B$ is the line $C$
that is perpendicular to both $A$ and $B$ and passes through the origin.
Negating a concept therefore involves projecting a vector onto this normal
subspace, which can move it over large parts of the space, so in this sense
quantum negation is a non-local operation.
#### 3.1.3 Subspace Conditionals Following Van Rijsbergen
The material conditional $A\rightarrow B$ of classical logic is modelled by
the inclusion $A\subseteq B$ in set theory, and the direct quantum analogy of
this is the subspace inclusion $A\leq B$, which means that $A$ is contained in
$B$ and $A$ and $B$ are both subspaces. In terms of the corresponding
projection operators, $A\leq B$ if and only if $\pi_{B}\pi_{A}=\pi_{A}$, which
is to say that if we have already projected onto $A$, projecting onto $B$ does
nothing.
In information retrieval, however, ? (?, Ch 5) is trying to describe the
relationship between a query and a relevant document, modelled by the document
‘implying’ the query — in the sense that the relevant document contains at
least the information in the query and more. Arguing that typically, a query
is not exactly implied by any document, the notion of relevance-as-implication
is qualified by the Logical Uncertainty Principle of ? (?):
> Given any two sentences $x$ and $y$; a measure of the uncertainty of
> $y\rightarrow x$ relative to a given data set, is determined by the minimal
> extent to which we have to add information to the data set, to establish the
> truth of $y\rightarrow x$.
? (?, Ch 5) argues that a conditional operator that meets this criterion is
(the projection onto) the subspace $A^{\perp}\vee(A\wedge B)$. This
corresponds to the notion that an object supports the claim “birds fly” if it
is a bird that flies or if it is not a bird at all. (The fact that pigs don’t
fly doesn’t provide positive evidence that birds fly, but at least it does not
refute the claim!) Van Rijsbergen’s exploration is logical rather than
empirical: the conditional is shown to satisfy logical compatibility
properties explored by earlier authors, and in the 2004 work, applications in
information retrieval such as relevance feedback and clustering were proposed
but not demonstrated. Nonetheless, this work sets the stage for later
developments, including the use of error terms in the classification
experiments of ? (?) and ? (?) discussed below in Section 4.3.
### 3.2 Application 1: Orthogonal Negation
It turns out that these relatively simple observations can be applied directly
to vector models for language. For example, suppose a corpus contains several
uses of the word suit in the sense of lawsuit, and several in the sense of
garment. It follows that the vector learned from training for suit will
involve a superposition of these senses, so
$\mathit{suit}=\mathit{suit}_{\mathit{lawsuit}}+\mathit{suit}_{\mathit{garment}}.$
(1)
Now, suppose a user is searching for documents about suit in the sense of
garment. An effective way to meet this need is to project the query vector
from the term suit onto the subspace orthogonal to lawsuit. This technique was
explored in detail with various term-retrieval measures by ? (?), which
demonstrated that the immediate advantage of projection over Boolean negation
in IR experiments was that it removed not only documents containing the
unwanted term lawsuit, but other documents that use terms related to lawsuit.
This form of vector negation reduced the occurrence of synonyms and neighbours
of the negated terms by as much as 76% compared with standard Boolean methods
— a practical outcome of the non-local nature of the negation operator. Its
effectiveness in improving document retrieval was demonstrated by ? (?), with
relative increases of up to 10.8% in Mean Average Precision (MAP) over a
standard baseline obtained when using a re-ranking strategy involving
rendering the vector average of 5 relevant documents orthogonal to that of a
single irrelevant counterpart, and much greater increases when incorporating
more relevance judgments (e.g. relative increase of 66.46% with feedback from
40 relevant and 40 irrelevant documents). The application of a projection
operator to remove or emphasize aspects of a mixed semantic vector
representation follows naturally from the fact that such vectors are generated
as superpositions of vectors that represent occurrence in a (usually sense-
specific) context. For a more thorough exposition of these points, see ? (?,
Ch 7) and ? (?).
Although negation is something of a niche operation in search engines, this
example helps us to start motivating an answer to the question: “What can
quantum models do for artificial intelligence?” In the case of modelling
negation, quantum logic from the 1930’s has provided a direct and
computationally simple technique for performing logical operations in a vector
model search engine. Building more thoroughly on the same foundation, ? (?)
have recently used the rules of quantum logic (including negation) to define
objective functions for optimizing the representation of ontologies and
relationships between concepts as vector space embeddings. So from this point
of view, quantum theory has motivated and guided these successes. However, it
is partly a historical accident that the logic of projections and subspaces is
called quantum logic at all: the meet, join, and orthogonal complement
operations on a vector space were all discussed back in the 1860’s by ? (?),
and the logical structure could easily have been explored independently of its
application to quantum mechanics by Birkhoff and von Neumann in the 1930’s. So
perhaps the right question is not “Can quantum logic be applied to IR?” (it
can), but “Would it be better to refer to ‘vector logic’ rather than ‘quantum
logic’ throughout the literature?”
The interpretation of products between vectors as subspaces is also a key part
of geometric algebra, which can be used a fundamental model for modelling
several physical and computational operators (?). One further complication
here is that there are other logical systems with vectors, including those of
? (?), ? (?), and ? (?). The question of which vector logic works most
effectively for information retrieval or other word embedding systems and
applications is a research area that is likely to yield many interesting new
results, but which has not to date attracted as much attention as it deserves.
### 3.3 Application 2: Modeling Term Dependencies for Information Retrieval
Another example of the utility of quantum models for information retrieval was
provided by ? (?), who investigated whether an extension of the standard
unigram / bag-of-words language model to accommodate dependencies between
terms in the context of frequently occurring multiword expressions such as
climate change can improve retrieval performance. The issue of term
dependencies is a longstanding problem in information retrieval. When
constructing a probabilistic or geometric representation of a document, the
question arises as to how to address the compositional nature of frequently
occurring short phrases. A simple approach would involve treating the phrase
climate change as a unit — as though it were a single word. However,
information about the occurrence of the component terms climate and change
would then be lost. Furthermore, this approach provides no clarity about how
best to weight such composite terms, which occur with less frequency than
their unigram components.
To address these issues, Sordoni and colleagues developed a Quantum Language
Model (QLM), a generalization of prior language modeling approaches that
provides the means to model term dependencies without severing the connection
between the probability of observing a multiword expression and the
probabilities of observing its component terms. Here, we will provide a brief
account of the model and the accompanying empirical results. We refer the
interested reader to ? (?) for further implementation details such as the
selection of phrases to model, smoothing and the estimation of model
parameters.
Language modeling approaches to information retrieval can be represented
geometrically by mapping each term in the vocabulary to an independent basis
vector (familiar to readers with a machine learning background as a ‘one hot
encoding’ (?, Ch 2)). Projection operators for each term can then be defined
to recover the probability of observing a particular term in the context of a
set of language model parameters learned from a document. Projection operators
for single words are diagonal matrices with a single non-zero value
corresponding to the basis vector of the term concerned.
With the QLM, multiword expressions are represented as a weighted
superposition of the basis vectors corresponding to their component terms:
$\overrightarrow{climate\\_change}=\alpha\times\overrightarrow{climate}+\beta\times\overrightarrow{change}$
which are normalized to length one. The projections onto these vectors lead to
off-diagonal entries in the corresponding matrices, so the term dependency
cannot be represented as just a weighted sum of the ‘one-hot’ diagonal
elements. The coefficients chosen for this weighting encode the extent to
which observation of the composite phrase indicates the presence of each
component term. This is intuitively appealing, and provides the means to
heuristically weight the relative importance of the terms within a phrase such
that, for example, documents containing the term ‘climate’ in isolation are
more likely to be retrieved in response to the query ‘climate change’ than
those containing the term ‘change’ alone. Given a unit of text, model
parameters are learned using an approximation algorithm that attempts to find
the parameters that maximize the probability of the observed document or
query. Relevance ranking is performed on the basis of the divergence between
parameters learned from queries and documents. Evaluation of the QLM was
conducted using selection of 450 queries drawn from across four information
retrieval evaluation sets. The best performance for each of the four sets was
obtained by a variant of the QLM, outperforming a unigram language model
baseline, with statistically significant improvements over then state-of-the-
art approaches using Markov Random Fields (MRF) (?) for the two larger web-
based sets with relative increases in MAP for the best-performing QLM as
compared with the best-performing MRF of 5.5% and 5.2%.
In summary, using quantum theoretic ideas directly in the traditional vector
model for information retrieval has provided several concrete opportunities,
including an account of conditionals and implications, negation and
disjunction, and term dependencies, some of which have improved performance in
retrieval experiments.
## 4 Categories, Hypernyms and Implication: More Advanced Structures in
Vector Spaces
Having presented examples of how vector spaces can provide a rich set of tools
for problems relevant to AI and ML, it is now time to move onto more advanced
vector space structures — what insights do they provide? Representing
categories and hypernym relationships will help us to motivate the use of
structures in vector spaces that are more complex than the individual points
referenced by vectors that we explored in the previous section.
### 4.1 Negation and ‘Parts’ of Vectors
The example above that uses negation to remove unwanted meanings of ambiguous
words carries an important conceptual lesson: a single vector can represent
multiple meanings in a high dimensional space, and specific ones can sometimes
be recovered explicitly using a method such as orthogonal projection to a
subspace.
This runs counter to a common belief expressed in the literature, that one
point can represent only one thing. As an example of this line of thinking,
consider ? (?) who start their widely cited paper with this statement:
> There is rising interest in vector-space word embeddings and their use in
> NLP, especially given recent methods for their fast estimation at very large
> scale. Nearly all this work, however, assumes a single vector per word type
> — ignoring polysemy and thus jeopardizing their usefulness for downstream
> tasks.
The effectiveness of vector negation for uncovering different word senses in a
single vector demonstrates a flaw in the claim that a single vector per word
ignores polysemy, even though the clustering results described by ? (?) are
still valuable. Mathematically, this claim would be equivalent to the claim
that a single vector per particle cannot represent a superposition of pure
states, a statement which quantum mechanics has shown to be flawed in the
quantum realm. Nonetheless, other authors have echoed this claim, including
for example ? (?):
> The prevailing objective of representing each word type as a single point in
> the semantic space has a major limitation: it ignores the fact that words
> can have multiple meanings and conflates all these meanings into a single
> representation.
The assumption that a vector is equivalent to a point and therefore cannot
represent something with internal structure or ingredients goes back to the
beginning: “A point is that which has no parts” is Book I, Definition I of
Euclid’s Elements. However, even in Euclidean space, a vector can also be
thought of as an arrow or straight path from the origin to a particular point.
This straight path certainly can have ‘parts’; in particular, each coordinate
represents a length along a particular axis (in Fourier analysis, a basic
harmonic function; in quantum mechanics, a basis state, etc.).
The notion that vectors have parts is much more obvious in quantum theory,
because as introduced in Section 2.3, the vectors in quantum mechanics are
functions: and of course, a function can have many different parts. Fourier
analysis itself is a particularly canonical way of breaking a function down
into a sum of different parts. From this point of view, the mistaken belief
that a vector can only represent one thing would be unlikely ever to have
arisen. Vectors can represent many types of things in many ways, and high-
dimensional vectors in particular can accurately represent many ingredients in
such a way that the ingredients can be clearly recognized and sometimes
recovered from the combined representation. The mathematical foundations
behind this claim are analyzed more thoroughly by ? (?) and ? (?).
Despite its mistaken foundations, work motivated by this understanding of
vector models has provided useful results. The perspective of ? (?) that we
criticised above still created word vectors by clustering contexts of word
occurrences (as pioneered by ? (?)) and built a representation for each sense.
Similarly, the survey of disambiguation techniques by ? (?), and the smoothed
Gaussian representations of ? (?) have both provided valuable contributions to
the field. Still, the stated mathematical motivation for such work is not
well-founded, and a quantum perspective on the mathematics naturally avoids
this mistake, and provides a richer formulation that can potentially be
leveraged to achieve novel results.
### 4.2 Subspaces, Disjunctions, and Generalization
Even though a single vector can represent several senses of a polysemous word,
it is important to realise that individual vectors cannot mathematically
represent everything that can be expressed in a vector space. The case of
linear subspaces, described in Section 3, is an obvious example. A
$k$-dimensional subspace in an $n$-dimensional space typically requires at
least $\min(k,n-k)$ vectors to describe it. The $n-k$ representation itself
provides an interesting insight: it is useful when $k>\frac{n}{2}$, which
makes it more convenient to express a subspace in terms of a normal projection
to the subspace. This technique is most commonly encountered in 3 dimensions
where the normal to a plane is just a line (for example see again Figure 3,
where the normal $C$ could also be used to define the plane $A\vee B$, because
$p\in A\vee B$ if and only if $p\cdot C=0$).
The link between disjunctions and categories or other ‘natural kinds’ is that
a more general category like mammal arises as a disjunction of examples like
cat, dog, mouse, elephant (?, Ch 3). In lattice theory, a disjunction is a
‘least common ancestor’, characterized by being the most specific element
available that subsumes (implies) all of the constituents (?, Ch 8). Since
subspaces are the natural representation of disjunctions in quantum logic, it
is tempting to assume they are a good representation for categories of word
vectors. This intuition has worked in some cases: for example, a combined
representation built using quantum subspace disjunction with limited numbers
of inputs has been shown to perform well at the task of geometrically-mediated
analogical inference, generally recovering more therapeutic relationships
between drugs and types of cancer than comparable superposition-based
approaches (?) and in information retrieval, subspaces have been shown to be
particularly effective for modelling negated disjunctions — that is, removing
many unwanted areas of meaning from a query vector (?). However, a
mathematical problem arises with positive disjunctions of many inputs: linear
sums of many similar vectors tend to extrapolate and overgeneralize (?).
Specifically, a disjunction of $k$ non-degenerate vectors always leads to a
$k$-dimensional subspace, however similar or different the initial vectors
are. For example, back in Figure 3, $A$ and $B$ would generate the exactly
same plane $A\vee B$ whether they are close together or far apart, so long as
they are not identical. This also means that in an $n$ dimensional space, $n$
very slightly different examples would nearly always generate the whole space.
So while subspaces have been shown to work well for some forms of semantic
generalization, we should expect this to be an incomplete and eventually over-
general model.
### 4.3 Density Matrices, Positive Operators, and Hyponymy
Quantum mechanics is already familiar with the problem that individual vectors
and even subspaces cannot represent everything we encounter in practice. A
single particle in a superposition of pure states can be represented as a
single vector, but an ensemble of particles cannot. Instead, ensembles are
modelled using a density matrix or density operator, which takes the form
$\rho=\sum_{j}P_{j}\ket{\Psi_{j}}\bra{\Psi_{j}}.$
In quantum mechanics, a system of particles that can be expressed as a density
matrix but not a state vector is called a mixed state. The motivation for
generalizing from states to density matrices is discussed in detail by ? (?,
Ch 6), and ? (?, Ch 6). Mathematically, the formalism is general enough to
represent all the relevant probability distributions, which an individual
state vector $\ket{\Psi}$ with corresponding density operator
$\rho=\ket{\Psi}\bra{\Psi}$ cannot.
Density matrices are positive operators, in the sense that
$\langle\Psi|\rho|\Psi\rangle\geq 0$ for all vectors $\Psi$. This can be used
to induce an ordering on density operators, defining $A\sqsubseteq
B\Longleftrightarrow B-A$ is positive. This is called the Löwner ordering, and
when the operators are restricted to projection onto subspaces, the Löwner
ordering becomes equivalent to the quantum logic introduced in Section 3.1.
The Löwner ordering is used by ? (?) and ? (?) to represent graded hyponymy
(where, for example, dog may be said to be a strict hyponym of mammal but a
graded hyponym of pet, because not all dogs are pets). In cases where $A$ and
$B$ are incomparable in this ordering, the systems proposed in these papers
work by finding positive operators $D$ and $E$ (an error term) such that
$A+D=B+E$. This avoids the problem with subspaces whereby any part of $A$ that
is not part of $B$ adds whole new dimensions: instead, if $A$ is nearly
subsumed by $B$, the error term $E$ will be correspondingly small. The smaller
$E$, the more strict the hyponymic relationship. In further work, ? (?)
extends this system to produce a graded form of negation, which can be applied
to more situations than the orthogonal negation of ? (?).
This example is critical, and illustrates the main point of this section:
while vectors are powerful representations and can represent many ingredients
at once, they cannot represent (for example) all the probability distributions
necessary for quantum mechanics. Exploring the ways these shortcomings have
been addressed within quantum theory suggests potentially fruitful research
avenues for those using vectors to model concepts in language.
## 5 Products of Vectors and Semantic Composition
The summary of categorization and negation in the previous section highlights
one of the key places where quantum techniques are useful in AI. They provide
us with a range of operators for exploring, manipulating, and generating
representations of concepts using semantic vector models. Some of these are
familiar, some are novel, and all open up potential new avenues for research
in AI. In this section we draw attention to the problem of semantic
composition. There are many ways to compose vectors — so far in this paper we
have discussed the vector sum, and various operators related to subspaces and
projections. More complicated structures are available, and semantic
composition is often represented in vector models using the tensor product and
its offshoots.
### 5.1 Tensor Products and Entanglement
The tensor product is one of the most significant methods used for composing
vectors. After exploring addition and subtraction, the next product structure
on vectors defined by ? (?, Ch 2) is the forerunner of today’s tensor product
in coordinate form: if $a=\sum a_{r}e_{r}$ and $b=\sum b_{s}e_{s}$, then their
product $[ab]$ is defined as $\sum a_{r}b_{s}[e_{r}e_{s}]$. By varying the
rules for interpreting and identifying the basic terms $[e_{r}e_{s}]$,
Grassmann showed that such product structures can be used to represent the
inner (scalar) product and combinatorial (exterior) product. Note that the
order matters here: unlike the sum of two vectors or the product of real or
complex numbers, this product between two vectors is not commutative. This
makes it more complicated, but also opens opportunities. For example,
generating a document vector as a weighted sum of term-vectors is surprisingly
effective for information retrieval, but a commutative sum that fails to take
word-order into account is unsuitable for building a number of important
applications (e.g. a conversational agent or any other dialogue system).
We have already noted that vectors can represent operators as well as states —
starting with the (co)vectors in the dual space $V^{*}$ that act as linear
maps from $V$ to its ground field (the ground field being the number system
used for coordinates, most commonly the real or complex numbers). Given this,
Grassmann’s product operator can be used to represent a bilinear map from
$V\times V$ to the ground field — for $\alpha,\beta\in V^{*}$ and $u,v\in V$,
we define $[\alpha\beta](u\times v)=\alpha(u)\beta(v)$, and it is easy to see
that this map is linear in all of its arguments. In Dirac notation, a product
$\bra{\phi_{i}\phi_{2}}$ of the bra vectors $\bra{\phi_{1}}$ and
$\bra{\phi_{2}}$ would map the product of two ket vectors
$\ket{\psi_{i}\psi_{2}}$ to
$\braket{\phi_{1}}{\psi_{1}}\braket{\phi_{2}}{\psi_{2}}$.
Today the space of all possible linear combinations of products of basis
vectors of two Hilbert spaces is called their tensor product, written using
the symbol $\otimes$, so that the tensor product of vector spaces $U$ and $V$
is written $U\otimes V$. It is defined more formally as an equivalence class
of mappings whereby (for example) if $E(A,B)\rightarrow U$ is a bilinear map
from (the Cartesian product of) $A$ and $B$ to $U$, then this corresponds to a
unique linear mapping from $A\otimes B$ to $U$, and it turns out that all
bilinear maps from $A\times B$ to $U$ can be represented in this manner.
Linear mappings from one space to another have a similar correspondence: the
space $L(A,B)$ of linear transformations from $A$ to $B$ is naturally
isomorphic to $A^{*}\otimes B$ (?, Ch. 16).
In a given coordinate basis, the tensor product of two vectors can be
represented as a matrix with the same coordinates as their outer product. Just
as the inner product or scalar product of two column vectors $u$, $v$ can be
written as $u^{T}v$, their outer product can be written $uv^{T}$. An example
with vectors in $\mathbb{R}^{3}$ is the following:
$u=\left(\begin{array}[]{c}1\\\ 0\\\ -2\end{array}\right)\qquad
v=\left(\begin{array}[]{c}2\\\ -1\\\ 3\end{array}\right)\qquad
u^{T}v=2+0-6=-4\qquad uv^{T}=\left(\begin{array}[]{ccc}2&-1&3\\\ 0&0&0\\\
-4&2&-6\end{array}\right)$
While this is intuitive and familiar, the mental identification of tensors
with matrices does lead to gaps. For example, the notation $u^{T}v$ for the
scalar product in coordinate matrix form encourages us to think of row vectors
as dual to column vectors, but matrix multiplication alone does not enable us
to map a $(n,1)$ column vector to a $(1,n)$ column vector. (Remember that
matrices of size $(p,q)$ can only be multiplied on the right by matrices of
size $(q,r$), resulting in a matrix of size $(p,r)$: so a matrix of size
$(n,1)$ can only be multiplied on the right to give a matrix of size $(1,n)$
if $n=1$, with a similar argument holding for left-multiplication.) So if we
were to identify tensors with such matrix representations, we would be tricked
into believing that tensors cannot be used to map vectors to covectors, which
is not the case.
More generally, tensors form an algebra of multilinear maps. If $V$ is a
vector space and $V^{*}$ is its dual, then the tensor space
$V\otimes\ldots\otimes V\otimes V^{*}\otimes\ldots\otimes V^{*}$ consists of
multilinear functions from $V^{*}\times\ldots\times V^{*}\times
V\times\ldots\times V$ to the ground field. (The $*$-symbols are deliberately
switched between the first and the second expression: the elements of $V^{*}$
act on elements of $V$, and vice versa.) Such a product with $p$ copies of $V$
and $q$ copies of $V^{*}$ is called a tensor of type $(p,q)$, and the number
$p+q$ is sometimes called the rank or the arity of the tensor. For example,
the density matrices of Section 4.3, being the product of a bra and a ket
vector, are tensors of type (1, 1) and rank 2.
Tensor algebra has been used extensively in many fields before computer
science. For example, it has been in use in differential geometry since the
early 1900’s, partly because so many geometric concepts can be expressed
easily using tensors, including vector fields, differential forms, metrics,
volume integrals, complex structures and Hamiltonian dynamics. Because of
this, some of the most readable and thorough introductions to tensor algebra
are from standard texts in differential geometry (for example, ? (?, Ch 5, 6),
? (?, Ch 2)). Continuum mechanics, which models the behavior of materials and
liquids as a continuous mass, also makes extensive use of tensor algebra to
extend Hooke’s law to high dimensions in the modelling of liquids, elasticity
of materials, and other important phenomena (via the stress-strain tensor —
see ? (?), which includes a good introduction to matrix and tensor algebra for
mechanics). Finally, as we have seen, in quantum mechanics the tensor product
naturally arises when we move towards representing composite systems. More
recently, in machine learning, tensors have made a sustained contribution
through the introduction of systems such as TensorFlow for training neural
networks (?, Ch 12). Tensor products are also used to describe the very
general notion of ‘processes happening simultaneously’ in physics, sometimes
generalizing the notion of tensor product to mathematical settings beyond
linear algebra (?, ?).
In summary, tensors have become an invaluable part of practical mathematics in
many fields. However, this very multidisciplinarity can result in confusion
around terminology. For example, a matrix can be used to represent a rank-2
tensor, but then the definition of the term rank is different from the
traditional definition in linear algebra, where the rank of a matrix is the
number of linearly independent rows and columns (a key theorem in linear
algebra guarantees that the row rank and the column rank are the same, so it
makes sense just to talk about the rank of a matrix (?, Ch 5)). This notion of
rank also generalizes to tensor algebra, and the rank of a tensor is also used
to mean the number of linearly-independent generators for the tensor. In some
fields (including machine learning), the difference between vector spaces and
their duals does not yet play nearly as important a role as it does in
differential geometry, and so this difference is often glossed over, and
instead of talking about tensors of type $(p,q)$, only the tensor rank $r=p+q$
is discussed. In particular, it is often said that the rank-2 tensors are
equivalent to matrices, which we have seen is an algebraically incomplete
representation (in the example above, a linear mapping from vectors to
covectors can be represented as a rank-2 tensor of type $(1,1)$, but no matrix
can be found that maps column vectors to row vectors using multiplication).
This teaching may change gradually, because used carefully, the distinction
between vectors and covectors can be a benefit in machine learning as well.
For example, ? (?) used precisely this approach to make a useful distinction
between domain and functional similarities in distributional semantics.
### 5.2 Tensor Products and Entanglement in Quantum Mechanics
In quantum mechanics, superposition becomes incomplete for describing the
state of systems as soon as we consider composites involving more than one
particle. Consider, for example, particles with wavefunctions $\ket{\phi}$ and
$\ket{\psi}$, each represented as a superposition
$\sum_{1}^{m}a_{i}\ket{\phi_{i}}$ and $\sum_{1}^{n}b_{j}\ket{\psi_{j}}$, where
$\ket{\phi_{i}}$ and $\ket{\psi_{j}}$ are the eigenstates for some observables
$A$ and $B$. In cases where the particles can be observed simultaneously, the
result of measuring $A$ and $B$ together can be any combination of
$\ket{\phi_{i}},\ket{\psi_{j}}$, and there are $mn$ such possibilities.
Therefore the combined system of $\ket{\phi_{i}}$ and $\ket{\psi_{j}}$ can
consist of any linear combination of these eigenstates, so it follows that the
Hilbert space of possible states for this system has $mn$ dimensions. This
deceptively natural conclusion should be contrasted with the classical
situation, where the state space for the combined system would be the
Cartesian product of the state spaces for the individual systems, having
dimension $m+n$. For a step-by-step example with spin states of 2- and
3-dimensional particles, see ? (?, §8.4). The combined state where measuring
$A$ results in $\ket{\phi_{i}}$ and $B$ results in $\ket{\psi_{j}}$ is written
as $\ket{\phi_{i}\psi_{j}}$, and the state that represents the combination of
the superpositions $\sum a_{i}\ket{\phi_{i}}$ and $\sum b_{j}\ket{\psi_{j}}$
is $\sum a_{a}b_{j}\ket{\phi_{i}\psi_{j}}$, which is exactly the same as
Grassmann’s definition of the product of two vectors above, just rewritten in
Dirac notation.
The mathematical behavior of tensor products is directly responsible for the
famous phenomenon of quantum entanglement. Entanglement is perhaps easiest to
introduce via its opposite phenomenon; separability. A composite state is
considered separable if it is possible to write it as a tensor product of two
distinct states (i.e. as $\ket{\phi}\otimes\ket{\psi}$ for some $\ket{\phi}$
and $\ket{\psi}$), in which case it is generally considered to be non-
interacting. In contrast, an entangled state cannot be decomposed in this
manner.
How does this come about? Mathematically, the answer is most obvious if we
consider the dimensions involved. If $U$ is of dimension $m$ and $V$ of
dimension $n$, their tensor product $U\otimes V$ has dimension $mn$, which of
course is typically greater than $m+n$. Immediately it follows that not every
element of $U\otimes V$ can be written as some individual product $u\otimes
v$, because there are only $m+n$ degrees of freedom for choosing the $u$ and
$v$. In order to generate any element of $U\otimes V$, we may need to
superpose many different individual products of the form $u\otimes v$, just as
in the example system above where the product state must be written as a
linear combination of states like $\ket{\phi_{i}}\otimes\ket{\psi_{j}}$. A
typical way of explaining the difference algebraically is that $u\otimes v$ is
not a standard element of $U\otimes V$, but it is a standard generator or
basis vector for $U\otimes V$. In matrix algebra, this corresponds to the
result that any rank-1 matrix (in the sense of matrix rank above!) can be
written as the outer product of a single row vector and a single column
vector.
With superposition and quantum disjunction, we saw earlier that a quantum
system might be represented as the sum of its parts without being identical
with or contained in any of those parts. The tensor products allows for even
richer combinations where the product is not contained in any of the
ingredients, and cannot even be broken down into a combination of one simple
ingredient from each part. A vector $u$ in the subspace $A+B$ can at least be
represented as the sum $a+b$ for some $a\in A,b\in B$, and this decomposition
is easily obtained using projection operators. On the other hand, a tensor
$u\in A\otimes B$ cannot even be decomposed like this: at its most general, it
must be written as a linear combination of the form $u=\sum a_{i}\otimes
b_{i}$ (and the minimum possible $i$ is the tensor generalization of the rank
of a matrix). While quantum logic can be described as non-distributive and
non-local, tensor products and entanglement demonstrate non-separability (?,
§4.1.1).
The concept of entangled quantum systems lies at the center of some of the
most interpretationally fraught aspects of quantum mechanics (?, ?). For
example, if a system of two particles is entangled, but then one of its
constituent particles is measured and thus observed to be in an eigenstate,
what does this mean for the combined system and for the other particle? While
physical implications of entanglement are conceptually challenging, the
concept is well-enshrined in modern physics: creating and observing
entanglement between qubits (quantum bits) is one of the key necessary and
challenging processes in quantum computing, a point that we will return to in
Section 6. First however, it is worth highlighting some of the ways in which
this theoretical phenomenon has been exploited in language models in AI.
### 5.3 Explicit Composition with Vectors and Tensors in AI
The use of tensor products in AI is often traced to the work of ? (?), and its
uses have become much more widespread more recently. During the later decades
of the 20th century, vector models were often described as “bag of words”
models because of the commutative nature of the vector sum operation they
applied. This choice of operator meant that the same representation could be
used for a collection of words in a document, irrespective of the order in
which the words appear. By contrast, discrete logical models used in formal
semantics have for many decades been quite explicit about the ways words
should be combined, but were often notably silent about what those words mean
in themselves (see ? (?, ?) for surveys of this methodological difference
between traditions). This history of two modelling frameworks led to an
unnecessary and unfortunate gap: there are many interesting product operations
such as the tensor product between vectors that are well-established in linear
algebra, but for years there were relatively little awareness of these
alternatives in language research. Since the early 2000’s, this situation has
changed markedly.
In a pioneering case for AI, tensor products were introduced at least as early
as the work of ? (?), where the tensor product was used to bind a variable to
its value, and the term ‘binding’ for products of vectors has been used in
various language and cognitive models involving vectors since the work of ?
(?). In the first decade of the current century, the use of tensor algebra for
combining word vectors was explored using a quantum formalism by ? (?),
whereby a sentence $(w_{1},\dots,w_{n})$ would be represented as the tensor
product $w_{1}\otimes\ldots\otimes w_{n}$. Taking this in explicitly
linguistic directions, product operations on vectors were introduced to model
grammatical and semantic composition (?, ?). Particularly interesting examples
included the work of ? (?) on noun composition, and ? (?) on adjective-noun
combinations. ? (?) used a combination of two vector spaces (a large-window
space capturing domain similarities, and a narrow-window function space
capturing functional similarities) to combine relational similarity with two-
word compositional behavior.
Inspired by Smolensky’s work on binding operators, the model of ? (?) used the
tensor product as a means to generate vector representations of phrases,
noting that one desirable property of the tensor product in this context is
that it does not commute, so the product
$\mathit{Djokovic}\otimes\mathit{beat}\otimes\mathit{Murray}$ is not equal to
the product $\mathit{Murray}\otimes\mathit{beat}\otimes\mathit{Djokovic}$.
Another useful mathematical property is that the inner product of two tensors
is equal to the product of the inner products of their related constituents,
e.g. $(\mathit{beat}\otimes\mathit{Murray})\cdot(\mathit{defeated}\otimes\
\mathit{Andy})=(\mathit{beat}\cdot\mathit{defeated})\times(\mathit{Andy}\cdot\mathit{Murray})$.
This means that the similarity between a pair of phrase tensors can be
calculated without the need to explicitly represent them (which would require
$n^{2}$ dimensions for $n$ dimensional vectors), and, with normalized vectors,
this is the product of the cosine similarities between the component vectors
for terms that occupy the identical position in the sentence structure.
However, the grammatical structure of related sentences may not be identical,
and as such their composite representations cannot be easily compared with
this method.
A solution to overcome this limitation of the tensor product approach was
proposed by ? (?). They made use of category theory (?) to demonstrate that
vector spaces and certain types of grammar (exemplified in this work by
pregroup grammars (?)) fall under the same category type, compact closed
categories (see Appendix Appendix: Category Theory for Quantum AI. This
realization provided a means to map between the grammatical and the vector
representation, and thus demonstrated that the rules for grammatical
composition can be applied to a compositional model in vector space. The
mathematical roots of this work are directly related to quantum mechanics due
to the work of ? (?), which developed Categorical Quantum Mechanics, an
axiomatic presentation of quantum theory using compact closed categories,
which makes the relationship with the grammatical work of Lambek particularly
clear. (An online appendix is provided for readers unfamiliar with category
theory.) This point will be revisited in Section 6 because it is especially
relevant to implementation on quantum computers. Since its introduction by ?
(?), the Distributional Compositional Categorical model has become known as
DisCoCat.
? (?) provided an implementation and evaluation of this approach, deriving
word representations from the British National Corpus. To do so, different
rules of composition were assigned to words with different grammatical types
in accordance with the categorical type of their pregroup, with ‘atomic’ types
such as nouns represented by distributional term-by-context-term vectors, and
‘adjoint’ types generated compositionally. For example, verb representations
were generated as the sum of the tensor products of the noun vectors
corresponding to their subject and object across every occurrence in the
corpus (intransitive verbs and adjectives can also be accommodated). Vector
representations of sentences can then be compared: for example, the vector
representation of the sentence “Djokovic beat Murray” is composed by pointwise
multiplication of the tensor representing the verb “beat” and the tensor
composed from the vectors for “Djokovic” and “Murray”:
$(\mathit{Djokovic}\otimes\mathit{Murray})\odot\mathit{beat}$. The model was
evaluated for its ability to estimate the similarities between short phrases,
and correlated with human judgment of similarity as well as, and better than,
the best available models on datasets concerning verb-noun and subject-verb-
object relationships. The authors report Spearman correlations of 0.17 and
0.21 for the verb-noun and subject-verb-object sets respectively with the
compositional model, as compared with 0.17 for both sets with the
multiplicative model of ? (?).
In methodologically similar work, words such as verbs and adjectives that take
arguments have been represented as tensor products in matrix form. For
example, ? (?) used this approach to model the action of adjectives upon
nouns, and ? (?) took it to the logical destination of representing each
internal node in a parse tree as a matrix operator acting upon its input
arguments. This area has become known as Compositional Distributional
Semantics, summarized by ? (?), and work in this area has continued, an
example being the work of ? (?) on sentence entailment in this framework,
where an F1-score of 0.86 is reported on recognizing semantic entailments like
“robin flies $\models$ bird moves”.
As with vector logics, the tensor product as used in quantum mechanics is only
one of the composition operations that can be used with vectors, albeit an
important one. One implication of using the tensor product is the inevitable
explosion in dimensionality that arises: a tensor product takes two
$n$-dimensional vectors and makes an $n^{2}$-dimensional tensor, which is a
problem for scalability on classical hardware, and creates products that are
not obviously comparable with their inputs. Various algebraic structures,
sometimes known as Vector Symbolic Architectures (?), have been used to
address this problem. They make use of circular correlation and circular
convolution, to roll the tensor product coordinates back into coordinates in
the initial space. The use of these structures for reasoning in continuous
models has been described by ? (?).
Entanglement has also been used in compositional modelling for language
concepts, motivated partly by the study of entanglement for word-association
in cognitive models (?, ?). For example, ? (?) explore the entangled
representation of transitive verbs using tensors in the categorical framework
of ? (?). In an explicit use of entangled superpositions for reasoning and
inference, predication-based semantic analysis was developed by ? (?), and
makes use of vector binding operations to represent a concept $A$ occurring in
a relation of the form $R(A,B)$ as the sum of the products $r\otimes b$ for
each relation in which it occurs. (For example, with the relational triple
‘aspirin TREATS headache’, the concept vector for aspirin gets incremented
with the bound product vector $\mathit{TREATS}\otimes\mathit{headache}$.) In
this framework, concepts become represented as superpositions of products of
pairs, and these cannot be expressed as any product of single ingredients $r$
and $b$. This uses some of the same mathematics that in quantum mechanics
leads to entanglement, including the use of complex vectors as a ground field
in some implementations (?, ?). (In an alternative approach to modelling
relations using complex numbers, the work of ? (?) explicitly uses the
complexified representation $A+iB$ to form pairs, and axioms of quantum logic
to formulate the compositional representation of relations.)
From the point of view of language and relationships, the notion of a general
relation being derived from several varied examples makes intuitive sense.
Each of the pairs (Leto, Artemis ), (Henry VIII, Elizabeth I ), (Lord Byron,
Ada Lovelace ), (Darth Vader, Princess Leia ), (Debbie Reynolds, Carrie Fisher
), is an example of a parent-child relationship, and all of these could be
combined into a thorough and very varied notion of parenthood. Because of this
variety, it would be surprising if there was any one ‘prototypical parent’ and
‘prototypical child’ that combines to make the relation of ‘prototypical
parenthood’. Thankfully, the mathematical structures used for tensor products
that give rise to entanglement enable us to represent a relation that is a
rich combination of different example pairs, even for relations where no
single pair of prototypical ingredients exists (?, §5).
### 5.4 Implicit Composition in Deep Neural Networks
Something the methods described in the previous section have in common is that
they encode some explicit syntactic structure: a role / value binding, or a
relationship in a grammatical parse tree. This raises the question: can models
go beyond the bag-of-words drawbacks and encode more order-dependent language
structures without using this traditional syntactic machinery? A recent and
comprehensive survey of this topic is provided by ? (?).
During the same years that compositional distributional semantics has been
developed, neural networks have made great strides in artificial intelligence,
particular the use of networks with several intervening layers, hence the term
‘deep learning’ (?). In some cases, work on deep learning and compositional
semantics has been explicitly combined: for example, ? (?) describe Recursive
Neural Networks used for training. These should not be confused with the the
simpler and more standard Recurrent Neural Networks (RNNs), in which the
output of a single neuron depends on its inputs and its previous state. RNNs
have been used for many sequence-modelling tasks, as have their more
sophisticated cousins, LSTMs (Long Short-Term Memory cells, where prior state
may be stored for longer and updated based on learned importance) (?, Ch. 15).
Later, attention-based networks have been introduced, where the attention
mechanism is designed to learn when pairs of inputs depend crucially on one
another, a capability that has demonstrably improved machine translation by
making sure that the translated output represents all of the given input even
when their word-orders do not correspond exactly (?). This has led to rapid
advances in the field, including the contextualized BERT (?) and ELMo (?)
models. For example, the ELMo model reports good results on traditional NLP
tasks including question answering, coreference resolution, semantic role
labelling, and part-of-speech tagging, and the authors attribute this success
to the model’s different neural-network layers implicitly representing several
different kinds of linguistic structure. The survey and experiments of ? (?)
evaluate three such neural networks on a range of tasks related to
composition, concluding that each network has strengths and weaknesses, that
the results are a stepping stone rather than and endpoint, and that developing
consensus around how such tasks should be designed, tested and shared is a
crucial task in itself.
Even without the cost of encoding order information, ? (?) demonstrated a deep
averaging network, showing that network depth could in come cases compensate
for the lack of syntactic sophistication in unordered models — with
considerable computational savings. At the time of writing, this has developed
into a very open research question: do neural networks need extra linguistic
information as inputs to properly understand language, or can they actually
recover such information as a byproduct of training on raw text input? Does a
complete NLP system need components for tokenization, part-of-speech tagging,
syntactic parsing, named entity recognition, and so on, or can some or all of
these be replaced by a single vector language model? If so, how do we describe
the way meaningful units are composed into larger meaningful structures in
such a model?
Tensor networks are one of the possible mathematical answers to this question.
Their use in libraries such as TensorFlow has become ubiquitous (see (?) and
numerous papers and packages referenced therein), though as cautioned above,
users should be wary of the equivalence between the use of matrices and
multidimensional arrays and tensor algebra that is often assumed in software
documentation. More explicit evidence is presented by work that continues to
build upon Smolensky’s introduction of tensors to AI: for example ? (?)
present evidence that the sequence-composition effects of Recurrent Neural
Networks (RNNs) can be approximated by Tensor Product Decomposition Networks,
at least in cases where using this structure provides measurable benefits over
bag-of-words models (see also ? (?) for a more detailed presentation). It has
also been shown that Tensor Product Networks can encode grammatical structure
more effectively than LSTMs for generating image captions (?), achieving for
example a BLEU-4 score of 0.305 compared the CNN-LSTM’s result of 0.292.
Tensor product networks have also been used to construct an attention
mechanism from which grammatical structure can be recovered by unbinding role-
filler tensor compositions (?). Explicitly quantum networks for natural
language processing are described by ? (?). The range of challenges and
application opportunities in AI for compositional vector representations is by
now much-studied and valued, and quantum-inspired tensor networks already
successfully combine and extend many of the mathematical features that are
core to AI today.
## 6 Physical Implementations on Quantum Computers
Quantum computing has regularly made the front page of scientific news from
2019 to 2021. This section gives a glimpse of opportunities this opens for
quantum mathematics in AI.
With AI based on classical computation, vectors and matrices are already
ubiquitous. While the algebra and geometry behind them contains many riches
that overlap with quantum mechanics, studying this overlap in detail is an
investment that many researchers might consider esoteric and risky — there are
more mainstream state-of-the-art ways to make dramatic progress in AI and
machine learning without learning quantum theory and tensor algebra. But what
if there was a platform for computation where vectors could be represented in
exponentially smaller memory, and instead of being an operation with quadratic
cost, the tensor product was just the most suitable mathematical
representation of a natural system? Quantum computers may soon provide this.
As outlined in Section 5.2, the tensor product arises naturally in quantum
mechanics. Computations using quantum circuits can be used to manipulate
tensor products (for example, from separable to entangled states (?, Ch 4)),
but no ‘work’ is required to compute the tensor product of a 2-qubit system in
the first place: the wavefunction or state of a 2-qubit system is represented
as the tensor product of the individual qubit systems, irrespective of any
computational operation we perform on these qubits (?, §2).
Perhaps the biggest computational promise that follows from the tensor product
is the potential for representing exponentially more information. A classical
register of $n$ physical bits can represent $n$ bits of information (choices
between 0 and 1). In quantum computing, these combinations correspond to basis
vectors — if the state vector of a qubit is represented as a vector in
$\mathbb{C}^{2}$ with basis vectors $\\{\ket{0},\ket{1}\\}$, then a register
with $n$ qubits is represented as a state vector in
$\bigotimes_{1}^{n}\mathbb{C}^{2}$, which has dimension $2^{n}$. Instead of a
state of the whole system, each of those $2^{n}$ combinations represents a
basis vector for a coordinate, and if each of these coordinates can be
accessed, the capacity of the memory grows exponentially instead of linearly
in the number of qubits in the register! In AI applications, it is easy to see
the appeal of this — an embedding vector of 256 dimensions using a 4-byte
floating point number for each coordinate requires 1KB of RAM, which can in
theory be represented by a tensor product of 8 qubits. Of course, there are
huge challenges with this idea, starting with the physical fact that we could
never observe such continuous coordinates directly, only their probabilistic
quantization to a pure state upon measurement. Nonetheless, the promise of
exponential quantum memory has led to ingenious research ideas (particularly
the ‘bucket-brigade’ qRAM protocol of (?)), and the idea is even presented in
an early-reader board book for children (?). There are still practical
physical challenges and theoretical caveats to any premature claims that
‘quantum algorithms give exponential improvements’ (?). Still, if the promise
of exponential qRAM is obvious and compelling enough to excite the youngest
readers, it is hard to imagine that no breakthroughs will happen in the next
few decades.
Still, by the end of 2019, most of the papers published on quantum algorithms
were, from a physical point-of-view, advanced thought-experiments or
simulations: they present mathematical techniques or simulated results on
classical hardware, not experimental results from quantum computers. Shor’s
quantum algorithm for prime factorization was published in 1994 (?, Ch 9), but
at the time of writing, the largest integer factorized on a quantum computer
appears to be 291,311 (?). There are very particular problems that can already
be solved much faster on a quantum computer than a classical computer,
including the problem used by (?) which was to simulate and predict the output
of a pseudo-random quantum circuit. However, this task was proposed in order
to demonstrate quantum advantages, rather than for some existing use-case, a
critique expounded by ? (?).
There are several research developments that are poised to take advantage of
quantum computers in AI, in areas including deep learning in particular (?),
machine learning in general (?), and language processing including parsing
(?), but it remains normal for such papers to develop algorithms and sometimes
results on real datasets without actually implementing them on quantum
machines. Even papers that describe physical implementation in detail
including those in quantum optical neural networks (?) and single photon image
classification (?) report results on high-end classical hardware and proposals
for quantum hardware. This is not a negative criticism, just a description of
the research frontier: in celestial mechanics, the orbit of a satellite was
studied and predicted for centuries before spacecraft could actually be
launched.
In 2020 and 2021, this has been changing: AI experiments on quantum hardware
have been successfully carried out. The single-photon classification
experiment of ? (?) uses a quantum-mechanical platform with a single qubit.
For comparison, the simulated work of ? (?) classifies all ten digits, whereas
? (?) build a classifier that only distinguishes zeros and ones: this
exemplifies the sort of tradeoff researchers have made in using real-but-
limited quantum computing resources. In the work recently reported by ? (?),
an actual quantum neural network is trained and shown to learn faster and more
effectively than a classical network (as measured by Fisher information and
effective dimension), showing as much as a 250% improvement over classical
training using the ibmq_montreal 27-qubit hardware.
In natural language processing on quantum computers, the most dramatic
development so far is perhaps the demonstration of a working Quantum NLP
system on one of IBM’s quantum devices (?, ?). Mathematically speaking, this
work has several key ingredients. A key part is the use of compact closed
categories as an axiomatic model for quantum mechanics (?), which enables the
DisCoCat model from ? (?) to use the same mathematical language. Diagrammatic
calculus, developed over some years and explained thoroughly by ? (?), enables
quantum structures for sentences that combine their syntactic structure (in
terms of combinations) and semantic content (in terms of vectors) to be
represented together. And the paper of ? (?) explains in detail how this
enables a quantum compiler to break these expressions down into quantum gates,
compiled into quantum circuits and run on a quantum computer. The sentences
used to train the model are still simple toy examples such as ‘Alice loves
Bob’ and ‘Bob who is cute loves Alice who is rich’ — as with prime
factorization and image recognition, the implementation using qubits in
quantum computers does not yet rival the scale of NLP on classical hardware —
but the first implementation of a compositional NLP system on quantum hardware
has been accomplished successfully, and given the level of investment in both
quantum computing and NLP, more will follow.
## 7 Other Areas Related to AI and Quantum Theory
This final section refers briefly to other areas related to both AI and
quantum theory that, to avoid excessive length, have not been emphasized in
this paper, but which are likely to be important avenues for future work. Many
more papers can be found on this stream of topics than we are able to list
here. We encourage interested readers to investigate the references provided
in this section for further links to other papers and more detailed avenues.
### 7.1 Quantum Search and Automated Problem Solving
Quantum search is a large algorithmic topic in itself. In this context, we
mean ‘search’ in the sense of ‘database search’ or ‘tree search’, rather than
the search engines of information retrieval: quantum search algorithms are
typically designed to find a particular element that solves a given problem in
computationally faster time than classical algorithms (?, Ch 6). A canonical
example of such a speedup is Grover’s search algorithm, which locates a unique
element out of $n$ choices in $\sqrt{n}$ time (?, Ch 9). The details are not
covered here, though the process itself is a hallmark example of a quantum
oracle that manipulates bases step-by-step in such a way that the states of
the non-solutions cancel one another out and the state of the solution
eventually ‘sticks out’ after at most $\sqrt{n}$ repetitions.
Grover’s search algorithm has been used as a building block for approaching
other higher-level problems. These include various forms of tree search (?)
and challenges that can be formulated as symbol manipulation problems, such as
a block puzzle (?). For an overview see ? (?, Ch 10–12). It is worth noting
that core data structure operations such as tree search have been investigated
for basic hardware components such as qRAM (?) while they are also being
developed for higher-level problem solving in quantum AI: for example, ? (?)
analyse the importance of reducing the number of lookup operations in the
‘bucket brigade’ qRAM protocol error-rates (demonstrating in the process some
of the great difficulties in implementing a real qRAM).
This evolution is historically different from the development of classical
computing, in which basic data structures were reliably and readily available
decades before the heyday of machine learning. (This is not to say that there
are no more developments to be made in classical data structures, but that the
machine learning practitioner often has no reason to wonder whether a
dictionary lookup is from a binary search tree vs. a hashmap — they ‘just
work’.) By contrast, in quantum computing, these innovations are happening at
the same time and facing similar challenges — quantum heuristics are sometimes
needed not only to expedite computation, but to give more reliable results,
because reducing the number of operations reduces errors. While such details
might be seen as frustratingly low-level in most of machine learning today,
this is an area where quantum computing and quantum AI may cross-fertilize
each other’s early development.
### 7.2 Quantum Probability and Cognition
The geometric and algebraic structures of quantum theory affect probability as
well as logic. In quantum theory, the probability of finding a system in state
$\ket{\psi}$ to be in the pure basis state $\ket{\lambda_{i}}$ is equal to
$|\braket{\lambda_{i}}{\psi}|^{2}$, which is called the Born rule. For
normalized unit vectors, the scalar product $\braket{\lambda_{i}}{\psi}$ is
well-known to be the cosine of the angle between the two vectors — so in
quantum theory, probability is geometrically related to angles between
vectors. This contrasts with so-called classical probability where
probabilities are obtained from ratios between volumes (?, Ch 2)(?, Ch 2).
Quantum probability behaves differently from classical probability: for
example, conditional probability depends on the spectrum of the observable in
question (?, p. 35), and the fact that a measurement of one of the variables
was performed beforehand (?, p. 165).
Quantum probability has been used to explain and accurately model order-
effects on attitudes, where asking people the same questions in a different
order has been shown to give different answers (?, Ch 3), resulting in various
disjunction effects in decision making. For example, preferences made with
more information have been shown to be different from those made with less
information, irrespective of what information has been learned in the
meantime, a violation of the Sure Thing Principle (?). In semantics, such
models have been used to model compositional behavior including
nonseparability in the human lexical representations (?).
The affordances of quantum probabilities in cognitive modelling have been
exploited by some authors to support more human-like models of automated
reasoning and decision making. For example, ? (?) made use of the structural
nature of quantum probability in an agent based model capable of representing
phenomena such as cognitive dissonance and attitude change in a social
context. Similarly, a stream of work in developing quantum-like Bayesian
network models (?) has resulted in sophisticated methods for modelling the
evolution of the beliefs of a decision maker.
### 7.3 Quantum Probability and Language Modelling
A key part of the language modeling work of ? (?) was the use of off-diagonal
elements in a matrix to model the joint probability of bigrams (where the on-
diagonal elements model unigram probabilities). The mathematics of this
particular construction is presented intuitively and explored thoroughly in
the statistical algebra of ? (?). This leads to a surprising and clear
correspondence between Formal Concept Analysis (?), in which objects and
attributes are represented in a lattice, and quantum probability. The basic
construction is to take the cross table (the matrix showing which objects
possess which attributes), and multiply it by its transpose, which gives a
matrix whose off-diagonal elements capture the overlaps between objects. This
conditional dependence information is reflected in the partial trace, which is
a basis-dependent mapping from the product space $V^{\star}\otimes V$ back to
$V$. ? (?) argues that the partial trace is the quantum analogy of the
classical process of marginalizing a joint distribution — but whereas in
classical probability, marginal probabilities retain no information about the
overlaps in the original joint distribution, in the quantum case, some of this
information is preserved. This richness may lead to quantum probability
becoming a more common tool of choice for language and concept models.
### 7.4 Real and Complex Numbers in AI
One place where physics, classical logic, and machine learning use quite
different fundamental mathematical building blocks is in the choice of number
field. Machine learning and AI tend to use real vectors because the basic
features are measurements; classical logic and computer science have used
binary numbers thanks largely to the enormous influence of George ? (?, Ch 2);
whereas physics most prevalently uses complex vectors, not just in quantum
mechanics but also in electrodynamics.
So far, the reasons for not using complex and binary numbers in machine
learning seem to be twofold (and reasonable): there are no widespread and
immediately intuitive interpretations for them; and the field is progressing
very quickly without them. However, there are computational benefits to using
complex numbers. For example, the convolution operator, sometimes used for
vector binding, is just the addition of phase angles, and is thus $O(n)$
rather than the $O(n\log n)$ of the real convolution operation optimized using
Fast Fourier Transforms (see ? (?)). And in several experiments on automatic
inference, we have ourselves found that using binary and complex vectors
sometimes yields much better results than their real-valued counterparts, for
reasons not yet properly understood (?). There are a few machine learning
papers on using complex numbers and even quaternions and octonions in machine
learning, for example ? (?). As quantum computing becomes more mainstream, we
should expect the use of complex numbers to become more widespread.
### 7.5 Complex Time Evolution
Imaginary and complex numbers are particularly important in quantum mechanics
for representing the momentum operator
$\hat{p}=-i\hbar\frac{\partial}{\partial x}$ and the resulting Hamiltonian in
the Schrödinger wave equation. This technique has roots in harmonic analysis,
particularly with the simple form of the plane wavefunction
$\Psi(x,t)=Ae^{i(kx-\omega t)}$. Harmonic functions can be written in this way
thanks to the identity $e^{i\theta}=\cos{\theta}+i\sin{\theta}$, and the
technique for representing time evolution could potentially be applied to
situations where harmonic functions are used to model word position as
attention moves along a sentence (?).
### 7.6 Again, Why Quantum Theory?
Throughout this paper we have introduced and surveyed several areas where
mathematics used in quantum theory is also fruitful in AI. However, we are yet
to answer the questions “What is quantum mathematics?” and “Why is it
important in AI?”, delaying them to this point. This has enabled us to
demonstrate the many and diverse ways in which mathematics has been abstracted
from quantum theory and applied to AI. But there is no clear characterization
of what precisely makes mathematics ‘quantum’.
This is well-illustrated by contrast with the other great pillar of 20th
century physics, general relativity. The mathematics of general relativity can
be defined crisply as the differential geometry of Lorentzian manifolds, which
are smooth 4-manifolds with a pseudo-Riemannian metric of signature (1, 3) —
that is, there must be 4 dimensions, and the metric must treat three of them
interchangeably (the spacelike dimensions) and one with the opposite sign (the
timelike dimension), giving a local distance metric that takes the form
$dt^{2}-dx^{2}-dy^{2}-dz^{2}$ (choosing units so that the speed of light is
1).
After years of searching, we are confident that there is no similarly clear
characterization of quantum mathematics. That disjunctions are non-
distributive or products non-separable have been proposed, but can be rejected
on the grounds that such mathematical structures are prominent in many other
areas, and many quantum models do not explicitly use these properties.
So the question “What makes mathematics distinctly quantum?” remains a matter
for discussion. In our opinion, quantum mathematics as a whole is the
mathematics of self-adjoint operators in complex Hilbert space. In theory,
complex numbers are needed to guarantee that roots of polynomials exist, which
guarantees that eigenvalues and eigenvectors exist. The self-adjoint
requirement guarantees that these eigenvalues are real numbers, which is
necessary for them to represent physical properties. This mathematical
argument is standard: in this case, we are following particularly the
narrative of ? (?, Ch 6), a similar presentation is given by ? (?), and indeed
? (?, Ch 1, 2).
Much of this mathematical apparatus is general, and has been used in a wide
array of fields beyond quantum theory. In this paper we have demonstrated a
number of these overlaps with AI. However, with this point comes an important
caveat: most AI applications are only using a part of quantum mathematics,
since few use complex numbers. For example, the information retrieval
applications discussed in Section 3 are explained explicitly in terms of
quantum logic (in the case of ? (?)) and quantum probability (in the case of ?
(?)), but use only real numbers throughout. Rather than a criticism, this
could become a recommendation: quantum-inspired AI systems should
systematically consider and investigate the use of complex numbers.
Instead of embodying a distinct piece of mathematics, the motivation for
developing quantum-inspired models in AI and related areas has sometimes been
the obvious shortcomings of so-called ‘classical’ models. Quantum theory
acknowledges that the future is not wholly predictable, that combined systems
cannot always be separated into atomic parts, and that the order in which
observations are made sometimes affects the outcome. These points are often
obscured by the oft-repeated claim “classical physics is the physics of
everyday life”, even though quantum physics is more like daily life in these
respects (points discussed in more detail by ? (?) and ? (?)). As these areas
have been explored in detail, with hindsight it is easy to understand why
quantum structures were seen as promising alternatives to reductionist
symbolic determinism: and once this approach is considered, mathematical
overlaps between quantum mechanics and other application areas are easy to
find. Without the advent of quantum computing, these overlaps may have been
considered a mathematical curiosity: a good motivation perhaps for individual
system designs so long as they perform well in practice. But now with quantum
computing coming of age, the importance of quantum mathematics in computer
science is set to increase, because it will be key to answering the question
“What’s the most effective way to implement this?”
## 8 Conclusion
Quantum theory and artificial intelligence use much of the same mathematics,
and the importance of the overlapping areas has grown with the ubiquitous use
of vectors in AI. As well as simple superposition and similarity operations,
quantum theory offers well-studied formulations of logical inference, concept
combination, and probability, many of which have been demonstrated to give
good results on various AI tasks, including examples from natural language
processing, image processing, reasoning and inference. The mathematics used in
these applications is often well-developed thanks to its use in other fields,
and in particular, Grassmannian and tensor algebra offers more in terms of
duality and symmetry than is typically used in AI. Some of these algebraic
operations are costly on classical hardware, but much cheaper or even free in
quantum computation, which in 2021 is rapidly becoming more practical. The
authors hope that this paper helps AI researchers familiar with vectors and
their uses to turn more often to quantum theory as a rich source of
mathematical motivation for advances today, and to approach the advent of
quantum computing with confidence and excitement.
## Appendix: Category Theory for Quantum AI
Category theory is one of the less familiar branches of mathematics to many
researchers in AI, and this appendix is designed as a very brief supplement.
Interested readers are referred particularly to the survey article of ? (?)
and the graduate text of ? (?) for a proper explanation.
Much of modern mathematics can be thought of as the study of objects with some
defined structure, along with maps that preserve this structures. Standard
examples well-known by the middle of the 20th century include sets and maps,
groups and homomorphisms, topological spaces and continuous maps,
differentiable manifolds and differentiable maps, and of course, vector spaces
and linear maps.
The abstract notion of a category was developed in the 20th century to reason
about such structures in a common fashion. A category consists of a collection
of objects, and for every pair of objects $A$ and $B$, a collection of
morphisms from $A$ to $B$. Morphisms must be composable in sequence in an
associative manner (so that $f\circ(g\circ h)=(f\circ g)\circ h$), and there
must be an identity morphism $\mathrm{id}$ so that $\mathrm{id}\circ
f=f\circ\mathrm{id}=f$.
In many categories, the morphisms are some kind of function. These include
Set, the category of sets and maps, and Vect, the category of vector spaces
and linear maps. There are also important cases where the morphisms are not
functions, including the category of sets and relations, which is often called
Rel. A relation between two sets $A$ and $B$ is a subset of their Cartesian
product $A\times B$, intuitively a ‘many to many correspondence’, and
relations can also be composed in a way that satisfies the definition of a
category (?, §0.1.3). Because of the potential for one-to-many
correspondences, if we think of ‘applying a morphism’ as ‘evolving in time’,
the category Rel can be used as a model for ‘nondeterministic classical
physics’, because a single state representing the present can evolve into more
than one possible state representing the future.
Following the pattern of abstract algebra, further definitions are introduced
and their properties and consequences explored. Some of these (here described
intuitively but not precisely) are as follows. A functor is itself a mapping
between two categories that maps objects to objects and morphisms to morphisms
in a regular composable fashion. The opposite or dual category $C^{op}$ has
the same objects as $C$ but ‘reversed’ morphisms: if we define the set of
morphisms from $A$ to $B$ in the category $C$ as $C(A,B)$, then the set of
morphisms from $B$ to $A$ in $C^{op}$ is defined by $C^{op}(B,A)=C(A,B)$. This
is simple but initially confusing, especially with the most obvious category
Set of sets and functions — how can a function from $A$ to $B$ be considered a
morphism from $B$ to $A$? In this surprising case, the answer is just ‘the
definition satisfies identity and associativity, so it forms a category’.
However, with the categories of Rel of sets and relations, and FVect of
(finite dimensional) vector spaces and linear maps, the notion of dual
categories becomes much more concrete, as the objects and morphisms themselves
have duals, not only the category as a whole. Much of the power in categorical
quantum mechanics arises from the use of category theory to explore the
properties and consequences of duality in these settings.
A closed category is one where the morphisms themselves have the structure of
objects in the category. For example, the mappings from a finite set $A$ to a
finite set $B$ themselves form a finite set (an element of $B$ is selected for
each element of $A$, so the number of such possibilities is $|B|^{|A|}$), so
finite sets are a closed category. A monoidal category is one where objects
from the same category can be composed with one another ‘in parallel’ using an
operator written $\otimes$ (as well as morphisms being composed ‘in
sequence’), and there needs to be an identity object $I$ that satisfies basic
compatibility conditions with the identity morphism. This gives the objects of
the category the structure of a monoid, which is a group without the
requirement of each element having an inverse. In a monoidal category, an
object $A$ may have a dual object $A^{\star}$, with canonical morphisms from
$I$ to $A^{\star}\otimes A$ and vice versa. If these morphisms contract to the
identity in given ways, the category is called compact.
A dagger category is one where each morphism $f:A\rightarrow B$ has a
corresponding dagger morphism $f^{\dagger}:B\rightarrow A$, with algebraic
properties generalized from those of the adjoint operator in linear algebra.
Dagger categories were used by ? (?) for describing quantum information
protocols, using the term strongly compact closed categories. The term ‘dagger
category’ was introduced as part of a more general exploration of their
mathematical properties by ? (?).
The connection of notation and terminology with linear algebra is quite
deliberate: finite dimensional complex vector spaces form a monoidal category
whose monoidal product is the tensor product, and whose dual objects are the
dual vector spaces. Hence the notation $A^{\star}$ for the dual of $A$, and
$A\otimes B$ for the monoidal category product, are just the same definitions
as given in linear algebra.
The relationship with quantum mechanics, called categorical quantum mechanics,
is due particularly to ? (?), and has become part of the mathematical backbone
of work on understanding quantum processes by ? (?) and in related works.
Categorical quantum mechanics seeks to describe quantum mechanics and quantum
processes in terms of mathematical properties like those outlined above.
This contributes to quantum mathematics in two ways. First, the results of
quantum mechanics can be linked even more directly to key mathematical
concepts. For example, ? (?, Ch. 4) uses the constrasting monoidal structures
of the categories Set and FHilb (finite dimensional Hilbert Spaces and bounded
linear maps) to explain why we should expect that state can be uniformly
copied in classical computing but not in quantum computing; and they relate
categorical properties to aspects of entanglement and quantum teleportation
throughout. Second, understanding such phenomena at a more abstract
mathematical level can lead to applications in other areas and cross-
fertilization. For example, the category of sets and relations (more general
than the category of sets and maps, because a relation can be many-to-many)
sometimes behaves much more like the category of vector spaces than that of
sets, particularly in relation to entanglement, which the statistical algebra
of ? (?) uses to describe concepts in a formal concept lattice. And of
particular importance, the use of category theory links the pregroup grammars
of Lambek calculus to the composed vector representations of ? (?), the
algebraic representation of pronouns (?, ?), and the gate-based quantum
circuit implementation of ? (?). Here, for example, the notion of monoidal
products and duals is used to give a categorical model for a linguistic notion
such as “a transitive verb operates on two nounphrases” and a vector notion
such as “a type (0, 2) tensor operates on two type (1, 0) vectors”, in such a
way that the verb and noun representations can be implemented on a quantum
computer.
While category theory can sound very abstract and technical, it gives clearer
insights into why apparently similar systems sometimes behave differently, and
why some apparently very different systems share common structures. Today
especially, it provides an ideal opportunity for mathematicians with a
background in algebra to contribute to technologies for computing, security,
and communication.
## References
* Aaronson Aaronson, S. (2013). Quantum computing since Democritus. Cambridge University Press.
* Aaronson Aaronson, S. (2015). Read the fine print. Nature Physics, 11(4), 291–293.
* Abbas et al. Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., and Woerner, S. (2021). The power of quantum neural networks. Nature Computational Science, 1(6), 403–409.
* Abramsky and Coecke Abramsky, S., and Coecke, B. (2004). A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004., pp. 415–425. IEEE.
* Abramsky and Coecke Abramsky, S., and Coecke, B. (2009). Categorical quantum mechanics. Handbook of quantum logic and quantum structures, 2, 261–325.
* Aerts and Czachor Aerts, D., and Czachor, M. (2004). Quantum aspects of semantic analysis and symbolic artificial intelligence. J. Phys. A: Math. Gen., 37, L123–L132.
* Aerts et al. Aerts, D., Durt, T., Grib, A., Van Bogaert, B., and Zapatrin, R. (1993). Quantum structures in macroscopic reality. International Journal of Theoretical Physics, 32(3), pp–489.
* Aerts and Sozzo Aerts, D., and Sozzo, S. (2011). Quantum structure in cognition: Why and how concepts are entangled. In International Symposium on Quantum Interaction, pp. 116–127. Springer.
* Arunachalam et al. Arunachalam, S., Gheorghiu, V., Jochym-O’Connor, T., Mosca, M., and Srinivasan, P. V. (2015). On the robustness of bucket brigade quantum ram. New Journal of Physics, 17(12), 123010.
* Arute et al. Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J. C., Barends, R., Biswas, R., Boixo, S., Brandao, F. G., Buell, D. A., et al. (2019). Quantum supremacy using a programmable superconducting processor. Nature, 574(7779), 505–510.
* Bankova et al. Bankova, D., Coecke, B., Lewis, M., and Marsden, D. (2019). Graded hyponymy for compositional distributional semantics. Journal of Language Modelling, 6(2), 225–260.
* Baroni et al. Baroni, M., Bernardi, R., Zamparelli, R., et al. (2014). Frege in space: A program for compositional distributional semantics. Linguistic Issues in language technology, 9(6), 5–110.
* Baroni and Zamparelli Baroni, M., and Zamparelli, R. (2010). Nouns are vectors, adjectives are matrices: Representing adjective-noun constructions in semantic space. In Proceedings of the 2010 Conference on Empirical Methods in Natural Language Processing (EMNLP).
* Basile et al. Basile, P., Caputo, A., and Semeraro, G. (2011). Negation for Document Re-ranking in Ad-hoc Retrieval, pp. 285–296. Springer, Berlin, Heidelberg.
* Bernhardt Bernhardt, C. (2019). Quantum Computing for Everyone. Mit Press.
* Biamonte et al. Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., and Lloyd, S. (2017). Quantum machine learning. Nature, 549(7671), 195–202.
* Birkhoff and von Neumann Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843.
* Bishop and Goldberg Bishop, R. L., and Goldberg, S. I. (1968). Tensor analysis on manifolds. Macmillan, republished by Dover (2012).
* Bollobás Bollobás, B. (1998). Modern Graph Theory. No. 184 in Graduate Texts in Mathematics. Springer-Verlag.
* Boole Boole, G. (1854). An Investigation of the Laws of Thought. Macmillan. Dover edition, 1958.
* Bradley Bradley, T.-D. (2020). At the Interface of Algebra and Statistics. Ph.D. thesis, City University of New York.
* Bruza et al. Bruza, P., Kitto, K., and McEvoy, D. (2008). Entangling words and meaning. In Quantum Interaction: Proceedings of the Second Quantum Interaction Symposium (QI-2008), pp. 118–124. College Publications.
* Bruza et al. Bruza, P. D., Kitto, K., Ramm, B. J., and Sitbon, L. (2015). A probabilistic framework for analysing the compositionality of conceptual combinations. Journal of Mathematical Psychology, 67, 26–38.
* Bruza et al. Bruza, P. D., Widdows, D., and Woods, J. (2009). A quantum logic of down below. In Engesser, K., Gabbay, D. M., and Lehmann, D. (Eds.), Handbook of quantum logic and quantum structures. Elsevier.
* Busemeyer and Bruza Busemeyer, J., and Bruza, P. (2012). Quantum Models of Cognition and Decision. Cambridge University Press.
* Camacho-Collados and Pilehvar Camacho-Collados, J., and Pilehvar, M. T. (2018). From word to sense embeddings: A survey on vector representations of meaning. Journal of Artificial Intelligence Research, 63, 743–788.
* Clark and Pulman Clark, S., and Pulman, S. (2007). Combining symbolic and distributional models of meaning.. In AAAI Spring Symposium: Quantum Interaction, pp. 52–55.
* Clarke Clarke, D. (2012). A context-theoretic framework for compositionality in distributional semantics. Computational Linguistics, 38(1), 41–71.
* Coecke et al. Coecke, B., de Felice, G., Meichanetzidis, K., and Toumi, A. (2020). Foundations for near-term quantum natural language processing. arXiv preprint, arXiv:2012.03755.
* Coecke and Kissinger Coecke, B., and Kissinger, A. (2017). Picturing quantum processes. Cambridge University Press.
* Coecke et al. Coecke, B., Sadrzadeh, M., and Clark, S. (2010). Mathematical foundations for a compositional distributional model of meaning. arXiv preprint, arXiv:1003.4394.
* Cohen and Widdows Cohen, T., and Widdows, D. (2015). Embedding probabilities in predication space with hermitian holographic reduced representations. In Ninth International Symposium on Quantum Interaction, pp. 245–257. Springer.
* Cohen et al. Cohen, T., Widdows, D., Vine, L. D., Schvaneveldt, R., and Rindflesch, T. C. (2012). Many paths lead to discovery: Analogical retrieval of cancer therapies. In Sixth International Symposium on Quantum Interaction.
* Devlin et al. Devlin, J., Chang, M.-W., Lee, K., and Toutanova, K. (2018). BERT: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint, arXiv:1810.04805.
* Dirac Dirac, P. (1930). The principles of quantum mechanics (4th edition, 1958, reprinted 1982 edition). Clarendon Press, Oxford.
* Dorst et al. Dorst, L., Fontijne, D., and Mann, S. (2010). Geometric algebra for computer science: an object-oriented approach to geometry. Elsevier.
* Fearnley-Sander Fearnley-Sander, D. (1979). Hermann Grassmann and the creation of linear algebra. The American Mathematical Monthly, 86(10), 809–817.
* Ferrie Ferrie, C. (2018). Quantum Information for Babies. Sourcebooks Explore.
* Fischbacher and Sbaiz Fischbacher, T., and Sbaiz, L. (2020). Single-photon image classification. arXiv preprint, arXiv:2008.05859.
* Ganter and Wille Ganter, B., and Wille, R. (1999). Formal Concept Analysis: Mathematical Foundations. Springer.
* Garg et al. Garg, D., Ikbal, S., Srivastava, S. K., Vishwakarma, H., Karanam, H., and Subramaniam, L. V. (2019). Quantum embedding of knowledge for reasoning. In Advances in Neural Information Processing Systems, pp. 5594–5604.
* Géron Géron, A. (2019). Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems. O’Reilly Media.
* Giovannetti et al. Giovannetti, V., Lloyd, S., and Maccone, L. (2008). Quantum random access memory. Physical review letters, 100(16), 160501.
* Grassmann Grassmann, H. (1862). Extension Theory. History of Mathematics Sources. American Mathematical Society, London Mathematical Society. Translated by Lloyd C. Kannenberg (2000).
* Greenstein and Zajonc Greenstein, G., and Zajonc, A. G. (1997). The Quantum Challenge: modern research on the Foundations of Quantum Mechanics. Jones and Bartlett Publishers, Sudbury, Massachusetts.
* Grefenstette and Sadrzadeh Grefenstette, E., and Sadrzadeh, M. (2011). Experimental support for a categorical compositional distributional model of meaning. In Proceedings of the 2011 Conference on Empirical Methods in Natural Language Processing (EMNLP).
* Hastie et al. Hastie, T., Tibshirani, R., and Friedman, J. H. (2001). The Elements of Statistical Learning. Springer Series in Statistics.
* Heunen and Vicary Heunen, C., and Vicary, J. (2019). Categories for Quantum Theory: an introduction. Oxford University Press.
* Hinton Hinton, G. E. (1990). Preface to the special issue on connectionist symbol processing. Artificial Intelligence, 46(1-2), 1–4.
* Huang et al. Huang, Q., Deng, L., Wu, D., Liu, C., and He, X. (2019). Attentive tensor product learning. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 33, pp. 1344–1351.
* Huang et al. Huang, Q., Smolensky, P., He, X., Deng, L., and Wu, D. (2017). Tensor product generation networks for deep NLP modeling. arXiv preprint, arXiv:1709.09118.
* Hupkes et al. Hupkes, D., Dankers, V., Mul, M., and Bruni, E. (2020). Compositionality decomposed: how do neural networks generalise?. Journal of Artificial Intelligence Research, 67, 757–795.
* Isham Isham, C. J. (1995). Lectures on Quantum Theory. Imperial College Press, London.
* Iyyer et al. Iyyer, M., Manjunatha, V., Boyd-Graber, J., and Daumé III, H. (2015). Deep unordered composition rivals syntactic methods for text classification. In Proceedings of the 53rd annual meeting of the Association for Computational Linguistics (ACL), pp. 1681–1691. |
# Gravity at cosmological distances:
Explaining the accelerating expansion without dark energy
Junpei Harada<EMAIL_ADDRESS>Health Sciences University of
Hokkaido, 1757 Kanazawa, Tobetsu-cho, Ishikari-gun, Hokkaido 061-0293, Japan
(August 15, 2023)
###### Abstract
Three theoretical criteria for gravitational theories beyond general
relativity are considered: obtaining the cosmological constant as an
integration constant, deriving the energy conservation law as a consequence of
the field equations, rather than assuming it, and not necessarily considering
conformally flat metrics as vacuum solutions. Existing theories, including
general relativity, do not simultaneously fulfill all three criteria. To
address this, a new gravitational field equation is proposed that satisfies
these criteria. From this equation, a spherically symmetric exact solution is
derived, which is a generalization of the Schwarzschild solution. It
incorporates three terms: the Schwarzschild term, the de Sitter term, and a
newly discovered term, which is proportional to $\displaystyle r^{4}$ in a
radial coordinate, that becomes significant only at large distances. The
equation is further applied to cosmology, deriving an equation for the scale
factor. It then presents a solution that describes the transition from
decelerating to accelerating expansion in a matter-dominated universe. This is
achieved without the need for negative pressure as dark energy or the positive
cosmological constant. This provides a novel explanation for the current
accelerating expansion of the universe.
## I Introduction
In certain gravitational theories beyond general relativity, the cosmological
constant $\displaystyle\Lambda$ is derived as a constant of integration. This
feature provides a notable theoretical advantage over the Einstein equations
in general relativity. Therefore, it is pertinent to establish the following
theoretical criteria for gravitational theories:
1. 1.
The cosmological constant $\displaystyle\Lambda$ is obtained as a constant of
integration.
In the Einstein equations, the presence or absence of the cosmological
constant $\displaystyle\Lambda$ is fixed from the beginning. Therefore,
general relativity does not meet this criterion. However, the trace-free
Einstein equations, denoted by $\displaystyle R_{\mu\nu}-Rg_{\mu\nu}/4=8\pi
G(T_{\mu\nu}-Tg_{\mu\nu}/4)$, which were initially investigated by Einstein
himself, do satisfy this criterion only if the conservation law,
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is assumed as an additional
assumption [1, 2]. The need for this assumption is theoretically a
disadvantage. Hence, it is appropriate to require the second theoretical
criterion:
1. 2.
The conservation law, $\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is derived
as a consequence of the gravitational field equations, rather than being
assumed.
General relativity fulfills the second criterion due to the Bianchi identity
but fails to satisfy the first criterion. On the other hand, the trace-free
Einstein equations fulfill the first criterion but not the second one.
Conformal gravity [3] and Cotton gravity [4, 5] satisfy both the first and the
second criteria. In these theories, the gravitational field equation does not
include the cosmological constant; it arises as a constant of integration.
Additionally, the conservation law,
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is derived from the field
equations, as in general relativity, due to the Bianchi identity.
Unfortunately, in these theories, any conformal flat metric serves as a vacuum
solution. This may be a potential disadvantage, as it allows for unphysical
solutions. For example, in cosmology, the conformally flat Friedmann-Lemaître-
Robertson-Walker metric is a vacuum solution even if the scale factor
$\displaystyle a(t)$ is an arbitrary function of $\displaystyle t$, and in
this case, it conflicts with observations. Therefore, it is reasonable to
consider the third theoretical criterion:
1. 3.
A conformally flat metric is not necessarily a solution in vacuum.
To date, no known theory simultaneously satisfies all three criteria. It
remains uncertain whether such a theory is even possible. Hence, the following
questions arise: Does a gravitational field equation satisfying the three
criteria exist? If so, what is its form? What are the physical implications of
such an equation?
This paper provides answers to these questions. First, a new gravitational
field equation is proposed, which satisfies all three criteria. Subsequently,
from this equation, a spherically symmetric solution is derived, which is a
generalization of the Schwarzschild solution. The solution contains three
terms: the Schwarzschild term $\displaystyle(\propto 1/r)$, the de Sitter term
$\displaystyle(\propto r^{2})$, and a newly discovered term
$\displaystyle(\propto r^{4})$ that only becomes significant at large
distances, being negligible at small distances.
The equation is further applied to cosmology. By assuming isotropy and spatial
homogeneity of the universe, an equation of motion for the scale factor is
derived. The solution to this equation exhibits a significant property: even
in the absence of dark energy or the cosmological constant, with only matter
present, the universe undergoes a transition from decelerating to accelerating
expansion. In fact, in this theory, the accelerating expansion naturally and
inevitably emerges as a consequence of the gravitational field equation,
rather than being attributed to negative pressure. This offers a novel
explanation for the current accelerating expansion of the universe.
This paper is organized as follows. In Sec. II, we present the gravitational
field equation that satisfies the three criteria mentioned earlier. Section
III explores a generalized solution of the Schwarzschild solution. In Sec.
IV.1, we derive the equation of motion for the scale factor, which serves as a
generalization of the Friedmann equation. In Sec. IV.2, we present a solution
that describes a transition from decelerating to accelerating expansion in a
matter-dominated universe. Finally, Sec. V provides a summary and conclusions.
Throughout this paper, we set $\displaystyle c=8\pi G=1$, although
$\displaystyle 8\pi G$ is explicitly stated in some cases. The covariant
derivative uses the Levi-Civita connection, and the metric signature is
$\displaystyle(-,+,+,+)$.
## II Gravitational field equation
Two different approaches satisfying the first criterion mentioned in the
Introduction are known.
The first approach involves demanding that the gravitational field equation be
traceless, as originally proposed by Einstein. However, while this approach
satisfies the first criterion, it fails to meet the second criterion, thus
requiring us to consider an alternative approach.
The second approach employs derivatives of the curvature tensors instead of
the curvature tensor itself. This approach includes conformal gravity, Cotton
gravity, and Yang’s gravitational field equation [6]. In a previous study, the
author explored a scenario in which the gravitational field equation possesses
the same symmetry as $\displaystyle\nabla_{\mu}R^{\mu}{}_{\nu\rho\sigma}$.
While this approach satisfies the first and the second criteria, it was found
to fail to fulfill the third criterion. Therefore, alternative symmetries need
to be considered in place of
$\displaystyle\nabla_{\mu}R^{\mu}{}_{\nu\rho\sigma}$.
Based on these observations, we consider the following scenario. We examine
two possible totally symmetric derivatives of the curvature:
$\displaystyle\displaystyle\nabla_{\rho}R_{\mu\nu}+\nabla_{\mu}R_{\nu\rho}+\nabla_{\nu}R_{\rho\mu},$
(1a)
$\displaystyle\displaystyle(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})R.$
(1b) Here, $\displaystyle\nabla_{\mu}$ represents the covariant derivative,
$\displaystyle R_{\mu\nu}$ is the Ricci tensor, and $\displaystyle R$ is the
Ricci scalar. These two terms, Eqs. (1a) and (1b), are linearly independent,
allowing for a linear combination of (1a) and (1b) to serve as the left-hand
side of the gravitational field equation.
A similar representation can be employed for the right-hand side of the
gravitational field equation, which comprises two potential terms:
$\displaystyle\displaystyle\nabla_{\rho}T_{\mu\nu}+\nabla_{\mu}T_{\nu\rho}+\nabla_{\nu}T_{\rho\mu},$
(1c)
$\displaystyle\displaystyle(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})T.$
(1d)
Here, $\displaystyle T_{\mu\nu}$ is the energy-momentum tensor, and
$\displaystyle T$ denotes its trace.
Hence, the gravitational field equation can be expressed as follows:
$\displaystyle\displaystyle
a(\nabla_{\rho}R_{\mu\nu}+\nabla_{\mu}R_{\nu\rho}+\nabla_{\nu}R_{\rho\mu})$
$\displaystyle\displaystyle\qquad+b(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})R$
$\displaystyle\displaystyle=c(\nabla_{\rho}T_{\mu\nu}+\nabla_{\mu}T_{\nu\rho}+\nabla_{\nu}T_{\rho\mu})$
$\displaystyle\displaystyle\qquad+d(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})T,$
(2)
where $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, and
$\displaystyle d$ are coefficients.
The coefficients $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, and
$\displaystyle d$ can be determined as follows. By multiplying Eq. (2) by
$\displaystyle g^{\nu\rho}$, we obtain
$2(a+3b)\partial_{\mu}R=2c\nabla_{\lambda}T^{\lambda}{}_{\mu}+(c+6d)\partial_{\mu}T,$
(3)
where we have used the identity $\displaystyle
2\nabla_{\mu}R^{\mu}{}_{\nu}=\partial_{\nu}R$.
To ensure that the conservation law
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$ is satisfied identically, we can
derive the following conditions from Eq. (3):
$\displaystyle\displaystyle a+3b$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 0,$ (4a) $\displaystyle\displaystyle c+6d$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle 0.$ (4b)
We also impose the condition that every solution of the Einstein equations
satisfies Eq. (2). By substituting $\displaystyle
T_{\mu\nu}=R_{\mu\nu}-Rg_{\mu\nu}/2$ (with $\displaystyle 8\pi G=1$) and
$\displaystyle T=-R$ into the right-hand side of Eq. (2), we obtain
$\displaystyle\displaystyle(a-c)(\nabla_{\rho}R_{\mu\nu}+\nabla_{\mu}R_{\nu\rho}+\nabla_{\nu}R_{\rho\mu})$
$\displaystyle\displaystyle+\left(b+\frac{c}{2}+d\right)(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})R=0.$
(5)
From this equation, we can obtain the following conditions:
$\displaystyle\displaystyle a-c$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 0,$ (6a) $\displaystyle\displaystyle
b+\frac{c}{2}+d$ $\displaystyle\displaystyle=$ $\displaystyle\displaystyle 0.$
(6b)
Using Eqs. (4a), (4b), (6a), and (6b) (three of them are linearly
independent), we can determine the coefficients $\displaystyle a$,
$\displaystyle b$, $\displaystyle c$, and $\displaystyle d$ as follows:
$\displaystyle\displaystyle a=1,$ $\displaystyle\displaystyle b=-\frac{1}{3},$
$\displaystyle\displaystyle c=1,$ $\displaystyle\displaystyle d=-\frac{1}{6},$
(7)
where we have set $\displaystyle a=1$ as a normalization.
Here, it is convenient to define the tensor $\displaystyle H_{\mu\nu\rho}$ as
$\displaystyle\displaystyle H_{\mu\nu\rho}$ $\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\nabla_{\rho}R_{\mu\nu}+\nabla_{\mu}R_{\nu\rho}+\nabla_{\nu}R_{\rho\mu}$
(8)
$\displaystyle\displaystyle-\frac{1}{3}(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})R,$
which is totally symmetric in $\displaystyle\mu$, $\displaystyle\nu$, and
$\displaystyle\rho$. It satisfies
$\displaystyle\displaystyle g^{\nu\rho}H_{\mu\nu\rho}=0.$ (9)
Another convenient definition is
$\displaystyle\displaystyle T_{\mu\nu\rho}$ $\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\nabla_{\rho}T_{\mu\nu}+\nabla_{\mu}T_{\nu\rho}+\nabla_{\nu}T_{\rho\mu}$
(10)
$\displaystyle\displaystyle-\frac{1}{6}(g_{\mu\nu}\partial_{\rho}+g_{\nu\rho}\partial_{\mu}+g_{\rho\mu}\partial_{\nu})T,$
which is also totally symmetric in $\displaystyle\mu$, $\displaystyle\nu$, and
$\displaystyle\rho$. It satisfies
$\displaystyle\displaystyle
g^{\nu\rho}T_{\mu\nu\rho}=2\nabla_{\nu}T^{\nu}{}_{\mu}.$ (11)
Consequently, we obtain the gravitational field equation expressed by third-
order totally symmetric tensors,
$H_{\mu\nu\rho}=8\pi GT_{\mu\nu\rho},$ (12)
where we explicitly show $\displaystyle 8\pi G=1$.
Multiplying Eq. (12) by $\displaystyle g^{\nu\rho}$ and using Eqs. (9) and
(11), we can confirm that the conservation law,
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is satisfied as
$g^{\nu\rho}H_{\mu\nu\rho}=16\pi G\nabla_{\nu}T^{\nu}{}_{\mu}=0.$ (13)
Here, we provide three remarks on Eq. (12). First, every solution of the
Einstein equations satisfies Eq. (12). This means the following: by
substituting $\displaystyle 8\pi GT_{\mu\nu}=G_{\mu\nu}$ into the right-hand
side of Eq. (12), we can confirm that Eq. (12) is satisfied. Furthermore, by
substituting $\displaystyle 8\pi GT_{\mu\nu}=G_{\mu\nu}+\Lambda g_{\mu\nu}$,
where $\displaystyle\Lambda$ is nonvanishing, into the right-hand side of Eq.
(12), we can confirm that Eq. (12) is still satisfied. Thus, Eq. (12) does not
distinguish between $\displaystyle 8\pi GT_{\mu\nu}=G_{\mu\nu}$ and
$\displaystyle 8\pi GT_{\mu\nu}=G_{\mu\nu}+\Lambda g_{\mu\nu}$. This implies
that the cosmological constant $\displaystyle\Lambda$ arises as an integration
constant. Second, as shown in Eq. (13), the conservation law,
$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$, is satisfied without being
assumed. Third, it should be noted that the vanishing of the Weyl tensor
$\displaystyle C_{\mu\nu\rho\sigma}$ does not mean the vanishing of
$\displaystyle H_{\mu\nu\rho}$. Therefore, a conformally flat spacetime is not
necessarily a vacuum solution of $\displaystyle H_{\mu\nu\rho}=0$.
Consequently, the gravitational field equation (12) simultaneously satisfies
all three criteria stated in the Introduction.
## III Spherically symmetric static vacuum solution
We consider the Schwarzschild-like metric given by
$ds^{2}=-e^{\nu(r)}dt^{2}+e^{-\nu(r)}dr^{2}+r^{2}d\Omega^{2},$ (14)
where $\displaystyle d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta d\phi^{2}$.
Substituting this into Eq. (8), we find that the component of $\displaystyle
H_{\mu\nu\rho}$ is expressed as follows:
$\displaystyle\displaystyle-2H^{1}{}_{11}=-\frac{8}{r^{3}}$
$\displaystyle\displaystyle+e^{\nu}\left(\nu^{\prime\prime\prime}+3\nu^{\prime}\nu^{\prime\prime}+(\nu^{\prime})^{3}-\frac{2\nu^{\prime\prime}}{r}-\frac{2(\nu^{\prime})^{2}}{r}-\frac{2\nu^{\prime}}{r^{2}}+\frac{8}{r^{3}}\right),$
(15)
where a prime denotes the derivative with respect to $\displaystyle r$. The
other components vanish except for $\displaystyle H^{0}{}_{01}$,
$\displaystyle H^{2}{}_{12}$, and $\displaystyle H^{3}{}_{13}$, which are
proportional to $\displaystyle H^{1}{}_{11}$. By making the substitution
$y(r)=\left(\nu^{\prime}-\frac{2}{r}\right)e^{\nu},$ (16)
Eq. (15) can be simplified to
$-2H^{1}{}_{11}=y^{\prime\prime}-\frac{6y}{r^{2}}-\frac{8}{r^{3}}.$ (17)
The solution to the equation $\displaystyle H^{1}{}_{11}=0$ is given by
$y(r)=-\frac{2}{r}+\frac{c_{1}}{r^{2}}+c_{2}r^{3},$ (18)
where $\displaystyle c_{1}$ and $\displaystyle c_{2}$ are constants of
integration. Therefore, Eq. (16) can be rewritten as
$\left(\nu^{\prime}-\frac{2}{r}\right)e^{\nu}=-\frac{2}{r}+\frac{c_{1}}{r^{2}}+c_{2}r^{3}.$
(19)
This equation can be solved as
$e^{\nu}=1-\frac{c_{1}}{3r}+c_{3}r^{2}+\frac{c_{2}}{2}r^{4},$ (20)
where $\displaystyle c_{3}$ is a constant of integration.
If we rename the constants as $\displaystyle c_{1}=6M$, $\displaystyle
c_{3}=-\Lambda/3$, and $\displaystyle c_{2}=-2\lambda/5$, then the solution is
given by
$-g_{00}=1/g_{11}=e^{\nu}=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}-\frac{\lambda}{5}r^{4}.$
(21)
This solution is exact. The term $\displaystyle 2M/r$ represents the
Schwarzschild term. The term $\displaystyle\Lambda r^{2}/3$ corresponds to the
de Sitter term, indicating that the cosmological constant
$\displaystyle\Lambda$ arises as a constant of integration, as expected. The
last term, $\displaystyle\lambda r^{4}/5$, is a newly discovered term that
does not emerge from the Einstein equations. When $\displaystyle\lambda$
vanishes (or when $\displaystyle r$ is sufficiently small to ignore the term
$\displaystyle\lambda r^{4}/5$), Eq. (21) is the Schwarzschild–de Sitter
metric, thus remaining consistent with observations. The term
$\displaystyle\lambda r^{4}/5$ only becomes significant at large distances,
such as cosmological distances, and can be ignored at small distances.
We present the curvature invariants as follows:
$\displaystyle\displaystyle R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{48M^{2}}{r^{6}}+\frac{8\Lambda^{2}}{3}+\frac{48M\lambda}{5r}$
(22a) $\displaystyle\displaystyle\quad+8\Lambda\lambda
r^{2}+\frac{212\lambda^{2}r^{4}}{25},$ $\displaystyle\displaystyle
R^{\mu\nu}R_{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 4\Lambda^{2}+12\Lambda\lambda
r^{2}+10\lambda^{2}r^{4},$ (22b) $\displaystyle\displaystyle R$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle 4\Lambda+6\lambda
r^{2},$ (22c) $\displaystyle\displaystyle
C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{48M^{2}}{r^{6}}+\frac{48M\lambda}{5r}+\frac{12\lambda^{2}r^{4}}{25}.$
(22d)
Hence, we observe that $\displaystyle\lambda$ contributes to both the Ricci
tensor and the Weyl tensor. On the other hand, the $\displaystyle M$ only
contributes to the Weyl tensor, while the cosmological constant
$\displaystyle\Lambda$ contributes solely to the Ricci tensor.
## IV Accelerating universe
In this section, we apply our gravitational field equation to cosmology.
First, we derive the equation of motion for the scale factor. Then, we find a
solution that describes the accelerating expansion of the universe.
### IV.1 Equation of motion for the scale factor
We assume that the universe is isotropic and spatially homogeneous. This
assumption leads us to choose a spacetime coordinate system where the metric
takes the Friedmann-Lemaître-Robertson-Walker metric [7, 8, 9, 10, 11, 12],
$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right).$
(23)
Here, $\displaystyle a(t)$ represents the scale factor, and $\displaystyle k$
is a constant that represents the curvature of three-dimensional space. The
requirements of isotropy and spatial homogeneity dictate that the components
of the energy-momentum tensor take the form
$T^{\mu}{}_{\nu}={\rm diag}(-\rho(t),p(t),p(t),p(t)),$ (24)
and its trace is
$T\equiv T^{\mu}{}_{\mu}=-\rho(t)+3p(t).$ (25)
The energy conservation law gives
$0=-\nabla_{\mu}T^{\mu}{}_{0}=\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p),$ (26)
where $\displaystyle\dot{\rho}\equiv d\rho/dt$ and $\displaystyle\dot{a}\equiv
da/dt$.
We now focus on the gravitational field equation (12). For the Friedmann-
Lemaître-Robertson-Walker metric, the components of $\displaystyle
H_{\mu\nu\rho}$, defined by Eq. (8), are given by
$\displaystyle\displaystyle H^{1}{}_{01}$
$\displaystyle\displaystyle=H^{2}{}_{02}=H^{3}{}_{03}=-\frac{1}{3}H^{0}{}_{00}$
$\displaystyle\displaystyle=-4\left(\frac{\dot{a}}{a}\right)^{3}-\frac{\dddot{a}}{a}+5\frac{\dot{a}\ddot{a}}{a^{2}}-4k\frac{\dot{a}}{a^{3}}$
$\displaystyle\displaystyle=\frac{d}{dt}\left[2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}+\frac{2k}{a^{2}}\right],$
(27)
where dots denote time derivatives. The remaining components of $\displaystyle
H_{\mu\nu\rho}$ vanish.
We also require the components of the tensor $\displaystyle T_{\mu\nu\rho}$ as
defined by Eq. (10). From Eqs. (23)–(25), we obtain the following expressions:
$T^{0}{}_{00}=-\frac{1}{2}(5\dot{\rho}+3\dot{p})$ (28)
and
$\displaystyle\displaystyle T^{1}{}_{01}$
$\displaystyle\displaystyle=T^{2}{}_{02}=T^{3}{}_{03}$
$\displaystyle\displaystyle=-2\frac{\dot{a}}{a}(\rho+p)+\frac{1}{6}(\dot{\rho}+3\dot{p}).$
(29)
Using Eq. (26), we find that
$\displaystyle\displaystyle
T^{1}{}_{01}=T^{2}{}_{02}=T^{3}{}_{03}=-T^{0}{}_{00}/3.$ (30)
The remaining components of $\displaystyle T_{\mu\nu\rho}$ vanish.
The gravitational field equation (12) is therefore
$\frac{d}{dt}\left[2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}+\frac{2k}{a^{2}}\right]=8\pi
G\frac{d}{dt}\left[\frac{1}{6}(5\rho+3p)\right].$ (31)
By integrating this equation, we obtain
$2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}+\frac{2k}{a^{2}}=\frac{4\pi
G}{3}(5\rho+3p)+{\rm const},$ (32)
where $\displaystyle{\rm const}$ is a constant of integration. If we rename it
as $\displaystyle\Lambda/3$, then we can find that the Friedmann equations
$\displaystyle\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{8\pi
G}{3}\rho-\frac{k}{a^{2}}+\frac{\Lambda}{3},$ (33)
$\displaystyle\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3},$ (34)
satisfy Eq. (32). Thus, in our gravitational theory, the cosmological constant
$\displaystyle\Lambda$ is indeed a constant of integration, as expected.
By substituting $\displaystyle\Lambda/3$ for const in Eq. (32), we obtain the
following equation for the scale factor:
$2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}=\frac{4\pi
G}{3}(5\rho+3p)-\frac{2k}{a^{2}}+\frac{\Lambda}{3}.$ (35)
This is the equation of motion for the scale factor in our gravity theory,
which is a generalization of the Friedmann equation in general relativity. It
should be noted that the Friedmann equations, Eqs. (33) and (34), with certain
$\displaystyle\rho$ and $\displaystyle p$, satisfy Eq. (35) with the same
$\displaystyle\rho$ and $\displaystyle p$. However, the inverse is not
necessarily true; Eq. (35), with certain $\displaystyle\rho$ and
$\displaystyle p$, does not necessarily satisfy Eqs. (33) and (34) with the
same $\displaystyle\rho$ and $\displaystyle p$. Given $\displaystyle p$ as a
function of $\displaystyle\rho$, we can solve Eq. (26) to find
$\displaystyle\rho$ as a function of $\displaystyle a$. Then, using the
obtained $\displaystyle\rho$ as a function of $\displaystyle a$, we can solve
Eq. (35) to determine $\displaystyle a$ as a function of $\displaystyle t$.
### IV.2 Accelerating expansion
#### IV.2.1 Preliminary
In general relativity, Eq. (34) indicates that the accelerating expansion of
the universe $\displaystyle(\ddot{a}>0)$ requires either a positive
cosmological constant $\displaystyle\Lambda$ or a negative
$\displaystyle\rho+3p$ (representing dark energy). Therefore, in a matter-
dominated universe ($\displaystyle p=0$) with $\displaystyle\Lambda=0$, Eq.
(34) implies that $\displaystyle-\ddot{a}/a\propto\rho$, indicating
decelerating expansion $\displaystyle(\ddot{a}<0)$.
However, in our gravitational theory, the result differs significantly from
general relativity. For a matter-dominated universe ($\displaystyle p=0$),
with spatial flatness ($\displaystyle k=0$) and $\displaystyle\Lambda=0$, Eq.
(35) yields $\displaystyle 2(\dot{a}/a)^{2}-\ddot{a}/a\propto\rho$. This does
not necessarily imply that $\displaystyle\ddot{a}<0$, because $\displaystyle
2(\dot{a}/a)^{2}$ is positive. In the following, by solving the equation of
motion for the scale factor, we will demonstrate the existence of a solution
that describes the accelerating expansion in a matter-dominated universe.
Figure 1: The ratio of the scale factors $\displaystyle a(t)/a_{0}$ is shown
as a function of time $\displaystyle t$ (present is $\displaystyle t=0$) in
Gyr. The Hubble constant is assumed to be $\displaystyle H_{0}=68$ km s-1
Mpc-1. The four dashed lines represent Eq. (46) for $\displaystyle\Omega_{\rm
m}=(0.1,0.3,0.5,0.7)$. These lines clearly demonstrate that even in the
absence of dark energy or the cosmological constant, and when only matter is
present, the universe undergoes a transition from decelerating to accelerating
expansion. The time of this transition, for each case of
$\displaystyle\Omega_{\rm m}=(0.1,0.3,0.5,0.7)$, is $\displaystyle-11.4$,
$\displaystyle-6.2$, $\displaystyle-3.7$, and $\displaystyle-1.5$ Gyr,
respectively. The negative sign of the transition time indicates that the
transition occurred in the past $\displaystyle(t<0)$. The age of the universe
$\displaystyle t_{0}$, for each case of $\displaystyle\Omega_{\rm
m}=(0.1,0.3,0.5,0.7)$, is $\displaystyle 21.4,14.9,12.4$, and $\displaystyle
11.0$ Gyr, respectively. If we impose the constraint of $\displaystyle
t_{0}>13.0$ Gyr, which is motivated by observations, the range for
$\displaystyle\Omega_{\rm m}$ is determined to be $\displaystyle\Omega_{\rm
m}<0.44$. The solid blue line represents the Einstein–de Sitter universe with
$\displaystyle\Omega_{\rm m}=1$, while the solid orange line represents the
$\displaystyle\Lambda$CDM model following the Friedmann equation with
$\displaystyle\Omega_{\rm m}=0.3$ and $\displaystyle\Omega_{\Lambda}=0.7$. The
age of the universe $\displaystyle t_{0}$ is 9.6 Gyr for the Einstein–de
Sitter universe, and the age is 13.9 Gyr for the $\displaystyle\Lambda$CDM
with $\displaystyle\Omega_{\rm m}=0.3$ and
$\displaystyle\Omega_{\Lambda}=0.7$.
#### IV.2.2 An accelerating solution
For simplicity, we assume that the universe is spatially flat
$\displaystyle(k=0)$ and $\displaystyle\Lambda=0$. For a matter-dominated
universe ($\displaystyle p=0$), Eq. (26) yields
$\rho(t)=\rho_{0}\left(\frac{a(t)}{a_{0}}\right)^{-3},$ (36)
where $\displaystyle a_{0}$ represents the scale factor at the present time,
and $\displaystyle\rho_{0}$ is the matter density at the present time.
Substituting Eq. (36), along with $\displaystyle p=k=\Lambda=0$, into Eq.
(35), we obtain
$2\left(\frac{\dot{a}}{a}\right)^{2}-\frac{\ddot{a}}{a}=\frac{5}{2}H_{0}^{2}\Omega_{\rm
m}\left(\frac{a}{a_{0}}\right)^{-3},$ (37)
where $\displaystyle H_{0}$ is the Hubble constant. The density parameter
$\displaystyle\Omega_{\rm m}$, defined as
$\Omega_{\rm m}\equiv\frac{\rho_{0}}{\rho_{\rm
c}},\quad\rho_{c}\equiv\frac{3H_{0}^{2}}{8\pi G},$ (38)
represents the ratio of matter density to the critical density
$\displaystyle\rho_{\rm c}$. It should be noted that in our gravity theory,
$\displaystyle\Omega_{\rm m}$ does not necessarily satisfy
$\displaystyle\Omega_{\rm m}+\Omega_{\rm others}=1$, where
$\displaystyle\Omega_{\rm others}$ represents the density parameter for other
components, if they exist. Even if $\displaystyle\Omega_{\rm others}=0$,
$\displaystyle\Omega_{\rm m}$ is not necessarily equal to 1, because the
Friedmann equations are not necessarily satisfied. Therefore, we assume
$\displaystyle\Omega_{\rm m}\leq 1$.
The solution to Eq. (37) is given by
$\displaystyle\displaystyle t+t_{0}=\frac{2}{3H_{0}\sqrt{\Omega_{\rm
m}}}\left(\frac{a}{a_{0}}\right)^{3/2}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};A\left(\frac{a}{a_{0}}\right)^{5}\right),$
(39)
where $\displaystyle t_{0}$ and $\displaystyle A$ are two constants of
integration, and $\displaystyle{}_{2}F_{1}\left(a,b;c;x\right)$ is the
hypergeometric function. The scale factor $\displaystyle a(t)$ reaches zero at
$\displaystyle t=-t_{0}$. By substituting $\displaystyle t=0$ (representing
present time) in Eq. (39), we obtain
$\displaystyle\displaystyle t_{0}=\frac{2}{3H_{0}\sqrt{\Omega_{\rm
m}}}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};A\right),$ (40)
where we have used $\displaystyle a_{0}=a(t=0)$.
We also need to determine the constant $\displaystyle A$. This can be done as
follows. Differentiating Eq. (39) with respect to $\displaystyle t$, we have
$\displaystyle\displaystyle 1=\frac{1}{H_{0}\sqrt{\Omega_{\rm
m}}\sqrt{1-A(a/a_{0})^{5}}}\left(\frac{a}{a_{0}}\right)^{1/2}\frac{\dot{a}}{a_{0}}.$
(41)
Using Eqs. (39) and (41) to eliminate $\displaystyle 1/(H_{0}\sqrt{\Omega_{\rm
m}})$, we find that the Hubble parameter $\displaystyle H(t)\equiv\dot{a}/a$
is given by
$\displaystyle\displaystyle
H(t)=\frac{2\sqrt{1-A(a/a_{0})^{5}}}{3(t+t_{0})}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};A\left(\frac{a}{a_{0}}\right)^{5}\right).$
(42)
Therefore, the Hubble constant $\displaystyle H_{0}=H(t=0)$ is
$\displaystyle\displaystyle
H_{0}=\frac{2\sqrt{1-A}}{3t_{0}}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};A\right).$
(43)
By substituting Eq. (40) into Eq. (43), we obtain
$\sqrt{(1-A)\Omega_{\rm m}}=1.$ (44)
This yields
$A=1-\frac{1}{\Omega_{\rm m}}=\frac{\Omega_{\rm m}-1}{\Omega_{\rm m}}.$ (45)
Taking the ratio between Eqs. (39) and (40) and using Eq. (45), we find that
the scale factor $\displaystyle a(t)$ satisfies
$\displaystyle\displaystyle\frac{t+t_{0}}{t_{0}}=\frac{{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};\frac{\Omega_{\rm
m}-1}{\Omega_{\rm
m}}\left(\frac{a}{a_{0}}\right)^{5}\right)}{{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};\frac{\Omega_{\rm
m}-1}{\Omega_{\rm m}}\right)}\left(\frac{a}{a_{0}}\right)^{3/2}.$ (46)
Here, the age of the universe $\displaystyle t_{0}$ is given by
$\displaystyle\displaystyle t_{0}=\frac{2}{3H_{0}\sqrt{\Omega_{\rm
m}}}{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};\frac{\Omega_{\rm
m}-1}{\Omega_{\rm m}}\right).$ (47)
These equations, Eqs. (46) and (47), are fundamental equations. Using the
observed value of the Hubble constant $\displaystyle H_{0}$, Eq. (47)
determines $\displaystyle t_{0}$ as a function of $\displaystyle\Omega_{\rm
m}$. Subsequently, using $\displaystyle t_{0}$ as a function of
$\displaystyle\Omega_{\rm m}$, Eq. (46) determines $\displaystyle a(t)$ as a
function of $\displaystyle t$ and $\displaystyle\Omega_{\rm m}$.
In the special case where $\displaystyle\Omega_{\rm m}=1$, Eqs. (46) and (47)
simplify to
$\displaystyle\displaystyle\frac{a}{a_{0}}=\left(\frac{t+t_{0}}{t_{0}}\right)^{2/3},$
$\displaystyle\displaystyle t_{0}=\frac{2}{3H_{0}}.$ (48)
These equations represent the Einstein–de Sitter universe, and thus Eqs. (46)
and (47) include the result derived from general relativity as a special case.
In general cases where $\displaystyle\Omega_{\rm m}\not=1$, Fig. 1 illustrates
the behavior of $\displaystyle a(t)/a_{0}$ as a function of time. The figure
demonstrates that even in the absence of dark energy or the cosmological
constant, and with only matter present, the universe undergoes a transition
from decelerating to accelerating expansion.
This transition, from deceleration to acceleration, occurs at $\displaystyle
t=-t_{\star}$, which is the time when the acceleration $\displaystyle\ddot{a}$
reaches zero. By performing a straightforward calculation, we can determine
the time $\displaystyle t_{\star}$ as follows:
$t_{\star}=t_{0}\left[1-\left(\frac{\Omega_{\rm m}}{4(1-\Omega_{\rm
m})}\right)^{3/10}\frac{{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};-\frac{1}{4}\right)}{{}_{2}F_{1}\left(\frac{3}{10},\frac{1}{2};\frac{13}{10};\frac{\Omega_{\rm
m}-1}{\Omega_{\rm m}}\right)}\right],$ (49)
where $\displaystyle{}_{2}F_{1}(3/10,1/2;13/10;-1/4)\approx 0.97383$. This
equation indicates that $\displaystyle t_{\star}$ is positive if
$\displaystyle\Omega_{\rm m}$ is less than 0.8. A positive $\displaystyle
t_{\star}$ (or equivalently negative $\displaystyle-t_{\star}$) implies that
the transition from deceleration to acceleration occurred in the past
($\displaystyle t<0$). Therefore, we can conclude that in our gravitational
theory, even in the absence of dark energy, the current universe is in an
accelerating phase if $\displaystyle\Omega_{\rm m}<0.8$. The transition time
($\displaystyle t=-t_{\star}$) for typical values of $\displaystyle\Omega_{\rm
m}$ is provided in the caption of Fig. 1.
## V Summary and conclusions
In this study, we have set three theoretical criteria for gravitational
theories, as outlined in the Introduction:
1. 1.
The gravitational field equations should not explicitly contain the
cosmological constant $\displaystyle\Lambda$, but it can emerge as a constant
of integration.
2. 2.
The conservation law $\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=0$ should be
derived as a consequence of the field equations, rather than being introduced
as an additional assumption.
3. 3.
A conformally flat metric should not necessarily be a vacuum solution.
These criteria impose stringent restrictions on gravitational theories, and so
far, no theory has been known to fulfill all three criteria. In this paper, we
have presented the gravitational field equation, Eq. (12), which satisfies all
three criteria. Our construction provides an explicit model, and while it is a
unique model that the author could find, it may not be the only one. These
criteria and their fulfillment are summarized in Table. 1.
Table 1: A summary of typical gravitational theories and their fulfillment against the three criteria. Criterion | GR111General relativity. | TFE222Trace-free Einstein equations. | CG333Conformal gravity. | Cotton444Cotton gravity. | This work
---|---|---|---|---|---
1 | No | Yes | Yes | Yes | Yes
2 | Yes | No | Yes | Yes | Yes
3 | Yes | Yes | No | No | Yes
Additionally, we have derived a spherically symmetric solution that
generalizes the Schwarzschild solution. This solution consists of three terms:
the Schwarzschild term $\displaystyle(\propto 1/r)$, the de Sitter term
$\displaystyle(\propto r^{2})$, and a newly discovered term
$\displaystyle(\propto r^{4})$. The $\displaystyle r^{4}$ term only becomes
significant at large distances while being negligible at short distances. This
indicates that gravity described by Eq. (12) differs from general relativity
primarily at large distances, such as cosmological distances.
Motivated by this observation, we have applied our gravitational equations to
cosmology. Assuming the isotropy and spatial homogeneity of the universe, we
have derived an equation for the scale factor, Eq. (35), which serves as a
generalization of the Friedmann equation. Through our analysis, we have
demonstrated that even in the absence of dark energy or the cosmological
constant, the universe undergoes a transition from a decelerating phase to an
accelerating phase. Thus, in our gravitational theory, the current
accelerating expansion is a natural and inevitable consequence in a matter-
dominated universe.
###### Acknowledgements.
This work was supported by JSPS KAKENHI Grant No. JP22K03599.
## References
* Ellis _et al._ [2011] G. F. R. Ellis, H. van Elst, J. Murugan, and J.-P. Uzan, On the trace-free Einstein equations as a viable alternative to general relativity, Class. Quant. Grav. 28, 225007 (2011), arXiv:1008.1196 [gr-qc] .
* Ellis [2014] G. F. R. Ellis, The trace-free Einstein equations and inflation, Gen. Rel. Grav. 46, 1619 (2014), arXiv:1306.3021 [gr-qc] .
* Mannheim and Kazanas [1989] P. D. Mannheim and D. Kazanas, Exact vacuum solution to conformal Weyl gravity and galactic rotation curves, Astrophys. J. 342, 635 (1989).
* Harada [2021] J. Harada, Emergence of the Cotton tensor for describing gravity, Phys. Rev. D 103, L121502 (2021), arXiv:2105.09304 [gr-qc] .
* Mantica and Molinari [2023] C. A. Mantica and L. G. Molinari, Codazzi tensors and their space-times and Cotton gravity, Gen. Rel. Grav. 55, 62 (2023), arXiv:2210.06173 [gr-qc] .
* Yang [1974] C. N. Yang, Integral Formalism for Gauge Fields, Phys. Rev. Lett. 33, 445 (1974).
* Friedmann [1922] A. Friedmann, On the curvature of space, Z. Phys. 10, 377 (1922).
* Friedmann [1924] A. Friedmann, On the possibility of a world with constant negative curvature of space, Z. Phys. 21, 326 (1924).
* Lemaître [1927] G. Lemaître, A homogeneous universe of constant mass and growing radius accounting for the radial velocity of extragalactic nebulae, Annales Soc. Sci. Bruxelles A 47, 49 (1927).
* Lemaître [1931a] G. Lemaître, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae, Mon. Not. Roy. Astron. Soc. 91, 483 (1931a).
* Lemaître [1931b] G. Lemaître, The expanding universe, Mon. Not. Roy. Astron. Soc. 91, 490 (1931b).
* Robertson [1935] H. P. Robertson, Kinematics and world-structure, Astrophys. J. 82, 284 (1935).
|
# A note on the global regularity results for strongly nonhomogeneous
$p,q$-fractional problems and applications
Jacques Giacomoni 1 111e-mail<EMAIL_ADDRESS>Deepak Kumar
2222e-mail<EMAIL_ADDRESS>and Konijeti Sreenadh 2333 e-mail:
<EMAIL_ADDRESS>
${}^{1}\,$Université de Pau et des Pays de l’Adour, LMAP (UMR E2S-UPPA CNRS
5142)
Bat. IPRA, Avenue de l’Université F-64013 Pau, France
${}^{2}\,$Department of Mathematics, Indian Institute of Technology Delhi,
Hauz Khaz, New Delhi-110016, India
###### Abstract
In this article, we communicate with the glimpse of the proofs of new global
regularity results for weak solutions to a class of problems involving
fractional $(p,q)$-Laplacian, denoted by
$(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}$, for $s_{2},s_{1}\in(0,1)$ and
$1<p,q<\infty$. We also obtain the boundary Hölder continuity results for the
weak solutions to the corresponding problems involving at most critical growth
nonlinearities. These results are almost optimal. Moreover, we establish Hopf
type maximum principle and strong comparison principle. As an application to
these new results, we prove the Sobolev versus Hölder minimizer type result,
which provides the multiplicity of solutions in the spirit of seminal work
[2].
###### Résumé
Dans cette note, nous présentons de nouveaux résultats de régularité
Höldérienne des solutions faibles d’une classe de problèmes faisant intervenir
des opérateurs de diffusion fractionnaire non linéaires et non homogènes de la
forme $(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}$ avec $s_{2},s_{1}\in(0,1)$
et $1<p,q<\infty$. Précisément, nous obtenons des résultats de régularité
intérieure et près du bord pour les solutions faibles de ces problèmes alors
que la nonlinéarité du membre de droite est de croissance critique au sens de
l’injection de Sobolev. Ce résultat étend les principaux résultats de
régularité intérieure de [1] où le cas de l’opérateur homogène
$(-\Delta)^{s_{1}}_{p}$ est investi, améliore de façon optimale et complète
ceux de [8].
Nous établissons par ailleurs un lemme de Hopf et un principe de comparaison
fort pour cette classe de problèmes. Nous appliquons ensuite ces résultats
pour démontrer la propriété que les minimiseurs locaux de l’énergie associée
dans $C^{\alpha}(\overline{\Omega})$ avec $\alpha\in(0,s_{1})$ sont aussi
minimiseurs locaux dans $W^{s_{1},p}_{0}(\Omega)$ dans l’esprit de l’article
pionnier [2]. Ceci conduit à des nouveaux résultats de muliplicité de
solutions pour ces problèmes non locaux et fortement non homogènes.
Keywords: Fractional $(p,q)$-Laplacian, non-homogeneous nonlocal operator,
Hölder continuity up to the boundary, maximum principle, strong comparison
principle.
2010 Mathematics Subject Classification: 35J60, 35R11, 35B45, 35D30.
## 1 Introduction
In this note we study the Hölder continuity results and maximum principle for
weak solutions to the following problem:
$\begin{array}[]{rllll}(-\Delta)^{s_{1}}_{p}u+(-\Delta)^{s_{2}}_{q}u=f\quad\text{in
}\Omega,\end{array}$ ($\mathcal{P}$)
where $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with $C^{1,1}$
boundary, $2\leq q,p<\infty$, $0<s_{2}\leq s_{1}<1$ and $f\in
L^{\infty}_{\mathrm{loc}}(\Omega)$. The fractional $p$-Laplace operator
$(-\Delta)^{s}_{p}$ is defined as
${(-\Delta)^{s}_{p}u(x)}=2\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}dy.$
The leading differential operator,
$(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}$, in problem ($\mathcal{P}$) is
known as the fractional $(p,q)$-Laplacian. The operator is non-homogeneous in
the sense that for any $t>0$, there does not exist any $\sigma\in\mathbb{R}$
such that
$((-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q})(tu)=t^{\sigma}((-\Delta)^{s_{1}}_{p}u+(-\Delta)^{s_{2}}_{q}u)$
holds for all $u\in W^{s_{1},p}(\Omega)\cap W^{s_{2},q}(\Omega)$.
The regularity results and maximum principles for the equations involving the
homogeneous nonlocal operators are well known, see [1, 4, 9, 10, 12], whereas
the regularity issues for the problems involving the fractional
$(p,q)$-Laplacian is still developing and only few continuity results are
available, see for instance, [7, 6]. The strong non-homogeneity, in this case,
feature creates an additional difficulty while handling the distance function
in order to prove the boundary behavior of the weak solutions.
In the present work, we obtain interior regularity results for local weak
solutions, which improves the regularity results of [7] for larger class of
exponents. Our proof of the improved local Hölder continuity result (see
Theorem 2.1) relies on Moser’s iteration technique to obtain suitable
embedding for Besov spaces into the Hölder spaces. Here, we stress that we do
not assume any order relation on the exponents $p$ and $q$. Subsequently, we
establish the asymptotic behavior of the fractional $q$-Laplacian
($(-\Delta)_{q}^{s_{2}}$) of the distance function $d^{s_{1}}$ near the
boundary, which in turn gives us almost optimal (and optimal in some cases,
see Remark 3) boundary behavior of the weak solution. This coupled with the
interior Hölder regularity result of Theorem 2.1 provides the almost optimal
Hölder continuity result. As a consequence of this, we obtain the Hopf type
maximum principle for non-negative solutions. Additionally, under the
restriction that the fractional $q$-Laplacian of the subsolution is bounded
from below, we prove a strong comparison principle. In the last section, as an
application to these results, we obtain the multiplicity results for problem
involving the subcritical nonlinearity by establishing Sobolev versus Hölder
type minimization result for nonlinearities with atmost critical growth.
Complete proofs of the regularity main results and other applications (in
particular to singular problems) can be found in [7].
## 2 Preliminaries and main results
We denote $[t]^{p-1}:=|t|^{p-2}t$, for all $p>1$ and $t\in\mathbb{R}$. For
$(\ell,s)\in\\{(p,s_{1}),(q,s_{2})\\}$ and for $S_{1}\times
S_{2}\subset\mathbb{R}^{N}\times\mathbb{R}^{N}$, we set
$\displaystyle A_{\ell}(u,v,S_{1}\times S_{2})=\int_{S_{1}\times
S_{2}}\frac{[u(x)-u(y)]^{\ell-1}(v(x)-v(y))}{|x-y|^{N+\ell
s}}\leavevmode\nobreak\ dxdy.$
We define the distance function as
$d(x):=\mathrm{dist}(x,\mathbb{R}^{N}\setminus\Omega)$ and a neighborhood of
the boundary as $\Omega_{\varrho}:=\\{x\in\Omega\ :\ d(x)<\varrho\\}$, for
$\varrho>0$.
We will follow the notation $p^{*}_{s_{1}}:=Np/(N-ps_{1})$ if $N>ps_{1}$,
otherwise an arbitrarily large number.
### 2.1 Function Spaces
For $E\subset\mathbb{R}^{N}$, $p\in[1,\infty)$ and $s\in(0,1)$, the fractional
Sobolev space $W^{s,p}(E)$ is defined as
$\displaystyle W^{s,p}(E):=\left\\{u\in
L^{p}(E):[u]_{W^{s,p}(E)}<\infty\right\\}$
endowed with the norm $\|u\|_{W^{s,p}(E)}:=\|u\|_{L^{p}(E)}+[u]_{W^{s,p}(E)}$,
where
$\displaystyle[u]_{W^{s,p}(E)}:=\left(\int_{E}\int_{E}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\right)^{1/p}.$
For any (proper) subset $E$ of $\mathbb{R}^{N}$, we have
$\displaystyle W^{s,p}_{0}(E):=\\{u\in W^{s,p}(\mathbb{R}^{N})\ :\
u=0\quad\mbox{in }\mathbb{R}^{N}\setminus E\\}$
which is a uniformly convex Banach space when equipped with the norm
$[\cdot]_{W^{s,p}(\mathbb{R}^{N})}$ (hereafter, it will be denoted by
$\|\cdot\|_{W^{s,p}_{0}(E)}$). Next, we define
$\mathcal{W}(E):=W^{s_{1},p}(E)\cap W^{s_{2},q}(E)$
equipped with the norm
$\|\cdot\|_{\mathcal{W}(E)}:=\|\cdot\|_{W^{s_{1},p}(E)}+\|\cdot\|_{W^{s_{2},q}(E)}$.
The space $\mathcal{W}_{0}(E)$ is defined analogously. We say that
$u\in\mathcal{W}_{\rm loc}(E)$ if $u\in\mathcal{W}(E^{\prime})$, for all
$E^{\prime}\Subset E$. Note that for $1<q\leq p<\infty$, $0<s_{2}<s_{1}<1$ and
the domain $E$ with Lipschitz boundary, $W^{s_{1},p}_{0}(E)$ coincides with
the space $X_{p,s_{1}}$, as defined in [8]. Indeed, from [8, Lemma 2.1], we
have
$\displaystyle\|u\|_{W^{s_{2},q}_{0}(E)}\leq
C\|u\|_{W^{s_{1},p}_{0}(E)},\quad\text{for all }\;u\in W^{s_{1},p}_{0}(E),$
for some $C=C(|E|,\;N,\;p,\;q,\;s_{1},\;s_{2})>0$. Additionally, we define
$\displaystyle\widetilde{W}^{s,p}(\Omega):=\bigg{\\{}u\in L^{p}_{\rm
loc}(\mathbb{R}^{N})\ :\ \exists\Omega^{\prime}\Supset\Omega\mbox{ s.t. }u\in
W^{s,p}(\Omega^{\prime}),\
\int_{\mathbb{R}^{N}}\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}dx<\infty\bigg{\\}}.$
###### Definition 1.
Let $u:\mathbb{R}^{N}\to\mathbb{R}$ be a measurable function and
$0<m,\alpha<\infty$. We define the tail space and the nonlocal tail,
respectively, as below:
$\displaystyle L^{m}_{\alpha}(\mathbb{R}^{N})=\bigg{\\{}u\in L^{m}_{\rm
loc}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\frac{|u(x)|^{m}dx}{(1+|x|)^{N+\alpha}}<\infty\bigg{\\}},\
\
T_{m,\alpha}(u;x_{0},R)=\left(R^{\alpha}\int_{B_{R}(x_{0})^{c}}\frac{|u(y)|^{m}dy}{|x_{0}-y|^{N+\alpha}}\right)^{\frac{1}{m}}.$
Set $T_{m,\alpha}(u;R)=T_{m,\alpha}(u;0,R)$. We will follow the notation
$T_{p-1}(u;x,R):=T_{p-1,s_{1}p}(u;x,R)$ and
$T_{q-1}(u;x,R):=T_{q-1,s_{2}q}(u;x,R)$, unless otherwise stated.
### 2.2 Statements of main results
In this subsection, we state our main results. We start with the definition of
local weak solution.
###### Definition 2 (Local weak solution).
A function $u\in\mathcal{W}_{\rm loc}(\Omega)\cap
L^{p-1}_{s_{1}p}(\mathbb{R}^{N})\cap L^{q-1}_{s_{2}q}(\mathbb{R}^{N})$ is said
to be a local weak solution of problem ($\mathcal{P}$) if
$\displaystyle
A_{p}(u,\phi,\mathbb{R}^{N}\times\mathbb{R}^{N})+A_{q}(u,\phi,\mathbb{R}^{N}\times\mathbb{R}^{N})=\int_{\Omega}f\phi
dx,$
for all $\phi\in\mathcal{W}(\Omega)$ with compact support contained in
$\Omega$.
Our first main theorem is the following higher local Hölder continuity result.
###### Theorem 2.1.
Suppose that $(q-p+2)s_{2}<2$. Let $u\in\mathcal{W}_{\rm loc}(\Omega)\cap
L^{p-1}_{s_{1}p}(\mathbb{R}^{N})\cap L^{q-1}_{s_{2}q}(\mathbb{R}^{N})$ be a
locally bounded local weak solution to problem ($\mathcal{P}$). Then, for
every $\sigma\in(0,\Theta)$, $u\in C^{0,\sigma}_{\mathrm{loc}}(\Omega)$, where
$\displaystyle\Theta\equiv\Theta(p,s_{1},q,s_{2})=\begin{cases}\min\\{1,ps_{1}/(p-1)\\}\mbox{
if }qs_{2}<ps_{1}+2(1-s_{1}),\\\ \min\\{1,qs_{2}/(q-1)\\}\mbox{ if
}ps_{1}<qs_{2}.\end{cases}$
Moreover, for $B_{2\bar{R}_{0}}(x_{0})\Subset\Omega$ with
$\bar{R}_{0}\in(0,1)$, there holds
$\displaystyle[u]_{C^{\sigma}(B_{\bar{R}_{0}/2}(x_{0}))}\leq
C\big{(}K_{2}(u)(\|u\|_{W^{s_{1},p}(B_{\bar{R}_{0}}(x_{0}))}+1)\big{)}^{j_{\infty}}$
where $C=C(N,s_{1},p,s_{2},q,\sigma)>0$ is a constant and $K_{2}$ is given by
$\displaystyle
K_{2}=1+T_{p-1}(u;x_{0},\bar{R}_{0})^{p-1}+T_{q-1}(u;x_{0},\bar{R}_{0})^{q-1}+\|u\|^{\frac{(\ell_{1}+j_{\infty})(\ell_{1}-1)}{\ell_{1}-2}}_{L^{\infty}(B_{\bar{R}_{0}}(x_{0}))}+\|u\|^{q-1}_{L^{\infty}(B_{R_{0}})}+\|f\|_{L^{\infty}(B_{\bar{R}_{0}}(x_{0}))}$
with $\ell_{1}=\max\\{p,q\\}$ and $j_{\infty}\in\mathbb{N}$ depends only on
$N,\sigma$ and $(p,s_{1})$ or $(q,s_{2})$.
###### Remark 1.
We remark that for $1<q\leq p<\infty$, the conclusion of Theorem 2.1 holds for
some $\sigma<\min\\{\frac{ps_{1}}{p-1},\frac{qs_{2}}{q-1}\\}$. See Theorem 2.1
and Corollary 2.1 of [6] for details.
Next, we have the following global Hölder continuity result.
###### Theorem 2.2.
Suppose that $(q-p+2)s_{2}<2$. Let $u\in\mathcal{W}_{0}(\Omega)$ be a solution
to problem ($\mathcal{P}$) with $f\in L^{\infty}(\Omega)$. Then, for every
$\sigma\in(0,s_{1})$, $u\in C^{0,\sigma}(\overline{\Omega})$. Moreover,
$\displaystyle\|u\|_{C^{\sigma}(\overline{\Omega})}\leq C,$ (2.1)
where $C=C(\Omega,N,p,s_{1},q,s_{2},\sigma,\|f\|_{L^{\infty}(\Omega)})>0$ is a
constant (which depends as a non-decreasing function of
$\|f\|_{L^{\infty}(\Omega)}$).
###### Corollary 2.3.
Suppose that $2\leq q\leq p<\infty$. Let $u\in W^{s_{1},p}_{0}(\Omega)$ be a
solution to problem ($\mathcal{P}$) with $f(x):=f(x,u)$, a Carathéodory
function satisfying $|f(x,t)|\leq C_{0}(1+|t|^{p^{*}_{s_{1}}-1})$, where
$C_{0}>0$ is a constant. Then, $u\in C^{0,\sigma}(\overline{\Omega})$, for all
$\sigma\in(0,s_{1})$, and (2.1) holds.
Now, we mention our strong comparison theorem.
###### Theorem 2.4 (Strong Comparison principle).
Suppose that $1<q\leq p<\infty$. Let $u,v\in W^{s_{1},p}_{0}(\Omega)\cap
C(\overline{\Omega})$ be such that $0<v\leq u$ in $\Omega$ with $u\not\equiv
v$, and for some $K,K_{1}>0$, the following holds:
$(-\Delta)_{p}^{s_{1}}v+(-\Delta)_{q}^{s_{2}}v\leq(-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u\leq
K\quad\mbox{and }(-\Delta)_{q}^{s_{2}}v\geq-K_{1},\ \ \ \mbox{weakly in
}\Omega.$
Then $u>v$ in $\Omega$. Moreover, for $s_{1}\neq q^{\prime}s_{2}$,
$\frac{u-v}{d^{s_{1}}}\geq C>0$ in $\Omega$.
## 3 Hölder regularity results
We first recall some boundedness results.
###### Proposition 3.1 (Local boundedness).
Suppose $1<q\leq p<\infty$. Let $u\in W^{s_{1},p}_{\rm loc}(\Omega)\cap
L^{p-1}_{s_{1}p}(\mathbb{R}^{N})\cap L^{q-1}_{s_{2}q}(\mathbb{R}^{N})$ be a
local weak solution to the problem ($\mathcal{P}$). Then, $u\in
L^{\infty}_{\mathrm{loc}}(\Omega)$, and the following holds
$\displaystyle\|u\|_{L^{\infty}(B_{r/2}(x_{0}))}\leq
C\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}}|u|^{p}dx\right)^{1/p}+T_{p-1}(u;x_{0},\frac{r}{2})+T_{q-1}(u;x_{0},\frac{r}{2})^{\frac{q-1}{p-1}}+\|f\|_{L^{\infty}(B_{r})}^{1/(p-1)}+1,$
where $C(N,p,q,s_{1})>0$ is a constant.
Proceeding similar to [3, Theorem 3.3] and noticing that the terms
corresponding to the fractional $q$-Laplacian will be non-negative (using the
similar inequality, as in the fractional $p$-Laplacian, for different
$G_{\beta}$ in there), we can prove the following boundedness property.
###### Theorem 3.2.
Let $1<q\leq p<\infty$. Let $u\in W^{s_{1},p}_{0}(\Omega)$ be a weak solution
to problem ($\mathcal{P}$) with $f(x):=f(x,u)$ satisfying $|f(x,t)|\leq
C_{0}(1+|t|^{p^{*}_{s_{1}}-1})$, for all $t\in\mathbb{R}$ and a.e.
$x\in\Omega$, where $C_{0}>0$ is a constant. Then, $u\in L^{\infty}(\Omega)$.
###### Remark 2.
We remark that, as in [3, Remark 3.4], the quantity
$\|u\|_{L^{\infty}(\Omega)}$ depends only on the constants $C_{0}$, $N$, $p$,
$s_{1}$, $\|u\|_{W^{s_{1},p}_{0}(\Omega)}$ and the constant $M>0$ satisfying
$\int_{\\{|u|\geq M\\}}|u|^{p^{*}_{s_{1}}}<\epsilon$, for given
$\epsilon\in(0,1)$.
###### Corollary 3.3.
Suppose that $1<q\leq p<\infty$. Let $u_{\varepsilon}\in
W^{s_{1},p}_{0}(\Omega)$, for $\varepsilon\in(0,1)$, be the family of weak
solution to problem ($\mathcal{P}$) with
$f_{\epsilon}(x):=f_{\epsilon}(x,u_{\epsilon})$ satisfying
$|f_{\epsilon}(x,t)|\leq C_{0}(1+|t|^{p^{*}_{s_{1}}-1})$, for all
$t\in\mathbb{R}$ and a.e. $x\in\Omega$, where $C_{0}$ is independent of
$\epsilon$. Assume that the sequence
$\\{\|u_{\varepsilon}\|_{W^{s_{1},p}_{0}(\Omega)}\\}_{\varepsilon}$ is bounded
and $u_{\varepsilon}\to u_{0}$ in $L^{p^{*}_{s_{1}}}(\Omega)$, as
$\varepsilon\to 0$. Then the sequence
$\\{\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}\\}_{\varepsilon}$ is also
bounded.
### 3.1 Interior regularity
Let $u:\mathbb{R}^{N}\to\mathbb{R}$ be a measurable function and
$h\in\mathbb{R}^{N}$, then we define
$\displaystyle
u_{h}(x)=u(x+h),\quad\delta_{h}u(x)=u_{h}(x)-u(x),\quad\delta^{2}_{h}u(x)=\delta_{h}(\delta_{h}u(x))=u_{2h}(x)+u(x)-2u_{h}(x).$
For $1\leq m<\infty$ and $u\in L^{m}(\mathbb{R}^{N})$, we set
$\displaystyle[u]_{\mathcal{B}^{\beta,m}_{\infty}(\mathbb{R}^{N})}:=\sup_{|h|>0}\bigg{\|}\frac{\delta^{2}_{h}u}{|h|^{\beta}}\bigg{\|}_{L^{m}(\mathbb{R}^{N})}\mbox{
for }\beta\in(0,2).$
Now, we prove our improved interior Hölder regularity result for local weak
solutions.
Sketch of the proof of Theorem 2.1: We first consider the case
$\sigma\in(0,s_{1})$. For $qs_{2}<ps_{1}+2(1-s_{1})$, we claim that, for every
$4h_{0}<R\leq R_{0}-5h_{0}$, there holds:
$\displaystyle\sup_{0<|h|<h_{0}}\bigg{\|}\frac{\delta^{2}_{h}u}{|h|^{s_{1}}}\bigg{\|}^{m+1}_{L^{m+1}(B_{R-4h_{0}})}\leq
CK_{2}(u,m)\Big{(}\sup_{0<|h|<h_{0}}\bigg{\|}\frac{\delta^{2}_{h}u}{|h|^{s_{1}}}\bigg{\|}^{m}_{L^{m}(B_{R+4h_{0}})}+1\Big{)},$
(3.1)
where $m\geq p$, $h_{0}=R_{0}/10$, $C=C(N,h_{0},p,m,s_{1})>0$ (which depends
inversely on $h_{0}$) and
$\displaystyle
K_{2}(u,m,R_{0}):=1+T_{p-1}(u;R_{0})^{p-1}+T_{q-1}(u;R_{0})^{q-1}+\|u\|^{\frac{m(p-1)}{p-2}}_{L^{\infty}(B_{R_{0}})}+\|u\|^{q-1}_{L^{\infty}(B_{R_{0}})}+\|f\|_{L^{\infty}(B_{R_{0}})}.$
Indeed, for $2\leq q\leq p<\infty$, (3.1) is proved in [7, Proposition 3.9].
For the other case, we set $S_{1}=\\{(x,y)\in B_{R}\times B_{R}\;:\;|x-y|\leq
1\\}$ and $S_{2}=(B_{R}\times B_{R})\setminus S_{1}$. Then, the proof, in this
case, runs similarly by noting the following (using the same notations of the
Proposition),
$\displaystyle\left(\iint_{S_{1}}+\iint_{S_{2}}\right)\frac{|u(x)-u(y)|^{q-2}}{|x-y|^{N+qs_{2}}}\big{|}\eta^{\frac{p}{2}}(x)-\eta^{\frac{p}{2}}(y)\big{|}^{2}\frac{|\delta_{h}u(x)|^{\beta+1}}{|h|^{1+\nu\beta}}dxdy$
$\displaystyle\leq
C\|u\|^{q-p+1}_{L^{\infty}(B_{R_{0}})}\iint_{S_{1}}\frac{|u(x)-u(y)|^{p-2}}{|x-y|^{N+ps_{1}+\alpha-2}}\frac{|\delta_{h}u(x)|^{\beta}}{|h|^{1+\nu\beta}}dxdy+C\|u\|^{q-1}_{L^{\infty}(B_{R_{0}})}\int_{B_{R}}\frac{|\delta_{h}u(x)|^{\beta}}{|h|^{1+\nu\beta}}dx$
$\displaystyle\leq
CK_{2}(u)[u]^{m}_{W^{\frac{s_{1}(p-2-\epsilon)}{p-2},m}(B_{R+h_{0}})}+CK_{2}(u)\Big{(}\int_{B_{R}}\frac{|\delta_{h}u(x)|^{\frac{\beta
m}{m-p+2}}}{|h|^{\frac{(1+\nu\beta)m}{m-p+2}}}dx+1\Big{)}$
where we have used Hölder’s and Young’s inequality together with the fact that
$qs_{2}\leq ps_{1}+\alpha$ with $\alpha<2(1-s_{1})$ and
$\epsilon\in(0,\frac{2-\alpha}{s_{1}}-2)$. Thus, $\tilde{I}_{11}(q)$ (hence
$I_{11}(q)$) is estimated as similar to $\tilde{I}_{11}(p)$. Set
$\displaystyle s_{1}-\sigma>\frac{N}{p+i_{\infty}},\quad
h_{0}=\frac{\bar{R}_{0}}{64i_{\infty}}\quad\mbox{for some
}i_{\infty}\in\mathbb{N}$
and define the following sequences
$\displaystyle m_{i}=p+i,\quad
R_{i}=\frac{7\bar{R}_{0}}{8}-4(2i+1)h_{0}\quad\mbox{for all
}i=0,\dots,i_{\infty}.$
We take $\psi\in C_{c}^{\infty}(B_{(5\bar{R}_{0})/8})$ such that
$0\leq\psi\leq 1,\quad\psi=1\mbox{ in
}B_{\bar{R}_{0}/2},\quad|\nabla\psi|\mbox{ and }|\nabla^{2}\psi|\leq C.$
Using the discrete Leibniz rule on $\delta^{2}_{h}$, we obtain
$\displaystyle[u\psi]_{\mathcal{B}^{s_{1},m_{i_{\infty}}}_{\infty}(\mathbb{R}^{N})}\leq
C\Big{[}\sup_{0<|h|<h_{0}}\bigg{\|}\frac{\delta^{2}_{h}u}{|h|^{s_{1}}}\bigg{\|}_{L^{m_{i_{\infty}}}(B_{(3\bar{R}_{0})/4})}+\|u\|_{L^{m_{i_{\infty}}}(B_{(3\bar{R}_{0})/4})}\Big{]}.$
The first term of the right hand side on the above expression is estimated on
account of (3.1). Therefore, employing the embedding result of the Besov
spaces into the Hölder spaces, we get that $u\in
C^{0,\sigma}_{\mathrm{loc}}(\Omega)$, for all $\sigma\in(0,s_{1})$. For the
case $2\leq p<q$ and $ps_{1}<qs_{2}$, we proceed exactly as above by
interchanging the role of $(p,s_{1})$ with $(q,s_{2})$ and the corresponding
spaces. In this case, $\tilde{I}_{11}(p)$ is estimated as above by choosing
$\epsilon>0$ such that $\epsilon<\frac{2}{s_{2}}-q+p-2$.
The higher regularity result follows by using the above almost $s_{1}$ (or
$s_{2}$)-Hölder continuity result and proceeding on the similar lines of the
proof of [1, Theorem 5.2] (with minor modification as in the proof above).
$\square$
### 3.2 Boundary regularity and maximum principle
In this subsection, we prove the boundary behavior of the weak solutions. For
$\alpha,\rho>0$ and $\kappa\geq 0$, we set
$\displaystyle d_{e}(x)=\begin{cases}d(x)&\mbox{if }x\in\Omega,\\\
-d(x)&\mbox{if }x\in(\Omega^{c})_{\rho},\\\
-\rho&\mbox{otherwise},\end{cases}\quad\overline{w}_{\rho}(x)=\begin{cases}(d_{e}(x)+\kappa^{1/\alpha})_{+}^{\alpha}\quad&\mbox{if
}x\in\Omega\cup(\Omega^{c})_{\rho},\\\ 0\qquad&\mbox{otherwise},\end{cases}$
where
$(\Omega^{c})_{\rho}:=\\{x\in\Omega^{c}:\mathrm{dist}(x,\partial\Omega)<\rho\\}$.
Sketch of the proof of Theorem 2.2: We proceed as below.
* (a)
By flattening the boundary $\partial\Omega$ and using suitable
$C^{1,1}(\mathbb{R}^{N},\mathbb{R}^{N})$ diffeoemorphisms, we prove that:
there exist $\kappa_{1},\varrho_{1}>0$ such that for all
$\kappa\in[0,\kappa_{1})$ and $\varrho\in(0,\varrho_{1})$,
$\displaystyle(-\Delta)_{p}^{s_{1}}\overline{w}_{\rho}\begin{cases}\geq
C_{1}(d+\kappa^{1/\alpha})^{-(ps_{1}-\alpha(p-1))}\quad\mbox{for all
}\alpha\in(0,s_{1}),\\\ =h\quad\mbox{for all }\alpha\in[s_{1},1)\mbox{ with
}\alpha\neq p^{\prime}s_{1}\end{cases}\quad\mbox{weakly in }\Omega_{\varrho},$
where $C_{1}>0$ is a constant and $h\in L^{\infty}(\Omega_{\varrho_{1}})$
(both are independent of $\kappa\in(0,1)$). Further, for all $\kappa>0$ and
$\alpha\in(0,s_{1})$,
$\overline{w}_{\rho}\in\widetilde{W}^{s_{1},p}(\Omega_{\varrho_{1}})$, and for
$k=0$, $\overline{w}_{\rho}\in\widetilde{W}^{s_{1},p}(\Omega_{\varrho_{1}})$,
whenever $\alpha>s_{1}-1/p$.
* (b)
For $\Gamma>1$, $\max\\{s_{1}-1/p,s_{2}-1/q\\}<\alpha<s_{1}$ and $\varrho>0$
(sufficiently small), we have, weakly in $\Omega_{\varrho}$,
$\displaystyle(-\Delta)_{p}^{s_{1}}(\Gamma
d^{\alpha})+(-\Delta)_{q}^{s_{2}}(\Gamma d^{\alpha})$ $\displaystyle\geq
C_{5}\Gamma^{p-1}d^{-(ps_{1}-\alpha(p-1))}-\Gamma^{q-1}\|h\|_{L^{\infty}(\Omega_{\varrho})}$
$\displaystyle\geq C_{6}\Gamma^{p-1}d^{-(ps_{1}-\alpha(p-1))}.$
Then, employing the weak comparison principle in $\Omega_{\varrho}$, for
suitable $\Gamma$, we get that $u\leq\Gamma d^{\alpha}$ in $\Omega$.
Subsequently, we perform a similar process for $-u$ also.
* (c)
The proof of the Hölder continuity can be completed by taking into account
Theorem 2.1 and the boundary behavior presented in Step (b). $\square$
###### Remark 3.
We remark that in Theorem 2.2, the choice of $\sigma$ can be optimal (that is,
$\sigma=s_{1}$) for the case $s_{1}=s_{2}$ or $s_{1}>q^{\prime}s_{2}$. Indeed,
for $s_{1}=s_{2}$, we can show that the barrier function as constructed in [9,
Lemma 4.3] satisfies $(-\Delta)_{q}^{s_{1}}w\geq 0$ weakly in
$B_{r}(e_{N})\setminus\overline{B_{1}}$. Thus, for appropriate choice of
$\Gamma>1$, the Step (b) above can be improved. Similar arguments apply to the
case $s_{1}>q^{\prime}s_{2}$ with a careful reading of the proof of [7, Lemma
3.12].
Proof of Corollary 2.3: When $f(x):=f(x,u)$, on account of Theorem 3.2 and
Remark 2, we observe that
$|f(x,u)|\leq C_{0}(1+|u|^{p^{*}_{s_{1}}-1})\leq
C_{0}(1+\|u\|_{L^{\infty}(\Omega)}^{p^{*}_{s_{1}}-1})=:K>0.$
Thus, the required result, in this case, follows from Theorem 2.2. $\square$
Next we state our strong maximum principle. The proof is contained in [6] and
done by proving that continuous weak super-solutions are viscosity super-
solutions.
###### Theorem 3.4.
Suppose that $1<q\leq p<\infty$. Let $g\in C(\mathbb{R})\cap BV_{\rm
loc}(\mathbb{R})$ and let $u\in W^{s_{1},p}_{0}(\Omega)\cap
C(\overline{\Omega})$ be such that
$\displaystyle(-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u+g(u)\geq
g(0)\quad\mbox{weakly in }\Omega.$
Further, assume that $u\not\equiv 0$ with $u\geq 0$ in $\Omega$. Then, there
exists $c_{1}>0$ such that $u\geq c_{1}{\rm
dist}(\cdot,\partial\Omega)^{s_{1}}$ in $\Omega$.
Sketch of the proof of Theorem 2.4: By continuity and the fact that
$u\not\equiv v$, we can find $x_{0}\in\Omega$, $\rho,\epsilon>0$ such that
$B_{\rho}(x_{0})\subset\Omega$ and
$\displaystyle\sup_{B_{\rho}(x_{0})}v<\inf_{B_{\rho}(x_{0})}u-\epsilon/2.$
(3.2)
For $\Gamma>1$ and for all $x\in\mathbb{R}^{N}$, we define
$\displaystyle w_{\Gamma}(x)=\begin{cases}\Gamma v(x)&\mbox{if }x\in
B_{\rho/2}^{c}(x_{0})\\\ u(x)&\mbox{if }x\in B_{\rho/2}(x_{0}).\end{cases}$
Taking into account the nonlocal super-position principle [8, Lemma 2.5], we
have, weakly in $\Omega\setminus B_{\rho}(x_{0})$,
$\displaystyle(-\Delta)_{p}^{s_{1}}w_{\Gamma}+(-\Delta)_{q}^{s_{2}}w_{\Gamma}$
$\displaystyle\leq(-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u+(\Gamma^{p-1}-1)K+(\Gamma^{p-1}-\Gamma^{q-1})K_{1}-C_{1}\epsilon^{p-1}-C_{2}\epsilon^{q-1}.$
We can choose $\Gamma>1$ (close to $1$) to employ the weak comparison
principle ([8, Proposition 2.6]), consequently, we get $w_{\Gamma}\leq u$ in
$\Omega$. Hence, using (3.2) and Theorem 3.4, we obtain $u\geq\Gamma v>v$ in
$\Omega$, and $\frac{u-v}{d^{s_{1}}}\geq\frac{(\Gamma-1)v}{d^{s_{1}}}\geq C>0$
in $\Omega$. $\square$
## 4 Applications
We consider the problem ($\mathcal{P}$) with the choice $f(x):=f(x,u)$, where
$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carathéodory function satisfying
the following:
* (A1)
$|f(x,t)|\leq C_{0}(1+|t|^{r-1})$, for a.a. $x\in\Omega$ and all
$t\in\mathbb{R}$, where $C_{0}>0$ is a constant and $r\in(1,p^{*}_{s_{1}}]$.
* (A2)
For a.a. $x\in\Omega$, $f(x,t)t\leq 0$, for all $t\in[-\varsigma,\varsigma]$
($\varsigma>0$) and $f(x,t)t\geq-c_{1}t^{p}$ ($c_{1}>0$) for all
$t\in\mathbb{R}$.
* (A3)
For $F(x,t):=\int_{0}^{t}f(x,\tau)d\tau$,
$\lim_{|t|\to\infty}\frac{F(x,t)}{|t|^{p}}=\infty$ uniformly for a.a.
$x\in\Omega$.
* (A4)
Let $r\in(p,p^{*}_{s_{1}})$, there exists
$\nu\in\big{(}(r-p)\max\\{N/(ps_{1}),1\\},p^{*}_{s_{1}}\big{)}$ such that
$\displaystyle\lim_{|t|\to\infty}\frac{f(x,t)t-pF(x,t)}{|t|^{\nu}}>0\quad\mbox{uniformly
a.e. }x\in\Omega.$
One example for $f$ satisfying (A1)-(A4) is given by
$f(x,u)=-c_{1}|t|^{p-2}t+|t|^{r-2}t$. The Euler functional
$\mathcal{J}:W^{s_{1},p}_{0}(\Omega)\rightarrow\mathbb{R}$ associated to
problem ($\mathcal{P}$) is given by
$\mathcal{J}(u)=\frac{1}{p}\|u\|_{W^{s_{1},p}_{0}(\Omega)}^{p}+\frac{1}{q}\|u\|_{W^{s_{2},q}_{0}(\Omega)}^{q}-\int_{\Omega}F(x,u)dx.$
First we prove the following Sobolev versus Hölder minimizer result.
###### Theorem 4.1.
Suppose that $2\leq q\leq p<\infty$ and (A1) holds. Let $u_{0}\in
W^{s,p}_{0}(\Omega)$, then for all $\alpha\in(0,s_{1})$, the following are
equivalent
1. (i)
there exists $\sigma>0$ such that $\mathcal{J}(u_{0}+v)\geq\mathcal{J}(u_{0})$
for all $v\in W^{s_{1},p}_{0}(\Omega)$,
$\|v\|_{W^{s_{1},p}_{0}(\Omega)}\leq\sigma$,
2. (ii)
there exists $\omega>0$ such that $\mathcal{J}(u_{0}+v)\geq\mathcal{J}(u_{0})$
for all $v\in W^{s_{1},p}_{0}(\Omega)\cap C^{0,\alpha}(\overline{\Omega})$
with $\|v\|_{C^{\alpha}(\overline{\Omega})}\leq\omega$.
Proof. From $(ii)$ and the density argument, we get that
$\langle\mathcal{J}^{\prime}(u_{0}),\phi\rangle=0$ for all $\phi\in
W^{s_{1},p}_{0}(\Omega)$. Consequently, Theorem 2.2 implies that $u_{0}\in
C^{0,\alpha}(\overline{\Omega})$. To prove $(i)$, on the contrary assume that
there exists $\tilde{u}_{n}\in W^{s_{1},p}_{0}(\Omega)$ such that
$\tilde{u}_{n}\to u_{0}$ in $W^{s_{1},p}_{0}(\Omega)$ and
$\mathcal{J}(\tilde{u}_{n})<\mathcal{J}(u_{0})$ for all $n\in\mathbb{N}$. Set
$\mathcal{K}(v)=\frac{1}{p^{*}_{s_{1}}}\int_{\Omega}|v|^{p^{*}_{s_{1}}},\
\varepsilon_{n}:=\mathcal{K}(\tilde{u_{n}}-u_{0})\ \mbox{and }S_{n}:=\\{u\in
W^{s_{1},p}_{0}(\Omega)\ :\ \mathcal{K}(u-u_{0})\leq\varepsilon_{n}\\}.$
By the continuous embedding $W^{s_{1},p}_{0}(\Omega)\hookrightarrow
L^{p^{*}_{s_{1}}}(\Omega)$, we see that $\varepsilon_{n}\to 0$ and hence
$S_{n}$ is a closed convex subset of $W^{s_{1},p}_{0}(\Omega)$. Next, for all
$t\in\mathbb{R}$ and $k>0$, set $[t]_{k}=\mathrm{sign}(t)\min\\{|t|,k\\}$ and
$f_{k}(x,t):=f(x,[t]_{k})$ with $F_{k}(x,t):=\int_{0}^{t}f_{k}(x,\tau)d\tau$.
Then, on account of the Lebesgue dominated convergence theorem, for fixed
$n\in\mathbb{N}$ and
$\sigma_{n}\in(0,\mathcal{J}(u_{0})-\mathcal{J}(\tilde{u}_{n}))$, there exists
$k_{n}>\|u_{0}\|_{L^{\infty}(\Omega)}+1$ such that
$\displaystyle\Big{|}\int_{\Omega}F_{n}(x,\tilde{u}_{n})dx-\int_{\Omega}F(x,\tilde{u}_{n})dx\Big{|}<\sigma_{n},$
where $F_{n}=F_{k_{n}}$. Furthermore, we define
$\displaystyle\mathcal{J}_{n}(u)=\frac{1}{p}\|u\|_{W^{s_{1},p}_{0}(\Omega)}^{p}+\frac{1}{q}\|u\|_{W^{s_{2},q}_{0}(\Omega)}^{q}-\int_{\Omega}F_{n}(x,u)\leavevmode\nobreak\
dx.$
From the structure of the function $F_{n}$, it is clear that there exists a
minimizer $u_{n}\in S_{n}$ for $\mathcal{J}_{n}$. Moreover, by the choice of
$\sigma_{n}$ and $k_{n}$, we see that
$\displaystyle\mathcal{J}_{n}(u_{n})\leq\mathcal{J}_{n}(\tilde{u}_{n})\leq\mathcal{J}(\tilde{u}_{n})+\sigma_{n}<\mathcal{J}(u_{0})=\mathcal{J}_{n}(u_{0}).$
(4.1)
It is clear that $\mathcal{J}_{n}$ is Gǎteaux differentiable at $u_{n}$.
Therefore, there exists $\mu_{n}\leq 0$ such that
$(\mathcal{P}_{n})\left\\{\begin{array}[]{rllll}(-\Delta)^{s_{1}}_{p}u_{n}+(-\Delta)^{s_{2}}_{q}u_{n}&=f_{n}(x,u_{n})+\mu_{n}|u_{n}-u_{0}|^{p^{*}_{s_{1}}-2}(u_{n}-u_{0})\;\text{
in }\Omega,\\\ u_{n}&=0\quad\text{ in
}\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$
If $\inf_{n}\mu_{n}:=l>-\infty$, then from the fact that $u_{0}\in
L^{\infty}(\Omega)$, we have
$\displaystyle|f_{n}(x,u_{n})+\mu_{n}|u_{n}-u_{0}|^{p^{*}_{s_{1}}-2}(u_{n}-u_{0})|\leq
C(1+|u_{n}|^{p^{*}_{s_{1}}-1}).$
If $\inf_{n}\mu_{n}:=-\infty$, there exists $M>0$ (independent of $n$) such
that
$f_{n}(x,t)+\mu_{n}|t-u_{0}(x)|^{p^{*}_{s_{1}}-2}(t-u_{0}(x))<0\quad\mbox{for
a.a. }x\in\Omega,\mbox{ and all }t\in(M,\infty).$
This implies that $u_{n}\leq M$ for all $n\in\mathbb{N}$. Further, since
$\mathcal{J}^{\prime}(u_{0})=0$, we take
$w=|u_{n}-u_{0}|^{\kappa-1}(u_{n}-u_{0})$ as a test function and using [11,
Lemma 2.3] (consequently, the difference of terms involving $A_{q}$, below, is
non-negative), we have
$\displaystyle(C\kappa^{p-1})^{-1}\|(u_{n}-u_{0})^{\frac{p-1+\kappa}{p}}\|^{p}_{W^{s_{1},p}_{0}(\Omega)}$
$\displaystyle\leq
A_{p}(u_{n},[u_{n}-u_{0}]^{\kappa})-A_{p}(u_{0},[u_{n}-u_{0}]^{\kappa})$
$\displaystyle\quad+A_{q}(u_{n},[u_{n}-u_{0}]^{\kappa})-A_{q}(u_{n},[u_{n}-u_{0}]^{\kappa})$
$\displaystyle=\int_{\Omega}(f_{n}(x,u_{n})-f(x,u_{0}))[u_{n}-u_{0}]^{\kappa}+\mu_{n}\int_{\Omega}|u_{n}-u_{0}|^{p^{*}_{s_{1}}-1+\kappa}.$
Noting the uniform bound $\|u_{n}\|_{L^{\infty}(\Omega)}\leq M$ and using
Hölder’s inequality, and subsequently passing to the limit $\kappa\to\infty$,
we obtain
$\displaystyle-\mu_{n}\|u_{n}-u_{0}\|^{p^{*}_{s_{1}}-1}_{L^{\infty}(\Omega)}\leq
C$
where $C>0$ is a constant independent of $n$. Thus, in all the cases, we
obtain
$|f_{n}(x,u_{n})+\mu_{n}|u_{n}-u_{0}|^{p^{*}_{s_{1}}-2}(u_{n}-u_{0})|\leq
C(1+|u_{n}|^{p^{*}_{s_{1}}-1})\quad\mbox{for all }n\in\mathbb{N}.$
Moreover, from the construction of $u_{n}$, it is clear that
$\\{\|u_{n}\|_{W^{s_{1},p}_{0}(\Omega)}\\}_{n}$ remains bounded. Then,
applying Corollary 3.3, we have $\|u_{n}\|_{L^{\infty}(\Omega)}\leq C$, where
$C>0$ is a constant independent of $n$. Consequently, from Corollary 2.3, we
deduce that $\|u_{n}\|_{C^{\alpha}(\overline{\Omega})}\leq C$, for some
positive constant $C$ independent of $n$, and all $\alpha\in(0,s_{1})$.
Therefore, by Arzela-Ascoli’s theorem, $u_{n}\to u_{0}$ in
$C^{0,\alpha}(\overline{\Omega})$, for all $\alpha<s_{1}$. Thus, for
sufficiently large $n$, we have
$\|u_{n}-u_{0}\|_{C^{\alpha}(\overline{\Omega})}\leq\omega$, and since $u_{n}$
is uniformly bounded in $L^{\infty}(\Omega)$,
$\mathcal{J}_{n}(u_{n})=\mathcal{J}(u_{n})$, for sufficiently large $n$. This
along with (4.1) contradicts the fact that $u_{0}$ is a minimizer for
$\mathcal{J}$ in $W^{s_{1},p}_{0}(\Omega)\cap
C^{0,\alpha}(\overline{\Omega})$.
The proof of the other implication is standard. $\square$
###### Theorem 4.2.
Suppose that $2\leq q\leq p<\infty$. Then, there exist at least three non-
trivial solutions $u\in W^{s_{1},p}_{0}(\Omega)\cap
C^{0,\alpha}(\overline{\Omega})$, for all $\alpha\in(0,s_{1})$, to problem
($\mathcal{P}$) with $f(x):=f(x,u)$ satisfying (A1)-(A4). Moreover, if $u$ is
non-negative and $s_{1}\neq q^{\prime}s_{2}$, then $u\geq cd^{s_{1}}$ in
$\Omega$.
Sketch of the Proof: We consider the truncation of the nonlinear term as
$f_{\pm}(x,t)=f(x,\pm t^{\pm})$ with
$F_{\pm}(x,t):=\int_{0}^{t}f_{\pm}(x,\tau)d\tau$ and the corresponding Euler
functionals as $\mathcal{J}_{\pm}$. By using Theorem 4.1, we can prove that
$0$ is a local minimizer for $\mathcal{J}_{+}$ in $W^{s_{1},p}_{0}(\Omega)$
topology and it satisfies the mountain pass geometry. Thus, we obtain a
positive solution $u_{+}$ to problem ($\mathcal{P}$). Similar procedure yields
a negative solution $u_{-}$. Subsequently, by using topological tools (such as
critical groups and Morse theory, see [5] for the linear operator case), we
establish a third solution of undetermined sign nature. $\square$
## References
* [1] L. Brasco, E. Lindgren and A. Schikorra, Higher Hölder regularity for the fractional $p$-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.
* [2] H. Brezis and L. Nirenberg, $H^{1}$ versus $C^{1}$ local minimizers, C.R.A.S., 317 (1993), 465-472.
* [3] W. Chen, S. Mosconi and M. Squassina, Nonlocal problems with critical Hardy nonlinearity, J. Funct. Anal. 275 (2018) 3065-3114.
* [4] L. M. Del Pezzo and A. Quaas, A Hopf’s lemma and a strong minimum principle for the fractional $p$-Laplacian, J. Differ. Equ. 263 (2017), 765-778.
* [5] F.G. Düzgün and A. Iannizzotto, Three nontrivial solutions for nonlinear fractional Laplacian equations, Adv. Nonlinear Anal., 7 (2018), 211-226.
* [6] J. Giacomoni, D. Kumar, K. Sreenadh, Global regularity results for non-homogeneous growth fractional problems, to appear in J. Geometric Anal., doi: 10.1007/s12220-021-00837-4.
* [7] J. Giacomoni, D. Kumar and K. Sreenadh, Interior and boundary regularity results for strongly nonhomogeneous $p,q$-fractional problems, Adv. Calc. Var. (2021), doi:10.1515/acv-2021-0040
* [8] D. Goel, D. Kumar and K. Sreenadh, Regularity and multiplicity results for fractional $(p,q)$-Laplacian equation, Commun. Contemp. Math. 22 (2020), no. 8, 37 pp.
* [9] A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam. 32 (2016), 1353-1392.
* [10] A. Iannizzotto, S. Mosconi and M. Squassina, Fine boundary regularity for the fractional $p$-Laplacian, J. Functional Analysis 279 (2020), no. 8, 108659.
* [11] A. Iannizzotto, S. Mosconi and M. Squassina, Sobolev versus Hölder minimizers for the degenerate fractional $p$-Laplacian, Nonlinear Anal. 191 (2020), 111635.
* [12] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
|
# Spatial Analysis and Synthesis Methods:
Subjective and Objective Evaluations Using Various Microphone Arrays in the
Auralization of a Critical Listening Room
Alan Pawlak, Hyunkook Lee, Aki Mäkivirta, and Thomas Lund The research
presented in this paper was funded by Genelec Oy and the University of
Huddersfield.A. Pawlak and H. Lee are with the Applied Psychoacoustics Lab
(APL), University of Huddersfield, HD1 3DH Huddersfield, United Kingdom
(e-mail<EMAIL_ADDRESS>h.lee@hud.ac.uk)A. Mäkivirta and T. Lund are
with Genelec OY, 74100 Iisalmi, Finland (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Parametric sound field synthesis methods, such as the Spatial Decomposition
Method (SDM) and Higher-Order Spatial Impulse Response Rendering (HO-SIRR),
are widely used for the analysis and auralization of sound fields. This paper
studies the performances of various sound field synthesis methods in the
context of the auralization of a critical listening room. The influence on the
perceived spatial and timbral fidelity of the following factors is considered:
the rendering framework, direction of arrival (DOA) estimation method,
microphone array structure, and use of a dedicated center reference microphone
with SDM. Listening tests compare the synthesized sound fields to a reference
binaural rendering condition. Several acoustic parameters are measured to gain
insights into objective differences between methods. A high-quality pressure
microphone improves the SDM framework’s timbral fidelity. Additionally, SDM
and HO-SIRR show similarities in spatial fidelity. Performance variation
between SDM configurations is influenced by the DOA estimation method and
microphone array construction. The binaural SDM (BSDM) presentations display
temporal artifacts impacting sound quality.
###### Index Terms:
Spatial Audio, Binaural Rendering, Spatial Decomposition Method (SDM), Higher-
Order Spatial Impulse Response Rendering (HO-SIRR), Binaural Room Impulse
Responses (BRIR), Auralization, Microphone Arrays, Subjective Audio
Evaluation, MUSHRA, Direction of Arrival (DOA), Time Difference of Arrival
(TDOA), Pseudo Intensity Vectors (PIV)
## I Introduction
Directional sound field analysis reveals spatial reflection patterns in
enclosed spaces. Its goals include: (i) pinpointing problematic room
reflections [1], (ii) understanding perceptual impacts of room features [1],
(iii) identifying reflections contributing to the desirable perceptual
attributes [2], (iv) assessing the impact of reflection regions on objective
metrics [3], (v) advancing psychoacoustic research [2], and (vi) forming the
foundation for parametric auralisation systems [4, 5].
Over the past two decades, the principles of spatial analysis have driven
significant growth in parametric spatial audio rendering. This trend began
with the introduction of the Spatial Impulse Response Rendering (SIRR) [4],
which utilizes first-order spherical harmonics (SPH) for sound field analysis
and synthesis in the time-frequency domain. A subsequent, more straightforward
approach known as the Spatial Decomposition Method (SDM) was introduced by
Tervo et al. [5]. This method operates in the time domain and interprets each
sample in an impulse response as an image source characterized by both
pressure and direction. The public availability of SDM as a MATLAB toolbox [6]
has made it a popular choice for analyzing enclosed spaces, auralization, and
research [7, 8, 9, 10, 11, 12, 13]. As object-based audio gained prominence,
the Reverberant Spatial Audio Object (RSAO) was developed, parameterizing
spatial room impulse response (SRIR) into a concise set of coefficients. This
was aimed at enabling reverberation synthesis within audio object renderers
[14].
Later, the HO-SIRR, a higher-order adaptation of the SIRR, was introduced
[15], offering enhanced spatial resolution through the use of higher-order
SPH. Concurrently, the Ambisonic SDM (ASDM) was introduced, allowing for the
upscaling of the first-order ambisonics (FOA) to higher-order (HOA).
Subsequent innovations include the binaural versions of SDM (BSDM) [16] and
HO-SIRR[17], an enhanced ASDM known as 4D-ASDM [18], and the Reproduction and
Parameterisation of Array Impulse Responses (REPAIR) [19].
Despite the rapid developments, the SDM and SIRR (and HO-SIRR) remain
predominant. Previous studies, such as [20, 5, 15, 19], have evaluated these
methods primarily through subjective experiments with loudspeakers employing
simulated environments. However, some acoustic simulation methods, like the
Image Source Method (ISM) [21] may favor methods like the SDM due to shared
underlying assumptions. Moreover, inconsistencies arise, such as those between
the SDM and HO-SIRR evaluations, which may stem from different microphone
array configurations or direction-of-arrival (DOA) estimation techniques used.
Previous studies have not comprehensively compared all these methods under
real-life conditions, which this study intends to do.
Additionally, the experimental setups in prior studies appear to be somewhat
limited. While SDM has been used with various microphone arrays, only two
studies have examined how the array affects SDM’s auralization quality [22]
and DOA accuracy [16]. Despite commonly using signals from any omnidirectional
microphone in arrays, no perceptual differences were found between signals
from Ambisonics and a central omni microphone, although the research lacked
methodological detail [23].
In this paper111This paper extends our initial study [24], offering a detailed
analysis with more subjects and source positions, objective metrics, in-lab
experiments (replacing the previous remote setup due to COVID-19), refined
Eigenmike em32 impulse response measurements, and revising evaluated systems.,
we present a comprehensive perceptual evaluation to test the hypothesis that
different synthetic binaural room impulse responses (BRIRs) produced using
various parametric spatial audio rendering techniques—specifically SDM, BSDM,
and HO-SIRR—will differ in spatial and timbral fidelities compared to a
reference BRIR recorded with the KU100 dummy head. Additionally, building on
[5] and [22], we investigate whether a greater number of sensors in an array
leads to improved localization performance. To this end, we employ both an
Eigenmike em32—equipped with 32 omni capsules on a rigid sphere with a 42 mm
radius—and a compact microphone array with six omni microphones for SDM.
Further, we examine the impact of direction of arrival (DOA) estimation
algorithms in SDM, focusing on those based on time difference of arrival
(TDOA) and pseudo intensity vectors (PIVs), and assess the influence of using
a dedicated omnidirectional microphone at the array’s center as a pressure
signal in SDM.
The study is confined to a room that complies with recommendations in ITU-R
1116-3 [25] since our main focus is on the spatial analysis and auralization
of critical listening or sound mixing room. We operate under the assumption
that our findings will be replicable in similar rooms adhering to this widely
recognized standard. Our study utilizes source positions from six
orientations, aligned with industry standards like the ITU-R BS.2051-2 [26]
and Dolby Atmos 7.1.4 configurations. The aim is to ground the discussion in
scientific rigor amidst the rapid expansion of parametric spatial audio
reproduction techniques, delineating current standings, identifying gaps, and
defining a system for optimal spatial data capture and accurate reproduction
in the context of critical listening room.
The paper is structured as follows: Section II introduces the foundational
theories of the direction of arrival estimation, with a focus on time
difference of arrival (TDOA) and Pseudo Intensity Vectors (PIVs) which are
crucial to the SDM and SIRR methods. Section III elaborates on the specifics
of the SDM and SIRR methods. Section IV details our methodology and
experimental design. Section V showcases our perceptual study results, based
on the MUSHRA methodology. Section VI reports on objective metrics. Section
VII synthesizes our key findings. Section VIII concludes the paper.
## II Direction of Arrival Estimation Methods
Estimating the direction of arrival (DOA) is a fundamental aspect of
parametric sound field synthesis methods, as it determines the wave’s origin
and propagation direction. Various methods have been developed for DOA
estimation, including Time Difference of Arrival (TDOA)-based methods like the
Generalized Cross Correlation (GCC), subspace methods such as ESPRIT or MUSIC,
beamforming approaches, and Pseudo Intensity Vectors (PIVs), with
comprehensive tutorials available in [27]. This section will focus on two
approaches particularly relevant to SDM and SIRR: TDOA and PIVs.
### II-A Time Difference of Arrival (TDOA)
The time difference of arrival is frequently used to determine the source’s
direction of arrival. It measures the time lag of a signal across multiple
sensors. Knowing the sensors’ relative positions and these time lags allows
estimation of the source’s origin direction. Common TDOA estimation techniques
include cross-correlation with weightings such as GCC-SCOT and GCC-PHAT [28].
While the Spatial Decomposition Method (SDM) can be combined with any
appropriate direction of arrival (DOA) estimation algorithm, the foundational
paper utilized a least squares method for DOA estimation via TDOA using GCC
with no weighting [5]. As this algorithm was incorporated into a popular
toolbox [6], it is often linked with the original SDM. For detailed equations
and further technical specifics, please refer to [5, 29, 28, 30].
### II-B Pseudo-Intensity-Vectors (PIVs)
Pseudo Intensity Vectors (PIVs) offer a viable alternative to TDOA-based DOA
estimation, building upon historical sound intensity measurement methodologies
[31, 32, 1, 33]. The term ”Pseudo Intensity Vectors” was first introduced by
Jarrett et. al. [34] and refers to the sound intensity vectors computed from
the zeroth and first-order eigenbeams.
The subjective results obtained by McCormack et al.[15] and Ahrens [35]
implied that broadband DOA estimation using PIVs is inferior to other SDM and
SIRR configurations in auralization. However, Zaunschirm et al. showed that
using this method for bandwidth between 200 Hz and just below the microphone
array’s spatial aliasing frequency resulted in SDM rendering nearly identical
to the binaural reference in terms of auditory image width, distance, and
diffuseness [36]. Bassuet stressed minimizing microphone directivity effects
in broadband PIV-based DOA estimation, suggesting a 100 Hz–5 kHz filter range
where the Soundfield first-order Ambisonics microphone exhibits consistent
directivity characteristics [37].
Alternatively, the PIV can be represented in the frequency domain, as done in
SIRR and HO-SIRR [4, 15], the PIV is computed for each frequency bin, averaged
over time windows. This method allows for the calculation of the diffuseness
coefficient $\psi(\omega)$, indicating the balance between the pseudo-
intensity vector magnitude and the energy density (details in [4, 15]).
## III Parametric Sound Reproduction Methods
### III-A Spatial Decomposition Method (SDM)
The Spatial Decomposition Method (SDM), introduced in 2013, utilizes the image
source model for parametric spatial encoding, treating impulse response
samples as broadband image sources [5]. The resulting metadata can be utilized
for loudspeaker reproduction or binaural reproduction using Head-Related
Transfer Functions (HRTFs). SDM’s process involves a microphone array for
direction of arrival (DOA) estimation and an omnidirectional microphone for
pressure signal capture. The method comprises two stages: spatial analysis
using DOA estimation algorithms (TDOA or PIVs) and synthesis, which utilizes
DOA data and pressure signals to create directional output signals using
techniques like Vector Base Amplitude Panning (VBAP) [5, 38, 10], K-Nearest
Neighbour (KNN) mapping [38, 8, 35], or Ambisonics [39]. Post-equalization to
mitigate spectral whitening and optimized loudspeaker grids are also utilized
for enhanced sound reproduction [8, 40].
Binaural Spatial Decomposition Method (BSDM), offers improvements for binaural
reproduction [16]. It includes the rotation of the DOA matrix for various head
orientations and post-processing techniques for direct sound enforcement and
DOA quantization. BSDM also features RTMod+AP equalization for better echo
density and decay, building on the previous post-equalization method designed
for loudspeaker reproduction [8]. Later, updates to the publicly available
toolbox have introduced features such as impulse response denoising and band-
limited spatial analysis [41].
SDM’s performance hinges on the microphone array configuration and window
size. Historically, SDM has employed a variety of microphone arrays, from GRAS
50VI probes to custom arrays with varying spacings [5, 7, 42, 8, 22, 23, 9,
43, 16]. Optimal array design, guided by research, focuses on specific
microphone spacings to minimize DOA errors and perceptual discrepancies [5,
22, 16]. PIV-based DOA estimation allows flexibility in microphone array
choice, but it requires an array capable of producing high-quality first-order
spherical harmonics [16]. In this method, windowing aims at smoothing the DOA
while in TDOA-based estimation, the window size is key to performance,
ensuring a balance between temporal and spatial resolution. Larger windows
enhance estimation robustness but increase the risk of multiple reflections,
conflicting with the single-reflection assumption [5]. Historically, window
sizes were chosen arbitrarily [8, 42, 7, 38]. Recent research by Amengual Garí
et al. on optimal window size for DOA estimation in a simulated setup suggests
that for a 100 mm spaced array, 36 or 64 samples at 48 kHz is most effective
[16].
### III-B Spatial Impulse Response Rendering (SIRR)
Spatial Impulse Response Rendering (SIRR) is the first parametric spatial
encoding method for SRIR, focusing on emulating key perceptual features like
Interaural Time Difference (ITD), Interaural Level Difference (ILD), and
Interaural Coherence (IC) [4]. Distinct from the SDM, SIRR operates in the
time-frequency domain using spherical harmonic input.
The processing in SIRR utilizes the Short Time Fourier Transform (STFT) with a
Hann window, using Pseudo Intensity Vectors (PIVs)—as outlined in Section
II-B—to determine the direction of arrival (DOA) and diffuseness for each
time-frequency bin. In the synthesis stage, the pressure response is divided
into non-diffuse and diffuse parts, guided by the estimated diffuseness
coefficient. The directional part is panned using Vector-Based Amplitude
Panning (VBAP), while the diffuse component is decorrelated and distributed
uniformly among the loudspeakers.
SIRR was extended to Higher-Order Spatial Impulse Response Rendering (HO-SIRR)
[15], which employs higher-order spherical harmonics and beamforming to divide
the sound field into uniform sectors. Each sector undergoes separate analysis,
allowing for a more precise estimation. This is especially valuable in
challenging situations, for instance, when sound events from two distinct
directions reach the array simultaneously. During the synthesis stage, the
directional components from all sectors are panned via VBAP to their
respective channels and then combined. Meanwhile, the diffuse components are
re-encoded into spherical harmonics, decoded to loudspeaker signals, and then
decorrelated.
The binaural variant of HO-SIRR, as detailed by Hold et al. [17], addresses
coloration and HRTF resolution in virtual speaker binauralization. This
approach integrates HRTFs in the synthesis phase of HO-SIRR, rendering
directional components according to arrival directions and diffuse components
by their sector steering directions. Objective analyses have demonstrated a
reduction in coloration compared to the traditional HO-SIRR method.
## IV Experimental Design
### IV-A Evaluation Method
The main goal of this study is to determine which synthetic binaural room
impulse response (BRIR) yields an auralization most perceptually similar to
that produced by a BRIR recorded with the KU100 dummy head. To evaluate this
similarity, we adopted the fidelity attribute, as defined by [44]. Zieliński
described it as the ”trueness of reproduction quality to that of the
original”. The experiment had two dependent variables: (i) spatial fidelity
and (ii) timbral fidelity.
The experiment used the MUSHRA methodology [45], in which participants rated
the similarity of test sounds to a reference on a continuous scale from 1.0
(”Extremely different”) to 5.0 (”Same”), with intermediate values of 2.0
(”Very different”), 3.0 (”Different”), and 4.0 (”Slightly different”). This
scale has been used in analogous subjective studies employing the MUSHRA
methodology [46, 5, 47, 23].
The listening experiment took place in the ITU-R BS.1116-compliant listening
room (6.2m x 5.6m x 3.4m) at the University of Huddersfield’s Applied
Psychoacoustics Laboratory (APL). The HULTI-GEN Version 2 software provided
the test interface [48]. The study was structured around a multifactor design,
focusing on two attributes (ATTR): Spatial Fidelity and Timbral Fidelity. The
evaluation for each attribute was divided into six sessions, each dedicated to
a different source position (POSITION). The average session duration was
approximately 15 minutes. Furthermore, within each session, there were three
specific trials (ITEM), each assessing a distinct type of program material.
During each trial, participants rated 10 test conditions (SYSTEM), denoted as
A through J (Table III). Before the experiment, subjects were provided with a
detailed instruction sheet. The purpose of the document was to familiarise
subjects with the procedure and introduce them to the definitions of spatial
and timbral fidelity and methodology.
### IV-B Measurement of Spatial Room Impulse Responses
Impulse response measurements were conducted in an ITU-R BS.1116-compliant
listening room (6.2m x 5.6m x 3.4m; RT 0.25s; NR 12) at the University of
Huddersfield’s Applied Psychoacoustics Laboratory (APL). The loudspeakers used
for the measurements were Genelec 8040A, offering a free field frequency
response within $\pm$ 2.0 dB across a range from 48 Hz to 20 kHz. The
microphone systems used for the measurements were as follows.
* •
Neumann KU100 dummy head microphone
* •
mhAcoustics Eigenmike em32 (referred to as em32 hereon)
* •
A 6OM1 open microphone array, comprising six omnidirectional Line Audio OM1
microphones. These are arranged in a three-dimensional grid with each
microphone pair spaced 100 mm apart along the X, Y, and Z axes, closely
mimicking the GRAS 50VI intensity probe array as in [5]
Binaural impulse responses acquired using the KU100 were used to create
reference stimuli for the listening tests. Impulse responses with the em32
were used for rendering stimuli for the SDM and HO-SIRR methods. The
particular microphone system was chosen for its high number of capsules (32),
which might enhance the direction of arrival (DOA) estimation [5]. In
contrast, the open mic array was used for SDM and chosen based on studies
suggesting its optimal performance in terms of DOA error and perceptual
quality [16, 22].
For the KU100 and 6OM1, the Merging Horus audio interface served as the AD/DA
converter and microphone preamp. The measurements performed with Eigenmike
em32 involved the use of the Eigenmike Microphone Interface Box (EMIB) as the
recording device and Merging Horus as the playback device. To counter
potential impulse response distortions due to clock mismatch between devices
[49], word clock was used for synchronization, with the Merging Horus device
set as master.
The Exponential Sine Sweep (ESS) was used as an excitation signal as described
in [49], with the following parameters: a sample rate of 48 kHz, frequency
range of 20 Hz to 20 kHz, sweep length of 20 seconds, and fade in/out of 10
ms.
The acoustic centers of the microphone systems were positioned at a height of
127.5 cm from the floor, aligning with the heights of the acoustic axes of the
zero elevation loudspeakers, which form the zero elevation plane of the
system. Although our measurements encompassed a complete Dolby 7.1.4 setup,
conforming to the 4+7+0 loudspeaker layout as per [26], the study primarily
focused on a subset of these configurations, as detailed in Table I. For
azimuth angle measurements, defined as the angle relative to the zero azimuth,
each loudspeaker in the zero elevation level was placed at a distance of 2.00
m ($\pm 0.02\ \text{m}$) from the center of the microphone array. For the
elevation angle measurements, the loudspeakers were placed at a distance of
1.92 m ($\pm 0.1\ \text{m}$) from the center of the microphone array.
TABLE I: Tested Loudspeaker Positions in the ITU-R 4+7+0 Layout Position | Azimuth | Elevation
---|---|---
Front center | $0\degree$ | $0\degree$
Front left | $30\degree$ | $0\degree$
Side left | $90\degree$ | $0\degree$
Back left | $135\degree$ | $0\degree$
Upper front left | $45\degree$ | $45\degree$
Upper back left | $135\degree$ | $45\degree$
### IV-C Test Conditions and Variables
In this study, SDM conditions were generated using the SDM Toolbox222SDM
Toolbox (version 1.3001, Updated 22 Apr 2018). [6], while PIV-based DOA
estimation algorithm was adapted from Zaunschirm et al.[39]. Given BSDM’s
growing relevance [50, 12, 11, 51, 52], we included two conditions utilizing
the available toolbox333BinauralSDM (version 0.5, commit 965da5c). [41].
HO-SIRR condition was rendered via its MATLAB implementation444HO-SIRR (commit
085f20d). [53]. Both SDM utilizing PIV-based DOA analysis and HO-SIRR
conditions, employed spherical harmonics from the Eigenmike em32, with SDM
additionally using Line Audio OM1 as a pressure signal in one of the
conditions.
Settings for SDM, HO-SIRR, and BSDM adhered to the recommended configurations
in their respective toolboxes [41, 6, 53]. The exception was the window size,
which was standardized at 64 samples across all frameworks, following Amengual
Gari et al.’s recommendations [16]. Additionally, post-equalization [8] was
disabled in the SDM Toolbox.
Both BSDM and PIV-based methods applied band-limited DOA estimation (200
Hz-2400 Hz), aligned with the spatial aliasing frequency of the 6OM1 array.
BSDM’s mixing time was set at 38 ms, based on ISM simulations of our study
room.
We ensured that the anchor demonstrated impairments in both spatial and
timbral fidelity. Specifically, for azimuthal sources at zero elevation
positions, the anchor employed a KU100 BRIR that was offset by an additional
60 to 90 degrees from the source position being evaluated. For sources at
elevated positions, given the limited number of measured positions, a BRIR
corresponding to a diametrically opposite position was chosen, 180 degrees
from the source position being evaluated. To introduce a timbral fidelity
impairment, we applied a 3.5kHz low-pass filter [45].
### IV-D Synthesis of Binaural Room Impulse Responses
To facilitate a subjective experiment, we employed binaural rendering
techniques for SDM, BSDM, and HO-SIRR to incorporate the KU100 BRIR as a
ground truth reference. Consequently, we employed a dataset of 2702 KU100
head-related impulse responses (HRIRs) sampled on the Lebedev grid [54]. The
synthesis process entailed rendering virtual loudspeaker signals at points on
the Lebedev grid using the evaluated systems. For SDM, this was achieved using
K-nearest neighbor allocation, while HO-SIRR used VBAP. The final stage
involved convolving these virtual loudspeaker signals with corresponding HRTFs
from the SOFA file [55] and summing them to produce the final BRIRs.
### IV-E Programme Material
Accurate binaural sound-field reproduction requires anechoic audio, free from
original recording environment effects. This is achieved by convolving the
anechoic material with binaural room impulse responses (BRIR), simulating room
acoustics. The anechoic samples employed in our study are as follows: ”Bongo”
(Track 26, 12.780s-–20.330s) and ”Danish Speech” (Track 9, 0.590s-–9.240s)
from Bang & Olufsen’s ”Music for Archimedes” [56, 57], and Handel/Harty’s
”No.6 Water Music Suite” (Track 9, 0.480s-–8.380s) from Denon’s anechoic
orchestral collection [58].
### IV-F Reproduction Configuration and Calibration
The study employed a reproduction system that included AKG K702 headphones and
a Merging Technologies Horus audio interface. These headphones were fitted
with an inverse filter originally developed for Lee et al.’s research [59],
designed to correct spectral coloration and replicate the response in a KU100
dummy head’s ear. The filter’s creation involved placing AKG K702 headphones
on a KU100 dummy head and conducting five measurements for each ear in both
left/right and right/left positions, then repeating the process with another
pair of headphones, resulting in 40 impulse responses in total. The filter
design followed the high-pass-regularized least-mean-square (LMS) inversion
approach [60], identified as a perceptually sound inversion algorithm [61].
Test stimuli were uniformly normalized to -26 LUFS and the headphone amp gain
was set uniformly throughout the experiment. LAeq loudness for KU100 BRIR at
+30° azimuth, measured via miniDSP EARS, averaged 70.7dB (Bongo), 77.5dB
(Speech), and 73.5dB (Orchestra).
### IV-G Test Subjects
In total, 14 participants took part in the study. These participants ranged in
age from 20 to 47 years. They were a mixture of staff members, postgraduate
students, and undergraduate researchers affiliated with the Applied
Psychoacoustics Laboratory (APL) at the University of Huddersfield. All
participants claimed to have normal hearing abilities and had previously taken
part in listening experiments. Prior to the formal listening test, the
participants were asked to complete a short 10-minute familiarization phase
for each attribute (ATTR).
### IV-H Objective Metrics
The present study considered four objective metrics: Interaural Level
Difference ($\mathrm{ILD}$), Interaural Time Difference ($\mathrm{ITD}$),
reverberation time ($\mathrm{T30_{mid}}$), early Interaural Cross-Correlation
Coefficient ($\mathrm{IACC_{E3}}$) and late IACC ($\mathrm{IACC_{L3}}$). To
evaluate accuracy, the Mean Absolute Error (MAE) quantifies the error’s
magnitude, while the Mean Signed Difference (MSD) detects systematic biases,
providing insights into the extent and direction of the system’s performance
deviations.
Each BRIR was energy-normalized to allow fair comparison. This involved
identifying the onset of direct sound, indicated by the earliest sound arrival
in either channel and calculating the Root Mean Square (RMS) value over a 2.5
ms segment starting from this point [62]. This RMS value, representing the
direct sound’s energy, was then used to normalize the entire BRIR, thus
minimizing gain differences and establishing a consistent baseline for
objective analysis.
#### IV-H1 Interaural Level Difference (ILD)
The ILD is a major cue for horizontal sound localization, accentuated by head
shadowing when sources are off-center, with the Just Noticeable Difference
(JND) ranging from 1 to 2 dB [63]. Accounting for the precedence effect, ILD
was derived from the first 2.5 ms of BRIRs over 39 equivalent rectangular
bandwidth (ERB) bands and averaged as follows:
$\mathrm{ILD}_{\text{avg}}=\frac{1}{N}\sum_{f=1}^{N}20\log_{10}\left(\frac{\hat{y}_{L}(f)}{\hat{y}_{R}(f)}\right)$
(1)
where $\mathrm{ILD}_{\text{avg}}$ is the average ILD over $N$ ERB bands, with
$\hat{y}_{L}(f)$ and $\hat{y}_{R}(f)$ as the RMS values for left and right
channels at each frequency band $f$.
#### IV-H2 Interaural Time Difference (ITD)
The ITD is the second of two auditory cues, critical for lateral sound
localization, with JND of about 40 $\mu s$ for frontal sources and
approximately 100 $\mu s$ for lateral sources [64]. The ITD can extend up to
700 $\mu s$ for azimuth angles up to 90 degrees. ITD estimation is based on
the peak time lag of the interaural cross-correlation function:
$\text{ITD}=\arg\max_{-1\ \text{ms}<\tau<1\ \text{ms}}(|\text{IACF}(\tau)|)$
(2)
#### IV-H3 Reverberberation Time ($\text{RT}_{60}$)
The $\text{RT}_{60}$, was calculated in octave bands in accordance with the
standards outlined in [65]. The value presented was derived based on a 30 dB
evaluation range and subsequently averaged across the 500 Hz and 1 kHz octave
bands, resulting in $\text{T30}_{mid}$. The JND for reverberation time is
established at 5% of the $\text{RT}_{60}$ [65]. With a 0.25-second
reverberation time in the auralized room, the JND in the present scenario is
roughly 12.5 ms.
#### IV-H4 Interaural Cross Correlation Coefficient (IACC)
The IACC is a commonly used metric of spatial impression (SI) in concert
halls, with a JND of 0.075 [65]. The Apparent Source Width (ASW) and Listener
Envelopment (LEV) are best estimated by averaging the IACC across 500 Hz, 1
kHz, and 2 kHz octave bands, resulting in $\text{IACC}_{E3}$ and
$\text{IACC}_{L3}$ metrics [66]. In the present study, we calculated
$\text{1-IACC}_{E3}$ and $\text{1-IACC}_{L3}$, which are positively correlated
with ASW and LEV. The IACC is computed as follows:
$\text{IACC}=\max_{-1\ \text{ms}<\tau<1\ \text{ms}}|\text{IACF}(\tau)|$ (3)
## V Subjective Evaluation Results
TABLE II: Results of the aggregated Friedman test for different factors and attributes ATTR | Factor | Test Statistic ($\chi^{2}$) | DF | p-value | sig.
---|---|---|---|---|---
Spatial | SYSTEM | 101 | 9 | $<$0.0001 | ****
Spatial | ITEM | 2.43 | 2 | 0.297 | ns
Spatial | POSITION | 21.4 | 5 | 0.0007 | ***
Timbral | SYSTEM | 106 | 9 | $<$0.0001 | ****
Timbral | ITEM | 0.531 | 2 | 0.767 | ns
Timbral | POSITION | 11.5 | 5 | 0.0417 | *
Figure 1: Subjective evaluation results showing median spatial and timbral
fidelity scores on similarity scale (1-5), for various systems (SYSTEM).
Scores are compiled from multiple source positions (POSITION) and program
materials (ITEM), with 95% non-parametric confidence intervals.
Following ITU guidelines [45], a post-screening process was implemented to
identify and potentially exclude assessors who had rated the hidden reference
below 4.5 (90% of the 5.0 maximum score) for over 15% of test items. This was
done to ensure that only reliable subjects were considered. Ultimately, none
of the assessors met these exclusion criteria.
To verify the normality of the ratings given to each system for each
combination of factors, the data were grouped by SYSTEM, ATTR, ITEM, and
POSITION. Consistently, the ratings assigned to systems ’A’ (Reference) and
’J’ (Anchor) were found not to follow a normal distribution. Excluding these,
non-normal distributions were found in 6.25% of cases for the spatial fidelity
and approximately 6.94% for the timbral fidelity attribute. Considering these
results, we opted for a non-parametric statistical approach in subsequent
analyses.
TABLE III: Systems under test Label | Test system (SYSTEM)
---|---
A | Anchor
B | BSDM 6OM1 Omni
C | BSDM em32 Omni
D | SDM em32
E | SDM em32 Omni
F | SDM 6OM1 Omni
G | HO-SIRR diffuse
H | SDM PIV
I | SDM PIV Omni
J | KU100 (Reference)
Figure 2: Subjective evaluation results showing median for spatial and timbral
fidelity scores on similarity scale (1-5), across different systems (SYSTEM)
and source positions (POSITION), aggregated over all program materials (ITEM).
The graph includes 95% non-parametric confidence intervals. Figure 3:
Subjective evaluation results showing median for spatial and timbral fidelity
scores on similarity scale (1-5), across different systems (SYSTEM) and
program materials (ITEM), aggregated over all source positions (POSITION). The
graph includes 95% non-parametric confidence intervals.
Friedman’s tests were conducted to examine the main effects of various
factors—SYSTEM, ITEM, POSITION—on both spatial and timbral fidelity
attributes. The detailed statistical outcomes of these tests, including test
statistics and significance levels, are summarized in Table II.
These results underscore a significant influence of the SYSTEM on both spatial
and timbral attributes, indicating variations in the capability of the
rendering methods to replicate spatial and timbral fidelity. Interestingly,
the specific program material (ITEM) did not have a significant impact on
participant ratings, while the source position (POSITION) demonstrated a
noticeable effect, especially in spatial fidelity.
First, we present the aggregated results across various ITEMs and POSITIONs in
Figure 1, giving an overview of the overall system performance. Given the
significant effect of POSITION and the non-significant effect of ITEM from our
Friedman test findings, we further delved into Position-Based Performance
(Figure 2) and potential subtle influences of Program Material (Figure 3).
Disparities between rendering systems were analyzed using the Wilcoxon signed-
rank test with Holm-Bonferroni correction [67]. Additionally, we calculated
the effect size ($r$) using the method described in [68], which involves
converting the p-value to a z-score and then to $r$, based on the total number
of observations (N).
Figure 1 shows the ratings for spatial and timbral Fidelity of various
systems. Low anchor (A), consistently scored at the lower end of the scale
with a median of 1.0, indicating ’Extremely Different’. In contrast, KU100 (J)
achieved a median score of 5.0 (’Same’) implying that subjects had no problem
in identifying the hidden reference. Systems BSDM 6OM1 Omni (B) to SDM PIV
Omni (I) varied, with spatial fidelity ratings mainly between 3 and 4 and
timbral fidelity also clustering between 3 and 4.2.
Looking more closely at spatial fidelity, SDM em32 Omni (E), HO-SIRR (G), and
SDM PIV Omni (I) displayed comparable performance (medians between 3.9 and 4).
However, SDM PIV (H) with a median of 3.8 was slightly inferior to both SDM
em32 Omni (E) and HO-SIRR (G). SDM 6OM1 Omni (F) trailed behind these three
systems. Additionally, BSDM 6OM1 Omni (B), BSDM em32 Omni (C), and SDM 6OM1
Omni (F) were grouped closely together (medians between 3.7 to 3.8), though
BSDM em32 Omni (C) was outperformed by both HO-SIRR (G) and SDM PIV (H).
Notably, SDM em32 (D) performance was significantly lower than the rest
(median 3.2).
In terms of timbral fidelity, SDM em32 Omni (E) and SDM PIV Omni (I) were
almost indistinguishable (medians between 4.1 and 4.2), with SDM em32 Omni (E)
outperforming HO-SIRR (G) and SDM PIV (H). SDM 6OM1 Omni (F), HO-SIRR (G), and
SDM PIV (H) were on par with each other (medians between 3.9 and 4), whereas
BSDM 6OM1 Omni (B) and BSDM em32 Omni (C), while similar to each other
(medians 3.7), lagged behind the preceding group, and even more so when
compared to SDM em32 Omni (E) and SDM PIV Omni (I). Again, SDM em32 (D) scored
substantially lower than BSDM 6OM1 Omni (B) and BSDM em32 Omni (C).
### V-A Dedicated center Omnidirectional Microphone
The impact of using a dedicated center omnidirectional microphone can be
analyzed by comparing SDM em32 Omni (E) and SDM em32 (D)—utilizing TDOA-based
DOA, as well as SDM PIV Omni (I) and SDM PIV (H)—utilizing PIV-based DOA.
These conditions used SRIR from em32 (and obtained SPH), with and without
center omnidirectional microphone.
The dedicated omnidirectional microphone (SDM em32 Omni, E) notably enhanced
the em32 array’s performance in the context of SDM’s TDOA-based DOA variant
(SDM em32, D), with significant improvements in overall results for both
spatial and timbral fidelity, exhibiting a large effect size ($p<0.001$,
$r>0.7$). The same trend was also observed for individual positions and
stimuli, however, for certain positions with medium effect size, for instance,
+45°, +45° (Spatial) and +135°, +45° (Timbral) with $p<0.005$ and $r=0.4$.
Improvements were also noticeable with a dedicated center microphone in SDM’s
PIV-based DOA estimation variant, but they were less pronounced. SDM PIV (H)
used a zeroth-order eigenbeam, and SDM PIV Omni (I), which utilized a
dedicated omni microphone, showed significant enhancement in spatial and
timbral fidelity, albeit with a small effect size ($p<0.009$, $r<0.3$).
Interestingly, for the +135° source position, the improvement was significant
with a large effect size ($p<0.001$, $r=0.61$). Across most source positions,
the improvement for both attributes was observed with a small effect size
($p<0.04$, $r<0.26$), except for Bongo and Speech in spatial fidelity.
### V-B Microphone Array and DOA Estimation Method
This research evaluated TDOA and PIV-based DOA estimation methods in SDM for
spatial data capture and auralization. It focused on TDOA-based DOA using em32
and 6OM1 arrays, represented by SDM em32 Omni (E) and SDM 6OM1 Omni (F), and
PIV-based DOA with first-order SPH, denoted as SDM PIV Omni (I), all utilizing
a central pressure signal from the array.
As depicted in Figure 1, our results indicate that SDM em32 Omni (E), which
relies on TDOA-based DOA estimation, exhibited performance on par with SDM PIV
Omni (I)—utilizing a PIV-based DOA estimation—when the identical
omnidirectional pressure signals were used. The comparison revealed no
substantial differences in either spatial or timbral fidelity. Conversely, SDM
6OM1 Omni (F)—also based on TDOA but incorporating an array of only six
microphones, was found to perform significantly worse than SDM em32 Omni (E)
with a small effect size ($p<0.001$, $r<0.26$) and SDM PIV Omni (I) with
medium effect size ($p<0.001$, $0.3<r<0.36$) for both spatial and timbral
fidelity.
Upon examining the specific source positions (Figure 2), we observed a similar
pattern. SDM 6OM1 Omni (F) generally underperformed compared to SDM em32 Omni
(E) and SDM PIV Omni (I). This was particularly pronounced for source
positions at +30° and +90°, where SDM 6OM1 Omni (F) demonstrated significant
deviation from SDM em32 Omni (E) ($p<0.007$, medium and large effect size:
$0.41<r<0.61$) and SDM PIV Omni (I) ($p<0.001$, large effect: $0.66<r<0.69$)
concerning spatial fidelity. On the other hand, it was observed that SDM PIV
Omni (I) outperformed SDM em32 Omni (E) at +30° and +45°, +45° source
positions with medium effect size ($p<0.013$, $0.38<r<0.46$) in terms of
spatial fidelity. Regarding timbral fidelity, SDM em32 Omni (E) and SDM PIV
Omni (I) notably outperformed SDM 6OM1 Omni (F) at the +135° position
($p<0.001$, large effect: $0.5<r<0.54$), with SDM PIV Omni (I) also showing
superior performance at the +30° and +90° source positions with medium effect
size ($p=0.011$, r = 0.39).
An analysis of stimulus-dependent performance revealed that there were no
significant differences between SDM em32 Omni (E) and SDM PIV Omni (I), except
in two specific scenarios. SDM PIV Omni (I) exhibited superior spatial
fidelity over SDM em32 Omni (E) for Orchestra stimuli, with a small effect
size ($p<0.045$, $r=0.22$). For Speech stimuli, SDM PIV Omni (I) also showed
greater timbral fidelity than SDM em32 Omni (E), again with a small effect
size ($p<0.028$, $r=0.24$). In contrast, SDM 6OM1 Omni (F) underperformed in
both spatial and timbral fidelity for Speech compared to SDM em32 Omni (E) and
SDM PIV Omni (I), with a medium effect size ($p<0.005$, $0.3<r<0.36$).
However, the exception was the comparison of timbral fidelity between SDM PIV
Omni (I) and SDM 6OM1 Omni (F), where SDM PIV Omni (I) demonstrated a large
effect size ($p<0.001$, $r=0.57$). Additionally, for Orchestra stimuli, SDM
PIV Omni (I) outperformed SDM 6OM1 Omni (F) in both spatial and timbral
fidelity with medium ($p<0.001$, $r=0.36$) and small ($p<0.022$, $r=0.24$)
effect sizes, respectively.
### V-C Spatial Encoding Systems
This section analyzes the impact of different rendering systems on spatial
encoding efficacy, comparing SDM’s performance in SDM em32 (D), SDM em32 Omni
(E), SDM 6OM1 Omni (F), SDM PIV (H), and SDM PIV Omni (I), with HO-SIRR (G),
and BSDM’s optimizations in BSDM 6OM1 Omni (B) and BSDM em32 Omni (C).
It is noteworthy that no enhancement in spatial fidelity was observed between
SDM 6OM1 Omni (F) and BSDM 6OM1 Omni (B). Intriguingly, SDM 6OM1 Omni (F)
generally outperformed BSDM 6OM1 Omni (B) in terms of timbral fidelity, albeit
with a modest effect size ($p=0.001$, $r=0.2$), considering the overall
results. A similar trend was evident for BSDM em32 Omni (C) and SDM em32 Omni
(E), indicating that the optimizations employed by BSDM frequently led to
degradation of these attributes, exhibiting small ($p<0.001$, $r=0.25$) and
medium ($p<0.001$, $r=0.49$) effect sizes for spatial and timbral fidelity,
respectively.
Looking at how performance varies with position, BSDM 6OM1 Omni (B) generally
matched the performance of SDM 6OM1 Omni (F) in terms of spatial and timbral
fidelity, with the exception of the +30° position. At this position, SDM 6OM1
Omni (F) outperformed BSDM 6OM1 Omni (B) with a large effect size ($p<0.001$,
$r=0.54$) in terms of timbral fidelity. Similarly, at +30° BSDM 6OM1 Omni (B)
underperformed compared to systems SDM em32 Omni (E) to SDM PIV Omni (I) with
a large effect size ($p<0.001$, $r>0.52$) in terms of timbral fidelity. In
terms of spatial fidelity at +30°, BSDM 6OM1 Omni (B) did not show a
significant difference when compared to SDM 6OM1 Omni (F), yet HO-SIRR (G),
SDM PIV (H), and SDM PIV Omni (I) outperformed BSDM 6OM1 Omni (B) with a large
effect at +90° ($p<0.001$, $r>0.5$). Additionally, at the +135°, +45°
position, BSDM 6OM1 Omni (B) underperformed with a medium effect size
($p<0.016$, $r=0.37$) in terms of both spatial and timbral fidelity compared
to SDM 6OM1 Omni (F).
(a) Spatial Fidelity
(b) Timbral Fidelity
Figure 4: Principal Component Analysis (PCA) of spatial and timbral fidelity
scores for evaluated systems (SYSTEM), considering the median spatial and
timbral fidelity scores across different program materials (ITEM) and source
positions (SYSTEM). Systems are clustered in a two-dimensional space by the
first two principal components, highlighting the similarities in their
fidelity scores.
Similar observations were made for BSDM em32 Omni (C) and SDM em32 Omni (E).
Both systems performed similarly for most positions, except at the +90° and
+135°, +45° source positions in terms of both spatial and timbral fidelity. At
these positions, SDM em32 Omni (E) outperformed BSDM em32 Omni (C) with large
($p<0.001$, $0.51<r<0.65$) and medium ($p<0.002$, $0.44<r<0.48$) effect sizes
in terms of spatial fidelity. However, in terms of timbral fidelity, the
differences were less pronounced, with the main differences observed at the
+135°, +45° source position. Moreover, BSDM em32 Omni (C) underperformed for
spatial fidelity when compared with systems SDM em32 Omni (E) to SDM PIV Omni
(I) at +135°, +45° and +90° with large ($p<0.001$, $r>0.5$) and medium effect
sizes ($p<0.02$, $0.35<r<0.45$).
Examining the results for individual stimuli reveals an interesting pattern
regarding spatial fidelity. BSDM 6OM1 Omni (B) and BSDM em32 Omni (C)
performed comparably to systems utilizing SDM (E, F, H, and I) and HO-SIRR (G)
for Orchestra and Speech stimuli in terms of spatial fidelity. However, their
performance significantly deteriorated for the Bongo stimulus, with a large
effect size ($p<0.001$, $r>0.5$). A similar trend was observed for timbral
fidelity, though BSDM 6OM1 Omni (B) and BSDM em32 Omni (C) occasionally
underperformed compared to systems SDM em32 Omni (E), SDM 6OM1 Omni (F), SDM
PIV (H), and SDM PIV Omni (I) for Orchestra and Speech stimuli with small to
large effect sizes.
While we previously concluded that systems utilizing SDM–SDM em32 Omni (E) and
SDM PIV Omni (I) tend to perform better than SDM 6OM1 Omni (F), this
observation suggests the impact of the microphone array used. When examining
the performance differences between SDM and HO-SIRR (G) in terms of spatial
fidelity, we observe no significant difference between HO-SIRR (G), SDM em32
Omni (E), SDM PIV (H), and SDM PIV Omni (I). However, HO-SIRR (G) shows
significantly better performance compared to SDM 6OM1 Omni (F) with a small
effect size ($p<0.009$, $r=0.16$). On the other hand, it performed worse than
SDM em32 Omni (E) and SDM PIV Omni (I) in terms of timbral fidelity, but with
a small effect size ($p<0.018$, $0.15<r<0.27$). Interestingly, HO-SIRR (G) and
H (SDM PIV) did not demonstrate a significant difference in terms of spatial
and timbral fidelity, which may imply that the use of the zeroth-order
eigenbeam as the pressure signal in SDM PIV (H) and HO-SIRR (G) is a
contributing factor.
The results for individual source positions implied that HO-SIRR (G) performed
similarly to SDM em32 Omni (E) and SDM PIV Omni (I) in most of the evaluated
source positions, with the occasional exceptions when it underperformed, for
instance at +135° source position with large effect size ($p=0.001$, $r>0.5$).
Conversely, at +45°, +45° source position, HO-SIRR (G) outperformed SDM em32
Omni (E) with a medium effect size for spatial fidelity ($p=0.024$, $r>0.34$).
Interestingly, the performance of HO-SIRR (G) mirrored that of SDM PIV (H) in
spatial and timbral fidelity for individual stimuli, with only occasional
differences. For instance, regarding timbral fidelity, HO-SIRR (G) was
outperformed by SDM em32 Omni (E) and SDM PIV Omni (I) for the Orchestra
stimulus with a medium effect size ($p<0.004$, $r>0.31$).
### V-D Principal Component Analysis
In the spatial analysis and auralization of a critical listening room, SDM and
HO-SIRR systems exhibited similar performance, particularly in terms of
spatial fidelity. Systems such as SDM em32 Omni (E), HO-SIRR (G), SDM PIV Omni
(I), and SDM PIV (H) demonstrated comparable results, though with some
variations. With respect to timbral fidelity, SDM em32 Omni (E) and SDM PIV
Omni (I), which utilize TDOA-based analysis with Eigenmike em32 and PIV-based
analysis with a dedicated pressure signal, closely aligned with the reference.
This alignment is supported by Principal Component Analysis (PCA), as shown in
Figure 4. The PCA clusters these systems together based on their median
spatial and timbral fidelity scores across all positions and stimuli. The
analysis revealed that the first two principal components accounted for about
77% of the variance in spatial fidelity and 80% in timbral fidelity.
### V-E Correlation between Spatial and Timbral fidelity
Figure 5: Correlation between median scores of spatial and timbral fidelity
across all source positions and stimuli.
To explore the link between spatial and timbral fidelity, we calculated
Spearman’s correlation based on median scores across all positions and
stimuli, as depicted in Figure 5.
The weakest correlations between spatial and timbral fidelity were observed in
SDM em32 Omni (E, $\rho=0.41$), SDM em32 (D), and SDM 6OM1 Omni (F,
$\rho=0.45$), followed by SDM PIV Omni (I, $\rho=0.52$). A significant
increase in correlation was noted for HO-SIRR (G, $\rho=0.59$) and SDM PIV (H,
$\rho=0.6$), with BSDM em32 Omni (C, $\rho=0.72$) and BSDM 6OM1 Omni (B,
$\rho=0.77$) exhibiting the strongest positive correlations.
The strong correlation in BSDM-rendered conditions (B and C) indicates a
uniform impact of artifacts on both spatial and timbral fidelity, possibly due
to reverb equalization effects. Conversely, the pronounced correlation between
spatial and timbral fidelity in HO-SIRR (G) and SDM PIV (H) implies an effect
resulting from the use of a zeroth-order spherical harmonic as a pressure
signal within these systems.
## VI Objective Evaluation Results
(a) ILD computed from BRIRs across 39 ERB bands, averaged above and below 1.5
kHz as detailed in [69].
(b) Interaural cross-correlation coefficients: Early ($\text{1-IACC}_{E3}$)
and Late ($\text{1-IACC}_{L3}$).
(c) ITD calculated using maximum interaural cross-correlation (MaxIACCr)
method.
(d) Mid-frequency reverberation time $\text{T30}_{mid}$ as defined in [65].
Figure 6: Objective metrics presented using mean absolute error (MAE) and mean
signed difference (MSD) using the reference KU100 (J) as the ground truth.
### VI-A Interaural Level Difference (ILD)
Figure 6a illustrates the mean absolute error (MAE) and mean signed difference
(MSD) for ILD across systems (B through I), categorized by the ERB bands above
and below 1500 Hz used for averaging. At higher frequencies, SDM em32 (D), SDM
em32 Omni (E), and SDM 6OM1 Omni (F) showed notable deviation with high MAEs
and negative MSDs, underscoring a tendency to underestimate ILDs beyond the
established JND [63]. HO-SIRR (G) also tended to underestimate ILD but to a
lesser extent. In contrast, lower frequencies exhibited reduced MAEs for most
systems, except HO-SIRR (G), and MSDs near zero, indicating minimal bias
except for the consistent underestimation by SDM em32 (D), SDM em32 Omni (E),
and SDM 6OM1 Omni (F). Overall, all systems faced more difficulty with higher
frequency ILDs, implying challenges in capturing spatial cues accurately.
However, performance improved at frequencies below 1500 Hz, with SDM em32 (D)
being the exception.
### VI-B Interaural Time Difference (ITD)
Figure 6c demonstrates ITD accuracy with MAE and MSD values, indicating that
BSDM 6OM1 Omni (B) and BSDM em32 Omni (C) consistently underestimate ITD,
whereas SDM em32 (D), SDM em32 Omni (E), and SDM 6OM1 Omni (F) overestimate,
particularly SDM 6OM1 Omni (F) with the highest MAE of 34.7 $\mu s$. HO-SIRR
(G) has minor overestimations, while SDM PIV (H) and SDM PIV Omni (I) maintain
the highest accuracy with slight underestimation tendencies. Given the JND for
ITD ranges from 40 $\mu s$ frontally to 100 $\mu s$ laterally [64], most
systems’ deviations fall within perceptually insignificant limits. Notably,
there is an inconsistency in MSD and MAE magnitudes and a suggested ILD-ITD
trade-off in systems SDM em32 (D) through HO-SIRR (G).
### VI-C Reverberberation Time
Figure 6d shows that all systems tend to overestimate reverberation time, as
indicated by positive MSD values. Most systems’ errors exceed the JND of 12.5
ms, which may affect perception, except for BSDM 6OM1 Omni (B) and BSDM em32
Omni (C) which maintain minimal perceptible errors with MAEs of 12.4 and 14.7
ms. These results align with Amengual et al.’s findings, confirming that RTMod
maintains synthesized BRIR’s reverberation time within the JND [16]. On the
other hand, HO-SIRR (G), with an MAE of 108 ms, significantly overshoots the
JND, hinting at a noticeable impact on spatial fidelity.
### VI-D Interaural Cross Correlation Coefficient (IACC)
Figure 6b presents MAE and MSD for systems using $\text{1-IACC}_{E3}$ and
$\text{1-IACC}_{L3}$. Evaluated systems vary in accuracy, with
$\text{1-IACC}_{E3}$ MAE ranging from 0.0427 to 0.0836 and
$\text{1-IACC}_{L3}$ MAE from 0.0637 to 0.134. Errors for $\text{1-IACC}_{E3}$
mostly fall within the JND of 0.075, implying that the errors are unlikely to
be perceptible. Low MSDs suggest non-systematic errors, except for SDM em32
(D) and SDM PIV Omni (I), which show a bias in $\text{1-IACC}_{E3}$ estimates,
potentially affecting ASW.
In contrast, MAEs for $\text{1-IACC}_{L3}$ surpass the JND of 0.075 for all
but HO-SIRR (G) and SDM PIV Omni (I), with several systems showing negative
MSDs, hinting at consistent underestimation and possible LEV impact. HO-SIRR
(G) and SDM PIV Omni (I), however, remain within JND bounds for both MAE and
MSD, suggesting minimal LEV alteration.
## VII Discussion
McCormack et al. [15] found that the post-equalized Spatial Decomposition
Method (SDM) using pseudo-intensity vectors (PIVs) underperforms compared to
spatial impulse response rendering (SIRR) in simulated environments. Improved
performance of SDM is observed when PIV-based direction of arrival (DOA)
estimation incorporates band-limitation [36, 39]. More recently, McCormack et
al. [19] found that SDM, utilizing time difference of arrival (TDOA) based DOA
estimation, shows comparable performance to Higher-Order SIRR (HO-SIRR) for
stationary stimuli in a small simulated room, but exhibits reduced performance
for transient stimuli.
Our study finds similar spatial fidelity for HO-SIRR, and SDM employing TDOA
and PIVs for DOA estimation. Although objective measurements showed HO-SIRR to
overestimate reverberation time by more than 100 ms, potentially impacting
spatial fidelity for non-stationary stimuli, subjective evaluations did not
reflect this. The present study stands out by using real-life measurements,
contrasting with previous studies’ simulated conditions. This offers a more
realistic context, directly relevant to practical room acoustics applications.
The performance of SDM with optimizations for binaural rendering (BSDM)
depends on the stimulus type. The Bongo stimulus adversely impacted spatial
and timbral fidelity, suggesting that BSDM’s equalization quality might be
compromised by suboptimal reverberation time estimation in low-reverb
environments [50]. While objective measurements confirm RTmod’s efficacy,
showing the reverberation time’s mean absolute error (MAE) within the just
noticeable difference (JND) range, the observed poor performance may be linked
to artifacts related to RTmod, appearing in transient sounds [16].
BSDM significantly underestimated $\text{1-IACC}_{L3}$, a key metric for
listener envelopment [66]. This underestimation surpasses the established JND
[65], suggesting a potential reduction in listener envelopment relative to the
reference.
Amengual et al. observed that BSDM’s perceived plausibility was rated
similarly to that of real loudspeakers [16]. However, in an ITU-R
BS.1116-3-compliant listening room, BSDM shows lower consistency and
robustness compared to the standard, unoptimized SDM. This finding, alongside
insights from our pilot study—which focused exclusively on DOA enforcement for
direct sound and band-limited DOA estimation—suggests that more fundamental
optimizations from BSDM could lead to a more robust SDM framework, especially
when compared to the full BSDM including RTmod+AP optimization.
Both TDOA and PIVs can be equally effective DOA estimators in SDM when a
sufficient number of microphones is utilized for TDOA estimation. In band-
limited PIV-based DOA estimation, the condition utilizing an omnidirectional
pressure signal demonstrates superior performance compared to the TDOA-based
algorithm with a six-microphone array. McCormack et al. [15] and Ahrens [22]
conducted studies that did not incorporate band limitation in the DOA
estimation with the PIV method. In their findings, Ahrens demonstrated that
the SDM variant utilizing the PIV method, without band limitation, tended to
cause a larger perceived difference in the presented auralizations when
compared to most other array geometries, with the dummy head reference serving
as the baseline for comparison. Similarly, McCormack et al. observed that the
SDM utilizing PIV-based DOA estimation without band limitation, significantly
underperformed in comparison to any variant of the SIRR. Zaunschirm et al.
[36] found that applying band limitation to PIV-based DOA estimation
significantly enhances SDM performance. Their effective frequency range, from
200 Hz to just below the spatial aliasing frequency of the microphone array,
led to SDM results closely mirroring the binaural reference in terms of image
width, distance, and diffuseness. These findings align with our study,
reinforcing the critical role of band limitation in PIV-based DOA estimation
for optimal spatial analysis and synthesis.
We did not expect SDM em32 Omni (E) to match in performance with SDM PIV Omni
(I) given the complexity of the time difference of arrival estimation using
spherical microphone arrays and the rigid sphere design of em32 [27]. Although
the larger number of microphones in em32 suggests a potential for improved
TDOA estimation, objective measurements indicate higher errors in ILD, ITD,
and reverberation time, aligning em32 more with SDM 6OM1 Omni (E) rather than
SDM PIV Omni (I). Previous applications of SDM employed small microphone
separations with the smallest inter-mic distance of 17.7 mm [8]. The 26.5 mm
inter-mic distance in em32 may suit well for TDOA estimation. Optimizing the
TDOA model for microphones on rigid spheres [70] could improve estimation
accuracy in em32.
SDM em32 Omni (E) with 32 sensors outperforms the SDM 6OM1 Omni (F) with six
sensors. In contrast, Ahrens’ study using different array configurations,
including a 12-microphone array, found that a six-microphone array, especially
with radii of 50 mm and 100 mm, produced the smallest perceptual differences
to the dummy head reference [22]. The superior performance of the em32 in our
study might be attributed to the larger number of microphones in em32 aiding
in localization [5], as well as a higher spatial aliasing frequency which
could positively impact the TDOA estimation [71].
Using the em32 array with a dedicated pressure signal—obtained from a separate
high-quality omnidirectional microphone—seems to give substantial benefits for
SDM. Amengual Gari et al.’s [23] find the benefit of an omnidirectional
microphone over an Ambisonic microphone being small, suggesting the influence
of equipment and setup on the auralization quality to be more significant.
The spatial and timbral fidelity similarities between HO-SIRR (G) and SDM PIV
(H) methods, along with the improvements observed between SDM PIV (H) and SDM
PIV Omni (I), suggest that adding a dedicated pressure microphone may improve
SIRR methods, as noted in [4].
## VIII Conclusion
The study evaluated spatial analysis and synthesis methods, specifically SDM,
BSDM, and HO-SIRR, along with their variations, in creating synthetic binaural
room impulse responses (BRIRs) of an ITU-R BS.1116 compliant listening room.
The objective was to achieve auralization perceptually similar to the KU100
dummy head in terms of spatial fidelity and timbral fidelity, using
Zieliński’s fidelity attribute [44]. Certain SDM configurations exhibited
spatial fidelity levels comparable to those of HO-SIRR, while BSDM suffered
from artifacts in the temporal structure of stimuli. A larger number of
microphone capsules enhanced the performance of SDM methods, and a separate
pressure signal improved timbre. The results are expected to more generally
describe the performance of synthetic binaural room impulse responses in small
rooms, such as critical listening rooms and audio mixing rooms. Future work
can include the study of other rooms, additional subjective dimensions and new
systems like 4D-ASDM [18] or REPAIR [19].
## Acknowledgments
The research presented in this paper was funded by Genelec Oy and the
University of Huddersfield. The authors would like to thank everyone who took
part in the listening test.
## References
* [1] A. Abdou and R. W. Guy, “Spatial information of sound fields for room-acoustics evaluation and diagnosis,” _The Journal of the Acoustical Society of America_ , vol. 100, no. 5, pp. 3215–3226, Nov. 1996.
* [2] B. N. Gover, J. G. Ryan, and M. R. Stinson, “Measurements of directional properties of reverberant sound fields in rooms using a spherical microphone array,” _The Journal of the Acoustical Society of America_ , vol. 116, no. 4, pp. 2138–2148, Oct. 2004.
* [3] Y. Yamasaki and T. Itow, “Measurement of spatial information in sound fields by closely located four point microphone method.” _Journal of the Acoustical Society of Japan (E)_ , vol. 10, no. 2, pp. 101–110, 1989.
* [4] J. Merimaa and V. Pulkki, “Spatial impulse response rendering I: Analysis and synthesis,” _Journal of the Audio Engineering Society_ , vol. 53, no. 12, pp. 1115–1127, 2005.
* [5] S. Tervo, J. Pätynen, A. Kuusinen, and T. Lokki, “Spatial decomposition method for room impulse responses,” _Journal of the Audio Engineering Society_ , vol. 61, no. 1/2, pp. 17–28, 2013.
* [6] S. Tervo, “SDM Toolbox,” https://uk.mathworks.com/matlabcentral/fileexchange/56663-sdm-toolbox, Sep. 2023.
* [7] J. Pätynen, S. Tervo, and T. Lokki, “Analysis of concert hall acoustics via visualizations of time-frequency and spatiotemporal responses,” _The Journal of the Acoustical Society of America_ , vol. 133, no. 2, pp. 842–857, 2013.
* [8] S. Tervo, J. Pätynen, N. Kaplanis, M. Lydolf, S. Bech, and T. Lokki, “Spatial analysis and synthesis of car audio system and car cabin acoustics with a compact microphone array,” _Journal of the Audio Engineering Society_ , vol. 63, no. 11, pp. 914–925, 2015.
* [9] S. V. A. Garí, J. Pätynen, and T. Lokki, “Physical and Perceptual Comparison of Real and Focused Sound Sources in a Concert Hall,” _Journal of the Audio Engineering Society_ , vol. 64, no. 12, pp. 1014–1025, Dec. 2016.
* [10] S. V. Amengual Garí, D. Eddy, M. Kob, and T. Lokki, “Real-time auralization of room acoustics for the study of live music performance,” in _Jahrestagung Für Akustik, DAGA 2016_ , 2016.
* [11] L. Müller and J. Ahrens, “Perceptual Differences for Modifications of the Elevation of Early Room Reflections,” in _Audio Engineering Society Conference: AES 2022 International Audio for Virtual and Augmented Reality Conference_. Audio Engineering Society, Aug. 2022.
* [12] F. Bederna, L. Müller, and J. Ahrens, “Perceptual Detection Thresholds for Alterations of the Azimuth of Early Room Reflections,” in _Jahrestagung Für Akustik, DAGA 2023_ , Hamburg, Germany, Mar. 2023.
* [13] O. C. Gomes, N. Meyer-Kahlen, W. Lachenmayr, and T. Lokki, “Perceptual Consequences of Direction and Level of Early Reflections in a Chamber Music Hall,” in _Jahrestagung Für Akustik, DAGA 2022_ , 2022.
* [14] P. Coleman, A. Franck, P. Jackson, R. J. Hughes, L. Remaggi, and F. Melchior, “Object-based reverberation for spatial audio,” _Journal of the Audio Engineering Society_ , vol. 65, no. 1/2, pp. 66–77, 2017.
* [15] L. McCormack, V. Pulkki, A. Politis, O. Scheuregger, and M. Marschall, “Higher-Order Spatial Impulse Response Rendering: Investigating the Perceived Effects of Spherical Order, Dedicated Diffuse Rendering, and Frequency Resolution,” _Journal of the Audio Engineering Society_ , vol. 68, no. 5, pp. 338–354, Jun. 2020.
* [16] S. V. Amengual Garí, J. M. Arend, P. T. Calamia, and P. W. Robinson, “Optimizations of the spatial decomposition method for binaural reproduction,” _Journal of the Audio Engineering Society_ , vol. 68, no. 12, pp. 959–976, 2021.
* [17] C. Hold, L. McCormack, and V. Pulkki, “Parametric binaural reproduction of higher-order spatial impulse responses,” in _Proceedings of the 24th International Congress on Acoustics_ , Oct. 2022, p. 5.
* [18] E. Hoffbauer and M. Frank, “Four-Directional Ambisonic Spatial Decomposition Method With Reduced Temporal Artifacts,” _Journal of the Audio Engineering Society_ , vol. 70, no. 12, pp. 1002–1014, Dec. 2022.
* [19] L. Mccormack, N. Meyer-Kahlen, and A. Politis, “Spatial Reconstruction-Based Rendering of Microphone Array Room Impulse Responses,” _Journal of the Audio Engineering Society_ , vol. 71, no. 5, pp. 267–280, 2023.
* [20] V. Pulkki and J. Merimaa, “Spatial impulse response rendering II: Reproduction of diffuse sound and listening tests,” _Journal of the Audio Engineering Society_ , vol. 54, no. 1/2, pp. 3–20, 2006.
* [21] J. B. Allen and D. A. Berkley, “Image method for efficiently simulating small-room acoustics,” _The Journal of the Acoustical Society of America_ , vol. 65, no. 4, pp. 943–950, 1979.
* [22] J. Ahrens, “Perceptual evaluation of binaural auralization of data obtained from the spatial decomposition method,” in _2019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA)_. IEEE, 2019, pp. 65–69.
* [23] S. V. Amengual Garí, W. Lachenmayr, and E. Mommertz, “Spatial analysis and auralization of room acoustics using a tetrahedral microphone,” _The Journal of the Acoustical Society of America_ , vol. 141, no. 4, pp. EL369–EL374, 2017.
* [24] A. Pawlak, H. Lee, A. Mäkivirta, and T. Lund, “Subjective evaluation of spatial analysis and synthesis methods using different microphone arrays,” in _2021 Immersive and 3D Audio: From Architecture to Automotive (I3DA)_. IEEE, 2021, pp. 1–7.
* [25] ITU-R BS.1116-3, “Methods for the Subjective Assessment of Small Impairments in Audio Systems,” Feb. 2015.
* [26] ITU-R BS.2051-3, “Advanced sound system for programme production,” May 2022\.
* [27] D. P. Jarrett, E. A. Habets, and P. A. Naylor, _Theory and Applications of Spherical Microphone Array Processing_ , ser. Springer Topics in Signal Processing. Cham: Springer International Publishing, 2017, vol. 9.
* [28] C. Knapp and G. Carter, “The generalized correlation method for estimation of time delay,” _IEEE Transactions on Acoustics, Speech, and Signal Processing_ , vol. 24, no. 4, pp. 320–327, Aug. 1976.
* [29] L. Zhang and X. Wu, “On cross correlation based-discrete time delay estimation,” in _Proceedings.(ICASSP’05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005._ , vol. 4. IEEE, 2005, pp. iv–981.
* [30] J. Yli-Hietanen, K. Kalliojarvi, and J. Astola, “Low-complexity angle of arrival estimation of wideband signals using small arrays,” in _Proceedings of 8th Workshop on Statistical Signal and Array Processing_. IEEE, 1996, pp. 109–112.
* [31] Y. Yamasaki, S. Yamasuda, H. Saitow, K. Yasukawa, and T. Itow, “An application of digital techniques to obtain the spatial information of a room,” _The Journal of the Acoustical Society of America_ , vol. 64, no. S1, pp. S149–S149, 1978.
* [32] K. Sekiguchi, S. Kimura, and T. Hanyuu, “Analysis of sound field on spatial information using a four-channel microphone system based on regular tetrahedron peak point method,” _Applied Acoustics_ , vol. 37, no. 4, pp. 305–323, 1992.
* [33] S. Tervo, T. Korhonen, and T. Lokki, “Estimation of reflections from impulse responses,” _Building Acoustics_ , vol. 18, pp. 159–173, 2011.
* [34] D. P. Jarrett, E. A. Habets, and P. A. Naylor, “3D source localization in the spherical harmonic domain using a pseudointensity vector,” in _2010 18th European Signal Processing Conference_. IEEE, 2010, pp. 442–446.
* [35] J. Ahrens, “Auralization of omnidirectional room impulse responses based on the spatial decomposition method and synthetic spatial data,” in _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2019, pp. 146–150.
* [36] M. Zaunschirm, M. Frank, and F. Zotter, “BRIR synthesis using first-order microphone arrays,” in _Audio Engineering Society Convention 144_. Audio Engineering Society, 2018\.
* [37] A. Bassuet, “New acoustical parameters and visualization techniques to analyze the spatial distribution of sound in music spaces,” _Building Acoustics_ , vol. 18, no. 3-4, pp. 329–347, 2011.
* [38] J. Pätynen, S. Tervo, and T. Lokki, “Amplitude panning decreases spectral brightness with concert hall auralizations,” in _Audio Engineering Society Conference: 55th International Conference: Spatial Audio_. Audio Engineering Society, 2014\.
* [39] M. Zaunschirm, M. Frank, and F. Zotter, “Binaural rendering with measured room responses: First-order ambisonic microphone vs. dummy head,” _Applied Sciences_ , vol. 10, no. 5, p. 1631, 2020.
* [40] O. Puomio, J. Pätynen, and T. Lokki, “Optimization of virtual loudspeakers for spatial room acoustics reproduction with headphones,” _Applied Sciences_ , vol. 7, no. 12, p. 1282, 2017.
* [41] S. V. Amengual Garí, “BinauralSDM,” https://github.com/facebookresearch/BinauralSDM, Jun. 2023.
* [42] S. Tervo, P. Laukkanen, J. Pätynen, and T. Lokki, “Preferences of critical listening environments among sound engineers,” _Journal of the Audio Engineering Society_ , vol. 62, no. 5, pp. 300–314, 2014.
* [43] S. V. A. Garí, W. O. Brimijoin, H. G. Hassager, and P. W. Robinson, “Flexible binaural resynthesis of room impulse responses for augmented reality research,” in _EAA Spatial Audio Signal Processing Symposium_ , 2019, pp. 161–166.
* [44] S. K. Zielinski, F. Rumsey, R. Kassier, and S. Bech, “Comparison of basic audio quality and timbral and spatial fidelity changes caused by limitation of bandwidth and by down-mix algorithms in 5.1 surround audio systems,” _Journal of the Audio Engineering Society_ , vol. 53, no. 3, pp. 174–192, 2005.
* [45] ITU-R BS.1534-3, “Method for the subjective assessment of intermediate quality level of audio systems,” 2014.
* [46] K. Hiyama, S. Komiyama, and K. Hamasaki, “The minimum number of loudspeakers and its arrangement for reproducing the spatial impression of diffuse sound field,” in _Audio Engineering Society Convention 113_. Audio Engineering Society, 2002.
* [47] F. Menzer and C. Faller, “Investigations on modeling BRIR tails with filtered and coherence-matched noise,” in _Audio Engineering Society Convention 127_. Audio Engineering Society, 2009.
* [48] D. Johnson and H. Lee, “Huddersfield Universal Listening Test Interface Generator (HULTI-GEN) Version 2,” in _Audio Engineering Society Convention 149_. Audio Engineering Society, 2020.
* [49] A. Farina, “Advancements in impulse response measurements by sine sweeps,” in _Audio Engineering Society Convention 122_. Audio Engineering Society, 2007.
* [50] H. Helmholz, I. Ananthabhotla, P. T. Calamia, and S. V. Amengual Gari, “Towards the Prediction of Perceived Room Acoustical Similarity,” in _Audio Engineering Society Conference: AES 2022 International Audio for Virtual and Augmented Reality Conference_. Audio Engineering Society, Aug. 2022.
* [51] T. Surdu, C. Schneiderwind, P. Popp, L. Treybig, and S. Werner, “A. LI. EN: An Audiovisual Dataset of different Acoustical Impulse Responses Measured in a Living Room Environment,” in _Jahrestagung Für Akustik, DAGA 2023_ , Hamburg, Germany, 2023\.
* [52] N. Meyer-Kahlen, S. V. A. Garí, I. Ananthabhotla, and P. Calamia, “A Two-Dimensional Threshold Test for Reverberation Time and Direct-to-Reverberant Ratio,” in _2023 Immersive and 3D Audio: From Architecture to Automotive (I3DA)_. IEEE, 2023, pp. 1–8.
* [53] L. McCormack, “HO-SIRR,” https://github.com/leomccormack/HO-SIRR, Aug. 2023.
* [54] B. Bernschütz, “A spherical far field HRIR/HRTF compilation of the Neumann KU 100,” in _Proceedings of the 40th Italian (AIA) Annual Conference on Acoustics and the 39th German Annual Conference on Acoustics (DAGA) Conference on Acoustics_. AIA/DAGA, 2013, p. 29.
* [55] P. Majdak, F. Zotter, F. Brinkmann, J. De Muynke, M. Mihocic, and M. Noisternig, “Spatially Oriented Format for Acoustics 2.1: Introduction and Recent Advances,” _Journal of the Audio Engineering Society_ , vol. 70, no. 7/8, pp. 565–584, 2022.
* [56] Bang & Olufsen, “Music for Archimedes,” [Audio CD], Denmark, 1992.
* [57] V. Hansen and G. Munch, “Making recordings for simulation tests in the Archimedes project,” _Journal of the Audio Engineering Society_ , vol. 39, no. 10, pp. 768–774, 1991.
* [58] Various Artists, “Anechoic orchestral music recording,” [Audio CD] Denon, PG-6006, Nippon Columbia Co., Ltd, Japan, 1988.
* [59] H. Lee, M. Frank, and F. Zotter, “Spatial and timbral fidelities of binaural ambisonics decoders for main microphone array recordings,” in _Audio Engineering Society Conference: 2019 AES International Conference on Immersive and Interactive Audio_. Audio Engineering Society, 2019.
* [60] O. Kirkeby and P. A. Nelson, “Digital filter design for inversion problems in sound reproduction,” _Journal of the Audio Engineering Society_ , vol. 47, no. 7/8, pp. 583–595, 1999.
* [61] Z. Schärer and A. Lindau, “Evaluation of equalization methods for binaural signals,” in _Audio Engineering Society Convention 126_. Audio Engineering Society, 2009.
* [62] P. Zahorik, “Direct-to-reverberant energy ratio sensitivity,” _The Journal of the Acoustical Society of America_ , vol. 112, no. 5, pp. 2110–2117, Nov. 2002.
* [63] J. Blauert, _Spatial Hearing: The Psychophysics of Human Sound Localization_. MIT press, 1997.
* [64] A. Andreopoulou and B. F. G. Katz, “Identification of perceptually relevant methods of inter-aural time difference estimation,” _The Journal of the Acoustical Society of America_ , vol. 142, no. 2, pp. 588–598, Aug. 2017.
* [65] ISO 3382-1, “Acoustics — Measurement of room acoustic parameters. Part 1: Performance spaces,” 2009.
* [66] T. Hidaka, L. L. Beranek, and T. Okano, “Interaural cross-correlation, lateral fraction, and low- and high-frequency sound levels as measures of acoustical quality in concert halls,” _The Journal of the Acoustical Society of America_ , vol. 98, no. 2, pp. 988–1007, Aug. 1995.
* [67] S. Holm, “A simple sequentially rejective multiple test procedure,” _Scandinavian journal of statistics_ , pp. 65–70, 1979.
* [68] A. Field, J. Miles, and Z. Field, _Discovering Statistics Using R_. Sage publications, 2012.
* [69] H. Lee and D. Johnson, “3D microphone array comparison: Objective measurements,” _Journal of the Audio Engineering Society_ , vol. 69, no. 11, pp. 871–887, 2021.
* [70] J. Nikunen and T. Virtanen, “Time-difference of arrival model for spherical microphone arrays and application to direction of arrival estimation,” in _2017 25th European Signal Processing Conference (EUSIPCO)_. IEEE, 2017, pp. 1255–1259.
* [71] J. Benesty, J. Chen, and Y. Huang, _Microphone Array Signal Processing_ , ser. Springer Topics in Signal Processing. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008, vol. 1.
|
# Error estimates for finite element discretizations of the instationary
Navier-Stokes equations
Boris Vexler Chair of Optimal Control, Technical University of Munich, School
of Computation Information and Technology, Department of Mathematics ,
Boltzmannstraße 3, 85748 Garching b. Munich, Germany (vexler@tum.de). and
Jakob Wagner Chair of Optimal Control, Technical University of Munich, School
of Computation Information and Technology, Department of Mathematics ,
Boltzmannstraße 3, 85748 Garching b. Munich, Germany<EMAIL_ADDRESS>ORCID: https://orcid.org/0000-0001-8510-9790.
###### Abstract.
In this work we consider the two dimensional instationary Navier-Stokes
equations with homogeneous Dirichlet/no-slip boundary conditions. We show
error estimates for the fully discrete problem, where a discontinuous Galerkin
method in time and inf-sup stable finite elements in space are used. Recently,
best approximation type error estimates for the Stokes problem in the
$L^{\infty}(I;L^{2}(\Omega))$, $L^{2}(I;H^{1}(\Omega))$ and
$L^{2}(I;L^{2}(\Omega))$ norms have been shown. The main result of the present
work extends the error estimate in the $L^{\infty}(I;L^{2}(\Omega))$ norm to
the Navier-Stokes equations, by pursuing an error splitting approach and an
appropriate duality argument. In order to discuss the stability of solutions
to the discrete primal and dual equations, a specially tailored discrete
Gronwall lemma is presented. The techniques developed towards showing the
$L^{\infty}(I;L^{2}(\Omega))$ error estimate, also allow us to show best
approximation type error estimates in the $L^{2}(I;H^{1}(\Omega))$ and
$L^{2}(I;L^{2}(\Omega))$ norms, which complement this work.
###### Key words and phrases:
Navier-Stokes, transient, instationary, finite elements, discontinuous
Galerkin, error estimates, best approximation, fully discrete
###### 1991 Mathematics Subject Classification:
35Q30, 65M60, 65M15, 65M22, 76D05, 76M10
## 1\. Introduction
In this paper, we consider the instationary Navier-Stokes equations in two
space dimensions with homogeneous boundary conditions, i.e.,
$\left\\{\begin{aligned} \partial_{t}u-\nu\Delta u+(u\cdot\nabla)u+\nabla
p&=f&&\text{in }I\times\Omega,\\\ \nabla\cdot u&=0&&\text{in
}I\times\Omega,\\\ u(0)&=u_{0}&&\text{in }\Omega,\\\ u&=0&&\text{on
}I\times\partial\Omega.\end{aligned}\right.$ (1)
Here $\nu>0$ denotes the viscosity, $I=(0,T]\subset\mathbb{R}$ a bounded,
half-open interval for some fixed finite endtime $T>0$, and
$\Omega\subset\mathbb{R}^{2}$ a bounded convex polygonal domain. The equations
are discretized in time by a discontinuous Galerkin (dG) method, i.e., the
solution is approximated by piecewise polynomials in time, defined on
subintervals of $I$, without any requirement of continuity at the time nodes,
see, e.g., [24, 49]. The parameter indicating the time discretization will be
denoted by $k$ and corresponds to the length of the largest subinterval in the
partition of $I$. To discretize in space, we use inf-sup stable pairs of
finite element spaces for the velocity and pressure components. The parameter
indicating the spatial discretization will be denoted by $h$ and corresponds
to the largest diameter of cells in the mesh.
Due to the variational formulation of the dG time discretization, this
discretization scheme is particularly suited for treating optimal control
problems. See, e.g., [13, 14] for optimal control problems governed by the
Navier-Stokes equations, also [42, 37] for optimal control of general
parabolic problems. While in [13, 14] the focus was put on low order schemes,
recently the authors of [2] analyzed dG schemes of arbitrary order for the
Navier-Stokes equations. Another advantage of the dG time discretization is
the fact, that the maximal parabolic regularity, exhibited by parabolic
problems, is preserved on the discrete level, and moreover can be extended to
the limiting cases $L^{1}$ and $L^{\infty}$ in time, at the expense of a
logarithmic factor, see [35, Theorems 11 & 12]. The natural energy norm for
the Navier-Stokes equations is the norm of the space
$L^{\infty}(I;L^{2}(\Omega)^{2})\,\cap\,L^{2}(I;H^{1}(\Omega)^{2})$. Indeed by
formally testing (1) with the solution $u$, one obtains
$\|u\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u\|_{L^{2}(I;H^{1}(\Omega))}\leq
C\left(\|u_{0}\|_{L^{2}(\Omega)}+\|f\|_{L^{1}(I;L^{2}(\Omega))}\right).$
This bound is also preserved on the discrete level, i.e., holds for the fully
discrete solution $u_{kh}$, see Theorem 4.13. Our main goal in writing this
paper, was the investigation of the discretization error in terms of the
$L^{\infty}(I;L^{2}(\Omega))$-norm. In particular, such estimates are required
for the analysis of optimal control problems, subject to state constraints
pointwise in time, see [42], as the corresponding Lagrange multiplier in this
case is a measure in time. There have been numerous approaches in the
literature, deriving error estimates for the Navier-Stokes equations. In [13,
Theorem 4.7], the same combination of dG-cG discretization schemes was used,
and for $f\in L^{2}(I;L^{2}(\Omega)^{2})$ an estimate
$\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq C(\sqrt{k}+h)$ (2)
was shown. Additional terms arise, when changes in the spatial mesh on
different time intervals are permitted. Under the much stricter assumption
$f\in W^{1,\infty}(I;L^{2}(\Omega)^{2})$ and for the implicit Euler time
discretization, in [31] the estimate
$\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq C(k+h^{2})$ (3)
was shown. It was extended to the Crank-Nicholson scheme in [32] under the
assumption of $f\in W^{2,\infty}(I;L^{2}(\Omega)^{2})$, yielding an error
estimate at the time nodes of order $\mathcal{O}(k^{2}+h^{2})$. Beyond the
references mentioned above, there are many works analyzing the stationary
equations, e.g., [27, 45], semidiscrete equations in space, e.g., [9, 20], or
semidiscrete equations in time, e.g., [23, 46, 4]. Fully discrete error
estimates for stabilized discretization schemes can be found e.g., in [2, 34,
18, 5]. The error estimates are most often derived assuming the necessary
regularity of $u$, such that the discretization schemes can exhibit their full
approximative power. The main drawback of the results in the literature is the
fact, that usually the errors in $L^{\infty}(I;L^{2}(\Omega))$ and
$L^{2}(I;H^{1}(\Omega))$ are estimated in a combined fashion. Thus the
estimate always has to account for the spatial error in the $H^{1}$ norm,
yielding an order reduction for the estimate of the error in the spatial
$L^{2}$ norm. The goal of this paper is to prove an error estimate which can
be formulated as a best approximation type error estimate, which thus
estimates the error in the $L^{\infty}(I;L^{2}(\Omega))$ norm in an isolated
manner. Such an estimate for the instationary Stokes equations, has been
derived recently in [7]. More specifically, for velocity and pressure fields
$(w,r)$, solving the instationary Stokes equations on the continuous level,
and their fully discrete approximations $(w_{kh},r_{kh})$, it holds
$\|w-w_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\ln\left(\frac{T}{k}\right)\left(\inf_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\in
V_{kh}}\|w-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|w-R^{S}_{h}(w,r)\|_{L^{\infty}(I;L^{2}(\Omega))}\right),$
where $V_{kh}$ is the space of discretely divergence free space-time finite
element functions. The operator $R^{S}_{h}$ denotes a stationary Stokes
projection, see [7] or Section 5 for a formal definition. Using this Stokes
result, we will show that a best approximation type result also holds for the
nonlinear Navier-Stokes equations. This main result is stated in Theorem 5.6.
In Corollary 5.8, for $f\in L^{\infty}(I;L^{2}(\Omega)^{2})$, we then obtain
an estimate in terms of $\mathcal{O}(l_{k}(k+h^{2}))$, where $l_{k}$ denotes a
logarithmic term depending on $k$. This result provides a better order of
convergence compared to the estimate (2), shown in [13]. The order of
convergence in the estimate (3), that was presented in [31] is comparable, but
requires a much stronger regularity assumption. The main tools used in this
paper for proving the proposed $L^{\infty}(I;L^{2}(\Omega))$ error estimate
are an error splitting approach and a bootstrapping argument, to apply the
corresponding error estimate for the Stokes equations to the first part of the
error. In order to apply such an argument, understanding the precise
regularity of the occuring nonlinear term $(u\cdot\nabla)u$ is crucial. The
second part of the error will be estimated by a duality argument. This is
possible due to the variational nature of the dG time discretization. We will
derive a stability result for a discrete dual equation. For this result, a
specially adapted version of a discrete Gronwall lemma, Lemma 4.11, will be
presented. For the analysis of the discrete dual problem, we require an
estimate in the $L^{2}(I;H^{1}(\Omega))$ norm, which is why we formulate this
result first. We shall then show the error estimate in
$L^{\infty}(I;L^{2}(\Omega))$ with the tools presented above. The proof of the
error estimate in the $L^{2}(I;L^{2}(\Omega))$ norm is then straightforward
and concludes our work. Summarizing, our main results read
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|u-u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}}$
$\displaystyle\leq C\left(\sqrt{k}+h\right)$ $\displaystyle\text{if }f\in
L^{2}(I;L^{2}(\Omega)^{2})$ $\displaystyle\text{ and }u_{0}\in V,$
$\displaystyle\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}$ $\displaystyle\leq
C\ln(T/k)^{2}\left(k+h^{2}\right)$ $\displaystyle\text{if }f\in
L^{\infty}(I;L^{2}(\Omega)^{2})$ $\displaystyle\text{ and }u_{0}\in V\cap
H^{2}(\Omega)^{2},$ $\displaystyle\|u-u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle\leq C\left(k+h^{2}\right)$ $\displaystyle\text{if }f\in
L^{2}(I;L^{2}(\Omega)^{2})$ $\displaystyle\text{ and }u_{0}\in V,$
and can be found in Theorems 4.16, 5.6 and 5.7 and Corollaries 5.8 and 5.9.
All three results are, up to logarithmic terms, optimal in terms of order of
convergence and in terms of required regularity. The structure of this paper
will be as follows. First, we fix some notation and function spaces in Section
2. We proceed in Section 3 by stating the appropriate weak formulations of (1)
with and without pressure and recall some known regularity results. We
conclude the section with an analysis of the regularity properties of the
nonlinear term $(u\cdot\nabla)u$. Section 4 will be devoted to the spatial and
temporal discretizations. We present a version of a discrete Gronwall lemma,
which allows us to treat problems with right hand sides that are only $L^{1}$
in time, and show stability results for fully discrete primal equations. With
this result, we derive an error estimate for the Navier-Stokes equations in
$L^{2}(I;H^{1}(\Omega))$. Lastly, in Section 5, we show stability results for
a discrete dual problem, which in the end allows us to show the error
estimates in the $L^{\infty}(I;L^{2}(\Omega))$ and $L^{2}(I;L^{2}(\Omega))$
norms.
## 2\. Preliminary
For a convex, polygonal domain $\Omega\subset\mathbb{R}^{2}$, $1\leq
p\leq\infty$ and $k\in\mathbb{N}$, we denote by $L^{p}(\Omega)$,
$W^{k,p}(\Omega)$, $H^{k}(\Omega)$ and $H^{1}_{0}(\Omega)$ the usual Lebesgue
and Sobolev spaces. The inner product on $L^{2}(\Omega)$ will be denoted by
$\left(\cdot,\cdot\right)_{\Omega}$. The space $L^{2}_{0}(\Omega)$ is the
subspace of $L^{2}(\Omega)$, consisting of all functions, that have zero mean.
For $s\in\mathbb{R}\backslash\mathbb{N}$, $s>0$ the fractional order
Sobolev(-Slobodeckij) space $W^{s,p}(\Omega)$ is defined, see, e.g., [21], as
$W^{s,p}(\Omega):=\left\\{\,v\in W^{\lfloor
s\rfloor,p}(\Omega):\sum_{|\alpha|=\lfloor
s\rfloor}\iint_{\Omega\times\Omega}\dfrac{|D^{\alpha}v(x)-D^{\alpha}v(y)|^{p}}{|x-y|^{(s-\lfloor
s\rfloor)p+2}}dxdy<+\infty\,\right\\}.$
In case $p=2$ we again use the notation $H^{s}(\Omega)$. Note that in this
case $H^{s}(\Omega)$ can equivalently be obtained via real or complex
interpolation of the integer degree spaces $H^{k}(\Omega)$. This is due to the
fact, that in the Hilbert space setting, all resulting Bessel potential spaces
$H^{s}_{2}(\Omega)$, Besov spaces $B^{s}_{2,2}(\Omega)$ and
Sobolev-(Slobodeckij) spaces $H^{s}(\Omega)$ coincide, see [50, pp. 12,39].
For $X$ being any function space over $\Omega$, we denote by $X^{*}$ its
topological dual space, and abbreviate the duality pairing by
$\left\langle\cdot,\cdot\right\rangle_{\Omega}$. We will also use the notation
$H^{1}_{0}(\Omega)^{*}=H^{-1}(\Omega)$. The structure of the Stokes and
Navier-Stokes equations requires also the definition of some vector valued
spaces, consisting of divergence free vector fields. We denote by
$\nabla\cdot$ the divergence operator and introduce the spaces
$V:=\overline{\left\\{\,v\in C^{\infty}_{0}(\Omega)^{2}:\ \nabla\cdot
v=0\,\right\\}}^{H^{1}(\Omega)}\quad\text{and}\quad
H:=\overline{\left\\{\,v\in C^{\infty}_{0}(\Omega)^{2}:\ \nabla\cdot
v=0\,\right\\}}^{L^{2}(\Omega)}.$
Note that instead of the definition via closures, in the case of $\Omega$
being bounded and Lipschitz, these spaces are alternatively characterized in
the following way, see [48, Chapter 1, Theorems 1.4 & 1.6]:
$V=\left\\{\,v\in H^{1}_{0}(\Omega)^{2}:\ \nabla\cdot
v=0\,\right\\}\quad\text{and}\quad H=\left\\{\,v\in L^{2}(\Omega)^{2}:\
\nabla\cdot v=0,\ u\cdot n=0\text{ on }\partial\Omega\,\right\\},$
where by $u\cdot n$ we denote the normal trace of the vector field $u$. To
improve readability, whenever vectorial spaces like $H^{1}(\Omega)^{2}$ would
arise in the subscript of some norm, we shall drop the outer superscript
$(\cdot)^{2}$. For a Banach space $X$ and $I=(0,T]$ we denote by $L^{p}(I;X)$
the Bochner space of $X$ valued functions, for which the following norm is
finite
$\|v\|_{L^{p}(I;X)}=\left(\int_{I}\|v(t)\|_{X}^{p}\ dt\right)^{1/p},$
with the usual convention when $p=+\infty$. It holds
$L^{p}(I;L^{p}(\Omega))\cong L^{p}(I\times\Omega)$, and for $p=2$, we denote
the inner product by $\left(\cdot,\cdot\right)_{I\times\Omega}$. Whenever $X$
is separable and $1\leq p<\infty$, it holds $(L^{p}(I;X))^{*}\cong
L^{p^{*}}(I;X^{*})$, where $1/p+1/p^{*}=1$. The duality pairing for such
spaces will be denoted by
$\left\langle\cdot,\cdot\right\rangle_{I\times\Omega}$. By $W^{k,p}(I;X)$ and
$H^{k}(I;X)$ for $k\in\mathbb{N}$ we denote the spaces of functions $v$
satisfying $\partial_{t}^{j}v\in L^{p}(I;X)$, $j=0,...,k$.
## 3\. Navier-Stokes equations
We start by recalling some regularity results for the Navier-Stokes equations,
and we are going to prove some additional results, especially adapted to the
situation considered in this paper. Throughout this paper, we shall always
assume the convexity of $\Omega$. We will state explicitly, whenever results
also hold in a more general setting. It is a well known result, that for $f\in
L^{2}(I;V^{*})+L^{1}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in H$, there exists a
unique weak solution, i.e., the following Proposition holds, see [48, Chapter
3, Theorem 3.1 & Remark 3.1]:
###### Proposition 3.1.
Let $f\in L^{2}(I;V^{*})+L^{1}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in H$. Then
there exists a unique weak solution $u\in L^{2}(I;V)\cap
C(\bar{I};L^{2}(\Omega)^{2})$ of (1), satisfying
$\left\langle\partial_{t}u,v\right\rangle_{\Omega}+\nu\left(\nabla u,\nabla
v\right)_{\Omega}+\left((u\cdot\nabla)u,v\right)_{\Omega}=\left(f,v\right)_{\Omega}\qquad\text{for
all }v\in V$ (4)
in the sense of distributions on $I$, and $u(0)=u_{0}$. Moreover, there holds
an estimate
$\|u\|_{L^{\infty}(I;L^{2}(\Omega))}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{\nu}}\|u\|_{L^{2}(I;V)}\leq
C\left(\|u_{0}\|_{H}+\|f\|_{L^{2}(I;V^{*})+L^{1}(I;L^{2}(\Omega))}\right).$
Note that the constant $C$ above only depends on $\nu$ and $\Omega$, but is
independent of $T$, see [47, Theorems V.1.4.2, V.1.5.3, V.3.1.1]. It is well
known, that under the assumptions of Proposition 3.1, the nonlinearity
satisfies for $1\leq s,q<2$:
$(u\cdot\nabla)u\in
L^{s}(I;L^{q}(\Omega)^{2})\quad\text{whenever}\quad\frac{1}{s}+\frac{1}{q}\geq\frac{3}{2},$
(5)
see [47, Lemma V.1.2.1]. Equation 4 is the weak formulation of (1) in
divergence free spaces. The proof of the above proposition relies heavily on
the fact, that the trilinear form $c(\cdot,\cdot,\cdot)$ defined by
$c:{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{1}_{0}(\Omega)^{2}\times
H^{1}_{0}(\Omega)^{2}\times H^{1}_{0}(\Omega)^{2}}\rightarrow\mathbb{R},\quad
c\left(u,v,w\right)=\left((u\cdot\nabla)v,w\right)_{\Omega},$
posesses the properties summarized in the following lemma.
###### Lemma 3.2.
Let $\Omega\subset\mathbb{R}^{2}$ be an open Lipschitz domain, then there
holds the estimate
$\displaystyle\|v\|_{L^{4}(\Omega)}$ $\displaystyle\leq
C\|v\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|v\|_{H^{1}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }v\in H^{1}(\Omega),$ (6)
due to which, the trilinear form $c\left(\cdot,\cdot,\cdot\right)$ satisfies
for all $u,v,w\in H^{1}_{0}(\Omega)^{2}$:
$\displaystyle c\left(u,v,w\right)$
$\displaystyle\leq\|u\|_{L^{4}(\Omega)}\|\nabla
v\|_{L^{2}(\Omega)}\|w\|_{L^{4}(\Omega)},$ $\displaystyle c\left(u,v,u\right)$
$\displaystyle\leq C\|u\|_{L^{2}(\Omega)}\|\nabla u\|_{L^{2}(\Omega)}\|\nabla
v\|_{L^{2}(\Omega)}.$
Let further $\nabla\cdot u=0$. Then it holds
$c\left(u,v,w\right)=-c\left(u,w,v\right)\quad\text{and}\quad
c\left(u,v,v\right)=0.$
###### Proof.
The estimate for the $L^{4}(\Omega)$ norm can be found in [26, Lemma II.3.2]
and [1, Theorem 3]. The properties of the trilinear forms are then
consequences of Hölder’s inequality and integration by parts, and are shown,
e.g., in [26, Lemma IX.2.1]. ∎
In what follows, we often consider the trilinear form $c$ integrated in time,
which we denote by
$c\left(\\!\left(u,v,w\right)\\!\right):=\int_{I}c\left(u,v,w\right)\ dt.$
In analyzing the Navier-Stokes equations, the instationary Stokes equations
frequently arise as an auxiliary problem. For initial data $u_{0}\in H$ and
right hand side $f\in
L^{1}(I;L^{2}(\Omega)^{2}){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+L^{2}(I;V^{*})}$,
there exists a unique solution $w\in L^{2}(I;V)\cap
L^{\infty}(I;L^{2}(\Omega)^{2})$ to the Stokes equations
$\left\\{\begin{aligned}
\left\langle\partial_{t}w,v\right\rangle_{\Omega}+\nu\left(\nabla w,\nabla
v\right)_{\Omega}&=\left(f,v\right)_{\Omega}\qquad\text{for all }v\in V,\\\
w(0)&=u_{0},\end{aligned}\right.$ (7)
where the first line of (7) is understood in the sense of distributions on
$I$. We introduce the Stokes operator $A\colon D(A)\to H$ defined by
$\left(Aw,v\right)_{\Omega}=\nu\left(\nabla w,\nabla
v\right)_{\Omega}\qquad\text{for all }v\in V,$
with domain $D(A):=\left\\{\,v\in V:\Delta v\in L^{2}(\Omega)^{2}\,\right\\}$
and the projection operator $\mathbb{P}:L^{2}(\Omega)^{2}\to H$, defined by
$\left(\mathbb{P}w,v\right)_{\Omega}=\left(w,v\right)_{\Omega}\qquad\text{for
all }v\in H,$
which is called the Helmholtz or Leray projection. Note that with the vector-
valued Laplacian
$-\Delta\colon D(\Delta)\to L^{2}(\Omega)^{2},$
with domain $D(\Delta):=\left\\{\,v\in H^{1}_{0}(\Omega)^{2}:\Delta v\in
L^{2}(\Omega)^{2}\,\right\\}$, the Stokes operator also satisfies the
representation
$A=-\mathbb{P}\Delta.$
For convex $\Omega$, the domains of the operators introduced above satisfy the
representations
$D(\Delta)=H^{1}_{0}(\Omega)^{2}\cap H^{2}(\Omega)^{2}\qquad\text{and}\qquad
D(A)=V\cap H^{2}(\Omega)^{2},$
see [17] for the $H^{2}$ regularity of the Stokes operator. The Stokes
operator $A$ generates an analytic semigroup in $H$, see [7, 40], also [6] for
a detailled general analysis. One important feature of the Stokes problem is
the maximal parabolic regularity, which indicates, that both the time
derivative $\partial_{t}w$ and the Stokes operator $Aw$ individually inherit
certain regularity properties of $f$, see Proposition 3.4 below. For
homogeneous initial data, this consequence of the analyticity of the semigroup
has been shown in [19], see also [47, Chapter IV, Theorem 1.6.3]. Since our
analysis should also treat inhomogeneous initial data $u_{0}$, we need to
define the proper spaces for the initial data:
$V_{1-1/s}:=\left\\{\,v\in H:\ \|v\|_{V_{1-1/s}}<\infty\,\right\\},$
see also [7] and [6, Chapter 1, Section 3.3], where
$\|v\|_{V_{1-1/s}}:=\left(\int_{I}\|A\exp(-tA)v\|_{H}^{s}\
dt\right)^{1/s}+\|v\|_{H}.$
###### Remark 3.3.
Instead of using the spaces $V_{1-1/s}$ explicitly as a requirement for the
initial data, we can make use of the following imbedding results: For $1<s\leq
2$ it holds $V\hookrightarrow V_{1-1/s}$ and for $1<s<\infty$, it holds $V\cap
D(\Delta)\hookrightarrow V_{1-1/s}$, see [7, Remarks 2.8, 2.9].
The Stokes problem exhibits the following regularity properties, see [7,
Proposition 2.6].
###### Proposition 3.4 (Maximal parabolic regularity).
Let $1<s<\infty$, $f\in L^{s}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V_{1-1/s}$.
Then the solution $w$ of the Stokes equations (7) satisfies
$\|\partial_{t}w\|_{L^{s}(I;L^{2}(\Omega))}+\|Aw\|_{L^{s}(I;L^{2}(\Omega))}\leq
C\left(\|f\|_{L^{s}(I;L^{2}(\Omega))}+\|u_{0}\|_{V_{1-1/s}}\right).$
In the setting of Proposition 3.4, the Stokes problem (7) formulated in
divergence free spaces is equivalent to the following velocity-pressure
formulation: Find $(w,r)\in\left[L^{2}(I;H^{1}_{0}(\Omega)^{2})\cap
C(\bar{I};L^{2}(\Omega)^{2})\right]\times L^{s}(I;L^{2}_{0}(\Omega))$
satisfying
$\left\langle\partial_{t}w,v\right\rangle_{I\times\Omega}+\nu\left(\nabla
w,\nabla v\right)_{I\times\Omega}-\left(r,\nabla\cdot
v\right)_{I\times\Omega}+\left(q,\nabla\cdot
w\right)_{I\times\Omega}=\left(f,v\right)_{I\times\Omega}$ (8)
for all $(v,q)\in\left[L^{2}(I;H^{1}_{0}(\Omega)^{2})\cap
L^{\infty}(I;L^{2}(\Omega)^{2})\right]\times L^{2}(I;L^{2}_{0}(\Omega))$ and
$w(0)=u_{0}$, see [7, Theorem 2.10]. Furthermore, the maximal parabolic
regularity results of Proposition 3.4 imply the following estimate for the
pressure:
$\|r\|_{L^{s}(I;L^{2}(\Omega))}\leq
C\left(\|f\|_{L^{s}(I;L^{2}(\Omega))}+\|u_{0}\|_{V_{1-1/s}}\right).$
Without additional smoothness assumptions, the maximal parabolic regularity
for the Stokes problem does not immediately extend to the Navier-Stokes
equations. A partial result can be obtained by considering that the maximal
parabolic regularity of the Stokes operator was recently extended to the
$L^{p}(\Omega)$ setting, where $p$ in general depends on the smoothness of
$\Omega$. For a general Lipschitz domain $\Omega$, [25, Theorem 1.6] shows
that for some $\varepsilon>0$ and any $p$, such that
$|1/p-1/2|<1/4+\varepsilon$, the maximal parabolic regularity holds. With
this, we obtain the following result for the Navier-Stokes equations.
###### Theorem 3.5.
Let $f\in L^{s}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V_{1-1/s}$ for some $s>1$
and $u\in L^{2}(I;V)\cap L^{\infty}(I;L^{2}(\Omega)^{2})$ the unique solution
of (4). Then for $\gamma:=\min\\{s,4/3\\}$ it holds
$\partial_{t}u\ ,\ Au\in L^{\gamma}(I;L^{4/3}(\Omega)^{2}).$
Further, there exists a unique $p\in L^{\gamma}(I;L^{2}_{0}(\Omega))$, such
that
$\left\langle\partial_{t}u,v\right\rangle_{\Omega}+\nu\left(\nabla u,\nabla
v\right)_{\Omega}+\left((u\cdot\nabla)u,v\right)_{\Omega}-\left(p,\nabla\cdot
v\right)_{\Omega}+\left(\nabla\cdot
u,q\right)_{\Omega}=\left(f,v\right)_{\Omega}$
for all $v\in H^{1}_{0}(\Omega)^{2},q\in L^{2}_{0}(\Omega)$ in the sense of
distributions on $I$, and $u(0)=u_{0}$.
###### Proof.
Due to (5) it holds $(u\cdot\nabla)u\in L^{4/3}(I;L^{4/3}(\Omega)^{2})$. Since
$f\in L^{s}(I;L^{2}(\Omega))$, this implies $\tilde{f}:=f-(u\cdot\nabla)u\in
L^{\gamma}(I;L^{4/3}(\Omega)^{2})$. Since $|3/4-1/2|=1/4$, [25, Theorem 1.6]
shows that the Stokes problem posesses maximal parabolic regularity in
$L^{4/3}(\Omega)$, which applied to $u_{0}$ and $\tilde{f}$ then yields
$\partial_{t}u,Au\in L^{\gamma}(I;L^{4/3}(\Omega)^{2}).$The existence of a
pressure can then be proven as in [7, Theorem 2.10]. ∎
The above result only recovers parts of the regularity available for the
Stokes problem. Using $H^{2}$ regularity, we will now show improved regularity
results for the Navier-Stokes equations. Note that from now on we explicitly
require the convexity of $\Omega$, whereas Propositions 3.1 and 3.4 and
Theorem 3.5 also hold for general Lipschitz domains.
###### Theorem 3.6 ($H^{2}$ regularity).
Let $u_{0}\in V$ and $f\in L^{2}(I;L^{2}(\Omega)^{2})$. Then the weak solution
$u$ to the Navier-Stokes equations (4) satisfies the improved regularity
$u\in L^{2}(I;H^{2}(\Omega)^{2})\cap H^{1}(I;L^{2}(\Omega)^{2})\hookrightarrow
C({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};H^{1}(\Omega)^{2}),$
and there exist constants $C_{1},C_{2}>0$, depending on $\nu,\Omega$ but
independent of $T$, such that there hold the bounds
$\displaystyle\|u\|_{L^{\infty}(I;H^{1}(\Omega))}$ $\displaystyle\leq
C_{1}\left(\|u_{0}\|_{V}+\|f\|_{L^{2}(I\times\Omega)}\right)\exp\left(C_{2}\left(\|u_{0}\|_{H}^{4}+\|f\|_{L^{1}(I;L^{2}(\Omega))}^{4}\right)\right),$
$\displaystyle\|u\|_{L^{2}(I;H^{2}(\Omega))}$ $\displaystyle\leq
C_{1}\left(\|u_{0}\|_{V}+\|f\|_{L^{2}(I\times\Omega)}\right)\left(1+\|u_{0}\|_{H}^{2}+\|f\|_{L^{1}(I;L^{2}(\Omega))}^{2}\right)\exp\left(C_{2}\left(\|u_{0}\|_{H}^{4}+\|f\|_{L^{1}(I;L^{2}(\Omega))}^{4}\right)\right).$
###### Proof.
The proof of this result for $C^{2}$ domains can be found in [48, Chapter 3,
Theorem 3.10]. Instead of a $C^{2}$ boundary, we can also use the $H^{2}$
regularity for the Stokes operator on convex, polygonal domains, see, e.g.,
[17, Theorem 5.5] or [33, Theorem 2], to obtain the claimed regularity. The
norm bounds are obtained by the Gronwall lemma. ∎
Note that contrary to the $H^{2}$ regularity for the instationary Stokes
problem, for the Navier-Stokes problem, even the proofs of the regularities
$\partial_{t}u,Au\in L^{2}(I;L^{2}(\Omega)^{2})$, that are contained in the
above result, require convexity of the domain. We shall state a corresponding
estimate for $\|\partial_{t}u\|_{L^{2}(I\times\Omega)}$ after discussing the
regularity of the nonlinearity.
###### Remark 3.7.
By Hölder’s inequality, with this $H^{2}$ regularity result, and the imbedding
$H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$, we immediately obtain
$(u\cdot\nabla)u\in L^{2}(I;L^{2}(\Omega)^{2}).$
The regularity $u\in
C({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};H^{1}(\Omega)^{2})$
almost yields that $u$ is uniformly bounded over the whole space time
cylinder. However, since in two dimensions $H^{1}(\Omega)\not\hookrightarrow
L^{\infty}(\Omega)$, the boundedness in space has to be shown via an
additional argument.
We first show, that the nonlinearity actually posesses more regularity, than
what was claimed in Remark 3.7.
###### Theorem 3.8.
Let the assertions of Theorem 3.6 be satisfied, i.e., $u_{0}\in V$ and $f\in
L^{2}(I;L^{2}(\Omega)^{2})$, and let $u$ be the unique solution to the Navier-
Stokes equations (4). Then the nonlinearity satisfies
$(u\cdot\nabla)u\in L^{s}(I;L^{2}(\Omega)^{2})\qquad\text{for any }1\leq
s<\infty,$
and for any $0<\delta\leq\min\\{2,s\\}$ there holds the estimate
$\|(u\cdot\nabla)u\|_{L^{s}(I;L^{2}(\Omega))}\leq
C\|u\|_{L^{2}(I;H^{2}(\Omega))}^{\frac{\delta}{s}}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2-\frac{\delta}{s}},$
where the norms of $u$ on the right hand side can be estimated by Theorem 3.6.
###### Proof.
The proof follows the ideas of [47, Theorem V.1.8.2], however we will use
interpolation spaces instead of fractional powers of the Stokes operator. With
Hölder’s inequality and the Sobolev imbedding
$H^{1+\frac{\delta}{s}}(\Omega)\hookrightarrow C(\overline{\Omega})$, it holds
for any $s<\infty$ and $\delta>0$:
$\|(u\cdot\nabla)u\|_{L^{2}(\Omega)}^{s}\leq\|u\|_{L^{\infty}(\Omega)}^{s}\|\nabla
u\|_{L^{2}(\Omega)}^{s}\leq
C\|u\|_{H^{1+\frac{\delta}{s}}(\Omega)}^{s}\|\nabla u\|_{L^{2}(\Omega)}^{s}.$
Since $\delta\leq s$, we can express the space
$H^{1+\frac{\delta}{s}}(\Omega)$ as interpolation space
$[H^{1}(\Omega),H^{2}(\Omega)]_{\delta/s}$, and obtain from [11, Theorem 1]
the estimate
$\|u\|_{H^{1+\frac{\delta}{s}}(\Omega)}\leq
C\|u\|_{H^{1}(\Omega)}^{1-\frac{\delta}{s}}\|u\|_{H^{2}(\Omega)}^{\frac{\delta}{s}}.$
All in all, we see that
$\|(u\cdot\nabla)u\|_{L^{2}(\Omega)}^{s}\leq
C\|u\|_{H^{2}(\Omega)}^{\delta}\|u\|_{H^{1}(\Omega)}^{2s-\delta},$
which is integrable in time, since $u\in L^{\infty}(I;H^{1}(\Omega)^{2})\cap
L^{2}(I;H^{2}(\Omega)^{2})$, by Theorem 3.6 and $\delta\leq 2$. With Hölder’s
inequality, we obtain the proposed estimate, which concludes the proof. ∎
###### Remark 3.9.
The previous result $(u\cdot\nabla)u\in L^{s}(I;L^{2}(\Omega)^{2})$ shows,
that the Navier-Stokes equations inherit the maximal parabolic regularity of
the Stokes problem, in cases where $f\in L^{s}(I;L^{2}(\Omega)^{2})$ and
$u_{0}\in V_{1-1/s}$ for some $2\leq s<\infty$. Especially it also holds
$Au,\partial_{t}u\in L^{s}(I;L^{2}(\Omega)^{2})$ with
$\|\partial_{t}u\|_{L^{s}(I;L^{2}(\Omega))}+\|Au\|_{L^{s}(I;L^{2}(\Omega))}\leq
C\left(\|u_{0}\|_{V_{1-1/s}}+\|f\|_{L^{s}(I;L^{2}(\Omega))}+\|(u\cdot\nabla)u\|_{L^{s}(I;L^{2}(\Omega))}\right).$
With Theorem 3.8, and slightly higher regularity of the data, we obtain the
boundedness of $u$ in the space-time cylinder:
###### Theorem 3.10.
Let $\varepsilon>0$, $f\in L^{2+\varepsilon}(I;L^{2}(\Omega)^{2})$ and
$u_{0}\in V_{1-1/(2+\varepsilon)}$. Then the unique weak solution $u$ to (1)
satisfies additionally
$u\in C(\bar{I}\times\bar{\Omega})^{2},$
and for $\varepsilon$ sufficiently small, there holds the estimate
$\|u\|_{L^{\infty}(I\times\Omega)}\leq
C\left(\|u_{0}\|_{V_{1-1/(2+\varepsilon)}}+\|f\|_{L^{2+\varepsilon}(I;L^{2}(\Omega))}+\|u\|_{L^{2}(I;H^{2}(\Omega))}^{\varepsilon}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2-\varepsilon}\right),$
where the norms of $u$ on the right hand side can be estimated by Theorem 3.6.
###### Proof.
Applying Theorem 3.8, we observe, that it holds especially $(u\cdot\nabla)u\in
L^{2+\varepsilon}(I;L^{2}(\Omega)^{2})$. For $\varepsilon$ small enough, such
that $\delta:=\varepsilon(2+\varepsilon)\leq 2$, there moreover holds
$\|(u\cdot\nabla)u\|_{L^{2+\varepsilon}(I;L^{2}(\Omega))}\leq
C\|u\|_{L^{2}(I;H^{2}(\Omega))}^{\varepsilon}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2-\varepsilon}.$
By a bootstrapping argument, since $f\in
L^{2+\varepsilon}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(\Omega)^{2}})$
and $u_{0}\in V_{1-1/\varepsilon}$, we can apply the maximal parabolic
regularity of Proposition 3.4, and obtain together with $H^{2}$ regularity
$\partial_{t}u\in
L^{2+\varepsilon}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(\Omega)^{2}}),\qquad
u\in
L^{2+\varepsilon}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{2}(\Omega)^{2}}).$
Hence $u\in W^{1,2+\varepsilon}(I;L^{2}(\Omega)^{2})\cap
L^{2+\varepsilon}(I;H^{2}(\Omega)^{2})$. With the definitions
$p:=2+\varepsilon$, $s:=\frac{1}{2+\varepsilon/2}$,
$\theta:=1-\frac{1}{2+\varepsilon/4}=\frac{1}{2}+\frac{2\varepsilon}{8+\varepsilon}$
it holds $1/p<s<1/2$ and $1/2<\theta<1-s$. We can thus apply [3, Theorem 3],
see also [22, Lemma 2.11 b)], and obtain
$u\in
C^{s-1/p}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};(L^{2}(\Omega),H^{2}(\Omega))_{\theta,1}^{2}).$
Applying [8, Theorems 3.4.1 & 6.4.5], and [21, Theorem 4.57], we moreover have
$\displaystyle u\in
C^{s-1/p}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I};(L^{2}(\Omega),H^{2}(\Omega))_{\theta,2}^{2}})=C^{\frac{\varepsilon}{8+6\varepsilon+\varepsilon^{2}}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};H^{1+\frac{4\varepsilon}{8+\varepsilon}}(\Omega)^{2})\hookrightarrow
C(\bar{I}\times\bar{\Omega})^{2}.$
Note that here we have used, that for the $L^{2}(\Omega)$ case, all
corresponding Sobolev-Slobodeckij, Besov, Bessel-potential and interpolation
spaces coincide, see, e.g., [41]. For an overview over the topic of function
spaces, see also [50]. From the used embeddings, we moreover have the proposed
estimate. This concludes the proof. ∎
###### Corollary 3.11.
Let the assertions of Theorem 3.10 be satisfied, i.e., $f\in
L^{2+\varepsilon}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V_{1-1/(2+\varepsilon)}$
for some $\varepsilon>0$. Then the nonlinear term in the Navier-Stokes
equations satisfies
$(u\cdot\nabla)u\in L^{\infty}(I;L^{2}(\Omega)^{2}),$
and there holds the estimate
$\|(u\cdot\nabla)u\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\left(\|u_{0}\|_{V_{1-1/(2+\varepsilon)}}+\|f\|_{L^{2+\varepsilon}(I;L^{2}(\Omega))}+\|u\|_{L^{2}(I;H^{2}(\Omega))}^{\frac{2}{2+\varepsilon}}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2-\frac{2}{2+\varepsilon}}\right)\|u\|_{L^{\infty}(I;H^{1}(\Omega))}.$
where the norms of $u$ on the right hand side can be estimated by Theorem 3.6.
###### Proof.
This is a direct consequence of Theorems 3.6 and 3.10 and application of
Hölder’s inequality. ∎
In the formulation of equation 4, we have used divergence free test functions.
Whenever we want to test the equation with functions, that are not divergence
free, we have to consider an alternative, equivalent formulation, that
includes the pressure. The following theorem guarantees, that we can freely
switch between the two formulations, when $u_{0}\in V$ and $f\in
L^{2}(I;L^{2}(\Omega)^{2})$.
###### Theorem 3.12.
Let the assertions of Theorem 3.6 be satisfied, i.e., $u_{0}\in V$ and $f\in
L^{2}(I;L^{2}(\Omega)^{2})$. Then there exists a unique solution $(u,p)$ with
$u\in L^{2}(I;H^{2}(\Omega)^{2})\cap
C({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};V){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},\
\partial_{t}u\in L^{2}(I;L^{2}(\Omega)^{2})}\quad\text{and}\quad p\in
L^{2}(I;H^{1}(\Omega)\cap L^{2}_{0}(\Omega)),$
such that $u(0)=u_{0}$ and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}u,v\right)_{I\times\Omega}}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}+\left((u\cdot\nabla)u,v\right)_{I\times\Omega}-\left(p,\nabla\cdot
v\right)_{I\times\Omega}+\left(q,\nabla\cdot
u\right)_{I\times\Omega}=\left(f,v\right)_{I\times\Omega},$ (9)
for all $(v,q)\in L^{2}(I;H^{1}_{0}(\Omega)^{2})\times
L^{2}(I;L^{2}_{0}(\Omega))$. Further it holds
$\|p\|_{L^{2}(I;H^{1}(\Omega))}\leq
C\left(\|u_{0}\|_{V}+\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u\|_{L^{2}(I;H^{2}(\Omega))}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}\right),$
where the norms of $u$ on the right hand side can be estimated by Theorem 3.6.
###### Proof.
This result can be shown using Theorem 3.8 with the choice $s=2$, together
with a bootstrapping argument. Moving $(u\cdot\nabla)u\in
L^{2}(I;L^{2}(\Omega)^{2})$ to the right hand side and applying Proposition
3.4 yields $\partial_{t}u,Au\in L^{2}(I;L^{2}(\Omega)^{2})$. The pressure is
then obtained by following the same steps as [7, Theorem 2.10, Corollary
2.11], where its uniqueness is given by its construction. ∎
## 4\. Discretization
After discussing the continuous formulation of the Navier-Stokes equations, we
now turn towards their discretization.
### 4.1. Spatial discretization
Let $\\{\mathcal{T}_{h}\\}$ denote a family of quasi-uniform triangulations of
$\overline{\Omega}$ consisting of closed simplices. The index $h$ denotes the
maximum meshsize. We discretize the velocity $u$ by a discrete function space
$U_{h}\subset H^{1}_{0}(\Omega)^{2}$ and the pressure $p$ by the discrete
space $M_{h}\subset L^{2}_{0}(\Omega)$, where $(U_{h},M_{h})$ satisfy the
discrete, uniform LBB-condition
$\sup_{v_{h}\in U_{h}}\dfrac{(\nabla\cdot v_{h},q_{h})_{\Omega}}{\|\nabla
v_{h}\|_{L^{2}(\Omega)}}\geq\beta\|q_{h}\|_{L^{2}(\Omega)}\qquad\text{for all
}q_{h}\in M_{h},$
with a constant $\beta>0$ independent of $h$. Throughout this work, we will
assume the following approximation properties of the spaces $U_{h}$ and
$M_{h}$. This assumption is valid, e.g., for Taylor-Hood and MINI finite
elements, even on shape regular meshes, see [7, Assumption 7.2].
###### Assumption 4.1.
There exist interpolation operators $i_{h}\colon H^{2}(\Omega)^{2}\cap
H^{1}_{0}(\Omega)^{2}\to U_{h}$ and $r_{h}\colon L^{2}(\Omega)\to M_{h}$, such
that
$\displaystyle\|\nabla(v-i_{h}v)\|_{L^{2}(\Omega)}$ $\displaystyle\leq
ch\|\nabla^{2}v\|_{L^{2}(\Omega)}$ $\displaystyle\text{for all }v\in
H^{2}(\Omega)^{2}\cap H^{1}_{0}(\Omega)^{2},$
$\displaystyle\|q-r_{h}q\|_{L^{2}(\Omega)}$ $\displaystyle\leq ch\|\nabla
q\|_{L^{2}(\Omega)}$ $\displaystyle\text{for all }q\in H^{1}(\Omega).$
The (vector valued) discrete Laplacian $\Delta_{h}:U_{h}\to U_{h}$ is defined
by
$\left(\nabla u_{h},\nabla
v_{h}\right)_{\Omega}=-\left(\Delta_{h}u_{h},v_{h}\right)_{\Omega}\quad\text{for
all }v_{h}\in U_{h}.$
We introduce the space $V_{h}$ of discretely divergence free functions as
$V_{h}:=\left\\{v_{h}\in U_{h}:\ \left(\nabla\cdot
v_{h},q_{h}\right)_{\Omega}=0\ \text{ for all }q_{h}\in M_{h}\right\\}.$
The $L^{2}$ projection onto this space will be denoted by
$\mathbb{P}_{h}\colon{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(\Omega)^{2}}\to
V_{h}$, satisfying
$\left(\mathbb{P}_{h}v,\phi_{h}\right)_{\Omega}=\left(v,\phi_{h}\right)_{\Omega}\quad\text{for
all }\phi_{h}\in V_{h},$
and allows us to introduce the discrete Stokes operator $A_{h}\colon V_{h}\to
V_{h},A_{h}v_{h}=-\mathbb{P}_{h}\Delta_{h}v_{h}$. Having defined these
discrete spaces and operators, we can now consider the Ritz projection for the
stationary Stokes problem. For any $(w,r)\in H^{1}_{0}(\Omega)^{2}\times
L^{2}(\Omega)$, the projection
$\left(R^{S}_{h}(w,r),R^{S,p}_{h}(w,r)\right)\in U_{h}\times M_{h}$ is defined
by
$\displaystyle\left(\nabla\left(w-R^{S}_{h}(w,r)\right),\nabla\phi_{h}\right)_{\Omega}-\left(r-R^{S,p}_{h}(w,r),\nabla\cdot\phi_{h}\right)_{\Omega}$
$\displaystyle=0\quad\text{for all }\phi_{h}\in U_{h}$ (10)
$\displaystyle\left(\nabla\cdot\left(w-R^{S}_{h}(w,r)\right),\psi_{h}\right)_{\Omega}$
$\displaystyle=0\quad\text{for all }\psi_{h}\in M_{h}.$
In case that $w$ is discretely divergence free, i.e., $\left(\nabla\cdot
w,\psi_{h}\right)_{\Omega}=0$ for all $\psi_{h}\in M_{h}$, it holds
$R_{h}^{S}(w,r)\in V_{h}$. Note that the space $V_{h}$ is in general not a
subspace of the space $V$ of pointwise divergence free functions. This means,
that on the discrete space $V_{h}$, the form $c\left(\cdot,\cdot,\cdot\right)$
does not posess the anti symmetry properties shown in Lemma 3.2. Hence we
define, as in [31, 16], an anti symmetric variant:
$\hat{c}:{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{1}_{0}(\Omega)^{2}\times
H^{1}_{0}(\Omega)^{2}\times
H^{1}_{0}(\Omega)^{2}}\to\mathbb{R},\quad\hat{c}\left(u,v,w\right)=\frac{1}{2}c\left(u,v,w\right)-\frac{1}{2}c\left(u,w,v\right).$
(11)
Analogously to the trilinear form
$c\left(\\!\left(\cdot,\cdot,\cdot\right)\\!\right)$, we will use the notation
$\hat{c}\left(\\!\left(u,v,w\right)\\!\right):=\int_{I}\hat{c}\left(u,v,w\right)\
dt.$
###### Remark 4.2.
Note that, due to Lemma 3.2, we can equivalently replace
$c(\cdot,\cdot,\cdot)$ in the continuous formulation of the Navier-Stokes
equations (4) by $\hat{c}(\cdot,\cdot,\cdot)$, as the two forms coincide on
$V$.
By its definition, $\hat{c}\left(\cdot,\cdot,\cdot\right)$ now has the
following antisymmetric properties on the space $V_{h}$, which will later
allow us to show the stability of the fully discrete solutions.
###### Lemma 4.3.
The trilinear form $\hat{c}\left(\cdot,\cdot,\cdot\right)$ satisfies
$\displaystyle\hat{c}\left(u_{h},v_{h},w_{h}\right)$ $\displaystyle\leq
C\|\nabla u_{h}\|_{L^{2}(\Omega)}\|\nabla v_{h}\|_{L^{2}(\Omega)}\|\nabla
w_{h}\|_{L^{2}(\Omega)}$ $\displaystyle\text{for all }u_{h},v_{h},w_{h}\in
U_{h},$ $\displaystyle\hat{c}\left(u_{h},v_{h},w_{h}\right)$
$\displaystyle=-\hat{c}\left(u_{h},w_{h},v_{h}\right)$ $\displaystyle\text{for
all }u_{h},v_{h},w_{h}\in U_{h},$
$\displaystyle\hat{c}\left(u_{h},v_{h},v_{h}\right)$ $\displaystyle=0$
$\displaystyle\text{for all }u_{h},v_{h}\in U_{h}.$
###### Proof.
The last two identities are a direct consequence of the definition of
$\hat{c}\left(\cdot,\cdot,\cdot\right)$. The first estimate follows from Lemma
3.2 and the imbedding $H^{1}_{0}(\Omega)\hookrightarrow L^{4}(\Omega)$. ∎
Note that due to the above lemma, formally we are still allowed to switch the
second and third argument of $\hat{c}\left(\cdot,\cdot,\cdot\right)$. The
original form $c\left(\cdot,\cdot,\cdot\right)$ however had a strict
disctinction between the two arguments, as it contains the gradient of the
second argument, but only the function values of the third argument. This is
of importance when estimating the form in terms of its arguments. In
$\hat{c}\left(\cdot,\cdot,\cdot\right)$ gradients occur in the second and
third argument, thus switching the arguments does not allow us to obtain
improved estimates. For this reason, we state the following lemma, which
allows us to switch the second and third arguments of
$c\left(\cdot,\cdot,\cdot\right)$ by introducing an additional term, even if
the first argument is not (pointwise) divergence free.
###### Lemma 4.4.
Let
$u,v,w\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{1}_{0}(\Omega)^{2}}$.
Then it holds
$c\left(u,v,w\right)=-c\left(u,w,v\right)-\left(\nabla\cdot u,v\cdot
w\right)_{\Omega}.$
###### Proof.
The proof is simply an application of integration by parts, and can also be
seen, e.g., from [2, Equation 2.9]. ∎
We conclude this subsection on the space discretization by recalling some
important interpolation estimates. On the continuous level, applying (6) to
the first order derivatives, and using $H^{2}$ regularity, yields
$\|\nabla w\|_{L^{4}(\Omega)}\leq C\|\nabla
w\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|Aw\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w\in H^{2}(\Omega)^{2}.$ (12)
Using the Stokes operator on the right hand side, instead of second order
derivatives, allows us to translate this result to the discrete setting. This
is facilitated by the following result, showing that for discretely divergence
free functions, the discrete Laplacian $\Delta_{h}$ can be bounded in terms of
the discrete Stokes operator $A_{h}$:
$\|\Delta_{h}w_{h}\|_{L^{2}(\Omega)}\leq
C\|A_{h}w_{h}\|_{L^{2}(\Omega)}\qquad\text{for all }w_{h}\in V_{h},$ (13)
see [30, Corollary 4.4] or [28, Lemma 4.1]. With this, we can translate (12)
to the discrete setting, by considering for some fixed $w_{h}\in V_{h}$ the
solution $w\in H^{1}_{0}(\Omega)^{2}$ to the continuous problem
$\left(\nabla w,\nabla
v\right)_{\Omega}=\left(-\Delta_{h}w_{h},v\right)_{\Omega}\qquad\text{for all
}v\in H^{1}_{0}(\Omega)^{2}.$
By the stability of the Poisson Ritz projection in $W^{1,4}(\Omega)$, (12) and
(13), we then obtain the discrete version of (12).
$\|\nabla w_{h}\|_{L^{4}(\Omega)}\leq C\|\nabla
w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w_{h}\in V_{h}.$ (14)
Analogously, the Gagliardo-Nirenberg inequality,
$\|w\|_{L^{\infty}(\Omega)}\leq
C\|w\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|Aw\|_{L^{2}(\Omega)}^{\frac{1}{2}},$
(15)
which is a consequence of [1, Theorem 3] together with $H^{2}$ regularity, has
a discrete analogon. It can be shown using the standard discrete Gagliardo-
Nirenberg inequality
$\|w_{h}\|_{L^{\infty}(\Omega)}\leq
C\|w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|\Delta_{h}w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w_{h}\in U_{h},$
which was proven in [29, Lemma 3.3]. The proof stated there for smooth domains
remains the same for convex domains. The discrete version of (15) is then
again obtained by applying (13) and reads
$\|w_{h}\|_{L^{\infty}(\Omega)}\leq
C\|w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w_{h}\in V_{h}.$ (16)
Straightforward calculations, using the definition of $A_{h}$, also give
$\|\nabla w_{h}\|_{L^{2}(\Omega)}\leq
C\|w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w_{h}\in V_{h}.$ (17)
With these considerations regarding the spatial discretization, we can now
consider the fully discrete Navier-Stokes equations by also discretizing in
time.
### 4.2. Temporal discretization
For discretization in time, we employ the discontinuous Galerkin method of
order q (dG(q)), which is also used, e.g., in [7, 16, 24]. The time interval
$I=(0,T]$ is partitioned into $M$ half-open sub-intervals
$I_{m}=(t_{m-1},t_{m}]$ with $0=t_{0}<t_{1}<t_{2}<...<t_{M}=T$. We denote each
timestep by $k_{m}=t_{m}-t_{m-1}$ and for fixed $M$ the maximal timestep by
$k:=\max_{1\leq m\leq M}k_{m}$, as well as the minimal one by
$k_{\min}:=\min_{1\leq m\leq M}k_{m}$. If we want to emphasize that $I_{m}$
belongs to a discretization level $k$, we denote it by $I_{m,k}$. We make some
standard assumptions on the properties on the time discretization:
1. (1)
There are constants $C,\beta>0$ independent of $k$, such that
$k_{\min}\geq Ck^{\beta}.$
2. (2)
There is a constant $\kappa>0$ independent of $k$, such that for all
$m=1,2,...,M-1$
$\kappa^{-1}\leq\frac{k_{m}}{k_{m-1}}\leq\kappa.$
3. (3)
It holds $k\leq\frac{T}{4}$.
A dG(q) function with values in a given Banach space $\mathcal{B}$ is then
given as a function in the space
$X^{q}_{k}(\mathcal{B}):=\left\\{\,v\in L^{2}(I;\mathcal{B}):\
v|_{I_{m}}\in\mathcal{P}_{q}(I_{m};\mathcal{B})\quad\text{for all }1\leq m\leq
M\,\right\\},$
where on each $I_{m}$ the space $\mathcal{P}_{q}(I_{m};\mathcal{B})$ is given
as the space of polynomials in time up to degree $q$ with values in
$\mathcal{B}$:
$\mathcal{P}_{q}(I_{m};\mathcal{B})=\left\\{\,v\in L^{2}(I_{m};\mathcal{B}):\
\exists\ v_{0},...,v_{q}\in\mathcal{B}\ \text{s.t.}\
v=\sum_{j=0}^{q}v_{j}t^{j}\,\right\\}.$
Note that no continuity is required at the time nodes $t_{m}$, which is why we
use the following standard notations for one sided limits and jump terms:
$v_{m}^{+}:=\lim_{\varepsilon\to 0^{+}}v(t_{m}+\varepsilon),\qquad
v_{m}^{-}:=\lim_{\varepsilon\to
0^{+}}v(t_{m}-\varepsilon),\qquad[v]_{m}:=v_{m}^{+}-v_{m}^{-}.$
We introduce the compact notations
$V_{kh}:=X^{q}_{k}(V_{h}),\qquad U_{kh}:=X^{q}_{k}(U_{h}),\qquad
M_{kh}:=X^{q}_{k}(M_{h}),\qquad Y_{kh}:=U_{kh}\times M_{kh}.$
Having defined these dG spaces, we introduce the following projection operator
in time: $\pi_{\tau}\colon C(I;L^{2}(\Omega))\to X^{q}_{k}(L^{2}(\Omega))$
defined by
$\left\\{\begin{aligned}
\left(\pi_{\tau}v-v,\varphi\right)_{I_{m}\times\Omega}&=0\qquad\quad\text{for
all }\varphi\in\mathcal{P}_{q-1}(I_{m};L^{2}(\Omega)),\text{ if }q>0,\\\
\pi_{\tau}v(t_{m}^{-})&=v(t_{m}^{-}),\end{aligned}\right.$ (18)
for all $m=1,2,...,M$. In case $q=0$, the projection operator is defined
solely by the second condition.
###### Remark 4.5.
In this paper we will restrict ourselves to the two lowest cases $q=0$ and
$q=1$. Since we work in a setting of low regularity of the right hand side
$f$, the error estimates would not benefit from higher order schemes. Also,
since the Navier-Stokes equations already pose a challenging large system to
solve, higher order schemes in many applications are not feasible from the
standpoint of computational cost.
We can now introduce the time-discretized formulation of the Navier-Stokes
equations. We define the time-discrete bilinear form for the transient Stokes
equations as in [7] by
$\mathfrak{B}(u,v):=\sum_{m=1}^{M}\left(\partial_{t}u,v\right)_{I_{m}\times\Omega}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}+\sum_{m=2}^{M}\left([u]_{m-1},v^{+}_{m-1}\right)_{\Omega}+\left(u_{0}^{+},v_{0}^{+}\right)_{\Omega}.$
Since we frequently will test some discrete equations with their respective
solutions, let us recall the following lemma.
###### Lemma 4.6.
For any $v_{k}\in X_{k}^{q}(L^{2}(\Omega))$ it holds
$\displaystyle\left(\partial_{t}v_{k},v_{k}\right)_{I_{m}\times\Omega}+\left([v_{k}]_{m-1},v_{kh,m-1}^{+}\right)_{\Omega}$
$\displaystyle=\frac{1}{2}\left(\|v_{k,m}^{-}\|_{L^{2}(\Omega)}^{2}+\|[v_{k}]_{m-1}\|_{L^{2}(\Omega)}^{2}-\|v_{k,m-1}^{-}\|_{L^{2}(\Omega)}^{2}\right),$
$\displaystyle-\left(v_{k},\partial_{t}v_{k}\right)_{I_{m}\times\Omega}-\left(v_{k,m}^{-},[v_{k}]_{m}\right)_{\Omega}$
$\displaystyle=\frac{1}{2}\left(\|v_{k,m-1}^{+}\|_{L^{2}(\Omega)}^{2}+\|[v_{k}]_{m}\|_{L^{2}(\Omega)}^{2}-\|v_{k,m}^{+}\|_{L^{2}(\Omega)}^{2}\right).$
###### Proof.
For the first equality, we can express the integral over the time derivatives
via
$\left(\partial_{t}v_{k},v_{k}\right)_{I_{m}\times\Omega}=\frac{1}{2}\|v_{k,m}^{-}\|^{2}-\frac{1}{2}\|v_{k,m-1}^{+}\|^{2}.$
Writing $v_{k,m-1}^{+}=[v]_{k,m-1}+v_{k,m-1}^{-}$ and recombining terms gives
the first identity. The proof of the second equality works completely
anologous. ∎
The fully discrete formulation of the transient Navier-Stokes equations, using
the anti symmetrized trilinear form $\hat{c}\left(\cdot,\cdot,\cdot\right)$,
introduced in (11), is now given as: Find $u_{kh}\in V_{kh}$, such that
$\mathfrak{B}(u_{kh},v_{kh})+\hat{c}\left(\\!\left(u_{kh},u_{kh},v_{kh}\right)\\!\right)=\left(f,v_{kh}\right)_{I\times\Omega}+\left(u_{0},v_{kh,0}^{+}\right)_{\Omega}\quad\text{for
all }v_{kh}\in V_{kh}.$ (19)
Since in general $V_{h}\not\subset V$, the discrete solution $u_{kh}\in
V_{kh}$ is not divergence free, and thus, we are not allowed to use it as a
test function for the divergence-free continuous formulation (4). Because of
this, we introduce an equivalent formulation with pressure: Find
$(u_{kh},p_{kh})\in Y_{kh}$, such that for all $(v_{kh},q_{kh})\in Y_{kh}$ it
holds
$B((u_{kh},p_{kh}),(v_{kh},q_{kh}))+\hat{c}\left(\\!\left(u_{kh},u_{kh},v_{kh}\right)\\!\right)=\left(f,v_{kh}\right)_{I\times\Omega}+\left(u_{0},v_{kh,0}^{+}\right)_{\Omega},$
(20)
where the mixed bilinear form $B$ is defined by
$\displaystyle B((u,p),(v,q)):=$
$\displaystyle\sum_{m=1}^{M}\left(\partial_{t}u,v\right)_{I_{m}\times\Omega}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}+\sum_{m=2}^{M}\left([u]_{m-1},v^{+}_{m-1}\right)_{\Omega}+\left(u_{0}^{+},v_{0}^{+}\right)_{\Omega}$
$\displaystyle-\left(\nabla\cdot v,p\right)_{I\times\Omega}+\left(\nabla\cdot
u,q\right)_{I\times\Omega}.$
###### Theorem 4.7.
The two formulations (19) and (20) are equivalent in the sense that, if
$u_{kh}\in V_{kh}$ satisfies (19), then there exists $p_{kh}\in M_{kh}$ such
that $(u_{kh},p_{kh})$ solves (20). Conversely, if $(u_{kh},p_{kh})$ satisfies
(20), then $u_{kh}$ is an element of $V_{kh}$ and satisfies (19).
###### Proof.
This can be shown by using the same arguments as [7, Proposition 4.3]. ∎
### 4.3. Stokes error estimates
Before further analyzing the fully discrete Navier-Stokes equations, let us
recall the discrete formulation of the Stokes equations (8), and the available
error estimates that we wish to extend to the Navier-Stokes equations in this
work. The fully discrete formulation of the instationary Stokes equation
reads: Find $(w_{kh},r_{kh})\in Y_{kh}$ satisfying
$B((w_{kh},r_{kh}),(\phi_{kh},\psi_{kh}))=\left(u_{0},\phi_{kh,0}^{+}\right)_{\Omega}+\left(f,\phi_{kh}\right)_{I\times\Omega}\qquad\text{for
all }(\phi_{kh},\psi_{kh})\in Y_{kh}.$ (21)
In the recent contributions [7, 36], best approximation type error estimates
for the Stokes problem were shown in the norms of
$L^{\infty}(I;L^{2}(\Omega))$, $L^{2}(I;L^{2}(\Omega))$ and
$L^{2}(I;H^{1}(\Omega))$. There hold the following results, formulated in
terms of best approximation error terms and the errors of the projection in
time $\pi_{\tau}$, defined in (18), and the Stokes Ritz projection
$R_{h}^{S}$, defined in (10).
###### Proposition 4.8 ([36, Theorems 6.1& 6.3]).
Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$, let $(w,r)$ and
$(w_{kh},r_{kh})$ be the continuous and fully discrete solutions to the Stokes
problems (8) and (21). Then for any $\chi_{kh}\in V_{kh}$, there holds
$\displaystyle\|w-w_{kh}\|_{L^{2}(I\times\Omega))}$ $\displaystyle\leq
C\left(\|w-\chi_{kh}\|_{L^{2}(I\times\Omega)}+\|w-\pi_{\tau}w\|_{L^{2}(I\times\Omega)}+\|w-R_{h}^{S}(w,r)\|_{L^{2}(I\times\Omega)}\right)$
$\displaystyle\|\nabla(w-w_{kh})\|_{L^{2}(I\times\Omega))}$ $\displaystyle\leq
C\left(\|\nabla(w-\chi_{kh})\|_{L^{2}(I\times\Omega)}+\|\nabla(w-\pi_{\tau}w)\|_{L^{2}(I\times\Omega)}+\|\nabla(w-R_{h}^{S}(w,r))\|_{L^{2}(I\times\Omega)}\right).$
###### Proposition 4.9 ([7, Corollary 6.4]).
Let $f\in L^{s}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V_{1-1/s}$ for some $s>1$,
and let $(w,r)$ and $(w_{kh},r_{kh})$ be the continuous and fully discrete
solutions to the Stokes equations (8) and (21). Then for any $\chi_{kh}\in
V_{kh}$, there holds
$\|w-w_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\left(\ln\frac{T}{k}\right)\left(\|w-\chi_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|w-R_{h}^{S}(w,r)\|_{L^{\infty}(I;L^{2}(\Omega))}\right).$
Having shown these error estimates for the Stokes equations, the natural
question arises, whether these results can be extended to the Navier-Stokes
equations. We first give a positive answer for the $L^{2}(I;H^{1}(\Omega))$
error in Theorem 4.16. The main result of this work is the error estimate in
the $L^{\infty}(I;L^{2}(\Omega))$ norm, presented in Theorem 5.6. With the
same techniques, the proof of the $L^{2}(I;L^{2}(\Omega))$ error estimate is
then straightforward, and we state the result in Theorem 5.7.
### 4.4. Existence and stability results
We begin our analysis of the fully discrete Navier-Stokes equations by
presenting some stability results, followed by an existence result of fully
discrete solutions. To show these, let us first recall the following version
of a discrete Gronwall lemma, which is stated in [32, Lemma 5.1] for a
constant timestep $k$, but its proof can easily be adapted to the setting of
variable timesteps $k_{m}$.
###### Lemma 4.10.
Let $\\{k_{n}\\},\\{a_{n}\\},\\{b_{n}\\},\\{c_{n}\\},\\{\gamma_{n}\\}$ be
sequences of nonnegative numbers and $B\geq 0$ a constant, such that for each
$n\in\mathbb{N}_{0}$ it holds
$a_{n}+\sum_{m=0}^{n}k_{m}b_{m}\leq\sum_{m=0}^{n}k_{m}\gamma_{m}a_{m}+\sum_{m=0}^{n}k_{m}c_{m}+B,$
and $k_{m}\gamma_{m}<1$ for all $m\in\\{0,...,n\\}$. Then with
$\sigma_{m}:=(1-k_{m}\gamma_{m})^{-1}$ it holds
$a_{n}+\sum_{m=0}^{n}k_{m}b_{m}\leq\exp\left(\sum_{m=0}^{n}k_{m}\sigma_{m}\gamma_{m}\right)\cdot\left(\sum_{m=0}^{n}k_{m}c_{m}+B\right).$
###### Proof.
By choosing $\widetilde{k}=1$, $\widetilde{\gamma}_{m}=k_{m}\gamma_{m}$ and
$\widetilde{c}_{m}=k_{m}c_{m}$, the quantities with $\widetilde{\cdot}$
correspond to the notation of [32, Lemma 5.1]. The result then directly
follows from this redefinition. ∎
Continuous and discrete Gronwall lemmas are stated in many different forms in
the literature, see, e.g., [38] and the references therein for an overview
over different generalizations of the original lemma. Note that, since the sum
on the right hand side of the assumed bound goes up to $n$, the additional
assumption $k_{m}\gamma_{m}<1$ is needed. This is not the case in explicit
forms of discrete Gronwall lemmas, where the sum on the right goes only up to
$n-1$, which is the form most often considered in the literature, e.g., [43].
In the context of dG timestepping schemes, the sequences $\\{a_{n}\\}$ and
$\\{b_{n}\\}$ often correspond to squared norms at time nodes or over
subintervals. It therefore seems natural to also state the following version
of a Gronwall lemma, where we also include a sum over non-squared
contributions. To the best of the authors knowledge, a result like this has
not been explicitly used in the literature. In later sections of this work,
this lemma will facilitate the analysis of discrete problems with right hand
sides that are $L^{1}$ in time. Here the norms of the solution occur in a non
squared contribution. The following lemma shows, that the weights of the
squared sum enter exponentially into the estimate, whereas the weights of the
linear sum enter linearly in the estimate for $x_{n}$. It can be understood as
an adaptation of Bihari’s inequality to the discrete setting, see [10, 39]. To
improve readability, we drop the explicit mentioning of the timesteps $k_{m}$,
as we can always apply a transformation as done in the proof of Lemma 4.10.
###### Lemma 4.11.
Let $\\{x_{n}\\},\\{b_{n}\\},\\{c_{n}\\},\\{d_{n}\\},\\{\gamma_{n}\\}$ be
sequences of nonnegative numbers and $B\geq 0$ a constant, such that for each
$n\in\mathbb{N}_{0}$ it holds
$x_{n}^{2}+\sum_{m=0}^{n}b_{m}\leq\sum_{m=0}^{n}\gamma_{m}x_{m}^{2}+\sum_{m=0}^{n}d_{m}x_{m}+\sum_{m=0}^{n}c_{m}+B,$
and $\gamma_{m}<1$ for all $m\in\\{0,...,n\\}$. Then with
$\sigma_{m}:=(1-\gamma_{m})^{-1}$ it holds
$x_{n}^{2}+\sum_{m=0}^{n}b_{m}\leq
2\exp\left(2\sum_{m=0}^{n}\sigma_{m}\gamma_{m}\right)\cdot\left[\left(\sum_{m=0}^{n}d_{m}\right)^{2}+\sum_{m=0}^{n}c_{m}+B\right].$
###### Proof.
For $n\in\mathbb{N}_{0}$ define $\delta_{n}\geq 0$ such that
$x_{n}^{2}+\sum_{m=0}^{n}b_{m}+\delta_{n}=\sum_{m=0}^{n}\gamma_{m}x_{m}^{2}+\sum_{m=0}^{n}d_{m}x_{m}+\sum_{m=0}^{n}c_{m}+B,$
(22)
and set $X_{n}:=\sqrt{x_{n}^{2}+\sum_{m=0}^{n}b_{m}+\delta_{n}}$. We will show
$X_{n}^{2}\leq
2\exp\left(2\sum_{m=0}^{n}\sigma_{m}\gamma_{m}\right)\cdot\left[\left(\sum_{m=0}^{n}d_{m}\right)^{2}+\sum_{m=0}^{n}c_{m}+B\right]\qquad\text{for
all }n\in\mathbb{N}_{0},$
from which the assertion follows by the definition of $X_{n}$. Note that by
assumption it holds
$X_{n}^{2}-X_{n-1}^{2}=\gamma_{n}x_{n}^{2}+d_{n}x_{n}+c_{n}\geq
0\qquad\text{for all }n\in\mathbb{N},$
and thus the sequences $\\{X_{n}^{2}\\}$ and $\\{X_{n}\\}$ are monotonically
increasing. By $X_{n}\geq x_{n}$ and the monotonicity of $X_{n}$, from (22) we
obtain
$\displaystyle X_{n}^{2}$
$\displaystyle\leq\sum_{m=0}^{n}\gamma_{m}X_{m}^{2}+\sum_{m=0}^{n}d_{m}X_{m}+\sum_{m=0}^{n}c_{m}+B$
$\displaystyle\leq\left(\sum_{m=0}^{n}\gamma_{m}X_{m}+\sum_{m=0}^{n}d_{m}\right)X_{n}+\sum_{m=0}^{n}c_{m}+B.$
(23)
Let us now recall, that for $a,b>0$, an estimate $x^{2}\leq ax+b$ implies
$x\leq a+\sqrt{b}$. To see this, we start by computing the roots of the
quadratic polynomial, yielding $x\leq\frac{a}{2}+\sqrt{\frac{a^{2}}{4}+b}$.
The root of the polynomial can then be estimated by
$\frac{a}{2}+\sqrt{\frac{a^{2}}{4}+b}\leq a+\sqrt{b},$ which can be shown by
subtracting $a/2$ and squaring both sides. Applied to equation 23, we have
thus for every $n\in\mathbb{N}_{0}$ the estimate
$X_{n}\leq\sum_{m=0}^{n}\gamma_{m}X_{m}+\sum_{m=0}^{n}d_{m}+\sqrt{\sum_{m=0}^{n}c_{m}+B}=\sum_{m=0}^{n}\gamma_{m}X_{m}+\sum_{m=0}^{n}\tilde{c}_{m}+\tilde{B},$
where we have defined $\tilde{B}:=0$ as well as
$\tilde{c}_{0}:=\left(d_{0}+\sqrt{c_{0}+B}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\geq
0},\quad\text{and}\quad\tilde{c}_{m}:=\left(d_{m}+\sqrt{\sum_{j=0}^{m}c_{j}+B}-\sqrt{\sum_{j=0}^{m-1}c_{j}+B}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\geq
0}\text{ for }m>0.$
We can thus apply Lemma 4.10 to $X_{n}$ and obtain after resubstituting the
tilded quantities:
$X_{n}\leq\exp\left(\sum_{m=0}^{n}\gamma_{m}\sigma_{m}\right)\cdot\left(\sum_{m=0}^{n}\tilde{c}_{m}+\tilde{B}\right)\leq\exp\left(\sum_{m=0}^{n}\gamma_{m}\sigma_{m}\right)\cdot\left(\sum_{m=0}^{n}d_{m}+\sqrt{\sum_{m=0}^{n}c_{m}+B}\right).$
Squaring and estimating the square of the sum yields the result. ∎
With these technical lemmas, we can show stability of the discrete solutions
in different norms under different assumptions on $f$. Solutions to the
discrete problem also satisfy the same energy bounds as the weak solutions,
i.e., there holds the following proposition, see [16, Lemma 5.1, Theorem A.1].
###### Proposition 4.12 (Stability of discrete Navier-Stokes).
Let $f\in
L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)^{2}})$,
$u_{0}\in H$ and $u_{kh}\in V_{kh}$ satisfy (19). Then there hold the bounds
$\displaystyle\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}\leq
C\left(\|u_{0}\|_{H}+\|f\|_{L^{2}(I;H^{-1}(\Omega))}\right),\quad\text{ if
}q=0,$
$\displaystyle\left(\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\right)^{\frac{1}{2}}+\|u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}\leq
C\left(\|u_{0}\|_{H}+\|f\|_{L^{2}(I;H^{-1}(\Omega))}\right),\quad\text{ if
}q\geq 1,$
with a constant $C$ depending on the domain $\Omega$ and the viscosity $\nu$.
Note that for the case $q\geq 1$, contrary to the continuous setting of
Proposition 3.1, due to the exponent $\frac{1}{2}$ on the left hand side, the
above proposition states a bound for
$\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}$, which depends on the squared norms
of the data $u_{0}$ and $f$. For the two low order cases $q=0$ and $q=1$, and
$f\in L^{2}(I;L^{2}(\Omega)^{2})$, [16, Lemma 5.1] also shows an estimate of
the form
$\max_{1\leq m\leq
M}\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}+\|u_{kh}\|_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(I;H^{1}(\Omega))}}\leq
C_{1}e^{C_{2}T}\left(\|u_{0}\|_{H}+\|f\|_{L^{2}(I;L^{2}(\Omega))}\right).$
With our discrete Gronwall estimate Lemma 4.11, we can generalize this result
to $f\in L^{1}(I;L^{2}(\Omega)^{2})$. Furthermore, by using the version of
Gronwall’s lemma presented here, contrary to [16], the bound does not grow
exponentially in $T$. This is in agreement with the result of Proposition 3.1
in the continuous setting. We obtain the following result, which now yields an
estimate for the norms of $u_{kh}$, that is linear in terms of the data.
###### Theorem 4.13.
Let $f\in
L^{1}(I;L^{2}(\Omega)^{2})+L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)^{2}})$,
$u_{0}\in H$ and $u_{kh}\in V_{kh}$ satisfy equation 19 for either $q=0$ or
$q=1$. Then there holds the bound
$\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{\nu}}\|u_{kh}\|_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(I;H^{1}(\Omega))}}\leq
C\left(\|u_{0}\|_{H}+\|f\|_{L^{1}(I;L^{2}(\Omega))+L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)})}\right),$
with a constant $C$ only depending on $\Omega,\nu$. If $f\in
L^{1}(I;L^{2}(\Omega)^{2})$, then $C$ only depends on $\Omega$.
###### Proof.
We first prove the result for $f\in L^{1}(I;L^{2}(\Omega)^{2})$ and remark at
the end, which modifications are needed to also cover the case, where $f$ is
partly in $L^{2}(I;H^{-1}(\Omega)^{2})$. Note that in the special case of
$f\in L^{2}(I;H^{-1}(\Omega)^{2})$ and $q=0$, this theorem is precisely the
first statement of [16, Lemma 5.1]. For notational simplicity, we use the
convention $u_{kh,0}^{-}:=u_{0}$ and accordingly
$[u_{kh}]_{0}=u_{kh,0}^{+}-u_{0}$. By testing (19) with $u_{kh}|_{I_{m}}$ and
using Lemma 4.3 we arrive on each time interval at:
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}u_{kh},u_{kh}\right)_{I_{m}\times\Omega}}+\nu\left(\nabla
u_{kh},\nabla
u_{kh}\right)_{I_{m}\times\Omega}+\left([u_{kh}]_{m-1},u_{kh,m-1}^{+}\right)_{\Omega}=\left(f,u_{kh}\right)_{I_{m}\times\Omega}.$
Applying Lemma 4.6 gives
$\frac{1}{2}\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
u_{kh}\|_{L^{2}(I_{m}\times\Omega)}^{2}=\frac{1}{2}\|u_{kh,m-1}^{-}\|_{L^{2}(\Omega)}^{2}+\left(f,u_{kh}\right)_{I_{m}\times\Omega}.$
Multiplying by two and summing up the identity over the intervals $1,...,n$,
we obtain
$\|u_{kh,n}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+2\nu\|\nabla
u_{kh}\|^{2}_{L^{2}(I_{m}\times\Omega)}=\|u_{kh,0}^{-}\|_{L^{2}(\Omega)}^{2}+2\sum_{m=1}^{n}\left(f,u_{kh}\right)_{I_{m}\times\Omega}.$
(24)
From this point on, we have to treat the two cases $q=0$ and $q=1$ separately.
Case 1: If $q=0$, then it holds
$\|u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}=\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}$.
Thus with Hölder’s inequality, the terms in the last sum of (24) can be
estimated as
$\left(f,u_{kh}\right)_{I_{m}\times\Omega}\leq\|f\|_{L^{1}(I_{m};L^{2}(\Omega))}\|u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\leq\|f\|_{L^{1}(I_{m};L^{2}(\Omega))}\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}.$
Hence, from (24) we obtain the following
$\|u_{kh,n}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+2\nu\|\nabla
u_{kh}\|^{2}_{L^{2}(I_{m}\times\Omega)}\leq\|u_{0}\|_{L^{2}(\Omega)}^{2}+2\sum_{m=1}^{n}\|f\|_{L^{1}(I_{m};L^{2}(\Omega))}\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}.$
An application of Lemma 4.11 proves the assertion.
Case 2: If $q=1$, then the $L^{\infty}(I_{m})$ norm can be estimated by the
evaluation at the two endpoints of the interval:
$\|u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\leq\max\\{\|u_{kh,m}^{-}\|_{L^{2}(\Omega)},\|u_{kh,m-1}^{+}\|_{L^{2}(\Omega)}\\}.$
With triangle inequality we can estimate the right sided limit in terms of a
left sided limit and a jump:
$\|u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\leq\max\\{\|u_{kh,m}^{-}\|_{L^{2}(\Omega)},\|u_{kh,m-1}^{-}\|_{L^{2}(\Omega)}+\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}\\}.$
Defining
$x_{n}^{2}:=\|u_{kh,n}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}$
for $n=1,...,M$ yields from (24) the estimate
$x_{n}^{2}+2\nu\sum_{m=1}^{n}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\leq\|u_{0}\|_{L^{2}(\Omega)}^{2}+2\sum_{m=1}^{n}\|f\|_{L^{1}(I_{m};L^{2}(\Omega))}(x_{m}+x_{m-1}),$
or after an index shift
$x_{n}^{2}+2\nu\sum_{m=1}^{n}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\leq\|u_{0}\|_{L^{2}(\Omega)}^{2}+2\|f\|_{L^{1}(I_{1};L^{2}(\Omega))}\|u_{0}\|+2\sum_{m=1}^{n}\|f\|_{L^{1}(I_{m}\cup
I_{m+1};L^{2}(\Omega))}\cdot x_{m}.$
In order to shorten the notation, we have added here a term
$\|f\|_{L^{1}(I_{n+1};L^{2}(\Omega))}x_{n}$, where in case $n=M$ we use the
convention $I_{M+1}=\emptyset$. Again we can apply Lemma 4.11, which yields
the estimate
$x_{n}^{2}+2\nu\sum_{m=1}^{n}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\leq
2\left[\left(2\sum_{m=1}^{n}\|f\|_{L^{1}(I_{m}\cup
I_{m+1};L^{2}(\Omega))}\right)^{2}+\|u_{0}\|_{L^{2}(\Omega)}^{2}+2\|f\|_{L^{1}(I_{1};L^{2}(\Omega))}\|u_{0}\|\right].$
Using Young’s inequality, to estimate the term
$2\|f\|_{L^{1}(I_{1};L^{2}(\Omega))}\|u_{0}\|$, concludes the proof. In case
$f=f_{1}+f_{2}$, with
$f_{1}\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}(I;L^{2}(\Omega)^{2})}$,
$f_{2}\in
L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)^{2}})$,
the contribution of $f_{1}$ can be treated as above. For the $f_{2}$
contribution, in (24) one applies Hölder’s and Young’s inequalities, and
absorbs the $\|\nabla u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}$ term to the left.
Note that in both cases $q=0,1$, in the estimates before the application of
Lemma 4.11, no $x_{m}^{2}$ terms are summed on the right hand side, thus no
exponential dependency is introduced in the final estimate. ∎
The techniques presented above now allow us to show the existence of solutions
to the discrete equations (19).
###### Theorem 4.14.
Let $u_{0}\in H$ and $f\in
L^{1}(I;L^{2}(\Omega)^{2})+L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)^{2}})$.
Let further either $q=0$ or $q=1$. Then there exists a solution $u_{kh}\in
V_{kh}$ of (19).
###### Proof.
The existence can be shown by applying a standard fixpoint argument, using the
stability result of Theorem 4.13. ∎
###### Remark 4.15.
Note that uniqueness of solutions $u_{kh}$ to (19) is not known at this point.
We shall show this later in Theorem 4.19, as we first need to obtain a first
convergence result, that allows us to apply Lemma 4.11.
### 4.5. $L^{2}(I;H^{1}(\Omega))$ error estimates
With the stability results presented thus far, we can show a first error
estimate in the norm of $L^{2}(I;H^{1}(\Omega))$. For this norm, [16, Theorem
5.2] shows a result, estimating the error for the Navier-Stokes equations in
terms of the error of a Stokes problem. The discrete Stokes problem there is
defined with right hand side $\partial_{t}u-\nu\Delta u$, which corresponds to
$f-\nabla p-(u\cdot\nabla)u$, i.e., the pressure is included on the right.
This means, that when applying the corresponding orthogonality relations, a
pressure term remains. In the following result, we use a different right hand
side for the discrete Stokes problem, yielding the following error estimate
for the Navier-Stokes equations in $L^{2}(I;H^{1}(\Omega))$.
###### Theorem 4.16.
Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$, $u_{0}\in V$ and let $(u,p)$,
$(u_{kh},p_{kh})$ solve the continuous and fully discrete Navier-Stokes
equations (9) and (20) respectively. Further let $k$ be small enough. Then for
any $\chi_{kh}\in V_{kh}$, there holds
$\displaystyle\|\nabla(u-u_{kh})\|_{L^{2}(I\times\Omega)}$ $\displaystyle\leq
C\left(\|\nabla(u-\chi_{kh})\|_{L^{2}(I\times\Omega)}+\|\nabla(u-\pi_{\tau}u)\|_{L^{2}(I\times\Omega)}+\|\nabla(u-R_{h}^{S}(u,p))\|_{L^{2}(I\times\Omega)}\right).$
###### Proof.
The proof uses the same arguments as the one of [16, Theorem 5.2]. We repeat
the main steps, in order to motivate, why due to the choice of our
instationary Stokes projection, no error term for the pressures arises on the
right hand side. We denote by $e:=u-u_{kh}$ the error, which we want to
estimate, and introduce $(\tilde{u}_{kh},\tilde{p}_{kh})\in Y_{kh}$ as the
instationary Stokes-projection of $(u,p)$, i.e., the solution to
$B((\tilde{u}_{kh},\tilde{p}_{kh}),(\phi_{kh},\psi_{kh}))=\left(f,v_{kh}\right)_{I\times\Omega}-c\left(\\!\left(u,u,v_{kh}\right)\\!\right)+\left(u_{0},v_{kh,0}^{+}\right)_{\Omega}\quad\text{for
all }(\phi_{kh},\psi_{kh})\in Y_{kh}.$ (25)
Note that for the definition of this equation, we can equivalently choose $c$
or $\hat{c}$, as we have $u$ with $\nabla\cdot u=0$ in its first argument.
There exists a solution $(\tilde{u}_{kh},\tilde{p}_{kh})$ to this discrete
problem, since the assumptions on $f$ and $u_{0}$ yield $(u\cdot\nabla)u\in
L^{2}(I;L^{2}(\Omega)^{2})$, see Theorem 3.8. Furthermore, due to [16, Theorem
4.10], the solution $\tilde{u}_{kh}$ to the above discrete Stokes problem
satisfies a bound
$\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}\leq C,$ (26)
with $C$ depending on $f,u_{0}$, but independent of $k,h$. We split the
velocity error into two components $e=\xi+\eta_{kh}$ where
$\xi:=u-\tilde{u}_{kh}$ and $\eta_{kh}:=\tilde{u}_{kh}-u_{kh}$. The error in
the pressure will be denoted by $r:=p-p_{kh}$, and we will split it into the
same contributions, $r=\omega+\kappa_{kh}$, with $\omega:=p-\tilde{p}_{kh}$
and $\kappa_{kh}:=\tilde{p}_{kh}-p_{kh}$. The error of the Stokes projection
$\xi$ is then bounded by Proposition 4.8 since by Theorem 3.8 it holds
$f-(u\cdot\nabla)u\in L^{2}(I;L^{2}(\Omega)^{2})$. It remains to estimate
$\eta_{kh}$. By definition, we have the following Galerkin orthogonalities:
for all $(\phi_{kh},\psi_{kh})\in Y_{kh}$ it holds
$\displaystyle
B((u-u_{kh},p-p_{kh}),(\phi_{kh},\psi_{kh}))+\hat{c}\left(\\!\left(u,u,\phi_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh},u_{kh},\phi_{kh}\right)\\!\right)$
$\displaystyle=0,$ $\displaystyle B((\xi,\omega),(\phi_{kh},\psi_{kh}))$
$\displaystyle=0.$
Subtracting the two and choosing
$(\phi_{kh},\psi_{kh})=(\eta_{kh},\kappa_{kh})$ yields
$B((\eta_{kh},\kappa_{kh}),(\eta_{kh},\kappa_{kh}))+\hat{c}\left(\\!\left(u,u,\eta_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh},u_{kh},\eta_{kh}\right)\\!\right)=0.$
(27)
The two trilinear forms in (27) satisfy the relation
$\hat{c}\left(\\!\left(u,u,\eta_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh},u_{kh},\eta_{kh}\right)\\!\right)=\hat{c}\left(\\!\left(\xi,u,\eta_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\tilde{u}_{kh},\xi,\eta_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\eta_{kh},\tilde{u}_{kh},\eta_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(u_{kh},\eta_{kh},\eta_{kh}\right)\\!\right),$
Due to Lemma 4.3, the last term on the right hand side vanishes, and for the
remaining ones, we obtain by Hölder’s and Young’s inequalities
$\displaystyle\hat{c}\left(\\!\left(\xi,u,\eta_{kh}\right)\\!\right)$
$\displaystyle\leq\sum_{m=1}^{M}\|\xi\|_{L^{2}(I_{m};H^{1}(\Omega))}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))}$
$\displaystyle\leq\sum_{m=1}^{M}C\|\xi\|_{L^{2}(I_{m};H^{1}(\Omega))}^{2}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2}+\frac{\nu}{4}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))}^{2},$
$\displaystyle\hat{c}\left(\\!\left(\xi,\tilde{u}_{kh},\eta_{kh}\right)\\!\right)$
$\displaystyle\leq\sum_{m=1}^{M}\|\xi\|_{L^{2}(I_{m};H^{1}(\Omega))}\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))}$
$\displaystyle\leq\sum_{m=1}^{M}C\|\xi\|_{L^{2}(I_{m};H^{1}(\Omega))}^{2}\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}^{2}+\frac{\nu}{4}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))},$
$\displaystyle\hat{c}\left(\\!\left(\eta_{kh},\tilde{u}_{kh},\eta_{kh}\right)\\!\right)$
$\displaystyle\leq\sum_{m=1}^{M}\int_{I_{m}}\|\eta_{kh}\|_{H^{1}(\Omega)}^{\frac{3}{2}}\|\eta_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|\tilde{u}_{kh}\|_{H^{1}(\Omega)}\
dt$
$\displaystyle\leq\sum_{m=1}^{M}C\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}^{4}k_{m}\|\eta_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}+\frac{\nu}{4}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega)}^{2}.$
Due to $u_{kh}$ and $\tilde{u}_{kh}$ both being discretely divergence free,
all pressure contributions in (27) vanish, and with the above estimates, after
absorbing all $\frac{\nu}{4}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))}$ terms
to the left, we obtain
$\displaystyle\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}\eta_{kh},\eta_{kh}\right)_{I_{m}\times\Omega}}+\frac{\nu}{4}\left(\nabla\eta_{kh},\nabla\eta_{kh}\right)_{I_{m}\times\Omega}+\sum_{m=1}^{M}\left([\eta_{kh}]_{m-1},\eta_{kh,m-1}^{+}\right)_{\Omega}$
$\displaystyle\leq
C\left(\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2}+\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}^{2}\right)\|\xi\|_{L^{2}(I;H^{1}(\Omega))}^{2}+C\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}^{4}\sum_{m=1}^{M}k_{m}\|\eta_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}.$
To abbreviate notation we have used here $[\eta_{kh}]_{0}:=\eta_{kh,0}^{+}$.
Choosing $k$ small enough, and using the boundedness of
$\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}$, due to (26) as well as of
$\|u\|_{L^{\infty}(I;H^{1}(\Omega))}$, shown in Theorem 3.6, allows us to
apply the discrete Gronwall Lemma 4.11 as in Theorem 4.13, which concludes the
proof. ∎
### 4.6. Uniqueness and strong stability of discrete solutions
We next show the uniqueness and stability of the discrete solution $u_{kh}$
in stronger norms, comparable to the continuous result of Theorem 3.6. These
results will be obtained, by applying the discrete Gronwall Lemma 4.11, to
which end we need the two following technical lemmas, guaranteeing, that the
coefficients in the Gronwall lemma become arbitrarily small, uniformly in $m$,
as $(k,h)\to 0$.
###### Lemma 4.17.
Let $I\subset\mathbb{R}$ a bounded interval, $\varepsilon>0$ and $\zeta\in
L^{1}(I)$. Then
$\sup_{x\in\bar{I}}\int_{x}^{x+\varepsilon}|\zeta(s)|\
ds\xrightarrow{\varepsilon\to 0}0.$
To keep notation simple, we formally extend $\zeta$ to $[0,T+\varepsilon]$ by
values $0$ such that the integration is well defined.
###### Proof.
The proof relies on dominated convergence and Dini’s theorem, see, e.g., [12,
p. 125]. We first define for fixed $\varepsilon$ the function
$\sigma_{\varepsilon}:\bar{I}\to\mathbb{R},x\mapsto\int_{x}^{x+\varepsilon}|\zeta(s)|\
ds$. Note that for each $\varepsilon>0$ this function $\sigma_{\varepsilon}$
is continuous. To see this, let $x\in I$ be fixed and let $y\to x$. W.l.o.g.
let $y>x$:
$|\sigma_{\varepsilon}(x)-\sigma_{\varepsilon}(y)|\leq\left|\int_{x}^{y}|\zeta(s)|\
ds+\int_{x+\varepsilon}^{y+\varepsilon}|\zeta(s)|\
ds\right|=\left|\int_{I}\chi_{x,y}(s)|\zeta(s)|\ ds\right|$
where $\chi_{x,y}(s)=1$ if $s\in(x,y)\cup(x+\varepsilon,y+\varepsilon)$, and
$0$ otherwise. For a fixed $s$, the integrand $\chi_{x,y}(s)|\zeta(s)|\to 0$
as $y\to x$, thus the integrand converges pointwise to $0$. By the dominated
convergence theorem, this means that the integral converges to $0$. For fixed
$\varepsilon>0$ this shows the continuity of $\sigma_{\varepsilon}$. Moreover,
by the same argument we can show, that for fixed $x$ and $\varepsilon\to 0$ it
holds $\sigma_{\varepsilon}(x)\to 0$. Thus the sequence $\sigma_{\varepsilon}$
converges pointwise to $0$. Note moreover, that for $\delta>\varepsilon$, and
fixed $x$, it holds $\sigma_{\delta}(x)\geq\sigma_{\varepsilon}(x)$, as the
integration covers a larger interval. We thus have shown, that when
$\varepsilon\to 0$ monotonically, also this pointwise convergence of
$\sigma_{\varepsilon}(x)$ is monotone. Hence we can apply Dini’s theorem, see
[12, p. 125], to obtain
$\|\sigma_{\varepsilon}\|_{L^{\infty}(I)}\rightarrow 0,\quad\text{ as
}\varepsilon\rightarrow 0.$
∎
###### Lemma 4.18.
Let $u\in L^{2}(I;H^{1}_{0}(\Omega)^{2})$, and $u_{kh}\in U_{kh}$ such that
$\|u-u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}\to 0$ as $(k,h)\to 0$. Then it holds
$\displaystyle\sup_{1\leq m\leq M}\int_{I_{m,k}}\|\nabla
u\|_{L^{2}(\Omega)}^{2}\ dt\xrightarrow{(k,h)\to
0}0\quad\text{and}\quad\sup_{1\leq m\leq M}\int_{I_{m,k}}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}^{2}\ dt\xrightarrow{(k,h)\to 0}0.$
###### Proof.
We first show the statement for $u$. To this end, note that for $k=\max_{1\leq
m\leq M}|I_{m,k}|$
$\sup_{1\leq m\leq M}\int_{I_{m,k}}\|\nabla u\|_{L^{2}(\Omega)}^{2}\
dt\leq\sup_{x\in\bar{I}}\int_{x}^{x+k}\|\nabla u\|_{L^{2}(\Omega)}^{2}\
dt\xrightarrow{k\to 0}0$
by Lemma 4.17. In order to show the result for the discrete solution $u_{kh}$,
we cannot directly apply the previous lemma, as $u_{kh}$ depends on $k$. We
thus insert insert $\pm u$ and apply the triangle inequality to obtain
$\displaystyle\sup_{1\leq m\leq M}\int_{I_{m,k}}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}^{2}\ dt$ $\displaystyle\leq 2\sup_{1\leq m\leq
M}\int_{I_{m,k}}\|\nabla u\|_{L^{2}(\Omega)}^{2}\
dt+2\int_{I}\|\nabla(u-u_{kh})\|_{L^{2}(\Omega)}^{2}\ dt.$
With the claim for $u$ and $u_{kh}\to u$ in $L^{2}(I;H^{1}(\Omega))$, we have
shown the claim for $u_{kh}$. ∎
With these technical lemmas, we can show two results, the uniqueness of
solutions $u_{kh}$ and their boundedness in stronger norms. We first obtain
the following uniqueness result.
###### Theorem 4.19.
Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$. Then for $(k,h)$ small
enough, the solution $u_{kh}$ to (19) is unique.
###### Proof.
To show uniqueness, we assume two solutions $u_{kh}^{1},u_{kh}^{2}$ and define
$e_{kh}:=u_{kh}^{1}-u_{kh}^{2}$. Testing the equations for $m=1,...,M$ with
$e_{kh}\chi_{m}$, where $\chi_{m}$ denotes the characteristic function of the
interval $I_{m}$, and subtracting them leads to
$\mathfrak{B}(e_{kh},e_{kh}\chi_{m})+\hat{c}\left(\\!\left(u_{kh}^{1},u_{kh}^{1},e_{kh}\chi_{m}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh}^{2},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)=0$
Adding and subtracting
$\hat{c}\left(\\!\left(u_{kh}^{1},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)$
and applying Lemma 4.3 yields
$\mathfrak{B}(e_{kh},e_{kh}\chi_{m})+\hat{c}\left(\\!\left(e_{kh},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)=0.$
Applying Lemma 4.6, the bilinear form can be written as
$\mathfrak{B}(e_{kh},e_{kh}\chi_{m})=\frac{1}{2}\left(\|e_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\|[e_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}-\|e_{kh,m-1}^{-}\|_{L^{2}(\Omega)}^{2}\right)+\nu\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}.$
From the definition of $\hat{c}$, the estimates of Lemma 3.2, and Hölder’s
inequality in time, we obtain
$\displaystyle\hat{c}\left(\\!\left(e_{kh},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)$
$\displaystyle=\frac{1}{2}\left[\left((e_{kh}\cdot\nabla)u_{kh}^{2},e_{kh}\right)_{I_{m}\times\Omega}-\left((e_{kh}\cdot\nabla)e_{kh},u_{kh}^{2}\right)_{I_{m}\times\Omega}\right]$
$\displaystyle\leq C\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}$
$\displaystyle\quad+C\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{\frac{3}{2}}\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}.$
We can apply Young’s inequality and obtain
$\displaystyle\hat{c}\left(\\!\left(e_{kh},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)$
$\displaystyle\leq\frac{\nu}{2}\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}+C\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\left(1+\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right)\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}.$
where the first summand can be absorbed into
$\mathfrak{B}(e_{kh},e_{kh}\chi_{m})$. Thus on each time interval, there holds
the bound
$\displaystyle\frac{1}{2}\left(\|e_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\|[e_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}-\|e_{kh,m-1}^{-}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\right)$ $\displaystyle\leq
C\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\left(1+\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right)\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}.$
Multiplying the derived inequality by 2 and summing up the inequalities for
$m=1,...,n$ yields for any $n\in\\{1,...,M\\}$
$\displaystyle\|e_{kh,n}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\left(\|[e_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\right)$ $\displaystyle\leq
C\sum_{m=1}^{n}\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\left(1+\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right)\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}.$
Here we have used $e_{kh,0}^{-}=0$ as $u_{kh}^{1}$ and $u_{kh}^{2}$ satisfy
the same initial condition. As in the proof of Theorem 4.13,
$\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}$ can be bounded by evaluations
of one-sided limits at the time nodes. Due to $f\in
L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$, Theorem 4.16 holds, and thus
Lemma 4.18 yields $\|\nabla u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\to
0$ uniformly in $m$ for $(k,h)\to 0$. Together with Theorem 4.13, bounding
$\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}$, we can choose the
discretization fine enough, such that
$C(1+\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2})\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}<\frac{1}{2}$ for all $m$. All in
all, as in Theorem 4.13, an application of the Gronwall Lemma 4.11 shows that
$e_{kh}=0$, concluding the proof of uniqueness. ∎
We now turn toward proving the stability of $u_{kh}$ in stronger norms. The
proof follows the steps of the continuous result, shown in [48, Chapter 3,
Theorem 3.10], using the discrete analogons of the inequalities presented in
Section 4.1, and the discrete Gronwall Lemma 4.11. For the application of the
lemma, we need coefficients, that become small, as $k\to 0$. These
coefficients depend on $u_{kh}$, and thus on $k,h$, hence we apply Theorem
4.16 together with Lemma 4.18, in order to have coefficients, that converge to
0 uniformly in $m$. There holds the following.
###### Theorem 4.20.
Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$, then for $(k,h)$ small
enough, the unique solution $u_{kh}\in V_{kh}$ to (19) satisfies
$\displaystyle\|u_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}+\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}\leq$
$\displaystyle
C_{1}\exp{\left(C_{2}\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}\|\nabla
u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}^{2}\right)}$
$\displaystyle\times\left(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{H^{1}(\Omega)}\right),$
with constants $C_{1},C_{2}$ independent of $k,h$. The
$L^{\infty}(I;L^{2}(\Omega))$ and $L^{2}(I;H^{1}(\Omega))$ norms of $u_{kh}$
can be estimated by the results of Theorem 4.13.
###### Proof.
For $m=1,...,M$, we test (19) with $A_{h}u_{kh}|_{I_{m}}$, which yields with
the convention $[u_{kh}]_{0}:=u_{kh,0}^{+}-u_{0}$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}u_{kh},A_{h}u_{kh}\right)_{I_{m}\times\Omega}}+\nu\left(\nabla
u_{kh},\nabla A_{h}u_{kh}\right)_{I_{m}\times\Omega}$
$\displaystyle+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left([u_{kh}]_{m-1},A_{h}u_{kh,m-1}^{+}\right)_{\Omega}}+\hat{c}\left(\\!\left(u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}\right)\\!\right)$
$\displaystyle=\left(f,A_{h}u_{kh}\right)_{I_{m}\times\Omega},$
From the above identity, for $m=2,...,M$, the definition of $A_{h}$ gives
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}\nabla
u_{kh},\nabla
u_{kh}\right)_{I_{m}\times\Omega}}+\nu\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$
(28)
$\displaystyle+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left([\nabla
u_{kh}]_{m-1},\nabla
u_{kh,m-1}^{+}\right)_{\Omega}}+\hat{c}\left(\\!\left(u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}\right)\\!\right)$
$\displaystyle=\left(f,A_{h}u_{kh}\right)_{I_{m}\times\Omega}.$
Here the terms containing time derivatives and jumps can be combined according
to Lemma 4.6. Some care has to be taken on the first time interval. It holds
$\displaystyle\left(u_{0},A_{h}u_{kh,0}^{+}\right)_{\Omega}$
$\displaystyle=-\left(u_{0},\mathbb{P}_{h}\Delta_{h}u_{kh,0}^{+}\right)_{\Omega}=-\left(\mathbb{P}_{h}u_{0},\Delta_{h}u_{kh,0}^{+}\right)_{\Omega}=\left(\nabla\mathbb{P}_{h}u_{0},\nabla
u_{kh,0}^{+}\right)_{\Omega}$
Thus on the first time interval, with the same arguments as in the proof of
Lemma 4.6, it holds
$\left(\partial_{t}u_{kh},A_{h}u_{kh}\right)_{I_{1}\times\Omega}+\left(u_{kh,0}^{+}-u_{0},A_{h}u_{kh,0}^{+}\right)_{\Omega}=\frac{1}{2}\left(\|\nabla
u_{kh,1}^{-}\|_{L^{2}(\Omega)}^{2}+\|\nabla(u_{kh,0}^{+}-\mathbb{P}_{h}u_{0})\|_{L^{2}(\Omega)}^{2}-\|\nabla\mathbb{P}_{h}u_{0}\|_{L^{2}(\Omega)}^{2}\right).$
For a more compact notation, we shall write $[\nabla
u_{kh}]_{0}:=\nabla(u_{kh,0}^{+}-\mathbb{P}_{h}u_{0})$. Note the need to
include the projection on the right hand side. With this slight abuse of
notation, (28) also holds on the first time interval. Since $u_{0}\in V$ we
can use the stability of $\mathbb{P}_{h}$ in $H^{1}$ for continuously
divergence free functions, see [36, Lemma 5.4], yielding an estimate
$\|\nabla\mathbb{P}_{h}u_{0}\|_{L^{2}(\Omega)}\leq C\|\nabla
u_{0}\|_{L^{2}(\Omega)}.$ (29)
For the terms of (28) involving the right hand side $f$, it holds
$\left(f,A_{h}u_{kh}\right)_{I_{m}\times\Omega}\leq\|f\|_{L^{2}(I_{m};L^{2}(\Omega))}\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}\leq
C\|f\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}+\frac{\nu}{4}\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2},$
where we can absorb the $A_{h}u_{kh}$ term. We now turn towards estimating the
trilinear form remaining in (28), which poses the main difficulty of this
proof. Due to Lemma 4.4, the trilinear term satisfies the expression
$\hat{c}\left(\\!\left(u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}\right)\\!\right)=\left((u_{kh}\cdot\nabla)u_{kh},A_{h}u_{kh}\right)_{I_{m}\times\Omega}+\frac{1}{2}\left(\nabla\cdot
u_{kh},u_{kh}\cdot A_{h}u_{kh}\right)_{I_{m}\times\Omega},$
which we can estimate with Hölder’s inequality by
$\hat{c}\left(\\!\left(u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}\right)\\!\right)\leq
C\int_{I_{m}}\|u_{kh}\|_{L^{\infty}(\Omega)}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}\|A_{h}u_{kh}\|_{L^{2}(\Omega)}\ dt.$
Using (16) and Young’s inequality, we obtain the estimate
$\hat{c}\left(\\!\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}}\right)\\!\right)\leq\int_{I_{m}}C\|u_{kh}\|_{L^{2}(\Omega)}^{2}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}^{4}+\frac{\nu}{4}\|A_{h}u_{kh}\|_{L^{2}(\Omega)}^{2}\
dt.$
The latter term can be absorbed, hence we proceed by discussing the first one.
It holds
$\int_{I_{m}}C\|u_{kh}\|_{L^{2}(\Omega)}^{2}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}^{4}\ dt\leq
C\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\|\nabla
u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}.$
Let us introduce
$\gamma_{m}:=C\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$. All in all, we have derived from
(28) an estimate of the form
$\displaystyle\frac{1}{2}\left(\|\nabla
u_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\|[\nabla
u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}-\|\nabla
u_{kh,m-1}^{-}\|_{L^{2}(\Omega)}^{2}\right)+\frac{1}{2}\nu\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$
$\displaystyle\leq\|f\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}+\gamma_{m}\|\nabla
u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2},$
with the usual modifications on the first time interval. Multiplying this by 2
and summing up from $m=1,...,n$ yields for any $n\in\\{1,...,M\\}$ an estimate
of the form
$\displaystyle\|\nabla
u_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\left(\|[\nabla
u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+\nu\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\right)$
$\displaystyle\leq C\|\nabla
u_{0}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\left(\|f\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}+2\gamma_{m}\|\nabla
u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right),$
where due to (29) we have dropped the projection $\mathbb{P}_{h}$ of the
initial data on the right hand side. Since $f\in L^{2}(I;L^{2}(\Omega)^{2})$
and $u_{0}\in V$, Theorem 4.16 shows that $u_{kh}\to u$ in
$L^{2}(I;H^{1}(\Omega))$ as $(k,h)\to 0$. Hence, due to Lemma 4.18, and
Theorem 4.13, it holds $\gamma_{m}\to 0$ uniformly, as $(k,h)\to 0$. This
implies that by following the same steps as the proof of Theorem 4.13, we
obtain the result as a consequence of the discrete Gronwall Lemma 4.11. ∎
## 5\. Error estimates
### 5.1. Duality based best approximation type estimates
In this concluding section of our work, we will show the main results, which
are the $L^{\infty}(I;L^{2}(\Omega))$ and $L^{2}(I;L^{2}(\Omega))$ error
estimates. Their proofs are based on duality arguments, and hence we begin
this section by showing two stability results for discrete dual equations. Let
us first motivate the specific dual problem, that we will consider. Due to the
nonlinear structure of the Navier-Stokes equations, there does not hold a
Galerkin orthogonality with respect to the bilinear form $B$, i.e., for
solutions $(u,p)$ of (9) and their discrete counterparts $(u_{kh},p_{kh})$ of
(20), it holds for test functions $(v_{kh},q_{kh})\in Y_{kh}$:
$B((u-u_{kh},p-p_{kh}),(v_{kh},q_{kh}))=-\hat{c}\left(\\!\left(u,u,v_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(u_{kh},u_{kh},v_{kh}\right)\\!\right),$
where the right hand side is nonzero in general. Since we want to use this
orthogonality relation after testing the dual equation with $u-u_{kh}$, we
need to reformulate the trilinear terms, such that $u-u_{kh}$ occurs linearly.
To this end, we use the identity
$\hat{c}\left(\\!\left(u,u,v_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh},u_{kh},v_{kh}\right)\\!\right)=\hat{c}\left(\\!\left(\overline{uu_{kh}},u-u_{kh},v_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(u-u_{kh},\overline{uu_{kh}},v_{kh}\right)\\!\right),$
(30)
where we linearize around the average of continuous and discrete solutions to
the Navier-Stokes equations
$\overline{uu_{kh}}:=\frac{1}{2}(u+u_{kh}).$ (31)
With these considerations, we have the following lemma:
###### Lemma 5.1.
Let $(u,p)$ be a solution to the Navier-Stokes equations (9), and
$(u_{kh},p_{kh})$ their discrete approximation solving (20). Let further
$\overline{uu_{kh}}$ denote the average of $u$ and $u_{kh}$ as defined by
(31). Then for any $(v_{kh},q_{kh})\in Y_{kh}$, it holds
$B((u-u_{kh},p-p_{kh}),(v_{kh},q_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},u-u_{kh},v_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(u-u_{kh},\overline{uu_{kh}},v_{kh}\right)\\!\right)=0.$
###### Proof.
This result is an immediate consequence of the definitions of solutions to (9)
and (20), together with the identity (30) ∎
This motivates the choice of $\overline{uu_{kh}}$ as linearization point for
setting up a dual equation:
Find $(z_{kh},\varrho_{kh})\in Y_{kh}$ such that for all
$(\phi_{kh},\psi_{kh})\in Y_{kh}$ it holds
$B((\phi_{kh},\psi_{kh}),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left\langle
g,\phi_{kh}\right\rangle_{I\times\Omega},$ (32)
where the right hand side $g$ will be chosen appropriately, see the proofs of
Theorems 5.6 and 5.7. Note that the right hand side of (32) implicitly
prescribes the final data $z_{kh,M}^{+}=0$. This dual equation will help us in
deriving the sought error estimates, which we will do in the following
section.
###### Remark 5.2.
To analyze this dual problem, it will be convenient, to have dual
represenations of $\mathfrak{B}$ and $B$ at hand, which are obtained by
partial integration on each $I_{m}$ and rearranging the terms. Note that in
this representation, the time derivative is applied to the second argument. It
holds
$\hskip
39.83385pt\mathfrak{B}(u,v)=-\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(u,\partial_{t}v\right)_{I_{m}\times\Omega}}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}-\sum_{m=1}^{M-1}\left(u_{m}^{-},[v]_{m}\right)_{\Omega}+\left(u_{M}^{-},v_{M}^{-}\right)_{\Omega},$
(33) $\displaystyle B((u,p),(v,q))=$
$\displaystyle-\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(u,\partial_{t}v\right)_{I_{m}\times\Omega}}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}-\sum_{m=1}^{M-1}\left(u_{m}^{-},[v]_{m}\right)_{\Omega}+\left(u_{M}^{-},v_{M}^{-}\right)_{\Omega}$
(34) $\displaystyle+\left(\nabla\cdot
u,q\right)_{I\times\Omega}-\left(\nabla\cdot v,p\right)_{I\times\Omega},$
see also [35, 7].
###### Remark 5.3.
Similar to the discrete Navier-Stokes equations, we can consider an equivalent
formulation for the discrete dual equation in discretely divergence free
spaces: Find $z_{kh}\in V_{kh}$ satisfying
$\mathfrak{B}(\phi_{kh},z_{kh})+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left\langle
g,\phi_{kh}\right\rangle_{I\times\Omega}\qquad\text{for all }\phi_{kh}\in
V_{kh}.$ (35)
With the same argument as [7, Proposition 4.3], there holds: If $z_{kh}\in
V_{kh}$ solves (35), then there exists $\varrho_{kh}\in M_{kh}$ such that
$(z_{kh},\varrho_{kh})\in Y_{kh}$ solves (32). If $(z_{kh},\varrho_{kh})\in
Y_{kh}$ solves (32), then $z_{kh}\in V_{kh}$ and it solves (35).
We first show unique solvability of the discrete dual problem, and the
stability in $L^{\infty}(I;L^{2}(\Omega))\cap L^{2}(I;H^{1}(\Omega))$. Since
both $u$ and $u_{kh}$ occur in the formulation of the discrete problem, we
need both results of Lemma 4.18 to hold true.
###### Theorem 5.4.
Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$, $u_{0}\in V$ and $u$,$u_{kh}\in
L^{\infty}(I;L^{2}(\Omega)^{2})\cap L^{2}(I;H^{1}(\Omega)^{2})$ be solutions
to the weak and fully discretized Navier-Stokes equations (4) and (19) for
either $q=0$ or $q=1$. Then for $(k,h)$ small enough, problem (35) has a
unique solution $z_{kh}\in V_{kh}$ for any $g\in L^{1}(I;L^{2}(\Omega)^{2})$,
and there holds the bound
$\|z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|z_{kh}\|_{L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{1}(\Omega)})}\leq
K\left(\|\overline{uu_{kh}}\|_{L^{2}(I;H^{1}(\Omega))\cap
L^{\infty}(I;L^{2}(\Omega))}\right)\|g\|_{L^{1}(I;L^{2}(\Omega))},$
where $K:[0,+\infty)\to[0,+\infty)$ is a strictly monotonically increasing,
continuous nonlinear function, independent of $k,h$.
###### Proof.
On the continuous level and for $g\in L^{2}(I;H^{-1}(\Omega)^{2})$, the proof
of a corresponding estimate can be found in [15, Proposition 2.7]. We adapt it
to the discrete setting and to $g\in L^{1}(I;L^{2}(\Omega)^{2})$, making use
of the previously derived discrete Gronwall Lemma 4.11. We only have to prove
the norm bound, since problem (32) is a quadratic system of linear equations,
thus is solvable, if it is injective. The norm bound yields, that for right
hand side $g=0$, $z_{kh}=0$ is the only solution, thus the norm bound implies
existence and uniqueness. For ease of notation, we use the convention
$[z_{kh}]_{M}=-z_{kh,M}^{-}$. Testing equation 35 with $z_{kh}\chi_{I_{m}}$,
$m=1,...,M$, where by $\chi_{I_{m}}$ we denote the indicator function of the
subinterval $I_{m}$, yields:
$\displaystyle-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(z_{kh}\chi_{I_{m}},\partial_{t}z_{kh}\right)_{I_{m}\times\Omega}}$
$\displaystyle+\nu\left(\nabla z_{kh}\chi_{I_{m}},\nabla
z_{kh}\right)_{I\times\Omega}-\left(z_{kh,m}^{-},[z_{kh}]_{m}\right)_{\Omega}$
$\displaystyle+\hat{c}\left(\\!\left(\overline{uu_{kh}},z_{kh}\chi_{I_{m}},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)\\!=\\!\left(g,z_{kh}\chi_{I_{m}}\right)_{I\times\Omega}.$
Applying Lemma 4.6 and writing the inner product as norm yields
$\displaystyle\frac{1}{2}\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$ (36)
$\displaystyle+\hat{c}\left(\\!\left(\overline{uu_{kh}},z_{kh}\chi_{I_{m}},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle=\frac{1}{2}\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}^{2}+\\!{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(g,z_{kh}\right)_{I_{m}\times\Omega}}.$
We proceed by estimating the trilinear forms. According to its definition
(11), the first term vanishes, and for the second one, it holds
$\displaystyle\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle=\frac{1}{2}c\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)-\frac{1}{2}c\left(\\!\left(z_{kh}\chi_{I_{m}},z_{kh},\overline{uu_{kh}}\right)\\!\right).$
After applying Hölder’s inequality in space, we obtain
$\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)\leq\frac{1}{2}\int_{I_{m}}\left(\|z_{kh}\|_{L^{4}(\Omega)}^{2}\|\nabla\overline{uu_{kh}}\|_{L^{2}(\Omega)}+\|z_{kh}\|_{L^{4}(\Omega)}\|\overline{uu_{kh}}\|_{L^{4}(\Omega)}\|\nabla
z_{kh}\|_{L^{2}(\Omega)}\right)\ dt.$
After estimating the $L^{4}$ norms by Lemma 3.2 and applying Hölder’s
inequality in time, we arrive at
$\displaystyle\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)\leq$
$\displaystyle C\Big{(}\|\nabla
z_{kh}\|_{L^{2}(I_{m}\times\Omega)}\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\|\nabla\overline{uu_{kh}}\|_{L^{2}(I_{m}\times\Omega)}$
$\displaystyle+\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}\|\nabla
z_{kh}\|_{L^{2}(I_{m}\times\Omega)}^{\frac{3}{2}}\|\overline{uu_{kh}}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}\|\nabla\overline{uu_{kh}}\|_{L^{2}(I_{m}\times\Omega)}^{\frac{1}{2}}\Big{)}.$
An application of Young’s inequality yields
$\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)\leq\frac{\nu}{2}\|\nabla
z_{kh}\|_{L^{2}(I_{m}\times\Omega)}^{2}+C\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\left(1+\|\overline{uu_{kh}}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right)\|\nabla\overline{uu_{kh}}\|_{L^{2}(I_{m}\times\Omega)}^{2}.$
In order to abbreviate the notation, we introduce
$\gamma_{m}:=C\|\nabla\overline{uu_{kh}}\|_{L^{2}(I_{m}\times\Omega)}^{2}\left(1+\|\overline{uu_{kh}}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right).$We
insert the above estimate into (36), absorb terms, and multiply by 2, which
yields
$\displaystyle\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)}^{2}+\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$
$\displaystyle\leq\gamma_{m}\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}+\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}^{2}+2{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(g,z_{kh}\right)_{I_{m}\times\Omega}}.$
After an application of Hölder’s inequality, we obtain
$\displaystyle\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)}^{2}+\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$
$\displaystyle\leq\gamma_{m}\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}+\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}^{2}$
(37)
$\displaystyle\quad+2\|g\|_{L^{1}(I_{m};L^{2}(\Omega))}\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}.$
Since we have assumed $q=0$ or $q=1$, we can estimate the
$\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}$ terms by evaluations at the
right and left endpoints:
$\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}=\max\\{\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)},\|z_{kh,m}^{-}\|_{L^{2}(\Omega}\\}$.
With triangle inequality there hold the following estimates:
$\displaystyle\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}$
$\displaystyle\leq\max\\{\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)},\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}+\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}\\},$
$\displaystyle\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}$
$\displaystyle\leq\max\\{\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)}^{2},2\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}^{2}+2\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}\\}.$
We introduce
$x_{n}^{2}:=\|z_{kh,n-1}^{+}\|_{L^{2}(\Omega)}^{2}+\sum_{m=n}^{M}\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$. Note that by the terminal
condition for $z_{kh}$, it holds $x_{M+1}=0$. Then after summing up (37) from
$m=n$ to $m=M$, we have:
$x_{n}^{2}\leq\sum_{m=n}^{M}2\gamma_{m}(x_{m}^{2}+x_{m+1}^{2})+2\|g\|_{L^{1}(I_{m};L^{2}(\Omega))}(x_{m}+x_{m+1}).$
Shifting indices, we arrive at
$x_{n}^{2}\leq\sum_{m=n}^{M}2(\gamma_{m}+\gamma_{m-1})x_{m}^{2}+2\|g\|_{L^{1}(I_{m}\cup
I_{m-1};L^{2}(\Omega))}x_{m},$
where for $n=1$ we use the convention $I_{0}=\emptyset$. Hence we are in the
setting of Lemma 4.11, where formally we have to introduce an index
transformation $\tilde{n}=M-n$. In order to apply the lemma, we need to verify
$2(\gamma_{m}+\gamma_{m-1})<1$ for all $m$. Due to $f\in
L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$, Theorem 4.16 shows
$\|u-u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}\to 0$ as $(k,h)\to 0$ and we can apply
Lemma 4.18 and obtain with triangle inequality, that $\sup_{1\leq m\leq
M}\|\nabla\overline{uu_{kh}}\|^{2}_{L^{2}(I_{m};L^{2}(\Omega))}\to 0$ as
$(k,h)\to 0$. Together with the bounds from Proposition 3.1 and Theorem 4.13
for $\|u\|_{L^{\infty}(I;L^{2}(\Omega))}$ and
$\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}$, we obtain that $\gamma_{m}\to 0$
uniformly in $n$ for $(k,h)\to 0$. Thus we can choose the discretization fine
enough, such that $\gamma_{m}<1/8$, and thus we obtain from Lemma 4.11
$\displaystyle\|z_{kh,n-1}^{+}\|_{L^{2}(\Omega)}^{2}+\sum_{m=n}^{M}\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$ (38) $\displaystyle\leq
C_{1}\exp\left(C_{2}\|\nabla\overline{uu_{kh}}\|^{2}_{L^{2}(I\times\Omega)}\left(1+\|\overline{uu_{kh}}\|^{2}_{L^{\infty}(I,L^{2}(\Omega))}\right)\right)\|g\|_{L^{1}(I;L^{2}(\Omega))}^{2}.$
∎
The next theorem, similar to Theorem 4.20, states a stability result for the
discrete dual solution in stronger norms, whenever the right hand side
posesses more regularity.
###### Theorem 5.5.
Let the assumptions of Theorem 5.4 hold true, i.e., let $f\in
L^{2}(I;L^{2}(\Omega)^{2})$, $u_{0}\in V$ and $u$,$u_{kh}$ be solutions to the
weak and fully discretized Navier-Stokes equations (4) and (19) for either
$q=0$ or $q=1$. Then for $(k,h)$ small enough, for any $g\in
L^{2}(I;L^{2}(\Omega)^{2})$, the unique solution $z_{kh}\in V_{kh}$ to (35)
satisfies the bound
$\|z_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}+\|A_{h}z_{kh}\|_{L^{2}(I;L^{2}(\Omega))}\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{u,u_{kh}}}\|g\|_{L^{2}(I\times\Omega)},$
with a constant $C_{u,u_{kh}}$ depending on $u,u_{kh}$ in the form
$C_{u,u_{kh}}=C_{1}\exp\left(C_{2}\left(\|Au\|_{L^{2}(I;L^{2}(\Omega))}^{2}\|u\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}+\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}^{2}\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}\right)\right),$
where the norms of $u,u_{kh}$ are bounded by Proposition 3.1 and Theorems 3.6,
4.20 and 4.13, due to the assumptions on $f$ and $u_{0}$.
###### Proof.
The proof follows the same steps as the proof the stability of $u_{kh}$ in
stronger norms, presented in Theorem 4.20. We begin by testing the dual
equation with $A_{h}z_{kh}\chi_{I_{m}}$ for arbitrary $m=1,...,M$, where
$\chi_{I_{m}}$ denotes the characteristic function of the time interval
$I_{m}$. The nonlinear terms that occur for the discrete dual equation are of
the form
$\hat{c}\left(\\!\left(\overline{uu_{kh}},A_{h}z_{kh}\chi_{I_{m}},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(A_{h}z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right).$
By Hölder’s inequality in space, (17) and the discrete Gagliardo-Nirenberg
inequality (16), these terms can be estimated by
$\displaystyle
C\int_{I_{m}}\|z_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}z_{kh}\|_{L^{2}(\Omega)}^{\frac{3}{2}}\left(\|\nabla\overline{uu_{kh}}\|_{L^{2}(\Omega)}+\|\overline{uu_{kh}}\|_{L^{\infty}(\Omega)}\right)dt.$
Applying the continuous and discrete Gagliardo-Nirenberg inequalities (15) and
(16), and (17) to $\overline{uu_{kh}}$, it remains
$\displaystyle
C\int_{I_{m}}\|z_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}z_{kh}\|_{L^{2}(\Omega)}^{\frac{3}{2}}\left(\|u\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|Au\|_{L^{2}(\Omega)}^{\frac{1}{2}}+\|u_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}u_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\right)dt.$
An application of Young’s inequality, absorbing terms and summing over the
subintervals allows us to conclude the proof. By Theorem 5.4 it holds
$z_{kh}\in L^{\infty}(I;L^{2}(\Omega)^{2})$ with a bound independent on $k,h$
and linear in $\|g\|_{L^{1}(I;L^{2}(\Omega))}$. Further, the terms involving
$u,u_{kh}$ are summable, since the $L^{\infty}(I;L^{2}(\Omega))$ norms of
$u,u_{kh}$ remain bounded via Proposition 3.1 and Theorem 4.13, and the
$Au,A_{h}u_{kh}$ terms are summable by Theorem 3.6 and Theorem 4.20, where the
latter holds true for $(k,h)$ small enough, due to $f\in
L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$. ∎
We now turn towards showing the main result of our work, i.e., the error
estimate for the Navier-Stokes equations in the $L^{\infty}(I;L^{2}(\Omega))$
norm. As in the proof of Theorem 4.16, we will split the error $u-u_{kh}$ into
an error for a Stokes problem, and a remainder term, which we will estimate
using the discrete dual equation (35), to which we can apply the results of
Theorem 5.4. Similar to the result for the Stokes equations of Proposition
4.9, the error estimate will consist of two terms with the first one being a
best approximation error, and the second one being the error of the stationary
Stokes Ritz projection introduced in (10). This result estimates the
$L^{\infty}(I;L^{2}(\Omega))$ norm in an isolated fashion, and thus does not
suffer from an order reduction, which is observed in results that estimate the
error norm combined with the $L^{2}(I;H^{1}(\Omega))$ norm.
###### Theorem 5.6.
Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$. Let $(u,p)$ be the
unique solution to the Navier-Stokes equations (9), and $(u_{kh},p_{kh})$ the
corresponding solution to the discretized equations (20) for a discontinuous
Galerkin method in time with order $q=0$ or $q=1$, for sufficiently small
discretization parameters $(k,h)$. Then there holds
$\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\left(\ln\frac{T}{k}\right)\left(\inf_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\in
V_{kh}}\|u-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u-R_{h}^{S}(u,p)\|_{L^{\infty}(I;L^{2}(\Omega))}\right).$
###### Proof.
We use the same notation as in the proof of Theorem 4.16 and denote the
velocity error by $e:=u-u_{kh}$ and the pressure error by $r:=p-p_{kh}$. We
consider the Stokes projection $(\tilde{u}_{kh},\tilde{p}_{kh})$ of $(u,p)$,
solving (25), and the induced splitting of the error into $e=\xi+\eta_{kh}$,
$r=\omega+\kappa_{kh}$, where $\xi=u-\tilde{u}_{kh}$,
$\eta_{kh}=\tilde{u}_{kh}-u_{kh}$, $\omega=p-\tilde{p}_{kh}$ and
$\kappa_{kh}=\tilde{p}_{kh}-p_{kh}$. We immediately obtain the estimate
$\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\left(\ln\frac{T}{k}\right)\left(\inf_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\in
V_{kh}}\|u-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u-R_{h}^{S}(u,p)\|_{L^{\infty}(I;L^{2}(\Omega))}\right),$
by Proposition 4.9, since by Theorem 3.8 it holds $f-(u\cdot\nabla)u\in
L^{2}(I;L^{2}(\Omega)^{2})$. Thus to finalize the proof, we have to estimate
the $L^{2}(\Omega)$ norm of $\eta_{kh}$ pointwise in time. To this end, as in
[7, Theorem 6.2], we fix $\tilde{t}\in I$ and construct $\theta\in
C^{\infty}_{0}(I)$ in such a way, that $\operatorname{supp}\theta\subset
I_{m}$ where $m$ is chosen such that $\tilde{t}\in I_{m}$, and
$\left(\eta_{kh}(\tilde{t})\theta,\phi_{kh}\right)_{I\times\Omega}=\left(\eta_{kh}(\tilde{t}),\phi_{kh}(\tilde{t})\right)_{\Omega}\
\text{for all }\phi_{kh}\in U_{kh},\quad\|\theta\|_{L^{1}(I)}\leq C\text{
independent of }\tilde{t}\text{ and }k.$
For the construction of such a function $\theta$ serving the purpose of a
regularized Dirac measure, we refer to [44, Appendix A.5]. We then define the
dual solution $z_{kh}\in V_{kh}$ such that for all $\phi_{kh}\in V_{kh}$, it
satisfies
$\mathfrak{B}(\phi_{kh},z_{kh})+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left(\eta_{kh}(\tilde{t})\theta,\phi_{kh}\right)_{I\times\Omega}.$
By Theorem 5.4 we have the existence, uniqueness and regularity of $z_{kh}$,
satisfying the bound
$\|z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|\nabla
z_{kh}\|_{L^{2}(I\times\Omega)}\leq
K\left(\|\overline{uu_{kh}}\|_{L^{2}(I;H^{1}(\Omega))\cap
L^{\infty}(I;L^{2}(\Omega))}\right)\|\eta_{kh}(\tilde{t})\theta\|_{L^{1}(I;L^{2}(\Omega))},$
(39)
for $(k,h)$ small enough. From Remark 5.3, we obtain the existence of an
associated pressure $\varrho_{kh}\in M_{kh}$, such that for all
$(\phi_{kh},\psi_{kh})\in Y_{kh}$, it holds
$B((\phi_{kh},\psi_{kh}),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left(\eta_{kh}(\tilde{t})\theta,\phi_{kh}\right)_{I\times\Omega}.$
Choosing the specific test functions
$(\phi_{kh},\psi_{kh})=(\eta_{kh},\kappa_{kh})\in Y_{kh}$, we have
$\displaystyle\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=\left(\eta_{kh}(\tilde{t})\theta,\eta_{kh}\right)_{I\times\Omega}$
$\displaystyle=B((\eta_{kh},\kappa_{kh}),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},\eta_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\eta_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle=B((e-\xi,r-\omega),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},e-\xi,z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(e-\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle=B((e,r),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},e,z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(e,\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle\phantom{==}-B((\xi,\omega),(z_{kh},\varrho_{kh}))-\hat{c}\left(\\!\left(\overline{uu_{kh}},\xi,z_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$
For $\xi$ we can use the Galerkin orthogonality with respect to $B$, i.e.,
from (25), we obtain
$B((\xi,\omega),(\phi_{kh},\psi_{kh}))=0\qquad\text{for all
}(\phi_{kh},\psi_{kh})\in Y_{kh}.$
Furthermore, due to Lemma 5.1 it holds
$B((e,r),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},e,z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(e,\overline{uu_{kh}},z_{kh}\right)\\!\right)=0.$
Thus we see directly
$\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}=-\hat{c}\left(\\!\left(\overline{uu_{kh}},\xi,z_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$
We want to make use of the $L^{\infty}(I;L^{2}(\Omega))$ estimate of $\xi$ and
thus have to move it to an argument of $\hat{c}$ that has no spatial gradient
applied to it. Since $\hat{c}$ was obtained by anti-symmetrizing $c$, there
are gradients in the second and third argument of $\hat{c}$, which is why we
revert to the original trilinear form $c$. The first argument has no gradient,
and thus
$\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}=-\frac{1}{2}c\left(\\!\left(\overline{uu_{kh}},\xi,z_{kh}\right)\\!\right)+\frac{1}{2}c\left(\\!\left(\overline{uu_{kh}},z_{kh},\xi\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$
(40)
Lemma 4.4 allows us to switch $\xi$ to the third argument, and we thus have
$\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}=\frac{1}{2}\left(\nabla\cdot\overline{uu_{kh}},z_{kh}\cdot\xi\right)_{I\times\Omega}+c\left(\\!\left(\overline{uu_{kh}},z_{kh},\xi\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$
After application of Hölder’s inequality, we obtain
$\displaystyle\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}\leq$ $\displaystyle
C\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}\|1\|_{L^{4}(I;L^{\infty}(\Omega))}$
(41)
$\displaystyle\times\left(\|z_{kh}\|_{L^{4}(I;L^{4}(\Omega))}\|\nabla\overline{uu_{kh}}\|_{L^{2}(I;L^{4}(\Omega))}+\|\nabla
z_{kh}\|_{L^{2}(I\times\Omega)}\|\overline{uu_{kh}}\|_{L^{4}(I;L^{\infty}(\Omega))}\right).$
By Theorem 3.6 and Theorem 4.20, the solutions $u$,$u_{kh}$ satisfy the bound
$\|\overline{uu_{kh}}\|_{L^{\infty}(I;H^{1}(\Omega))}+\|u\|_{L^{2}(I;H^{2}(\Omega))}+\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}\leq
C,$
with a constant depending on the data. Hölder’s inequality and the continuous
and discrete Gagliardo-Nirenberg inequalities (15) and (16) yield moreover
$\|\overline{uu_{kh}}\|_{L^{4}(I;L^{\infty}(\Omega))}\leq
C\left(\|u\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}\|Au\|_{L^{2}(I;L^{2}(\Omega))}^{\frac{1}{2}}+\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}^{\frac{1}{2}}\right).$
From (12) and (14), we further obtain
$\|\nabla\overline{uu_{kh}}\|_{L^{2}(I;L^{4}(\Omega))}\leq C\left(\|\nabla
u\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}\|Au\|_{L^{2}(I;L^{2}(\Omega))}^{\frac{1}{2}}+\|\nabla
u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}^{\frac{1}{2}}\right).$
With the above estimates, we thus obtain from (41)
$\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}\leq
CT^{\frac{1}{4}}\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}\left(\|z_{kh}\|_{L^{4}(I;L^{4}(\Omega))}+\|\nabla
z_{kh}\|_{L^{2}(I\times\Omega)}\right).$
With Lemma 3.2, we further obtain the estimate
$\|z_{kh}\|_{L^{4}(I;L^{4}(\Omega))}\leq\|z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|\nabla
z_{kh}\|_{L^{2}(I\times\Omega)}^{\frac{1}{2}}}.$
Hence, applying (39) yields
$\|\eta_{kh}(\tilde{t})\|_{L^{2}(\Omega)}^{2}\leq
CT^{\frac{1}{4}}\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}\|\eta_{kh}(\tilde{t})\theta\|_{L^{1}(I;L^{2}(\Omega))}.$
(42)
By definition of $\theta$ we can estimate
$\|\eta_{kh}(\tilde{t})\theta\|_{L^{1}(I;L^{2}(\Omega))}$ by
$C\|\eta_{kh}(\tilde{t})\|_{L^{2}(\Omega)}$. This allows us to divide (42) by
$\|\eta_{kh}(\tilde{t})\|_{L^{2}(\Omega)}$ which shows the bound
$\displaystyle\|\eta_{kh}(\tilde{t})\|_{L^{2}(\Omega)}\leq
CT^{\frac{1}{4}}\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}.$
As $\tilde{t}\in I$ was arbitrary, this shows
$\displaystyle\|\eta_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}$ $\displaystyle\leq
CT^{\frac{1}{4}}\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}.$
Making use of the previously derived bound for
$\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}$ and triangle inequality for
$e=\xi+\eta_{kh}$ concludes the proof. ∎
The above theorem is the main result of this work, and the development of the
discrete Gronwall lemma and analysis of the dual problem were the key
ingredients, in order to prove it. With these techniques established, it is
now straightforward, to also prove an error estimate in the
$L^{2}(I;L^{2}(\Omega))$ norm. If we follow the steps of the proof of Theorem
5.6 up to (40), we then have to estimate the occuring trilinear form in terms
of $\|\xi\|_{L^{2}(I;L^{2}(\Omega))}$, i.e., the $L^{2}(I;L^{2}(\Omega))$ norm
of the Stokes error. This implies that we need to estimate the occuring
trilinear terms by stronger norms of the dual state $z_{kh}$, which are
bounded by the results presented in Theorem 5.5 With these considerations, we
can show the following theorem.
###### Theorem 5.7.
Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$. Further let $(u,p)$
and $(u_{kh},p_{kh})$ be the continuous and fully discrete solutions to the
Navier-Stokes equations (9) and (20), for sufficiently small discretization
parameters $(k,h)$. Then for any $\chi_{kh}\in V_{kh}$, there holds
$\displaystyle\|u-u_{kh}\|_{L^{2}(I\times\Omega)}$ $\displaystyle\leq
C\left(\|u-\chi_{kh}\|_{L^{2}(I\times\Omega)}+\|u-\pi_{\tau}u\|_{L^{2}(I\times\Omega)}+\|u-R_{h}^{S}(u,p)\|_{L^{2}(I\times\Omega)}\right).$
###### Proof.
To deduce the error estimate for the Navier-Stokes equations from the
corresponding Stokes result of Proposition 4.8, we follow a duality argument
similar to the proof of Theorem 5.6. We split the error in the same fashion
$u-u_{kh}=u-\tilde{u}_{kh}+\tilde{u}_{kh}-u_{kh}$ where
$\xi:=u-\tilde{u}_{kh}$ is estimated by the Stokes result, and for
$\eta_{kh}=\tilde{u}_{kh}-u_{kh}$ we consider $z_{kh}$ a solution to the dual
problem
$B((\phi_{kh},\psi_{kh}),(z_{kh},\rho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left(\eta_{kh},\phi_{kh}\right)_{I\times\Omega}.$
As before, we test the dual equation with $\eta_{kh}$ and obtain after
elimination of terms by applying Galerkin orthogonalities and Lemma 4.4:
$\|\eta_{kh}\|_{L^{2}(I\times\Omega)}^{2}=\frac{1}{2}\left(\nabla\cdot\overline{uu_{kh}},z_{kh}\cdot\xi\right)_{I\times\Omega}+c\left(\\!\left(\overline{uu_{kh}},z_{kh},\xi\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$
Applying Hölder’s inequality yields
$\displaystyle\|\eta_{kh}\|_{L^{2}(I\times\Omega)}^{2}\leq$ $\displaystyle
C\|\xi\|_{L^{2}(I\times\Omega)}\|1\|_{L^{4}(I;L^{\infty}(\Omega))}$
$\displaystyle\times\left(\|\overline{uu_{kh}}\|_{L^{4}(I;L^{\infty}(\Omega))}\|\nabla
z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|z_{kh}\|_{L^{4}(I;L^{\infty}(\Omega))}\|\nabla\overline{uu_{kh}}\|_{L^{\infty}(I;L^{2}(\Omega))}\right).$
By Proposition 3.1, Theorem 3.6, Theorem 4.13 and Theorem 4.20 all terms
containing $u$, $u_{kh}$ are bounded. Together with the discrete Gagliardo-
Nirenberg inequality (16) and Young’s inequality, we obtain
$\displaystyle\|\eta_{kh}\|_{L^{2}(I\times\Omega)}^{2}\leq
C(u,u_{kh})\|\xi\|_{L^{2}(I\times\Omega)}T^{\frac{1}{4}}\left(\|A_{h}z_{kh}\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla
z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|z_{kh}\|_{L^{2}(I;L^{2}(\Omega))}\right).$
Using Theorem 5.5, we can bound the norms of $z_{kh}$ and obtain
$\displaystyle\|\eta_{kh}\|_{L^{2}(I\times\Omega)}^{2}\leq
C(u,u_{kh})T^{\frac{1}{4}}\|\xi\|_{L^{2}(I\times\Omega)}\|\eta_{kh}\|_{L^{2}(I\times\Omega)}.$
Canceling terms concludes the proof. ∎
### 5.2. Explicit orders of convergence
Using the same arguments from before, instead of the best approximation type
estimate, we can also directly use the error estimate for the Stokes
projection, shown in [7, Theorem 7.4] and [36, Corollaries 6.2& 6.4] to obtain
the following corollaries, yielding explicit orders of convergence.
###### Corollary 5.8.
Let 4.1 be fulfilled. Further let $f\in L^{\infty}(I;L^{2}(\Omega)^{2})$,
$u_{0}\in V\cap H^{2}(\Omega)^{2}$ and let $u$, $u_{kh}$ be the continuous and
fully discrete solutions to the Navier-Stokes equations (4) and (19)
respectively. Then there holds
$\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega)))}\leq
C\left(\ln\frac{T}{k}\right)^{2}(k+h^{2})\left(\|f\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u_{0}\|_{V\cap
H^{2}(\Omega)^{2}}+\|(u\cdot\nabla)u\|_{L^{\infty}(I;L^{2}(\Omega))}\right),$
where the constants $C$ depend continuously on
$\|f\|_{L^{2}(I;L^{2}(\Omega))}$ and $\|u_{0}\|_{V}$. The last term
$\|(u\cdot\nabla)u\|_{L^{\infty}(I;L^{2}(\Omega))}$ can be bounded in terms of
$\|f\|_{L^{\infty}(I;L^{2}(\Omega))}$ and $\|u_{0}\|_{V\cap
H^{2}(\Omega)^{2}}$ by Corollary 3.11.
###### Proof.
From Corollary 3.11 and Remark 3.3, we obtain $(u\cdot\nabla)u\in
L^{\infty}(I;L^{2}(\Omega)^{2})$. Hence this result is a direct consequence of
Theorem 5.6 and [7, Theorem 7.4]. ∎
###### Corollary 5.9.
Let 4.1 hold true. Further let $f\in L^{2}(I;L^{2}(\Omega)^{2})$, $u_{0}\in V$
and let $u$, $u_{kh}$ be the continuous and fully discrete solutions to the
Navier-Stokes equations (4) and (19) respectively. Then there hold the
estimates
$\displaystyle\|\nabla(u-u_{kh})\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle\leq C\left(k^{\frac{1}{2}}+h\right),\text{ and}$
$\displaystyle\|(u-u_{kh})\|_{L^{2}(I;L^{2}(\Omega))}$ $\displaystyle\leq
C\left(k+h^{2}\right),$
where the constants $C$ depend continuously on
$\|f\|_{L^{2}(I;L^{2}(\Omega))}$ and $\|u_{0}\|_{V}$.
###### Proof.
This result is a direct consequence of Theorem 4.16, Theorem 5.7 and [36,
Corollaries 6.2 & 6.4]. ∎
## References
* [1] R. Adams and J. Fournier, Cone conditions and properties of Sobolev spaces, Journal of Mathematical Analysis and Applications, 61 (1977), pp. 713–734.
* [2] N. Ahmed and G. Matthies, Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier–Stokes equations, IMA Journal of Numerical Analysis, 41 (2021), pp. 3113–3144.
* [3] H. Amann, Linear parabolic problems involving measures, Real Academia de Ciencias Exactas, Fisicas y Naturales. Revista. Serie A, Matematicas, 95 (2001), pp. 85–119.
* [4] D. Arndt, H. Dallmann, and G. Lube, Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem, Numerical Methods for Partial Differential Equations, 31 (2015), pp. 1224–1250.
* [5] , Quasi-optimal error estimates for the incompressible Navier-Stokes problem discretized by finite element methods and pressure-correction projection with velocity stabilization, arXiv:1609.00807, (2016).
* [6] A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkhäuser Basel, 1994.
* [7] N. Behringer, B. Vexler, and D. Leykekhman, Fully discrete best-approximation-type estimates in $L^{\infty}(I;L^{2}(\Omega)^{d})$ for finite element discretizations of the transient Stokes equations, IMA Journal of Numerical Analysis, 43 (2022), pp. 852–880.
* [8] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, vol. 223 of Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 1976.
* [9] C. Bernardi and G. Raugel, A conforming finite element method for the time-dependent Navier–Stokes equations, SIAM Journal on Numerical Analysis, 22 (1985), pp. 455–473.
* [10] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Mathematica Academiae Scientiarum Hungaricae, 7 (1956), pp. 81–94.
* [11] H. Brezis and P. Mironescu, Gagliardo-Nirenberg inequalities and non-inequalities: The full story, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 35 (2018), pp. 1355–1376.
* [12] M. Brokate and G. Kersting, Measure and Integral, Springer International Publishing, 2015.
* [13] E. Casas and K. Chrysafinos, A discontinuous Galerkin time-stepping scheme for the velocity tracking problem, SIAM Journal on Numerical Analysis, 50 (2012), pp. 2281–2306.
* [14] , Analysis of the velocity tracking control problem for the 3d evolutionary Navier–Stokes equations, SIAM Journal on Control and Optimization, 54 (2016), pp. 99–128.
* [15] E. Casas and K. Kunisch, Well-posedness of evolutionary Navier-Stokes equations with forces of low regularity on two-dimensional domains, ESAIM: Control, Optimisation and Calculus of Variations, 27 (2021). |
# Dynamical tidal Love numbers of Kerr-like compact objects
Sumanta Chakraborty 0000-0003-3343-3227<EMAIL_ADDRESS>School of Physical
Sciences, Indian Association for the Cultivation of Science, Kolkata-700032,
India Elisa Maggio 0000-0002-1960-8185<EMAIL_ADDRESS>Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), D-14476
Potsdam, Germany Michela Silvestrini
<EMAIL_ADDRESS>Dipartimento di Fisica, “Sapienza”
Università di Roma & Sezione INFN Roma1, P.A. Moro 5, 00185, Roma, Italy
Paolo Pani 0000-0003-4443-1761<EMAIL_ADDRESS>Dipartimento di Fisica,
“Sapienza” Università di Roma & Sezione INFN Roma1, P.A. Moro 5, 00185, Roma,
Italy
###### Abstract
We develop a framework to compute the tidal response of a Kerr-like compact
object in terms of its reflectivity, compactness, and spin, both in the static
and the frequency-dependent case. Here we focus on the low-frequency regime,
which can be solved fully analytically. We highlight some remarkable novel
features, in particular: i) Even in the zero-frequency limit, the tidal Love
numbers (TLNs) depend on the linear-in-frequency dependence of the object’s
reflectivity in a nontrivial way. ii) Intriguingly, the static limit of the
frequency-dependent TLNs is discontinuous, therefore the static TLNs differ
from the static limit of the (phenomenologically more interesting) frequency-
dependent TLNs. This shows that earlier findings regarding the static TLNs of
ultracompact objects correspond to a measure-zero region in the parameter
space, though the logarithmic behavior of the TLNs in the black hole limit is
retained. iii) In the non-rotating case, the TLNs _generically vanish_ in the
zero-frequency limit (just like for a black hole), except when the
reflectivity is ${\cal R}=1+\,{\cal O}(M\omega)$, in which case they vanish
with a model-dependent scaling, which is generically logarithmic, in the
black-hole limit. The TLNs initially grow with frequency, for any nonzero
reflectivity, and then display oscillations and resonances tied up with the
quasi-normal modes of the object. iv) For rotating compact objects, the TLNs
decrease when the reflectivity decreases or the rotation parameter increases.
Our results lay the theoretical groundwork to develop model-independent tests
of the nature of compact objects using tidal effects in gravitational-wave
signals.
## I Introduction
According to the theory of general relativity (GR), a black hole (BH) has
vanishing tidal susceptibility. Namely, when immersed in a quasi-stationary
tidal field (for example the one produced by a binary companion at a large
distance) its induced multipole moments are all zero. This special property is
quantified by the vanishing of the static tidal Love numbers (TLNs) [1, 2, 3,
4, 5, 6, 7, 8, 9, 10] and can be related to some “hidden” symmetries of the
Kerr solution in GR [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. It is a quite
unique and fragile feature [21, 22]: it does not hold true for any other
object, even within GR [22, 23, 24, 25, 22, 26, 27, 28], for BHs in modified
gravity [22, 29], or in higher dimensions [30, 31, 32, 33, 34, 35, 36]. Its
violation can therefore be seen as a smoking gun for deviations from the
standard paradigm involving BHs in GR [22, 28], especially for supermassive
compact objects [37, 38, 39, 40] which, in the standard paradigm, can only be
BHs. So far the TLNs of compact objects other than Kerr BHs have been computed
on a case-by-case basis, including boson stars [22, 23, 24], gravastars [25,
22, 26], anisotropic stars [27], and other simple exotic compact objects
(ECOs) [41, 28] with stiff equation of state at the surface [22]. Furthermore,
they have been mostly computed for nonspinning objects and for static tidal
fields, with the notable exceptions of Refs. [9, 42, 43, 44, 45].
In the context of gravitational-wave tests of the nature of compact objects
based on their tidal response (see [28, 46] for some reviews), it would be
useful to have a framework to compute the TLNs from the general properties of
the object and not restricted to some specific underlying model. The scope of
this work is to provide such a general framework for the TLNs of a Kerr-like
compact object, by carefully taking into account the reflectivity of the
object, the role of the angular momentum, and the possible time dependence of
the tidal field. We exploit a better understanding of the role of the
reflectivity of ECOs, recently investigated in the context of their
quasinormal modes (QNMs) in the ringdown [47, 48]. While our framework is
valid for any frequency, here we focus on the low-frequency regime which is
amenable to analytical treatment using matching asymptotics.
In II we present our setup, which is based on considering a Kerr metric with
generic boundary conditions at a finite distance from the would-be horizon and
assuming GR (at least) in the exterior of the compact object. We use the
formalism developed in Ref. [9] (see also [43, 49]) to derive a master
equation for the TLNs, paying particular attention to the object’s
reflectivity, ${\cal R}$. We emphasize that the reflectivity of any compact
object is more naturally defined with respect to plane wave modes near the
horizon. It is therefore convenient to define the reflectivity in terms of a
master function describing linear perturbations that are wave-like near the
horizon. For this reason, we use the Detweiler function rather than the more
standard Teukolsky function, following closely Ref. [48].
Another important aspect of our analysis is the static limit, particularly for
the non-rotating case. As we will demonstrate, quite counter-intuitively, the
$\omega\rightarrow 0$ limit of the frequency-dependent TLNs for a non-rotating
compact object with $\mathcal{R}\neq 0$ is discontinuous111A similar
discontinuity was discussed in [50] in the context of the magnetic TLNs of
fluid stars. and does not coincide with the strictly static TLNs derived in
[22]. This happens because the solutions of the Teukolsky equation for an ECO,
admitting outgoing behavior near the horizon, with $\omega\neq 0$, do not tend
to the respective solutions in the zero-frequency limit222Note that in the BH
limit only the in-going solution to the Teukolsky equation matters, which has
a continuous zero-frequency limit. Thus, for BHs such a discontinuity does not
arise.. Hence the strictly static TLNs of a non-rotating compact object must
be determined in a separate manner and do not follow from a continuous
$\omega\rightarrow 0$ limit. Even more interesting is the result that the
zero-frequency limit of the frequency-dependent TLNs depends on the small-
frequency limit of the reflectivity, which we model as
${\cal R}(\omega)={\cal R}_{0}+iM\omega{\cal R}_{1}+{\cal
O}(M^{2}\omega^{2})\,,$ (1)
where $\omega$ is the frequency of the tidal perturbation, $M$ is the object’s
mass, and the (complex) coefficients ${\cal R}_{i}$ generically depend on the
object’s properties. One of our main findings is that, due to some
cancellations, the coefficient ${\cal R}_{1}$ of the ${\cal O}(M\omega)$ term
in the above equation enters the TLNs in the _static_ limit ($\omega\to 0$).
We, therefore, argue that modeling the tidal deformability of a compact object
requires knowledge of the frequency dependence of the reflectivity, even when
restricting to the _static_ TLNs. Moreover, we find that, in the static limit
and for nonrotating objects, the _TLNs vanish identically_ (as in the BH
case), except when ${\cal R}_{0}=1$, in which case the corresponding nonzero
value of the TLNs depends on ${\cal R}_{1}$ and exhibits a logarithmic
behavior as a function of the compactness parameter as the BH limit is
approached. A logarithmic dependence was also found in Ref. [22] by strictly
static perturbations of perfectly reflecting, non-spinning objects. However,
as we shall discuss, beside the logarithmic dependence the general expression
of the TLN in the zero-frequency limit differs from the one found in Ref. [22]
in the strictly static case. This show that a strictly static limit cannot be
obtained in a continuous manner from the frequency-dependent TLNs, as the
solutions of the Teukolsky equation with $\omega\neq 0$ do not have a smooth
limit to the solutions of the Teukolsky equation with $\omega=0$. Since the
process of binary coalescence is always dynamic, it seems more reasonable to
determine the ”dynamical TLNs” and then take the $\omega\to 0$ in order to
arrive at the static TLNs (see [50] for a related discussion). Intriguingly,
the logarithmic behavior as a function of the compactness parameter for the
TLNs of compact objects is a generic result, which holds in the strictly
static case as well as in the dynamical contexts, but in the dynamical case it
depends on the choice of $\mathcal{R}_{1}$ as well. In particular, it is
possible to choose $\mathcal{R}_{1}$, such that the TLNs do not have a
logarithmic dependence.
In III we present a collection of results for the TLNs of a compact object in
terms of its reflectivity, compactness, and spin, both in the static and in
the frequency-dependent case. For the case of a non-rotating compact object,
we discuss the static limit in two different manners — (a) substituting
$\omega=0$ in the Teukolsky equation and then determining the (strictly
static) TLNs from its solution, which coincides with the results of [22]; (b)
solving the dynamical case and thus determining the dynamical TLNs, and then
taking $\omega\rightarrow 0$ limit. As mentioned above, the static limit of
the TLNs associated with a non-rotating compact object depends on
$\mathcal{R}_{0}$ and $\mathcal{R}_{1}$, and the TLNs are either zero or
vanish in the BH limit with a scaling that depends on the compactness and on
the model-dependent choice of ${\cal R}_{1}$. Though the non-zero TLNs
generically depend on the compactness in a logarithmic fashion, only when
setting $\omega=0$ from the beginning, do we recover the results of the
previous work [22].
Finally, in the more general spinning and frequency-dependent case we
determine the TLNs in the small frequency limit. We present the dependence of
the TLNs of rotating compact objects on the model parameters, showing that the
induced multipole moments decrease as the rotation increases or as the
reflectivity decreases. Interestingly, the dynamical TLNs in the non-rotating
case depict certain oscillatory and resonant behaviors which are seemingly
characteristics of the low-frequency QNMs [51, 47, 48] associated with the
compact object.
We conclude in IV, with a discussion of our main findings and possible
extensions. Several detailed calculations have been presented in the
appendices for the benefit of the reader. Throughout this work, we use
geometrized $G=1=c$ units, unless otherwise mentioned. We also work with the
signature convention such that flat Minkowski metric in the Cartesian
coordinate system reads $\eta_{\mu\nu}=\textrm{diag.}(-1,+1,+1,+1)$.
## II Setup
### II.1 Exterior geometry of Kerr-like compact objects
In this work, we analyze the tidal effects on a spinning and horizonless
compact object whose exterior spacetime is described by the Kerr metric. This
is an approximation since, beyond spherical symmetry, the external spacetime
of a compact object can have arbitrary multipole moments, even in GR. At the
same time, regularity requires that any multipolar deviation from the Kerr
metric should vanish in the BH limit at least within GR [52].
It is useful to express the Kerr metric in terms of the ingoing null
coordinates $(v,r,\theta,\tilde{\phi})$ (their relation to Boyer-Lindquist
coordinates $(t,r,\theta,\phi)$ is given in A) such that the line element of
the exterior Kerr spacetime reads
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\left(\Delta-a^{2}\sin^{2}\theta\right)}{\Sigma}dv^{2}+2dvdr$
$\displaystyle-$ $\displaystyle 2a\sin^{2}\theta
drd\tilde{\phi}-\frac{4Mar\sin^{2}\theta}{\Sigma}dvd\tilde{\phi}$
$\displaystyle+$
$\displaystyle{\Sigma}d\theta^{2}+\frac{\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}{\Sigma}\sin^{2}\theta d\tilde{\phi}^{2}\,,$
where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$, $\Delta=(r-r_{+})(r-r_{-})$, and
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$ are the locations of the horizons. The
parameters $M$ and $J\equiv aM$ in the line element are the total mass and the
angular momentum of the compact object, respectively.
The radius $r_{0}$ of the compact object is assumed to be only slightly larger
than the event horizon, located at $r=r_{+}$, and smaller than the photon
sphere. Such a scenario can be motivated by models of microscopic corrections
at the horizon scale (see [28] for an overview). For the purpose of this work,
we consider the radius of the compact object to be
$r_{0}=r_{+}(1+\epsilon)\,,$ (3)
where $0<\epsilon\ll 1$. The properties of the object’s interior are
parametrized by a complex and frequency-dependent reflectivity
$\mathcal{R}(\omega)$ at the surface333Note that, within our formalism,
$r_{0}$ can be any effective radius in the exterior geometry, as long as the
reflectivity $\mathcal{R}(\omega)$ refers to the surface at $r=r_{0}$. For
example, if the object deviates from Kerr only near the would-horizon, one
could take $r_{0}$ to be sufficiently large so that the hypothesis of a Kerr
background for $r>r_{0}$ is satisfied. of the compact object, as will be
discussed in detail in Section II.3.
### II.2 Tidal Love numbers (TLNs)
Here, we wish to provide the basic framework for computing the deformation of
a Kerr-like compact object when placed in an external tidal field. In
particular, we wish to study the TLNs, which encapsulate the linear response
of the compact object due to a perturbing tidal field. We first present the
Newtonian description of the tidal response and then generalize it to the
relativistic case.
#### II.2.1 Newtonian TLNs
The study of the tidal response of a self-gravitating object due to an
external tidal field was historically performed in the Newtonian framework,
where the total gravitational potential outside the object in the external
tidal field can be divided into two parts (see, e.g., [53]): (a) $U_{\rm
tidal}$, the gravitational potential due to the tidal field, and (b) $U_{\rm
body}$, the gravitational potential due to the compact object. The tidal
potential can be decomposed in the spherical harmonic basis as $U_{\rm
tidal}\sim\sum_{\ell m}\mathcal{E}_{\ell m}Y_{\ell m}r^{\ell}$, where
$\mathcal{E}_{\ell m}$ are the components of the tidal field associated with
the spherical harmonic decomposition, $\ell\geq 0$ is the angular number, and
$m$ is the azimuthal number (with $|m|\leq\ell$). In general,
$\mathcal{E}_{\ell m}$ will involve $\ell$ derivatives of the tidal potential.
Similarly, the Newtonian potential due to the central object can be expanded
in terms of its multipole moments, $I_{\ell m}$, as $U_{\rm
body}\sim\sum_{\ell m}I_{\ell m}Y_{\ell m}r^{-\ell-1}$. At the linear level,
the deviation $\delta I_{\ell m}$ from the background multipole moments (those
of a Kerr metric [54] in our case) are proportional to the tidal field
$\mathcal{E}_{\ell m}$ and its time derivatives,
$\displaystyle\delta I_{\ell m}\propto r_{0}^{2\ell+1}\left(2k_{\ell
m}\mathcal{E}_{\ell m}-\tau_{0}\nu_{\ell m}\dot{\mathcal{E}}_{\ell
m}+...\right)~{}.$ (4)
Here, $k_{\ell m}$ are the TLNs (related to the conservative response),
$\tau_{0}$ is a characteristic timescale associated with the tidal field,
$\nu_{\ell m}$ is associated with the tidal dissipation of the compact object
(sometimes called tidal heating in different contexts [55, 56, 57, 58, 59]),
and the dots denote higher-order derivatives with respect to the Boyer-
Lindquist time coordinate. Note that the proportionality factor due to the
linear relation between the multipole and tidal moments depends on the nature
of the compact object and its size, namely on $r_{0}$. Given the above
multipolar expansion, the total potential outside the compact object can be
expressed as
$\displaystyle U_{\rm total}$ $\displaystyle=U_{\rm body}+U_{\rm
tidal}=-\frac{M}{r}+{\rm intrinsic~{}moments}$ $\displaystyle+\int
d\omega\sum_{\ell m}\mathcal{E}_{\ell m}r^{\ell}\left[1+F_{\ell
m}(\omega)\left(\frac{r_{0}}{r}\right)^{2\ell+1}\right]Y_{\ell m}~{},$ (5)
where the frequency-dependent response function $F_{\ell m}(\omega)$ reads
$\displaystyle F_{\ell m}(\omega)=2k_{\ell m}(\omega)+i\omega\tau_{0}\nu_{\ell
m}(\omega)\,,$ (6)
and the (conservative) Newtonian TLNs are given by $k_{\ell
m}=(1/2)\textrm{Re}F_{\ell m}$.
#### II.2.2 Relativistic TLNs
A gauge-invariant, relativistic extension of the Newtonian TLNs can be
achieved with the use of Weyl scalars in the Newman-Penrose formalism [60].
The Weyl scalar which describes the behavior of the gravitational waves that
propagate towards asymptotic infinity is $\Psi_{4}$ and hence is the one we
will use in this work. In the Kerr-like background geometry, the Weyl scalar
$\Psi_{4}$ can be decomposed as [61]
$\rho^{4}\Psi_{4}=\int d\omega\sum_{\ell m}R_{\ell m}(r)~{}_{-2}S_{\ell
m}(\theta)e^{i(m\phi-\omega v)}\,,$ (7)
where $\rho=-(r-ia\cos\theta)^{-1}$, $R_{\ell m}(r)$ is the radial Teukolsky
function, and ${}_{-2}S_{\ell m}(\theta)e^{im\phi}$ is the spin-weighted
spheroidal harmonic with spin $s=-2$.
The fact that $\Psi_{4}$ is able to describe the tidal effects can be seen by
computing it in the Newtonian limit. In this case, the only nontrivial metric
functions read $g_{00}=-(1+2c^{-2}U_{\rm total})$ and $g_{ij}=(1+2c^{-2}U_{\rm
total})\delta_{ij}$, where $U_{\rm total}$ is the Newtonian potential
including the tidal effects and we have restored the speed of light for the
moment. In the Newtonian limit, obtained as $c\to\infty$, the computation of
$\Psi_{4}$ reveals the following behavior [9]:
$\displaystyle\lim_{c\rightarrow\infty}c^{2}\Psi_{4}\propto$
$\displaystyle\int d\omega\sum_{\ell m}\mathcal{E}_{\ell
m}r^{\ell-2}\left[1+F_{\ell
m}(\omega)\left(\frac{r_{0}}{r}\right)^{2\ell+1}\right]~{}_{-2}Y_{\ell m}\,,$
(8)
where ${}_{-2}Y_{\ell m}$ are the spin-weighted spherical harmonics. The above
behavior is valid in the intermediate region $r_{0}\ll r\ll 1/\omega$, i.e.
sufficiently far away both from the central object and the tidal source. As
evident, the radial part of the above equation is identical to the one derived
from the Newtonian potential in 5, with $F_{\ell m}$ being the tidal response
of the compact object to the external tidal field, modulo the external power
of the radial coordinate $r$. In 5, the external radial dependence scales as
$r^{\ell}$, while in II.2.2 the radial dependence behaves as $r^{\ell-2}$.
This difference can be accounted for by noting that in the Newtonian limit
$\Psi_{4}\sim\bar{m}^{i}\bar{m}^{j}\nabla_{i}\nabla_{j}U_{\rm total}$, where
$\bar{m}^{i}$ is one of the complex vector in the Newman-Penrose formalism,
$\nabla_{i}$ is the three-dimensional covariant derivative, and $U_{\rm
total}$ has been defined earlier. The null vector $\bar{m}^{i}$ has a radial
dependence $\sim(1/r)$ and is the origin of the $(1/r^{2})$ difference between
the Newtonian potential and the Weyl scalar.
The radial dependence in II.2.2 is valid also in the relativistic case in the
intermediate region $r_{0}\ll r\ll 1/\omega$. In this region, we can extract
the tidal response function $F_{\ell m}(\omega)$ from $\Psi_{4}$ and define
the relativistic TLNs in a gauge invariant manner as
$\displaystyle k_{\ell m}=\frac{1}{2}\text{Re}F_{\ell m}~{}.$ (9)
In general, the determination of the TLNs from $\Psi_{4}$ is achieved through
the following procedure: (i) find $\Psi_{4}$ in the near-horizon region and
apply the appropriate boundary conditions at $r_{0}$ (which will be discussed
in Section II.3), (ii) determine $\Psi_{4}$ in an intermediate region, which
is far away from both the compact object and the source of the tidal field,
(iii) for each $(\ell,m)$, determine the tidal response as the coefficient of
$r^{-\ell-3}$ in II.2.2, (iv) determine the real part of the tidal response,
which provides the corresponding TLN through 9. In the following, we perform
each of these steps and provide the basic set-up for obtaining the TLNs. For
brevity, we shall omit the $(\ell,m)$ subscripts in the radial and angular
functions in the Weyl scalar.
The radial function $R(r)$ satisfies the (source-free) radial Teukolsky
equation [61, 62, 63], which in the Kinnersley null tetrad [64] and in the
ingoing null coordinate system, reads
$\displaystyle\frac{d^{2}R(r)}{dr^{2}}+\left(\frac{2iP_{+}-1}{r-r_{+}}-\frac{2iP_{-}+1}{r-r_{-}}-2i\omega\right)\frac{dR(r)}{dr}$
$\displaystyle+\left(\frac{4iP_{-}}{(r-r_{-})^{2}}-\frac{4iP_{+}}{(r-r_{+})^{2}}+\frac{A_{-}+iB_{-}}{(r-r_{-})(r_{+}-r_{-})}\right.$
$\displaystyle\qquad-\left.\frac{A_{+}+iB_{+}}{(r-r_{+})(r_{+}-r_{-})}\right)R(r)=0\,,$
(10)
where the quantities $P_{\pm}$, $A_{\pm}$ and $B_{\pm}$ are defined as
$\displaystyle P_{\pm}$ $\displaystyle=$
$\displaystyle\frac{am-2r_{\pm}M\omega}{r_{+}-r_{-}}\,,$ (11) $\displaystyle
A_{\pm}$ $\displaystyle=$ $\displaystyle E_{\ell
m}-2-2(r_{+}-r_{-})P_{\pm}\omega$ (12)
$\displaystyle-(r_{\pm}+2M)r_{\pm}\omega^{2}\,,$ $\displaystyle B_{\pm}$
$\displaystyle=$ $\displaystyle 2r_{\pm}\omega\,,$ (13)
and $E_{\ell m}=\ell(\ell+1)+\mathcal{O}(a\omega)$. We would like to emphasize
that the structure of the above differential equation is different in the
static limit of the non-rotating case, i.e., for $a=0=\omega$. This will have
important implications for the static limit and for the dependence of the TLNs
on the compactness parameter $\epsilon$, which we will discuss in a later
section.
An analytical expression for the TLNs can be derived from the small-frequency
approximation of the radial Teukolsky equation in 10. The details of the
small-frequency expansion can be found in B. The near horizon limit of the
solution yields (see B for a derivation),
$\displaystyle
R(r)\sim\frac{\mathcal{A}}{(r_{+}-r_{-})^{4}}\Delta^{2}+\mathcal{B}e^{2i\bar{\omega}r_{*}}~{},\quad
r\to r_{+}~{},$ (14)
where, $\bar{\omega}=\omega-m\Omega_{\rm H}$, $\Omega_{\rm H}$ being the
angular velocity of the horizon, and $r_{*}$ is the tortoise coordinate
defined as $dr_{*}/dr=\left(r^{2}+a^{2}\right)/\Delta$. The unknown functions
$\mathcal{A}$ and $\mathcal{B}$ depend on the frequency and are to be
determined from the boundary conditions, which will be discussed in Section
II.3. After imposing these boundary conditions, the Teukolsky equation can be
solved up to the intermediate region, far from both the compact object and the
source of the tidal field, yielding the response function of the compact
object to the external tidal field. Finally, the TLN of the $(\ell,m)$ mode
reads in the small-frequency approximation (see B for a derivation),
$\displaystyle k_{\ell
m}=\textrm{Re}\Bigg{[}-i\frac{P_{+}}{2}\left(\frac{(\ell+2)!(\ell-2)!}{(2\ell)!(1+2\ell)!}\right)$
$\displaystyle\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)$
$\displaystyle\qquad\times\left\\{\frac{1-\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}{1+\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}\right\\}$
$\displaystyle\Bigg{]}~{},$ (15)
where,
$\displaystyle\Gamma_{1}=\frac{(\ell+2)!}{(\ell-2)!}\frac{\left(3+2iP_{+}\right)_{\ell-2}}{\left(-1-2iP_{+}\right)_{\ell+2}}~{}.$
(16)
In the expression for $\Gamma_{1}$, we have introduced the Pochhammer symbol,
$z_{n}\equiv z(z+1)\times\cdots\times(z+n-1)$. A similar expression (related
to the imaginary part of $F_{\ell m}$) gives the dissipative response
$\nu_{\ell m}$.
As expected, the tidal response depends on the ratio ${\cal B}/{\cal A}$ and,
for $\mathcal{B}=0$ (i.e., in the BH limit), the response function becomes
imaginary, leading to vanishing TLNs. However, for compact objects other than
BHs, $\mathcal{B}$ is non-zero, and hence it is uniquely determined by the
boundary conditions at $r=r_{0}$, which we now discuss.
### II.3 Reflectivity of compact objects
As shown in Section II.2, the TLNs of Kerr-like compact objects computed in
II.2.2 depend on the unknown constants of integration, $\mathcal{A}$ and
$\mathcal{B}$, whose ratio is fixed by the boundary conditions at $r=r_{0}$.
The radial Teukolsky function in the near horizon regime has two contributions
[63]: (a) an ingoing part, which behaves as $\Delta^{2}$ and (b) an outgoing
part, with radial dependence $e^{2i\bar{\omega}r_{*}}$. Thus, from 14, one may
identify the constant $\mathcal{A}$ to be associated with the ingoing mode at
the horizon, while the constant $\mathcal{B}$ is associated with the outgoing
mode at the horizon. For a BH, regularity of the perturbation at $r\sim r_{+}$
imposes no outgoing modes at the horizon since the BH absorbs everything and
hence one should set $\mathcal{B}=0$.
For non-BH objects, be it a neutron star, an ECO [41, 28, 46] or a quantum
corrected BH [65, 66, 67], there would be nontrivial reflection by the
object’s interior, and hence there would be outgoing modes for which
$\mathcal{B}\neq 0$. Overall, the properties of the compact object are
embedded into a single quantity, defined as the reflectivity of the object.
BHs have zero reflectivity, ordinary neutron stars have almost perfect
reflectivity [47] (due to viscosity they have tiny interaction with
gravitational waves, but see [68] for a recent analysis of the out-of-
equilibrium tidal dynamics), while non-BH compact objects will in general have
nonzero and frequency-dependent reflectivity. We parametrize the latter by a
complex number ${\cal R}(\omega)$, which might generically introduce a phase
shift in the reflected wave relative to the incident one.
Loosely speaking, the reflectivity is the ratio of the amplitudes of the
outgoing and the ingoing waves and one may consider the ratio
$(\mathcal{B}/\mathcal{A})$ as the reflectivity. However, this would not be a
correct assessment, since the outgoing and ingoing modes in terms of the
radial Teukolsky function in the $(v,r,\theta,\tilde{\phi})$ coordinate are
_not_ plane waves and hence the energy carried by ingoing and outgoing modes
is not $|{\cal A}|^{2}$ and $|{\cal B}|^{2}$, but rather a more complicated
function of the frequency [63]. The way around this difficulty is to express
the radial Teukolsky equation as a Schrödinger-like equation with a real
potential, which is constant in the near horizon regime, in which case the
mode functions near the horizon are exact plane waves. This is achieved by the
Detweiler function, $X$, which is defined as a linear combination of the
radial Teukolsky function $R^{(t)}$ in the Boyer-Lindquist coordinates and its
derivative as [69, 47]
$\displaystyle X=\frac{\sqrt{r^{2}+a^{2}}}{\Delta}\left[\alpha
R^{(t)}+\frac{\beta}{\Delta}\frac{dR^{(t)}}{dr}\right]\,,$ (17)
where the quantities $\alpha$ and $\beta$ are functions of $r$ and $\omega$,
their explicit dependence can be found in [69, 47]. With this transformation,
it turns out that the Detweiler function satisfies the following Schrödinger-
like equation,
$\displaystyle\frac{d^{2}X}{dr_{*}^{2}}-V(r,\omega)X=0~{},$ (18)
where the potential is purely real and has the following asymptotics: at the
horizon $V(r\to r_{+},\omega)\to-\bar{\omega}^{2}$, and at infinity
$V(r\to\infty,\omega)\to-\omega^{2}$. Hence, it is possible to define purely
outgoing and ingoing wave modes $\sim e^{\pm i\bar{\omega}r_{*}}$ at the
horizon and $\sim e^{\pm i\omega r_{*}}$ at infinity. Using these the
Detweiler function near the boundary of the compact object reads,
$X\sim
e^{-i\bar{\omega}(r_{*}-r^{0}_{*})}+\mathcal{R}(\omega)e^{i\bar{\omega}(r_{*}-r_{*}^{0})}~{},\quad
r_{*}\sim r_{*}^{0}\,,$ (19)
where $r_{*}^{0}$ is the location of the surface of the object in the tortoise
coordinate. Thus, we can define the reflectivity $\mathcal{R}(\omega)$ of the
compact object in a straightforward manner as
$\displaystyle\mathcal{R}(\omega)=\left[\frac{1-\frac{i}{\bar{\omega}}\left(\frac{1}{X}\frac{dX}{dr_{*}}\right)}{1+\frac{i}{\bar{\omega}}\left(\frac{1}{X}\frac{dX}{dr_{*}}\right)}\right]_{r_{*}^{0}}~{},$
(20)
which is in terms of the Detweiler function and its first radial derivative
near the surface of the compact object. To connect the ratio
$(\mathcal{B}/\mathcal{A})$, appearing in II.2.2, to the reflectivity derived
from the Detweiler function we first have to express the radial Teukolsky
function in the $(v,r,\theta,\tilde{\phi})$ coordinates, in terms of the
Boyer-Lindquist coordinates $(t,r,\theta,\phi)$. This has been performed
explicitly in A. Besides, one can use the transformation between the Teukolsky
function and the Detweiler function, in order to determine the connection
between the Detweiler reflectivity $\mathcal{R}$, with the Teukolsky
reflectivity $(\mathcal{B}/\mathcal{A})$ (for a detailed expression, see A).
Finally, the above relation can be inverted, and the ratio
$\mathcal{B}/\mathcal{A}$ can be expressed in terms of the Detweiler
reflectivity $\mathcal{R}$, which when substituted back into II.2.2, will
provide the TLNs in terms of the Detweiler reflectivity. In what follows we
will consider some generic form of the reflectivity, focusing on its small-
frequency expansion, as in 1, and shall derive the dependence of the TLNs on
the model parameters and frequency. In particular, our interest will be on the
scaling of the TLN with the parameter $\epsilon$, providing the departure of
the surface of the compact object from the BH horizon.
## III TLNs of compact objects
Having set up the basic equations and boundary conditions, we compute the TLNs
for (generically rotating) compact objects, both in the static and frequency-
dependent case. As we shall show, our general framework provides results for
static perturbations and non-spinning objects, which does _not_ coincide with
the strictly static limit, more details will be discussed in Section III.4.
Here we will elucidate some general properties of the dynamical TLNs for
rotating and non-rotating compact objects.
### III.1 The BH limit
To set the stage, we start from the BH limit of our results, which corresponds
to ${\cal R}=0$ and $\epsilon\to 0$. We discuss the TLNs of BHs in the linear-
frequency approximation and comment on the behavior of the TLNs when terms of
$\mathcal{O}(\omega^{2})$ are included in the analysis.
At linear order in the frequency, one can directly use the analytical results
derived in Sec. II.2. It is evident from II.2.2 that the tidal response
function of a rotating BH reads
$\displaystyle F^{\rm rot,BH}_{\ell m}$
$\displaystyle=-i\left(\frac{am-2r_{+}M\omega}{r_{+}-r_{-}}\right)\frac{(\ell-2)!(\ell+2)!}{(2\ell)!(2\ell+1)!}$
$\displaystyle\prod_{j=1}^{\ell}\left[j^{2}+4\left(\frac{am-2r_{+}M\omega}{r_{+}-r_{-}}\right)^{2}\right]\,,$
(21)
which agrees with previous results [9, 43] (see also [49]). For a
Schwarzschild BH, the tidal response function becomes
$F^{\rm
nonrot,BH}_{\ell}=2iM\omega\frac{(\ell-2)!(\ell+2)!}{(2\ell)!(2\ell+1)!}\prod_{j=1}^{\ell}\left[j^{2}+16M^{2}\omega^{2}\right]\,.$
(22)
Note that in the non-rotating case, the tidal response function does not
depend on the azimuthal number $m$, but only on the angular number $\ell$. For
$\ell=2$, the quadrupolar response function for the Schwarzschild BH takes the
form:
$F^{\rm nonrot,BH}_{2}=\frac{i}{15}M\omega+{\cal O}(\omega^{2}M^{2})\,.$ (23)
As evident in all of these cases, the response function in the small-frequency
regime is a purely imaginary quantity and hence the TLNs, which are the
conservative part of the response function and hence are given by the real
part of the response function, identically vanishes. Our analysis is
consistent with the findings of Ref. [9] and shows that for BHs the TLNs
identically vanish in the small frequency limit.
In the above expressions we have kept terms up to linear order in $\omega M$
and hence it is interesting to address what happens to the TLNs of BHs when
terms $\mathcal{O}(\omega^{2}M^{2})$ are taken into account. The starting
point is the radial Teukolsky equation where we keep all the terms up to
$\mathcal{O}(\omega^{2}M^{2})$ in the near-horizon approximation (for a
derivation, see C),
$\displaystyle\frac{d^{2}R}{dz^{2}}$
$\displaystyle+\left[\frac{2iP_{+}-1}{z}-\frac{1+2iP_{+}+2i\omega[2M+(r_{+}-r_{-})]}{(1+z)}\right]\frac{dR}{dz}+\Bigg{[}-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+2i\omega[4M-(r_{+}-r_{-})]}{(1+z)^{2}}$
$\displaystyle\qquad-\frac{\ell(\ell+1)-2}{z(1+z)}+\frac{2ma\omega}{z(1+z)}\left(1-\frac{E_{1}}{2m}\right)-\frac{2i\omega
r_{+}}{z(1+z)}-\frac{\omega^{2}a^{2}(1+E_{2})}{z(1+z)}\Bigg{]}R=0~{},$ (24)
where the rescaled radial coordinate $z$ is defined in 81, and $E_{1}$ and
$E_{2}$ are the coefficients of $\mathcal{O}(a\omega)$ and
$\mathcal{O}(a^{2}\omega^{2})$ terms in the expansion of the angular
eigenvalue in the angular Teukolsky equation, whose explicit forms can be
obtained from [70]. The first derivative term, namely $dR/dz$, does not depend
on $\omega^{2}$, while the coefficient of the radial perturbation $R$ does
only through the combination $a^{2}\omega^{2}$. Thus, for a Schwarzschild BH,
any correction of $\mathcal{O}(\omega^{2}M^{2})$ is absent and the TLNs of a
Schwarzschild BH vanish as
$\displaystyle k_{2}^{\textrm{nonrot, BH}}=0+\mathcal{O}(M^{3}\omega^{3})~{}.$
(25)
The same holds for slowly rotating BHs to linear order in the spin since any
possible correction to the TLNs is proportional to
$\mathcal{O}(a^{2}\omega^{2})$. For a generically rotating BH, there can be
non-trivial effects due to the $\mathcal{O}(a^{2}\omega^{2})$ term in the
Teukolsky equation. In other words, while it is guaranteed that a BH has
vanishing static TLNs, it could have non-trivial dynamical TLNs. We wish to
explore this interesting problem in a future work.
### III.2 Non-spinning case
We present here the TLN of a non-rotating ultracompact object, which can be
obtained by taking the limit $a\to 0$ in II.2.2. The resulting expression
depends on the frequency, the mass of the compact object, the parameter
$\epsilon$, and the reflectivity. In this limit $P_{+}=-2M\omega$, thus the
expression for the TLN associated with the $\ell=2$ mode simplifies to,
$\displaystyle k_{2}$
$\displaystyle=\textrm{Re}\Bigg{[}\left(\frac{iM\omega}{30}\right)\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)$
$\displaystyle\qquad\qquad\times\left\\{\frac{1-\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}{1+\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}\right\\}\Bigg{]}~{},$
(26)
where (see B for a derivation)
$\displaystyle\Gamma_{1}=\frac{3i(1-2iM\omega)}{M\omega(1+4M^{2}\omega^{2})(1+16M^{2}\omega^{2})}~{}.$
(27)
As we will demonstrate later, at frequencies satisfying the condition
$|1+(\mathcal{B}/\mathcal{A})\Gamma_{1}|\simeq 0$, there are resonances in the
TLNs. To see this explicitly, one must use the relation between the Teukolsky
reflectivity $(\mathcal{B}/\mathcal{A})$ with the Detweiler reflectivity
$\mathcal{R}(\omega)$, which in the non-rotating case reads (see B for
details),
$\displaystyle\frac{\mathcal{B}}{\mathcal{A}}$
$\displaystyle=\left(\frac{2M\omega}{3}\right)\frac{(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})}{2-iM\omega}$
$\displaystyle\qquad\times\mathcal{R(\omega)}e^{8\pi
M\omega-2i\omega\left(r_{*}^{0}-2M\right)}~{}.$ (28)
In the next sections, we will discuss the zero-frequency limit of the above
expressions and hence determine the static TLNs from the dynamical ones.
#### III.2.1 Zero-frequency limit
In this section, we discuss the zero-frequency limit of the TLNs of static and
spherically symmetric compact objects. Before proceeding, it is worth
emphasizing that we need to set the rotation parameter $a\to 0$ and then the
frequency $\omega\to 0$ to determine the static TLN for a non-rotating compact
object. As we will show in detail, the zero-frequency limit of the frequency-
dependent TLN differs from the strictly static TLN, as the branch of the
solution to the Teukolsky equation with outgoing behavior close to the horizon
becomes ill-defined in this limit. Therefore, we do not expect the TLN in the
zero frequency limit to match with the results derived in Ref. [22], but
nonetheless the zero-frequency limit seems more natural to consider as the
coalescence of binary compact objects is frequency-dependent.
Given the expression for $\Gamma_{1}$ in 27 and the relation between the
Teukolsky and Detweiler reflectivities in III.2, we can easily obtain the
zero-frequency limit of both of these expressions, by keeping the leading
order terms in $M\omega$. This yields
$\Gamma_{1}=(3i/M\omega)+6+\mathcal{O}(M\omega)$, while the Teukolsky
reflectivity reduces to
$\displaystyle\frac{\mathcal{B}}{\mathcal{A}}$
$\displaystyle=\frac{iM\omega\mathcal{R}}{3}+\frac{M^{2}\omega^{2}\mathcal{R}}{6}\left(3+16i\pi+8\epsilon+8\ln\epsilon\right)$
$\displaystyle+\mathcal{O}(M^{3}\omega^{3})~{}.$ (29)
Thus, keeping terms up to linear order in $M\omega$, we obtain
$\displaystyle\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma_{1}=-\mathcal{R}+\frac{iM\omega\mathcal{R}}{2}\left(7+16i\pi+8\epsilon+8\ln\epsilon\right)\,,$
(30)
and hence the TLN in the zero frequency limit becomes,
$\displaystyle k_{2}$
$\displaystyle=\textrm{Re}\Bigg{\\{}\left(\frac{iM\omega}{30}\right)$
$\displaystyle\times\frac{1+\mathcal{R(\omega)}\left[1-\frac{iM\omega}{2}\left(7+16i\pi+8\epsilon+8\ln\epsilon\right)\right]}{1-\mathcal{R}(\omega)\left[1-\frac{iM\omega}{2}\left(7+16i\pi+8\epsilon+8\ln\epsilon\right)\right]}\Bigg{\\}}~{}.$
(31)
The above equation shows one of our main results: except for the single case
where the above expression has a pole (to be discussed below), the TLN
identically vanishes in the zero-frequency limit for any other reflectivity,
just as in the BH case. This includes all cases of partial reflection444It
might come as a surprise that the TLNs are zero if ${\cal R}\neq 0$ in the
zero-frequency limit. For example one could argue that a perfect-fluid star
has nonzero TLNs and viscosity (which effectively corresponds to $|{\cal
R(\omega)}|^{2}<1$) cannot drastically change the TLNs. However, note that
viscosity would affect the reflectivity starting at ${\cal O}(M\omega)$,
leaving ${\cal R}_{0}$ unaffected. Thus, our formalism also accounts for the
TLNs of a neutron star, wherein ${\cal R}_{1}$ depends on the equation of
state., i.e., $|{\cal R}(\omega)|^{2}<1$. In the exceptional case where
$\mathcal{R}(\omega)=1+iM\omega\mathcal{R}_{1}$, III.2.1 has a pole and the
TLN for the $\ell=2$ mode, in the zero-frequency limit, becomes
$\displaystyle
k_{2}=\frac{2}{15}\textrm{Re}\Bigg{[}\frac{1}{-2\mathcal{R}_{1}+\left\\{7+16i\pi+8(\epsilon+\ln\epsilon)\right\\}}\Bigg{]}~{}.$
(32)
The above result does not correspond to the strictly static case, as we must
keep terms $\mathcal{O}(M\omega)$ in the combination
$(\mathcal{B}/\mathcal{A})\Gamma_{1}$ as well as the reflectivity
$\mathcal{R}$ must be frequency-dependent (i.e., $\mathcal{R}_{1}\neq 0$) in
order to have a non-zero TLN. Note that, the magnitude of the reflectivity
reads
$|\mathcal{R}(\omega)|=1-M\omega\,\textrm{Im}\mathcal{R}_{1}+\mathcal{O}(\omega^{2})$.
Thus, if $\mathcal{R}_{1}$ is real, the object is perfectly reflecting, i.e.,
$|\mathcal{R}(\omega)|=1$. In general, if $\mathcal{R}_{1}$ is complex with
$\textrm{Im}\mathcal{R}_{1}>0$, the reflectivity is smaller than unity. Note
that $\textrm{Im}\mathcal{R}_{1}<0$ is not physical since it corresponds to a
reflectivity larger than unity.
Thus, the TLN associated with the $\ell=2$ mode depends on the frequency-
dependent part of the reflectivity and in general displays a logarithmic
behavior with $\epsilon$. Whenever $|\ln\epsilon|\gg|{\cal R}_{1}|/4$ in the
$\epsilon\to 0$ limit, the TLN vanishes as
$k_{2}=\frac{1}{60\ln\epsilon}\,.$ (33)
Although in this case we obtain a logarithmic scaling as found in [22] for
static TLNs of perfectly reflecting ECOs, the numerical coefficients differ
from those obtained in the zero-frequency limit of dynamical TLNs. Thus, the
static limit of the dynamical TLN does not coincide with the strictly static
case. This is because the solution of the Teukolsky equation in the zero-
frequency case differs from the one with linear-in-frequency terms. We will
discuss these aspects in detail in Sec. III.4.
The variation of the dynamical TLN in the zero-frequency limit with $\epsilon$
is presented in 1, for $\mathcal{R}_{0}=1$, and three different choices of
$\mathcal{R}_{1}$. The plot clearly demonstrates the dependence of the static
TLN on $\log\epsilon$, for constant values of $\mathcal{R}_{1}$. Also, 1 shows
that the magnitude of the static TLN depends heavily on $\epsilon$ and also
mildly on $\mathcal{R}_{1}$, as suggested by 32.
Figure 1: Absolute value of the quadrupolar TLN, $|k_{2}|$, the zero-frequency
limit and as a function of the compactness parameter $\epsilon$, in a
logarithmic scale, and for $\mathcal{R}_{0}=1$ and various choices of the
linear-in-frequency reflectivity $\mathcal{R}_{1}$. For larger and positive
values of $\mathcal{R}_{1}$ the absolute value of the TLN decreases, while for
negative choices of $\mathcal{R}_{1}$ the TLN increases. Overall, as
$\epsilon$ becomes larger, the TLN also increases. The plot shows the generic
logarithmic dependence of the static TLN on the compactness parameter
$\epsilon$.
#### III.2.2 Frequency dependence of the TLNs
Figure 2: Quadrupolar TLN as a function of the frequency, for three cases —
(a) BHs, with $\mathcal{R}=0$, (b) for a compact object, with
$\mathcal{R}_{0}=1$ and (c) another compact object with $\mathcal{R}_{0}=-1$.
In the latter two cases, involving non-rotating compact objects we have taken
$\epsilon=10^{-40}$ and $\mathcal{R}_{1}=1$. As evident, only for the choice
$\mathcal{R}_{0}=1$, the TLN goes to a finite and negative value in the zero-
frequency limit, while even for $\mathcal{R}_{0}=-1$ (which still corresponds
to a perfectly reflecting object), the TLN vanishes in the zero-frequency
limit. In both of these cases, the TLN increases with an increase in
frequency, and in general, the frequency-dependent TLN does not vanish.
Having discussed the static limit of the dynamical TLNs for non-rotating
compact objects, in this section, we will discuss the properties of the
dynamical TLN associated with the $\ell=2$ mode. By combining 27 and III.2,
the expression for the TLN associated with the $\ell=2$ mode of a non-rotating
compact object reads,
$\displaystyle k_{2}$
$\displaystyle=\textrm{Re}\Bigg{[}\left(\frac{iM\omega}{30}\right)\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)$
$\displaystyle\qquad\qquad\qquad\times\left(\frac{1+\mathcal{R}(\omega)G(\omega)}{1-\mathcal{R}(\omega)G(\omega)}\right)\Bigg{]}~{},$
(34)
where we have introduced the frequency-dependent quantity $G(\omega)$,
$\displaystyle G(\omega)$ $\displaystyle\equiv 2\frac{\exp\left[8\pi
M\omega-4iM\omega\left(\epsilon+\ln\epsilon\right)\right]}{2-iM\omega}$
$\displaystyle\qquad\times\left(\frac{1-2iM\omega+16M^{2}\omega^{2}-32iM^{3}\omega^{3}}{1+2iM\omega+16M^{2}\omega^{2}+32iM^{3}\omega^{3}}\right)~{}.$
(35)
Note that $G(\omega)\to 1$ in the zero-frequency limit, so one recovers
III.2.1. The frequency dependence of the TLN for a non-rotating compact
object, both in the perfectly reflecting and partially absorbing cases have
been depicted in 2 and 3, respectively. As evident, except for the case
$\mathcal{R}_{0}=1$, the TLN identically vanishes in the limit
$\omega\rightarrow 0$, which is consistent with III.2.1. Moreover, the TLN
associated with the $\ell=2$ mode in the zero-frequency limit is negative for
$\mathcal{R}_{0}=1$ and a positive choice of $\mathcal{R}_{1}$ and small
$\epsilon$, which is consistent with III.2.1. We will now discuss the features
associated with the frequency-dependent TLNs.
In all the cases involving either a perfectly reflecting compact object with
$\mathcal{R}_{0}=\pm 1$ (see 2), or a partially reflecting one with
$\mathcal{R}_{0}<1$ (see 3), the TLN associated with the $\ell=2$ mode
increases with frequency, and we observe oscillations. 3 also suggests that
the $\ell=2$ TLN decreases at higher frequencies, but our low-frequency
analysis, presented here, eventually breaks down, and thus the high-frequency
results should be taken with a grain of salt.
Figure 3: The TLN of a compact object as a function of the frequency for a
non-spinning and partially as well as perfectly reflecting compact object with
$\epsilon=10^{-40}$ and $\ell=2$. As evident, except for the case
$\mathcal{R}_{0}=1$, the TLN in the zero-frequency limit identically vanishes.
On the other hand, for $\mathcal{R}_{0}=1$, there are oscillations at non-zero
frequencies, while for $\mathcal{R}_{0}<1$, there are also resonances, which
drift to higher frequencies as the reflectivity decreases. See the main text
for more discussion.
We will now depict the one-to-one correspondence between the oscillation
frequencies as well as the resonances with the real part of the QNMs of the
compact object. Following Ref. [47], the QNMs in the small-$\epsilon$ limit
can be written as,
$\omega_{R}\sim-\frac{\pi(q+1)}{2|r_{*}^{0}|}+m\Omega_{\rm H}\,,$ (36)
where $r_{*}^{0}\sim M[1+(1-\chi^{2})^{-1/2}]\ln\epsilon$, $\chi=a/M$ is the
dimensionless spin, $q=2n-1\ (2n)$ for perfectly reflecting objects with
$\mathcal{R}_{0}=1$ ($\mathcal{R}_{0}=-1$), and $n\geq 1$ is the overtone
number. As discussed in Refs. [47, 51, 71], the above analytical approximation
for the real part of the QNMs is also valid for partially absorbing objects,
with $q=2n-1\ (2n)$ for compact objects with $\mathcal{R}_{0}>0$
($\mathcal{R}_{0}<0$). In the case of a non-spinning compact object with
$\mathcal{R}_{0}>0$ and $\epsilon=10^{-40}$, the analytical approximation for
the real part of the QNMs in 36 yields the following frequencies
$M\omega_{R}=0.017\ (n=1),0.034\ (n=2),0.051\ (n=3),0.068\ (n=4),0.085\
(n=5)$, etc. Intriguingly, at precisely these frequencies the oscillations in
the dynamical TLN occur, as shown in 3. The amplitudes of the oscillations and
resonances are then related to the absolute value of $\mathcal{R}_{0}$ and the
factor $G(\omega)$. As evident from III.2.2, for partially reflective compact
objects, close to the point $\mathcal{R}(\omega)G(\omega)\simeq 1$, the TLN
for the $\ell=2$ mode displays resonances. This can be verified from 3 as
well, which shows the appearance of resonances at two different frequencies
for the choices of reflectivities considered here. One can easily verify that
the locations of these resonances are exactly at the solutions of the equation
$\mathcal{R}(\omega)G(\omega)\simeq 1$, as predicted by the analytical
results. In particular, for smaller values of the reflectivity, these
resonances appear at larger values of $M\omega$, and hence, as the
reflectivity becomes very small, the resonance frequencies will become so
large that the small-frequency approximation will break down. We also observe
that the amplitudes of the resonances become larger for smaller
reflectivities, since the overall multiplication factor involving $M\omega$ in
the TLN becomes larger.
The above summarizes the behavior of the TLNs for non-rotating compact objects
both in the dynamical context, as well as in the zero-frequency limit. We now
turn our attention to the case of spinning compact objects.
### III.3 Spinning case
Having discussed the features of the TLN in the case of a nonspinning compact
object, in this section, we will concentrate on determining the TLN in the
spinning case. The general expression has been provided in II.2.2, but we will
quote some results associated with the $\ell=2$ mode since that is the most
relevant one. For this case, the tidal response function can be determined
using 88, 91 and 92 in B, which reads,
$\displaystyle k_{2m}$
$\displaystyle=\textrm{Re}\Bigg{[}-\frac{iP_{+}}{60}\left(1+4P_{+}^{2}\right)\left(1+P_{+}^{2}\right)$
$\displaystyle\qquad\qquad\times\left\\{\frac{1+\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\frac{6i\left(1+iP_{+}\right)}{P_{+}\left(1+4P_{+}^{2}\right)\left(1+P_{+}^{2}\right)}}{1-\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\frac{6i\left(1+iP_{+}\right)}{P_{+}\left(1+4P_{+}^{2}\right)\left(1+P_{+}^{2}\right)}}\right\\}\Bigg{]}~{},$
(37)
where $P_{+}=\\{(am-2M\omega r_{+})/(r_{+}-r_{-})\\}$ as defined in 11. Note
that, in the non-rotating case, it was possible to derive a relation between
the Teukolsky reflectivity $(\mathcal{B}/\mathcal{A})$ and the Detweiler
reflectivity $\mathcal{R}$ in a closed form, see III.2. However, in the
rotating case, the analytical expression for such a relation is cumbersome and
we will not report it. Following this procedure, we express the TLN in III.3
in terms of the more physical Detweiler reflectivity. Note that, unlike the
non-rotating case, the TLN for a rotating compact object depends on the
azimuthal number $m$ through $P_{+}$. Moreover, even when the
$\omega\rightarrow 0$ limit is taken, due to the non-zero rotation parameter,
$P_{+}$ remains non-zero and hence the TLN may not vanish. However, the exact
zero-frequency limit depends on the relation between the Teukolsky and the
Detweiler function in the appropriate limits. In what follows, we will depict
the dependence of the TLN with the frequency as well as with $\epsilon$,
capturing the departure of the surface of the compact object from the would-be
horizon, for various choices of the reflectivity and rotation parameter.
The dependence of the TLN on $\epsilon$, the parameter which determines the
distance between the reflective surface and the would-be horizon, for a
rotating compact object, at zero-frequency is presented in 4 and 5,
respectively. In 4, we have plotted the absolute value of the TLN against
$\epsilon$ for a perfectly reflecting compact object, with
$\mathcal{R}_{0}=1=\mathcal{R}_{1}$, and for various choices of the
dimensionless rotation parameter. Interestingly, the zero-frequency TLN
decreases with increasing rotation, i.e., faster-rotating compact objects are
more difficult to deform than the slowly rotating ones. Furthermore, the TLN
decreases as the reflectivity of the compact object decreases, as expected
(cf. 5). This is because, the $\mathcal{R}_{0}\rightarrow 0$ limit corresponds
to the Kerr BH, having zero TLN at least for small frequencies, and hence it
is expected that a smaller zero-frequency reflectivity leads to a smaller TLN.
We would like to point out that the oscillatory behavior of the TLN with
$\epsilon$ originates from the term involving
$\exp[-2i\omega(r_{0}^{*}-r_{+})]$, appearing in the connection between the
Teukolsky reflectivity $(\mathcal{B}/\mathcal{A})$ and the Detweiler
reflectivity $\mathcal{R}$. Further, as evident from 5, the oscillation length
scale does not depend on the reflectivity, i.e., all the curves in the figure
are in phase with each other. However, the amplitude depends on the
reflectivity and decreases for smaller reflectivity. Note that both of these
plots are in the zero-frequency limit, and hence it is clear that, unlike the
case of a non-rotating ECO where the TLN generically vanishes in the zero-
frequency limit, for a rotating ECO the zero-frequency TLN is in general non-
zero.
Figure 4: The zero-frequency limit of the frequency-dependent TLN as a
function of $\epsilon$, which determines the distance between the surface of
the compact object and the would-be horizon for a spinning and perfectly
reflecting compact object ($\mathcal{R}_{0}=1=\mathcal{R}_{1}$) with $\ell=2$
and $m=2$. We considered three possible choices of the dimensionless rotation
parameter $\chi$. It is clear that, increasing the rotation, the zero-
frequency TLN decreases. Figure 5: Zero-frequency limit of the TLN as a
function of $\epsilon$ for a spinning compact object with $\chi=0.5$,
$\ell=2$, $m=2$ and for three different choices of the reflectivity. The plot
shows that a decrease in the zero-frequency reflectivity $\mathcal{R}_{0}$
decreases the TLN, while the oscillation pattern remains the same.
Besides the zero-frequency limit, we have also studied the dependence of the
$\ell=2$ TLN on the frequency. In 6, we show this dependence for different
choices of the spin, for a perfectly reflecting compact object with
$\mathcal{R}_{0}=1=\mathcal{R}_{1}$ and $\epsilon=10^{-10}$. The plot
explicitly demonstrates the presence of resonances in the structure of the
TLN, and these precisely appear where the denominator of III.3 vanishes. The
resonance frequencies associated with the dynamical TLNs also correspond to
the QNM frequencies of the rotating object, as discussedi n the previous
section. In the dynamical context, for higher frequencies, the behavior of the
TLN is very similar to that of a non-rotating compact object, involving
oscillations and resonances at specific frequencies. Thus, one can safely
argue that the frequency-dependent TLN of a compact object has similar
behavior in both the non-rotating and rotating case. In particular, rapidly
rotating compact objects have larger TLNs at higher frequencies, but very
small TLNs for low-frequency tidal fields. Therefore, at lower frequencies
faster-rotating objects are difficult to be tidally deformed, while at
slightly higher frequencies faster-rotating objects are the easiest to be
tidally deformed. On a similar note, 7 depicts the variation of the $\ell=2$
TLN with the frequency for various partially reflecting rotating compact
objects. In this case, the TLN also decreases with reflectivity, implying that
objects with smaller reflectivity are difficult to deform. This result is in
complete agreement with the fact that BHs, which are objects with no
reflectivity, cannot be tidally deformed. Thus, we have provided all the
details regarding the TLNs of a rotating compact object in the dynamical
context, as well as in the zero frequency limit, depicting non-trivial
behavior.
Figure 6: TLN of a spinning and perfectly reflecting
$(\mathcal{R}_{0}=1=\mathcal{R}_{1})$ compact object as a function of the
frequency with $\epsilon=10^{-10}$, $\ell=2$ and $m=2$. We observe resonances
in the spectrum of the TLN as in the case of non-rotating compact objects.
Interestingly, with an increase in rotation, the TLN decreases significantly,
validating our claim that faster-rotating objects are difficult to deform
tidally at lower frequencies. Figure 7: The absolute value of the TLN as a
function of the frequency for spinning compact objects with three different
choices of the reflectivity and the following choices of the physical
parameters: $\chi=0.5$, $\epsilon=10^{-10}$, $\ell=2$ and $m=2$. Following our
expectations, a decrease in the reflectivity results in a decrease in the TLN,
ultimately vanishing in the BH limit.
### III.4 Static TLNs for a non-rotating compact object
As discussed in the previous sections, the strictly static TLN for a non-
rotating compact object needs to be derived from scratch. One of the prime
reasons is the fact that the outgoing solution of the Teukolsky equation
derived in B is ill-behaved in the limit $P_{+}\to 0$ (as we are interested in
non-rotating objects with $a=0$, and the static limit correspond to $\omega\to
0$). Thus, the static limit for a non-rotating compact object needs to be
studied by taking the zero-rotation and zero-frequency limit of the Teukolsky
equation, presented in 10, which reads,
$\displaystyle\dfrac{d^{2}R_{\ell}}{dr^{2}}$
$\displaystyle+\left(-\frac{1}{r-r_{+}}-\frac{1}{r-r_{-}}\right)\dfrac{dR_{\ell}}{dr}$
$\displaystyle+\left(-\frac{1}{r-r_{+}}+\frac{1}{r-r_{-}}\right)\frac{\gamma_{\ell}}{(r_{+}-r_{-})}R_{\ell}=0~{},$
(38)
where, we have defined $\gamma_{\ell}\equiv(\ell+2)(\ell-1)$. Introducing, the
variable $z$ as in B, in the present scenario we obtain $z=\frac{r}{2M}-1$, in
terms of which the above differential equation reduces to,
$\displaystyle\dfrac{d^{2}R_{\ell}}{dz^{2}}-\left(\frac{1}{z}+\frac{1}{1+z}\right)\dfrac{dR_{\ell}}{dz}+\left(-\frac{1}{z}+\frac{1}{1+z}\right)\gamma_{\ell}R_{\ell}=0~{}.$
(39)
This equation has the following explicit solution,
$\displaystyle
R_{\ell}=z(1+z)\left[c_{1}P_{\ell}^{2}(1+2z)+c_{2}Q_{\ell}^{2}(1+2z)\right]~{},$
(40)
where, $P_{\ell}^{2}(x)$ and $Q_{\ell}^{2}(x)$ are the associated Legendre
polynomials. To see explicitly how these solutions differ from the zero-
frequency and zero-rotation limit of B, we use the relations between the
associated Legendre polynomials and the hypergeometric functions,
$\displaystyle
P_{\ell}^{2}(1+2z)=\frac{\Gamma(\ell+3)}{2\Gamma(\ell-1)}z(1+z)\,_{2}F_{1}(3+\ell,2-\ell;3;-z)~{}.$
(41)
As evident, the above hypergeometric function follows from the first solution
in B, under the limit $P_{+}\to 0$. While, for the other associated Legendre
polynomial,
$\displaystyle Q_{\ell}^{2}(1+2z)$
$\displaystyle=\frac{\sqrt{\pi}}{2^{\ell-1}}\frac{\Gamma(\ell+3)}{2\Gamma(\ell+\frac{3}{2})}\frac{z(1+z)}{(1+2z)^{\ell+3}}$
$\displaystyle\times\,_{2}F_{1}\left(2+\frac{\ell}{2},\frac{3}{2}+\frac{\ell}{2};\ell+\frac{3}{2};\frac{1}{(1+2z)^{2}}\right)~{},$
(42)
which has no analog in the static limit of B for a non-rotating system. This
is why the solution to the Teukolsky equation in the strictly static situation
differs from the static limit of the dynamical solution for the perturbation
of a reflecting compact object.
The above solution for the radial part of the Teukolsky function
$r^{4}\Psi_{4}$ holds for any values of $z$, and hence we can start by
considering the near-horizon, i.e., $z\to 0$ limit, as well as the far zone,
which corresponds to the large-$z$ limit of it. Let us first explore the far-
zone region. In this case, the associated Legendre polynomial
$P_{\ell}^{2}(1+2z)$ can be expanded as,
$\displaystyle P_{\ell}^{2}(1+2z)\Big{|}_{\rm
far}\simeq\frac{\Gamma(\ell+3)}{2\Gamma(\ell-1)}z^{2}$
$\displaystyle\Big{[}\frac{\Gamma(3)\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell)}z^{-3-\ell}$
$\displaystyle+\frac{\Gamma(3)\Gamma(1+2\ell)}{\Gamma(\ell+3)\Gamma(1+\ell)}z^{\ell-2}\Big{]}~{}.$
(43)
Even though the ratio $\\{\Gamma(-1-2\ell)/\Gamma(-\ell)\\}$ has a finite
limit for integer values of $\ell$, the term $\Gamma(2-\ell)$ diverges for any
$\ell\geq 2$, and hence, the coefficient of the term $z^{-3-\ell}$ identically
vanishes. Therefore, for $P_{\ell}^{2}$, we obtain the following result in the
far-zone limit,
$\displaystyle P_{\ell}^{2}(1+2z)\Big{|}_{\rm
far}=\frac{\Gamma(1+2\ell)}{\Gamma(\ell-1)\Gamma(1+\ell)}\left(\frac{r}{2M}\right)^{\ell}~{},$
(44)
such that for $\ell=2$, we obtain,
$\displaystyle P_{2}^{2}(1+2z)\Big{|}_{\rm
far}=3\left(\frac{r}{M}\right)^{2}~{},$ (45)
which matches exactly with the expressions derived in Ref. [72]. On a similar
footing, the other associated Legendre polynomial $Q_{\ell}^{2}$ in the far-
zone reads,
$\displaystyle Q_{\ell}^{2}(1+2z)\Big{|}_{\rm
far}=\frac{\sqrt{\pi}\Gamma(3+\ell)}{2^{2\ell+2}\Gamma(\ell+\frac{3}{2})}z^{-\ell-1}~{}.$
(46)
Thus, among the two independent solutions of the Teukolsky equation,
$P_{\ell}^{2}$ provides the tidal field, while $Q_{\ell}^{2}$ yields the
response of the non-rotating compact object to the tidal field. Note that, for
$\ell=2$, the far-zone limit of $Q_{2}^{2}(1+2z)$ reduces to $(8/5)(M/r)^{3}$,
which is again consistent with [72]. Therefore, the radial part of the
Teukolsky function, from 40, can be expressed as,
$\displaystyle R_{\ell}\Big{|}_{\rm far}$
$\displaystyle=c_{1}\frac{\Gamma(1+2\ell)}{\Gamma(\ell-1)\Gamma(1+\ell)}\left(\frac{r}{2M}\right)^{\ell+2}$
$\displaystyle\qquad\times\left[1+\mathcal{F}_{\ell}\left(\frac{r}{2M}\right)^{-2\ell-1}\right]~{}.$
(47)
Here, $\mathcal{F}_{\ell}$ is the tidal response function, which is purely
real, and thus the TLN becomes,
$\displaystyle
k_{\ell}=\frac{c_{2}}{c_{1}}\frac{\sqrt{\pi}\Gamma(3+\ell)}{2^{2\ell+3}\Gamma(\ell+\frac{3}{2})}\frac{\Gamma(\ell-1)\Gamma(1+\ell)}{\Gamma(1+2\ell)}~{}.$
(48)
Therefore the determination of the TLN boils down to determining the ratio
$(c_{2}/c_{1})$. For BHs, one can show that at the horizon, the associated
Legendre polynomials $P_{\ell}^{2}$ read
$\displaystyle P_{\ell}^{2}(1+2z)\Big{|}_{\rm near}$
$\displaystyle=\frac{\Gamma(\ell+3)}{2\Gamma(\ell-1)}z(1+z)$
$\displaystyle=\frac{\Gamma(\ell+3)}{2\Gamma(\ell-1)}\left(\frac{r}{2M}\right)^{2}\left(1-\frac{2M}{r}\right)~{},$
(49)
which is well-behaved at $r=2M$. While the other branch of the associated
Legendre polynomial becomes,
$\displaystyle Q_{\ell}^{2}(1+2z)\Big{|}_{\rm near}$
$\displaystyle=\frac{\sqrt{\pi}}{2^{\ell-1}}\frac{\Gamma(\ell+3)}{2\Gamma(\ell+\frac{3}{2})}\frac{z(1+z)}{(1+2z)^{\ell+3}}$
$\displaystyle\times\frac{\Gamma(\ell+\frac{3}{2})\Gamma(-2)}{\Gamma(\frac{\ell}{2}-\frac{1}{2})\Gamma(\frac{\ell}{2})}~{},$
(50)
which is ill-behaved due to the appearance of $\Gamma(-2)$ term. Thus,
regularity at the horizon demands $c_{2}=0$ in 40 and hence the BH TLNs
identically vanish. This is consistent with the static limit of the dynamical
TLNs associated with the BH spacetimes, as the term which is ill-behaved in
the static limit is also ill-behaved on the horizon, and hence does not appear
in the BH spacetime. For compact objects other than BHs, on the other hand,
one has to impose appropriate boundary conditions, e.g., in the perfectly
reflecting case one may impose the condition that the Zerilli and the Regge-
Wheeler function should vanish on the surface. Thus, to compare the above
result derived solely from the Teukolsky equation in the zero-frequency and
zero-rotation limit with those in the literature, we must connect the
Teukolsky function with the metric perturbations in the axial and polar
sectors. This is what we do next.
In the non-rotating case, the radial part of the Teukolsky function can be
decomposed into axial and polar parts, $R_{\ell}=R_{\ell}^{\rm
axial}+R_{\ell}^{\rm polar}$, each of which can be expressed in terms of the
Regge-Wheeler and the Zerilli functions respectively, such that in the zero-
frequency limit we obtain (see D for details),
$\displaystyle\frac{R_{\ell}^{\textrm{axial}}}{\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}$
$\displaystyle=\frac{r^{3}}{8}\Bigg{[}V^{\rm axial}\Psi^{\rm RW}_{\ell}+W^{\rm
axial}\left(\dfrac{d\Psi_{\ell}^{\rm RW}}{dr_{*}}\right)\Bigg{]}~{},$ (51)
$\displaystyle\frac{R_{\ell}^{\textrm{polar}}}{\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}$
$\displaystyle=\frac{r^{3}}{8}\left[V^{\rm polar}\Psi^{\rm Z}_{\ell}+W^{\rm
polar}\left(\dfrac{d\Psi^{\rm Z}_{\ell}}{dr_{*}}\right)\right]~{},$ (52)
where, we have introduced
$\displaystyle V^{\rm axial}$ $\displaystyle=V_{\rm
RW}=\frac{(r-2M)\left\\{(\gamma_{\ell}+2)r-6M\right\\}}{r^{4}}~{},$ (53)
$\displaystyle W^{\rm axial}$ $\displaystyle=\frac{2(r-3M)}{r^{2}}~{},$ (54)
$\displaystyle V^{\rm polar}$ $\displaystyle=V_{\rm
Z}=\frac{(r-2M)}{r^{4}(\gamma_{\ell}r+6M)^{2}}\Big{[}\gamma_{\ell}^{2}(\gamma_{\ell}+2)r^{3}$
$\displaystyle+6M\gamma_{\ell}^{2}r^{2}+36M^{2}\gamma_{\ell}r+72M^{3}\Big{]}~{},$
(55) $\displaystyle W^{\rm polar}$
$\displaystyle=\frac{2\gamma_{\ell}r^{2}-6\gamma_{\ell}Mr-12M^{2}}{r^{2}(\gamma_{\ell}r+6M)}~{}.$
(56)
Here, $V_{\rm RW}$ is the Regge-Wheeler and $V_{\rm Z}$ is the Zerilli
potential. The above relations connect the radial part of the Teukolsky
function with the Regge-Wheeler and the Zerilli functions in the zero-
frequency limit. For our purpose, we wish to use the relations connecting the
Regge-Wheeler and the Zerilli functions with the axial metric perturbation
$h_{0}$ and the polar metric perturbation $H_{0}$, respectively, so that the
Teukolsky function can be directly related to the metric perturbations in the
zero-frequency case. We start with the polar sector, involving the functions
$V^{\rm polar}$ and $W^{\rm polar}$, which can be simplified further as
$\displaystyle V^{\rm polar}\Psi^{\rm Z}_{\ell}$ $\displaystyle+W^{\rm
polar}\left(\dfrac{d\Psi^{\rm
Z}_{\ell}}{dr_{*}}\right)=\frac{(r-2M)}{r^{4}(\gamma_{\ell}r+6M)^{2}}\left[\gamma_{\ell}^{2}(\gamma_{\ell}+2)r^{3}+6M\gamma_{\ell}^{2}r^{2}+36M^{2}\gamma_{\ell}r+72M^{3}\right]\Psi^{\rm
Z}_{\ell}$
$\displaystyle+\frac{2\gamma_{\ell}r^{2}-6\gamma_{\ell}Mr-12M^{2}}{r^{2}(\gamma_{\ell}r+6M)}\left(\frac{r-2M}{r}\right)\frac{d\Psi^{\rm
Z}_{\ell}}{dr}$
$\displaystyle=\frac{2(r-2M)}{r^{2}}\Bigg{[}\frac{\gamma_{\ell}r^{2}-3\gamma_{\ell}Mr-6M^{2}}{r(\gamma_{\ell}r+6M)}\frac{d\Psi^{\rm
Z}_{\ell}}{dr}+\left(\frac{\gamma_{\ell}^{2}(\frac{\gamma_{\ell}}{2}+1)r^{2}+3M\gamma_{\ell}^{2}r+18M^{2}\gamma_{\ell}+36\frac{M^{3}}{r}}{r(\gamma_{\ell}r+6M)^{2}}\right)\Psi^{\rm
Z}_{\ell}\Bigg{]}$ $\displaystyle=\frac{2(r-2M)}{r^{2}}H_{0}~{},$ (57)
where, we have used the relation between the polar metric perturbation $H_{0}$
and the Zerilli function $\Psi^{\rm Z}_{\ell}$ [22]. Therefore, the radial
part of the Teukolsky equation associated with the polar perturbation becomes,
$\displaystyle R_{\ell}^{\rm
polar}=\frac{r(r-2M)}{4}\sqrt{\gamma_{\ell}(\gamma_{\ell}+2)}H_{0}~{}.$ (58)
In the $z$ coordinate system, the above relation simply becomes $R_{\ell}^{\rm
polar}\propto z(1+z)H_{0}$. Thus, from 40, we can express the polar metric
perturbation $H_{0}$ in terms of the associated Legendre polynomials
$P_{\ell}^{2}$, $Q_{\ell}^{2}$, and the arbitrary constants linked with them.
Thus, imposing for example Dirichlet boundary conditions on the Zerilli
function at the surface of the compact object (note that this is the same
boundary condition imposed in [22] for a perfectly-reflective mirror), i.e.,
setting $\Psi_{\ell}^{\rm Z}(r_{*}^{0})=0$, we can determine the ratio of the
arbitrary constants, which in turn will yield the TLNs. For the $\ell=2$ mode,
we obtain
$\displaystyle k_{2}^{\rm polar}=\frac{8}{5\left(7+3\ln\epsilon\right)}~{},$
(59)
which exactly matches with [22]. Thus, the TLNs in the strictly zero-frequency
case can indeed be obtained by solving the Teukolsky equation in the non-
rotating scenario. Therefore, the formalism developed here can be reconciled
with earlier results in the literature [22].
For the axial case, we note that in the zero frequency limit the metric
perturbation $h_{0}$ can be expressed as,
$\displaystyle h_{0}$ $\displaystyle=\dfrac{d}{dr_{*}}\left(r\Psi_{\ell}^{\rm
RW}\right)=r\dfrac{d\Psi_{\ell}^{\rm
RW}}{dr_{*}}+\left(\frac{r-2M}{r}\right)\Psi_{\rm RW}~{},$ (60) $\displaystyle
h_{0}^{\prime}$ $\displaystyle=\left(\frac{r^{2}}{r-2M}\right)V_{\rm
RW}\Psi_{\rm RW}+2\dfrac{d\Psi_{\ell}^{\rm RW}}{dr_{*}}$
$\displaystyle\qquad+\Psi_{\ell}^{\rm RW}\left(1+\frac{2M}{r^{2}}\right)~{},$
(61)
where the prime denotes the derivative with respect to the radial coordinate
$r$. Given the above expressions for the metric perturbation $h_{0}$ and its
derivative $h_{0}^{\prime}$ in terms of the Regge-Wheeler function and its
first derivative, we can consider the following combination
$h_{0}^{\prime}-(2h_{0}/r)$, which does not involve any derivatives of the
Regge-Wheeler function, such that,
$\displaystyle\Psi_{\ell}^{\rm
RW}=\frac{r^{3}}{\gamma_{\ell}}\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)~{}.$
(62)
Having expressed the Regge-Wheeler function in terms of $h_{0}$ and its
derivative, we can use 60 to write down the derivative of the Regge-Wheeler
function as well in a similar manner,
$\displaystyle\dfrac{d\Psi_{\ell}^{\rm RW}}{dr_{*}}$
$\displaystyle=\frac{h_{0}}{r}-\left(\frac{r-2M}{r^{2}}\right)\Psi_{\rm RW}$
$\displaystyle=\frac{h_{0}}{r}-\left(\frac{r(r-2M)}{\gamma_{\ell}}\right)\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)~{}.$
(63)
Such that, we obtain the following combination of the Regge-Wheeler function
and its derivative, to read
$\displaystyle V^{\rm axial}\Psi^{\rm RW}_{\ell}$ $\displaystyle+W^{\rm
axial}\left(\dfrac{d\Psi_{\ell}^{\rm
RW}}{dr_{*}}\right)=\frac{(r-2M)\left\\{(\gamma_{\ell}+2)r-6M\right\\}}{r\gamma_{\ell}}\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)$
$\displaystyle\qquad\qquad+\frac{2(r-3M)}{r^{2}}\left[\frac{h_{0}}{r}-\left(\frac{r(r-2M)}{\gamma_{\ell}}\right)\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)\right]$
$\displaystyle=(r-2M)\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)+\frac{2(r-3M)}{r^{3}}h_{0}=\frac{(r-2M)}{r^{2}}\dfrac{dh_{0}}{dr}-\frac{2M}{r^{3}}h_{0}~{}.$
(64)
Therefore, the axial part of the Teukolsky radial function can be expressed in
terms of the metric perturbation $h_{0}$ as,
$\displaystyle R_{\ell}^{\textrm{axial}}$
$\displaystyle=\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}\left[\frac{r(r-2M)}{8}\dfrac{dh_{0}}{dr}-\frac{M}{4}h_{0}\right]~{}.$
(65)
The situation for generic choices of $\ell$ is more involved, as the axial
metric perturbation $h_{0}$ depends on the MeijerG function, with complicated
analytical properties. Thus, we will demonstrate the validity of the above
equation for the $\ell=2$ case, which will also be consistent with the results
in earlier literature. For the $\ell=2$ mode, the solution of the Teukolsky
equation reads
$\displaystyle R_{\ell}^{\textrm{axial}}$
$\displaystyle=c_{1}\left(\frac{3r^{2}}{M^{2}}\right)\left(r-2M\right)^{2}$
$\displaystyle+c_{2}\left(\frac{r-2M}{M}\right)\left(3r^{2}-6Mr-2M^{2}\right)$
$\displaystyle+c_{2}\left(\frac{3r^{2}}{2M^{2}}\right)\left(r-2M\right)^{2}\ln\left(1-\frac{2M}{r}\right)~{}.$
(66)
When substituted in 65, the above solution for the Teukolsky radial function
yields the following expression for the axial metric perturbation $h_{0}$,
$\displaystyle h_{0}$
$\displaystyle=c_{1}r^{2}(r-2M)+\frac{c_{2}}{24M^{5}r}\Big{[}4M^{4}+4M^{3}r+6M^{2}r^{2}$
$\displaystyle+6Mr^{3}\ln\left(1-\frac{2M}{r}\right)-3r^{4}\ln\left(1-\frac{2M}{r}\right)\Big{]}~{}.$
(67)
One can check that the above expression for the axial metric perturbation
$h_{0}$ satisfies the perturbed Einstein’s equations in the static limit.
Again, given this solution, one can impose Dirichlet boundary condition on the
Regge-Wheeler function at the surface of the compact object, by setting
$\Psi^{\rm RW}_{2}(r_{*}^{0})=0$. This in turn will relate the two unknown
coefficients $c_{1}$ and $c_{2}$ and one arrives at the following expression
for the TLN in terms of the compactness parameter $\epsilon$,
$\displaystyle k_{2}^{\rm
axial}=\frac{32}{5\left(25+12\ln\epsilon\right)}~{}.$ (68)
Again this coincides with the corresponding expression presented in [22] and
has the distinctive $\log\epsilon$ structure. Thus, the static limit of the
Teukolsky equation and the corresponding solution indeed provide the TLNs with
logarithmic dependence, consistent with earlier literature. In summary, the
TLNs in the $\omega=0$ case differ from the dynamical TLNs in the $\omega\to
0$ limit in numerical coefficients, though in both cases the logarithmic
dependence of the TLNs on $\epsilon$ remains. Thus, the strictly zero
frequency case is a set of measure zero, as it corresponds to a single
disjoint point in the frequency plane.
## IV Discussion
TLNs provide a unique opportunity to test the nature of compact objects, as
well as the underlying gravitational theory, since only for BHs in GR, the
TLNs identically vanish. Earlier results regarding the TLNs of ECOs depicted
an astounding behavior, namely the TLNs are non-zero and behave as
$\log\epsilon$ in the BH limit, where $\epsilon$ is the difference between the
surface of the ECO and the location of the would-be horizon. This suggested
that even if $\epsilon$ is small, maybe of the Planckian order, the effect on
the TLNs are significant. However, such a result was derived in the context of
a static tidal field for non-rotating ECOs, while in practice all the
deformations are dynamic and all compact objects are rotating. Thus, it is
important to see if such a logarithmic behavior persists even in the dynamical
context for rotating ECOs. The inclusion of rotation and dynamics
automatically urges one to employ the Teukolsky formalism and it turns out
that the Weyl scalars entering the Teukolsky formalism are closely related to
the Newtonian potential in the appropriate limits. Thus, the Weyl scalars
provide the desired quantity that mimics the role of Newtonian potential in
the relativistic context and the Teukolsky equation governs the response
function of a compact object under an external tidal field. In this work, we
have solved the Teukolsky equation with a generic reflective boundary
condition at the surface of the ECO. This in turn leads to a modified tidal
response function, and hence on a TLN depending on the reflectivity of the
ECO. Since the mode functions are not plane waves in the Teukolsky formalism,
we converted the reflectivity in the Teukolsky formalism to the Detweiler
formalism using appropriate transformations, and presented the TLNs in terms
of the parameters of the ECO and the Detweiler reflectivity, ${\cal R}$. To
our surprise, it turns out that for generic reflectivity the zero-frequency
TLNs of a nonspinning compact object identically vanish and non-zero TLNs can
be obtained if and only if $\mathcal{R}(\omega\to 0)=1$. Intriguingly, the
$\log\epsilon$ behavior of the TLNs persists even in the zero-frequency limit
of the dynamical TLNs, but it differs in numerical coefficients from the
strictly static case and depends on the frequency-dependent part of the
reflectivity. Moreover, as the rotation of the compact object increases the
TLNs decrease sharply, implying that fast-rotating objects are difficult to
deform by applying external tidal forces. On the other hand, as the
reflectivity increases, the TLNs also increase, implying that the more
reflective an object is, the higher is its deformation under an external tidal
field. Further, we observe certain oscillation patterns in the TLNs, depending
on the parameter $\epsilon$. Also, both in the non-rotating and the rotating
case, there can be resonances in the TLNs, which happens at higher frequencies
for smaller reflectivities. Hence the dynamics of the TLNs are highly non-
trivial, as they depend on the parameter $\epsilon$ in an oscillatory manner,
while the amplitude of the TLNs depend crucially on the reflectivity and the
rotation of the compact object.
The situation in the zero-frequency limit needs an elaborate discussion.
Naively one would expect to arrive at the same result obtained in [22] for
static perturbations also from the zero-frequency limit of the dynamical TLNs.
However, this turns out not to be the case, because the solutions of the
Teukolsky equation in the frequency-dependent case become singular in the
zero-frequency limit for non-rotating compact objects. Therefore, the
$\omega=0$ is an isolated point, which cannot be arrived at by taking
$\omega\to 0$, suggesting that the strictly static behavior of the TLNs is a
set of measure zero in the phase space. In particular, the TLNs generically
vanish in the zero-frequency limit, except for the following reflectivity of
the compact object: $\mathcal{R}=1+i\omega
M\mathcal{R}_{1}+\mathcal{O}(M^{2}\omega^{2})$. Thus, the reflectivity must
have a frequency dependence, for the zero-frequency TLNs to be non-zero.
Moreover, the parameter $\mathcal{R}_{1}$ explicitly appears in the
reflectivity, and can potentially modify the dependence on $\epsilon$. The
$\log\epsilon$ behavior, though generic, can be eliminated by an appropriate
choice of $\mathcal{R}_{1}$. Thus, we arrive at the following conclusion:
static TLNs correspond to an idealized situation since the inspiral of a
binary system is _always_ dynamical. Moreover, the static limit of the
dynamical TLNs does _not_ coincide with the TLNs derived in the strictly
static case. Finally, when they are non-zero the TLNs vanish in the BH limit
with a typical logarithmic behavior as a function of the distance between the
surface of the compact object and the would-be horizon. This makes the
detection of non-zero TLNs for ECOs a tantalizing possibility for future
generations of GW detectors [22, 57].
There are several future directions of exploration. For example, whether such
behavior for the dynamical TLNs is observed also in the effective field theory
description, remains to be seen, see, e.g., two recent papers [73, 45]. The
first one claims the existence of logarithmic divergences in the field theory,
which must be renormalized to arrive at a finite TLN [73], while the second
one arrives at a $\log(r/2M)$ term in the TLN [45]. Both of these logarithmic
features are very different from the one discussed here in the context of
static TLNs. A detailed comparison with our results deserves further study.
Also, the decomposition of the TLNs in the axial and polar sectors is not very
clear in the context of Weyl scalars, governed by the Teukolsky equation.
Furthermore, it could be interesting to reassess the detectability of the TLNs
from binary coalescence signals, as the prospects are optimistic owing to the
logarithmic scaling [22, 57]. From the theoretical standpoint, our analysis
could be extended to background geometries with fewer symmetries than Kerr
[27], although that would require going beyond the Teukolsky formalism.
Finally, it is possible to map our general framework for TLNs into the
membrane paradigm [74, 75, 76], which can be used to describe ECOs as recently
developed in Ref. [48] (see also [77, 66] for recent work on the subject).
Work along this direction is underway [78].
###### Acknowledgements.
Research of S.C. is funded by the INSPIRE Faculty fellowship from the DST,
Government of India (Reg. No. DST/INSPIRE/04/2018/000893). E.M. acknowledges
funding from the Deutsche Forschungsgemeinschaft (DFG) - project number:
386119226. P.P. acknowledges the financial support provided under the European
Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480, under MIUR
PRIN (Grant 2020KR4KN2 “String Theory as a bridge between Gauge Theories and
Quantum Gravity”) and FARE (GW-NEXT, CUP: B84I20000100001, 2020KR4KN2)
programs, and support from the Amaldi Research Center funded by the MIUR
program “Dipartimento di Eccellenza” (CUP: B81I18001170001).
## Appendix A Connecting the near-horizon behavior of the Teukolsky function
to the Detweiler variable
The first step of connecting the near-horizon behavior of the radial Teukolsky
function in the $(v,r,\theta,\tilde{\phi})$ coordinate with the Detweiler
function, is to transform the radial Teukolsky function in the Boyer-Lindquist
coordinate system. For that the radial function in the Boyer-Lindquist
coordinate will have an additional factor of $e^{-i\omega r_{*}}$ (as,
$v=t+r_{*}$) and also a factor of $e^{im\bar{r}_{*}}$ (as,
$\tilde{\phi}=\phi+\bar{r}_{*}$) in the near horizon limit. Therefore the
radial Teukolsky function in the $(t,r,\theta,\phi)$ coordinate system
becomes,
$\,{}_{-2}R_{\ell
m}^{(t)}=\mathcal{A}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\omega
r_{*}}e^{im\bar{r}_{*}}+\mathcal{B}e^{-i\omega
r_{*}}e^{im\bar{r}_{*}}e^{2i\bar{\omega}r_{*}}$ (69)
where the tortoise coordinate $r_{*}$ and the azimuthal tortoise coordinate
$\bar{r}_{*}$ are given by,
$\displaystyle\frac{dr_{*}}{dr}=\frac{r^{2}+a^{2}}{\Delta}~{};\qquad\frac{d\bar{r}_{*}}{dr}=\frac{a}{\Delta}~{}.$
(70)
In the near horizon limit, both of these tortoise coordinates get related to
each other, such that,
$\displaystyle
d\bar{r}_{*}=\frac{a}{\Delta}dr=\frac{a}{r_{+}^{2}+a^{2}}\frac{r_{+}^{2}+a^{2}}{\Delta}dr=\frac{a}{r_{+}^{2}+a^{2}}dr_{*}~{}.$
(71)
Hence the radial Teukolsky function in the Boyer-Lindquist coordinate system
can expressed solely in terms of the tortoise coordinate $r_{*}$ and hence the
radial function becomes,
$\,{}_{-2}R_{\ell
m}^{(t)}=\mathcal{A}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}+\mathcal{B}e^{i\bar{\omega}r_{*}}~{},$
(72)
where,
$\displaystyle\bar{\omega}=\omega-\frac{am}{r_{+}^{2}+a^{2}}=\omega-m\Omega_{\rm
H}~{}.$ (73)
Here, $\Omega_{\rm H}$ is the angular velocity of the horizon. The above
expression for the radial Teukolsky function in the Boyer-Lindquist coordinate
has been used in the main text.
The next task is to transform the near-horizon radial Teukolsky function in
the Boyer-Lindquist coordinate to Detweiler function, through 17. Since the
transformation is singular on the horizon, the near horizon limit must be
taken with care. For this purpose, we expand the quantities $\mathcal{A}$ and
$\mathcal{B}$ in the near horizon regime as, $\mathcal{A}=A_{0}$ and
$\mathcal{B}=B_{0}+B_{1}\Delta+B_{2}\Delta^{2}$, where $A_{0}$, $B_{0}$,
$B_{1}$ and $B_{2}$ are constants, independent of the radial coordinate. Thus,
the radial Teukolsky function in the Boyer-Lindquist coordinate becomes
$\displaystyle\,_{-2}R_{\ell
m}^{(t)}=A_{0}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)e^{i\bar{\omega}r_{*}}~{}.$
(74)
Of course, the quantities $A_{0}$, $B_{0}$, $B_{1}$ and $B_{2}$ cannot be
arbitrary, rather they will be fixed by the Teukolsky equation. One
substitutes the above expression for the radial perturbation in the radial
Teukolsky equation and demands that the radial Teukolsky equation will be
satisfied at all orders in $\Delta$, thereby uniquely fixing the above
quantities. In what follows, we will choose these quantities to have precisely
those values, which are consistent with the radial Teukolsky equation.
Moving forward, the determination of the Detweiler function requires
determining the derivative of the radial Teukolsky function with respect to
$r$ in the Boyer-Lindquist coordinate, which yields,
$\displaystyle\dfrac{d}{dr}\left({}_{-2}R^{(t)}_{\ell m}\right)$
$\displaystyle=A_{0}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}\left(-\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta}\right)+A_{0}\frac{2\Delta}{(r_{+}-r_{-})^{3}}e^{-i\bar{\omega}r_{*}}$
$\displaystyle+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)e^{i\bar{\omega}r_{*}}\left(\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta}\right)+\left\\{B_{1}(r_{+}-r_{-})+2B_{2}\Delta(r_{+}-r_{-})\right\\}e^{i\bar{\omega}r_{*}}~{}.$
(75)
Substituting the above expression for the radial derivative of the radial
Teukolsky function, along with the radial Teukolsky function itself in 17, we
obtain the following expression for the Detweiler function,
$\,{}_{-2}X_{\ell m}$
$\displaystyle=\sqrt{r_{+}^{2}+a^{2}}\left[\frac{\alpha}{\Delta}R_{-2}^{(t)}+\frac{\beta}{\Delta^{2}}\dfrac{dR_{-2}^{(t)}}{dr}\right]$
$\displaystyle=\sqrt{r_{+}^{2}+a^{2}}\Bigg{[}\frac{\alpha}{\Delta}\left\\{A_{0}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)e^{i\bar{\omega}r_{*}}\right\\}$
$\displaystyle\quad+\frac{\beta}{\Delta^{2}}\Bigg{\\{}A_{0}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}\left(-\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta}\right)+A_{0}\frac{2\Delta}{(r_{+}-r_{-})^{3}}e^{-i\bar{\omega}r_{*}}$
$\displaystyle\quad+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)e^{i\bar{\omega}r_{*}}\left(\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta}\right)+\left\\{B_{1}(r_{+}-r_{-})+2B_{2}\Delta(r_{+}-r_{-})\right\\}e^{i\bar{\omega}r_{*}}\Bigg{\\}}\Bigg{]}$
$\displaystyle=\sqrt{r_{+}^{2}+a^{2}}\left[\frac{\alpha\Delta}{(r_{+}-r_{-})^{4}}+\frac{2\beta}{\Delta(r_{+}-r_{-})^{3}}-\frac{\beta}{\Delta}\left(\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{(r_{+}-r_{-})^{4}}\right)\right]A_{0}e^{-i\bar{\omega}r_{*}}$
$\displaystyle\quad+\sqrt{r_{+}^{2}+a^{2}}\Bigg{[}B_{0}\frac{\alpha}{\Delta}+B_{1}\alpha+B_{2}\alpha\Delta+B_{1}(r_{+}-r_{-})\frac{\beta}{\Delta^{2}}+2B_{2}(r_{+}-r_{-})\frac{\beta}{\Delta}$
$\displaystyle\quad+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)\left(\frac{i\beta\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta^{3}}\right)\Bigg{]}e^{i\bar{\omega}r_{*}}$
$\displaystyle\equiv\mathcal{A}_{X}e^{-i\bar{\omega}r_{*}}+\mathcal{B}_{X}e^{i\bar{\omega}r_{*}}~{},$
(76)
where the two radial coordinate dependent quantities, namely $\mathcal{A}_{X}$
and $\mathcal{B}_{X}$, sitting in front of the ingoing and outgoing solution
near the horizon can be expressed as,
$\displaystyle\mathcal{A}_{X}$
$\displaystyle=\frac{\sqrt{r_{+}^{2}+a^{2}}}{(r_{+}-r_{-})^{4}}\left[\alpha\Delta+\left\\{2(r_{+}-r_{-})-i\bar{\omega}(r_{+}^{2}+a^{2})\right\\}\frac{\beta}{\Delta}\right]A_{0}$
(77) $\displaystyle\mathcal{B}_{X}$
$\displaystyle=\sqrt{r_{+}^{2}+a^{2}}\Bigg{[}B_{0}\left\\{\frac{\alpha}{\Delta}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta^{3}}\right\\}+B_{1}\left\\{\alpha+(r_{+}-r_{-})\frac{\beta}{\Delta^{2}}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta^{2}}\right\\}$
$\displaystyle\qquad+B_{2}\left\\{\alpha\Delta+2(r_{+}-r_{-})\frac{\beta}{\Delta}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta}\right\\}\Bigg{]}~{}.$
(78)
Thus, the reflectivity of the compact object, whose surface is very close to
the BH horizon, can be determined by taking the ratio
$(\mathcal{B}_{X}/\mathcal{A}_{X})$, which reads,
$\displaystyle\mathcal{R}$ $\displaystyle=\lim_{r\rightarrow
r_{0}}(r_{+}-r_{-})^{4}\left[\alpha\Delta+\left\\{2(r_{+}-r_{-})-i\bar{\omega}(r_{+}^{2}+a^{2})\right\\}\frac{\beta}{\Delta}\right]^{-1}\Bigg{\\{}\Bigg{[}\frac{B_{0}}{A_{0}}\left\\{\frac{\alpha}{\Delta}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta^{3}}\right\\}$
$\displaystyle\qquad+\frac{B_{1}}{A_{0}}\left\\{\alpha+(r_{+}-r_{-})\frac{\beta}{\Delta^{2}}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta^{2}}\right\\}+\frac{B_{2}}{A_{0}}\left\\{\alpha\Delta+2(r_{+}-r_{-})\frac{\beta}{\Delta}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta}\right\\}\Bigg{]}\Bigg{\\}}~{}.$
(79)
Note that the coefficients $A_{0}$, $A_{1}$, $B_{0}$, $B_{1}$ and $B_{2}$,
appearing in the above expression for the Detweiler reflectivity, are to be
fixed by demanding that the radial function with the above coefficients, as in
74, follows the Teukolsky equation. Moreover, the reflectivity depends on the
quantities $\alpha$ and $\beta$ appearing in the transformation from the
Teukolsky to the Detweiler functions and have very specific behavior near the
horizon, which must also be taken into account. All in all, the above limit
needs to be taken carefully, since there can be various divergent
contributions canceling each other, yielding a finite value for the
reflectivity $\mathcal{R}$ as a function of the object parameters, frequency,
the ratio $\mathcal{B}/\mathcal{A}$ and of course $\epsilon$. This is the
result we have used in the main text.
## Appendix B Analytical expression for the TLN in the small-frequency limit
In this appendix, we provide the derivation of the analytical expression for
the TLN for rotating spacetimes in the small-frequency regime, i.e.,
$M\omega\ll 1$. The starting point is the radial Teukolsky equation in the
$(v,r,\theta,\phi)$ coordinate, as presented in 10. As emphasized in the main
text, besides the small frequency approximation, we also require the near-
horizon limit in order to arrive at an analytical expression for the
frequency-dependent TLN. The near-horizon expansion of 10 at the leading order
in the frequency reads [9]
$\displaystyle\dfrac{d^{2}\,_{-2}R^{(v)}_{\ell
m}}{dz^{2}}+\left[\frac{2iP_{+}-1}{z}-\frac{2iP_{+}+1}{z+1}\right]\dfrac{d\,_{-2}R^{(v)}_{\ell
m}}{dz}+\left[\frac{4iP_{+}}{(1+z)^{2}}-\frac{4iP_{+}}{z^{2}}-\frac{\ell(\ell+1)-2}{z(1+z)}\right]\,_{-2}R^{(v)}_{\ell
m}=0~{},$ (80)
where we have defined the quantity $z$ as,
$\displaystyle z\equiv\frac{r-r_{+}}{r_{+}-r_{-}}~{};\qquad
P_{+}=\frac{am-2M\omega
r_{+}}{r_{+}-r_{-}}=-\frac{2Mr_{+}\bar{\omega}}{r_{+}-r_{-}}~{},$ (81)
and the expression for $P_{+}$ follows from 11. Further, we have introduced
the following definition for rescaled frequency $\bar{\omega}$ as,
$\bar{\omega}\equiv\omega-m\Omega_{\rm H}$, where $\Omega_{\rm H}$ is the
angular velocity of the horizon. The above equation can be explicitly solved
in terms of the Hypergeometric functions, yielding,
$\displaystyle\,_{-2}R^{(v)}_{\ell m}$
$\displaystyle=\mathcal{A}z^{2}(1+z)^{2}~\,{}_{2}F_{1}\left(2-\ell,3+\ell,3+2iP_{+};-z\right)$
$\displaystyle\qquad\qquad+\mathcal{B}z^{-2iP_{+}}(1+z)^{2}~\,{}_{2}F_{1}\left(-\ell-2iP_{+},1+\ell-2iP_{+},-1-2iP_{+};-z\right)~{},$
(82)
where, $\mathcal{A}$ and $\mathcal{B}$ are the arbitrary constants of
integration. Therefore, close to the horizon we can use the property that
$\lim_{x\rightarrow 0}\,{}_{2}F_{1}(a,b;c;x)=1$, in order to obtain the
following expression for the radial Teukolsky function:
$\,{}_{-2}R^{(v)}_{\ell m}\Big{|}_{\rm near-horizon}$
$\displaystyle=\mathcal{A}z^{2}+\mathcal{B}z^{-2iP_{+}}$
$\displaystyle=\frac{\mathcal{A}}{(r_{+}-r_{-})^{4}}\Delta^{2}+\mathcal{B}\exp\left[2i\left(\frac{r_{+}^{2}+a^{2}}{r_{+}-r_{-}}\right)\bar{\omega}\ln
z\right]~{}.$ (83)
In order to arrive at the first term in the second line, we have used the
relation between $z$ and the radial coordinate $r$, while the second term has
been obtained by relating the tortoise coordinate $r_{*}$ to the near-horizon
coordinate $z$. The determination of the arbitrary constants $\mathcal{A}$ and
$\mathcal{B}$ are to be performed by imposing appropriate boundary conditions
on the radial Teukolsky wave function close to the horizon.
The asymptotic limit of the hypergeometric functions, on the other hand, can
be used to determine the radial part of the rescaled and perturbed Weyl scalar
$\rho^{4}\Psi_{4}$ in the intermediate regime. This region is far from the
surface of the compact object ($r\gg r_{+}$) but is much closer to the compact
object than the distance of the compact object from the source of the tidal
field ($r\ll b$, where $b$ is the distance between the compact object and the
source of the tidal field). In this region, the radial part of the perturbed
and rescaled Weyl scalar, obtained in B, becomes,
$\,{}_{-2}R^{(v)}_{\ell m}\Big{|}_{\rm far}$
$\displaystyle=\mathcal{A}z^{4}\left\\{\frac{\Gamma(3+2iP_{+})\Gamma(1+2\ell)}{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}z^{\ell-2}+\frac{\Gamma(3+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})}z^{-\ell-3}\right\\}$
$\displaystyle\qquad\qquad+\mathcal{B}z^{2-2iP_{+}}\left\\{\frac{\Gamma(-1-2iP_{+})\Gamma(1+2\ell)}{\Gamma(1+\ell-2iP_{+})\Gamma(\ell-1)}z^{\ell+2iP_{+}}+\frac{\Gamma(-1-2iP_{+})\Gamma(-1-2\ell)}{\Gamma(-\ell-2iP_{+})\Gamma(-2-\ell)}z^{2iP_{+}-1-\ell}\right\\}$
$\displaystyle=z^{4}\Bigg{[}\left\\{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(1+2\ell)}{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(1+2\ell)}{\Gamma(1+\ell-2iP_{+})\Gamma(\ell-1)}\right\\}z^{\ell-2}$
$\displaystyle\qquad\qquad+\left\\{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(-1-2\ell)}{\Gamma(-\ell-2iP_{+})\Gamma(-2-\ell)}\right\\}z^{-\ell-3}\Bigg{]}$
$\displaystyle=z^{\ell+2}\left\\{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(1+2\ell)}{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(1+2\ell)}{\Gamma(1+\ell-2iP_{+})\Gamma(\ell-1)}\right\\}\Bigg{[}1+\mathcal{F}_{\ell}z^{-2\ell-1}\Bigg{]}~{}.$
(84)
A comparison of $\,{}_{-2}R^{(v)}_{\ell m}$, derived above, depicting the
radial part of the rescaled and perturbed Weyl scalar $\rho^{4}\Psi_{4}$ in
the intermediate zone, with II.2.2 reveals that one can identify the tidal
response function $\mathcal{F}_{\ell}$ (note that the response function being
independent of the azimuthal number $m$, we have omitted it from the
subscript) to have the following expression (see also [67])
$\displaystyle\mathcal{F}_{\ell}$
$\displaystyle=\frac{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(-1-2\ell)}{\Gamma(-\ell-2iP_{+})\Gamma(-2-\ell)}}{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(1+2\ell)}{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(1+2\ell)}{\Gamma(1+\ell-2iP_{+})\Gamma(\ell-1)}}$
$\displaystyle=\frac{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})\Gamma(1+2\ell)}\left[\frac{1+\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{2}}{1+\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}\right]~{},$
(85)
where, we have introduced two quantities $\Gamma_{1}$ and $\Gamma_{2}$,
dependent on the frequency and the angular number $\ell$ as,
$\displaystyle\Gamma_{1}$
$\displaystyle=\frac{\Gamma(-1-2iP_{+})\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}{\Gamma(1+\ell-2iP_{+})\Gamma(3+2iP_{+})\Gamma(\ell-1)}~{},$
(86) $\displaystyle\Gamma_{2}$
$\displaystyle=\frac{\Gamma(-1-2iP_{+})\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})}{\Gamma(-\ell-2iP_{+})\Gamma(3+2iP_{+})\Gamma(-2-\ell)}~{}.$
(87)
The real part of the tidal response $\mathcal{F}_{\ell}$ provides the TLN.
Note that the above expression for the tidal response function can be further
simplified, along with the quantities $\Gamma_{1}$ and $\Gamma_{2}$, using
various identities involving Gamma functions with imaginary arguments. In
particular, we can express the term outside the square bracket in B as
$\displaystyle\frac{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})\Gamma(1+2\ell)}$
$\displaystyle=\frac{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})\Gamma(1+\ell-2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})\Gamma(1+\ell-2iP_{+})\Gamma(1+2\ell)}$
$\displaystyle=\frac{\Gamma(3+\ell)\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(1+2\ell)}\left[\frac{2\pi
P_{+}}{\sinh(2\pi
P_{+})}\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)\right]\left[\frac{\sin\left(-\pi\ell+2i\pi
P_{+}\right)}{\pi}\right]$
$\displaystyle=\left(-1\right)^{\ell}\frac{\Gamma(3+\ell)\Gamma(1+\ell)\Gamma(-1-2\ell)\Gamma(2+2\ell)}{\Gamma(1+\ell)\Gamma(-\ell)(\ell-1)\ell\Gamma(1+2\ell)\Gamma(2+2\ell)}\left[2iP_{+}\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)\right]$
$\displaystyle=\left(-1\right)^{\ell}\frac{\pi}{\sin(2\pi+2\pi\ell)}\left(\frac{(\ell+2)!(\ell-2)!}{(2\ell)!(1+2\ell)!}\right)\frac{\sin(\pi+\pi\ell)}{\pi}\left[2iP_{+}\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)\right]$
$\displaystyle=-iP_{+}\left(\frac{(\ell+2)!(\ell-2)!}{(2\ell)!(1+2\ell)!}\right)\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)$
(88)
where, along with the results — (a) $\sin(ix)=i\sinh(x)$, (b)
$\sin(2x)=2\sin(x)\cos(x)$, and (c) $\Gamma(1+n)=n!$, we have also used the
following identities involving $\Gamma$ functions:
$\displaystyle\Gamma(z)\Gamma(1-z)$ $\displaystyle=\frac{\pi}{\sin(\pi
z)}~{},$ (89) $\displaystyle|\Gamma(1+n+ib)|^{2}$ $\displaystyle=\frac{\pi
b}{\sinh(\pi b)}\prod_{j=1}^{n}\left(j^{2}+b^{2}\right)~{};\qquad
j\in\mathbb{N}\,.$ (90)
The functions $\Gamma_{1}$ and $\Gamma_{2}$ can also be expressed as,
$\displaystyle\Gamma_{1}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{\Gamma(-1-2iP_{+})\Gamma(1+\ell+2iP_{+})}{\Gamma(1+\ell-2iP_{+})\Gamma(3+2iP_{+})}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{\left(\ell+2iP_{+}\right)\left(\ell-1+2iP_{+}\right)\times\cdots\times\left(3+2iP_{+}\right)}{\left(\ell-2iP_{+}\right)\left(\ell-1-2iP_{+}\right)\times\cdots\times\left(-2iP_{+}\right)\left(-1-2iP_{+}\right)}\,,$
(91) $\displaystyle\Gamma_{2}$
$\displaystyle=\frac{\Gamma(-1-2iP_{+})\Gamma(2-\ell)\Gamma(\ell-1)\Gamma(-\ell+2iP_{+})}{\Gamma(-\ell-2iP_{+})\Gamma(3+2iP_{+})\Gamma(-2-\ell)\Gamma(3+\ell)}\frac{\Gamma(3+\ell)}{\Gamma(\ell-1)}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{\pi}{\sin\left[\pi(\ell-1)\right]}\frac{\sin\left[\pi(\ell+3)\right]}{\pi}\frac{\Gamma(-1-2iP_{+})\Gamma(-\ell+2iP_{+})}{\Gamma(-\ell-2iP_{+})\Gamma(3+2iP_{+})}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{\left(-2-2iP_{+}\right)\left(-3-2iP_{+}\right)\times\cdots\times\left(-\ell-2iP_{+}\right)}{\left(2+2iP_{+}\right)\left(1+2iP_{+}\right)\left(2iP_{+}\right)\times\cdots\times\left(-\ell+2iP_{+}\right)}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{(-1)^{\ell-1}\left(2+2iP_{+}\right)\left(3+2iP_{+}\right)\times\cdots\times\left(\ell+2iP_{+}\right)}{(-1)^{\ell+2}\left(2+2iP_{+}\right)\left(-1-2iP_{+}\right)\left(-2iP_{+}\right)\times\cdots\times\left(\ell-2iP_{+}\right)}=-\Gamma_{1}\,,$
(92)
where, we have used the result,
$\sin[\pi(3+\ell)]=\sin[\pi(\ell-1)]\cos(4\pi)+\sin[4\pi]\cos[\pi(\ell-1)]=\sin[\pi(\ell-1)]$.
Therefore, the tidal response function depends on $\Gamma_{1}$ alone. This
result, along with the expression for $\Gamma_{1}$ has been presented in the
main text.
Having discussed the most general situation with arbitrary choices for $\ell$
and $a$, let us briefly touch upon the interesting limit, with $\ell=2$ and
$a=0$. In this case, it follows that $P_{+}=-2M\omega$ and the expression for
$\Gamma_{1}$ simplifies to,
$\displaystyle\Gamma_{1}$
$\displaystyle=24\frac{\Gamma(-1+4iM\omega)}{\Gamma(3+4iM\omega)}=\frac{24}{(2+4iM\omega)(1+4iM\omega)4iM\omega(-1+4iM\omega)}\simeq\frac{3i}{M\omega}~{}.$
(93)
Moreover, we also obtain,
$\displaystyle\Gamma_{2}=24\frac{\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)}\simeq-\frac{3i}{M\omega}~{}.$
(94)
Thus, in the non-rotating case, with $\ell=2$, we obtain the following
expression for the response function
$\displaystyle F_{2}$
$\displaystyle=\frac{8iM\omega}{5!}\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)\frac{1+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\frac{\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)}}{1+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\frac{\Gamma(-1+4iM\omega)}{\Gamma(3+4iM\omega)}}$
$\displaystyle=\frac{8iM\omega}{5!}\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)\frac{\Gamma(3+4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)}$
$\displaystyle\qquad\times\Bigg{[}\frac{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{\Gamma(3+4iM\omega)+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma(-1+4iM\omega)}\Bigg{]}~{}.$
(95)
Further simplification can be achieved by using the following identity,
$\displaystyle\Gamma(3-4iM\omega)$
$\displaystyle=(2-4iM\omega)(1-4iM\omega)(-4iM\omega)(-1-4iM\omega)(-2-4iM\omega)\Gamma(-2-4iM\omega)$
$\displaystyle=-16iM\omega\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)\Gamma(-2-4iM\omega)~{},$
(96)
so that the response function becomes,
$\displaystyle F_{2}$
$\displaystyle=-\frac{1}{240}\frac{\Gamma(3+4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(-2-4iM\omega)}$
$\displaystyle\qquad\qquad\qquad\times\Bigg{[}\frac{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{\Gamma(3+4iM\omega)+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma(-1+4iM\omega)}\Bigg{]}~{}.$
(97)
Note that the above expression is in terms of the Teukolsky reflectivity
$(\mathcal{B}/\mathcal{A})$, which needs to be converted to the Detweiler
reflectivity $\mathcal{R}$. The steps involved in performing this modification
will be detailed in A, while here we simply quote the final expression for the
response function of the $\ell=2$ mode, in the non-rotating limit, in terms of
the Detweiler reflectivity,
$\displaystyle F_{2}$
$\displaystyle=-\frac{1}{240}\frac{\Gamma(3+4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(-2-4iM\omega)}$
$\displaystyle\qquad\qquad\qquad\times\Bigg{[}\frac{e^{4iM\omega(\epsilon+\ln\epsilon)}(2-iM\omega)\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)+\widetilde{\mathcal{R}}\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{e^{4iM\omega(\epsilon+\ln\epsilon)}\Gamma(2-iM\omega)\Gamma(3+4iM\omega)+\widetilde{\mathcal{R}}\Gamma(-1+4iM\omega)}\Bigg{]}~{},$
(98)
where the frequency dependent quantity $\widetilde{\mathcal{R}}$ is related to
the Detweiler reflectivity $\mathcal{R}$ through the following relation,
$\displaystyle\widetilde{\mathcal{R}}=16e^{8\pi
M\omega}M\omega(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})\mathcal{R}~{}.$
(99)
Therefore, a comparison between the response function in B, obtained by using
the Teukolsky function, and the one in B, determined by the Detweiler
function, reveals the following connection between the two reflectivities,
$\displaystyle\frac{\mathcal{B}}{\mathcal{A}}$
$\displaystyle=\frac{2}{3}\frac{e^{8\pi
M\omega}M\omega\mathcal{R}(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})}{e^{4iM\epsilon\omega}\epsilon^{4iM\omega}(2-iM\omega)}$
$\displaystyle=e^{8\pi
M\omega-4iM\omega\left(\epsilon+\ln\epsilon\right)}\frac{M\omega}{3}\frac{(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})}{1-\frac{iM\omega}{2}}\mathcal{R}~{}.$
(100)
Note that, the tortoise coordinate $r^{*}_{0}$ at the location of the
reflective surface can be expressed in terms of the parameter $\epsilon$,
which depicts the shift of the reflective surface from the would-be horizon as
$\displaystyle
r^{*}_{0}=2M\left(1+\epsilon\right)+2M\ln\epsilon=2M\left(1+\epsilon+\ln\epsilon\right)~{},$
(101)
and hence the relation between the Teukolsky and the Detweiler reflectivity
takes the following form,
$\displaystyle\frac{\mathcal{B}}{\mathcal{A}}=e^{8\pi
M\omega-2i\omega\left(r^{*}_{0}-2M\right)}\frac{(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})}{1-\frac{iM\omega}{2}}\left(\frac{M\omega\mathcal{R}}{3}\right)~{}.$
(102)
In the limit of $\omega\rightarrow 0$, the Teukolsky and the Detweiler
reflectivity gets related in a simple form, which reads,
$(\mathcal{B}/\mathcal{A})=(iM\omega\mathcal{R}/3)$. We will use this
particular form in the main text while discussing the static limit of the TLN
for a non-rotating BH.
## Appendix C Teukolsky equation with quadratic-in-frequency corrections
We start by simplifying 10 by keeping terms up to
$\mathcal{O}(M^{2}\omega^{2})$. For this purpose, we note the following
identities among the parameters of 10,
$\displaystyle P_{+}-P_{-}$ $\displaystyle=-2M\omega~{},\qquad
P_{-}=P_{+}+2M\omega~{},$ (103) $\displaystyle A_{+}-A_{-}$
$\displaystyle=-2(r_{+}-r_{-})(P_{+}-P_{-})\omega-(r_{+}+2M)r_{+}\omega^{2}+(r_{-}+2M)r_{-}\omega^{2}$
$\displaystyle=-\omega^{2}(r_{+}-r_{-})(r_{+}+r_{-}-2M)~{},$ (104)
$\displaystyle B_{+}-B_{-}$ $\displaystyle=2\omega(r_{+}-r_{-})~{}.$ (105)
Besides, note that $E_{\ell
m}=\ell(\ell+1)+E_{1}a\omega+E_{2}a^{2}\omega^{2}$, where $E_{1}$ and $E_{2}$
depends on the angular and azimuthal numbers $\ell$ and $m$, along with the
mass of the BH. Changing the radial coordinate $r$ to a new coordinate $z$,
defined in 81, and using the above identities, the coefficient of $(dR/dz)$
term in the radial Teukolsky equation reads,
$\displaystyle\textrm{Coefficient~{}of}~{}\frac{dR}{dz}$
$\displaystyle=\frac{2iP_{+}-1}{z}-\frac{1+2iP_{+}+2i\omega\\{2M+(r_{+}-r_{-})(1+z)\\}}{(1+z)}\,,$
(106)
which in the near-horizon approximation ($z\approx 0$), reduces to,
$\displaystyle\textrm{Coefficient~{}of}~{}\frac{dR}{dz}\Bigg{|}_{\rm
near~{}horizon}$
$\displaystyle=\frac{2iP_{+}-1}{z}-\frac{1+2iP_{+}+2i\omega\\{2M+(r_{+}-r_{-})\\}}{(1+z)}~{}.$
(107)
Note that there are no terms of $\mathcal{O}(M^{2}\omega^{2})$ in the
coefficient of $(dR/dz)$. Similarly, the coefficient of the radial Teukolsky
function $R$, in the radial Teukolsky equation, becomes,
$\displaystyle\textrm{Coefficient~{}of}~{}R$
$\displaystyle=-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+8iM\omega}{(1+z)^{2}}-\frac{A_{+}+iB_{+}}{z(1+z)}+\frac{(A_{-}-A_{+})+i(B_{-}-B_{+})}{(1+z)}$
$\displaystyle=-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+8iM\omega}{(1+z)^{2}}-\frac{\ell(\ell+1)-2+(E_{1}-2m)a\omega+2i\omega
r_{+}+\omega^{2}(-r_{+}^{2}+2Mr_{+}+E_{2}a^{2})}{z(1+z)}$
$\displaystyle\qquad+\frac{\omega^{2}(r_{+}-r_{-})(r_{+}+r_{-}-2M)-2i\omega(r_{+}-r_{-})}{(1+z)}$
$\displaystyle=-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+2i\omega\\{4M-(1+z)(r_{+}-r_{-})\\}+\omega^{2}(r_{+}-r_{-})(r_{+}+r_{-}-2M)(1+z)}{(1+z)^{2}}$
$\displaystyle\qquad-\frac{\ell(\ell+1)-2}{z(1+z)}+\frac{2ma\omega}{z(1+z)}\left(1-\frac{E_{1}}{2m}\right)-\frac{2i\omega
r_{+}}{z(1+z)}-\frac{\omega^{2}(-r_{+}^{2}+2Mr_{+}+E_{2}a^{2})}{z(1+z)}~{},$
(108)
which depends on terms involving $M^{2}\omega^{2}$. Using the result that
$r_{+}+r_{-}=2M$, as well as $r_{+}-2M=-r_{-}$ and $r_{+}r_{-}=a^{2}$, along
with the near-horizon approximation, the coefficient of the Teukolsky radial
function in the Teukolsky equation becomes,
$\displaystyle\textrm{Coefficient~{}of}~{}R\Bigg{|}_{\rm near~{}horizon}$
$\displaystyle=-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+2i\omega\\{4M-(r_{+}-r_{-})\\}}{(1+z)^{2}}$
$\displaystyle\qquad-\frac{\ell(\ell+1)-2}{z(1+z)}+\frac{2ma\omega}{z(1+z)}\left(1-\frac{E_{1}}{2m}\right)-\frac{2i\omega
r_{+}}{z(1+z)}-\frac{\omega^{2}a^{2}(1+E_{2})}{z(1+z)}~{}.$ (109)
These expressions for the coefficients of $(dR/dz)$ and $R$ term in the radial
Teukolsky equation have been used in the main text.
## Appendix D Zero-rotation limit of the Detweiler function
In this appendix, we will discuss the perturbation of compact objects having
zero rotation. In this limit, we first write down the relation between the
Detweiler and the Teukolsky function with $s=-2$, which reads [79],
$\,{}_{-2}X^{0}_{\ell m}$
$\displaystyle=\left\\{\left(\frac{r}{r^{2}-2Mr}\right)\alpha_{-2}^{0}\right\\}\,_{-2}R_{\ell
m}^{0}+\left\\{\left(\frac{r}{\left(r^{2}-2Mr\right)^{2}}\right)\beta_{-2}^{0}\right\\}\dfrac{\,{}_{-2}R_{\ell
m}^{0}}{dr}$
$\displaystyle=\left\\{\left(\frac{1}{r-2M}\right)\alpha_{-2}^{0}\right\\}\,_{-2}R_{\ell
m}^{0}+\left\\{\left(\frac{1}{\left(r-2M\right)^{3}}\right)\beta_{-2}^{0}\right\\}\dfrac{\,{}_{-2}R_{\ell
m}^{0}}{dr_{*}}\,,$ (110)
where $\alpha_{-2}^{0}$ and $\beta_{-2}^{0}$ are the coefficients necessary to
make the potential for the Detweiler function real with zero rotation and
$r_{*}$ is the tortoise coordinate. For $s=-2$ and in the zero rotation limit,
the functions $\alpha_{-2}^{0}$ and $\beta_{-2}^{0}$ read as follows [79],
$\displaystyle\alpha_{-2}^{0}$
$\displaystyle=\frac{\kappa(A_{1}+iA_{2})+|\kappa|^{2}}{\sqrt{2}|\kappa|\sqrt{A_{1}+\textrm{Re}\kappa}}~{},$
(111) $\displaystyle\beta_{-2}^{0}$ $\displaystyle=\frac{i\kappa
B_{2}}{\sqrt{2}|\kappa|\sqrt{A_{1}+\textrm{Re}\kappa}}~{},$ (112)
where, we have introduced the quantities $\kappa$, $A_{1}$, $A_{2}$ and
$B_{2}$, defined as,
$\displaystyle\kappa$ $\displaystyle=4\gamma_{\ell}(\gamma_{\ell}+2)+48i\omega
M~{},\qquad\gamma_{\ell}=\left(\ell+2\right)\left(\ell-1\right)~{},$ (113)
$\displaystyle A_{1}$
$\displaystyle=4\left[\frac{8r^{8}\omega^{4}}{(r^{2}-2Mr)^{2}}+\frac{8r^{4}\omega^{2}}{(r^{2}-2Mr)}\left(\frac{M^{2}}{(r^{2}-2Mr)}-\gamma_{\ell}\right)-\frac{4r^{2}\omega^{2}}{(r^{2}-2Mr)}\left(3r^{2}+2Mr\right)+12r^{2}\omega^{2}+\gamma_{\ell}(\gamma_{\ell}+2)\right]~{},$
(114) $\displaystyle A_{2}$
$\displaystyle=4\left[-\frac{24r^{5}\omega^{3}}{(r^{2}-2Mr)}+\frac{4\gamma_{\ell}(r-M)r^{2}\omega}{(r^{2}-2Mr)}+4\omega
r\gamma_{\ell}+12\omega M\right]~{},$ (115) $\displaystyle B_{2}$
$\displaystyle=32r^{6}\omega^{3}+16r^{2}\omega(r^{2}-2Mr)\left\\{\frac{2M^{2}}{(r^{2}-2Mr)}-\gamma_{\ell}\right\\}-8M\omega
r(r^{2}-2Mr)~{}.$ (116)
Further, the Teukolsky function $\,{}_{-2}R_{\ell m}^{0}$ can be related to
the Regge-Wheeler and Zerilli function in the zero-rotation limit. The first
step in this direction is to decompose the Teukolsky function into polar and
axial parts,
$\,{}_{-2}R_{\ell m}^{0}=\,_{-2}R_{\ell
m}^{0,~{}\textrm{axial}}+\,_{-2}R_{\ell m}^{0,~{}\textrm{polar}}~{}.$ (117)
Since the Detweiler function depends on the Teukolsky function in a linear
manner, it follows that such a decomposition will hold for the Detweiler
function as well,
$\,{}_{-2}X_{\ell m}^{0}=\,_{-2}X_{\ell
m}^{0,~{}\textrm{axial}}+\,_{-2}X_{\ell m}^{0,~{}\textrm{polar}}~{}.$ (118)
The axial and polar parts of the Teukolsky function can then be expressed in
terms of the Regge-Wheeler and Zerilli functions, respectively, as
$\,{}_{-2}R_{\ell m}^{0,~{}\textrm{axial}}$
$\displaystyle=\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left[V^{\rm
axial}\Psi^{\rm RW}_{\ell m}+\left(W^{\rm
axial}+2i\omega\right)\left(\dfrac{d\Psi_{\ell m}^{\rm
RW}}{dr_{*}}+i\omega\Psi_{\ell m}^{\rm RW}\right)\right]$
$\displaystyle=\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left[\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}\Psi^{\rm RW}_{\ell
m}+\left(W^{\rm axial}+2i\omega\right)\left(\dfrac{d\Psi_{\ell m}^{\rm
RW}}{dr_{*}}\right)\right]~{},$ (119) $\,{}_{-2}R_{\ell
m}^{0,~{}\textrm{polar}}$
$\displaystyle=-\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left[\left\\{V^{\rm
polar}+i\omega\left(W^{\rm polar}+2i\omega\right)\right\\}\Psi^{\rm Z}_{\ell
m}+\left(W^{\rm polar}+2i\omega\right)\left(\dfrac{d\Psi^{\rm Z}_{\ell
m}}{dr_{*}}\right)\right]~{},$ (120)
where we have introduced four functions of the radial coordinate $r$, namely
$V_{\rm axial}$, $V_{\rm polar}$, $W_{\rm axial}$ and $W_{\rm polar}$, each of
which can be defined as,
$\displaystyle V^{\rm axial}$
$\displaystyle=\frac{(r-2M)\left\\{(\gamma_{\ell}+2)r-6M\right\\}}{r^{4}}=V_{\rm
RW}~{},$ (121) $\displaystyle W^{\rm axial}$
$\displaystyle=\frac{2(r-3M)}{r^{2}}~{},$ (122) $\displaystyle V^{\rm polar}$
$\displaystyle=\frac{(r-2M)}{r^{4}(\gamma_{\ell}r+6M)^{2}}\left[\gamma_{\ell}^{2}(\gamma_{\ell}+2)r^{3}+6M\gamma_{\ell}^{2}r^{2}+36M^{2}\gamma_{\ell}r+72M^{3}\right]=V_{\rm
Z}~{},$ (123) $\displaystyle W^{\rm polar}$
$\displaystyle=\frac{2\gamma_{\ell}r^{2}-6\gamma_{\ell}Mr-12M^{2}}{r^{2}(\gamma_{\ell}r+6M)}~{}.$
(124)
Therefore, one can directly relate the axial and polar parts of the Detweiler
function to the Regge-Wheeler and the Zerilli function, respectively,
$\,{}_{-2}X_{\ell m}^{\rm axial}$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\,_{-2}R_{\ell
m}^{0,~{}\textrm{axial}}+\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\dfrac{\,{}_{-2}R_{\ell
m}^{0,~{}\textrm{axial}}}{dr_{*}}$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left[\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}\Psi^{\rm RW}_{\ell
m}+\left(W^{\rm axial}+2i\omega\right)\left(\dfrac{d\Psi^{\rm RW}_{\ell
m}}{dr_{*}}\right)\right]\Bigg{]}$
$\displaystyle+\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\frac{d}{dr_{*}}\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left[\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}\Psi^{\rm RW}_{\ell
m}+\left(W^{\rm axial}+2i\omega\right)\left(\dfrac{d\Psi^{\rm RW}_{\ell
m}}{dr_{*}}\right)\right]\Bigg{]}$ $\displaystyle\equiv A_{\rm RW}(r)\Psi^{\rm
RW}_{\ell m}+B_{\rm RW}(r)\left(\frac{\Psi^{\rm RW}_{\ell m}}{dr}\right)~{},$
(125)
where, we have defined the two functions $A_{\rm RW}$ and $B_{\rm RW}$, in the
following expressions,
$\displaystyle A_{\rm RW}(r)$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}$
$\displaystyle\qquad+f(r)\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\frac{d}{dr}\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}\Bigg{]}$
$\displaystyle\qquad+\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\left(\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\right)\left(W^{\rm
axial}+2i\omega\right)V_{\rm RW}~{},$ (126) $\displaystyle B_{\rm RW}(r)$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left(W^{\rm
axial}+2i\omega\right)f(r)$
$\displaystyle\qquad+f(r)\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}$
$\displaystyle\qquad\qquad+\frac{d}{dr}\left\\{\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left(W^{\rm
axial}+2i\omega\right)\right\\}\Bigg{]}~{}.$ (127)
Hence, we can express, the ratio of the derivative of the axial part of the
Detweiler function with the Detweiler function itself in terms of the same
ratio for the Regge-Wheeler function as,
$\displaystyle\frac{\left(\frac{\,{}_{-2}X_{\ell m}^{\rm
axial}}{dr}\right)}{\,{}_{-2}X_{\ell m}^{\rm axial}}=\frac{\left[A_{\rm
RW}^{\prime}+B_{\rm RW}\left(\frac{r^{2}V_{\rm
RW}}{(r-2M)^{2}}\right)\right]+\left[A_{\rm RW}+B_{\rm RW}^{\prime}-B_{\rm
RW}\left(\frac{2M}{r(r-2M)}\right)\right]\frac{\Psi_{\rm
RW}^{\prime}}{\Psi_{\rm RW}}}{A_{\rm RW}+B_{\rm RW}\frac{\Psi_{\rm
RW}^{\prime}}{\Psi_{\rm RW}}}~{}.$ (128)
The above ratio can also be expressed in terms of the tortoise coordinate
$r_{*}$,
$\displaystyle\frac{1}{f}\frac{\left(\frac{\,{}_{-2}X_{\ell m}^{\rm
axial}}{dr_{*}}\right)}{\,{}_{-2}X_{\ell m}^{\rm
axial}}=\frac{f(r)\left[A_{\rm RW}^{\prime}+B_{\rm RW}\left(\frac{r^{2}V_{\rm
RW}}{(r-2M)^{2}}\right)\right]+\left[A_{\rm RW}+B_{\rm RW}^{\prime}-B_{\rm
RW}\left(\frac{2M}{r(r-2M)}\right)\right]\frac{\frac{d\Psi_{\rm
RW}}{dr_{*}}}{\Psi_{\rm RW}}}{f(r)A_{\rm RW}+B_{\rm RW}\frac{\frac{d\Psi_{\rm
RW}}{dr_{*}}}{\Psi_{\rm RW}}}~{}.$ (129)
Along identical lines, for the polar sector, we obtain,
$\,{}_{-2}X_{\ell m}^{\rm polar}=-A_{\rm Z}(r)\Psi^{\rm Z}_{\ell m}-B_{\rm
Z}(r)\left(\frac{\Psi^{\rm Z}_{\ell m}}{dr}\right)~{},$ (130)
where the functions $A_{\rm Z}$ and $B_{\rm Z}$ are defined as,
$\displaystyle A_{\rm Z}(r)$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left\\{V^{\rm
polar}+i\omega\left(W^{\rm polar}+2i\omega\right)\right\\}$
$\displaystyle\qquad+f(r)\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\frac{d}{dr}\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left\\{V^{\rm
polar}+i\omega\left(W^{\rm polar}+2i\omega\right)\right\\}\Bigg{]}$
$\displaystyle\qquad+\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\left(\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\right)\left(W^{\rm
polar}+2i\omega\right)V_{\rm Z}~{},$ (131) $\displaystyle B_{\rm Z}(r)$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left(W^{\rm
polar}+2i\omega\right)f(r)$
$\displaystyle\qquad+f(r)\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left\\{V^{\rm
polar}+i\omega\left(W^{\rm polar}+2i\omega\right)\right\\}$
$\displaystyle\qquad\qquad+\frac{d}{dr}\left\\{\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left(W^{\rm
polar}+2i\omega\right)\right\\}\Bigg{]}~{}.$ (132)
Therefore, the ratio of the derivative of the polar sector of the Detweiler
function with respect to the tortoise coordinate and the Detweiler function
itself becomes,
$\displaystyle\frac{\left(\frac{\,{}_{-2}X_{\ell m}^{\rm
polar}}{dr_{*}}\right)}{\,{}_{-2}X_{\ell m}^{\rm
polar}}=\frac{f(r)\left[A_{\rm Z}^{\prime}+B_{\rm Z}\left(\frac{r^{2}V_{\rm
Z}}{(r-2M)^{2}}\right)\right]+\left[A_{\rm Z}+B_{\rm Z}^{\prime}-B_{\rm
Z}\left(\frac{2M}{r(r-2M)}\right)\right]\frac{\frac{d\Psi_{\rm
Z}}{dr_{*}}}{\Psi_{\rm Z}}}{f(r)A_{\rm Z}+B_{\rm Z}\frac{\frac{d\Psi_{\rm
Z}}{dr_{*}}}{\Psi_{\rm Z}}}~{}.$ (133)
These results have been used in the main text, as well as while discussing the
static limit of the TLNs for non-rotating compact objects.
## References
* [1] See Section V of T. Damour’s contribution to Ref. [Deruelle:1984hq] for the first published calculation of the TLNs of black holes.
* [2] T. Binnington and E. Poisson, “Relativistic theory of tidal Love numbers,” Phys. Rev. D80 (2009) 084018, arXiv:0906.1366 [gr-qc].
* [3] T. Damour and A. Nagar, “Relativistic tidal properties of neutron stars,” Phys. Rev. D80 (2009) 084035, arXiv:0906.0096 [gr-qc].
* [4] N. Gurlebeck, “No-hair theorem for Black Holes in Astrophysical Environments,” Phys. Rev. Lett. 114 no. 15, (2015) 151102, arXiv:1503.03240 [gr-qc].
* [5] E. Poisson, “Tidal deformation of a slowly rotating black hole,” Phys. Rev. D91 no. 4, (2015) 044004, arXiv:1411.4711 [gr-qc].
* [6] P. Pani, L. Gualtieri, A. Maselli, and V. Ferrari, “Tidal deformations of a spinning compact object,” Phys. Rev. D92 no. 2, (2015) 024010, arXiv:1503.07365 [gr-qc].
* [7] P. Landry and E. Poisson, “Tidal deformation of a slowly rotating material body. External metric,” Phys. Rev. D91 (2015) 104018, arXiv:1503.07366 [gr-qc].
* [8] A. Le Tiec, M. Casals, and E. Franzin, “Tidal Love Numbers of Kerr Black Holes,” Phys. Rev. D 103 no. 8, (2021) 084021, arXiv:2010.15795 [gr-qc].
* [9] H. S. Chia, “Tidal deformation and dissipation of rotating black holes,” Phys. Rev. D 104 no. 2, (2021) 024013, arXiv:2010.07300 [gr-qc].
* [10] A. Le Tiec and M. Casals, “Spinning Black Holes Fall in Love,” Phys. Rev. Lett. 126 no. 13, (2021) 131102, arXiv:2007.00214 [gr-qc].
* [11] L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, “Static response and Love numbers of Schwarzschild black holes,” JCAP 04 (2021) 052, arXiv:2010.00593 [hep-th].
* [12] P. Charalambous, S. Dubovsky, and M. M. Ivanov, “Hidden Symmetry of Vanishing Love Numbers,” Phys. Rev. Lett. 127 no. 10, (2021) 101101, arXiv:2103.01234 [hep-th].
* [13] P. Charalambous, S. Dubovsky, and M. M. Ivanov, “On the Vanishing of Love Numbers for Kerr Black Holes,” JHEP 05 (2021) 038, arXiv:2102.08917 [hep-th].
* [14] L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, “Ladder symmetries of black holes. Implications for love numbers and no-hair theorems,” JCAP 01 no. 01, (2022) 032, arXiv:2105.01069 [hep-th].
* [15] R. Berens, L. Hui, and Z. Sun, “Ladder symmetries of black holes and de Sitter space: love numbers and quasinormal modes,” JCAP 06 (2023) 056, arXiv:2212.09367 [hep-th].
* [16] J. Ben Achour, E. R. Livine, S. Mukohyama, and J.-P. Uzan, “Hidden symmetry of the static response of black holes: applications to Love numbers,” JHEP 07 (2022) 112, arXiv:2202.12828 [gr-qc].
* [17] P. Charalambous, S. Dubovsky, and M. M. Ivanov, “Love symmetry,” JHEP 10 (2022) 175, arXiv:2209.02091 [hep-th].
* [18] T. Katagiri, M. Kimura, H. Nakano, and K. Omukai, “Vanishing Love numbers of black holes in general relativity: From spacetime conformal symmetry of a two-dimensional reduced geometry,” Phys. Rev. D 107 no. 12, (2023) 124030, arXiv:2209.10469 [gr-qc].
* [19] M. M. Ivanov and Z. Zhou, “Vanishing of Black Hole Tidal Love Numbers from Scattering Amplitudes,” Phys. Rev. Lett. 130 no. 9, (2023) 091403, arXiv:2209.14324 [hep-th].
* [20] V. De Luca, J. Khoury, and S. S. C. Wong, “Nonlinearities in the tidal Love numbers of black holes,” Phys. Rev. D 108 no. 2, (2023) 024048, arXiv:2305.14444 [gr-qc].
* [21] R. A. Porto, “The Tune of Love and the Nature(ness) of Spacetime,” Fortsch. Phys. 64 (2016) 723–729, arXiv:1606.08895 [gr-qc].
* [22] V. Cardoso, E. Franzin, A. Maselli, P. Pani, and G. Raposo, “Testing strong-field gravity with tidal Love numbers,” Phys. Rev. D95 no. 8, (2017) 084014, arXiv:1701.01116 [gr-qc].
* [23] N. Sennett, T. Hinderer, J. Steinhoff, A. Buonanno, and S. Ossokine, “Distinguishing Boson Stars from Black Holes and Neutron Stars from Tidal Interactions in Inspiraling Binary Systems,” Phys. Rev. D96 no. 2, (2017) 024002, arXiv:1704.08651 [gr-qc].
* [24] R. F. P. Mendes and H. Yang, “Tidal deformability of boson stars and dark matter clumps,” Class. Quant. Grav. 34 no. 18, (2017) 185001, arXiv:1606.03035 [astro-ph.CO].
* [25] P. Pani, “I-Love-Q relations for gravastars and the approach to the black-hole limit,” Phys. Rev. D92 no. 12, (2015) 124030, arXiv:1506.06050 [gr-qc].
* [26] N. Uchikata, S. Yoshida, and P. Pani, “Tidal deformability and I-Love-Q relations for gravastars with polytropic thin shells,” Phys. Rev. D94 no. 6, (2016) 064015, arXiv:1607.03593 [gr-qc].
* [27] G. Raposo, P. Pani, M. Bezares, C. Palenzuela, and V. Cardoso, “Anisotropic stars as ultracompact objects in General Relativity,” Phys. Rev. D99 no. 10, (2019) 104072, arXiv:1811.07917 [gr-qc].
* [28] V. Cardoso and P. Pani, “Testing the nature of dark compact objects: a status report,” Living Rev. Rel. 22 no. 1, (2019) 4, arXiv:1904.05363 [gr-qc].
* [29] V. Cardoso, M. Kimura, A. Maselli, and L. Senatore, “Black Holes in an Effective Field Theory Extension of General Relativity,” Phys. Rev. Lett. 121 no. 25, (2018) 251105, arXiv:1808.08962 [gr-qc].
* [30] K. Chakravarti, S. Chakraborty, S. Bose, and S. SenGupta, “Tidal Love numbers of black holes and neutron stars in the presence of higher dimensions: Implications of GW170817,” Phys. Rev. D 99 no. 2, (2019) 024036, arXiv:1811.11364 [gr-qc].
* [31] K. Chakravarti, S. Chakraborty, K. S. Phukon, S. Bose, and S. SenGupta, “Constraining extra-spatial dimensions with observations of GW170817,” Class. Quant. Grav. 37 no. 10, (2020) 105004, arXiv:1903.10159 [gr-qc].
* [32] V. Cardoso, L. Gualtieri, and C. J. Moore, “Gravitational waves and higher dimensions: Love numbers and Kaluza-Klein excitations,” Phys. Rev. D 100 no. 12, (2019) 124037, arXiv:1910.09557 [gr-qc].
* [33] P. Charalambous and M. M. Ivanov, “Scalar Love numbers and Love symmetries of 5-dimensional Myers-Perry black hole,” arXiv:2303.16036 [hep-th].
* [34] M. J. Rodriguez, L. Santoni, A. R. Solomon, and L. F. Temoche, “Love Numbers for Rotating Black Holes in Higher Dimensions,” arXiv:2304.03743 [hep-th].
* [35] R. Dey, S. Chakraborty, and N. Afshordi, “Echoes from braneworld black holes,” Phys. Rev. D 101 no. 10, (2020) 104014, arXiv:2001.01301 [gr-qc].
* [36] R. Dey, S. Biswas, and S. Chakraborty, “Ergoregion instability and echoes for braneworld black holes: Scalar, electromagnetic, and gravitational perturbations,” Phys. Rev. D 103 no. 8, (2021) 084019, arXiv:2010.07966 [gr-qc].
* [37] A. Maselli, P. Pani, V. Cardoso, T. Abdelsalhin, L. Gualtieri, and V. Ferrari, “From micro to macro and back: probing near-horizon quantum structures with gravitational waves,” arXiv:1811.03689 [gr-qc].
* [38] S. Datta, “Probing horizon scale quantum effects with Love,” Class. Quant. Grav. 39 no. 22, (2022) 225016, arXiv:2107.07258 [gr-qc].
* [39] P. Pani and A. Maselli, “Love in Extrema Ratio,” Int. J. Mod. Phys. D 28 no. 14, (2019) 1944001, arXiv:1905.03947 [gr-qc].
* [40] G. A. Piovano, A. Maselli, and P. Pani, “Constraining the tidal deformability of supermassive objects with extreme mass ratio inspirals and semianalytical frequency-domain waveforms,” Phys. Rev. D 107 no. 2, (2023) 024021, arXiv:2207.07452 [gr-qc].
* [41] G. F. Giudice, M. McCullough, and A. Urbano, “Hunting for Dark Particles with Gravitational Waves,” JCAP 1610 no. 10, (2016) 001, arXiv:1605.01209 [hep-ph].
* [42] G. Creci, T. Hinderer, and J. Steinhoff, “Tidal response from scattering and the role of analytic continuation,” Phys. Rev. D 104 no. 12, (2021) 124061, arXiv:2108.03385 [gr-qc]. [Erratum: Phys.Rev.D 105, 109902 (2022)].
* [43] D. Consoli, F. Fucito, J. F. Morales, and R. Poghossian, “CFT description of BH’s and ECO’s: QNMs, superradiance, echoes and tidal responses,” JHEP 12 (2022) 115, arXiv:2206.09437 [hep-th].
* [44] M. V. S. Saketh, Z. Zhou, and M. M. Ivanov, “Dynamical Tidal Response of Kerr Black Holes from Scattering Amplitudes,” arXiv:2307.10391 [hep-th].
* [45] M. Perry and M. J. Rodriguez, “Dynamical Love Numbers for Kerr Black Holes,” arXiv:2310.03660 [gr-qc].
* [46] E. Maggio, P. Pani, and G. Raposo, “Testing the nature of dark compact objects with gravitational waves,” arXiv:2105.06410 [gr-qc].
* [47] E. Maggio, V. Cardoso, S. R. Dolan, and P. Pani, “Ergoregion instability of exotic compact objects: electromagnetic and gravitational perturbations and the role of absorption,” Phys. Rev. D99 no. 6, (2019) 064007, arXiv:1807.08840 [gr-qc].
* [48] E. Maggio, L. Buoninfante, A. Mazumdar, and P. Pani, “How does a dark compact object ringdown?,” Phys. Rev. D 102 no. 6, (2020) 064053, arXiv:2006.14628 [gr-qc].
* [49] R. P. Bhatt, S. Chakraborty, and S. Bose, “Addressing issues in defining the Love number for black holes,” arXiv:2306.13627 [gr-qc].
* [50] P. Pani, L. Gualtieri, T. Abdelsalhin, and X. Jiménez-Forteza, “Magnetic tidal Love numbers clarified,” Phys. Rev. D 98 no. 12, (2018) 124023, arXiv:1810.01094 [gr-qc].
* [51] E. Maggio, P. Pani, and V. Ferrari, “Exotic Compact Objects and How to Quench their Ergoregion Instability,” Phys. Rev. D96 no. 10, (2017) 104047, arXiv:1703.03696 [gr-qc].
* [52] G. Raposo, P. Pani, and R. Emparan, “Exotic compact objects with soft hair,” Phys. Rev. D99 no. 10, (2019) 104050, arXiv:1812.07615 [gr-qc].
* [53] E. Poisson and C. Will, Gravity: Newtonian, Post-Newtonian, Relativistic. Cambridge University Press, Cambridge, UK, 1953.
* [54] R. Hansen, “Multipole moments of stationary space-times,” J.Math.Phys. 15 (1974) 46–52.
* [55] J. B. Hartle, “Tidal Friction in Slowly Rotating Black Holes,” Phys. Rev. D8 (1973) 1010–1024.
* [56] S. A. Hughes, “Evolution of circular, nonequatorial orbits of Kerr black holes due to gravitational wave emission. II. Inspiral trajectories and gravitational wave forms,” Phys. Rev. D64 (2001) 064004, arXiv:gr-qc/0104041 [gr-qc]. [Erratum: Phys. Rev.D88,no.10,109902(2013)].
* [57] A. Maselli, P. Pani, V. Cardoso, T. Abdelsalhin, L. Gualtieri, and V. Ferrari, “Probing Planckian corrections at the horizon scale with LISA binaries,” Phys. Rev. Lett. 120 no. 8, (2018) 081101, arXiv:1703.10612 [gr-qc].
* [58] S. Datta, R. Brito, S. Bose, P. Pani, and S. A. Hughes, “Tidal heating as a discriminator for horizons in extreme mass ratio inspirals,” Phys. Rev. D 101 no. 4, (2020) 044004, arXiv:1910.07841 [gr-qc].
* [59] S. Chakraborty, S. Datta, and S. Sau, “Tidal heating of black holes and exotic compact objects on the brane,” Phys. Rev. D 104 no. 10, (2021) 104001, arXiv:2103.12430 [gr-qc].
* [60] E. Newman and R. Penrose, “An Approach to gravitational radiation by a method of spin coefficients,” J. Math. Phys. 3 (1962) 566–578.
* [61] S. A. Teukolsky, “Rotating black holes - separable wave equations for gravitational and electromagnetic perturbations,” Phys. Rev. Lett. 29 (1972) 1114–1118.
* [62] S. A. Teukolsky, “Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations,” Astrophys. J. 185 (1973) 635–647.
* [63] S. A. Teukolsky and W. H. Press, “Perturbations of a rotating black hole. III \- Interaction of the hole with gravitational and electromagnet ic radiation,” Astrophys. J. 193 (1974) 443–461.
* [64] W. Kinnersley, “Type d vacuum metrics,” Journal of Mathematical Physics 10 no. 7, (1969) 1195–1203, https://doi.org/10.1063/1.1664958. https://doi.org/10.1063/1.1664958.
* [65] N. Oshita, Q. Wang, and N. Afshordi, “On Reflectivity of Quantum Black Hole Horizons,” JCAP 04 (2020) 016, arXiv:1905.00464 [hep-th].
* [66] S. Chakraborty, E. Maggio, A. Mazumdar, and P. Pani, “Implications of the quantum nature of the black hole horizon on the gravitational-wave ringdown,” Phys. Rev. D 106 no. 2, (2022) 024041, arXiv:2202.09111 [gr-qc].
* [67] S. Nair, S. Chakraborty, and S. Sarkar, “Dynamical Love numbers for area quantized black holes,” Phys. Rev. D 107 no. 12, (2023) 124041, arXiv:2208.06235 [gr-qc].
* [68] J. L. Ripley, A. Hegade K. R., and N. Yunes, “Probing internal dissipative processes of neutron stars with gravitational waves during the inspiral of neutron star binaries,” arXiv:2306.15633 [gr-qc].
* [69] S. Detweiler, “On resonant oscillations of a rapidly rotating black hole,” Proceedings of the Royal Society of London Series A 352 (July, 1977) 381–395.
* [70] E. Berti, V. Cardoso, and M. Casals, “Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions,” Phys. Rev. D 73 (2006) 024013, arXiv:gr-qc/0511111. [Erratum: Phys.Rev.D 73, 109902 (2006)].
* [71] E. Maggio, M. van de Meent, and P. Pani, “Extreme mass-ratio inspirals around a spinning horizonless compact object,” Phys. Rev. D 104 no. 10, (2021) 104026, arXiv:2106.07195 [gr-qc].
* [72] T. Hinderer, “Tidal Love numbers of neutron stars,” Astrophys. J. 677 (2008) 1216–1220, arXiv:0711.2420 [astro-ph]. Erratum: ibid. 697, 964 (2009). |
17116 LABEL:LastPageDec. 16, 2019Feb. 18, 2021
# A Formal Proof of the Irrationality of $\zeta(3)$
Assia Mahboubi and Thomas Sibut-Pinote LS2N UFR Sciences et Techniques, 2
rue de la Houssinière, BP 92208 44322 Nantes Cedex 3 France
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.
This paper presents a complete formal verification of a proof that the
evaluation of the Riemann zeta function at 3 is irrational, using the Coq
proof assistant. This result was first presented by Apéry in 1978, and the
proof we have formalized essentially follows the path of his original
presentation. The crux of this proof is to establish that some sequences
satisfy a common recurrence. We formally prove this result by an _a
posteriori_ verification of calculations performed by computer algebra
algorithms in a Maple session. The rest of the proof combines arithmetical
ingredients and asymptotic analysis, which we conduct by extending the
Mathematical Components libraries.
###### Key words and phrases:
formal proof, number theory, irrationality, creative telescoping, symbolic
computation, Coq, Apéry’s recurrences, Riemann zeta function
This work was supported in part by the project FastRelax ANR-14-CE25-0018-01.
## 1\. Introduction
In 1978, Apéry proved that $\zeta(3)$, which is the sum
$\sum_{i=1}^{\infty}\frac{1}{i^{3}}$ now known as the _Apéry constant_ , is
irrational. This result was the first dent in the problem of the irrationality
of the evaluation of the Riemann zeta function at _odd_ positive integers. As
of today, this problem remains a long-standing challenge of number theory.
Zudilin [Zud01] showed that at least one of the numbers
$\zeta(5)$,$\zeta(7)$,$\zeta(9),\zeta(11)$ must be irrational. Ball and Rivoal
[Riv00, BR01] established that there are infinitely many irrational odd zeta
values. Fischler, Sprang and Zudilin proved [FSZ19] that there are
asymptotically more than any power of $\log(s)$ irrational values of the
Riemann zeta function at odd integers between $3$ and $s$. But today
$\zeta(3)$ is the only _known_ such value to be irrational.
Van der Poorten reports [vdP79] that Apéry’s announcement of this result was
at first met with wide skepticism. His obscure presentation featured “a
sequence of unlikely assertions” without proofs, not the least of which was an
enigmatic recurrence (Lemma 6) satisfied by two sequences $a$ and $b$. It took
two months of collaboration between Cohen, Lenstra, and Van der Poorten, with
the help of Zagier, to obtain a thorough proof of Apéry’s theorem:
###### Theorem 1 (Apéry, 1978).
The constant $\zeta(3)$ is irrational.
In the present paper, we describe a formal proof of this theorem inside the
Coq proof assistant [The20], using the Mathematical Components libraries
[coq19]. This formalization follows the structure of Apéry’s original proof.
However, we replace the manual verification of recurrence relations by an
automatic discovery of these equations, using symbolic computation. For this
purpose, we use Maple packages to perform calculations outside the proof
assistant, and we verify a posteriori the resulting claims inside Coq. By
combining these verified results with additional formal developments, we
obtain a complete formal proof of Theorem 1, formalized using the Coq proof
assistant without additional axiom. In particular, the proof is entirely
constructive, and does _not_ rely on the axiomatic definition of real numbers
provided in Coq’s standard library. A previous paper [CMSPT14] reported on the
implementation of the cooperation between a computer algebra system and a
proof assistant used in the formalization. The present paper is self-
contained: it includes a summary of the latter report, and provides more
details about the rest of the formal proof. In particular, it describes the
formalization of an upper bound on the asymptotic behavior of $lcm(1,...,n)$,
the least common multiple of the integers from $1$ to $n$, a part of the proof
which was missing in the previous report.
The rest of the paper is organized as follows. We first describe the
background formal theories used in our development (Section 2). We then
outline the proof of Theorem 1 (Section 3). We summarize the algorithms used
in the Maple session, the data this session produces and the way this data can
be used in formal proofs (Section 4). We then describe the proof of the
consequences of Apéry’s recurrence (Section 5). Finally, we present an
elementary proof of the bound on the asymptotic behavior of the sequence
$lcm(1,...,n)$, which is used in this irrationality proof (Section 6), before
commenting on related work and concluding (Section 7).
The companion code to the present article can be found in the following
repository:
https://github.com/math-comp/apery.
## 2\. Preliminaries
This section provides some hints about the representation of the different
natures of numbers at stake in this proof in the libraries backing our formal
development. It also describes a few extensions we devised for these libraries
and sets some notations used throughout this paper. Most of the material
presented here is related to the Mathematical Components libraries [coq19,
GAA+13].
### 2.1. Integers
In Coq, the set $\mathbb{N}$ of natural numbers is usually represented by the
type nat:
⬇
Inductive nat := O | S : nat -> nat.
This type is defined in a prelude library, which is automatically imported by
any Coq session. It models the elements of $\mathbb{N}$ using a unary
representation: Coq’s parser reads the number $2$ as the term S (S O). The
structural induction principle associated with this inductive type coincides
with the usual recurrence scheme on natural numbers. This is convenient for
defining elementary functions on natural numbers, like comparison or
arithmetical operations, and for developing their associated theory. However,
the resulting programs are usually very naive and inefficient implementations,
which should only be evaluated for the purpose of small scale computations.
The set $\mathbb{Z}$ of integers can be represented by gluing together two
copies of type nat, which provides a signed unary representation of integers:
⬇
Inductive int : Set := Posz of nat | Negz of nat.
If the term n : nat represents the natural number $n\in\mathbb{N}$, then the
term (Posz n) : int represents the integer $n\in\mathbb{Z}$ and the term (Negz
n) : int represents the integer $-(n+1)\in\mathbb{Z}$. In particular, the
constructor Posz : nat -> int implements the embedding of type nat into type
int, which is invisible on paper because it is just the inclusion
$\mathbb{N}\subset\mathbb{Z}$. In order to mimic the mathematical practice,
the constant Posz is declared as a _coercion_ , which means in particular that
unless otherwise specified, this function is hidden from the terms displayed
by Coq to the user (in the current goal, in answers to search queries, etc).
The Mathematical Components libraries provide formal definitions of a few
elementary concepts and results from number theory, defined on the type nat.
For instance, they provide the theory of Euclidean division, a boolean
primality test, the elementary properties of the factorial function, of
binomial coefficients, etc. In the rest of the paper, we use the standard
mathematical notations $n!$ and $\binom{n}{m}$ for the corresponding formal
definition of the factorial and of the binomial coefficients respectively.
These libraries also define the $p$-adic valuation $v_{p}(n)$ of a number $n$:
if $p$ is a prime number, it is the exponent of $p$ in the prime decomposition
of $n$. However, we had to extend the available basic formal theory with a few
extra standard results, like the formula giving the $p$-adic valuation of
factorials:
###### Lemma 2.
For any $n\in\mathbb{N}$ and for any prime number $p$:
$v_{p}(n!)=\sum_{i=1}^{\left\lfloor{\log_{p}n}\right\rfloor}\left\lfloor{\frac{n}{p^{i}}}\right\rfloor.$
Incidentally, the formal version of this formula is a typical example of the
slight variations one may introduce in a mathematical statement, in order to
come up with a formal sentence which is not only correct and faithful to the
original mathematical result, but also a tool which is easy to use in
subsequent formal proofs. First, although the fraction in the original
statement of Lemma 2 may suggest that rational numbers play a role here,
$\left\lfloor{\frac{n}{m}}\right\rfloor$ is in fact exactly the quotient of
the Euclidean division of $n$ by $m$. In the rest of the paper, for
$n,m\in\mathbb{N}$ and $m$ non-zero, we thus write
$\left\lfloor{\frac{n}{m}}\right\rfloor$ for the quotient of the Euclidean
division of $n$ by $m$. Perhaps more interestingly, the formal statement of
Lemma 2 rather corresponds to the following variant:
$\textrm{For any prime }p\textrm{ and any }j,n\in\mathbb{N},\textrm{ such that
}n<p^{j+1},v_{p}(n!)=\sum_{i=1}^{j}\left\lfloor{\frac{n}{p^{i}}}\right\rfloor.$
Adding an extra variable to generalize the upper bound of the sum is a better
option because it will ease unification when this formula is applied or used
for rewriting. Moreover, we do not really need to introduce logarithms here:
indeed, $\left\lfloor{\log_{p}n}\right\rfloor$ is used to denote the largest
power of $p$ smaller than $n$. For this purpose, we could use the function
trunc_log : nat -> nat -> nat provided by the Mathematical Components
libraries, which computes the greatest exponent $\alpha$ such that
$n^{\alpha}\leq m$, in other words $\left\lfloor{\log_{n}m}\right\rfloor$.
Better yet, since the summand is zero when the index $i$ exceeds this value,
we can simplify the side condition on the extra variable and require only that
$n<p^{j+1}$.
The basic theory of binomial coefficients present in the Mathematical
Components libraries describes their role in elementary enumerative
combinatorics. However, when viewing binomial coefficients as a sequence which
is a certain solution of a recurrence system, it becomes natural to extend
their domain of definition to integers: we thus developed a small library
about these generalized binomial coefficients. We also needed to extend these
libraries with properties of multinomial coefficients. For
$n,k_{1},\dotsc,k_{l}\in\mathbb{N}$, with $k_{1}+\dotsb+k_{l}=n$, the
coefficient of $x_{1}^{k_{1}}\dotsm x_{l}^{k_{l}}$ in the formal expansion of
$\left(x_{1}+\dotsb+x_{l}\right)^{n}$ is called a _multinomial coefficient_
and denoted $\binom{n}{k_{1},\dotsc,k_{l}}$. Its value is
$\frac{n!}{k_{1}!\dotsc k_{l}!}$ or equivalently,
$\prod\limits_{i=1}^{l}\binom{k_{1}+\dots+k_{i}}{k_{i}}$. The latter formula
provides for free the fact that multinomial coefficients are non-negative
integers and we use it in our formal definition: for (l : seq nat) a finite
sequence $l_{1},\dots,l_{s}$ of natural numbers, then (multinomial l) is the
multinomial coefficient $\binom{l_{1}+\dots+l_{s}}{l_{1},\dotsc,l_{s}}$:
⬇
Definition multinomial (l : seq nat) : nat :=
\prod_(0 <= i < size l) (binomial (\sum_(0 <= j < i.+1) l‘_j) l‘_i).
From this definition, we prove formally the other characterizations, as well
as the generalized Newton formula describing the expansion of
$\left(x_{1}+\dotsb+x_{l}\right)^{n}$.
### 2.2. Rational numbers, algebraic numbers, real numbers
In the Mathematical Components libraries, rational numbers are represented
using a dependent pair. This type construct, also called $\Sigma$-type, is
specific to dependent type theory: it makes possible to define a type that
decorates a data with a proof that a certain property holds on this data. The
Mathematical Components libraries also include a construction of the algebraic
closure for countable fields, and thus a construction of
${\overline{\mathbb{Q}}}$, algebraic closure of $\mathbb{Q}$, the field of
rational numbers. The corresponding type, named algC is equipped with a
structure of (partially) ordered, algebraically closed field [GAA+13].
Slightly abusing notation, we denote by
${\overline{\mathbb{Q}}}\cap\mathbb{R}$ the subset of
${\overline{\mathbb{Q}}}$ containing elements with a zero imaginary part and
we call such elements real algebraic numbers.
Almost all the irrational numbers involved in the present proof are real
algebraic numbers, and more precisely, they are of the form $r^{\frac{1}{n}}$
for $r$ a non-negative rational number $r$ and for $n\in\mathbb{N}$. The only
place where these numbers play a role is in auxiliary lemmas for the proof of
the asymptotic behavior of the sequence $(\ell_{n})_{n\in\mathbb{N}}$, where
$\ell_{n}$ is the least common multiple of integers between $1$ and $n$. They
appear in inequalities expressing signs and estimations.
It might come as a surprise that we used the type algC of algebraic (complex)
numbers to cast these quantities, although we do not actually need imaginary
complex numbers. But this choice proved convenient due to the fact that the
type algC features both a definition of $n$-th roots, and a clever choice of
partial order. Indeed, although ${\overline{\mathbb{Q}}}$ cannot be ordered as
a field, it is equipped with a binary relation, denoted $\leq$, which
coincides with the real order relation on
${\overline{\mathbb{Q}}}\cap\mathbb{R}$:
$\forall x,y\in{\overline{\mathbb{Q}}},\;x\leq y\Leftrightarrow
y-x\in\mathbb{R}_{\geq 0}.$
In particular, for any $z\in{\overline{\mathbb{Q}}}$:
$0\leq z\Leftrightarrow z\in{\mathbb{R}_{\geq 0}}\textrm{ and }z\leq
0\Leftrightarrow z\in{\mathbb{R}_{\leq 0}}.$
Moreover, the type algC is equipped with a function n.-root :algC -> algC,
defined for any (n : nat), such that (n.-rootz) is the $n$-th (complex) root
of z with minimal non-negative argument. Crucially, when (z : algC) represents
a non-negative real number, (n.-root z) coincides with the definition of the
real $n$-th root, and thus the following equivalence holds:
⬇
Lemma rootC_ge0 (n : nat) (z : algC) : n > 0 -> (0 <= n.-root z) = (0 <= z).
The shape of Lemma rootC_ge0 is typical of the style pervasive in the
Mathematical Components libraries, where equivalences between decidable
statements are stated as boolean equalities. It expresses that for an
algebraic number $x$, that is for $x\in{\overline{\mathbb{Q}}}$, we have
$x^{\frac{1}{n}}\in\mathbb{R}_{\geq 0}$ if and only if $x\in\mathbb{R}_{\geq
0}$.
The one notable place at which we need to resort to a larger subset of the
real numbers is the definition of the number $\zeta(3)$, if only because as of
today, it is not even known whether $\zeta(3)$ is algebraic or transcendental.
This number is actually defined as the limit of the partial sums
$\sum_{m=1}^{n}\frac{1}{m^{3}}$, so we start our formal study by establishing
the existence of this limit.
More precisely, we show that this sequence of partial sums is a _Cauchy
sequence_ , with an explicit modulus of convergence.
A _Cauchy sequence_ is a sequence
$(x_{n})_{n\in\mathbb{N}}\in\mathbb{Q}^{\mathbb{N}}$ together with a modulus
of convergence $m_{x}\ :\ \mathbb{Q}\rightarrow\mathbb{N}$ such that if
$\varepsilon$ is a positive rational number, then any two elements of index
greater than $m_{x}(\varepsilon)$ are separated at most by $\varepsilon$.
###### Proposition 3.
The sequence $z_{n}=\sum_{m=1}^{n}\frac{1}{m^{3}}$ is a Cauchy sequence.
Two sequences $x$ and $y$ are _Cauchy equivalent_ if $x$ and $y$ are both
Cauchy sequences, and if eventually $|x_{n}-y_{n}|<\varepsilon$, for any
$\varepsilon>0$. Real numbers could be constructed formally by introducing a
quotient type, whose element are the equivalent classes of the latter
relation. But this is rather irrelevant for this formalization, which involves
explicit sequences and their asymptotic properties, rather than real numbers.
For this reason, the formal statements in this formal proof only involve
sequences of rational numbers, and a type of Cauchy sequences which pairs such
a sequence with a proof that it has the Cauchy property. For instance, for two
Cauchy sequences $x$ and $y$, we write $x<y$ if there is a rational number
$\varepsilon>0$ such that eventually $x_{n}+\varepsilon\leq y_{n}$.
We benefited from the formal library about Cauchy sequences developed by Cohen
[Coh12]. This library defines Cauchy sequences of elements in a totally
ordered field, and introduces a type (creal F) of Cauchy sequences over the
totally ordered field F, given as a parameter. We thus use the instance (creal
rat). The infix notation ==, in the notation scope CR, denotes the equivalence
of Cauchy sequences, as in the statement (x == y)%CR, which states that the
two Cauchy sequences x, y are equivalent. The library implements a _setoid_ of
field operations over this type [BCP03, Soz09], so as to facilitate
substitutions for equivalents in formulas. In addition, the library provides a
tactic called bigenough, which eases formal proofs by allowing a dose of
laziness. This tactic is specially useful in proofs that a certain property on
sequences is eventually true, which involve constructing effective moduli of
convergence.
The formal statement corresponding to Theorem 1 is thus:
⬇
Theorem zeta_3_irrational : ~ exists (r : rat), (z3 == r%:CR)%CR.
where the postfix notation r%:CR denotes the Cauchy sequence whose elements
are all equal to the rational number (r : rat). The term z3 is the Cauchy
sequence corresponding to the partial sums $(z_{n})_{n\in\mathbb{N}}$, that
is, the dependent pair of this sequence with a proof of Property 3. The formal
statement thus expresses that no constant rational sequence can be Cauchy
equivalent to $(z_{n})_{n\in\mathbb{N}}$. Interestingly, a long-lasting typo
has marred the formal statement of theorem zeta_3_irrational in the
corresponding Coq libraries, until writing the revised version of the present
paper. Until then, the (inaccurate) statement was indeed:
⬇
Theorem incorrect_zeta_3_irrational : ~ exists (r : rat), (z3 = r%:CR)%CR.
Replacing == by = changes the statement completely, as it now expresses that
there is no constant sequence of rationals equal to z3: and this is trivially
true. The typo was already present in the version of the code that we made
public for our previous report [CMSPT14], and the typo has remained unnoticed
since. Yet fortunately, the _proof script_ was right, and actually described a
correct proof of the stronger statement zeta_3_irrational.
### 2.3. Notations
In this section, we provide a few hints on the notations used in the formal
statements corresponding to the paper proof, so as to make precise the meaning
of the statements we have proved formally. Indeed, this development makes
heavy use of the notation facilities offered by the Coq proof assistant, so as
to improve the readability of formulas. For instance, notation scopes allow to
use the same infix notation for a relation on type nat, and in this case (x <
y) unfolds to (ltn x y) : bool, or for a relation on type creal rat), and in
that case (x < y) unfolds to (lt_creal x y) : Prop, the comparison predicate
described in Section 2.2. Notation scopes can be made explicit using post-
fixed tags: (x < y)%N is interpreted in the scope associated with natural
numbers, and (x < y)%CR, in the scope associated with Cauchy sequences.
Generic notations can also be shared thanks to type-class like mechanisms. The
Mathematical Components libraries feature a hierarchy of algebraic structures
[GGMR09], which organizes a corpus of theories and notations shared by all the
instances of a same structure. This hierarchy implements inheritance and
sharing using Coq’s mechanisms of coercions and of canonical structures
[MT13]. Each structure in the hierarchy is modeled by a dependent record,
which packages a type with some operations on this type and with requirements
on these operations. In order to equip a given type with a certain structure,
one has to endow this type with enough operations and properties, following
the signature prescribed by the structure. For example, these structures are
all _discrete_ , which means that they require a boolean equality test. In
turn, all instances of all these structures share the same infix notation (x
== y) for the latter boolean equality test between x and y: this notation
makes sense for x,y in type nat,int, rat, alC, etc. because all these types
are instances of the same structure. For instance, although type rat is a
dependent pair (see Section 2.2), the boolean comparison test only needs to
work with the data: by Hedberg’s theorem [Hed98], the proof stored in the
proof component can be made irrelevant. Note that the situation is different
for the type (creal rat) of Cauchy sequences. The formal statement of theorem
zeta_3_irrational (see Section 2.2) uses the same == infix symbol, but in a
different scope, in which it refers to Prop-valued Cauchy equivalence. Indeed,
this relation cannot be turned constructively into a boolean predicate, as the
comparison of Cauchy sequences is not effective. The type algC of algebraic
numbers by contrast enjoys the generic version of the notation, as ordered
fields only require a partial order relation.
Partial order, but also units of a ring, and inverse operations are examples
of operations involved in some structures of the hierarchy, that make sense
only on a subset of the elements of the carrier. In the dependent type theory
implemented by Coq, it would be possible to use a dependent pair in order to
model the source type of such an inverse operation. Instead, as a rule of
thumb, the signature of a given structure avoids using rich types as the
source types of their operations but rather “curry” the specification. For
instance, the source type of the inverse operation in the structure of ring
with units is the carrier type itself, but the signature of this structure
also has a boolean predicate, which selects the units in this carrier type.
The inverse operation has a default behavior outside units and the equations
of the theory that involve inverses are typically guarded with invertibility
conditions. Hence although the expression x^-1 * x is syntactically well-
formed for any term x of an instance of ring with units, it can be rewritten
to 1 only when x is known to be invertible.
The readability of formulas also requires dealing in a satisfactory manner
with the inclusion of the various collections of numbers at stake, that are
represented with distinct types, for instance:
$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset{\overline{\mathbb{Q}}}.$
The implementation of the inclusion $\mathbb{N}\subset\mathbb{Z}$ was
mentioned in Section 2.1. The canonical embedding of type int is available in
the generic theory of rings, but unfortunately, it cannot be declared as a
coercion, and eluded in formal statements: the type of the corresponding
constant would violate the uniform inheritance condition prescribed by Coq’s
coercion mechanism [The20]. Its formal definition hence comes with a generic
postfix notation _%:~R, modeled as a reminiscence of the syntax of notation
scopes and used to cast an integer as an element of another ring. The latter
embedding is pervasive in formulas expressing the recurrence relations
involved in this proof. Indeed, these recurrence relations feature polynomial
coefficients in their indices and relate the rational elements of their
solutions. See for instance Equation 2.
### 2.4. Computations
Using the unary representation of integers described in Section 2.1, the
command:
⬇
Compute 100*1000.
which asks Coq to evaluate this product, triggers a stack overflow. For the
purpose of running computations inside Coq’s logic, on integers of a medium
size, an alternate data-structure is required, together with less naive
implementations of the arithmetical operations. The present formal proof
requires this nature of computations at several places, for instance in order
to evaluate sequences defined by a recurrence relation at a few particular
values. For these computations, we used the binary representation of integers
provided by the ZArith library included in the standard distribution of Coq,
together with the fast reduction mechanism included in Coq’s kernel [GL02].
These two ingredients are also used behind the scene by tactics implementing
verified decision procedures. For instance, we make extensive use of proof
commands dedicated to the normalization of algebraic expressions like the
field tactic for rational fractions, and the ring tactic for polynomials
[GM05]. The field tactic generates proof obligations describing sufficient
conditions for the simplifications it made. In our case, these conditions in
turn are solved using the lia decision procedure for linear arithmetics
[Bes07].
These tactics work by first converting formulas in the goal into instances of
appropriate data structures, suitable for larger scale computation. This pre-
processing, hidden to the user, is performed by extra-logical code that is
part of the internal implementation of these tactics. The situation is
different when a computational step in a proof requires the evaluation of a
formula at a given argument, and when both the formula and the argument are
described using proof-oriented, inefficient representations. In that case, for
instance for evaluating terms in a given sequence, we used the CoqEAL library
[CDM13], which provides an infrastructure automating the conversion between
different data-structures and algorithms used to model the same mathematical
objects, like different representations of integers or different
implementations of a matrix product. Note that although the CoqEAL library
itself depends on a library for big numbers, which provides direct access in
Coq to Ocaml’s library for arbitrary-precision, arbitrary-size signed
integers, the present proof does not need this feature.
## 3\. Outline of the proof
There exists several other proofs of Apéry’s theorem. Notably, Beukers [Beu79]
published an elegant proof, based on integrals of pseudo-Lengendre
polynomials, shortly after Apéry’s announcement. According to Fischler’s
survey [Fis04], all these proofs share a common structure. They rely on the
asymptotic behavior of the sequence $\ell_{n}$, the least common multiple of
integers between $1$ and $n$, and they proceed by exhibiting two sequences of
rational numbers $a_{n}$ and $b_{n}$, which have the following properties:
1. (1)
For a sufficiently large $n$:
$a_{n}\in\mathbb{Z}\quad\textrm{ and }2\ell_{n}^{3}b_{n}\in\mathbb{Z};$
2. (2)
The sequence $\delta_{n}=a_{n}\zeta(3)-b_{n}$ is such that:
$\limsup_{n\rightarrow\infty}|2\delta_{n}|^{\frac{1}{n}}\leq(\sqrt{2}-1)^{4};$
3. (3)
For an infinite number of values $n$, $\delta_{n}\neq 0$.
Altogether, these properties entail the irrationality of $\zeta(3)$. Indeed,
if we suppose that there exists $p,q\in\mathbb{Z}$ such that
$\zeta(3)=\frac{p}{q}$, then $2q\ell_{n}^{3}\delta_{n}$ is an integer when $n$
is large enough. One variant of the Prime Number theorem states that
$\ell_{n}=e^{n\,(1+o(1))}$ and since $(\sqrt{2}-1)^{4}e^{3}<1$, the sequence
$2q\ell_{n}^{3}\delta_{n}$ has a zero limit, which contradicts the third
property. Actually, the Prime Number theorem can be replaced by a weaker
estimation of the asymptotic behavior of $\ell_{n}$, that can be obtained by
more elementary means.
###### Lemma 4.
Let $\ell_{n}$ be the least common multiple of integers $1,\dots,n$, then
$\ell_{n}=O(3^{n}).$
Since we still have $(\sqrt{2}-1)^{4}3^{3}<1$, this observation [Han72, Fen05]
is enough to conclude. Section 6 discusses the formal proof of Lemma 4, an
ingredient which was missing at the time of writing the previous report on
this work [CMSPT14].
In our formal proof, we consider the pair of sequences proposed by Apéry in
his proof [Apé79, vdP79]:
$a_{n}=\sum_{k=0}^{n}{\tbinom{n}{k}}^{2}{\tbinom{n+k}{k}}^{2},\qquad
b_{n}=a_{n}z_{n}+\sum_{k=1}^{n}\sum_{m=1}^{k}\frac{(-1)^{m+1}{\tbinom{n}{k}}^{2}{\tbinom{n+k}{k}}^{2}}{2m^{3}\binom{n}{m}\binom{n+m}{m}}$
(1)
where $z_{n}$ denotes $\sum_{m=1}^{n}\frac{1}{m^{3}}$, as already used in
Proposition 3.
By definition, $a_{n}$ is a positive integer for any $n\in\mathbb{N}$. The
integrality of $2\ell_{n}^{3}b_{n}$ is not as straightforward, but rather easy
to see as well: each summand in the double sum defining $b_{n}$ has a
denominator that divides $2\ell_{n}^{3}$. Indeed, after a suitable re-
organization in the expression of the summand, using standard properties of
binomial coefficients, this follows easily from the following slightly less
standard property:
###### Lemma 5.
For any integers $i,j,n$ such that $1\leq j\leq i\leq n$, $j\binom{i}{j}\text{
divides }\ell_{n}$.
###### Proof 3.1.
For $i,j,n$ such that $1\leq j\leq i\leq n$, the proof goes by showing that
for any prime $p$, the $p$-adic valuation of $j\binom{i}{j}$ is at most that
of $\ell_{n}$. Let us fix a prime number $p$. Let $t_{p}(i)$ be the largest
integer $e$ such that $p^{e}\leq i$. By definition, and since $i\leq n$, we
thus have $p^{t_{p}(i)}|\ell_{n}$ and so $t_{p}(i)\leq v_{p}(\ell_{n})$. Hence
it suffices to prove that $v_{p}(\binom{i}{j})\leq t_{p}(i)-v_{p}(j)$. Using
Lemma 2, and because $j\leq i<p^{t_{p}(i)+1}$, we have:
$v_{p}(\binom{i}{j})=\sum_{k=1}^{t_{p}(i)}\left\lfloor{\frac{i}{p^{k}}}\right\rfloor-(\sum_{k=1}^{t_{p}(i)}\left\lfloor{\frac{j}{p^{k}}}\right\rfloor+\sum_{k=1}^{t_{p}(i)}\left\lfloor{\frac{(i-j)}{p^{k}}}\right\rfloor)$
Remember that for $a,b\in\mathbb{N}$, $\left\lfloor{\frac{a}{b}}\right\rfloor$
is just $a$ modulo $b$. Now for $1\leq k\leq v_{p}(j)$, and because $p^{k}|j$,
we have
$\left\lfloor{\frac{i}{p^{k}}}\right\rfloor=\left\lfloor{\frac{j}{p^{k}}}\right\rfloor+\left\lfloor{\frac{(i-j)}{p^{k}}}\right\rfloor$,
and thus:
$\displaystyle v_{p}(\binom{i}{j})$ $\displaystyle=$
$\displaystyle\sum_{k=v_{p}(j)+1}^{t_{p}(i)}\left\lfloor{\frac{i}{p^{k}}}\right\rfloor-(\sum_{k=v_{p}(j)+1}^{t_{p}(i)}\left\lfloor{\frac{j}{p^{k}}}\right\rfloor+\sum_{k=v_{p}(j)+1}^{t_{p}(i)}\left\lfloor{\frac{(i-j)}{p^{k}}}\right\rfloor)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{t_{p}(i)-v_{p}(j)}\left\lfloor{\frac{i}{p^{v_{p}(j)+k}}}\right\rfloor-(\sum_{k=1}^{t_{p}(i)-v_{p}(j)}\left\lfloor{\frac{j}{p^{v_{p}(j)+k}}}\right\rfloor+\sum_{k=1}^{t_{p}(i)-v_{p}(j)}\left\lfloor{\frac{(i-j)}{p^{v_{p}(j)+k}}}\right\rfloor)$
Now for any $1\leq k\leq t_{p}(i)-v_{p}(j)$, we have:
$\left\lfloor{\frac{i}{p^{v_{p}(j)+k}}}\right\rfloor\leq\left\lfloor{\frac{j}{p^{v_{p}(j)+k}}}\right\rfloor+\left\lfloor{\frac{(i-j)}{p^{v_{p}(j)+k}}}\right\rfloor+1$
Summing both sides for $k$ from $1$ to $t_{p}(i)-v_{p}(j)$ and using the
previous identity for $v_{p}(\binom{i}{j})$ eventually proves that
$v_{p}(\binom{i}{j})\leq t_{p}(i)-v_{p}(j)$, which concludes the proof.
The rest of the proof is a study of the sequence
$\delta_{n}=a_{n}\zeta(3)-b_{n}$. It not difficult to see that $\delta_{n}$
tends to zero, from the formulas defining the sequences $a$ and $b$, but we
also need to prove that it does so fast enough to compensate for
$\ell_{n}^{3}$, while being positive. In his original proof, Apéry derived the
latter facts by combining the definitions of the sequences $a$ and $b$ with
the study of a mysterious recurrence relation. Indeed, he made the surprising
claim that Lemma 6 holds:
###### Lemma 6.
For $n\geq 0$, the sequences $(a_{n})_{n\in\mathbb{N}}$ and
$(b_{n})_{n\in\mathbb{N}}$ satisfy the _same_ second-order recurrence:
$(n+2)^{3}y_{n+2}-(17n^{2}+51n+39)(2n+3)y_{n+1}+(n+1)^{3}y_{n}=0.$ (2)
Equation 2 is a typical example of a linear recurrence equation with
polynomial coefficients and standard techniques [Sal03, vdP79] can be used to
study the asymptotic behavior of its solutions. Using this recurrence and the
initial conditions satisfied by $a$ and $b$, one can thus obtain the two last
properties of our criterion, and conclude with the irrationality of
$\zeta(3)$. For the purpose of our formal proof, we devised an elementary
version of this asymptotic study, mostly based on variations on the
presentation of van der Poorten [vdP79]. We detail this part of the proof in
Section 5.
Using only Equation 2, even with sufficiently many initial conditions, it
would not be easy to obtain the first property of our criterion, about the
integrality of $a_{n}$ and $b_{n}$ for a large enough $n$. In fact, it would
also be difficult to prove that the sequence $\delta$ tends to zero: we would
only know that it has a finite limit, and how fast the convergence is. By
contrast, it is fairly easy to obtain these facts from the explicit closed
forms given in Formula 1.
The proof of Lemma 6 was by far the most difficult part in Apéry’s original
exposition. In his report [vdP79], van der Poorten describes how he, with
other colleagues, devoted significant efforts to this verification after
having attended the talk in which Apéry exposed his result for the first time.
Actually, the proof of Lemma 6 boils down to a routine calculation using the
two auxiliary sequences $U_{n,k}$ and $V_{n,k}$, themselves defined in terms
of $\lambda_{n,k}=\binom{n}{k}^{2}\binom{n+k}{k}^{2}$ (with $\lambda_{n,k}=0$
if $k<0$ or $k>n$):
$\displaystyle U_{n,k}$ $\displaystyle=$ $\displaystyle
4(2n+1)(k(2k+1)-(2n+1)^{2})\lambda_{n,k},$ $\displaystyle V_{n,k}$
$\displaystyle=$ $\displaystyle
U_{n,k}\left(\sum_{m=1}^{n}\frac{1}{m^{3}}+\sum_{m=1}^{k}\frac{(-1)^{m-1}}{2m^{3}\binom{n}{m}\binom{n+m}{m}}\right)+\frac{5(2n+1)k(-1)^{k-1}}{n(n+1)}\binom{n}{k}\binom{n+k}{k}$
The key idea is to compute telescoping sums for $U$ and $V$. For instance, we
have:
$U_{n,k}-U_{n,k-1}=(n+1)^{3}\lambda_{n+1,k}-(34n^{3}+51n^{2}+27n+5)\lambda_{n,k}+n^{3}\lambda_{n-1,k}$
(3)
Summing Equation 3 on $k$ shows that the sequence $a$ satisfies the recurrence
relation of Lemma 6. A similar calculation proves the analogue for $b$, using
telescoping sums of the sequence $V$.
Not only is the statement of Formula 2 difficult to discover: even when this
recurrence is given, finding the suitable auxiliary sequences $U$ and $V$ by
hand is a difficult task. Moreover, there is no other known way of proving
Lemma 6 than by exhibiting this nature of certificates. Fortunately, the
sequences $a$ and $b$ belong in fact to a class of objects well known in the
fields of combinatorics and of computer-algebra. Following seminal work of
Zeilberger’s [Zei90], algorithms have been designed and implemented in
computer-algebra systems, which are able to obtain linear recurrences for
these sequences. For instance the Maple package Mgfun (distributed as part of
the Algolib [alg13] library) implements these algorithms, among others. Basing
on this implementation, Salvy wrote a Maple worksheet [Sal03] that follows
Apéry’s original method but interlaces Maple calculations with human-written
parts. In particular, this worksheet illustrates how parts of this proof,
including the discovery of Apéry’s mysterious recurrence, can be performed by
symbolic computations. Our formal proof of Lemma 6 follows an approach similar
to the one of Salvy. It is based on calculations performed using the Algolib
library, and certified a posteriori. This part of the formal proof is
discussed in Section 4.1.
## 4\. Algorithms, Recurrences and Formal Proofs
This section quotes and summarizes an earlier publication [CMSPT14],
describing a joint work with Chyzak and Tassi.
Lemma 6 is the bottleneck in Apéry’s proof. Both sums $a_{n}$ and $b_{n}$ in
there are instances of _parameterized summation_ : they follow the pattern
$F_{n}=\sum_{k=\alpha(n)}^{\beta(n)}f_{n,k}$ in which the summand $f_{n,k}$,
potentially the bounds, and thus the sum, depend on a parameter $n$. This
makes it appealing to resort to the algorithmic paradigm of _creative
telescoping_ , which was developed for this situation in computer algebra.
### 4.1. Recurrences as a data structure for sequences
A fruitful idea from the realm of computer algebra is to represent sequences
not explicitly, such as the univariate $(n!)_{n}$ or the bivariate
$(\binom{n}{k})_{n,k}$, but by a system of linear recurrences of which they
are solutions such as $\\{u_{n+1}=(n+1)u_{n}\\}$ or
$\\{u_{n+1,k}=\frac{n+1}{n+1-k}u_{n,k},u_{n,k+1}=\frac{n-k}{k+1}u_{n,k}\\}$,
accompanied with sufficient initial conditions. Sequences which can be
represented in such a way are called _$\partial$ -finite_. The finiteness
property of their definition makes algorithmic most operations under which the
class of $\partial$-finite sequences is stable.
In the specific bivariate case which interests us, let $S_{n}$ be the shift
operator in $n$ mapping a sequence $(u_{n,k})_{n,k}$ to $(u_{n+1,k})_{n,k}$
and similarly, let $S_{k}$ map $(u_{n,k})_{n,k}$ to $(u_{n,k+1})_{n,k}$.
Linear recurrences canceling a sequence $f$ can be seen as elements of a non-
commutative ring of polynomials with coefficients in $\mathbb{Q}(n,k)$, and
with the two indeterminates $S_{n}$ and $S_{k}$, with the action $(P\cdot
f)_{n,k}=\sum_{(i,j)\in I}p_{i,j}(n,k)f_{n+i,k+j}$, where subscripts denote
evaluation. For example for $f_{n,k}=\binom{n}{k}$, the previous recurrences
once rewritten as equalities to zero can be represented as $P\cdot f=0$ for
$P=S_{n}-\frac{n+1}{n+1-k}$ and $P=S_{k}-\frac{n-k}{k+1}$, respectively.
Computer algebra gives us algorithms to produce canceling operators for
operations such as the addition or product of two $\partial$-finite sequences,
using for both its inputs and output a Gröbner basis as a canonical way to
represent the set of their canceling operators, which gives some uniqueness
guarantees.
The case of summing a sequence $(f_{n,k})$ into a parameterized sum
$F_{n}=\sum_{k=0}^{n}f_{n,k}$ is more involved: it follows the method of
_creative telescoping_ [Zei91], in two stages. First, an _algorithmic_ step
determines pairs $(P,Q)$ satisfying
$P\cdot f=(S_{k}-1)Q\cdot f$ (4)
with $P\in\mathbb{Q}(n)[S_{n}]$ and $Q\in\mathcal{A}$. To continue with our
example $f_{n,k}=\binom{n}{k}$, we could choose $P=S_{n}-2$ and $Q=S_{n}-1$.
Second, a _systematic_ but not fully algorithmic step follows: summing (4) for
$k$ between 0 and $n+\deg P$ yields
$(P\cdot F)_{n}=(Q\cdot f)_{k=n+\deg P+1}-(Q\cdot f)_{k=0}.$ (5)
Continuing with our binomial example, summing (4) for $k$ from 0 to $n+1$ (and
taking special values into account) yields
$\sum_{k=0}^{n+1}\binom{n+1}{k}-2\sum_{k=0}^{n}\binom{n}{k}=0$, a special form
of (5) with right-hand side canceling to zero. This tells us that the sequence
$(\sum_{k=0}^{n}\binom{n}{k})_{n\in\mathbb{N}}$ is a solution of the same
recurrence $P=S_{n}-2$ as $(2^{n})_{n\in\mathbb{N}}$: a simple check of
initial values gives us the identity $\forall
n\in\mathbb{N},2^{n}=\sum_{k=0}^{n}\binom{n}{k}$. The formula (5) in fact
assumes several hypotheses that hold not so often in practice; this will be
formalized by Equation (6) below.
### 4.2. Apéry’s sequences are $\partial$-finite constructions
The sequences $a$ and $b$ in (1) are $\partial$-finite: they have been
announced to be solutions of (2). But more precisely, they can be viewed as
constructed from “atomic” sequences by operations under which the class of
$\partial$-finite sequences is stable. This is summarized in Table 1.
step | explicit form | GB | operation | input(s)
---|---|---|---|---
1 | $c_{n,k}={\binom{n}{k}}^{2}{\binom{n+k}{k}}^{2}$ | $C$ | direct |
2 | $a_{n}=\sum_{k=1}^{n}c_{n,k}$ | $A$ | creative telescoping | $C$
3 | $d_{n,m}=\frac{(-1)^{m+1}}{2m^{3}\binom{n}{m}\binom{n+m}{m}}$ | $D$ | direct |
4 | $s_{n,k}=\sum_{m=1}^{k}d_{n,m}$ | $S$ | creative telescoping | $D$
5 | $z_{n}=\sum_{m=1}^{n}\frac{1}{m^{3}}$ | $Z$ | direct |
6 | $u_{n,k}=z_{n}+s_{n,k}$ | $U$ | addition | $Z$ and $S$
7 | $v_{n,k}=c_{n,k}u_{n,k}$ | $V$ | product | $C$ and $U$
8 | $b_{n}=\sum_{k=1}^{n}v_{n,k}$ | $B$ | creative telescoping | $V$
Table 1. Construction of $a_{n}$ and $b_{n}$: At each step, the Gröbner basis
named in column GB, which annihilates the sequence given in explicit form, is
obtained by the corresponding operation _on ideals_ , with input(s) given on
the last column.
In this table, Gröbner bases are systems of recurrence operators: at each line
in the table, the sequence given in explicit form is a solution of the system
of recurrences described by the operators in the Gröbner basis column. Note
that in fact none of these results rely on the _specific_ sequences in the
explicit form: at each step, a new Gröbner basis is obtained from known ones,
the ones that are cited in the input column. The table can also be read
bottom-up for the purpose of verification: the Gröbner basis obtained at a
given step can be verified using _only_ the Gröbner bases obtained at some
previous steps, all the way down to $C$ and $D$. These operators describe a
more general class of (germs of) sequences than just the explicit sequences
used in this table, thus initial conditions are needed to describe a precise
sequence.
### 4.3. Provisos and sound creative telescoping
We illustrate the process of verifying candidate new recurrences using known
ones on the example of Pascal’s triangle rule. One can “almost prove” Pascal’s
triangle rule using only the following recurrences, satisfied by binomial
coefficients:
$u_{n+1,k}=\frac{n+1}{n+1-k}u_{n,k}\quad\textrm{ and }\quad
u_{n,k+1}=\frac{n-k}{k+1}u_{n,k}.$
Indeed, we have:
$\binom{n+1}{k+1}-\binom{n}{k+1}-\binom{n}{k}=\left(\frac{n+1}{n-k}\frac{n-k}{k+1}-\frac{n-k}{k+1}-1\right)\binom{n}{k}=0\times\binom{n}{k}=0,$
_but this requires $k\neq-1$ and $k\neq n$_. Therefore, this does not prove
Pascal’s rule for all $n$ and $k$. The phenomenon is general: computer algebra
is unable to take denominators into account. This incomplete modeling of
sequences by algebraic objects may cast doubt on these computer-algebra
proofs, in particular when it comes to the output of creative-telescoping
algorithms.
By contrast, in our formal proofs, we augmented the recurrences with provisos
that restrict their applicability. In this setting, we validate a candidate
identity like the Pascal triangle rule by a normalization modulo the elements
of a Gröbner basis plus a verification that this normalization only involves
legal instances of the recurrences. In the case of creative telescoping, Eq.
(4) takes the form:
$(n,k)\notin\Delta\Rightarrow(P\cdot f_{\\_,k})_{n}=(Q\cdot f)_{n,k+1}-(Q\cdot
f)_{n,k},$ (6)
where $\Delta\subset\mathbb{Z}^{2}$ guards the relation and where $f_{\\_,j}$
denotes the univariate sequence obtained by specializing the second argument
of $f$ to $j$. Thus our formal analogue of Eq. (5) takes this restriction into
account and has the shape
$\begin{split}(P\cdot F)_{n}&=\Bigl{(}(Q\cdot f)_{n,n+\beta+1}-(Q\cdot
f)_{n,\alpha}\Bigr{)}+\sum_{i=1}^{r}\sum_{j=1}^{i}p_{i}(n)\,f_{n+i,n+\beta+j}\\\
&+\sum_{\alpha\leq k\leq n+\beta\ \wedge\ (n,k)\in\Delta}(P\cdot
f_{\\_,k})_{n}-(Q\cdot f)_{n,k+1}+(Q\cdot f)_{n,k},\end{split}$ (7)
for $F$ the sequence with general term
$F_{n}=\sum_{k=\alpha}^{n+\beta}f_{n,k}$ and where
$P=\sum\limits_{i=0}^{r}p_{i}(n)S_{n}^{i}$.
The last term of the right-hand side, which we will call the singular part,
witnesses the possible partial domain of validity of relation (6). Thus the
operator $P$ is a valid recurrence for the sequence $F$ if the right-hand side
of Eq. (7) normalizes to zero, at least outside of an algebraic locus that
will guard the recurrence.
### 4.4. Generated Operators, hand-written provisos, and formal proofs
For each step in Table 1, we make use of the data computed by the Maple
session in a systematic way, using pretty-printing code to express this data
in Coq. As mentioned in Section 4.3, we manually annotate each operator
produced by the computer-algebra program with provisos and turn it this way
into a conditional recurrence predicate on sequences. In our formal proof,
each step in Table 1 consists in proving that some conditional recurrences on
a composed sequence can be proved from some conditional recurrences known for
the arguments of the operation.
These steps are far from automatic, mainly because the singular part yields
terms which have to be reduced manually through trial-and-error using a
Gröbner basis of annihilators for $f$, but also because we have to show that
some rational-function coefficients of the remaining instances of $f$ are
zero. This is done through a combination of the field and lia proof commands,
helped by some factoring of denominators pre-obtained in Maple.
### 4.5. Composing closures and reducing the order of $B$
In order to complete the formal proof of Lemma 6, we verify formally that each
sequence involved in the construction of $a_{n}$ and $b_{n}$ is a solution of
the corresponding Gröbner system of annotated recurrence, starting from
$c_{n}$, $d_{n}$, and $z_{n}$ all the way to the the final conclusions. This
proves that $a_{n}$ is a solution of the recurrence (2) but only provides a
recurrence of order four for $b_{n}$. We then prove that $b$ also satisfies
the recurrence (2) using four evaluations $b_{0},b_{1},b_{2},b_{3}$.
## 5\. Consequences of Apéry’s recurrence
In this section, we detail the elementary proofs of the properties obtained as
corollaries of Lemma 6. We recall, from Section 3, that these properties
describe the asymptotic behavior of the sequence
$\delta_{n}=a_{n}\zeta(3)-b_{n}$, with:
$a_{n}=\sum_{k=0}^{n}{\tbinom{n}{k}}^{2}{\tbinom{n+k}{k}}^{2},\qquad
b_{n}=a_{n}z_{n}+\sum_{k=1}^{n}\sum_{m=1}^{k}\frac{(-1)^{m+1}{\tbinom{n}{k}}^{2}{\tbinom{n+k}{k}}^{2}}{2m^{3}\binom{n}{m}\binom{n+m}{m}}$
(8)
Throughout the section, we use the vocabulary and notations of Cauchy
sequences numbers, as introduced in Section 2.2. For instance, we have:
###### Lemma 7.
For any $\varepsilon$, eventually $|z_{n}-\frac{b_{n}}{a_{n}}|<\varepsilon$.
###### Proof 5.1.
Easy from the definition of $z$, $a$ and $b$.
###### Corollary 8.
The sequence $(\frac{b_{n}}{a_{n}})_{n\in\mathbb{N}}$ is a Cauchy sequence,
which is Cauchy equivalent to $(z_{n})_{n\in\mathbb{N}}$.
The formal statement corresponding to Lemma 7 is:
⬇
Lemma z3seq_b_over_a_asympt : {asympt e : n / |z3seq n - b_over_a_seq n| < e}.
where b_over_a_seq n represents $\frac{b_{n}}{a_{n}}$. The notation {asympt e
: i / P}, used in this formal statement, comes from the external library for
Cauchy sequences [Coh12]. In the expression {asympt e :i / P}, asympt is a
keyword, and both e and i are names for variables bound in the term P. This
expression unfolds to the term (asympt1 (fun e i => P)), a dependent pair that
ensures the existence of an explicit witness that property P asymptotically
holds:
⬇
Definition asympt1 R (P : R -> nat -> Prop) :=
{m : R -> nat | forall (eps : R) (i : nat), 0 < eps -> m eps <= i -> P eps i}.
The formalization of Corollary 8 then comes in three steps: first the proof
that $(\frac{b_{n}}{a_{n}})_{n\in\mathbb{N}}$ is a Cauchy sequence, as
formalized by the creal_aiom predicate:
⬇
Corollary creal_b_over_a_seq : creal_axiom b_over_a_seq.
This formal proof is a one-liner, because the corresponding general argument,
a sequence that is asymptotically close to a Cauchy sequence will itself
satisfy the Cauchy property, is already present in the library. Then, the
latter proof is used to forge an inhabitant of the type of rational Cauchy
sequences, which just amounts to pairing the sequence b_over_a_seq : nat ->
rat with the latter proof:
⬇
Definition b_over_a : {creal rat} := CReal creal_b_over_a_seq.
Now we can state the proof of equivalence between the two Cauchy sequences,
i.e., between the two corresponding terms of type {creal rat}:
⬇
Fact z3_eq_b_over_a : (z3 == b_over_a)%CR.
The proof of the latter fact is again a one-liner, with no additional
mathematical content added to lemma creal_b_over_a_seq, but it provides access
to automation based on setoid rewriting facilities
The Mathematical Components libraries do not cover any topic of analysis, and
even the most basic definitions of transcendental functions like the
exponential or the logarithm are not available. However, it is possible to
obtain the required properties of the sequence $\delta$ by very elementary
means, and almost all these elementary proofs can be inferred from a careful
reading and a combination of Salvy’s proof [Sal03] and of van der Poorten’s
description [vdP79].
Following van der Poorten, we introduce an auxiliary sequence
$(w_{n})\in\mathbb{Q}^{n}$, defined as:
$w_{n}=\begin{vmatrix}b_{n+1}&a_{n+1}\\\
b_{n}&a_{n}\end{vmatrix}=b_{n+1}a_{n}-a_{n+1}b_{n}.$
The sequence $w$ is called a _Casoratian_ : as $a$ and $b$ are solutions of a
same linear recurrence relation (2) of order 2, this can be seen as a discrete
analogue of the Wronskian for linear differential systems. For example, $w$
satisfies a recurrence relation of order 1, which provides a closed form for
$w$:
###### Lemma 9.
For $n\geq 2$, $w_{n}=\frac{6}{(n+1)^{3}}$.
###### Proof 5.2.
Since $a$ and $b$ satisfy the recurrence relation 1, $w$ satisfies the
relation:
$\forall k\in\mathbb{N},(k+2)^{3}w_{k+1}-(k+1)^{3}w_{k}=0.$
The result follows from the computation of $w_{0}$.
From this formula, we can obtain the positivity of the sequence $\delta$, and
an evaluation of its asymptotic behavior in terms of the sequence $a$.
###### Corollary 10.
For any $n\in\mathbb{N}$ such that $2\leq n$,
$0<\zeta(3)-\frac{b_{n}}{a_{n}}$.
The formal statement corresponding to Corollary 10 is the following:
⬇
Lemma lt0_z3_minus_b_over_a (n : nat) : 2 <= n -> (0%:CR < z3 - (b_over_a_seq
n)%:CR)%CR.
Note the postfix CR tag which enforces that inside the corresponding
parentheses, notations are interpreted in the scope associated with Cauchy
sequences: in particular, the order relation (on Cauchy sequences) is the one
described in Section 2.2.
Term (b_over_a_seq n : rat) is the rational number $\frac{a_{n}}{b_{n}}$,
which is casted as a Cauchy real, the corresponding constant sequence, using
the postfix %:CR, so as to be subtracted to the Cauchy sequence z3. This proof
in particular benefits from setoid rewriting using equivalences like
z3_eq_b_over_a, the formal counterpart of Corollary 8.
###### Proof 5.3.
Since $\zeta(3)$ is Cauchy equivalent to
$(\frac{b_{n}}{a_{n}})_{n\in\mathbb{N}}$, it is sufficient to show that for
any $k<l$, we have $0<\frac{b_{l}}{a_{l}}-\frac{b_{k}}{a_{k}}$. Thus it is
sufficient to observe that for any $k$, we have
$0<\frac{b_{k+1}}{a_{k+1}}-\frac{b_{k}}{a_{k}}$, which follows from Lemma 9.
###### Corollary 11.
$\zeta(3)-\frac{b_{n}}{a_{n}}=\mathcal{O}(\frac{1}{a_{n}^{2}})$.
###### Proof 5.4.
Since $\zeta(3)=\frac{b}{a}$, it is sufficient to show that there exists a
constant $K$, such that for any $k<l$,
$\frac{b_{l}}{a_{l}}-\frac{b_{k}}{a_{k}}\leq\frac{K}{a_{k}^{2}}$. But since
$a$ is an increasing sequence, Lemma 9 proves that for any $k<l$,
$\frac{b_{l}}{a_{l}}-\frac{b_{k}}{a_{k}}\leq\sum_{i=k}^{l-1}\frac{w_{i}}{a_{i}a_{i+1}}\leq\sum_{i=k}^{l-1}\frac{6}{(i+1)^{3}a_{k}^{2}}\leq\frac{K}{a_{k}^{2}}$,
for any $K$ greater than $6\cdot\zeta(3)$.
The last remaining step of the proof is to show that the sequence $a$ grows
fast enough. The elementary version of Lemma 12 is based on a suggestion by F.
Chyzak.
###### Lemma 12.
$33^{n}=\mathcal{O}(a_{n})$.
###### Proof 5.5.
Consider the auxiliary sequence $\rho_{n}=\frac{a_{n+1}}{a_{n}}$. Since
$\rho_{51}$ is greater than $33$, we only need to show that the sequence
$\rho$ is increasing. For the sake of readability, we denote $\mu_{n}$ and
$\nu_{n}$ the fractions coefficients of the recurrence satisfied by $a$,
obtained from Equation 2 after division by its leading coefficient. Thus $a$
satisfies the recurrence relation:
$a_{n+2}-\mu_{n}a_{n+1}+\nu_{n}=0.$
For $n\in\mathbb{N}$, we also introduce the function
$h_{n}(x)=\mu_{n}+\frac{\nu_{n}}{x}$, so that $\rho_{n+1}=h_{n}(\rho_{n})$.
The polynomial $P_{n}(x)=x^{2}-\mu_{n}x+\nu_{n}$ has two distinct roots
$x_{n}^{\prime}<x_{n}$, and the formula describing the roots of polynomials of
degree 2 show that $0<x_{n}^{\prime}<1<x_{n}$ and that the sequence $x_{n}$ is
increasing. But since $h_{n}(x)-x=-\frac{P_{n}(x)}{x}$, for $1<x<x_{n}$, we
have $h_{n}(x)>x$. A direct recurrence shows that for any $n\geq 2$,
$\rho_{n}\in[1,x_{n}]$, which concludes the proof.
In the formal proof of Lemma 12, the computation of $\rho_{51}$ was made
possible by using the CoqEAL library, as already mentioned in Section 2.4.
This proof also requires a few symbolic computations that are a bit tedious to
perform by hand: in these cases, we used Maple as an oracle to massage
algebraic expressions, before formally proving the correctness of the
simplification. This was especially useful to study the roots $x_{n}^{\prime}$
and $x_{n}$ of $P_{n}$.
We can now conclude with the limit of the sequence $\ell_{n}^{3}\delta_{n}$,
under the assumption that $\ell_{n}=\mathcal{O}(3^{n})$.
###### Corollary 13.
$\lim\limits_{n\to\infty}(\ell_{n}^{3}\delta_{n})=0$.
###### Proof 5.6.
Immediate, since $\delta_{n}=\mathcal{O}(\frac{1}{a_{n}})$ by Corollary 11,
and $\ell_{n}^{3}=\mathcal{O}((3^{3})^{n})$, and $3^{3}<33$.
In the next Section, we describe the proof of the last remaining assumption,
about the asymptotic behavior of $\ell_{n}$.
## 6\. Asymptotics of $lcm(1,...,n)$
For any integer $1\leq n$, let $\ell_{n}$ denote the least common multiple
$lcm(1,...,n)$ of the integers no greater than $n$. By convention, we set
$\ell_{0}=1$. The asymptotic behavior of the sequence $(\ell_{n})$ is a
classical corollary of the Prime Number Theorem. A sufficient estimation for
the present proof can actually but obtained as a direct consequence, using an
elementary remark about the $p$-adic valuations of $\ell_{n}$.
###### Remark 14.
For any prime number $p$, the integer $p^{v_{p}(\ell_{n})}$ is the highest
power of $p$ not exceeding $n$, so that:
$v_{p}(\ell_{n})=\left\lfloor{\log_{p}(n)}\right\rfloor.$
###### Proof 6.1.
Noticing that $v_{p}(lcm(a,b))=\max(v_{p}(a),v_{p}(b))$, we see by induction
on $n$ that $v_{p}(\ell_{n})=\max\limits_{i=1}^{n}v_{p}(i)$. Recall from
Section 2.1 that $\lfloor\log_{p}(n)\rfloor$ is a notation for the greatest
integer $\alpha$ such that $p^{\alpha}\leq n$. Since
$\alpha=v_{p}(p^{\alpha})$, we have $\alpha\leq v_{p}(\ell_{n})$. Now suppose
that $v_{p}(\ell_{n})=v_{p}(i)$ for some $i\in\\{1,\dotsc,n\\}$. Then
$i=p^{v_{p}(i)}q$ with $gcd(p,q)=1$ so that
$p^{v_{p}(\ell_{n})}=p^{v_{p}(i)}\leq i\leq n$ and thus
$v_{p}(\ell_{n})\leq\alpha$. This proves that $v_{p}(\ell_{n})=\alpha$.
By Remark 14, $\ell_{n}$ can hence be written as $\prod_{p\leq
n}p^{\left\lfloor{\log_{p}(n)}\right\rfloor}$ and therefore:
$\ln(\ell_{n})=\sum_{p\leq
n}\ln(p^{\left\lfloor{\log_{p}(n)}\right\rfloor})\leq\sum_{p\leq n}\ln(n).$
If $\pi(n)$ is the number of prime numbers no greater than $n$, we hence have:
$\ln(\ell_{n})\leq\pi(n)\ln(n).$
The Prime Number theorem states that $\pi(n)\sim\frac{n}{\ln(n)}$; we can thus
conclude that:
$\ell_{n}=\mathcal{O}(e^{n}).$
Note that this estimation is in fact rather precise, as in fact:
$\ell_{n}\sim e^{n\,(1+o(1))}.$
J. Avigad and his co-authors provided the first machine-checked proof of the
Prime Number theorem [ADGR07], which was considered at the time as a
formalization _tour de force_. Their formalization is based on a proof
attributed to A. Selberg and P. Erdös. Although the standard proofs of this
theorem use tools from complex analysis like contour integrals, their choice
was guided by the corpus of formalized mathematics available for the Isabelle
proof assistant, or the limits thereof. Although less direct, the proof by A.
Selberg and P. Erdös is indeed more elementary and avoids complex analysis
completely.
### 6.1. Statement, Notations and Outline
In order to prove Corollary 13 in Section 5, we only need to resort to Lemma
4, i.e., to the fact that:
$\ell_{n}=\mathcal{O}(3^{n}).$
This part of the proof was left as an assumption in our previous report
[CMSPT14]. This weaker description of the asymptotic behavior of $(\ell_{n})$
was in fact known before the first proofs of the Prime Number theorem but our
formal proof is a variation on an elementary proof proposed by Hanson [Han72].
The idea of the proof is to replace the study of $\ell_{n}$ by that of another
sequence $C(n)$. The latter is defined as a multinomial coefficient depending
on elements of a fast-growing sequence $\alpha$. The fact that
$\prod_{i=1}^{n}\alpha_{i}^{1/\alpha_{i}}<3$ independently of $n$ then allows
to show that $C(n)=\mathcal{O}(3^{n})$.
### 6.2. Proof
Define the sequence $(\alpha_{n})_{n\in\mathbb{N}}$ by $\alpha_{0}=2$, and
$\alpha_{n+1}=\alpha_{1}\alpha_{2}\dotsm\alpha_{n}+1$ for $n\geq 1$. By an
induction on $n$, this is equivalent to
$\alpha_{n+1}=\alpha_{n}^{2}-\alpha_{n}+1$. For $n,k\in\mathbb{N}$, let
$C(n,k)=\frac{n!}{\lfloor n/\alpha_{1}\rfloor!\lfloor
n/\alpha_{2}\rfloor!\dotsm\lfloor n/\alpha_{k}\rfloor!}.$
As soon as $\alpha_{k}\geq n$, $C(n,k)$ is independent of $k$ and we denote
$C(n)=C(n,k)$ for all such $k$. Hanson directly defines $C(n)$ as a limit, but
we found this to be inconvenient to manipulate in the proof. Moreover, most
inequalities stated on $C(n)$ actually hold for $C(n,k)$ with little or no
more hypotheses. The following technical lemma will be useful in the study of
this sequence.
###### Lemma 15.
For $k\in\mathbb{N}$,
$\sum_{i=1}^{k}\frac{1}{\alpha_{i}}=\frac{\alpha_{k+1}-2}{\alpha_{k+1}-1}<1\textrm{
and thus for }x\in\mathbb{Q}\textrm{ with }x\geq 1,\lfloor
x\rfloor>\sum_{i=1}^{k}\left\lfloor\frac{x}{\alpha_{i}}\right\rfloor.$
###### Proof 6.2.
The proof is done by induction and relies on the fact that if $a\in\mathbb{Q}$
and $m\in\mathbb{N}^{+}$, we have
$\left\lfloor\frac{a}{m}\right\rfloor=\left\lfloor\frac{\lfloor
a\rfloor}{m}\right\rfloor.$
Notice that by Lemma 15,
$\sum\limits_{i=0}^{k}\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor<n$ and
thus:
$C(n,k)=\binom{\sum\limits_{i=0}^{k}\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor}{\lfloor
n/\alpha_{1}\rfloor,\lfloor n/\alpha_{2}\rfloor,\dotsc,\lfloor
n/\alpha_{k}\rfloor}\frac{n!}{\left(\sum\limits_{i=0}^{k}\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor\right)!}.$
In particular, $C(n,k)\in\mathbb{N}$. The goal is now to show that
$\ell_{n}\leq C(n)<K\cdot 3^{n}$ for some $K$.
In the following, for $n,k,p\in\mathbb{N}$ and $p$ prime, we denote
$\beta_{p}(n,k)$ for $v_{p}(C(n,k))$.
###### Lemma 16.
For all $n,k\in\mathbb{N}$, with $1\leq n$ and $p$ prime,
$\beta_{p}(n,k)\geq\lfloor\log_{p}(n)\rfloor=v_{p}(\ell_{n})$. Therefore
$C(n,k)\geq\ell_{n}$ for all $n$.
###### Proof 6.3.
The proof uses Lemma 2.
$\displaystyle\beta_{p}(n,k)$
$\displaystyle=v_{p}(n!)-\sum_{i=1}^{k}v_{p}(\lfloor n/\alpha_{i}\rfloor!)$
$\displaystyle=\sum_{i=1}^{\lfloor\log_{p}(n)\rfloor}\lfloor
n/p^{i}\rfloor-\sum_{i=1}^{k}\sum_{j=1}^{\lfloor\log_{p}(\lfloor\frac{n}{\alpha_{i}}\rfloor)\rfloor}\left\lfloor\frac{n}{\alpha_{i}p^{j}}\right\rfloor$
$\displaystyle=\sum_{i=1}^{\lfloor\log_{p}(n)\rfloor}\left(\lfloor
n/p^{i}\rfloor-\sum_{j=1}^{k}\left\lfloor\frac{\lfloor\frac{n}{p^{i}}\rfloor}{\alpha_{j}}\right\rfloor\right)$
$\displaystyle\geq\sum_{i=1}^{\lfloor\log_{p}(n)\rfloor}1\textrm{ (because
$\sum\frac{1}{\alpha_{i}}<1$ by Lemma\leavevmode\nobreak\ \ref{suminv})}.$
Since $\ell_{n}=\prod\limits_{p\leq n}p^{\lfloor\log_{p}(n)\rfloor}$ from
Remark 14, we get $\ell_{n}\leq C(n,k)=\prod\limits_{p\leq
n}p^{\beta_{p}(n,k)}$.
###### Lemma 17.
For $i\geq 1$ and $n\geq\alpha_{i}$,
$\frac{\left(\frac{n}{\alpha_{i}}\right)^{\frac{n}{\alpha_{i}}}}{\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor^{\lfloor\frac{n}{\alpha_{i}}\rfloor}}<\left(\frac{10\,n}{\alpha_{i}}\right)^{\frac{\alpha_{i}-1}{\alpha_{i}}}.$
###### Proof 6.4.
If $n=\alpha_{i}$, we have $1<\sqrt{10}\leq
10^{\frac{\alpha_{i}-1}{\alpha_{i}}}$, hence the result. Otherwise
$n>\alpha_{i}$: let us write $n=b\alpha_{i}+r$, with $0\leq r<\alpha_{i}$ the
Euclidean division of $n$ by $\alpha_{i}$. We have:
$\frac{n-\alpha_{i}+1}{\alpha_{i}}=b-1+\frac{r+1}{\alpha_{i}}\leq b.$
Since $b=\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor$, it follows that
$\frac{n-\alpha_{i}+1}{\alpha_{i}}\leq\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor$.
Now for $\frac{1}{2}\leq x$, the function $x^{x}$ is increasing, thus since
$\frac{1}{2}\leq\frac{1}{\alpha_{i}}\leq\frac{n-\alpha_{i}+1}{\alpha_{i}}$, we
deduce that
$\left(\frac{n-\alpha_{i}+1}{\alpha_{i}}\right)^{\frac{n-\alpha_{i}+1}{\alpha_{i}}}\leq\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor^{\lfloor\frac{n}{\alpha_{i}}\rfloor}$,
and we hence have:
$\displaystyle\frac{\left(\frac{n}{\alpha_{i}}\right)^{\frac{n}{\alpha_{i}}}}{\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor^{\lfloor\frac{n}{\alpha_{i}}\rfloor}}$
$\displaystyle\leq\frac{\left(\frac{n}{\alpha_{i}}\right)^{\frac{n}{\alpha_{i}}}}{\left(\frac{n-\alpha_{i}+1}{\alpha_{i}}\right)^{\frac{n-\alpha_{i}+1}{\alpha_{i}}}}={\left(1+\frac{\alpha_{i}-1}{{n-\alpha_{i}+1}}\right)^{\frac{n-\alpha_{i}+1}{\alpha_{i}-1}}}^{\frac{\alpha_{i}-1}{\alpha_{i}}}\left(\frac{n}{\alpha_{i}}\right)^{\frac{\alpha_{i}-1}{\alpha_{i}}}.$
The first operand in the last expression is of the shape
$\left({\left({{1+\frac{1}{x}}}\right)^{x}}\right)^{\frac{\alpha_{i}-1}{\alpha_{i}}}$,
where $x$ is a positive rational. We showed using only elementary properties
of rational numbers, like the binomial formula or the summation formula for
geometric progressions, that for any such positive rational number $x$,
$\left(1+\frac{1}{x}\right)^{x}<10$, hence the result. Note that we only
needed that there exist a constant $K>0$ such that
$\left(1+\frac{1}{x}\right)^{x}<K$.
Of course, using elementary real analysis allows for the tighter bound $e$,
which was used in Hanson’s paper, but this bound is irrelevant for the final
result. We can now establish the following bound on $C(n,k)$:
###### Lemma 18.
For $k\geq 1$ and $n\geq 2$,
$C(n,k)<\frac{n^{n}}{{\lfloor\frac{n}{\alpha_{1}}\rfloor}^{\lfloor\frac{n}{\alpha_{1}}\rfloor}\dotsm{\lfloor\frac{n}{\alpha_{k}}\rfloor}^{\lfloor\frac{n}{\alpha_{k}}\rfloor}}.$
###### Proof 6.5.
First observe that if $m=m_{1}+\dots+m_{k}$ where $m$ and the $m_{i}$ are (not
all zero) non-negative integers, we have because of the multinomial theorem:
$\displaystyle\left(m_{1}+\dots+m_{k}\right)^{m}\geq\binom{m}{m_{1},\dots,m_{k}}m_{1}^{m_{1}}\dotsm
m_{k}^{m_{k}}.$ (9)
Let $k\geq 1$, and define:
$t=\sum_{i=1}^{k}\left\lfloor\frac{n}{\alpha_{i}}\right\rfloor.$
Then $t<n$ by Lemma 15. We have:
$\displaystyle C(n,k)$ $\displaystyle=n\cdot(n-1)\dotsm(t+1)\binom{t}{\lfloor
n/\alpha_{1}\rfloor,\lfloor n/\alpha_{2}\rfloor,\dots,\lfloor
n/\alpha_{k}\rfloor}.$ (10)
Because of equation (9), we know that
$\displaystyle\binom{t}{\lfloor n/\alpha_{1}\rfloor,\lfloor
n/\alpha_{2}\rfloor,\dots,\lfloor n/\alpha_{k}\rfloor}$
$\displaystyle\leq\frac{t^{t}}{\lfloor n/\alpha_{1}\rfloor^{\lfloor
n/\alpha_{1}\rfloor}\lfloor n/\alpha_{2}\rfloor^{\lfloor
n/\alpha_{2}\rfloor}\dots\lfloor n/\alpha_{k}\rfloor^{\lfloor
n/\alpha_{k}\rfloor}}.$ (11)
From equations (10) and (11) we deduce that
$\displaystyle C(n,k)$ $\displaystyle<\frac{n^{n}}{\lfloor
n/\alpha_{1}\rfloor^{\lfloor n/\alpha_{1}\rfloor}\lfloor
n/\alpha_{2}\rfloor^{\lfloor n/\alpha_{2}\rfloor}\dots\lfloor
n/\alpha_{k}\rfloor^{\lfloor n/\alpha_{k}\rfloor}}.$
###### Lemma 19.
Let $k\geq 3$, $n\in\mathbb{N}$. If $\alpha_{k}\leq n$ then
$k<\lfloor\log_{2}{\lfloor\log_{2}n\rfloor}\rfloor+2.$
###### Proof 6.6.
First observe by a simple induction that for all $k\geq 3$,
$\alpha_{k}>2^{2^{k-2}}+1$ so that
$k-2<\lfloor\log_{2}\left(\lfloor\log_{2}\alpha_{k}\rfloor\right)\rfloor\leq\lfloor\log_{2}\left(\lfloor\log_{2}n\rfloor\right)\rfloor$.
###### Lemma 20.
Let $k\geq 1$, $n\in\mathbb{N}$. If $\alpha_{k}\leq n$,
$C(n,k)<\frac{n^{n}(\frac{10\,n}{\alpha_{1}})^{\frac{\alpha_{1}-1}{\alpha_{1}}}(\frac{10\,n}{\alpha_{2}})^{\frac{\alpha_{2}-1}{\alpha_{2}}}\dots(\frac{10\,n}{\alpha_{k}})^{\frac{\alpha_{k}-1}{\alpha_{k}}}}{\left(\frac{10\,n}{\alpha_{1}}\right)^{\frac{10\,n}{\alpha_{1}}}\left(\frac{10\,n}{\alpha_{1}}\right)^{\frac{10\,n}{\alpha_{1}}}\dots\left(\frac{10\,n}{\alpha_{k}}\right)^{\frac{10\,n}{\alpha_{k}}}}.$
###### Proof 6.7.
The result is straightforward by combining Lemmas 17 and 18.
###### Lemma 21.
Let $w_{k}=\prod\limits_{i=1}^{k}\alpha_{i}^{\frac{1}{\alpha_{i}}}$, $k\geq
1$. Then $w_{k}$ is increasing and there exists $w\in\mathbb{R}$, with
$w<2.98,$
such that $w_{k}<w$.
###### Proof 6.8.
The sequence $w_{k}$ is increasing because
${\alpha_{i}}^{\frac{1}{\alpha_{i}}}>1$ (because $\alpha_{i}>1$). Since
$\alpha_{i}^{2}>\alpha_{i+1}>(\alpha_{i}-1)^{2}$, one can see that for $i\geq
3$,
$\alpha_{i+1}^{\frac{1}{\alpha_{i+1}}}<\sqrt{\alpha_{i}^{\frac{1}{\alpha_{i}}}}$,
so that for all $k\geq 1$ and $l\geq 0$,
$w_{k+l}\leq\prod\limits_{i=1}^{k}\alpha_{i}^{\frac{1}{\alpha_{i}}}\cdot\alpha_{k+1}^{\frac{1}{\alpha_{k+1}}\sum_{i=0}^{l}\frac{1}{2^{i}}}\leq
w_{k}\cdot{\alpha_{k+1}^{\frac{2}{\alpha_{k+1}}}}.$
We establish by an elementary external computation verified in Coq that
$\alpha_{1}^{\frac{1}{\alpha_{1}}}<\frac{283}{200}$,
$\alpha_{2}^{\frac{1}{\alpha_{2}}}<\frac{1443}{1000}$,
$\alpha_{3}^{\frac{1}{\alpha_{3}}}<\frac{1321}{1000}$,
$\alpha_{4}^{\frac{1}{\alpha_{4}}}<\frac{273}{250}$ and
$\alpha_{5}^{\frac{1}{\alpha_{5}}}<\frac{201}{200}$. From the bound above with
$k=4$ we get
$w<w_{4}\cdot\alpha_{5}^{\frac{2}{\alpha_{5}}}\leq\frac{5949909309448377}{2\cdot
10^{15}}<2.98$.
###### Remark 22.
For $k\geq 1$, we have
$\displaystyle\frac{\alpha_{1}-1}{\alpha_{1}}+\frac{\alpha_{2}-1}{\alpha_{2}}+\dots+\frac{\alpha_{k}-1}{\alpha_{k}}=k-1+\frac{1}{\alpha_{k+1}-1}.$
###### Proof 6.9.
It is a direct consequence of Lemma 15.
Note that the statement of Remark 22 actually corrects a typo in the original
paper.
###### Theorem 23.
If $\alpha_{k}\leq n<\alpha_{k+1}$,
$C(n,k)=C(n)<(10\,n)^{k-\frac{1}{2}}w^{n+1}.$
###### Proof 6.10.
From Lemma 20, recall that we have
$\displaystyle C(n,k)$
$\displaystyle<\frac{n^{n}(\frac{10\,n}{\alpha_{1}})^{\frac{\alpha_{1}-1}{\alpha_{1}}}(\frac{10\,n}{\alpha_{2}})^{\frac{\alpha_{2}-1}{\alpha_{2}}}\dots(\frac{10\,n}{\alpha_{k}})^{\frac{\alpha_{k}-1}{\alpha_{k}}}}{\left(\frac{n}{\alpha_{1}}\right)^{\frac{n}{\alpha_{1}}}\left(\frac{n}{\alpha_{2}}\right)^{\frac{n}{\alpha_{2}}}\dots\left(\frac{n}{\alpha_{k}}\right)^{\frac{n}{\alpha_{k}}}}$
$\displaystyle=\frac{n^{n}(10\,n)^{\left(\sum\limits_{i=1}^{k}\frac{\alpha_{i}-1}{\alpha_{i}}\right)}\left(\prod_{i=1}^{k}\alpha_{i}^{\frac{1}{\alpha_{i}}}\right)^{n}}{n^{n\sum_{i=1}^{k}\frac{1}{\alpha_{i}}}\prod_{i=1}^{k}\alpha_{i}^{\frac{\alpha_{i}-1}{\alpha_{i}}}}.$
It can be seen using Lemma 15 that:
$n^{n\left(1-\sum_{i=1}^{k}\frac{1}{\alpha_{i}}\right)}\leq n.$
Thus
$\displaystyle C(n,k)$
$\displaystyle<n\frac{(10\,n)^{k-1+\frac{1}{\alpha_{k+1}-1}}w_{k}^{n}}{\prod_{i=1}^{k}\alpha_{i}^{\frac{\alpha_{i}-1}{\alpha_{i}}}}\textrm{
(by Remark\leavevmode\nobreak\ \ref{obs1})}$ $\displaystyle\leq
n\frac{(10\,n)^{k-1+\frac{1}{\alpha_{k+1}-1}}w^{n}}{\prod_{i=1}^{k}\alpha_{i}^{\frac{\alpha_{i}-1}{\alpha_{i}}}}\textrm{
because $w_{k}\leq w$.}$
Since $n<\alpha_{k+1}=1+\prod_{i=1}^{k}\alpha_{i}$,
$n\leq\prod_{i=1}^{k}\alpha_{i}$, and we have
$\prod_{i=1}^{k}\alpha_{i}^{\frac{\alpha_{i}-1}{\alpha_{i}}}=\frac{\prod\limits_{i=1}^{k}\alpha_{i}}{w_{k}}\geq\frac{n}{w_{k}}.$
Thus
$\displaystyle C(n,k)$
$\displaystyle<{(10\,n)^{k-1+\frac{1}{\alpha_{k+1}-1}}w^{n}w_{k}}$
$\displaystyle\leq{(10\,n)^{k-\frac{1}{2}}w^{n+1}}\textrm{ as
$\alpha_{k+1}\geq 3$ and $w_{k}\leq w$}.$
We can now prove Lemma 4:
###### Proof 6.11.
We have:
$\ell_{n}/3^{n}\leq
C(n,k)/3^{n}=(10\,n)^{k-\frac{1}{2}}\left(\frac{w}{3}\right)^{n+1}.$
Remembering that $k<\lfloor\log_{2}{\lfloor\log_{2}n\rfloor}\rfloor+2$ and
$w<3$, it is elementary to show that the quantity on the right is eventually
decreasing to $0$ and therefore bounded, which proves the result. We once
again make use of the fact that $\left(1+\frac{1}{x}\right)^{x}$ is bounded in
the course of this elementary proof.
## 7\. Conclusion
We are not aware of a comprehensive, reference, formal proof library on the
topic of number theory, in any guise. The most comprehensive work in this
direction is probably the Isabelle/HOL library on analytic number theory
contributed by Eberl [Ebe19b], which covers a substantial part of an
introductory textbook by Apostol [Apo76]. This library is based on an
extensive corpus in complex analysis initially formalized by Harrison in the
HOL-Light prover, and later ported to the Isabelle/HOL prover by Paulson and
Li. Formal proofs also exists of a few emblematic results. The elementary fact
that $\sqrt{2}$ is irrational was used as an example problem in a comparative
study of the styles of various theorem provers [Wie06], including Coq. The
Prime Number theorem was proved formally for the first time by Avigad _et al_.
[ADGR07], using the Isabelle/HOL prover and later by Harrison [Har09] with the
HOL-Light prover. Shortly after the submission of the first version of the
present paper, Eberl verified [Ebe19a] Beuker’s proof of Apéry’s theorem
[Beu79], using the Isabelle/HOL prover, and relying on the Prime Number
theorem to derive the asymptotic properties of $\ell_{n}$. Bingham was the
first to formalize a proof that $e$ is transcendental [Bin11], with the HOL-
Light prover. Later, Bernard _et al._ formally proved the transcendence of
both $\pi$ and $e$ in Coq [BBRS15].
Some of the ingredients needed in the present proof are however not specific
to number theory. For instance, we here use a very basic, but sufficient,
infrastructure to represent asymptotic behaviors. But “big Oh”, also called
Bachman-Landau, notations have been discussed more extensively by Avigad _et
al._ [ADGR07] in the context of their formal proof of the Prime Number
Theorem, and by Boldo _et al._ [BCF+10], for the asymptotic behavior of real-
valued continuous functions. Affeldt _et al._ designed a sophisticated
infrastructure for equational reasoning in Coq with Bachman-Landau notation
[ACR18], which relies on a non-constructive choice operator. Another example
of such a secondary topic is the theory of multinomial coefficients, which is
also relevant to combinatorics, and which is also defined by Hivert in his Coq
library Coq-Combi [Hiv]. However, up to our knowledge this library does not
feature a proof of the generalized Newton identity.
Harrison [Har15] presented a way to produce rigorous proofs from certificates
produced by Wilf-Zeilberger certificates, by seeing sequences as limits of
complex functions. His method applies to the sequence $a$, and can in
principle prove that it satisfies the recurrence equation (2). However, this
method does not allow for a proof that $b$ satisfies the recurrence relation
(2), because the summand is itself a sum but not a hypergeometric one. Up to
our knowledge, there is no known way today to justify the output of the
efficient algorithms of creative telescoping used here without handling a
trace of provisos.
The idea to use computer algebra software (CAS) as an oracle outputting a
certificate to be checked by a theorem prover, dubbed a _skeptic’s_ approach,
was first introduced by Harrison and Théry [HT98]. It is based on the
observation that CAS are very efficient albeit not always correct, while
theorem provers are sound but slow. This technique thus takes the best of both
worlds to produce reliable proofs requiring large scale computations. In the
case of Coq, this viewpoint is especially fruitful since the kernel of the
proof assistant includes efficient evaluation mechanisms for the functional
programs written inside the logic [GL02]. Notable successes based on this idea
include the use of Pocklington certificates to check primality inside Coq
[GTW06] or external computations of commutative Gröbner bases, with
applications for instance in geometry [Pot10]. Delahaye and Mayero proposed
[DM05] to use CAS to help experimenting with algebraic expressions inside a
proof assistant, before deciding what to prove and how to prove it.
Unfortunately, their tool was not usable in our case, where algebraic
expressions are made with operations that come from a hierarchy of structures.
Organizing the cooperation of a CAS and a proof assistant sheds light on their
respective differences and drawbacks. The initial motivation of this work was
to study the algorithms used for the automatic discovery and proof of
recurrences. Our hope was to be able to craft an automated tool providing
formal proofs of recurrences, by using the output of these algorithms, in a
skeptical way. This plan did not work and Section 4 illustrates the impact of
confusing the rational fractions manipulated by symbolic computations with
their evaluations, which should be guarded by conditions on the denominators.
On the other hand, proof assistants are not yet equipped to manipulate the
large expressions imported during the cooperation, even those which are of a
small to moderate size for the standards of computer algebra systems. For
instance, we have highlighted in our previous report [CMSPT14] the necessity
to combine two distinct natures of data-structures in our libraries: one
devoted to formal proofs, which may use computation inside the logic to ease
bureaucratic steps, and one devoted to larger scale computations, which
provides a fine-grained control on the complexity of operations. The later was
crucial for the computations involved in the normalization of ring expressions
during the a posteriori verification of computer-algebra produced recurrences.
But it was also instrumental in our proof of Lemma 12.
Incidentally, our initial formal proof of Lemma 21 also involved this nature
of calculations, with rather larger numbers111For the current standards of
proof assistants.. Indeed, the proof requires bounding the five first values
of sequence $\alpha_{n}^{1/a_{n}}$ and the straightforward strategy involves
intermediate computations with integers with about $4160$ decimal digits.
Following a suggestion by one of the anonymous referees, we now use a less
naive formula to bound $\alpha_{5}$. This dramatically reduces the size of the
numbers involved, to the price of some additional manual bureaucracy in the
proof script, mostly for evaluating binomial coefficients without an
appropriate support at the level of libraries.
The Coq proof assistant is not equipped with a code generation feature akin to
what is offered, for instance, in the Isabelle/HOL prover [HN18]. In
principle, it is possible to plug in Isabelle/HOL formal developments the
result of computations executed by external, generated programs that are
verified down to machine code. In the present formal proofs, computation are
instead carried _inside_ the logic, using the Calculus of Inductive
constructions as a programming language. Such an approach is possible in Coq
because its proof-checker includes an efficient mechanism for evaluating these
functional programs [GL02]. Automating the correctness proofs of the program
transformations required for the sake of efficiency is on-going research. In
the present work, we have used the CoqEAL library [CDM13], which is itself
based on a plugin for parametricity proofs [KL12]. The variant proposed by
Tabareau _et al._ [TTS18] might eventually help improving the extensibility of
the refinement framework, which is a key issue.
The interfaces of proof assistants are also notoriously less advanced than
those of modern computer algebra. For instance, reasoning by transitivity on
long chains of equalities/inequalities is often cumbersome in Coq, because of
the limited support for selecting terms in an expression and for reasoning by
transitivity. The Lean theorem prover [dMKA+15] features a `calc` environment
for proofs on transitive relations which might be used as a first step in this
direction.
On several occasions in this work, we wrote more elementary versions of the
proofs than what we had found in the texts we were formalizing. We partly
agree with Avigad _et al._ [ADGR07] when they write that this can be both
frustrating and enjoyable: on one hand, it can illustrate the lack of
mathematical libraries for theorems which mathematicians would find simple,
such as elementary analysis for studying the asymptotics of sequences as in
Section 5. Ten years later, “the need for elementary workarounds” is still
present, despite his fear that it would “gradually fade, and with it, alas,
one good reason for investing time in such exercises”[ADGR07]. On the other
hand, this need gives an opportunity to better understand the minimal scope of
mathematical theories used in a proof, with the help of a computer. For
instance, it was not clear to us at first that we could manage to completely
avoid the need to define transcendental functions, or to avoid defining the
constant $e$, base of the natural logarithm, to formalize Hanson’s paper
[Han72]. This minimality however comes at the price of arguably more
pedestrian calculations.
## Acknowledgment
We wish to thank the anonymous reviewers, Alin Bostan, Frédéric Chyzak,
Georges Gonthier, Marc Mezzarobba, and Bruno Salvy for their comments and
suggestions. We also thank Cyril Cohen, Pierre Roux and Enrico Tassi for their
help, in particular with the libraries this work depends on.
## References
* [ACR18] Reynald Affeldt, Cyril Cohen, and Damien Rouhling. Formalization techniques for asymptotic reasoning in classical analysis. J. Formalized Reasoning, 11(1):43–76, 2018.
* [ADGR07] Jeremy Avigad, Kevin Donnelly, David Gray, and Paul Raff. A formally verified proof of the prime number theorem. ACM Trans. Comput. Logic, 9(1), December 2007.
* [alg13] Algolib. http://algo.inria.fr/libraries/, 2013. Version 17.0. For Maple 17.
* [Apé79] Roger Apéry. Irrationalité de $\zeta(2)$ et $\zeta(3)$. Astérisque, 61, 1979. Société Mathématique de France.
* [Apo76] Tom M. Apostol. Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, 1976.
* [BBRS15] Sophie Bernard, Yves Bertot, Laurence Rideau, and Pierre-Yves Strub. Formal proofs of transcendence for e and $\pi$ as an application of multivariate and symmetric polynomials. CoRR, abs/1512.02791, 2015.
* [BCF+10] Sylvie Boldo, François Clément, Jean-Christophe Filliâtre, Micaela Mayero, Guillaume Melquiond, and Pierre Weis. Formal proof of a wave equation resolution scheme: the method error. CoRR, abs/1005.0824, 2010.
* [BCP03] Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in type theory. J. Funct. Program., 13(2):261–293, 2003.
* [Bes07] Frédéric Besson. Fast reflexive arithmetic tactics the linear case and beyond. In TYPES’06, LNCS, pages 48–62, Berlin, Heidelberg, 2007. Springer-Verlag.
* [Beu79] F. Beukers. A note on the irrationality of $\zeta(2)$ and $\zeta(3)$. Bull. London Math. Soc., 11(3):268–272, 1979.
* [Bin11] Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4(1):71–84, 2011.
* [BR01] Keith Ball and Tanguy Rivoal. Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Invent. Math., 146(1):193–207, 2001.
* [CDM13] Cyril Cohen, Maxime Dénès, and Anders Mörtberg. Refinements for Free! In Certified Programs and Proofs, pages 147 – 162, Melbourne, Australia, December 2013.
* [CMSPT14] Frédéric Chyzak, Assia Mahboubi, Thomas Sibut-Pinote, and Enrico Tassi. A Computer-Algebra-Based Formal Proof of the Irrationality of $\zeta$(3). In ITP - 5th International Conference on Interactive Theorem Proving, Vienna, Austria, 2014.
* [Coh12] Cyril Cohen. Construction of real algebraic numbers in Coq. In ITP, volume 7406 of LNCS. Springer, August 2012.
* [coq19] Mathematical Components Libraries. http://math-comp.github.io/math-comp/, 2019.
* [DM05] David Delahaye and Micaela Mayero. Dealing with algebraic expressions over a field in Coq using Maple. J. Symb. Comput., 39(5):569–592, May 2005.
* [dMKA+15] Leonardo de Moura, Soonho Kong, Jeremy Avigad, Floris van Doorn, and Jakob von Raumer. The Lean Theorem Prover (System Description), pages 378–388. Springer International Publishing, Cham, 2015.
* [Ebe19a] Manuel Eberl. The irrationality of $\zeta(3)$. Archive of Formal Proofs, December 2019. http://isa-afp.org/entries/Zeta_3_Irrational.html, Formal proof development.
* [Ebe19b] Manuel Eberl. Nine chapters of analytic number theory in isabelle/hol. In John Harrison, John O’Leary, and Andrew Tolmach, editors, 10th International Conference on Interactive Theorem Proving, ITP 2019, September 9-12, 2019, Portland, OR, USA, volume 141 of LIPIcs, pages 16:1–16:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
* [Fen05] Bei-ye Feng. A simple elementary proof for the inequality $d_{n}<3^{n}$. Acta Math. Appl. Sin. Engl. Ser., 21(3):455–458, 2005.
* [Fis04] Stéphane Fischler. Irrationalité de valeurs de zêta (d’après Apéry, Rivoal, $\dots$). In Séminaire Bourbaki. Volume 2002/2003. Exposés 909–923, pages 27–62, ex. Paris: Société Mathématique de France, 2004\.
* [FSZ19] Stéphane Fischler, Johannes Sprang, and Wadim Zudilin. Many odd zeta values are irrational. Compositio Mathematica, 155(5):938–952, 2019.
* [GAA+13] Georges Gonthier, Andrea Asperti, Jeremy Avigad, Yves Bertot, Cyril Cohen, François Garillot, Stéphane Le Roux, Assia Mahboubi, Russell O’Connor, Sidi Ould Biha, Ioana Pasca, Laurence Rideau, Alexey Solovyev, Enrico Tassi, and Laurent Théry. A Machine-Checked Proof of the Odd Order Theorem. In Sandrine Blazy, Christine Paulin, and David Pichardie, editors, ITP 2013, 4th Conference on Interactive Theorem Proving, volume 7998 of LNCS, pages 163–179, Rennes, France, July 2013. Springer.
* [GGMR09] François Garillot, Georges Gonthier, Assia Mahboubi, and Laurence Rideau. Packaging Mathematical Structures. working paper or preprint, March 2009.
* [GL02] Benjamin Grégoire and Xavier Leroy. A compiled implementation of strong reduction. In Proceedings of the Seventh ACM SIGPLAN International Conference on Functional Programming, ICFP ’02, pages 235–246, New York, NY, USA, 2002. ACM.
* [GM05] Benjamin Grégoire and Assia Mahboubi. Proving equalities in a commutative ring done right in Coq. In TPHOLs 2005, volume 3603 of LNCS, pages 98–113, Oxford, United Kingdom, August 2005. Springer.
* [GTW06] Benjamin Grégoire, Laurent Théry, and Benjamin Werner. A Computational Approach to Pocklington Certificates in Type Theory, pages 97–113. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006.
* [Han72] Denis Hanson. On the product of the primes. Canad. Math. Bull., 15:33–37, 1972.
* [Har09] John Harrison. Formalizing an analytic proof of the Prime Number Theorem. Journal of Automated Reasoning, 43:243–261, 2009.
* [Har15] John Harrison. Formal proofs of hypergeometric sums (dedicated to the memory of andrzej trybulec). Journal of Automated Reasoning, 55:223–243, 2015.
* [Hed98] Michael Hedberg. A coherence theorem for Martin-Löf’s type theory. Journal of Functional Programming, 8(4):413–436, July 1998.
* [Hiv] Florent Hivert. Coq-combi. https://github.com/hivert/Coq-Combi.
* [HN18] Lars Hupel and Tobias Nipkow. A verified compiler from isabelle/hol to cakeml. In Amal Ahmed, editor, Programming Languages and Systems - 27th European Symposium on Programming, ESOP 2018, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2018, Thessaloniki, Greece, April 14-20, 2018, Proceedings, volume 10801 of Lecture Notes in Computer Science, pages 999–1026. Springer, 2018.
* [HT98] John Harrison and L. Théry. A skeptic’s approach to combining HOL and Maple. J. Automat. Reason., 21(3):279–294, 1998.
* [KL12] Chantal Keller and Marc Lasson. Parametricity in an impredicative sort. In Patrick Cégielski and Arnaud Durand, editors, Computer Science Logic (CSL’12) - 26th International Workshop/21st Annual Conference of the EACSL, CSL 2012, September 3-6, 2012, Fontainebleau, France, volume 16 of LIPIcs, pages 381–395. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2012.
* [MT13] Assia Mahboubi and Enrico Tassi. Canonical Structures for the working Coq user. In Sandrine Blazy, Christine Paulin, and David Pichardie, editors, ITP 2013, 4th Conference on Interactive Theorem Proving, volume 7998 of LNCS, pages 19–34, Rennes, France, July 2013. Springer.
* [Pot10] Loïc Pottier. Connecting Gröbner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics. CoRR, abs/1007.3615, 2010.
* [Riv00] Tanguy Rivoal. La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris Sér. I Math., 331(4):267–270, 2000.
* [Sal03] Bruno Salvy. An Algolib-aided version of Apéry’s proof of the irrationality of $\zeta(3)$. http://algo.inria.fr/libraries/autocomb/Apery2-html/apery.html, 2003\.
* [Soz09] Matthieu Sozeau. A new look at generalized rewriting in type theory. J. Formalized Reasoning, 2(1):41–62, 2009.
* [The20] The Coq Development Team. The Coq Proof Assistant, version 8.11.0, January 2020.
* [TTS18] Nicolas Tabareau, Éric Tanter, and Matthieu Sozeau. Equivalences for free: univalent parametricity for effective transport. PACMPL, 2(ICFP):92:1–92:29, 2018.
* [vdP79] Alfred van der Poorten. A proof that Euler missed: Apéry’s proof of the irrationality of $\zeta(3)$. Math. Intelligencer, 1(4):195–203, 1979. An informal report.
* [Wie06] Freek Wiedijk. The Seventeen Provers of the World: Foreword by Dana S. Scott (Lecture Notes in Computer Science / Lecture Notes in Artificial Intelligence). Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006.
* [Zei90] Doron Zeilberger. A holonomic systems approach to special functions identities. J. Comput. Appl. Math., 32(3):321–368, 1990.
* [Zei91] Doron Zeilberger. The method of creative telescoping. J. Symbolic Comput., 11(3):195–204, 1991.
* [Zud01] V. V. Zudilin. One of the numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is irrational. Uspekhi Mat. Nauk, 56(4(340)):149–150, 2001.
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# A unified class of null proportion estimators with plug-in FDR control
Sebastian Döhler Iqraa Meah
###### Abstract
Since the work of Storey et al., (2004), it is well-known that the performance
of the Benjamini-Hochberg (BH) procedure can be improved by incorporating
estimators of the number (or proportion) of null hypotheses, yielding an
adaptive BH procedure which still controls FDR. Several such plug-in
estimators have been proposed since then, for some of these, like Storey’s
estimator, plug-in FDR control has been established, while for some others,
e.g. the estimator of Pounds and Cheng, (2006), some gaps remain to be closed.
In this work we introduce a unified class of estimators, which encompasses
existing and new estimators and unifies proofs of plug-in FDR control using
simple convex ordering arguments. We also show that any convex combination of
such estimators once more yields estimators with guaranteed plug-in FDR
control. Additionally, the flexibility of the new class of estimators also
allows incorporating distributional informations on the $p$-values. We
illustrate this for the case of discrete tests, where the null distributions
of the $p$-values are typically known. In that setting, we describe two
generic approaches for adapting any estimator from the general class to the
discrete setting while guaranteeing plug-in FDR control. While the focus of
this paper is on presenting the generality and flexibility of the new class of
estimators, we also include some analyses on simulated and real data.
False discovery rate,
Adaptive BH procedure,
Convex ordering of random variables,
Discrete hypothesis testing,
Multiple hypothesis testing,
###### keywords:
###### Contents
1. 1 Introduction
1. 1.1 Background
2. 1.2 Contributions
2. 2 Framework
1. 2.1 Distributional assumptions
2. 2.2 FDR control for plug-in estimates
3. 3 A unified class of plug-in estimators
4. 4 Homogeneous estimators
1. 4.1 Numerical results
2. 4.2 More details on the Pounds and Cheng estimator
5. 5 Adjusted estimators for discrete $p$-values
1. 5.1 Transformations of discrete $p$-values
2. 5.2 Adjusting the rescaling constants
3. 5.3 A randomization approach
4. 5.4 Simulation results
5. 5.5 Real data analysis
6. 6 Discussion
7. A Auxiliary definitions and results
8. B Complements to Section 4.1
9. C Complements to Section 4.2
10. D Additional Figures for simulated data of Section 4
11. E Upper and lower bounds for the inverse moment of the uniform sum distribution
## 1 Introduction
### 1.1 Background
When many statistical tests are performed simultaneously, an ubiquituous way
to account for the false rejections is the false discovery rate (FDR), that is
the expected proportion of errors among the rejections. The seminal Benjamini
and Hochberg, (1995) procedure (abbreviated in the sequel as BH procedure)
works by rejecting $H_{(1)},\ldots,H_{(\hat{k})}$, where $\hat{k}$ is
determined in the following step-up manner
$\displaystyle\hat{k}$
$\displaystyle=\max\left\\{\ell\in\\{0,\ldots,m\\}:p_{(\ell)}\leq\frac{\ell}{m}\cdot\alpha\right\\},$
(1)
where $p_{(1)}\leq\ldots\leq p_{(m)}$ denote the ordered $p$-values, and
$H_{(1)},\ldots,H_{(m)}$ the corresponding null hypotheses and $p_{(0)}:=0$.
According to results in Benjamini and Hochberg, (1995) and Benjamini and
Yekutieli, (2001), this procedure guarantees that
$\textnormal{FDR}\leq\pi_{0}\alpha$ when the $p$-values are independent or
positively dependent, while for arbitrarily dependent $p$-values the Benjamini
and Yekutieli, (2001) procedure is available. The simplicity of the BH
procedure and its many useful theoretical properties have made it an
indispensable tool in modern high dimensional data analysis, see e.g.
Benjamini, (2010). Much work has gone into analyzing, extending and adapting
this procedure to various settings.
In this context, Storey et al., (2004) showed that the plug-in BH procedure
$\displaystyle\hat{k}$
$\displaystyle=\max\left\\{\ell\in\\{0,\ldots,m\\}:p_{(\ell)}\leq\frac{\ell}{\widehat{m}_{0}}\cdot\alpha\right\\},$
(2)
obtained by replacing $m$ in (1) by an estimate $\widehat{m_{0}}$ of $m_{0}$
still provides so called adaptive or plug-in FDR control at level $\alpha$
while allowing for more power. Classical examples of such estimates were
proposed by Storey et al., (2004)
$\displaystyle\widehat{m}_{0}^{\text{Storey}}$
$\displaystyle=\frac{1+\sum_{i=1}^{m}{\mathbf{1}\\{p_{i}>\lambda\\}}}{1-\lambda},$
(3) where $\lambda\in[0,1)$ is a tuning parameter, and by Pounds and Cheng,
(2006) $\displaystyle\widehat{m}_{0}^{\text{PC,2006}}$
$\displaystyle=1\wedge\left(2\sum_{i=1}^{m}p_{i}\right).$ (4)
While the focus of this work is on plug-in FDR control, estimates of $m_{0}$
(or equivalently $\pi_{0}=m_{0}/m$) can also be used for FDR estimation
purposes (see Storey, (2002)). Thus, there is a large body of literature on
this topic and numerous methods for establishing plug-in FDR control are
available, see e.g. Benjamini et al., (2006); Sarkar, (2008); Blanchard and
Roquain, (2009); Heesen and Janssen, (2016); Ditzhaus and Janssen, (2019) and
references therein. In this paper, we use a classical condition proposed by
Blanchard and Roquain, (2009) for establishing plug-in FDR control for a
unified class of $m_{0}$-estimators, see Section 2.2 for more details.
Previous work on plug-in FDR control has focused on continuous test
statistics, for which the null $p$-values are distributed according to the
uniform distribution. Considering the abundance of super-uniform $p$-values in
real life applications, the uniformity assumption is often violated which may
lead to undesirable conservatism of the $m_{0}$-estimators, see Section 5 for
more details. Super-uniform $p$-values can be observed when testing composite
null hypotheses or when dealing with discrete tests, the latter being the
setting of our interest in this work.
Discrete tests often originate when the tests are based on counts or
contingency tables: for example in clinical studies, the efficacy or safety of
drugs is determined by counting patients who survive a certain period, or
experience a certain type of adverse drug reaction after being treated, see
e.g. Chavant et al., (2011); and also in biology, where the genotype effect on
the phenotype can be analyzed by knocking out genes and counting the number of
individuals with a changed phenotype, see e.g. Muñoz-Fuentes et al., (2018).
In discrete testing, each $p$-value is super-uniform and (potentially) has its
own support, thus producing heterogeneous $p$-values. Pounds and Cheng, (2006)
recognized the need for developing methods tailored to discrete $p$-values and
introduced $\widehat{m}_{0}^{\text{PC,2006}}$ as a simple and robust
$m_{0}$-estimate in this setting. They did not, however, provide a proof of
plug-in FDR control, not even in the uniform setting. Further works adressing
the discreteness and heterogeneity include Chen et al., (2018) who introduced
a $m_{0}$-estimator for discrete $p$-values based on averaging Storey type
estimators for plug-in control. Biswas and Chattopadhyay, (2020) pointed out
an error in the proof of Chen et al., (2018) and provided a corrected version.
However it is unclear whether this estimator actually provides an improvement
over the classical (uniform) Storey estimator in practice. Thus, there are
gaps to be filled on $m_{0}$-estimation both for the uniform and discrete
case.
### 1.2 Contributions
In this paper we address some of the gaps and limitations mentioned above by
introducing a simple and flexible class of $m_{0}$ estimators which has the
following properties:
* •
Plug-in FDR control is guaranteed for all estimators contained in this class
under independence of $p$-values. We give a unified proof using simple convex
ordering arguments (for the reader’s convenience we restate some definitions
and classical results on stochastic and convex ordering in Appendix A).
* •
It provides a simple and flexible generic formulation which is useful for
designing new estimators. In particular, we obtain a simple modification of
$\widehat{m}_{0}^{\text{PC,2006}}$ with guaranteed plug-in FDR control.
* •
Additional distributional information on the $p$-values like heterogeneity and
super-uniformity can be incorporated easily into estimators from the class. In
particular, the estimators of this class can be used in conjunction with
classical discrete $p$-value transformations like the mid-p transformation.
* •
Combining several weighted estimators from the class preserves plug-in FDR
control.
The paper is organized as follows: Section 2 presents the statistical setting
and restates a classical sufficient criterion for plug-in FDR control. Section
3 introduces the new class of estimators, and presents the main mathematical
results on plug-in FDR control, followed by some numerical results in Section
4. In Section 5 we present approaches for adjusting estimators to
discreteness, and investigate their performance on simulated and real data.
The paper concludes with a discussion in Section 6. Technical details –
including classical results on stochastic and convex ordering– and further
analyses are deferred to the Appendices.
## 2 Framework
### 2.1 Distributional assumptions
We use a classical setting for multiple testing encompassing homogeneous and
heterogeneous nulls, see e.g. Döhler et al., (2018). We observe $X$, defined
on an abstract probabilistic space, valued in an observation space
$(\mathcal{X},\mathfrak{X})$ and generated by a distribution $P$ that belongs
to a set $\mathcal{P}$ of possible distributions. We consider $m$ null
hypotheses for $P$, denoted $H_{0,i}$, $1\leq i\leq m$, and we denote the
corresponding set of true null hypotheses by $\mathcal{H}_{0}(P)=\\{1\leq
i\leq m\>:\>\mbox{$H_{0,i}$ is satisfied by $P$}\\}$. We also denote by
$\mathcal{H}_{1}(P)$ the complement of $\mathcal{H}_{0}(P)$ in
$\\{1,\dots,m\\}$ and by $m_{0}(P)=m_{0}=|\mathcal{H}_{0}(P)|$ the number of
true nulls.
We assume that there exists a set of $p$-values that is a set of random
variables $\\{p_{i}(X),1\leq i\leq m\\}$, valued in $[0,1]$. We introduce the
following dependence assumptions between the $p$-values:
All the $p$-values $\\{p_{i}(X)$, $1\leq i\leq m\\}$ are mutually independent
in the model $\mathcal{P}$. (Indep)
The (maximum) null cumulative distribution function (c.d.f) of each $p$-value
is denoted
$F_{i}(t)=\sup_{P\in\mathcal{P}\>:\>i\in\mathcal{H}_{0}(P)}\\{\mathbf{P}_{X\sim
P}(p_{i}(X)\leq t)\\},\>\>t\in[0,1],\>\>1\leq i\leq m.$ (5)
We assume that the set of c.d.f $\mathcal{F}=\\{F_{i},1\leq i\leq m\\}$ is
known and we consider the following possible situations for the functions in
$\mathcal{F}$:
For all $i\in\\{1,\dots,m\\}$, $F_{i}$ is continuous on $[0,1]$ (Cont)
$\displaystyle\begin{array}[]{c}\mbox{For all $i\in\\{1,\dots,m\\}$, there
exists some finite set $\mathcal{A}_{i}\subset[0,1]$ such that}\\\
\mbox{$F_{i}$ is a step function, right continuous, that jumps only at some
points of $\mathcal{A}_{i}$.}\end{array}$ (Discrete)
The case (Discrete) typically arises when for all $P\in\mathcal{P}$ and
$i\in\\{1,\dots,m\\}$, $\mathbf{P}_{X\sim P}(p_{i}(X)\in\mathcal{A}_{i})=1$
for some given finite sets $\mathcal{A}_{i}\subset[0,1]$. Throughout the
paper, we will assume that we are either in the case (Cont) or (Discrete) and
we denote $\mathcal{A}=\cup_{i=1}^{m}\mathcal{A}_{i}$, with by convention
$\mathcal{A}_{i}=[0,1]$ when (Cont) holds. We will also make use of the
following classical assumption:
$\displaystyle\mbox{For all $i\in\\{1,\dots,m\\}$, $F_{i}(t)\leq t$ for all
$t\in[0,1]$}.$ (SuperUnif)
In this paper we will always assume that the $p$-values are mutually
independent ((Indep) holds) and super-uniform under the null ((SuperUnif)
holds).
### 2.2 FDR control for plug-in estimates
The following Theorem is a central result for plug-in FDR control, providing a
sufficient condition based on bounding the inverse moment of the estimator
$\widehat{m}_{0}$ by the inverse of $m_{0}$. Our presentation follows
Blanchard and Roquain, (2009), similar results can be found in Benjamini et
al., (2006); Sarkar, (2008); Zeisel et al., (2011).
###### Theorem 2.1.
Let $\widehat{m}_{0}=\widehat{m}_{0}(p_{1},\ldots,p_{m})$ be a coordinatewise
non-decreasing function of the $p$-values $(p_{1},\ldots,p_{m})$. Assume that
$(p_{1},\ldots,p_{m})$ are mutually independent (Indep) and (SuperUnif). For
$h\in\mathcal{H}_{0}$, denote by $p_{0,h}$ the set of $p$-values where $p_{h}$
has been replaced by $0$. If
$\displaystyle\mathbb{E}\left(\frac{1}{\widehat{m}_{0}(p_{0,h})}\right)$
$\displaystyle\leq\frac{1}{m_{0}}$ (IMC)
holds for all $h\in\mathcal{H}_{0}$, then the plug-in BH procedure given by
(2) controls FDR at level $\alpha$.
Throughout this paper, we will only consider coordinatewise non-decreasing
estimators and will always assume that the $p$-values are mutually
independent. Thus, when additionally (SuperUnif) holds true, the inverse
moment criterion (IMC) is sufficient for establishing plug-in FDR control in
our proofs. We mostly present results in term of the absolute number of null
hypotheses $m_{0}$, but clearly equivalent statements using the proportion of
null hypotheses $\pi_{0}=m_{0}/m$ hold, and in some cases we will present
results in terms of $\pi_{0}$ instead of $m_{0}$.
## 3 A unified class of plug-in estimators
In this section, we introduce a new class of estimators for $m_{0}$ (or
equivalently $\pi_{0}$) that mathematically guarantees plug-in FDR control. It
is based on sums of suitably transformed $p$-values, allowing us to recover
classical estimators, such as the Storey (3) and the PC (slightly modified)
(4) estimators, and also to define new estimators. We first present a general
result for single estimators and then show that plug-in FDR control is also
preserved for convex combinations of such estimators. To start, assume that
the $p$-values are transformed by certain functions $g\in\mathcal{G}$, with
$\displaystyle\mathcal{G}$ $\displaystyle=\\{g:[0,1]\rightarrow[0,1]:\text{
$g$ is non-decreasing and $\mathbb{E}g(U)>0$, where
$U\sim\mathcal{U}[0,1]$}\\}.$ (8)
Accordingly we define the class of estimators $\mathcal{F}_{0}$ as
$\displaystyle\mathcal{F}_{0}$
$\displaystyle=\left\\{\widehat{m}_{0}:[0,1]^{m}\rightarrow[0,\infty)|\>\widehat{m}_{0}(p_{1},\ldots,p_{m})=\frac{1}{{\nu(g)}}\left(1+\sum_{i=1}^{m}g(p_{i})\right),g\in\mathcal{G}\right\\},$
(9)
where $\nu(g)=\mathbb{E}g(U)$ for any $g\in\mathcal{G}$ with
$U\sim\mathcal{U}[0,1]$ (for brevity we sometimes omit the $g$ in $\nu$ when
there is no ambiguity concerning the function $g$). The class
$\mathcal{F}_{0}$ contains the classical estimator
$\widehat{m}_{0}^{\text{Storey}}$ (3) by taking
$g(u)={\mathbf{1}\\{u>\lambda\\}}$ and $\nu=1-\lambda$. It also contains a
slightly modified version $\widehat{m}_{0}^{\text{PC,new}}$ of the classical
estimator $\widehat{m}_{0}^{\text{PC,2006}}$ (4) obtained from taking $g(u)=u$
with $\nu=1/2$, i.e.
$\displaystyle\widehat{m}_{0}^{\text{PC,new}}$
$\displaystyle=2+2\sum_{i=1}^{m}p_{i}.$ (10)
In Section 4 we will introduce some additional estimators and discuss
$\widehat{m}_{0}^{\text{PC,new}}$ in more detail. The rationale behind the
definitions of the classes $\mathcal{G}$ and $\mathcal{F}_{0}$ is two-fold.
Requiring that $g$ is non-decreasing ensures that $\widehat{m}_{0}$ is
coordinatewise non-decreasing, allowing us to apply Theorem 2.1. The quantity
$g(p_{i})/\nu$ can be interpreted as the (local) contribution of $p_{i}$ to
the estimate of $m_{0}$. If we expect large $p$-values to provide evidence for
null hypotheses, then it seems reasonable to require $g$ to be non-decreasing.
Rescaling $g(p_{i})$ by $\nu=\mathbb{E}g(U)$ is a simple way of ensuring that
$\sum_{i=1}^{m}g(p_{i})/\nu$ is conservatively biased in the sense that
$\mathbb{E}(\sum_{i=1}^{m}g(p_{i})/\nu)\geq m_{0}$ in any constellation of
null and alternative hypotheses. This type of conservativeness may however not
be strong enough for plug-in control. As our main result – Proposition 3.2
below – shows, simply adding $1/\nu$ as a ’safety margin’ to the above
estimate is enough for ensuring plug-in FDR control.
In some situations $p$-values under the null may be heterogeneous, i.e. the
$p$-values may have different distributions under the null, so that using an
individual transformation for each $p$-value may be helpful. To this end, we
introduce the following richer and more flexible class of estimators.
$\begin{split}\mathcal{F}=\biggl{\\{}\widehat{m}_{0}:[0,1]^{m}&\rightarrow[0,\infty)\
\bigg{|}\
\widehat{m}_{0}(p_{1},\ldots,p_{m})=\frac{1}{\min(\nu_{1},\ldots,\nu_{m})}\\\
&+\sum_{i=1}^{m}\frac{g_{i}(p_{i})}{\nu_{i}},\quad\text{with
$g_{i}\in\mathcal{G}$ and $\nu_{i}=\mathbb{E}[g_{i}(U)]$,
$U\sim\mathcal{U}[0,1]$ for all $i$}\biggr{\\}}.\end{split}$ (11)
We state our main result on plug-in FDR control for this more general class
below. Clearly, $\mathcal{F}_{0}\subset\mathcal{F}$, so that the results
stated for $\mathcal{F}$ also hold for $\mathcal{F}_{0}$. We now present an
upper bound in the convex order for transformed uniform random variables in
terms of Bernoulli random variables, which is the main technical tool we use
for proving plug-in FDR control for the class $\mathcal{F}$.
###### Lemma 3.1.
For any $g\in\mathcal{G}$ we have
$g(U)\leqslant_{\text{cx}}\mathbf{Bin}(1,\nu)$ and $\nu=\mathbb{E}g(U)$, where
$U\sim\mathcal{U}[0,1]$, and the notation $\leqslant_{\text{cx}}$ denotes the
convex ordering (see Definition A.2).
###### Proof.
For $U\sim\mathcal{U}[0,1]$ define $X=g(U)$, so that $\mathbb{E}(X)=\nu$. Let
$l_{X}=\inf_{x\in[0,1]}g(x)$, $u_{X}=\sup_{x\in[0,1]}g(x)$ be the lower and
upper endpoints of the support of $X$, and define a two-point distribution $Y$
concentrated on $\\{l_{X},u_{X}\\}$ by $P(Y=l_{X})=(u_{X}-\nu)/(u_{X}-l_{X})$
and $P(Y=u_{X})=(\nu-l_{X})/(u_{X}-l_{X})$. By Lemma A.3 we have
$X\leqslant_{\text{cx}}Y$.
Now let $Z\sim\mathbf{Bin}\left(1,\nu\right)$ and denote the distribution
function of $Y$ and $Z$ by $F$ and $G$. Clearly,
$\mathbb{E}Y=\nu=\mathbb{E}Z$. Since $[l_{X},u_{X}]\subset[0,1]$ and both $Y$
and $Z$ are two-point distributions, the function $G-F$ posesses one crossing
point on $[0,1)$. Indeed, for $t\in[0,l_{X}),\quad F(t)=0$ while $G(t)=1-\nu$
so that $G-F$ is positive, and for $t\in[u_{X},1),F(t)=1$ while $G(t)=1-\nu$
so that $G-F$ is negative. For $t\in[l_{X},u_{X})$, $G-F$ can be positive or
negative depending on $\nu$. Overall, the sign sequence of $G-F$ is $+,-$ so
that Lemma A.4 implies that $Y\leqslant_{\text{cx}}Z$ and the claim follows. ∎
The following proposition is our main result on plug-in FDR control.
###### Proposition 3.2.
Assume that $p_{1},\ldots,p_{m}$ are mutually independent and (SuperUnif)
holds. Then (IMC) holds true for any estimator
$\widehat{m}_{0}\in\mathcal{F}$, where $\mathcal{F}$ is defined by (11). In
particular, the BH plug-in procedure (2) using $\widehat{m}_{0}$ controls FDR
at level $\alpha$.
###### Proof.
Since $\widehat{m}_{0}$ is coordinatewise non-decreasing, it is sufficient to
verify (IMC). For any $h\in\mathcal{H}_{0}$, monotonicity and super-uniformity
give us $\widehat{m}_{0}(p_{0,h})\geqslant_{\text{st}}1/\nu+S_{0}$, where
$\nu=min_{l\in\mathcal{H}_{0}\setminus\\{h\\}}\nu_{l}$, and
$S_{0}=\sum_{\ell\in\mathcal{H}_{0}\setminus\\{h\\}}g_{\ell}(U_{\ell})/\nu_{\ell}$
with $(U_{\ell})_{\ell\in\mathcal{H}_{0}}$ i.i.d random variables distributed
according to $\mathcal{U}[0,1]$. By Lemma 3.1 we have
$g_{\ell}(U_{\ell})\leqslant_{\text{cx}}\mathbf{Bin}(1,\nu_{\ell})$ and Lemma
A.6 gives
$\mathbf{Bin}(1,\nu_{i})/\nu_{i}\leqslant_{\text{cx}}\mathbf{Bin}(1,\nu)/\nu$.
Since the convex ordering is preserved under convolutions (see Lemma A.5) we
obtain $\nu S_{0}\leqslant_{\text{cx}}\mathbf{Bin}(m_{0}-1,\nu)$. Finally, the
mapping $x\mapsto\nu/(1+x)$ is convex on $[0,\infty)$ and therefore from the
Definition A.2 of $\leqslant_{\text{cx}}$ we obtain that
$\displaystyle\mathbb{E}\left(\frac{1}{\widehat{m}_{0}(p_{0,h})}\right)$
$\displaystyle\leq\mathbb{E}\left(\frac{1}{\frac{1}{\nu}+S_{0}}\right)=\mathbb{E}\left(\frac{\nu}{1+\nu
S_{0}}\right)\leq\mathbb{E}\left(\frac{\nu}{1+\mathbf{Bin}(m_{0}-1,\nu)}\right)\leq\frac{1}{m_{0}},$
(12)
where the last bound is a well-known result for the inverse moment of Binomial
distributions (see e.g. A.7 in Appendix) so that (IMC) is proved. The
statement on plug-in FDR control now follows from Theorem 2.1. ∎
By taking $g(u)={\mathbf{1}\\{u>\lambda\\}}$ we have $\nu
S_{0}\sim\mathbf{Bin}(m_{0}-1,\nu)$ and therefore the second inequality from
the right in (12) can be replaced by an equality. Thus, it may be tempting to
conclude that $\widehat{m}_{0}^{\text{Storey}}$ is optimal. In the case of a
Dirac-Uniform constellation of $p$-values (see Blanchard and Roquain, (2009))
this is indeed true, since $\nu\widehat{m}_{0}^{\text{Storey}}\sim
1+\mathbf{Bin}(m_{0}-1,\nu)$ and therefore the left inequality in (12) can
also be replaced by an equality. In more general settings however, other
choices of $g$ may be better, as the results in Section 4.1 show.
We highlight that introducing a general class of estimators as (11) with
Proposition 3.2 allows a unified proof of plug-in FDR control for known
estimators like $\widehat{m}_{0}^{\text{Storey}}$ and
$\widehat{m}_{0}^{\text{PC,new}}$ and also for new estimators that we will
define in Section 4. Additionally, the classes $\mathcal{F}_{0}$ and
$\mathcal{F}$ possess stability properties that make it possible to combine
various plug-in estimators while maintaining FDR control.
###### Proposition 3.3.
Let $\widehat{m}_{1},\widehat{m}_{2}\in\mathcal{F}$, where $\mathcal{F}$ is
defined by (11) and let $\lambda\in[0,1]$. Then the BH plug-in procedure (2)
using $\widehat{m}_{0}=\lambda\widehat{m}_{1}+(1-\lambda)\widehat{m}_{2}$
controls FDR at level $\alpha$.
###### Proof.
We show that $\widehat{m}_{0}$ satisfies (IMC). Let
$\widehat{m}_{1},\widehat{m}_{2}\in\mathcal{F}$ have the representation
$\displaystyle\widehat{m}_{1}$
$\displaystyle=\frac{1}{\nu}+\sum_{i=1}^{m}\frac{g_{i}(p_{i})}{\nu_{i}}\qquad\text{and}\qquad\widehat{m}_{2}=\frac{1}{\mu}+\sum_{i=1}^{m}\frac{h_{i}(p_{i})}{\mu_{i}},$
where $\nu=min(\nu_{1},\ldots,\nu_{m})$ and $\mu=min(\mu_{1},\ldots,\mu_{m})$
so that $\displaystyle\widehat{m}_{0}$
$\displaystyle=\frac{\lambda}{\nu}+\frac{1-\lambda}{\mu}+\sum_{i=1}^{m}\frac{\lambda
g_{i}(p_{i})}{\nu_{i}}+\frac{(1-\lambda)h_{i}(p_{i})}{\mu_{i}}$
and define weights
$\kappa_{i}=\frac{\lambda\mu_{i}}{\lambda\mu_{i}+(1-\lambda)\nu_{i}}$ and
transformations $f_{i}=\kappa_{i}g_{i}+(1-\kappa_{i})h_{i}$. Clearly,
$\kappa_{i}\in[0,1]$ and $f_{i}\in\mathcal{G}$ and we introduce
$\epsilon_{i}=\mathbb{E}(f_{i})=\kappa_{i}\nu_{i}+(1-\kappa_{i})\mu_{i}$. From
the above definitions we obtain with some straightforward algebra
$\displaystyle\lambda$
$\displaystyle=\frac{\kappa_{i}\nu_{i}}{\epsilon_{i}}\qquad\text{and}\qquad
1-\lambda=\frac{(1-\kappa_{i})\mu_{i}}{\epsilon_{i}}$ (13) which yields
$\displaystyle\frac{\lambda}{\nu_{i}}g_{i}$
$\displaystyle+\frac{(1-\lambda)}{\mu_{i}}h_{i}=\frac{\kappa_{i}}{\epsilon_{i}}g_{i}+\frac{(1-\kappa_{i})}{\epsilon_{i}}h_{i}=\frac{f_{i}}{\epsilon_{i}}.$
(14) From (13) we have $\displaystyle\frac{\lambda}{\nu}$
$\displaystyle=\max\left(\frac{\lambda}{\nu_{1}},\ldots,\frac{\lambda}{\nu_{m}}\right)=\max\left(\frac{\kappa_{1}}{\epsilon_{1}},\ldots,\frac{\kappa_{m}}{\epsilon_{m}}\right)\qquad\text{and}$
$\displaystyle\frac{1-\lambda}{\mu}$
$\displaystyle=\max\left(\frac{1-\lambda}{\mu_{1}},\ldots,\frac{1-\lambda}{\mu_{m}}\right)=\max\left(\frac{1-\kappa_{1}}{\epsilon_{1}},\ldots,\frac{1-\kappa_{m}}{\epsilon_{m}}\right)$
so that the sub-additivity of the $\max$ function now yields the bound
$\displaystyle\frac{\lambda}{\nu}$
$\displaystyle+\frac{1-\lambda}{\mu}\geq\frac{1}{\epsilon},$ (15)
where $\epsilon=\min(\epsilon_{1},\ldots,\epsilon_{m})$. Combining (14) and
(15) now gives us
$\displaystyle\widehat{m}_{0}$
$\displaystyle=\frac{\lambda}{\nu}+\frac{1-\lambda}{\mu}+\sum_{i=1}^{m}\frac{\lambda
g_{i}(p_{i})}{\nu_{i}}+\frac{(1-\lambda)h_{i}(p_{i})}{\mu_{i}}\geq\frac{1}{\epsilon}+\sum_{i=1}^{m}\frac{f_{i}(p_{i})}{\epsilon_{i}}:=\widetilde{m}_{0}$
with $\widetilde{m}_{0}\in\mathcal{F}$ and from Proposition 3.2 we know that
(IMC) holds true for $\widetilde{m}_{0}$ and therefore also for
$\widehat{m}_{0}$. ∎
The proof shows that $\mathcal{F}$ is “almost” convex in the sense that
whenever equality holds in (15) we have
$\widehat{m}_{0}=\widetilde{m}_{0}\in\mathcal{F}$. If
$\widehat{m}_{1},\widehat{m}_{2}\in\mathcal{F}_{0}$, i.e. each estimator uses
only a single transformation function then it is easy to see that equality
holds in (15) which leads to the following result:
###### Proposition 3.4.
The class of estimators $\mathcal{F}_{0}$ given by (9) is convex. In
particular this implies that for any
$\widehat{m}_{1},\widehat{m}_{2}\in\mathcal{F}_{0}$ and $\lambda\in[0,1]$ the
BH plug-in procedure (2) controls FDR at level $\alpha$ for the estimator
$\widehat{m}_{0}=\lambda\widehat{m}_{1}+(1-\lambda)\widehat{m}_{2}$.
## 4 Homogeneous estimators
In this section we focus on the class of homogeneous estimators
$\mathcal{F}_{0}$ given by (9), i.e. on estimators of the form
$\displaystyle\widehat{m}_{0}$
$\displaystyle=\widehat{m}_{0}(p_{1},\ldots,p_{m})=\frac{1}{\nu}\left(1+\sum_{i=1}^{m}g(p_{i})\right),$
where $g\in\mathcal{G}$ and $\nu=\nu(g)=\mathbb{E}g(U)>0$, with
$U\sim\mathcal{U}[0,1]$. As mentioned before, this class includes the
classical estimator $\widehat{m}_{0}^{\text{Storey}}$ (3) and the new
estimator $\widehat{m}_{0}^{\text{PC,new}}$ (10), and also gives the scientist
freedom to define new estimators with proven plug-in FDR control thanks to
Proposition 3.2 There are many conceivable ways in which this can be done. As
an ad hoc example, we define a polynomial estimator of degree $r\geq 0$ and
thresholding parameter $\lambda\in[0,1)$ by taking $\widehat{m}_{0}$ as above
in (9) with $g(u)=g_{r,\lambda}(u)=u^{r}\cdot{\mathbf{1}\\{u>\lambda\\}}$, so
that $\nu=\frac{1-\lambda^{r+1}}{r+1}$. This gives us
$\displaystyle\widehat{m}_{0}^{\text{Poly}}(r,\lambda)=\frac{r+1}{1-\lambda^{r+1}}+\frac{r+1}{1-\lambda^{r+1}}\sum_{i=1}^{m}p_{i}^{r}\cdot{\mathbf{1}\\{p_{i}>\lambda\\}}.$
(16)
It is easily seen that the classical estimators
$\widehat{m}_{0}^{\text{Storey}}$ and $\widehat{m}_{0}^{\text{PC,new}}$ are
particular instances of $\widehat{m}_{0}^{\text{Poly}}(r,\lambda)$ with $r=0$
for $\widehat{m}_{0}^{\text{Storey}}$ and $r=1$ and $\lambda=0$ for
$\widehat{m}_{0}^{\text{PC,new}}$. Taking $r=1$ and $\lambda>0$ yields a
hybrid estimator which combines $\widehat{m}_{0}^{\text{Storey}}$ and
$\widehat{m}_{0}^{\text{PC,new}}$ which has the potential to combine the
strengths of both methods. For all estimators
$\widehat{m}_{0}^{\text{Poly}}(r,\lambda)$ plug-in FDR control follows
immediately from Proposition 3.2. For illustrational purposes we effectively
only use $r$ as a parameter and set the thresholding parameter to the
classical value of $\lambda=1/2$ throughout the paper. These examples are
primarily meant to illustrate the freedom and flexibility Proposition 3.2
allows for the class $\mathcal{F}_{0}$ and should not be interpreted as
recommendations for optimal choices. These examples are investigated further
in the following section.
### 4.1 Numerical results
Here we compare the performance of several estimators from the class
$\mathcal{F}_{0}$ in a Gaussian one-sided testing setting. We assume that we
observe $X_{1},\dots,X_{m}$ independent random variables with
$X_{i}\sim\mathcal{N}(0,1)$ for $i\in\mathcal{H}_{0}$ and
$X_{i}\sim\mathcal{N}(\mu,1)$ for $i\in\mathcal{H}_{1}$, with $\mu>0$, and we
test $H_{0,i}:\mu=0$ vs. $H_{1,i}:\mu>0$. For a given signal strength $\mu>0$
under the alternative, closed-form expressions for the expectation and
variance of $\widehat{m}_{0}$ are available, see Appendix B for more details.
Thus we can numerically compare the mean squared error (MSE) of estimators
from $\mathcal{F}_{0}$.
For this analysis, we fix $m=10\>000$ and first compare the MSE w.r.t. to the
signal strength $\mu$ of the alternatives with a fixed proportion of true
nulls $\pi_{0}=0.6$. Then, we compare the MSE w.r.t. the proportion of true
nulls $\pi_{0}$ with a fixed signal strength $\mu=1.5$. The considered
estimators for this comparison are $\widehat{m}_{0}^{\text{Storey}}$ (3),
$\widehat{m}_{0}^{\text{PC,new}}$ (10),
$\widehat{m}_{0}^{\text{Poly}}(1,1/2)$, and
$\widehat{m}_{0}^{\text{Poly}}(2,1/2)$ (see (16) for both). The MSE is
evaluated in terms of $\pi_{0}$ for better readability and displayed in Figure
1. The qualitative comparison between the estimators remains consistent across
both panels of Figure 1: $\widehat{m}_{0}^{\text{PC,new}}$ has the poorest
performance, characterized by the largest MSE, followed by
$\widehat{m}_{0}^{\text{Storey}}$. While the polynomial approach shows some
benefits, the improvement is not particularly remarkable except for small to
moderate values of $\pi_{0}$. For larger values of $\pi_{0}$ or $\mu$, there
are no noticeable differences in performance.
Figure 1: MSE against signal strength $\mu\in[0,3]$ with $\pi_{0}=0.6$(left)
and MSE against $\pi_{0}\in[0.1,0.9]$ with $\mu=1.5$(right)
### 4.2 More details on the Pounds and Cheng estimator
While FDR control for $\widehat{m}_{0}^{\text{Storey}}$ is a classical result
following from Theorem 2.1 (Blanchard and Roquain, (2009); Benjamini et al.,
(2006)), much less is known about the validity of
$\widehat{m}_{0}^{\text{PC,2006}}$ as a plug-in estimator. Indeed, Pounds and
Cheng, (2006) introduced their estimator (4) primarily to obtain a robust
estimate of FDR. To the best of our knowlegde, the only related result on
plug-in FDR control was obtained by Zeisel et al., (2011), who defined the
following modified version of $\widehat{m}_{0}^{\text{PC,2006}}$:
$\displaystyle\widehat{m}_{0}^{\text{PC,ZZD}}$
$\displaystyle=C(m)\cdot\min\left[m,\max\left(s(m),2\cdot\sum_{i=1}^{m}p_{i}\right)\right],$
(17)
where the correction factors $C(m)$ and $s(m)$ are chosen in such a way so
that (IMC) holds. However, determining these factors is non-trivial and
requires extensive use of numerical integration and approximations methods
(see Supplement B in Zeisel et al., (2011) for further details) so that no
simple representation of $C(m)$ and $s(m)$ is available (for selected values
of $m$, Table S1 in Zeisel et al., (2011) lists values for the correction
factors).
By contrast, our new modification (10) is extremely simple and, as we show in
Section 5, can be adapted easily to e.g. discrete tests, thus confirming a
conjecture in Pounds and Cheng, (2006). Its validity for plug-in FDR control
follows directly from Proposition 3.2 and involves no sophisticated asymptotic
or numerical approximations. Supplementary material in Appendix C shows that
the two versions of the PC estimator behave more or less identically.
Nevertheless, we argue in favor of using $\widehat{m}_{0}^{\text{PC,new}}$
since it is both conceptually and computationally much simpler than
$\widehat{m}_{0}^{\text{PC,ZZD}}$.
## 5 Adjusted estimators for discrete $p$-values
In this section we assume – additionally to mutual independence of the
$p$-values – that the null distribution functions $F_{1},\ldots,F_{m}$ are
known. As a particular application we consider the setting of discrete
$p$-values (see Section 1.1 for more detailed references). The classical plug-
in estimators, like the Storey, (2002) estimator defined in (3), were
developed for uniformly distributed $p$-values under the nulls, and can thus
suffer of an inflated bias when computed under (SuperUnif) assumption.
To illustrate the problem, we compare the bias of an arbitrary estimator
$\widehat{m}_{0}\in\mathcal{F}_{0}$ under the uniform setting with the bias
under the super-uniform setting. In the classical uniform case, considering
marginally independent $p$-values $p_{i}\sim X_{0}$ for $i\in\mathcal{H}_{0}$,
and $p_{i}\sim X_{1}$ for $i\in\mathcal{H}_{1}$, for some variables
$X_{0},X_{1}$ defined on $[0,1]$, the bias is seen to be
$\displaystyle\textnormal{Bias}[\widehat{m}_{0}]$
$\displaystyle=\mathbb{E}[\widehat{m}_{0}]-m_{0}=\frac{1}{\nu}(1+m_{1}\mathbb{E}[g(X_{1})]).$
(18) In contrast, under the super-uniform setting, still considering
independent $p$-values under the null and the alternative, the bias is
$\displaystyle\textnormal{Bias}[\widehat{m}_{0}]$
$\displaystyle=\frac{1}{\nu}(1+m_{1}\mathbb{E}[g(X_{1})])+\frac{1}{\nu}m_{0}(\mathbb{E}[g(X_{0})]-\nu).$
(19)
Recall that $\nu=\mathbb{E}[g(U)]$ with $U\sim\mathcal{U}[0,1]$, thus under
super-uniformity $\mathbb{E}[g(X_{0})]\geq\nu$ (see the characterization of
the usual stochastic order in Appendix A), which shows that an additional
source of conservativeness is present in this case. In general, practitioners
use classical estimators without worrying about $p$-values distributions,
ingenuously expecting the estimator to perform according to the “uniform” bias
(18) when in fact it often performs according to the “super-uniform” bias
(19). This motivates the need for a correction in the estimator that will aim
at deflating (19).
Super-uniformity does not solely appear in the discrete setting, it also
occurs e.g. when testing composite nulls, however in the discrete setting
additional information on the $p$-values c.d.f (as defined by (5)) may be
available and leveraged to correct the over-conservativeness. In this section,
we present two such approaches that incorporate the available knowledge of
$F_{i}$ – the $p$-value c.d.f under the null – in the estimators. The standard
way of defining $p$-values for discrete tests leads to distribution functions
that satisfy (Discrete) and (SuperUnif). As we later introduce transformed
$p$-values with transformed distribution, for clearer distinguishability the
c.d.f associated to these standard discrete $p$-values are denoted by
$F^{\text{sd}}_{1},\ldots,F^{\text{sd}}_{m}$ (where the upper-script “sd”
denotes “standard discrete”).
### 5.1 Transformations of discrete $p$-values
In order to reduce the individual conservatism of $p$-values caused by super-
uniformity, various transformation of discrete $p$-values have been proposed,
see e.g. Habiger, (2015). Perhaps the most popular transformation is the so-
called mid-$p$-value. For the realization $x$ of the random variable $X$, let
$p(x)$ be the (realized) standard $p$-value. Now define the mid-$p$-value
(Rubin-Delanchy et al.,, 2019) $q(x)$ given the observation $x$ as
$\displaystyle q(x)$ $\displaystyle=p(x)-\frac{1}{2}P_{0}(p(X)=p(x)),$ (20)
where $P_{0}$ denotes the distribution of $X$ under the null (for simplicity,
we assume that such a unique distribution exists). Transforming the $p$-value
through (20) helps to mimic the behavior of a uniform random variable in
expectation. Indeed, we always have $\mathbb{E}[q(X)]=1/2$, see Berry and
Armitage, (1995) for more details. However, the distribution of the
mid-$p$-value is no longer super-uniform but shrunk toward 0 as displayed in
Figure 2. In what follows, we denote by
$F^{\text{mid}}_{1},\ldots,F^{\text{mid}}_{m}$ the distribution functions of
the mid-$p$-values associated with the distribution functions
$F^{\text{sd}}_{1},\ldots,F^{\text{sd}}_{m}$ of the standard $p$-values. In
Section 5.2 we show how the distribution functions of standard discrete or
mid-$p$-values can be used in $m_{0}$-estimators introduced in Section 3,
while preserving plug-in FDR control.
Another transformation to reduce the conservativeness of discrete $p$-values
uses so-called randomized $p$-values which are defined in our context by
$\displaystyle r(x,u)$ $\displaystyle=p(x)-u\cdot P_{0}(p(X)=p(x)),$ (21)
where $u$ is the realization of a uniform random variable
$U\sim\mathcal{U}[0,1]$, independent of $X$. Alternatively to the notation
$r(x,u)$, we will also use (with a slight abuse) $r(p,u)$, where $p=p(x)$ is
the standard $p$-value obtained from observation $x$. Randomized $p$-values
and mid-$p$ values are related via the conditional expectation on the
observations $q(x)=\mathbb{E}_{U}[r(x,U)|X=x]$. Randomization leads to an
(unconditional) uniform behavior, however at the cost of introducing an
additional source of randomness which makes its use controversial for
decisions on individual hypotheses, see e.g. Habiger and Pena, (2011) for a
discussion. We show in Section 5.3 that for estimation purposes however,
randomized $p$-values can be beneficial for obtaining an efficient non-
randomized estimator.
Figure 2: Distribution functions of a standard $p$-value and its associated
mid-$p$-value.The standard $p$-value has support $\\{0.3,0.55,0.7,0.9,1\\}$
(orange rug plot), the c.d.f (orange solid line) is super-uniform, i.e. below
the uniform c.d.f (gray line). The support of the associated mid-$p$-value
$\\{0.15,0.465,0.625,0.8,0.95\\}$ (green rug plot) is shifted to the left, but
the probabilities (given by the jumps in the green c.d.f) remain the same.
### 5.2 Adjusting the rescaling constants
The first approach for adjusting estimators to discrete $p$-values is tailored
to estimators from the class $\mathcal{F}_{0}$ and adjusts the rescaling
constant $\nu$ in $\widehat{m}_{0}$. In fact, this approach is not limited to
the discrete setting, and can also be applied for arbitrary $p$-value
distributions.
###### Proposition 5.1.
Assume that $p_{1},\ldots,p_{m}$ are mutually independent and the null
distribution functions $F_{1},\ldots,F_{m}$ are known. For any
$\widehat{m}_{0}\in\mathcal{F}_{0}$ (9) with
$\displaystyle\widehat{m}_{0}(p_{1},\ldots,p_{m})$
$\displaystyle=\frac{1}{\nu(g)}\left(1+\sum_{i=1}^{m}g(p_{i})\right)$ (22)
define the adjusted estimator
$\displaystyle\widehat{m}^{\text{adj}}_{0}(p_{1},\ldots,p_{m})$
$\displaystyle=\frac{1}{min(\nu^{\text{adj}}_{1},\ldots,\nu^{\text{adj}}_{m})}+\sum_{i=1}^{m}\frac{g(p_{i})}{\nu^{\text{adj}}_{i}}$
(23)
where $\nu^{\text{adj}}_{i}=\mathbb{E}_{p_{i}\sim F_{i}}[g(p_{i})]$, is the
expectation of the transformed p-value taken w.r.t. $F_{i}$. Then the BH plug-
in procedure (2) using $\widehat{m}^{\text{adj}}_{0}$ controls FDR at level
$\alpha$.
###### Proof.
Without loss of generality we assume that $\nu^{\text{adj}}_{i}>0$, otherwise
the sum and minimum in (23) is to be taken over the index set
$\\{i:\nu^{\text{adj}}_{i}>0\\}$.
For any $i\in\\{1,\dots,m\\}$ define $g_{i}:[0,1]\rightarrow[0,1]$ by
$g_{i}(y)=g\circ F^{-1}_{i}(y)$ for $y\in(0,1]$, where
$F^{-1}_{i}(y)=\inf\\{x\in\mathbb{R}:F_{i}(x)\geq y\\}$ is the generalized
inverse of $F_{i}$, and set $g_{i}(0)=g(0)$. Since $g\in\mathcal{G}$ and
$F^{-1}_{i}$ are both nondecreasing, so is $g_{i}$. For $i\in\mathcal{H}_{0}$,
with $U\sim\mathcal{U}[0,1]$, we have $p_{i}\sim F^{-1}_{i}(U)$ by Proposition
2 in Embrechts and Hofert, (2013), so that $g_{i}(U)\sim g(p_{i})$, which
implies that $\mathbb{E}[g_{i}(U)]=\mathbb{E}_{p_{i}\sim
F_{i}}[g(p_{i})]=\nu^{\text{adj}}_{i}$, so that (23) belongs to the class
$\mathcal{F}$.
Now let $z_{1},\dots z_{m}$ be independent random variables with
$z_{i}\sim\mathcal{U}[0,1]$ for $i\in\mathcal{H}_{0}$ and
$z_{i}\sim\delta_{0}$ for $i\in\mathcal{H}_{1}$ (Dirac-Uniform configuration).
Since $g_{i}(U)\sim g(p_{i})$ for $i\in\mathcal{H}_{0}$, we have
$\displaystyle\widetilde{m}^{\text{adj}}_{0}(z_{1},\ldots,z_{m})$
$\displaystyle=\frac{1}{min(\nu^{\text{adj}}_{1},\ldots,\nu^{\text{adj}}_{m})}+\sum_{i\in\mathcal{H}_{0}}\frac{g_{i}(z_{i})}{\nu^{\text{adj}}_{i}}$
$\displaystyle\sim\frac{1}{min(\nu^{\text{adj}}_{1},\ldots,\nu^{\text{adj}}_{m})}+\sum_{i\in\mathcal{H}_{0}}\frac{g(p_{i})}{\nu^{\text{adj}}_{i}}\leq\widehat{m}^{\text{adj}}_{0}(p_{1},\ldots,p_{m})\quad\mbox{(a.s.)}$
Since $\widetilde{m}^{\text{adj}}_{0}(z_{1},\ldots,z_{m})\in\mathcal{F}$, by
Proposition 3.2 (IMC) holds for
$\widetilde{m}^{\text{adj}}_{0}(z_{1},\ldots,z_{m})$, and since
$\widetilde{m}^{\text{adj}}_{0}(z_{1},\ldots,z_{m})\leq\widehat{m}^{\text{adj}}_{0}(p_{1},\ldots,p_{m})$
(a.s.), (IMC) also holds for
$\widehat{m}^{\text{adj}}_{0}(p_{1},\ldots,p_{m})$. ∎
Following Proposition 5.1, we define the discrete-uniform estimator using (23)
with standard discrete $p$-values $p_{1},\ldots,p_{m}$ and their distribution
functions $F^{\text{sd}}_{1},\ldots,F^{\text{sd}}_{m}$
$\displaystyle\widehat{m}_{0}^{\text{du}}(p_{1},\ldots,p_{m})$
$\displaystyle=\frac{1}{min(\nu^{\text{du}}_{1},\ldots,\nu^{\text{du}}_{m})}+\sum_{i=1}^{m}\frac{g(p_{i})}{\nu^{\text{du}}_{i}},$
(24)
where $\nu^{\text{du}}_{i}=\mathbb{E}_{p_{i}\sim
F^{\text{sd}}_{i}}[g(p_{i})]$.
###### Corollary 5.1.
Assume that $p_{1},\ldots,p_{m}$ are mutually independent and (SuperUnif)
holds with null distribution functions
$F^{\text{sd}}_{1},\ldots,F^{\text{sd}}_{m}$ that are known. Then the BH plug-
in procedure (2) using $\widehat{m}_{0}^{\text{du}}$ as in (24)controls FDR at
level $\alpha$. Moreover $\widehat{m}_{0}^{\text{du}}\leq\widehat{m}_{0}$
(a.s.), where $\widehat{m}_{0}$ is the base non-adjusted estimator (22).
The last statement of the corollary shows that for standard discrete
$p$-values the estimator $\widehat{m}_{0}^{\text{du}}$ is guaranteed to
perform better than $\widehat{m_{0}}$. This follows from the fact that
$\nu^{\text{du}}_{1},\ldots,\nu^{\text{du}}_{m}\geq\nu$ because $g$ is non-
decreasing and (SuperUnif) holds (see Appendix A).
For classical estimators, the adjusted rescaling constants can be computed
easily, using
* •
$\nu^{\text{du-Storey}}_{i}=1-F^{\text{sd}}_{i}(\lambda)$;
* •
$\nu^{\text{du-PC}}_{i}=\sum_{x\in\mathcal{A}_{i}}x\cdot P(p_{i}=x)$, where
$\mathcal{A}_{i}$ denotes the support of $F^{\text{sd}}_{i}$.
Similarly to (24), we define a mid $p$-value estimator using (23) with
mid-$p$-values $q_{1},\ldots,q_{m}$ and their distribution functions
$F^{\text{mid}}_{1},\ldots,F^{\text{mid}}_{m}$
$\displaystyle\widehat{m}_{0}^{\text{mid}}(q_{1},\ldots,q_{m})$
$\displaystyle=\frac{1}{min(\nu^{\text{mid}}_{1},\ldots,\nu^{\text{mid}}_{m})}+\sum_{i=1}^{m}\frac{g(q_{i})}{\nu^{\text{mid}}_{i}}$
(25)
where $\nu^{\text{mid}}_{i}=\mathbb{E}_{q_{i}\sim
F^{\text{mid}}_{i}}[g(q_{i})]$ is the expectation taken under the null using
the mid $p$-value distribution function $F^{\text{mid}}_{i}$. For
$\widehat{m}_{0}^{\text{Storey}}$ we have $\nu^{\text{mid-
Storey}}_{i}=1-F^{\text{mid}}_{i}(\lambda)\leq
1-F^{\text{sd}}_{i}(\lambda)=\nu^{\text{du-Storey}}_{i}$ and
$g(q_{i})={\mathbf{1}\\{q_{i}>\lambda\\}}\leq{\mathbf{1}\\{p_{i}>\lambda\\}}=g(p_{i})$
so that $\widehat{m}_{0}^{\text{mid}}$ can be smaller or larger than
$\widehat{m}_{0}^{\text{du}}$, depending on the specific constellation. In the
case of the PC estimator we have $g(x)=x$ and since $\mathbb{E}q_{i}=1/2$ for
any mid-$p$-value (see Berry and Armitage, (1995)) we have
$\nu^{\text{mid}}_{1}=\ldots=\nu^{\text{mid}}_{m}=1/2$ so that in this case
the mid-$p$ estimator has a particularly simple representation. Combining this
with the fact that $q_{i}\leq p_{i}$ $(a.s.)$ gives us the following result.
###### Corollary 5.2.
Assume that $p_{1},\ldots,p_{m}$ are mutually independent and super-uniform
under the null (i.e. (SuperUnif) holds), and let $q_{1},\ldots,q_{m}$ denote
the corresponding mid-$p$-values. Then the mid-$p$ estimator of
$\widehat{m}_{0}^{\text{PC,new}}$ is given by
$\displaystyle\widehat{m}_{0}^{\text{mid-PC}}(q_{1},\ldots,q_{m})$
$\displaystyle=2+2\cdot\sum_{i=1}^{m}q_{i}$
and the BH plug-in procedure (2) using $\widehat{m}_{0}^{\text{mid-
PC}}(q_{1},\ldots,q_{m})$ controls FDR at level $\alpha$. Moreover,
$\widehat{m}_{0}^{\text{mid-PC}}\leq\widehat{m_{0}}$ $(a.s.)$, where
$\widehat{m}_{0}$ is the base non-adjusted estimator (22).
This result implies that for the PC estimator with discrete data we can simply
use $2+2\cdot\sum_{i=1}^{m}q_{i}$ instead of the more conservative
$2+2\cdot\sum_{i=1}^{m}p_{i}$ estimator without losing plug-in FDR control. We
point out that the mid-$p$-values are used exclusively for estimating $m_{0}$
in the plug-in procedure defined by (2) while the (ordered) standard discrete
$p$-values $p_{(k)}$ are used in the final decision step.
### 5.3 A randomization approach
Here we describe an approach related to Dickhaus et al., (2012) who argue for
using randomization methods in estimating $m_{0}$ on discrete data. For any
estimator $\widehat{m}_{0}$, not necessarily belonging to $\mathcal{F}_{0}$
define the associated expected randomized estimator as
$\displaystyle\widehat{m}_{0}^{\text{rand}}(p_{1},\ldots,p_{m})$
$\displaystyle=\left[\mathbb{E}_{(U_{1},\ldots,U_{m})}\left(\frac{1}{\widehat{m}_{0}(r(p_{1},U_{1}),\ldots,r(p_{m},U_{m}))}\right)\right]^{-1}$
(26)
where $U_{1},\ldots,U_{m}\sim\mathcal{U}[0,1]$ denote i.i.d uniform random
variables independent of $(p_{1},\ldots,p_{m})$. Thus, for fixed
$(p_{1},\ldots,p_{m})$ this estimator is obtained by taking the expectation
over the randomized $p$-values associated with $(p_{1},\ldots,p_{m})$. In most
cases (26) is analytically intractable, we therefore use Monte-Carlo
approximation of $\widehat{m}_{0}^{\text{rand}}$ obtained by averaging a large
number of simulations of
$\widetilde{m}_{0}(r(p_{1},U_{1}),\ldots,r(p_{m},U_{m}))$ (the vector
$(U_{1},\ldots,U_{m})$ is simulated many times, while $(p_{1},\ldots,p_{m})$
is kept fixed). Again, this approach comes with guaranteed FDR plug-in
control.
###### Corollary 5.3.
Assume that $p_{1},\ldots,p_{m}$ are mutually independent and super-uniform
under the null (i.e. (SuperUnif) holds) and let $\widehat{m}_{0}$ satisfy the
conditions of Theorem 2.1. Then the BH plug-in procedure (2) using
$\widehat{m}_{0}^{\text{rand}}(p_{1},\ldots,p_{m})$ defined by (26) controls
FDR at level $\alpha$. Moreover
$\widehat{m}_{0}^{\text{rand}}\leq\widehat{m}_{0}$ (a.s.).
###### Proof.
The proof uses Theorem 2.1. First, we show that
$\widehat{m}_{0}^{\text{rand}}(p_{1},\ldots,p_{m})$ is coordinatewise non-
decreasing. For fixed $(u_{1},\ldots,u_{m})\in[0,1]^{m}$ each (realized)
randomized $p$-value $r_{i}=r(p_{i},u_{i})$ is non-decreasing in $p_{i}$.
Since $\widehat{m}_{0}\in\mathcal{F}$ is coordinatewise non-decreasing in
$(p_{1},\ldots,p_{m})$, the function
$1/\widehat{m}_{0}(r(\cdot,u_{1}),\ldots,r(\cdot,u_{m}))$ is coordinatewise
decreasing for all $(u_{1},\ldots,u_{m})\in[0,1]^{m}$ and so is its
expectation which implies that $\widehat{m}_{0}^{\text{rand}}$ is
coordinatewise non-decreasing. To establish (IMC), we denote for
$h\in\mathcal{H}_{0}$ by $r_{0,h}$ the set of randomized $p$-values
$(r_{1},\ldots,r_{m})$, where $r_{h}$ has been replaced by $0$. By the
definition of $\widehat{m}_{0}^{\text{rand}}$ we have
$\displaystyle\mathbb{E}_{(p_{1},\ldots,p_{m})}\left[\frac{1}{\widehat{m}_{0}^{\text{rand}}(p_{0,h})}\right]$
$\displaystyle=\mathbb{E}_{(p_{1},\ldots,p_{m})}\left[\mathbb{E}_{(U_{1},\ldots,U_{m})}\frac{1}{\widehat{m}_{0}(r_{0,h})}\right]=\mathbb{E}_{(r_{1},\ldots,r_{m})}\left[\frac{1}{\widehat{m}_{0}(r_{0,h})}\right].$
where the second equality follows from the fact that for super-uniform
$p$-value $p_{h}=0$, the associated randomized $p$-value $r(p_{h},u)=0$ (a.s)
by Definition (21). Since the $(r_{1},\ldots,r_{m})$ are mutually independent
and uniform under the null and $\widehat{m}_{0}\in\mathcal{F}$, the bound
(IMC) for $\widehat{m}_{0}(r_{0,h})$ now follows since $\widehat{m}_{0}$
satisfies the conditions of Theorem 2.1. Therefore, the r.h.s. of the last
equation can be bounded by $1/m_{0}$ and plug-in FDR control for
$\widehat{m}_{0}^{\text{rand}}$ now follows from Theorem 2.1. To see that the
last statement of the corollary holds true, observe that since
$\widehat{m}_{0}$ is coordinatewise non-decreasing and $r(p_{i},U_{i})\leq
r(p_{i},0)=p_{i}$ we have
$\widehat{m}_{0}(r(p_{1},U_{1}),\ldots,r(p_{m},U_{m}))\leq\widehat{m}_{0}(p_{1},\ldots,p_{m})$
and therefore the r.h.s. of (26) is bounded by
$\widehat{m}_{0}(p_{1},\ldots,p_{m})$ $(a.s.)$. ∎
Dickhaus et al., (2012) argue for using randomized $p$-values in (essentially)
Storey’s estimator, i.e. applying $\widehat{m}_{0}^{\text{Storey}}$ to
$(r_{1},\ldots,r_{m})$ instead of $(p_{1},\ldots,p_{m})$ which yields a random
estimate that should provide a better estimate for $m_{0}$. They show that
plugging this estimator into the Bonferroni procedure yields asymptotic
control of the Familywise Error Rate (FWER) under certain assumptions. They
also point out that if fully reproducible results are desired it may be more
appropriate to work with the conditional expectation w.r.t. randomization,
i.e. using
$\mathbb{E}_{(U_{1},\ldots,U_{m})}\left(\widehat{m}_{0}^{\text{Storey}}(r_{1},\ldots,r_{m})\right)$.
Corollary 5.3 shows that we can obtain similar guarantees w.r.t. to plug-in
FDR control in a finite-sample setting for any estimator $\widehat{m}_{0}$
satisfying the conditions of Theorem 2.1 and in particular for
$\widehat{m}_{0}\in\mathcal{F}_{0}$ by using conditional expectation w.r.t.
randomization. The slightly complicated form of (26) is a natural consequence
of Theorem 2.1, but if the variance of
$\widehat{m}_{0}(r(p_{1},U_{1}),\ldots,r(p_{m},U_{m}))$ w.r.t.
$U_{1},\ldots,U_{m}$ is small we have the approximation
$\widehat{m}_{0}^{\text{rand}}(p_{1},\ldots,p_{m})\approx\mathbb{E}_{(U_{1},\ldots,U_{m})}\widehat{m}_{0}(r(p_{1},U_{1}),\ldots,r(p_{m},U_{m}))$.
### 5.4 Simulation results
In this section, we analyze how the discrete adjustments can improve the base
estimators on simulated data. More specifically, we follow Döhler et al.,
(2018) by simulating a two-sample problem in which a vector of $m=500$
independent binary responses is observed for $N=25$ subjects in both groups.
The goal is to test the $m$ null hypotheses $H_{0,i}$: ’$p_{1i}=p_{2i}$’,
$i=1,...,m$ where $p_{1i}$ and $p_{2i}$ are the success probabilities for the
$i^{th}$ binary response in group A and B respectively. Thus, for each
hypothesis $i$, the data can be summarized by a $2\times 2$ contingency table,
and we use (two-sided) Fisher’s exact tests (FETs) for testing $H_{0i}$. The
$m$ hypotheses are split in three groups of size $m_{1}$, $m_{2}$, and $m_{3}$
such that $m=m_{1}+m_{2}+m_{3}$. Then, the binary responses are generated as
i.i.d Bernoulli of probability 0.01 ($\mathbf{Bin}(1,0.01)$) at $m_{1}$
positions for both groups, i.i.d $\mathbf{Bin}(1,0.10$) at $m_{2}$ positions
for both groups, and i.i.d $\mathbf{Bin}(1,0.10)$ at $m_{3}$ positions for one
group and i.i.d $\mathbf{Bin}(1,p_{3})$ at $m_{3}$ positions for the other
group. Thus, null hypotheses are true for $m_{1}+m_{2}$ positions, while they
are false for $m_{3}$ positions (set $\mathcal{H}_{1}$). We interpret $p_{3}$
as the strength of the signal and set it to 0.4, while
$\pi_{1}=\frac{m_{3}}{m}$, corresponds to the proportion of signal. Also,
$m_{1}$ and $m_{2}$ are both taken equal to $\frac{m-m_{3}}{2}$.
We first compare the base estimators $\widehat{m}_{0}^{\text{Storey}}$ (3),
$\widehat{m}_{0}^{\text{PC,new}}$ (10), and
$\widehat{m}_{0}^{\text{Poly}}(2,1/2)$ (16) with their standard discrete
rescaled versions. Figure 3 displays the estimation results for a grid of true
$\pi_{0}\in\\{0.1,\dots,0.9\\}$. We can see that incorporating discreteness
leads to considerable improvements for all estimators over the entire range of
$\pi_{0}$ values. This is particularly relevant for large values of $\pi_{0}$
where the base estimators may lead to a strong deterioration in the power of
the plug-in BH procedure. On another note, among the base estimators we can
see that $\widehat{m}_{0}^{\text{Poly}}(2,1/2)$ performs poorly compared to
the results of Section 4.1. This seems plausible since a large portion of
$p$-values are equal to 1 in the discrete setting in contrast to the Gaussian
setting. For these $p$-values, the contribution in the
$\widehat{m}_{0}^{\text{Poly}}(2,1/2)$ estimator is equal to the constant
$\nu=\frac{3}{1-1/2^{3}}=\frac{24}{7}$ which is much larger than the
corresponding contribution of $\nu=2$ in the Storey estimator.
Figure 3: Comparison between base estimators
$\widehat{m}_{0}^{\text{Storey}}$, $\widehat{m}_{0}^{\text{PC,new}}$ and
$\widehat{m}_{0}^{\text{Poly}}(2,1/2)$ and their standard discrete rescaled
versions on simulated data.
In a second step, we compare the different discrete adjustments
$\widehat{m}_{0}^{\text{du}}$ as in (24), $\widehat{m}_{0}^{\text{mid}}$ as in
(25), and $\widehat{m}_{0}^{\text{rand}}$ as in (26) in Figure 4, where we
display the estimation results for three values of true
$\pi_{0}\in\\{0.2,0.5,0.7\\}$. We can see that there are no relevant
differences between the different adjustments. Therefore, there is no strong
reason to advocate a specific type of adjustment since they yield similar
outcomes.
Figure 4: Comparison between discrete adjustments
$\widehat{m}_{0}^{\text{du}}$, $\widehat{m}_{0}^{\text{mid}}$ and
$\widehat{m}_{0}^{\text{rand}}$ on simulated data.
### 5.5 Real data analysis
Finally, we compare the performance of base and discrete estimators on three
different datasets. The first dataset consists of data provided by the
International Mice Phenotyping Consortium (IMPC) (Karp et al.,, 2017), which
coordinates studies on the genotype influence on mouse phenotype. This dataset
includes, for each of the $m=266952$ studied genes, the counts of normal and
abnormal phenotypes thus providing multiple two by two contingency tables,
which can be analysed using FETs. Then we analyze the methylation dataset for
cytosines of Arabidopsis in Lister et al. (2008) which is part of the
R-package fdrDiscreteNull of Chen and Doerge, (2015). This dataset contains
$m=3525$ counts for a biological entity under two different biological
conditions or treatments also analyzed using FETs. Finally, the third dataset,
provided by the Regulatory Agency in the United Kingdom, includes adverse drug
reactions due to medicines and healthcare products. It contains the number of
reported cases of amnesia as well as the total number of adverse events
reported for each of the $m=2446$ drugs in the database. For more details we
refer to Heller and Gur, (2011) and to the accompanying R-package discreteMTP
of Heller et al., (2012), which also contains the data. Heller and Gur, (2011)
investigate the association between reports of amnesia and suspected drugs by
performing for each drug (one-sided) FETs.
From the results in Table 1 we can see that taking discreteness into account
is always beneficial, regardless of the adjustment used. Depending on the type
of discreteness and the amount of signal contained in the data, adjusting for
discreteness can provide a great improvement in some cases. Indeed, as the
example of the IMPC data shows, base estimators may not be able to recognize
the presence of any alternatives. However, the discrete estimators clearly
suggest that a considerable amount of alternatives is present.
Table 1: $\pi_{0}$-estimates for base estimators and adjusted discrete estimators on three different datasets containing discrete data. | | Dataset
---|---|---
Adjustment | Estimator | IMPC | Arabidopsis | Pharmacovigilance
standard (none) | Storey | 1.26 | 0.67 | 1.79
| PC | 1.26 | 0.73 | 1.79
| Poly$(2,1/2)$ | 2.16 | 0.75 | 2.97
rescaled (du) | Storey | 0.63 | 0.59 | 1.05
| PC | 0.63 | 0.64 | 1.04
| Poly$(2,1/2)$ | 0.63 | 0.57 | 1.10
rescaled (mid) | Storey | 0.63 | 0.63 | 1.03
| PC | 0.63 | 0.64 | 1.05
| Poly$(2,1/2)$ | 0.63 | 0.58 | 1.11
randomized | Storey | 0.63 | 0.58 | 1.08
| PC | 0.63 | 0.64 | 1.06
| Poly$(2,1/2)$ | 0.63 | 0.56 | 1.14
## 6 Discussion
In this paper we introduce a unified class of $m_{0}$-estimators with
mathematical guarantees on plug-in FDR control in a classical setting. We also
describe some general approaches for adjusting $m_{0}$-estimators constructed
for continuous $p$-values to discrete $p$-values. While we show that these
results are useful both from a methodological viewpoint and for practical
purposes, there a numerous possibilities for further investigations, some of
which we describe now.
While we focus on FDR control in this paper, it is clear, that our new
estimators can also be used for FDR estimation, see Storey, (2002). For the
discrete estimators from Section 5 that are uniformly better than their
classical counterparts, this implies that the corresponding FDR estimators are
uniformly better es well.
In Section 4, we describe the performance of various polynomial estimators in
a one-sided Gaussian testing framework with the aim of illustrating the
flexibility of our Proposition 3.2 on plug-in FDR control. The numerical
results in Section 4.1 show that $\widehat{m}_{0}^{\text{Poly}}(2,1/2)$
performs uniformly better in terms of MSE than the other estimators in this
specific framework. In a second step, it might be interesting to study whether
optimal estimators can be derived, either within the whole class
$\mathcal{F}_{0}$ or perhaps within some sub-class like polynomial estimators.
This way, it may eventually be possible to obtain more efficient estimators in
practice, or at least give the user some guidance for choosing estimators from
the class $\mathcal{F}_{0}$.
We would also like to point out some connections to the work of Heesen and
Janssen, (2016), who split the unit interval into an estimation region on
which an estimator of $m_{0}$ is constructed and a rejection region on which
the BH procedure is run. Thus these estimators do not use all available
$p$-values, in contrast to our approach. Heesen and Janssen, (2016) derive a
general sufficient criterion, similar to Theorem 2.1, for finite sample plug-
in FDR control which they apply to Storey-type estimators and histogram-type
estimators (see MacDonald et al., (2019)). In contrast to our approach, the
transformation function $g$ applied to $p$-values need not be monotone,
however it is unclear whether e.g. smooth functions fit into this framework.
Their approach also accommodates “dynamization”, which allows data-dependent
tuning of parameters, see MacDonald et al., (2019). We may wonder if this
approach can be used for the estimators of our class, but as this question
exceeds the scope of this paper, we leave it for future research.
While we use polynomial estimators as simple examples that include both the
classical Storey and Pounds and Cheng estimators, other choices are
conceivable. Taking, for instance, certain kernel-type transformation
functions would lead to estimators that are advantageous from an asymptotic
viewpoint, see Neuvial, (2013). We leave this topic for future research.
In Section 5 we illustrate how information on the null distribution functions
of discrete $p$-values can be used to obtain more efficient
$m_{0}$-estimators. This information can be seen as a special case of
auxiliary covariates, for which it is well-known that their incorporation into
multiple testing procedures e.g. by weighting, can be highly beneficial (see
Ignatiadis et al., (2016); Durand, (2019)). We would like to mention that our
methods for $m_{0}$-estimation are not limited to the special case of discrete
$p$-values but should be able to accomodate these types of heterogeneity as
well. Thus, they might also be useful in the settings described above for
obtaining more efficient $m_{0}$-estimators.
While we assume independence of the $p$-values throughout this paper, it is
well-known that dependence may adversely affect the performance of
$m_{0}$-estimators or may require re-adjustments of tuning parameters like
$\lambda$ in $\widehat{m}_{0}^{\text{Storey}}$ (see e.g. Blanchard and
Roquain, (2009)). Thus, it might be interesting to investigate the behaviour
of our new estimators under various types of dependency.
Constructing multiple testing procedures for discrete data that provide finite
sample plug-in FDR control is challenging. In this paper we make some progress
by obtaining improved discrete estimators for $m_{0}$. While using these
discrete estimators in the plug-in BH procedure provide more power than using
classical estimators (based on uniformly distributed $p$-values under the
null), it is still not ideal because the discreteness is ignored in the
rejection stage of the procedure. Döhler et al., (2018) propose discrete
variants of the standard (i.e. non plug-in) BH procedure. They also sketch a
possible plug-in method based on combining this procedure with estimators of
$m_{0}$, but caution that it comes without mathematical guarantees. Thus, as
MacDonald et al., (2019) pointed out, it still remains an open problem to
develop procedures that integrate discreteness of the data in both the
estimation of $m_{0}$ and the rejection of $p$-values.
## Acknowledgements
This work is part of project DO 2463/1-1, funded by the Deutsche
Forschungsgemeinschaft. The authors thank Etienne Roquain for insightful
discussions and helpful suggestions which significantly improved the
manuscript.
## Appendix A Auxiliary definitions and results
In this appendix we recall some definitions and results of stochastic ordering
following the presentation in [SS], to which we also refer the reader for
further details. We also recall a well-known bound on the inverse moment of
the Binomial distribution.
###### Definition A.1 (Stochastic order).
Let $X$ and $Y$ be two random variables such that
$\displaystyle\mathbf{P}(X>x)\leq\mathbf{P}(Y>x)\quad\text{for all
}x\in(-\infty,\infty),$
Then $X$ is said to be smaller than $Y$ in the usual stochastic order denoted
by $X\leqslant_{\text{st}}Y$.
An equivalent characterization of the stochastic order is that
$X\leqslant_{\text{st}}Y$ $\Leftrightarrow$
$\mathbb{E}[g(X)]\leq\mathbb{E}[g(Y)]$, for all non-decreasing functions
$g:\mathbb{R}\rightarrow\mathbb{R}$ for which the expectations exist (see
(1.A.7) in [SS])).
###### Definition A.2 (Convex order).
Let $X$ and $Y$ be two random variables such that
$\displaystyle\mathbb{E}(\phi(X))\leq\mathbb{E}(\phi(Y))\quad\text{for all
convex functions }\phi:\mathbb{R}\rightarrow\mathbb{R},$
provided the expectations exist. Then $X$ is said to be smaller than $Y$ in
the convex order denoted as $X\leqslant_{\text{cx}}Y$.
The next results follows from the definition of convex ordering, see Chapter 3
of [SS].
###### Lemma A.3 (Theorem 3.A.24 in [SS]).
Let $X$ be a random variable with mean $\mathbb{E}X$. Denote the left (right)
endpoint of the support of $X$ by $l_{X}\left[u_{X}\right]$. Let $Z$ be a
random variable such that
$\mathbf{P}\left\\{Z=l_{X}\right\\}=\left(u_{X}-\mathbb{E}X\right)/\left(u_{X}-l_{X}\right)$
and $\mathbf{P}\left\\{Z=u_{X}\right\\}=(\mathbb{E}X-$
$\left.l_{X}\right)/\left(u_{X}-l_{X}\right)$. Then
$\mathbb{E}X\leqslant_{\text{cx}}X\leqslant_{\text{cx}}Z$
where $\mathbb{E}X$ denotes a random variable that takes on the value
$\mathbb{E}X$ with probability 1 (the left handside just restates Jensen’s
inequality).
###### Lemma A.4 (Theorem 3.A.44 in [SS]).
Let $X$ and $Y$ be two random variables with equal means, density functions
$f$ and $g$, distribution functions $F$ and $G$, and survival functions
$\bar{F}$ and $\bar{G}$, respectively. Denote by $S^{-}(a)$ the number of sign
changes for function $a$. Then $X\leq_{\mathrm{cx}}Y$ if any of the following
conditions hold:
$\displaystyle S^{-}(g-f)=2\mbox{ and the sign sequence is }+,-,+;$
$\displaystyle S^{-}(\bar{F}-\bar{G})=1\mbox{ and the sign sequence is }+,-;$
$\displaystyle S^{-}(G-F)=1\mbox{ and the sign sequence is }+,-.$
###### Proposition A.5 (Theorem 3.A.12 d) in [SS]).
Let $X_{1},X_{2},\ldots,X_{m}$ be a set of independent random variables and
let $Y_{1},Y_{2},\ldots,Y_{m}$ be another set of independent random variables.
If $X_{i}\leqslant_{\text{cx}}Y_{i}$ for $i=1,2,\ldots,m$, then
$\sum_{j=1}^{m}X_{j}\leqslant_{\text{cx}}\sum_{j=1}^{m}Y_{j}.$
That is, the convex order is closed under convolutions.
###### Lemma A.6 (Example 3.A.48 in [SS]).
Let $X$ and $Y$ be Bernoulli random variables with parameters $p$ and $q$,
respectively, with $0<p\leq q\leq 1$. Then
$\frac{X}{p}\geq_{\mbox{cx}}\frac{Y}{q}.$
###### Lemma A.7 (Inverse moment for the Binomial distribution).
Let $B_{1},\ldots,B_{k}\sim\mathbf{Bin}(1,q)$. Then
$\mathbb{E}[1/(1+\sum_{i=1}^{k}B_{i})]\leq 1/((k+1)q)$.
###### Proof.
See e.g. Benjamini et al., (2006). ∎
## Appendix B Complements to Section 4.1
In the context of Gaussian one-sided testing described in Section 4.1, let
$\widehat{m}_{0}(p_{1},\ldots,p_{m})=\frac{1}{\nu}\left(1+\sum_{i=1}^{m}g(p_{i})\right)\in\mathcal{F}_{0}$.
Define $X_{0}\sim g(p_{i})$ for $i\in\mathcal{H}_{0}$ and $X_{1}\sim g(p_{i})$
where $i\in\mathcal{H}_{1}$. Then we have
$\displaystyle\textnormal{Bias}(\widehat{m}_{0})$
$\displaystyle=\frac{1}{\nu}\mathbb{E}\left(1+\sum_{i\in\mathcal{H}_{1}}g(p_{i})\right)=(1+(m-m_{0})\cdot\mathbb{E}X_{1})/\nu,$
$\displaystyle\textnormal{Var}(\widehat{m}_{0})$
$\displaystyle=\frac{1}{\nu^{2}}\textnormal{Var}\left(\sum_{i\in\mathcal{H}_{0}}g(p_{i})+\sum_{i\in\mathcal{H}_{1}}g(p_{i})\right)=(m_{0}\cdot\textnormal{Var}(X_{0})+(m-m_{0})\cdot\textnormal{Var}(X_{1}))/\nu^{2},\qquad\text{with}$
$\displaystyle\textnormal{Var}(X_{0})$
$\displaystyle=\int_{0}^{1}g(u)^{2}du-\left[\int_{0}^{1}g(u)du\right]^{2},$
$\displaystyle\textnormal{Var}(X_{1})$
$\displaystyle=\int_{0}^{1}g(u)^{2}f_{1}(u)du-\left[\int_{0}^{1}g(u)f_{1}(u)du\right]^{2}.$
where $f_{1}(t)=\exp\left(-\mu\cdot\Phi^{-1}(t)-\mu^{2}/2\right)$ denotes the
density of the $p$-values under the alternative.
## Appendix C Complements to Section 4.2
Here we present some numerical results, comparing the performance of
$\widehat{m}_{0}^{\text{PC,new}}$ (see (10)) and
$\widehat{m}_{0}^{\text{PC,ZZD}}$ (see (17)) for $m=500$, where the correction
factors $C(500)=1.011709$ and $s(500)=98$ are taken from Table S1 in Zeisel et
al., (2011).
We first analyze the two estimators on simulated data in a one-sided Gaussian
testing setting where we observe realizations of independent rv’s
$X_{1},\dots,X_{m_{0}}\sim N(0,1)$ and $X_{m_{0}+1},\dots,X_{500}\sim
N(1.5,1)$ for $1000$ Monte-Carlo simulation runs and a varying range of
$m_{0}=50,100,\dots,450$. We obtain $500$ $p$-values by testing the null
hypotheses $H_{0,i}:\mu=0$ vs. the alternatives $H_{1,i}:\mu>0$ simultaneously
for all $i\in\\{1,\dots,500\\}$ and calculate
$\widehat{m}_{0}^{\text{PC,new}}$ and $\widehat{m}_{0}^{\text{PC,ZZD}}$ as
well as the number of rejections obtained from the plug-in BH procedure in (2)
with $\alpha=0.05$.
|
---|---
Figure 5: Box plots for the difference between
$\widehat{m}_{0}^{\text{PC,new}}$ and $\widehat{m}_{0}^{\text{PC,ZZD}}$ for
point estimation (left) and rejection numbers for the plug-in BH procedures
(right) against a range of true $m_{0}=50,100,\dots,450$.
Figure 5 shows that over a wide range of true $m_{0}$ values,
$\widehat{m}_{0}^{\text{PC,new}}$ and $\widehat{m}_{0}^{\text{PC,ZZD}}$ yield
comparable results both w.r.t. the point estimates and for the number of
rejections. In fact, $\widehat{m}_{0}^{\text{PC,new}}$ appears to be slightly
more efficient than $\widehat{m}_{0}^{\text{PC,ZZD}}$.
Another comparison can be obtained when we assume that the signal under the
alternative is strong and that most hypotheses are nulls. In this case we have
$2\sum_{i=1}^{m}p_{i}\approx 2\sum_{i\in\mathcal{H}_{0}}p_{i}=:S$ so that we
can use the Central Limit Theorem to quantify the probability that
$\widehat{m}_{0}^{\text{PC,new}}$ is more conservative than
$\widehat{m}_{0}^{\text{PC,ZZD}}$
$\displaystyle\mathbf{P}(\widehat{m}_{0}^{\text{PC,new}}>\widehat{m}_{0}^{\text{PC,ZZD}})$
$\displaystyle=\mathbf{P}(S>m\cdot
C(m)-2)\approx\overline{\Phi}\left(\sqrt{\frac{3}{m_{0}}}\cdot(m\cdot
C(m)-(m_{0}+2))\right).$
Figure 6 shows that this probability, for various values of the true $m_{0}$,
is quite small and even under the complete null ($m_{0}=500$) it is bounded by
$1/3$.
Figure 6: Approximate probabilities
$\mathbf{P}(\widehat{m}_{0}^{\text{PC,new}}>\widehat{m}_{0}^{\text{PC,ZZD}})$
for various values of true $m_{0}$, with fixed $m=500$.
## Appendix D Additional Figures for simulated data of Section 4
We provide additional results on simulated data in the Gaussian one-sided
testing setting described in Section 4.1, with $m=10000$ and $\mu=1.5$. Figure
7 displays estimation results for $\pi_{0}$ over $1000$ Monte-Carlo
replications. They are in line with the analytical comparisons of the MSE
provided in Figure 1. Alongside, we also provide results on power, defined as
the ratio of the number of true discoveries to the number of alternatives, for
the corresponding plug-in BH (abbreviated in ABH for adaptive BH) procedures
using each of the estimators, the raw BH and oracle plug-in BH (using the true
$m_{0}$). The procedures are run for a fixed level $\alpha=0.05$. The power
enhancement among the different plug-in estimators’ is not striking except
perhaps for very small values of $\pi_{0}$ were where we recover the same
performance ranking as in Figure 1. For larger values of $\pi_{0}$, the
differences in power is not perceptible anymore, every procedure behaves
poorly as there is less and less signal.
|
---|---
Figure 7: Estimation results (left panel) for
$\widehat{m}_{0}^{\text{Storey}}$, $\widehat{m}_{0}^{\text{PC,new}}$,
$\widehat{m}_{0}^{\text{Poly}}(1,1/2)$, and
$\widehat{m}_{0}^{\text{Poly}}(2,1/2)$, and power results (right panel) for
the associated plug-in BH procedures on simulated data.
## Appendix E Upper and lower bounds for the inverse moment of the uniform
sum distribution
The Pounds and Cheng estimator is closely related to the sum of independent
uniform random variables. This distribution plays a role in various contexts
and is also known as the Irwin-Hall distribution (for more details, see
Johnson et al., (1970)). As an auxiliary result, we give lower and upper
bounds for the inverse moment of this distribution.
###### Lemma E.1 (Inverse moments for Erlang distributions).
Let $E_{1},\ldots,E_{k}\sim\mathcal{E}(1)$ be independent exponentially
distributed random variables. Then $\mathbb{E}[1/\sum_{i=1}^{k}E_{i}]\leq
1/(k-1).$
###### Proof.
Since $X=\sum_{i=1}^{k}E_{i}$ is Gamma-distributed with shape $\alpha=k$ and
inverse scale parameter $\beta=1$ then $1/X$ is Inverse-gamma distributed with
mean $\beta/(\alpha-1)$, see Gelman et al., (2013). ∎
###### Proposition E.2.
(Inverse moment for sums of uniforms) For $k\geq 2$ let
$U_{1},U_{2},\ldots,U_{k}\sim\mathcal{U}[0,1]$ iid. Then we have
$\displaystyle\frac{2}{k}$
$\displaystyle\leq\mathbb{E}\left(\frac{1}{\sum_{i=1}^{k}U_{i}}\right)\leq\frac{2}{k-1}$
(27)
###### Proof.
Let $E_{1},E_{2},\ldots,E_{k}\sim\mathcal{E}(1)$ iid. From Theorems 3.A.24 and
3.A.46 in [SS] we have for $i=1,\ldots,,k$
$\displaystyle 1$
$\displaystyle\leqslant_{\text{cx}}2U_{i}\leqslant_{\text{cx}}E_{i}$ and since
the convex ordering is preserved under convolutions (see [SS] Theorem 3.A.12.)
we obtain $\displaystyle k$
$\displaystyle\leqslant_{\text{cx}}\sum_{i=1}^{k}2U_{i}\leqslant_{\text{cx}}\sum_{i=1}^{k}E_{i}.$
Together with the convexity of the mapping $x\mapsto 1/x$ on $(0,1)$ this
yields $\displaystyle\frac{1}{k}$
$\displaystyle\leq\mathbb{E}\left(\frac{1}{\sum_{i=1}^{k}2U_{i}}\right)\leq\mathbb{E}\left(\frac{1}{\sum_{i=1}^{k}E_{i}}\right)\leq\frac{1}{k-1},$
where the last inequality follows from Lemma E.1. ∎
## References
* Benjamini, (2010) Benjamini, Y. (2010). Simultaneous and selective inference: Current successes and future challenges. Biometrical Journal, 52(6):708–721.
* Benjamini and Hochberg, (1995) Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal statistical society: series B (Methodological), 57(1):289–300.
* Benjamini et al., (2006) Benjamini, Y., Krieger, A. M., and Yekutieli, D. (2006). Adaptive linear step-up procedures that control the false discovery rate. Biometrika, 93(3):491–507.
* Benjamini and Yekutieli, (2001) Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of statistics, pages 1165–1188.
* Berry and Armitage, (1995) Berry, G. and Armitage, P. (1995). Mid-p confidence intervals: a brief review. Journal of the Royal Statistical Society Series D: The Statistician, 44(4):417–423.
* Biswas and Chattopadhyay, (2020) Biswas, A. and Chattopadhyay, G. (2020). Modified estimator for the proportion of true null hypotheses under discrete setup with proven fdr control by the adaptive benjamini-hochberg procedure. arXiv preprint arXiv:2009.03803.
* Blanchard and Roquain, (2009) Blanchard, G. and Roquain, É. (2009). Adaptive false discovery rate control under independence and dependence. Journal of Machine Learning Research, 10(12).
* Chavant et al., (2011) Chavant, F., Favrelière, S., Lafay-Chebassier, C., Plazanet, C., and Pérault-Pochat, M.-C. (2011). Memory disorders associated with consumption of drugs: updating through a case/noncase study in the french pharmacovigilance database. British journal of clinical pharmacology, 72(6):898–904.
* Chen and Doerge, (2015) Chen, X. and Doerge, R. (2015). fdrdiscretenull: False discovery rate procedure under discrete null distributions. R package version, 1.
* Chen et al., (2018) Chen, X., Doerge, R. W., and Heyse, J. F. (2018). Multiple testing with discrete data: proportion of true null hypotheses and two adaptive fdr procedures. Biometrical Journal, 60(4):761–779.
* Dickhaus et al., (2012) Dickhaus, T., Straßburger, K., Schunk, D., Morcillo-Suarez, C., Illig, T., and Navarro, A. (2012). How to analyze many contingency tables simultaneously in genetic association studies. Statistical applications in genetics and molecular biology, 11(4).
* Ditzhaus and Janssen, (2019) Ditzhaus, M. and Janssen, A. (2019). Variability and stability of the false discovery proportion. Electronic Journal of Statistics, 13(1):882–910.
* Döhler et al., (2018) Döhler, S., Durand, G., and Roquain, E. (2018). New fdr bounds for discrete and heterogeneous tests. Electronic Journal of Statistics, 12(1):1867–1900.
* Durand, (2019) Durand, G. (2019). Adaptive p-value weighting with power optimality.
* Embrechts and Hofert, (2013) Embrechts, P. and Hofert, M. (2013). A note on generalized inverses. Mathematical Methods of Operations Research, 77(3):423–432.
* Gelman et al., (2013) Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian data analysis. Chapman and Hall/CRC Texts in Statistical Science Series. CRC, Boca Raton, Florida, third edition.
* Habiger, (2015) Habiger, J. D. (2015). Multiple test functions and adjusted p-values for test statistics with discrete distributions. Journal of Statistical Planning and Inference, 167:1–13.
* Habiger and Pena, (2011) Habiger, J. D. and Pena, E. A. (2011). Randomised p-values and nonparametric procedures in multiple testing. Journal of nonparametric statistics, 23(3):583–604.
* Heesen and Janssen, (2016) Heesen, P. and Janssen, A. (2016). Dynamic adaptive multiple tests with finite sample fdr control. Journal of Statistical Planning and Inference, 168:38–51.
* Heller and Gur, (2011) Heller, R. and Gur, H. (2011). False discovery rate controlling procedures for discrete tests. arXiv preprint arXiv:1112.4627, pages 00092–17.
* Heller et al., (2012) Heller, R., Gur, H., and Yaacoby, S. (2012). discretemtp: Multiple testing procedures for discrete test statistics. R package version 0.1-2.
* Ignatiadis et al., (2016) Ignatiadis, N., Klaus, B., Zaugg, J. B., and Huber, W. (2016). Data-driven hypothesis weighting increases detection power in genome-scale multiple testing. Nature methods, 13(7):577–580.
* Johnson et al., (1970) Johnson, N. L., Kotz, S., and Johnson, N. L. (1970). Continuous univariate distributions.
* Karp et al., (2017) Karp, N. A., Mason, J., Beaudet, A. L., Benjamini, Y., Bower, L., Braun, R. E., Brown, S. D., Chesler, E. J., Dickinson, M. E., Flenniken, A. M., et al. (2017). Prevalence of sexual dimorphism in mammalian phenotypic traits. Nature communications, 8(1):1–12.
* MacDonald et al., (2019) MacDonald, P. W., Liang, K., and Janssen, A. (2019). Dynamic adaptive procedures that control the false discovery rate.
* Muñoz-Fuentes et al., (2018) Muñoz-Fuentes, V., Cacheiro, P., Meehan, T. F., Aguilar-Pimentel, J. A., Brown, S. D., Flenniken, A. M., Flicek, P., Galli, A., Mashhadi, H. H., Hrabě de Angelis, M., et al. (2018). The international mouse phenotyping consortium (impc): a functional catalogue of the mammalian genome that informs conservation. Conservation genetics, 19:995–1005.
* Neuvial, (2013) Neuvial, P. (2013). Asymptotic results on adaptive false discovery rate controlling procedures based on kernel estimators. Journal of Machine Learning Research, 14(44):1423–1459.
* Pounds and Cheng, (2006) Pounds, S. and Cheng, C. (2006). Robust estimation of the false discovery rate. Bioinformatics, 22(16):1979–1987.
* Rubin-Delanchy et al., (2019) Rubin-Delanchy, P., Heard, N. A., and Lawson, D. J. (2019). Meta-analysis of mid-p-values: some new results based on the convex order. Journal of the American Statistical Association, 114(527):1105–1112.
* Sarkar, (2008) Sarkar, S. K. (2008). On methods controlling the false discovery rate. Sankhyā: The Indian Journal of Statistics, Series A (2008-), pages 135–168.
* Shaked and Shantikumar, (2007) Shaked, M. and Shantikumar, J. G. (2007). Stochastic orders. Springer.
* Storey, (2002) Storey, J. D. (2002). A direct approach to false discovery rates. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 64(3):479–498.
* Storey et al., (2004) Storey, J. D., Taylor, J. E., and Siegmund, D. (2004). Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66(1):187–205.
* Zeisel et al., (2011) Zeisel, A., Zuk, O., and Domany, E. (2011). Fdr control with adaptive procedures and fdr monotonicity. The Annals of applied statistics, pages 943–968.
|
# Logarithmic Smoothing for Pessimistic Off-Policy Evaluation, Selection and
Learning
Otmane Sakhi
Criteo AI Lab, Paris, France
<EMAIL_ADDRESS>
Imad Aouali
CREST, ENSAE
Criteo AI Lab, Paris, France
<EMAIL_ADDRESS>Pierre Alquier
ESSEC Business School, Singapore
<EMAIL_ADDRESS>Nicolas Chopin
CREST, ENSAE
<EMAIL_ADDRESS>
###### Abstract
This work investigates the offline formulation of the contextual bandit
problem, where the goal is to leverage past interactions collected under a
behavior policy to evaluate, select, and learn new, potentially better-
performing, policies. Motivated by critical applications, we move beyond point
estimators. Instead, we adopt the principle of _pessimism_ where we construct
upper bounds that assess a policy’s worst-case performance, enabling us to
confidently select and learn improved policies. Precisely, we introduce novel,
fully empirical concentration bounds for a broad class of importance weighting
risk estimators. These bounds are general enough to cover most existing
estimators and pave the way for the development of new ones. In particular,
our pursuit of the tightest bound within this class motivates a novel
estimator (LS), that _logarithmically smooths_ large importance weights. The
bound for LS is provably tighter than all its competitors, and naturally
results in improved policy selection and learning strategies. Extensive policy
evaluation, selection, and learning experiments highlight the versatility and
favorable performance of LS.
## 1 Introduction
In decision-making under uncertainty, offline contextual bandit [15, 46]
presents a practical framework for leveraging past interactions with an
environment to optimize future decisions. This comes into play when we possess
logged data summarizing an agent’s past interactions [9]. These interactions,
typically captured as context-action-reward tuples, hold valuable insights
into the underlying dynamics of the environment. Each tuple represents a
single round of interaction, where the agent observes a context (including
relevant features), takes an action according to its current policy, often
called _behavior policy_ , and receives a reward that depends on both the
observed context and the taken action. This framework is prevalent in
interactive systems like online advertising, music streaming, and video
recommendation. In online advertising, for instance, the user’s profile is the
context, the recommended product is the action, and the click-through rate
(CTR) is the expected reward. By learning from past interactions, the
recommender system tailors product suggestions to individual preferences,
maximizing engagement and ultimately, business success.
To optimize future decisions without requiring real-time deployments, this
framework presents us with three tasks: off-policy evaluation (OPE) [15], off-
policy selection (OPS) [31], and off-policy learning (OPL) [57]. OPE estimates
the risk: the _negative of expected reward_ that a _target policy_ would
achieve, essentially predicting its performance if deployed. OPS selects the
best-performing policy from a finite set of options, and OPL finds the optimal
policy within an infinite class of policies. In general, OPE is an
intermediary step for OPS and OPL since its primary goal is policy comparison.
A significant amount of research in OPE has centered around Inverse Propensity
Scoring (IPS) estimators [23, 15, 16, 17, 62, 18, 55, 56, 37, 31, 44]. These
estimators rely on importance weighting to address the discrepancy between the
target policy and the behavior policy. While unbiased under mild conditions,
IPS induces high variance. To mitigate this, regularization techniques have
been proposed for IPS [9, 37, 56, 5, 20] trading some bias for reduced
variance. However, these estimators can still deviate from the true risk,
undermining their reliability for decision-making, especially in critical
applications. In such scenarios, practitioners need estimates that cover the
true risk with high confidence. To address this, several approaches focused on
constructing either asymptotic [9, 48, 14] or finite sample [31, 20], high
probability, empirical upper bounds on the risk. These bounds evaluate the
performance of a policy in the worst-case scenario, adopting the principle of
pessimism [26].
If this principle is used in OPE, it is central in OPS and OPL, where
strategies are inspired by, or directly derived from, upper bounds on the risk
[57, 34, 31, 49, 5, 61, 20]. Examples for OPS include Kuzborskij et al. [31]
who employed an Efron-Stein bound for self-normalized IPS, or Gabbianelli et
al. [20] that based their analysis on an upper bound constructed with the
Implicit Exploration estimator. Focusing on OPL, Swaminathan and Joachims [57]
exploited the empirical Bernstein bound [35] alongside the Clipping estimator
to motivate sample variance penalization. This work was recently improved by
either modifying the penalization [61] or analyzing the problem from the PAC-
Bayesian lens [34]. The latter was further explored by Sakhi et al. [49],
Aouali et al. [5], Gabbianelli et al. [20] resulting in tight, tractable PAC-
Bayesian bounds that can be directly optimized.
Existing _pessimistic_ OPE, OPS, and OPL approaches often involve analyzing
the concentration properties of a pre-defined risk estimator. We propose a
different approach: deriving concentration bounds for a broad class of
regularized IPS estimators and then identifying the estimator with the best
concentration properties. This allows us to design a tailored estimator, named
Logarithmic Smoothing (LS), which achieves the tightest concentration
inequality. LS enjoys several desirable properties. It concentrates at a sub-
Gaussian rate and has a finite variance, and these benefits are achieved
without requiring the estimator itself to be bounded. Its concentration upper
bound allow us to evaluate the worst-case risk of any policy, enables us to
derive a simple OPS strategy that directly minimizes our estimator akin to
Gabbianelli et al. [20], and achieves state-of-the-art learning guarantees for
OPL when analyzed within the PAC-Bayesian framework akin to [34, 49, 5, 20].
This paper is structured as follows. Section 2 introduces the necessary
background. In Section 3, we provide unified risk bounds for a broad class of
regularized IPS estimators, for which LS enjoys the tightest upper bound. In
Section 4, we analyze LS for OPS and OPL, and we further extend the analysis
within the PAC-Bayesian framework. Extensive experiments in Section 5
highlight the favorable performance of LS, and Section 6 provides concluding
remarks.
## 2 Setting and background
Offline contextual bandit. Let $\mathcal{X}\subset\mathbb{R}^{d}$ be the
_context space_ , which is a compact subset of $\mathbb{R}^{d}$, and let
$\mathcal{A}=[K]$ be a finite _action set_. An agent’s actions are guided by a
_stochastic_ and _stationary_ policy $\pi\in\Pi$ within a policy space $\Pi$.
Given a context $x\in\mathcal{X}$, $\pi(\cdot|x)$ is a probability
distribution over the action set $\mathcal{A}$; $\pi(a|x)$ is the probability
that the agent selects action $a$ in context $x$. Then, an agent interacts
with a contextual bandit over $n$ rounds. In round $i\in[n]$, the agent
observes a context $x_{i}\sim\nu$ where $\nu$ is a distribution with support
$\mathcal{X}$. After this, the agent selects an action
$a_{i}\sim\pi_{0}(\cdot|x_{i})$, where $\pi_{0}$ is the _behavior policy_ of
the agent. Finally, the agent receives a stochastic cost $c_{i}\in[-1,0]$ that
depends on the observed context $x_{i}$ and the taken action $a_{i}$. This
cost $c_{i}$ is sampled from a cost distribution $p(\cdot|x_{i},a_{i})$. This
leads to $n$-sized logged data,
$\mathcal{D}_{n}=(x_{i},a_{i},c_{i})_{i\in[n]},$ where tuples
$(x_{i},a_{i},c_{i})$ for $i\in[n]$ are i.i.d. The expected cost of taking
action $a$ in context $x$ is $c(x,a)=\mathbb{E}_{c\sim
p(\cdot|x,a)}\left[c\right]$, and the costs are negative because they are
interpreted as the negative of rewards. The performance of a policy
$\pi\in\Pi$ is evaluated through its _risk_ , which aggregates the expected
costs $c(x,a)$ over all possible contexts $x\in\mathcal{X}$ and taken actions
$a\in\mathcal{A}$ by policy $\pi$, such as
$\displaystyle R(\pi)$
$\displaystyle=\mathbb{E}_{x\sim\nu,a\sim\pi(\cdot|x),c\sim p(\cdot\mid
x,a)}\left[c\right]=\mathbb{E}_{x\sim\nu,a\sim\pi(\cdot|x)}\left[c(x,a)\right]\,.$
(1)
The main goal is to use logged dataset $\mathcal{D}_{n}$ to enhance future
decision-making without necessitating live deployments. This often entails
three tasks: OPE, OPS, and OPL. First, OPE is concerned with constructing an
estimator $\hat{R}_{n}(\pi)$ of the risk $R(\pi)$ of a fixed _target policy_
$\pi$ and study its deviation, aspiring for $\hat{R}_{n}(\pi)$ to concentrate
well around $R(\pi)$. Second, OPS focuses on selecting the best performing
policy $\hat{\pi}_{n}^{\textsc{s}}$ from a _predefined_ and _finite_
collection of target policies $\pi_{1},\ldots,\pi_{m}$, effectively seeking to
determine $\operatorname*{argmin}_{k\in[m]}R(\pi_{k})$. Third, OPL aims to
find a policy $\hat{\pi}_{n}^{\textsc{l}}$ within the _potentially infinite
policy space $\Pi$_ that achieves the lowest risk, essentially aiming to find
$\operatorname*{argmin}_{\pi\in\Pi}R(\pi)$. In general, both OPS and OPL rely
on OPE’s initial estimation of the risk.
Regularized IPS. Our work focuses on the inverse propensity scoring (IPS)
estimator [23]. IPS approximates the risk of a policy $\pi$, $R(\pi)$, by
adjusting the contribution of each sample in logged data according to its
_importance weight (IW)_ , which is the ratio of the probability of an action
under the target policy $\pi$ to its probability under the behavior policy
$\pi_{0}$,
$\displaystyle\hat{R}_{n}(\pi)=\frac{1}{n}\sum_{i=1}^{n}w_{\pi}(x_{i},a_{i})c_{i}\,,$
(2)
where for any
$(x,a)\in\mathcal{X}\times\mathcal{A}\,,w_{\pi}(x,a)=\pi(a|x)/\pi_{0}(a|x)$
are the IWs. IPS is unbiased under the coverage assumption (see for example
Owen [38, Chapter 9]). However, it can suffer high variance, which tends to
scale linearly with IWs [59]. This issue becomes particularly pronounced when
there is a significant discrepancy between the target policy $\pi$ and the
behavior policy $\pi_{0}$. To mitigate this, a common strategy consists in
applying a regularization function
$h:[0,1]^{2}\times[-1,0]\rightarrow]-\infty,0]$ to $\pi(a|x)$, $\pi_{0}(a|x)$
and $c$. This function $h$ is designed to reduce the estimator’s variance at
the cost of introducing some bias. Formally, the function $h$ needs to satisfy
the condition (C1) defined as
$\displaystyle h\,\text{satisfies
{(C1)}}\iff\forall(p,q,c)\in[0,1]^{2}\times[-1,0],\quad pc/q\leq h(p,q,c)\leq
0.$
With such function $h$, the regularized IPS estimator reads
$\displaystyle\hat{R}^{h}_{n}(\pi)=\frac{1}{n}\sum_{i=1}^{n}h\left(\pi(a_{i}|x_{i}),\pi_{0}(a_{i}|x_{i}),c_{i}\right)=\frac{1}{n}\sum_{i=1}^{n}h_{i}\,,$
(3)
where $h_{i}=h\left(\pi(a_{i}|x_{i}),\pi_{0}(a_{i}|x_{i}),c_{i}\right)$. We
recover standard IPS in (2) when $h(p,q,c)=pc/q$. Numerous regularization
functions $h$ were studied in the literature. For example,
$\displaystyle
h(p,q,c)=\min(p/q,M)c\,,M\in\mathbb{R}^{+}\implies\text{Clipping
\cite[citep]{[\@@bibref{Number}{bottou2013counterfactual}{}{}]} }\,,$ (4)
$\displaystyle
h(p,q,c)=pc/q^{\alpha}\,,\alpha\in[0,1]\implies\text{Exponential Smoothing
\cite[citep]{[\@@bibref{Number}{aouali23a}{}{}]}}\,,$ $\displaystyle
h(p,q,c)=pc/(q+\gamma)\,,\gamma\geq 0\implies\text{Implicit Exploration
\cite[citep]{[\@@bibref{Number}{gabbianelli2023importance}{}{}]}}\,.$
Other IW regularizations include Harmonic [37] and Shrinkage [56]. By imposing
(C1) on $h$, we ensure that our estimator implements a form of pessimism [26]
that holds in expectation, as we have
$\displaystyle h\,\text{satisfies
{(C1)}}\implies\forall\pi\in\Pi,\,R(\pi)\leq\mathbb{E}[\hat{R}^{h}_{n}(\pi)]\,.$
(5)
This ensures regularized IPS covers the true risk in expectation, and it’s the
only requirement to derive our result: a family of high-probability bounds
that hold for regularized IPS with $h$ satisfying (C1).
## 3 Pessimistic off-policy evaluation
OPE often relies on point estimates of risk, which is insufficient in critical
applications, where it is also essential to measure the confidence level of
our estimation. Pessimistic OPE addresses this by using high-probability upper
bounds to assess the worst-case risk of policies [9, 14, 31]. This section
focuses on deriving tight upper bounds on the risk. We achieve this by
analyzing a general family of bounds applicable to regularized IPS in (3) with
any $h$ satisfying (C1).
### 3.1 Preliminaries and unified risk bounds
Let $\pi\in\Pi$, we define the empirical $\ell$-th moment of regularized IPS
$\hat{R}^{h}_{n}(\pi)$ as
$\displaystyle\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}h_{i}^{\ell}\,.$ (6)
Moreover, for $\lambda>0$, we define the function $\psi_{\lambda}:\to$ as
$\displaystyle\psi_{\lambda}(x)=\left(1-\exp(-\lambda x)\right)/\lambda\,.$
$\displaystyle\text{In particular, }\,\psi_{\lambda}(x)\leq x\,\text{ for any
}\,x\in\,.$ (7)
###### Proposition 1.
(Empirical moments risk bound). Let $\pi\in\Pi$, $L\geq 1$, $\delta\in(0,1]$
and $\lambda>0$. Then it holds with probability at least $1-\delta$ that
$\displaystyle
R(\pi)\leq\psi_{\lambda}\Big{(}\hat{R}^{h}_{n}(\pi)+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}\,,$ (8)
where $\psi_{\lambda}$ and $\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)$ are defined
in (7) and (6), respectively, and recall that $\psi_{\lambda}(x)\leq x$.
We provide detailed proof in Section F.1, leveraging Chernoff bounds with a
careful analysis of the moment-generating function. This leads to the first
empirical, high-order moment bound for offline contextual bandits, offering
several advantages. First, the bound applies to any regularization function
$h$ that meets the mild condition (C1). This allows us to design a tailored
regularization function $h$ that minimizes the bound. Second, this bound does
not assume the existence of theoretical moments since it only incorporates
empirical ones. Third, it is fully empirical and tractable, enabling efficient
implementation of pessimism. Finally, the value of $L$ controls the number of
moments used, allowing us to balance bound tightness and computational cost.
Precisely, for sufficiently small values of $\lambda$, higher values of $L$
lead to tighter bounds. However, this can come at the cost of increased
computational complexity for evaluating the bound. This is stated formally as
follows.
###### Proposition 2 (Impact of $L$ on the bound’s tightness).
Let $\pi\in\Pi$, $\delta\in(0,1]$, $\lambda>0$, and $L\geq 1$. Let
$U^{\lambda,h}_{L}(\pi)=\psi_{\lambda}\big{(}\hat{R}^{h}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)\big{)}$
be the bound in Equation 8. Then,
$\displaystyle\lambda\leq\min_{i\in[n]}\left\\{\frac{2L+2}{(2L+1)|h_{i}|}\right\\}\implies
U^{\lambda,h}_{L+1}(\pi)\leq U^{\lambda,h}_{L}(\pi)\,.$ (9)
From (9), the bound in Equation 8, $U^{\lambda,h}_{L}(\pi)$, becomes a
decreasing function of $L$ when $\lambda\leq\min_{i\in[n]}(1/|h_{i}|)$,
suggesting that for sufficiently small $\lambda$, the tightest bound is
achieved by taking $L\to\infty$. That said, we focus particularly on two
instances, $L=1$ resulting in an empirical second-moment bound, and
$L\rightarrow\infty$ which yields a tight bound motivating our novel
estimator.
### 3.2 Global clipping
###### Corollary 3 (Empirical second-moment risk bound with $L=1$).
Let $\pi\in\Pi$, $\delta\in(0,1]$ and $\lambda>0$. Then it holds with
probability at least $1-\delta$ that
$\displaystyle
R(\pi)\leq\psi_{\lambda}\left(\hat{R}^{h}_{n}(\pi)+\frac{\lambda}{2}\hat{\mathcal{M}}^{h,2}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\right)\,.$ (10)
This is a direct consequence of Equation 8 when $L=1$. The bound holds for any
$h$ satisfying (C1). Thus we search for a function $h_{*,1}$ that minimizes
bound in (10). This function $h_{*,1}$ writes
$\displaystyle h_{*,1}(p,q,c)=-\min(p|c|/q,1/\lambda)\,.$ (11)
In particular, if we assume that costs are binary, $c\in\\{-1,0\\}$, then
$h_{*,1}$ corresponds to Clipping in (4) with parameter $M=1/\lambda$. This is
because $-\min(|c|p/q,1/\lambda)=\min\left(p/q,\frac{1}{\lambda}\right)c\,$
when $c$ is binary. This motivates the widely used Clipping estimator [9].
However, this also suggests that the standard way of clipping (as in (4)) is
only optimal111Here, optimality of a function $h$ is defined with respect to
our bound with $L=1$ (Corollary 3). for binary costs. In general, the cost
should also be clipped (as in (11)). Finally, with a suitable choice of
$\lambda=\mathcal{O}(1/\sqrt{n})$, our bound in Corollary 3, using clipping
(i.e., $h=h_{*,1}$), outperforms the existing empirical Bernstein bound [57],
which was specifically derived for clipping. Therefore, Corollary 3 not only
applies to a broad range of regularization functions $h$, but also provides
tighter concentration guarantees than specialized bounds. Section F.3 gives
the the proof to find $h_{*,1}$ and formal comparisons are provided in Section
E.2.1.
### 3.3 Logarithmic smoothing
###### Corollary 4 (Empirical infinite-moment bound with $L\to\infty$).
Let $\pi\in\Pi$, $\delta\in(0,1]$ and $\lambda>0$. Then it holds with
probability at least $1-\delta$ that
$\displaystyle
R(\pi)\leq\psi_{\lambda}\left(-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)+\frac{\ln(1/\delta)}{\lambda n}\right)\,.$ (12)
Setting $L\to\infty$ in Equation 8 results in the bound in Corollary 4, which
has different properties than Corollary 3. This result can also be derived by
applying [1, Lemma 1.3]. In Section F.5, we prove that the function
$h_{*,\infty}$ that minimizes this bound is $h_{*,\infty}(p,q,c)=pc/q$. This
corresponds to the standard (non-regularized) IPS in (2). This differs from
the $L=1$ bound in Corollary 3 that favored clipping. This shows the impact of
the moment order $L$ on the optimal function $h$. Applying the bound in
Corollary 4 with the optimal $h_{*,\infty}$ leads to the following empirical
upper bound of $R(\pi)$
$\displaystyle
R(\pi)\leq\psi_{\lambda}\left(\hat{R}^{\lambda}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\right)\,.$ (13)
While we set $h_{*,\infty}(p,q,c)=pc/q$ (no initial IW regularization), a
novel regularized IPS (satisfies (C1)), called Logarithmic Smoothing (LS),
still emerges as
$\displaystyle\hat{R}^{\lambda}_{n}(\pi)=-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
w_{\pi}(x_{i},a_{i})c_{i}\right)\,.$ (14)
The LS estimator in (14) is defined for any non-negative $\lambda\geq 0$, and
its bound in (13) holds for any positive $\lambda>0$. In particular,
$\lambda=0$ recovers the standard (non-regularized) IPS estimator in (2) and
$\lambda>0$ introduces a bias-variance trade-off by smoothing logarithmically
the IWs. This estimator can be interpreted as a soft, differentiable variant
of clipping with parameter $1/\lambda$. Precisely, once the IWs are small
compared to $1/\lambda$, we recover the IPS estimator. Even if the LS
estimator is not bounded, $\lambda>0$ ensures a finite variance and sub-
Gaussianity of LS (Section E.1 discusses the properties of this estimator). We
focus in this section on the tightness of the resulting bound in (13). Recall
that when $\lambda$ is small enough, the bound is decreasing in $L$
(Proposition 2). Thus the bound with $L\to\infty$ is smaller than the bound
with $L=1$. In fact, this condition on $\lambda$ is not necessary when
comparing the bounds evaluated at their optimal regularization function $h$,
as stated in the following proposition.
###### Proposition 5.
Let $U^{\lambda}_{L}(\pi)=\min_{h}U^{\lambda,h}_{L}(\pi)$ for any $\pi\in\Pi$.
Then, for any $\lambda>0$, it holds that for any $L\geq
1\,,\,U^{\lambda}_{L}(\pi)\leq U^{\lambda}_{1}(\pi)$. In particular, for any
$\lambda>0$, it holds that $U^{\lambda}_{\infty}(\pi)\leq
U^{\lambda}_{1}(\pi)$.
Section F.6 proves this result. Note that the bound of LS in (13) corresponds
to $U^{\lambda}_{\infty}(\pi)$. Thus Proposition 5 shows that, irrespective of
the value of $\lambda$, bound of LS is tighter than that obtained with the
optimal $h$ when $L=1$ (i.e, the bound in Corollary 3 evaluated at the
clipping function $h_{*,1}$). It is also provably tighter than existing bounds
in the literature (Section E.2). That said, we use it to derive our
pessimistic OPS and OPL strategies.
## 4 Off-policy selection and learning
### 4.1 Off-policy selection
Let $\Pi_{\textsc{s}}=\\{\pi_{1},...,\pi_{m}\\}$ be a finite set of policies.
In OPS, the goal is to find $\pi_{*}^{\textsc{s}}\in\Pi_{\textsc{s}}$ that
satisfies
$\displaystyle\pi_{*}^{\textsc{s}}=\operatorname*{argmin}_{\pi\in\Pi_{\textsc{s}}}R(\pi)=\operatorname*{argmin}_{k\in[m]}R(\pi_{k})\,.$
(15)
As we do not have access to the true risk, we use a data-driven selection
strategy that guarantees the identification of policies of performance close
to that of $\pi_{*}^{\textsc{s}}$. Precisely, for $\lambda>0$, we search for
$\displaystyle\hat{\pi}_{n}^{\textsc{s}}=\operatorname*{argmin}_{\pi\in\Pi_{\textsc{s}}}\hat{R}^{\lambda}_{n}(\pi)=\operatorname*{argmin}_{k\in[m]}\hat{R}^{\lambda}_{n}(\pi_{k})\,.$
(16)
To derive our strategy in (16), we minimize the bound of LS in (13), employing
pessimism [26]. Fortunately, in our case, this boils down to minimizing
$\hat{R}^{\lambda}_{n}(\pi)$, since the other terms in the bound are
independent of the target policy $\pi$. This allows us to avoid computing
complex statistics [57, 31] and does not require access to the behavior policy
$\pi_{0}$. As we show next, it also ensures low suboptimality.
###### Proposition 6.
(Suboptimality of our selection strategy in (16)).Let $\lambda>0$ and
$\delta\in(0,1]$. Then, it holds with probability at least $1-\delta$ that
$\displaystyle 0\leq
R(\hat{\pi}_{n}^{\textsc{s}})-R(\pi_{*}^{\textsc{s}})\leq\lambda\mathcal{S}_{\lambda}(\pi_{*}^{\textsc{s}})+\frac{2\ln(2|\Pi_{\textsc{s}}|/\delta)}{\lambda
n}\,,$ (17)
where
$\mathcal{S}_{\lambda}(\pi)=\mathbb{E}\left[(w_{\pi}(x,a)c)^{2}/\left(1-\lambda
w_{\pi}(x,a)c\right)\right]$, $\pi_{*}^{\textsc{s}}$ and
$\hat{\pi}_{n}^{\textsc{s}}$ are defined in (15) and (16).
$\mathcal{S}_{\lambda}(\pi)$ measures the discrepancy between $\pi$ and
$\pi_{0}$. As for the IX selection strategy [20], our derived suboptimality
bound only requires coverage of the optimal actions (support of the optimal
policy $\pi^{s}_{*}$), and improves on IX suboptimality, matching the minimax
suboptimality lower bound of pessimistic methods [33, 26, 27]. Section G.1
provides a proof of this suboptimality bound, and we discuss in Section E.3
how this suboptimality improves upon existing strategies. By selecting
$\lambda^{s}_{n}=\sqrt{2\ln(2|\Pi_{\textsc{s}}|/\delta)/n}$ for LS, we achieve
a suboptimality scaling of $\mathcal{O}(1/\sqrt{n})$,
$\displaystyle 0\leq
R(\hat{\pi}_{n}^{\textsc{s}})-R(\pi_{*}^{\textsc{s}})\leq\left(1+\mathcal{S}_{\lambda^{s}_{n}}(\pi^{\textsc{s}}_{*})\right)\sqrt{2\ln(2|\Pi_{\textsc{s}}|/\delta)/n},$
(18)
which ensures finding the optimal policy with sufficient samples.
Additionally, the multiplicative constant is smaller when $\pi_{0}$ is close
to $\pi_{*}^{\textsc{s}}$, confirming the known observation that it is easier
to identify the best policy if it is similar to the behavior policy $\pi_{0}$.
### 4.2 Off-policy learning
Similar to how we extended the evaluation bound in Corollary 4 (which applies
to a single fixed target policy) to OPS (where it applies to a finite set of
target policies), we can further derive bounds for an infinite policy class
$\Pi$, enabling OPL. Several approaches have been proposed in previous work,
primarily based on replacing the finite union bound over policies with more
sophisticated uniform-convergence arguments. This was used by [57], which
derived a variance-sensitive bound scaling with the covering number [63].
Since these approaches incorporate a complexity term that depends only on the
policy class, the resulting pessimistic learning strategy (which minimizes the
upper bound) would be similar to the selection strategy adopted earlier,
leading, for a fixed $\lambda$, to
$\displaystyle\hat{\pi}_{n}^{\textsc{l}}=\operatorname*{argmin}_{\pi\in\Pi}\hat{R}^{\lambda}_{n}(\pi)+\frac{\mathcal{C}(\Pi)}{\lambda
n}=\operatorname*{argmin}_{\pi\in\Pi}\hat{R}^{\lambda}_{n}(\pi).$ (19)
where $\mathcal{C}(\Pi)$ is a complexity measure [63]. This learning strategy
is straightforward because it involves a smooth estimator that can be
optimized using first-order methods and does not require second-order
statistics. However, analyzing this approach is more challenging because the
complexity measure $\mathcal{C}(\Pi)$ varies depending on the policy class
considered, is often intractable [49] and can only be upper bounded with
problem dependent constants [27].
Instead of the method described above, we derive PAC-Bayesian generalization
bounds [36, 10] that apply to arbitrary policy classes. This framework has
been shown to provide strong performance guarantees for OPL in practical
scenarios [49, 5]. The PAC-Bayesian framework analyzes the performance of
policies by viewing them as randomized predictors [34]. Specifically, let
$\mathcal{F}(\Theta)=\\{f_{\theta}:\mathcal{X}\rightarrow[K],\theta\in\Theta\\}$
be a set of parameterized predictors that associate the context $x$ with the
action $f_{\theta}(x)\in[K]$. Let $\mathcal{P}(\Theta)$ be the set of all
probability distributions on $\Theta$. Each distribution
$Q\in\mathcal{P}(\Theta)$ defines a policy $\pi_{Q}$ by setting the
probability of action $a$ given context $x$ as the probability that a random
predictor $f_{\theta}\sim Q$ maps $x$ to action $a$, that is,
$\textstyle\pi_{Q}(a|x)=\mathbb{E}_{\theta\sim
Q}\left[\mathbbm{1}\left[f_{\theta}(x)=a\right]\right]\,,$
$\textstyle\forall(x,a)\in\mathcal{X}\times\mathcal{A}\,.$ (20)
This characterization is not restrictive as any policy can be represented in
this form [49]. Deriving PAC-Bayesian generalization bounds with this policy
definition requires the regularized IPS to be linear in the target policy
$\pi$ [34, 5, 20]. Our estimator LS in (14) is non-linear in $\pi$. Therefore,
for this PAC-Bayesian analysis, we introduce a linearized variant of LS,
called LS-LIN, and defined as
$\displaystyle\hat{R}^{\lambda\textsc{-lin}}_{n}(\pi)=-\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{\lambda}\log\left(1-\frac{\lambda
c_{i}}{\pi_{0}(a_{i}|x_{i})}\right),$ (21)
which smooths the impact of the behavior propensity $\pi_{0}$ instead of the
IWs $\pi/\pi_{0}$. We provide in the following a core result of this section,
the PAC-Bayesian bound that defines our learning strategy.
###### Proposition 7.
(PAC-Bayes learning bound for $\hat{R}^{\lambda{\textsc{-lin}}}_{n})$. Given a
prior $P\in\mathcal{P}(\Theta)$, $\delta\in(0,1]$ and $\lambda>0$, the
following holds with probability at least $1-\delta$:
$\displaystyle\forall Q\in\mathcal{P}(\Theta),\quad
R(\pi_{Q})\leq\psi_{\lambda}\left(\hat{R}^{\lambda{\textsc{-lin}}}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}\right)\,,$ (22)
where $\mathcal{KL}(Q||P)$ is the Kullback-Leibler divergence from $P$ to $Q$.
PAC-Bayes bounds hold uniformly for all distributions
$Q\in\mathcal{P}(\Theta)$ and replace the complexity measure
$\mathcal{C}(\Pi)$ with the divergence $\mathcal{KL}(Q||P)$ from a reference
_prior_ distribution $P$. Extensive research focuses on identifying the best
strategies for choosing this prior $P$ [39]. While these bounds hold for any
fixed prior $P$, in practice, it is typically set to the distribution inducing
the behavior policy $\pi_{0}$, meaning $P$ satisfies $\pi_{0}=\pi_{P}$. This
leads to an intuitive learning principle: by minimizing the upper bound, we
seek policies with good empirical risk that do not deviate significantly from
$\pi_{0}$.
Our bound can also be obtained using the truncation method from Alquier [1,
Corollary 2.5]. This bound surpasses the already tight PAC-Bayesian bounds
derived for Clipping [49], Exponential Smoothing [5], and IX [20], resulting
in the tightest known generalization bound in OPL. Section G.2 gives formal
proof of this bound and comparisons with existing PAC-Bayesian bounds can be
found in Appendix E.4. For a fixed $\lambda$ and a fixed prior $P$, we derive
a learning strategy that minimizes the upper bound for a subset
$\mathcal{L}(\Theta)\subseteq\mathcal{P}(\Theta)$ of distributions, seeking
$\displaystyle
Q_{n}=\operatorname*{argmin}_{Q\in\mathcal{L}(\Theta)}\left\\{\hat{R}^{\lambda\textsc{-lin}}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)}{\lambda
n}\right\\}\,,\quad\text{and setting
}\hat{\pi}_{n}^{\textsc{l}}=\pi_{Q_{n}}\,.$ (23)
(23) is tractable and can be efficiently optimized for various policy classes
[49, 5]. Below, we analyze its suboptimality compared to the best policy in
the chosen class,
$\pi_{Q^{*}}=\operatorname*{argmin}_{Q\in\mathcal{L}(\Theta)}R(\pi_{Q})$.
###### Proposition 8.
(Suboptimality of the learning strategy in (23)). Let $\lambda>0$,
$P\in\mathcal{L}(\Theta)$ and $\delta\in(0,1]$. Then, it holds with
probability at least $1-\delta$ that
$\displaystyle 0\leq
R(\hat{\pi}_{n}^{\textsc{l}})-R(\pi_{Q^{*}})\leq\lambda\mathcal{S}^{{\textsc{lin}}}_{\lambda}(\pi_{Q^{*}})+\frac{2\left(\mathcal{KL}(Q^{*}||P)+\ln(2/\delta)\right)}{\lambda
n},$ (24)
where
$\mathcal{S}^{{\textsc{lin}}}_{\lambda}(\pi)=\mathbb{E}\left[\pi(a|x)c^{2}/(\pi^{2}_{0}(a|x)-\lambda\pi_{0}(a|x)c)\right]$
and $\hat{\pi}_{n}^{\textsc{l}}$ is defined in (23).
Our suboptimality bound only requires coverage of the optimal actions (support
of the optimal policy $\pi_{Q_{*}}$). This bound matches the minimax
suboptimality lower bound of pessimistic learning with deterministic policies
[27]. Section G.3 provides a proof of Proposition 8, while Section E.5
discusses the suboptimality bound further and proves that it is better than
the IX learning strategy of [20, Section 5]. Setting
$\lambda^{l}_{n}=2/\sqrt{n}$ guarantees us a suboptimality that scales with
$\mathcal{O}(1/\sqrt{n})$ as
$\textstyle 0\leq
R(\hat{\pi}_{n}^{\textsc{l}})-R(\pi_{Q^{*}})\leq(2\mathcal{S}^{{\textsc{lin}}}_{\lambda^{l}_{n}}(\pi_{Q^{*}})+\mathcal{KL}(Q^{*}||P)+\ln(2/\delta))/{\sqrt{n}}.$
By setting the reference $P$ to the distribution inducing $\pi_{0}$, we find
that the learning suboptimality is reduced when the behavior policy $\pi_{0}$
is close to the optimal policy $\pi_{Q^{*}}$. This is similar to the
suboptimality for our selection strategy. The suboptimality upper bound
reflects a common intuition in the OPL literature: pessimistic learning
algorithms converge faster when $\pi_{0}$ is close to $\pi_{Q^{*}}$.
## 5 Experiments
Our experimental setup follows the classical multiclass to bandit conversion
used in most prior studies [17, 57]. We consider a contextual bandit problem
such that for every context $x$, its true label in the classification dataset
is denoted by $\rho(x)\in[K]$, and represents the action with the highest
average reward. The reward $r$ for playing action $a$ for context $x$ is
modelled as Bernoulli with probability
$p_{x}=\epsilon+\mathbbm{1}\left[a=\rho(x)\right](1-2\epsilon)$, with
$\epsilon$ a noise parameter. This ensures an average reward of $1-\epsilon$
for the optimal action and $\epsilon$ for all other actions. This procedure
helps us build a logged bandit feedback dataset of the form
$\\{x_{i},a_{i},c_{i}\\}_{i\in[n]}$, where $c_{i}=-r_{i}$ is the corresponding
cost.
Table 1: Bound’s tightness $(\lvert U(\pi)/R(\pi)-1\rvert)$ with varying number of samples of the kropt dataset. Number of samples | SN-ES | cIPS-EB | IX | cIPS-L=1 (Ours) | LS (Ours)
---|---|---|---|---|---
$2^{8}$ | 1.000 | 0.917 | 0.373 | 0.364 | 0.362
$2^{9}$ | 1.000 | 0.732 | 0.257 | 0.289 | 0.236
$2^{10}$ | 0.794 | 0.554 | 0.226 | 0.240 | 0.213
$2^{11}$ | 0.649 | 0.441 | 0.171 | 0.197 | 0.159
$2^{12}$ | 0.472 | 0.327 | 0.126 | 0.147 | 0.117
$2^{13}$ | 0.374 | 0.204 | 0.062 | 0.077 | 0.054
$2^{14}$ | 0.257 | 0.138 | 0.041 | 0.049 | 0.035
### 5.1 Off-policy evaluation and selection experiments
For both evaluation and selection, we adopt the same experimental design as
[31] to facilitate the comparison. We consider exponential target policies
$\pi(a|x)\propto\exp(\frac{1}{\tau}f(a,x))$, with $\tau$ a temperature
controlling the policy’s entropy and $f(a,x)$ the score of the item $a$ for
the context $x$. We use this to define ideal policies as
$\pi^{\texttt{ideal}}(a|x)\propto\exp(\frac{1}{\tau}\mathbb{I}\\{\rho(x)=a\\})$,
and also create faulty, mismatching policies for which the peak is shifted to
another, wrong action for a set of faulty actions $F\subset[K]$. To recreate
real world scenarios, we also consider policies directly learned from logged
bandit feedback, of the form
$\pi_{\theta^{\texttt{IPS}}}(a|x)\propto\exp(\frac{1}{\tau}x^{t}\theta_{a}^{\texttt{IPS}})$
and
$\pi_{\theta^{\texttt{SN}}}(a|x)\propto\exp(\frac{1}{\tau}x^{t}\theta_{a}^{\texttt{SN}})$,
with their parameters learned by respectively minimizing the IPS [23] and SN
[58] empirical risks. More details on the definition of the different policies
are given in Appendix H. Finally, 11 real multiclass classification datasets
are chosen from the UCI ML Repository [7] (See Table 3 in Section H.1.1) with
various number of samples, dimensions and action space sizes to conduct our
experiments.
(OPE) Tightness of the bounds. Evaluating the worst case performance of a
policy is done through evaluating risk upper bounds [9, 31]. This means that a
better evaluation will solely depend on the tightness of the bounds used. To
this end, given a policy $\pi$, we are interested in bounds with a small
relative radius $\lvert U(\pi)/R(\pi)-1\rvert$. We compare our newly derived
bounds (cIPS-L=1 for $U^{\lambda}_{1}$ and LS for $U^{\lambda}_{\infty}$ both
with $\lambda=1/\sqrt{n}$) to empirical evaluation bounds of the literature:
SN-ES: the Efron Stein bound for Self Normalized IPS [31], cIPS-EB: Empirical
Bernstein for Clipping [57] and the recent IX: Implicit Exploration bound
[20]. The first experiment uses the kropt dataset with $\epsilon=0.2$,
collects bandit feedback with faulty behavior policy (with $\tau=0.25$) to
evaluate an ideal policy ($\tau=0.1$), and explores how the relative radiuses
of the considered bounds shrink while varying the number of datapoints. Table
1 compiles the results of the experiments and suggest that the log smoothing
bound is tighter than its competitors no matter the size of the feedback
collected. The second experiments uses all 11 datasets, with different
behavior policies ($\tau_{0}\in\\{0.2,0.25,0.3\\}$) and different noise levels
($\epsilon\in\\{0.,0.1,0.2\\}$) to evaluate ideal policies with different
temperatures ($\tau\in\\{0.1,0.3,0.5\\}$), defining $\sim 300$ different
scenarios to validate our findings. We plot in Figure 1 the cumulative
distribution of the relative radius of the considered bounds. We observe that
while cIPS-L=1 and IX can be comparable, the LS bound is tighter than all its
competitors. We also provide detailed results in Section H.1.2 that further
confirm the superiority of the LS bound.
(OPS) Find the best, avoid the worst policy. Policy selection aims at
identifying the best policy among a set of finite candidates. In practice, we
are interested in finding policies that improve on $\pi_{0}$ and avoid
policies that perform worse than $\pi_{0}$. To replicate real world scenarios,
we design an experiment where $\pi_{0}$ is a faulty policy ($\tau=0.2$), that
collects noisy ($\epsilon=0.2$) interaction data, some of which is used to
learn $\pi_{\theta^{\texttt{IPS}}},\pi_{\theta^{\texttt{SN}}}$, and that we
add to our discrete set of policies
$\Pi_{k=4}=\\{\pi_{0},\pi^{\texttt{ideal}},\pi_{\theta^{\texttt{IPS}}},\pi_{\theta^{\texttt{SN}}}\\}$.
The goal is to measure the ability of our selection strategies to choose from
$\Pi_{k=4}$, better performing policies than $\pi_{0}$. We thus define three
possible outcomes: a strategy can select worse performing policies, better
performing or the best policy. Our goal in these experiments is to empirically
validate the pitfalls of point estimators while confirming the benefits of
using the pessimism principle. To this end, we compare _pessimistic_ selection
strategies to policy selection using the classical point estimators IPS [23]
and SN [58]. The comparison is conducted on the 11 UCI datasets with 10
different seeds resulting in 110 scenarios. We plot in Figure 1 the percentage
of time each method selected the best policy, a better or a worse policy than
$\pi_{0}$. While risk estimators can identify the best policy, they are
unreliable as they can choose worse performing policies than $\pi_{0}$, a
catastrophic outcome in critical applications. Pessimistic selection is more
conservative, as it avoids poor performing policies completely and empirically
confirms that tighter upper bounds result in better selection strategies: LS
upper bound is less conservative and finds best policies the most (comparable
to SN) while never selecting poor performing policies. Fine grained results
(for each dataset) can be found in Section H.1.3.
Figure 1: Results for OPE and OPS experiments.
### 5.2 Off-policy learning experiments
We follow the successful off policy learning paradigm based on directly
minimizing PAC-Bayesian risk generalization bounds [49, 5] as it comes with
guarantees of improvement and avoids hyper-parameter tuning. For comparable
results, we use the same 4 datasets (described in Section H.2, Table 7) as in
[49, 5] and adopt the LGP: Linear Gaussian Policies [49] as our class of
parametrized policies. For each dataset, we use behavior policies trained on a
small fraction of the data in a supervised fashion, combined with different
inverse temperature parameters $\alpha\in\\{0.1,0.3,0.5,0.7,1.\\}$ to cover
cases of diffused and peaked behavior policies. These policies generate for 10
different seeds, 10 logged bandit feedback datasets resulting in 200 different
scenarios to test our learning approaches. In the PAC-Bayesian OPL paradigm,
we minimize the empirical upper bounds $UB_{n}$ directly and obtain the
learned policy as the bound’s minimizer $\hat{\pi}_{UB_{n}}$ (as in (23)).
With $\hat{\pi}_{UB_{n}}$ obtained, we are interested in two quantities: The
guaranteed risk $\mathcal{GR}_{UB_{n}}=UB_{n}(\hat{\pi}_{UB_{n}})$ by the
bound. It is the value of the bound $UB_{n}$ at its minimizer
$\hat{\pi}_{UB_{n}}$. This quantity reflects the worst case performance of the
learned policy. A lower value implies stronger guarantees on the policy’s
performance. We are also interested in the true risk of the minimizer of the
bound $R(\hat{\pi}_{UB_{n}})$ as it translates the performance of the obtained
policy acting on unseen data. As this learning paradigm is based on optimizing
tractable, generalization bounds, we only compare our approach to methods that
provide them. Precisely, we compare our LS-LIN learning strategy in (23) to
strategies based on minimizing off-policy PAC Bayesian bounds from the
literature: clipped IPS (cIPS) and Control Variate clipped IPS (cvcIPS) in
[49], Exponential Smoothing (ES) in [5] and Implicit Exploration (IX) in [20].
The results are summarized in Table 2 where we compute:
$\textstyle
rI(x)=(R(\pi_{0})-x)/(R(\pi_{0})-R(\pi^{*}))=(R(\pi_{0})-x)/(R(\pi_{0})+1)\,,$
the improvement of $R(\pi_{0})$ achieved by minimizing the different bounds in
terms of $x\in\\{\mathcal{GR},R\\}$, relative to an ideal improvement. This
metric helps us normalize the results, and we report its average over 200
different scenarios, with results in bold being significantly better. More
detailed results can be found in Section H.2.4. We observe that the LS-LIN
PAC-Bayesian bound improves substantially on its competitors in terms of the
guaranteed risk, and also obtains the best performing policies (on par with
the IX PAC-Bayesian bound).
Table 2: OPL: Relative Improvement of $\mathcal{GR}$ and $R$ of the bounds averaged over 200 scenarios. | cIPS | cvcIPS | ES | IX | LS-LIN (Ours)
---|---|---|---|---|---
$rI(\mathcal{GR}_{UB_{n}})$ | 14.48% | 21.28% | 7.78% | 24.74% | 26.31%
$rI(R(\hat{\pi}_{UB_{n}}))$ | 28.13% | 33.64% | 29.44% | 36.70% | 36.76%
## 6 Conclusion
Motivated by the _pessimism_ principle, we have derived novel, empirical risk
upper bounds tailored for the regularized IPS family of estimators. Minimizing
these bounds within this family unveiled Logarithmic Smoothing, a simple
estimator with good concentration properties. With its tight upper bound, LS
confidently evaluates a policy, and shows provably better guarantees for both
selecting and learning policies than all competitors. Our upper bounds remain
broadly applicable, only requiring _negative costs_. While this condition does
not impact importance weighting estimators, it does not hold for doubly robust
estimators. Extending our approach to derive empirical bounds for this type of
estimators presents a nontrivial, yet interesting task to explore in future
work. Another potential extension would be to relax the i.i.d. assumption of
the contextual bandit problem to address, the general offline Reinforcement
Learning setting. This direction will introduce a more challenging estimation
task and requires developing new concentration bounds.
## References
* Alquier [2006] Pierre Alquier. _Transductive and Inductive Adaptative Inference for Regression and Density Estimation_. Theses, ENSAE ParisTech, December 2006. URL https://pastel.hal.science/tel-00119593.
* Alquier [2024] Pierre Alquier. User-friendly introduction to PAC-Bayes bounds. _Foundations and Trends® in Machine Learning_ , 17(2), 2024.
* Aouali et al. [2022a] Imad Aouali, Amine Benhalloum, Martin Bompaire, Achraf Ait Sidi Hammou, Sergey Ivanov, Benjamin Heymann, David Rohde, Otmane Sakhi, Flavian Vasile, and Maxime Vono. Reward Optimizing Recommendation using Deep Learning and Fast Maximum Inner Product Search. In _Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_ , KDD ’22, page 4772–4773, New York, NY, USA, 2022a. Association for Computing Machinery. ISBN 9781450393850. doi: 10.1145/3534678.3542622. URL https://doi.org/10.1145/3534678.3542622.
* Aouali et al. [2022b] Imad Aouali, Achraf Ait Sidi Hammou, Sergey Ivanov, Otmane Sakhi, David Rohde, and Flavian Vasile. Probabilistic Rank and Reward: A Scalable Model for Slate Recommendation, 2022b.
* Aouali et al. [2023] Imad Aouali, Victor-Emmanuel Brunel, David Rohde, and Anna Korba. Exponential Smoothing for Off-Policy Learning. In _Proceedings of the 40th International Conference on Machine Learning_ , pages 984–1017. PMLR, 2023.
* Aouali et al. [2024] Imad Aouali, Victor-Emmanuel Brunel, David Rohde, and Anna Korba. Bayesian off-policy evaluation and learning for large action spaces. _arXiv preprint arXiv:2402.14664_ , 2024.
* Asuncion and Newman [2007] A. Asuncion and D. J. Newman. UCI machine learning repository, 2007. URL http://www.ics.uci.edu/$∼$mlearn/{MLR}epository.html.
* Bang and Robins [2005] Heejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference models. _Biometrics_ , 61(4):962–973, 2005.
* Bottou et al. [2013] Léon Bottou, Jonas Peters, Joaquin Quiñonero-Candela, Denis X Charles, D Max Chickering, Elon Portugaly, Dipankar Ray, Patrice Simard, and Ed Snelson. Counterfactual reasoning and learning systems: The example of computational advertising. _Journal of Machine Learning Research_ , 14(11), 2013.
* Catoni [2007] Olivier Catoni. PAC-Bayesian supervised classification: The thermodynamics of statistical learning. _IMS Lecture Notes Monograph Series_ , page 1–163, 2007. ISSN 0749-2170. doi: 10.1214/074921707000000391. URL http://dx.doi.org/10.1214/074921707000000391.
* Chen et al. [2019] Minmin Chen, Alex Beutel, Paul Covington, Sagar Jain, Francois Belletti, and Ed H. Chi. Top-K Off-Policy Correction for a REINFORCE Recommender System. In _Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining_ , WSDM ’19, page 456–464, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9781450359405. doi: 10.1145/3289600.3290999. URL https://doi.org/10.1145/3289600.3290999.
* Chernozhukov et al. [2019] Victor Chernozhukov, Mert Demirer, Greg Lewis, and Vasilis Syrgkanis. Semi-parametric efficient policy learning with continuous actions. _Advances in Neural Information Processing Systems_ , 32, 2019.
* Cief et al. [2024] Matej Cief, Jacek Golebiowski, Philipp Schmidt, Ziawasch Abedjan, and Artur Bekasov. Learning action embeddings for off-policy evaluation. In _European Conference on Information Retrieval_ , pages 108–122. Springer, 2024.
* Dai et al. [2020] Bo Dai, Ofir Nachum, Yinlam Chow, Lihong Li, Csaba Szepesvari, and Dale Schuurmans. Coindice: Off-policy confidence interval estimation. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, _Advances in Neural Information Processing Systems_ , volume 33, pages 9398–9411. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper_files/paper/2020/file/6aaba9a124857622930ca4e50f5afed2-Paper.pdf.
* Dudík et al. [2011] Miroslav Dudík, John Langford, and Lihong Li. Doubly robust policy evaluation and learning. In _Proceedings of the 28th International Conference on International Conference on Machine Learning_ , ICML’11, page 1097–1104, 2011.
* Dudík et al. [2012] Miroslav Dudík, Dumitru Erhan, John Langford, and Lihong Li. Sample-efficient nonstationary policy evaluation for contextual bandits. In _Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence_ , UAI’12, page 247–254, Arlington, Virginia, USA, 2012. AUAI Press.
* Dudik et al. [2014] Miroslav Dudik, Dumitru Erhan, John Langford, and Lihong Li. Doubly robust policy evaluation and optimization. _Statistical Science_ , 29(4):485–511, 2014.
* Farajtabar et al. [2018] Mehrdad Farajtabar, Yinlam Chow, and Mohammad Ghavamzadeh. More robust doubly robust off-policy evaluation. In _International Conference on Machine Learning_ , pages 1447–1456. PMLR, 2018.
* Flynn et al. [2023] Hamish Flynn, David Reeb, Melih Kandemir, and Jan Peters. PAC-Bayes Bounds for Bandit Problems: A Survey and Experimental Comparison. _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 45(12):15308–15327, 2023. doi: 10.1109/TPAMI.2023.3305381.
* Gabbianelli et al. [2023] Germano Gabbianelli, Gergely Neu, and Matteo Papini. Importance-Weighted Offline Learning Done Right. _arXiv preprint arXiv:2309.15771_ , 2023.
* Gilotte et al. [2018] Alexandre Gilotte, Clément Calauzènes, Thomas Nedelec, Alexandre Abraham, and Simon Dollé. Offline a/b testing for recommender systems. In _Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining_ , pages 198–206, 2018.
* Hagerup and Rüb [1990] Torben Hagerup and Christine Rüb. A Guided Tour of Chernoff Bounds. _Inf. Process. Lett._ , 33(6):305–308, 1990. URL http://dblp.uni-trier.de/db/journals/ipl/ipl33.html#HagerupR90.
* Horvitz and Thompson [1952] Daniel G Horvitz and Donovan J Thompson. A generalization of sampling without replacement from a finite universe. _Journal of the American statistical Association_ , 47(260):663–685, 1952.
* Ionides [2008] Edward L Ionides. Truncated importance sampling. _Journal of Computational and Graphical Statistics_ , 17(2):295–311, 2008.
* Jeunen and Goethals [2021] Olivier Jeunen and Bart Goethals. Pessimistic reward models for off-policy learning in recommendation. In _Fifteenth ACM Conference on Recommender Systems_ , pages 63–74, 2021.
* Jin et al. [2021] Ying Jin, Zhuoran Yang, and Zhaoran Wang. Is pessimism provably efficient for offline rl? In _International Conference on Machine Learning_ , pages 5084–5096. PMLR, 2021.
* Jin et al. [2023] Ying Jin, Zhimei Ren, Zhuoran Yang, and Zhaoran Wang. Policy learning "without” overlap: Pessimism and generalized empirical Bernstein’s inequality, 2023.
* Kallus and Zhou [2018] Nathan Kallus and Angela Zhou. Policy evaluation and optimization with continuous treatments. In _International conference on artificial intelligence and statistics_ , pages 1243–1251. PMLR, 2018.
* Kingma and Ba [2014] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. _arXiv preprint arXiv:1412.6980_ , 2014.
* Krishnamurthy et al. [2020] Akshay Krishnamurthy, John Langford, Aleksandrs Slivkins, and Chicheng Zhang. Contextual bandits with continuous actions: Smoothing, zooming, and adapting. _Journal of Machine Learning Research_ , 21(137):1–45, 2020.
* Kuzborskij et al. [2021] Ilja Kuzborskij, Claire Vernade, Andras Gyorgy, and Csaba Szepesvári. Confident off-policy evaluation and selection through self-normalized importance weighting. In _International Conference on Artificial Intelligence and Statistics_ , pages 640–648. PMLR, 2021.
* Lattimore and Szepesvari [2019] Tor Lattimore and Csaba Szepesvari. _Bandit Algorithms_. Cambridge University Press, 2019.
* Li et al. [2015] Lihong Li, Remi Munos, and Csaba Szepesvari. Toward Minimax Off-policy Value Estimation. In Guy Lebanon and S. V. N. Vishwanathan, editors, _Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics_ , volume 38 of _Proceedings of Machine Learning Research_ , pages 608–616, San Diego, California, USA, 09–12 May 2015. PMLR. URL https://proceedings.mlr.press/v38/li15b.html.
* London and Sandler [2019] Ben London and Ted Sandler. Bayesian counterfactual risk minimization. In _International Conference on Machine Learning_ , pages 4125–4133. PMLR, 2019.
* Maurer and Pontil [2009] Andreas Maurer and Massimiliano Pontil. Empirical Bernstein bounds and sample variance penalization. _arXiv preprint arXiv:0907.3740_ , 2009.
* McAllester [1998] David A. McAllester. Some PAC-Bayesian theorems. In _Proceedings of the Eleventh Annual Conference on Computational Learning Theory_ , COLT’ 98, page 230–234, New York, NY, USA, 1998. Association for Computing Machinery. ISBN 1581130570. doi: 10.1145/279943.279989. URL https://doi.org/10.1145/279943.279989.
* Metelli et al. [2021] Alberto Maria Metelli, Alessio Russo, and Marcello Restelli. Subgaussian and differentiable importance sampling for off-policy evaluation and learning. _Advances in Neural Information Processing Systems_ , 34:8119–8132, 2021.
* Owen [2013] Art B. Owen. _Monte Carlo theory, methods and examples_. https://artowen.su.domains/mc/, 2013.
* Parrado-Hernández et al. [2012] Emilio Parrado-Hernández, Amiran Ambroladze, John Shawe-Taylor, and Shiliang Sun. PAC-Bayes Bounds with Data Dependent Priors. _Journal of Machine Learning Research_ , 13(112):3507–3531, 2012. URL http://jmlr.org/papers/v13/parrado12a.html.
* Peng et al. [2023] Jie Peng, Hao Zou, Jiashuo Liu, Shaoming Li, Yibao Jiang, Jian Pei, and Peng Cui. Offline policy evaluation in large action spaces via outcome-oriented action grouping. In _Proceedings of the ACM Web Conference 2023_ , pages 1220–1230, 2023.
* Robins and Rotnitzky [1995] James M Robins and Andrea Rotnitzky. Semiparametric efficiency in multivariate regression models with missing data. _Journal of the American Statistical Association_ , 90(429):122–129, 1995.
* Sachdeva et al. [2020] Noveen Sachdeva, Yi Su, and Thorsten Joachims. Off-policy bandits with deficient support. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, pages 965–975, 2020.
* Sachdeva et al. [2023] Noveen Sachdeva, Lequn Wang, Dawen Liang, Nathan Kallus, and Julian McAuley. Off-policy evaluation for large action spaces via policy convolution. _arXiv preprint arXiv:2310.15433_ , 2023.
* Saito and Joachims [2022] Yuta Saito and Thorsten Joachims. Off-Policy Evaluation for Large Action Spaces via Embeddings. _arXiv preprint arXiv:2202.06317_ , 2022.
* Saito et al. [2023] Yuta Saito, Qingyang Ren, and Thorsten Joachims. Off-policy evaluation for large action spaces via conjunct effect modeling. In _international conference on Machine learning_ , pages 29734–29759. PMLR, 2023.
* Sakhi [2023] Otmane Sakhi. _Offline Contextual Bandit : Theory and Large Scale Applications_. Theses, Institut Polytechnique de Paris, December 2023. URL https://theses.hal.science/tel-04417936.
* Sakhi et al. [2020a] Otmane Sakhi, Stephen Bonner, David Rohde, and Flavian Vasile. BLOB: A Probabilistic model for recommendation that combines organic and bandit signals. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, pages 783–793, 2020a.
* Sakhi et al. [2020b] Otmane Sakhi, Louis Faury, and Flavian Vasile. Improving Offline Contextual Bandits with Distributional Robustness, 2020b.
* Sakhi et al. [2023a] Otmane Sakhi, Pierre Alquier, and Nicolas Chopin. PAC-Bayesian Offline Contextual Bandits with Guarantees. In _International Conference on Machine Learning_ , pages 29777–29799. PMLR, 2023a.
* Sakhi et al. [2023b] Otmane Sakhi, David Rohde, and Nicolas Chopin. Fast Slate Policy Optimization: Going Beyond Plackett-Luce. _Transactions on Machine Learning Research_ , 2023b. ISSN 2835-8856. URL https://openreview.net/forum?id=f7a8XCRtUu.
* Sakhi et al. [2023c] Otmane Sakhi, David Rohde, and Alexandre Gilotte. Fast Offline Policy Optimization for Large Scale Recommendation. _Proceedings of the AAAI Conference on Artificial Intelligence_ , 37(8):9686–9694, Jun. 2023c. doi: 10.1609/aaai.v37i8.26158. URL https://ojs.aaai.org/index.php/AAAI/article/view/26158.
* Seldin et al. [2011] Yevgeny Seldin, Peter Auer, John Shawe-taylor, Ronald Ortner, and François Laviolette. PAC-Bayesian analysis of contextual bandits. In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K.Q. Weinberger, editors, _Advances in Neural Information Processing Systems_ , volume 24. Curran Associates, Inc., 2011. URL https://proceedings.neurips.cc/paper/2011/file/58e4d44e550d0f7ee0a23d6b02d9b0db-Paper.pdf.
* Seldin et al. [2012] Yevgeny Seldin, Nicolò Cesa-Bianchi, Peter Auer, François Laviolette, and John Shawe-Taylor. Pac-bayes-bernstein inequality for martingales and its application to multiarmed bandits. In Dorota Glowacka, Louis Dorard, and John Shawe-Taylor, editors, _Proceedings of the Workshop on On-line Trading of Exploration and Exploitation 2_ , volume 26 of _Proceedings of Machine Learning Research_ , pages 98–111, Bellevue, Washington, USA, 02 Jul 2012. PMLR. URL https://proceedings.mlr.press/v26/seldin12a.html.
* Shrivastava and Li [2014] Anshumali Shrivastava and Ping Li. Asymmetric LSH (ALSH) for Sublinear Time Maximum Inner Product Search (MIPS). In Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K.Q. Weinberger, editors, _Advances in Neural Information Processing Systems_ , volume 27. Curran Associates, Inc., 2014. URL https://proceedings.neurips.cc/paper_files/paper/2014/file/310ce61c90f3a46e340ee8257bc70e93-Paper.pdf.
* Su et al. [2019] Yi Su, Lequn Wang, Michele Santacatterina, and Thorsten Joachims. Cab: Continuous adaptive blending for policy evaluation and learning. In _International Conference on Machine Learning_ , pages 6005–6014. PMLR, 2019.
* Su et al. [2020] Yi Su, Maria Dimakopoulou, Akshay Krishnamurthy, and Miroslav Dudík. Doubly robust off-policy evaluation with shrinkage. In _International Conference on Machine Learning_ , pages 9167–9176. PMLR, 2020.
* Swaminathan and Joachims [2015a] Adith Swaminathan and Thorsten Joachims. Batch learning from logged bandit feedback through counterfactual risk minimization. _The Journal of Machine Learning Research_ , 16(1):1731–1755, 2015a.
* Swaminathan and Joachims [2015b] Adith Swaminathan and Thorsten Joachims. The self-normalized estimator for counterfactual learning. _advances in neural information processing systems_ , 28, 2015b.
* Swaminathan et al. [2017] Adith Swaminathan, Akshay Krishnamurthy, Alekh Agarwal, Miro Dudik, John Langford, Damien Jose, and Imed Zitouni. Off-policy evaluation for slate recommendation. _Advances in Neural Information Processing Systems_ , 30, 2017.
* Taufiq et al. [2024] Muhammad Faaiz Taufiq, Arnaud Doucet, Rob Cornish, and Jean-Francois Ton. Marginal density ratio for off-policy evaluation in contextual bandits. _Advances in Neural Information Processing Systems_ , 36, 2024.
* Wang et al. [2023] Lequn Wang, Akshay Krishnamurthy, and Aleksandrs Slivkins. Oracle-efficient pessimism: Offline policy optimization in contextual bandits. _arXiv preprint arXiv:2306.07923_ , 2023.
* Wang et al. [2017] Yu-Xiang Wang, Alekh Agarwal, and Miroslav Dudık. Optimal and adaptive off-policy evaluation in contextual bandits. In _International Conference on Machine Learning_ , pages 3589–3597. PMLR, 2017.
* Zhou [2002] Ding-Xuan Zhou. The covering number in learning theory. _J. Complex._ , 18(3):739–767, sep 2002. ISSN 0885-064X. doi: 10.1006/jcom.2002.0635. URL https://doi.org/10.1006/jcom.2002.0635.
## Appendix A Limitations
This work develops theoretically grounded and practical pessimistic approaches
for the offline contextual bandit setting. Even if the proposed algorithms are
general, and provably better than competitors, they still suffer from the
intrinsic limitations of importance weighting estimators. Specifically, our
method, as presented, will perform poorly in _extremely_ large action spaces.
However, these limitations can be mitigated by incorporating additional
structure as in Saito and Joachims [44], Saito et al. [45]. Another limitation
arises from the offline contextual bandit setting itself, which assumes i.i.d.
observations. While this assumption is valid in simple scenarios, it becomes
unsuitable once we want to capture the long term effect of interventions.
Extending our results to the more general, reinforcement learning setting
would be an interesting research direction as it comes with a challenging
estimation task and will require developing new concentration bounds.
## Appendix B Broader impact
Our work contributes to the development of theoretically grounded and
practical pessimistic approaches for the offline contextual bandit setting.
The derived algorithms can improve the robustness of decision-making processes
by prioritizing safety and minimizing uncertainty associated risks. By
leveraging pessimistic strategies, we ensure that decisions are made with a
conservative bias, thereby potentially improving outcomes in high-stakes
environments where the cost of errors is substantial. Although our framework
and algorithms have broad, potentially good applications, their specific
social impacts will solely depend on the chosen application domain.
## Appendix C Extended related work
Offline contextual bandits. Contextual bandit is a widely adopted framework
for online learning in uncertain environments [32]. However, some real-world
applications present challenges for existing online algorithms, and thus
offline methods that leverage historical data to optimize decision-making have
gained traction [9]. Fortunately, large datasets summarizing past interactions
are often available, allowing agents to improve their policies offline [57].
Our work explores this offline approach, known as offline (or off-policy)
contextual bandits [15]. In this setting, off-policy evaluation (OPE)
estimates policy performance using historical data, mimicking real-time
evaluations. Depending on the application, the goal might be to find the best
policy within a predefined finite set (off-policy selection (OPS)) or the
optimal policy overall (off-policy learning (OPL)).
Off-policy evaluation. In recent years, OPE has experienced a noticeable surge
of interest, with numerous significant contributions [15, 16, 17, 62, 18, 55,
56, 37, 31, 44, 47, 25]. The literature on OPE can be broadly classified into
three primary approaches. The first, referred to as the direct method (DM)
[25, 6], involves the development of a model designed to approximate expected
costs for any context-action pair. This model is subsequently employed to
estimate the performance of the policies. This approach is often designed for
specific applications such as large-scale recommender systems [47, 25, 4]. The
second approach, known as inverse propensity scoring (IPS) [23, 16], aims to
estimate the costs associated with the evaluated policies by correcting for
the inherent preference bias of the behavior policy within the dataset. While
IPS maintains its unbiased nature when operating under the assumption that the
evaluation policy is absolutely continuous with respect to the behavior
policy, it can be susceptible to high variance and substantial bias when this
assumption is violated [42]. In response to the variance issue, various
techniques have been introduced, including clipping [24, 9], shrinkage [56],
power-mean correction [37], implicit exploration [20], self-normalization
[58], among others [21]. The third approach, known as doubly robust (DR) [41,
8, 15, 17, 18], combines elements from both the direct method (DM) and inverse
propensity scoring (IPS). This work focuses on regularized IPS.
Off-policy selection and learning. as in OPE, three key approaches dominate:
DM, IPS and DR in OPS and OPL. In OPS, all these methods share the same core
objective: identifying the policy with the highest estimated reward from a
finite set of candidates. However, they differ in their reward estimation
techniques, as discussed in the OPE section above. In contrast, in OPL, DM
either deterministically selects the action with the highest estimated reward
or constructs a distribution based on these estimates. IPS and DR, on the
other hand, employ gradient descent for policy learning [57], updating a
parameterized policy denoted by $\pi_{\theta}$ as
$\theta_{t+1}\leftarrow\theta_{t}-\nabla_{\theta}R(\pi_{\theta})$ for each
iteration $t$. Since the true risk $R$ is unknown,
$\nabla_{\theta}R(\pi_{\theta})$ is unknown and needs to be estimated using
techniques like IPS or DR.
Pessimism in offline contextual bandits. Most OPE studies directly use their
point estimators of the risk in OPE, OPS and OPL. However, point estimators
can deviate from the true value of the risk, rendering them unreliable for
decision-making. Therefore, and to increase safety, alternative approaches
focus on constructing bounds on the risk. These bounds, either asymptotic [9,
48, 14] or finite sample [31, 20], aim to evaluate a policy’s worst-case
performance, adhering to the principle of _pessimism in face of uncertainty_
[26]. The principle of pessimism transcends OPE, influencing both OPS and OPL.
In these domains, strategies are predominantly inspired by, or directly
derived from, upper bounds on the true risk [57, 34, 31, 49, 5, 61]. Consider
OPS: [31] leveraged an Efron-Stein bound for the self-normalized IPS
estimator, while [20] anchored their analysis on a bound constructed with the
Implicit Exploration estimator. Shifting focus to OPL, [57] combined the
empirical Bernstein bound [35] with the clipping estimator, motivating sample
variance penalization for policy learning. Recent advancements include
modifications to the penalization term [61] to be scalable and efficient.
PAC-Bayes extension. The PAC-Bayesian paradigm [36, 10] (see Alquier [2] for a
recent introduction) provides a rich set of tools to prove generalization
bounds for different statistical learning problems. The classical (online)
contextual bandit problem received a lot of attention from the PAC-Bayesian
community with the seminal work of Seldin et al. [52, 53]. It is just recently
that these tools were adapted to the offline contextual bandit setting, with
[34] that introduced a clean and scalable PAC-Bayesian perspective to OPL.
This perspective was further explored by [19, 49, 5, 20], leading to the
development of tight, tractable PAC-Bayesian bounds suitable for direct
optimization.
Large action space extension. While regularization techniques can improve IPS
properties, they often fall short when dealing with extremely large action
spaces. Additional assumptions regarding the structure of the contextual
bandit problem become necessary. For example, Saito and Joachims [44]
introduced the Marginalized IPS (MIPS) framework and estimator. MIPS leverages
auxiliary information about the actions in the form of action embeddings.
Roughly speaking, MIPS assumes access to embeddings $e_{i}$ within logged data
and defines the risk estimator as
$\displaystyle\hat{R}_{n}^{\text{MIPS}}(\pi)=\frac{1}{n}\sum_{i=1}^{n}\frac{\pi\left(e_{i}\mid
x_{i}\right)}{\pi_{0}\left(e_{i}\mid
x_{i}\right)}c_{i}=\frac{1}{n}\sum_{i=1}^{n}w\left(x_{i},e_{i}\right)c_{i}\,,$
where the logged data
$\mathcal{D}_{n}=\left\\{\left(x_{i},a_{i},e_{i},r_{i}\right)\right\\}_{i=1}^{n}$
now includes action embeddings for each data point. The marginal importance
weight
$\displaystyle w(x,e)=\frac{\pi(e\mid x)}{\pi_{0}(e\mid
x)}=\frac{\sum_{a}p(e\mid x,a)\pi(a\mid x)}{\sum_{a}p(e\mid x,a)\pi_{0}(a\mid
x)}$
is a key component of this approach. Compared to IPS and DR, MIPS achieves
significantly lower variance in large action spaces [44] while maintaining
unbiasedness if the action embeddings directly influence costs $c$. This
necessitates informative embeddings that capture the causal effects of actions
on costs. However, high-dimensional embeddings can still lead to high variance
for MIPS, similar to IPS. Additionally, high bias can arise if the direct
effect assumption is violated and embeddings fail to capture these causal
effects. This bias is particularly present when performing action feature
selection for dimensionality reduction. Recent work proposes learning such
embeddings directly from logged data [40, 43, 13], or loosen this assumption
[60, 45]. Our proposed importance weight regularization can be potentially
combined with these estimators under their respective assumptions on the
underlying structure of the contextual bandit problem, extending our approach
to large action spaces, and we posit that this will be beneficial when, for
example, the action embedding dimension is high. Another line of research in
large action spaces is more interested with the learning problem, precisely
solving the optimization issues arising from policies defined on large action
spaces. Indeed, naive optimization tends to be slow and scales linearly with
the number of actions $K$ [11]. Recent work [51, 50] solve this by leveraging
fast maximum inner product search [54, 3] in the training loop, reducing the
optimization complexity to _logarithmic_ in the action space size. These
methods however require a linear objective on the target policy. Luckily, our
PAC-Bayesian learning objective is linear in the policy and its optimization
is amenable to such acceleration.
Continuous action space extension. While research has predominantly focused on
discrete action spaces, a limited number of studies have tackled the
continuous case [28, 12, 61]. For example, [28] explored non-parametric
evaluation and learning of continuous action policies using kernel smoothing,
while [12] investigated the semi-parametric setting. Recently, [61] leveraged
the smoothing approach from [30] to extend their discrete OPL method to
continuous actions. Our work can either use the densities directly, or be
similarly extended to continuous actions through a well-defined discretization
of the space. Imagine a scenario with infinitely many actions, where policies
are defined by density functions. For any context $x$, $\pi(a\mid x)$
represents the density function that maps actions $a$ to probabilities. The
discretization process transforms the original contextual bandit problem
characterized by the density-based policy class $\Pi$ into an OPL problem
defined by a discrete, mass-based policy class $\Pi_{K}$ (for a finite number
of actions $K$). Each policy within $\Pi_{K}$ approximates a policy in $\Pi$
through a smoothing process.
## Appendix D Useful lemmas
In the following, and for any quantity $Z$, all expectations are computed
w.r.t to the distribution of the data when playing actions under the behaviour
policy $\pi_{0}$, as in:
$\mathbb{E}\left[Z\right]=\mathbb{E}_{x\sim\nu,a\sim\pi_{0}(\cdot|x),c\sim
p(\cdot\mid x,a)}\left[Z\right].$
A lot of the results derived in the paper are based on the use of the well
known Chernoff Inequality, that we state below for a sum of i.i.d. random
variables:
###### Lemma 9.
(Chernoff Inequality for a sum of i.i.d. random variables.) Let
$a\in\mathbbm{R}$, $n\in\mathbbm{N}^{*}$ and $\\{X_{i},i\in[n]\\}$ a
collection of $n$ i.i.d. random variables. The following concentration bounds
on the right tail of $\sum_{i\in[n]}X_{i}$ hold for any $\lambda\geq 0$:
$\displaystyle
P\left(\sum_{i\in[n]}X_{i}>a\right)\leq\left(\mathbbm{E}\left[\exp\left(\lambda
X_{1}\right)\right]\right)^{n}\exp(-\lambda a)$
This result is classical in the literature [22] and we omit its proof. We will
also need the following lemma, that states the monotonous nature of a key
function in our analysis, and that we take the time to prove.
###### Lemma 10.
Let $L\geq 1$ and $f_{L}$ be the following function:
$f_{L}(x)=\frac{\log(1+x)-\sum_{\ell=1}^{L}\frac{(-1)^{\ell-1}}{\ell}x^{\ell}}{(-1)^{L}x^{L+1}}.$
We have that $f_{L}$ is a decreasing function in $\mathbbm{R}^{+}$ for all
$L\in\mathbbm{N}^{*}$.
###### Proof.
Let $L\geq 1$ and $f_{L}$ be the following function:
$f_{L}(x)=\frac{\log(1+x)-\sum_{\ell=1}^{L}\frac{(-1)^{\ell-1}}{\ell}x^{\ell}}{(-1)^{L}x^{L+1}}.$
Let $x\in\mathbbm{R}^{+}$, we have the following identity holding $\forall
t>0$ and $\forall n\geq 0$:
$\displaystyle\frac{1+(-1)^{n}t^{n+1}}{1+t}=\sum_{k=0}^{n}(-1)^{k}t^{k}\iff\frac{1}{1+t}=\sum_{k=0}^{n}(-1)^{k}t^{k}+\frac{(-1)^{n+1}t^{n+1}}{1+t}.$
(25)
Recall the integral form of the log function:
$\log(1+x)=\int_{0}^{x}\frac{1}{1+t}dt.$
We integrate both sides of the Equality (25) and show that the numerator of
$f_{L}(x)$ is equal to:
$\log(1+x)-\sum_{k=1}^{K}\frac{(-1)^{k-1}}{k}x^{k}=(-1)^{K}\int_{0}^{x}\frac{t^{K}}{1+t}dt.$
This result enables us to rewrite the function $f_{L}$ as:
$f_{L}(x)=\frac{1}{x^{L+1}}\int_{0}^{x}\frac{t^{L}}{1+t}dt.$
Using the change of variable $t=ux$, we obtain:
$f_{L}(x)=\int_{0}^{1}\frac{u^{L}}{1+xu}dt$
which is clearly decreasing for in $\mathbbm{R}^{+}$. This ends the proof. ∎
Finally, we also state the important change of measure lemma:
###### Lemma 11.
Change of measure. Let $g$ be a function of the parameter $\theta$ and data
$\mathcal{D}_{n}$, for any distribution Q that is P continuous, for any
$\delta\in(0,1]$, we have with probability $1-\delta$ :
$\displaystyle\mathbbm{E}_{\theta\sim
Q}[g(\theta,\mathcal{D}_{n})]\leq\mathcal{KL}(Q||P)+\ln\frac{\Psi_{g}}{\delta}$
(26) with $\Psi_{g}=\mathbbm{E}_{\mathcal{D}_{n}}\mathbbm{E}_{\theta\sim
P}[e^{g(\theta,\mathcal{D}_{n})}]$.
Lemma 11 is the backbone of a multitude of PAC-Bayesian bounds. It is proven
in many references, see for example [2] or Lemma 1.1.3 in [10]. With this
result, the recipe of constructing a generalisation bound reduces to choosing
an adequate function $g$ for which we can control $\Psi_{g}$.
## Appendix E Additional results and discussions
### E.1 A study of logarithmic smoothing estimator
Recall the form of the Logarithmic Smoothing estimator, defined for any
$\lambda\geq 0$:
$\displaystyle\hat{R}^{\lambda}_{n}(\pi)=-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
w_{\pi}(x_{i},a_{i})c_{i}\right)\,.$ (27)
Our estimator $\hat{R}^{\lambda}_{n}(\pi)$, is defined for a non-negative
$\lambda\geq 0$. In particular, $\lambda=0$ recovers the unbiased IPS
estimator in (2) and $\lambda>0$ introduces a bias variance trade-off. This
estimator can be interpreted as Logarithmic Soft Clipping, and have a similar
behavior than Clipping of Bottou et al. [9]. Indeed, $1/\lambda$ plays a
similar role to the clipping parameter $M$, as for any $i\in[n]$, we have:
$\displaystyle w_{\pi}(x_{i},a_{i})c_{i}\ll\frac{1}{\lambda}$
$\displaystyle\implies-\frac{1}{\lambda}\log\left(1-\lambda
w_{\pi}(x_{i},a_{i})c_{i}\right)\approx w_{\pi}(x_{i},a_{i})c_{i}.$
$\displaystyle w_{\pi}(x_{i},a_{i})c_{i}<M$
$\displaystyle\implies\min\left(w_{\pi}(x_{i},a_{i}),M\right)c_{i}=w_{\pi}(x_{i},a_{i})c_{i}.$
LS can be seen as a smooth, differentiable version of clipping. We plot the
graph of the two functions in Figure 2. One can observe that once $\lambda>0$,
LS exhibits a bias-variance trade-off, with a declining bias with
$\lambda\rightarrow 0$. This is different than Clipping as no bias is suffered
once $M$ is bigger than the support of $x$, this comes however with the price
of suffering the full variance of IPS. In the following, we study the bias-
variance trade-off that emerges with the new Logarithmic Smoothing estimator.
Figure 2: Comparison of Logarithmic Smoothing and Clipping.
We begin by defining the bias and variance of $\hat{R}^{\lambda}_{n}(\pi)$:
$\displaystyle\mathcal{B}^{\lambda}(\pi)=\mathbb{E}\left[\hat{R}^{\lambda}_{n}(\pi)\right]-R(\pi)\,,$
$\displaystyle\mathcal{V}^{\lambda}(\pi)=\mathbb{E}\left[\left(\hat{R}^{\lambda}_{n}(\pi)-\mathbb{E}\left[\hat{R}^{\lambda}_{n}(\pi)\right]\right)^{2}\right]\,.$
(28)
Moreover, for any $\lambda\geq 0$, we define the following quantity
$\displaystyle\mathcal{S}_{\lambda}(\pi)=\mathbb{E}\left[\frac{w_{\pi}(x,a)^{2}c^{2}}{1-\lambda
w_{\pi}(x,a)c}\right]\,,$ (29)
that will be essential in studying the properties of this estimator akin to
the coverage ratio used for the IX-estimator [20]. In the following, we study
the properties of our estimator $\hat{R}^{\lambda}_{n}(\pi)$ in (14). We start
with bounding its mean squared error (MSE), which involves bounding its bias
and variance.
###### Proposition 12 (Bias-variance trade-off).
Let $\pi\in\Pi$ and $\lambda\geq 0$. Then we have that $\displaystyle
0\leq\mathcal{B}^{\lambda}(\pi)\leq\lambda\mathcal{S}_{\lambda}(\pi)\,,\quad\text{and}\quad\mathcal{V}^{\lambda}(\pi)\leq\mathcal{S}_{\lambda}(\pi)/n\,.$
Moreover, it holds that for any $\lambda>0$:
$\mathcal{V}^{\lambda}(\pi)\leq\frac{\lvert
R(\pi)\rvert}{n\lambda}\leq\frac{1}{n\lambda}.$
###### Proof.
Let us start with bounding the bias. We have for any $\lambda\geq 0$:
$\displaystyle\mathcal{B}^{\lambda}(\pi)$
$\displaystyle=\mathbb{E}\left[\hat{R}^{\lambda}_{n}(\pi)\right]-R(\pi)$
$\displaystyle=\mathbb{E}\left[-\frac{1}{\lambda}\log(1-\lambda
w_{\pi}(x,a)c)-w_{\pi}(x,a)c\right]\quad(\text{IPS is unbiased}).$
Using $\log(1+x)\leq x$ for any $x\geq 0$ proves that the bias is positive.
For its upper bound, we use the following inequality
$\log(1+x)\geq\frac{x}{1+x}$ holding for $x\geq 0$:
$\displaystyle\mathcal{B}^{\lambda}(\pi)$
$\displaystyle=\mathbb{E}\left[-\frac{1}{\lambda}\log(1-\lambda
w_{\pi}(x,a)c)-w_{\pi}(x,a)c\right]$
$\displaystyle\leq\mathbb{E}\left[\frac{w_{\pi}(x,a)c}{1-\lambda
w_{\pi}(x,a)c}-w_{\pi}(x,a)c\right]=\lambda\mathbb{E}\left[\frac{(w_{\pi}(x,a)c)^{2}}{1-\lambda
w_{\pi}(x,a)c}\right]=\lambda\mathcal{S}_{\lambda}(\pi).$
Now focusing on the variance, we have:
$\displaystyle\mathcal{V}^{\lambda}(\pi)$
$\displaystyle=\mathbb{E}\left[\left(\hat{R}^{\lambda}_{n}(\pi)-\mathbb{E}\left[\hat{R}^{\lambda}_{n}(\pi)\right]\right)^{2}\right]$
$\displaystyle\leq\frac{1}{n\lambda^{2}}\mathbb{E}\left[\log(1-\lambda
w_{\pi}(x,a)c)^{2}\right].$
We use the following inequality $\log(1+x)\leq x/\sqrt{x+1}$ holding for
$x\geq 0$ to obtain our result:
$\displaystyle\mathcal{V}^{\lambda}(\pi)$
$\displaystyle\leq\frac{1}{n}\mathcal{S}_{\lambda}(\pi).$
Notice that once $\lambda>0$, we have:
$\displaystyle\mathcal{S}_{\lambda}(\pi)$
$\displaystyle=\mathbb{E}\left[\frac{w_{\pi}(x,a)^{2}c^{2}}{1-\lambda
w_{\pi}(x,a)c}\right]\leq\frac{1}{\lambda}\mathbb{E}\left[w_{\pi}(x,a)\lvert
c\rvert\right]=\frac{\lvert R(\pi)\rvert}{\lambda},$
resulting in a finite variance whenever $\lambda>0$:
$\displaystyle\mathcal{V}^{\lambda}(\pi)$ $\displaystyle\leq\frac{\lvert
R(\pi)\rvert}{n\lambda}\leq\frac{1}{n\lambda}.$
∎
$\lambda=0$ recovers the IPS estimator in (2), with zero bias and variance
bounded by $\mathbb{E}\left[w^{2}(x,a)c^{2}\right]/n$. When $\lambda>0$, a
bias-variance trade-off emerges. The bias is always non-negative as we still
recover an estimator that verifies (C1). The bias is capped at
$\lambda\mathcal{S}_{\lambda}(\pi)$, which diminishes to zero when $\lambda$
is small and goes to $|R(\pi)|$ as $\lambda$ increases. Conversely, the
variance decreases with a higher $\lambda$. Notably, $\lambda>0$ ensures
finite variance bounded by $1/\lambda n$, despite the estimator being
unbounded. This is different from previous regularizations that relied on
bounded functions to ensure finite variance.
While prior evaluations of estimators often relied on bias and variance
analysis, Metelli et al. [37] argued for studying the non-asymptotic
concentration rate of the estimators, advocating for sub-Gaussianity as a
desired property. Even if our estimator is not bounded, we prove in the
following that it is sub-Gaussian.
###### Proposition 13.
(Sub-Gaussianity). Let $\pi\in\Pi$, $\delta\in(0,1]$ and $\lambda>0$. Then the
following inequalities holds with probability at least $1-\delta$:
$\displaystyle
R(\pi)-\hat{R}^{\lambda}_{n}(\pi)\leq\frac{\ln(2/\delta)}{\lambda
n}\,,\qquad\text{and}\qquad\hat{R}^{\lambda}_{n}(\pi)-R(\pi)\leq\lambda\mathcal{S}_{\lambda}(\pi)+\frac{\ln(2/\delta)}{\lambda
n}\,.$ In particular, setting
$\lambda=\lambda_{*}=\sqrt{\ln(2/\delta)/n\mathbb{E}\left[w_{\pi}(x,a)^{2}c^{2}\right]}$
yields that
$\displaystyle|R(\pi)-\hat{R}^{\lambda_{*}}_{n}(\pi)|\leq\sqrt{2\sigma^{2}\ln(2/\delta)}\,,$
$\displaystyle\text{where
}\,\sigma^{2}=2\mathbb{E}\left[w_{\pi}(x,a)^{2}c^{2}\right]/n\,.$ (30)
###### Proof.
Let $\pi\in\Pi$, $\lambda>0$ and $\delta>0$. To prove sub-Gaussianity, we need
both upper bounds and lower bounds on $R(\pi)$ using
$\hat{R}^{\lambda}_{n}(\pi)$. For the upper bound, we can use the bound of
Corollary 4, and recall that $\psi_{\lambda}(x)\leq x$ for all $x$. We then
obtain with a probability $1-\delta$:
$\displaystyle
R(\pi)\leq\psi_{\lambda}\left(\hat{R}^{\lambda}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\right)\implies
R(\pi)-\hat{R}^{\lambda}_{n}(\pi)\leq\frac{\ln(1/\delta)}{\lambda n}.$
For the lower bound on the risk, we go back to our Chernoff Lemma 9, and use
the collection of i.i.d. random variable, that for any $i\in[n]$, are defined
as:
$\bar{X}_{i}=-\frac{1}{\lambda}\log\left(1-\lambda
w_{\pi}(x_{i},a_{i})c_{i}\right).$
This gives for $a\in\mathbbm{R}$:
$\displaystyle
P\left(\sum_{i\in[n]}\bar{X}_{i}>a\right)\leq\left(\mathbbm{E}\left[\exp\left(\lambda\bar{X}_{1}\right)\right]\right)^{n}\exp(-\lambda
a)$ $\displaystyle
P\left(\sum_{i\in[n]}\bar{X}_{i}>a\right)\leq\left(\mathbbm{E}\left[\frac{1}{1-\lambda
w_{\pi}(x,a)c}\right]\right)^{n}\exp(-\lambda a)$
Solving for $\delta=\left(\mathbbm{E}\left[\frac{1}{1-\lambda
w_{\pi}(x,a)c}\right]\right)^{n}\exp(-\lambda a)$, we get:
$\displaystyle
P\left(\frac{1}{n}\sum_{i\in[n]}\bar{X}_{i}>\frac{1}{\lambda}\log\left(\mathbbm{E}\left[\frac{1}{1-\lambda
w_{\pi}(x,a)c}\right]\right)+\frac{\ln(1/\delta)}{\lambda n}\right)\leq\delta$
The complementary event holds with at least probability $1-\delta$:
$\displaystyle\hat{R}^{\lambda}_{n}(\pi)\leq\frac{1}{\lambda}\log\left(\mathbbm{E}\left[\frac{1}{1-\lambda
w_{\pi}(x,a)c}\right]\right)+\frac{\ln(1/\delta)}{\lambda n},$
which implies using the inequality $\log(x)\leq x-1$ for all $x>0$:
$\displaystyle\hat{R}^{\lambda}_{n}(\pi)-R(\pi)$
$\displaystyle\leq\frac{1}{\lambda}\log\left(\mathbbm{E}\left[\frac{1}{1-\lambda
w_{\pi}(x,a)c}\right]\right)-R(\pi)+\frac{\ln(1/\delta)}{\lambda n}$
$\displaystyle\leq\frac{1}{\lambda}\left(\mathbbm{E}\left[\frac{1}{1-\lambda
w_{\pi}(x,a)c}\right]-1\right)-R(\pi)+\frac{\ln(1/\delta)}{\lambda n}$
$\displaystyle\leq\mathbbm{E}\left[\frac{w_{\pi}(x,a)c}{1-\lambda
w_{\pi}(x,a)c}-w_{\pi}(x,a)c\right]+\frac{\ln(1/\delta)}{\lambda n}$
$\displaystyle\leq\lambda\mathbbm{E}\left[\frac{w_{\pi}(x,a)^{2}c^{2}}{1-\lambda
w_{\pi}(x,a)c}\right]+\frac{\ln(1/\delta)}{\lambda
n}=\lambda\mathcal{S}_{\lambda}(\pi)+\frac{\ln(1/\delta)}{\lambda n},$
which proves the lower bound on the risk. As both results hold with high
probability, we use a union argument to have them both holding for probability
at least $1-\delta$:
$\displaystyle
R(\pi)-\hat{R}^{\lambda}_{n}(\pi)\leq\frac{\ln(2/\delta)}{\lambda
n}\,,\qquad\text{and}\qquad\hat{R}^{\lambda}_{n}(\pi)-R(\pi)\leq\lambda\mathcal{S}_{\lambda}(\pi)+\frac{\ln(2/\delta)}{\lambda
n}\,,$
which implies that:
$\displaystyle\lvert
R(\pi)-\hat{R}^{\lambda}_{n}(\pi)\rvert\leq\lambda\mathcal{S}_{\lambda}(\pi)+\frac{\ln(2/\delta)}{\lambda
n}\,\leq\lambda\mathbb{E}\left[w_{\pi}(x,a)^{2}c^{2}\right]+\frac{\ln(2/\delta)}{\lambda
n}\,.$
This means that setting
$\lambda=\lambda_{*}=\sqrt{\ln(2/\delta)/n\mathbb{E}\left[w_{\pi}(x,a)^{2}c^{2}\right]}$
yields a sub-Gaussian concentration:
$|R(\pi)-\hat{R}^{\lambda_{*}}_{n}(\pi)|\leq
2\sqrt{\frac{\mathbb{E}\left[w_{\pi}(x,a)^{2}c^{2}\right]\ln(2/\delta)}{n}}\,.$
This ends the proof. ∎
From (30), $\hat{R}^{\lambda_{*}}_{n}(\pi)$ is sub-Gaussian with variance
proxy $\sigma^{2}=2\mathbb{E}\left[\omega(x,a)^{2}c^{2}\right]/n$, which is
lower that the variance proxy of the Harmonic estimator of Metelli et al.
[37].
### E.2 OPE: Formal comparisons of the bounds
#### E.2.1 Comparison with empirical Bernstein
We begin by comparing the Second Moment Bound with Swaminathan and Joachims
[57]’s bound as they both manipulate similar quantities. The bound of [57]
uses the Empirical Bernstein bound of [35] applied to the Clipping Estimator.
We recall its expression below for a parameter $M>0$:
$\hat{R}^{M}_{n}(\pi)=\frac{1}{n}\sum_{i=1}^{n}\min\left\\{\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})},M\right\\}c_{i}.$
We also give below the Empirical Bernstein Bound applied to this estimator:
###### Proposition.
[Empirical Bernstein for Clipping of [57]] Let $\pi\in\Pi$, $\delta\in(0,1]$
and $M>0$. Then it holds with probability at least $1-\delta$ that
$\displaystyle
R(\pi)\leq\hat{R}^{M}_{n}(\pi)+\sqrt{\frac{2\hat{V}^{M}_{n}(\pi)\ln(2/\delta)}{n}}+\frac{7M\ln(2/\delta)}{3(n-1)}\,,$
(31) with $\hat{V}^{M}_{n}(\pi)$ the empirical variance of the clipping
estimator.
We are usually interested in the case where $\pi$ and $\pi_{0}$ are different,
leading to substantial importance weights. In this practical scenario, the
variance and the second moment are of the same magnitude of $M$. Indeed, one
can see it from the following equality:
$\displaystyle\underbrace{\hat{V}^{M}_{n}(\pi)}_{\mathcal{O}(M)}$
$\displaystyle=\underbrace{\hat{\mathcal{M}}^{M,2}_{n}(\pi)}_{\mathcal{O}(M)}-\underbrace{\left(\hat{R}^{M}_{n}(\pi)\right)^{2}}_{\mathcal{O}(\bar{c}^{2})}$
$\displaystyle\approx\underbrace{\hat{\mathcal{M}}^{M,2}_{n}(\pi)}_{\mathcal{O}(M)}\quad(M\gg\bar{c}^{2}=o(1).)$
This means that in practical scenarios, the empirical variance and the
empirical second moment are approximately the same. Recall that the Second
Moment Bound works for any regularizer $h$, As Clipping satisfies (C1), we
give the Second Moment Upper of Corollary 3 with Clipping below:
$\displaystyle\psi_{\lambda}\Big{(}\hat{R}^{M}_{n}(\pi)+\frac{\lambda}{2}\hat{\mathcal{M}}^{M,2}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}\ $
$\displaystyle\leq\hat{R}^{M}_{n}(\pi)+\frac{\lambda}{2}\hat{\mathcal{M}}^{M,2}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\quad(\psi_{\lambda}(x)\leq x,\forall x)$
$\displaystyle\leq\hat{R}^{M}_{n}(\pi)+\frac{\lambda}{2}\hat{\mathcal{M}}^{M,2}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}.$
Choosing a
$\lambda\approx\sqrt{2\ln(1/\delta)/(n\hat{\mathcal{M}}^{M,2}_{n}(\pi))}$
gives us an upper bound that is close to:
$\displaystyle\hat{R}^{M}_{n}(\pi)+\frac{\lambda}{2}\hat{\mathcal{M}}^{M,2}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}$
$\displaystyle\approx\hat{R}^{M}_{n}(\pi)+\sqrt{\frac{2\hat{\mathcal{M}}^{M,2}_{n}(\pi)\ln(1/\delta)}{n}}$
$\displaystyle\approx\hat{R}^{M}_{n}(\pi)+\sqrt{\frac{2\hat{V}^{M}_{n}(\pi)\ln(1/\delta)}{n}}$
$\displaystyle\leq\hat{R}^{M}_{n}(\pi)+\sqrt{\frac{2\hat{V}^{M}_{n}(\pi)\ln(2/\delta)}{n}}+\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\frac{7M\ln(2/\delta)}{3(n-1)}}.$
This means that in practical scenarios, and with a good choice of
$\lambda\sim\mathcal{O}(1/\sqrt{n})$, the Second Moment bound would be better
than the Empirical Bernstein bound, and this difference will be even greater
when $M\gg 1$. This is aligned with our experiments, where we see that the new
Second Moment bound is much tighter in practice. This also confirms that the
Logarithmic smoothing bound is even tighter, because it is smaller than the
Second Moment bound as stated in Proposition 5.
#### E.2.2 Comparison with the IX bound
We now attack the recently derived IX bound in Gabbianelli et al. [20] and
show that our newly proposed bound dominates it in all scenarios.
###### Proposition 14.
(Comparison with IX [20]) Let $\pi\in\Pi$, $\delta\in]0,1]$ and $\lambda>0$,
the IX bound from [20] states that we have with at least probability
$1-\delta$: $\displaystyle
R(\pi)\leq\hat{R}^{IX-\lambda}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda n}$ (32)
with:
$\hat{R}^{IX-\lambda}_{n}(\pi)=\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})+\lambda/2}c_{i}.$
Let $U^{\lambda}_{IX}(\pi)$ be the upper bound given of (32), we have for any
$\lambda>0$: $\displaystyle U^{\lambda}_{\infty}(\pi)\leq
U^{\lambda}_{IX}(\pi)\,.$ (33)
###### Proof.
Let $\pi\in\Pi$, $\delta\in]0,1]$ and $\lambda>0$. Recall that
$U^{\lambda}_{\infty}(\pi)=\psi_{\lambda}\Big{(}\hat{R}^{\lambda}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}$. We have:
$\displaystyle\psi_{\lambda}\Big{(}\hat{R}^{\lambda}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}$
$\displaystyle\leq\hat{R}^{\lambda}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\quad(\forall x,\psi_{\lambda}(x)\leq x)$
$\displaystyle\leq-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
w_{\pi}(x_{i},a_{i})c_{i}\right)+\frac{\ln(1/\delta)}{\lambda n}.$
Using the inequality $\log(1+x)\geq\frac{x}{1+x/2}$ for all $x>0$, we get:
$\displaystyle U^{\lambda}_{\infty}(\pi)$
$\displaystyle\leq-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
w_{\pi}(x_{i},a_{i})c_{i}\right)+\frac{\ln(1/\delta)}{\lambda n}$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\frac{w_{\pi}(x_{i},a_{i})}{1-\lambda
w_{\pi}(x_{i},a_{i})c_{i}/2}c_{i}+\frac{\ln(1/\delta)}{\lambda
n}\quad\left(\log(1+x)\geq\frac{x}{1+x/2}\right)$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})-\lambda\pi(a_{i}|x_{i})c_{i}/2}c_{i}+\frac{\ln(1/\delta)}{\lambda
n}$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})+\lambda/2}c_{i}+\frac{\ln(1/\delta)}{\lambda
n}\quad\left(-\pi(a_{i}|x_{i})c_{i}\leq 1\text{ and }c_{i}\leq 0\right)$
$\displaystyle\leq\hat{R}^{IX-\lambda}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}=U^{\lambda}_{IX}(\pi),$
which ends the proof. ∎
The result states the dominance of the LS bound compared to IX. The proof of
this result also gives us insight on when the LS bound will be much tighter
than IX. Indeed, to obtain the IX bound, LS bound is loosened through 3 steps:
1. 1.
The use of $\psi_{\lambda}(x)\leq x,\forall x$.
2. 2.
The use of $\log(1+\lambda x)\geq\frac{\lambda x}{1+\lambda x/2},\forall x\geq
0$.
3. 3.
The use of $-\pi(a_{i}|x_{i})c_{i}\leq 1,\forall i\in[n]$.
The two first inequalities are loose when $\lambda\sim 1/\sqrt{n}$ is not too
small, which means that LS will be much better in problems with few samples.
The third inequality is loose when $\pi$ is not a peaked policy or the cost is
way less than 1. Even if LS bound is always smaller than IX, LS will give way
better result if the number of samples is small, and/or the policy evaluated
is diffused.
### E.3 OPS: Formal comparison with IX suboptimality
Let us begin by stating results from the IX work [20]. Recall that the IX
estimator is defined for any $\lambda>0$, by:
$\hat{R}^{IX-\lambda}_{n}(\pi)=\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})+\lambda/2}c_{i}.$
Let $\Pi_{\textsc{s}}=\\{\pi_{1},...,\pi_{m}\\}$ be a finite set of predefined
policies. In OPS, the goal is to find
$\pi_{*}^{\textsc{s}}\in\Pi_{\textsc{s}}$ that satisfies
$\textstyle\pi_{*}^{\textsc{s}}=\operatorname*{argmin}_{\pi\in\Pi_{\textsc{s}}}R(\pi)=\operatorname*{argmin}_{k\in[m]}R(\pi_{k})\,.$
for $\lambda>0$, the selection strategy suggested in Gabbianelli et al. [20]
was to search for:
$\displaystyle\hat{\pi}_{n}^{\textsc{s},IX}=\operatorname*{argmin}_{\pi\in\Pi_{\textsc{s}}}\hat{R}^{IX-\lambda}_{n}(\pi)=\operatorname*{argmin}_{k\in[m]}\hat{R}^{IX-\lambda}_{n}(\pi_{k})\,.$
(34)
###### Proposition 15.
(Suboptimality of the IX selection strategy. Let $\lambda>0$ and
$\delta\in(0,1]$. Then, it holds with probability at least $1-\delta$ that
$\displaystyle 0\leq
R(\hat{\pi}_{n}^{\textsc{s},IX})-R(\pi_{*}^{\textsc{s}})\leq\lambda\mathcal{C}_{\lambda/2}(\pi_{*}^{\textsc{s}})+\frac{2\ln(2|\Pi_{\textsc{s}}|/\delta)}{\lambda
n}\,,$ (35) where
$\mathcal{C}_{\lambda}(\pi)=\mathbb{E}\left[\frac{\pi(a|x)}{\pi^{2}_{0}(a|x)+\lambda\pi_{0}(a|x)}\lvert
c\rvert\right].$
Both suboptimalities (LS and IX) have the same form, they only depend on two
different quantities ($\mathcal{S}_{\lambda}$ and $\mathcal{C}_{\lambda}$
respectively). For a $\pi\in\Pi$ and $\lambda>0$, If we can identify when
$\mathcal{S}_{\lambda}(\pi)\leq\mathcal{C}_{\lambda/2}(\pi)$, then we can
prove that the sub-optimality of LS selection strategy is better than the one
of IX. Luckily, this is always the case, and it is stated formally below.
###### Proposition 16.
Let $\pi\in\Pi$ and $\lambda>0$. We have:
$\displaystyle\mathcal{S}_{\lambda}(\pi)\leq\mathcal{C}_{\lambda/2}(\pi).$
(36)
###### Proof.
Let $\pi\in\Pi$ and $\lambda>0$, we have:
$\displaystyle\mathcal{C}_{\lambda/2}(\pi)-\mathcal{S}_{\lambda}(\pi)$
$\displaystyle=\mathbb{E}\left[\frac{\pi(a|x)}{\pi^{2}_{0}(a|x)+\frac{\lambda}{2}\pi_{0}(a|x)}\lvert
c\rvert-\frac{w_{\pi}(x,a)^{2}c^{2}}{1-\lambda w_{\pi}(x,a)c}\right]$
$\displaystyle=\mathbb{E}\left[\frac{\pi(a|x)}{\pi^{2}_{0}(a|x)+\frac{\lambda}{2}\pi_{0}(a|x)}\lvert
c\rvert-\frac{\pi(a|x)^{2}c^{2}}{\pi^{2}_{0}(a|x)-\lambda\pi_{0}(a|x)\pi(a|x)c}\right]$
$\displaystyle=\mathbb{E}\left[\pi(a|x)\lvert
c\rvert\left(\frac{1}{\pi^{2}_{0}(a|x)+\frac{\lambda}{2}\pi_{0}(a|x)}-\frac{\pi(a|x)|c|}{\pi^{2}_{0}(a|x)+\lambda\pi_{0}(a|x)\pi(a|x)\lvert
c\rvert}\right)\right]$ $\displaystyle=\mathbb{E}\left[\pi(a|x)\lvert
c\rvert\left(\frac{\pi^{2}_{0}(a|x)\left(1-\pi(a|x)|c|\right)+\frac{\lambda}{2}\pi_{0}(a|x)\pi(a|x)\lvert
c\rvert}{(\pi^{2}_{0}(a|x)+\frac{\lambda}{2}\pi_{0}(a|x))(\pi^{2}_{0}(a|x)+\lambda\pi_{0}(a|x)\pi(a|x)\lvert
c\rvert)}\right)\right]$ $\displaystyle\geq 0.$
∎
This means that the suboptimality of LS selection strategy is better bounded
than the one of IX. Our experiments confirm that the LS selection strategy is
better than IX in practical scenarios.
##### Minimax optimality of our selection strategy.
As discussed in Gabbianelli et al. [20], pessimistic algorithms tend to have
the property that their regret scales with the minimax sample complexity of
estimating the value of the optimal policy [26]. For the case of multi-armed
bandit (one context $x$), this estimation minimax sample complexity is proved
by Li et al. [33] and is of the rate
$\mathcal{O}(\mathbb{E}[w_{\pi^{*}}(x,a)^{2}c^{2}])$, with $\pi^{*}$ being the
optimal policy. Our bound matches the lower bound proved by Li et al. [33],
as:
$\displaystyle\mathcal{S}_{\lambda}(\pi^{*})=\mathbb{E}\left[\frac{w_{\pi^{*}}(x,a)^{2}c^{2}}{1-\lambda
w_{\pi^{*}}(x,a)c}\right]\leq\mathbb{E}\left[w_{\pi^{*}}(x,a)^{2}c^{2}\right],$
which is not the case for the suboptimality of IX, that only matches it in the
deterministic setting with binary costs, as:
$\displaystyle\mathcal{C}_{\lambda}(\pi^{*})$
$\displaystyle=\mathbb{E}\left[\frac{\pi^{*}(a|x)}{\pi^{2}_{0}(a|x)+\lambda\pi_{0}(a|x)}\lvert
c\rvert\right]\leq\mathbb{E}\left[\frac{\pi^{*}(a|x)}{\pi^{2}_{0}(a|x)}\lvert
c\rvert\right]=\mathbb{E}\left[\left(\frac{\pi^{*}(a|x)}{\pi_{0}(a|x)}\right)^{2}c^{2}\right],$
with the last inequality only holding when $\pi^{*}$ is deterministic and the
costs are binary. For deterministic policies and the general contextual
bandit, we invite the reader to see a formal proof of the minimax lower bound
of pessimism in Jin et al. [27, Theorem 4.4], matched for both IX and LS.
### E.4 OPL: Formal comparison of PAC-Bayesian bounds
As it is easier to work with linear estimators within the PAC-Bayesian
framework, we define the following estimator of the risk
$\hat{R}^{p-\textsc{LIN}}_{n}(\pi)$, with the help of a function
$p:\mathbbm{R}\rightarrow\mathbbm{R}$ as:
$\hat{R}^{p-\textsc{LIN}}_{n}(\pi)=\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{p(\pi_{0}(a_{i}|x_{i}))}c_{i}$
with the only condition on $p$ to be
$\\{\boldsymbol{C^{\textsc{LIN}}_{1}}:\forall x,p(x)\geq x\\}$. This condition
helps us control the impact of actions with low probabilities under $\pi_{0}$.
This risk estimator encompasses well known risk estimators depending on the
choice of $p$.
Now that we defined the family of estimators covered by our analysis, we
attack the problem of deriving generalization bounds. We derive our empirical
high order bound expressed in the following:
###### Proposition 17.
Empirical High Order PAC-Bayes bound:
Let $L\geq 1$. Given a prior $P$ on $\mathcal{F}_{\Theta}$, $\delta\in(0,1]$
and $\lambda>0$, the following bound holds with probability at least
$1-\delta$ uniformly for all distribution $Q$ over $\mathcal{F}_{\Theta}$:
$\displaystyle
R(\pi_{Q})\leq\psi_{\lambda}\left(\hat{R}^{p-\textsc{LIN}}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{p-\textsc{LIN},\ell}_{n}(\pi_{Q})\right)$
(37) with: $\displaystyle\hat{\mathcal{M}}^{p-\textsc{LIN},\ell}_{n}(\pi_{Q})$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{\pi_{Q}(a_{i}|x_{i})}{p(\pi_{0}(a_{i}|x_{i}))^{\ell}}c_{i}^{\ell}$
$\displaystyle\psi_{\lambda}$
$\displaystyle=x:\rightarrow\frac{1-\exp(-\lambda x)}{\lambda}.$
###### Proof.
Let $L\geq 1$, we have from Lemma 10, and for any positive random variable
$X\geq 0$ and $\lambda>0$:
$f_{2L-1}(0)=\frac{1}{2L}\geq f_{2L-1}(\lambda X)=-\frac{\log(1+\lambda
X)-\sum_{\ell=1}^{2L-1}\frac{(-1)^{\ell-1}}{\ell}(\lambda X)^{\ell}}{(\lambda
X)^{2L}}$
which is equivalent to:
$\displaystyle\sum_{\ell=1}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\leq\log(1+\lambda X)$
$\displaystyle\iff\exp\left(\sum_{\ell=1}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\right)\leq 1+\lambda X$
$\displaystyle\implies\mathbbm{E}\left[\exp\left(\sum_{\ell=1}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\right)\right]\leq 1+\mathbbm{E}\left[\lambda X\right]$
$\displaystyle\implies\mathbbm{E}\left[\exp\left(\sum_{\ell=1}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\right)\right]\leq\exp\left(\log(1+\mathbbm{E}\left[\lambda
X\right])\right)$
$\displaystyle\implies\mathbbm{E}\left[\exp\left(\lambda(X-\frac{1}{\lambda}\log(1+\mathbbm{E}\left[\lambda
X\right]))+\sum_{\ell=2}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\right)\right]\leq 1.$
For any $X\leq 0$, we can inject $-X\geq 0$ to obtain:
$\displaystyle\forall X\leq
0,\quad\mathbbm{E}\left[\exp\left(\lambda\left(-\frac{1}{\lambda}\log(1+\mathbbm{E}\left[\lambda
X\right])-X\right)-\sum_{k=2}^{2K}\frac{1}{k}(\lambda X)^{k}\right)\right]\leq
1.$ (38)
Let:
$\displaystyle
d_{\theta}(a|x)=\mathbbm{1}\left[f_{\theta}(x)=a\right]\,,\forall(x,a)\in\mathcal{X}\times\mathcal{A}\,,$
it means that:
$\displaystyle\pi_{Q}(a|x)=\mathbb{E}_{\theta\sim
Q}\left[d_{\theta}(a|x)\right]\,,$
$\displaystyle\forall(x,a)\in\mathcal{X}\times\mathcal{A}\,.$
Let $\lambda>0$. The adequate function $g$ we are going to use in combination
with Lemma 11 is:
$\displaystyle g(\theta,\mathcal{D}_{n})$
$\displaystyle=\sum_{i=1}^{n}\lambda\left(-\frac{1}{\lambda}\log(1+\lambda
R^{p-\textsc{LIN}}(d_{\theta}))-\frac{d_{\theta}(a_{i}|x_{i})}{p(\pi_{0}(a_{i}|x_{i}))}c_{i}\right)-\sum_{\ell=2}^{2L}\frac{1}{\ell}\left(\lambda\frac{d_{\theta}(a_{i}|x_{i})}{p(\pi_{0}(a_{i}|x_{i}))}c_{i}\right)^{\ell}$
$\displaystyle=\sum_{i=1}^{n}\lambda\left(-\frac{1}{\lambda}\log(1+\lambda
R^{p-\textsc{LIN}}(d_{\theta}))-\frac{d_{\theta}(a_{i}|x_{i})}{p(\pi_{0}(a_{i}|x_{i}))}c_{i}\right)-\sum_{\ell=2}^{2L}\frac{d_{\theta}(a_{i}|x_{i})}{\ell}\left(\frac{\lambda}{p(\pi_{0}(a_{i}|x_{i}))}c_{i}\right)^{\ell}.$
By exploiting the i.i.d. nature of the data and exchanging the order of
expectations ($P$ is independent of $\mathcal{D}_{n}$), we can naturally prove
using (38) that:
$\displaystyle\Psi_{g}=\mathbbm{E}_{P}\left[\prod_{i=1}^{n}\mathbbm{E}\left[\exp\left(\lambda\left(-\frac{1}{\lambda}\log(1+\lambda
R^{p-\textsc{LIN}}(d_{\theta}))-\frac{d_{\theta}(a_{i}|x_{i})}{p(\pi_{0}(a_{i}|x_{i}))}c_{i}\right)-\sum_{k=2}^{2K}\frac{1}{k}\left(\lambda\frac{d_{\theta}(a_{i}|x_{i})}{p(\pi_{0}(a_{i}|x_{i}))}c_{i}\right)^{k}\right)\right]\right]\leq
1,$
as we have :
$\frac{d_{\theta}(a_{i}|x_{i})}{p(\pi_{0}(a_{i}|x_{i}))}c_{i}\leq
0\quad\forall i.$
Injecting $\Psi_{g}$ in Lemma 11, rearranging terms and using that
$\hat{R}^{p-\textsc{LIN}}_{n}(\pi)$ has positive bias concludes the proof. ∎
Similarly to the OPE section, we use this general bound to obtain a PAC-
Bayesian Empirical Second Moment bound and the PAC-Bayesian LS-LIN bound. That
we state directly below:
##### Empirical second moment bound.
With $L=1$, we obtain the following:
###### Corollary 18.
Second Moment Upper bound:
Given a prior $P$ on $\mathcal{F}_{\Theta}$, $\delta\in(0,1]$ and $\lambda>0$.
The following bound holds with probability at least $1-\delta$ uniformly for
all distribution $Q$ over $\mathcal{F}_{\Theta}$: $\displaystyle
R(\pi_{Q})\leq\psi_{\lambda}\left(\hat{R}^{p}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}+\frac{\lambda}{2}\hat{\mathcal{M}}^{p-\textsc{LIN},2}_{n}(\pi_{Q})\right).$
(39)
##### Log Smoothing PAC-Bayesian Bound.
With $L\rightarrow\infty$, we obtain the following:
###### Proposition 19.
$\hat{R}^{\lambda-\textsc{LIN}}_{n}$ PAC-Bayes bound:
Given a prior $P$ on $\mathcal{F}_{\Theta}$, $\delta\in(0,1]$ and $\lambda>0$,
the following bound holds with probability at least $1-\delta$ uniformly for
all distribution $Q$ over $\mathcal{F}_{\Theta}$: $\displaystyle
R(\pi_{Q})\leq\psi_{\lambda}\left(\hat{R}^{\lambda-\textsc{LIN}}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}\right).$ (40) with:
$\displaystyle\hat{R}^{\lambda\textsc{-lin}}_{n}(\pi)=-\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{\lambda}\log\left(1-\frac{\lambda
c_{i}}{\pi_{0}(a_{i}|x_{i})}\right).$
Following the same proof schema as of the OPE section, we can demonstrate that
the Log Smoothing PAC-Bayesian bound dominates the Empirical Second moment
PAC-Bayesian bound $L=1$. However, we use the bound of $L=1$ as an
intermediary to state the dominance of the Log Smoothing PAC-Bayesian bound.
Indeed, we can easily compare the result obtained with $L=1$ to previously
derived PAC-Bayesian bounds for off-policy learning. We start by writing down
the conditional Bernstein bound of Sakhi et al. [49] holding for the (linear)
cIPS ($p:x\rightarrow\max(x,\tau)$). For a policy $\pi_{Q}$ and a $\lambda>0$,
we have:
$\displaystyle R(\pi_{Q})$
$\displaystyle\leq\hat{R}^{\tau}_{n}(\pi_{Q})+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\sqrt{\frac{\mathcal{KL}(Q||P)+\ln\frac{4\sqrt{n}}{\delta}}{2n}}}}+\frac{\mathcal{KL}(Q||P)+\ln\frac{2}{\delta}}{\lambda
n}+\lambda{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{g\left(\lambda/\tau\right)\mathcal{V}_{n}^{\tau}(\pi_{Q})}}.$
(c-Bern) $\displaystyle R(\pi_{Q})$
$\displaystyle\leq\hat{R}^{\tau}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}+\frac{\lambda}{2}\hat{\mathcal{S}}^{\tau}_{n}(\pi_{Q}).$ (L = 1)
We can observe that the previously derived conditional Bernstein bound has
several terms that make it less tight:
* •
It has an additional, strictly positive square root KL divergence term.
* •
The multiplicative factor $g(\lambda/\tau)$ is always bigger than $1/2$, and
diverges when $\tau\rightarrow 0$.
* •
With enough data ($n\gg 1$), we also have:
$\hat{\mathcal{S}}^{\tau}_{n}(\pi_{Q})\approx\mathbbm{E}\left[\frac{\pi_{Q}(a|x)}{\max\\{\pi_{0}(a|x),\tau\\}^{2}}c(a,x)^{2}\right]\leq\mathbbm{E}\left[\frac{\pi_{Q}(a|x)}{\max\\{\pi_{0}(a|x),\tau\\}^{2}}\right]\approx\mathcal{V}_{n}^{\tau}(\pi_{Q}).$
These observations confirm that the new bound derived with $L=1$ is tighter
than what was previously proposed for cIPS, especially when $n\gg 1$. As our
bound can work for other estimators, we also compare it to a recently proposed
PAC-Bayes bound in Aouali et al. [5] for the exponentially-smoothed estimator
($p:x\rightarrow x^{\alpha}$) with $\alpha\in[0,1]$:
$\displaystyle R(\pi_{Q})$
$\displaystyle\leq\hat{R}^{\alpha}_{n}(\pi_{Q})+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\sqrt{\frac{\mathcal{KL}(Q||P)+\ln\frac{4\sqrt{n}}{\delta}}{2n}}}}+\frac{\mathcal{KL}(Q||P)+\ln\frac{2}{\delta}}{\lambda
n}+\frac{\lambda}{2}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\mathcal{V}_{n}^{\alpha}(\pi_{Q})}}+\hat{\mathcal{S}}^{\alpha}_{n}(\pi_{Q})\right).$
($\alpha$-Smooth) $\displaystyle R(\pi_{Q})$
$\displaystyle\leq\hat{R}^{\alpha}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}+\frac{\lambda}{2}\hat{\mathcal{S}}^{\alpha}_{n}(\pi_{Q}).$ (L = 1)
We can clearly see that the previously proposed bound for the exponentially
smoothed estimator has two additional positive quantities that makes it less
tight than our bound. In addition, computing our bound does not rely on
expectations under $\pi_{0}$ (contrary to the previous bounds that have
$\mathcal{V}_{n}$) which alleviates the need to access the logging policy and
reduce the computations.
This demonstrates the superiority of $L=1$ compared to existing variance
sensitive PAC-Bayesian bounds. It means that $L\rightarrow\infty$ is even
better. We can also prove that the Log smoothing PAC-Bayesian Bound is better
than the one of IX in Gabbianelli et al. [20]. Indeed, using
$\log(1+x)\geq\frac{x}{1+x/2}$ for all $x\geq 0$, we have for any
$P,Q\in\mathcal{P}(\Theta)$ and $\lambda>0$:
$\displaystyle\psi_{\lambda}\left(\hat{R}^{\lambda-\textsc{LIN}}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}\right)$
$\displaystyle\leq\hat{R}^{\lambda-\textsc{LIN}}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}$
$\displaystyle\leq-\frac{1}{n}\sum_{i=1}^{n}\frac{\pi_{Q}(a_{i}|x_{i})}{\lambda}\log\left(1-\frac{\lambda
c_{i}}{\pi_{0}(a_{i}|x_{i})}\right)+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n}$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\frac{\pi_{Q}(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})-\lambda
c_{i}/2}+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda n}$
$\displaystyle\leq\hat{R}^{\lambda-\textsc{IX}}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)+\ln\frac{1}{\delta}}{\lambda
n},$ (IX-bound)
which proves that our bound is better than the IX bound. This means that our
PAC-Bayesian bound is better than all existing PAC-Bayesian off-policy
learning bounds.
### E.5 OPL: Formal comparison with IX PAC-Bayesian learning suboptimality
Let us begin by stating results from the IX work [20]. Recall that the IX
estimator is defined for any $\lambda>0$, by:
$\hat{R}^{\textsc{IX}-\lambda}_{n}(\pi)=\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})+\lambda/2}c_{i},$
and that we used the linearized version of the LS estimator, LS-LIN defined
as:
$\displaystyle\hat{R}^{\lambda\textsc{-lin}}_{n}(\pi)=-\frac{1}{n}\sum_{i=1}^{n}\frac{\pi(a_{i}|x_{i})}{\lambda}\log\left(1-\frac{\lambda
c_{i}}{\pi_{0}(a_{i}|x_{i})}\right).$
Let $\Theta$ be a parameter space and $\mathcal{P}(\Theta)$ be the set of all
probability distribution on $\Theta$. Our goal is to find the best policy in a
chosen class $\mathcal{L}(\Theta)\subset\mathcal{P}(\Theta)$:
$\pi_{Q^{*}}=\operatorname*{argmin}_{Q\in\mathcal{L}(\Theta)}R(\pi_{Q}).$
For $\lambda>0$ and a prior $P\in\mathcal{P}(\Theta)$, the PAC-Bayesian
learning strategy suggested in Gabbianelli et al. [20] is to find in
$\mathcal{L}(\Theta)\subset\mathcal{P}(\Theta)$:
$\displaystyle\hat{\pi}^{\textsc{IX}}_{Q_{n}}=\operatorname*{argmin}_{Q\in\mathcal{L}(\Theta)}\left\\{\hat{R}^{\textsc{IX}-\lambda}_{n}(\pi_{Q})+\frac{\mathcal{KL}(Q||P)}{\lambda
n}\right\\}.$
This learning strategy suffers from a suboptimality bounded in the result
below:
###### Proposition 20.
(Suboptimality of the IX PAC-Bayesian learning strategy from [20]. Let
$\lambda>0$ and $\delta\in(0,1]$. Then, it holds with probability at least
$1-\delta$ that $\displaystyle 0\leq
R(\hat{\pi}^{\textsc{IX}}_{Q_{n}})-R(\pi_{Q^{*}})\leq\lambda\mathcal{C}_{\lambda/2}(\pi_{Q^{*}})+\frac{2\left(\mathcal{KL}(Q^{*}||P)+\ln(2/\delta)\right)}{\lambda
n}\,,$ where
$\mathcal{C}_{\lambda}(\pi)=\mathbb{E}\left[\frac{\pi(a|x)}{\pi^{2}_{0}(a|x)+\lambda\pi_{0}(a|x)}\lvert
c\rvert\right].$
Similarly for PAC-Bayesian learning, both suboptimalities (LS and IX) have the
same form, they only depend on two different quantities
($\mathcal{S}^{\textsc{LIN}}_{\lambda}$ and $\mathcal{C}_{\lambda}$
respectively). For a $\pi\in\Pi$ and $\lambda>0$, If we can identify when
$\mathcal{S}^{\textsc{LIN}}_{\lambda}(\pi)\leq\mathcal{C}_{\lambda/2}(\pi)$,
then we can prove that the sub-optimality of LS PAC-Bayesian learning strategy
is better than the one of IX in certain cases. Luckily, this is always the
case, and it is stated formally below.
###### Proposition 21.
Let $\pi\in\Pi$ and $\lambda>0$. We have:
$\displaystyle\mathcal{S}^{\textsc{LIN}}_{\lambda}(\pi)\leq\mathcal{C}_{\lambda/2}(\pi).$
(41)
###### Proof.
Let $\pi\in\Pi$ and $\lambda>0$, and recall that:
$\mathcal{S}^{{\texttt{LIN}}}_{\lambda}(\pi)=\mathbb{E}\left[\frac{\pi(a|x)c^{2}}{\pi^{2}_{0}(a|x)-\lambda\pi_{0}(a|x)c}\right].$
We have:
$\displaystyle\mathcal{C}_{\lambda/2}(\pi)-\mathcal{S}^{\textsc{LIN}}_{\lambda}(\pi)$
$\displaystyle=\mathbb{E}\left[\frac{\pi(a|x)}{\pi^{2}_{0}(a|x)+\frac{\lambda}{2}\pi_{0}(a|x)}\lvert
c\rvert-\frac{\pi(a|x)c^{2}}{\pi^{2}_{0}(a|x)-\lambda\pi_{0}(a|x)c}\right]$
$\displaystyle=\mathbb{E}\left[\pi(a|x)\lvert
c\rvert\left(\frac{1}{\pi^{2}_{0}(a|x)+\frac{\lambda}{2}\pi_{0}(a|x)}-\frac{|c|}{\pi^{2}_{0}(a|x)+\lambda\pi_{0}(a|x)\lvert
c\rvert}\right)\right]$ $\displaystyle=\mathbb{E}\left[\pi(a|x)\lvert
c\rvert\left(\frac{\pi^{2}_{0}(a|x)\left(1-|c|\right)+\frac{\lambda}{2}\pi_{0}(a|x)\lvert
c\rvert}{(\pi^{2}_{0}(a|x)+\frac{\lambda}{2}\pi_{0}(a|x))(\pi^{2}_{0}(a|x)+\lambda\pi_{0}(a|x)\lvert
c\rvert)}\right)\right]$ $\displaystyle\geq 0.$
∎
Similarly, this means that the suboptimality of LS-LIN PAC-Bayesian learning
strategy is also, better bounded than the one of IX.
##### Minimax optimality of our learning strategy.
From Jin et al. [27, Theorem 4.4] we can state that the minimax suboptimality
lower bound, in the case of deterministic optimal policies is of the rate
$\mathcal{O}(1/\sqrt{nC^{*}})$ with
$\inf_{x\in\mathcal{X}}\pi_{0}(\pi^{*}(x)|x)>C^{*}$. Our bound as well as IX
bound match this minimax lower bound, as:
$\displaystyle\mathcal{S}^{\texttt{LIN}}_{\lambda}(\pi^{*})$
$\displaystyle=\mathbb{E}_{x,c}\left[\frac{c^{2}}{\pi_{0}(\pi^{*}(x)|x)-\lambda
c}\right]\leq\frac{1}{C^{*}}$ $\displaystyle\mathcal{C}_{\lambda}(\pi^{*})$
$\displaystyle=\mathbb{E}_{x,c}\left[\frac{\lvert
c\rvert}{\pi_{0}(\pi^{*}(x)|x)+\lambda}\right]\leq\frac{1}{C^{*}}.$
One can see that for both, selecting a
$\lambda^{*}=\sqrt{\frac{2\left(\mathcal{KL}(Q^{*}||P)+\ln(2/\delta)\right)C^{*}}{n}},$
gets you the desired bound, matching this minimax rate.
## Appendix F Proofs of OPE
### F.1 Proof of high order empirical moments bound (Proposition 1)
###### Proposition.
(Empirical moments risk bound). Let $\pi\in\Pi$, $L\geq 1$, $\delta\in(0,1]$
and $\lambda>0$. Then it holds with probability at least $1-\delta$ that
$\displaystyle
R(\pi)\leq\psi_{\lambda}\Big{(}\hat{R}^{h}_{n}(\pi)+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}\,,$ where $\psi_{\lambda}$ and $\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)$
are defined in (7) and (6), respectively, and recall that
$\psi_{\lambda}(x)\leq x$.
###### Proof.
Let $L\in\mathbbm{N}^{*}$, $\lambda>0$ and $X\geq 0$ a positive random
variable. We have $2L-1\geq 1$, and with the decreasing nature of $f_{(2L-1)}$
(Lemma 10), we also have:
$\displaystyle f_{(2L-1)}(0)\geq f_{2L-1}(\lambda X)$
$\displaystyle\iff\frac{1}{2L}\geq-\frac{\log(1+\lambda
X)-\sum_{l=1}^{2L-1}\frac{(-1)^{\ell-1}}{k}(\lambda X)^{\ell}}{(\lambda
X)^{2L}}$ $\displaystyle\iff\sum_{\ell=1}^{2L}\frac{(-1)^{\ell-1}}{k}(\lambda
X)^{\ell}\leq\log(1+\lambda X)$
$\displaystyle\iff\exp\left(\sum_{\ell=1}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\right)\leq 1+\lambda X$
$\displaystyle\implies\mathbbm{E}\left[\exp\left(\sum_{\ell=1}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\right)\right]\leq 1+\lambda\mathbbm{E}\left[X\right]$
$\displaystyle\implies\mathbbm{E}\left[\exp\left(\sum_{\ell=1}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\right)\right]\leq\exp\left(\left(\log(1+\lambda\mathbbm{E}\left[X\right]\right)\right)$
$\displaystyle\implies\mathbbm{E}\left[\exp\left(\lambda(X-\frac{1}{\lambda}\log\left(1+\lambda\mathbbm{E}\left[X\right]\right))+\sum_{\ell=2}^{2L}\frac{(-1)^{\ell-1}}{\ell}(\lambda
X)^{\ell}\right)\right]\leq 1.$
For any $X\leq 0$, we can inject $-X\geq 0$ to obtain:
$\displaystyle\forall X\leq
0,\quad\mathbbm{E}\left[\exp\left(\lambda\left(-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[X\right]\right)-X\right)-\sum_{\ell=2}^{2L}\frac{1}{\ell}(\lambda
X)^{\ell}\right)\right]\leq 1.$ (42)
The result in Equation (42) will be combined with Chernoff Inequality (Lemma
9) to finally prove our bound. Let $\lambda>0$, for our problem, we define the
random variable $X_{i}$ to use in the Chernoff Inequality as:
$\displaystyle X_{i}$
$\displaystyle=-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)-h_{i}-\sum_{\ell=2}^{2L}\frac{1}{\ell}(\lambda
h_{i})^{\ell}.$
For any $a\in\mathbbm{R}$, this gives us the following:
$\displaystyle
P\left(\sum_{i\in[n]}X_{i}>a\right)\leq\left(\mathbbm{E}\left[\exp\left(\lambda
X_{1}\right)\right]\right)^{n}\exp(-\lambda a)$ $\displaystyle
P\left(-\frac{n}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)-\sum_{i\in[n]}\left(h_{i}+\sum_{\ell=2}^{2L}\frac{1}{\ell}(\lambda
h_{i})^{\ell}\right)>a\right)\leq\left(\mathbbm{E}\left[\exp\left(\lambda
X_{1}\right)\right]\right)^{n}\exp(-\lambda a)$ $\displaystyle
P\left(-\frac{n}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)-\sum_{i\in[n]}\left(h_{i}+\sum_{\ell=2}^{2L}\frac{1}{\ell}(\lambda
h_{i})^{\ell}\right)>a\right)\leq\exp(-\lambda a)\quad(\text{Use of
Equation~{}\eqref{app:mgf_new}})$
Solving for $\delta=\exp(-\lambda a)$, we get:
$\displaystyle
P\left(-\frac{n}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)-\sum_{i\in[n]}\left(h_{i}+\sum_{\ell=2}^{2L}\frac{1}{\ell}(\lambda
h_{i})^{\ell}\right)>\frac{\ln(1/\delta)}{\lambda}\right)\leq\delta$
$\displaystyle
P\left(-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)-\frac{1}{n}\sum_{i\in[n]}\left(h_{i}+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell}}{\ell}h_{i}^{\ell}\right)>\frac{\ln(1/\delta)}{\lambda
n}\right)\leq\delta$ $\displaystyle
P\left(-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)-\hat{R}^{h}_{n}(\pi)-\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)>\frac{\ln(1/\delta)}{\lambda
n}\right)\leq\delta$ $\displaystyle
P\left(-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)>\hat{R}^{h}_{n}(\pi)+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)\frac{\ln(1/\delta)}{\lambda
n}\right)\leq\delta.$
This means that the following, complementary event will hold with probability
at least $1-\delta$:
$\displaystyle-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)\leq\hat{R}^{h}_{n}(\pi)+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)\frac{\ln(1/\delta)}{\lambda
n}.$
$\psi_{\lambda}$ being a non-decreasing function, applying it to the two sides
of this inequality gives us:
$\displaystyle\mathbbm{E}\left[h\right]\leq\psi_{\lambda}\Big{(}\hat{R}^{h}_{n}(\pi)+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}.$
Finally, $h$ satisfies (C1), this means that the bound is also an upper bound
on the true risk, giving:
$\displaystyle
R(\pi)\leq\psi_{\lambda}\Big{(}\hat{R}^{h}_{n}(\pi)+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}\,,$
which concludes the proof. ∎
### F.2 Proof of the impact of $L$ on the bound’s tightness (Proposition 2)
###### Proposition (Impact of $L$ on the bound’s tightness).
Let $\pi\in\Pi$, $\delta\in(0,1]$, $\lambda>0$, and $L\geq 1$. Let
$U^{\lambda,h}_{L}(\pi)=\psi_{\lambda}\left(\hat{R}^{h}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}+\sum_{\ell=2}^{2L}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)\right)$
be the upper bound in Equation 8. Then,
$\displaystyle\lambda\leq\min_{i\in[n]}\left\\{\frac{2L+2}{(2L+1)|h_{i}|}\right\\}\implies
U^{\lambda,h}_{L+1}(\pi)\leq U^{\lambda,h}_{L}(\pi)\,.$ (43) which implies
that: $\lambda\leq\min_{i\in[n]}\left\\{\frac{1}{|h_{i}|}\right\\}\implies
U^{\lambda,h}_{L}(\pi)\text{ is a decreasing function w.r.t $L$.}$
###### Proof.
We want to prove the implication (43) from which the condition on the
decreasing nature of our bound will follow. Indeed, Let us suppose that (43)
is true, we have:
$\displaystyle\lambda\leq\min_{i\in[n]}\left\\{\frac{1}{|h_{i}|}\right\\}$
$\displaystyle\implies\forall L\geq
1,\quad\lambda\leq\min_{i\in[n]}\left\\{\frac{2L+2}{(2L+1)|h_{i}|}\right\\}$
$\displaystyle\implies\forall L\geq 1,\quad U^{\lambda,h}_{L+1}(\pi)\leq
U^{\lambda,h}_{L}(\pi)\quad(\text{Using \eqref{app_eq:K_increase}})$
$\displaystyle\implies U^{\lambda,h}_{L}(\pi)\text{ is a decreasing function
w.r.t $L$.}$
Now let us prove the implication in (43). We have for any $L\geq 1$:
$\displaystyle U^{\lambda,h}_{L+1}(\pi)\leq U^{\lambda,h}_{L}(\pi)$
$\displaystyle\iff\sum_{\ell=2L+1}^{2L+2}\frac{\lambda^{\ell-1}}{\ell}\hat{\mathcal{M}}^{h,\ell}_{n}(\pi)\leq
0$
$\displaystyle\iff\frac{\lambda^{2L}}{n}\sum_{i=1}^{n}h_{i}^{2L+1}\left(\frac{1}{2L+1}+\frac{\lambda
h_{i}}{2L+2}\right)\leq 0$
As $h_{i}\leq 0$, we can ensure this inequality by choosing a $\lambda$ that
verifies:
$\displaystyle\forall
i\in[n],\quad\lambda\leq\left\\{\frac{2L+2}{(2L+1)|h_{i}|}\right\\}\iff\lambda\leq\min_{i\in[n]}\left\\{\frac{2L+2}{(2L+1)|h_{i}|}\right\\}$
which concludes the proof. ∎
### F.3 Proof of the optimality of global clipping for Corollary 3
###### Proposition.
Optimal $h$ for $L=1$:
Let $\lambda>0$. The function $h$ that minimizes the bound for $L=1$, giving
the tightest result is: $\displaystyle\forall i,\quad
h_{i}=h(\pi(a_{i}|x_{i}),\pi_{0}(a_{i}|x_{i}),c_{i}))=-\min\left\\{\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})}|c_{i}|,\frac{1}{\lambda}\right\\}$
This means that when the costs are binary, we obtain the classical Clipping
estimator of parameter $1/\lambda$:
$h_{i}=\min\left\\{\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})},\frac{1}{\lambda}\right\\}c_{i}.$
###### Proof.
We want to look for the value of $h$ that minimizes the bound. Formally, by
fixing all variables of the bound, this problem reduces to:
$\displaystyle\operatorname*{argmin}_{h\in\text{{(C1)}}}\hat{R}^{h}_{n}(\pi)+\frac{\lambda}{2}\hat{\mathcal{M}}^{h,2}_{n}(\pi)$
$\displaystyle=\operatorname*{argmin}_{h\in\text{{(C1)}}}\frac{1}{n}\sum_{i=1}^{n}\left(h_{i}+\frac{\lambda}{2}h_{i}^{2}\right).$
The objective decomposes across data points, so we can solve it for every
$h_{i}$ independently. Let us fix a $j\in[n]$, the following problem:
$\displaystyle\operatorname*{argmin}_{h_{j}\in\mathbbm{R}}\hat{R}^{h}_{n}(\pi)+\frac{\lambda}{2}\hat{\mathcal{M}}^{h,2}_{n}(\pi)=\operatorname*{argmin}_{h_{j}\in\mathbbm{R}}\left\\{h_{j}+\frac{\lambda}{2}h_{j}^{2}\right\\}$
$\displaystyle\text{subject to}\quad
h_{j}\geq\frac{\pi(a_{j}|x_{j})}{\pi_{0}(a_{j}|x_{j})}c_{j}$
is strongly convex in $h_{j}$. We write the KKT conditions for $h_{j}$ to be
optimal; there exists $\alpha^{*}$ that verifies:
$\displaystyle 1+\lambda h_{j}-\alpha^{*}=0$ (44) $\displaystyle\alpha^{*}\geq
0$ (45)
$\displaystyle\alpha^{*}\left(\frac{\pi(a_{j}|x_{j})}{\pi_{0}(a_{j}|x_{j})}c_{j}-h_{j}\right)=0$
(46) $\displaystyle
h_{j}\geq\frac{\pi(a_{j}|x_{j})}{\pi_{0}(a_{j}|x_{j})}c_{j}$ (47)
We study the two following two cases:
##### Case 1:
$h_{j}\leq-\frac{1}{\lambda}:$
we have $\alpha^{*}=1+\lambda h_{j}\leq 0\implies\alpha^{*}=0$, meaning that:
$h_{j}=-\frac{1}{\lambda}$
##### Case 2:
$h_{j}>-\frac{1}{\lambda}:$
we have $\alpha^{*}=1+\lambda h_{j}>0$, which combined to condition (36)
gives:
$h_{j}=\frac{\pi(a_{j}|x_{j})}{\pi_{0}(a_{j}|x_{j})}c_{j}.$
The two results combined mean that we always have:
$h_{j}\geq-\frac{1}{\lambda},\text{ and whenever
}h_{j}>-\frac{1}{\lambda}\implies
h_{j}=\frac{\pi(a_{j}|x_{j})}{\pi_{0}(a_{j}|x_{j})}c_{j}.$
We deduce that $h_{j}$ has the following form:
$\displaystyle h_{j}$
$\displaystyle=h(\pi(a_{j}|x_{j}),\pi_{0}(a_{j}|x_{j}),c_{j})=-\min\left\\{\frac{\pi(a_{j}|x_{j})}{\pi_{0}(a_{j}|x_{j})}\left|c_{j}\right|,\frac{1}{\lambda}\right\\}$
(48) $\displaystyle\alpha^{*}$
$\displaystyle=1-\lambda\min\left\\{\frac{\pi(a_{j}|x_{j})}{\pi_{0}(a_{j}|x_{j})}\left|c_{j}\right|,\frac{1}{\lambda}\right\\}$
(49)
These values verify the KKT conditions. As the problem is strongly convex,
$h_{j}$ has a unique possible value and must be equal to equation (38). The
form of $h_{j}$ is a global clipping that includes the cost in the function as
well. In the case where the cost function $c$ is binary:
$\forall i\quad c_{i}\in\\{-1,0\\},$
we recover the classical Clipping with parameter $1/\lambda$ as an optimal
solution for $h$:
$h_{j}=\min\left\\{\frac{\pi(a_{j}|x_{j})}{\pi_{0}(a_{j}|x_{j})},\frac{1}{\lambda}\right\\}c_{j}.$
∎
### F.4 Proof of the $L\rightarrow\infty$ bound (Corollary 4)
###### Proposition (Empirical Logarithmic Smoothing bound with $L\to\infty$).
Let $\pi\in\Pi$, $\delta\in(0,1]$ and $\lambda>0$. Then it holds with
probability at least $1-\delta$ that $\displaystyle
R(\pi)\leq\psi_{\lambda}\Big{(}-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)+\frac{\ln(1/\delta)}{\lambda n}\Big{)}\,.$
Taking the limit of $L$ naively recovers this form of the bound, but imposes a
condition on $\lambda$ for the bound to converge. We instead, take another
path of proof that does not impose any condition on $\lambda$, developed
below.
###### Proof.
Recall that for the proof of the Empirical moments bounds, we used the
following random variable defined with $\lambda>0$:
$\displaystyle X_{i}$
$\displaystyle=-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)-h_{i}-\sum_{\ell=2}^{2L}\frac{1}{\ell}(\lambda
h_{i})^{\ell},$
combined with Chernoff Inequality (Lemma 9) to prove our bound. If we take the
limit $L\rightarrow\infty$ for our random variable, we obtain the following
random variable:
$\displaystyle\tilde{X}_{i}$
$\displaystyle=-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)+\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)$ $\displaystyle=\frac{1}{\lambda}\log\left(\frac{1-\lambda
h_{i}}{1-\lambda\mathbbm{E}\left[h\right]}\right).$
We use the random variable $\tilde{X}_{i}$ with the Chernoff Inequality. For
any $a\in\mathbbm{R}$, we have:
$\displaystyle
P\left(\sum_{i\in[n]}\tilde{X}_{i}>a\right)\leq\left(\mathbbm{E}\left[\exp\left(\lambda\tilde{X}_{1}\right)\right]\right)^{n}\exp(-\lambda
a)$ $\displaystyle
P\left(-\frac{n}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)+\sum_{i\in[n]}\left(\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)\right)>a\right)\leq\left(\mathbbm{E}\left[\exp\left(\lambda\tilde{X}_{1}\right)\right]\right)^{n}\exp(-\lambda
a)$
On the other hand, we have:
$\mathbbm{E}\left[\exp\left(\lambda\tilde{X}_{1}\right)\right]=\frac{\mathbbm{E}\left[1-\lambda
h_{i}\right]}{1-\lambda\mathbbm{E}\left[h\right]}=1.$
Using this equality and solving for $\delta=\exp(-\lambda a)$, we get:
$\displaystyle
P\left(-\frac{n}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)+\sum_{i\in[n]}\left(\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)\right)>\frac{\ln(1/\delta)}{\lambda}\right)\leq\delta$
$\displaystyle
P\left(-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)+\frac{1}{n}\sum_{i\in[n]}\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)>\frac{\ln(1/\delta)}{\lambda n}\right)\leq\delta$
This means that the following, complementary event will hold with probability
at least $1-\delta$:
$\displaystyle-\frac{1}{\lambda}\log\left(1-\lambda\mathbbm{E}\left[h\right]\right)\leq-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)+\frac{\ln(1/\delta)}{\lambda n}.$
$\psi_{\lambda}$ being a non-decreasing function, applying it to the two sides
of this inequality gives us:
$\displaystyle\mathbbm{E}\left[h\right]\leq\psi_{\lambda}\Big{(}-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)+\frac{\ln(1/\delta)}{\lambda n}\Big{)}.$
As $h$ satisfies (C1), we obtain the required inequality:
$\displaystyle
R(\pi)\leq\psi_{\lambda}\Big{(}-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda}\log\left(1-\lambda
h_{i}\right)+\frac{\ln(1/\delta)}{\lambda n}\Big{)}.$
and conclude the proof. ∎
### F.5 Proof of the optimality of IPS for Corollary 4
###### Proposition.
Optimal $h$ for $L\rightarrow\infty$:
Let $\lambda>0$. The function $h$ that minimizes the bound for
$L\rightarrow\infty$, giving the tightest result is: $\displaystyle\forall
i,\quad
h_{i}=h(\pi(a_{i}|x_{i}),\pi_{0}(a_{i}|x_{i}),c_{i}))=\frac{\pi(a_{i}|x_{i})}{\pi_{0}(a_{i}|x_{i})}c_{i}$
###### Proof.
The proof of this proposition is quite simple. The function:
$f(x)=-\log\left(1-\lambda x\right)$
is increasing. This means that the lowest possible value of $h_{i}$ ensures
the tightest result. As our variables $h_{i}$ verifies (C1), we recover IPS as
an optimal choice for this bound. ∎
### F.6 Proof of $U^{\lambda}_{\infty}$ tightness (Proposition 5)
###### Proposition.
Let $\pi\in\Pi$, and $\lambda>0$, we define:
$U^{\lambda}_{L}(\pi)=\min_{h}U^{\lambda,h}_{L}(\pi).$ Then, for any
$\lambda>0$, it holds that for any $L>1$: $U^{\lambda}_{L}(\pi)\leq
U^{\lambda}_{1}(\pi).$ In particular, $\forall\lambda>0$: $\displaystyle
U^{\lambda}_{\infty}(\pi)\leq U^{\lambda}_{1}(\pi)\,,$ (50)
###### Proof.
Let $\pi\in\Pi$, $\lambda>0$ and
$U^{\lambda}_{L}(\pi)=\min_{h}U^{\lambda,h}_{L}(\pi).$
We already proved that:
$\displaystyle
U^{\lambda}_{1}(\pi)=U^{\lambda,h_{*,1}}_{1}(\pi)=\psi_{\lambda}\Big{(}\hat{R}^{h_{*,1}}_{n}(\pi)+\frac{\lambda}{2}\hat{\mathcal{M}}^{h_{*,1},2}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}$
with:
$h_{*,1}(p,q,c)=-\min(|c|p/q,1/\lambda),$
and that:
$\displaystyle
U^{\lambda}_{\infty}(\pi)=\psi_{\lambda}\Big{(}\hat{R}^{\lambda}_{n}(\pi)+\frac{\ln(1/\delta)}{\lambda
n}\Big{)}.$
From Proposition 2, we have that for any $h$:
$\lambda\leq\min_{i\in[n]}\left\\{\frac{1}{|h_{i}|}\right\\}\implies
U^{\lambda,h}_{L}(\pi)\text{ is a decreasing function w.r.t $L$.}$
It appears that the optimal function $h_{*,1}$ respects this condition, as by
definition:
$\min_{i\in[n]}\left\\{\frac{1}{|(h_{*,1})_{i}|}\right\\}\geq\lambda,$
meaning that:
$U^{\lambda,h_{*,1}}_{L}(\pi)\text{ is a decreasing function w.r.t $L$.}$
This result suggests that the Empirical Second Moment bound, evaluated in its
optimal function $h_{*,1}$, is always bigger than bounds with additional
moments (evaluated in the same $h_{*,1}$). This leads us to the result wanted,
as for any $L>1$:
$\displaystyle U^{\lambda}_{L}(\pi)=\min_{h}U^{\lambda,h}_{L}(\pi)\leq
U^{\lambda,h_{*,1}}_{L}(\pi)\leq
U^{\lambda,h_{*,1}}_{1}(\pi)=U^{\lambda}_{1}(\pi).$
In particular, we get:
$\displaystyle U^{\lambda}_{\infty}(\pi)\leq U^{\lambda}_{1}(\pi),$
which ends the proof. ∎
## Appendix G Proofs of OPS and OPL
### G.1 OPS: Proof of suboptimality bound (Proposition 6)
###### Proposition.
(Suboptimality of our selection strategy in (16)). Let $\lambda>0$ and
$\delta\in(0,1]$. Then, it holds with probability at least $1-\delta$ that
$\displaystyle 0\leq
R(\hat{\pi}_{n}^{\textsc{s}})-R(\pi_{*}^{\textsc{s}})\leq\lambda\mathcal{S}_{\lambda}(\pi_{*}^{\textsc{s}})+\frac{2\ln(2|\Pi_{\textsc{s}}|/\delta)}{\lambda
n}\,,$ where $\pi_{*}^{\textsc{s}}$ and $\hat{\pi}_{n}^{\textsc{s}}$ are
defined in (15) and (16), and
$\mathcal{S}_{\lambda}(\pi)=\mathbb{E}\left[(w_{\pi}(x,a)c)^{2}/\left(1-\lambda
w_{\pi}(x,a)c\right)\right].$ In addition, our upper bound is always finite
as:
$\displaystyle\lambda\mathcal{S}_{\lambda}(\pi)=\lambda\mathbb{E}\left[\frac{(w_{\pi}(x,a)c)^{2}}{1-\lambda
w_{\pi}(x,a)c}\right]\leq\min\left\\{|R(\pi)|,\lambda\mathbb{E}\left[(w_{\pi}(x,a)c)^{2}\right]\right\\}\leq|R(\pi)|.$
###### Proof.
To prove this bound on the suboptimality of our selection method, we need both
an upper bound and a lower bound on the true risk using the LS estimator.
Luckily, we already have derived them in Proposition 13. For a fixed
$\lambda$, taking a union of the two bounds over the cardinal of the finite
policy class $\lvert\Pi_{s}\rvert$, we get the following holding with
probability at least $1-\delta$ for all $\pi\in\Pi_{s}$:
$\displaystyle
R(\pi)-\hat{R}^{\lambda}_{n}(\pi)\leq\frac{\ln(2\lvert\Pi_{s}\rvert/\delta)}{\lambda
n}\,,\qquad\text{and}\qquad\hat{R}^{\lambda}_{n}(\pi)-R(\pi)\leq\lambda\mathcal{S}_{\lambda}(\pi)+\frac{\ln(2\lvert\Pi_{s}\rvert/\delta)}{\lambda
n}\,.$
As $\hat{\pi}_{n}^{\textsc{s}}\in\Pi_{s}$ and by definition of
$\hat{\pi}_{n}^{\textsc{s}}$ (minimizer of $\hat{R}^{\lambda}_{n}(\pi)$), we
have:
$\displaystyle
R(\hat{\pi}_{n}^{\textsc{s}})\leq\hat{R}^{\lambda}_{n}(\hat{\pi}_{n}^{\textsc{s}})+\frac{\ln(2\lvert\Pi_{s}\rvert/\delta)}{\lambda
n}$
$\displaystyle\leq\hat{R}^{\lambda}_{n}(\hat{\pi}_{*}^{\textsc{s}})+\frac{\ln(2\lvert\Pi_{s}\rvert/\delta)}{\lambda
n}.$
Using the lower bound on the risk of $R(\hat{\pi}_{*}^{\textsc{s}})$, we have:
$\displaystyle R(\hat{\pi}_{n}^{\textsc{s}})$
$\displaystyle\leq\hat{R}^{\lambda}_{n}(\hat{\pi}_{*}^{\textsc{s}})+\frac{\ln(2\lvert\Pi_{s}\rvert/\delta)}{\lambda
n}$ $\displaystyle\leq
R(\hat{\pi}_{*}^{\textsc{s}})+\lambda\mathcal{S}_{\lambda}(\hat{\pi}_{*}^{\textsc{s}})+\frac{2\ln(2\lvert\Pi_{s}\rvert/\delta)}{\lambda
n}.$
which gives us the suboptimality upper bound:
$\displaystyle 0\leq |
# A Generic Topological Criterion for Flat Bands in Two Dimensions
Alireza Parhizkar Joint Quantum Institute, University of Maryland, College
Park, MD 20742, USA Victor Galitski Joint Quantum Institute, University of
Maryland, College Park, MD 20742, USA Center for Computational Quantum
Physics, The Flatiron Institute, New York, NY 10010, United States
###### Abstract
Mutually distorted layers of graphene give rise to a moiré pattern and a
variety of non-trivial phenomena. We show that the continuum limit of this
class of models is equivalent to a $(2+1)$-dimensional field theory of Dirac
fermions coupled to two classical gauge fields. We further show that the
existence of a flat band implies an effective dimensional reduction in the
field theory, where the time dimension is “removed.” The resulting two-
dimensional Euclidean theory contains the chiral anomaly. The associated
Atiyah-Singer index theorem provides a self-consistency condition for the
existence of flat bands. In particular, it reproduces a series of quantized
magic angles known to exist in twisted bilayer graphene in the chiral limit
where there is a particle-hole symmetry. We also use this criterion to prove
that an external magnetic field splits this series into pairs of magnetic
field-dependent magic angles associated with flat moiré-Landau bands. The
topological criterion we derive provides a generic practical method for
finding flat bands in a variety of material systems including but not limited
to moiré bilayers.
Moiré phenomena in twisted bilayer graphene [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
and other systems [11, 12] have been an active topic of research in recent
years. Of particular interest for the former are flat bands, where localized
electrons exhibit a variety of strongly-correlated phases. In this Letter we
draw a universal picture for the occurrence of the flat bands through emergent
gauge fields and the anomaly.
Consider a bilayer graphene system with a small twist and/or strain applied to
the layers. As emphasized in Ref. [11], both can be described in terms of a
diffeomorphism:
$\bm{\mathbf{\xi}}_{\omega}\equiv\epsilon\omega(\mathbf{r})\mathbf{r}$ for
strain and
$\bm{\mathbf{\xi}}_{\theta}\equiv\epsilon\theta(\mathbf{r})\hat{z}\times\mathbf{r}$
for a twist (parameterized by $\epsilon\ll 1$). With
$\bm{\mathbf{\xi}}_{\omega}$ and $\bm{\mathbf{\xi}}_{\theta}$ being orthogonal
to each other and thus forming a basis, all possible flows in two dimensions
can be described as a linear combination of these two fields. Therefore, a
bilayer system with a general infinitesimal deformation can be achieved by
applying the diffeomorphism
$\mathbf{r}\rightarrow\mathbf{r}+\bm{\mathbf{\xi}}_{\omega,\theta}/2$ to one
layer and
$\mathbf{r}\rightarrow\mathbf{r}-\bm{\mathbf{\xi}}_{\omega,\theta}/2$ to the
other, as in Fig. 1. A finite deformation is reached by successively applying
such diffeomorphisms. The time dependent moiré fields $\omega(t,\mathbf{r})$
and $\theta(t,\mathbf{r})$ can be further looked at as bosonic phonon-like
degrees of freedom for the bilayer system.
Figure 1: Each blue hexagon designates the Brillouin zone of a single layer of
graphene that has gone under either a strain, $\bm{\mathbf{\xi}}_{\omega}$, or
a twist, $\bm{\mathbf{\xi}}_{\theta}$. In (a) the Brillouin zones are
expanded/shrunken with respect to the undeformed Brillouin zone (green dotted
hexagon), where as in (b) they are rotated with respect to each other. Each
pair gives rise to its moiré reciprocal lattice (shown by red hexagonals).
Also $K$ points are designated by black dots.
Focusing on only one valley, $K$, in the Brillouin zone, an electron that hops
from the $K$-point of one layer to that of the other, can do so along three
different momentum vectors that will form the moiré reciprocal lattice vectors
of the bilayer system. This is because the deformation has separated the
equivalent $K$-points in three different ways (along
$\mathbf{q}^{\theta}_{1,2,3}$ for twist and $\mathbf{q}^{\omega}_{1,2,3}$ for
strain as depicted in Fig. 1) that are related to each other by a $2\pi/3$
rotation. For the general deformation these vectors are given by
$\mathbf{q_{1}}\equiv
2K_{D}(\sinh\frac{\omega}{2}\hat{x}-\sin\frac{\theta}{2}\hat{y})$ with
$\mathbf{q}_{2,3}$ derived by successive $2\pi/3$ rotations of
$\mathbf{q}_{1}$, while $K_{D}$ is the distance between $K$-points of the
undeformed Brillouin zone and its center.
The characteristic length of the superstructure, i.e. the moiré lattice
constant, $L$, can be read off of the magnitude of the moiré reciprocal
lattice vectors,
$|\mathbf{q}|=2K_{D}(\sinh^{2}\frac{\omega}{2}+\sin^{2}\frac{\theta}{2})^{1/2}$,
which is equal to the distance between the corresponding $K$-points (denoted
by $k$ and $k^{\prime}$ in Fig. 1): $L=4\pi/3|\mathbf{q}|$. If the electron
hops from one layer to the other without changing its position, it acquires a
phase determined only by the moiré reciprocal lattice vectors. The dynamics of
this electron is given by the following Hamiltonian density [8],
$\displaystyle
h(\mathbf{r})\\!=\\!\\!\left[\\!\begin{array}[]{cc}-i\nu\not{\partial}&T(\mathbf{r})\\\
T^{\dagger}(\mathbf{r})&-i\nu\not{\partial}\end{array}\\!\right]\\!,\
T(\mathbf{r})\\!\equiv\\!\\!\left[\\!\begin{array}[]{cc}vV(\mathbf{r})&uU^{*}(-\mathbf{r})\\\
uU(\mathbf{r})&vV(\mathbf{r})\end{array}\\!\right]\\!,$ (5)
where $-i\nu\not{\partial}$ is the Hamiltonian density for a single layer of
graphene with $\not{\partial}\equiv\sigma^{i}\partial_{i}$, and
$T(\mathbf{r})$ is the interlayer tunneling matrix and encodes the periodic
profile of the moiré pattern, Fig. 3.
$V(x)=\sum_{j}e^{-i\mathbf{q}_{j}\cdot\mathbf{r}}$ takes the electron from one
sublattice (either $A$ or $B$) to the same sublattice on the other layer,
while $U(x)=\sum_{j}e^{-i\mathbf{q}_{j}\cdot\mathbf{r}}e^{i(j-1)2\pi/3}$ takes
it to the opposing sublattice, with $v$ and $u$ being the corresponding
tunneling amplitudes and $j=1,2,3$.
Using the algebra of gamma matrices,
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}$, with $\eta_{\mu\nu}$ as the
metric, and the unitary transformation,
$\displaystyle h(\mathbf{r})\rightarrow\Omega h(\mathbf{r})\Omega^{\dagger}\ \
\ \text{with}\ \ \
\Omega=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}-\mathds{1}&\mathds{1}\\\
\sigma^{z}&\sigma^{z}\end{array}\right]\,,$ (8)
we can write the Hamiltonian density (5) in terms of Dirac fermions coupled to
non-fluctuating gauge fields as follows
$\displaystyle
H=\int\\!d^{2}x\Big{[}\,\bar{\psi}i\nu\gamma^{a}\left(\partial_{a}+i\mathcal{A}_{a}\gamma_{5}+i\mathcal{S}_{a}i\gamma_{3}\right)\psi$
$\displaystyle-\bar{\psi}\gamma^{0}\mathcal{A}_{0}\gamma_{5}\psi-\bar{\psi}\gamma^{0}\mathcal{S}_{0}i\gamma_{3}\psi\Big{]}\,,$
(9)
with $\bar{\psi}\equiv\psi^{\dagger}\gamma^{0}$. Or in the even tidier action
formulation,
$S=\int\\!d^{3}x\,\bar{\psi}i\not{D}\psi,\quad\\!\not{D}\equiv\gamma^{\mu}\left(\partial_{\mu}+i\mathcal{A}_{\mu}\gamma_{5}+i\mathcal{S}_{\mu}i\gamma_{3}\right),$
(10)
where $a=1,2$, $\mu=0,1,2$ and the field components are,
$\displaystyle\mathcal{A}_{0}=-\frac{v}{\nu}\text{Re}[V(\mathbf{r})],\ \
\mathcal{S}_{0}=\frac{v}{\nu}\text{Im}[V(\mathbf{r})],$ (11)
$\displaystyle\mathcal{A}_{1}=\frac{u}{2\nu}\text{Re}[U(\mathbf{r})+U(-\mathbf{r})],\
\mathcal{A}_{2}=\frac{u}{2\nu}\text{Im}[U(\mathbf{r})+U(-\mathbf{r})],$
$\displaystyle\mathcal{S}_{1}=\frac{u}{2\nu}\text{Im}[U(\mathbf{r})-U(-\mathbf{r})],\
\mathcal{S}_{2}=\frac{u}{2\nu}\text{Re}[U(-\mathbf{r})-U(\mathbf{r})].$
The total field strength associated with $\not{D}$ is given by
$F_{\mu\nu}=\gamma_{5}F^{\mathcal{A}}_{\mu\nu}+i\gamma_{3}F^{\mathcal{S}}_{\mu\nu}+2i\gamma_{5}\gamma_{3}\mathcal{A}_{\mu}S_{\nu}$,
with $F_{\mu\nu}^{\mathcal{A},\mathcal{S}}$ being the field strengths
generated by $\mathcal{A}_{\mu}$ and $\mathcal{S}_{\mu}$ respectively.
Looking at the action (10), we see that the bilayer problem has transformed
into that of Dirac fermions moving in a $(2+1)$ dimensional spacetime and
acted upon by two axial-vector fields: A chiral gauge field,
$\mathcal{A}_{\mu}$, and a spin field, $\mathcal{S}_{\mu}$. Note that
$\gamma_{3}$ measures the spin along the direction normal to the material
plane. The gauge fields, (11), are periodic with periodicity $\sqrt{3}L$,
hence their corresponding field strengths are proportional to $1/L$ and also
periodic with the same period, while the distance between each minimum and its
neighboring maximum is $L$ (see Fig. 2 for example). In particular the spatial
part of the field strength
$F^{\mathcal{A}}_{12}=\partial_{2}\mathcal{A}_{1}-\partial_{1}\mathcal{A}_{2}$
is given by
$\displaystyle B(\mathbf{r})\equiv
F^{\mathcal{A}}_{12}=\frac{u}{\nu}\sum_{j}\hat{\mathbf{q}}^{\theta}_{j}\cdot\bm{\mathbf{\nabla}}(\mathbf{q}_{j}\cdot\mathbf{r})\,\sin\\!{\left(\mathbf{q}_{j}\cdot\mathbf{r}\right)}\,,$
(12)
where $\hat{\mathbf{q}}^{\theta}_{j}\cdot\bm{\mathbf{\nabla}}$ is the
derivative along the unit vector
$\hat{\mathbf{q}}^{\theta}_{j}\equiv\mathbf{q}^{\theta}_{j}/|q^{\theta}_{j}|$.
That of $\mathcal{S}_{a}$ is given by
$\sum_{j}\hat{\mathbf{q}}^{\theta}_{j}\cdot\bm{\mathbf{\nabla}}(\mathbf{q}_{j}\cdot\mathbf{r})\,\cos\\!{\left(\mathbf{q}_{j}\cdot\mathbf{r}\right)}$.
Figure 2: (a) The magnetic field $B$ created by $\mathcal{A}_{a}$ felt by
$\psi_{+}$ while its negative is exerted upon the $\psi_{-}$. (b) The field
strength associated with the spin field. (c) Vector fields $\mathcal{A}_{a}$
(blue) and $\mathcal{S}_{a}$ (red). The black line designates two magnetic
regions related by parity. Parallel sides are identified with each other and
the whole system can be reconstructed by sewing these together. The
$\mathcal{S}_{a}$ field is zero everywhere on the green dashed hexagon. (d)
The chiral scalar potential $\mathcal{A}_{0}$ as experienced by $\psi_{+}$ and
(e) that of $\mathcal{S}_{0}$. Except for (c) the fields are zero on the black
curves. Note how $\mathcal{S}_{0}$ coincides with the magnetic field in (a)
and also how $\mathcal{A}_{0}$ coincides with field strength in (b), in
particular $\mathcal{S}_{0}$ vanishes on edge of each magnetic region. Also
note that a fermion configuration localized on the edge of each magnetic
region will be perpendicular to $\mathcal{S}_{a}$ (see (c) for example).
Consider the following chiral transformations associated with action (10)
$\displaystyle\left\\{\\!\begin{array}[]{cc}\bar{\psi}\rightarrow\bar{\psi}e^{i\alpha\gamma_{5}}\\\
\psi\rightarrow e^{i\alpha\gamma_{5}}\psi\end{array}\\!\right\\}\\!,\
\left\\{\\!\begin{array}[]{cc}\bar{\psi}\rightarrow\bar{\psi}e^{i\alpha
i\gamma_{3}}\\\ \psi\rightarrow e^{i\alpha
i\gamma_{3}}\psi\end{array}\\!\right\\}\\!,\
\left\\{\\!\begin{array}[]{cc}\bar{\psi}\rightarrow\bar{\psi}e^{i\alpha\Gamma}\\\
\psi\rightarrow e^{i\alpha\Gamma}\psi\end{array}\\!\right\\}\\!,$ (19)
with $\Gamma\equiv\gamma_{0}\gamma_{3}\gamma_{5}$. Each one of the above
chiral transformations can become a symmetry of the action (10) under
additional constraints: $\psi\rightarrow e^{i\alpha\gamma_{5}}\psi$,
$\psi\rightarrow e^{i\alpha i\gamma_{3}}\psi$, and $\psi\rightarrow
e^{i\alpha\Gamma}\psi$ are promoted to symmetries if $\mathcal{S}_{\mu}=0$,
$\mathcal{A}_{\mu}=0$, and $\mathcal{A}_{0}=\mathcal{S}_{0}=0$ respectively.
Chiral particles can be defined with respect to each of these symmetries by
using projection operators, e.g.
$\psi_{\pm}\equiv\frac{1}{2}(1\pm\gamma_{5})\psi$. Since $\gamma_{5}$ and
$\gamma_{3}$ do not commute we cannot simultaneously create fermions with
definite $\gamma_{5}$ and $\gamma_{3}$ handedness, in contrast to $\Gamma$
which commutes with both. So we can, for example, have
$\psi_{{\uparrow\\!\downarrow}}^{\circlearrowright,\circlearrowleft}\equiv\frac{1}{4}(1\pm
i\gamma_{3})(1\pm\Gamma)\psi$.
Our interest here is focused, more than anything else, on flat bands, which
can be looked at as a class of modes covering the whole Brillouin zone at
constant energy,
$\bar{\psi}_{k}\gamma^{0}\partial_{0}\psi_{k}=\mu\psi^{\dagger}_{k}\psi_{k}$,
within which the electrons are therefore localized $\partial E_{k}/\partial
k=0$. If we only consider this class, we eliminate the time dependence from
the action entirely and reduce the $(2+1)$ dimensional theory to its $(2+0)$
dimensional version. Specifically, in the case of $v=0$, which supports exact
flat bands [4], we have
$I=\int\\!\mathcal{D}\bar{\psi}\psi\exp\\!\left\\{\int\\!d^{2}x\,\bar{\psi}i\gamma^{a}\left(\partial_{a}+i\mathcal{A}_{a}\gamma_{5}+i\mathcal{S}_{a}i\gamma_{3}\right)\psi\right\\}\\!.$
(20)
Using any of the chiral projections,
$\psi^{\circlearrowright,\circlearrowleft}_{\pm\ \text{or}\
{\uparrow\\!\downarrow}}$, we can break the above path-integral further down,
for instance, to
$\displaystyle
I=\int\\!\mathcal{D}\bar{\psi}_{\pm}\psi_{\pm}\exp\\!\Bigg{\\{}\int\\!d^{2}x\Big{[}\bar{\psi}_{\pm}i\gamma^{a}\left(\partial_{a}\pm
i\mathcal{A}_{a}\right)\psi_{\pm}$ $\displaystyle\quad\
+\bar{\psi}_{\mp}\gamma^{a}\epsilon_{ab}\mathcal{S}^{b}\psi_{\pm}\Big{]}\Bigg{\\}}\,,$
(21)
where the path-integral is over the four field variables $\bar{\psi}_{\pm}$
and $\psi_{\pm}$, while $\psi_{\pm}$ fermions are coupled to
$\pm\mathcal{A}_{a}$.
In this form the anomaly residing in the theory given by the path integral
(20) takes the familiar shape of the chiral anomaly in two dimensional
Euclideanized spacetime. In a path integral such as above the gauge field
$\mathcal{A}_{a}$ has an index associated with it that is directly given by
the chiral anomaly [13, 14]. But the index must be an integer number which as
we will see is only possible for certain values of $\theta$ and $\omega$ that
coincide with the magic angle. This consistency condition can therefore tell
us whether a flat band exists or not. Before proceeding to a more detailed
investigation, it is worth mentioning that this reasoning, following the
reduction from Eq. (10) to Eq. (21), is generalizable to other more
complicated systems such as multilayer graphene in which case we should use
higher dimensional gamma matrices to accommodate for the additional layers.
Since the gauge potentials are periodic the path integral can be divided into
equivalent patches sewn together by an integration over all field
configurations on the boundaries.
$\displaystyle
I=\int\prod_{\triangledown}\mathcal{D}\bar{\psi}_{\partial\triangledown}\psi_{\partial\triangledown}\,I_{\triangledown}\left[\bar{\psi}_{\partial\triangledown},\psi_{\partial\triangledown}\right]\,,$
(22) $\displaystyle\text{with}\quad
I_{\triangledown}\left[\bar{\psi}_{\partial\triangledown},\psi_{\partial\triangledown}\right]=\int_{\bar{\psi}_{\partial\triangledown},\psi_{\partial\triangledown}}\mathcal{D}\bar{\psi}_{\triangledown}\psi_{\triangledown}e^{iS_{\triangledown}}\,,$
(23)
where ${\partial\triangledown}$ designates the boundary of the patches and the
configuration residing on it, while $\triangledown$ designates the patch
itself. $I_{\triangledown}$ is the path-integral over all configurations on
one patch, $[\bar{\psi}_{\triangledown},\psi_{\triangledown}]$, that go to
$[\bar{\psi}_{\partial\triangledown},\psi_{\partial\triangledown}]$ on the
boundary. Also $S_{\triangledown}$ is the same action before but with an
integral only over $\triangledown$. The patches are chosen so that the action
$S_{\triangledown}$ is the same in all $I_{\triangledown}$. While satisfying
this property, we choose $\triangledown$ so that $\mathcal{S}_{a}$ will either
vanish on or be perpendicular to ${\partial\triangledown}$. This way the edge
configurations along ${\partial\triangledown}$ will have an additional chiral
symmetry with respect to $e^{i\alpha\gamma_{5}}$ and the edge mode residing on
${\partial\triangledown}$ will obtain no phase from $\mathcal{S}_{a}$.
If the fermions remain confined within one patch, which should be the case
when they are localized, we can exclude from the path-integration those
configurations that connect different patches together. The probability
density and current are here given by $\psi^{\dagger}\psi$ and
$j^{a}\equiv\bar{\psi}\gamma^{\mu}\psi$ respectively. We expect the excluded
configurations to be those with a nonzero flow of probability current,
$j^{a}$, out of $\triangledown$:
$\int_{\triangledown}\partial_{0}\psi^{\dagger}\psi=\int_{\triangledown}\partial_{a}j^{a}=\int_{\partial\triangledown}\hat{n}_{\partial\triangledown}^{a}\cdot
j_{a}\neq 0$. In that case the problem is reduced, from the initial path-
integral $I$, to segregated path-integrals of
$I_{\triangledown}=\int\mathcal{D}\bar{\psi}_{\triangledown}\psi_{\triangledown}\exp\\{iS_{\triangledown}\\}$.
If, moreover, there is a flat-band then the transition amplitude and therefore
the path-integral, from any state to any other state within the flat-band
would be time independent
$\left<\zeta\right|e^{iHt}\left|\chi\right>\equiv\int^{\zeta}_{\chi}\mathcal{D}\bar{\psi}\psi
e^{iS}=\left<\zeta\
\vphantom{\chi}\right|\left.\chi\vphantom{\zeta}\right>\,.$ (24)
This allows us to reduce the theory to $(2+0)$ dimensions as in Eq. (10) to
Eq. (21):
$\displaystyle
I_{\triangledown}=\int_{\triangledown}\\!\mathcal{D}\bar{\psi}_{\pm}\psi_{\pm}\exp\\!\Bigg{\\{}\int_{\triangledown}\\!d^{2}x\Big{[}\bar{\psi}_{\pm}i\gamma^{a}\left(\partial_{a}\pm
i\mathcal{A}_{a}\right)\psi_{\pm}$ $\displaystyle\quad\
+\bar{\psi}_{\mp}\gamma^{a}\epsilon_{ab}\mathcal{S}^{b}\psi_{\pm}\Big{]}\Bigg{\\}}\,,$
(25)
where we have removed the $\triangledown$ sign from the fermionic field
variables (and brought it under the path-integral sign instead) in order to
avoid clutter.
Focusing only on $I_{\triangledown}$, we see however, that not all gauge field
configurations fit within the boundaries ${\partial\triangledown}$; only those
with a complete integer index,
$n_{\circlearrowright}-n_{\circlearrowleft}\in\mathbb{Z}$. One way to observe
this is first to notice that a continuous chiral rotation of
$\psi_{+}\rightarrow e^{i2\pi\Gamma}\psi_{+}=\psi_{+}$ takes the spinor field
to itself while leaving the action unchanged. However, the theory (25) is
subject to chiral anomaly, namely, the Jacobian of our chiral transformation,
$J_{5}$, is non-trivial,
$\displaystyle I_{\triangledown}=$
$\displaystyle\int_{\triangledown}\\!\left[\mathcal{D}\bar{\psi}_{+}\psi_{+}\right]\mathcal{D}\bar{\psi}_{-}\psi_{-}e^{iS_{\triangledown}}\longrightarrow$
(26)
$\displaystyle\int_{\triangledown}\\!\left[\mathcal{D}\bar{\psi}_{+}\psi_{+}J_{5}\right]\mathcal{D}\bar{\psi}_{-}\psi_{-}e^{iS_{\triangledown}}=I_{\triangledown}e^{i2\pi(n_{\circlearrowright}-n_{\circlearrowleft})}\,.$
The last equality above comes from knowing that the Jacobian of chiral
transformation is connected to the Atiyah-Singer index [15, 13, 14]. Chiral
transformation discriminates between right and left handed modes,
$n_{\circlearrowright}-n_{\circlearrowleft}$, hence the path-integral (which
yields the determinant of the Dirac operator) obtains a phase associated with
this difference. This phase encodes the winding number of the gauge field
associated with the Dirac operator—an integer number which in two dimensional
spacetime is written as,
$\frac{1}{2\pi}\int_{\triangledown}d^{2}x\,\epsilon^{ab}\partial_{a}\mathcal{A}_{b}=n_{\circlearrowright}-n_{\circlearrowleft}\,.$
(27)
But since the field variables do not change by a complete rotation, in Eq.
(26), the path integral must also remain the same:
$I_{\triangledown}=I_{\triangledown}e^{i2\pi(n_{\circlearrowright}-n_{\circlearrowleft})}$.
Thus, if $e^{i2\pi(n_{\circlearrowright}-n_{\circlearrowleft})}\neq 1$, then
the only way that the initial and the transformed path-integrals can be equal
is for them to vanish, $I_{\triangledown}=0$. This zero valued partition
function means that the state is unrealizable. In contrast, the flat-band can
be realized if $n_{\circlearrowright,\circlearrowleft}\in\mathbb{Z}$.
What we have discussed so far applies generally to all deformations of any
bilayer system that shares the symmetries of graphene. Let us now focus on
uniform twist for which
$\mathbf{q}_{1}=\mathbf{q}_{1}^{\theta}=q^{\theta}(-1,0)$ and
$\mathbf{q}_{2,3}=\mathbf{q}_{2,3}^{\theta}=q^{\theta}(\pm\sqrt{3}/2,1/2)$
with $q^{\theta}=2K_{D}\sin(\theta/2)=4\pi/3L$. The gauge fields
$\mathcal{A}_{a}$ and $\mathcal{S}_{a}$ generated by uniform twist are
divergenceless with their corresponding field strengths proportional to $u/\nu
L$. The gauge potentials and field strengths are shown in Fig. 2. We need to
calculate the minimum value of $L$ corresponding to $n_{\circlearrowright}=1$.
Using Fujikawa’s method of calculating anomalies [13] we find,
$\displaystyle
n_{\circlearrowright}\\!=\\!\frac{1}{4\pi}\int_{\triangledown}\\!d^{2}x\,\epsilon^{ab}\partial_{a}\mathcal{A}_{b}=\frac{1}{4\pi}\int_{\triangledown}\\!d^{2}x\,B=\frac{3\sqrt{3}u}{4\pi\nu}L\,,$
(28)
which is equal to $1$ for $L=L_{0}\equiv 4\pi\nu/3\sqrt{3}u$. Considering
$L=a/2\sin(\theta/2)$, where $a=2.46\text{\AA}$ is the graphene lattice
constant, the first magic angle is given by $\theta\approx 3\sqrt{3}au/4\pi
v\approx 1.1^{\circ}$ with $u=0.11\text{eV}$ and $\nu K_{D}=9.9\text{eV}$.
To develop semi-classical intuition, let us first assume that the spin current
term $\mathcal{S}_{a}\bar{\psi}\gamma^{a}\gamma_{3}\psi$ is disregardable.
Then we are left with only a magnetic field with strength proportional to
$u/\nu L$ acting with opposite signs on $\psi_{\pm}$ fermions that are
completely decoupled from each other. If the magnetic field was constant
across the material then the electrons would have been subject to Landau
localization rotating around a fixed center, in a semi-classical picture, and
forming Landau levels. Since the moiré effective magnetic field is
inhomogeneous, the semi-classical picture changes to that of drifting
fermions—rotating around a cycling center. See Fig. 3. For localization to be
possible, the drifting fermion should be able to fit into one magnetic region.
The smallest rotating fermion, according to the uncertainty principle, has a
size of $\ell_{B}\propto 1/\sqrt{\bar{B}}$ and, since here the average
magnetic field $\bar{B}$ is proportional to $1/L$, it expands with $\sqrt{L}$.
But the size of the magnetic regions, or $\triangledown$, is proportional to
$L$ which grows faster than $\ell_{B}\propto\sqrt{L}$. Thus $\triangledown$
gets bigger faster than the smallest possible electron and eventually can
catch one, at which moment one electron has just been trapped inside the
magnetic region and can complete a cycle there without getting out of it.
Figure 3: On the left, a generic electron (red trajectory) drifting in the
magnetic region while another electron (green trajectory) is moving close to
the edge of one magnetic region. A right or left handed fermion belongs only
to either of the magnetic regions but since the sides are identified a right
handed fermion in one region is the left handed fermion in the other. Having
one electron in both regions is similar to having two electrons in one region
and forgetting about the other. On the right the twisted bilayer graphene at
first magic angle with blue dots denoting AA stacking and yellow/green dots
AB/BA stacking. The distance between the neighboring equivalent dots is equal
to $L$.
At that exact moment we expect to have an edge mode on the boundary of the
magnetic region. Appropriately, for this mode the spin current term is indeed
disregardable since $\mathcal{S}_{a}$ and $\bar{\psi}\gamma^{a}\gamma_{3}\psi$
are perpendicular to each other at ${\partial\triangledown}$. But if there is
an edge mode it means that the number of right or left handed fermions that
reside in the magnetic region must at least be one,
$n_{\circlearrowright,\circlearrowleft}=1$. This again leads us to Eq. (28)
and the magic angle. This also can be seen in terms of unit of flux and a
restricted type of Landau quantization. The size of the unit flux is given by
$\ell_{B}=\sqrt{L\nu/u}$. Therefore, $n=L^{2}/2\pi\ell_{B}^{2}=uL/2\pi\nu$
will be the degeneracy of the restricted Landau level, at least for large
$n\in\mathbb{Z}$. Thus we expect a series of magic angles connected to each
other by steps of $\delta L\approx 2\pi\nu/u$.
Reformulating the theory in terms of Dirac fermions (10) has other merits as
well. For example, the application of an external electromagnetic field yields
the same theory (10) but now with the Dirac operator carrying an additional
external gauge field, $A_{\mu}$,
$\not{D}\equiv\gamma^{\mu}\left(\partial_{\mu}+iA_{\mu}+i\mathcal{A}_{\mu}\gamma_{5}+i\mathcal{S}_{\mu}i\gamma_{3}\right)\,.$
Upon projecting the Dirac fermions into
$\psi_{\pm}\equiv\frac{1}{2}(1\pm\gamma_{5})\psi$ as before, we see that the
chiral fermions are now coupled to the shifted gauge fields
$A_{a}\pm\mathcal{A}_{a}$. For example, if $A_{a}$ is due to a constant
magnetic field $H$, Eq. (28) reads
$\displaystyle
n_{\circlearrowright}=\frac{1}{4\pi}\int_{\triangledown}\\!d^{2}x\,\left(H\pm
B\right)=\frac{H}{4\pi}\frac{3\sqrt{3}}{4}L^{2}\pm\frac{3\sqrt{3}u}{4\pi\nu}L\,.$
(29)
For $u^{2}/\nu^{2}>4\pi H/3\sqrt{3}$, the requirement
$n_{\circlearrowright}=\pm 1$ has more than one solution in contrast to Eq.
(28). Thus, an external magnetic field splits each magic angle into
$L=\pm\frac{u}{\nu}\frac{2}{H}\pm\sqrt{\left(\frac{u}{\nu}\frac{2}{H}\right)^{2}\pm\frac{16\pi}{3\sqrt{3}H}}\,,$
(30)
with the magic angles given by $\theta=2\arcsin(L/2)$, while each combination
of pluses and minuses above yields a solution to Eq. (29). For a small
external magnetic field $H\ll(3\sqrt{3}/4\pi)u^{2}/\nu^{2}\approx
140\text{mT}$ the positive solutions can be written as,
$L_{1}^{\pm}=L_{0}\pm\frac{4\pi^{2}}{27}\frac{\nu^{3}}{u^{3}}H\,,\ \
L_{2}^{\pm}=\frac{4}{H}\frac{u}{\nu}\pm
L_{0}-\frac{4\pi^{2}}{27}\frac{\nu^{3}}{u^{3}}H\,.$ (31)
with $L_{0}\equiv 4\pi\nu/3\sqrt{3}u$, being the magic angle in the absence of
the external magnetic field. In the limit $H\rightarrow 0$ we regain the
previous magic angle from $L_{1}^{\pm}$, in addition to having
$L_{2}^{\pm}\rightarrow\infty$ that happens when the bilayer is untwisted,
$\theta\rightarrow 0$, and the moiré reciprocal lattice, which would have a
vanishing size, is in fact flat.
Let us also consider applying a uniform strain on top of the already existing
uniform twist. The moiré pattern will rotate by
$\arccos(q^{\theta}/|\mathbf{q}|)$ and its length, $L$, will shrink
accordingly. In this case, using Eq. (12), the magnetic field generated by
$\mathcal{A}_{a}$ and its corresponding index are given by
$B=q^{\theta}\frac{u}{\nu}\sum_{j}\sin(\mathbf{q}_{j}\cdot\mathbf{r})$ and
$n_{\circlearrowright}=(q^{\theta}/|\mathbf{q}|)L/L_{0}$ respectively.
Comparing this with Eq. (28) we see that the flat-band now happens at,
$L=L_{0}\sqrt{1+\frac{\sinh^{2}(\omega/2)}{\sin^{2}(\theta/2)}}\,.$ (32)
Now consider the case of finite $AA$ hoppings by gradually increasing $v$ from
zero. This is equivalent to reintroducing $\mathcal{S}_{0}$ and
$\mathcal{A}_{0}$ to the Lagrangian. The former vanishes on the edges of
magnetic regions (see Fig. 2) and therefore will have no effect on the edge
mode. On the other hand, the $\mathcal{A}_{0}$ term acts as an electric
potential for either $\psi_{\pm}$. An edge mode trajectory is perpendicular to
constant $\mathcal{A}_{0}$ curves, so even though $\mathcal{A}_{0}$ might
redistribute the density of the mode along the edge, it will do little to
deviate it out of the edge. Therefore, the first magic angle is robust against
a non-vanishing $v$. Like a mass term, $\gamma^{0}\mathcal{A}_{0}$ does not
anti-commute with the generator of the chiral transformation, $\Gamma$. Since
we have $\\{\not{D},\Gamma\\}\propto v$ the $v\rightarrow 0$ ensures that the
action is invariant under $\psi\rightarrow e^{i\alpha\Gamma}\psi$ and that for
each positive eigenvalue of $\not{D}$ there is a negative one with the same
magnitude. Deviating form the $v=0$ limit breaks particle-hole symmetry,
shifts the eigenvalues of $\not{D}$, disconnects the number of left- and
right-handed zero modes $n_{\circlearrowright,\circlearrowleft}$ from the
Jacobian of the transformation, and gives the formerly flat-band a curvature.
In this situation we can define the magic angle, where the flatness of the
band is approximate, through the flux felt by $\psi_{\pm}$, e.g.
$\int_{\triangledown}F_{12}/4\pi=\int_{\triangledown}\mathcal{A}_{a}\bar{\psi}_{\partial\triangledown}\gamma^{a}\psi_{\partial\triangledown}=1$,
but this implies an exact flat-band upon the existence of a well-defined
$n_{\circlearrowright,\circlearrowleft}$ which can be found only at $v=0$.
As we have seen in this Letter, although anomalies are not present in all
dimensions, it is still possible to conjure them in specific situations. In
particular, a flat band can be described through an anomaly in the timeless
version of its hosting theory. We saw that the dimensionally reduced theory
and thus the flat band are not always realizable. In the case of bilayer
graphene, the obstruction comes from the chiral anomaly and the need to
satisfy an index condition, which in turn confirms the topological nature of
the flat band. The Dirac field theory form, Eq. (10), of the bilayer moiré
lattice problem allows many generalizations including the finite temperature
case, the presence of complex inhomogeneous external fields and general
deformations, and interaction effects in the spirit of Refs. [16, 17] where
the interplay of anomaly with interactions are discussed. Of particular
interest are quantum Hall phenomena and unconventional superconducting pairing
associated with the moiré gauge fields.
This work was supported by the National Science Foundation under Grant No.
DMR-2037158, the U.S. Army Research Office under Contract No. W911NF1310172,
and the Simons Foundation.
## References
* Lopes dos Santos _et al._ [2007] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Graphene bilayer with a twist: Electronic structure, Phys. Rev. Lett. 99, 256802 (2007).
* Bistritzer and MacDonald [2011] R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proceedings of the National Academy of Sciences 108, 12233 (2011), https://www.pnas.org/content/108/30/12233.full.pdf .
* San-Jose _et al._ [2012] P. San-Jose, J. González, and F. Guinea, Non-abelian gauge potentials in graphene bilayers, Phys. Rev. Lett. 108, 216802 (2012).
* Tarnopolsky _et al._ [2019] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Phys. Rev. Lett. 122, 106405 (2019).
* Cao _et al._ [2018a] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature 556, 43 (2018a).
* Cao _et al._ [2018b] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, _et al._ , Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature 556, 80 (2018b).
* Andrei and MacDonald [2020] E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nature materials 19, 1265 (2020).
* Balents [2019] L. Balents, General continuum model for twisted bilayer graphene and arbitrary smooth deformations, SciPost Phys. 7, 48 (2019).
* Sboychakov _et al._ [2015] A. O. Sboychakov, A. L. Rakhmanov, A. V. Rozhkov, and F. Nori, Electronic spectrum of twisted bilayer graphene, Phys. Rev. B 92, 075402 (2015).
* Zou _et al._ [2018] L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Band structure of twisted bilayer graphene: Emergent symmetries, commensurate approximants, and wannier obstructions, Phys. Rev. B 98, 085435 (2018).
* Parhizkar and Galitski [2022a] A. Parhizkar and V. Galitski, Strained bilayer graphene, emergent energy scales, and moiré gravity, Phys. Rev. Research 4, L022027 (2022a).
* Parhizkar and Galitski [2022b] A. Parhizkar and V. Galitski, Moiré gravity and cosmology (2022b).
* Fujikawa and Suzuki [2004] K. Fujikawa and H. Suzuki, _Path integrals and quantum anomalies_ , 122 (Oxford University Press on Demand, 2004).
* Atiyah and Singer [1963] M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bulletin of the American Mathematical Society 69, 422 (1963).
* Fujikawa [1979] K. Fujikawa, Path-integral measure for gauge-invariant fermion theories, Phys. Rev. Lett. 42, 1195 (1979).
* Rylands _et al._ [2021] C. Rylands, A. Parhizkar, A. A. Burkov, and V. Galitski, Chiral anomaly in interacting condensed matter systems, Phys. Rev. Lett. 126, 185303 (2021).
* Rylands _et al._ [2022] C. Rylands, A. Parhizkar, and V. Galitski, Non-abelian bosonization in a $(3+1)$-d kondo semimetal via quantum anomalies, Phys. Rev. B 105, 195108 (2022).
|
# If our aim is to build morality into an artificial agent, how might we begin
to go about doing so?
Reneira Seeamber Department of Computing, Imperial College London, London,
SW7 2BX, United Kingdom Cosmin Badea Department of Computing, Imperial
College London, London, SW7 2BX, United Kingdom
###### Abstract
As Artificial Intelligence (AI) becomes pervasive in most fields, from
healthcare to autonomous driving, it is essential that we find successful ways
of building morality into our machines, especially for decision-making.
However, the question of what it means to be moral is still debated,
particularly in the context of AI. In this paper, we highlight the different
aspects that should be considered when building moral agents, including the
most relevant moral paradigms and challenges. We also discuss the top-down and
bottom-up approaches to design and the role of emotion and sentience in
morality. We then propose solutions including a hybrid approach to design and
a hierarchical approach to combining moral paradigms. We emphasize how
governance and policy are becoming ever more critical in AI Ethics and in
ensuring that the tasks we set for moral agents are attainable, that ethical
behavior is achieved, and that we obtain good AI.
If our aim is to build morality into an artificial agent, or machine, how
might we begin to go about doing so? This is a crucial question that is of
particular relevance at present, in the context of the flurry of recent
activity in the field, including recent but already famous applications such
as ChatGPT and Bard.
Let us begin by defining key terms in this question, namely ‘morality’ and
‘artificial agent’. Morality can be defined as the differentiation of
intentions, decisions, and actions into those that are good (or right) and
those that are bad (or wrong) [1]. An artificial agent, or machine, we will
consider to be a system that can intelligently perform certain tasks by making
decisions.
Deciding whether or not we should build morality into artificial intelligence
(AI) is determined by many factors, including whether we want the AI to act as
a moral agent or adviser and whether we want to build autonomy into it. Going
down the latter route brings with it an obvious need for morality to be
implemented, but the fundamentals of what it is to be ’good’ have long been
debated and no clear consensus has been reached[6].
To begin addressing this dilemma, we can look at the three main philosophical
theories of morality in AI Ethics: virtue ethics, utilitarianism, and
deontology[6]. Deontology emphasizes the imperative moral value of certain
actions or rules, rather than their consequences. Virtue ethics, on the other
hand, focuses on the character of the individual who is acting, rather than on
specific rules or actions. It emphasizes the importance of developing virtuous
traits, such as honesty and courage and developing a virtuous character, which
allows individuals to make morally good choices. Utilitarianism is a
consequentialist theory that focuses on actions that maximize utility,
commonly measured as happiness or pleasure, for the greatest number of people
[6].
Existing research has investigated different approaches to developing good AI
and important obstacles and potential solutions have been identified. For
instance, the Interpretation Problem, introduced by Badea and Artus (2022)[2]
is the idea that any symbolic representation we might use for AI necessarily
gives rise to a plethora of possible interpretations, and thus ambiguity and
mistakes by the AI. There are ways to mitigate its effects, such as building
character into the agents, thus creating virtuous agents by using values as a
tether between what the AI understands and what we mean to communicate to it
[2]. There are also more practical questions, such as those around what role
agents would play in a clinical context, with proposals including them acting
alongside the human experts, perhaps relying on such experts as moral
exemplars. [3]. Additionally, there are arguments around the use of particular
moral paradigms (such as utilitarianism, deontology and virtue ethics) in
building ethical AI and resolving contemporary moral dilemmas like breaking
bad news to a patient [4] or antimicrobial resistance and use [5].
Furthermore, the variety of potential moral paradigms and lack of agreement
has also led to a perceived necessity for a moral paradigm-agnostic decision-
making framework, which Badea (2022) proposes as MARS [6]. However, building
decision-making agents is itself a difficult process, and deciding on how to
do it, or Meta-decision-making for AI [7], is a new and very promising field
of research, which we aim to contribute to here. An ontology for this process
has been proposed before, based on three steps, in Badea and Gilpin (2022). In
this paper, we continue previous attempts to identify methods and challenges
for building good AI, and to propose solutions that overcome these challenges.
With this work, we examine canonical approaches to AI to provide a broader
perspective on implementing morality. Having identified many different
challenges and potential ways of dealing with them, we found ourselves in the
position of wanting to take a step back and focus on the techniques of
building ethical AI. Thus, we examined existing methodology from the
literature and identified top-down and bottom-up approaches from classical AI
[9]. Having identified these canonical approaches, we aimed to apply them to
the task of mitigating the issues discussed above. Hence we set out with the
goal of coming up with and categorizing such methods, and continuing this line
of thought in attempting to examine the potential technical approaches to
implementing the suggestions above and constructing working ethical AI.
## 1 THE TOP-DOWN APPROACH ALONE MIGHT BE INSUFFICIENT
An important aspect to consider in designing AI is the influence of culture
and how it affects morals. Massachusetts Institute of Technology (MIT) created
the Moral Machine, an online simulation where the public can decide for
themselves which is the ‘right’ action to take given variations of the famous
‘Trolley dilemma’ [10]. The dilemma originally involved choosing between
allowing a group of five people to die or choosing to pull a lever and
sacrificing one person instead, while the Moral Machine creates variations on
this with different demographics in each of the two groups. Global data from
the MIT study revealed variations in how people respond to such moral
dilemmas, given their demographic. For example, people in Western countries
were more likely to favor saving younger people than the elderly compared with
people in Eastern countries. This casts doubt on the possibility of using a
top-down approach alone because implementing such approaches (utilitarianism,
deontology) might be nigh impossible, given that there may never be a global
consensus between humans on developing a universal set of ‘rules’ or ‘duties’.
Noothigattu et al. (2018) proposed a voting-based system, whereby data from
the moral machine is collected and used to train multiple models to learn
societal preference for alternative outcomes across multiple ethical dilemmas.
These models are then aggregated to form a model that considers the collective
outcome of all voters, thus satisfying the utilitarian approach. This method
is limited in that self-reported preferences have been shown to vary from
real-life decisions [11]. Additionally, it may be argued that few AI
developers would have to deal with ethical dilemmas, such as the Trolley
problem, in their AI application and the majority of humans have not or will
not have to make such a decision. Instead, we should be looking towards more
realistic dilemmas, such as social media advertising and whether it could lead
to accessibility to violence or compulsive behavior formation. It is up for
debate whether a set of deducible principles could be created and whether a
model needs more than simply a set of internal coherence constraints, as it
could easily satisfy constraints but still produce undesirable outcomes. In
reality, multiple goals and rules come into conflict with each other,
producing complexity. Developing rules that a machine will understand in the
way we desire is another issue in itself, which may stem from us not knowing
explicitly what it is we want them to understand [2].
In Plato’s dialogue Euthyphro, Socrates asked Euthyphro, “Is the pious loved
by the Gods because it is pious or is it pious because it is loved by the
Gods?”. This is proving very relevant to AI morality as we, humans, may begin
to see ourselves as the Gods, or creators, of AI. Thus, is it what we
(subjectively) believe morality to be that AI should consider as moral, or
should we leave it to the AI to determine what is moral for itself,
potentially then leading us to reassess what we see as moral? For instance,
the aggregation of societal preferences has the potential to produce a morally
better system than that of any individual alone, and may even enable us to
then identify general principles that underlie our own decision-making [7].
## 2 EMOTION, SENTIENCE AND MORALITY
It has been debated, especially following Jeremy Bentham’s and John Stuart
Mill’s formulation of ‘the greater good for the greatest number’, whether the
stoic characteristic of machines limits their ability to carry out moral
decision-making because of their incapacity to care. The dual-process theory
of moral judgment, which proposes the idea of dichotomous thinking, with a
fast, instinctive and emotional process driving us alongside a slower and more
logical, reason-guided process [8], has led to the question: Does this
emotional capacity of humans lead us to moral subjectivity? Or does it allow
us to decide between what is wrong or right? Thinking of humans as having two
different systems may play an important role in developing AI machines that
have both reason and the capacity to care. ‘Cognitive emotion’ could be built
into machines, whereby one allows for weights to be added to, for example,
‘human life’, thereby producing somewhat similar caring systems for human life
[8].
However, the innate stoicism of machines may be advantageous in reducing the
moral biases we have as humans, one of the limiting factors as to why it is
hard to come to a consensus [8]. The idea of ‘embodiment’ and whether
artificial moral decision-making should be extended with emotional and sensory
components is interesting. However, various issues come to light when
extensive sensor systems are involved. For example, these are more vulnerable
to attack, and people may become worried about privacy issues.
Whether or not a sentient agent is achievable remains to be explored. The
symbiosis between technology and humans will become ever more important, and
similar to a jury determining someone’s innocence, or a group of doctors
discussing how to proceed with the most complex medical problems, there are
simply certain decisions that require several inputs before coming to a
conclusion [3]. It is therefore imperative that any system taking on
responsibilities that would otherwise be discussed by a group of people should
have a notification state. This state would ensure that a board of human
overseers intervene and is involved in the final decision when conflicts
arise.
## 3 PROPOSING A HYBRID APPROACH
Figure 1: A hierarchical structure could be developed for building ethical AI
by combining virtue ethics, deontology and utilitarianism.
How do we, as humans, learn our morals and values? Why do they differ so much
globally as we have seen with the Moral Machine experiment? Children grow to
imitate and follow what they have been bought up to see. It is the complex
interaction between nature and nurture that ultimately shapes our values, as
is becoming more notable in the field of behavioral epigenetics.
With the advent of apprenticeship learning, i.e. inverse reinforcement
learning, it would be reasonable to try and develop similar artificial systems
that would learn by viewing the behaviors of humans [9]. As opposed to
reinforcement learning which uses rewards and punishments to learn certain
behaviors, inverse reinforcement learning would involve an AI system observing
human behaviors and figuring out the goal of the behavior itself. This paves a
way for coding the complexity of human ethical values without having to
explicitly code every rule for every possible scenario. However, this
highlights the need once again for stringent thought into the data AI should
learn from. A child growing up with abusive parents would possibly learn that
it is right to behave in such a way. An excellent example of how this type of
exploitation could affect AI is Tay bot, which learned racial remarks from
Twitter trolls [12].
Regarding the data an AI should learn from, we could look to the actions of
the moral exemplars [3], and thus to the agent-based ethics approach proposed
by Aristotle: virtue ethics. This emphasizes an agent’s moral character in
their actions, rather than emphasizing duties (deontology) or consequences
(utilitarianism). Aristotle himself stresses that virtue is gained through
habit, thus a virtuous person consistently embodies the right actions because
they have built the right character.
For instance, in a similar manner to how we have courts and ordered decision-
making processes in place to judge one’s character with a rigorous review
panel, it may be possible to have an equivalent scrutiny panel for AI. We
could use this as part of a hierarchical structure for building moral AI, in
which we could combine choice aspects of different moral paradigms, such as
virtue ethics, deontology and utilitarianism, an example of which we provide
in (Figure 1).
Furthermore, tying back to one of the first points regarding the differences
in moral ideologies across different cultures, it is important to consider the
principles that can be taken from other philosophers, for example, Confucian
or Buddhist virtues. The four pillars of Confucian ethics (Yen, love and
empathy; Yi, righteousness; Li, veneration and comity; Zhi, wisdom) could also
be considered when choosing virtuous people for data collection as the
‘teachers’, or moral exemplars, of the AI.
Hagendorff (2022) performed a reductionist clustering approach to meta-studies
on AI ethics guidelines to distill four fundamental AI virtues: Justice,
honesty, responsibility and care [13]. The author suggests that these four
virtues are behind the majority of practical AI ethical principles discussed
in the literature today, such as fairness, transparency, accountability and
social cohesion. Notably, prudence and fortitude were identified as second-
order virtues that aim to counteract factors, such as innate bias and peer
influence. This provides an important starting point for addressing how we
might define and identify moral exemplars for developing ethical AI.
As shown in Figure 1, following the selection of ‘teachers’, or moral
exemplars, a refinement process will take place: data will be chosen through
the selection of actions that comply with very specific rules imposed that are
decided upon a priori, as constituting duties in a deontological fashion. This
could be followed by a selection of valid actions which exhibit consequences
desired according to utilitarianism or other frameworks, such as being desired
by the majority as suggested by Noothigattu et al. (2018) [11].
It is necessary to highlight an important hypothetical scenario described by
Wallach and Allen, regarding AI-based trading systems that lead to overly high
oil prices [8]. This would lead to other automated systems moving to coal as
an alternative source of energy to reduce costs – a massive influx in coal
demand and the need for full-time production could lead to an explosion and
power outage. As one can see, the channels of multiple automated decision
systems in interaction and under the influence of one another could lead to
catastrophic human and economic consequences. It is with that, that there has
to be a higher-order framework in place to assess all potential interactions
between AI systems, for example, the AI influencing the trade of oil with an
AI deciding which power source to use.
This ties into the self-awareness embodiment of AI and awareness of other
systems as well as humans. Hence, the cognitive simulation theory seems
another way to help avoid such scenarios where AI could be programmed through
internal simulations of actions with predictions of their consequences,
instead of logic statements [14]. In this paper, a multi-layer architecture
involving a controller that generates several actions to be performed to reach
a certain goal is proposed. These actions are then sent to an ethical layer
that simulates each action using the current state of the environment, the
human involved and the robot. Internal states of the human and robot are
retrieved and used in conjunction to evaluate the best action to take.
With this, it is important to determine the tests and simulations that must be
carried out even before AI systems are given higher moral authority. Devin
Gonier (2018) introduced the notion of ’tricking’ AI models based on the
Hawthorne effect - the observer affects the outcome of an experiment [15].
Based on this, if we were to constantly observe a model, the outcome should be
in our favor. We could have two scenarios. The computer thinks it moves from a
simulation to the real world, but in reality, it stays in simulation mode.
Conversely, the model could think it’s in a simulation but actually in the
real world. Ultimately, keeping the model in a ’doubt’ mode enables control.
Could we then have parallel systems running that would each take a vote on the
best action? If the AI working on trading systems could predict that high oil
prices would result in decisions to move away from oil by other logically-
programmed machines, it would know to avoid such a scenario. Such a framework
could be combined with the inverse reinforcement learning approach where the
AI learns from observing actions from exemplars, and figuring out the goals
and internal states of impacted agents. In doing so, an extensive learned base
could be created to enable internal simulations of actions and accurate
predictions of consequences without the need for explicit rules. Such a
process would take time and experience to learn, hence the introduction of
regulatory sandboxes, which will be discussed in the next section, are a key
way to ensure a controlled application of ethical AI.
## 4 AI GOVERNANCE PRINCIPLES
Ultimately, the development of moral machines heavily relies on stringent
rules in place for deploying such systems. At the very least, it should be
proven that accountability has been covered, the system is transparent for
inspection, cannot be easily manipulated (i.e. strong security measures have
been employed to reduce the chance of hackers intercepting) and that the AI
has a high level of predictability that can match the majority of humans.
Regarding opacity and accountability, it may be possible to have parallel
functioning systems, the black box - referring to complex models that are not
highly interpretable to humans - and it’s more transparent shadow that aims to
conduce step-by-step thought processes. Prospector is one such example,
whereby researchers have developed an interactive visual representation of
predictive models that aims to help us understand how and why data points are
predicted, and the predictive value of different features [16]. Notably, the
authors of this paper stress that we need to move away from solely evaluating
performance metrics, such as accuracy.
Research is being done to ensure AI is more transparent, for example,
gradient-weighted class activation mapping is one method that attempts to
explain convoluted neural networks used in histology or magnetic resonance
imaging labeling. Maps are created to demonstrate the most relevant areas of
images used in the classification. Such techniques enable histopathologists or
radiologists to gain some insight into the decision-making process. Whilst
this provides information on how a model made a decision, current explanatory
methods typically do not justify whether a decision was appropriate or not.
Notably, Ghassemi, Oakden-Rayner and Beam discussed the idea that humans
unknowingly use heat maps to understand whether the areas identified are
justifiable based on intuition, and thus, bias can be introduced. Humans do
tend to over-trust computers, and explainability methods could reduce the
ability to identify errors and/or unreasonably increase our confidence in
algorithmic decisions [17].
Indeed, detailed logs must be made available by the AI to help understand any
decision and to further guide its development. As suggested by Mitchell et al.
(2019), documentation detailing the performance of models in different
contexts (e.g. age, race, sex) should be provided [18]. Increasing numbers of
publications relating to frameworks for dataset generation and AI development
are becoming available and a compilation of these academic works must be
acknowledged in any regulatory documents going forward.
Incorporating these techniques and tools into the AI development process can
help us work towards creating AI systems that reflect the world we want to
live in. By fostering transparency, accountability, and fairness, we can
ensure that AI systems adhere to the principles of virtue ethics in the age of
AI.
Various governments, including the UK, US and Japan, have leaned towards the
use of ‘regulatory sandboxes’, defined as a testbed for innovation in a
controlled real-life environment prior to launch. The 2023 UK policy paper, ”A
pro-innovation approach to AI regulation”, highlights the use of sandboxes to
encourage innovation and provide an adaptable response to regulating AI [19].
This contrasts with proposals for EU legislation (the EU AI Act or the
Liability Directive) which take a different approach with more comprehensive
regulations, including the General Data Protection Regulation (GDPR) that
applies to AI development and data sharing across EU member states [20]. GDPR
emphasizes data subject rights, data minimization, and transparency. The UK’s
Data Protection Act 2018 incorporates the GDPR into UK law, ensuring similar
data protection standards for AI development and data sharing. Nevertheless,
Spain has piloted the first regulatory sandbox on AI in the EU to aid in
generating future guidelines [20]. Many countries have also adopted a sector-
specific approach to regulations, impacting AI development and data sharing in
various industries, such as finance, energy and healthcare.
As technology constantly evolves, guidelines will also have to evolve with it.
There is no set of rules that can foresee all future problems that arise. This
is also why it is important to have interdisciplinary research involving
engineers, philosophers, neuroscientists, psychologists and policymakers to
ensure the best possible outcomes. It may be that regulations require
companies to involve a multidisciplinary team during development. University
education may also need to introduce varied modules (i.e. ML courses include a
philosophy and ethics module, and medical degrees include ML models) to
prepare future generations for such collaborative tasks. Regulatory workshops
should be made more regular and accessible to inform and guide companies
during the development process. Either way, there is still much yet to debate
regarding human ethics before machines can become the moral entities we want
them to become. If we are to take the bottom-up approach to train AI, we must
ensure the data is a reflection of the world we want to live in.
Encouraging open discussions among individuals and communities with different
views will be important in decision-making. This could be done by creating
national and international forums, where society can share experiences and
develop recommendations for AI governance. Such forums can provide a platform
for dialogue, consensus-building, and the exchange of best practices. By
fostering a culture of dialogue and debate, society can collectively engage
with moral dilemmas to understand and overcome the nuances and complexities.
Additionally, individuals can further develop the ability to critically
analyze moral dilemmas through the promotion of moral education and critical
thinking from an early age, enabling them to understand different viewpoints
and appreciate the diversity of moral values across cultures and traditions.
We must also ensure that decision-making processes are inclusive and represent
the diversity of society. This will help to avoid the domination of a single
moral perspective and to embrace multiple moral views.
## 5 TOWARDS MORAL AI
In conclusion, we have looked at the matter of building moral machines from
different angles, such as those of technical approaches, culture, emotion,
sentience, and governance. Our main contribution is the proposal of a hybrid
approach using a combination of top-down and bottom-up and a further
combination of different moral paradigms to get around the different obstacles
and limitations discussed. We have also discussed some of the practical issues
and techniques relevant to implementing our approach.
Ultimately, determining what constitutes good morals is an ongoing and
collaborative effort that requires input from various stakeholders and an
openness to learn, grow, and adapt. By engaging in these processes, society
can work towards establishing shared moral values that contribute to the
greater good. It is in doing so, that we may be able to start to answer some
of the questions we, as humans, hold regarding morality and what it is to act
with moral grounding. This collaborative approach can help create a more
responsible, inclusive, and beneficial AI landscape for all.
## 6 ACKNOWLEDGMENTS
Reneira Seeamber was supported by the UKRI CDT in AI for Healthcare
http://ai4health.io (Grant No. EP/S023283/1). For the purpose of open access,
the authors have applied a Creative Commons Attribution (CC BY) license to the
Accepted Manuscript.
## REFERENCES
* [1] M. Khatibi and F. Khormaei, ”Morality and Ethics: A Brief Review.” Journal Of Life Science And Biomedicine. 6, 66-70 (2016,5) (journal)
* [2] C. Badea and G. Artus, ”Morality, Machines, and the Interpretation Problem: A Value-based, Wittgensteinian Approach to Building Moral Agents.” Artificial Intelligence XXXIX. pp. 124-137 (2022) (journal)
* [3] S. Hindocha and C. Badea, ”Moral exemplars for the virtuous machine: the clinician’s role in ethical artificial intelligence for healthcare.” AI And Ethics. 2, 167-175 (2022,2), https://doi.org/10.1007/s43681-021-00089-6 (journal)
* [4] B. Post, C. Badea, A. Faisal, and S. Brett, ”Breaking bad news in the era of artificial intelligence and algorithmic medicine: an exploration of disclosure and its ethical justification using the hedonic calculus.” AI And Ethics. (2022,10), https://doi.org/10.1007/s43681-022-00230-z (journal)
* [5] W. Bolton, C. Badea, P. Georgiou, A. Holmes and T. Rawson, ”Developing moral AI to support decision-making about antimicrobial use.” Nature Machine Intelligence. 4, 912-915 (2022,11), Number: 11 Publisher: Nature Publishing Group (journal)
* [6] C. Badea, ”Have a Break from Making Decisions, Have a MARS: The Multi-valued Action Reasoning System.” Artificial Intelligence XXXIX. pp. 359-366 (2022) (journal)
* [7] C. Badea and L. Gilpin, ”Establishing Meta-Decision-Making for AI: An Ontology of Relevance, Representation and Reasoning.” (arXiv,2022,10), http://arxiv.org/abs/2210.00608, arXiv:2210.00608 (journal)
* [8] W. Wallach and C. Allen, ”Moral machines.” (Oxford Univ. Press,2009) (book)
* [9] D. Abel, J. MacGlashan and M. Littman, ”Reinforcement Learning As a Framework for Ethical Decision Making.” (2016) AAAI Workshop: AI, Ethics, and Society. (conference paper)
* [10] E. Awad, S. Dsouza, R. Kim, J. Schulz, J. Henrich, A. Shariff, J. Bonnefon and I. Rahwan, ”The Moral Machine experiment.” Nature. 563, 59-64 (2018,11), ISBN: 0028-0836 (journal)
* [11] R. Noothigattu, S. Gaikwad, E. Awad, S. Dsouza, I. Rahwan, P. Ravikumar and A. Procaccia, ”A Voting-Based System for Ethical Decision Making.” Proceedings Of The AAAI Conference On Artificial Intelligence. 32 (2018,4) (journal)
* [12] G. Neff and P. Nagy, ”Talking to bots: symbiotic agency and the case of Tay.” International Journal Of Communication (Online). pp. 4915 (2016,10) (journal)
* [13] T. Hagendorff, ”A Virtue-Based Framework to Support Putting AI Ethics into Practice.” Philosophy & Technology. 35, 55 (2022,6), https://doi.org/10.1007/s13347-022-00553-z (journal)
* [14] D. Vanderelst and A. Winfield, ”An architecture for ethical robots inspired by the simulation theory of cognition.” Cognitive Systems Research. 48 pp. 56-66 (2018,5), http://dx.doi.org/10.1016/j.cogsys.2017.04.002, ISBN: 1389-0417 (journal)
* [15] Devin Gonier, ”Morality and Artificial Intelligence: The Science and Beyond.” TEDxAustinCollege. (2018,10), https://www.youtube.com/watch?v=2_8kgV-EfTA (video)
* [16] J. Krause, A. Perer, and K. Ng, ”Interacting with Predictions.”(ACM,2016,5), http://dl.acm.org/citation.cfm?id=2858529 (journal)
* [17] M. Ghassemi, L. Oakden-Rayner and A. Beam, ”The false hope of current approaches to explainable artificial intelligence in health care.” The Lancet (British Edition). 3, e745-e750 (2021,11), https://dx.doi.org/10.1016/S2589-7500(21)00208-9, ISBN: 2589-7500 (journal)
* [18] M. Mitchell et al. ”Model Cards for Model Reporting.” (ACM,2019,1), http://dl.acm.org/citation.cfm?id=3287596 (journal)
* [19] Department for Science, Innovation & Technology A pro-innovation approach to AI regulation. (2023), https://www.gov.uk/government/publications/ai-regulation-a-pro-innovation-approach (policy document)
* [20] European Commission, “Liability Rules for Artificial Intelligence.” https://commission.europa.eu/business-economy-euro/doing-business-eu/contract-rules/digital-contracts/liability-rules-artificial-intelligence_en (accessed 2023) (URL)
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# Krylov Complexity and Dynamical Phase Transition in the quenched LMG model
Pedro H. S. Bento QPequi Group, Institute of Physics, Federal University of
Goiás, 74.690-900, Goiânia, Brazil Adolfo del Campo Department of Physics and
Materials Science, University of Luxembourg, L-1511 Luxembourg, G. D.
Luxembourg Donostia International Physics Center, E-20018 San Sebastián,
Spain Lucas C. Céleri QPequi Group, Institute of Physics, Federal University
of Goiás, 74.690-900, Goiânia, Brazil
###### Abstract
Investigating the time evolution of complexity in quantum systems entails
evaluating the spreading of the system’s state across a defined basis in its
corresponding Hilbert space. Recently, the Krylov basis has been identified as
the one that minimizes this spreading. In this study, we develop a numerical
exploration of the Krylov complexity in quantum states following a quench in
the Lipkin-Meshkov-Glick model. Our results reveal that the long-term averaged
Krylov complexity acts as an order parameter when the quench originates from a
zero magnetic field. It effectively discriminates between the two dynamic
phases induced by the quench, sharing a critical point with the conventional
order parameter. Additionally, we examine the inverse participation ratio and
Shannon entropy in both the Krylov basis and the energy basis. A matching
dynamic behavior is observed in both bases when the initial state possesses a
specific symmetry. This behavior is analytically elucidated by establishing
the equivalence between the Krylov basis and the pre-quench energy eigenbasis.
## I Introduction
Quantifying the complexity of physical phenomena has been a long-standing area
of interest [1]. Various measures, such as the entropy generated during the
time evolution of a quantum state [2], algorithmic randomness [3], and quantum
Kolmogorov complexity [4], have been employed to characterize complexity.
While many existing complexity measures can capture certain features, some
fail in encompassing other aspects or even exhibit ambiguity due to the
specific basis choices involved in their definition.
There is a widely accepted notion that chaotic systems exhibit greater
complexity compared to integrable ones, attributed in part to phenomena such
as the butterfly effect [5, 6]. In the realm of quantum systems, this effect
is typically characterized by the exponential growth of the noncommutativity
of local operators over time, as quantified by out-of-time order correlators.
However, this exponential increase is not consistently observed in chaotic
systems and may even manifest in nonchaotic ones [7, 8, 9], rendering this
feature elusive.
A novel measure of complexity has recently been proposed, relying on the
spread of operators across a specifically ordered basis known as the Krylov
basis [10]. This basis provides the visualization of dynamics as a single
particle moving in a semi-infinite chain, with hopping terms corresponding to
Lanczos coefficients obtained during the computation of the Krylov basis [11].
A large body of literature has followed, addressing fundamental properties of
this measure [12, 13, 14, 15, 16, 17, 18, 19, 20], its generalizations for
other kinds of evolution [21, 22, 23, 24, 25, 26], and applications, e.g., to
characterize phases of matter [27, 28] and for quantum control [29, 30]. The
Krylov complexity involves a choice of inner product between operators,
raising questions about the optimal inner product for minimizing complexity.
This challenge was addressed with the extension of the Krylov complexity for
quantum state evolution presented in Ref. [31]. It was shown that the Krylov
basis minimizes the proposed complexity definition. As a result, the Krylov
complexity of quantum states, also known as the spread complexity, emerges as
an unambiguous measure, eliminating the need for an inner product choice. It
has been explored in the SYK model [10, 13], the XXZ model [15], the Su-
Schrieffer-Heeger model [28], evolutions governed by dynamical symmetry groups
associated with specific Lie algebrass [16, 18, 23], and random matrix models
[18, 31], among others. Additionally, it has been applied to analyze the
transition from integrability to a chaotic regime [20, 32].
The exploration of dynamics following a global quench in quantum many-body
systems is widely used to probe a system in nonequilibrium statistical
mechanics [33]. Global quenches induce rapid departures from equilibrium,
along with coherence in the energy spectrum, producing intriguing effects,
such as dynamical phase transitions that delve into the criticality of quantum
many-body system dynamics [34, 35]. Dynamical quantum phase transitions
(DQPTs) arise in quenched many-body systems and manifest as cusps in the
Loschmidt echo or survival probability, i.e., non-analytic behavior in the
time domain for certain initial states. This paper contributes to their study
by investigating the evolution of Krylov complexity in quantum states
following a quench in the paradigmatic Lipkin-Meshkov-Glick (LMG) model [36].
The LMG model, known to exhibit dynamical phase transitions depending on the
quench [37], reveals that the long-time averaged Krylov complexity serves as
an order parameter for the dynamical phase transition when the quench starts
at null magnetic field. It exhibits distinct behavior over time in both
dynamical phases induced by the quench, and remarkably mirrors the
oscillations and revivals seen in one of the components of the total
magnetization, a conventional dynamical order parameter in spin systems.
In order to understand this fact, we conduct a numerical study of the inverse
participation ratio (IPR) and the Shannon entropy in both the Krylov basis and
the pre-quench energy basis over time. Remarkably, these two quantities
display identical behavior in both bases. Finally, by deriving the Krylov
basis for the case of a quench starting at null magnetic field, we
analytically demonstrate its equivalence to the pre-quench energy basis. This
analytical result explains the identical temporal behavior of IPR and Shannon
entropy in these two bases and also why Krylov complexity is an order
parameter for the considered model.
The paper is structured as follows: Section II provides a review of key
concepts related to dynamical phase transitions while Section III introduces
the theory of Krylov complexity of quantum states. The goal of these two
sections is twofold: setting the notation of the paper and make it self-
contained. Section IV offers an overview of the dynamical phase transition in
the LMG model and Section V presents our results. We present our conclusions
in Sec. VI.
## II Dynamical Quantum Phase Transition
Let us consider a quantum many-body system described by a Hamiltonian $H(h)$,
which depends on an externally controlled parameter $h$. At the initial time,
the system is prepared in the ground state
$|\psi(0)\rangle\equiv|\psi_{0}\rangle$ of the Hamiltonian $H(h_{0})\equiv
H_{0}$. After a sudden change, quench, during which the value of $h$ changes
from $h_{0}$ to $h_{f}$, the system is left to evolve under $H(h_{f})\equiv
H_{f}$. In this context, the central quantity in the theory of dynamical
quantum phase transition is the return probability amplitude
$\displaystyle\mathcal{G}(t)=\langle\psi_{0}|\psi_{t}\rangle=\langle\psi_{0}|e^{-iH_{f}t}|\psi_{0}\rangle,$
(1)
also know as the survial amplitude [38, 39, 40, 41] and the Loschmidt
amplitude [42]. The characterization of the DQPT relies on a formal analogy
between $\mathcal{G}(t)$ and a special case of the boundary partition function
$Z_{B}=\langle\psi_{1}|e^{-RH}|\psi_{2}\rangle$, with $R$ being the distance
between the boundaries $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ [43].
Therefore, formally, $\mathcal{G}(t)$ is the boundary partition function with
$R=it$ and $|\psi_{1}\rangle=|\psi_{2}\rangle=|\psi_{0}\rangle$. The
corresponding probability
$\displaystyle\mathcal{L}(t)=|\mathcal{G}(t)|^{2}.$ (2)
is called return probability, survival probability, or Loschmidt echo.
The DQPT is then defined in terms of the Fisher, or Lee-Yang zeros [44, 45] of
this partition function. Consider the return amplitude with complex time $t\to
z=t+i\tau$. A transition occurs every time the zeros of $\mathcal{G}(z)$
crosses the real-time axis of the complex plane, and these crossings indicate
the critical times of the dynamics. This analogy with the equilibrium quantum
phase transition leads us to another important quantity in the dynamical case,
a quantity that can be understood as the dynamical version of the free energy,
the rate function
$\displaystyle r(t)=-\frac{1}{N}\log\left[\mathcal{L}(t)\right],$ (3)
with $N$ being the size of the system. At the critical times, $\mathcal{L}(t)$
is zero and the rate function becomes nonanalytic. It is important to observe
that this behaviour only emerges in the thermodynamic limit where
$N\rightarrow\infty$ and the Fisher zeros accumulate in a line or in a
continuous region [34].
The above-described phenomenon was termed DPT-II to differentiate it from
another related phenomenon, known as DPT-I. DPT-II was initially investigated
in the transverse field Ising model [42], and since then, it has been explored
in various quantum many-body systems. These encompass short to long-range
interacting quantum spin chains [46, 47, 37, 48, 49], two-dimensional spin
systems [50, 51], nonintegrable systems [52], and optical systems [53], to
name a few examples. For an in-depth review of DPT-II, covering both
theoretical studies and experimental realizations, see Ref. [34].
In the transverse field Ising model, the condition for the Fisher zeros to
cross the real-time axis in the complex plane is that the quench crosses the
critical point of the equilibrium quantum phase transition (QPT). Such a
connection was thought to be a general feature of quantum many-body systems
presenting a DQPT. However, several exceptions appeared later on, highlighting
the nonexistence of a one-to-one correspondence between dynamical and
equilibrium QPTs [34].
While DPT-II is investigated through the lens of the return probability, DPT-I
is featured by the Landau order parameter of the system. This order parameter,
a quantity whose derivative undergoes nonanalytic changes at a critical point
of the quench parameter, serves to distinguish dynamical phases. Typically,
the order parameter involves the time average of some physical quantity, such
as the magnetization in spin chains. It is noteworthy that this dynamical
critical point might not align with the critical point of the equilibrium QPT.
DPT-I has been examined in various systems and we point the reader to Refs.
[35, 54] for comprehensive reviews on both its theoretical and experimental
aspects.
Connections between DPT-I and DPT-II were also explored. As an example, in
Ref. [55], a link is established between microscopic probabilities and the
order parameter in the XXZ model when the initial state exhibits broken
symmetry. In the transverse field Ising model with power-law decaying
interaction, Ref. [48] demonstrates that DPT-II occurs only when crossing the
dynamical critical point of DPT-I. The study also reveals that the
$\mathbb{Z}_{2}$ type symmetry, explicitly broken in the initial state, is
restored in a long-time limit and at the critical times of DPT-II.
More recently, a connection between DPT-I and DPT-II and the excited-state
quantum phase transition (ESQPT) in quantum many-body systems with infinite-
range interactions has been established [56]. The authors in Ref. [56] define
a generalized microcanonical ensemble by introducing three noncommuting
charges, which considers the presence of dynamical order parameters. In the
thermodynamic limit, these order parameters are non-zero if the energy $E$ is
less than the ESQPT critical energy $E_{c}$, but they always become zero if
$E>E_{c}$. Concerning DPT-II, it was demonstrated that non-analyticities in
the rate function $r(t)$ only occur if the energy of the system after the
quench is greater than $E_{c}$, being prohibited if $E<E_{c}$. The same
theoretical framework was applied to a finite-range interacting system in Ref.
[57].
The main focus of the present study is to investigate the complexity of the
dynamics associated with DPT-I in the paradigmatic Lipkin-Meshkov-Glick (LMG)
model [36].
## III Krylov Complexity of Quantum States
Now we introduce a notion of complexity of the quantum dynamics based on how
deep $|\psi_{t}\rangle$ spreads over the system Hilbert space. For this
reason, this quantity is also referred as spread complexity [31, 28].
Let us start by writing the evolved state in the form
$\displaystyle|\psi_{t}\rangle=\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}H^{n}|\psi_{0}\rangle.$
(4)
The successive applications of $H$ to the initial state generates the set of
quantum states
$\displaystyle\\{|\psi_{0}\rangle,H|\psi_{0}\rangle,H^{2}|\psi_{0}\rangle,\cdots\\}=\\{H^{n}|\psi_{0}\rangle\\}_{n\geq
0},$ (5)
which describes the spreading of the initial state over the Hilbert space
during time evolution. The subspace spanned by
$H^{n}|\psi_{0}\rangle\\}_{n\geq 0}$ is known as the Krylov space. The measure
of complexity that we consider in this work was introduced in Ref. [31] and it
is given by
$\displaystyle
C_{\mathcal{B}}(t)=\sum_{n}c_{n}|\langle\psi_{t}|B_{n}\rangle|^{2}$ (6)
where $\mathcal{B}$ represents a basis in the Hilbert space, whose elements
are denoted as ${|B_{n}\rangle}_{n}$. Intuitively, we expect that complex
dynamics lead to larger spreads over this basis. A basis independent
complexity measure can be obtained from $C_{\mathcal{B}}(t)$ by performing a
special minimization process, resulting in
$\displaystyle C(t)=\min_{\mathcal{B}}C_{\mathcal{B}}(t).$ (7)
Of course, it is always possible to construct a basis at the initial time such
that $|\psi_{0}\rangle=|B_{0}\rangle$ has a non-zero overlap only with one
state of the basis, thereby minimizing complexity. Instead, we consider a
functional minimization of (6) which takes into account the spread of the
state. For this purpose, it is natural to look at the set
$\\{H^{n}|\psi_{0}\rangle\\}_{n\geq 0}$ spanning the Krylov space, since it
includes only the portion of the Hilbert space actually visited by the system
over time evolution.
The Krylov complexity, a precedent of the spread complexity, was first
introduced in Ref. [10] in the context of operator evolution, i.e., in the
Heisenberg picture. Based on this concept, an universal hypothesis for the
maximum growth of local operators in quantum many-body systems was presented.
Later, Balasubramanian and collaborators [31] extended the idea to the
evolution of quantum states, in the Schrödinger picture, and proved that the
basis which minimizes (6) for $c_{n}=n$ is the so-called Krylov basis. The
Krylov basis is generated performing Gram-Schmidt orthogonalization on the set
$\\{H^{n}|\psi_{0}\rangle\\}_{n\geq 0}$. We denote the Krylov basis as
$\mathcal{K}$ and its elements as $\\{|K_{n}\rangle\\}_{n}$.
Alternatively, the Krylov basis can be generated using the Lanczos algorithm
[11, 58], which is a well known recursive method used to generate orthogonal
basis. Starting with the initial state as the first Krylov state
$|K_{0}\rangle=|\psi_{0}\rangle$, the next state is obtained as
$|K_{1}\rangle=\frac{1}{b_{1}}H|K_{0}\rangle$ with $b_{1}=\langle
K_{1}|K_{1}\rangle^{1/2}$ being the normalization constant. The subsequent
states $\\{|K_{n}\rangle\\}_{n\geq 2}$ are generated via the following
recursion method
$\displaystyle|A_{n}\rangle=H|K_{n-1}\rangle-a_{n}|K_{n-1}\rangle-
b_{n-1}|K_{n-2}\rangle$ (8)
$\displaystyle|K_{n}\rangle=b_{n}^{-1}|A_{n}\rangle.$ (9)
The constants $a_{n}$ and $b_{n}$, with $b_{0}=0$, are called Lanczos
coefficients and they are defined as
$\displaystyle a_{n}=\langle K_{n}|H|K_{n}\rangle,\ \ \ \ b_{n}=\langle
K_{n}|K_{n}\rangle^{1/2}.$ (10)
Isolating the first term in the right hand side of Eq. (8),
$\displaystyle
H|K_{n-1}\rangle=a_{n}|K_{n-1}\rangle+b_{n}|K_{n}\rangle+b_{n-1}|K_{n-2}\rangle,$
(11)
we observe that the hamiltonian is tridiagonal in the Krylov basis
$\\{|K_{n}\rangle\\}_{n}$.
The authors of Ref. [31] proved that the Krylov basis minimizes Eq. (6) in a
very specific way. Formally, let
$\displaystyle
S_{\mathcal{B}}=\left(C_{\mathcal{B}}^{(0)},C_{\mathcal{B}}^{(1)},C_{\mathcal{B}}^{(2)},\cdots\right)$
(12)
be the sequence of derivatives of $C_{\mathcal{B}}(t)$ calculated at $t=0$,
that is
$\displaystyle C^{(m)}_{\mathcal{B}}\equiv
C_{\mathcal{B}}^{(m)}(0)=\frac{d^{m}}{dt^{m}}C_{\mathcal{B}}(t)\Bigr{|}_{t=0},\
\ \ \ m=0,1,2,\cdots$ (13)
We say that $S_{\mathcal{B}_{1}}<S_{\mathcal{B}_{2}}$ if there is some $k$
such that $C^{(m)}_{\mathcal{B}_{1}}=C^{(m)}_{\mathcal{B}_{2}}$ for $m<k$ and
$C^{(m)}_{\mathcal{B}_{1}}<C^{(m)}_{\mathcal{B}_{2}}$ for $m=k$. Thus, for any
basis $\mathcal{B}$, $S_{\mathcal{K}}\leq S_{\mathcal{B}}$ with the equality
corresponding to the case $\mathcal{B}=\mathcal{K}$ [31]. This is the
minimization that we referred above as a functional minimization.
The Lanczos coefficients determine the matrix representation of the generator
of evolution in Krylov space. Their specific role in quantum dynamics is still
subject to investigation. The operator growth problem can be seen as a hopping
single-particle in an one-dimensional semi-infinite chain with the Lanczos
coefficients representing the hopping terms and the complexity in this case is
the expectation value of the _lattice index_ $n$ of the hopping particle; see,
e.g., Ref. [12]. This interpretation is inspired by the fact that in the
Krylov basis, the Hamiltonian has a tridiagonal form similar to the one of a
tight-binding model, which describes the hopping of electrons in solids.
## IV DPT-I in the LMG model
Let us consider a spin chain described by the Hamiltonian
$\displaystyle H(h)=-\frac{J}{N}S_{z}^{2}-hS_{x},$ (14)
where $S_{\alpha}=\sum_{i=1}^{N}\sigma^{\alpha}_{i}/2$ ($\alpha=x,y,z$) are
collective spin operators, with $\sigma^{\alpha}_{i}$ denoting the $\alpha$
Pauli matrix acting on the $i$-th site of a chain of $N=2j$ sites and total
angular momentum $j$. This Hamiltonian is a particular case of the well-known
Lipkin-Meshkov-Glick model [36]. The constant $J$ denotes the ferromagnetic
coupling between the spins in the $z$ direction while $h$ represents the
strength of magnetic field applied along the $x$-axis. This Hamiltonian
describes an ensemble of $N$ spin-$1/2$ systems subject to all-to-all pairwise
interactions in the presence of a magnetic field.
Considering a quench $h_{0}=0\rightarrow h_{f}>0$, we show in Fig. 1(a) the
bifurcation diagram of Hamiltonian (14) for the time-averaged magnetization
(order parameter for this model)
$\displaystyle\overline{S_{z}}=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\langle\psi(t)|S_{z}|\psi(t)\rangle
dt,$ (15)
as a function of the quench parameter $h$, which characterizes DPT-I in this
model. The initial state was taken to be the collective spin aligning with the
$-z$ direction, the south pole of the Bloch sphere representation. This state
corresponds to one of the two-fold degenerate ground states of $H_{0}$ [59],
denoted as $|\downarrow\rangle_{z}$. Throughout this paper, we use $J=1$. The
order parameter $\overline{S_{z}}$ delineates two distinct dynamical phases: a
dynamical ferromagnetic phase for $h<1/2$, where magnetization oscillates
around a finite value depending on the initial state, leading to
$\overline{S_{z}}\neq 0$, and a dynamical paramagnetic phase for $h\geq 1/2$,
where magnetization oscillates around zero, yielding $\overline{S_{z}}=0$. The
critical point separating these phases is at $h_{c}=1/2$.
Figure 1: Magnetization as an order parameter. Panel (a) shows the time-
averaged magnetization $\overline{S_{z}}$ as a function of quench
$h_{0}=0\rightarrow h_{f}=h$. The initial state is
$|\psi_{0}\rangle=|\downarrow\rangle_{z}$. In the thermodynamic limit,
$\overline{S_{z}}$ signals the dynamical phase transition at the critical
point $h=1/2$, thus identifying two dynamical critical phases. Panels (b) and
(c) show the magnetization $S_{z}(t)$ as a function of time for a quench in
the ferromagnetic phase ($h=0.3$) and a quench crossing the critical point to
the paramagnetic phase ($h_{f}=0.8$), respectively. In both phases, the
thermodynamic limit prevents the magnetization from revivals.
The DPT-I in this model has been extensively studied [48, 37, 60]. The
dynamical critical point, denoted as $h_{c}$, can be determined analytically
for the Hamiltonian (14), resulting in $h_{c}=(h_{0}+J)/2$ [61]. As the system
size increases, the behavior of $\overline{S_{z}}$ converges towards the mean-
field solution, highlighting the exact nature of the mean-field solution for
the LMG model in the thermodynamic limit [59].
An essential aspect of DPT-I in this context is the symmetry of the initial
state. This critical phenomenon is only observed when the initial state of the
dynamics exhibits broken symmetry [48]. The LMG model feature spin-flip
symmetry [62], represented by the operator $S=e^{i\pi(S_{x}-j)}$. This
symmetry dictates that $H$ does not couple standard eigenbasis
$|j,m_{z}\rangle$ states (eigenvectors of $S^{2}$ and $S_{z}$) with even and
odd $m_{z}$. Under these conditions, a quench within the same dynamical phase
produce oscillations in the system around a broken-symmetry effective state,
while a quench crossing the dynamical critical point produces oscillations
around a symmetric effective state. These oscillations are characterized by
long-lived steady states and they are usually linked to prethermalization
[63].
Figures 1(b) and 1(c) illustrate instances of magnetization behavior over
time, depicting a quench within the same phase and crossing the dynamical
critical point, respectively. It is evident that the observed revivals in
these figures arise from finite-size effects within the system. As the system
size $N$ increases, these revivals disappear. In the following section, we
demonstrate that the time-averaged Krylov complexity can also serve as an
order parameter for this system.
## V Results
We start by discussing the behaviour of the spread complexity and its long-
time average. In order to understand why Krylov complexity can be taken as an
order parameter for the DPT-I in the LMG model, we proceed with the analysis
of two other quantities, the inverse participation ratio and the Shannon
entropy, considering two distinct basis, the energy and the Krylov basis.
### V.1 Krylov Complexity of quantum states in the LMG model
Figure 2: Krylov complexity as an order parameter. Panel (a) shows the
normalized time-averaged Krylov complexity $\overline{C}$ as a function of $h$
considering $h_{0}=0$ and initial state
$|\psi_{0}\rangle=|\downarrow\rangle_{z}$. $\overline{C}$ exhibits exactly the
same transition as $\overline{S_{z}}$ at the critical point $h_{c}=1/2$.
Panels (b) and (c) show the Krylov complexity $C_{\mathcal{K}}(t)$ as a
function of time for a quenches in the ferromagnetic phase ($h_{f}=0.3$) and
across the critical point, to the paramagnetic phase ($h_{f}=0.8$). In both
phases, the thermodynamic limit also prevents the complexity from revivals.
We turn to the spread complexity given in Eq. (6). We argue that the long-time
average of the Krylov complexity
$\displaystyle\overline{C}=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}C_{\mathcal{K}}(t)dt$
(16)
behaves as a dynamical order parameter in the DPT-I of this model.
Using the same setup described in the last section, Fig. 2(a) shows the
normalized time-averaged complexity as function of the quench intensity $h$.
Remarkably, $\overline{C}$ exhibits the same qualitative behaviour as the
dynamical order parameter $\overline{S_{z}}$, with the only difference being
the numerical values assumed in both dynamical phases.
Similar to the magnetization $S_{z}(t)$, the Krylov complexity exhibits
distinct behavior in each dynamical phase over time. This characteristic is
depicted in panels (b) and (c) of Fig. 2. Extensive numerical calculations
suggest that, in the thermodynamic limit, and for $h_{0}=0$, the complexity
oscillates around a finite value, consistently remaining below unity for a
quench within the ferromagnetic phase. Conversely, it oscillates around the
unit value for a quench into the paramagnetic phase, thus crossing the
critical point $h_{c}=1/2$. Based on these findings, we can argue that we use
the time-averaged Krylov complexity to indicate a DPT-I.
We additionally examined how the time-averaged complexity varies concerning
the ground-state manifold. The LMG model possesses a two-fold degenerate
ground state, representing the north and south poles in the Bloch sphere
representation of the collective spin variables $S_{\alpha},\ (\alpha=x,y,z)$.
Interestingly, both of these initial ground states lead to the same pattern in
the behavior of the Krylov complexity over time. Consequently, the long-time
average $\overline{C}$ as a function of $h$ exhibits a qualitative similarity
to $\overline{S_{z}}$.
Given that DPT-I is associated with a dynamic symmetry break, it is reasonable
to anticipate that the sensitivity of the Krylov basis extends to the symmetry
of the model. Therefore, a deeper relationship between the Krylov basis and
the energy one should exist. This aspect will be investigated in the
concluding part of this section.
### V.2 Inverse Participation Ratio
In order to analyze the relation between the energy basis and the Krylov
basis, we first consider the inverse participation ratio (IPR) in each one of
these bases. The IPR is given by
$\displaystyle IPR(t)=\sum_{k}|\langle
k|\psi(t)\rangle|^{4}=\sum_{k}p_{k}^{2}(t),$ (17)
for some basis whose elements we generically denote as $\\{|k\rangle\\}$. As
its name suggests, the IPR measures how many states of the chosen basis
effectively participate in the course of the time evolution of the system.
Considering the same protocol as before, we compute the IPR for two distinct
bases: the pre-quench energy eigenbasis (the eigenvectors of $H_{0}$) and the
Krylov basis. Two instances of the results are shown in Fig. 3. Interestingly,
the IPR computed in both bases are identical. However, this coincidence only
happens when we start with $h_{0}=0$, when the initial energy spectra is
doubly degenerate. Extensive numerical analysis shows that they are different
otherwise. Another property of IPR that we observed numerically is the
independence with respect to which ground-state is taken as initial state,
thus giving the same result also for the second ground-state
$|\psi_{0}\rangle=|\uparrow\rangle_{z}$.
Figure 3: Inverse Participation Ratio. Panel (a) shows IPR for a quench
$h=0.3$, while panel (b) shows the same quantity, but for a quench crossing
the critical point ($h=0.8$). We take $N=200$ in both panels. Exactly the same
behaviour is observed if the initial state is taken as the second ground-
state, $|\uparrow\rangle_{z}$.
The reason for this match will be analytically explored latter in this
article. However, before presenting this analysis, let us consider another
interesting quantity, the Shannon entropy.
### V.3 $\mathcal{K}$-Entropy - Shannon entropy in the Krylov basis
Considering a probability distribution $p_{n}$, the Shannon entropy
$\displaystyle\mathcal{E}(t)=-\sum_{n}p_{n}(t)\log(p_{n}(t))$ (18)
is a measure of the uncertainty of the system —or the classical information
contained in the system. We are here interested in the entropy associated with
both bases discussed in the previous section: the Krylov basis, for which
$p_{n}(t)=|\langle K_{n}|\psi(t)\rangle|^{2}$ and the initial energy basis,
whose probabilities are $p_{n}(t)=|\langle E^{0}_{n}|\psi(t)\rangle|^{2}$. The
Shannon entropy in the Krylov basis was introduce in Ref. [12] and is known
also as the $\mathcal{K}$-entropy. It is a measure of complexity of the
dynamics. The authors of Ref. [31] argued that the complexity defined as the
exponential of this quantity measures the minimum Hilbert space dimension
required to store the probability distribution $p_{n}(t)$.
By comparing the Shannon entropy in both basis we again observe a perfect
match. Also, such feature only occurs when we start at $h_{0}=0$. An example
is shown in Fig. 4 for the same parameters used in Fig. 3.
Figure 4: Shannon entropy. Panel (a) shows the results for the quench $h=0.3$,
while panel (b) considers the case $h=0.8$. In both cases the initial state is
$|\psi_{0}\rangle=|\downarrow\rangle_{z}$ and $N=200$ as in the previous
calculations. Similar results are obtained considering the second ground-state
of the LMG model for $h_{0}=0$, which we denote by $|\uparrow\rangle$.
We note that the probabilities $p_{n}(t)=|\langle
E^{0}_{n}|\psi(t)\rangle|^{2}$ are the populations of the density matrix in
the energy eigenbasis and, thus, the entropy associated with such distribution
is the diagonal entropy, proposed as the thermodynamic entropy for closed
quantum systems [64]. Since $p_{n}(t)=|\langle K_{n}|\psi(t)\rangle|^{2}$ can
be seen as the probability density associated to site $n$ in the semi-infinite
Krylov lattice, naturally the corresponding Shannon entropy
$\mathcal{E}_{\mathcal{K}}(t)$ can be interpreted as the uncertainty in the
spreading of the initial state through the Krylov subspace.
The numerical results described above indicates a deeper match, beyond average
spread. The goal of the next subsection is to investigate why this happens.
### V.4 Derivation of the Krylov basis
Figure 5: Lanczos coefficients. Comparison between the numerical (dots) and
analytical (yellow lines) coefficients for $N=200$ and
$|\psi_{0}\rangle=|\downarrow\rangle_{z}$. They feature perfect agreement
showing that $b_{m_{z}}=c_{+}(m_{z})$ and attesting that the Krylov states are
the usual angular momentum states $|j,m_{z}\rangle$ if the initial state is
equal to any one of the ground-states.
Let us consider quenches starting from $h_{0}=0$. In this case, it is possible
to derive analytical results involving the Krylov basis. The eigenstates of
the pre-quench Hamiltonian
$\displaystyle H_{0}=-\frac{1}{2j}S_{z}^{2},$ (19)
are the usual angular momentum basis $|j,m_{z}\rangle$, with the index
$m_{z}\in\\{-j,-j+1,\cdots,j-1,j\\}$ specifying the $2j+1$ spin projections.
Of course, the ground-states denoted by $|\uparrow\rangle_{z}$ and
$|\downarrow\rangle_{z}$ mentioned previously are examples of these
eigenstates. Then, the quench is performed in the magnetic field $h$ such that
the post-quench Hamiltonian can be written in terms of ladder operators
$\displaystyle
H_{f}=-\frac{1}{2j}S_{z}^{2}-\frac{h_{f}}{2}\left(S_{+}+S_{-}\right).$ (20)
Now we observe that the set of operators $\\{S_{z},S_{+},S_{-}\\}$ are the
operators defining the well-known $SU(2)$ algebra, thus the following
commutation relations hold
$\displaystyle[S_{z},S_{\pm}]=\pm S_{\pm},\ \ \ \ [S_{+},S_{-}]=2S_{z}.$ (21)
The action of the Hamiltonian given in Eq. (20) on one of the basis states
$|j,m_{z}\rangle$ is
$\displaystyle
H_{f}|j,m_{z}\rangle=c_{0}|j,m_{z}\rangle+c_{+}|j,m_{z}+1\rangle+c_{-}|j,m_{z}-1\rangle$
(22)
where $c_{0}=-m_{z}^{2}/2j$, $c_{+}=-\frac{h}{2}\sqrt{j(j+1)-m_{z}(m_{z}+1)}$
and $c_{-}=-\frac{h}{2}\sqrt{j(j+1)-m_{z}(m_{z}-1)}$. Comparing Eqs. (22) with
Eq. (11), we immediately see that the states satisfying the Lanczos algorithm
for $|\psi_{0}\rangle=|j,m_{z}\rangle$ are precisely the set
$\\{|j,m_{z}\rangle\\}$ up to a factor $\pm 1$, that is
$\displaystyle|K_{i}\rangle=\pm|j,m_{z}\rangle.$ (23)
Therefore, we conclude that the Krylov states are proportional to the pre-
quench energy eigenstates. This explains why both the IPR and the Shannon
entropy are the same in both bases.
The action of $H_{f}$ on the state $|j,m_{z}\rangle$, Eq. (22), provides us an
analytical expression for the Lanczos coefficients, which correspond to the
constant $c_{-}$. For convenience, we relabel the index $m_{z}$ as
$m_{z}\rightarrow-j+m_{z}$ such that now $m_{z}$ runs through the set
$\\{0,1,\cdots,2j\\}$. With this change, the constant $c_{-}$ becomes
$\displaystyle c_{-}(m_{z})=\frac{h_{f}}{2}\sqrt{m_{z}(2j-m_{z}+1)},$ (24)
where we absorbed the minus sign in the state $|j,m_{z}\rangle$. Figure 5
shows the perfect agreement between the Lanczsos coefficients calculated
numerically using the Hamiltonian (14) and the Lanczos algorithm (9) and the
analytical expression for $c_{-}$, proving that $c_{-}(m_{z})=b_{m_{z}}$.
Since $|K_{m_{z}}\rangle=\pm|j,m_{z}\rangle$, the Krylov basis must share the
same symmetries as the pre-quench energy basis. This fact turns the Krylov
basis sensitive to the break or restoration of the spin-flip symmetry if a
quench is performed in the LMG model, which explains the ability of the Krylov
complexity to characterize DPT-I. Moreover, note that we can write the
expression of the Krylov complexity in the form
$\displaystyle
C_{\mathcal{K}}(t)=\sum_{m_{z}=0}^{2j}m_{z}|\langle\psi_{t}|K_{m_{z}}\rangle|^{2}.$
(25)
Considering now the expression of $S_{z}(t)$,
$\displaystyle S_{z}(t)=\langle\psi_{t}|\hat{S}_{z}|\psi_{t}\rangle.$ (26)
and employing the completeness of the angular momentum basis,
$\sum_{m_{z}=0}^{2j}|j,-j+m_{z}\rangle\langle j,-j+m_{z}|=\mathbb{I}$, we can
readily show that
$\displaystyle C_{\mathcal{K}}(t)=S_{z}(t)+j,$ (27)
thus confirming that
$\displaystyle\overline{C}=\overline{S_{z}}+j.$ (28)
The above discussion proves that the Krylov complexity must have the same time
behaviour as the magnetization. We thus conclude that its time average is an
order parameter for this model.
Finally, we note that when $|\psi_{0}\rangle=|j,m_{z}\rangle$ for any $m_{z}$,
$\dim{(\mathcal{K})}=\dim{(\mathcal{H})}$, i.e., the dimension of the Krylov
subspace is equal to the dimension of the Hilbert space.These initial states
explore the whole Hilbert space, showing some kind of ergodicity. We emphasize
that this is not an feature of all quantum systems.
## VI Conclusion
Understanding the complex nature of the temporal evolution of many-body
quantum systems holds fundamental significance across various research fields.
This study contributes to this area by examining the Krylov complexity (spread
complexity) and its connection to the dynamical phase transition within the
LMG model. In essence, we show that the time-averaged Krylov complexity serves
as an order parameter in this context. This conclusion stems from a numerical
investigation encompassing not only the Krylov complexity but also the inverse
participation ratio and Shannon entropy within the Krylov basis. These
analysis suggest a deep relation between the Krylov basis and the energy
eigenbasis. Subsequently, an analytical study establishes that the time-
averaged Krylov complexity effectively signals the dynamical phase transition
in this model by means of the equivalence between the Krylov and the energy
bases.
It is important to note that a thorough numerical investigation reveals that
this connection is valid exclusively in instances where the symmetry of the
model is broken. Our analytical analysis further substantiates this
conclusion. This prompts questions about the how Krylov complexity behaves
under general changes of symmetry. Addressing this question will certain
deepen our comprehension of the dynamics inherent in many-body systems.
Our findings also provide insights into the thermodynamics of critical systems
of this nature. While the thermodynamics of equilibrium phase transitions is
well-established, the same cannot be said for its dynamic counterpart. Our
results indicate that, in the case of the LMG model, the $\mathcal{K}$-entropy
serves as the quantum thermodynamic entropy. This assertion is grounded in the
definition of diagonal entropy [64] and the gauge theory of quantum
thermodynamics outlined in Ref.[65]. Furthermore, this observation aligns with
earlier investigations into the thermodynamics of dynamical quantum phase
transitions, where a thermodynamic entropy in quantum phase space not only
signals the transition but also, on average, exhibits a monotonic increase
over time [66]. Consequently, our results suggest a profound connection
between dynamical critical behavior and the process of thermalization.
An intriguing observation emerges when considering the interconnectedness of
symmetry and thermalization in thermodynamics, suggesting a profound link
between the inquiries addressed in the preceding paragraphs.
Whether the time-averaged Krylov complexity can function as an order parameter
for DPT-I in systems beyond the LMG model remains an open question.
Additionally, delving into the broader applicability of the connection between
Krylov and energy bases is crucial. Addressing these questions promises to
provide valuable insights into the underlying nature of this dynamic critical
behavior.
###### Acknowledgements.
This work was supported by the National Institute for the Science and
Technology of Quantum Information (INCT-IQ), Grant No. 465469/2014-0, by the
National Council for Scientific and Technological Development (CNPq), Grants
No 308065/2022-0, and by Coordination of Superior Level Staff Improvement
(CAPES).
## References
* Zurek [2018] W. Zurek, _Complexity, Entropy And The Physics Of Information_ (CRC Press, 2018).
* Lloyd and Pagels [1988] S. Lloyd and H. Pagels, Complexity as thermodynamic depth, Annals of Physics 188, 186 (1988).
* Zurek [1989] W. H. Zurek, Algorithmic randomness and physical entropy, Phys. Rev. A 40, 4731 (1989).
* Berthiaume _et al._ [2001] A. Berthiaume, W. van Dam, and S. Laplante, Quantum kolmogorov complexity, Journal of Computer and System Sciences 63, 201 (2001).
* Lorenz [1993] E. N. Lorenz, _The Essence of Chaos_ (University of Washington Press, Seattle, Washington, 1993).
* Maldacena _et al._ [2016] J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics 2016, 10.1007/jhep08(2016)106 (2016).
* Fine _et al._ [2014] B. V. Fine, T. A. Elsayed, C. M. Kropf, and A. S. de Wijn, Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2, Phys. Rev. E 89, 012923 (2014).
* Rozenbaum _et al._ [2020] E. B. Rozenbaum, L. A. Bunimovich, and V. Galitski, Early-time exponential instabilities in nonchaotic quantum systems, Phys. Rev. Lett. 125, 014101 (2020).
* Hashimoto _et al._ [2020] K. Hashimoto, K.-B. Huh, K.-Y. Kim, and R. Watanabe, Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator, Journal of High Energy Physics 2020, 68 (2020).
* Parker _et al._ [2019] D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
* Lanczos [1950] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. of Research of the Nation Bureau of Standards 45, 255 (1950).
* Barbón _et al._ [2019] J. L. F. Barbón, E. Rabinovici, R. Shir, and R. Sinha, On the evolution of operator complexity beyond scrambling, Journal of High Energy Physics 2019, 264 (2019).
* Rabinovici _et al._ [2021] E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Operator complexity: a journey to the edge of Krylov space, J. High Energ. Phys. 2021 (6), 62.
* Dymarsky and Smolkin [2021] A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104, L081702 (2021).
* Rabinovici _et al._ [2022a] E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov localization and suppression of complexity, Journal of High Energy Physics 2022, 10.1007/jhep03(2022)211 (2022a).
* Caputa _et al._ [2022] P. Caputa, J. M. Magan, and D. Patramanis, Geometry of krylov complexity, Phys. Rev. Res. 4, 013041 (2022).
* Rabinovici _et al._ [2022b] E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity from integrability to chaos, Journal of High Energy Physics 2022, 10.1007/jhep07(2022)151 (2022b).
* Hörnedal _et al._ [2022] N. Hörnedal, N. Carabba, A. S. Matsoukas-Roubeas, and A. del Campo, Ultimate speed limits to the growth of operator complexity, Communications Physics 5, 207 (2022).
* Mück and Yang [2022] W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nuclear Physics B 984, 115948 (2022).
* Hashimoto _et al._ [2023] K. Hashimoto, K. Murata, N. Tanahashi, and R. Watanabe, Krylov complexity and chaos in quantum mechanics (2023), arXiv:2305.16669 [hep-th] .
* Bhattacharya _et al._ [2022] A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Operator growth and krylov construction in dissipative open quantum systems, Journal of High Energy Physics 2022, 10.1007/jhep12(2022)081 (2022).
* Liu _et al._ [2023] C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5, 033085 (2023).
* Hörnedal _et al._ [2023] N. Hörnedal, N. Carabba, K. Takahashi, and A. del Campo, Geometric operator quantum speed limit, wegner hamiltonian flow and operator growth, Quantum 7, 1055 (2023).
* Bhattacharjee _et al._ [2023a] B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, Operator growth in open quantum systems: lessons from the dissipative syk, Journal of High Energy Physics 2023, 10.1007/jhep03(2023)054 (2023a).
* Bhattacharya _et al._ [2023] A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, On krylov complexity in open systems: an approach via bi-lanczos algorithm (2023), arXiv:2303.04175 [quant-ph] .
* Bhattacharjee _et al._ [2023b] B. Bhattacharjee, P. Nandy, and T. Pathak, Operator dynamics in lindbladian syk: a krylov complexity perspective (2023b), arXiv:2311.00753 [quant-ph] .
* Ballar Trigueros and Lin [2022] F. Ballar Trigueros and C.-J. Lin, Krylov complexity of many-body localization: Operator localization in krylov basis, SciPost Physics 13, 10.21468/scipostphys.13.2.037 (2022).
* Caputa and Liu [2022] P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Physical Review B 106, 10.1103/physrevb.106.195125 (2022).
* Takahashi and del Campo [2023] K. Takahashi and A. del Campo, Shortcuts to adiabaticity in krylov space (2023), arXiv:2302.05460 [quant-ph] .
* Bhattacharjee [2023] B. Bhattacharjee, A lanczos approach to the adiabatic gauge potential (2023), arXiv:2302.07228 [quant-ph] .
* Balasubramanian _et al._ [2022] V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
* Scialchi _et al._ [2023] G. F. Scialchi, A. J. Roncaglia, and D. A. Wisniacki, Integrability to chaos transition through krylov approach for state evolution (2023), arXiv:2309.13427 [quant-ph] .
* Eisert _et al._ [2015] J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nature Physics 11, 124–130 (2015).
* [34] M. Heyl, Dynamical quantum phase transitions: a review, Rep. Prog. Phys. 81, 054001 (2018).
* Marino _et al._ [2022] J. Marino, M. Eckstein, M. S. Foster, and A. M. Rey, Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments, Reports on Progress in Physics 85, 116001 (2022).
* Lipkin _et al._ [1965] H. Lipkin, N. Meshkov, and A. Glick, Validity of many-body approximation methods for a solvable model: (i). exact solutions and perturbation theory, Nuclear Physics 62, 188 (1965).
* Žunkovič _et al._ [2016] B. Žunkovič, A. Silva, and M. Fabrizio, Dynamical phase transitions and loschmidt echo in the infinite-range xy model, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374, 20150160 (2016).
* Muga _et al._ [2007] G. Muga, R. Mayato, and I. Egusquiza, _Time in Quantum Mechanics_, Lecture Notes in Physics (Springer Berlin Heidelberg, 2007).
* Muga _et al._ [2012] G. Muga, A. Ruschhaupt, and A. Campo, _Time in Quantum Mechanics - Vol. 2_ , Lecture Notes in Physics (Springer Berlin Heidelberg, 2012).
* del Campo [2011] A. del Campo, Long-time behavior of many-particle quantum decay, Phys. Rev. A 84, 012113 (2011).
* del Campo [2016] A. del Campo, Exact quantum decay of an interacting many-particle system: the calogero–sutherland model, New Journal of Physics 18, 015014 (2016).
* Heyl _et al._ [2013] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field ising model, Phys. Rev. Lett. 110, 135704 (2013).
* LeClair _et al._ [1995] A. LeClair, G. Mussardo, H. Saleur, and S. Skorik, Boundary energy and boundary states in integrable quantum field theories, Nuclear Physics B 453, 581–618 (1995).
* Yang and Lee [1952] C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. i. theory of condensation, Phys. Rev. 87, 404 (1952).
* Lee and Yang [1952] T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. ii. lattice gas and ising model, Phys. Rev. 87, 410 (1952).
* Andraschko and Sirker [2014] F. Andraschko and J. Sirker, Dynamical quantum phase transitions and the loschmidt echo: A transfer matrix approach, Phys. Rev. B 89, 125120 (2014).
* Campbell [2016] S. Campbell, Criticality revealed through quench dynamics in the lipkin-meshkov-glick model, Phys. Rev. B 94, 184403 (2016).
* Žunkovič _et al._ [2018] B. Žunkovič, M. Heyl, M. Knap, and A. Silva, Dynamical quantum phase transitions in spin chains with long-range interactions: Merging different concepts of nonequilibrium criticality, Phys. Rev. Lett. 120, 130601 (2018).
* Halimeh _et al._ [2021] J. C. Halimeh, M. Van Damme, L. Guo, J. Lang, and P. Hauke, Dynamical phase transitions in quantum spin models with antiferromagnetic long-range interactions, Phys. Rev. B 104, 115133 (2021).
* Hashizume _et al._ [2022] T. Hashizume, I. P. McCulloch, and J. C. Halimeh, Dynamical phase transitions in the two-dimensional transverse-field ising model, Phys. Rev. Res. 4, 013250 (2022).
* Schmitt and Kehrein [2015] M. Schmitt and S. Kehrein, Dynamical quantum phase transitions in the kitaev honeycomb model, Phys. Rev. B 92, 075114 (2015).
* Karrasch and Schuricht [2013] C. Karrasch and D. Schuricht, Dynamical phase transitions after quenches in nonintegrable models, Phys. Rev. B 87, 195104 (2013).
* Puebla [2020] R. Puebla, Finite-component dynamical quantum phase transitions, Phys. Rev. B 102, 220302 (2020).
* Defenu _et al._ [2023] N. Defenu, A. Lerose, and S. Pappalardi, Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions (2023), arXiv:2307.04802 [cond-mat.quant-gas] .
* Heyl [2014] M. Heyl, Dynamical quantum phase transitions in systems with broken-symmetry phases, Phys. Rev. Lett. 113, 205701 (2014).
* [56] Á. L. Corps and A. Relaño, Dynamical and excited-state quantum phase transitions in collective systems. Phys. Rev. B 106, 024311 (2022).
* [57] Á. L. Corps and A. Relaño, Theory of dynamical phase transitions in quantum systems with symmetry-breaking eigenstates. Phys. Rev. Lett. 130, 100402 (2023).
* V. Viswanath [1994] G. M. V. Viswanath, _The Recursion Method: Application to Many-Body Dynamics_ (Springer-Verlag, 1994).
* Dusuel and Vidal [2005] S. Dusuel and J. Vidal, Continuous unitary transformations and finite-size scaling exponents in the lipkin-meshkov-glick model, Phys. Rev. B 71, 224420 (2005).
* Lerose _et al._ [2019] A. Lerose, B. Žunkovič, J. Marino, A. Gambassi, and A. Silva, Impact of nonequilibrium fluctuations on prethermal dynamical phase transitions in long-range interacting spin chains, Phys. Rev. B 99, 045128 (2019).
* Sciolla and Biroli [2013] B. Sciolla and G. Biroli, Quantum quenches, dynamical transitions, and off-equilibrium quantum criticality, Phys. Rev. B 88, 201110 (2013).
* Ribeiro _et al._ [2008] P. Ribeiro, J. Vidal, and R. Mosseri, Exact spectrum of the lipkin-meshkov-glick model in the thermodynamic limit and finite-size corrections, Phys. Rev. E 78, 021106 (2008).
* Mori _et al._ [2018] T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Thermalization and prethermalization in isolated quantum systems: a theoretical overview, Journal of Physics B: Atomic, Molecular and Optical Physics 51, 112001 (2018).
* Polkovnikov [2011] A. Polkovnikov, Microscopic diagonal entropy and its connection to basic thermodynamic relations, Annals of Physics 326, 486–499 (2011).
* Céleri and Rudnicki [2021] L. C. Céleri and L. u. Rudnicki, Gauge invariant quantum thermodynamics: consequences for the first law, arXiv:2104.10153 (2021).
* Goes _et al._ [2020] B. O. Goes, G. T. Landi, E. Solano, M. Sanz, and L. C. Céleri, Wehrl entropy production rate across a dynamical quantum phase transition, Phys. Rev. Res. 2, 033419 (2020).
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# Addressing Training Bias in Machine Learning Visualization Recommendation
Systems by Identifying Trends in Training Data
Improving VizML Through a Statistical Analysis of the Plotly Community Feed
Allen Tu<EMAIL_ADDRESS>University of Maryland, College ParkCollege ParkMDUSA ,
Priyanka Mehta<EMAIL_ADDRESS>University of Maryland, College ParkCollege
ParkMDUSA , Alexander Wu<EMAIL_ADDRESS>University of Maryland, College
ParkCollege ParkMDUSA , Nandhini Krishnan<EMAIL_ADDRESS>University of
Maryland, College ParkCollege ParkMDUSA and Amar Mujumdar<EMAIL_ADDRESS>University of Maryland, College ParkCollege ParkMDUSA
## Abstract
Machine learning is a promising approach to visualization recommendation due
to its high scalability and representational power. Researchers can create a
neural network to predict visualizations from input data by training it over a
corpus of datasets and visualization examples. However, these machine learning
models can reflect trends in their training data that may negatively affect
their performance. Our research project aims to address training bias in
machine learning visualization recommendation systems by identifying trends in
the training data through statistical analysis.
## 1\. Introduction
Our project focuses on VizML (10.1145/3290605.3300358, 1), a machine learning
recommendation system that is trained over 2.3 million dataset-visualization
pairs from the open-source Plotly Community Feed. VizML is adept at providing
interpretable measures of feature importance, and its models can easily be
integrated into existing visualization systems due to their learned
understanding of design choices. VizML recommendations are quantifiably
similar to visualizations created by human analysts, and its performance
exceeds that of other visualization recommendation systems.
However, VizML’s creators acknowledge that it suffers from significant
training bias. In other words, VizML models reflect trends in the Plotly
Community Feed that heavily influence its performance. The authors do not
elaborate any further, and we did not find any existing discussions about
trends in the Plotly Community Feed. Furthermore, bias in machine learning
recommendation systems seems to be an underexplored topic as a whole.
The sheer size of VizML models poses several challenges for offsetting their
bias. Since we are limited in both time and computing power, we are not able
to experiment by training multiple full VizML models. We cannot create a
similar dataset for testing, and it is difficult to adapt other open-source
datasets to the required input format. Additionally, the creators of VizML do
not explain their findings about training bias, so we do not have any prior
knowledge of the trends that we are searching for. Thus, performing a
statistical analysis of the Plotly Community Feed is our optimal approach.
In this paper, we perform a statistical analysis to identify trends in the
training data. Then, we evaluate the significance of these trends and connect
them to the performance of the model. Finally, we develop methods to address
our findings and improve the model. Although our results are specific to
VizML, our process is generalizable and can be used to improve any machine
learning visualization recommendation model.
Our insight is that we can address training bias in machine learning
visualization recommendation systems by identifying trends in the training
data through statistical analysis. In our findings, we identified several
trends in the Plotly Community Feed that influence recommendations by the
VizML model. Some of these trends are conscious design decisions that improve
the readability of visualizations, while others demonstrate how Plotly users
tend to create charts based on convenience rather than effectiveness. We can
use our findings to improve the model by counteracting detrimental trends and
improving in areas of deficiency.
The next section of our paper will focus largely on related work. We discuss
machine learning visualization recommendation systems, the VizML model, and
the current state of research on its training bias. Then, we detail our
overall approach and how we collected and processed our data. Our findings are
centered around our statistical analysis of the Plotly Community Feed. We
identify significant trends in the data and connect them to the performance of
the model. In our conclusion, we summarize our results and explore avenues for
addressing the training bias.
## 2\. Related Work
Visualizing data is often the first step of analysis; however, visualization
tools can be difficult and tedious to use because they require the user to
manually specify options via code or menus. Recent computer science research
has led to sophisticated systems that automatically suggest tailored
visualizations based on the selected data and the user’s needs. These
recommenders can make creating visualizations more accessible to people
without an analytical background as well as assist analysts in producing them
more efficiently.
Current visualization recommenders, such as VisC
(10.1007/978-3-319-07233-3_58, 2) and Voyager 2 (2016-voyager, 3), rely on
large sets of predefined questions and rules. While these systems are
effective to an extent, they are costly to create in terms of labor because
they require domain experts to manually write constraints. Additionally, this
process hinders the system’s ability to identify trends, incorporate
contextual information, and adapt to user behavior (10.1145/3092931.3092937,
4).
A machine learning approach solves many of these limitations. Instead of
relying on a set of manually specified constraints, a machine learning model
generates predictions based on training data, so improvements can be made
relatively easily by retraining the model or fine-tuning its design. These
neural networks are also much more effective at capturing complex
relationships and can be trained on vast amounts of high-dimensional data.
### 2.1. VizML
Figure 1. “Diagram of data processing and analysis flow in VizML, starting
from (1) the original Plotly Community Feed API endpoints, proceeding to (2)
the deduplicated dataset-visualization pairs, (3a) features describing each
individual column, pair of columns, and dataset, (3b) design choices extracted
from visualizations, (4) task-specific models trained on these features, and
(5) potential recommended design choices” (10.1145/3290605.3300358, 1).
VizML (10.1145/3290605.3300358, 1), a more recent project, learns the
relationships between datasets and the visualizations analysts create from
them by training on 2.3 million dataset-visualization pairs from the Plotly
Community Feed. VizML models are adept at providing interpretable measures of
feature importance and can easily be integrated into existing visualization
systems. They are also trained on an extremely large and diverse corpus that
was created based on real visual analysis by analysts on their own datasets,
so their representational ability far exceeds that of older recommenders like
Draco-Learn (Draco, 5) and DeepEye (DeepEye, 6).
However, it is not without limitations. VizML’s creators acknowledge that the
model suffers from significant training bias. In other words, VizML models
reflect trends in the Plotly Community Feed that may significantly affect its
performance. The authors do not elaborate any further, and we did not find any
existing discussions about trends in the Plotly Community Feed. Furthermore,
bias in machine learning recommendation systems seems to be an underexplored
topic as a whole. Our research project aims to provide insights into VizML’s
training bias through a statistical analysis of the training data.
## 3\. Approach
The goal of our project is to address training bias in the VizML model. We
begin by collecting and processing training data from the Plotly Community
Feed. Then, we conduct a statistical analysis of the dataset-visualization
pairs by evaluating the frequency of each chart type. We discover and
interpret trends in the training data and connect them to the training bias of
the VizML model. Finally, we explore methods for addressing VizML’s training
bias based on our findings, avenues for further statistical analysis, and
possible expansions for the VizML model in our conclusion. These steps are
generalizable and can be used to improve any machine learning visualization
recommendation system.
Our insight is that we can address training bias in machine learning
visualization recommendation systems by identifying trends in the training
data. In our statistical analysis, we identified several trends in the Plotly
Community Feed that influence recommendations by the VizML model. Some of
these trends are conscious design decisions that improve the readability of
visualizations, while others demonstrate how Plotly users tend to create
charts based on convenience rather than effectiveness. We can use our findings
to improve the model by counteracting detrimental trends and improving in
areas of deficiency.
### 3.1. Data Collection
The VizML model is trained on the Plotly Community Feed dataset, which
consists of over 2.3 million dataset-visualization pairs. Analyzing the entire
corpus was unrealistic given our available time and computing power, so we
acquired a 1,000 example subset of the training data from the VizML GitHub
repository. Then, we loaded the data into a Pandas DataFrame inside of a
Jupyter Notebook for exploration and statistical analysis.
Figure 2. Pandas DataFrame of the processed dataset.
We found that the dataset is organized into four columns. fid, the example’s
unique identifier, and layout, the formatting specifier of the chart, are not
relevant to our analysis, so we discarded them from the DataFrame entirely.
The remaining two columns are chart_data, which specifies the user-created
chart, and table_data, which contains the user-submitted data that the chart
was created from. We convert each entry in these columns from the JSON format
into lists and dictionaries that are easily interpreted by most Python
packages and libraries.
Figure 3. Scatter plot in chart_data. ‘Temperature’ and ‘RH’ are plotted
across a single predictor variable. Regression lines for each response
variable are drawn over the data points.
Each row in chart_data is a list of chart objects represented by dictionaries.
The entries in the dictionary specify the properties of the chart object. All
dictionaries are required to include the name of the object, the type of
chart, and the xsrc and ysrc of the predictor and response variables,
respectively, in the corresponding table_data entry. The user can also specify
optional fields. For example, they can fix the color of the object to
turquoise, or they can update mode to add an automatically calculated
regression line to a ‘scatter’ type object.
### 3.2. Statistical Analysis
The vast majority of the chart_type objects across our entire 1,000 example
dataset only have the four required fields: name, type, xsrc, and ysrc. name
is not useful for our purposes because VizML generates them based on the
column labels in the data. VizML also relies on the user to select the data
query, so it is difficult to measure the impact of xsrc and ysrc on model bias
through only a statistical analysis of the training data. The remaining field
is type, which most significantly influences the VizML model because it
specifies the type of chart that is created from the input data query.
By analyzing the trends in the type field, we will be able to draw insights
regarding the types of visualizations that the VizML model is likely to
recommend. Our statistical analysis focuses on the overall frequency and
complexity of charts typical of our subset of the Plotly Community Feed. We
determined the frequency of each of the 15 chart types in our dataset and
identified trends in the preferences of Plotly users. For each of the three
most common chart types, we evaluated trends in their complexity by measuring
the number of response variables or categories visualized by each of their
corresponding examples.
## 4\. Findings
### 4.1. Frequency of Chart Types
First, we calculate the frequency of each type of chart in our dataset. Each
row in chart_data maps to exactly one chart type. It is worth noting that line
charts are initialized with the line field instead of the type field, and they
are distinct from scatter plots with a line overlaid on the point
distribution. Charts initialized without a type field default to scatter
plots.
Figure 4. Frequency of each chart type in the dataset. Scatter, line, and bar
plots account for 39.6%, 31.0%, and 17.9% of the examples, respectively. The
remaining twelve less common chart types account for 11.4% of the examples.
We find that scatter plots, line plots, and bar plots are by far the most
common chart types in the Plotly Community Feed. We expected this because
these chart types are simple and effective at visualizing trends in data, and
they are often the first visualizations that analysts reach for in their
initial exploration of the data. VizML is almost guaranteed to recommend one
of these three chart types for every selected data query.
It is unlikely for the model to ever recommend any of the other twelve chart
types, which account for only 11.4% of the dataset all together. For example,
the fourth most common chart type is the heatmap, which has a relative
frequency of 0.031. The scatter plot has a relative frequency of 0.396, so it
is almost 13 times more likely for a Plotly user to create a scatter plot than
a heatmap. Since many of these chart types fill somewhat niche roles, this is
also expected.
However, a notably underrepresented chart type is the histogram. It is a
ubiquitous, effective chart type that is commonly used by analysts to
visualize datasets typical of the Plotly Community Feed. Most analysts would
likely expect its relative frequency to be much higher than 0.006; in fact, we
expected it to be the fourth most common chart type.
A possible explanation is that a histogram can be relatively difficult to set
up in Plotly. If the user is not satisfied with the automatically generated
bins, they have to specify them manually. This process requires some deeper
knowledge of the data, and additional coding is needed to manually specify
bins over a wide range. Thus, Plotly users may find it more convenient to
choose a scatter or line plot instead, even if they believe a histogram may be
more effective at visualizing their data.
### 4.2. Analysis of Scatter Plots
Figure 5. Frequency of the number of response variables across all scatter
plots in the dataset. One and two response variables account for 37.1% and
27.0% of the examples, respectively. We discarded the extreme outliers 61 and
216 response variables for readability.
Figure 6. Boxplot of the number of response variables across all scatter plots
in the dataset. The quartiles of the distribution are 1, 2, and 4 response
variables.
Next, we examined the number of objects in each chart to gauge their
complexity. Each object in a scatter plot represents a different response
variable. We found that 64.1% scatter plots in the Plotly Community Feed have
one or two response variables, but it is normal for them to have up to four.
It is unlikely for scatter plots to have five or more response variables,
although an extreme outlier that was omitted from the visualizations has 216.
The preference clearly leans towards simple scatter plots with less response
variables. Plotly users are unlikely to select more than four response
variables even if more data is available. A possible explanation is that a
scatter plot can become unreadable if the density of datapoints is too high,
so using too many response variables damages the effectiveness of the
visualization. Additionally, the plotly.scatter method creates single response
variable scatter plots by default, so users may prefer creating these simple
scatter plots out of convenience.
### 4.3. Analysis of Line Plots
Figure 7. Frequency of the number of lines across all line plots in the
dataset. 3, 4, 7, and 8 lines account for 18.1%, 37.2%, 18.8%, and 25.9% of
the examples, respectively. We discarded the extreme outlier 50 lines for
readability.
Figure 8. Boxplot of the number of lines across all line plots in the dataset.
The quartiles of the distribution are 4, 4, and 8 lines.
Each object in a line plot represents a single line. We found that most line
plots in the Plotly Community Feed have four to eight lines, but an extreme
outlier that was omitted from the visualization has 50. Our analysis indicates
it is unlikely for Plotly users to create a line plot with only one or two
lines; in fact, none were present in our 1,000 example subset.
In a previous segment of our analysis, we found that many scatter plots with
only one or two response variables use the mode field to overlay a regression
line. A possible explanation is that Plotly users prefer simpler, less
cluttered graphs. If a scatter plot has too many response variables, the
visual density of the data points and regression lines may make it difficult
to read. To solve this problem, the user can omit the data points entirely by
creating a line plot instead. Our analysis provides statistical evidence that
supports the theory that Plotly users prefer scatter plots for visualizing one
or two lines and line plots for visualizing three or more lines.
### 4.4. Analysis of Bar Plots
Figure 9. Frequency of the number of categories across all bar plots in the
dataset. 1 category accounts for 42.5% of the examples, while 2, 16, 18, and
24 categories account for 8.1%, 9.7%, 7.3%, and 9.7% of the examples,
respectively.
Figure 10. Boxplot of the number of categories across all barplots in the
dataset. The quartiles of the distribution are 1, 2, and 16 categories.
Each object in a bar plot represents a single category, or bar. 105, or 42.5%,
of the bar plots only have one category. We found this to be abnormal because
a primary purpose of a bar plot is to visually compare different categories.
The representational capacity of a single bar is limited because there is no
other data to compare it to. One category bar graphs essentially represent
only a single number or proportion, and most analysts would agree that it is
more effective to use another format to represent it. Thus, the high relative
frequency of single category bar graphs is a significant indicator of
potential bias in the Plotly Community Feed.
Interestingly, our distribution also indicates that it is just as common for
Plotly users to visualize two categories as it is for them to visualize 16 to
20 categories. Upon further statistical analysis, we determined that Plotly
users tend to create bar plots that either visualize a single category or
every available category in their table. A possible explanation is that these
are the two most convenient approaches when using the plotly.bar method. The
user can select one xsrc and ysrc pair to create a single bar, or they can
select their entire dataset and quickly create a bar for every category.
A more powerful visualization often lies in the middle ground. For example, it
may be more effective to show only the three most significant categories with
over 250 counts each without the other 17 categories with less than 10 counts
each. We found that bar plots with two to seven categories are more likely to
have selected only a subset of the available categories in their table. This
requires some preliminary knowledge of the dataset as well as additional
coding, which may explain why these examples are relatively less frequent.
Therefore, our statistical analysis suggests that users who picked one or all
of the categories may have done so out of convenience rather than
effectiveness.
## 5\. Conclusion
We found that the Plotly Community Feed heavily prefers a small subset of the
available chart types. Data queries with one to two response variables are
nearly always used to create scatter plots, while those with three to seven
response variables are sometimes used in line plots. Categorical data queries
are almost exclusively used to create bar plots. Since it uses the Plotly
Community Feed as training data, VizML strongly reflects these trends.
Recommendations made by the model given any data query are almost guaranteed
to be scatter, line, or bar plots; it is unlikely for the remaining twelve
chart types to be recommended at all.
VizML’s significant bias toward charts typical of its training data is not
necessarily negative. It recommends scatter, line, and bar plots because they
are the essential charts that the vast majority of Plotly users would only
create. Some preferences, such as choosing between line and scatter plots
based on the number of response variables, are practical design choices that
can increase the effectiveness of generated charts. Recommendations by VizML
are quantifiably similar to visualizations created by human analysts, and the
model’s performance exceeds that of other visualization recommendation
systems.
Figure 11. “Consensus-Adjusted Recommendation Score of three ML-based, two
rule-based, and two human predictors when predicting consensus visualization
type. Error bars show 95% bootstrapped confidence intervals, with 105
bootstraps. The mean minimum achievable score is the lower dashed line, while
the highest achieved CARS is the upper dotted line” (10.1145/3290605.3300358,
1).
However, accurately predicting human behavior is not always positive either.
For example, the histogram is a common and effective chart type, but it is
severely underrepresented in our dataset. Since histograms are relatively
difficult to set up in Plotly, users may choose to use a scatter or line plot
instead, even if they believe a histogram would be more effective. Similarly,
we found that 42.5% of bar plots in the dataset visualize a single category,
limiting their representational power. Further analysis also revealed that
users tend to create bar plots with either one or every category – the two
approaches that are the fastest to perform in Plotly.
Our insight is that some trends in the Plotly Community Feed are the result of
convenience rather than effectiveness. Scatter, line, and bar plots are
heavily preferred because they are common charts that are easy and simple to
create. Histograms can require more setup than line plots, so they are
neglected almost completely. Users tend to choose one or all of the categories
when creating a bar plot because determining which categories are significant
requires an intermediary step of analysis.
The goal of a visualization recommendation system is to recommend a high
quality visualization from a given data query, but the VizML model is
influenced by how easy or convenient it is to make certain visualizations in
Plotly. The user does not save any time because the recommendation process is
automatic, and the generated visualization may have been of higher quality if
there were more training examples that required more effort to manually
specify. Therefore, VizML’s training bias can be detrimental because human
behavior is influenced by factors beyond optimizing performance. Addressing
these weaknesses in the VizML model may lower its score on consensus-adjusted
evaluators; however, the resulting visualizations recommended by the model
will ultimately be of higher quality.
### 5.1. Avenues for Further Analysis
10,000 and 100,000 example subsets of the Plotly Community Feed, as well as
the entire 2.3 million example dataset, are available on the VizML GitHub
repository. Evaluating these larger datasets could reveal trends beyond the
ones we discovered in our 1,000 example subset. The scope of our statistical
analysis can also be extended to include layout and table_data, as well as the
remaining fields in chart_data. Some of the relationships in these areas only
affect a small demographic of charts, but they may have implications that
relate to our findings.
Due to our limited time and computing resources, we were not able to perform
experiments using the VizML model itself. A promising expansion is to
exaggerate different characteristics in the training data. Training VizML
using these variations could provide insight into how to affect biases in the
model. For example, experimenting with the proportion of chart types may
address the underrepresentation of histograms. Another experiment could
measure the performance of VizML on datasets that are not typical of the
Plotly Community Feed to identify areas for improvement.
### 5.2. Improving VizML
Our statistical analysis concluded that the Plotly Community Feed contains a
large number of examples that may have been influenced by convenience rather
than effectiveness. For example, the proportion of bar plots that visualize a
single category is abnormally high. We can identify and clean examples with
potentially unwanted characteristics like these, reducing their frequency in
the training data and their influence on the VizML model. They can easily be
replaced with better unused examples from the Plotly Community Feed.
Some chart types such as histograms are underrepresented by the training data
and will not be recommended by VizML. We can increase their relative frequency
by adding unused examples with these chart types from the Plotly Community
Feed. Since the training data is no longer a random sample of the population,
experimentation is necessary for finding the appropriate number of additional
examples. The new model will recommend these previously underrepresented chart
types, so the training bias is in part addressed. If VizML is configured to
recommend multiple visualizations, this will also help diversify the chart
types in the output.
Many chart types, like heatmaps and contour plots, fill niche purposes and are
ill-suited for a general use recommender like VizML. Fortunately, their
combined effect on the model’s performance is negligible due their low
relative frequency. A possible avenue for expansion is to train a VizML model
to specifically recommend these more complex visualizations. Approximately 7%
of our 1,000 example subset use these chart types, so a sizable training
dataset could be built by selecting all useful examples from a larger subset.
## References
* (1) Kevin Hu et al. “VizML: A Machine Learning Approach to Visualization Recommendation” In _Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems_ , CHI ’19 Glasgow, Scotland Uk: Association for Computing Machinery, 2019, pp. 1–12 DOI: 10.1145/3290605.3300358
* (2) Taissa Abdalla Filgueiras Sousa and Simone Diniz Junqueira Barbosa “Recommender System to Support Chart Constructions with Statistical Data” In _Human-Computer Interaction. Theories, Methods, and Tools_ Cham: Springer International Publishing, 2014, pp. 631–642
* (3) Kanit Wongsuphasawat et al. “Voyager: Exploratory Analysis via Faceted Browsing of Visualization Recommendations” In _IEEE Trans. Visualization & Comp. Graphics (Proc. InfoVis)_, 2016 URL: http://idl.cs.washington.edu/papers/voyager
* (4) Manasi Vartak et al. “Towards Visualization Recommendation Systems” In _SIGMOD Rec._ 45.4 New York, NY, USA: Association for Computing Machinery, 2017, pp. 34–39 DOI: 10.1145/3092931.3092937
* (5) Dominik Moritz et al. “Formalizing Visualization Design Knowledge as Constraints: Actionable and Extensible Models in Draco”, 2018 DOI: 10.31219/osf.io/3eg9c
* (6) Yuyu Luo, Xuedi Qin, Nan Tang and Guoliang Li “DeepEye: Towards Automatic Data Visualization”, 2018, pp. 101–112 DOI: 10.1109/ICDE.2018.00019
|
# Realization of all logic gates and memory latch in the SC-CNN cell of the
simple nonlinear MLC circuit
P. Ashokkumar<EMAIL_ADDRESS>PG & Research Department of Physics, Nehru
Memorial College (Autonomous), Affiliated to Bharathidasan University,
Puthanampatti, Tiruchirappalli - 621 007, India. M. Sathish Aravindh
<EMAIL_ADDRESS>PG & Research Department of Physics, Nehru Memorial
College (Autonomous), Affiliated to Bharathidasan University, Puthanampatti,
Tiruchirappalli - 621 007, India. Department of Nonlinear Dynamics, School of
Physics, Bharathidasan University, Tiruchirappalli - 620 024, India. A.
Venkatesan<EMAIL_ADDRESS>PG & Research Department of Physics, Nehru
Memorial College (Autonomous), Affiliated to Bharathidasan University,
Puthanampatti, Tiruchirappalli - 621 007, India. M. Lakshmanan
<EMAIL_ADDRESS>Department of Nonlinear Dynamics, School of Physics,
Bharathidasan University, Tiruchirappalli - 620 024, India.
###### Abstract
We investigate the State-Controlled Cellular Neural Network (SC-CNN) framework
of Murali-Lakshmanan-Chua (MLC) circuit system subjected to two logical
signals. By exploiting the attractors generated by this circuit in different
regions of phase-space, we show that the nonlinear circuit is capable of
producing all the logic gates, namely OR, AND, NOR, NAND, Ex-OR and Ex-NOR
gates available in digital systems. Further the circuit system emulates three-
input gates and Set-Reset flip-flop logic as well. Moreover, all these logical
elements and flip-flop are found to be tolerant to noise. These phenomena are
also experimentally demonstrated. Thus our investigation to realize all logic
gates and memory latch in a nonlinear circuit system paves the way to replace
or complement the existing technology with a limited number of hardware.
> Exploiting the hopping of trajectories in different wells of nonlinear
> systems, it is found that these systems possess the ability to produce all
> kinds of logic operations. Specifically in the present work, we show that if
> one uses two square waves in an aperiodic manner as input to the SC-CNN
> based simple nonlinear Murali-Lakshmanan-Chua(MLC) circuit, the response
> produces all the logic gates and RS flip-flop which are also experimentally
> demonstrated. Our investigation to realize all the logic gates and memory
> latch in a nonlinear circuit system paves the way to replace or complement
> the existing technology with a limited number of hardware components.
## I Introduction
Logic gates are building blocks of any digital circuit and computer
architecture. In general, following Boolean algebra, the logic operations are
performed by converting given two inputs into a single logical output. For any
logic operation, both the inputs and outputs have two states, namely ’ON’ or
’TRUE’ and ’OFF’ or ’FALSE’ states Mano (2003). The reliability of logic
operations depends on the reliable operations of systems chosen. Since the
current demand for miniaturization of logic devices, speed of computation and
low-power consumption devices, it is inevitable to design an appropriate
system which is able to produce noise-immune gates. As a result, for the past
two decades or more, several schemes are being proposed such as DNA/RNA
computing Adleman (1994); Collier _et al._ (1999), quantum computing Ladd
_et al._ (2010); Nielsen and Chuang (2002), nano computing Bachtold _et al._
(2001) and nonlinear dynamics based computing in order to replace or
complement the existing computer architecture based on silicon chipsHopfield
(1982); Sinha and Ditto (1998, 1999); Prusha and Lindner (1999); Murali _et
al._ (2009a, b); Sinha _et al._ (2009); Guerra _et al._ (2010); Worschech
_et al._ (2010); Zamora Munt and Masoller (2010); Zhang, Song, and He (2010);
Bulsara _et al._ (2010); Singh and Sinha (2011); Dari _et al._ (2011a, b);
Storni _et al._ (2012); Roychowdhury (2015); Kohar _et al._ (2017);
Venkatesh, Venkatesan, and Lakshmanan (2017a, b); Neves, Voit, and Timme
(2017); Kia, Lindner, and Ditto (2017); Murali _et al._ (2018); Manaoj
Aravind, Murali, and Sinha (2018); Sathish Aravindh, Venkatesan, and
Lakshmanan (2018); Sathish Aravindh _et al._ (2020). Among these methods,
nonlinear dynamics based computing can make reliable and re-configurable
computer architecture because the underlying nonlinear systems posses a large
number of basic functions.
Utilizing the flexibility of nonlinear dynamical systems for storing,
communicating and processing of information in computer architecture has been
an active area of research in nonlinear dynamics. In this connection, Hopfield
had constructed a memory device using artificial neural network to store and
retrieve information Hopfield (1982). Sinha and Ditto proposed a chaos-
computing scheme to emulate different logic elementsSinha and Ditto (1998,
1999). In an optimal window of noise, it was observed the possibility of
occurrence of logic behavior and this phenomenon is termed as logical
stochastic resonance(LSR)Murali _et al._ (2009a, b); Sinha _et al._ (2009);
Bulsara _et al._ (2010). The present authors employed strange nonchaotic
attractors to build dynamical logic gates Sathish Aravindh, Venkatesan, and
Lakshmanan (2018); Sathish Aravindh _et al._ (2020); Sathish Aravindh,
Venkatesan, and Lakshmanan (2020). Besides, self-sustained oscillators can
function as latches and registers if Boolean logic states are associated with
the phases of the oscillator signalsRoychowdhury (2015). Heteroclinic
computing is another nonlinear phenomenon based computation using a collective
system of nonlinear oscillators Ashwin and Borresen (2004, 2005).
Not only nonlinear dynamics based computing complements existing silicon based
technology, there also exist several efforts to extend computation techniques
to other domains such as opticalSingh and Sinha (2011), chemicalSinha _et
al._ (2009); de Lacy Costello _et al._ (2009); Stevens _et al._ (2012),
physicalSathish Aravindh, Venkatesan, and Lakshmanan (2018), mechanicalMahboob
_et al._ (2011), biologicalGerstung, Timmer, and Fleck (2009); Ando _et al._
(2011), molecularCollier _et al._ (1999); Kompa and Levine (2001) and other
areas of scienceMotoike _et al._ (2001); Sinha, Munakata, and Ditto (2002);
Norrell and Socolar (2009); Zhang, Song, and He (2010); Mozeika, Saad, and
Raymond (2010); Zhang _et al._ (2012); Miyamoto _et al._ (2010). Instead of
needing multiple hardware for different types of computations, nonlinear
dynamical systems can act as processors of a flexibly configured and
reconfigured device to produce different logic gates Worschech _et al._
(2010). In practice, the generation of nonidealitic and ambient noise
restricts the ability to obtain different logic gates in these systems Murali
_et al._ (2009b). Thus it is essential to choose appropriate nonlinear-
dynamics based computing systems to overcome these odds.
Further, most of the previous studies have focused to produce OR(NOR) and
AND(NAND) logic gates Sinha and Ditto (1998, 1999); Prusha and Lindner (1999);
Murali _et al._ (2009a, b); Kohar and Sinha (2012); Kohar, Murali, and Sinha
(2014); Kohar _et al._ (2017); Venkatesh, Venkatesan, and Lakshmanan (2017a,
b); Sathish Aravindh, Venkatesan, and Lakshmanan (2018); Sathish Aravindh _et
al._ (2020). Obtaining Ex-OR (simply XOR) and Ex-NOR (XNOR) in dynamical
systems are equally important since these gates are the basis of ubiquitous
bit-by-bit addition Storni _et al._ (2012). A half adder consists of an XOR
gate and AND gate. Other uses of XOR gate include subtractor, comparator and
controlled inverter. XOR and XNOR gates are usually obtained by the
concatenation of NOR and NAND gates.
In the literature, it has been shown that in several nonlinear-dynamics based
concepts like chaos computing Ditto, Murali, and Sinha (2008), heteroclinic
network for computation Ashwin and Borresen (2004, 2005); Bick and Rabinovich
(2009); Schittler Neves and Timme (2012); Neves, Voit, and Timme (2017), etc.
the richness of nonlinear dynamics can be exploited to obtain flexible and
reconfigurable logic gates including XOR gates. In this connection, Sinha _et
al._ have demonstrated the flexible parallel implementation of logic gates
using chaotic elements Sinha, Munakata, and Ditto (2002). Peng _et al._ have
explored piecewise-linear systems to construct all dynamical logic gates Peng
_et al._ (2008, 2010). Campos-Canton _et al._ have reported electronics
experiments to obtain NOR, NAND and XOR gates in a piece-wise linear system
Campos-Cantón _et al._ (2010). In ref.Cafagna and Grassi (2006); Campos-
Cantón _et al._ (2012a) the equation of the plane in analytical geometry has
been used to build a contribution of SR flip-flop and basic logic gates
Cafagna and Grassi (2006); Campos-Cantón _et al._ (2012a). Storni _et al._
have investigated LSR by extending the analysis to a three well potential to
realize XOR logic gates Storni _et al._ (2012). In ref.Canton, Martienz, and
Duron (2017), the authors have proposed a method and circuit for integrating a
programmable matrix in the field of reconfigurable logic gates employing a
nonlinear system and an efficient programmable rewiring Canton, Martienz, and
Duron (2017). Also Campos-Canton _et al._ have reported a parameterized method
to design multivibrator circuit via Chua’s circuit system Campos-Cantón _et
al._ (2012b).
Further, Murali _et al._ have shown that if one applies two low amplitude
square waves as inputs to a two-state system, the response of the system
produces a logical output (NOR/OR) with a probability controlled by the
interplay between noise and bistable dynamics of the system. That is the
interplay of nonlinearity and noise produces a flexible and realizable logic
behavior. The authors have termed this phenomenon as Logical Stochastic
Resonance (LSR) Bulsara _et al._ (2010). For the past few years LSR has been
realized in many nonlinear systems such as a nanoscale device Guerra _et al._
(2010), resonant tunnel diodes Worschech _et al._ (2010), a vertical cavity
surface emitting laser Zamora Munt and Masoller (2010); Zhang, Song, and He
(2010), a polarization bistable laser Singh and Sinha (2011), a chemical
system Bulsara _et al._ (2010), synthetic gene networks Dari _et al._
(2011a), and so on. Recently, two of the present authors along with Venkatesh
employed coupled dynamical systems to build dynamical logic gates by altering
the value of the logic inputs Venkatesh, Venkatesan, and Lakshmanan (2017b,
a).
Also Gupta _et al._ have examined the possibility of noise free LSR by driving
a two-stable system with periodic forcing instead of random noise Gupta _et
al._ (2011). Kohar _et al._ have found that periodic forcing enhances the LSR
in noisy bistable systems with periodic forcing instead of random noise Kohar,
Murali, and Sinha (2014). By replacing noise with high-frequency harmonics,
Venkatesh _et al._ realized logic operations AND, OR and RS flip-flop in the
MLC circuit and they termed this phenomenon as Logical Vibrational Resonance
(LVR). LVR has been realized in a two-potential system Venkatesh and
Venkatesan (2016); Venkatesh, Venkatesan, and Lakshmanan (2017b, a). Besides
logic gates, LSR and LVR can be used to realize SR flip-flops Gui _et al._
(2020a, b); Vincent _et al._ (2021); Murali _et al._ (2021). Then the
question arises whether LSR and LVR can produce all the logic gates including
XOR gate when one employs a bistable system. One finds that practically this
is not possible. The reason for this shortcoming is the bistable nonlinearity
of the system. For this case all the logic gates are obtained by assuming that
for the input state $(0,0)$, the response of the bistable system resides in
the left well and for the input state $(1,1)$ it is in the right well and for
the other input state $(0,1)/(1,0)$ it may be any one of the wells depending
upon the logic operations chosen. With these assumptions, bistable nonlinear
systems are well suited for producing OR/NOR and AND/NAND logic gates. However
these assumptions are not sufficient enough to obtain the other logic gates,
namely the XOR and XNOR gates. For example, an XOR gate admits a high logic
output only if the inputs are at different logic levels [(0,1) or (1,0)] and
low logic output only if the inputs are at the same logic level [(0,0) or
(1,1)]. For this logic gate case, the assumption for obtaining logic gates in
bistable nonlinear systems leads to a possible loss of information and further
the bistable nonlinear systems are unable to hold the condition for XOR/XNOR
logic gates. Thus, the bistable system is unable to produce XOR/XNOR gates. In
this paper, we propose a solution by considering a three-well potential
problem so as to implement all the logic elements including XOR gates.
It was also shown in several studies that the three well potential nonlinear
systems which exhibit logical behaviors are better suited for computational
purpose than the logical gates generated by the traditional bistable nonlinear
systems when one considers aspects such as noise interference, waveform
smoothness and bit error rate Lu _et al._ (2019); Liu and Liu (2014). Thus it
is important to ask the question whether one can realize all the logic gates
and memory latches in a single nonlinear circuit system. We address this issue
in this paper.
We consider a State-Controlled Cellular Neural Network (SC-CNN) based Murali-
Lakshmanan-Chua’s (MLC) circuit Günay (2010, 2010); Swathy and Thamilmaran
(2014). This circuit is constructed by using two CNN cells and external forces
including sinusoidal force, biasing and noise. The existing MLC consists of
Chua’s diode and a linear resistor, a linear capacitor, and an inductor. The
discrete inductor restricts the circuit for fabrications of ICs. Further, CNN
is well suited when extending the analysis to the coupled system. Thus SC-CNN
MLC is more advantageous than existing MLC in the aspects of hardware
realization. We show that this nonautonomous oscillator when subjected to two
aperiodic logical signals produces all the logic gates available in digital
electronics. We also show that these logic gates are tolerant for an optimal
range of noise intensity. Besides, it is reported that the fundamental logical
behaviors such as OR/AND/XOR gates can be observed through one of the state
variables of the circuit and for the realization of the complimentary logical
operations NOR/NAND/XNOR gates the other state variable can be utilized.
Further, we report the possibility of the occurrence of both high active SR
flip-flop and low SR flip-flop as well because of the parallelism innate in
this circuit. One can also further realize three-input gates in the circuit.
This paper is organized as follows. We discuss the dynamical mechanism for the
implementation of logical gates in the three well potential system and present
a basic study of the SC-CNN cell of the Murali-Lakshmanan-Chua’s (MLC) circuit
in Sec.III and Sec.III, respectively. We present the experimental realization
of CNN cell of the MLC circuit in Sec.IV. We also describe the experimental
and numerical results for the realization of OR/NOR and AND/NAND gates,
Exclusive-OR (XOR) and Exclusive-NOR (XNOR) logic gates, Set-Reset memory
latch, effect of noise and three input gates in Sec.V. Finally, we conclude
our analysis in Sec.VI.
## II Dynamical mechanism for the implementation of logical gates in three
well potential system
Consider a general nonlinear system
$\displaystyle\ddot{x}$
$\displaystyle=-\dfrac{dV(x)}{dx}-h\dot{x}+E+I+\sqrt{D}\xi(t)+fsin(\omega
t),~{}~{}\dot{x}=\dfrac{dx}{dt}$ (1a) Eq.(1a) can also be rewritten as
$\displaystyle\dot{x}$ $\displaystyle=y,$ $\displaystyle\dot{y}$
$\displaystyle=-\dfrac{dV(x)}{dx}-h\dot{x}+E+I+\sqrt{D}\xi(t)+fsinz,$
$\displaystyle\dot{z}$ $\displaystyle=\omega.$ (1b)
Figure 1: Schematic diagram of mechanism for obtaining all the Logic Gates in
a triple well system.
Here $V(x)$ is a potential function of the nonlinear system. Assume that the
potential $V(x)$ is three well. It has three potential wells and two potential
barriers as shown in Fig.1. Here $E$, $I$, $\xi(t)$ and $fsin(\omega t)$ are
the bias value, logical input value, Gaussian white noise with intensity
‘$D$’, and the amplitude of external periodic forcing, respectively. The
system (1) is driven by two square wave signals, namely $I_{1}$ and $I_{2}$,
which encode the logical inputs and the response of the system (1) is
considered as the logical output. For example, the inputs $I_{1}$ and $I_{2}$
take the value $+\Delta$ for the binary logic number ‘1’ and for $-\Delta$ the
binary logical value is ‘0’. Then, the four possible combinations viz. (0,0),
(0,1)/(1,0) and (1,1) of the input streams $I_{1},I_{2}$ are merged into three
distinct values $+2\Delta,0,-2\Delta$ as $I=I_{1}+I_{2}$. Thus, the resultant
input signal $I$ is a three level aperiodic wave form. Depending on the logic
input, the potential function of the system assumes three different forms. The
output of logic gates is given by the well in which the state variable ‘x’
resides. More precisely, for OR gate if ‘x’ resides in the left well, we set
the logic output as ‘0’, and ‘1’ if it resides in any of the center-well or
right-well. In the XOR gate, the ON output is possible if one and only if one
of the inputs to the gate is in the ON state. That is if both the inputs are
ON states or both are OFF, an OFF output results. In this circumstance, a
bistable system is not able to hold the three input states. Hence when we use
a bistable nonlinear system for these kind of cases, it leads to an effective
loss of information. To take care of this problem, we propose a solution by
generalizing the chosen system from a bistable one to a three well potential
system as described above. In particular we extend the scope of implementing
the logic gates by increasing the numbers of potential wells in order to
extend the possible input-output associations. For example, for the XOR gate
we set the output to be logical ’1’ if the state value of the system lies
between the two local maxima of the potential function, and it is assumed to
be the logical value ’0’ otherwise. For the XNOR gate we set the output to be
logical value ’0’ if the state of the system resides between the two local
maxima of the potential function of the system and it is to be ’1’ if it
resides anywhere in the other two potential wells. The truth table (see Table
1) for different input-output combinations is defined and summarized in
Table.2.
Table 1: Truth table of logic gates Logic Gates | (0,0) | (0,1)/(1,0) | (1,1)
---|---|---|---
AND | 0 | 0 | 1
NAND | 1 | 1 | 0
OR | 0 | 1 | 1
NOR | 1 | 0 | 0
XOR | 0 | 1 | 0
XNOR | 1 | 0 | 1
Table 2: Definitions of the outputs for obtaining all logic gates Logic Gates | Left well | Center well | Right well
---|---|---|---
AND | OFF | OFF | ON
NAND | ON | ON | OFF
OR | OFF | ON | ON
NOR | ON | OFF | OFF
XOR | OFF | ON | OFF
XNOR | ON | OFF | ON
## III SC-CNN cell of Murali-Lakshmanan-Chua’s (MLC) circuit
Cellular Neural Network (CNN), introduced by Chua and Yang in 1988 Chua and
Yang (1988), is a n-dimensional array of circuit elements of analog components
such as OP-amps, resistors, and capacitors $(n=1,2,3...)$. It is constructed
by a large number of intercoupled identical dynamical systems called cells.
These cells or nodes are essentially modeled by nonlinear ordinary
differential equations. CNN is a relatively simple structure and is easy to be
implemented by appropriate electronic circuit. Thus it is a powerful tool for
the emulation and implementation of nonlinear dynamical systems having complex
dynamics. A generalization of the CNN paradigm in which the CNN cells locally
share their outputs as well as their state variables with each other was
introduced by Arena Arena _et al._ (1995). This generalization exploits an
analog architecture of CNN known as State Controlled-CNN (SC-CNN). Many
studies have been reported on designing and implementation of chaotic circuits
in terms of SC-CNNs Günay (2010); Swathy and Thamilmaran (2014); Luo and Wang
(2016). The advantages of a SC-CNN are that they are inductorless and are RC
based circuitry only, thereby leading to the realization of hardware and VLSI
implementations Manganaro, Arena, and Fortuna (2012). The other significant
applications of these kind of CNNs are parallel computing and low-power
consumption. With these motivating facts, in this paper we consider an SC-CNN
frame work of the well known MLC circuit. This circuit is essentially
perturbed by two aperiodic square waves and as a consequence we show that the
resultant output exhibits parallel logic elements and memory latch. In order
to discuss this framework further, we first consider the MLC circuit as shown
in Fig.2.
Figure 2: Experimental realization of periodically driven Murali- Lakshmanan-
Chua circuit.
Figure 3: Panel (a) shows the realization of CNN based MLC circuit using five
op-amps (OA1-OA5) and 18 Resistors plus two capacitors. The resistors are
$R11-R17,R21-R27,Rn1-Rn4$. The capacitors are chosen as $C_{1}=C_{2}=10nF$.
Here $F(t)$ is the input driving signal and panel (b) shows the circuit for
generating the driving signal $F(t)$. Here we use four op-amps (OA6-OA7) and 8
resistors $R31-R38$.
It is an established fact in Murali, Lakshmanan, and Chua (1994); Lakshmanan
and Rajasekar (2003) that the normalized form of the well known MLC circuit
equation can be written as
$\displaystyle\dot{x}$ $\displaystyle=y-h(x),$ $\displaystyle\dot{y}$
$\displaystyle=-\beta(1+\nu)y-\beta x+E+\sqrt{D}\xi(t)+I+f\sin(\omega t).$
(2a) or equivalently as $\displaystyle\dot{x}$ $\displaystyle=y-h(x),$
$\displaystyle\dot{y}$ $\displaystyle=-\beta(1+\nu)y-\beta
x+E+\sqrt{D}\xi(t)+I+f\sin z,$ $\displaystyle\dot{z}$ $\displaystyle=\omega.$
(2b)
Here the overdot indicates time differentiation.
where
$h(x)=\left\\{\begin{array}[]{ll}bx+(a-b),&x>1,\\\ ax,&|x|\leq 1,\\\
bx-(a-b),&x<-1.\end{array}\right.$ (3)
The relationships between the various circuit variables and circuit parameters
and the above dynamical variables and parameters can be obtained from
Ref.Murali, Lakshmanan, and Chua (1994); Lakshmanan and Murali (1996);
Lakshmanan and Rajasekar (2003). Earlier studies on the dimensionless version
of the circuit when the parameters were fixed at
$a=-1.02,\>b=-0.55,\>\gamma=0.015,\>\beta=1.0$ and $\omega=1.0$ were made. The
quantity $f$ was varied in these studies Murali, Lakshmanan, and Chua (1994);
Lakshmanan and Rajasekar (2003). The circuit exhibits various underlying
dynamical features including period-doubling route to chaos, intermittent
route, strange non-chaotic attractors (SNA), etc Lakshmanan and Murali (1996);
Lakshmanan and Rajasekar (2003). Three of the present authors have also shown
that when the quasiperiodically driven MLC circuit is subjected to two
aperiodic and logical square waves, it reproduces logical response in both the
SNAs and chaotic regimes of the circuit. The authors have shown that how these
attractors are tolerant to noise, so that even one can emulate different logic
functions Sathish Aravindh _et al._ (2020).
Following the work of Arena Arena _et al._ (1995), the SC-CNN associated with
the MLC circuit can be generalized by the following dimensionless nonlinear
state equations,
$\displaystyle\dot{x}_{j}$ $\displaystyle=$ $\displaystyle-
x_{j}+a_{j}y_{j}+G_{0}+G_{s}+i_{j},$ $\displaystyle\dot{y}_{j}$
$\displaystyle=$ $\displaystyle 0.5\times(|x_{j}+1|-|x_{j}-1|).$ (4)
In Eq.(4), j is the cell index, $x_{j}$ and $y_{j}$ are the state variables
and the cell outputs, respectively. Each $a_{j}$ represents a constant
parameter and $i_{j}$ is the threshold value. Also in Eq.(4) $G_{0}$ and
$G_{s}$ are linear combinations of the outputs and state variables,
respectively, of the connected cells. A dynamic model of two generalized CNN
cells corresponding to Eq.(4) can be defined as follows:
$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle-
x_{1}+a_{1}y_{1}+a_{12}y_{2}+\sum_{k=1}^{2}s_{1k}x_{k}+i_{1},$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle-
x_{2}+a_{21}y_{1}+a_{2}y_{2}+\sum_{k=1}^{2}s_{2k}x_{k}+i_{2},$ (5)
where $x_{1}$ and $x_{2}$ are state variables, and $y_{1}$ and $y_{2}$ are the
corresponding outputs. The MLC circuit equation defined by Eq.(2) can be
derived from Eq.(4), by assuming : $x=x_{1}$, $y=x_{2}$, $a_{1}=b-a$,
$a_{12}=a_{21}=a_{2}=0$, $s_{11}=1-b$, $s_{12}=1$, $s_{21}=-\beta$,
$s_{22}=1-\beta(1+\nu)$, $i_{1}=0$ and $i_{2}=F(t)$.
Consequently, from Eq.(4) the SC-CNN based MLC circuit model is organized as
below :
$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle-
x_{1}+a_{1}y_{1}+s_{11}x_{1}+s_{12}x_{2},$ $\displaystyle\dot{x}_{2}$
$\displaystyle=$ $\displaystyle-x_{2}+s_{21}x_{1}+s_{22}x_{2}+F(t),$
$\displaystyle y_{1}$ $\displaystyle=$ $\displaystyle
0.5\times(|x_{1}+1|)-(|x_{1}-1|).$ (6)
Here $F(t)=E+\sqrt{D}\xi(t)+I+fsin(\omega t)$ and $I=I_{1}+I_{2}$,
corresponding to two logic inputs, $E$ is the bias and the remaining
parameters are fixed at $a=-1.02$, $b=-0.55$, $\nu=0.015$, $\beta=1.0$ and
$\omega=1.0$. The MLC circuit in the framework of SC-CNN has been studied
numerically, experimentally and analytically in ref.Günay (2010). The
standard MLC circuit (Fig.2) consists of a nonlinear resistor which has the
three-segment piece-wise characteristics of Chua’s diode, a linear resistor, a
linear inductor, and a linear capacitor with sinusoidal voltage source. On the
other hand the SC-CNN model of MLC circuit (see Fig.3) mainly consists of a
few OP-AMPs along with RC based circuits. Thus the SC-CNN model of the MLC
circuit is an inductor-less purely RC based circuit, which leads to the
realization of hardware easily. In the present paper, we consider this circuit
and investigate the effect of two aperiodic and logical square wave signals on
the SC-CNN of MLC circuit.
## IV Experimental realization of CNN cell of MLC
Fig.3 represents the complete experimental setup for two SC-CNN cells which
generate the dynamics of the SC-CNN based MLC circuit. It is clearly obvious
from Fig.3, that the circuit consists of a CNN with two cells, corresponding
to the dynamical variables $x_{1}\&x_{2}$ in Eq.(6). The two state variables
$x_{1}$ and $x_{2}$ of Eq(6) are associated with the voltages across the two
capacitors $C_{1}$ and $C_{2}$, respectively. The nonlinearity is simply
implemented by taking into account the saturation of op-amps. The circuit
realization of the SC-CNN based MLC circuit is achieved with the cell
components:
$R_{11}=207K\Omega,\>R_{12}=66K\Omega,\>R_{13}=100K\Omega,\>R_{14}=100K\Omega,\>R_{15}=1K\Omega,\>R_{16}=100K\Omega,\>R_{17}=100K\Omega,\>R_{N1}=220K\Omega,\>R_{N2}=3M\Omega,\>R_{N3}=180K\Omega,\>R_{N4}=16K\Omega,\>R_{21}=100K\Omega,\>R_{22}=6666.6K\Omega,\>R_{23}=100K\Omega,\>R_{24}=100K\Omega,\>R_{25}=1K\Omega,\>R_{26}=100K\Omega,\>R_{27}=100K\Omega,R_{31}-R_{38}=10K\Omega,~{}C_{1}=10nF,\>C_{2}=10nF,\>$
and active element $IC741$ type voltage op-amps with $\pm 12V$ supply
voltages. All the experimental results are obtained by using Agilent (33220A)
function generators and Agilent digital oscilloscope (DSO 7014B). The
amplitude of the forcing signal $f$ is used to study the dynamics of the CNN
based model.
## V Experimental and Numerical results
### V.1 Three level logic input
Now, we analyze the response of the system (6) to deterministic logic input
signal $I$. Specifically, we drive the system (6) with a low/moderate
amplitude signal $I=I_{1}+I_{2}$ with two square waves of strengths $I_{1}$
and $I_{2}$ encoding two logic inputs. The inputs can be either 0 or 1, giving
rise to four distinct logic input sets $(I_{1},I_{2}):(0,0),(0,1),(1,0)$ and
$(1,1)$. For a logical ‘$0$’, we set $I_{1}=I_{2}=-\Delta$, whereas for a
‘$1$’, we set $I_{1}=I_{2}=+\Delta$, where $\Delta$ represents a
small/moderate intensity input signal. As $I=I_{1}+I_{2}$, the input signal
sets (0,1) and (1,0) provide the same input signal $I$. As a consequence, the
input signals take the values $-2\Delta,~{}0,$ and $+2\Delta$ corresponding to
the input sets $(0,0),~{}(0,1)$ or $(1,0),$ and $(1,1)$, respectively.
Figure 4: Panels: (a)-(b) correspond to the two different logic inputs of
$I_{1}$ and $I_{2}$, respectively. Panel (c) shows a combination of two input
signals $I_{1}+I_{2}$. Input $I_{1}=I_{2}=-0.2$ when the logic input is
${}^{\prime}0^{\prime}$ and $I_{1}=I_{2}=+0.2$ when the logic input is
${}^{\prime}1^{\prime}$. The ’3’ level square waves with $-0.4$ corresponding
to the input set $(0,0)$, $0$ for $(0,1)/(1,0)$ set and $+0.4$ for $(1,1)$
input set.
For example, if $\Delta=0.2$ then both the inputs $I_{1}=I_{2}=-0.2$ for
logical input ‘$0$’ and values $0.2$ when it is ‘$1$’. Figs.4(a) & 4(b)
correspond to input signals $I_{1}$ & I2, Fig.4(c) is a three-level square
wave. Here, as $I=I_{1}+I_{2}$, the amplitude value $-0.4$ of the wave is for
input sets $(0,0)$, 0 for input sets $(0,1)$ or $(1,0)$ and $+0.4$ for the
input set $(1,1)$. Then the output of the appropriate logic gate is determined
by the well in which the dynamical variable $x_{1}$ or $x_{2}$ resides. More
specifically, for the OR/NOR and AND/NAND logic operations, the state
variables hop between the left and right/center wells depending on the input
streams, whereas for the XOR/XNOR logic gates, the state variables leap among
all the three wells equally.
Figure 5: Phase-space of attractor of Eq.6 for OR logic behavior. It is having
three segments: Left segment $D_{-}$ ($x_{1}<-1$), Center segment
$D_{0}$($-1\leq x_{1}\leq 1$) and Right segment $D_{+}$($x_{1}>+1$)
### V.2 Realization of OR/NOR and AND/NAND logic gates
Two typical logical operations in digital circuits are OR and AND or the
complementary gates NOR and NAND. To realize these gates, at least two inputs
need to be converted into a single output. For example, in the case of OR
logic operation, at least one of the two inputs is an ’ON’ state, so as to get
an ’ON’ output, while for AND logic operation, it is essential to be in the
’ON’ for both the inputs, so as to obtain an ’ON’ output(see Table.2). The
logical response of a desired type can be extracted by finding the solution of
state variables $x_{1}$ and $x_{2}$ from (6). One can define or extract an
appropriate gate by considering the state variables $x_{1}$ and $x_{2}$ in the
phase space. For example, in our system (6), the system oscillates in three
regions: namely 1) $D_{-}$ region - the state variable $x_{1}$ resides in the
region $x_{1}<-1$, 2) $D_{0}$ region - the state variable $x_{1}$ oscillates
between $-1\leq x_{1}\leq+1$ and 3) $D_{+}$ region - the state variable
$x_{1}$ exists for $x_{1}>1$.
Figure 6: Panel (a) shows a combination of two input signals $I=I_{1}+I_{2}$.
Input $I_{1}+I_{2}=-0.4$ when the logic input is ${}^{\prime}0^{\prime}$ and
$I_{1}+I_{2}=+0.4$ when the logic input is ${}^{\prime}1^{\prime}$. Panel (b)
is the constant bias, when it varies from $0.01$ to $-0.01$. Panels (c) and
(e) represent the corresponding dynamical responses OR/AND and NOR/NAND logic
gates of the system $x_{1}(t)$ and $x_{2}(t)$, respectively, under periodic
forcing with $f=0.1$. Panels (d) and (f) are the corresponding logic outputs
of panels (c) and (e), respectively.
Next to obtain the OR logic operation, when the state variable $x_{1}$ is in
the $D_{-}$ region of the system (6), the output state is assumed to be the
logical output value ’0’ and when the state variable $x_{1}$ resides in any
one of the regions $D_{0}$ and $D_{+}$, the output is assumed to be the
logical value ’1’. With this input-output association, we solve equation (6)
numerically by fixing the parameters as $I_{1}=I_{2}=0.2$, $E=0.01,f=0.1$ and
the remaining parameter values are assigned the same values as given in
section III.
Now, we analyze the attractors of the system (6). In Fig.5, it is obvious that
for input signal $I$ which corresponds to either (1,1) or (0,1)/(1,0) states,
the response of the system ‘$x_{1}$’ hops either in the $D_{+}$ region or in
the positive side of $D_{0}$ region of the phase space and when the input
signal is in the (0,0) state, the value of the state variable ‘$x_{1}$’, is in
other segment, namely the $D_{-}$ region. The logical output is ‘$1$’ when
$x_{1}(t)>0$ and it is ‘0’ for $x_{1}(t)<0$. As a consequence, for the two
sets of input streams (1,1) and (0,1)/(1,0), the attractor of the system is
bounded in $x_{1}(t)>0$ region and its is $x_{1}(t)<0$ region for the (0,0)
state. This confirms the fact that the dynamical attractor is although
chaotic, it behaves as the logical OR operation. It is also interesting to
note in Fig.5 that when $x_{1}(t)>0$, the $x_{2}(t)$ variable of the attractor
is bounded in the region $x_{2}(t)<0$. That is the response of the $x_{2}$
variable turns out to be the inverted output of the $x_{1}$ variable. As a
result, these two variables $x_{1}(t)$ and $x_{2}(t)$ produce logic gates OR
and NOR in parallel in this circuit.
These results are clearly illustrated in the time trajectory plot as shown in
Figs.6. In Fig.6(a) the three level diagram for the logic input
$I=I_{1}+I_{2}$ is shown and as the bias value ‘$E$’ is changed from $0.01$ to
$-0.01$ (see Fig.6(b)) the response of the system morphs from OR gate to the
other logical AND gate in the $x_{1}(t)$ regime of Fig.6(c) and the
corresponding logic output as shown in Fig.6(d). For this case, it is found
that for the (1,1) state, the attractor is in the $D_{+}$ region of phase
space while for (0,1) /(1,0) and (0,0) state it is either in the negative
$D_{0}$ region or in the $D_{-}$ region. Hence, switching the bias value from
+0.01 to -0.01 morphs logical OR gate to logical AND gate (see Figs.6(c) &
6(d)). In a similar fashion the other state variable $x_{2}(t)$ mimics the
inverted output signal of $x_{1}(t)$, thereby produces a clean logical NOR for
bias value $E=0.01$. It is further observed that the response of state
variable $x_{2}(t)$ of the system morphs from NOR logic to NAND logic (see
Figs.6(e) & 6(f)) when the threshold switches from the value of 0.01 to -0.01.
Thus the system (6) produces logic OR/AND through one of state variable
$x_{1}$ while its complementary logic function NOR/NAND is realized via the
other state variable $x_{2}$.
Figure 7: Realization of the OR/NOR logic gates in experimental electronic
circuits. Panels (a) & (e) and (b) & (f) are the inputs $I_{1}$ and $I_{2}$
(if the logic input is 0, it corresponds to $-100mV$, and $+100mV$ is
considered as logic input 1). Panels (c) & (d) clearly indicate the OR logic
output (when $v_{1}(t)>0$ it is considered as logic input 1, and $v_{1}(t)<0$
its logic input is considered as 0). Panels (g) & (h) yield the complementary
NOR logic gate. The bias voltage is fixed as $+10mV$ and the experimental
output was observed in Agilent DSO 7014B. Figure 8: Realization of the
AND/NAND logic gates in experimental electronic circuits. Panels (a) & (e)
and (b) & (f) are the inputs $I_{1}$ and $I_{2}$ (if the logic input is 0, it
corresponds to $-100mV$, and $+100mV$ is considered as logic input 1). Panels
(c) & (d) clearly indicate the AND logic output (when $v_{1}(t)>0$ it is
considered as logic input 1, and $v_{1}(t)<0$ is considered as the input 0).
Panels (g) & (h) yield the complementary NAND logic gate. The bias voltage is
fixed as $-10mV$ and the experimental output was observed in Agilent DSO
7014B.
Next we verify the numerical results discussed earlier in the electronic
circuit analog of the nonlinear system described by Eq.(6) and ascertain its
robustness in experiments. The complete circuit realization for the two SC-CNN
cells generating the dynamics of SC-CNN MLC circuit is shown in Fig.3. As
pointed out earlier, Fig.3 consists of two cells, each corresponding to one
dynamical variable in Eqs.(6). The two state variables $x_{1}$ and $x_{2}$ of
Eqs.(6) are associated with the voltages $v_{1}$ and $v_{2}$ across the two
capacitors $C_{1}$ and $C_{2}$, respectively. Now we fix $I_{1}=100mV$,
$I_{2}=100mV$ and $bias=+10mV$ corresponding to dimensionless units of the
circuit parameter that we discussed earlier. The changes in the dynamics of
the circuit under the effect of input streams are obtained by measuring the
voltages $v_{1}$ and $v_{2}$ across the capacitors $C_{1}$ and $C_{2}$,
respectively. The logic input signals $I_{1}$ and $I_{2}$ are $+100mV$ when
the logic input is ‘1’ and $-100mV$ for ‘0’ bias voltage $bias=10mV$. For this
case, the response of the circuit exhibits OR gate when we measure the voltage
$v_{1}$ across $C_{1}$ and NOR gate when we measure the voltage $v_{2}$ across
$C_{2}$ (see Fig.7). In a similar way, for $bias=-10mV$, the system gives the
AND gate when the voltage $v_{1}$ is measured across $C_{1}$ and NAND gate
when the voltage $v_{2}$ is measured across $C_{2}$ (see Fig.8). The logical
output ‘$Q$’ in Figs.7(d)/7(h) & 8(d)/8(h) are obtained as $v_{1}/v_{2}$ by
feeding the output response to an appropriate comparator circuit Campos-Cantón
_et al._ (2010). Further, we can quantify the process of obtaining a given
logic output by calculating $P(logic)$ as shown in Fig.9 in which it is
observed that the fundamental logic operation OR is realized in an optimal
width of forcing amplitude ‘$f$’. It is interesting to note that the logic
operation is realized for subthreshold input which results in an optimal
window of forcing value where $P(logic)$ tends to $1$.
Figure 9: The probability distribution of obtaining logical behavior for
different values of $f$ with bias value $E=0.01$ for OR/NOR gates and
$E=-0.01$ for AND/NAND gates.
### V.3 Realization of XOR and XNOR logic gates
The logic gate XOR is quite different from the previously discussed gates AND,
OR, NAND and NOR. XOR gates admit a logic output of ‘1’ or high logic level if
the inputs are at different logic levels, that is either of 0 and 1 or 1 and
0. Conversely, the output will be a ‘0’ or a low logic if the inputs are at
the same logic levels 0 and 0 or 1 and 1. The function of XOR gate is to start
with a regular ‘OR’ gate. However, the output is inhibited from going to ‘1’
or high when both the inputs are high or ‘1’ and it takes logic low or ‘0’
even when both the inputs are high or ‘1’. For these conditions, a bistable
nonlinear system fails to satisfy them. Since the system (6) has a three
segment piece-wise continuous function, it results in the gradient of a triple
well energy potential. As a result, the present system (6) has the ability to
produce XOR and XNOR gates, apart from producing the fundamental logic gates
OR,AND, NOR and NAND. To realize XOR, we set the output to be logical ‘1’ if
the state variable $x_{1}$ lies in between $-1.5\leq x_{1}\leq+1.5$ and the
output is assumed to be logic ‘0’ if the state variable $x_{1}$ resides
anywhere else, that is, when the attractor of system (6) resides in the
$D_{0}$ region, the response output is assumed to be high or ’1’ and ’0’
otherwise, when the attractor may be in the $D_{+}$ region or $D_{-}$ region
(see Fig.10).
Figure 10: Figure shows the phase-space section of XOR logic behavior. It is
having three segments: Left segment $D_{-}$ ($x_{1}<-1.5$), Center segment
$D_{0}$($-1.5\leq x_{1}\leq 1.5$) and Right segment $D_{+}$($x_{1}>+1.5$).
Figure 11: Panel (a) shows a combination of two input signals $I_{1}+I_{2}$.
Input $I_{1}=I_{2}=-0.2$ when the logic input is ${}^{\prime}0^{\prime}$ and
$I_{1}=I_{2}=+0.2$ when the logic input is ${}^{\prime}1^{\prime}$. Panels (b)
& (d) represent the corresponding dynamical responses in the form of XOR and
XNOR logic gates of the system through the variable $x_{1}(t)$ and $x_{2}(t)$
under periodic forcing $f=0.16$, with fixed bias parameter $E=0.01$, and
panels (c) & (d) represent the logic outputs of panels (b) & (d). Figure 12:
Realization of the XOR/XNOR logic gates in experimental electronic circuits.
Panels (a) & (e) and (b) & (f) are the inputs of $I_{1}$ and $I_{2}$ (if the
logic input is 0, it corresponds to $-100mV$, and $+100mV$ is considered as
logic input 1). Panels (c) & (d) clearly indicate the XOR logic outputs (when
$D_{-}\leq v_{1}(t)\geq D_{+}$ is considered as logic input 0, and $D_{-}\geq
v_{1}(t)\leq D_{+}$ its logic input is considered as 1). Panels (g) & (h)
yield the complementary XNOR logic gate. The bias voltage is fixed as $+10mV$
and the experimental output was observed in Agilent DSO 7014B.
With this input-output correspondence, we solve the Eq.(6) numerically by
fixing the system parameters as $I_{1}=I_{2}=0.2$, $E=0.01$ and $f=0.16$. It
is clearly shown in the phase space diagram Fig.10 that for the (0,0) state
the attractor of the system resides in the $D_{-}$ region of the phase space,
for the input (0,1) or (1,0) states, it hops in between -1 to +1 in the
$D_{0}$ region and for (1,1) state it is in the $D_{+}$ region. Thus for the
state (0,1) or (1,0) the output response of the system is considered to be
logical value ‘1’ since the state variable $x_{1}$ of the system hops in the
region $D_{0}$ and for the other states (0,0) or (1,1) the response of the
output is considered to be the logical value ‘0’ because the state variable
$x_{1}$ oscillates either in the $D_{-}$ or in $D_{+}$ region depending on the
input streams. Hence, the output response of the system admits a high logic
output for inputs of different logic levels, either of 0 and 1 or 1 and 0, and
it is a low logic output for the same logic levels, namely 0 and 0 or 1 and 1.
Thus for the above set of parameter values, the system (6) admits logic XOR
gate. It is also clearly demonstrated in phase diagram (see Fig.10) that the
response of the state variable $x_{2}$ turns out to be the inverted output of
the other state variable $x_{1}$. As a consequence, when the state variable
$x_{1}$ produces the logical XOR gate, the other state variable $x_{2}$
produces the complementary gate, namely the logical XNOR gate. Thus the
present system has the ability to produce XOR gate in one of the state
variables $x_{1}$ and XNOR gate in other state variable $x_{2}$ parallely
without altering any system parameter. These results are also illustrated in
the time trajectory plots in Fig.11. In Figs.12(c), 12(d)& 12(g), 12(h), it is
demonstrated how the response of the system variables $x_{1}$ and $x_{2}$
behave as logical XOR and XNOR operations, respectively, under different input
streams as shown in Figs.12(a),12(b),12(e) and 12(f).
To substantiate our numerical simulations, we carried out the experimental
realization of XOR and XNOR gates by using the analog circuit given in Fig.3.
For this purpose, we fix the parameter values as $I_{1}=-100mV$, $I_{2}=100mV$
and $bias=+10mV$ corresponding to the dimensionless parameters. The logical
XOR and XNOR responses are obtained by measuring the voltages $v_{1}$ and
$v_{2}$ across the capacitors $C_{1}$ and $C_{2}$, respectively. It is obvious
from Fig.12 that the signals measured across $C_{1}$ and $C_{2}$ behave as
logical XOR and logical XNOR operations under different three-level input
streams. Thus our experimental study confirms the fact that the considered
circuit system has the potential to produce not only the fundamental gates
OR/NOR and AND/NAND, but it can also emulate the logical XOR and XNOR
operations as well. The logical outputs ‘$Q$’ in Figs.12(d)/12(h) are obtained
as $v_{1}/v_{2}$ by feeding the output response to an appropriate comparator
circuit Campos-Cantón _et al._ (2010). Also, we can quantify the process of
obtaining a given logic output by calculating $P(logic)$ as shown in Fig.13,
where it is observed that the fundamental logic operation XOR is realized in
an optimal width of forcing amplitude ‘$f$’. It is interesting to note that
the logic operations are realized for subthreshold input which results in an
optimal window of forcing value where $P(logic)$ tends to $1$.
Figure 13: The probability distribution of obtaining logical behavior for
different values of $f$ with bias value $E=0.01$ for XOR/XNOR gates.
### V.4 Set-Reset Memory latch
Table 3: Truth Table of Set-Reset (SR) flip-flop $(I_{1})$ Set | $(I_{2})$ Reset | Output | State
---|---|---|---
0 | 0 | Q | Last state
0 | 1 | 0 | Reset
1 | 0 | 1 | Set
1 | 1 | ? | Not allowed
Figure 14: Set-Reset circuit realization for generating the driving signal
$F(t)$. Here we use four op-amps (OA6-OA9) and 14 resistors
$R51-R64=10K\Omega$
Figure 15: Panel (a) shows a 3-level subtraction of two logic inputs
$I=I_{1}-I_{2}$, $I=0.1$ for $(1,0)$, $I=0.0$ for $(0,0)/(1,1)$ and $I=-0.1$
for $(0,1)$ sets. Panels (b) and (d) are the desired S-R latch regime of
$x_{1}(t)$ and $x_{2}(t)$ under periodic forcing $f=0.1$, with fixed
parameters $\Delta=0.2$ and $E=0.0$ and panels (c) and (e) represent the
corresponding logic outputs of panels (b) and (d). Figure 16: Realization of
the Set-Reset memory latch in experimental electronic circuit. Panels (a) &
(e) and (b) & (f) are the inputs of $I_{1}$ and $I_{2}$ (if the logic input is
0, it corresponds to $-100mV$ and $+100mV$ is considered as the logic input
1). Panels (c) & (d) show the Set-Reset memory latch regime and corresponding
complementary memory latch output is shown in the panels (g) & (h). The
experimental output was observed in an Agilent DSO 7014B without bias. Figure
17: The probability distribution of obtaining logical behavior for different
values of $f$ with absence of bias value $E=0.00$ for SR memory latch.
Apart from designing logic gates many efforts have been made to exploit
nonlinear systems to construct memory devices. Chaos based SR flip-flop from
two cross coupled Chua’s circuit, SR flip-flop using reconfigurable analog
block, set-reset latch in noise assisted bistable system, R-S flip-flop from
two-cross coupled quasiperiodically driven MLC circuit are few examples for
constructing memory latch using nonlinear systemsCafagna and Grassi (2006);
Campos-Cantón _et al._ (2012a); Venkatesh, Venkatesan, and Lakshmanan
(2017b); Sathish Aravindh _et al._ (2020). In the present work, we
demonstrate how the simple SC-CNN based periodically driven MLC circuit (6)
can produce a consistent and complete RS flip-flop.
In this case, the input streams should be modified. It is quite obvious from
the truth table of an SR flip-flop that (0,1) and (1,0) input sets are quite
different, since these sets produce different output states (see Table.3). As
a result we encode the inputs in a different way. Here the first input $I_{1}$
takes ‘1’ when the logic input is ‘+1’ while it is ‘0’ when the logic input is
‘0’. But the second input $I_{2}$ takes the value ‘1’ when the logic is ‘0’
while ‘0’ for logic ‘+1’. This could be achieved by applying the NOT operation
to $I_{2}$. For this case also the input streams
$(I_{1},I_{2}):(0,0),(0,1),(1,0),$ and $(1,1)$ represent to values of
$0,-1,1,$ and $0$ values. Here, $(1,1)$ set is a restricted one. For example,
if $\Delta=0.2$, both the inputs $I_{1}=I_{2}=$ -0.2 for logical input ‘$0$’
and the values $0.2$ for logical input ‘$1$’. The input signal $I=I_{1}-I_{2}$
is thus a three-level wave form : $-0.4$ for (0,1), $0$ for (0,0)/(1,1) and
$0.4$ for (1,0) inputs sets.
When we apply above mentioned input streams (Figs.15(a) and 15(b)) to the
system (6), it is possible to realize the S-R flip-flop in the circuit. As
usual, the logical output can be obtained by assuming that the output is to be
the value ‘$1$’ for $x>0$ and 0 if $x<0$. S-R flip-flop operation is clearly
shown in Figs.15(c) & 15(d) and 15(g) & 15(h). When $I_{1}$ and $I_{2}$ inputs
are in the low states, it is found that the system response $x_{1}(t)$ remains
unchanged. When $I_{1}$ is in the low state and $I_{2}$ is in the high state,
it is observed that the response of the system $x_{1}(t)<0$ and hence it is
assumed that the logical input is low. This makes the system as a latch with
reset condition. When the input $I_{1}$ is in the high state while $I_{2}$ is
low, it is realized that the output of the system is $x_{1}(t)>0$ and thus it
is assigned the logical value ‘$1$’. In a similar way, the output of the
dynamical variable $x_{2}(t)$ (Fig.15(g)), provides the inverted output of
Fig.15(c) and it behaves as an active high RS flip-flop. Usually in digital
circuits, active low and active high RS flip flops are obtained by two cross
coupled NAND and NOR gates. But in the present case, these two active high and
low RS flip flops are obtained through two dynamical variables of the circuit.
Hence the considered circuit is not only providing logical gates but also acts
as a sequential circuit to provide RS flip flop operations of various
categories such as active low and active high RS flip flops. To realize the
Set-Rest latch experimentally, we design the system (6) with two inputs
$I_{1}$ and $I_{2}$. In the case of logic gates as discussed earlier, two
inputs are added by an op-amp summing amplifier, whereas in the case of Set-
Reset latch case, the two inputs are subtracted by an op-amp subtracting
amplifier. In the circuit of Fig.14, the signal $F(t)$ is generated by a set
of op-amp summing amplifier by adding the resulting signal $I(t)$, external
bias voltage, external noise signal and sinusoidal signal. Fig.16 verifies the
Set-Reset latch behavior in our electronic circuit. The logical output ‘$Q$’
in Figs.16(d)/16(h) are obtained as $v_{1}/v_{2}$ by feeding the output
response to an appropriate comparator circuit Campos-Cantón _et al._ (2010).
Further, we can quantify the process of obtaining a given logic output by
calculating $P(logic)$ as shown in Fig.17 where it is observed the Set-Reset
memory latch is realized in an optimal width of forcing amplitude ‘$f$’. It is
interesting to note that the logic operations which are realized for
subthreshold input results in an optimal window of forcing value where
$P(logic)$ tends to $1$.
Figure 18: Panels (a)-(c) show the three different logic inputs of $I_{1}$, $I_{2}$ and $I_{3}$, respectively. Input $I_{1}=I_{2}=I_{3}=-0.2$ when the logic input is ${}^{\prime}0^{\prime}$ and $I_{1}=I_{2}=I_{3}=+0.2$ when the logic input is ${}^{\prime}1^{\prime}$. Panel (d) represents the corresponding dynamical response OR logic gate of the system $x(t)$ under periodic forcing with $f=0.1$ and bias $E=0.25$ and panel (e) represents the corresponding logic output of panel (d). Figure 19: Realization of the OR logic gates in experimental electronic circuit. Panels (a)-(c) and (e)-(g) show the three different logic inputs $I_{1},I_{2}\&I_{3}$ (if the logic input is 0, it is considered as $-100mV$ and $+100mV$ is considered as the logic input 1). Panels (d) and (h) clearly indicate the OR logic output (when $x_{1}>0$ it is considered as logic input 1, and $x_{1}<0$ the logic input is considered as 0). The bias voltage is fixed as $100mV$ and the experimental output was observed in an Agilent DSO 7014B. Table 4: The truth table for realization of three input OR gate $I_{1}~{}I_{2}~{}I_{3}$ | Left well | Center well | Right well
---|---|---|---
0 0 0 | ON | OFF | OFF
0 0 1 | OFF | ON | ON
0 1 0 | OFF | ON | ON
0 1 1 | OFF | ON | ON
1 0 0 | OFF | ON | ON
1 0 1 | OFF | ON | ON
1 1 0 | OFF | ON | ON
1 1 1 | OFF | ON | ON
### V.5 Logic responses even in three-input configuration
Next we consider the question whether we can extend the scope of this
nonlinear circuit system to three inputs or higher inputs?. The answer is yes
and we have found that without altering any of the parameters, the system (6)
admits logic responses even when feeding three-inputs. In particular, the
system produces logic behavior when we feed three inputs $I_{1}$, $I_{2}$ and
$I_{3}$.
To be specific, for obtaining the OR gate, we fix the numerical parameters as
$I_{1}=I_{2}=I_{3}=0.2$, $E=0.25$ and $f=0.1$ (experimental parameters:
$I_{1}=100mV$, $I_{2}=100mV$, $I_{3}=100mV$, $f=2.65KHz$, and $bias=100mV$).
The first three panels of Figs.18 & 19 show the three different inputs
$I=I_{1}+I_{2}+I_{3}$ while the fourth panel corresponds to logical OR
dynamics. Hence for states (0,0,0) the output response of the system is
assumed to be the logical ‘0’ since the state variable $x_{1}$ oscillates in
the $D_{-}$ region and for other states the output response is to be logical
’1’ value because the state variable $x_{1}$ oscillates in the $D_{+}$ and
$D_{0}$ regions [see Figs.18(d) & 19(d)] and also the corresponding logic
output represent in Figs.18(h) & 19(e). The logical output ‘$Q$’ in
Figs.19(d)/19(h) are obtained as $v_{1}/v_{2}$ by feeding the output response
to an appropriate comparator circuit(see Table.4). Switching the bias voltage
to $E=-0.25(bias=-100mV)$, one can obtain three-input AND gate. However,
three-input XOR and XNOR are not possible since the circuit is a three-segment
piecewise linear system. To realize the three-input XOR and XNOR gates, one
has to employ a minimum of four-well potential nonlinear system.
Figure 20: The probability distribution of obtaining logical behavior for
different values of $D$ with bias value $E=0.01$ for OR/NOR gates and
$E=-0.01$ for AND/NAND gates.
### V.6 Effect of noise
Now, we investigate the effect of noise on the logic behavior of the circuit
system (6). In particular, we ascertain the robustness of the logic response
with respect to ambient noise. That is, we have to analyze whether the logic
response persists even in the presence of noise or not. For this propose, we
calculate the probability P(logic) of getting logic gates for different noise
strength levels, with strength $D$. Essentially P(logic) denotes the ratio of
the total number of successful runs to the total number of runs. If the system
exhibits the desired logic output in response to all the logic inputs,
P(logic) is assigned a value ‘1’, otherwise it is treated as ‘0’. Our
numerical simulation, P(logic) for the circuit (6), is obtained by sampling
1000 runs of the given input set and this process is repeated for 500 such
sets. It is clearly indicated in Fig.20, that for noise strength $0<D<0.447$,
the system exhibits logic behavior. Beyond $D>0.447$, the system loses its
logic behavior. Thus, it is clearly demonstrated that the robustness of
logical response of the system continues even in the presence of noise
originating due to electronic components or any other external factors.
## VI Conclusion
We have investigated the effect of logic square wave signals on the SC-CNN of
MLC circuit system. Exploiting the hopping of attractors generated by this
circuit, we have found that the circuit can emulate all basic logic operations
corresponding to the logic gates OR, AND, NOR, NAND, XOR and XNOR and memory
latch element RS flip-flop. It is also shown that all these logic behaviors
are tolerant to noise. We have extended the scope of obtaining the logic gates
by even feeding higher order inputs, specifically three-inputs. Results
obtained by numerical simulations are in good agreement with experimental
demonstration. Our study throws some useful light in replacing the existing
computer technology with a minimal hardware.
It is clearly evident from our studies that the SC-CNN model of the MLC
circuit can function as logic gates as well as memory latch even for small-
amplitudes of the input signal. As a result, it consumes only a small power.
Further, the same circuit can be used to produce a logic output corresponding
to three-input logic. Thus this circuit may be used as two-input and three-
input logic gates as well. The low active RS flip-flop and the high active RS
flip-flop are constructed by cross-coupling of two NAND gates and cross-
coupling of two NOR gates. Hence, the same MLC circuit can produce low active
RS flip-flop through one state variable and high active RS flip-flop through
another state variable. Further, the circuit can potentially operate in noisy
environments.
Thus, the circuit can function not only in a noisy atmosphere but can also be
capable of utilizing its resources by configuring the circuit into two-input,
as well as three input gates and also as a memory device depending on the
requirement. For instance, if we need to run a program, the circuit will mimic
as logic gates of two inputs or three inputs, and on the other hand, for
processing of data, the circuit will mimic as memory device which enables one
to utilize it optimally. Apart from the computer architecture, the circuit
morphs into a memory latch which may lead to acting as an electronic
switching-de bouncer.
In this paper, we have proved that the underlying chaotic trajectories, which
hop in different quadrants of the phase-space, induce parallel logic
operations and memory latch in the SC-CNN of the MLC circuit and this feature
may be considered for the replacement of currently existing processors and
also as a remedy for Moore’s lawKia, Lindner, and Ditto (2017). Further one
has to study the important features of this circuit such as access clock time,
access energies and cell feature size and compare them with the existing
technology, provide technically feasibility of the circuit for design and
development of dynamic architecture. We hope to pursue these aspects in the
near feature.
## AUTHOR’S CONTRIBUTIONS
All authors contributed equally.
## Acknowledgment
P.A. and A.V. acknowledge the DST-SERB for providing financial support for a
research project under Grant No.EMR/2017/002813. M.S. sincerely thanks the
Council of Scientific & Industrial Research, India for providing a fellowship
under SRF Scheme No.08/711(0001)2K19-EMR-I. A.V. also acknowledges the DST-
FIST for supporting experimental work under Grant
No.SR/FST/College-2018-372(C). M.L. acknowledges the DST-SERB Distinguished
Fellowship program under Grant No.SB/DF/04/2017 for financial support.
## DATA AVAILABILITY
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
## REFERENCES
## References
* Mano (2003) M. M. Mano, _Computer system architecture_ (Prentice-Hall of India, 2003).
* Adleman (1994) L. M. Adleman, “Molecular computation of solutions to combinatorial problems,” Science 266, 1021–1024 (1994).
* Collier _et al._ (1999) C. Collier, E. Wong, M. Belohradskỳ, F. Raymo, J. Stoddart, P. Kuekes, R. Williams, and J. Heath, “Electronically configurable molecular-based logic gates,” Science 285, 391–394 (1999).
* Ladd _et al._ (2010) T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464, 45–53 (2010).
* Nielsen and Chuang (2002) M. A. Nielsen and I. Chuang, _Quantum computation and quantum information_ (American Association of Physics Teachers, 2002).
* Bachtold _et al._ (2001) A. Bachtold, P. Hadley, T. Nakanishi, and C. Dekker, “Logic circuits with carbon nanotube transistors,” Science 294, 1317–1320 (2001).
* Hopfield (1982) J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. 79, 2554–2558 (1982).
* Sinha and Ditto (1998) S. Sinha and W. L. Ditto, “Dynamics based computation,” Phys. Rev. Lett. 81, 2156 (1998).
* Sinha and Ditto (1999) S. Sinha and W. L. Ditto, “Computing with distributed chaos,” Phys. Rev. E 60, 363 (1999).
* Prusha and Lindner (1999) B. S. Prusha and J. F. Lindner, “Nonlinearity and computation: Implementing logic as a nonlinear dynamical system,” Phys. Lett. A 263, 105–111 (1999).
* Murali _et al._ (2009a) K. Murali, I. Rajamohamed, S. Sinha, W. L. Ditto, and A. R. Bulsara, “Realization of reliable and flexible logic gates using noisy nonlinear circuits,” Appl. Phys. Lett. 95, 194102 (2009a).
* Murali _et al._ (2009b) K. Murali, S. Sinha, W. L. Ditto, and A. R. Bulsara, “Reliable logic circuit elements that exploit nonlinearity in the presence of a noise floor,” Phys. Rev. Lett. 102, 104101 (2009b).
* Sinha _et al._ (2009) S. Sinha, J. Cruz, T. Buhse, and P. Parmananda, “Exploiting the effect of noise on a chemical system to obtain logic gates,” EPL (Europhysics Letters) 86, 60003 (2009).
* Guerra _et al._ (2010) D. N. Guerra, A. R. Bulsara, W. L. Ditto, S. Sinha, K. Murali, and P. Mohanty, “A noise-assisted reprogrammable nanomechanical logic gate,” Nano Lett. 10, 1168–1171 (2010).
* Worschech _et al._ (2010) L. Worschech, F. Hartmann, T. Kim, S. Höfling, M. Kamp, A. Forchel, J. Ahopelto, I. Neri, A. Dari, and L. Gammaitoni, “Universal and reconfigurable logic gates in a compact three-terminal resonant tunneling diode,” Appl. Phys. Lett. 96, 042112 (2010).
* Zamora Munt and Masoller (2010) J. Zamora Munt and C. Masoller, “Numerical implementation of a VCSEL-based stochastic logic gate via polarization bistability,” Opt. Express 18, 16418–16429 (2010).
* Zhang, Song, and He (2010) L. Zhang, A. Song, and J. He, “Effect of colored noise on logical stochastic resonance in bistable dynamics,” Phys. Rev. E 82, 051106 (2010).
* Bulsara _et al._ (2010) A. R. Bulsara, A. Dari, W. L. Ditto, K. Murali, and S. Sinha, “Logical stochastic resonance,” J. Chem. Phys 375, 424–434 (2010).
* Singh and Sinha (2011) K. P. Singh and S. Sinha, “Enhancement of “logical” responses by noise in a bistable optical system,” Phys. Rev. E 83, 046219 (2011).
* Dari _et al._ (2011a) A. Dari, B. Kia, A. R. Bulsara, and W. Ditto, “Creating morphable logic gates using logical stochastic resonance in an engineered gene network,” Europhys. Lett. 93, 18001 (2011a).
* Dari _et al._ (2011b) A. Dari, B. Kia, X. Wang, A. R. Bulsara, and W. Ditto, “Noise-aided computation within a synthetic gene network through morphable and robust logic gates,” Phys. Rev. E 83, 041909 (2011b).
* Storni _et al._ (2012) R. Storni, H. Ando, K. Aihara, K. Murali, and S. Sinha, “Manipulating potential wells in logical stochastic resonance to obtain XOR logic,” Phys. Lett. A 376, 930–937 (2012).
* Roychowdhury (2015) J. Roychowdhury, “Boolean computation using self-sustaining nonlinear oscillators,” Proceedings of the IEEE 103, 1958–1969 (2015).
* Kohar _et al._ (2017) V. Kohar, B. Kia, J. F. Lindner, and W. L. Ditto, “Implementing boolean functions in hybrid digital-analog systems,” Phys. Rev. Appl. 7, 044006 (2017).
* Venkatesh, Venkatesan, and Lakshmanan (2017a) P. R. Venkatesh, A. Venkatesan, and M. Lakshmanan, “Implementation of dynamic dual input multiple output logic gate via resonance in globally coupled Duffing oscillators,” Chaos 27, 083106 (2017a).
* Venkatesh, Venkatesan, and Lakshmanan (2017b) P. R. Venkatesh, A. Venkatesan, and M. Lakshmanan, “Design and implementation of dynamic logic gates and RS flip-flop using quasiperiodically driven Murali–Lakshmanan–Chua circuit,” Chaos 27, 033105 (2017b).
* Neves, Voit, and Timme (2017) F. S. Neves, M. Voit, and M. Timme, “Noise-constrained switching times for heteroclinic computing,” Chaos 27, 033107 (2017).
* Kia, Lindner, and Ditto (2017) B. Kia, J. F. Lindner, and W. L. Ditto, “Nonlinear dynamics as an engine of computation,” Phil. Trans. R. Soc. A 375, 20160222 (2017).
* Murali _et al._ (2018) K. Murali, S. Sinha, V. Kohar, B. Kia, and W. L. Ditto, “Chaotic attractor hopping yields logic operations,” PloS one 13, e0209037 (2018).
* Manaoj Aravind, Murali, and Sinha (2018) V. Manaoj Aravind, K. Murali, and S. Sinha, “Coupling induced logical stochastic resonance,” Phys. Lett. A 382, 1581–1585 (2018).
* Sathish Aravindh, Venkatesan, and Lakshmanan (2018) M. Sathish Aravindh, A. Venkatesan, and M. Lakshmanan, “Strange nonchaotic attractors for computation,” Phys. Rev. E 97, 052212 (2018).
* Sathish Aravindh _et al._ (2020) M. Sathish Aravindh, R. Gopal, A. Venkatesan, and M. Lakshmanan, “Realisation of parallel logic elements and memory latch in a quasiperiodically-driven simple nonlinear circuit.” Pramana 94 (2020), 10.1007/s12043-020-1939-4.
* Sathish Aravindh, Venkatesan, and Lakshmanan (2020) M. Sathish Aravindh, A. Venkatesan, and M. Lakshmanan, “Route to logical strange nonchaotic attractors with single periodic force and noise,” Chaos 30, 093137 (2020).
* Ashwin and Borresen (2004) P. Ashwin and J. Borresen, “Encoding via conjugate symmetries of slow oscillations for globally coupled oscillators,” Phys. Rev. E 70, 026203 (2004).
* Ashwin and Borresen (2005) P. Ashwin and J. Borresen, “Discrete computation using a perturbed heteroclinic network,” Phys. Lett. A 347, 208–214 (2005).
* de Lacy Costello _et al._ (2009) B. de Lacy Costello, R. Toth, C. Stone, A. Adamatzky, and L. Bull, “Implementation of glider guns in the light-sensitive belousov-zhabotinsky medium,” Phys. Rev. E 79, 026114 (2009).
* Stevens _et al._ (2012) W. M. Stevens, A. Adamatzky, I. Jahan, and B. de Lacy Costello, “Time-dependent wave selection for information processing in excitable media,” Phys. Rev. E 85, 066129 (2012).
* Mahboob _et al._ (2011) I. Mahboob, E. Flurin, K. Nishiguchi, A. Fujiwara, and H. Yamaguchi, “Interconnect-free parallel logic circuits in a single mechanical resonator,” Nat. Commun. 2, 1–7 (2011).
* Gerstung, Timmer, and Fleck (2009) M. Gerstung, J. Timmer, and C. Fleck, “Noisy signaling through promoter logic gates,” Phys. Rev. E 79, 011923 (2009).
* Ando _et al._ (2011) H. Ando, S. Sinha, R. Storni, and K. Aihara, “Synthetic gene networks as potential flexible parallel logic gates,” EPL (Europhysics Letters) 93, 50001 (2011).
* Kompa and Levine (2001) K. Kompa and R. Levine, “A molecular logic gate,” Proceedings of the National Academy of Sciences 98, 410–414 (2001).
* Motoike _et al._ (2001) I. N. Motoike, K. Yoshikawa, Y. Iguchi, and S. Nakata, “Real-time memory on an excitable field,” Phys. Rev. E 63, 036220 (2001).
* Sinha, Munakata, and Ditto (2002) S. Sinha, T. Munakata, and W. L. Ditto, “Flexible parallel implementation of logic gates using chaotic elements,” Phys. Rev. E 65, 036216 (2002).
* Norrell and Socolar (2009) J. Norrell and J. E. Socolar, “Boolean modeling of collective effects in complex networks,” Phys. Rev. E 79, 061908 (2009).
* Mozeika, Saad, and Raymond (2010) A. Mozeika, D. Saad, and J. Raymond, “Noisy random boolean formulae: A statistical physics perspective,” Phys. Rev. E 82, 041112 (2010).
* Zhang _et al._ (2012) H. Zhang, Y. Xu, W. Xu, and X. Li, “Logical stochastic resonance in triple-well potential systems driven by colored noise,” Chaos 22, 043130 (2012).
* Miyamoto _et al._ (2010) S. Miyamoto, K. Nishiguchi, Y. Ono, K. M. Itoh, and A. Fujiwara, “Resonant escape over an oscillating barrier in a single-electron ratchet transfer,” Phys. Rev. B 82, 033303 (2010).
* Kohar and Sinha (2012) V. Kohar and S. Sinha, “Noise-assisted morphing of memory and logic function,” Phys. Lett. A 376, 957–962 (2012).
* Kohar, Murali, and Sinha (2014) V. Kohar, K. Murali, and S. Sinha, “Enhanced logical stochastic resonance under periodic forcing,” Commun. Nonlinear Sci. Numer. Simul. 19, 2866–2873 (2014).
* Ditto, Murali, and Sinha (2008) W. L. Ditto, K. Murali, and S. Sinha, “Chaos computing: ideas and implementations,” Philos. Trans. Royal Soc. A 366, 653–664 (2008).
* Bick and Rabinovich (2009) C. Bick and M. I. Rabinovich, “Dynamical origin of the effective storage capacity in the brain’s working memory,” Phys. Rev. Lett. 103, 218101 (2009).
* Schittler Neves and Timme (2012) F. Schittler Neves and M. Timme, “Computation by switching in complex networks of states,” Phys. Rev. Lett. 109, 018701 (2012).
* Peng _et al._ (2008) H. Peng, Y. Yang, L. Li, and H. Luo, “Harnessing piecewise-linear systems to construct dynamic logic architecture,” Chaos 18, 033101 (2008).
* Peng _et al._ (2010) H. Peng, F. Liu, L. Li, Y. Yang, and X. Wang, “Dynamic logic architecture based on piecewise-linear systems,” Phys. Lett. A 374, 1450–1456 (2010).
* Campos-Cantón _et al._ (2010) I. Campos-Cantón, J. Pecina-Sánchez, E. Campos-Cantón, and H. C. Rosu, “A simple circuit with dynamic logic architecture of basic logic gates,” Int. J. Bifurc. Chaos 20, 2547–2551 (2010).
* Cafagna and Grassi (2006) D. Cafagna and G. Grassi, “Chaos-based SR flip–flop via Chua’s circuit,” Int. J. Bifurc. Chaos 16, 1521–1526 (2006).
* Campos-Cantón _et al._ (2012a) I. Campos-Cantón, E. Campos-Cantón, H. C. Rosu, and E. Castellanos-Velasco, “SET-RESET flip-flop circuit with a simple output logic,” Circuits, Systems, and Signal Processing 31, 753–760 (2012a).
* Canton, Martienz, and Duron (2017) E. C. Canton, M. G. Martienz, and R. R. R. Duron, “Method and circuit for integrating a programmable matrix in the field of reconfigurable logic gates employing a non-lineal system and an efficient programmable rewiring,” (2017), uS Patent 9,793,897.
* Campos-Cantón _et al._ (2012b) E. Campos-Cantón, R. Femat, J. G. Barajas-Ramírez, and I. Campos-Cantón, “A multivibrator circuit based on chaos generation,” Int. J. Bifurc. Chaos 22, 1250011 (2012b).
* Gupta _et al._ (2011) A. Gupta, A. Sohane, V. Kohar, K. Murali, and S. Sinha, “Noise-free logical stochastic resonance,” Phys. Rev. E 84, 055201 (2011).
* Venkatesh and Venkatesan (2016) P. Venkatesh and A. Venkatesan, “Vibrational resonance and implementation of dynamic logic gate in a piecewise-linear murali–lakshmanan–chua circuit,” Commun. Nonlinear Sci. Numer. Simul. 39, 271–282 (2016).
* Gui _et al._ (2020a) R. Gui, Y. Wang, Y. Yao, and G. Cheng, “Enhanced logical vibrational resonance in a two-well potential system,” Chaos Solitons Fractals 138, 109952 (2020a).
* Gui _et al._ (2020b) R. Gui, H. Zhang, G. Cheng, and Y. Yao, “Set–reset latch logic operation in a bistable system under suprathreshold and subthreshold signals,” Chaos 30, 023119 (2020b).
* Vincent _et al._ (2021) U. E. Vincent, P. V. McClintock, I. A. Khovanov, and S. Rajasekar, “Vibrational and stochastic resonances in driven nonlinear systems,” Phil. Trans. R. Soc. A 279, 20200226 (2021).
* Murali _et al._ (2021) K. Murali, S. Rajasekar, M. V. Aravind, V. Kohar, W. Ditto, and S. Sinha, “Construction of logic gates exploiting resonance phenomena in nonlinear systems,” Phil. Trans. R. Soc. A 379, 20200238 (2021).
* Lu _et al._ (2019) S. Lu, Z. Dai, Y. Liu, G. Liu, H. Yang, and F. Wang, “Decreasing bit error rate by using enhanced tristable logical stochastic resonance in heavy noise condition,” Chin. J. Phys. 58, 179–189 (2019).
* Liu and Liu (2014) Y. Liu and W. Liu, “A new design for reconfigurable xor function based on cellular neural networks,” Connection Science 26, 403–413 (2014).
* Günay (2010) E. Günay, “Mlc circuit in the frame of cnn,” Int. J. Bifurc. Chaos 20, 3267–3274 (2010).
* Swathy and Thamilmaran (2014) P. Swathy and K. Thamilmaran, “Dynamics of sc-cnn based variant of mlc circuit: An experimental study,” Int. J. Bifurc. Chaos 24, 1430008 (2014).
* Chua and Yang (1988) L. O. Chua and L. Yang, “Cellular neural networks: Theory,” IEEE Transactions on circuits and systems 35, 1257–1272 (1988).
* Arena _et al._ (1995) P. Arena, S. Baglio, L. Fortuna, and G. Manganaro, “Chua’s circuit can be generated by cnn cells,” IEEE Trans. Circuits Syst. I. Fundam. Theory Appl. 42, 123–125 (1995).
* Luo and Wang (2016) T. Luo and Z. Wang, “Dynamics and sc-cnn circuit implementation of a periodically forced non-smooth mechanical system,” Nonlinear Dyn. 85, 23–45 (2016).
* Manganaro, Arena, and Fortuna (2012) G. Manganaro, P. Arena, and L. Fortuna, _Cellular neural networks: chaos, complexity and VLSI processing_ , Vol. 1 (Springer Science & Business Media, 2012).
* Murali, Lakshmanan, and Chua (1994) K. Murali, M. Lakshmanan, and L. O. Chua, “The simplest dissipative nonautonomous chaotic circuit,” IEEE Trans. Circuits Syst. I. Fundam. Theory Appl. 41, 462–463 (1994).
* Lakshmanan and Rajasekar (2003) M. Lakshmanan and S. Rajasekar, _Nonlinear Dynamics: Integrability, Chaos and Patterns_ (Springer-Verlag Berlin Heidelberg, 2003).
* Lakshmanan and Murali (1996) M. Lakshmanan and K. Murali, _Chaos in Nonlinear Oscillators: Controlling and Synchronization_ , Vol. 13 (World scientific, 1996).
|
# Reconfigurable multiplex setup for high throughput electrical
characterisation at cryogenic temperature
X Bian1<EMAIL_ADDRESS>H J Joyce2, C G Smith3, M J Kelly2, G A
D Briggs1 J A Mol4<EMAIL_ADDRESS>1Department of Materials, University of
Oxford, Oxford, OX1 3PH, UK
2Centre for Advanced Photonics and Electronics, Department of Engineering,
University of Cambridge, CB3 0FA, UK
3 Department of Physics, Cavendish Laboratory, University of Cambridge, CB3
0HE, UK
4 School of Physics and Astronomy, Queen Mary University, London, E1 4NS, UK
###### Abstract
In this paper, we present a reconfigurable multiplex (MUX) setup that
increases the throughput of electrical characterisation at cryogenic
temperature. The setup separates the MUX circuitry from quantum device under
test (qDUT), allowing qDUT chips to be exchanged easily and MUX chips to be
reused. To interface with different types of qDUTs, board-level designs are
incorporated to allow interconnects flexibly routed into different topology.
MUXs are built based on a multiple level selective gating (MLSG) scheme, where
the number of multiplexed output channels (interconnects) is exponentially
dependent on the number of control lines. In the prototype setup presented in
this paper, with 14 out of 44 existing wires from room temperature, 4 MUXs at
cryogenic temperature can supply in total 128 interconnects to interface with
qDUTs. We validate the MUX setup operation and assess the various limits
existed by measuring k$\Omega$ resistors made of $\mu$m-size graphene ribbons.
We further demonstrate the setup by performing charge transport measurement on
128 nm-size graphene quantum devices in a single cooling down.
## I Introduction
To develop new technologies that exceed the limits of classic electronics,
different types of quantum devices have been proposed and demonstrated
(Zwanenburg et al., 2013; Sarma et al., 2015; Watson et al., 2018; Xiang et
al., 2016). However, most candidates are still limited to small scale circuit
operation and lack a viable way to scale up. A bottleneck is the limited
throughput of electrical characterisation at cryogenic temperature, which
slows down the optimisation process towards high reproducibility. To overcome
the problem requires large numbers of qDUTs to be tested in a single cooling
down and constraints from two aspects need to be dealt with. One is the
various constraints (e.g. cooling power, physical space) imposed by operating
at cryogenic temperature, particularly the limited number of wires allowed
from room temperature. The other is the inherent complexity to interface with
qDUTs. To operate a qDUT typically requires multiple signal lines with
individual tunability. For example, 8 signal lines are needed to operate an
electrically defined double quantum dot, while only a few tens of wires are
available in a typical crystat setup, thus only a few devices can be tested in
a single cooling down.
There have been some attempts made towards addressing this throughput issue so
far. One approach is to build cryogenic wafer scale probe station. As reported
recently, quantum dot type devices were characterised at a wafer temperature
as low as 1.9K, which however only allows certain device characteristics to be
measured (Pillarisetty et al., 2019). Other key metrics requiring lower
temperature and high magnetic field still rely on measurement inside mK
dilution refrigerators, of which the throughput is still limited. An
alternative approach is to incorporate the MUX circuitry to increase the
number of interconnects at cryogenic temperature, which has the advantage to
be compatible with mK dilution refrigerator. Two initial works have the MUX
circuitry integrated on chip (Al-Taie et al., 2013; Ward et al., 2013), which
makes them less suitable for fast prototype, test and optimisation. Besides
the additional complexity involved in fabrication, changes in qUDT design are
difficult to be accommodated. For instance, adding an extra gate electrode in
quantum dot device to define a quantum point contact for probing charge
occupancy will require a completely new design. Two more recent solutions
(Wuetz et al., 2020; Pauka et al., 2019) have implemented the MUX circuitry
off-chip instead. However, they are still with their own drawbacks. Both
solutions are still with limited flexibility to alter the interconnect
topology to meet different wiring requirements. Secondly, both off-chip
solutions rely on a digital clock driven serial in parallel out (SIPO) shift
register to control MUXs. As discussed by themselves, the complexity arises
naturally from the requirement to match digital signal delay at cryogenic
temperature and to decouple digital noise.
In this paper, we present a cryogenic MUX setup that can be easily configured
to meet different wiring requirements. The MUXs are built based on the MLSG
method as previously used in (Al-Taie et al., 2013; Ward et al., 2013), which
does not rely on a digital clock signal to function. We implement the MUXs
off-chip, easing the exchange of qDUT chips and reusing the MUX chips. To
accommodate changes made to qDUTs, we also incorporate schemes to allow
interconnects to be routed into different topology at the qDUT side. Moreover,
the MUX operation is supported by a shared control line arrangement. Such
arrangement not only facilitates efficient scaling up the total number of
interconnects, but also simplifies the simultaneous control of multiple
independent signals supplied from different MUXs. As a result, qDUTs that
require multiple signal lines to operate can be interfaced easily. We first
validate and assess the operation of MUX setup by measuring $\mu$m-size
graphene ribbons. We further demonstrate the setup by performing transport
measurement on 128 nm-size graphene quantum devices. Lastly, we discuss the
potential improvements that can further increase the number of interconnects
and estimate that it can supply more than 1000 interconnects if all the
improvements are implemented.
## II Multiplex measurement setup details
Figure 1a shows the schematics of the MUX setup and gives an overview of how
signals are multiplexed and routed at cryogenic temperature. The setup
consists of two print circuit boards (PCBs), of which one is for mounting MUX
chips and the other is for mounting qDUT chips. The MUX PCB and qDUT PCB are
connected via swoppable board to board connectors. The MUX PCB comprises N
MUXs and each MUX is connected to a input signal line from room temperature
converting it to M individually addressable output channels. The MLSG method
(Al-Taie et al., 2013; Ward et al., 2013) used to implement signal
multiplexing ensures great scalability, as the number of multiplexed output
channels grows exponentially with the number of control lines. For a K-level
base-2 MUX, 2K control lines are needed for addressing gates and the multiplex
ratio M is equal to 2K. Physically, base-2 MUX corresponds to each channel at
current level being split into two channels at next level, thus leading to the
exponential increase. Control lines are supplied to the MUX PCB and shared
among all the MUX chips, adding another layer of scalability. Thus, N + 2K
wires from room temperature are able to supply in total N $\times$ 2K
interconnects at cryogenic temperature. To scale up the total number of
interconnects, it can be accomplished either by using MUXs of more levels or
by increasing the number of MUXs. For every extra level added to MUXs, two
more control lines are required and the total number of interconnects doubles.
For every K-level base-2 MUX added to the setup, only one extra signal line is
required and the total number of interconnects increases by 2K.
As the control lines are shared among MUXs, the same multiplexed output
channel of different MUXs can be simultaneously selected and set to supply
completely independent signals. Such a shared control line arrangement
provides a very simple way to control multiple MUXs to interface with qDUTs.
For a setup with N MUXs, it is able to interface with qDUTs requiring up to N
independent signals to operate. Certainly, qDUTs that need fewer independent
signals to operate can also be interfaced. The reconfiguration of the setup to
meet different wiring requirements is straightforward and can be accomplished
solely at the qDUT side. Figure 1b shows how the interconnects supplied from
the MUX PCB are routed into different topology in different qDUT PCBs. The
qDUT PCB 1, 2 and 3 can be used to interface with qDUTs that require 2, 4, and
8 signal lines to operate respectively. Interconnects supplied from different
MUXs are routed via different inner layer tracks, allowing multiple tier wire
bonding to be implemented.
Figure 1: (a) Schematics of the MUX setup. The setup comprises two PCBs, of
which one is for mounting MUX chips and the other is for mounting qDUT chips.
Two PCBs are connected via swoppable board to board connectors. The
interconnects can be regrouped differently at the qDUT PCB side to meet
different wiring requirements. Each MUX (MUX[1], MUX[2], …, MUX[N]) is
supplied with a input signal (S[1], S[2], …, S[N]) from room temperature and
converts it to M individually addressable outputs. MUXs are controlled by the
shared control lines CRL[1:2K]. With 2K control lines, each MUX can supply up
to 2K output channels, i.e. M = 2K. (b) Cross section view of the MUX setup.
MUX chips are wire bonded to the top side of MUX PCB. Interconnects supplied
from MUX chips are connected to the female connectors at the backside of MUX
PCB through the PCB vias. qDUT chips are wire bonded to the backside of qDUT
PCB, which are then connected through PCB vias with the male connectors at the
top side of qDUT PCBs. The MUX PCB stay unchanged once being installed and
different qDUT PCBs can be used for mounting different qDUTs. qDUT PCB 1, 2, 3
can be used to interface with qDUTs that require 2, 4, 8 signals to operate
respectively. The multiple layer PCBs used here allows multiple tier wire
bonding to be used.
For the prototype demonstrated in this paper, 5-level base-2 MUXs with a
multiplex ratio of 32 = 25 are used. With 4 MUXs installed, 128 interconnects
are available at cryogenic temperature and only 14 wires (10 for addressing
gates and 4 for inputs) from room temperature are used. Figure 2 shows the
schematics of the 5-level MUXs, each chip consists of two MUXs sharing 10
addressing gates. As compared with the MUXs demonstrated previously in (Al-
Taie et al., 2013), an extra layer of 20 nm Al2O3 is added in our MUXs (see
Figure 2d). This extra layer of dielectric serves two purposes here. One is to
improve the GaAs surface quality by a self-cleaning reaction (Hinkle et al.,
2008), so to minimise the variation in MUX characteristics induced by surface
trapped charges. More importantly, it will allow us to operate the MUXs over a
extended signal range as compared with the original demonstration (Al-Taie et
al., 2013). The signal range would otherwise be limited by the Schottky
junction formed between Au/Ti addressing gates and MUX conduction channels.
The fabrication procedure of MUXs is as follows. The MUXs are fabricated on
GaAs/AlGaAs substrate. Firstly, conduction channels (MESA) are defined by
standard wet etching process and Ohmic contacts are made for the input and
each output (Figure 2a). Secondly, 20 nm Al2O3 is deposited globally by atomic
layer deposition (ALD) process (only shown in cross section view Figure 2d)
and windows are opened over Ohmic contacts by hydrofluoric acid etching. Then,
photo definable polyimide is selectively deposited on top of the 20 nm Al2O3
layer to modulate the gate capacitance (Figure 2b). Lastly, 10 nm Ti and 150
nm Au are deposited by tilted rotatory thermal evaporation to ensure
continuous metalisation over the thick polyimide layer.
At cryogenic temperature, current conduction along channels is through the
2-dimensional electron gas (2DEG) formed at the interface of AlGaAs/GaAs.
Channel conduction is normally on and can be pinched off by applying a
negative voltage to addressing gates. Figure S2a shows the channel pinch off
process as the 2DEG is gradually depleted. As the potential difference exceeds
certain threshold $V_{T}$, the 2DEG underneath the gate becomes completely
depleted and conduction is lost. By modulating the gate capacitance with
selectively deposited polyimide, addressing gates will only affect channels
that are not covered by polyimide, if the gate voltage applied is within a
certain window (i.e. selective gating). As can be seen from Figure 2b, a pair
of addressing gates at each level are designed to be complementary, of which
one controls the odd channels and the other controls even channels. Therefore,
each output channel can be individually selected by a specific gate
combination. For example, output channel 22 can be selected by activating
(applying pinch off voltage) addressing gate 1L/2R/3L/4L/5R (Figure 2c).
Figure 2: MUXs are fabricated on GaAs substrate and each chip consists of two
MUXs. (a) The conduction channels (MESAs) are first defined by wet etching,
then Ohmic contacts are made for the input and each output. The isolated Ohmic
contacts (and MESAs underneath) are used as bond pads for addressing gates.
(b) 20 nm Al2O3 is deposited globally (only shown in cross section view) and
is etched to expose Ohmic contacts. Then polyimide is selectively deposited
over the MESAs, which is used for modulating gate capacitance, such that
channels can be selectively pinched off. (c) Au/Ti metal stack is deposited as
addressing gates. By activating addressing gates 1L/2R/3L/4L/5R, channel 22 is
selected. The relationship is clearly revealed in binary representation, if L
corresponds to 1 and R corresponds to 0, then 10110 in binary is essentially
22 in decimal. (d) Cross section view of the MUXs showing two adjacent
conduction channels, of which the gate capacitance is modulated by selectively
deposited polyimide.
Below we validate the proper operation of this 5-level base-2 MUX, as well as
assess the various limits that exist. For the MUX to operate properly, the
selected (open) channel should present as a low-resistance path over the
signal range of interest and the leakage current of all other unselected
(closed) channels should be negligible. As will be shown below, normally the
resistance of the selected channel is much smaller as compared with that of
qDUT, however it can become very large even dominant when the signal applied
to MUX input is above certain positive threshold, limiting the input signal
range at the positive side. As for the negative side, it is to ensure the
potential of active addressing gates stay sufficiently negative with respect
to the conduction channels. The potential difference has to exceed the
threshold voltage $V_{T}$ to maintain unselected channels at pinched off
state. For a specific gate voltage applied, the input signal range at the
negative side is thus limited. These two limits are not fundamental but need
to be dealt with carefully. In contrast, the number of output channels in a
single MUX is more of a fundamental limit. To ensure the leakage current being
negligible, it is required to have the effective resistance of all closed
channels in parallel to be much larger as compared with that of a qDUT. If the
off-resistance of each closed channel is $R_{off}$ and the maximum resistance
of a qDUT (resistance of a qDUT is usually gate and bias dependent) is
$R_{max}$, then the total number of channels M is limited to be $\sim$
$\frac{R_{off}}{10R_{max}}$ in practice.
Figure 3a shows a representative channel pinch off characteristics of a MUX.
It was measured by sweeping the voltage applied to a pair of addressing gates
at a same level simultaneously, while the MUX input was kept at $V_{sd}=-0.4$
V. All the channels are pinched off when the voltage applied to the addressing
gates reaches $V_{g}=-0.66$ V, which is equivalent to a threshold voltage
$V_{T}$ of $-0.26$ V. On the other hand, the inset shows the pinch off
characteristics when negative voltage is only applied to a single addressing
gate. Half of the channels without polyimide are pinched off at $V_{g}=-0.66$
V (not shown), while all other channels covered by polyimide are not pinched
off until $V_{g}=-7.2$ V. There is indeed a window of gate voltage that the
channels can be selectively pinched off. Figure 3b shows the leakage current
of 32 pinched off channels of a MUX, when two addressing gates of each level
were set to $-0.66$ V and the input signal was swept from $-0.4$ V to 0.4 V.
The leakage currents are plotted with 10 pA offset for clarity. As can be
seen, the total leakage current of 32 channels is less than 1 pA. By fitting
the total leakage current, we infer the off-resistance of each channel is
around 30 T$\Omega$. For a MUX with 256 channels, the effective resistance of
all closed channels is still above 100 G$\Omega$ and will allow us to measure
qDUTs up to $\sim$ 10 G$\Omega$ resistance. From the leakage current, we also
learn indirectly that the noise added by MUX circuitry is negligible, since no
significant change is observed as compared with the noise floor set by
electronics.
Figure 3c shows the pinch off characteristics of the same MUX, when the input
signal was set to $-0.5$ V and $-0.6$ V respectively. It is evident that more
negative gate voltages are required to pinch off channels. Generally, to
ensure active addressing gates pinch off all unselected channels completely,
input signal $V_{in}$ is limited to $V_{g}-V_{T}$ at the negative side, where
$V_{g}$ is the gate voltage applied and $V_{T}$ is the threshold voltage.
Therefore, to extend the negative limit of input signal $V_{in}$ is
essentially to extend the negative limit of gate voltage $V_{g}$ that can be
applied. As each addressing gate controls multiple channels, the potential
difference between channel and addressing gate also varies. It can be found
out that the smallest potential difference is $V_{g}-V_{in}$ and the biggest
potential difference is $V_{g}$ (see Figure S3a as an example). It is the
channels that have the biggest potential difference with respect to the active
addressing gate setting the negative limit of $V_{g}$. By incorporating 20 nm
Al2O3 layer, the negative limit of $V_{g}$ is substantially increased, which
would otherwise be set by the breakdown voltage of reversely biased Schottky
junction formed between addressing gates and channels. From the measurement to
obtain the pinch off voltage of channels that are covered by polyimide (Figure
3a inset), we learn that the voltage applied to addressing gate can be at
least as large as $-10$ V, since channels without polyimide are also being
gated up to $-10$ V in that measurement.
On the other hand, the input signal range at positive side is limited by
inactive addressing gates. Figure 3d shows the multiplexed measurement of 32
$\mu$m-size graphene ribbons, where active addressing gates were set to
$-0.66$ V and inactive addressing gates were left grounded. The bias voltage
$V_{sd}$ was supplied to the MUX input and swept over a range from $-0.4$ V to
$0.4$ V. All the graphene ribbons share a common drain, from which the current
was measured (Figure S4). As can be seen, the current measured flowing through
each MUX channel is saturated at high positive bias, which is not an intrinsic
characteristic of $\mu$m-size graphene ribbons but rather originates from the
MUX channel. As the input signal (bias voltage) increases at the positive
side, the inactive addressing gates (kept grounded) become effectively more
negative with respect to the open MUX channel, such that the 2DEG is gradually
depleted (Figure S2b) and any further increase of the bias voltage is dropped
across the depleted region. The input signal range at positive side is thus
limited. Each channel saturates at slightly different voltage, which is due to
the variation in the local carrier density of 2DEG and the gate capacitance
combined. To prevent the MUX channel from going into saturation and extend the
positive limit of the input signal, a positive voltage can be applied to
inactive addressing gates. Figure 3e shows the multiplexed measurement of the
same 32 $\mu$m-size graphene ribbons, but the inactive addressing gates were
set to 0.4 V. As can be seen, the currents are no longer saturated and show
linear current-voltage relationship as expected from $\mu$m-size graphene
ribbons. The positive voltage that can be applied to the inactive addressing
gates itself is limited. Without the dielectric layer, it is set by the turn-
on voltage of the Schottky junction ($\sim$ 500 mV), beyond which large gate
leakage current to unselected channels will be generated (see Figure S3b as an
example). By adding 20 nm Al2O3, the gate leakage can be effectively
suppressed until dielectric breakdown occurs. Although we didn’t test the
breakdown voltage at the positive side, we can safely estimate it to exceed 10
V based on two reasons. First, unlike Schottky junction, metal dielectric
interface has little dependence on voltage polarity, so we expect it to be
comparable to the negative side. Second, based on the typical breakdown
electric field reported for ALD grown Al2O3 on GaAs substrate (Wu et al.,
2007), dielectric breakdown voltage for 20 nm Al2O3 can be as large as 20 V at
room temperature and can be a few times higher at cryogenic temperature.
The dotted line in Figure 3e shows the current trace of a open MUX channel
measured separately without any qDUT connected, which is fitted to have an on-
resistance of 862 $\Omega$. It is much smaller as compared with the resistance
of a typical qDUT, which is at least a few tens of k$\Omega$. Figure 3f shows
the current trace of a graphene qDUT measured with different positive voltages
applied to inactive addressing gates. As can be seen, when no voltage is
applied, the current trace shows a plateau above 0.2 V. As the voltage
increases to 0.2 V, the onset of saturation is shifted to higher voltage. When
the accumulation voltage is set to 0.4 V, the plateau disappears completely.
This current plateau is due to 2DEG saturation and can be wrongly attributed
to resonant transport behaviour, if the MUX is not operated properly.
Figure 3: (a) Pinch off characteristics of MUX channels that are not covered
by 500 nm polyimide. It was measured with a input signal of $-0.4$ V and a
pair of addressing gates at the same level were swept from 0 V to $-0.8$ V.
All the 32 MUX channels are completely pinched off when the gate voltage
reaches -0.66V. Inset shows the pinch off characteristics of MUX channels that
are covered by 500nm polyimide. It was measured with the same input signal of
-0.4V, however only one addressing gate was used to pinch off channels. Half
of channels that are not covered by polyimide are pinched off at -0.66V, while
the other half are pinched off at $-7.2$ V. (b) The leakage currents of all 32
channels, when a pair of gates at each level were set to pinch off channels
respectively. The leakage currents are plotted with 10 pA offset for clarity.
(c) Pinch off characteristics of MUX channels when the input signal was set to
$-0.5$ V and $-0.6$ V respectively. Evidently, more negative gate voltage is
required to pinch off channels. (d) Multiplexed measurement of a nm-size
graphene device, where various positive voltages are applied to inactive
gates. As can be seen, the current saturation in MUX channel can obscure the
intrinsic characteristics of the qDUT and lead to measurement artefact, if the
MUX is not operated properly.
## III Measuring graphene nanogaps
We further demonstrate the MUX setup by performing quantum transport
measurement on nm-size graphene nano-gaps. Graphene nanogap has been developed
to contact single molecule and to study quantum transport across the single
molecular junction (Lau et al., 2014). As compared with conventional
approaches based on scanning probe or metal break junction, it allows easy
integration of a backgate to tune the molecular orbitals. Moreover, the
reduced electric field screening of graphene allows multiple molecular
orbitals to be accessed for transport study. Most importantly, its planar
device structure holds the prospect to scale up and build functional molecular
circuit. Given all these advantages, it does have a disadvantage that it is
difficult to collect sufficient device statistics, thereby to provide feedback
for optimisation. This problem has to be overcome before building any
functional molecular circuit becomes possible. Take a work done previously in
our group for example (Limburg et al., 2018), which assessed the effectiveness
of different anchor groups to form molecular junctions. In total it took three
people two years of experiments to collect statistics on 1000 devices. In
comparison, the multiplexed measurement of 128 devices only took 5 days to
finish. Same amount of statistics will only need 40 days, which is roughly 20
times speed up.
Graphene nano-gaps used here were fabricated using the electroburning
process(Lau et al., 2014). A difference in device structure is that a local
back gate is used instead (Figure 4a). Additionally, local back gate and drain
are shared among the graphene devices Figure S4, such that only one
multiplexed interconnect instead of three is needed for each device to perform
transport measurement. Figure 4b shows a typical electroburning process. At
each burning cycle, the bias voltage, applied between the source(S) and drain
(D) across the graphene nano-constriction, is ramped up at a rate of 5V/s.
Upon the sudden decrease of the current resulting from the breakdown of
graphene constriction, it is ramped down at a faster rate of 50V/s. By
repeated controlled breakdown, a nanogap forms in the end and shows sub-nA
tunneling current (Figure 4c). By fitting into the Simmons model, it can be
inferred that the gap size is around 1.2 nm.
Not all the graphene nano-gaps are suitable for contacting molecules,
significant percentages of nanogaps have carbon debris in between. At
cryogenic temperature it may behave like a quantum dot and show resonant
transport features that are indistinguishable from molecule and an example is
as shown Figure 4d. If a nanogap like that is used for contacting molecules,
any features appear after molecule deposition cannot be simply attributed to
the molecule, therefore should be screened before molecule deposition. In
comparison, Figure 4e shows a different nanogap, which does not show any
Coulomb Diamond alike features and is suitable for contacting molecules.
Usually we perform electroburning at multiple back gate (BG) voltages, which
makes it more likely to form a featureless gap like Figure 4e. However, for
the purpose of demonstrating MUX setup, i.e. to easily distinguish different
gaps being measured, all the graphene nano-gaps measured were fabricated with
gate floated. We measured 128 graphene nano-gaps in total, 32 of them are
shown in Figure 5 and all other 96 of them are shown in Figure S5. In Figure 5
top panel shows the tunneling curves measured at the end of electroburing
process at room temperature and bottom panel shows the stability diagrams
measured at 2.9K in puck tester (Figure S1). The tunnelling curves are plotted
in the same scale, where the bias range is from $-0.4$ V to 0.4 V and the
current range is from $-500$ pA to 500 pA. The stability diagram is plotted
with the bias range from $-0.4$ V to $0.4$ V and a back gate voltage range
from $-4$ V to $4$ V.
Figure 4: (a) Schematics of a graphene nano-constriction, which is used to
fabricate a graphene nano-gap by electroburning process. Inset shows the SEM
of a graphene nano-constriction. (b) Electroburning curves of the graphene
nano-constriction. At each burning cycle, the bias voltage $V_{sd}$ is ramped
up at rate of 5V/s to break the graphene, upon the deteciton of a sudden
current drop, the voltage is ramped down at a rate of 50V/s. This controlled
breakdown of graphene nano-constriction leads to the formation of nm-size
graphene nanogap. (c) From the tunneling curve at the end of electroburning,
we fit it with Simmons model and infer the gap size to be 1.2 nm. (d)
Stability diagram of a graphene nanogap measured at 2.9 K in puck tester. It
shows resonant tunneling features, which are indistinguishable from intrinsic
charge transport features across molecular junctions, such that this nanogap
is not suitable for contacting molecules. (e) In contrast, stability diagram
of a different graphene nanogap, which does not show any particular features
and is suitable for contacting molecules. Figure 5: Top: IV curves of 32
graphene nanogaps measured at the end of electroburning process at room
temperature. They were measured with a bias voltage $V_{sd}$ range from $-0.4$
V to $0.4$ V and zero back gate voltage $V_{bg}=0$ V. All the curves are
plotted in the same scale, the vertical axis is between -100pA to 100 pA.
Bottom: Stability diagram of the same 32 graphene nanogaps measured at 2.9 K
in puck tester. The measurement was performed with a bias voltage $V_{sd}$
range from $-0.4$ V to 0.4 V and a back gate voltage $V_{bg}$ range from $-4$
V to 4 V. The 32 stability diagrams are plotted in a same colorbar, where
different shades of blue corresponds to the range from $-500$ pA (blue) to 0
pA (white) and red corresponds to 500 pA (red) to 0 pA (white).
## IV Conclusion and Outlook
The prototype setup demonstrated in this paper can supply in total 128
interconnects at cryogenic temperature, which uses only 14 out of 44 existing
wires from room temperature. As can be seen from Figure S1b, the 2 MUX chips
(4 MUXs) are still sparsely placed on the PCB. Even with our very primitive 2
layer PCB (minimum track size 0.2 mm, drill size 0.3 mm), we can easily fit 1
more MUX chip (as shown in Figure S6). This will give us in total 192
interconnects, of which 10 shared control lines and 6 input signal lines are
used. If multiple layer PCB is used and all the tracks placed in inner layers,
there will be space for mounting up to 9 MUX dies (18 MUXs). Another aspect
that can easily be improved is the PCB track size and the footprint of board
to board connector (Figure S1c). If the size is reduced by half, we can add
one more level to the MUX and fit it roughly into the same space. With 9 MUX
dies mounted, they can supply 1152 interconnects in total with only 30 (12+18)
wires used.
To conclude, we have successfully demonstrated a MUX setup to increase the
throughput of electrical characterisation at cryogenic temperature. We have
also assessed various limits that exist in MUX operation. By implementing the
MUX circuitry off-chip, the qDUT chips can be easily exchanged. Different qDUT
PCBs are designed to meet the wiring requirements of different qDUTs.
Moreover, we have incorporated a shared control signal arrangement at board
level to support the MUX operation. Such implementation will not only provide
an efficient to scale up the total number of interconnects, but also will
allow us to easily control multiple MUXs simultaneously to interface with
qDUTs requiring multiple signal lines to operate.
## V Acknowlegement
JAM was supported through the UKRI Future Leaders Fellowship, Grant No.
MR/S032541/1, with in-kind support from the Royal Academy of Engineering.
## References
* Zwanenburg et al. (2013) F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, Reviews of Modern Physics 85, 961 (2013).
* Sarma et al. (2015) S. D. Sarma, M. Freedman, and C. Nayak, npj Quantum Information 1 (2015).
* Watson et al. (2018) T. F. Watson, S. G. J. Philips, E. Kawakami, D. R. Ward, P. Scarlino, M. Veldhorst, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, et al., Nature 555, 633 (2018).
* Xiang et al. (2016) D. Xiang, X. Wang, C. Jia, T. Lee, and X. Guo, Chemical Reviews 116, 4318 (2016).
* Pillarisetty et al. (2019) R. Pillarisetty, N. Kashani, P. Keys, R. Kotlyar, F. Luthi, D. Michalak, K. Millard, J. Roberts, J. Torres, O. Zietz, et al., in _2019 IEEE International Electron Devices Meeting (IEDM)_ (IEEE, 2019).
* Al-Taie et al. (2013) H. Al-Taie, L. W. Smith, B. Xu, P. See, J. P. Griffiths, H. E. Beere, G. A. C. Jones, D. A. Ritchie, M. J. Kelly, and C. G. Smith, Applied Physics Letters 102, 243102 (2013).
* Ward et al. (2013) D. R. Ward, D. E. Savage, M. G. Lagally, S. N. Coppersmith, and M. A. Eriksson, Applied Physics Letters 102, 213107 (2013).
* Wuetz et al. (2020) B. P. Wuetz, P. L. Bavdaz, L. A. Yeoh, R. Schouten, H. van der Does, M. Tiggelman, D. Sabbagh, A. Sammak, C. G. Almudever, F. Sebastiano, et al., npj Quantum Information 6 (2020).
* Pauka et al. (2019) S. J. Pauka, K. Das, J. M. Hornibrook, G. C. Gardner, M. J. Manfra, M. C. Cassidy, and D. J. Reilly, arXiv (2019), eprint 1908.07685v1.
* Hinkle et al. (2008) C. L. Hinkle, A. M. Sonnet, E. M. Vogel, S. McDonnell, G. J. Hughes, M. Milojevic, B. Lee, F. S. Aguirre-Tostado, K. J. Choi, H. C. Kim, et al., Applied Physics Letters 92, 071901 (2008).
* Wu et al. (2007) Y. Q. Wu, H. C. Lin, P. D. Ye, and G. D. Wilk, Applied Physics Letters 90, 072105 (2007).
* Lau et al. (2014) C. S. Lau, J. A. Mol, J. H. Warner, and G. A. D. Briggs, Phys. Chem. Chem. Phys. 16, 20398 (2014).
* Limburg et al. (2018) B. Limburg, J. O. Thomas, G. Holloway, H. Sadeghi, S. Sangtarash, I. C.-Y. Hou, J. Cremers, A. Narita, K. Müllen, C. J. Lambert, et al., Advanced Functional Materials 28, 1803629 (2018).
## VI Supplemantary Material
Figure S1: (a) Measurement setup used in this paper. The puck is compatbile
with both 2.9K puck tester and mK dilution fridge of Oxford Instrument. (b)
MUX PCB: the two red rectangles are for mouting MUX chip. With this PCB, we
can install 4 MUXs in total (2 MUXs in each chip). 2 layer PCBs are used here
and most of board area is occupied by the PCB tracks. If multiple layer PCBs
are used and all the tracks put into the inner layer, there is enough space to
accomodate up to 9 MUX chips. The board to board connector used here is of
0.4mm pitch size and each contsist of 80 pins. (c) The MUX PCBs are made with
primitive technology, of which the minimal track width is 0.2mm (equivalent to
a pitch size of 0.4mm). If PCBs are made with a track width of 0.1mm, MUXs
with 64 output channles roughly fit into the same space . (d) qDUT PCB used in
this setup. Figure S2: (a) Schematics showing the channel pinch off process.
As the gate voltage becomes more negative with repsect to the MUX channel, the
2DEG is gradually depleted. (b) Schematis showing the 2DEG saturation process.
As the input signal increases and 2DEG becomes more positive with respect to
addressing gate, which acts effectively as if a negative gate voltage was
applied, thus the 2DEG is depleted closer to the input signal side. The
current saturation arises a result, since any further increase of input signal
is dropped over the depleted region. (c) A positive voltage applied to the
gate can reduce the potential difference between addressing gate and channel,
so to prevent the 2DEG going to saturation. The signal range at the positive
side can thus be extended. Figure S3: Schematics of a 2-level base-2 MUX,
where addressing gate 1L and 2L are activated to select channel 3. (a) The
negative limit of input signal is set by the negative voltage applied to
addressing gate 1L and 2L. The gate voltage itself is limited by the channel
that has the biggest potential difference with respect to active addressing
gate. Here it is between channel 0 and gate 2L, of which the potential
difference is Vin. (b) The positive limit of input signal is set by the
inactive addressing gates. To prevent the 2DEG going to saturation, a positive
voltage can be applied to 1R and 2R to extend the input signal range at
positive side. However, the positive voltage that can be applied is also
limited by the channel that has the biggest potential difference with respect
t inactive addressing gate. Here it is between channel 1 and addressing gate
2R. Figure S4: Silicon substrate with electrodes for fabricating graphene
used in this paper. Both $\sim$ $\mu$m size graphene ribbons and 100nm
graphene nanoconstrictions are made with this device structure, where all the
devices of a column share a common drain and a common local back gate. The
local back gate is covered is 10nm HfO2 grown by ALD. Figure S5: Stability
diagrams of all other 96 graphene nanogaps which were measured in single
cooling down. The measurement was performed at 2.9K in puck tester. The
measurement was performed with a bias voltage Vsd range from -0.4V to 0.4V and
a back gate voltage Vbg range from -4V to 4V. The 32 stability diagrams are
plotted in a same colorbar, where different shades of blue corresponds to the
range from -500 pA (blue) to 0 pA (white) and red corresponds to 500 pA (red)
to 0 pA (white). Figure S6: A different MUX PCB that can mount 3 MUX chips.
In total 192 interconnects can be controlled using 16 wires from room
temperature.
|
* [16] F. Tröltzsch, Optimal Control of Partial Differential Equations, Vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010, theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. doi:10.1090/gsm/112.
* [17] E. Casas, F. Tröltzsch, First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM J. Control Optim. 48 (2) (2009) 688–718. doi:10.1137/080720048.
* [18] F. H. Clarke, Optimization and Nonsmooth Analysis, Vol. 5 of Classics Appl. Math., SIAM, 1990. doi:10.1137/1.9781611971309.
* [19] G. Wachsmuth, D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional, ESAIM: Control Optim. Calc. Var. 17 (2011) 858–886. doi:10.1051/cocv/2010027.
* [20] E. Casas, F. Tröltzsch, Optimal sparse boundary control for a semilinear parabolic equation with mixed control-state constraints, Control and Cybernetics 48 (1) (2019) 89–124.
* [21] E. Casas, Second order analysis for bang-bang control problems of PDEs, SIAM J. Control Optim. 50 (4) (2012) 2355–2372. doi:10.1137/120862892.
* [22] M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Verlag, Basel, 2009. doi:10.1007/978-3-7643-9982-5.
* [23] J. F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, Berlin, Heidelberg, 2000. doi:10.1007/978-1-4612-1394-9.
* [24] D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer-Verlag, Berlin, Heidelberg, 1990. doi:10.1007/BFb0085564.
* [25] T. Bayen, J. F. Bonnans, F. J. Silva, Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations, Trans. Amer. Math. Soc. 366 (2014) 2063–2087. doi:10.1090/S0002-9947-2013-05961-2.
* [26] E. Casas, Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints, ESAIM: Control Optim. Calc. Var. 14 (3) (2008) 575–589. doi:10.1051/cocv:2007063.
* [27] D. Wachsmuth, Adaptive regularization and discretization of bang-bang optimal control problems, Electron. Trans. Numer. Anal. 40 (2013) 249–267.
* [28] K. Deckelnick, M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls, Comput. Optim. Appl. 51 (2012) 931–939. doi:10.1007/s10589-010-9365-z.
* [29] G. Alberti, S. Bianchini, G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5) XII (4) (2013) 863–902. doi:10.2422/2036-2145.201107_006.
* [30] U. Felgenhauer, On stability of bang-bang type controls, SIAM Journal on Control and Optimization 414 (6) (2003) 1843–1867. doi:10.1137/S0363012901399271.
* [31] C. Christof, C. Meyer, Sensitivity analysis for a class of $H^{1}_{0}$-elliptic variational inequalities of the second kind, Set-Valued and Variational Analysis 27 (2019) 469–502. doi:10.1007/s11228-018-0495-2. |
$\displaystyle\phi_{1,4}\times\phi_{1,4}$ $\displaystyle\sim\mathbb{1}$
$\displaystyle\phi_{1,4}\times\phi_{2,2}\sim\phi_{2,2}$
$\displaystyle\phi_{1,4}\times\phi_{2,1}$ $\displaystyle\sim\phi_{2,1}$
$\displaystyle\phi_{2,2}\times\phi_{2,2}\sim\mathbb{1}+\phi_{1,2}+\phi_{1,3}+\phi_{1,4}$
$\displaystyle\phi_{2,2}\times\phi_{2,1}$
$\displaystyle\sim\phi_{1,2}+\phi_{1,3}$
$\displaystyle\phi_{2,1}\times\phi_{2,1}\sim\mathbb{1}+\phi_{1,4}.$
The non trivial OPE coefficients are
$\displaystyle C^{(1,3)}_{(1,2),(1,2)}=$
$\displaystyle\frac{2}{3}\sqrt{\frac{\Gamma(4/5)\Gamma(2/5)^{3}}{\Gamma(1/5)\Gamma(3/5)^{3}}}$
$\displaystyle C^{(1,3)}_{(1,3),(1,3)}=$ $\displaystyle
C^{(1,3)}_{(1,2),(1,2)}$ $\displaystyle C^{(1,3)}_{(1,2),(1,4)}=$
$\displaystyle 3/7$ $\displaystyle C^{(1,2)}_{(2,1),(2,2)}=$ $\displaystyle
1/2$ $\displaystyle C^{(1,2)}_{(2,2),(2,2)}=$ $\displaystyle
3/2\,C^{(1,3)}_{(1,2),(1,2)}$ $\displaystyle C^{(1,3)}_{(2,1),(2,2)}=$
$\displaystyle 3/4$ $\displaystyle C^{(1,3)}_{(2,2),(2,2)}=$ $\displaystyle
1/4\,C^{(1,3)}_{(1,2),(1,2)}$ $\displaystyle C^{(1,4)}_{(2,2),(2,2)}=$
$\displaystyle 1/56$ $\displaystyle C^{(1,4)}_{(2,1),(2,1)}=$ $\displaystyle
7/8\,C^{(1,3)}_{(1,2),(1,2)}.$
### $M(3,7)$
The fusion rules of the $M(3,7)$ model are:
$\displaystyle\phi_{1,2}\times\phi_{1,2}\sim\mathbb{1}+i\phi_{1,3}$
$\displaystyle\phi_{1,2}\times\phi_{1,3}$ $\displaystyle\sim
i\phi_{1,2}+\phi_{1,4}$
$\displaystyle\phi_{1,2}\times\phi_{1,4}\sim\phi_{1,3}+i\phi_{1,5}$
$\displaystyle\phi_{1,2}\times\phi_{1,5}$ $\displaystyle\sim
i\phi_{1,4}+\phi_{1,6}$
$\displaystyle\phi_{1,2}\times\phi_{1,6}\sim\phi_{1,5}$
$\displaystyle\phi_{1,3}\times\phi_{1,3}$
$\displaystyle\sim\mathbb{1}+i\phi_{1,3}+\phi_{1,5}$
$\displaystyle\phi_{1,3}\times\phi_{1,4}\sim\phi_{1,2}+i\phi_{1,4}+\phi_{1,6}$
$\displaystyle\phi_{1,3}\times\phi_{1,5}$
$\displaystyle\sim\phi_{1,3}+i\phi_{1,5}$
$\displaystyle\phi_{1,3}\times\phi_{1,6}\sim\phi_{1,4}$
$\displaystyle\phi_{1,4}\times\phi_{1,4}$
$\displaystyle\sim\mathbb{1}+i\phi_{1,3}+\phi_{1,5}$
$\displaystyle\phi_{1,4}\times\phi_{1,5}\sim i\phi_{1,2}+\phi_{1,4}$
$\displaystyle\phi_{1,4}\times\phi_{1,6}$ $\displaystyle\sim\phi_{1,3}$
$\displaystyle\phi_{1,5}\times\phi_{1,5}\sim\mathbb{1}+i\phi_{1,3}$
$\displaystyle\phi_{1,5}\times\phi_{1,6}$ $\displaystyle\sim\phi_{1,2}$
$\displaystyle\phi_{1,6}\times\phi_{1,6}\sim\mathbb{1}.$
We have indicated with a factor of $i$ the OPE coefficients which are purely
imaginary, to distinguish them from the real ones. In (C.2), we give the value
of the non trivial OPE coefficients between primaries in the even sector:
$\displaystyle C_{333}=$
$\displaystyle\,i\sqrt{-\frac{\Gamma(1/7)^{3}\Gamma(3/7)^{3}\Gamma(5/7)^{3}\Gamma(8/7)}{\Gamma(-1/7)\Gamma(2/7)^{3}\Gamma(4/7)^{3}\Gamma(6/7)^{3}}}$
$\displaystyle\approx$ $\displaystyle\quad 1.97409i,$ $\displaystyle C_{355}=$
$\displaystyle\,i\sqrt{-\frac{\Gamma(-5/7)^{2}\Gamma(1/7)\Gamma(3/7)^{3}\Gamma(5/7)\Gamma(8/7)\Gamma(11/7)^{2}}{\Gamma(-4/7)^{2}\Gamma(-1/7)\Gamma(2/7)\Gamma(4/7)^{3}\Gamma(6/7)\Gamma(12/7)^{2}}}$
$\displaystyle\approx$ $\displaystyle\quad 0.979627i,$ (C.1) $\displaystyle
C_{335}=$
$\displaystyle\,-\sqrt{\frac{\Gamma(-5/7)\Gamma(-2/7)\Gamma(3/7)\Gamma(5/7)^{2}\Gamma(6/7)\Gamma(8/7)^{2}}{\Gamma(-1/7)^{2}\Gamma(1/7)\Gamma(2/7)^{2}\Gamma(4/7)\Gamma(9/7)\Gamma(12/7)}}$
$\displaystyle\approx$ $\displaystyle\quad-0.113565.$
These constants (including one nontrivial relative sign between $C_{333}$ and
$C_{355}$) can be determined from the four-point functions
$\langle\phi_{1,3}\,\phi_{1,3}\,\phi_{1,3}\,\phi_{1,3}\rangle$ and
$\langle\phi_{1,3}\,\phi_{1,3}\,\phi_{1,3}\,\phi_{1,5}\rangle$, which we
computed using the Coulomb gas method [35, 36].
### $M(3,5)$
Here we give the fusion rules of the $M(3,5)$ model.
$\displaystyle\phi_{1,2}\times\phi_{1,2}\sim\mathbb{1}+i\phi_{1,3}$
$\displaystyle\phi_{1,2}\times\phi_{1,3}$ $\displaystyle\sim
i\phi_{1,2}+\phi_{1,4}$ (C.2)
$\displaystyle\phi_{1,2}\times\phi_{1,4}\sim\phi_{1,3}$
$\displaystyle\phi_{1,3}\times\phi_{1,3}$ $\displaystyle\sim\mathbb{1}$ (C.3)
$\displaystyle\phi_{1,3}\times\phi_{1,4}\sim\phi_{1,2}$
$\displaystyle\phi_{1,4}\times\phi_{1,4}$ $\displaystyle\sim\mathbb{1}.$ (C.4)
## Appendix D Review of $\mathcal{PT}$ Symmetry
In this appendix we review in detail the $\mathcal{PT}$ symmetry present in
the $M(3,7)$ flow and what impact it has on the spectrum.
An obvious consequence of imaginary OPE coefficients in the $M(3,7)$ CFT is
that the QFT Hamiltonian (6.3) is not a Hermitian operator. Nevertheless such
a non-Hermitian Hamiltonian can have a real spectrum bounded from below
provided it is $\mathcal{PT}$-symmetric, see e.g. [38] for a comprehensive
review. We now summarize the main facts about $\mathcal{PT}$-symmetry which
are relevant for the analysis of the $M(3,7)+i\phi_{1,3}+\phi_{1,5}$ theory
studied in the main text. The discrete symmetry is represented on the Hilbert
space of the theory by an antiunitary involution operator
($(\mathcal{PT})^{2}=\mathbb{1}$ and
$\mathcal{PT}\lambda\mathcal{PT}=\lambda^{*}$ for any $\lambda\in\mathbb{C}$).
There are two conditions that define unbroken $\mathcal{PT}$ symmetry. The
first is the usual definition of a symmetry of a quantum system, i.e.
$[\mathcal{PT},H]=0.$ (D.1)
The second is that any eigenstate of the Hamiltonian $\ket{\psi}$ is also an
eigenstate of $\mathcal{PT}$ 222222In the case of a linear symmetry operator,
equations (D.1) and (D.2) are famously equivalent: any commuting
(diagonalizable) linear operators can be simultaneously diagonalized. However
this theorem does not hold for antilinear operators, hence the two distinct
conditions. with eigenvalue one:232323The eigenvalues of antiunitary
involutions are pure phases $e^{i\alpha},\alpha\in\mathbb{R}$, and if the
Hamiltonian and the $\mathcal{PT}$ operator are simultaneously diagonalizable,
one can always define Hamiltonian eigenstates such that their eigenvalue under
$\mathcal{PT}$ is one. Indeed, given an eigenvector of the Hamiltonian
$\ket{\psi}$ such that $\mathcal{PT}\ket{\psi}=e^{i\alpha}\ket{\psi}$, then
$\ket{\psi^{\prime}}\equiv e^{i\alpha/2}\ket{\psi}$ will have eigenvalue one:
$\mathcal{PT}\ket{\psi^{\prime}}=\mathcal{PT}e^{i\alpha/2}\ket{\psi}=e^{-i\alpha/2}\mathcal{PT}\ket{\psi}=e^{i\alpha/2}\ket{\psi}=\ket{\psi^{\prime}}.$
$\mathcal{PT}\ket{\psi}=\ket{\psi},\qquad(H\ket{\psi}=E\ket{\psi}).$ (D.2)
Assuming (D.1) and (D.2) it is straightforward to show that any eigenvalue $E$
of the Hamiltonian must be real:
$E\lambda\ket{\psi}=H\mathcal{PT}\ket{\psi}=\mathcal{PT}H\ket{\psi}=\mathcal{PT}E\ket{\psi}=\mathcal{PT}E\mathcal{PT}\mathcal{PT}\ket{\psi}=E^{*}\lambda\ket{\psi}.$
We now argue that the $M(3,7)$ CFT has unbroken $\mathcal{PT}$ in the sense of
(D.1) and (D.2). First, we note that the eigenvalues of the CFT Hamiltonian on
the cylinder are real. Second, the CFT states are eigenstates of the
$\mathcal{PT}$ operator defined in 6, with eigenvalues given in Tab. 3 (the
descendants of a given conformal family have the same eigenvalues as their
primary operator). Together these two facts imply (D.1), since:
$\mathcal{PT}H_{\text{CFT}}\ket{\mathcal{O}_{\text{CFT}}}=\mathcal{PT}E_{\mathcal{O}}\ket{\mathcal{O}_{\text{CFT}}}=E_{\mathcal{O}}(-1)^{\text{PT}_{\mathcal{O}}}\ket{\mathcal{O}_{\text{CFT}}}=H_{CFT}\mathcal{PT}\ket{\mathcal{O}_{\text{CFT}}},$
Where $E_{\mathcal{O}}=(\Delta_{\mathcal{O}}-c/12)/R$. Since this is true for
all CFT states, the commutator vanishes identically. 242424One may ask why we
want to prove $\mathcal{PT}$ symmetry in the UV CFT, when we already have a
Hermitian Hamiltonian and therefore a real spectrum. However this is necessary
to show the full deformed Hamiltonian $H_{\text{CFT}}+\sum_{i}V_{i}$ is
$\mathcal{PT}$ invariant, and isn’t a trivial fact since there exists
Hermitian Hamiltonians without $\mathcal{PT}$ symmetry.
Now we discuss the deformations to the UV CFT. The primary $\phi_{1,5}$ is
even under $\mathcal{PT}$ while $\phi_{1,3}$ is odd, so adding to the CFT
Hamiltonian $g_{5}\phi_{1,5}+ig_{3}\phi_{1,3}$ will maintain condition (D.1)
along the RG flow (with $g_{3},g_{5}\in\mathbb{R}$). Therefore, a real
spectrum along the RG flow is now equivalent to condition (D.2) being
satisfied. It is clear that close to the UV fixed point, conformal
perturbation theory will always predict real energy levels, so we expect (D.2)
to hold for small coupling. However, at strong coupling (D.2) may no longer
hold (see for example an illustrative two dimensional quantum mechanical toy
model in [38]), in which case the spectrum will develop complex eigenvalues.
Complex eigenvalues come in conjugate pairs, and the corresponding Hamiltonian
eigenstates are related under $\mathcal{PT}$: 252525The phases of the H
eigenstates $\ket{E}$ and $\ket{E^{*}}$ can always be chosen such that (D.3)
holds.
$\mathcal{PT}\ket{E}=\ket{E^{*}},$ (D.3)
where $H\ket{E}=E\ket{E},\quad H\ket{E^{*}}=E^{*}\ket{E^{*}}$. Equation (D.3)
can be shown to hold with a very similar argument to the one used to show the
reality of the spectrum under unbroken $\mathcal{PT}$ symmetry:
$H\mathcal{PT}\ket{E}=\mathcal{PT}H\ket{E}=\mathcal{PT}E\ket{E}=E^{*}\mathcal{PT}\ket{E}\implies\mathcal{PT}\ket{E}\propto\ket{E^{*}},$
Where we have only used (D.1) and the properties of the $\mathcal{PT}$
operator.
In summary, the Hamiltonian of our QFT is $\mathcal{PT}$ invariant, which
implies either a real spectrum or complex conjugate pairs of eigenvalues. For
small values of the couplings, where conformal perturbation theory accurately
predicts the spectrum, $\mathcal{PT}$ symmetry is unbroken, hence the spectrum
is real and energy eigenstates can be defined to be invariant under
$\mathcal{PT}$. Finally, in some regions of the two dimensional phase diagram
spanned by the two couplings, we may expect $\mathcal{PT}$ symmetry to be
broken. This entails pairs of eigenvalues becoming complex conjugates of each
other, and in that case the corresponding eigenvectors get mapped to each
other under $\mathcal{PT}$.
## References
* [1] A. L. Fitzpatrick and E. Katz, “Snowmass White Paper: Hamiltonian Truncation,” arXiv:2201.11696 [hep-th].
* [2] A. J. A. James, R. M. Konik, P. Lecheminant, N. J. Robinson, and A. M. Tsvelik, “Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods,” arXiv:1703.08421 [cond-mat.str-el].
* [3] J. Elias Miro and J. Ingoldby, “Hamiltonian Truncation with larger dimensions,” JHEP 05 (2022) 151, arXiv:2112.09049 [hep-th].
* [4] P. Giokas and G. Watts, “The renormalisation group for the truncated conformal space approach on the cylinder,” arXiv:1106.2448 [hep-th].
* [5] M. Hogervorst, S. Rychkov, and B. C. van Rees, “Truncated conformal space approach in $d$ dimensions: A cheap alternative to lattice field theory?,” Phys. Rev. D91 (2015) 025005, arXiv:1409.1581 [hep-th].
* [6] S. Rychkov and L. G. Vitale, “Hamiltonian truncation study of the $\phi^{4}$ theory in two dimensions,” Phys. Rev. D91 (2015) 085011, arXiv:1412.3460 [hep-th].
* [7] M. Lencses and G. Takacs, “Confinement in the $q$-state Potts model: an RG-TCSA study,” JHEP 09 (2015) 146, arXiv:1506.06477 [hep-th].
* [8] S. Rychkov and L. G. Vitale, “Hamiltonian truncation study of the $\phi^{4}$ theory in two dimensions. II. The $\mathbb{Z}_{2}$ -broken phase and the Chang duality,” Phys. Rev. D 93 no. 6, (2016) 065014, arXiv:1512.00493 [hep-th].
* [9] J. Elias-Miro, M. Montull, and M. Riembau, “The renormalized Hamiltonian truncation method in the large $E_{T}$ expansion,” JHEP 04 (2016) 144, arXiv:1512.05746 [hep-th].
* [10] J. Elias-Miro, S. Rychkov, and L. G. Vitale, “High-Precision Calculations in Strongly Coupled Quantum Field Theory with Next-to-Leading-Order Renormalized Hamiltonian Truncation,” JHEP 10 (2017) 213, arXiv:1706.06121 [hep-th].
* [11] J. Elias-Miro, S. Rychkov, and L. G. Vitale, “NLO Renormalization in the Hamiltonian Truncation,” Phys. Rev. D96 no. 6, (2017) 065024, arXiv:1706.09929 [hep-th].
* [12] T. Cohen, K. Farnsworth, R. Houtz, and M. A. Luty, “Hamiltonian Truncation Effective Theory,” SciPost Phys. 13 no. 2, (2022) 011, arXiv:2110.08273 [hep-th].
* [13] J. Elias Miro and J. Ingoldby, “Effective Hamiltonians and Counterterms for Hamiltonian Truncation,” JHEP 07 (2023) 052, arXiv:2212.07266 [hep-th].
* [14] M. K. Lájer and R. M. Konik, “Krylov Spaces for Truncated Spectrum Methodologies,” arXiv:2308.00277 [hep-th].
* [15] V. P. Yurov and Al. B. Zamolodchikov, “Truncated Conformal Space Approach to Scaling Lee-Yang Model,” Int.J.Mod.Phys. A5 (1990) 3221–3246.
* [16] V. P. Yurov and A. B. Zamolodchikov, “Correlation functions of integrable 2-D models of relativistic field theory. Ising model,” Int. J. Mod. Phys. A 6 (1991) 3419–3440.
* [17] G. Mussardo, Statistical Field Theory. Oxford Graduate Texts. Oxford University Press, 3, 2020.
* [18] S.-H. Shao, “What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetry,” arXiv:2308.00747 [hep-th].
* [19] N. Anand, A. L. Fitzpatrick, E. Katz, Z. U. Khandker, M. T. Walters, and Y. Xin, “Introduction to Lightcone Conformal Truncation: QFT Dynamics from CFT Data,” arXiv:2005.13544 [hep-th].
* [20] H. Chen, A. L. Fitzpatrick, E. Katz, and Y. Xin, “Giving Hamiltonian Truncation a Boost,” arXiv:2207.01659 [hep-th].
* [21] A. B. Zamolodchikov, “From tricritical Ising to critical Ising by thermodynamic Bethe ansatz,” Nucl. Phys. B 358 (1991) 524–546.
* [22] M. Lassig, G. Mussardo, and J. L. Cardy, “The scaling region of the tricritical Ising model in two-dimensions,” Nucl. Phys. B 348 (1991) 591–618.
* [23] A. B. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,” JETP Lett. 43 (1986) 730–732.
* [24] D. Karateev, “Two-point functions and bootstrap applications in quantum field theories,” JHEP 02 (2022) 186, arXiv:2012.08538 [hep-th].
* [25] H.-L. Xu and A. Zamolodchikov, “2D Ising Field Theory in a magnetic field: the Yang-Lee singularity,” JHEP 08 (2022) 057, arXiv:2203.11262 [hep-th].
* [26] F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories,” Nucl. Phys. B 915 (2017) 363–383, arXiv:1608.05499 [hep-th].
* [27] A. Cavaglià, S. Negro, I. M. Szécsényi, and R. Tateo, “$T\bar{T}$-deformed 2D Quantum Field Theories,” JHEP 10 (2016) 112, arXiv:1608.05534 [hep-th].
* [28] J. Elias-Miró. https://media.scgp.stonybrook.edu/presentations/2023/20231106_Elias-Miro.pdf.
* [29] P. Dorey, C. Dunning, and R. Tateo, “New families of flows between two-dimensional conformal field theories,” Nucl. Phys. B 578 (2000) 699–727, arXiv:hep-th/0001185.
* [30] M. Lencsés, A. Miscioscia, G. Mussardo, and G. Takács, “Multicriticality in Yang-Lee edge singularity,” JHEP 02 (2023) 046, arXiv:2211.01123 [hep-th].
* [31] M. Lencsés, A. Miscioscia, G. Mussardo, and G. Takács, “$\mathcal{PT}$ breaking and RG flows between multicritical Yang-Lee fixed points,” JHEP 09 (2023) 052, arXiv:2304.08522 [cond-mat.stat-mech].
* [32] O. A. Castro-Alvaredo, B. Doyon, and F. Ravanini, “Irreversibility of the renormalization group flow in non-unitary quantum field theory,” J. Phys. A 50 no. 42, (2017) 424002, arXiv:1706.01871 [hep-th].
* [33] J. Elias-Miró and E. Hardy, “Exploring Hamiltonian Truncation in $\bf{d=2+1}$,” Phys. Rev. D 102 no. 6, (2020) 065001, arXiv:2003.08405 [hep-th].
* [34] N. Anand, E. Katz, Z. U. Khandker, and M. T. Walters, “Nonperturbative dynamics of (2+1)d $\phi^{4}$-theory from Hamiltonian truncation,” JHEP 05 (2021) 190, arXiv:2010.09730 [hep-th].
* [35] V. S. Dotsenko and V. A. Fateev, “Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models,” Nucl. Phys. B 240 (1984) 312.
* [36] V. S. Dotsenko and V. A. Fateev, “Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c $<$ 1,” Nucl. Phys. B 251 (1985) 691–734.
* [37] P. Fonseca and A. Zamolodchikov, “Ising field theory in a magnetic field: Analytic properties of the free energy,” J. Stat. Phys. 110 (2003) 527–590, arXiv:hep-th/0112167 [hep-th].
* [38] C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rept. Prog. Phys. 70 (2007) 947, arXiv:hep-th/0703096. |
Institut für Logic and Computation, Technische Universität Wien, Austria and
<EMAIL_ADDRESS>by the
European Union Horizon 2020 Marie Skłodowska-Curie grant No 689176 and FWF
projects I 2982, Y544-N2, and W1255-N23. Research School of Computer Science,
The Australian National University, Australia Research School of Computer
Science, The Australian National University, Australia Research School of
Computer Science, The Australian National University, Australia Tim Lyon and
Alwen Tiu and Rajeev Góre and Ranald Clouston[100]Theory of Computation Proof
Theory [100]Theory of Computation Automated ReasoningMaribel Fernández and
Anca Muscholl 2 28th EACSL Annual Conference on Computer Science Logic (CSL
2020) CSL 2020 CSL 2020 January 13–16, 2020 Barcelona, Spain 152 28
# Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via
Nested Sequents
Tim Lyon Alwen Tiu Rajeev Goré Ranald Clouston
###### Abstract
We provide a direct method for proving Craig interpolation for a range of
modal and intuitionistic logics, including those containing a “converse”
modality. We demonstrate this method for classical tense logic, its extensions
with path axioms, and for bi-intuitionistic logic. These logics do not have
straightforward formalisations in the traditional Gentzen-style sequent
calculus, but have all been shown to have cut-free nested sequent calculi. The
proof of the interpolation theorem uses these calculi and is purely syntactic,
without resorting to embeddings, semantic arguments, or interpreted
connectives external to the underlying logical language. A novel feature of
our proof includes an orthogonality condition for defining duality between
interpolants.
###### keywords:
Bi-intuitionistic logic, Interpolation, Nested calculi, Proof theory,
Sequents, Tense logics
###### category:
## 1 Introduction
The Craig interpolation property for a logic $\mathbf{L}$ states that if
$A\Rightarrow B\in\mathbf{L}$, then there exists a formula $C$ in the language
of $\mathbf{L}$ such that $A\Rightarrow C\in\mathbf{L}$ and $C\Rightarrow
B\in\mathbf{L}$, and every propositional variable appearing in $C$ appears in
$A$ and $B$. This property has many useful applications: it can be used to
prove Beth definability [11]; in computer-aided verification it can be used to
split a large problem involving $A\Rightarrow B$ into smaller problems
involving $A\Rightarrow C$ and $C\Rightarrow B$ [17]; and in knowledge
representation (uniform) interpolation can be used to conceal or forget
irrelevant or confidential information in ontology querying [15]. Therefore,
demonstrating that a logic possesses the Craig interpolation property is of
practical value.
Interpolation can be proved semantically or syntactically. In the semantic
method, $\mathbf{L}$ is the set of valid formulae, thereby requiring a
semantics for $\mathbf{L}$. In the syntactic method, often known as Maehara’s
method [16], $\mathbf{L}$ is the set of theorems, thereby requiring a proof-
calculus. The syntactic approach constructs the interpolant $C$ by induction
on the (usually cut-free) derivation of $A\Rightarrow B$, and usually also
provides derivations witnessing $A\Rightarrow C$ and $C\Rightarrow B$.
Over the past forty years, Gentzen’s original sequent calculus has been
extended in many different ways to handle a plethora of logics. The four main
extensions are hypersequent calculi [2], display calculi [7], nested sequent
calculi [1, 10, 20], and labelled calculi [18]. Various interpolation results
have been found using these calculi but the only general methodology that we
know of is the recent work of Kuznets [13] with Lellman [14]. Although they
use extended sequent calculi, binary-relational Kripke semantical arguments
are crucial for their methodology, and extending their method to other
semantics is left as further work. They also construct the interpolants using
a language containing (interpreted) meta-level connectives which are external
to the logic at hand, and do not handle logics containing converse modalities
such as tense logic. Finally, their method does not yield derivations
witnessing $A\Rightarrow C$ and $C\Rightarrow B$.
We give a general, purely syntactic, methodology for proving Craig
interpolation using nested sequent calculi for a variety of propositional,
non-classical logics including normal tense logics, their extensions with path
axioms, and bi-intuitionistic logic. Our methodology does not utilise
semantics, does not embed one logic into another, and does not utilise logical
connectives which are external to the underlying logical language.
The first novelty of our approach is a generalisation of the notion of
interpolant from formulas to sets of sequents. The second is a notion of
orthogonality which gives rise to a notion of duality via cut: if two
interpolants are orthogonal, then the empty sequent is derivable from the
sequents in the interpolants using only the cut and the contraction rules.
This duality via cut allows us to relate our more general notion of
interpolants (as sets of sequents) to the usual notion of interpolants (as
formulas). Moreover, given a derivation of $A\Rightarrow B$, our orthogonality
condition not only allows us to construct the interpolant $C$, but also the
derivations witnessing $A\Rightarrow C$ and $C\Rightarrow B$. This fact shows
that our approach possesses a distinct complexity-theoretic advantage over the
semantic approach: to verify that $C$ is indeed the interpolant of
$A\Rightarrow B$, one need only check the derivations of $A\Rightarrow C$ and
$C\Rightarrow B$, which is a PTIME process. In the semantic approach, to
verify that $A\Rightarrow C$ and $C\Rightarrow B$ are indeed valid (and that
$C$ is in fact an interpolant of $A\Rightarrow B$) one must construct proofs
of the implications, which is generally much harder (e.g., finding a proof of
a validity in one of the tense logics presented in Sec. 3 is PSPACE complete).
Related work. Interpolation has been heavily investigated in the description
logic community, where it is used to hide or forget information [22]. In this
setting, the logic ALC is a syntactic variant of the multimodal normal modal
logic Kn while its extension with inverse roles, ALCI, is a variant of the
multimodal normal tense logic Ktn. Cate et al [22] utilise a complexity-
optimal tableau algorithm to prove interpolation for ALC via Maehara’s method.
They then embed ALCI into ALC and extend their interpolation result to ALCI.
By contrast, our methodology is direct: we obtain interpolation for the normal
tense logic Kt, and can then extract interpolation for the normal modal logic
K by simply observing that our nested sequent calculus obeys the separation
property: if the end-sequent $\vdash A\rightarrow B$ contains no occurrences
of the black (converse) modalities, then neither does the interpolant.
As mentioned earlier, the work of Kuznets et al. [6, 13, 14] on interpolation
for modal logics in nested sequent calculi is closest to ours. Our
construction of interpolants for tense logics shares some similarity with
theirs. One crucial difference is that our interpolants are justified purely
through syntactic and proof-theoretic means, whereas their interpolants are
justified via semantic arguments. Another important difference is that our
method extends to the bi-modal case and also (bi-)intuitionistic case, and it
is straightforward to adapt our work to the multi-modal case, e.g., using
nested sequent calculi as in [23]. Kowalski and Ono [12] showed interpolation
for bi-intuitionistic logic using a sequent calculus with analytic cut. In
contrast, our proof is based on a cut-free nested sequent calculus [8].
Outline of the paper. In Sec. 2 we give a brief overview of a typical
interpolation proof using the traditional sequent calculus, and highlight some
issues of extending it to nested sequent calculi, which motivates the
generalisation of the interpolation theorem we adopt in this paper. In Sec. 3
we show how the generalised notion of interpolants can be used to prove the
Craig interpolation theorem for classical tense logic and its extensions with
path axioms [9], covering all logics in the modal cube and more. We then show
how our approach can be extended to bi-intuitionistic logic in Sec. 4. In Sec.
5 we conclude and discuss future work.
## 2 Overview of our approach
We analyze a typical syntactic interpolation proof for Gentzen sequents,
highlight the issues of extending it to nested sequents, and motivate our
syntactic approach for interpolation.
Consider, for example, a two-sided sequent calculus for classical logic such
as G3c [24]. Interpolation holds when we can prove that for all
$\Gamma_{1},\Gamma_{2},\Delta_{1},\Delta_{2}$, if
${\Gamma_{1},\Gamma_{2}}\vdash{\Delta_{1},\Delta_{2}}$ is provable in G3c,
then so are both ${\Gamma_{1}}\vdash{\Delta_{1},C}$ and
${C,\Gamma_{2}}\vdash{\Delta_{2}}$, for some $C$ containing only propositional
variables common to both $\Gamma_{1},\Delta_{1}$ and $\Gamma_{2},\Delta_{2}$.
The inductive construction of $C$ can be encoded via inference rules over more
expressive sequents that specify the splitting of the contexts and the
interpolant constructed thus far. In G3c, we write
${\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{C}$ to
denote the sequent ${\Gamma_{1},\Gamma_{2}}\vdash{\Delta_{1},\Delta_{2}}$ with
its context split into $\Gamma_{1}\vdash\Delta_{1}$ and
$\Gamma_{2}\vdash\Delta_{2}$, and with $C$ the interpolant. Inference rules
for this extended sequent are similar to the usual ones, with variations
encoding the different ways the contexts may be split. For example, the
initial rule ${\Gamma,p}\vdash{p,\Delta}$ has the following four variants
corresponding to the four splittings of where $p$ can occur (with four
different interpolants!):
${\Gamma_{1},p\mid\Gamma_{2}}\vdash{p,\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\bot}\qquad{\Gamma_{1},p\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
p,\Delta_{2}}\mathrel{\|}{p}$
${\Gamma_{1}\mid\Gamma_{2},p}\vdash{p,\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\neg
p}\qquad{\Gamma_{1}\mid\Gamma_{2},p}\vdash{\Delta_{1}\mid
p,\Delta_{2}}\mathrel{\|}{\top}$
Branching rules, such as the right-introduction rule for $\land$, split into
two variants, depending on whether the principal formula is in the first or
the second partition of the context:
${\Gamma_{1}\mid\Gamma_{2}}\vdash{A\land
B,\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{C\lor
D}\lx@proof@logical@and{\Gamma_{1}\mid\Gamma_{2}}\vdash{A,\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{C}{\Gamma_{1}\mid\Gamma_{2}}\vdash{B,\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{D}$
${\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2},A\land
B}\mathrel{\|}{C\land
D}\lx@proof@logical@and{\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2},A}\mathrel{\|}{C}{\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2},B}\mathrel{\|}{D}$
Observe that the interpolants of the conclusion sequents are composed from the
interpolants of the premises, but with the main connectives dual to one
another: a disjunction in the $\land_{R_{1}}$ rule and a conjunction in the
$\land_{R_{2}}$ rule. These observations also apply for the other rules of
G3c, with a slight subtlety for the implication-left rule: see [24].
Interpolation for G3c can then be proved by a straightforward induction on the
height of proofs.
Below we discuss some issues with extending this approach to proving
interpolation for modal/tense logics and bi-intuitionistic logic using nested
sequent calculi, and how these issues lead to the generalisation of the
intermediate lemmas we need to prove (which amounts to an interpolation
theorem for sequents, rather than formulae).
### Classical modal and tense logics
A nested sequent [10] can be seen as a tree of traditional Gentzen-style
sequents. For classical modal logics, single-sided sequents suffice, so a
nested sequent in this case can be seen as a nested multiset: i.e. a multiset
whose elements can be formulae or multisets. Following the notation in [9], a
sequent nested inside another sequent is prefixed with a $\circ$, which is the
structural proxy for the $\Box$ modal operator. For example, the nested
sequent below first left, with two sub-sequents $\\{c,d\\}$ and $\\{e,f\\}$,
represents the formula shown second left:
${}\vdash{\\{a,b,\circ\\{c,d\\},\circ\\{e,f\\}\\}}\qquad a\lor
b\lor\square(c\lor d)\lor\square(e\lor f)\qquad{}\vdash{\Gamma,\lozenge
A,\circ\\{\Delta\\}}{}\vdash{\Gamma,\circ\\{A,\Delta\\}}\qquad{}\vdash{\lozenge\neg
p,\lozenge p,\circ\\{q\\}}{}\vdash{\lozenge\neg p,\circ\\{p,q\\}}$
Nested sequent calculi for modal logics [1, 9, 10] typically contain the
propagation rule for diamond shown third left above which “propagates” the $A$
into the scope of $\circ$, when read upwards. Propagation rules complicate the
adaptation of the interpolation proof from traditional Gentzen sequent
calculi. In particular, it is not sufficient to partition a context into two
disjoint multisets. That is, suppose a nested sequent
${}\vdash{\Gamma,\Delta}$ is provable, and we would like to construct an
interpolant $C$ such that ${}\vdash{\Gamma,C}$ and
${}\vdash{\overline{C},\Delta}$ are provable, where $\overline{C}$ is the
negation normal form of $\neg C$. Suppose the proof of
${}\vdash{\Gamma,\Delta}$ ends with a propagation rule, e.g., when
$\Gamma=\lozenge\neg p,\lozenge p$ and $\Delta=\circ\\{q\\}$ as shown above
far right. In this case, by induction, we can construct an interpolant $D$
such that the splittings ${}\vdash{\lozenge\neg p,D}$ and
${}\vdash{\overline{D},\circ\\{p,q\\}}$ of the premiss are provable, but it is
in general not obvious how to construct the desired interpolant $C$ for the
conclusion ${}\vdash{\lozenge\neg p,\circ\\{p,q\\}}$ from $D$. For this
example,$D$ should be $\square p$, $C$ should be $\square\bot$, which does not
mention $p$ at all.
The above issue with propagation rules suggests that we need to strengthen the
induction hypothesis to construct interpolants, i.e., by considering splitting
the sequent context at every sub-sequent in the nested sequent. For example,
the nested sequent ${}\vdash{\lozenge\neg p,\circ\\{p,q\\}}$ above should be
split into ${}\vdash{\lozenge\neg p,\circ\\{p\\}}$ and
${}\vdash{\circ\\{q\\}}$ when applying the induction hypothesis. Then,
$D=C=\square\bot$ is indeed an interpolant: both ${}\vdash{\lozenge\neg
p,\circ\\{p\\},\square\bot}$ and ${}\vdash{\lozenge\top,\circ\\{q\\}}$ are
provable. Nevertheless, employing a formula interpolant is not enough to push
through the inductive argument in general. Consider, for example, the nested
sequent ${}\vdash{\circ\\{p,\neg p\\}}$, which is provable with an identity
rule, and its partition ${}\vdash{\circ\\{p\\}}$ and ${}\vdash{\circ\\{\neg
p\\}}$. There is no formula $C$ such that both ${}\vdash{\circ\\{p\\},C}$ and
${}\vdash{\overline{C},\circ\\{\neg p\\}}$ are provable. One solution to this
problem is to generalise the interpolation statement to consider a nested
sequent as an interpolant: If a nested sequent ${}\vdash{\Gamma}$ is provable,
then for every ‘partitioning’ of ${}\vdash{\Gamma}$ into
${}\vdash{\Gamma_{1}}$ and ${}\vdash{\Gamma_{2}}$ (where the partitioning
applies to every sub-sequent in a nested sequent; the precise definition will
be given in subsequent sections), there exists ${}\vdash{\Delta}$ (the
interpolant), ${}\vdash{\Gamma_{1}^{\prime}}$ and
${}\vdash{\Gamma_{2}^{\prime}}$ such that
1. 1.
The propositional variables occuring in ${}\vdash{\Delta}$ are in both
${}\vdash{\Gamma_{1}}$ and ${}\vdash{\Gamma_{2}}$,
2. 2.
${}\vdash{\Gamma_{1}^{\prime}}$ splits into ${}\vdash{\Gamma_{1}}$ and
${}\vdash{\Delta}$, and ${}\vdash{\Gamma_{2}^{\prime}}$ splits into
${}\vdash{\Gamma_{2}}$ and ${}\vdash{\overline{\Delta}}$, where
${}\vdash{\overline{\Delta}}$ denotes the nested sequent ${}\vdash{\Delta}$
with all formula occurrences replaced with their negations, and
3. 3.
Both ${}\vdash{\Gamma_{1}^{\prime}}$ and ${}\vdash{\Gamma_{2}^{\prime}}$ are
provable.
For example, the nested sequent ${}\vdash{\circ\\{p,\neg p\\}}$, with
partitions ${}\vdash{\circ\\{p\\}}$ and ${}\vdash{\circ\\{\neg p\\}}$, has the
interpolant ${}\vdash{\Delta}={}\vdash{\circ\\{\neg p\\}}$ (hence
${}\vdash{\overline{\Delta}}={}\vdash{\circ\\{p\\}}$), and
${}\vdash{\Gamma_{1}^{\prime}}={}\vdash{\Gamma_{2}^{\prime}}={}\vdash{\circ\\{p,\neg
p\\}}.$
One remaining issue is that, since we now use a nested sequent as an
interpolant, the composition of interpolants needs to be adjusted as well.
Recall that in the construction of interpolants for G3c above, in the case
involving the right-introduction for $\land$, we constructed either $C\lor D$
or $C\land D$ as the interpolant for the conclusion. If $C$ and $D$ are nested
sequents, the expression $C\lor D$ or $C\land D$ would not be well-formed. To
solve this remaining issue, we generalise the interpolant further to be a set
of nested sequents.
Fitting and Kuznets [6] similarly generalise the notion of interpolants, but
instead of generalising interpolants to a set of (nested) sequents, they
introduce ‘meta’ connectives for conjunction and disjunction, applicable only
to interpolants, and justified semantically. Our notion of interpolants
requires no new logical operators or semantical notions.
### Propositional bi-intuionistic logic
Bi-intuitionistic logic is obtained from intuitionistic logic by adding a
subtraction (or exclusion) connective $-\\!\\!\\!<$ that is dual to
implication. Its introduction rules are the mirror images of those for
implication; in the traditional sequent calculus, these take the form:
${A-\\!\\!\\!<B}\vdash{\Delta}{A}\vdash{B,\Delta}\qquad{\Gamma}\vdash{\Delta,A-\\!\\!\\!<B}\lx@proof@logical@and{\Gamma}\vdash{\Delta,A}{\Gamma,B}\vdash{\Delta}$
However, as shown in [19], the cut rule cannot be entirely eliminated in a
sequent calculus featuring these rules, although they can be restricted to
analytic cuts [12]. In [8], Postniece et al. show how bi-intuitionistic logic
can be formalised in a nested sequent calculus. Although interpolation holds
for intuitionistic logic, it does not generalise straightforwardly to bi-
intuitionistic logic, and only very recently has interpolation for bi-
intuitionistic logic been shown [12]. The proof for the interpolation theorem
for intuitionistic logic is very similar to the proof of the same theorem for
classical logic; one simply needs to restrict the partitioning of the sequent
to the form ${\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}$ where
$\Delta_{1}$ is empty and $\Delta_{2}$ contains at most one formula
occurrence. Since the (nested) sequent calculus for bi-intuitionistic logic
uses multiple-conclusion (nested) sequents, the proof for intuitionistic logic
cannot be adapted to the bi-intutionistic case. The problem already shows up
in the very simple case involving the identity rule: suppose we have a proof
of the initial sequent ${p}\vdash{p}$ and we want to partition the sequent as
${\cdot\mid p}\vdash{p\mid\cdot}$. It is not possible to find an interpolant
$C$ such that ${\cdot}\vdash{p,C}$ and ${C,p}\vdash{\cdot}$ (otherwise, one
would be able to prove the excluded middle $p\lor(p\supset\bot)$, which is not
valid in bi-intuitionistic logic, using the cut formula $p\lor C$). In
general, the inductive construction of the interpolant for $A\supset B$ may
involve finding an interpolant $C$ for the problematic partition of the form
${\cdot\mid\Gamma}\vdash{\Delta\mid\cdot}$, where $\Delta$ is non-empty. This
case does not arise in the interpolation proof for intuitionistic logic in
[24], due to the restriction to single-conclusion sequents.
We show that the above issue with bi-intutionistic logic can be solved using
the same approach as in modal logic: simply extend the interpolant to a set of
nested sequents. In particular, for ${\cdot\mid p}\vdash{p\mid\cdot}$, the
generalised interpolation statement only requires finding an interpolating
sequent ${\Gamma}\vdash{\Delta}$ and its ‘dual’
${\Gamma^{\prime}}\vdash{\Delta^{\prime}}$ (see below) such that both
${\Gamma}\vdash{p,\Delta}$ and ${\Gamma^{\prime},p}\vdash{\Delta^{\prime}}$
are provable, which is achieved by letting $\Gamma=\\{p\\}$,
$\Delta=\\{\leavevmode\nobreak\ \\}$, $\Gamma^{\prime}=\\{\leavevmode\nobreak\
\\}$ and $\Delta^{\prime}=\\{p\\}.$
### Interpolating sequents and orthogonality
In a simplified form (e.g., sequent calculus), the generalised interpolation
result we show can be roughly summarised as follows: given a provable sequent
${\Gamma_{1},\Gamma_{2}}\vdash{\Delta_{1},\Delta_{2}}$, there exist two sets
of sequents $\mathcal{I}$ and $\mathcal{I}^{\prime}$ such that
1. 1.
For every sequent $({\Sigma}\vdash{\Theta})\in\mathcal{I}$, the sequent
${\Gamma_{1},\Sigma}\vdash{\Delta_{1},\Theta}$ is provable,
2. 2.
For every sequent
$({\Sigma^{\prime}}\vdash{\Theta^{\prime}})\in\mathcal{I}^{\prime}$, the
sequent ${\Gamma_{2},\Sigma^{\prime}}\vdash{\Delta_{2},\Theta^{\prime}}$ is
provable,
3. 3.
The propositional variables in $\mathcal{I}$ and $\mathcal{I}^{\prime}$ occur
in both ${\Gamma_{1}}\vdash{\Delta_{1}}$ and ${\Gamma_{2}}\vdash{\Delta_{2}}$,
and
4. 4.
The sequents in $\mathcal{I}$ and $\mathcal{I}^{\prime}$ are orthogonal to
each other, that is, the empty sequent ${}\vdash{}$ is derivable from all
sequents in $\mathcal{I}\cup{\mathcal{I}^{\prime}}$ using only the cut rule
and possibly structural rules (contraction and/or weakening).
The set $\mathcal{I}$ is taken to be the (sequent) interpolant.
Last, the orthogonality condition, can be seen as a generalisation of duality.
To see how this is the case, consider a degenerate case where
${\Gamma_{1},\Gamma_{2}}\vdash{\Delta_{1},\Delta_{2}}$ is a classical sequent
(e.g., in G3c). We show how one can convert a formula interpolant in the usual
definition (i.e., formula $C$ s.t. ${\Gamma_{1}}\vdash{C,\Delta_{1}}$ and
${\Gamma_{2},C}\vdash{\Delta_{2}}$ are provable) to a sequent interpolant
satisfying the four conditions above, and vice-versa. For the forward
direction, simply let $\mathcal{I}=\\{{}\vdash{C}\\}$ and
$\mathcal{I}^{\prime}=\\{{C}\vdash{}\\}$. It is easy to see that $\mathcal{I}$
is orthogonal to $\mathcal{I}^{\prime}.$ For the converse direction, suppose
we have a sequent interpolant $\mathcal{I}$ and its orthogonal
$\mathcal{I}^{\prime}$. We illustrate how one can construct a formula
interpolant $C$. To simplify the discussion, let us assume that
$\mathcal{I}=\\{({p,q}\vdash{r,s})\\}\quad{\mathcal{I}^{\prime}}=\\{({}\vdash{p}),\leavevmode\nobreak\
({}\vdash{q}),\leavevmode\nobreak\ ({r}\vdash{})\leavevmode\nobreak\
,({s}\vdash{\leavevmode\nobreak\ })\\}$
and that the following sequents are provable:
$(1)\leavevmode\nobreak\
{\Gamma_{1},p,q}\vdash{r,s,\Delta_{1}}\quad(2)\leavevmode\nobreak\
{\Gamma_{2}}\vdash{\Delta_{2},p}\quad(3)\leavevmode\nobreak\
{\Gamma_{2}}\vdash{\Delta_{2},q}\quad(4)\leavevmode\nobreak\
{\Gamma_{2},r}\vdash{\Delta_{2}}\quad(5)\leavevmode\nobreak\
{\Gamma_{2},s}\vdash{\Delta_{2}}$
Let $C=(p\land q)\supset(r\lor s).$ Then it is easy to see that
${\Gamma_{1}}\vdash{\Delta_{1},C}$ is provable given (1), and
${\Gamma_{2},C}\vdash{\Delta_{2}}$ is provable given (2) - (5). The formal
statement and the proof of the generalised interpolation theorem will be
discussed in detail in the next two sections.
### A note on notation
In what follows, we adopt a representation of nested sequents using restricted
labelled sequents where we use the labels and relational atoms to encode the
tree structure of a nested sequent. To clarify what we mean, consider the
following nested sequent for tense logic [9]:
$A,B,\circ\\{C,D\\},\circ\\{E,F,\bullet\\{G,H\\},\circ\\{I\\}\\}$
Graphically, the nested sequent can be represented as a tree (shown below
left) with two types of edges $\overset{\circ}{\rightarrow}$ and
$\overset{\bullet}{\rightarrow}$. Alternatively, the nested sequent can be
represented as the polytree shown below right with a single type of edge
$\overset{}{\rightarrow}$ and where the orientation of the edge encodes the
two types of structures $\circ{\\{}\\}$ and $\bullet{\\{}\\}$ of the nested
sequent (observe that the $\bullet$-edge from $E,F$ to $G,H$ in the left
diagram has been reversed in the right diagram).111A polytree is a directed
graph such that its underlying graph—the graph obtained by ignoring the
orientation of the edges—is a tree.
$\textstyle{A,B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\scriptstyle{\circ}$$\textstyle{C,D}$$\textstyle{E,F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\scriptstyle{\circ}$$\textstyle{G,H}$$\textstyle{I}$ | $\textstyle{A,B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C,D}$$\textstyle{E,F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G,H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I}$
---|---
In the latter representation, the structure of the nested sequent can be
encoded using a single binary relation: we label each node of the tree
corresponding to the nested sequent (as shown above left) with unique labels
$x$, $y$, $z$, $\ldots$, encode each edge $x\overset{\circ}{\rightarrow}y$
from a label $x$ to a label $y$ with a relation $Rxy$, and encode each edge
$x\overset{\bullet}{\rightarrow}y$ with a relation $Ryx$ [3]. The above nested
sequent can then be equivalently represented as a labelled sequent where
$\mathcal{R}=\\{Ruv,Ruw,Rxw,Rwy\\}$ and $R$ is a relational symbol:
${\mathcal{R}}\vdash{u:A,u:B,v:C,v:D,w:E,w:F,x:G,x:H,y:I}$
Inference rules in a nested sequent calculus can be trivially encoded as rules
in a restricted labelled calculus seen as a ‘data structure’ rather than a
proper labelled sequent calculus.
We stress that our labelled notation to represent nested sequents is just a
matter of presentation: the labelled representation is notationally simpler
for presenting inference rules and composing nested sequents. For instance,
the operation of merging two nested sequents with isomorphic shapes is simply
the union of the multiset of labelled formulae.
## 3 Interpolation for Tense Logics
$\overline{A}\vee\square\blacklozenge
A\qquad\overline{A}\vee\blacksquare\lozenge
A\qquad\lozenge(A\wedge\overline{B})\vee\lozenge\overline{A}\vee\square
B\qquad\blacklozenge(A\wedge\overline{B})\vee\blacklozenge\overline{A}\vee\blacksquare
B\qquad\square AA\qquad\blacksquare AA$
Figure 1: The minimal tense logic $\mathsf{Kt}$ consists of all classical
propositional tautologies, modus ponens, and is additionally extended with the
above axioms and inference rules.
As usual, we interpret $\square A$ as saying that $A$ holds at every point in
the immediate future, and $\lozenge A$ as saying that $A$ holds at some point
in the immediate future. Conversely, the $\blacksquare$ and $\blacklozenge$
modalities make reference to the past: $\blacksquare A$ says that $A$ holds at
every point in the immediate past, and $\blacklozenge A$ says that $A$ holds
at some point in the immediate past. Last, we take $\overline{p}$ to be the
negation of $p$, and use the notation $[?]\in\\{\square,\blacksquare\\}$ and
$\langle?\rangle\in\\{\lozenge,\blacklozenge\\}$.
We consider tense formulae in negation normal form (nnf) as this simplifies
our calculi while retaining the expressivity of the original language. The
language for the tense logics we consider is given via the following BNF
grammar:
$A::=p\ |\ \overline{p}\ |\ (A\wedge A)\ |\ (A\vee A)\ |\ (\square A)\ |\
(\lozenge A)\ |\ (\blacksquare A)\ |\ (\blacklozenge A).$
Since our language excludes an explicit connective for negation, we define it
formally below (Def. 3.1). Using the definition, we may define an implication
$A\rightarrow B$ to be $\overline{A}\vee B$.
###### Definition 3.1.
For a formula $A$, we define the _negation_ $\overline{A}$ recursively on the
structure of $A$: if $A=p$ then $\overline{A}:=\overline{p}$ and if
$A=\overline{p}$ then $\overline{A}:=p$. The clauses concerning the
connectives are as follows: (1) $\overline{B\land
C}:=\overline{B}\vee\overline{C}$, (2)
$\overline{[?]B}:=\langle?\rangle\overline{B}$, (3) $\overline{B\lor
C}:=\overline{B}\wedge\overline{C}$, and (4) $\overline{\langle?\rangle
B}:=[?]\overline{B}$.
Path axioms are of the form
$[?]_{1}[?]_{2}\cdots[?]_{n}\;\bar{p}\;\vee\langle?\rangle p$ (or,
equivalently, $\langle?\rangle_{1}\cdots\langle?\rangle_{n}p\to\langle?\rangle
p$) with $n\in\mathbb{N}$. See [23] for an overview of path axioms.
The tense logics we consider are all extentions of the minimal tense logic
$\mathsf{Kt}$ (Fig. 1) with path axioms. Thus, $\mathsf{Kt\Pi}$ is the minimal
extension of $\mathsf{Kt}$ with all axioms from the finite set $\mathsf{\Pi}$
of path axioms.
The calculus for $\mathsf{Kt}$, extended with a set of path axioms
$\mathsf{\Pi}$, is given in Fig. 2. Labelled sequents are defined to be
syntactic objects of the form ${\mathcal{R}}\vdash{\Gamma}$, where
$\mathcal{R}$ is a multiset of relational atoms of the form $Rxy$ and $\Gamma$
is a multiset of labelled formulae of the form $x:A$, with $A$ a tense formula
and labels from a countable set $\\{x,y,z,\ldots\\}$.
Note that the side conditions $x\mathcal{R}^{\mathsf{\Pi}}y$ and
$y\mathcal{R}^{\mathsf{\Pi}}x$ of the $\lozenge$ and $\blacklozenge$ rules,
respectively, depend on the set $\mathsf{\Pi}$ of path axioms added to
$\mathsf{Kt}$. The definition of the relation $\mathcal{R}^{\mathsf{\Pi}}$ is
founded upon various auxiliary concepts that fall outside the main scope of
this paper. We therefore refer the interested reader to App. A.6 where the
$\mathcal{R}^{\mathsf{\Pi}}$ relation as well as the concepts needed for its
definition are explicitly provided. See also [9, 23] for details.
###### Lemma 3.2.
The contraction rules $ctr$, the weakening rules $wk$ and $cut_{1}$ are
admissible, and all inference rules are invertible in $\mathsf{Kt\Pi L}$.
###### Proof 3.3.
See Fig. 6 for rules222Note that since $\mathsf{Kt\Pi L}$ uses one sided
sequents, we only consider instances of the rules where labelled formulae
occur solely on the right of the sequent arrow. and [9, 23] for details.
${\mathcal{R}}\vdash{x:\bar{p},x:p,\Delta}\qquad{\mathcal{R}}\vdash{x:A\lor
B,\Delta}{\mathcal{R}}\vdash{x:A,x:B,\Delta}\qquad{\mathcal{R}}\vdash{x:A\land
B,\Delta}\lx@proof@logical@and{\mathcal{R}}\vdash{x:A,\Delta}{\mathcal{R}}\vdash{x:B,\Delta}$
${\mathcal{R}}\vdash{x:\lozenge A,\Delta}{\mathcal{R}}\vdash{x:\lozenge
A,y:A,\Delta}\qquad\qquad{\mathcal{R}}\vdash{x:\square
A,\Delta}{\mathcal{R},Rxy}\vdash{y:A,\Delta}$
${\mathcal{R}}\vdash{x:\blacklozenge
A,\Delta}{\mathcal{R}}\vdash{x:\blacklozenge
A,y:A,\Delta}\qquad\qquad{\mathcal{R}}\vdash{x:\blacksquare
A,\Delta}{\mathcal{R},Ryx}\vdash{y:A,\Delta}$
Figure 2: The calculus $\mathsf{Kt\Pi L}$ for $\mathsf{Kt}$ extended with a
set of path axioms $\mathsf{\Pi}$.
###### Example 3.4.
Consider the formula
$\square\lozenge\overline{q}\rightarrow\square(\lozenge\overline{p}\lor\lozenge\lozenge
p)$, which is a theorem in the logic $\mathsf{Kt\Pi}$ with
$\mathsf{\Pi}=\\{\square\square\overline{p}\vee\lozenge p\\}$. A proof of this
formula is provided in Fig. 3.
${}\vdash{x:\lozenge\square
q\lor\square(\lozenge\overline{p}\lor\lozenge\lozenge
p)}{}\vdash{x:\lozenge\square
q,x:\square(\lozenge\overline{p}\lor\lozenge\lozenge
p)}{Rxy}\vdash{x:\lozenge\square q,y:\lozenge\overline{p}\lor\lozenge\lozenge
p}{Rxy}\vdash{x:\lozenge\square q,y:\lozenge\overline{p},y:\lozenge\lozenge
p}{Rxy}\vdash{x:\lozenge\square q,y:\square
q,y:\lozenge\overline{p},y:\lozenge\lozenge
p}{Rxy,Ryz}\vdash{x:\lozenge\square
q,z:q,y:\lozenge\overline{p},y:\lozenge\lozenge
p}{Rxy,Ryz}\vdash{x:\lozenge\square q,z:\square
q,z:q,y:\lozenge\overline{p},y:\lozenge\lozenge
p}{Rxy,Ryz,Rzw}\vdash{x:\lozenge\square
q,w:q,z:q,y:\lozenge\overline{p},y:\lozenge\lozenge
p}{Rxy,Ryz,Rzw}\vdash{x:\lozenge\square
q,w:q,z:q,y:\lozenge\overline{p},w:\overline{p},y:\lozenge\lozenge
p}{Rxy,Ryz,Rzw}\vdash{x:\lozenge\square
q,w:q,z:q,y:\lozenge\overline{p},w:\overline{p},y:\lozenge\lozenge
p,z:\lozenge p}{Rxy,Ryz,Rzw}\vdash{x:\lozenge\square
q,w:q,z:q,y:\lozenge\overline{p},w:\overline{p},y:\lozenge\lozenge
p,z:\lozenge p,w:p}$
Figure 3: A proof in $\mathsf{Kt\Pi L}$ where
$\mathsf{\Pi}=\\{\square\square\overline{p}\vee\lozenge p\\}$.
As stated in Sec. 2, we extend the notion of an interpolant to a set of nested
sequents. In our definition of interpolants, we are interested only in duality
via cut. In particular, the relational atoms (encoding the tree shape of a
nested sequent) are not explicitly represented in the interpolants since they
can be recovered from the contexts of the sequents in which the interpolants
are used. We therefore define a flat sequent to be a sequent without
relational atoms. For classical tense logic, a flat sequent is thus a multiset
of labelled formulas.
###### Definition 3.5.
An interpolant, denoted $\mathcal{I}$, is a set of flat sequents.
For example, the set below is an interpolant in our context:
$\\{({}\vdash{x:A,y:B,z:W}),({}\vdash{x:C,y:D}),({}\vdash{u:E})\\}.$
Since our interpolant is no longer a formula, we need to define the dual of an
interpolant in order to generalise the statement of the interpolation result
to sequents. We have informally explained in Sec. 2 that duality in this case
is defined via cut. Intuitively, given an interpolant $\mathcal{I}_{1}$, its
_dual_ is any set of nested sequents $\mathcal{I}_{2}$ such that the empty
sequent can be derived from $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ using cut
(possibly with contraction). For example, given
$\mathcal{I}_{1}=\\{({}\vdash{x:A,y:B}),({}\vdash{u:C}),({}\vdash{v:D})\\}$,
there are several candidates for its dual:
$\begin{array}[]{rcl}\mathcal{I}_{2}&=&\\{({}\vdash{x:\overline{A}}),\leavevmode\nobreak\
({}\vdash{y:\overline{B},u:\overline{C},v:\overline{D}})\\}\\\
\mathcal{I}_{3}&=&\\{({}\vdash{x:\overline{A},u:\overline{C},v:\overline{D}}),\leavevmode\nobreak\
({}\vdash{y:\overline{B}})\\}\\\
\mathcal{I}_{4}&=&\\{({}\vdash{x:\overline{A},u:\overline{C}}),({}\vdash{x:\overline{A},v:\overline{D}}),({}\vdash{y:\overline{B},u:\overline{C}}),\\{y:\overline{B},v:\overline{D}\\}\\}\end{array}$
The empty sequent can be derived from $\mathcal{I}_{1}\cup{\mathcal{I}_{i}}$,
for $i=2,3,4$ using cut (and contraction, in the case of $\mathcal{I}_{4}$).
In principle, any of the dual candidates to $\mathcal{I}_{1}$ can be used, but
to make the construction of the interpolants deterministic, our definition
below will always choose $\mathcal{I}_{4}$, as it is relatively
straightforward to define as a function of $\mathcal{I}_{1}.$
###### Definition 3.6.
For an interpolant
$\mathcal{I}=\\{{}\vdash{\Lambda_{1}},\ldots,{}\vdash{\Lambda_{n}}\\}$, the
orthogonal $({\mathcal{I}})^{\bot}$ is defined as
$({\mathcal{I}})^{\bot}=\\{({}\vdash{x_{1}:\overline{A_{1}},\ldots,x_{n}:\overline{A_{n}}})\mid\forall
i\in\\{1,\dots,n\\},x_{i}:A_{i}\in\Lambda_{i}\\}.$
For example, the orthogonal of
$\mathcal{I}=\\{({}\vdash{x:A,y:B}),({}\vdash{x:C,z:D})\\}$ is
$({\mathcal{I}})^{\bot}=\\{({}\vdash{x:\overline{A},x:\overline{C}}),({}\vdash{x:\overline{A},z:\overline{D}}),({}\vdash{y:\overline{B},x:\overline{C}}),({}\vdash{y:\overline{B},z:\overline{D}})\\}.$
###### Definition 3.7.
Let $\mathcal{I}$ be the interpolant
$\\{({}\vdash{\Delta_{1},y:B_{1,1},\ldots,y:B_{1,k_{1}}}),\ldots,({}\vdash{\Delta_{n},y:B_{n,1},\ldots,y:B_{n,k_{n}}})\\}$
where $y$ does not occur in $\Delta_{1},\ldots,\Delta_{n}$ and define
$[?]\mathcal{I}^{y}_{x}:=\\{({}\vdash{\Delta_{1},x:[?]\bigvee_{j=1}^{k_{1}}B_{1,j}}),\ldots,({}\vdash{\Delta_{n},x:[?]\bigvee_{j=1}^{k_{n}}B_{n,j}})\\}$
where an empty disjunction is $\bot$.
###### Definition 3.8.
We define an _interpolation sequent_ to be a syntactic object of the form
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}$,
where $\mathcal{R}$ is a set of relational atoms, $\Gamma_{i}$ and
$\Delta_{i}$ are multisets of labelled formulae (for $i\in\\{1,2\\}$), and
$\mathcal{I}$ is an interpolant. Note that in the interpolation calculus
$\mathsf{Kt\Pi LI}$, $\Gamma_{1}=\Gamma_{2}=\emptyset$ (see Fig. 4).
The vertical bar $\mid$ in an interpolation sequent marks where the sequent
will be partitioned, with the left partition serving as the antecedent and the
right partition serving as the consequent in the interpolation statement. For
example, the initial interpolation sequent shown below left splits into the
two sequents shown below right
$\mathcal{R}\vdash\Gamma\mid x:p,x:\overline{p},\Delta\mathrel{\|}\\{(\vdash
x:\top)\\}\qquad(\mathcal{R}\vdash\Gamma,x:\top)\qquad(\mathcal{R}\vdash
x:\bot,x:p,x:\overline{p},\Delta)$
where the first member $\Gamma$ of the split is placed in the left sequent and
the second member $x:p,x:\overline{p},\Delta$ is placed in the right sequent
(note that the relational atoms $\mathcal{R}$ are inherited by both sequents).
We think of the interpolant $x:\top$ as being _implied by_ the left sequent,
and so, we place it in the left sequent, and we think of the interpolant as
_implying_ the right sequent, so we place its negation (viz. $x:\bot$) in the
right sequent. Observe that an application of $cut_{1}$ between the two
sequents, yields $\mathcal{R}\vdash\Gamma,x:\overline{p},x:p,\Delta$ without
the interpolant. Performing a $cut_{1}$ in this way _syntactically
establishes_ (without evoking the semantics) that the interpolant is indeed an
interpolant (so long as the interpolant satisfies certain other properties;
cf. Lem. 3.12 below).
The interpolation calculus $\mathsf{Kt\Pi LI}$ (Fig. 4) uses interpolation
sequents. More importantly, the calculus succinctly represents our algorithm
for constructing interpolants. Most of the rules are straightforward
counterparts of the proof system in Fig. 2, except for the orthogonality rule
$orth.$ The orthogonality rule is arguably the most novel aspect of our
interpolation calculus, as it imposes a strong requirement on our generalised
notion of interpolants, that it must respect the underlying duality in the
logic. The key to the correctness of this rule is given in the Persistence
Lemma below, which shows that double-orthogonal transformation always retains
some sequents in the original interpolant.
###### Lemma 3.9 (Persistence).
If ${}\vdash{\Lambda}\in({({\mathcal{I}})^{\bot}})^{\bot}$, then there exists
a ${}\vdash{\Lambda^{\prime}}\in\mathcal{I}$ such that
$\Lambda^{\prime}\subseteq\Lambda.$
###### Proof 3.10.
Suppose otherwise, i.e., there exists
${}\vdash{\Lambda}\in({({\mathcal{I}})^{\bot}})^{\bot}$ such that for all
${}\vdash{\Lambda^{\prime}}\in\mathcal{I}$, we have
$\Lambda^{\prime}\not\subseteq\Lambda.$ Suppose
$\mathcal{I}=\\{{}\vdash{\Lambda_{1}},\ldots,{}\vdash{\Lambda_{n}}\\}.$ Then
for each $i$, there must be a labelled formula $x_{i}:A_{i}\in\Lambda_{i}$
such that $x_{i}:A_{i}\not\in\Lambda.$ Let
$\Theta=\\{x_{1}:A_{1},\ldots,x_{n}:A_{n}\\}.$ By construction, we must have
that $\Theta\cap\Lambda=\emptyset.$ However, by Def. 3.6, we have
$\overline{\Theta}\in({\mathcal{I}})^{\bot}$, and since
$\Lambda\in({({\mathcal{I}})^{\bot}})^{\bot}$, by Def. 3.6,
$\Theta\cap\Lambda\not=\emptyset$. Contradiction.
${\mathcal{R}}\vdash{\Gamma,x:\bar{p}\mid
x:{p},\Delta}\mathrel{\|}{\\{({}\vdash{x:p})\\}}\quad{\mathcal{R}}\vdash{\Gamma\mid
x:\bar{p},x:p,\Delta}\mathrel{\|}{\\{({}\vdash{x:\top})\\}}\quad{\mathcal{R}}\vdash{\Delta\mid\Gamma}\mathrel{\|}{({\mathcal{I}})^{\bot}}{\mathcal{R}}\vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$
${\mathcal{R}}\vdash{\Gamma\mid x:A\lor
B,\Delta}\mathrel{\|}{\mathcal{I}}{\mathcal{R}}\vdash{\Gamma\mid
x:A,x:B,\Delta}\mathrel{\|}{\mathcal{I}}\qquad{\mathcal{R}}\vdash{\Gamma\mid
x:A\land
B,\Delta}\mathrel{\|}{\mathcal{I}_{1}\cup\mathcal{I}_{2}}\lx@proof@logical@and{\mathcal{R}}\vdash{\Gamma\mid
x:A,\Delta}\mathrel{\|}{\mathcal{I}_{1}}{\mathcal{R}}\vdash{\Gamma\mid
x:B,\Delta}\mathrel{\|}{\mathcal{I}_{2}}$
${\mathcal{R}}\vdash{\Gamma\mid x:\lozenge
A,\Delta}\mathrel{\|}{\mathcal{I}}{\mathcal{R}}\vdash{\Gamma\mid x:\lozenge
A,y:A,\Delta}\mathrel{\|}{\mathcal{I}}\qquad{\mathcal{R}}\vdash{\Gamma\mid
x:\square
A,\Delta}\mathrel{\|}{\square\mathcal{I}^{y}_{x}}{\mathcal{R},Rxy}\vdash{\Gamma\mid
y:A,\Delta}\mathrel{\|}{\mathcal{I}}$
${\mathcal{R}}\vdash{\Gamma\mid x:\blacklozenge
A,\Delta}\mathrel{\|}{\mathcal{I}}{\mathcal{R}}\vdash{\Gamma\mid
x:\blacklozenge
A,y:A,\Delta}\mathrel{\|}{\mathcal{I}}\qquad{\mathcal{R}}\vdash{\Gamma\mid
x:\blacksquare
A,\Delta}\mathrel{\|}{\blacksquare\mathcal{I}^{y}_{x}}{\mathcal{R},Ryx}\vdash{\Gamma\mid
y:A,\Delta}\mathrel{\|}{\mathcal{I}}$
Figure 4: Calculus $\mathsf{Kt\Pi LI}$ for constructing interpolants for
$\mathsf{Kt}$ extended with path axioms $\mathsf{\Pi}$.
Given a formula $A$, we define the set of propositional variables $var(A)$ of
$A$ to be the set $\\{p\ |\ p\text{ or }\overline{p}\text{ in }A\\}$. This
notation extends straightforwardly to sets of formulae and interpolants.
We write ${\mathcal{R}}\Vdash{\Gamma,\Delta}$ to denote that the sequent
${\mathcal{R}}\vdash{\Gamma,\Delta}$ is provable in $\mathsf{Kt\Pi L}$.
Similarly, ${\mathcal{R}}\Vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$
denotes that the sequent
${\mathcal{R}}\vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$ is provable
in $\mathsf{Kt\Pi LI}.$
###### Definition 3.11.
A logic has the Craig interpolation property iff for every implication
$A\Rightarrow B$ in the logic, there is a formula $C$ such that (i)
$var(C)\subseteq var(A)\cap var(B)$ and (ii) $A\Rightarrow C$ and
$C\Rightarrow B$ are in the logic, where $\Rightarrow$ is taken to be the
implication connective of the logic.
We now establish that each tense logic $\mathsf{Kt\Pi}$ possess the Craig
interpolation property when the implication connective is taken to be
$\rightarrow$. To achieve this, we begin by showing that an interpolant
sequent can be constructed from any cut-free proof.
###### Lemma 3.12.
If ${\mathcal{R}}\Vdash{\Gamma,\Delta}$, then there exists an $\mathcal{I}$
such that ${\mathcal{R}}\Vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$,
$var(\mathcal{I})\subseteq var(\Gamma)\cap var(\Delta)$, and all labels
occuring in $\mathcal{I}$ also occur in $\mathcal{R},\Gamma$ or $\Delta$.
###### Proof 3.13.
Induction on the height of the proof of ${\mathcal{R}}\vdash{\Gamma,\Delta}$
and by using the rules of $\mathsf{Kt\Pi LI}$.
The next lemma establishes the correctness of the interpolants constructed
from our interpolation calculus in Fig. 4. Its proof can be found in App.
B.10.
###### Lemma 3.14.
For all $\mathcal{R},\Gamma,\Delta$ and $\mathcal{I}$, if
${\mathcal{R}}\Vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$, then
1. 1.
For all $({}\vdash{\Lambda})\in\mathcal{I},$ we have
${\mathcal{R}}\Vdash{\Gamma,\Lambda}$ and
2. 2.
For all $({}\vdash{\Theta})\in({\mathcal{I}})^{\bot}$, we have
${\mathcal{R}}\Vdash{\Theta,\Delta}.$
To prove Craig interpolation, we need to construct formula interpolants. Lem.
3.14 provides sequent interpolants, so the next step is to show how one can
derive a formula interpolant from a sequent interpolant. This is possible if
the formulas in an interpolant are all prefixed with the same label. In that
case, there is a straightforward interpretation of the interpolant as a
formula. More precisely, let
$\mathcal{I}=\\{({}\vdash{\Lambda_{1}}),\ldots,({}\vdash{\Lambda_{n}})\\}$,
where $\Lambda_{i}=\\{x:A_{i,1},\ldots,x:A_{i,k_{i}}\\}$ for all $1\leq i\leq
n$. Then, its formula interpretation is given by
$\bigwedge_{i=1}^{n}\bigvee_{j=1}^{k_{i}}A_{i,j}.$ Given such an interpolant
$\mathcal{I}$, we write $\bigwedge\bigvee\mathcal{I}$ to denote its formula
interpretation. The following lemma is a straightforward consequence of this
interpretation.
###### Lemma 3.15.
Let $\mathcal{I}=\\{{}\vdash{\Lambda_{1}},\ldots,{}\vdash{\Lambda_{n}}\\}$ be
an interpolant with $\Lambda_{i}=\\{x:A_{i,1},\ldots,x:A_{i,k_{i}}\\}$ for
each $1\leq i\leq n$. For any multiset of relational atoms $\mathcal{R}$ and
multiset of labelled formulae $\Gamma$, if
${\mathcal{R}}\Vdash{\Gamma,\Lambda}$ for all
${}\vdash{\Lambda}\in{\mathcal{I}}$, then
${\mathcal{R}}\Vdash{\Gamma,x:\bigwedge\bigvee\mathcal{I}}$.
However, the formula-interpolant derived in Lem. 3.15 gives only one-half of
the full picture, as one still needs to show that the orthogonal of a sequent
interpolant admits a dual interpretation as a formula. A key to this is the
following Duality Lemma that shows that orthogonality behaves like negation.
###### Lemma 3.16 (Duality).
Given an interpolant $\mathcal{I}$, the empty sequent is derivable from
$\mathcal{I}\cup({\mathcal{I}})^{\bot}$ using the $cut_{1}$ rule and the
contraction rule.
An interesting consequence of Duality Lemma is that it translates into duality
in the above formula interpretation as well, as made precise in the following
lemma.
###### Lemma 3.17.
Let $\mathcal{I}=\\{{}\vdash{\Lambda_{1}},\ldots,{}\vdash{\Lambda_{n}}\\}$ be
an interpolant with $\Lambda_{i}=\\{x:A_{i,1},\ldots,x:A_{i,k_{i}}\\}$ for
each $1\leq i\leq n$. For any multiset of relational atoms $\mathcal{R}$ and
multiset of labelled formulae $\Delta$, if
${\mathcal{R}}\Vdash{{\Theta},\Delta}$ for all
${}\vdash{\Theta}\in({\mathcal{I}})^{\bot}$, then
${\mathcal{R}}\Vdash{x:\overline{\bigwedge\bigvee\mathcal{I}},\Delta}$.
###### Proof 3.18.
Suppose
$({\mathcal{I}})^{\bot}=\\{({}\vdash{\Theta_{1}}),\ldots,({}\vdash{\Theta_{k}})\\}$
for some $k.$ By Lem. 3.16, we have a derivation $\Xi_{1}$ of the empty
sequent from assumptions $\mathcal{I}\cup({\mathcal{I}})^{\bot}$.
${}\vdash{}\vdots{}\vdash{\Lambda_{1}}\quad\cdots{}\vdash{\Lambda_{n}}\qquad{}\vdash{\Theta_{1}}\quad\cdots\quad{}\vdash{\Theta_{k}}$
Due to admissibility of weakening (Lem. 3.2), for each $\Lambda_{i}$, there is
a proof $\Psi_{i}$ of the sequent
${\mathcal{R}}\vdash{\Lambda_{i},x:\overline{\bigwedge\bigvee\mathcal{I}}}$.
Adding $x:\overline{\bigwedge\bigvee\mathcal{I}}$ to every leaf sequent in
$\Xi_{1}$ belonging to $\mathcal{I}$ gives us a derivation $\Xi_{2}$:
${\mathcal{R}}\vdash{x:\overline{\bigwedge\bigvee\mathcal{I}}}{\mathcal{R}}\vdash{(x:\overline{\bigwedge\bigvee\mathcal{I}})^{*}}\vdots[{\mathcal{R}}\vdash{x:F,\Lambda_{1}}]\quad\cdots\quad[{\mathcal{R}}\vdash{x:F,\Lambda_{n}}]\qquad{\mathcal{R}}\vdash{\Theta_{1}}\quad\cdots\quad{\mathcal{R}}\vdash{\Theta_{k}}$
where $F=\overline{\bigwedge\bigvee\mathcal{I}}$ and sequents in brackets are
provable, and where $*$ denotes multiple copies of sequents or rules. By the
assumption we know that each ${\mathcal{R}}\vdash{\Theta_{i},\Delta}$ is
provable, so by adding $\Delta$ to each premise sequent in $\Xi_{2}$, we get
the following proof:
${\mathcal{R}}\vdash{x:\overline{\bigwedge\bigvee\mathcal{I}},\Delta}{\mathcal{R}}\vdash{(x:\overline{\bigwedge\bigvee\mathcal{I}})^{*},\Delta^{*}}\vdots[{\mathcal{R}}\vdash{x:F,\Lambda_{1},\Delta}]\quad\cdots\quad[{\mathcal{R}}\vdash{x:F,\Lambda_{n},\Delta}]\qquad[{\mathcal{R}}\vdash{\Theta_{1},\Delta}]\quad\cdots\quad[{\mathcal{R}}\vdash{\Theta_{k},\Delta}]$
###### Theorem 3.19.
If ${}\Vdash{x:A\rightarrow B}$, then there exists a $C$ such that (i)
$var(C)\subseteq var(A)\cap var(B)$ and (ii) ${}\Vdash{x:A\rightarrow C}$ and
${}\Vdash{x:C\rightarrow B}$.
###### Corollary 3.20.
Every extension of the (minimal) tense logic $\mathsf{Kt}$ with a set
$\mathsf{\Pi}$ of path axioms has the Craig interpolation property.
###### Example 3.21.
Consider the formula given in example 3.4. By making use of its derivation in
Fig. 3, we can apply our interpolation algorithm as shown in Fig. 5 to
construct an interpolant for the formula.
${}\vdash{x:\lozenge\square q\mid
x:\square(\lozenge\overline{p}\lor\lozenge\lozenge
p)}\mathrel{\|}{\\{({}\vdash{x:\square\lozenge\lozenge\top})\\}}{Rxy}\vdash{x:\lozenge\square
q\mid y:\lozenge\overline{p}\lor\lozenge\lozenge
p}\mathrel{\|}{\\{({}\vdash{y:\lozenge\lozenge\top})\\}}{Rxy}\vdash{x:\lozenge\square
q\mid y:\lozenge\overline{p},y:\lozenge\lozenge
p}\mathrel{\|}{\\{({}\vdash{y:\lozenge\lozenge\top})\\}}{Rxy}\vdash{y:\lozenge\overline{p},y:\lozenge\lozenge
p\mid x:\lozenge\square
q}\mathrel{\|}{\\{({}\vdash{y:\square\square\bot})\\}}{Rxy}\vdash{y:\lozenge\overline{p},y:\lozenge\lozenge
p\mid x:\lozenge\square q,y:\square
q}\mathrel{\|}{\\{}({}\vdash{y:\square\square\bot})\\}{Rxy,Ryz}\vdash{y:\lozenge\overline{p},y:\lozenge\lozenge
p\mid x:\lozenge\square
q,z:q}\mathrel{\|}{\\{}({}\vdash{z:\square\bot})\\}{Rxy,Ryz}\vdash{y:\lozenge\overline{p},y:\lozenge\lozenge
p\mid x:\lozenge\square q,z:\square
q,z:q}\mathrel{\|}{\\{}({}\vdash{z:\square\bot)}\\}{Rxy,Ryz,Rzw}\vdash{y:\lozenge\overline{p},y:\lozenge\lozenge
p\mid x:\lozenge\square
q,w:q,z:q}\mathrel{\|}{\\{}({}\vdash{w:\bot})\\}{Rxy,Ryz,Rzw}\vdash{x:\lozenge\square
q,w:q,z:q\mid y:\lozenge\overline{p},y:\lozenge\lozenge
p}\mathrel{\|}{\\{}({}\vdash{w:\top})\\}{Rxy,Ryz,Rzw}\vdash{x:\lozenge\square
q,w:q,z:q\mid y:\lozenge\overline{p},w:\overline{p},y:\lozenge\lozenge
p}\mathrel{\|}{\\{}({}\vdash{w:\top})\\}{Rxy,Ryz,Rzw}\vdash{x:\lozenge\square
q,w:q,z:q\mid y:\lozenge\overline{p},w:\overline{p},y:\lozenge\lozenge
p,z:\lozenge
p}\mathrel{\|}{\\{}({}\vdash{w:\top})\\}{Rxy,Ryz,Rzw}\vdash{x:\lozenge\square
q,w:q,z:q\mid y:\lozenge\overline{p},w:\overline{p},y:\lozenge\lozenge
p,z:\lozenge p,w:p}\mathrel{\|}{\\{}({}\vdash{w:\top})\\}$ Figure 5: An
example of the construction of tense interpolants.
## 4 Interpolation for Bi-Intuitionistic Logic
The language for bi-intuitionistic logic $\mathsf{BiInt}$ is given via the
following BNF grammar:
$A::=p\ |\ \top\ |\ \bot\ |\ (A\wedge A)\ |\ (A\vee A)\ |\ (A\supset A)\ |\
(A-\\!\\!\\!<A)$
For an axiomatic definition of $\mathsf{BiInt}$ consult [21] and for a
semantic definition see [8, 19].
The calculus $\mathsf{BiIntL}$ for $\mathsf{BiInt}$ is given in Fig. 7. The
calculus makes use of sequents of the form
${\mathcal{R},\Gamma}\vdash{\Delta}$ with $\mathcal{R}$ a multiset of
relational atoms of the form $Rxy$, $\Gamma$ and $\Delta$ multisets of
labelled formulae of the form $x:A$ (where $A$ is a bi-intuitionistic
formula), and all labels are among a countable set $\\{x,y,z,\ldots\\}$. Note
that we need not restrict the consequent of sequents to at most one formula on
the right or left due to the eigenvariable condition imposed on the
$\supset_{R}$ and $-\\!\\!\\!<_{L}$ rules. Moreover, for a multiset
$\mathcal{R}$ of relational atoms or a multiset $\Gamma$ of labelled formulae,
we use the notation $\mathcal{R}[x/y]$ and $\Gamma[x/y]$ to represent the
multiset obtained by replacing each occurrence of the label $y$ for the label
$x$. The $monl$ and $monr$ rules are the natural way to capture monotonicity
when nested sequents are represented using labels.
${\mathcal{R},\Gamma,\Gamma^{\prime}}\vdash{\Delta}{\mathcal{R},\Gamma,\Gamma^{\prime},\Gamma^{\prime}}\vdash{\Delta}\quad{\mathcal{R},\Gamma}\vdash{\Delta,\Delta^{\prime}}{\mathcal{R},\Gamma}\vdash{\Delta,\Delta^{\prime},\Delta^{\prime}}\quad{\mathcal{R},\mathcal{R}^{\prime},\Gamma}\vdash{\Delta}{\mathcal{R},\mathcal{R}^{\prime},\mathcal{R}^{\prime},\Gamma}\vdash{\Delta}\quad{\mathcal{R},\Gamma}\vdash{\Delta,\Delta^{\prime}}{\mathcal{R},\Gamma}\vdash{\Delta}\quad{\mathcal{R},\Gamma,\Gamma^{\prime}}\vdash{\Delta}{\mathcal{R},\Gamma}\vdash{\Delta}$
${\mathcal{R},\mathcal{R}^{\prime},\Gamma}\vdash{\Delta}{\mathcal{R},\Gamma}\vdash{\Delta}\quad{\mathcal{R}}\vdash{\Gamma}\lx@proof@logical@and{\mathcal{R}}\vdash{\Gamma,x:A}{\mathcal{R}}\vdash{\Gamma,x:\overline{A}}\quad{\mathcal{R},\Gamma}\vdash{\Delta}\lx@proof@logical@and{\mathcal{R},\Gamma}\vdash{\Delta,x:A}{\mathcal{R},x:A,\Gamma}\vdash{\Delta}$
Figure 6: Admissible rules.
###### Lemma 4.1.
The calculus $\mathsf{BiIntL}$ enjoys the following: (1) admissibility of
$ctr$, $wk$ and $cut_{2}$ from Fig. 6; (2) invertibility of all inference
rules from Fig. 7; (3) if ${\mathcal{R},Rxy,\Gamma}\Vdash{\Delta}$, then
${\mathcal{R}[x/y],\Gamma[x/y]}\Vdash{\Delta[x/y]}$, and (4) If
${\Gamma}\Vdash{\Delta}$ where $\Gamma$ and $\Delta$ only contain formulae
solely labelled with $y$, then ${\Gamma[x/y]}\Vdash{\Delta[x/y]}$ for any
label $x$.
###### Proof 4.2.
For proofs of (1)-(4), see [19, Section 3]. Statement (4) follows from the
others.
${\mathcal{R},x:p,\Gamma}\vdash{\Delta,x:p}\qquad{\mathcal{R},\Gamma}\vdash{x:\top,\Delta}\qquad{\mathcal{R},\Gamma,x:\bot}\vdash{\Delta}$
${\mathcal{R},\Gamma,x:A\lor
B}\vdash{\Delta}\lx@proof@logical@and{\mathcal{R},\Gamma,x:A}\vdash{\Delta}{\mathcal{R},\Gamma,x:B}\vdash{\Delta}\qquad{\mathcal{R},\Gamma}\vdash{x:A\lor
B,\Delta}{\mathcal{R},\Gamma}\vdash{x:A,x:B,\Delta}\qquad{\mathcal{R},Rxy,x:A,\Gamma}\vdash{\Delta}{{\mathcal{R},Rxy,x:A,y:A,\Gamma}\vdash{\Delta}}$
${\mathcal{R},\Gamma}\vdash{x:A\land
B,\Delta}\lx@proof@logical@and{\mathcal{R},\Gamma}\vdash{x:A,\Delta}{\mathcal{R},\Gamma}\vdash{x:B,\Delta}\qquad{\mathcal{R},\Gamma,x:A\land
B}\vdash{\Delta}{\mathcal{R},\Gamma,x:A,x:B}\vdash{\Delta}\qquad{\mathcal{R},Rxy,\Gamma}\vdash{y:A,\Delta}{{\mathcal{R},Rxy,\Gamma}\vdash{x:A,y:A,\Delta}}$
${\mathcal{R},x:A-\\!\\!\\!<B,\Gamma}\vdash{\Delta}{\mathcal{R},Ryx,y:A,\Gamma}\vdash{y:B,\Delta}\quad{\mathcal{R},x:A\supset
B,\Gamma}\vdash{\Delta}\lx@proof@logical@and{\mathcal{R},x:A\supset
B,\Gamma}\vdash{x:A,\Delta}{\mathcal{R},x:B,\Gamma}\vdash{\Delta}$
${\mathcal{R},\Gamma}\vdash{x:A-\\!\\!\\!<B,\Delta}\lx@proof@logical@and{\mathcal{R},\Gamma}\vdash{x:A,\Delta}{\mathcal{R},x:B,\Gamma}\vdash{x:A-\\!\\!\\!<B,\Delta}\quad{\mathcal{R},\Gamma}\vdash{x:A\supset
B,\Delta}{\mathcal{R},Rxy,\Gamma,y:A}\vdash{y:B,\Delta}$
Figure 7: The calculus $\mathsf{BiIntL}$ for $\mathsf{BiInt}$ [19].
As in the case with tense logics, we define a generalised interpolant to be a
set of two-sided flat sequents. However, to ease the definition of orthogonal,
we shall use an encoding of two-sided sequents into single-sided sequents by
annotating the left-hand side occurrence of a formula with a $L$ and the
right-hand side occurrence with a $R$. In this way some results concerning
intuitionistic interpolants can be easily adapted from the classical
counterparts.
###### Definition 4.3.
A polarised formula is a formula annotated with $L$ (left-polarised) or $R$
(right-polarised). We write $A^{L}$ ($A^{R}$) for the left-polarised (right-
polarised) version of formula $A.$ A labelled polarised formula is a polarised
formula further annotated with a label. We write $x:A^{L}$ ($x:A^{R}$) to
denote a left-polarised (a right-polarised) formula labelled with $x.$ Given a
polarised formula $A^{L}$ (resp. $A^{R}$), its dual is defined as
$\overline{A^{L}}=A^{R}$ and $\overline{A^{R}}=A^{L}.$ That is, duality
changes polarities (the side where the formula occurs), but not the actual
formula.
###### Definition 4.4.
A polarised (flat) sequent is a single-sided (flat) sequent where all formulas
in the sequent are polarised. Given a two-sided sequent
$S={\mathcal{R},x_{1}:A_{1},\ldots,x_{m}:A_{m}}\vdash{y_{1}:B_{1},\ldots,y_{n}:B_{n}}$,
its corresponding polarised sequent is the following
${\mathcal{R}}\vdash{x_{1}:A_{1}^{L},\ldots,x_{m}:A_{m}^{L},y_{1}:B_{1}^{R},\ldots,y_{n}:B_{n}^{R}}.$
Given a two-sided sequent $S$, we denote with $\mathsf{P}(S)$ its encoding as
a polarised sequent. Conversely, given a polarised sequent $S$, we denote with
$\mathsf{T}(S)$ its two-sided counterpart. This notation extends to sets of
sequents by applying the encoding element-wise.
###### Definition 4.5.
An intuitionistic interpolant is a set of two-sided flat sequents. Given an
intuitionistic interpolant $\mathcal{I}$, its orthogonal
$({\mathcal{I}})^{\bot}$ is defined as
$\mathsf{T}(({\mathsf{P}(\mathcal{I})})^{\bot}).$
###### Example 4.6.
Let $\mathcal{I}=\\{({x:A}\vdash{y:B}),({}\vdash{u:C,v:D})\\}$. Then,
$({\mathcal{I}})^{\bot}$ is the set:
$\\{({x:A}\vdash{u:C}),({x:A}\vdash{v:D}),({}\vdash{u:C,y:B}),({}\vdash{v:D,y:B})\\}$
By defining orthogonality via the embedding into polarised sequents, the
Persistence Lemma for the intuitionistic case comes for free, by appealing to
Lem. 3.9. Note that for sequents ${\Gamma_{1}}\vdash{\Delta_{1}}$ and
${\Gamma_{2}}\vdash{\Delta_{2}}$, we write
${\Gamma_{1}}\vdash{\Delta_{1}}\subseteq{\Gamma_{2}}\vdash{\Delta_{2}}$ iff
$\Gamma_{1}\subseteq\Gamma_{2}$ and $\Delta_{1}\subseteq\Delta_{2}.$
###### Lemma 4.7 (Persistence).
If $\Lambda\in({({\mathcal{I}})^{\bot}})^{\bot}$, then there exists a
$\Lambda^{\prime}\in\mathcal{I}$ such that $\Lambda^{\prime}\subseteq\Lambda.$
###### Lemma 4.8 (Duality).
Given an intuitionistic interpolant $\mathcal{I}$, the empty sequent is
derivable from $\mathcal{I}\cup({\mathcal{I}})^{\bot}$ using the $cut_{2}$
rule and the contraction rule.
###### Proof 4.9.
Similar to Lem. 3.16.
###### Definition 4.10.
Let $\mathcal{I}$ be the interpolant below where $y$ does not occur in
$\Gamma_{1},\ldots,\Gamma_{n},\Delta_{1},\ldots,\Delta_{n}$:
$\begin{array}[]{ll}&\\{({\Gamma_{1},y:C_{1,1},\ldots,y:C_{1,k_{1}}}\vdash{\Delta_{1},y:D_{1,1},\ldots,y:D_{1,j_{1}}}),\ldots,\\\
&\hskip
5.0pt({\Gamma_{n},y:C_{n,1},\ldots,y:C_{n,k_{n}}}\vdash{\Delta_{n},y:D_{n,1},\ldots,y:D_{n,j_{n}}})\\}\\\
\end{array}$
The interpolants $-\\!\\!\\!<\mathcal{I}^{x}_{y}$ and
$\supset\mathcal{I}^{x}_{y}$ are shown below where empty conjunction denotes
$\top$ and empty disjunction denotes $\bot$:
$\displaystyle-\\!\\!\\!<\mathcal{I}^{x}_{y}$ $\displaystyle=$
$\displaystyle\\{({\Gamma_{1},x:\bigwedge^{k_{1}}_{i=1}C_{1,i}-\\!\\!\\!<\bigvee^{j_{1}}_{i=1}D_{1,i}}\vdash{\Delta_{1}}),\ldots,({\Gamma_{n},x:\bigwedge^{k_{n}}_{i=1}C_{n,i}-\\!\\!\\!<\bigvee^{j_{n}}_{i=1}D_{n,i}}\vdash{\Delta_{n}})\\}$
$\displaystyle\supset\mathcal{I}^{x}_{y}$ $\displaystyle=$
$\displaystyle\\{({\Gamma_{1}}\vdash{\Delta_{1},x:\bigwedge^{k_{1}}_{i=1}C_{1,i}\supset\bigvee^{j_{1}}_{i=1}D_{1,i}}),\ldots,({\Gamma_{n}}\vdash{\Delta_{n},x:\bigwedge^{k_{n}}_{i=1}C_{n,i}\supset\bigvee^{j_{n}}_{i=1}D_{n,i}})\\}$
The proof system $\mathsf{BiIntLI}$ for constructing intuitionistic
interpolants is given in Fig. 8.
${\mathcal{R},\Gamma_{1},x:p\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:p,\Delta_{2}}\mathrel{\|}{\\{{}\vdash{x:p}\\}}\qquad{\mathcal{R},\Gamma_{1}\mid
x:p,\Gamma_{2}}\vdash{\Delta_{1}\mid
x:p,\Delta_{2}}\mathrel{\|}{\\{{}\vdash{x:\bot}\\}}$
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:\top,\Delta_{2}}\mathrel{\|}{\\{{}\vdash{x:\bot}\\}}\qquad{\mathcal{R},\Gamma_{1}\mid
x:\bot,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\\{{}\vdash{x:\bot}\\}}$
${\mathcal{R},\Gamma_{2}\mid\Gamma_{1}}\vdash{\Delta_{2}\mid\Delta_{1}}\mathrel{\|}{({\mathcal{I}})^{\bot}}{\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}$
${\mathcal{R},Rxy,\Gamma_{1}\mid
x:A,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}{{\mathcal{R},Rxy,\Gamma_{1}\mid
x:A,y:A,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}}\qquad{{\mathcal{R},Rxy,\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
y:A,\Delta_{2}}\mathrel{\|}{\mathcal{I}}}{\mathcal{R},Rxy,\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:A,y:A,\Delta_{2}}\mathrel{\|}{\mathcal{I}}$
${\mathcal{R},\Gamma_{1}\mid x:A\lor
B,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}_{1}\cup\mathcal{I}_{2}}\lx@proof@logical@and{\mathcal{R},\Gamma_{1}\mid
x:A,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}_{1}}{\mathcal{R},\Gamma_{1}\mid
x:B,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}_{2}}$
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid x:A\land
B,\Delta_{2}}\mathrel{\|}{\mathcal{I}_{1}\cup\mathcal{I}_{2}}\lx@proof@logical@and{\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:A,\Delta_{2}}\mathrel{\|}{\mathcal{I}_{1}}{\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:B,\Delta_{2}}\mathrel{\|}{\mathcal{I}_{2}}$
${\mathcal{R},\Gamma_{1}\mid x:A\land
B,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}{\mathcal{R},\Gamma_{1}\mid
x:A,x:B,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}\qquad{\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:A\lor
B,\Delta_{2}}\mathrel{\|}{\mathcal{I}}{\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:A,x:B,\Delta_{2}}\mathrel{\|}{\mathcal{I}}$
${\mathcal{R},\Gamma_{1}\mid x:A\supset
B,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}_{1}\cup\mathcal{I}_{2}}\lx@proof@logical@and{\mathcal{R},\Gamma_{1}\mid
x:A\supset B,\Gamma_{2}}\vdash{\Delta_{1}\mid
x:A,\Delta_{2}}\mathrel{\|}{\mathcal{I}_{2}}{\mathcal{R},\Gamma_{1}\mid
x:B,\Gamma_{1}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}_{1}}$
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:A-\\!\\!\\!<B,\Delta_{2}}\mathrel{\|}{\mathcal{I}_{1}\cup\mathcal{I}_{2}}\lx@proof@logical@and{\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:A,\Delta_{2}}\mathrel{\|}{\mathcal{I}_{1}}{\mathcal{R},\Gamma_{1}\mid
x:B,\Gamma_{2}}\vdash{\Delta_{1},x:A-\\!\\!\\!<B\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}_{2}}$
${\mathcal{R},\Gamma_{1}\mid
x:A-\\!\\!\\!<B,\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{-\\!\\!\\!<\mathcal{I}^{y}_{x}}{\mathcal{R},Ryx,\Gamma_{1}\mid
y:A,\Gamma_{2}}\vdash{\Delta_{1}\mid
y:B,\Delta_{2}}\mathrel{\|}{\mathcal{I}}\qquad{\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid
x:A\supset
B,\Delta_{2}}\mathrel{\|}{\supset\mathcal{I}^{y}_{x}}{\mathcal{R},Rxy,\Gamma_{1}\mid
y:A,\Gamma_{2}}\vdash{\Delta_{1}\mid y:B,\Delta_{2}}\mathrel{\|}{\mathcal{I}}$
Figure 8: The calculus $\mathsf{BiIntLI}$ used to compute interpolants for
$\mathsf{BiInt}$. In $\supset_{R}$ and $-\\!\\!\\!<_{L}$, $y$ is fresh.
###### Lemma 4.11.
If ${\mathcal{R},\Gamma_{1},\Gamma_{2}}\Vdash{\Delta_{1},\Delta_{2}}$, then
there exists an $\mathcal{I}$ such that
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\Vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}$,
$var(\mathcal{I})\subseteq var(\Gamma_{1},\Delta_{1})\cap
var(\Gamma_{2},\Delta_{2})$, and all labels in $\mathcal{I}$ also occur in
$\mathcal{R},\Gamma_{1},\Delta_{1}$ or $\Gamma_{2},\Delta_{2}$.
###### Proof 4.12.
Induction on the height of the proof of
${\mathcal{R},\Gamma_{1},\Gamma_{2}}\vdash{\Delta_{1},\Delta_{2}}$ using rules
of $\mathsf{BiIntLI}$.
The main technical lemma below asserts that the interpolants constructed via
the proof system in Fig. 8 obey duality properties which are essential for
proving the main theorem (Thm. 4.17). The proof of this lemma can be found in
App. B.10.
###### Lemma 4.13.
For all $\mathcal{R},\Gamma_{1},\Gamma_{2},\Delta_{1},\Delta_{2}$ and
$\mathcal{I}$, if
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\Vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}$,
then
1. 1.
For all $({\Sigma}\vdash{\Theta})\in\mathcal{I}$, we have
${\mathcal{R},\Gamma_{1},\Sigma}\Vdash{\Theta,\Delta_{1}}$ and
2. 2.
For all $({\Lambda}\vdash{\Omega})\in({\mathcal{I}})^{\bot}$, we have
${\mathcal{R},\Gamma_{2},\Lambda}\Vdash{\Omega,\Delta_{2}}.$
Given a sequent $\Lambda$, we denote with $\Lambda^{L}$ (resp., $\Lambda^{R}$)
the multiset of labelled formulas on the left (resp. right) hand side of
$\Lambda$. The following two lemmas are counterparts of Lem. 3.15 and Lem.
3.17. Lem. 4.14 essentially states that in a specific case, an interpolant can
be interpreted straightforwardly as a conjunction of implications. Its proof
is given in App. B.10. The proof of Lem. 4.15 follows the same pattern as in
the proof of Lem. 3.17.
###### Lemma 4.14.
Let
$\mathcal{I}=\\{({\Sigma_{1}}\vdash{\Theta_{1}}),\ldots,({\Sigma_{n}}\vdash{\Theta_{n}})\\}$
be an interpolant with
$({\Sigma_{i}}\vdash{\Theta_{i}})=({x:C_{i,1},\ldots,x:C_{i,k_{i}}}\vdash{x:D_{i,1},\ldots,x:D_{i,j_{i}}})\mbox{
for each }1\leq i\leq n.$
If ${\Sigma_{i},\Gamma}\Vdash{\Theta_{i}}$, for all
$({\Sigma_{i}}\vdash{\Theta_{i}})\in\mathcal{I}$, and every formula in
$\Gamma$ is labelled with $x$, then
${\Gamma}\Vdash{x:\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m}}).$
###### Lemma 4.15.
Let
$\mathcal{I}=\\{({\Sigma_{1}}\vdash{\Theta_{1}}),\ldots,({\Sigma_{n}}\vdash{\Theta_{n}})\\}$
be an interpolant with
$({\Sigma_{i}}\vdash{\Theta_{i}})=({x:C_{i,1},\ldots,x:C_{i,k_{i}}}\vdash{x:D_{i,1},\ldots,x:D_{i,j_{i}}})\mbox{
for each }1\leq i\leq n.$
If ${\mathcal{R},\Sigma_{i},\Gamma}\Vdash{\Delta,\Theta_{i}}$ for all
$({\Sigma_{i}}\vdash{\Theta_{i}})\in({\mathcal{I}})^{\bot}$, then
${\mathcal{R},\Gamma,x:\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m})}\Vdash{\Delta.}$
###### Proof 4.16.
Follows from Lem. 4.8 and is similar to Lem. 3.17.
###### Theorem 4.17.
If ${}\Vdash{x:A\supset B}$, then there exists a $C$ such that (i)
$var(C)\subseteq var(A)\cap var(B)$ and (ii) ${}\Vdash{x:A\supset C}$ and
${}\Vdash{x:C\supset B}$.
###### Corollary 4.18.
The logic $\mathsf{BiInt}$ has the Craig interpolation property.
## 5 Conclusion and Future work
We have presented a novel approach to proving the interpolation theorem for a
range of logics possessing a nested sequent calculus. The key insight in our
approach is the generalisation of the interpolation theorem to allow sets of
sequents as interpolants. There is a natural definition of duality between
interpolants via cut. We have shown that our method can be used to prove
interpolation for logics for which interpolation was known to be difficult to
prove.
We intend on applying our approach to prove interpolation for bi-
intuitionistic linear logic (BiILL) [4]. Unlike tense logics and bi-
intuitionistic logic, there is no obvious Kripke semantics for BiILL, so
Kuznets et. al.’s approach is not immediately applicable, and it seems a
proof-theoretic approach like ours would offer some advantage. We conjecture
that the key insight in our work, i.e., the generalisation of interpolants to
sets of sequents and the use of orthogonality to define duality between
interpolants, can be extended to the linear logic setting; for example, via a
similar notion of orthogonality as in multiplicative linear logic [5].
## References
* [1] Kai Brünnler. Deep sequent systems for modal logic. Arch. Math. Log., 48(6):551–577, 2009.
* [2] Agata Ciabattoni, Nikolaos Galatos, and Kazushige Terui. From axioms to analytic rules in nonclassical logics. In Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science, LICS 2008, 24-27 June 2008, Pittsburgh, PA, USA, pages 229–240, 2008. URL: https://doi.org/10.1109/LICS.2008.39, doi:10.1109/LICS.2008.39.
* [3] Agata Ciabattoni, Tim Lyon, and Revantha Ramanayake. From display to labelled proofs for tense logics. In Sergei Artemov and Anil Nerode, editors, Logical Foundations of Computer Science, pages 120–139, Cham, 2018. Springer International Publishing.
* [4] Ranald Clouston, Jeremy E. Dawson, Rajeev Goré, and Alwen Tiu. Annotation-free sequent calculi for full intuitionistic linear logic. In Computer Science Logic 2013 (CSL 2013), CSL 2013, September 2-5, 2013, Torino, Italy, volume 23 of LIPIcs, pages 197–214. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. URL: https://doi.org/10.4230/LIPIcs.CSL.2013.197, doi:10.4230/LIPIcs.CSL.2013.197.
* [5] Vincent Danos and Laurent Regnier. The structure of multiplicatives. Arch. Math. Log., 28(3):181–203, 1989. URL: https://doi.org/10.1007/BF01622878, doi:10.1007/BF01622878.
* [6] Melvin Fitting and Roman Kuznets. Modal interpolation via nested sequents. Ann. Pure Appl. Logic, 166(3):274–305, 2015. URL: https://doi.org/10.1016/j.apal.2014.11.002, doi:10.1016/j.apal.2014.11.002.
* [7] Rajeev Goré. Substructural logics on display. Logic Journal of the IGPL, 6(3):451–504, 1998.
* [8] Rajeev Goré, Linda Postniece, and Alwen Tiu. Cut-elimination and proof-search for bi-intuitionistic logic using nested sequents. In Advances in Modal Logic 7, papers from the seventh conference on "Advances in Modal Logic," held in Nancy, France, 9-12 September 2008, pages 43–66, 2008. URL: http://www.aiml.net/volumes/volume7/Gore-Postniece-Tiu.pdf.
* [9] Rajeev Goré, Linda Postniece, and Alwen Tiu. On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics. Log. Methods Comput. Sci., 7(2):2:8, 38, 2011.
* [10] Ryo Kashima. Cut-free sequent calculi for some tense logics. Studia Logica, 53(1):119–136, 1994. URL: https://doi.org/10.1007/BF01053026, doi:10.1007/BF01053026.
* [11] Hitoshi Kihara and Hiroakira Ono. Interpolation properties, Beth definability properties and amalgamation properties for substructural logics. Journal of Logic and Computation, 20(4), August 2010.
* [12] Tomasz Kowalski and Hiroakira Ono. Analytic cut and interpolation for bi-intuitionistic logic. Rew. Symb. Logic, 10(2):259–283, 2017. URL: https://doi.org/10.1017/S175502031600040X, doi:10.1017/S175502031600040X.
* [13] Roman Kuznets. Multicomponent proof-theoretic method for proving interpolation properties. Ann. Pure Appl. Logic, 169(12):1369–1418, 2018.
* [14] Roman Kuznets and Björn Lellmann. Interpolation for intermediate logics via hyper- and linear nested sequents. In Proc. AiML 2018. Kings College Publications, 2018.
* [15] Carsten Lutz and Frank Wolter. Foundations for uniform interpolation and forgetting in expressive description logics. In IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16-22, 2011, pages 989–995, 2011. URL: https://doi.org/10.5591/978-1-57735-516-8/IJCAI11-170, doi:10.5591/978-1-57735-516-8/IJCAI11-170.
* [16] S. Maehara. Craig no interpolation theorem (in japanese). Suugaku, 12:235–237, 1960.
* [17] Kenneth L. McMillan. Interpolation and model checking. In Handbook of Model Checking, pages 421–446. Springer, 2018.
* [18] Sara Negri. Proof analysis in modal logic. J. Philosophical Logic, 34(5-6):507–544, 2005.
* [19] Luís Pinto and Tarmo Uustalu. A proof-theoretic study of bi-intuitionistic propositional sequent calculus. J. Log. Comput., 28(1):165–202, 2018. URL: https://doi.org/10.1093/logcom/exx044, doi:10.1093/logcom/exx044.
* [20] Francesca Poggiolesi. A cut-free simple sequent calculus for modal logic S5. Rew. Symb. Logic, 1(1):3–15, 2008. URL: https://doi.org/10.1017/S1755020308080040, doi:10.1017/S1755020308080040.
* [21] Cecylia Rauszer. An algebraic and Kripke-style approach to a certain extension of intuitionistic logic. Instytut Matematyczny Polskiej Akademi Nauk, 1980. URL: http://eudml.org/doc/268511.
* [22] Balder ten Cate, Enrico Franconi, and Inanç Seylan. Beth definability in expressive description logics. J. Artif. Intell. Res., 48:347–414, 2013.
* [23] Alwen Tiu, Egor Ianovski, and Rajeev Goré. Grammar logics in nested sequent calculus: Proof theory and decision procedures. In Advances in Modal Logic 9, papers from the ninth conference on "Advances in Modal Logic," held in Copenhagen, Denmark, 22-25 August 2012, pages 516–537, 2012. URL: http://www.aiml.net/volumes/volume9/Tiu-Ianovski-Gore.pdf.
* [24] Anne Troesltra and Helmut Schwichtenberg. Basic Proof Theory. CUP, 1996.
## Appendix A Definition of $\mathcal{R}^{{\sf\Pi}}$
In this appendix we define the relation $\mathcal{R}^{{\sf\Pi}}$ that is used
as a side condition in the $\lozenge$ and $\blacklozenge$ rules in
$\mathsf{Kt\Pi L}$. Concepts needed for the definition are defined first
followed by the definition of the relation.
###### Definition A.1 (Path Axiom Inverse [9]).
Let $\langle?\rangle^{-1}=\lozenge$ if $\langle?\rangle=\blacklozenge$, and
$\langle?\rangle^{-1}=\blacklozenge$, if $\langle?\rangle=\lozenge$. If $F$ is
a path axiom of the form below left, then we define the _inverse of $F$_
(denoted $I(F)$) to be the axiom below right:
$\langle?\rangle_{F_{1}}\cdots\langle?\rangle_{F_{n}}p\rightarrow\langle?\rangle_{F}p\qquad
I(F)=\langle?\rangle^{-1}_{F_{n}}\cdots\langle?\rangle^{-1}_{F_{1}}p\rightarrow\langle?\rangle^{-1}_{F}p$
Given a set of path axioms ${\sf\Pi}$, we define the _set of inverses_ to be
the set $I({\sf\Pi})=\\{I(F)|F\in{\sf\Pi}\\}$.
###### Definition A.2 (Composition of Path Axioms [9]).
Given two path axioms
${F=\langle?\rangle_{F_{1}}\cdots\langle?\rangle_{F_{n}}p\rightarrow\langle?\rangle_{F}p\qquad
G=\langle?\rangle_{G_{1}}\cdots\langle?\rangle_{G_{m}}p\rightarrow\langle?\rangle_{G}p}$
we say _$F$ is composable with $G$ at $i$_ iff
$\langle?\rangle_{F}=\langle?\rangle_{G_{i}}$.
We define the _composition_
$F\triangleright^{i}G=\langle?\rangle_{G_{1}}\cdots\langle?\rangle_{G_{i-1}}\langle?\rangle_{F_{1}}\cdots\langle?\rangle_{F_{n}}\langle?\rangle_{G_{i+1}}\cdots\langle?\rangle_{G_{m}}p\rightarrow\langle?\rangle_{G}p$
when $F$ is composable with $G$ at $i$.
Using these individual compositions, we define the following _set of
compositions_ :
$F\triangleright G=\\{F\triangleright^{i}G\ |\ \text{F is composable with G at
$i$}\\}$
###### Definition A.3 (Completion [9]).
The _completion_ of a set ${\sf\Pi}$ of path axioms, written ${\sf\Pi}^{*}$,
is the smallest set of path axioms containing ${\sf\Pi}$ such that:
* •
$\lozenge p\rightarrow\lozenge p,\blacklozenge p\rightarrow\blacklozenge
p\in{\sf\Pi}^{*}$;
* •
If $F,G\in{\sf\Pi}^{*}$ and $F$ is composable with $G$ for some $i$, then
$F\triangleright G\subseteq{\sf\Pi}^{*}$.
###### Definition A.4 (Propagation Graph).
Let ${\mathcal{R}}\vdash{\Gamma}$ be a labelled sequent where $N$ is the set
of labels occurring in the sequent. We define the _propagation graph
$PG({\mathcal{R}}\vdash{\Gamma})=(N,E,L)$_ to be the directed graph with the
set of nodes $N$, and where the set of edges $E$ and function $L$ that labels
edges with either a $\lozenge$ or $\blacklozenge$ are defined as follows: For
every $Rxy\in\mathcal{R}$, we have an edge $(x,y)\in E$ with $L(x,y)=\lozenge$
and an edge $(y,x)\in E$ with $L(y,x)=\blacklozenge$.
###### Definition A.5 (Path [9]).
A _path_ is a sequence of nodes and diamonds (labelling edges) of the form:
$x_{1},\langle?\rangle_{1},x_{2},\langle?\rangle_{2},\cdots,\langle?\rangle_{n-1},x_{n}$
in the propagation graph $PG(\Gamma)$ of a sequent $\Gamma$ such that $x_{i}$
is connected to $x_{i+1}$ by an edge labelled with $\langle?\rangle_{i}$. For
a given path
$\pi=x_{1},\langle?\rangle_{1},x_{2},\langle?\rangle_{2},\cdots,\langle?\rangle_{n-1},x_{n}$,
we define the _string of $\pi$_ to be the string of diamonds
$s(\pi)=\langle?\rangle_{1}\langle?\rangle_{2}\cdots\langle?\rangle_{n-1}$.
###### Definition A.6 (The Relation $\mathcal{R}^{{\sf\Pi}}$).
Let ${\sf\Pi}$ be a set of path axioms. For any two labels $x$ and $y$
occurring in a labelled sequent ${\mathcal{R}}\vdash{\Gamma}$,
$x\mathcal{R}^{{\sf\Pi}}y$ holds if and only if there exists a path $\pi$ in
$PG({\mathcal{R}}\vdash{\Gamma})$ such that $s(\pi)p\rightarrow\langle?\rangle
p\in{\sf\Pi}^{*}$.
## Appendix B Proofs
###### Lemma 3.16.
Given an interpolant $\mathcal{I}$, the empty sequent is derivable from
$\mathcal{I}\cup({\mathcal{I}})^{\bot}$ using the cut rule and the contraction
rule.
###### Proof B.1.
By induction on the size of $\mathcal{I}.$ Suppose
$\mathcal{I}=\\{({}\vdash{x_{1}:A_{1},\ldots,x_{n}:A_{n}})\\}\cup\mathcal{I}^{\prime}.$
Suppose
$({\mathcal{I}^{\prime}})^{\bot}=\\{{}\vdash{\Lambda_{1}},\ldots,{}\vdash{\Lambda_{k}}\\}.$
Then by the definition of orthogonal transformation:
$({\mathcal{I}})^{\bot}=\bigcup_{i=1}^{k}\bigcup_{j=1}^{n}\\{({}\vdash{\Lambda_{i},x_{j}:\overline{A_{j}}})\\}.$
So each $\Lambda_{i}\in({\mathcal{I}^{\prime}})^{\bot}$ corresponds to exactly
$n$ sequents in $({\mathcal{I}})^{\bot}$:
${}\vdash{x_{1}:\overline{A_{1}},\Lambda_{i}}\qquad\cdots\qquad{}\vdash{x_{n}:\overline{A_{n}},\Lambda_{i}}.$
From these, we can construct a derivaton $\Xi_{i}$ of ${}\vdash{\Lambda_{i}}$
using only cut and contraction:
${}\vdash{\Lambda_{i}}\vdots{}\vdash{x_{1}:A_{1},\ldots,x_{n}:A_{n}}\qquad{}\vdash{x_{1}:\overline{A_{1}},\Lambda_{i}}\quad\cdots\quad{}\vdash{x_{n}:\overline{A_{n}},\Lambda_{i}}$
By induction hypothesis, we have a derivation of ${}\vdash{}$ from
$\mathcal{I}^{\prime}\cup({\mathcal{I}^{\prime}})^{\bot}$:
${}\vdash{}\vdots\mathcal{I}^{\prime}\qquad{}\vdash{\Lambda_{1}}\cdots{}\vdash{\Lambda_{k}}$
By replacing every leaf ${}\vdash{\Lambda_{i}}$ with derivation $\Xi_{i}$, we
get a derivation of the empty sequent from assumptions $\mathcal{I}$ and
$({\mathcal{I}^{\prime}})^{\bot}$, using only cut and contraction as required.
###### Lemma 3.14.
For all $\mathcal{R},\Gamma,\Delta$ and $\mathcal{I}$, if
${\mathcal{R}}\Vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$, then
1. 1.
For all $({}\vdash{\Lambda})\in\mathcal{I},$ we have
${\mathcal{R}}\Vdash{\Gamma,\Lambda}$ and
2. 2.
For all $({}\vdash{\Theta})\in({\mathcal{I}})^{\bot}$, we have
${\mathcal{R}}\Vdash{\Theta,\Delta}.$
###### Proof B.2.
By induction on the height of the proof of
${\mathcal{R}}\vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$; we prove the
result for a representative set of cases since all others are simple or shown
similarly. The base cases are trivial, so we focus solely on the inductive
step.
### $orth$-rule
Suppose the proof of
${\mathcal{R}}\vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$ ends with the
$orth$-rule:
${\mathcal{R}}\vdash{\Delta\mid\Gamma}\mathrel{\|}{({\mathcal{I}})^{\bot}}{\mathcal{R}}\vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$
1. 1.
Let ${}\vdash{\Lambda}\in({\mathcal{I}})^{\bot}$: we need to establish that
${\mathcal{R}}\Vdash{\Lambda,\Delta}$. This follows immediately from the
induction hypothesis.
2. 2.
Let ${}\vdash{\Lambda}\in({({\mathcal{I}})^{\bot}})^{\bot}$: We need to
establish that ${\mathcal{R}}\Vdash{\Gamma,\Lambda}.$ By Lem. 3.9, there
exists a $\Lambda^{\prime}\in\mathcal{I}$ such that
$\Lambda^{\prime}\subseteq\Lambda$. By the IH, we have
${\mathcal{R}}\Vdash{\Gamma,\Lambda^{\prime}}$, and applying admissibility of
weakening (Lem. 3.2), we get ${\mathcal{R}}\Vdash{\Gamma,\Lambda}$ as
required.
### $\land$-rule
Suppose our proof ends with the inference:
${\mathcal{R}}\vdash{\Gamma\mid x:A\land
B,\Delta}\mathrel{\|}{\mathcal{I}_{1}\cup\mathcal{I}_{2}}\lx@proof@logical@and{\mathcal{R}}\vdash{\Gamma\mid
x:A,\Delta}\mathrel{\|}{\mathcal{I}_{1}}{\mathcal{R}}\vdash{\Gamma\mid
x:B,\Delta}\mathrel{\|}{\mathcal{I}_{2}}$
1. 1.
Let ${}\vdash{\Lambda}\in\mathcal{I}_{1}\cup\mathcal{I}_{2}$. By the induction
hypothesis, for any ${}\vdash{\Lambda_{1}}\in\mathcal{I}_{1}$ and
${}\vdash{\Lambda_{2}}\in\mathcal{I}_{2}$, we have that
${\mathcal{R}}\Vdash{\Gamma,\Lambda_{1}}$, and
${\mathcal{R}}\Vdash{\Gamma,\Lambda_{2}}$. Regardless of if
${}\vdash{\Lambda}\in\mathcal{I}_{1}$ or
${}\vdash{\Lambda}\in\mathcal{I}_{2}$, we achieve the desired conclusion.
2. 2.
Let ${}\vdash{\Lambda}\in({\mathcal{I}_{1}\cup\mathcal{I}_{2}})^{\bot}$. By
the induction hypothesis, for any
${}\vdash{\Lambda_{1}}\in({\mathcal{I}_{1}})^{\bot}$ and
${}\vdash{\Lambda_{2}}\in({\mathcal{I}_{2}})^{\bot}$,
${\mathcal{R}}\Vdash{x:A,\Delta,\Lambda_{1}}$ and
${\mathcal{R}}\Vdash{x:B,\Delta,\Lambda_{2}}$. Observe that by the definition
of orthogonal, there is a ${}\vdash{\Lambda_{1}}\in({\mathcal{I}_{1}})^{\bot}$
and ${}\vdash{\Lambda_{2}}\in({\mathcal{I}_{2}})^{\bot}$ such that
$({}\vdash{\Lambda})=({}\vdash{\Lambda_{1},\Lambda_{2}})$. Hence, by
admissibility of weakening (Lem. 3.2) we can derive
${{\mathcal{R}}\vdash{x:A,\Delta,\Lambda_{1},\Lambda_{2}}}$ and
${{\mathcal{R}}\vdash{x:B,\Delta,\Lambda_{1},\Lambda_{2}}}$, from which, an
application of the conjunction rule gives the desired conclusion.
### $\square$-rule
Suppose the proof of
${\mathcal{R}}\vdash{\Gamma\mid\Delta}\mathrel{\|}{\mathcal{I}}$ ends with a
$\square$ rule, i.e., $\Delta=\Delta^{\prime}\cup\\{x:\square A\\}$,
$\mathcal{I}=\square\mathcal{I}^{\prime y}_{x}$, and the proof has the form:
${\mathcal{R}}\vdash{\Gamma\mid x:\square
A,\Delta^{\prime}}\mathrel{\|}{\square\mathcal{I}^{\prime
y}_{x}}{\mathcal{R},Rxy}\vdash{\Gamma\mid
y:A,\Delta^{\prime}}\mathrel{\|}{\mathcal{I}^{\prime}}$
1. 1.
Let ${}\vdash{\Lambda}\in\square\mathcal{I}^{\prime y}_{x}$: we need to show
that ${\mathcal{R}}\Vdash{\Gamma,\Lambda}$. From the definition of
$\square\mathcal{I}^{\prime y}_{x}$, $\Lambda$ must be of the form
$\\{\Lambda^{\prime},x:\square(B_{1}\lor\cdots\lor B_{k})\\}$ where
$({}\vdash{\Lambda^{\prime},y:B_{1},\ldots,y:B_{k}})\in\mathcal{I}^{\prime}.$
The proof of ${\mathcal{R}}\vdash{\Gamma,\Lambda}$ is constructed as follows:
${\mathcal{R}}\vdash{\Gamma,\Lambda^{\prime},x:\square(B_{1}\lor\cdots\lor
B_{k})}{\mathcal{R},Rxy}\vdash{\Gamma,\Lambda^{\prime},y:B_{1}\lor\cdots\lor
B_{k}}{\mathcal{R},Rxy}\vdash{\Gamma,\Lambda^{\prime},y:B_{1},\ldots,y:B_{k}}$
with the premise derivable by the induction hypothesis.
2. 2.
Let ${}\vdash{\Lambda}\in({\square\mathcal{I}^{\prime y}_{x}})^{\bot}$: we
need to show that ${\mathcal{R}}\Vdash{\Lambda,x:\square A,\Delta^{\prime}}$.
In this case, $\Lambda$ could contain zero or more formulae of the form
$x:\overline{\square(B_{1}\lor\cdots\lor B_{k})}$ such that there exists
${}\vdash{\Lambda^{\prime}}\in\mathcal{I}^{\prime}$ with
$\\{y:B_{1},\ldots,y:B_{k}\\}\subseteq\Lambda^{\prime}.$ Let us refer to these
formulae as a boxed-interpolant formulae. We suppose for the sake of
simplicity that exactly one boxed-interpolant formula exists in $\Lambda$ and
that $k=2$; the general case is obtained similarly. We may therefore write
$\Lambda$ as $\\{{x:\overline{\Box(B_{1}\lor B_{2})}}\\}\cup\Theta$.
It follows from our assumptions then that there exists
$\Lambda^{\prime}\in\mathcal{I}^{\prime}$ with
$\\{y:B_{1},y:B_{2}\\}\subseteq\Lambda^{\prime}$. Hence, there must exist a
$\Lambda_{1}\in({\mathcal{I}^{\prime}})^{\bot}$ of the form
$\\{y:\overline{B_{1}}\\}\cup\Theta$ and a
$\Lambda_{2}\in({\mathcal{I}^{\prime}})^{\bot}$ of the form
$\\{y:\overline{B_{2}}\\}\cup\Theta$. By the induction hypothesis,
${\mathcal{R},Rxy}\Vdash{y:\overline{B}_{1},{\Theta},y:A,\Delta^{\prime}}$ and
${\mathcal{R},Rxy}\Vdash{y:\overline{B}_{2},{\Theta},y:A,\Delta^{\prime}}$.
Evoking weakening admissibility from Lem. 3.2, we have that
$\displaystyle{\mathcal{R},Rxy}\Vdash{y:\overline{B}_{1},{\Theta},x:\lozenge(\overline{B}_{1}\land\overline{B}_{2}),y:A,\Delta^{\prime}}$
$\displaystyle{\mathcal{R},Rxy}\Vdash{y:\overline{B}_{2},{\Theta},x:\lozenge(\overline{B}_{1}\land\overline{B}_{2}),y:A,\Delta^{\prime}}$
By using the $\land$ rule with $y:B_{1}$ and $y:B_{2}$ principal, we can
derive
${\mathcal{R},Rxy}\vdash{y:\overline{B}_{1}\land\overline{B}_{2},{\Theta},x:\lozenge(\overline{B}_{1}\land\overline{B}_{2}),y:A,\Delta^{\prime}}.$
The desired result is obtained as follows:
${\mathcal{R}}\vdash{{\Theta},x:\lozenge(\overline{B}_{1}\land\overline{B}_{2}),y:\square
A,\Delta^{\prime}}{\mathcal{R},Rxy}\vdash{{\Theta},x:\lozenge(\overline{B}_{1}\land\overline{B}_{2}),y:A,\Delta^{\prime}}{\mathcal{R},Rxy}\vdash{y:\overline{B}_{1}\land\overline{B}_{2},{\Theta},x:\lozenge(\overline{B}_{1}\land\overline{B}_{2}),y:A,\Delta^{\prime}}$
###### Theorem 3.19.
If ${}\Vdash{x:A\rightarrow B}$, then there exists a $C$ such that (i)
$var(C)\subseteq var(A)\cap var(B)$ and (ii) ${}\Vdash{x:A\rightarrow C}$ and
${}\Vdash{x:C\rightarrow B}$.
###### Proof B.3.
Assume that ${}\Vdash{x:A\rightarrow B}$, i.e. ${}\Vdash{x:\overline{A}\vee
B}$. By Lem. 3.2, we know that ${}\Vdash{x:\overline{A},x:B}$. Therefore, by
Lem. 3.12, we know that there exists an interpolant
$\mathcal{I}=\\{{}\vdash{\Lambda_{1}},\ldots,{}\vdash{\Lambda_{n}}\\}$ with
$\Lambda_{i}=\\{x:A_{i,1},\ldots,x:A_{i,k_{i}}\\}$ for each $1\leq i\leq n$
such that ${}\Vdash{x:\overline{A}\mid x:B}\mathrel{\|}{\mathcal{I}}$,
$var(\mathcal{I})\subseteq var(x:A)\cap var(x:B)$, and all formulae in
$\mathcal{I}$ are labelled with $x$. By Lem. 3.14, the following two
statements hold: (a) for all $(\vdash\Lambda)\in\mathcal{I}$,
${}\Vdash{x:\overline{A},\Lambda}$, (b) for all
$(\vdash\Theta)\in({\mathcal{I}})^{\bot}$, ${}\Vdash{\Theta,x:B}$. We now use
the interpolant $\mathcal{I}$ to construct a $C$ satisfying the conclusion of
the theorem.
The fact that all labelled formulae in $\mathcal{I}$ have the same label $x$,
along with statement (a) and Lem. 3.15, imply that
${}\Vdash{x:\overline{A},x:\bigwedge\bigvee\mathcal{I}}$. Moreover, statement
(b) and Lem. 3.17 imply that
${}\Vdash{x:\overline{\bigwedge\bigvee\mathcal{I}},x:B}$. Applying the $\lor$
rule in both cases implies that
${}\Vdash{x:\overline{A}\vee\bigwedge\bigvee\mathcal{I}}$ and
${}\Vdash{x:\overline{\bigwedge\bigvee\mathcal{I}}\vee B}$, thus giving the
desired result with $\bigwedge\bigvee\mathcal{I}$ the interpolant. Last, note
that since $\bigwedge\bigvee\mathcal{I}$ is our interpolant, and
$var(\mathcal{I})\subseteq var(x:A)\cap var(x:B)$ holds,
$var(\bigwedge\bigvee\mathcal{I})\subseteq var(A)\cap var(B)$ will be
satisfied as well.
###### Lemma B.4.
The following two rules are derivable in $\mathsf{BiIntL}$:
${\mathcal{R},Rxy,\Gamma,x:A\supset
B}\vdash{\Delta}\lx@proof@logical@and{\mathcal{R},Rxy,\Gamma,x:A\supset
B}\vdash{\Delta,y:A}{\mathcal{R},Rxy,\Gamma,y:B}\vdash{\Delta}$
${\mathcal{R},Ryx,\Gamma}\vdash{x:A-\\!\\!\\!<B,\Delta}\lx@proof@logical@and{\mathcal{R},Ryx,\Gamma}\vdash{\Delta,y:A}{\mathcal{R},Rxy,\Gamma,y:B}\vdash{x:A-\\!\\!\\!<B,\Delta}$
###### Proof B.5.
We prove the claim for the $\supset^{*}_{L}$ rule; the proof for
$-\\!\\!\\!<^{*}_{R}$ is similar. Assume that
${\mathcal{R},Rxy,\Gamma,x:A\supset B}\Vdash{\Delta,y:A}$ and
${\mathcal{R},Rxy,\Gamma,y:B}\Vdash{\Delta}$. By evoking admissibility of
weakening (Lem. 4.1), we obtain proofs of the following sequents:
$\displaystyle{\mathcal{R},Rxy,\Gamma,x:A\supset B,y:A\supset
B}\vdash{\Delta,y:A}\qquad$ $\displaystyle{\mathcal{R},Rxy,\Gamma,x:A\supset
B,y:B}\vdash{\Delta}$
By applying $\supset_{L}$ with the above sequents, we obtain the premise
below; the desired conclusion is derived with one application of $monl$:
${\mathcal{R},Rxy,\Gamma,x:A\supset
B}\vdash{\Delta}{\mathcal{R},Rxy,\Gamma,x:A\supset B,y:A\supset
B}\vdash{\Delta}$
###### Lemma B.6.
The rule
$\mathcal{R},\Gamma\vdash\Delta\mathcal{R},Rxx,\Gamma\vdash\Delta$
is admissible in $\mathsf{BiIntL}$.
###### Proof B.7.
We prove the result by induction on the height of the derivation of
$\mathcal{R},Rxx,\Gamma\vdash\Delta$. The base case is simple since if
$\mathcal{R},Rxx,\Gamma\vdash\Delta$ is an initial sequent, then so is
$\mathcal{R},\Gamma\vdash\Delta$. For all rules, with the exception of the
$monl$ and $monr$ rule, the inductive step is proven by applying the inductive
hypothesis followed by an application of the rule. We therefore focus on the
above two cases:
### $monl$-, $monr$-rules
If $Rxx$ is not principal in a $monl$ or $monr$ inference, then the case is
resolved by applying the inductive hypothesis followed by an application of
the rule. However, if $Rxx$ is principal, then the inferences are of the
following forms:
$\mathcal{R},Rxx,x:A,x:A,\Gamma\vdash\Delta$ $\mathcal{R},Rxx,x:A,\Gamma\vdash\Delta$ | $\mathcal{R},Rxx,\Gamma\vdash x:A,x:A,\Delta$ $\mathcal{R},Rxx,\Gamma\vdash x:A,\Delta$
---|---
Applying the inductive hypothesis to the premises, followed by an application
of $ctr$ admissibility (Lem. 4.1), gives the desired conclusion.
###### Lemma 4.13.
For all $\mathcal{R},\Gamma_{1},\Gamma_{2},\Delta_{1},\Delta_{2}$ and
$\mathcal{I}$, if
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\Vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}$,
then
1. 1.
For all $({\Sigma}\vdash{\Theta})\in\mathcal{I}$, we have
${\mathcal{R},\Gamma_{1},\Sigma}\Vdash{\Theta,\Delta_{1}}$ and
2. 2.
For all $({\Lambda}\vdash{\Omega})\in({\mathcal{I}})^{\bot}$, we have
${\mathcal{R},\Gamma_{2},\Lambda}\Vdash{\Omega,\Delta_{2}}.$
###### Proof B.8.
By induction on the height of proofs; we prove the result for a representative
set of cases since all others are simple or shown similarly. Moreover, the
base cases are trivial, so we focus solely on the inductive step, where we
show the cases for $\lor_{L}$ and $\supset_{R}$ since all other cases are
similar or simple.
### $\lor_{L}$-rule
Suppose that our derivation ends with the upmost $\lor_{L}$ in Fig. 8.
1. 1.
Let $({\Sigma}\vdash{\Theta})\in\mathcal{I}_{1}\cup\mathcal{I}_{2}$. By the
inductive hypothesis, we have that for all
$({\Sigma}\vdash{\Theta})\in\mathcal{I}_{1}$,
${\mathcal{R},\Gamma_{1},\Sigma}\Vdash{\Delta_{1},\Theta}$ and for all
$({\Sigma}\vdash{\Theta})\in\mathcal{I}_{2}$,
${\mathcal{R},\Gamma_{1},\Sigma}\Vdash{\Delta_{1},\Theta}$. Therefore,
regardless of if $({\Sigma}\vdash{\Theta})$ is in $\mathcal{I}_{1}$ or
$\mathcal{I}_{2}$, we have that
${\mathcal{R},\Gamma_{1},\Sigma}\Vdash{\Delta_{1},\Theta}$.
2. 2.
Let
$({\Sigma}\vdash{\Theta})\in({\mathcal{I}_{1}\cup\mathcal{I}_{2}})^{\bot}$. By
the inductive hypothesis, we have that for all
$({\Sigma}\vdash{\Theta})\in({\mathcal{I}_{1}})^{\bot}$,
${\mathcal{R},x:A,\Gamma_{2},\Sigma}\Vdash{\Delta_{2},\Theta}$ and for all
$({\Sigma}\vdash{\Theta})\in({\mathcal{I}_{2}})^{\bot}$,
${\mathcal{R},x:B,\Gamma_{2},\Sigma}\Vdash{\Delta_{2},\Theta}$. Observe that
by the definition of orthogonal, there will be a
$({\Sigma_{1}}\vdash{\Theta_{1}})$ and $({\Sigma_{2}}\vdash{\Theta_{2}})$ such
that
$({\Sigma_{1},\Sigma_{2}}\vdash{\Theta_{1},\Theta_{2}})\in({\mathcal{I}_{1}\cup\mathcal{I}_{2}})^{\bot}$.
Hence, by the inductive hypothesis and the admissibility of weakening (Lem.
4.1) we may derive
${\mathcal{R},x:A,\Gamma_{2},\Sigma_{1},\Sigma_{2}}\vdash{\Delta_{2},\Theta_{1},\Theta_{2}}\quad\mbox{and}\quad{\mathcal{R},x:B,\Gamma_{2},\Sigma_{1},\Sigma_{2}}\Vdash{\Delta_{2},\Theta_{1},\Theta_{2}}.$
One application of the $\lor_{L}$ rule gives the desired conclusion.
### $\supset_{R}$-rule
Assume that our derivation of
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid\Delta_{2}}\mathrel{\|}{\mathcal{I}}$
ends with a $\supset_{R}$ rule, where $\mathcal{I}\ =\
\supset{\mathcal{I}^{\prime}}^{y}_{x}$ and the last inference has the form:
${\mathcal{R},\Gamma_{1}\mid\Gamma_{2}}\vdash{\Delta_{1}\mid x:A\supset
B,\Delta_{2}}\mathrel{\|}{\supset\mathcal{I}^{\prime
y}_{x}}{\mathcal{R},Rxy,\Gamma_{1}\mid y:A,\Gamma_{2}}\vdash{\Delta_{1}\mid
y:B,\Delta_{2}}\mathrel{\|}{\mathcal{I}^{\prime}}$
1. 1.
Let $({\Sigma}\vdash{\Theta})\in\mathcal{I}$. We want to show that
${\mathcal{R},\Gamma_{1},\Sigma}\Vdash{\Delta_{1},\Theta}$. Note that
${\Sigma}\vdash{\Theta}$ is of the form:
${\Sigma}\vdash{x:C_{1}\wedge...\wedge C_{k}\supset D_{1}\vee...\vee
D_{n},\Theta^{\prime}},$
where $\Theta=x:C_{1}\wedge...\wedge C_{k}\supset D_{1}\vee...\vee
D_{n},\Theta^{\prime}$. This implies that there exists a
$({\Lambda}\vdash{\Omega})\in\mathcal{I}^{\prime}$ of the form
${\Sigma,y:C_{1},...,y:C_{k}}\vdash{y:D_{1},...,y:D_{n},\Theta^{\prime}}$
Therefore, the inductive hypothesis implies that the premise below is
derivable:
${\mathcal{R},\Gamma_{1},\Sigma}\vdash{x:C_{1}\wedge...\wedge C_{k}\supset
D_{1}\vee...\vee
D_{n},\Theta^{\prime},\Delta_{1}}{\mathcal{R},Rxy,\Gamma_{1},\Sigma,y:C_{1}\wedge...\wedge
C_{k}}\vdash{y:D_{1}\vee...\vee
D_{n},\Theta^{\prime},\Delta_{1}}{\mathcal{R},Rxy,\Gamma_{1},\Sigma,y:C_{1},...,C_{k}}\vdash{y:D_{1},...,y:D_{n},\Theta^{\prime},\Delta_{1}}$
The last $\supset_{R}$ inference may be applied because, by our assumption,
$y$ does not occur in
$\mathcal{R},\Gamma_{1},\Sigma,\Theta^{\prime},\Delta_{1}$.
2. 2.
Let $({\Sigma}\vdash{\Theta})\in({\mathcal{I}})^{\bot}$. Observe that $\Sigma$
will contain zero or more formulae of the form $x:C_{1}\land...\land
C_{k}\supset D_{1}\lor...\lor D_{n}$, which we refer to as an implication-
interpolant formula, such that there exists a sequent in
$\mathcal{I}^{\prime}$ of the form:
${\Sigma^{\prime},y:C_{1},...,y:C_{k}}\vdash{y:D_{1},...,y:D_{n},\Theta}.$
where $\Sigma^{\prime}$ is equal to $\Sigma$ minus all implication-interpolant
formulae. We assume for the sake of simplicity that $\Sigma$ contains one
implication-interpolant formula, that $k=2$, and $n=2$; the general case is
tedious, but shown similarly. ${\Sigma}\vdash{\Theta}$ is therefore of the
form:
${x:C_{1}\land...\land C_{k}\supset D_{1}\lor...\lor
D_{n},\Sigma^{\prime}}\vdash{\Theta}.$
It follows from our assumptions that there exists a
$({\Lambda}\vdash{\Omega})\in\mathcal{I}^{\prime}$ of the form
${\Sigma^{\prime},y:C_{1},y:C_{2}}\vdash{y:D_{1},y:D_{2},\Theta}.$
Hence, by the definition of the orthogonal, the following four sequents are
members of $({\mathcal{I}^{\prime}})^{\bot}$:
${\Sigma^{\prime}}\vdash{\Theta,y:C_{1}}\qquad{\Sigma^{\prime}}\vdash{\Theta,y:C_{2}}\qquad{y:D_{1},\Sigma^{\prime}}\vdash{\Theta}\qquad{y:D_{2},\Sigma^{\prime}}\vdash{\Theta}$
By the inductive hypothesis the following four claims hold:
$\displaystyle{\mathcal{R},Rxy,y:A,\Gamma_{2},\Sigma^{\prime}}\Vdash{\Theta,\Delta_{2},y:B,y:C_{1}}$
$\displaystyle{\mathcal{R},Rxy,y:A,\Gamma_{2},\Sigma^{\prime}}\Vdash{\Theta,\Delta_{2},y:B,y:C_{2}}$
$\displaystyle{\mathcal{R},Rxy,y:A,y:D_{1},\Gamma_{2},\Sigma^{\prime}}\Vdash{\Theta,\Delta_{2},y:B}$
$\displaystyle{\mathcal{R},Rxy,y:A,y:D_{2},\Gamma_{2},\Sigma^{\prime}}\Vdash{\Theta,\Delta_{2},y:B}$
Applying the $\land_{R}$ rule to the first two sequents with $y:C_{1}$ and
$y:C_{2}$ active, and the $\lor_{L}$ rule to the latter two sequents with
$y:D_{1}$ and $y:D_{2}$ active, gives derivations of the following two
sequents:
$\displaystyle{\mathcal{R},Rxy,y:A,\Gamma_{2},\Sigma^{\prime}}\vdash{\Theta,\Delta_{2},y:B,y:C_{1}\land
C_{2}}$ $\displaystyle{\mathcal{R},Rxy,y:A,y:D_{1}\lor
D_{2},\Gamma_{2},\Sigma^{\prime}}\vdash{\Theta,\Delta_{2},y:B}$
By admissibility of weakening (Lem. 4.1), we may weaken in $x:C_{1}\land
C_{2}\supset D_{1}\lor D_{2}$ on the left side of both sequents and apply the
$\supset^{*}_{L}$ rule (Lem. B.4) to obtain a derivation of:
${\mathcal{R},Rxy,y:A,x:C_{1}\land C_{2}\supset D_{1}\lor
D_{2},\Gamma_{2},\Sigma^{\prime}}\vdash{\Theta,\Delta_{2},y:B}$
An application of $\supset_{R}$ with $Rxy$, $y:A$, and $y:B$ principal gives
the desired result.
###### Lemma 4.14.
Let
$\mathcal{I}=\\{({\Sigma_{1}}\vdash{\Theta_{1}}),\ldots,({\Sigma_{n}}\vdash{\Theta_{n}})\\}$
be an interpolant with
$({\Sigma_{i}}\vdash{\Theta_{i}})=({x:C_{i,1},\ldots,x:C_{i,k_{i}}}\vdash{x:D_{i,1},\ldots,x:D_{i,j_{i}}})\mbox{
for each }1\leq i\leq n.$
If ${\Sigma_{i},\Gamma}\Vdash{\Theta_{i}}$, for all
$({\Sigma_{i}}\vdash{\Theta_{i}})\in\mathcal{I}$, and every formula in
$\Gamma$ is labelled with $x$, then
${\Gamma}\Vdash{x:\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m}}).$
###### Proof B.9.
By assumption,
${x:C_{i,1},\ldots,x:C_{i,k_{i}},\Gamma}\Vdash{x:D_{i,1},\ldots,x:D_{i,j_{i}}}$
for all $1\leq i\leq n$. By repeated application of $\land_{L}$ and
$\lor_{R}$, we obtain a derivation of the following sequent for each $1\leq
i\leq n$:
${x:\bigwedge_{m=1}^{k_{i}}C_{i,m},\Gamma}\vdash{x:\bigvee_{m=1}^{j_{i}}D_{i,m}}$
Let $y$ be a fresh variable and let $\Gamma^{\prime}=\Gamma[y/x]$, i.e., the
multiset of labelled formulae $\Gamma$ but with every formula labelled with
$y$ instead of $x$. By the admissibility of weakening (Lem. 4.1) and the
derivability of the above sequents, we may derive the following:
${Ryx,x:\bigwedge_{m=1}^{k_{i}}C_{i,m},\Gamma,\Gamma^{\prime}}\vdash{x:\bigvee_{m=1}^{j_{i}}D_{i,m}}$
for all $1\leq i\leq n$. We construct derivations from the derivations of the
above sequents by adding the following inferences to the bottom of each:
${\Gamma^{\prime}}\vdash{y:\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m}}{Ryx,x:\bigwedge_{m=1}^{k_{i}}C_{i,m},\Gamma^{\prime}}\vdash{x:\bigvee_{m=1}^{j_{i}}D_{i,m}}{Ryx,x:\bigwedge_{m=1}^{k_{i}}C_{i,m},\Gamma,\Gamma^{\prime}}\vdash{x:\bigvee_{m=1}^{j_{i}}D_{i,m}}$
for all $1\leq i\leq n$. By successively applying $\land_{R}$ we may derive
the following:
${\Gamma^{\prime}}\vdash{y:\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m}})$
By the fact that the above sequent is derivable and by statement (4) of Lem.
4.1, we obtain the desired conclusion:
${\Gamma}\Vdash{x:\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m})}$
###### Theorem 4.17.
If ${}\Vdash{x:A\supset B}$, then there exists a $C$ such that (i)
$var(C)\subseteq var(A)\cap var(B)$ and (ii) ${}\Vdash{x:A\supset C}$ and
${}\Vdash{x:C\supset B}$.
###### Proof B.10.
Suppose that ${}\Vdash{x:A\supset B}$. By Lem. 4.1 (2), (4) and Lem. B.6, this
implies that ${x:A}\Vdash{x:B}$. By Lem. 4.11, we know there exists an
$\mathcal{I}=\\{({\Sigma_{1}}\vdash{\Theta_{1}}),\ldots,({\Sigma_{n}}\vdash{\Theta_{n}})\\}$
with
$({\Sigma_{i}}\vdash{\Theta_{i}})=({x:C_{i,1},\ldots,x:C_{i,k_{i}}}\vdash{x:D_{i,1},\ldots,x:D_{i,j_{i}}})$
for each $1\leq i\leq n$ such that ${x:A\mid\cdot}\Vdash{\cdot\mid
x:B}\mathrel{\|}{\mathcal{I}}$, $var(\mathcal{I})\subseteq var(x:A)\cap
var(x:B)$, and $x$ is the only label occurring in $\mathcal{I}$. By Lem. 4.13,
the following two statements hold: (a) for all
$({\Sigma}\vdash{\Theta})\in\mathcal{I}$, we have
${x:A,\Sigma}\Vdash{\Theta}$, and (b) for all
$({\Sigma}\vdash{\Theta})\in({\mathcal{I}})^{\bot}$, we have
${\Sigma}\Vdash{\Theta,x:B}$.
We now use the interpolant $\mathcal{I}$ to construct a formula $C$ satisfying
the conditions specified in the claim of the theorem.
By the assumption that $\mathcal{I}$ only contains formulae with the label $x$
along with statement (a) and Lem. 4.14, we obtain the following:
${x:A}\Vdash{x:\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m})}$
Additionally, statement (b) and Lem. 4.15 imply the following:
${x:\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m})}\Vdash{x:B}$
By weakening admissibility (Lem. 4.1), we may weaken both sequents with $Ryx$
and apply $\supset_{R}$ to obtain the following:
${}\Vdash{y:A\supset\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m})}$
${}\Vdash{y:\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m})\supset
B}$
Evoking statement (4) of Lem. 4.1 allows us to replace the label $y$ with $x$.
Last, since $var(\mathcal{I})\subseteq var(x:A)\cap var(x:B)$, it is easy to
see that
$\bigwedge_{i=1}^{n}(\bigwedge_{m=1}^{k_{i}}C_{i,m}\supset\bigvee_{m=1}^{j_{i}}D_{i,m})$
is our formula interpolant for $A$ and $B$.
|
Moving bottom detection]Optimal control approach for moving bottom detection in one-dimensional shallow waters by surface measurements
R. Lecaros]R. Lecaros
[R. Lecaros]Departamento de Matemática, Universidad Técnica Federico Santa María, Santiago, Chile.
J. López-Ríos]J. López-Ríos
[J. López-Ríos](Corresponding Author)Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia
G. I. Montecinos]G. I. Montecinos
[G. I. Montecinos]Department of Mathematical Engineering, Universidad de La Frontera, Temuco, Chile
E. Zuazua]E. Zuazua
[E. Zuazua]Chair in Dynamics, Control and Numerics / Alexander von Humboldt-Professorship
Department of Data Science
Friedrich-Alexander-Universität Erlangen-Nürnberg
91058 Erlangen, Germany
Chair of Computational Mathematics
Fundación Deusto
Av. de las Universidades, 24
48007 Bilbao, Basque Country, Spain
Departamento de Matemáticas
Universidad Autónoma de Madrid
28049 Madrid, Spain
R. Lecaros, J. López-Ríos and G. Montecinos have been partially supported by the Math-Amsud project
CIPIF 22-MATH-01. R. Lecaros was partially supported by FONDECYT (Chile) Grant 1221892. J. López-Ríos acknowledges support by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, project 3752.
E. Zuazua has been funded by the Alexander von Humboldt-Professorship program, the Transregio 154 Project “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks" of the DFG, the ModConFlex Marie Curie Action, HORIZON-MSCA-2021-$d$N-01, the COST Action MAT-DYN-NET, grants PID2020-112617GB-C22 and TED2021-131390B-I00 of MINECO (Spain), and by the Madrid Goverment – UAM Agreement for the Excellence of the University Research Staff in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).
We consider the Boussinesq-Peregrine (<ref>) system as described by Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water regime, and study the inverse problem of determining the time and space variations of the channel bottom profile, from measurements of the wave profile and its velocity on the free surface. A well-posedness result within a Sobolev framework for (<ref>), considering a time dependent bottom, is presented. Then, the inverse problem is reformulated as a nonlinear PDE-constrained optimization one. An existence result of the minimum, under constraints on the admissible set of bottoms, is presented. Moreover, an implementation of the gradient descent approach, via the adjoint method, is considered. For solving numerically both, the forward (<ref>) and its adjoint system, we derive a universal and low-dissipation scheme, which contains non-conservative products. The scheme is based on the FORCE-$\alpha$ method proposed in [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form. Journal of Computational Physics, 416, 109545]. Finally, we implement this methodology to recover three different bottom profiles; a smooth bottom, a discontinuous one, and a continuous profile with a large gradient. We compare with two classical discretizations for (<ref>) and the adjoint system. These results corroborate the effectiveness of the proposed methodology to recover bottom profiles.
[2020]49K20; 76B15; 76B03; 65M08
§ INTRODUCTION
The problem of wave generation and the forces that originate them comprise a wide area of study, involving modeling through fundamental laws, numerical resolution of equations and laboratory scale tests. In the particular case of water in the ocean, it is of interest to study those waves produced by displacements and changes in the seabed, a task that has been developed from many points of view in the classical literature [7, 9, 14, 27] and is currently a very active area of research [6, 19, 30, 33].
In this paper we investigate, theoretically and numerically, the effect of ocean bottom motions on waves. More precisely, within the context of shallow water models, we analyze the inverse problem of finding a single bottom, which depends on the time variable, and which generates a specific wave. These types of problems have been considered before, in the context of unidirectional shallow water approximating models [5, 27] and in a broader sense, through wave generating devices (wave-makers) such as those considered in [25, 34, 35] for the water-waves equation.
The problem of generating waves through seafloor displacements is directly related to the generation of tsunamis by submarine earthquakes and plate displacements (see [9, 14, 29]). Since it is important to quantify both types of displacements, vertical and horizontal, in this work we follow an approach that allows considering both of them, as reported in [10, 14, 28]. Moreover, from the point of view of applications, there are man-made facilities for surfing away from the ocean [1], where a wave with specific characteristics is created by the unidirectional motion of an underwater rigid object, moving in a preset direction.
We put ourselves in the context of wave propagation in a shallow water channel, so that the dynamics correspond to the laws of conservation of momentum and mass [17]. In this general framework, the interaction between the rigid boundary, represented by the bottom of the channel, with the free surface can be well understood and characterized through terms representing a boundary condition or an external source in the Partial Differential Equation. In particular, we will use the shallow water approximation of the general system of water-waves, with a time-varying bottom, known as the Boussinesq-Peregrine (<ref>) model with varying bottom, as derived in [17] (see details in section <ref>).
The main objective of this work is to provide a general theoretical framework, based on conservation laws, to analyze the inverse problem of detecting a seafloor moving in time. Moreover, we propose an efficient computational procedure to fully determine the variations of the channel (ocean) bottom in time and space.
For that, we chose to study the (<ref>) system, which is general enough and contains the main characteristics of shallow water systems; it also includes terms corresponding to a time-varying bottom and in this context has not been studied before. We study three main aspects: first, the existence of solutions of the system in the framework of Sobolev spaces. Second, the theoretical identification of the bottom, as the solution to a minimization problem, involving the state equation and its adjoint system. Third, the implementation of a descent algorithm and a finite volume scheme discretization of the equation and its adjoint system.
Existence and uniqueness results for the (<ref>) system have been obtained in Sobolev spaces [15, 17], for a steady bottom. In particular, a classical energy approach has been employed in [15] applied to a symmetrizable hyperbolic system. We provide a well-posedness theorem for the local existence of solutions to (<ref>), when the bottom is time-dependent, which has not been reported so far. On the extensive literature concerning well posedness for the general system of water-waves, we mention Nalimov [26], Yosihara [40], Craig [8], S.Wu [38, 39] and Lannes [16].
Optimal control problems on water-waves have been studied in [12], in the context of the inverse problem of bottom identification through surface measurements, where the authors addressed the identifiability and set the problem of finding a unique bottom minimizing the $L^2$ norm in the horizontal variable of the Dirichlet-Neumann operator. Concerning the (<ref>) system and the bathymetry detection, we mention the work by Dutykh et al. [27], where the authors reduced the problem to the unidirectional wave propagation to get a Benjami-Bona-Mahony (BBM)-type equation and studied the optimization problem of generating the largest possible wave in the $L^2$ sense within a given bounded interval, by a constant velocity moving bottom.
Our numerical procedure to find the bottom profiles involves two hyperbolic systems: the system (<ref>) for modeling the wave problem, and its adjoint which contains non-conservative products. We propose a numerical scheme, for both (<ref>) and the adjoint system, which is universal in the sense that it applies for both conservative and non-conservative systems. We consider two additional schemes aimed at evaluating how a classical combination of conservative and non-conservative schemes behaves for solving the problem. We refer to them as reference schemes since they are viable discretizations that users can employ. The first one consists of using the conservative scheme of Rusanov [32] (also called Local Lax-Friedrich flux) for (<ref>) and the finite difference discretization for the adjoint system. The second scheme consists of the conservative scheme of Rusanov for (<ref>) and the non-conservative one, for the adjoint system.
A related formulation to the one we are studying is the controllability problem by a source, for the (<ref>) system. Alazard et al. [2] have provided a local exact controllability result, obtained for the full water-waves system by controlling a localized portion of the free surface. In [11], the authors proved the interior exact controllability of the 2d-Euler's system by injection of jet fluids through a rigid boundary. We can also mention the works on KdV [31], BBM [21, 41], and some Boussinesq–type systems [22], concerning some literature about controllability of asymptotic regimes and dispersive wave equations. Finally, even though the computation of the adjoint state in the optimal control problem approach follow classical ideas, it is worth to mention that this has not been done before in this context of moving bottoms.
The paper is organized as follows. In section <ref>, we present the (<ref>) system and the formulation of the optimal control problem. In section <ref>, we set the optimal control problem and prove the existence of a minimum. In section <ref>, a particular formulation for recovering a moving bottom is settled. In section <ref>, the coupled formulation approach and the numerical method for solving the inverse problem, are presented. In section <ref>, numerical results are reported. Finally, in section <ref>, conclusions are drawn.
§ THE MATHEMATICAL FRAMEWORK
We consider an ideal incompressible fluid in a two dimensional domain, (see Figure <ref>). We denote the horizontal independent variable by $x$ and the vertical variable by $y$, and assume that the line $y=1$ corresponds to the still water level and $H_0=1$ is a constant reference depth. Let $\zeta,b:[0,T]\times\R\rightarrow\R$ ($T>0$), be the surface and bottom parameterizations respectively. Let $\mu$ be a small parameter to be specified later and related to the shallow water regime and $\epsilon=O(\mu)$ which represents the small variations of the bottom and free surface, as explained in [17]. We assume that the total depth $h\equiv h(t,x)=1+\epsilon(\zeta-b)$, remains positive at all times $t$:
\begin{equation}
\label{ES34}
\exists h_{min}>0, \ \inf_{x\in\R}h\ge \hm,
\end{equation}
which is a necessary condition for the system (<ref>) to be valid.
For a given bottom $b(t,x)$, we are going to consider, on $[0,T]\times\R$, the one dimensional Boussinesq-Peregrine system with moving bottom for $(\zeta,V)$
\begin{equation}
\label{BP}
\tag{BP}
\left\{
\begin{aligned}
&\zeta_t+(hV)_x=b_t, \\
&\left(1-\frac{\mu}{3h}\partial_x(h^3\partial_x\cdot)\right)V_t+\zeta_x+\epsilon VV_x=-\frac{\epsilon}{2}b_{ttx}, \\
&\zeta(0,\cdot)=\zeta_0(\cdot), \ V(0,\cdot)=V_0(\cdot), \ \text{in }\R,
\end{aligned}
\right.
\end{equation}
with $V:[0,T]\times\R\rightarrow\R$ being the total depth averaged velocity. From now on, this will be our model to understand the wave-bottom interaction. Recall that $(\zeta,V)$ are the unknown free surface elevation and velocity, respectively, and $b$ is a given function, representing the topography of the moving bottom. Finally, $\mu$ and $\epsilon$ are dimensionless, small parameters, representing the shallowness of the channel.
Sketch of the physical domain and main notation
System (<ref>) can be derived from the general water waves equations under the shallow water assumption ($\mu\ll1$), assuming $\epsilon=O(\mu)$. See <cit.> for a two dimensional version of (<ref>), and [18], where many of the shallow water models are deduced from the general water waves system.
We now turn our attention to the optimization problem for system (<ref>). We assume that the bottom can change during a time interval over the entire domain; namely, the time-dependent bathymetry is given by $b=b(t,x)$. The detection of the bottom will be obtained as the solution to an optimization problem. That is, we will find, theoretically and numerically, the function $b$ to watch a prescribed wave and velocity $\overline{\zeta}$, $\overline{V}$, at some fixed time $T>0$ (see sections <ref> and <ref>). Mathematically, for fixed $\zeta_0$ and $V_0$, we minimize the functional
\begin{equation}
\end{equation}
where $\bar{\zeta},\bar{V}\in L^2((0,T);L^2(\R))$ are given, $(\zeta,V)$ is a solution to (<ref>) in $C([0,T];H^s\times H^{s+1})$, and the Sobolev space $H^s=H^s(\R)$, $s\ge 3/2$ (see <ref> for details).
Note the explicit way the time derivatives of $b$ appear in (<ref>), at the right-hand side in the $b_t,b_{ttx}$ terms. We start by formulating an existence and uniqueness theorem for (<ref>), so that the regularity of the bottom be explicit. This is needed when formulating the problem of finding $b$ by the optimal control approach.
Let $s>3/2$ and $(\zeta_0,V_0)\in \mathbf{X}^s$. Let $b\in W^{2,\infty}([0,\infty);H^{s+1})$ and assume (<ref>) is valid. Then, there exists $T_{BP}>0$, uniformly bounded from below with respect to $\epsilon$, such that system (<ref>) admits a unique solution $(\zeta,V)^T\in C([0,T_{BP}/\epsilon];\mathbf{X}^s)$ with initial condition $(\zeta_0,V_0)^T$.
Here we have introduced the space $\mathbf{X}^s$, defined as
\[ \mathbf{X}^s=H^s(\R)\times H^{s+1}(\R), \]
endowed with the norm
\[ \forall \mathbf{U}\in \mathbf{X}^s, \ |\mathbf{U}|_{\mathbf{X}^s}^2=|\zeta|_{H^s}^2+|V|_{H^s}^2+\mu|V_x|_{H^s}^2. \]
For the sake of clearness in the presentation, we leave the details of the proof for <ref>.
When the water waves equations are nondimensionalized, a rough analysis of their linearization around the rest state is performed. This shows the relevance of various dimensionless parameters, namely, the amplitude parameter $\epsilon$, the shallowness parameter $\mu$, and the topography parameter $\beta$ (see [17] for details). With the relevant physical dimensionless parameters introduced, asymptotic regimes are identified (the shallow water regime for instance) as conditions on these dimensionless parameters (e.g., $\mu<<1$ for the shallow water regime).
The asymptotic regime that corresponds to the system (<ref>) is precisely
\[ \{(\epsilon,\beta\,\mu):0\le\mu\le\mu_0, \ 0\le\beta\le\epsilon\le\mu\}. \]
That is the reason why we decided to keep the two parameters $\mu$ and $\epsilon$ in the formulation of (<ref>) above.
§ THE OPTIMAL CONTROL APPROACH
Following the explanations in the previous section, the process of minimizing the functional $J$ will allow us to design the bottom $b(t,x)$, given a specific surface profile and its velocity, $(\overline{\zeta},\overline{V})$. Therefore we will address two practical questions. The first is tsunami detection with surface measurements, where we also include the possibility of horizontal and vertical displacements caused by plate motions and underwater earthquakes [9]. Second, the design of moving underwater structures to generate specific waves for an entertainment purpose [27]. With this in mind, we present the following mathematical formulation, and the sufficient conditions for a bottom $b$ to exist, minimizing functional $J$.
The first thing we need is an existence and uniqueness of solutions theorem, for system (<ref>). Because the proof strategy is classical and follows arguments similar to those presented in [15, 17] for the nonlinear shallow water and green-naghdi systems, respectively, we postpone the proof to an appendix and concentrate on the optimization problem.
We consider the space $\mathbf{X}_T^s:=C([0,T/\epsilon];\mathbf{X}^s)$ endowed with its canonical norm. Let $\mathcal{F}\subset W^{2,\infty}(0,T;H^{s+1})$ be a nonempty, closed and convex set. Given $d_0>\frac{1}{2}$, $s\ge d_0+1$ and $(\zeta_0,V_0)\in \mathbf{X}^s$, let the following minimization problem related to (<ref>):
\begin{equation}
\label{P}
\tag{P}
\left\{
\begin{aligned}
&\text{Find $(\zeta,V,b)\in \mathbf{X}^s\times\mathcal{F}$ such that the functional} \\
& J(b)=\frac{1}{2}\int_0^T|\zeta-\bar{\zeta}|_{L^2}^2dt+\frac{1}{2}\int_0^T|V-\bar{V}|_{L^2}^2dt \\
&\text{is minimized, subject to } \ (\zeta,V,b) \ \text{satisfies } \eqref{BP}.
\end{aligned}
\right.
\end{equation}
Here $(\bar{\zeta},\bar{V})\in L^2(0,T;L^2)\times L^2(0,T;L^2)$ represents the desired state.
The set of optimal solutions of (<ref>) is defined by
\[ \mathcal{S}_{ad}=\{\omega=(\zeta,V,b)\in X^s\times\mathcal{F}: \omega \text{ is a solution of }\eqref{BP}\}. \]
Note that $b$ is not acting on the right–hand side of (<ref>) only (through $b_t$, $b_{ttx}$), but the spatial derivatives of the terms involving $h$. That is, $b$ acts on the source and coefficients.
Thanks to the remark <ref> in the appendix, we can take $\epsilon$ sufficiently small, so that the existence time of the solutions is guaranteed in a time interval containing $[0,T]$. This is important when passing to the limit in a minimizing sequence for proving the existence of a minimum for the functional $J$, as the next theorem shows.
Let $(\zeta_0,V_0)^T\in \mathbf{X}^s$ with $s\ge 1+\delta$, $\delta>0$. Let $T>0$ such that $\mathcal{F}$ is bounded in $W^{2,\infty}(0,T;H^{s+1})$. Then the optimal control problem (<ref>) has at least one global optimal solution $(\zeta^*,V^*,b^*)\in\mathcal{S}_{ad}$.
From Theorem <ref> in the appendix, we deduce that $\mathcal{S}_{ad}\ne\emptyset$. Let $\{\omega^n\}_{n\in\mathbb{N}}=\{(\zeta^n,V^n,b^n)\}_{n\in\mathbb{N}}\subset\mathcal{S}_{ad}$ a minimizing sequence of $J$. Let $\mathbf{U}^n(t,x)$ be the corresponding solution of problem (<ref>) with bottom $b^n(t,x)$. From the definition of $J$ and the assumption $\mathcal{F}$ bounded in $W^{2,\infty}(0,T;H^{s+1})$,
\[ \{b^n\} \text{ is bounded in } W^{2,\infty}(0,T;H^{s+1}). \]
We know $\mathcal{F}$ is weakly closed in $W^{2,\infty}(H^{s+1})$. Then, from Theorem <ref> and the corresponding estimate as in (<ref>), there exists $\omega^*=(\zeta^*,V^*,b^*)\in\mathcal{S}_{ad}$ such that, for some subsequence of $\{\omega^n\}$, still denoted $\{\omega^n\}_{n\in\mathbb{N}}$, we have weak convergences in $\mathbf{X}^s\times W^{2,\infty}(0,T;H^{s+1})$.
If $\mathbf{U}^{n,m}:=\mathbf{U}^n-\mathbf{U}^m$ we have (see Remark <ref>)
\begin{equation*}
\begin{pmatrix}
1 & 0 \\ 0 & \mt\cdot
\end{pmatrix}
\begin{pmatrix}
\zeta^{n,m}_t \\ V^{n,m}_t
\end{pmatrix}
\begin{pmatrix}
\epsilon V^n & h^n \\ h^n & \mt(\epsilon V^n\cdot)
\end{pmatrix}
\begin{pmatrix}
\zeta^{n,m}_x \\ V^{n,m}_x
\end{pmatrix}
\begin{pmatrix}
b^{n,m}_t+\epsilon b^{n,m}_xV^m+(\zeta_x^m-b_x^n)V^{n,m} \\ -\frac{\epsilon}{2}h^nb^{n,m}_{ttx}-\frac{\epsilon}{2}h^{n,m}b^m_{ttx}-\zeta_x^mh^{n,m}
\end{pmatrix}.
\end{equation*}
Then, multiplying last equation in $L^2$ by $\mathbf{U}^{n,m}$, integrating by parts and applying the Gronwall inequality (as is done in the proof of Proposition <ref> in the appendix), we obtain that $\mathbf{U}^{n,m}$ is a Cauchy sequence in $H^1$. Hence $\omega^*$ is solution of (<ref>), pointwisely; that is, $\omega^*\in\mathcal{S}_{ad}$. Therefore,
\[ \lim_{n\to+\infty}J(\omega_n)=\inf_{\omega\in\mathcal{S}_{ad}}J(\omega_n)\le J(\omega^*). \]
On the other hand, since $J$ is lower semicontinuous on $\mathcal{S}_{ad}$, we have $J(\omega^*)\le\liminf_{n\to+\infty}J(\omega_n)$.
Finally to implement the descent strategy, let us rewrite (<ref>) as
\begin{equation*}
\left\{
\begin{aligned}
&r_t+(hV)_x=0, \\
& V_t-\frac{\mu}{3h}\partial_x(h^3V_{tx})+r_x+\epsilon VV_x=-b_x-\frac{\epsilon}{2}b_{ttx},
\end{aligned}
\right.
\end{equation*}
with $r=\zeta-b$, $h=1+\epsilon r$. Then, the adjoint system is given by (see <ref>)
\begin{equation*}
\left\{
\begin{aligned}
&p_t+\epsilon Vp_x+q_x=\bar{\zeta}-\zeta, \\
&q_t-\frac{\mu}{3}\partial_x(h^2q_{tx})+hp_x+\epsilon Vq_x=\bar{V}-V, \\
&p(T)=0, \quad q(T)=0,
\end{aligned}
\right.
\end{equation*}
and the descent direction is
\[ -\nabla J(b)=-q_x-\frac{\epsilon}{2}q_{ttx}+(\bar{\zeta}-\zeta), \]
accordingly (see Theorem <ref> in the Appendix).
Because we are in the shallow water regime and $\mu<<1$, let us neglect the term $\mu\partial_x(h^3\partial_xV_t)$.
§ NUMERICAL RECONSTRUCTION OF A TIME DEPENDENT BATHYMETRY
Now we will consider the problem of numerically reconstructing the bathymetry $b(t,x)$ from measurements of $\zeta$ and $V$ at the surface. We will do so in the shallow water regime described in the previous section, in which the term involving $V_{txx}$ is neglected (since $\mu<<1$). As it was described above, we are looking for a function $b$ such that
\begin{eqnarray}
\label{eq:dJ}
\begin{array}{c}
\displaystyle
J(b) = \frac{1}{ 2} \int_{0}^T |\zeta - \bar{\zeta}|_{L^2}^2 + \frac{1}{ 2} \int_{0}^T|V - \bar{V}|_{L^2}^2 \;
\end{array}
\end{eqnarray}
is minimized, where $\zeta,V$ are constrained to the system
\begin{eqnarray}
\label{eq:direct-system}
\begin{array}{c}
\partial_t
\left[
\begin{array}{c}
r \\
\end{array}\right]
+ \partial_x
\left[
\begin{array}{c}
hV \\
r + \epsilon \frac{ V^2}{2}
\end{array}\right]
\left[
\begin{array}{c}
0 \\
- b_x - \frac{ \epsilon}{2} b_{ttx}
\end{array}\right]
\;
\end{array},
\end{eqnarray}
with $ r = \zeta - b$, $h = 1 + \epsilon r$.
In (<ref>), $\bar{\zeta}(0,x)$ is a given function. $\bar{V}$ is obtained after solving (<ref>) over the interval $[0,L]$ with appropriate boundary conditions and a given bathymetry $\bar{b}$ and initial condition $[r(0,x), V(0,x)]^T = [\bar{\zeta}(0,x)- \bar{b}(0,x), V_0]^T $ for $V_0$ a given initial velocity. That is, we use synthetic data for assessing the methodology.
The problem of finding the bathymetry is stated as a PDE-constraint optimization problem which is studied through the adjoint method. That is, a minimizing sequence $\{ b^k \}$ is generated via
\begin{eqnarray}
\begin{array}{c}
b^{k+1} = b^k - \lambda_b \cdot \nabla J(b^k) \;,
\end{array}
\end{eqnarray}
where $\lambda_b$ is a constant parameter and
\begin{eqnarray}
\label{eq:NablaJ}
\begin{array}{c}
\nabla J = q_x + \frac{1}{2} \epsilon q_{xtt} - (\bar{\zeta} - \zeta) \;,
\end{array}
\end{eqnarray}
with $p$ and $q$ adjoint variables related through the adjoint system
\begin{eqnarray}
\label{eq:dual-system}
\begin{array}{c}
\partial_t
\left[
\begin{array}{c}
p \\
\end{array}\right]
\left[
\begin{array}{cc}
\epsilon V & 1 \\
h & \epsilon V
\end{array}\right]
\partial_x
\left[
\begin{array}{c}
p \\
\end{array}\right]
\left[
\begin{array}{c}
\bar{\zeta}-\zeta \\
\bar{V}-V
\end{array}\right]
\;,
\end{array}
\end{eqnarray}
on $[0,L]$, endowed with $ p(T,x) = q(T,x) = 0$ and boundary conditions chosen consistently with that of (<ref>), namely transmissive or periodic boundary conditions. Notice that, this system is solved back in time from $t = T$ up to $t = 0$. Here $ \zeta = r + b^k \;,$ $r$ and $V$ satisfy the system
\begin{eqnarray}
\begin{array}{c}
\partial_t
\left[
\begin{array}{c}
r \\
\end{array}\right]
+ \partial_x
\left[
\begin{array}{c}
hV \\
r + \epsilon \frac{V^2}{2}
\end{array}\right]
\left[
\begin{array}{c}
0 \\
- b^k_x - \frac{ \epsilon}{2} b^k_{ttx}
\end{array}\right]
\;,
\end{array}
\end{eqnarray}
with $[r(0,x),V(0,x)]^T = [\bar{\zeta}(0,x) - \bar{b}(0,x),V_0]^T $ on $[0, L]$ and transmissible boundary condition.
§ NUMERICAL DISCRETIZATION
System (<ref>) can be expressed as
\begin{eqnarray}
\label{eq:general-state}
\begin{array}{c}
\partial_t \mathbf{U} + \partial_x \mathbf{F}(\mathbf{U}) = \mathbf{B}(x, \mathbf{U}) \;,
\\
\mathbf{U} = \mathbf{U}_0(x) \;,
\end{array}
\end{eqnarray}
where $\mathbf{U}_0(x)$ is the prescribed initial condition function. The system in its quasilinear version takes the form
\begin{eqnarray}
\label{eq:general-state-quasi}
\begin{array}{c}
\partial_t \mathbf{U} + \mathbf{A}(\mathbf{U}) \partial_x \mathbf{U} = \mathbf{B}(x, \mathbf{U}) \;,
\\
\mathbf{U} = \mathbf{U}_0(x) \;,
\end{array}
\end{eqnarray}
where $ \mathbf{A}(\mathbf{U} ) $ is the Jacobian matrix of $ \mathbf{F}(\mathbf{U})$ with respect to $\mathbf{U}$.
The adjoint system (<ref>) can be written as
\begin{eqnarray}
\label{eq:general-dual}
\begin{array}{c}
\partial_t \mathbf{P} +\mathbf{A}^T(\mathbf{U}) \partial_x \mathbf{P} = \mathbf{R}(x, \mathbf{U}, \mathbf{P}) \;,
\\
\mathbf{P}(T,x) = \mathbf{0} \;,
\end{array}
\end{eqnarray}
where $\mathbf{P}$ is the adjoint state. Both systems can be written in a unified way, in the sense of [24], for the state $\mathbf{W} = [\mathbf{Q}, \mathbf{P}]^T$ given by
\begin{eqnarray}
\label{eq:unified-system}
\begin{array}{c}
\partial_t \mathbf{W}
\mathbf{A_W}
\partial_x \mathbf{W}
\mathbf{S_W}
\;,
\end{array}
\end{eqnarray}
\begin{eqnarray}
\label{eq:unified-system-matrix}
\begin{array}{c}
\mathbf{A_W} =
\left[
\begin{array}{cc}
\mathbf{A}(\mathbf{U}) & \mathbf{0} \\
\mathbf{0} & \mathbf{A}^T(\mathbf{U})
\end{array}
\right]
\;, \
\mathbf{S_W}
\left[
\begin{array}{c}
\mathbf{B}(x, \mathbf{U})
\\
\mathbf{R}(x, \mathbf{U}, \mathbf{P})
\end{array}
\right]
\;.
\end{array}
\end{eqnarray}
We derive a numerical scheme for system (<ref>). In this type of applications, the numerical diffusion can increase due to small CFL coefficients which are used in the context of inverse problems, in contrast to the usual applications (direct problems) to conservation laws. In this paper, we explore the novel FORCE-$\alpha$ scheme reported in [37] for first order of accuracy and extended to higher orders in [23]. This scheme has the advantage of improving the numerical diffusion. The numerical scheme has the form
\begin{eqnarray}
\label{eq:one-step-unified-dual}
\begin{array}{c}
\mathbf{W}_i^{n+1} = \mathbf{W}_i^n - \omega \frac{\Delta t}{ \Delta x} ( \mathbf{D}_{i+\frac{1}{2} }^{-} + \mathbf{D}_{ i-\frac{1}{2} }^{+}) + \omega \Delta t \mathbf{S_W}_i \;,
\end{array}
\end{eqnarray}
\begin{eqnarray}
\begin{array}{c}
\mathbf{D}_{i+\frac{1}{2}}^{\pm}
\omega \mathbf{A}^\pm_{i+\frac{1}{2}}
\cdot (\mathbf{W}_{i+1}^n - \mathbf{W}_{i}^n ) \;,
\end{array}
\end{eqnarray}
here $ \omega $ is a parameter that controls forward and backward propagation in time, $ \omega = 1$ and $\omega= -1$, respectively. The matrix $\mathbf{A}^\pm_{i+\frac{1}{2}}$ are given by
\begin{eqnarray}
\begin{array}{c}
\mathbf{A}^\pm_{i+\frac{1}{2}}
= \frac{1}{2}\hat{\mathbf{A}}_{i+\frac{1}{2}} \pm \frac{1}{4} \frac{\alpha_F \Delta t}{\Delta x}( \hat{\mathbf{A}}_{i+\frac{1}{2}}^2 + (\frac{ \Delta t}{\alpha \Delta t} )^2 \mathbf{I} ) \;,
\end{array}
\end{eqnarray}
where $\mathbf{I}$ is the identity matrix, and
\begin{eqnarray}
\begin{array}{c}
\hat{\mathbf{A}}_{i+\frac{1}{2}} = \int_{0}^{1} \mathbf{A_W}( \Phi(s,\mathbf{W}_i^n, \mathbf{W}_{i+1}^n)) ds
\approx \sum_{k=1}^{3} w_1 \cdot \hat{\mathbf{A}}_{i+\frac{1}{2}} = \int_{0}^{1} \mathbf{A_W}( \Phi(\xi_k,\mathbf{W}_i^n, \mathbf{W}_{i+1}^n)) \;,
\end{array}
\end{eqnarray}
where $ \Phi(s,\mathbf{W}_i^n, \mathbf{W}_{i+1}^n) = \mathbf{W}_{i}^{n} + (\omega s-\frac{(\omega-1)}{2}) ( \mathbf{W}_{i+1}^{n} - \mathbf{W}_{i}^{n})$. The last term is approximated via the Gauss-Legendre quadrature rule characterized by $ \xi_1 = \frac{1}{2} (1 - \sqrt{\frac{3}{5}}) $, $ \xi_2 = \frac{1}{2} $, $ \xi_3 = \frac{1}{2} (1 + \sqrt{ \frac{3}{5} } ) $, $ w_1 = \frac{5}{18} $, $ w_2 = \frac{8}{18} $ and $ w_3 = \frac{5}{18}$.
The source term is discretized as
\begin{eqnarray}
\label{eq:source-term-unified}
\begin{array}{c}
\mathbf{S_W}_{i}
\displaystyle
\left[
\begin{array}{c}
\displaystyle
0 \\
\\
\displaystyle
- b^n_{x,i}
- \frac{\epsilon}{2} \biggl(
\frac{ ( b_{x,i}^{n+1}-2b_{x,i}^n+b_{x,i}^{n-1} ) }{\Delta t^2}
\biggr)
\\
\zeta_i^n - \bar{\zeta}(x_i,t^n) \\
\end{array}
\right]
\;,
\end{array}
\end{eqnarray}
\begin{eqnarray}
\label{eq:bx}
\begin{array}{c}
b^n_{x,i} = \frac{ b_{i+1}^n - b_{i-1}^n }{ 2 \Delta x} \;.
\end{array}
\end{eqnarray}
Notice that for shallow water applications, the approximation of spatial derivatives, for a $b(t,x)$ known, can be replaced by $ b^n_{x,i} = \frac{ b(x_i-\frac{\Delta x}{2}, t^n) - b(x_i+\frac{\Delta x}{2}, t^n) }{ \Delta x} \;.$ In this way the well-balanced property is ensured. However, in this work $b$ is only known in a discrete spatial and temporal location at each iteration, so we are limited to use (<ref>).
In (<ref>), $\omega = 1$ evolves the system forward in time and thus the first components of $\mathbf{W}$ solve (<ref>), whereas $\omega = -1$ solves (<ref>) backward in time, so the last components of $\mathbf{W}$ solve (<ref>). For solving the system backwards, we set to zero the variables of $\mathbf{W}$ associated to the adjoint state and froze the values of $\mathbf{W}$ associated to $\mathbf{U}$, that is, we keep the values obtained in the forward evaluation. This is needed because the hyperbolic system is not reversible in time, that means wave patterns in backward evolution may be different from forward ones. Since in the forward evolution, variables associated to the adjoint system do not influence the state variables we also set them to be zero. At each iteration of the present strategy, we extract components $r$, $p$ and $q$ to form $\nabla J$ given by (<ref>). We remark that the coupled formulation is only required for building the numerical scheme; in practice this still works as a classical solver.
Although the particular case of finding $b(t,x)$ is addressed, the framework is general enough for any source $b(t,x)$ as well. As stated in [23], this method is universal, since hyperbolic problems written in conservative and non-conservative form are solved with the scheme without any modification of the code. Since the present scheme is derived from the non-conservative FORCE-$\alpha$ method applied to both direct and adjoint systems through the Couple System Formulation, from now on, we referee this scheme FORCE-$\alpha$+CSF.
§ NUMERICAL RESULTS
In this section, we solve the following three test problems; smooth bottom profiles, discontinuous bottom profiles and smooth profiles with a large gradient. In order to compare the current scheme, we use the following reference schemes. The first one, presented in <ref>, is characterized by the use of the conservative Rusanov finite volume method for solving (<ref>) and finite difference approximation for handling (<ref>), from now on called Rusanov+FD. The second scheme, consists of the non-conservative FORCE-$\alpha$ scheme applied for solving (<ref>) only, and finite difference approximation, as in <ref>, for the adjoint system (<ref>), from now on FORCE-$\alpha$+FD. The choice of these reference schemes attempts to reproduce the discretization that users usually implement as a first choice when performing this type of tests.
The reference schemes do not consider strategies based on other types of discretizations. For instance, those in [20], where finite difference methods for both (<ref>) and adjoint systems are implemented. Neither those in [13], where a semidiscrete scheme is employed to solve (<ref>), which consists of a conservative flux for spatial discretization [27] and Runge-Kutta scheme for time discretization. They are used in combination with the Matlab Optimization Toolbox fmincon for solving the constrained optimization problem. The comparison with some of these methods is material for a future work.
The first reference scheme must be sensitive to numerical diffusion due to small CFL coefficients, whereas the second one must control this by implementing the FORCE-$\alpha$ on the state system.
To illustrate the performance of the present scheme and reference ones, the sequence of time $t=0.25$, $t = 0.5$, $t=0.75$ and iterations $0,1,2,4,8,$ are depicted. In all simulations, we set $L=20$, $\bar{\zeta} = 1$, $\Delta t = 0.01$, $100$ cells, $\alpha_F = 2$, $V_0 = 1.5$, $\varepsilon = 0.001$, $b^0 = 0.01$ and $\lambda_b = 0.71$.
§.§ Smooth bottom profile
This test aims at recovering the smooth bottom profile
\begin{eqnarray}
\label{eq:b-smooth}
\begin{array}{c}
\bar{b}(t,x) = 0.1(1 + t \cdot exp( - (x-10- 2.5 t)^2) )\;.
\end{array}
\end{eqnarray}
Systems (<ref>) and (<ref>) are solved with transmissible boundary conditions. Note that this profile consists of a soliton that moves to the right and the amplitude of the wave increases with the time.
Figure <ref> shows the results for the Rusanov+FD scheme. We observe some oscillations in the first iterations. We note that they do not disappear but reduce as the iterations increase. Figure <ref> shows the results for the FORCE-$\alpha$+FD scheme. We still observe some oscillations on the first iterations, but they almost disappear as the iterations increase. Figure <ref> shows the results for the FORCE-$\alpha$+CSF scheme. We note that oscillations are reduced at the initial iterations and disappear as the iterations increase. Figure <ref> shows the $L_\infty$ norm of $\nabla J$ against the number of iterations, to facilitate the visualization the plot is depicted in logarithmic scale. So, this measures the error between $b^k $ and $b^{k+1}$, which is the empirical convergence of the global algorithm. These results show that in order to obtain convergence, it is not only important how the state system is discretized, but also how the adjoint system is does so. This test reveals that a low-dissipation scheme is beneficial in the PDE-constraint optimization context and smooth variables.
Smooth bottom profile (<ref>): The $L_\infty$ norm of $\nabla J $ for $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells at $t = 1$, $\alpha_F = 2$.
(Dot line) Rusanov+FD scheme.
(Dash line) FORCE-$\alpha$+FD scheme.
(Full line) FORCE-$\alpha$+CSF scheme.
Smooth bottom profile (<ref>): Result for the reconstruction procedure resulting from Rusanov+FD. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
Smooth bottom profile (<ref>): Result for the reconstruction procedure resulting from non-conservative FORCE-$\alpha$+FD. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
Smooth bottom profile (<ref>): Result for the reconstruction procedure resulting from FORCE-$\alpha$+CSF. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
§.§ Discontinuous bottom profile
This test aims at recovering the discontinuous bottom profile
\begin{eqnarray}
\label{eq:b-discontinuous}
\begin{array}{c}
\bar{b}(t,x) = \left\{
\begin{array}{cc}
\frac{1}{4}, & 5 < x< 7, \\
0.3 t, & 7 < x< 10+4t, \\
0.1, & otherwise.
\end{array}
\right.
\end{array}
\end{eqnarray}
Systems (<ref>) and (<ref>) are solved with transmissible boundary conditions. This profile consists of two square waves, one is kept fix while the other increases in amplitude and moves on the right as the time advances.
Figure <ref> shows the results for the Rusanov+FD scheme. Despite the locations of discontinuities are recovered, the right amplitude of the waves are not recovered. Furthermore, spurious oscillation appear in both the direct and inverse problems. This approach is not suitable for recovering this type of bottom profile. It is well known that Rusanov scheme introduces large numerical diffusion when CFL coefficients are small. Figure <ref> shows the results for the FORCE-$\alpha$+FD. It can be seen that the procedure recovers the correct profile. This highlights the fact that a low dissipation scheme for the state system is suitable for achieving convergence of the global procedure. Figure <ref> shows the results for the FORCE-$\alpha$+CSF scheme. We note that global convergence is achieved. We observe that already in the second iteration of the descent step procedure the scheme is able to recover the main features of the profile. Figure <ref> shows the $L_\infty$ norm of $\nabla J$ against the number of iterations, to facilitate the visualization the plot is depicted in logarithmic scale. So, this measures the error between $b^k $ and $b^{k+1}$, which is the empirical convergence of the global algorithm. These results show that to obtain convergence, not only becomes important the form in which the state system is discretized but also how the adjoint system does so. As before, a low-dissipation scheme is beneficial in the PDE-constraint optimization context and discontinuous variables.
Discontinuous bottom profile (<ref>): The $L_\infty$ norm of $\nabla J $ for $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells at $t = 1$, $\alpha_F = 2$.
(Dot line) Rusanov+FD scheme.
(Dash line) FORCE-$\alpha$+FD scheme.
(Full line) FORCE-$\alpha$+CSF scheme.
Discontinuous bottom profile (<ref>): Result for the reconstruction procedure resulting from Rusanov+FD. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
Discontinuous bottom profile (<ref>): Result for the reconstruction procedure resulting from non-conservative FORCE-$\alpha$+FD. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
Discontinuous bottom profile (<ref>): Result for the FORCE-$\alpha$+CSF. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
§.§ Smooth bottom profile with large gradients
This test aims at recovering the smooth bottom profile with a large gradient
\begin{eqnarray}
\label{eq:b-large-gradient}
\begin{array}{c}
\bar{b}(t,x) = 0.15 t\left( 1 + \sin^4\left( \frac{ \pi ( x-10 - 4.5 t)}{5} \right) \right).
\end{array}
\end{eqnarray}
Systems (<ref>) and (<ref>) are solved with periodic boundary conditions. The profile consists of a triggered of pulses that moves periodically to the right, elevates with respect to an equilibrium $b=0$ and increments its amplitude as the time advances.
These type of tests are a challenge for the balance law (<ref>). They introduce stiffness on the source term and may induce large gradient on the other variables.
Figure <ref> shows the results for the Rusanov+FD scheme. Although the overall procedure converges, a phase mismatch of the amplitudes and asymmetries are observed in the first few iterations. Asymmetry is still present in the $8th$ iteration at $t = 0.5$. Figure <ref> shows the results for the FORCE-$\alpha$+FD. Asymmetries are observed in the initial iterations that disappear by the eighth iteration. Figure <ref> shows the results for the FORCE-$\alpha$+CSF scheme. In the first iterations, errors appear on the boundaries. In addition, asymmetries appear in the first iterations, which are not perceived after the fourth iteration. Figure <ref> shows the $L_\infty$ norm of $\nabla J$ against the number of iterations, to facilitate the visualization the plot is depicted in logarithmic scale. So, this measures the error between $b^k $ and $b^{k+1}$, which is the empirical convergence of the global algorithm. We observe that for smooth bottom profiles with large gradients, all methods have oscillations during the iterative process and asymmetries in the first iterations. By comparing with reference schemes, FORCE-$\alpha$+CSF depicts the best convergence. This test evidences that the methodology also applies to other types of boundary conditions than transmissive.
Smooth bottom profile with large gradient (<ref>): The $L_\infty$ norm of $\nabla J $ for $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells at $t = 1$, $\alpha_F = 2$.
(Dot line) Rusanov+FD scheme.
(Dash line) FORCE-$\alpha$+FD scheme.
(Full line) FORCE-$\alpha$+CSF scheme.
Smooth bottom profile with large gradient (<ref>): Result for the reconstruction procedure resulting from Rusanov+FD. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
Smooth bottom profile with large gradient (<ref>): Result for the reconstruction procedure resulting from FORCE-$\alpha$+FD. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
Smooth bottom profile with large gradient (<ref>): Result for the FORCE-$\alpha$+CSF. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells.
Feft: $t=0.25$, centered $t=0.5$, right: $t=0.75$.
§ CONCLUSIONS
In this paper we have considered the inverse problem of detecting a time-varying bottom through free surface measurements. This problem has applications in oceanography, in the study of tsunami generation by vertical and horizontal displacements of underwater plates. Also in man-made facilities to design surf pools.
To study the problem we used the system of conservation laws for an incompressible, non-viscous fluid, in the shallow water regime with time-varying bottom, known as the Boussinesq-Peregrine system (<ref>) (see [17]). The solution to (<ref>) is obtained through an iterative Picard scheme that generates a sequence of linearized systems. For each of them, energy estimates are obtained, where the symmetrization of (<ref>) is fundamental. Well-posedness follows classical ideas from energy estimates as in [15, 36].
The identification of the bottom was obtained as the minimum of a functional involving the $L^2$ norm of the solution of (<ref>). Existence of an optimal solution and formulation of a first order necessary condition, involving the adjoint system, were obtained.
The numerical identification of the bottom followed the methodology described above, through a descent method. Both, the state and adjoint equations, were discretized by using a unified finite volume scheme for non-conservative systems. Namely, we implemented the FORCE-$\alpha$ a universal, low-dissipation scheme applied to the coupled system (<ref>), in the classical fashion of forward and backward evolution that we called FORCE-$\alpha$+CSF, and contrasted with two reference solutions, namely Rusanov+FD and FORCE-$\alpha$+FD, assumed as standard discretizations.
Let us mention a few things in this regard. First, for a stable computation of (<ref>) and its adjoint system, small CFL coefficients are required. Second, the reference scheme Rusanov+FD uses the Rusanov method (see [32]) to solve BP and it is the one with the worst results. Possibly due to the fact that Rusanov scheme introduces the largest numerical diffusion among the three methods. Third, the universal, low-dissipation scheme for both the state and adjoint systems yields the best results, making clear the importance of discretizing the adjoint system in an appropriate manner.
Finally, test problems presented in this paper show that the bottoms have been successfully recovered, and therefore, low-dissipation schemes work as a correct numerical methodology in this type of inverse problems.
In the future, we will implement this strategy in more sophisticated models, such as the Green-Naghdi system (see [15]). Also, it would be interesting to understand how much of this strategy can be replicated for the class of equations coming from the water-waves theory in [17]. Finally, the current strategy leads us to the question of the exact or approximate controllability of these systems through a source term in the equation of state.
§ EXISTENCE AND UNIQUENESS OF SOLUTIONS
In this section we prove a well-posedness theorem for the Boussinesq–Peregrine system (<ref>). The idea is to write (<ref>) as a symmetrizable hyperbolic system (see [36]). Then, an iterative Picard scheme is carried out. For the resulting linear systems, an energy estimate is constructed which yields a solution to (<ref>). This procedure has been done for the Nonlinear Shallow Water [17] and Green–Naghdi equations [15], in the case of a stationary bottom.
After multiplying second equation in (<ref>) by $h$, the system can be written as
\begin{equation}
\label{ES31}
\left\{
\begin{aligned}
&\zeta_t+\epsilon(\zeta-b)_xV+hV_x=b_t, \\
&\left(h+\mu h\left(-\frac{1}{3h}\partial_x(h^3\partial_x\cdot)\right)\right)(V_t+\epsilon VV_x)+h\zeta_x=-\frac{\epsilon h}{2}b_{ttx}.
\end{aligned}
\right.
\end{equation}
Then, if one defines the operator
\begin{equation}
\label{Tgeneral}
\mt w=hw+\mu h\left(-\frac{1}{3h}\partial_x(h^3\partial_xw)\right),
\end{equation}
system (<ref>) becomes
\begin{equation*}
\begin{pmatrix}
1 & 0 \\ 0 & \mt\cdot
\end{pmatrix}
\begin{pmatrix}
\zeta_t \\ V_t
\end{pmatrix}
\begin{pmatrix}
\epsilon V & h \\ h & \mt(\epsilon V\cdot)
\end{pmatrix}
\begin{pmatrix}
\zeta_x \\ V_x
\end{pmatrix}
\begin{pmatrix}
b_t+\epsilon b_xV \\ -\frac{\epsilon}{2}hb_{ttx}
\end{pmatrix}.
\end{equation*}
\begin{equation}
\label{ES32}
\mathbf{S}(\mathbf{U})\partial_t\mathbf{U}+\mathbf{A}(\mathbf{U})\partial_x\mathbf{U}=\mathbf{B}(\mathbf{U}),
\end{equation}
with $\mathbf{U}=(\zeta,V)^T$,
\begin{equation*}
\mathbf{B}(\mathbf{U})=
\begin{pmatrix}
b_t+\epsilon b_xV \\ -\frac{\epsilon}{2}hb_{ttx}
\end{pmatrix}
\end{equation*}
\begin{equation}
\label{ES33}
\mathbf{S}(\mathbf{U})=
\begin{pmatrix}
1 & 0 \\ 0 & \mt\cdot
\end{pmatrix},
\quad
\mathbf{A}(\mathbf{U})=
\begin{pmatrix}
\epsilon V & h \\ h & \mt(\epsilon V\cdot)
\end{pmatrix},
\end{equation}
be two symmetric operators. Existence and uniqueness of a solution for system (<ref>) is guaranteed by the following theorem.
Let $s>3/2$ and $(\zeta_0,V_0)\in \mathbf{X}^s$. Let $b\in W^{2,\infty}([0,\infty);H^{s+1})$ and assume (<ref>) is valid. Then, there exists $T_{BP}>0$, uniformly bounded from below with respect to $\epsilon$, such that system (<ref>) admits a unique solution $(\zeta,V)^T\in \mathbf{X}_{T_{BP}}^s$ with initial condition $(\zeta_0,V_0)^T$.
This result is a consequence of applying a strategy similar to the one described in [36], Chapter 16. Namely, the solution is built as the limit of the following Picard iterative scheme
\begin{equation}\label{picard}
\left\{
\begin{aligned}
& \mathbf{S}(\mathbf{U}^n)\partial_t\mathbf{U}^{n+1}+\mathbf{A}(\mathbf{U}^n)\partial_x\mathbf{U}^{n+1}=\mathbf{B}(\mathbf{U}^n), \\
& \mathbf{U}^{n+1}|_{t=0}=\mathbf{U}_0.
\end{aligned}
\right.
\end{equation}
Now, to study (<ref>), we consider the initial value problem
\begin{equation}
\label{ES35}
\left\{
\begin{aligned}
& \mathbf{S}(\mathbf{\uu})\partial_t\mathbf{U}+\mathbf{A}(\mathbf{\uu})\partial_x\mathbf{U}=\mathbf{B}(\textbf{\uu}), \\
& \mathbf{U}|_{t=0}=\mathbf{U}_0,
\end{aligned}
\right.
\end{equation}
where $\mathbf{\uu}=(\zetau,\vu)^T$ is a reference state. For this system we provide an energy estimate. Before giving the proof, let us recall two lemmas describing some important properties of the operator $\mt$. Proofs can be found in [15].
Let $\zeta,b\in W^{1,\infty}(\R)$ such that (<ref>) is satisfied. Then the operator
\[ \mt:H^2(\R)\rightarrow L^2(\R) \]
is well defined, one-to-one and onto.
Let $d_0>\frac{1}{2}$ and $\zeta,b\in H^{d_0+1}(\R)$ be such that (<ref>) is satisfied.
* $\forall s\ge0$, $|\mt^{-1}f|_{H^s}+\sqrt{\mu}|\partial_x\mt^{-1}f|_{H^s}\le C(\frac{1}{h_{min}},|\zeta,b|_{H^{s\vee d_0+1}})|f|_{H^s}$.
* $\forall s\ge0$, $\sqrt{\mu}|\mt^{-1}\partial_xf|_{H^s}\le C(\frac{1}{h_{min}},|\zeta,b|_{H^{s\vee d_0+1}})|f|_{H^s}$.
Finally, we present the energy estimate for the linearized system (<ref>). The proof follows the ideas presented in [17] for the Saint-Venant equation, but in our case, the bottom also depends on the time variable.
Let $s>3/2$, $T>0$, and $b\in W^{2,\infty}((0,\infty);H^{s+1})$. Let also $\mathbf{\uu}=(\zetau,\vu)^T\in \mathbf{X}_{T}^s$ be such that $\partial_t\mathbf{\uu}\in \mathbf{X}_{T}^{s-1}$ satisfying the condition (<ref>) on $[0,T/\epsilon]$. Then for any $\mathbf{U}_0\in \mathbf{X}^s$, there exists a unique solution $\mathbf{U}=(\zeta,V)^T\in \mathbf{X}_{T}^s$ to (<ref>) such that for any $t\in[0,T/\epsilon]$
\begin{align}
\label{ES35_1}
|\mathbf{U}|_{\mathbf{X}^s}^2\le e^{\epsilon tC(\mathbf{\uu})}|\mathbf{U}_0|_{\mathbf{X}^s}^2+\epsilon C(\mathbf{\uu},T)\int_0^t(|b_t|_{H^s}^2+|b_{tt}|_{H^{s+1}}^2)d\tau.
\end{align}
Due to the approach of the inverse problem, as an optimal control problem, it is important to note that no assumption has been made on the size of $\epsilon$. On the contrary, thanks to the fact that $T_{BP}$ is uniformly bounded from below with respect to $\epsilon$, it allows one to conclude that if any smallness assumption is made on $\epsilon$, the existence time is large, of order $O(1/\epsilon)$.
\[ \mathbf{\uu}\in C([0,T/\epsilon];\mathbf{X}^s), \ \partial_t\mathbf{\uu}\in C([0,T/\epsilon];\mathbf{X}^{s-1}). \]
\[ \Lambda=(1-\partial_x^2)^{1/2}=(1+|D|^2)^{1/2} \]
be the fractional derivative where $D=\frac{1}{i}\partial_x$ stands for the Fourier multiplier. We use notation $\tu=\mathcal{T}(\mathbf{\uu})$.
By taking the $\Lambda^s$ operator in (<ref>) and using the definition of the commutator we obtain
\begin{align}
\label{ES36}
S(\mathbf{\uu})\partial_t\ls \mathbf{U}+\mathbf{A}(\mathbf{\uu})\partial_x\ls \mathbf{U}&=\ls B(\mathbf{\uu})-[\ls,\mathbf{S}(\mathbf{\uu})]\partial_t\mathbf{U}-[\ls,\mathbf{A}(\mathbf{\uu})]\partial_x\mathbf{U} \\
&=\ls \mathbf{B}(\mathbf{\uu})-[\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{B}(\mathbf{\uu}) \notag\\
&\phantom{=}+[\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{A}(\mathbf{\uu})\partial_x\mathbf{U}-[\ls,\mathbf{A}(\mathbf{\uu})]\partial_x\mathbf{U}. \notag
\end{align}
Since $\mathbf{S}$ is symmetric, we have
\begin{align*}
\frac{1}{2}\frac{d}{dt}(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})&=\frac{1}{2}([\partial_t,\mathbf{S}(\mathbf{\uu})]\ls \mathbf{U},\ls \mathbf{U})+(\mathbf{S}(\mathbf{\uu})\partial_t\ls\mathbf{U},\ls \mathbf{U}) \\
&=\frac{1}{2}([\partial_t,\tu]\ls V,\ls V)+(\mathbf{S}(\mathbf{\uu})\partial_t\ls \mathbf{U},\ls \mathbf{U}).
\end{align*}
Then, by replacing (<ref>), from last equality we obtain
\begin{multline*}
\frac{1}{2}\frac{d}{dt}(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})=\frac{1}{2}([\partial_t,\tu]\ls V,\ls V)+(\ls \mathbf{B}(\mathbf{\uu}),\ls \mathbf{U}) \\
-([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{B}(\mathbf{\uu}),\ls \mathbf{U})+([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{A}(\mathbf{\uu})\partial_x\mathbf{U},\ls \mathbf{U}) \\
-([\ls,\mathbf{A}(\mathbf{\uu})]\partial_x\mathbf{U},\ls \mathbf{U})-(\mathbf{A}(\mathbf{\uu})\partial_x\ls \mathbf{U},\ls \mathbf{U}).
\end{multline*}
Thus, by the symmetry of $\mathbf{A}$,
\begin{multline*}
\frac{1}{2}\frac{d}{dt}(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})=\frac{1}{2}([\partial_t,\tu]\ls V,\ls V)+(\ls \mathbf{B}(\mathbf{\uu}),\ls \mathbf{U}) \\
-([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{B}(\mathbf{\uu}),\ls \mathbf{U})+([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{A}(\mathbf{\uu})\partial_x\mathbf{U},\ls \mathbf{U}) \\
-([\ls,\mathbf{A}(\mathbf{\uu})]\partial_x\mathbf{U},\ls \mathbf{U})+\frac{1}{2}([\partial_x,\mathbf{A}(\mathbf{\uu})]\ls \mathbf{U},\ls \mathbf{U}).
\end{multline*}
We are going to estimate now each term of the right–hand side of this last equality in terms of $|\mathbf{U}|_{\mathbf{X}^s}$, taking into account the expressions for $\mathbf{S}$ and $\mathbf{A}$ from (<ref>).
§.§ Estimate of $([\partial_x,\mathbf{A}(\mathbf{\uu})]\ls \mathbf{U},\ls \mathbf{U})$
\begin{align*}
([\partial_x,\mathbf{A}(\mathbf{\uu})]\ls \mathbf{U},\ls \mathbf{U})&=(\epsilon\vu_x\ls\zeta,\ls\zeta)+(\hu_x\ls\zeta,\ls\zeta) \\
&\phantom{=}+(\hu_x\ls V,\ls V)+([\partial_x,\tu(\epsilon\vu\cdot)]\ls V,\ls V),
\end{align*}
with $\hu=1+\epsilon(\zetau-b)$.
From the definition of the operator $\mt$, one has
\begin{align*}
([\partial_x,\tu(\epsilon\vu\cdot)]\ls V,\ls V)&=([\partial_x,(\epsilon\vu\hu)]\ls V,\ls V)-\frac{\mu}{3}([\partial_x,\partial_x(\hu^3\partial_x(\epsilon\vu\cdot))]\ls V,\ls V) \\
&=([\partial_x,(\epsilon\vu\hu)]\ls V,\ls V)+\frac{\mu}{3}([\partial_x,\hu^3]\partial_x(\epsilon\vu\ls V),\ls V_x).
\end{align*}
\begin{equation*}
([\partial_x,\mathbf{A}(\mathbf{\uu})]\ls \mathbf{U},\ls \mathbf{U})\le \epsilon C(|\mathbf{\uu},b|_{H^{d_0}})|\mathbf{U}|_{\mathbf{X}^s}^2.
\end{equation*}
§.§ Estimate of $([\ls,\mathbf{A}(\mathbf{\uu})]\partial_x\mathbf{U},\ls \mathbf{U})$
\begin{align*}
([\ls,\mathbf{A}(\mathbf{\uu})]\partial_x\mathbf{U},\ls \mathbf{U})&=([\ls,\epsilon\vu]\zeta_x,\ls\zeta)+([\ls,\hu]V_x,\ls\zeta) \\
&\phantom{=}+([\ls,\hu]\zeta_x,\ls V)+([\ls,\tu(\epsilon\vu)]V_x,\ls V).
\end{align*}
From the definition of $\mt$
\begin{align*}
([\ls,\tu(\epsilon\vu)]V_x,\ls V)&=([\ls,\epsilon\hu\vu]V_x,\ls V)-\frac{\mu}{3}([\ls,\partial_x(\hu^3\partial_x(\epsilon\vu\cdot))]V_x,\ls V).
\end{align*}
Since that
\begin{align*}
[\ls,\partial_x(\hu^3\partial_x(\epsilon\vu\cdot))]V_x&=\ls(\partial_x(\hu^3\partial_x(\epsilon\vu V_x)))-\partial_x(\hu^3\partial_x(\epsilon\vu\ls V_x)) \\
&=\partial_x\left(\ls(\hu^3\partial_x(\epsilon\vu V_x))-\hu^3\partial_x\ls(\epsilon\vu V_x)+\hu^3\partial_x([\ls,\epsilon\vu]V_x)\right) \\
&=\partial_x\left([\ls,\hu^3]\partial_x(\epsilon\vu V_x)+\hu^3\partial_x([\ls,\epsilon\vu]V_x)\right),
\end{align*}
we have
\begin{align*}
-\frac{\mu}{3}([\ls,\partial_x(\hu^3\partial_x&(\epsilon\vu\cdot))]V_x,\ls V) \\
&=\frac{\mu}{3}([\ls,\hu^3]\partial_x(\epsilon\vu V_x),\ls V_x)+\frac{\mu}{3}(\partial_x([\ls,\epsilon\vu]V_x),\hu^3\ls V_x) \\
&=\frac{\mu}{3}([\ls,\hu^3]\partial_x(\epsilon\vu V_x),\ls V_x)+\frac{\mu}{3}([\ls,\epsilon\vu_x]V_x,\hu^3\ls V_x) \\
&\phantom{=}+\frac{\mu}{3}([\ls,\epsilon\vu]V_{xx},\hu^3\ls V_x).
\end{align*}
Then one finally has
\begin{equation*}
([\ls,\mathbf{A}(\mathbf{\uu})]\partial_x\mathbf{U},\ls\mathbf{U})\le \epsilon C(|\mathbf{\uu},b|_{H^{s\vee d_0+1}},\mu|\vu_x|_{H^s})|\mathbf{U}|_{\mathbf{X}^s}^2.
\end{equation*}
§.§ Estimate of $([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{A}(\mathbf{\uu})\partial_x\mathbf{U},\ls \mathbf{U})$
We have that
\[
\mathbf{S}^{-1}
\begin{pmatrix}
1 & 0 \\ 0 & \mt^{-1}
\end{pmatrix}.
\]
Then using the definition of $\mathbf{A}$ given in (<ref>)
\begin{align*}
([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{A}(\mathbf{\uu})\partial_x\mathbf{U},\ls \mathbf{U})&=([\ls,\tu]\tu^{-1}(\hu\zeta_x),\ls V) \\
&\phantom{=}+([\ls,\tu]\epsilon \vu V_x,\ls V).
\end{align*}
By the definition of the operator $\mt$
\begin{align*}
([\ls,\tu]\tu^{-1}\hu\zeta_x,\ls V)&=([\ls,\hu]\tu^{-1}\hu\zeta_x,\ls V)-\frac{\mu}{3}([\ls,\partial_x(\hu^3\partial_x\cdot)]\tu^{-1}\hu\zeta_x,\ls V) \\
&=([\ls,\hu]\tu^{-1}\hu\zeta_x,\ls V)-\frac{\mu}{3}(\partial_x[\ls,\hu^3]\partial_x\tu^{-1}\hu\zeta_x,\ls V) \\
&=([\ls,\hu]\tu^{-1}\hu\zeta_x,\ls V)+\frac{\mu}{3}([\ls,\hu^3]\partial_x\tu^{-1}\hu\zeta_x,\ls V_x).
\end{align*}
On the other hand
\begin{align*}
([\ls,\tu]\epsilon \vu V_x,\ls V)&=([\ls,\hu]\epsilon \vu V_x,\ls V)-\frac{\mu}{3}([\ls,\partial_x(\hu^3\partial_x\cdot)]\epsilon \vu V_x,\ls V) \\
&=([\ls,\hu]\epsilon \vu V_x,\ls V)+\frac{\mu}{3}([\ls,\hu^3]\partial_x(\epsilon \vu V_x),\ls V_x).
\end{align*}
Therefore by Lemma <ref>
\begin{equation*} ([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{A}(\mathbf{\uu})\partial_x\mathbf{U},\ls \mathbf{U})\le \epsilon C(|\mathbf{\uu},b|_{H^{s\vee d_0+1}})|\mathbf{U}|_{\mathbf{X}^s}^2.
\end{equation*}
§.§ Estimate of $([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{B}(\mathbf{\uu}),\ls \mathbf{U})$
\begin{align*}
([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{B}(\mathbf{\uu}),\ls \mathbf{U})=-\frac{\epsilon}{2}([\ls,\tu]\tu^{-1}(\hu b_{ttx}),\ls V).
\end{align*}
Then, by the definition of the operator $\mt$
\begin{align*}
-\frac{\epsilon}{2}([\ls,\tu]\tu^{-1}(\hu b_{ttx}),\ls V)&=-\frac{\epsilon}{2}([\ls,\hu]\tu^{-1}(\hu b_{ttx}),\ls V) \\
&\phantom{=}+\frac{\epsilon\mu}{6}([\ls,\partial_x(\hu^3\partial_x\cdot)]\tu^{-1}(\hu b_{ttx}),\ls V) \\
&=-\frac{\epsilon}{2}([\ls,\hu]\tu^{-1}(\hu b_{ttx}),\ls V) \\
&\phantom{=}-\frac{\epsilon\mu}{6}([\ls,\hu^3]\partial_x\tu^{-1}(\hu b_{ttx}),\ls V_x).
\end{align*}
Therefore from Lemma <ref> and the commutator properties (<cit.>)
\begin{align*}
|[\ls,\hu^3]\partial_x\tu^{-1}(\hu b_{ttx})|_{L^2}&\le C|\hu-1|_{H^s}|\partial_x\tu^{-1}(\hu b_{ttx})|_{H^{s-1}} \\
&\le C(|\hu-1|_{H^{s\vee d_0+1}})|\hu b_{ttx}|_{H^s} \\
&\le C(|\uu,b|_{H^{s\vee d_0+1}})|b_{tt}|_{H^{s+1}},
\end{align*}
\begin{equation*}
([\ls,\mathbf{S}(\mathbf{\uu})]\mathbf{S}^{-1}(\mathbf{\uu})\mathbf{B}(\mathbf{\uu}),\ls \mathbf{U})\le \epsilon C(|\mathbf{\uu},b|_{H^{s\vee d_0+1}})(|\mathbf{U}|_{\mathbf{X}^s}^2+|b_{tt}|_{H^{s+1}}^2).
\end{equation*}
§.§ Estimate of $(\ls \mathbf{B}(\mathbf{\uu}),\ls \mathbf{U})$
\begin{align*}
(\ls \mathbf{B}(\mathbf{\uu}),\ls \mathbf{U})&=([\ls,\epsilon b_x]\vu+\epsilon b_x\ls\vu,\ls\zeta)+(\ls b_t,\ls\zeta) \\
&\phantom{=}-\frac{\epsilon}{2}([\ls,\hu]b_{ttx}+\hu\ls b_{ttx},\ls V).
\end{align*}
\begin{equation*}
(\ls \mathbf{B}(\mathbf{\uu}),\ls \mathbf{U})\le \epsilon C(|\mathbf{\uu}|_{H^{s\vee d_0+1}},|b|_{H^{s+1}})(|\mathbf{U}|_{\mathbf{X}^s}^2+|b_{tt}|_{H^{s+1}}^2+\frac{1}{\epsilon}|b_t|_{H^s}^2).
\end{equation*}
§.§ Estimate of $([\partial_t,\tu]\ls V,\ls V)$
By the definition of the operator $\mt$
\begin{align*}
([\partial_t,\tu]\ls V,\ls V)&=(\partial_t\hu\ls V,\ls V)-\frac{\mu}{3}(\partial_x(\partial_t\hu^3\ls V_x),\ls V) \\
&=(\partial_t\hu\ls V,\ls V)+\frac{\mu}{3}(\partial_t\hu^3\ls V_x,\ls V_x).
\end{align*}
\begin{equation*}
([\partial_t,\tu]\ls V,\ls V)\le \epsilon C(|\mathbf{\uu},b|_{W^{1,\infty}(H^{d_0})})|\mathbf{U}|_{\mathbf{X}^s}^2.
\end{equation*}
From the definition of $\mathbf{S}$ given in (<ref>) and by (<ref>)
\begin{equation}
\label{ES37}
|\mathbf{U}|_{\mathbf{X}^s}^2\le C\left(\frac{1}{\hm}\right)(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})
\end{equation}
\begin{equation}
\label{ES38}
(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})\le C(|\hu|_{L^\infty},|\hu_x|_{L^\infty})|\mathbf{U}|_{\mathbf{X}^s}^2.
\end{equation}
Then, summarizing all the estimates above together with (<ref>)
\begin{equation}
\label{ES39}
\frac{d}{dt}(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})\le\epsilon c_1(\mathbf{\uu})((\mathbf{S}\ls \mathbf{U},\ls \mathbf{U})+\frac{1}{\epsilon}|b_t|_{H^s}^2+|b_{tt}|_{H^{s+1}}^2),
\end{equation}
with $c_1(\mathbf{\uu})=C(\frac{1}{\hm},|\mathbf{\uu}|_{\mathbf{X}_{T}^s},|\partial_t\mathbf{\uu}|_{\mathbf{X}_{T}^{s-1}},|b|_{L^\infty(H^{s+1})},|b|_{W^{1,\infty}(H^{d_0})})$.
By (<ref>), for any $\lambda\in\R$
\begin{align*}
e^{\epsilon\lambda t}&\frac{d}{dt}(e^{-\epsilon\lambda t}(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})) \\
&=-\epsilon\lambda(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})+\frac{d}{dt}(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U}) \\
&\le (\epsilon c_1(\mathbf{\uu})-\epsilon\lambda)(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})+\epsilon c_1(\mathbf{\uu})(\frac{1}{\epsilon}|b_t|_{H^s}^2+|b_{tt}|_{H^{s+1}}^2).
\end{align*}
Then, if $\lambda=\lambda(c_1(\mathbf{\uu}))$ is sufficiently large, by (<ref>) and the last inequality
\begin{align*}
e^{\epsilon\lambda t}\frac{d}{dt}(e^{-\epsilon\lambda t}(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U}))\le\epsilon c_1(\mathbf{\uu})(\frac{1}{\epsilon}|b_t|_{H^s}^2+|b_{tt}|_{H^{s+1}}^2).
\end{align*}
Integrating in $0\le t\le T/\epsilon$
\begin{equation*}
(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})
\le e^{\epsilon\lambda t}(\mathbf{S}(\mathbf{\uu})\ls \mathbf{U},\ls \mathbf{U})|_{t=0}+\epsilon c_1(\mathbf{\uu})\int_0^te^{\epsilon\lambda(t-\tau)}(\frac{1}{\epsilon}|b_t|_{H^s}^2+|b_{tt}|_{H^{s+1}}^2)d\tau,
\end{equation*}
or equivalently, by (<ref>), (<ref>)
\begin{align}
\label{ES310}
|\mathbf{U}|_{\mathbf{X}^s}^2\le e^{\epsilon\lambda t}|\mathbf{U}_0|_{\mathbf{X}^s}^2+\epsilon c_2(\mathbf{\uu})\int_0^t(\frac{1}{\epsilon}|b_t|_{H^s}^2+|b_{tt}|_{H^{s+1}}^2)d\tau,
\end{align}
with $c_2(\mathbf{\uu})=C(c_1(\mathbf{\uu}),T)$.
The fact that $T_{BP}$ is bounded from below by some $T>0$ independent of $\epsilon\in(0,1)$, follows from the analysis above (see [3]).
As was mentioned at the beginning of this section, existence and uniqueness of solutions for system (<ref>), follow classical ideas based on the energy estimate provided in Proposition <ref>. Indeed, to prove existence of solutions for system (<ref>) one regularizes the operators $\mathbf{S}$ and $\mathbf{A}$, to apply the Cauchy–Lipschitz Theorem for ODE. To prove existence of solutions for (<ref>) one uses the above energy estimate and proceed as in [36], chapter 16 (see also [15]).
§ FIRST ORDER NECESSARY CONDITION AND OPTIMALITY SYSTEM
We now proceed to derive a system of first order necessary optimality conditions for problem (<ref>). This is done in a straightforward manner by studying the Gateaux derivative of the functional $J(b)$. From (<ref>), we are considering system:
\begin{equation}
\label{s1}
\left\{
\begin{aligned}
&r_t+(hV)_x=0, \\
& V_t-\frac{\mu}{3h}\partial_x(h^3V_{tx})+r_x+\epsilon VV_x=-b_x-\frac{\epsilon}{2}b_{ttx},
\end{aligned}
\right.
\end{equation}
with $r=\zeta-b$, $h=1+\epsilon r$.
We denote $\Omega=\{(x,y)\in\R^2:\epsilon b\le y\le\epsilon\zeta\}$. Given $\lambda>0$ and $b\in C^1(W^{1,\infty}(\R))$, we define the domain $\Omega_\lambda=\Omega+\lambda b$. For $U:=(\zeta,V)^T$ we use notation $\delta U=\lim_{\lambda\rightarrow0}\frac{U^\lambda-U}{\lambda}$ where $U^\lambda$ is the solution of (<ref>) in the domain $\Omega_\lambda$.
Differentiating formally in system (<ref>) with respect to bottom variations, we obtain
\begin{equation}
\left\{
\label{s2}
\begin{aligned}
&\delta r_t+\partial_x(\delta hV+h\delta V)=0, \\
& \delta V_t-\frac{\mu}{3}\delta\left(\frac{1}{h}\partial_x(h^3V_{tx})\right)+\delta r_x+\epsilon\delta(VV_x)=-\delta b_x-\frac{\epsilon}{2}\delta b_{ttx}.
\end{aligned}
\right.
\end{equation}
Multiplying first and second equations in (<ref>) by test functions $p=p(t,x)$ and $q=q(t,x)$ respectively, such that $\lim_{|x|\rightarrow\infty}p(t,x)=0=\lim_{|x|\rightarrow\infty}q(t,x)$, and integrating in $x$
\begin{equation}
\left\{
\label{s3}
\begin{aligned}
&\int_0^T\int_\R\delta r_tp-\int_0^T\int_\R(\delta hV+h\delta V)p_x=0, \\
&\int_0^T\int_\R\delta V_tq-\frac{\mu}{3}\int_0^T\int_\R\delta\left(\frac{1}{h}\partial_x(h^3V_{tx})\right)q-\int_0^T\int_\R\delta r q_x \\
&+\epsilon\int_0^T\int_\R\delta VV_xq-\epsilon\int_0^T\int_\R\delta V(Vq)_x=\int_0^T\int_\R\delta bq_x+\frac{\epsilon}{2}\int_0^T\int_\R\delta b_{tt}q_x.
\end{aligned}
\right.
\end{equation}
Since it is possible to drop the $O(\mu^2)$ terms, under the extra assumptions $\lim_{|x|\rightarrow\infty}q_x(t,x)=0$ and $q_{xx}(T,x)=0$, one has
\begin{align*}
-\frac{\mu}{3}\int_0^T\int_\R\delta\left(\frac{1}{h}\partial_x(h^3V_{tx})\right)q=\frac{\mu}{3}\int_0^T\int_\R\delta V\partial_x(h^2q_{tx}).
\end{align*}
We have that $\delta r(0,x)=\delta V(0,x)=0$. On the control $b$ we are going to assume $\delta b(0,x)=\delta b(T,0)=0$ and $\delta b_t(0,x)=\delta b_t(T,x)=0$. Then (<ref>) becomes into
\begin{equation}
\label{s4}
\left\{
\begin{aligned}
&-\int_0^T\int_\R\delta r p_t+\int_\R\delta r(T)p(T)-\int_0^T\int_\R\delta hVp_x-\int_0^T\int_\R\delta Vhp_x=0, \\
&-\int_0^T\int_\R\delta Vq_t+\int_\R\delta V(T)q(T)+\int_0^T\int_\R\delta V\frac{\mu}{3}\partial_x(h^2q_{tx})-\int_0^T\int_\R\delta r q_x \\
&-\epsilon\int_0^T\int_\R\delta VVq_x=\int_0^T\int_\R\delta bq_x+\frac{\epsilon}{2}\int_0^T\int_\R\delta bq_{ttx}.
\end{aligned}
\right.
\end{equation}
Summing both equations in (<ref>) and having into account that $\delta h=\epsilon\delta r$, we have
\begin{multline}
\label{s5}
\int_0^T\int_\R\delta r(-p_t-\epsilon Vp_x-q_x) \\
+\int_0^T\int_\R\delta V(-hp_x-q_t+\frac{\mu}{3}\partial_x(h^2q_{tx})-\epsilon Vq_x)+\int_\R\delta r(T)p(T)+\int_\R\delta V(T)q(T) \\
=\int_0^T\int_\R\delta b(q_x+\frac{\epsilon}{2}q_{ttx}).
\end{multline}
Note that we have made some assumptions on the behavior of the bottom and its first time derivative on $0$ and $T$. This is not an issue since we are thinking in an underwater device that is at rest at the beginning and end of the physical experiment.
Considering the functional $J$ defined in (<ref>), with $\alpha=0$, we obtain
\begin{align}
\label{s6}
\delta J(b)[\delta b]&=\int_0^T\int_\R(\zeta-\bar{\zeta})\delta\zeta+\int_0^T\int_\R(V-\bar{V})\delta V \\
&=\int_0^T\int_\R(\zeta-\bar{\zeta})(\delta r+\delta b)+\int_0^T\int_\R(V-\bar{V})\delta V, \notag
\end{align}
as long as $b(\cdot,x)$ be compact supported in $[0,T]$.
Therefore, assuming
\begin{equation}
\label{s7}
\left\{
\begin{aligned}
&p_t+\epsilon Vp_x+q_x=\bar{\zeta}-\zeta, \\
&q_t-\frac{\mu}{3}\partial_x(h^2q_{tx})+hp_x+\epsilon Vq_x=\bar{V}-V, \\
&p(T)=0, \quad q(T)=0,
\end{aligned}
\right.
\end{equation}
from (<ref>) and (<ref>) we have
\begin{equation*}
\delta J(b)[\delta b]=
\int_0^T\int_\R\delta b(q_x+\frac{\epsilon}{2}q_{ttx}+(\zeta-\bar{\zeta}))dxdt.
\end{equation*}
On the other hand, by Theorem <ref> we know that there exists an optimal pair $(U^*,b^*)$ of (<ref>). Then, necessarily we have $\nabla J(b^*)=0$. Namely, if we consider $\delta U=U+U^*$ and $\delta b=b+b^*$ then
\[ 0=\nabla J(b^*)=q_x^*+\frac{\epsilon}{2}q_{ttx}^*+(\zeta^*-\bar{\zeta}). \]
Observe that (<ref>) can be written as
\begin{equation*}
\left(
\begin{pmatrix}
h & 0 \\ 0 & 1
\end{pmatrix}
\begin{pmatrix}
0 \\ -\frac{1}{3}\partial_x(h^2\partial_x\cdot)
\end{pmatrix}
\right)
\begin{pmatrix}
p_t \\ q_t
\end{pmatrix}
\begin{pmatrix}
h\epsilon V & h \\ h & \epsilon V
\end{pmatrix}
\begin{pmatrix}
p_x \\ q_x
\end{pmatrix}
\begin{pmatrix}
h(\bar{\zeta}-\zeta) \\ \bar{V}-V
\end{pmatrix}.
\end{equation*}
From this last expression and proceeding as we did to prove Theorem <ref>, it can be proved that there exists a $T>0$ such that $(p^*,q^*)\in C([0,T/\epsilon];X^s)$. Therefore, we can state now a theorem giving a system of first order necessary optimality conditions for problem (<ref>).
Let $(U^*,b^*)$ be an optimal solution for (<ref>). Then there exists $(p^*,q^*)\in C([0,T/\epsilon];X^s)$ satisfying the following optimality system, in a variational sense:
\begin{equation*}
\left\{
\begin{aligned}
&r_t^*+(h^*V^*)_x=0, \\
& V_t^*-\frac{\mu}{3h^*}\partial_x((h^*)^3V_{tx}^*)+r_x^*+\epsilon V^*V_x^*=-b_x^*-\frac{\epsilon}{2}b_{ttx}^*, \\
&\zeta(0)=\zeta_0, \ V(0)=V_0, \\
&p_t^*+\epsilon V^*p_x^*+q_x^*=\bar{\zeta}-\zeta^*, \\
& q_t^*-\frac{\mu}{3}\partial_x((h^*)^2\partial_xq_t^*)+h^*p_x^*+\epsilon V^*q_x^*=\bar{V}-V^*, \\
&p^*(T)=0, \ q^*(T)=0, \\
\end{aligned}
\right.
\end{equation*}
§ THE REFERENCE SCHEME RUSANOV+FD
In this section we build a solver for the PDE-constraint optimization problem in which system (<ref>) is solved with a conservative finite volume scheme, the Rusanov method, and system (<ref>) is discretized by a finite difference approach. Here, we propose the solver in a general approach, so we describe the method for (<ref>) and (<ref>) which are more general than (<ref>) and (<ref>), respectively.
System (<ref>) is solved by the finite volume formula
\begin{eqnarray}
\label{eq:one-step-formula}
\begin{array}{c}
\mathbf{U}_i^{n+1}
\mathbf{U}_i^{n} - \frac{\Delta t}{\Delta x} ( \mathbf{F}_{i+\frac{1}{2}}-\mathbf{F}_{i-\frac{1}{2}}) + \Delta t \mathbf{B}_i \;,
\end{array}
\end{eqnarray}
where $\mathbf{F}_{i+\frac{1}{2}}$ is a numerical flux, in this case the Rusanov flux is implemented, that corresponds to
\begin{eqnarray}
\begin{array}{c}
\mathbf{F}_{i+\frac{1}{2}}
=\frac{1}{2} ( \mathbf{F}( \mathbf{U}_{i+1}^{n}) + \mathbf{F}( \mathbf{U}_i^{n}))
-\frac{\lambda_{i+\frac{1}{2}}}{2} ( \mathbf{U}_{i+1}^{n} - \mathbf{U}_{i}^{n} ) \;;
\end{array}
\end{eqnarray}
here $\lambda_{i+\frac{1}{2}} = max(\lambda_L, \lambda_R)$ with
$$ \lambda_L = max\{ | \epsilon V_i^n - \sqrt{r_i^n} |, | \epsilon V_i^n + \sqrt{r_i^n} | \} $$
$$ \lambda_R = max\{ | \epsilon V_{i+1}^n - \sqrt{r_{i+1}^n} |, | \epsilon V_{i+1}^n + \sqrt{r_{i+1}^n} | \} \;.$$
For the source term we use central finite difference scheme for approximating spatial and time derivatives, that is, a similar approach to that used in (<ref>).
To solve the adjoint system (<ref>) we use
\begin{eqnarray}
\label{eq:discretize-adjoint}
\begin{array}{c}
- \mathbf{P}_i^{n+1} + \mathbf{P}_i^{n} - \Delta t \mathbf{A}(\mathbf{U}_i^{n+1}, \mathbf{P}_i^{n+1}) \frac{ ( \mathbf{P}_{i+1}^{n+1} - \mathbf{P}_{i-1}^{n+1} )}{2 \Delta x} = \Delta t \mathbf{R}( x_{i}, \mathbf{P}_i^{n+1}, \mathbf{U}_i^{n+1} ) \;.
\end{array}
\end{eqnarray}
Notice that the adjoint system is solved back from $ \mathbf{P}_i^{n_T} = \mathbf{0}$ , where $ n_T$ is the number of time iterations to reach the output time, $T$, of the simulation.
As commented above, these methods are functional and simple enough for problems in this paper. To extend them to other systems only expressions for $\mathbf{B}(x,\mathbf{U})$, $\mathbf{F}(\mathbf{U})$, $\mathbf{A}(\mathbf{U}, \mathbf{P})$ and $\mathbf{R}(x,\mathbf{U}, \mathbf{P})$ must be provided.
This complete the description of the solvers for the state and adjoint systems. They need to be combined to generate the minimizer sequence for a cost functional $J(b)$. Algorithm <ref> summarizes the global procedure to obtain $b$, iteratively. The iterations are stopped by fixing the maximum number iterations to $IterTotal = 17$.
Global algorithm for obtaining $b(t,x)$. Provide $b^0$, $it = 1$, $\lambda_b$, $IterTotal$ and $Tol$.
$b$ at each $x_i$ and $t^n$.
$it \leq IterTotal$
Step 1: Solve eq. (<ref>) using (<ref>).
This generates $\{ \mathbf{U}_{i}^{n} \}_{n=1}^{n_T}$, $i=1,...,N_x$, where $N_x$ is the number of cells.
Step 2: Solve (<ref>) using (<ref>), in this step $\{ \mathbf{P}_{i}^{n} \}_{n=1}^{n_T}$ is generated.
Step 3: Update $b^{k+1} = b^k -\lambda_b\nabla(b^k)$, where
$ \nabla J = q_x +\frac{1}{2}\epsilon q_{xtt} - (\bar{\zeta} - \zeta) \;.$
Step 4: Stop the algorithm once $ | \nabla (b^k,\mathbf{U}_i, \mathbf{P}_i)| < Tol$, where $Tol$ is some prescribed tolerance.
[1]
S. l. instant sport.
Accessed: 2023-09-23.
[2]
T. Alazard, P. Baldi, and D. Han-Kwan.
Control of water waves.
Journal of the European Mathematical Society, 20(3):657–745,
[3]
S. Alinhac and P. Gérard.
Opérateurs pseudo-différentiels et théoreme de
In Opérateurs pseudo-différentiels et théorème
de Nash-Moser. EDP Sciences, 2022.
[4]
B. Alvarez-Samaniego and D. Lannes.
A nash-moser theorem for singular evolution equations. application to
the serre and green-naghdi equations.
Indiana University mathematics journal, pages 97–131, 2008.
[5]
L. Baudouin, E. Cerpa, E. Crépeau, and A. Mercado.
On the determination of the principal coefficient from boundary
measurements in a kdv equation.
Journal of Inverse and Ill-posed Problems, 22(6):819–845,
[6]
S. Beizel, L. Chubarov, D. Dutykh, G. Khakimzyanov, and N. Y. Shokina.
Simulation of surface waves generated by an underwater landslide in a
bounded reservoir.
Russian Journal of Numerical Analysis and Mathematical
Modelling, 27(6):539–558, 2012.
[7]
A. Constantin.
Nonlinear water waves with applications to wave-current
interactions and tsunamis.
SIAM, 2011.
[8]
W. Craig.
An existence theory for water waves and the boussinesq and
korteweg-devries scaling limits.
Communications in Partial Differential Equations,
10(8):787–1003, 1985.
[9]
D. Dutykh and F. Dias.
Water waves generated by a moving bottom.
In Tsunami and Nonlinear waves, pages 65–95. Springer, 2007.
[10]
D. Dutykh, D. Mitsotakis, L. B. Chubarov, and Y. I. Shokin.
On the contribution of the horizontal sea-bed displacements into the
tsunami generation process.
Ocean Modelling, 56:43–56, 2012.
[11]
M. Fontelos and J. López-Ríos.
Controllability of surface gravity waves and the sloshing problem.
ESAIM: Control, Optimisation and Calculus of Variations, 29:65,
[12]
M. A. Fontelos, R. Lecaros, J. López-Ríos, and J. H. Ortega.
Bottom detection through surface measurements on water waves.
SIAM Journal on Control and Optimization, 55(6):3890–3907,
[13]
J.-M. Ghidaglia, A. Kumbaro, and G. Le Coq.
On the numerical solution to two fluid models via a cell centered
finite volume method.
European Journal of Mechanics-B/Fluids, 20(6):841–867, 2001.
[14]
T. Iguchi.
A mathematical analysis of tsunami generation in shallow water due to
seabed deformation.
In Proceedings of the Royal Society of Edinburgh-A-Mathematics,
volume 141, page 551. Cambridge Univ Press, 2011.
[15]
S. Israwi.
Large time existence for 1d green-naghdi equations.
Nonlinear Analysis: Theory, Methods & Applications,
74(1):81–93, 2011.
[16]
D. Lannes.
Well-posedness of the water-waves equations.
Journal of the American Mathematical Society, 18(3):605–654,
[17]
D. Lannes.
The water waves problem: mathematical analysis and asymptotics,
volume 188.
American Mathematical Soc., 2013.
[18]
D. Lannes and P. Bonneton.
Derivation of asymptotic two-dimensional time-dependent equations for
surface water wave propagation.
Physics of Fluids (1994-present), 21(1):016601, 2009.
[19]
R. Lecaros, J. López-Ríos, J. Ortega, and S. Zamorano.
The stability for an inverse problem of bottom recovering in
Inverse Problems, 36(11):115002, 2020.
[20]
J.-M. Lellouche, J.-L. Devenon, and I. Dekeyser.
Boundary control of burgers' equation—a numerical approach.
Computers & Mathematics with Applications, 28(5):33–44, 1994.
[21]
S. Micu.
On the controllability of the linearized benjamin–bona–mahony
SIAM Journal on Control and Optimization, 39(6):1677–1696,
[22]
S. Micu, J. H. Ortega, L. Rosier, and B.-Y. Zhang.
Control and stabilization of a family of boussinesq systems.
Discrete and Continuous Dynamical Systems, 24:273–313, 2009.
[23]
G. I. Montecinos.
A universal centred high-order method based on implicit taylor series
expansion with fast second order evolution of spatial derivatives.
Journal of Computational Physics, 443:110535, 2021.
[24]
G. I. Montecinos, J. C. López-Ríos, J. H. Ortega, and R. Lecaros.
A numerical procedure and coupled system formulation for the adjoint
approach in hyperbolic pde-constrained optimization problems.
IMA Journal of Applied Mathematics, 84(3):483–516, 2019.
[25]
S. Mottelet.
Controllability and stabilization of a canal with wave generators.
SIAM journal on control and optimization, 38(3):711–735, 2000.
[26]
V. Nalimov.
The cauchy-poisson problem.
Dynamika Splosh. Sredy, 18:104–210, 1974.
[27]
H. Nersisyan, D. Dutykh, and E. Zuazua.
Generation of 2d water waves by moving bottom disturbances.
IMA Journal of Applied Mathematics, 80(4):1235–1253, 2015.
[28]
M. Nosov and S. Kolesov.
Method of specification of the initial conditions for numerical
tsunami modeling.
Moscow University Physics Bulletin, 64(2):208–213, 2009.
[29]
M. Nosov and S. Skachko.
Nonlinear tsunami generation mechanism.
Natural Hazards and Earth System Sciences, 1(4):251–253, 2001.
[30]
M. Qi, Y. Kuai, and J. Li.
Numerical simulation of water waves generated by seabed movement.
Applied Ocean Research, 65:302–314, 2017.
[31]
L. Rosier.
Exact boundary controllability for the linear korteweg–de vries
equation on the half-line.
SIAM Journal on Control and Optimization, 39(2):331–351, 2000.
[32]
V. Rusanov.
Calculation of interaction of non–steady shock waves with obstacles,
ussr comp. math. math., 1, 267–279.
[33]
Y. Shen, C. N. Whittaker, E. M. Lane, W. Power, and B. W. Melville.
Interference effect on tsunami generation by segmented seafloor
Ocean Engineering, 245:110244, 2022.
[34]
P. Su, M. Tucsnak, and G. Weiss.
Stabilizability properties of a linearized water waves system.
Systems & Control Letters, 139:104672, 2020.
[35]
P. Su, M. Tucsnak, and G. Weiss.
Strong stabilization of small water waves in a pool.
IFAC-PapersOnLine, 54(9):378–383, 2021.
[36]
M. E. Taylor.
Partial differential equations. iii, volume 117 of applied
mathematical sciences, 1997.
[37]
E. F. Toro, B. Saggiorato, S. Tokareva, and A. Hidalgo.
Low-dissipation centred schemes for hyperbolic equations in
conservative and non-conservative form.
Journal of Computational Physics, 416:109545, 2020.
[38]
S. Wu.
Well-posedness in sobolev spaces of the full water wave problem in
Inventiones mathematicae, 130(1):39–72, 1997.
[39]
S. Wu.
Well-posedness in sobolev spaces of the full water wave problem in
Journal of the American Mathematical Society, 12(2):445–495,
[40]
H. Yosihara.
Capillary-gravity waves for an incompressible ideal fluid.
Kyoto Journal of Mathematics, 23(4):649–694, 1983.
[41]
X. Zhang and E. Zuazua.
Unique continuation for the linearized benjamin-bona-mahony equation
with space-dependent potential.
Mathematische Annalen, 325(3):543–582, 2003.
|
# Colour remote sensing of the impact of artificial light at night (II):
Calibration of DSLR-based images from the International Space Station
Alejandro Sánchez de Miguel<EMAIL_ADDRESS>Jaime Zamorano Martin Aubé
Jonathan Bennie Jesús Gallego Francisco Ocaña Donald R. Pettit William L.
Stefanov Kevin J. Gaston Environment and Sustainability Institute,
University of Exeter, Penryn, Cornwall TR10 9FE, U.K. Depto. Física de la
Tierra y Astrofísica. Instituto de Física de Partículas y del Cosmos
(IPARCOS), Universidad Complutense, Madrid, Spain Physics dept., CEGEP de
Sherbrooke, Sherbrooke, J1E 4K1, Canada Instituto de Astrofísica de
Andalucía, Glorieta de la Astronomía, s/n,C.P.18008 Granada, Spain Centre for
Geography and Environmental Science, University of Exeter, Penryn, Cornwall
TR10 9FE, UK Quasar SR for ESA, European Space Astronomy Centre, E-28691
Villanueva de la Cañada, Spain Astronaut Office, Flight Operations
Directorate, NASA Lyndon B. Johnson Space Center, Houston, Texas, USA
Exploration Science Office, Astromaterials Research and Exploration Science
Division, Exploration Integration and Science Directorate, NASA Lyndon B.
Johnson Space Center, Houston, Texas, USA
###### Abstract
Nighttime images taken with DSLR cameras from the International Space Station
(ISS) can provide valuable information on the spatial and temporal variation
of artificial nighttime lighting on Earth. In particular, this is the only
source of historical and current visible multispectral data across the world
(DMSP/OLS and SNPP/VIIRS-DNB data are panchromatic and multispectral in the
infrared but not at visible wavelengths). The ISS images require substantial
processing and proper calibration to exploit intensities and ratios from the
RGB channels. Here we describe the different calibration steps, addressing in
turn Decodification, Linearity correction (ISO dependent), Flat
field/Vignetting, Spectral characterization of the channels, Astrometric
calibration/georeferencing, Photometric calibration (stars)/Radiometric
correction (settings correction - by exposure time, ISO, lens transmittance,
etc) and Transmittance correction (window transmittance, atmospheric
correction). We provide an example of the application of this processing
method to an image of Spain.
###### keywords:
artificial lighting , light pollution , night , remote sensing , urban
††journal: Remote Sensing for Environment
## 1 Introduction
There is growing demand for colour imagery of the Earth at night. This has
particularly been driven by increasing recognition of the impacts of outdoor
artificial nighttime lighting (from streetlights and other sources) on the
natural environment, human health, and associated policy and public concerns
(e.g. Rich and Longcore [2013]; Hölker et al. [2010]; Falchi et al. [2011];
Gaston et al. [2012, 2015]; Gaston [2013, 2018]; Garcia-Saenz et al. [2018]).
These impacts are not only influenced by the spatial extent, the timing and
the intensity of that lighting, but also by its spectrum. This last
consideration limits the insights that can be obtained from the vast majority
of historical and current sources of geographic scale data on the occurrence
of such lighting, including data from the Defense Meteorological
Program/Operational Line-Scan System (DMSP/OLS; Elvidge et al. [1997]) and
from the Suomi-National Polar-Orbiting Partnership/Visible and Infrared
Imaging Radiometer Suite - Day/Night Band (SNPP/VIIRS-DNB; Elvidge et al.
[2013]).
By far the most important source of colour imagery of the Earth at night is
that obtained from the International Space Station (ISS) by the astronauts and
cosmonauts of the five space agencies that constitute the ISS consortium:
NASA, ESA, JAXA, CSA-ASC and ROSCOSMOS (Stefanov et al. [2017, 2019a]).
Between 2002 and 2018, more than 3 million images have been taken by the crew
of the ISS, of which 1.34 million were taken at night (sun elevation less than
0 degrees). The diurnal and nocturnal images from NASA, CSA-ASC and ESA are
stored in the NASA archive (http:/eol.jsc.nasa.gov; Stefanov et al. [2017,
2019a]). Some, but not all, of the images taken by JAXA and ROSCOSMOS are also
stored in the NASA archive. The scientific value of these images is immense,
as they constitute the only large public dataset with nighttime colour
information obtained from space over the last 17 years (Levin et al., 2020).
Due to the limitations of data from the VIIRS and DMSP satellites, this
dataset is key to determining change in visible artificial lighting over this
long period (Kyba et al. [2017]; Sánchez de Miguel et al. [2019]).
The camera bodies and lenses used by the astronauts to take images of the
Earth have mainly been from Nikon’s professional product line (see Table 1 for
those used for nocturnal photography). These have usually been unmodified,
with the exception of the Nikon D3S S/N:2007944 used on missions ISS030 and
ISS031 that had the infra-red filter removed. Most of the Nikon cameras used
have had a CMOS sensor, although the Kodak 760C and the Nikon D1 that were
used had CCD sensors (Waltham [2013]). All the cameras have a Bayer filter
that provides simultaneous images in three colours (two green, one red and one
blue).
The environmental conditions on the ISS are strictly controlled, with
temperature at 24ºC $\pm$ 2ºC, pressure around 745-721 mmHg, and relative
humidity around 60%. This means that the cameras have not only been operating
well within their nominal environmental ranges but effectively in constant
environmental conditions. Although cameras have been used during
extravehicular activity (EVA) or in external experiments, principally windows
in the cupola and elsewhere on the ISS enable astronauts to take images of the
Earth from a wide variety of angles. The ground track of the ISS covers nadir
latitudes from 51.6º N to 51.6º S, at an altitude of approximately 405 km,
although this can vary in the range of 330 to 435 km. Usually images have been
taken by astronauts using cameras that are handheld. Pettit [2009] developed a
prototype system that partially compensated for movement during image
acquisition and was used during mission ISS006. Later missions, like ISS030 to
ISS040, have employed a special tripod (Nightpod; Sabbatini [2014]). At least
for images held in the open NASA archive there is no information on their
acquisition beyond the metadata of the images themselves, thus it is not easy
and frequently impossible to attribute them to a specific window, tracking
device or operator. Only when an astronaut posts their own images, on Twitter
for example, can we have reasonable confidence in their authorship.
A growing number of science studies have attempted to use images taken from
the ISS in their raw or crudely enhanced forms [Venzke et al., 2009, Lockwood
and Hazlett, 2013, Anderson et al., 2010, Levin and Duke, 2012, Liu et al.,
2011, Metcalf, 2012, Kuechly et al., 2012, Mazor et al., 2013, Sánchez de
Miguel et al., 2014, Li et al., 2014, So, 2014, Kyba et al., 2014, Rybnikova
and Portnov, 2017, 2018, Xu et al., 2018, Levin et al., 2020]. Indeed, the
only research work we are aware of that made some attempt to address issues of
calibration is that of So [2014], which used a dark subtraction, and those in
which the current team has been involved [Zamorano et al., 2011, Sánchez de
Miguel, 2015, Garcia-Saenz et al., 2018, 2019, Hale and Arlettaz, 2019, Bará,
2020, Pauwels et al., 2019, Sánchez de Miguel et al., 2019, 2020]. The effects
of this lack of calibration can be very different from study to study and need
to be considered carefully. Unfortunately, in many cases calibration of images
taken by astronauts from the ISS will be necessary in order to provide
accurate representations of the colour composition of scenes, and this is not
a trivial task.
In a previous paper [Sánchez de Miguel et al., 2019], we described the use of
colour-colour diagrams to analyse images taken by astronauts on the ISS and to
estimate spatial and temporal variation in the spectrum of artificial lighting
emissions. In this paper we provide a methodology for calibrating such images.
The approach that we describe is also relevant to the calibration of images
taken with standard DSLR cameras for other purposes, including meteor science
(meteor photometry - Borovička et al. [2014]; meteor video photometry -
Madiedo et al. [2019]; meteor spectroscopy - Cheng and Cheng [2011]),
measurement of skyglow [Hänel et al., 2018], and more generally scientific
photography using DSLR cameras with antiblooming. In order to use well
characterized standard emitters, we use stars as calibration sources, as the
variability and stability of their spectra is well known. Indeed, laboratory
calibration sources typically have a precision of 0.2% [Wolfe, 1998], whilst
calibration by stars can attain precisions of 0.001% [Poddanỳ et al., 2010].
This technique also allows us to use multiple calibrations from onboard the
ISS that are stable in time and that account for the window through which
images were obtained. A similar technique has also been proposed for
calibration of data from the SNPP/VIIRS-DNB [Fulbright and Xiong, 2015] and
was suggested first by Zamorano et al. [2011].
Calibration of images of the earth acquired from the ISS requires a number of
steps that we will address in turn (Fig. 1): Decodification, Linearity
correction (ISO dependent), Flat field/Vignetting, Spectral characterization
of the channels*, Astrometric calibration*/georeferencing, Photometric
calibration (stars)*/ Radiometric correction (settings correction - by
exposure time, ISO, lens transmittance, etc) and Transmittance correction
(window transmittance, atmospheric correction). The steps marked with an
asterisk may not be required for some specific science cases (e.g.
differential evolution time - Meier [2018], Bará et al. [2019]; edge detection
- Kotarba and Aleksandrowicz [2016]). Neglecting any particular correction
should always explicitly be justified.
In this paper, we will focus primarily on calibration of images taken with the
Nikon D3S (for calibration equipment see Supplementary Information) because
this is the best studied case and the camera that has been used most
intensively for nocturnal photography at the ISS over the two last decades
(from mission ISS026 until mission ISS045). However, in general, with small
adjustments, the procedures described can be applied to any other digital
camera, and we will consider all of the cameras used at the ISS (Table 1) and
highlight the differences between them when it is relevant. The Nikon D3S has
a 12.1 megapixel sensor equivalent to a full frame (35mm) and was announced by
Nikon Corporation on 14 October 2009. It has interchangeable lenses using the
F-mount, an ISO range from 200 to 12800, and 14-bit A/D conversion.
## 2 Calibration steps
### 2.1 Step 1: Decodification
A DSLR camera obtains information as photons of light that produce electrons,
and these electrons are effectively stored in pixels of the CCD/CMOS chip. The
electrons are trapped by the nearest potential well (one per pixel) and
digitalized by an analogue conversion (DA Converter) sensor that measures the
electrical potential on the wells of the chip. Whilst in some professional
cameras a non-destructive read out is possible [Nakamura et al., 1995], most
CCD/CMOS chips are read by measuring the current when it passes though the DA
converter [Fowler and El Gamal, 1995, Fossum et al., 2000]. This information
is pre-processed and saved as a data file. This file is usually coded in a
proprietary format, and in the case of Nikon cameras, this is called NEF. This
is what is called a RAW file. The first step in the calibration process is the
decodification of this file, taking the data coded in the proprietary format
and transforming it into a readable open format, such as TIFF [Desk, 1992] or
FITS [Hanisch et al., 2001], without losing crucial information or changing
the physical meaning.
Even so-called RAW files from a DSLR tend to have undergone some pre-
processing. A pure ideal CCD/CMOS RAW file coming from an ideal sensor will
have three components: (i) the signal itself, (ii) an artificial signal (bias)
included on the sensor to avoid negative values in the process of
digitalization, and (iii) Gaussian noise plus an additive constant (dark
current) arising from the sensor’s sensitivity to heat. On our reference
camera, the Nikon D3s, and all subsequent versions except the Nikon D5, a
complex hardware treatment is applied to the signal [Koyama and Jiang, 2011,
Sánchez de Miguel, 2015]. This treatment means that we do not need to do any
bias or dark correction in these cases, but noise effects need to be
controlled. DSLR sensors are capable of a 16-bit dynamic range. However, in
practice on commercial models it is 14 or 12 bits, because saving 16-bit
information takes too much time. This means that some detailed information is
lost in the digitalization process with a loss of dynamic range.
The current most widely used software to undertake decodification is DCRAW
[Coffin, 2008] and the most conservative decodification options are: -6 that
means 16 bit extraction, -o 0 that means no color balance, -H 1 that means
nothing excluded, and -v to show any error in the decodification process.
### 2.2 Step 2: Linearity correction
While most scientific camera sensors force linearity between numbers of
incident photons and pixel signal strength as a key characteristic, commercial
DSLR cameras have sensors with an anti-blooming function. By using an overflow
channel, this system ensures that when a well of the chip is saturated then
excess electrons do not contaminate adjacent wells (as can happen with regular
CCD or CMOS sensors)[Sakai et al., 2009]. This produces a roughly linear
response in the pixel signal when high gain is used. Unfortunately, there are
no details available of the specific technology used for the Nikon DX sensors.
Currently, we have not detected differential absorption by chip substrate,
although this could be investigated further in future versions of the
processing pipeline.
Typically, the gain of the camera operates to an ISO standard [ISO 12232:2006
(2006), E]. Fig. 2 shows the effect of anti-blooming when a camera is exposed
to a stable light source across all of the different exposure times that the
camera is able to produce. This effect can be compensated by characterization
of this response for a particular ISO. Even if a camera is in principle
calibrated to the ISO standard, this still allows a 20% error, and there can
be differences on the different channels. Indeed, without corrections, bad
colour information (aka. band ratios, see Sánchez de Miguel et al. [2019]) can
be produced. As Metcalf [2012] found, radiance values can be underestimated,
and radiance ratios can appear variable even if they are stable.
Having characterised the response of the camera for a particular ISO, the
intensity values of an image that has been taken using this can be divided by
these corrected ones. Those over the range of the linear response will not
change (aka. under $\sim 17.000$ ADU) whilst those in the range of the non-
linear response will be linearized.
### 2.3 2.3 Step 3: Flat field
A wide range of lenses have been used with DSLRs on the ISS, from fish eyes to
telephotos. A few have been used for long periods, but others on just a single
mission. These lenses introduce various additional calibration issues. The
first of these is flat field/illumination/vignetting correction. This is a
very standard correction with DSLR imagery, although unfortunately not usually
addressed with imagery from the ISS (see [Burggraaff et al., 2019] for
standard corrections to consumer cameras). This correction consists of the
acquisition of shots from a uniform emitting source (aka. flat field), so that
any heterogeneity in the acquired image is the result of vignetting of the
lens and linearity effects. Distortion effects, such Barrel, Pincushion, and
Mustache can also be corrected. However, this should be avoided unless
absolutely necessary because of the challenges of combining such corrections
with those necessary due to the angular perspective at which many ISS images
are obtained as a consequence of the Earth’s curvature, and because performing
several successive distortions will amplify errors [Cardiel et al., 2002])
(see 2.7 Step 6).
Images of star fields from the ISS are not available for all combinations of
cameras and lenses that have been used to obtain images of the Earth. In order
to translate the light intensity measured using one lens to that which would
be measured using another the light transmission of the optical elements of
the latter needs to be considered. The f/ number can give a first order idea
of this in a standardised way. However, this is not sufficiently accurate for
many purposes, so instead the Transmission coefficient (T number) of the lens
needs to be measured. Table 2 provides these values, as well as the f/ number
for some of the most frequently used lenses on the ISS. Values in this table
have been calculated using the SaveStar Consulting S.L. lab by acquiring
images of a $4\pm 0.1$ lux illuminated lambertian surface. The light Source
was a tungsten filament.
Note that in general the apertures of lenses can be closed down to provide
sharper images. When this happens, the diaphragm rarely closes to exactly the
same position (except when fully open), so it is recommended that imagery is
taken using maximum apertures (explained in more detail in 2.5 Step 5c). On
bright sources this can be considered a compromise between sharpness of
images, signal to noise ratios, and photometric accuracy. Fortunately, this
has been done for most of the images taken at night from the ISS.
### 2.4 Step 4: Astrometric calibration
The astronauts do not only acquire images of cities at night. They also
acquire lots of other kinds of nocturnal images, including of auroras,
sunsets, and occasionally also star fields. The ”Cities at Night” project has
a NASA archive selection of these images[Cities at night collaboration, 2015].
They can be used for several proposes, such as calculating lens distortion and
lens transmittance, or as we do in this case, for radiometric calibration.
Once the characteristics of a camera and lens have been determined it is
necessary to use standard sources to calibrate the imagery of the Earth that
has been taken. The use for this purpose of starfield images taken from the
ISS has the great advantage that they were obtained with the equipment under
the same temperature, pressure and humidity conditions as the images of the
Earth. They will also have been acquired through windows with the same
reflection and transmission characteristics; light transmission through the
windows is very high except for some Zvezda windows (we can assume that
absorption in all bands is less than 5% in the visible regime, although
windows in the Destiny lab show significant transmission reduction beyond the
NIR), and special coatings have been used to avoid reflections (see 2.9 Step
8).
The first step in using starfield images as standard sources is astrometric
calibration, determining the coordinates of the stars in each image. This can
be done using standard astronomical methods. We use the software
Astrometry.net [Lang et al., 2010]. This automatically spatially calibrates an
image, so that each pixel has corresponding celestial coordinates. Moreover,
it extracts the sources and identifies them using a catalog. Stars are the
flux standard sources that we will subsequently use.
Astrometric calibration (also called the World Coordinate System) by
Astrometry.net provides direction, orientation and plate scale (transformation
between the apparent angular separation and linear separation at the focal
plane). The software does not require any additional input, and makes a blind
calibration using only the image. It is a powerful tool as there is no need to
know spacecraft attitude nor window orientation nor the direction in which an
astronaut was pointing the camera. It has also been used successfully in non-
stabilised high-altitude balloon observations where attitude is not controlled
[Ocaña, 2017, Ocaña et al., 2019].
### 2.5 Step 5a: Photometric calibration on board
To perform photometric calibration, we need a standard source. As there are no
electronic standard flux sources onboard of the ISS the most accurate method
of calibration is using the stars, as has been proposed for other satellites
[Fulbright and Xiong, 2015] and as is routinely done in astronomy.
We will focus on use of a selection of stars from Ducati [2002]. Even though
stars are very well characterized light sources, there are differences between
the filters that were used in making those measurements and those used in DSLR
cameras. To solve this problem we can use a statistical correction based on
the synthetic photometry technique [Straizys, 1996, Sánchez de Miguel et al.,
2017, 2019]:
$GN=V+0.1291(2)\times(B-V)-0.0051(2)$ (1)
$BN=GN+0.6123(7)\times(B-V)-0.0340(6)$ (2)
$RN=GN+0.0262(3)+0.5880(5)\times(R-V)$ (3)
where BN, RN and GN are the respective RGB bands of the DSLR image, and B, V
and R are Johnson bands (an astronomical standard); these relationships are
shown in Fig. 3. These corrections have been calculated for the spectral
response of the D3S although, as can been seen in Sánchez de Miguel et al.
[2019], all Nikon cameras have extremely similar spectral responses. Once
these corrections to the intensities of the stars are applied, the
transformation from data numbers to radiometry units is immediate by applying
a linear fit (see Fig. 4).
As the Ducati [2002] catalogue has few stars in the dim part of the
calibration and only 18 stars in total, for verification purposes we used the
Tycho catalogue [Hog et al., 2000], which contains many more stars.Fig. 5
shows the effect of the saturation of the brightest stars. Future analysis can
be done the new catalog Cardiel et al. [2021], specifically designed for DSLR
cameras.
By using these methods we get the correspondence between the radiance and the
data numbers in magnitudes. To convert this on the international system of
units we need to use:
$AB=-2.5\log_{10}(flux)-5\log_{10}(w)+2.401$ (4)
where AB is in $mag/arcsec^{2}$, flux in
$nW/cm^{2}/sr/$\mathrm{\SIUnitSymbolAngstrom}$$,and w is wavelength in
$\mathrm{\SIUnitSymbolAngstrom}$ [Sánchez de Miguel et al., 2017].
When we calibrate a starfield image taken from the ISS, we do not need to
apply any atmospheric correction, as is usually done when this technique is
used with starfield images taken from the ground, because the ISS is at 400 km
above the Earth and the density of the atmosphere there is negligible and for
this purpose is considered to be a vacuum (at this elevation pressure is even
higher than that of the regular vacuum labs on Earth, from $10^{-6}$ to
$10^{-9}$ torr [Finckenor and de Groh, 2017], compared to the 760 torr of the
atmosphere at sea level).
#### 2.5.1 Step 5b: Photometric calibration from the ground
As previously mentioned, some images of the Earth taken from the ISS have been
obtained using lenses for which starfield images are not available (for
example, because their focal lengths make it impossible to take sharp
starfield images given the speed of movement of the ISS). For those cases when
starfield calibration is not possible, the solution is to apply a standard
absolute photometry technique to achieve photometric calibration of the lens
from the ground.
For observations taken from the ground the light of the stars travels through
the atmosphere and suffers absorption, scattering, and dispersion depending on
the wavelength. The amount of extinction (the combined effect) depends on the
extinction coefficient of the atmosphere at this time and on the length of the
path of the light across the atmosphere. As a result, the flux density (in
units of $ergs/s/cm^{2}/$\mathrm{\SIUnitSymbolAngstrom}$$) measured on the
ground for a star is lower than the flux at the top of the atmosphere:
$F\mbox{(observed)}=F_{o}\cdot 10^{-0.4KX}$ (5)
where $F_{o}$ is the flux outside the atmosphere (which we know since we are
observing standard stars), K is the extinction coefficient for this
wavelength, and X the airmass calculated as X = sec z (this formula is only
usable for zenith angles up to about $60^{\circ}$ to $75^{\circ}$; for more
accurate version see 2.8 Step 7), z being the zenith angle (90 - elevation
above horizon). We call photometric nights those clear nights with constant
transparency (see definition of photometric night at
https://www.eso.org/sci/observing/phase2/ObsConditions.html). For these nights
the extinction coefficient is constant for the whole night. Observing several
standard stars through the night at different heights above the horizon we can
derive this coefficient and the zero point of the photometry for each
photometric band. The derived zero point of the photometric band is valid for
this observational setup (camera, lens, ISO, and f number) regardless of the
night of the observation, i.e. it depends only on the instrumentation. So, the
atmospheric effects only need to be considered on the particular night that
the calibration images are acquired.
#### 2.5.2 Step 5c: Setting adjustments
It is not possible to have images calibrated for all of the possible cameras
and settings. So, we scale the adjustments to 50 mm f/1.4. It is well known
that the repeatability of mechanical iris-type lens diaphragms is limited,
having a noticeable tolerance. The blades move back and forth during each
image acquisition, limiting the entrance pupil to approximately the same
position but not exactly so [Chylinski, 2012], creating so-called ‘aperture
flickering’. We have created a model to estimate the potential effect of
closing the shutter but the current one is a general correction with the f/
number:
$L_{0}=2\times f/^{-2}\>\mbox{ or }\>Tn$ (6)
$\mbox{Correction factor}=1/(ISO/100)\times C_{0}/T/B_{N}\times C_{1}/L_{0}$
(7)
where, ISO is the ISO 12232:2006 standard for digital photography, $C_{0}$ is
a correction for the sensitivity of the camera model, T is the exposure time,
$B_{N}$ is the correction for the bit rate, and $C_{1}$ is the colour
correction between different camera models. $L_{0}$ is the correction for the
aperture expressed as the f number or the Tn true transmission of the lens
when the shutter is fully open (see Table 2). Some of these settings can have
up to 15% error according to the ISO 12232:2006 standard.
## 3 Step 6: Georeferencing
Images of the earth from the ISS need to be georeferenced to establish the
link between image pixels and actual points on the ground. The time of
acquisition of each image is known but, although the orbit of the ISS is
precisely defined, unfortunately this does not provide sufficient information.
The time stamp data are not precise enough to define a nadir location (errors
may be more than 1000 km), and an image may be taken in a direction that
departs substantially from the nadir. A major citizen science program, “Cities
at Night” (http://www.citiesatnight.org,[Sánchez de Miguel et al., 2014]), is
being used to automate the georeferencing of large numbers of ISS images by
identifying the urban areas that these represent. More generally, and where
sufficient control points cannot automatically be identified, georeferencing
can be done manually, for which purpose we have used QGIS [Team et al., 2015]
and Global Mapper [Geographics, 2011]. Several Python libraries have been used
in this process, like NumPy [Oliphant, 2006], AstroPy [Robitaille et al.,
2013] and GDAL [Warmerdam, 2008]. The preferred sampling method is bilinear
for the pixels and thin plate spline for the coordinates. The number of
recommended control points can depend a lot on the inclination of the image
and lens. For nadir and 400 mm, 20 control points can be enough to reach RMSE
2.5 pixels using polynomial third degree fitting. For tilted images taken with
a 24 mm, for example, more than 100 points would be needed and with 400 points
can be visually indistinguishable from a comparison layer (VIIRS); the only
transformations able to rectify deformations on tilted images are Thin Plane
Splines that provide unrealistic values of RMSE - a conservative estimation
can be RMSE $\sim 4$ pixels, but more research is needed to systematise this
analysis.
### 3.1 Step 7: Atmospheric correction
Whilst no atmospheric correction is required for starfield images taken from
the ISS, this needs to be done for images taken of the Earth. The procedure
selected depends on the concept of airmass. In astronomy, one airmass is
equivalent to the volume of atmosphere in the direction of the zenith of an
observer on the ground (that is equivalent to the same volume of mass to an
observer located in orbit looking to the nadir at the same geographical
coordinates). Using this approach it is straightforward to estimate the
equivalent volume of atmosphere at different zenith angles or the
corresponding nadir angles. The formula X = sec z, as above, can be used up to
80 degrees of zenith angle considering a plane parallel approximation of
atmosphere with about 3% error and for higher angles several models are
available (e.g. Pickering [2002]). Then, we can atmospherically correct each
image using:
$I/I_{0}=10^{X_{h}(h)\times K/-2.5}$ (8)
where $X_{h}$ is the airmass function of the height, h is the height (in
metres), I is the observed intensity, $I_{0}$ the intensity with atmospheric
effect, and K is the extinction coefficient (derived from Harwit [1973]). The
extinction coefficient for 1 air mass can be calculated as:
$K=A_{rag}+A_{aer}+A_{oz}$ (9)
where $A_{rag}$ is Raleigh scattering, $A_{aer}$ the absorption by aerosols,
and $A_{oz}$ the absorption by ozone. Using the definitions compiled in García
Gil et al. [2012]:
$A_{ray}=9.4977\times
10^{-3}\left(\frac{1}{\lambda}\right)^{4}c^{2}\times\mbox{exp}\left(-\frac{H_{obs}}{7.996}\right)$
(10)
$c=0.23465+\left(\frac{1.076\times
10^{2}}{146-(1/\lambda^{2})}\right)+\left(\frac{0.93161}{41-(1/\lambda^{2})}\right)$
(11)
where $H_{obs}$ is the height of the observer, $\lambda$ is the wavelength in
microns, and ”c” is the air’s refraction index.
$A_{aer}(\lambda,h)=A_{o}\lambda^{-\Sigma}exp(-H_{obs}/H)$ (12)
where H is the density scale height for aerosols and $A_{o}$ is the total
optical thickness of atmospheric aerosols for $\lambda=1\mu m$, which depends
on the total content of particles and on their efficiency for scattering and
absorption and is taken to be 0.087 [Walker, 1988, Mohan et al., 1999].
$\Sigma$ is a parameter that depends on the size of the aerosol particles.
Each of these factors $(A_{rag},A_{aer},A_{oz})$ is wavelength dependent and
also depends on atmospheric conditions. By convention, by default we have
considered the aerosol content conditions of the AERONET [Holben et al., 1998]
Madrid station on the 29th March 2012 (AOD340 = 0.333, AOD380 = 0.305, AOD440
= 0.251, AOD500 = 0.208, AOD675 = 0.127, AOD870 = 0.084, AOD1020 = 0.066),
interpolated for the center of the Nikon bands. We use Rayleigh [1899] to
consider the Rayleigh scattering (equation 10), Stalin et al. [2008] to
consider the aerosols (equation 12) and Hayes and Latham [1975] to consider
the Ozone absorption ($A_{oz}$ factor). To correct for all of these effects
equation 8 should be applied to each pixel of the image. Currently, the
aerosol content correction is only accurate for a narrow field of view, for a
wide field of view different AODs would be needed per pixel. In the future,
these data could be obtained from VIIRs products [Huang et al., 2016].
Currently, no other nocturnal lights products provide multispectral aerosol
corrections and the only other nocturnal light product that provides some
correction of this nature is the dimensional panchromatic aerosol corrections
from VIIRS VNP46A2 [Román et al., 2018] that are still in the beta phase.
To determine the original location of the ISS we use a photogrammetric
technique. Considering a circle on geographical coordinates, we will obtain an
ellipse on the raw image coordinates. The axis of the ellipse is either
coincident with the direction of observation or perpendicular to the direction
of observation. Matching those two directions with the ISS ground track, we
obtain two potential solutions for the true nadir of the ISS. Most of the time
the minor axis indicates the direction of the true nadir, although, because of
lens aberrations and the curvature of the earth, when the ISS astronauts take
images very close to the nadir it can be the major axis that will align with
the true view direction (see Fig. 6). Most of the time, a visual inspection of
the image allows the identification of the true nadir, by looking at the
patterns of the streets and how the buildings block the light of the streets.
Once we know the true nadir, as the altitude of the ISS is also known, it is
straightforward to determine the tilt angle (we define the tilt angle as that
between the nadir and the view angle of the center of the image). Then, the
atmospheric extinction can be determined using the approach used in astronomy
because of the Helmholtz reciprocity law, the same extinction that an observer
has looking to the ISS from a location A is what an observer on the ISS has
looking to location A. As the lens used may have a very wide field of view,
the distance from the ISS to the ground can be significantly different from
one part of an image to another, so we calculate this extinction for each
pixel of the image.
In some ISS images of the Earth at night, high thin clouds, fog, etc are
apparent. These are challenging to address in the generic pipeline described
here, and may require more of a case-by-case approach, employing available
tools. So, currently, we recommend discarding any areas where the Point Spread
Function (PSF) of the emission sources is not homogeneous.
### 3.2 Step 8: Intercalibration
For a variety of reasons, there may be some systematic differences in the
extrapolation of the calibration with stars. One of the main unknowns in
analysing an image taken from the ISS is which window was used. Sometimes this
can be determined because of the date on which the image was obtained or
because the image was taken using the Nightpod, which is always used at the
main Cupola window. The most popular windows are the Cupola, Window
Observational Research Facility (WORF), and Window 7 of the Zvezda module. The
Cupola and WORF window have very similar transmission characteristics and so,
in principle, should not create differences in colour, although some ESA
contacts have reported lower transmission of the WORF. This effect is
explained mainly because of the Cupola scratch pane, that is not always in
place as it is removable. Window 7 of the Zvezda module is comparable although
it is even more transparent in principle as it is also UV and IR transparent
Pettit [2006]; more information about the transmission can be found in ESA
[2011] and Stefanov et al. [2017, 2019b]. Transmittance can also vary greatly
with the angle to the window at which an image was taken. It is typically
impossible to know what this angle was, although the geometry of the Cupola,
WORF, and Zvezda do not allow acquisitions with very shallow angles.
Apart from the different windows used, another reason why there may be
systematic differences between the extrapolation of the calibration with stars
with different lenses is the use of shutters that are not fully open.
For these reasons, and possibly others, we might want to intercalibrate
different images or calibrate them against another source, such as data from
SNPP/VIIRS-DNB. In a future paper we will focus on these techniques.
## 4 Spain/Madrid: a worked application
Artificial nighttime lighting has been more extensively studied across Spain
than almost any other country [Sánchezde Miguel et al., 2014, Kyba et al.,
2014, Estrada-García et al., 2016, Oriol et al., 2017, Ges et al., 2018, Tong
et al., 2020]. To provide a worked example of the calibration of an astronaut
photograph, we use a wide angle image of Madrid and its environs (Fig. 7)
taken from the ISS (ISS053-E-249189) on 19/11/2017 at 21:51:38 GMT (note: that
the database time is incorrect by 1 h in this exceptional case. This is known
because the nadir location that corresponds with the RAW EXIF information is
in the Pacific Ocean [lat -49.2, -126.2], which is clearly not possible. The 1
h corrected location is just $\sim 200$ km [lat 38.4,lon -6.4]) from the true
nadir [lat 37.49,lon -7.77]) and downloaded on 21/05/2020 from NASA’s Gateway
to Astronaut Photography of Earth (https://eol.jsc.nasa.gov/). Fig. 8 shows
the RAW image. This is extracted and separated into four different colour
images (Fig. 9). These are not linear at the high values, so linearity
correction is applied (Fig. 10). Fig. 11 shows the flat field correction that
needs to be applied given that the original image was taken with a 24mm lens.
The parameters of equation 7 are, according to the image metadata and camera
model reference Nikon D3s, T=1/10, C0=1, C1=1 for the Green channel, 1.12 for
the Red Channel and 0.95 for the Blue Channel, ISO=5000, BN=4, and Tn=1.050.
Photometric calibration was carried out for the colour images using the stars
(Fig. 12), followed by georeferencing (Fig. 13). Once we have the
transformation between geographic location and pixel, we can generate a map of
the view angle of the camera with respect to the ground (Fig. 14), which gives
the amount of atmosphere that the light goes through before hitting the
sensor. After this correction we have a calibrated image that is internally
radiometrically coherent and colour coherent (Fig. 15). This could now be
compared with other ISS images and intercalibrated with them using reference
points or compared with VIIRS images. 102 control points were used with an
RSME of $\sim 4$ pixels (due to the large deformation of the image, the pixels
closer to the nadir have higher precision, and points close to the horizon
less. Also, the ISS sampling is 3 times higher than that of the reference from
VIIRS, so it is natural that the image cannot have higher accuracy than the
pixel size of the reference layer).
Fig 16 and Fig 17 show how this calibrated ISS image compares to a VIIRS
image. The comparison is not simple as the sensors used have different
spectral sensitivity, different flyby times and were obtained at different
angles, and there are not sufficient calibrated ISS images yet fully to
determine any systematic differences between the emissions detected in ISS and
VIIRS data. fig.18 shows that in comparing the calibrated ISS image of Madrid
and VIIRS data the main differences are due to small georeferencing/resolution
issues, and light sources that are visible on one image but not the other
because of the time of acquisition or the tilt. Nonetheless, despite in the
present example there being a 1 month difference in the timing of acquisition
of the data, 4 h difference in the flyby time, spectral sensitivity
difference, and difference in spatial resolution, there is predominantly a
linear relationship between the emissions detected by the two images, and they
compare very well. There is an absolute offset between the two of a factor
1.92. This could be explained by several factors, including dimming of the
streetlights of Madrid after midnight.
## 5 Madrid: ISS-ISS intercalibration
In order to examine the performance of the calibration of ISS imagery, we
selected four images of Madrid (see fig. 19). All were acquired before a major
street lighting change took place in the city in 2014 [Robles et al., 2021].
File | Tilt | Exp T. | f/ | ISO | Lens | Cam | Time
---|---|---|---|---|---|---|---
iss030e292893 | 7.0 | 1/50 | 2.8 | 10000 | 180mm f/2.8 D | D3S | 2012:02:08 21:09:16.87
iss030e292895 | 4.0 | 1/50 | 2.8 | 10000 | 180mm f/2.8 D | D3S | 2012:02:08 21:09:02.37
iss031e095634 | 7.0 | 1/50 | 1.4 | 3200 | 85mm f/1.4 D | D3S | 2012:06:04 22:26:49
iss035e023371 | 29.0 | 1/40 | 3.2 | 51200 | 400mm f/2.8 | D3S | 2013:04:18 22:10:09.73
Table 1: Specifications of the images used for comparison.
The two first images were obtained a few seconds from each other, but with
small changes of view angle and exactly the same camera settings, so they are
probably as similar as images taken from the ISS are likely to be. The third
image, was obtained with a different lens, ISO and aperture settings, and was
acquired 1 h and 30 min later, but with a similar view angle. The fourth image
was again obtained with a different lens, ISO, aperture and view angle, still
in the same daily time frame, but more than one year apart. A 5 pixel Gaussian
blur was applied to each image to minimize effects of errors in
georeferencing.
R2 values for the pixel by pixel comparison of light intensities determined
from the four images varied between 0.88 and 0.98 (Table 2), with the majority
of data points lying close to the 1:1 line (Figure 20). The differences in
intensities between images increase with variation in how those images were
acquired. Given the challenges of nocturnal remote sensing that result from
the nature of the light sources [Tong et al., 2020], we conclude that these
are satisfactory results and comparable to findings for panchromatic data from
satellite sensors [Coesfeld et al., 2018].
Ref Pro | iss030e292893 | iss030e292895 | iss031e095634 | iss035e023371
---|---|---|---|---
iss030e292893 | 1 | 0.98 | 0.95 | 0.88
iss030e292895 | 0.98 | 1 | 0.96 | 0.88
iss031e095634 | 0.95 | 0.96 | 1 | 0.88
iss035e023371 | 0.88 | 0.88 | 0.88 | 1
Table 2: $R^{2}$ values corresponding to the plots fig 20.
## 6 Conclusion
Calibration of nighttime images of the Earth taken by astronauts aboard the
ISS is not, in general, an easy task. These images were never designed to
provide remote sensing data. However, if handled properly, they can provide
multispectral information on the distribution and change in nighttime lighting
that is not available from any other source, and at spatial resolutions that
are comparable or better than those obtained by the DMSP-OLS and SNPP/VIIRS-
DNB platforms. Reflection correction is not needed and we did not apply it
because most studies of light pollution are concerned with the total light
received. Also, in urban areas the vast majority of the surfaces are asphalt
and concrete, and such grey materials do not change significantly the spectral
characteristics of the light.
As we have laid out, the correct calibration of nighttime images of the Earth
from the ISS requires processing through a number of steps (Fig. 1). Whilst
some of these undoubtedly have more significant effects on the final image
than others, we recommend that all are carried out, as their influences on
different images can vary greatly.
Calibration of nighttime images of the Earth taken from the ISS opens up
enormous possibilities for studying the spatial occurrence of artificial
lighting and how this is changing with time (nightly, through the year, and
across years). Determining how the spatial and temporal variation covaries and
determining the influences on other factors (e.g. human health and
environmental impacts) is of great interest. As we have previously shown, by
using colour-colour diagrams, inferences can also be drawn as to how
differences and changes in the lighting technologies being used give rise to
these spatio-temporal dynamics [Sánchez de Miguel et al., 2019].
## 7 Acknowledgements
We thank R. Moore of the Image Science and Analysis Group, NASA Johnson Space
Center for information on ISS cameras and window properties. We thank Lucía
García for her help with improving some figures. We also thank the anonymous
reviewers for their constructive feedback. This work was supported by the
EMISSI@N project (NERC grant NE/P01156X/1), Fonds de Recherche du Québec :
Nature et Technologies (FRQNT), COST (European Cooperation in Science and
Technology) Action ES1204 LoNNe (Loss of the Night Network), the ORISON
project (H2020-INFRASUPP-2015-2), the Cities at Night project, FPU grant from
the Ministerio de Ciencia y Tecnologia and F. Sánchez de Miguel.
Cameras were tested at Laboratorio de Investigaciónn Científica Avanzada
(LICA), a facility of UCM-UPM funded by the Spanish program of International
Campus of Excellence Moncloa (CEI). We acknowledge the support of the Spanish
Network for Light Pollution Studies (MINECO AYA2011-15808-E) and also from
STARS4ALL, a project funded by the European Union H2020-ICT-2015-688135. This
work has been partially funded by the Spanish MICINN,
(AyA2018-RTI-096188-B-I00), and by the Madrid Regional Government through the
TEC2SPACE-CM Project (P2018/NMT-4291).
The ISS images are courtesy of the Earth Science and Remote Sensing Unit, NASA
Johnson Space Center. Thanks to S. Doran for helping us to locate the missing
image that we use as an example. We are very grateful to all members of the
crews of the ISS from all agencies, NASA, ESA, JAXA, CSA-ASC and ROSCOSMOS,
for their images.
## Description of author’s responsibilities
A.S.M., J.Z., J.G., M.A., D.R.P., J.B. and K.J.G. conceived the study, A.S.M.,
M.A. and J. Z. conducted the calibration procedures, A.S.M., F.O. and J.Z.
analyzed the results, D.R.P. acquired very many of the images used, and
designed and developed the basic techniques for the acquisition of images and
documented the ISS specifications, W.L.S. led the Earth Science and Remote
Sensing Unit, NASA Johnson Space Center and provided key technical
information, and A.S.M., K.J.G. and J.Z. wrote the original manuscript, A.S.M,
J.Z., J.G., W.L.S., J.B. and K.J.G. conducted the funding requests. All
authors reviewed the manuscript.
## Declaration of interest
A.S.M., J.Z. and K.J.G. are members of environmental organizations, including
Bird Life, Celfosc and the International Dark-Sky Association. A.S.M.
occasionally provides consultancy advice for the Instituto de Astrofisica de
Andalucia - CSIC and the company SaveStars Consulting S.L.
## References
* Anderson et al. [2010] Anderson, S.J., Tuttle, B.T., Powell, R.L., Sutton, P.C., 2010\. Characterizing relationships between population density and nighttime imagery for denver, colorado: issues of scale and representation. International Journal of Remote Sensing 31, 5733–5746.
* Bará et al. [2019] Bará, S., Rodríguez-Arós, Á., Pérez, M., Tosar, B., Lima, R.C., Sánchez de Miguel, A., Zamorano, J., 2019. Estimating the relative contribution of streetlights, vehicles, and residential lighting to the urban night sky brightness. Lighting Research & Technology 51, 1092–1107.
* Bará [2020] Bará, S., 2020. Niveis de iluminación de referencia para as diferentes zonas da rede natura 2000 de galicia, tecnoloxías de iluminación e tecnoloxías e procedementos de monitorización e seguimento. URL: https://doi.org/10.5281/zenodo.3981095, doi:10.5281/zenodo.3981095.
* Borovička et al. [2014] Borovička, J., Koten, P., Shrbenỳ, L., Štork, R., Hornoch, K., 2014. Spectral, photometric, and dynamic analysis of eight draconid meteors. Earth, Moon, and Planets 113, 15–31.
* Burggraaff et al. [2019] Burggraaff, O., Schmidt, N., Zamorano, J., Pauly, K., Pascual, S., Tapia, C., Spyrakos, E., Snik, F., 2019\. Standardized spectral and radiometric calibration of consumer cameras. Optics express 27, 19075–19101.
* Cardiel et al. [2002] Cardiel, N., Gorgas, J., Gallego, J., Serrano, A., Zamorano, J., Garcia-Vargas, M.L., Gomez-Cambronero, P., Filgueira, J.M., 2002\. Proper handling of random errors and distortions in astronomical data analysis, in: Astronomical Data Analysis II, International Society for Optics and Photonics. pp. 297–304.
* Cardiel et al. [2021] Cardiel, N., Zamorano, J., Bará, S., Sánchez de Miguel, A., Cabello, C., Gallego, J., García, L., González, R., Izquierdo, J., Pascual, S., et al., 2021\. Synthetic rgb photometry of bright stars: definition of the standard photometric system and ucm library of spectrophotometric spectra. Monthly Notices of the Royal Astronomical Society 504, 3730–3748.
* Cheng and Cheng [2011] Cheng, S., Cheng, S., 2011. Meteor spectral observation with dslr, normal lens and prism. WGN, Journal of the International Meteor Organization .
* Chylinski [2012] Chylinski, R., 2012. Time-lapse Photography: A Complete Introduction to Shooting, Processing, and Rendering Time-lapse Movies with a DSLR Camera. LearnTimelapse. com.
* Cities at night collaboration [2015] Cities at night collaboration, 2015. Dark skies catalog: Mix. URL: https://pmisson.carto.com/viz/fe1a723a-fa81-11e4-bb11-0e018d66dc29/public_map.
* Coesfeld et al. [2018] Coesfeld, J., Anderson, S.J., Baugh, K., Elvidge, C.D., Schernthanner, H., Kyba, C., 2018\. Variation of individual location radiance in viirs dnb monthly composite images. Remote Sensing 10, 1964\.
* Coffin [2008] Coffin, D., 2008. Dcraw: Decoding raw digital photos in linux. URL: http://cybercom. net/~ dcoffin/dcraw 81\.
* Crameri [2021] Crameri, F., 2021. Scientific colour maps. URL: https://doi.org/10.5281/zenodo.4491293, doi:10.5281/zenodo.4491293.
* Desk [1992] Desk, A.D., 1992. Tiff. URL: https://ftp.nice.ch/pub/next/developer/resources/libraries/_tiff.3.0b.s/tiff/doc/_TIFF6.pdf.
* Ducati [2002] Ducati, J., 2002. Vizier online data catalog: Catalogue of stellar photometry in johnson’s 11-color system. yCat URL: http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/237.
* Elvidge et al. [1997] Elvidge, C.D., Baugh, K.E., Kihn, E.A., Kroehl, H.W., Davis, E.R., 1997. Mapping city lights with nighttime data from the dmsp operational linescan system. Photogrammetric Engineering and Remote Sensing 63, 727–734.
* Elvidge et al. [2013] Elvidge, C.D., Baugh, K.E., Zhizhin, M., Hsu, F.C., 2013\. Why viirs data are superior to dmsp for mapping nighttime lights. Proceedings of the Asia-Pacific Advanced Network 35, 62.
* ESA [2011] ESA, 2011. Research Announcement for ISS Experiments Relevant To Study Global Climate Change Annex 1: Additional Technical Information on ISS Capabilities And Background Information. Technical Report. ESA. URL: http://wsn.spaceflight.esa.int/docs/201107/Annex1-ESA-ISS-CC-AO-Climate-Change.pdf.
* Estrada-García et al. [2016] Estrada-García, R., García-Gil, Manuel aFnd Acosta, L., Bará, S., Sanchez-de Miguel, A., Zamorano, J., 2016\. Statistical modelling and satellite monitoring of upward light from public lighting. Lighting Research & Technology 48, 810–822.
* Falchi et al. [2011] Falchi, F., Cinzano, P., Elvidge, C.D., Keith, D.M., Haim, A., 2011. Limiting the impact of light pollution on human health, environment and stellar visibility. Journal of environmental management 92, 2714–2722.
* Finckenor and de Groh [2017] Finckenor, M.M., de Groh, K., 2017\. A researcher’s guide to: Space environmental effects. National Aeronautics and Space Administration International Space Station Researcher’s Guide Series, NP-2015-03-015-JSC , 15.
* Fossum et al. [2000] Fossum, E.R., Mendis, S.K., Pain, B., Nixon, R.H., Zhou, Z., 2000. Active pixel sensor having intra-pixel charge transfer with analog-to-digital converter. US Patent 6,021,172.
* Fowler and El Gamal [1995] Fowler, B., El Gamal, A., 1995\. Cmos image sensor with pixel level a/d conversion. US Patent 5,461,425.
* Fulbright and Xiong [2015] Fulbright, J.P., Xiong, X., 2015\. Suomi-npp viirs day/night band calibration with stars, in: Earth Observing Systems XX, International Society for Optics and Photonics. p. 96071S.
* García Gil et al. [2012] García Gil, M., Francia Payàs, P., San Martí Páramo, R., Solano Lamphar, H., 2012. Contaminación lumínica: una visión desde el foco contaminante: el alumbrado artificial. Universitat Politècnica de Catalunya. Iniciativa Digital Politècnica.
* Garcia-Saenz et al. [2018] Garcia-Saenz, A., Sánchez-de Miguel, A., Espinosa, A., Valentin, A., Aragonés, N., Llorca, J., Amiano, P., Martín Sánchez, V., Guevara, M., Capelo, R., Tardón, A., Peiró-Perez, R.o., iménez Moleón, J.J., Roca-Barceló, A., Pérez-Gómez, B., Dierssen-Sotos, T., Fernández-Villa, T., Moreno-Iribas, C., Moreno, V., García-Pérez, J., Castaño Vinyals, G., Pollán, M., Aubé, M., Kogevinas, M., 2018\. Evaluating the association between artificial light-at-night exposure and breast and prostate cancer risk in spain (mcc-spain study). Environmental health perspectives 126, 047011.
* Garcia-Saenz et al. [2019] Garcia-Saenz, A., Sánchez de Miguel, A., Espinosa, A., Valentin, A., Aragonés, N., Llorca, J., Amiano, P., Martín Sánchez, V., Guevara, M., Capelo, R., Tardón, A., Peiró-Perez, A., Jiménez-Moleón, J., Roca-Barceló, A., Pérez-Gómez, B., Dierssen-Sotos, T., Fernández-Villa, T., Moreno-Iribas, C., Moreno, V., García-Pérez, J., Castaño-Vinyals, G., Pollán, M., Aubé, M., Kogevinas, M., 2019\. Artificial light at night (alan), blue light spectrum exposure and colorectal cancer risk in spain (mcc-spain study). Environmental Epidemiology 3, 212\.
* Gaston [2013] Gaston, K.J., 2013. A green light for efficiency. Nature 497, 560–561.
* Gaston [2018] Gaston, K.J., 2018. Lighting up the nighttime. Science 362, 744–746.
* Gaston et al. [2012] Gaston, K.J., Davies, T.W., Bennie, J., Hopkins, J., 2012\. Reducing the ecological consequences of night-time light pollution: options and developments. Journal of Applied Ecology 49, 1256–1266.
* Gaston et al. [2015] Gaston, K.J., Gaston, S., Bennie, J., Hopkins, J., 2015\. Benefits and costs of artificial nighttime lighting of the environment. Environmental Reviews 23, 14–23.
* Geographics [2011] Geographics, B.M., 2011. Global mapper. URL: http://www. bluemarblegeo. com/products/global-mapper. php .
* Ges et al. [2018] Ges, X., Bará, S., García-Gil, M., Zamorano, J., Ribas, S.J., Masana, E., 2018\. Light pollution offshore: Zenithal sky glow measurements in the mediterranean coastal waters. Journal of quantitative spectroscopy and radiative transfer 210, 91–100.
* Hale and Arlettaz [2019] Hale, J.D., Arlettaz, R., 2019\. Artificial lighting and biodiversity in switzerland. URL: http://www.darksky.ch/dss/wp-content/uploads/2019/02/Artificial-lighting-End-Report-V4.pdf.
* Hänel et al. [2018] Hänel, A., Posch, T., Ribas, S.J., Aubé, M., Duriscoe, D., Jechow, A., Kollath, Z., Lolkema, D.E., Moore, C., Schmidt, N., et al., 2018\. Measuring night sky brightness: methods and challenges. Journal of Quantitative Spectroscopy and Radiative Transfer 205, 278–290.
* Hanisch et al. [2001] Hanisch, R.J., Farris, A., Greisen, E.W., Pence, W.D., Schlesinger, B.M., Teuben, P.J., Thompson, R.W., Warnock, A., 2001\. Definition of the flexible image transport system (fits). Astronomy & Astrophysics 376, 359–380.
* Harwit [1973] Harwit, M., 1973. Astrophysical concepts. Wiley, New York.
* Hayes and Latham [1975] Hayes, D.S., Latham, D., 1975\. A rediscussion of the atmospheric extinction and the absolute spectral-energy distribution of vega. The Astrophysical Journal 197, 593–601.
* Hog et al. [2000] Hog, E., Fabricius, C., Makarov, V.V., Urban, S., Corbin, T., Wycoff, G., Bastian, U., Schwekendiek, P., Wicenec, A., 2000. The Tycho-2 catalogue of the 2.5 million brightest stars. Technical Report. NAVAL OBSERVATORY WASHINGTON DC.
* Holben et al. [1998] Holben, B.N., Eck, T.F., Slutsker, I.a., Tanre, D., Buis, J., Setzer, A., Vermote, E., Reagan, J.A., Kaufman, Y., Nakajima, T., et al., 1998\. Aeronet—a federated instrument network and data archive for aerosol characterization. Remote sensing of environment 66, 1–16.
* Hölker et al. [2010] Hölker, F., Moss, T., Griefahn, B., Kloas, W., Voigt, C.C., Henckel, D., Hänel, A., Kappeler, P.M., Völker, S., Schwope, A., et al., 2010\. The dark side of light: a transdisciplinary research agenda for light pollution policy. Ecology and Society 15.
* Huang et al. [2016] Huang, J., Kondragunta, S., Laszlo, I., Liu, H., Remer, L.A., Zhang, H., Superczynski, S., Ciren, P., Holben, B.N., Petrenko, M., 2016\. Validation and expected error estimation of suomi-npp viirs aerosol optical thickness and ångström exponent with aeronet. Journal of Geophysical Research: Atmospheres 121, 7139–7160.
* ISO 12232:2006 (2006) [E] ISO 12232:2006 (E), 2006. Photography — Digital still cameras — Determination of exposure index, ISO speed ratings, standard output sensitivity, and recommended exposure index. Standard. International Organization for Standardization. Geneva, CH.
* Kotarba and Aleksandrowicz [2016] Kotarba, A.Z., Aleksandrowicz, S., 2016\. Impervious surface detection with nighttime photography from the international space station. Remote Sensing of Environment 176, 295–307.
* Koyama and Jiang [2011] Koyama, H., Jiang, H., 2011\. Development of digital slr camera nikon d3s. JSPST 73, 94–98.
* Kuechly et al. [2012] Kuechly, H.U., Kyba, C.C., Ruhtz, T., Lindemann, C., Wolter, C., Fischer, J., Hölker, F., 2012. Aerial survey and spatial analysis of sources of light pollution in berlin, germany. Remote Sensing of Environment 126, 39–50.
* Kyba et al. [2014] Kyba, C., Garz, S., Kuechly, H., Sánchez de Miguel, A., Zamorano, J., Fischer, J., Hölker, F., 2014. High-resolution imagery of earth at night: new sources, opportunities and challenges. Remote Sensing 7, 1–23.
* Kyba et al. [2017] Kyba, C.C., Kuester, T., De Miguel, A.S., Baugh, K., Jechow, A., Hölker, F., Bennie, J., Elvidge, C.D., Gaston, K.J., Guanter, L., 2017\. Artificially lit surface of earth at night increasing in radiance and extent. Science advances 3, e1701528.
* Lang et al. [2010] Lang, D., Hogg, D.W., Mierle, K., Blanton, M., Roweis, S., 2010. Astrometry. net: Blind astrometric calibration of arbitrary astronomical images. The astronomical journal 139, 1782\.
* Levin and Duke [2012] Levin, N., Duke, Y., 2012. High spatial resolution night-time light images for demographic and socio-economic studies. Remote Sensing of Environment 119, 1–10.
* Levin et al. [2020] Levin, N., Kyba, C.C., Zhang, Q., de Miguel, A.S., Román, M.O., Li, X., Portnov, B.A., Molthan, A.L., Jechow, A., Miller, S.D., et al., 2020\. Remote sensing of night lights: A review and an outlook for the future. Remote Sensing of Environment 237, 111443.
* Li et al. [2014] Li, X., Ge, L., Chen, X., 2014\. Quantifying contribution of land use types to nighttime light using an unmixing model. IEEE Geoscience and Remote Sensing Letters 11, 1667–1671.
* Liu et al. [2011] Liu, Q., Sutton, P.C., Elvidge, C.D., 2011. Relationships between nighttime imagery and population density for hong kong. Proc. Asia-Pac. Adv. Netw 31, 79\.
* Lockwood and Hazlett [2013] Lockwood, J.P., Hazlett, R.W., 2013\. Volcanoes: global perspectives. John Wiley & Sons.
* Madiedo et al. [2019] Madiedo, J.M., Ortiz, J.L., Morales, N., Santos-Sanz, P., 2019\. Multiwavelength observations of a bright impact flash during the 2019 january total lunar eclipse. Monthly Notices of the Royal Astronomical Society 486, 3380–3387.
* Mazor et al. [2013] Mazor, T., Levin, N., Possingham, H.P., Levy, Y., Rocchini, D., Richardson, A.J., Kark, S., 2013. Can satellite-based night lights be used for conservation? the case of nesting sea turtles in the mediterranean. Biological Conservation 159, 63–72.
* Meier [2018] Meier, J.M., 2018. Temporal profiles of urban lighting: Proposal for a research design and first results from three sites in berlin. 10.14279/depositonce-8285 .
* Metcalf [2012] Metcalf, J.P., 2012. Detecting and characterizing nighttime lighting using multispectral and hyperspectral imaging. Ph.D. thesis. Monterey, California. Naval Postgraduate School. URL: https://apps.dtic.mil/dtic/tr/fulltext/u2/a578620.pdf.
* Sánchez de Miguel et al. [2014] Sánchez de Miguel, A., Castaño, J.G., Zamorano, J., Pascual, S., Ángeles, M., Cayuela, L., Martinez, G.M., Challupner, P., Kyba, C.C., 2014. Atlas of astronaut photos of earth at night. Astronomy & Geophysics 55, 4–36.
* Mohan et al. [1999] Mohan, V., Uddin, W., Sagar, R., Gupta, S., 1999\. Atmospheric extinction at devasthal, naini tal. Bulletin of the Astronomical Society of India , 601\.
* Nakamura et al. [1995] Nakamura, J.i., Kemeny, S., Fossum, E., 1995. Cmos active pixel image sensor with simple floating gate pixels. IEEE Transactions on Electron Devices 42, 1693–1694.
* Ocaña [2017] Ocaña, F., 2017. Techniques for near-earth interplanetary matter detection and characterisation from optical ground-based observatories. (Doctoral dissertation, Universidad Complutense de Madrid) URL: https://eprints.ucm.es/47590/.
* Ocaña et al. [2019] Ocaña, F., de Miguel, A.S., Project, D., et al., 2019. Balloon-borne video observations of geminids 2016. arXiv preprint arXiv:1911.10064 .
* Oliphant [2006] Oliphant, T.E., 2006. A guide to NumPy. volume 1. Trelgol Publishing USA.
* Oriol et al. [2017] Oriol, N., Colom, L.and Redondo, J., Oriol, J., Rius, J., 2017\. La luz de la ciudad. Grup d’Estudis sobre Energía, Territori i Societat, Departament de Geografía, Universitat Autònoma de Barcelona.
* Pauwels et al. [2019] Pauwels, J., Le Viol, I., Azam, C., Valet, N., Julien, J.F., Bas, Y., Lemarchand, C., de Miguel, A.S., Kerbiriou, C., 2019. Accounting for artificial light impact on bat activity for a biodiversity-friendly urban planning. Landscape and Urban planning 183, 12–25.
* Pedregosa et al. [2011] Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E., 2011\. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research 12, 2825–2830.
* Pettit [2006] Pettit, D., 2006. Use of the international space station for observations beneficial to international polar year 2007-2008.
* Pettit [2009] Pettit, D., 2009. Exploring the frontier: science of opportunity on the international space station. Proceedings of the American Philosophical Society 153, 381–402.
* Pickering [2002] Pickering, K.A., 2002. The southern limits of the ancient star catalog and the commentary of hipparchos. DIO 12, 3–27.
* Poddanỳ et al. [2010] Poddanỳ, S., Brát, L., Pejcha, O., 2010. Exoplanet transit database. reduction and processing of the photometric data of exoplanet transits. New Astronomy 15, 297–301.
* Rayleigh [1899] Rayleigh, L., 1899. Xxxiv. on the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 47, 375–384.
* Rich and Longcore [2013] Rich, C., Longcore, T., 2013\. Ecological consequences of artificial night lighting. Island Press.
* Robitaille et al. [2013] Robitaille, T.P., Tollerud, E.J., Greenfield, P., Droettboom, M., Bray, E., Aldcroft, T., Davis, M., Ginsburg, A., Price-Whelan, A.M., Kerzendorf, W.E., et al., 2013. Astropy: A community python package for astronomy. Astronomy & Astrophysics 558, A33.
* Robles et al. [2021] Robles, J., Zamorano, J., Pascual, S., Sánchez de Miguel, A., Gallego, J., Gaston, K.J., 2021\. Evolution of brightness and color of the night sky in madrid. Remote Sensing 13, 1511\.
* Román et al. [2018] Román, M.O., Wang, Z., Sun, Q., Kalb, V., Miller, S.D., Molthan, A., Schultz, L., Bell, J., Stokes, E.C., Pandey, B., et al., 2018\. Nasa’s black marble nighttime lights product suite. Remote Sensing of Environment 210, 113–143.
* Rybnikova and Portnov [2018] Rybnikova, N., Portnov, B.A., 2018\. Population-level study links short-wavelength nighttime illumination with breast cancer incidence in a major metropolitan area. Chronobiology international 35, 1198–1208.
* Rybnikova and Portnov [2017] Rybnikova, N.A., Portnov, B.A., 2017\. Remote identification of research and educational activities using spectral properties of nighttime light. ISPRS Journal of Photogrammetry and Remote Sensing 128, 212–222.
* Sabbatini [2014] Sabbatini, M., 2014. Nightpod-nodding mechanism for the iss. Technical Report. Technical Report Experiment Record.
* Sakai et al. [2009] Sakai, S., Tashiro, Y., Akahane, N., Kuroda, R., Mizobuchi, K., Sugawa, S., 2009\. A pixel-shared cmos image sensor using lateral overflow gate, in: 2009 Proceedings of ESSCIRC, IEEE. pp. 240–243.
* Sánchez de Miguel [2015] Sánchez de Miguel, A., 2015. Variacion espacial, temporal y espectral de la contaminacion lumınica y sus fuentes: Metodologıa y resultados. Ph.D. thesis. Universidad Complutense de Madrid. doi:10.5281/zenodo.1289932.
* Sánchez de Miguel et al. [2017] Sánchez de Miguel, A., Aubé, M., Zamorano, J., Kocifaj, M., Roby, J., Tapia, C., 2017\. Sky quality meter measurements in a colour-changing world. Monthly Notices of the Royal Astronomical Society 467, 2966–2979.
* Sánchez de Miguel et al. [2014] Sánchez de Miguel, A., Gomez Castano, J., Zamorano, J., Pascual, S., Angeles, M., Cayuela, L., Martin Martinez, G., Challupner, P., Kyba, C., 2014. Atlas of astronaut photos of earth at night. Astronomy & Geophysics 55, 36–36.
* Sánchez de Miguel et al. [2019] Sánchez de Miguel, A., Kyba, C.C., Aubé, M., Zamorano, J., Cardiel, N., Tapia, C., Bennie, J., Gaston, K.J., 2019\. Colour remote sensing of the impact of artificial light at night (i): The potential of the international space station and other dslr-based platforms. Remote sensing of environment 224, 92–103.
* Sánchez de Miguel et al. [2020] Sánchez de Miguel, A., Kyba, C.C., Zamorano, J., Gallego, J., Gaston, K.J., 2020. The nature of the diffuse light near cities detected in nighttime satellite imagery. Scientific Reports 10, 1–16.
* Sánchezde Miguel et al. [2014] Sánchezde Miguel, A., Zamorano, J., Castaño, J.G., Pascual, S., 2014\. Evolution of the energy consumed by street lighting in spain estimated with DMSP-OLS data. Journal of quantitative spectroscopy and radiative transfer 139, 109–117.
* So [2014] So, C.w., 2014. Observational studies of contributions of artificial and natural light factors to the night sky brightness measured through a monitoring network in hong kong. HKU Theses Online (HKUTO) .
* Stalin et al. [2008] Stalin, C., Hegde, M., Sahu, D., Parihar, P., Anupama, G., Bhatt, B., Prabhu, T., 2008. Night sky at the indian astronomical observatory during 2000-2008. Bull. Astr. Soc. India , 111–127.
* Stefanov et al. [2019a] Stefanov, W., Jones, L., Cameron, A., Vanderbloemen, L., Evans, C., 2019a. A researcher’s guide to: International space station earth observations. Johnson Space Center Publication .
* Stefanov et al. [2019b] Stefanov, W., Jones, L., Cameron, A., Vanderbloemen, L., Evans, C., 2019b. A researcher’s guide to: International Space Station Earth Observations (2nd Edition). Technical Report. Johnson Space Center Publication NP-2019-07-003-JSC. URL: http://wsn.spaceflight.esa.int/docs/201107/Annex1-ESA-ISS-CC-AO-Climate-Change.pdf.
* Stefanov et al. [2017] Stefanov, W.L., Evans, C.A., Runco, S.K., Wilkinson, M.J., Higgins, M.D., Willis, K., 2017\. Astronaut photography: Handheld camera imagery from low earth orbit. hsa , 847.
* Straizys [1996] Straizys, V., 1996. The method of synthetic photometry. Baltic Astronomy 5, 459–476.
* Team et al. [2015] Team, Q.D., et al., 2015. Qgis geographic information system. ver. 2.14. 22. open source geospatial foundation project.
* Tong et al. [2020] Tong, K.P., Kyba, C.C., Heygster, G., Kuechly, H.U., Notholt, J., Kollth, Z., 2020\. Angular distribution of upwelling artificial light in europe as observed by suomi–npp satellite. Journal of Quantitative Spectroscopy and Radiative Transfer , 107009.
* Venzke et al. [2009] Venzke, E., Sennert, S.K., Wunderman, R., 2009. Reports from the smithsonian’s global volcanism network, june 2009. Bulletin of Volcanology 71, 1211\.
* Walker [1988] Walker, M.F., 1988. The effect of solar activity on the v and b band sky brightness. Publications of the Astronomical Society of the Pacific 100, 496.
* Waltham [2013] Waltham, N., 2013. Ccd and cmos sensors, in: Observing photons in space. Springer, pp. 423–442.
* Warmerdam [2008] Warmerdam, F., 2008. The geospatial data abstraction library, in: Open source approaches in spatial data handling. Springer, pp. 87–104.
* Wolfe [1998] Wolfe, W.L., 1998. Introduction to radiometry. TT29, SPIE, Bellingham. URL: https://cds.cern.ch/record/1613508.
* Xu et al. [2018] Xu, Y., Knudby, A., Côté-Lussier, C., 2018. Mapping ambient light at night using field observations and high-resolution remote sensing imagery for studies of urban environments. Building and Environment 145, 104–114.
* Zamorano et al. [2011] Zamorano, J., de Miguel, A., Pascual, S., Castaño, J., Ramírez, P., Challupner, P., 2011\. Iss nocturnal images as a scientific tool against light pollution. LICA report URL: https://eprints.ucm.es/12729/1/iss_processing_8.pdf.
Model | Kodak DCS760 | Nikon D1 | Nikon D2Xs | Nikon D3 | Nikon D3S | Nikon D4 | Nikon D5 | Sony Alpha a7S II
---|---|---|---|---|---|---|---|---
Agencies | All | All | All | All | All | All | All | JAXA
Model launch | 22/03/2001 | 15/06/1999 | 01/06/2006 | 23/08/2007 | 4/10/2009 | 06/01/2012 | 05/01/2016 | 11/09/2015
In use at ISS | 2002–2007 | 2004–2006 | 2008–2014 | 2009–2012 | 2010-2015 | 2015–2017 | 2017- | 2018-
Megapixels | 6 | 3 | 13 | 13 | 13 | 17 | 21 | 12
Sensor format | APS-H | APS-C | APS-C | Full Frame | Full Frame | Full Frame | Full Frame | Full Frame
Sensor type | CCD | CCD | CMOS | CMOS | CMOS | CMOS | CMOS | CMOS
ISO Range | 80 - 400 | 200 – 1600 | 100 – 800 | 200 -6400 | 200 – 12800 | 100 – 12800 | 100 – 102400 | 100 – 102400
Image format | RAW | RAW | RAW | RAW | RAW + TIFF | RAW | RAW | RAW
Fitting | Nikon F | Nikon F | Nikon F | Nikon F | Nikon F | Nikon F | Nikon F | Sony E
Amplification | 1.3x | 1.5x | 1.5x | 1x | 1x | 1x | 1x | 1x
Live view | No | No | No | Yes | Yes | Yes | Yes | Yes
Video resolution | | | | | 1280 x 720 | 1920 x 1080 | 3840 x 2160 | 4K
Storage | PCM CIA | CF | CF | CF | CF | CF,XQD | Dual CF,XQD | SD/SDH/C/SDXC
Table 3: Main characteristics of the most popular cameras used for nighttime photography and video at the ISS. Other cameras have been used but rarely or never for this purpose or only for specific experiments (like Nikon D800E and Nikon D850). More info at: https://eol.jsc.nasa.gov/FAQ. Source: https://www.dpreview.com/. Table 4: Transmission coefficients for the most popular lenses used at the ISS or in light pollution research, relative to that of the 50mm f/1.4 lens. The values correspond to the ratio between the signal of surface brightness of lambertian reference source illuminated at 4 lux with a tungsten lamp, at 1/60 s and ISO 200 with a Nikon D5, for different lenses. This number is equivalent to the transmission stop (T-stop), but on a linear scale. Source: This work. Lens | Transmission coefficient |
---|---|---
Nikon 180mm f/2.8 | 0.297 |
Nikon 85mm f/1.4 | 1.070 |
Nikon 85mm f/1.8 | 0.765 |
Nikon 50mm f/1.8 | 0.750 |
Nikon 50mm f/1.4 | 1 |
Nikon 10.5mm f/2.8 | 0.295 |
Nikon 24mm f/1.4 | 1.050 |
Nikon 400mm f/2.8 | 0.215 |
Sigma 8mm f/3.5 | 0.187 |
Nikon 200mm f/4 | 0.136 |
Nikon 28mm f/1.4 | 1.630 |
Nikon 28mm f/2.8 | 0.326 |
Nikon 28-70mm f/2.8 | 0.324 |
Nikon 14-24mm f/2.8 | 0.307 |
Figure 1: Schema of the potential calibration paths. The red path requires the
calibration of the camera using stars. This calibration provides colour
coherence and absolute calibration based on stars. The black path indicates
what can be done if we do not have an absolute calibrated image with stars, we
can alternatively calibrate against the VIIRS-DNB. Some researchers have in
the past used the blue path, using just decoded RAW images or even JPG images.
We cannot recommend that procedure, as the images will not have internal
photometric coherence or colour coherence, they will only have geometric
coherence (the degree of incoherence will depend on the lens and dynamic range
of the image). Figure 2: Response of camera to a uniformly lit screen with
stabilized source across all ISOs and times of exposure (T) in seconds. The
dispersion of the dots is compatible with the errors of the stabilization
(0.5%). Response values expressed in dimensionless units (aka ADU). Data
calculated with a D5, although compatible with previously calculated data for
D3S. Source: this work.
Figure 3: Relationship between emissions in the RVB Johnson bands and the RGB
Nikon bands. The 1:1 line is given in red, and the fit to the data in green.
Figure 4: Comparison of the radiance of stars in the Ducati II/237 catalogue
with the values determined from calibration images taken with the D3S.
Reference (X axis) brightness is in AB magnitudes. Instrumental magnitudes on
Y axis. The measurements are in green, the blue line is the fit to all of the
data, and the red line the fit excluding outliers using RANSAC. Figure 5:
Comparison of the radiance of stars in the Tycho I/259 catalogue with the
values determined from calibration images taken with the D3S. Each colour
represents a different image, which was obtained using different camera
settings (exposure time, ISO). Longer exposure times and higher ISO lead to
detection of more stars, but also present more saturation problems for bright
stars, as exemplified by the image represented by the small red dots (more
details in Sánchez de Miguel [2015]). Figure 6: With an ideal nadir view, that
is typical of most observations from satellites, an imaginary circle seen from
the nadir (top right view), would always be a circle on the ground. But, from
a tilted observation, the circle would be transformed, approximately, into an
ellipse (top left view and bottom right view), where the minor axis would be
in the direction of observation and the major axis will point perpendicular to
it. If we intersect view direction with the ground track of the ISS, we can
find the true nadir point of observation (bottom left view). We have a first
guess of the nadir point of observation thanks to the clock of the camera, but
this can have an error as high as 500 km. The deformation is trivial to
calculate, for example at the centre of the image, once the image has been
georeferenced, using the reverse equations that give us the image coordinates
from the ground equations. This simple technique allows us to find the true
nadir. In a practical case, once the image is georeferenced, we do not use any
target to stabilize the deformation of the image, we can use the
georeferentiation formulas to estimate one at the centre of the image of the
size that we prefer. In our case, we define a circle of 0.1 km of radius at
the centre of the rectified image and we use the deformation formulas to
calculate the deformed circle (ellipse). Figure 7: Original JPG image of Spain
used as an exemplar. Madrid (the spider-like lit area) looks to be saturated,
although this is not actually the case. This is only one of the reasons not to
use JPG images, although geometrically it is equivalent to the RAW image,
colours, relative intensities, and other issues like gamma correction mean
that this is not recommended. Figure 8: Detail of the center of Madrid on the
RAW image of Spain, with the Bayer matrix and the 4 channels merged in one
single image. The images do not appear clear because they have not been
stretched and the different bands are entangled on the Bayer matrix, as
mentioned in the Introduction. It is common for commercial software to use
debayering algorithms that can produce artifacts and change the photometry, so
we do not use that approach but calibrate each image separately. Figure 9:
Zoomed detail of Madrid from the exemplar image. The RAW image is extracted
and separated into four different images. From top left to bottom right, Red
(R1), Green (G2), Green (G3), and Blue (B4) channels. Figure 10: Zoomed detail
of the exemplar image, after linearity correction of the four separated
channels. Figure 11: Flat field correction for a 24 mm lens. The intensity at
the centre corresponds to 100%, the green lines 85%, 45% and 33%, from inside
to outside. Values presented correspond to percentage of the transmission
compared with peak. Figure 12: Zoom in (not full picture) from fig 11 after
the flat field correction. Results of photometric calibration of the exemplar
image. The four extracted channels are now in values of $nW\cdot
sr^{-1}cm^{-2}$\mathrm{\SIUnitSymbolAngstrom}$^{-1}$. This image has not been
stretched, so shows a lot of noise from low intensities and cosmic rays.
Figure 13: Full picture rectified. The four channel images have to be
georeferenced to create correspondence between pixel location and geographical
location. This is the same image (Fig. 8), as before, but because the area of
the pixels closer to the horizon is larger than for the pixels closer to the
nadir, the centre of the image now is France, but this is just a perspective
effect. The four extracted channels are now in values of $nW\cdot
sr^{-1}cm^{-2}$\mathrm{\SIUnitSymbolAngstrom}$^{-1}$. This image has not been
stretched, so shows a lot of noise from low intensities and cosmic rays.
Figure 14: Angular distance to the horizon for the exemplar image. The X and Y
axis correspond to the coordinates of the image (see fig 13). These data,
along with the altitude and the true nadir, are the basis of the atmospheric
correction. Figure 15: Calibrated version of the exemplar image. Note that
artefacts can appear at the edges because the flat field correction enhances
noise in those areas. Also, the farther the pixels are from the nadir the
blurrier they are. This is because pixels that are closer to the horizon
correspond to larger areas than those closer to the nadir, so when the
rectification takes place, there is less information in the first ones and
errors propagate more than in the second ones. That is why, when several
images are available of the same area, it is better to choose those that were
acquired with longer focal lengths and closer to the nadir. Units in $nW\cdot
sr^{-1}cm^{-2}$. This image has been stretched, so does not show noise as
clearly as in the other cases. Figure 16: Zoom to Madrid region. Left is the
ISS Green band radiance calibrated with stars. Middle image is ISS green band
inter-calibrated with VIIRS. Right image is VIIRS comparison image. Units in
$nW\cdot sr^{-1}cm^{-2}$. The comparison is between the VIIRS October 2017
(November 2017 has a defect) image EOG average Elvidge et al. [2013] and ISS.
Figure 17: Left: Comparison between VIIRS and ISS Green band star calibrated.
Right: Comparison between ISS and VIIRS after background and slope offset. Red
line corresponds to 1:1 relationship. Black line, standard linear fit, blue
line RANSAC Pedregosa et al. [2011] linear fit. Slope Offset is calculated
ratio from $100nW\cdot sr^{-1}cm^{-2}$ ISS compared to corresponding VIIRS
equivalent intensity. Background offset corresponds to maximum density points.
Figure 18: Difference between the VIIRS and ISS image. The main differences
are due to time of acquisition of the images. In the ISS image the lights can
be seen of the main commercial centers of the Madrid area, the Sol square and
surroundings (center of image (a)), the commercial center Gran Plaza 2 (in the
west(b)), and the commercial center Plenilunio (in the East (c)) and Plaza
Norte (in the north (d)). Also, there is a small difference in the
georeferencing. The Santiago Bernabeu Stadium(e) does not appear on the ISS
image, probably because of the tilt effect (dark area in the center of the
image). Look-up table ”Roma” from Crameri [2021]. Figure 19: From left to
right and top to bottom, the images iss030e292893, iss030e292895,
iss031e095634 and iss035e023371. Intensity scale in $nW\cdot
sr^{-1}cm^{-2}\mbox{\AA}^{-1}$. The grid represents latitude and longitude.
Figure 20: Density plots of the comparison between the images iss030e292893,
iss030e292895, iss031e095634 and iss035e023371 on the Green band. The red
lines indicate the 1:1 relationship. Units log10($nW\cdot sr^{-1}\cdot
cm^{-2}\cdot\mbox{\AA}^{-1}$).
|
# Ill-posedness of the Novikov equation in the critical Besov space
$B^{1}_{\infty,1}(\mathbb{R})$
Jinlu Li1, Yanghai Yu2,111E-mail<EMAIL_ADDRESS>yuyanghai214@sina.com(Corresponding author<EMAIL_ADDRESS>and Weipeng
Zhu3
1 School of Mathematics and Computer Sciences, Gannan Normal University,
Ganzhou 341000, China
2 School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002,
China
3 School of Mathematics and Big Data, Foshan University, Foshan, Guangdong
528000, China
Abstract: It is shown that both the Camassa-Holm and Novikov equations are
ill-posed in $B_{p,r}^{1+1/p}(\mathbb{R})$ with
$(p,r)\in[1,\infty]\times(1,\infty]$ in [14] and well-posed in
$B_{p,1}^{1+1/p}(\mathbb{R})$ with $p\in[1,\infty)$ in [35]. Recently, the
ill-posedness for the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$
has been proved in [15]. In this paper, we shall solve the only left an
endpoint case $r=1$ for the Novikov equation. More precisely, we prove the
ill-posedness for the Novikov equation in $B^{1}_{\infty,1}(\mathbb{R})$ by
exhibiting the norm inflation phenomena.
Keywords: Novikov equation; Ill-posedness; Critical Besov space.
MSC (2010): 35Q53, 37K10.
## 1 Introduction
Vladimir Novikov [29] investigated the question of integrability for Camassa-
Holm type equations of the form
$(1-\partial_{x}^{2})u_{t}=P\left(u,u_{x},u_{xx},u_{xxx},\cdots\right),$
where $P$ is a polynomial of $u$ and its $x$-derivatives. Using as test for
integrability the existence of an infinite hierarchy of (quasi-) local higher
symmetries, he produced about 20 integrable equations with quadratic
nonlinearities that include the Camassa-Holm $(\mathrm{CH})$ equation
$\displaystyle(1-\partial_{x}^{2})u_{t}=uu_{xxx}+2u_{x}u_{xx}-3uu_{x}$ (1.1)
and the Degasperis-Procesi $(\mathrm{DP})$ equation
$\displaystyle(1-\partial_{x}^{2})u_{t}=uu_{xxx}+3u_{x}u_{xx}-4uu_{x}.$ (1.2)
Moreover, he produced about 10 integrable equations with cubic nonlinearities
that include the following new equation (we may call it the Novikov equation
(NE))
$\displaystyle(1-\partial_{x}^{2})u_{t}=u^{2}u_{xxx}+3uu_{x}u_{xx}-4u^{2}u_{x}.$
(1.3)
The Camassa-Holm equation was originally derived as a bi-Hamiltonian system by
Fokas and Fuchssteiner [12] in the context of the KdV model and gained
prominence after Camassa-Holm [2] independently re-derived it from the Euler
equations of hydrodynamics using asymptotic expansions. (1.1) is completely
integrable [2, 5] with a bi-Hamiltonian structure [4, 12] and infinitely many
conservation laws [2, 12]. Also, it admits exact peaked soliton solutions
(peakons) of the form $ce^{-|x-ct|}$ with $c>0$, which are orbitally stable
[8] and models wave breaking (i.e., the solution remains bounded, while its
slope becomes unbounded in finite time [3, 6, 7]). The Degasperis-Procesi
equation with a bi-Hamiltonian structure is integrable [10] and has traveling
wave solutions [21]. Although DP is similar to CH in several aspects, these
two equations are truly different. One of the novel features of DP different
from CH is that it has not only peakon solutions [10] and periodic peakon
solutions [36], but also shock peakons [22] and the periodic shock waves [11].
For the Novikov equation, Hone-Wang [20] derived the Lax pair which is given
by
$\displaystyle\left(\begin{array}[]{l}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\end{array}\right)_{x}=U(m,\lambda)\left(\begin{array}[]{l}\psi_{1}\\\
\psi_{2}\\\
\psi_{3}\end{array}\right),\quad\left(\begin{array}[]{l}\psi_{1}\\\
\psi_{2}\\\
\psi_{3}\end{array}\right)_{t}=V(m,u,\lambda)\left(\begin{array}[]{l}\psi_{1}\\\
\psi_{2}\\\ \psi_{3}\end{array}\right),$
where $m=u-u_{xx}$ and the matrices $U$ and $V$ are defined by
$\displaystyle U(m,\lambda)=\left(\begin{array}[]{ccc}0&\lambda m&1\\\
0&0&\lambda m\\\ 1&0&0\end{array}\right)\quad\text{and}\quad
V(m,u,\lambda)=\left(\begin{array}[]{lrr}\frac{1}{3\lambda^{2}}-uu_{x}&\frac{u_{x}}{\lambda}-\lambda
mu^{2}&u_{x}^{2}\\\
\frac{u}{\lambda}&-\frac{2}{3\lambda^{2}}&-\frac{u_{x}}{\lambda}-\lambda
mu^{2}\\\
-u^{2}&\frac{u}{\lambda}&\frac{1}{3\lambda^{2}}+uu_{x}\end{array}\right).$
NE possesses peakon traveling wave solutions [18, 19, 13], which on the real
line are given by the formula $u(x,t)=\pm\sqrt{c}e^{-|x-ct|}$ where $c>0$ is
the wave speed. In fact, NE possesses multi-peakon traveling wave solutions on
both the line and the circle. More precisely, on the line the $n$-peakon
$u(x,t)=\sum_{j=1}^{n}p_{j}(t)e^{-\left|x-q_{j}(t)\right|}$
is a solution to $\mathrm{NE}$ if and only if the positions
$\left(q_{1},\ldots,q_{n}\right)$ and the momenta
$\left(p_{1},\ldots,p_{n}\right)$ satisfy the following system of $2n$
differential equations:
$\left\\{\begin{aligned} \frac{\mathrm{d}q_{j}}{\mathrm{d}t}&=u^{2}(q_{j}),\\\
\frac{\mathrm{d}p_{j}}{\mathrm{d}t}&=-u(q_{j})u_{x}(q_{j})p_{j}.\end{aligned}\right.$
Furthermore, it has infinitely many conserved quantities. Like CH, the most
important quantity conserved by a solution $u$ to NE is its $H^{1}$-norm
$\|u\|_{H^{1}}^{2}=\int_{\mathbb{R}}(u^{2}+u_{x}^{2})\mathrm{d}x.$
The well-posedness of the Camassa-Holm type equations has been widely
investigated during the past 20 years. The local well-posedness for the Cauchy
problem of CH [23, 30, 9, 24] and NE [16, 17, 28, 31, 32, 33, 34] in Sobolev
and Besov spaces $B_{p,r}^{s}(\mathbb{R})$ with $s>\max\\{1+1/{p},3/{2}\\}$
and $(p,r)\in[1,\infty]\times[1,\infty)$ has been established. In our recent
papers [26, 27], we established the ill-posedness for CH in
$B^{s}_{p,\infty}(\mathbb{R})$ with $p\in[1,\infty]$ by proving the solution
map starting from $u_{0}$ is discontinuous at $t=0$ in the metric of
$B^{s}_{p,\infty}(\mathbb{R})$. Guo-Liu-Molinet-Yin [14] established the ill-
posedness for the Camassa-Holm type equations in $B_{p,r}^{1+1/p}(\mathbb{R})$
with $(p,r)\in[1,\infty]\times(1,\infty]$ by proving the norm inflation, which
implies that $B_{p,1}^{1+1/p}$ is the critical Besov space for both CH and NE.
Ye-Yin-Guo [35] obtained the local well-posedness for the Camassa-Holm type
equation in critical Besov spaces $B^{1+1/p}_{p,1}(\mathbb{R})$ with
$p\in[1,\infty)$. We should mention that the well-posedness for DP in
$B^{1}_{\infty,1}(\mathbb{R})$ has been established in our recent paper [25].
Very recently, Guo-Ye-Yin [15] obtained the ill-posedness for CH in
$B^{1}_{\infty,1}(\mathbb{R})$ by constructing a special initial data which
leads to the norm inflation. However, their initial data seems to be invalid
when proving the ill-posedness for NE in $B^{1}_{\infty,1}(\mathbb{R})$. To
the best of our knowledge, whether NE is well-posed or not in in
$B^{1}_{\infty,1}(\mathbb{R})$ is still an open problem. We shall present the
negative result in this paper.
Setting $\Lambda^{-2}=(1-\partial^{2}_{x})^{-1}$, then $\Lambda^{-2}f=G*f$
where $G(x)=\frac{1}{2}e^{-|x|}$ is the kernel of the operator $\Lambda^{-2}$.
We can transform the Novikov equation into the following transport type
equation
$\begin{cases}u_{t}+u^{2}u_{x}=\mathbf{P}_{1}(u)+\mathbf{P}_{2}(u),\\\
u(x,t=0)=u_{0}(x),\end{cases}$ (1.4)
where
$\mathbf{P}_{1}(u)=-\frac{1}{2}\Lambda^{-2}u_{x}^{3}\quad\text{and}\quad\mathbf{P}_{2}(u)=-\partial_{x}\Lambda^{-2}\left(\frac{3}{2}uu^{2}_{x}+u^{3}\right).$
(1.5)
We can now state our main result as follows.
###### Theorem 1.1.
For any $n\in\mathbb{Z}^{+}$ large enough, there exist $u_{0}$ and $T>0$ such
that the Novikov equation has a solution $u\in\mathcal{C}([0,T);H^{\infty})$
satisfying
$\displaystyle\|u_{0}\|_{B^{1}_{\infty,1}}\leq\frac{1}{\log\log
n}\quad\text{but}\quad\|u(t_{0})\|_{B^{1}_{\infty,1}}\geq{\log\log
n}\quad\text{with}\quad t_{0}\in\left(0,\frac{1}{\log n}\right].$
Since the norm inflation implies discontinuous of the data-to-solution at the
trivial function $u_{0}\equiv 0$, Theorem 1.1 demonstrates that
###### Corollary 1.1.
The Cauchy problem for the Novikov equation is ill-posed in
$B^{1}_{\infty,1}(\mathbb{R})$ in the sense of Hadamard.
This paper is structured as follows. In Section 2, we list some notations and
known results and recall some Lemmas which will be used in the sequel. In
Section 3 we present the proof of Theorem 1.1 by dividing it into several
parts: (1) Construction of initial data; (2) Key Estimation for Discontinuity;
(3) The Equation Along the Flow; (4) Norm inflation.
## 2 Preliminaries
Notation $C$ stands for some positive constant independent of $n$, which may
vary from line to line. The symbol $A\approx B$ means that $C^{-1}B\leq A\leq
CB$. We shall call a ball $B(x_{0},r)=\\{x\in\mathbb{R}:|x-x_{0}|\leq R\\}$
with $R>0$ and an annulus
$\mathcal{C}(0,r_{1},r_{2})=\\{x\in\mathbb{R}:0<r_{1}\leq|x|\leq r_{2}\\}$
with $0<r_{1}<r_{2}$. Given a Banach space $X$, we denote its norm by
$\|\cdot\|_{X}$. We shall use the simplified notation
$\|f,\cdots,g\|_{X}=\|f\|_{X}+\cdots+\|g\|_{X}$ if there is no ambiguity. We
will also define the Lipschitz space $C^{0,1}$ using the norm
$\|f\|_{C^{0,1}}=\|f\|_{L^{\infty}}+\|\partial_{x}f\|_{L^{\infty}}$. For
$I\subset\mathbb{R}$, we denote by $\mathcal{C}(I;X)$ the set of continuous
functions on $I$ with values in $X$. Sometimes we will denote $L^{p}(0,T;X)$
by $L_{T}^{p}X$. Let us recall that for all $f\in\mathcal{S}^{\prime}$, the
Fourier transform $\widehat{f}$, is defined by
$(\mathcal{F}f)(\xi)=\widehat{f}(\xi)=\int_{\mathbb{R}}e^{-ix\xi}f(x)\mathrm{d}x\quad\text{for
any}\;\xi\in\mathbb{R}.$
The inverse Fourier transform of any $g$ is given by
$(\mathcal{F}^{-1}g)(x)=\check{g}(x)=\frac{1}{2\pi}\int_{\mathbb{R}}g(\xi)e^{ix\cdot\xi}\mathrm{d}\xi.$
Next, we will recall some facts about the Littlewood-Paley decomposition and
the nonhomogeneous Besov spaces (see [1] for more details). Let
$\mathcal{B}:=\\{\xi\in\mathbb{R}:|\xi|\leq 4/3\\}$ and
$\mathcal{C}:=\\{\xi\in\mathbb{R}:3/4\leq|\xi|\leq 8/3\\}.$ Choose a radial,
non-negative, smooth function $\chi:\mathbb{R}\mapsto[0,1]$ such that it is
supported in $\mathcal{B}$ and $\chi\equiv 1$ for $|\xi|\leq 3/4$. Setting
$\varphi(\xi):=\chi(\xi/2)-\chi(\xi)$, then we deduce that $\varphi$ is
supported in $\mathcal{C}$. Moreover,
$\displaystyle\chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1\quad\mbox{ for any
}\xi\in\mathbb{R}.$
We should emphasize that the fact $\varphi(\xi)\equiv 1$ for $4/3\leq|\xi|\leq
3/2$ will be used in the sequel.
For every $u\in\mathcal{S^{\prime}}(\mathbb{R})$, the inhomogeneous dyadic
blocks ${\Delta}_{j}$ are defined as follows
$\displaystyle\Delta_{j}u=$ $\displaystyle 0,$ if $j\leq-2$;
$\displaystyle\Delta_{j}u=$
$\displaystyle\chi(D)u=\mathcal{F}^{-1}(\chi\mathcal{F}u),$ if $j=-1$;
$\displaystyle\Delta_{j}u=$
$\displaystyle\varphi(2^{-j}D)u=\mathcal{F}^{-1}\left(\varphi(2^{-j}\cdot)\mathcal{F}u\right),$
if $j\geq 0$.
Let $s\in\mathbb{R}$ and $(p,r)\in[1,\infty]^{2}$. The nonhomogeneous Besov
space $B^{s}_{p,r}(\mathbb{R})$ is defined by
$\displaystyle
B^{s}_{p,r}(\mathbb{R}):=\left\\{f\in\mathcal{S}^{\prime}(\mathbb{R}):\;\|f\|_{B^{s}_{p,r}(\mathbb{R})}:=\left\|2^{js}\|\Delta_{j}u\|_{L^{p}}\right\|_{\ell^{r}(j\geq-1)}<\infty\right\\}.$
The following Bernstein’s inequalities will be used in the sequel.
###### Lemma 2.1 (Lemma 2.1 in [1]).
Let $\mathcal{B}$ be a Ball and $\mathcal{C}$ be an annulus. There exist
constants $C>0$ such that for all $k\in\mathbb{N}\cup\\{0\\}$, any
$\lambda\in\mathbb{R}^{+}$ and any function $f\in L^{p}$ with $1\leq p\leq
q\leq\infty$, we have
$\displaystyle{\rm{supp}}\widehat{f}\subset\lambda\mathcal{B}\;\Rightarrow\;\|\partial_{x}^{k}f\|_{L^{q}}\leq
C^{k+1}\lambda^{k+(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^{p}},$
$\displaystyle{\rm{supp}}\widehat{f}\subset\lambda\mathcal{C}\;\Rightarrow\;C^{-k-1}\lambda^{k}\|f\|_{L^{p}}\leq\|\partial_{x}^{k}f\|_{L^{p}}\leq
C^{k+1}\lambda^{k}\|f\|_{L^{p}}.$
###### Lemma 2.2 (Lemma 2.100 in [1]).
Let $1\leq r\leq\infty$, $1\leq p\leq p_{1}\leq\infty$ and
$\frac{1}{p_{2}}=\frac{1}{p}-\frac{1}{p_{1}}$. There exists a constant $C$
depending continuously on $p,p_{1}$, such that
$\left\|\left(2^{j}\left\|[\Delta_{j},v\partial_{x}]f\right\|_{L^{p}}\right)_{j}\right\|_{\ell^{r}}\leq
C\left(\|\partial_{x}v\|_{L^{\infty}}\|f\|_{B_{p,r}^{1}}+\|\partial_{x}f\|_{L^{p_{2}}}\|\partial_{x}v\|_{B_{p_{1},r}^{0}}\right).$
## 3 Proof of Theorem 1.1
In this section, we prove Theorem 1.1.
### 3.1 Construction of Initial Data
Define a smooth cut-off function $\chi$ with values in $[0,1]$ which satisfies
$\displaystyle\chi(\xi)=\begin{cases}1,\quad\mathrm{if}\
|\xi|\leq\frac{1}{4},\\\ 0,\quad\mathrm{if}\ |\xi|\geq\frac{1}{2}.\end{cases}$
From now on, we set $\gamma:=\frac{17}{24}$ just for the sake of simplicity.
Letting
$\displaystyle n\in
16\mathbb{N}=\left\\{16,32,48,\cdots\right\\}\quad\text{and}\quad\mathbb{N}(n)=\left\\{k\in
8\mathbb{N}:\frac{n}{4}\leq k\leq\frac{n}{2}\right\\}.$
We introduce the following new notation which will be used often throughout
this paper
$\displaystyle\|f\|_{B^{k}_{\infty,1}\left(\mathbb{N}(n)\right)}=\sum_{j\in\mathbb{N}(n)}2^{k}\|\Delta_{j}f\|_{L^{\infty}},\quad
k\in\\{0,1\\}.$
Now, we can define the initial data $u_{0}$ by
$\displaystyle u_{0}$
$\displaystyle=n^{-\frac{1}{3}}\left(u^{\rm{H}}_{0}+u^{\rm{L}}_{0}\right),$
where
$\displaystyle u^{\rm{H}}_{0}$ $\displaystyle:=2^{-n}\log
n\sum_{\ell\in\mathbb{N}(n)}\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma),$
(3.6) $\displaystyle u^{\rm{L}}_{0}$
$\displaystyle:=\sum_{\ell\in\mathbb{N}(n)}\check{\chi}(x+2^{\ell+1}\gamma).$
(3.7)
Some Observations
1. 1.
Obviously,
$\displaystyle\mathrm{supp}\
\widehat{u^{\rm{L}}_{0}}\subset\left\\{\xi\in\mathbb{R}:\
|\xi|\leq\frac{1}{2}\right\\}.$ (3.8)
2. 2.
It is not difficult to check that
$\displaystyle\mathrm{supp}\
\mathcal{F}\left(\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\right)\subset\left\\{\xi\in\mathbb{R}:\
2^{\ell}\gamma-\frac{1}{2}\leq|\xi|\leq 2^{\ell}\gamma+\frac{1}{2}\right\\},$
(3.9)
which gives in turn
$\displaystyle\mathrm{supp}\
\mathcal{F}\left(\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\right)$
$\displaystyle\qquad\subset\left\\{\xi\in\mathbb{R}:\
2^{n}\gamma-2^{\ell}\gamma-\frac{1}{2}\leq|\xi|\leq
2^{n}\gamma+2^{\ell}\gamma+\frac{1}{2}\right\\}.$ (3.10)
Thus
$\displaystyle\mathrm{supp}\
\widehat{u^{\rm{H}}_{0}}\subset\left\\{\xi\in\mathbb{R}:\
\frac{4}{3}2^{n-1}\leq|\xi|\leq\frac{3}{2}2^{n-1}\right\\}.$ (3.11)
3. 3.
Since $\check{\chi}$ is a Schwartz function, we have
$\displaystyle|\check{\chi}(x)|+|\partial_{x}\check{\chi}(x)|\leq
C(1+|x|)^{-M},\qquad M\gg 1.$ (3.12)
4. 4.
By $\check{\chi}(0)=\frac{1}{2\pi}\int_{\mathbb{R}}\chi(x)\mathrm{d}x$, we
have
$\displaystyle\left\|\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\check{\chi}^{3}(x+2^{\ell+1}\gamma)\right\|_{L^{\infty}}\geq\check{\chi}^{3}(0)\geq\frac{1}{128\pi}.$
(3.13)
###### Lemma 3.1.
There exists a positive constant $C$ independent of $n$ such that
$\displaystyle
2^{n}\|u^{\rm{H}}_{0}\|_{L^{\infty}}+\|\partial_{x}u^{\rm{H}}_{0}\|_{L^{\infty}}\leq
C\log n,$ $\displaystyle\|u^{\rm{L}}_{0}\|_{C^{0,1}}\leq C,$
$\displaystyle\|u_{0}\|_{B^{1}_{\infty,1}}\leq Cn^{-\frac{1}{3}}\log n.$
###### Proof.
Due to (3.11)-(3.12), by Bernstein’s inequality, we have
$\displaystyle 2^{n}\|u^{\rm{H}}_{0}\|_{L^{\infty}}$
$\displaystyle+\|\partial_{x}u^{\rm{H}}_{0}\|_{L^{\infty}}+\log
n\|u^{\rm{L}}_{0}\|_{C^{0,1}}$ $\displaystyle\leq C\log
n\left\|\sum_{\ell\in\mathbb{N}(n)}\frac{1}{(1+|x+2^{\ell+1}\gamma|)^{M}}\right\|_{L^{\infty}}$
$\displaystyle\leq C\log n$
and
$\displaystyle\|u_{0}\|_{B^{1}_{\infty,1}}$ $\displaystyle\leq
Cn^{-\frac{1}{3}}\left(\|u^{\rm{H}}_{0}\|_{B^{1}_{\infty,1}}+\|u^{\rm{L}}_{0}\|_{B^{1}_{\infty,1}}\right)$
$\displaystyle\leq
Cn^{-\frac{1}{3}}\left(2^{n}\|u^{\rm{H}}_{0}\|_{L^{\infty}}+\|u^{\rm{L}}_{0}\|_{L^{\infty}}\right)$
$\displaystyle\leq Cn^{-\frac{1}{3}}\log n.$
This completes the proof of Lemma 3.1. ∎
### 3.2 Key Estimation for Discontinuity
The following Lemma is crucial for the proof of Theorem 1.1.
###### Lemma 3.2.
There exists a positive constant $c$ independent of $n$ such that
$\displaystyle\left\|u_{0}(\partial_{x}u_{0})^{2}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\geq
c(\log n)^{2},\qquad n\gg 1.$
###### Proof.
Obviously,
$\displaystyle n\cdot u_{0}(\partial_{x}u_{0})^{2}$
$\displaystyle=\underbrace{u^{\rm{L}}_{0}(\partial_{x}u^{\rm{H}}_{0})^{2}}_{=:\;\mathbf{I}_{1}}+\underbrace{u^{\rm{H}}_{0}(\partial_{x}u^{\rm{H}}_{0}+\partial_{x}u^{\rm{L}}_{0})^{2}}_{=:\;\mathbf{I}_{2}}+\underbrace{u^{\rm{L}}_{0}\left((\partial_{x}u^{\rm{L}}_{0})^{2}+2\partial_{x}u^{\rm{H}}_{0}\partial_{x}u^{\rm{L}}_{0}\right)}_{=:\;\mathbf{I}_{3}}.$
Next, we need to estimate the above three terms.
Estimation of $\mathbf{I}_{2}$. Using Lemma 3.1 yields
$\displaystyle\left\|\mathbf{I}_{2}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\leq
Cn\|\mathbf{I}_{2}\|_{L^{\infty}}\leq
n\|u^{\rm{H}}_{0}\|_{L^{\infty}}\|\partial_{x}u^{\rm{H}}_{0},\partial_{x}u^{\rm{L}}_{0}\|^{2}_{L^{\infty}}\leq
Cn2^{-n}(\log n)^{2}.$
Estimation of $\mathbf{I}_{3}$. Notice that the support conditions (3.11) and
(3.8), one has
$\displaystyle\Delta_{j}\mathbf{I}_{3}=0\quad\text{for}\;j\in\mathbb{N}(n)\quad\Rightarrow\quad\left\|\mathbf{I}_{3}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}=0.$
Estimation of $\mathbf{I}_{1}$. Now we focus on the estimation of
$\mathbf{I}_{1}$. Obviously,
$\displaystyle\partial_{x}u^{\rm{H}}_{0}$ $\displaystyle=-\gamma\log
n\sum_{\ell\in\mathbb{N}(n)}\sin\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)$
$\displaystyle\quad+2^{-n}\log
n\sum_{\ell\in\mathbb{N}(n)}\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\partial_{x}\Big{(}\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\Big{)}.$
We decompose $\mathbf{I}_{1}$ into three terms
$\displaystyle\mathbf{I}_{1}=(\log
n)^{2}\left(\mathbf{I}_{11}+\mathbf{I}_{12}-\mathbf{I}_{13}\right),$
where
$\displaystyle\mathbf{I}_{11}$
$\displaystyle=\gamma^{2}u^{\rm{L}}_{0}\left(\sum_{\ell\in\mathbb{N}(n)}\sin\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\right)^{2},$
$\displaystyle\mathbf{I}_{12}$
$\displaystyle=2^{-2n}u^{\rm{L}}_{0}\left(\sum_{\ell\in\mathbb{N}(n)}\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\partial_{x}\Big{(}\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\Big{)}\right)^{2},$
$\displaystyle\mathbf{I}_{13}$ $\displaystyle=2\gamma
2^{-n}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\sin\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)$
$\displaystyle\quad\times\sum_{\ell\in\mathbb{N}(n)}\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\partial_{x}\Big{(}\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\Big{)}.$
Easy computations give that
$\displaystyle\|\mathbf{I}_{12}\|_{L^{\infty}}$ $\displaystyle\leq
C2^{-2n}\|u^{\rm{L}}_{0}\|_{L^{\infty}}\left\|\sum_{\ell\in\mathbb{N}(n)}\partial_{x}\Big{(}\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\Big{)}\right\|^{2}_{L^{\infty}}$
$\displaystyle\leq
C2^{-2n}\left\|\sum_{\ell\in\mathbb{N}(n)}\frac{2^{\ell}}{(1+|x+2^{\ell+1}\gamma|)^{M}}\right\|^{2}_{L^{\infty}}$
$\displaystyle\leq C2^{-n}.$
Similarly, we have $\|\mathbf{I}_{13}\|_{L^{\infty}}\leq C2^{-\frac{n}{2}}.$
Thus
$\displaystyle\left\|\mathbf{I}_{12},\mathbf{I}_{13}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\leq
Cn2^{-\frac{n}{2}}.$
By the simple equality
$\sin^{2}(a)\cos^{2}(b)=\frac{1}{4}(1-\cos(2a))(1+\cos(2b))$, we break
$\mathbf{I}_{11}$ down into some easy-to-handle terms
$\displaystyle\mathbf{I}_{11}=\gamma^{2}\sum_{i=1}^{5}\mathbf{I}_{11i},\quad\text{where}$
$\displaystyle\mathbf{I}_{111}$
$\displaystyle=\frac{1}{4}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\check{\chi}^{2}(x+2^{\ell+1}\gamma),$
$\displaystyle\mathbf{I}_{112}$
$\displaystyle=\frac{1}{4}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\check{\chi}^{2}(x+2^{\ell+1}\gamma),$
$\displaystyle\mathbf{I}_{113}$
$\displaystyle=-\frac{1}{4}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\cos\big{(}2^{n+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\cdot\check{\chi}^{2}(x+2^{\ell+1}\gamma),$
$\displaystyle\mathbf{I}_{114}$
$\displaystyle=-\frac{1}{4}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\cos\big{(}2^{n+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\cdot\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\cdot\check{\chi}^{2}(x+2^{\ell+1}\gamma),$
$\displaystyle\mathbf{I}_{115}$
$\displaystyle=u^{\rm{L}}_{0}\sum_{\ell,j\in\mathbb{N}(n)\atop\ell\neq
j}\Big{(}\sin\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)$
$\displaystyle\quad\times\sin\left(2^{n}\gamma(x+2^{j+1}\gamma)\right)\cdot\cos\left(2^{j}\gamma(x+2^{j+1}\gamma)\right)\cdot\check{\chi}(x+2^{j+1}\gamma)\Big{)}.$
Notice that the support conditions (3.9) and (2), one has
$\displaystyle\Delta_{j}\mathbf{I}_{112}=\Delta_{j}\mathbf{I}_{113}=\Delta_{j}\mathbf{I}_{114}=0\quad\text{for}\;j\in\mathbb{N}(n),$
which implies directly that
$\displaystyle\left\|\mathbf{I}_{112},\mathbf{I}_{113},\mathbf{I}_{114}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}=0.$
Using Lemma 3.1, we have
$\displaystyle\|\mathbf{I}_{115}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle\leq Cn\|\mathbf{I}_{115}\|_{L^{\infty}}\leq
Cn\|u^{\rm{L}}_{0}\|_{L^{\infty}}\sum_{\ell,j\in\mathbb{N}(n)\atop\ell\neq
j}\left\|\check{\chi}(x+2^{\ell+1}\gamma)\cdot\check{\chi}(x+2^{j+1}\gamma)\right\|_{L^{\infty}}$
$\displaystyle\leq
Cn\sum_{j>\ell\in\mathbb{N}(n)}\left\|(1+|x+2^{j+1}\gamma|^{2})^{-M}(1+|x+2^{\ell+1}\gamma|^{2})^{-M}\right\|_{L^{\infty}}$
$\displaystyle\leq
Cn\sum_{j>\ell\in\mathbb{N}(n)}\left\|(1+|x|^{2})^{-M}(1+|x-(2^{j+1}-2^{\ell+1})\gamma|^{2}))^{-M}\right\|_{L^{\infty}}$
$\displaystyle\leq
Cn\sum_{j>\ell\in\mathbb{N}(n)}\left(\gamma(2^{j}-2^{\ell})\right)^{-2M}$
$\displaystyle\leq Cn2^{-\frac{nM}{2}},$
where we have separated $\mathbb{R}$ into two different regions
$\\{x:|x|\leq\gamma(2^{j}-2^{\ell})\\}$ and
$\\{x:|x|>\gamma(2^{j}-2^{\ell})\\}$.
Finally, we can break $\mathbf{I}_{111}$ down into two parts, where the first
part contributes the main part.
$\displaystyle\mathbf{I}_{111}$
$\displaystyle=\sum_{\ell\in\mathbb{N}(n)}\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\check{\chi}^{3}(x+2^{\ell+1}\gamma)$
$\displaystyle\quad+\sum_{\ell,j\in\mathbb{N}(n)\atop\ell\neq
j}\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\check{\chi}^{2}(x+2^{\ell+1}\gamma)\check{\chi}(x+2^{j+1}\gamma)$
$\displaystyle:=\mathbf{I}_{1111}+\mathbf{I}_{1112}.$
Due to (3.9) and the support condition of $\varphi$ and for all
$k\in\mathbb{Z}$
$\displaystyle\varphi(2^{-k}\xi)\equiv
1\quad\text{for}\quad\xi\in\mathcal{C}_{k}\equiv\left\\{\xi\in\mathbb{R}^{d}:\
\frac{4}{3}2^{k}\leq|\xi|\leq\frac{3}{2}2^{k}\right\\},$
we have
${\dot{\Delta}_{j}\mathbf{I}_{1111}=\mathcal{F}^{-1}\left(\varphi(2^{-j}\cdot)\mathcal{F}\mathbf{I}_{1111}\right)=}\begin{cases}\cos\big{(}2^{j+1}\gamma(x+2^{j+1}\gamma)\big{)}\check{\chi}^{3}(x+2^{j+1}\gamma),&\text{if}\;\ell=j,\\\
0,&\text{if}\;\ell\neq j,\end{cases}$
which combining (3.13) implies that
$\displaystyle\|\mathbf{I}_{1111}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle=\sum_{j\in\mathbb{N}(n)}\left\|\cos\big{(}2^{j+1}\gamma(x+2^{j+1}\gamma)\big{)}\check{\chi}^{3}(x+2^{j+1}\gamma)\right\|_{L^{\infty}}\geq
cn.$
Following the same procedure as $\mathbf{I}_{115}$, we get
$\displaystyle\|\mathbf{I}_{1112}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}\leq
Cn2^{-\frac{nM}{2}},$
Gathering all the above estimates, we obtain that for large enough $n$
$\displaystyle
n\|u_{0}(\partial_{x}u_{0})^{2}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle\geq\|\mathbf{I}_{1}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}-\|\mathbf{I}_{2}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle\geq C(\log
n)^{2}\left(\|\mathbf{I}_{1111}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}-\|\mathbf{I}_{12},\mathbf{I}_{13},\mathbf{I}_{115},\mathbf{I}_{1112}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}-n2^{-n}\right)$
$\displaystyle\geq cn(\log n)^{2}.$
This completes the proof of Lemma 3.2. ∎
###### Remark 3.1.
Setting $u_{0}=n^{-\frac{1}{2}}u^{\mathrm{H}}_{0}$ where $u^{\mathrm{H}}_{0}$
is given by (3.6) and following the same argument as the proof of Lemma 3.2,
we can establish
$\displaystyle\left\|u_{0}\right\|_{B^{1}_{\infty,1}(\mathbb{R})}\leq
C_{1}n^{-\frac{1}{2}}\log
n\quad\text{and}\quad\left\|(\partial_{x}u_{0})^{2}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\geq
c_{2}(\log n)^{2},\qquad n\gg 1.$
### 3.3 The Equation Along the Flow
Given a Lipschitz velocity field $u$, we may solve the following ODE to find
the flow induced by $u^{2}$:
$\displaystyle\quad\begin{cases}\frac{\mathrm{d}}{\mathrm{d}t}\phi(t,x)=u^{2}(t,\phi(t,x)),\\\
\phi(0,x)=x,\end{cases}$ (3.14)
which is equivalent to the integral form
$\displaystyle\phi(t,x)=x+\int^{t}_{0}u^{2}(\tau,\phi(\tau,x))\mathrm{d}\tau.$
(3.15)
Considering
$\displaystyle\quad\begin{cases}\partial_{t}v+u^{2}\partial_{x}v=P,\\\
v(0,x)=v_{0}(x),\end{cases}$ (3.16)
then, we get from (3.16) that
$\displaystyle\partial_{t}(\Delta_{j}v)+u^{2}\partial_{x}\Delta_{j}v$
$\displaystyle=R_{j}+\Delta_{j}P,$
with
$R_{j}=[u^{2},\Delta_{j}]\partial_{x}v=u^{2}\Delta_{j}\partial_{x}v-\Delta_{j}(u^{2}\partial_{x}v)$.
Due to (3.14), then
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left((\Delta_{j}v)\circ\phi\right)$
$\displaystyle=R_{j}\circ\phi+\Delta_{j}P\circ\phi,$
which means that
$\displaystyle\Delta_{j}v\circ\phi=\Delta_{j}v_{0}+\int^{t}_{0}R_{j}\circ\phi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}P\circ\phi\mathrm{d}\tau.$
(3.17)
### 3.4 Norm Inflation
For $n\gg 1$, we have for $t\in[0,1]$
$\displaystyle\|u\|_{C^{0,1}}\leq C\|u_{0}\|_{C^{0,1}}\leq
Cn^{-\frac{1}{3}}\log n.$
To prove Theorem 1.1, it suffices to show that there exists
$t_{0}\in(0,\frac{1}{\log n}]$ such that
$\displaystyle\|u(t_{0},\cdot)\|_{B^{1}_{\infty,1}}\geq\log\log n.$ (3.18)
We prove (3.18) by contraction. If (3.18) were not true, then
$\displaystyle\sup_{t\in(0,\frac{1}{\log
n}]}\|u(t,\cdot)\|_{B^{1}_{\infty,1}}<\log\log n.$ (3.19)
We divide the proof into two steps.
Step 1: Lower bounds for $(\Delta_{j}u)\circ\phi$
Now we consider the equation along the Lagrangian flow-map associated to
$u^{2}$. Utilizing (3.17) to (1.4) yields
$\displaystyle(\Delta_{j}u)\circ\phi$
$\displaystyle=\Delta_{j}u_{0}+\int^{t}_{0}R^{1}_{j}\circ\phi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}F\circ\phi\mathrm{d}\tau$
$\displaystyle\quad+\int^{t}_{0}\big{(}\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\big{)}\mathrm{d}\tau+t\Delta_{j}E_{0},$
where
$\displaystyle R^{1}_{j}=[u^{2},\Delta_{j}]\partial_{x}u,\qquad
F=-\Lambda^{-2}\left(\frac{1}{2}(\partial_{x}u)^{3}+\partial_{x}(u^{3})\right),$
$\displaystyle
E=-\frac{3}{2}\partial_{x}\Lambda^{-2}\left(u(\partial_{x}u)^{2}\right)\quad\text{with}\quad
E_{0}=-\frac{3}{2}\partial_{x}\Lambda^{-2}\left(u_{0}(\partial_{x}u_{0})^{2}\right).$
Due to Lemma 3.2, we deduce
$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E_{0}\|_{L^{\infty}}$
$\displaystyle\approx\sum_{j\in\mathbb{N}(n)}\|\Delta_{j}\partial_{x}E_{0}\|_{L^{\infty}}$
$\displaystyle\geq
C\sum_{j\in\mathbb{N}(n)}\left\|\Delta_{j}[u_{0}(\partial_{x}u_{0})^{2}]\right\|_{L^{\infty}}$
$\displaystyle\geq c(\log n)^{2}.$ (3.20)
Notice that (3.15), then we have for $t\in(0,\frac{1}{\log n}]$,
$\displaystyle\frac{1}{2}\leq|\partial_{x}\phi|\leq 2,$
thus
$\displaystyle\|f(t,\phi(t,x))\|_{L^{\infty}}=\|f(t,\cdot)\|_{L^{\infty}}.$
Then, using the commutator estimate from Lemma 2.2, we have
$\displaystyle\sum_{j\geq-1}2^{j}\|R^{1}_{j}\circ\phi\|_{L^{\infty}}$
$\displaystyle\leq C\sum_{j\geq-1}2^{j}\|R^{1}_{j}\|_{L^{\infty}}$
$\displaystyle\leq
C\|\partial_{x}(u^{2})\|_{B^{0}_{\infty,1}}\|u\|_{B^{1}_{\infty,1}}$
$\displaystyle\leq C\|u\|_{C^{0,1}}\|u\|^{2}_{B^{1}_{\infty,1}}$
$\displaystyle\leq Cn^{-\frac{1}{3}}(\log n)^{3}.$ (3.21)
Also, we have
$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}F\circ\phi\|_{L^{\infty}}$
$\displaystyle\leq C\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}F\|_{L^{\infty}}$
$\displaystyle\leq C\|(\partial_{x}u)^{3}+\partial_{x}(u^{3})\|_{L^{\infty}}$
$\displaystyle\leq C\|u\|^{3}_{C^{0,1}}\leq C\|u_{0}\|^{3}_{C^{0,1}}\leq
Cn^{-1}(\log n)^{3}.$ (3.22)
Combining (3.4)-(3.4) and using Lemmas 3.1-3.2 yields
$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|(\Delta_{j}u)\circ\phi\|_{L^{\infty}}$
$\displaystyle\geq
t\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E_{0}\|_{L^{\infty}}-\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\|_{L^{\infty}}$
$\displaystyle\quad-Cn^{-\frac{1}{3}}(\log
n)^{3}-C\|u_{0}\|_{B^{1}_{\infty,1}}$ $\displaystyle\geq
ct\log^{2}n-\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\|_{L^{\infty}}-Cn^{-\frac{1}{3}}(\log
n)^{3}.$
Step 2: Upper bounds for $\Delta_{j}E\circ\phi-\Delta_{j}E_{0}$
By easy computations,
$\partial_{x}\Lambda^{-2}[u^{2}u_{x}^{3}+2u\partial_{x}(u^{2}u_{x})u_{x}]=\partial_{x}\Lambda^{-2}[2u^{2}u_{x}^{3}+\partial_{x}(u^{3}u_{x}^{2})]=2\partial_{x}\Lambda^{-2}(u^{2}u_{x}^{3})+\Lambda^{-2}(u^{3}u_{x}^{2})-u^{3}u_{x}^{2},$
then we find that
$\displaystyle\partial_{t}E+u^{2}\partial_{x}E$
$\displaystyle=-\frac{3}{2}\left(\partial_{x}\Lambda^{-2}\partial_{t}(uu_{x}^{2})+u^{2}\partial_{x}^{2}\Lambda^{-2}(uu_{x}^{2})\right)$
$\displaystyle=\mathrm{J}+\frac{3}{2}\left(\partial_{x}\Lambda^{-2}[u^{2}u_{x}^{3}+2u\partial_{x}(u^{2}u_{x})u_{x}]+u^{3}u_{x}^{2}-u^{2}\Lambda^{-2}(uu_{x}^{2})\right)$
$\displaystyle=\mathrm{J}+\mathrm{K},$ (3.23)
where
$\displaystyle\mathrm{J}=-\frac{3}{2}\partial_{x}\Lambda^{-2}\left(\mathbf{P}_{1}(u)u^{2}_{x}+\mathbf{P}_{2}(u)u^{2}_{x}+2\partial_{x}\mathbf{P}_{1}(u)uu_{x}+2\partial_{x}\mathbf{P}_{2}(u)uu_{x}\right),$
$\displaystyle\mathrm{K}=\frac{3}{2}\left(2\partial_{x}\Lambda^{-2}(u^{2}u_{x}^{3})+\Lambda^{-2}(u^{3}u_{x}^{2})-u^{2}\Lambda^{-2}(uu_{x}^{2})\right).$
Utilizing (3.17) to (3.4) yields
$\displaystyle\Delta_{j}E\circ\phi-\Delta_{j}E_{0}=\int^{t}_{0}[u^{2},\Delta_{j}]\partial_{x}E\circ\phi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}(\mathrm{J}+\mathrm{K})\circ\phi\mathrm{d}\tau.$
Using the commutator estimate from Lemma 2.2, one has
$\displaystyle 2^{j}\|[u^{2},\Delta_{j}]\partial_{x}E\|_{L^{\infty}}$
$\displaystyle\leq
C(\|\partial_{x}(u^{2})\|_{L^{\infty}}\|E\|_{B^{1}_{\infty,\infty}}+\|\partial_{x}E\|_{L^{\infty}}\|u\|_{B^{1}_{\infty,\infty}})$
$\displaystyle\leq C\|u\|^{5}_{C^{0,1}}\leq Cn^{-\frac{5}{3}}(\log n)^{5}.$
Due to the facts
$\displaystyle\|\Lambda^{-2}f\|_{L^{\infty}}\leq\|f\|_{L^{\infty}}\quad\text{and}\quad\|\partial_{x}\Lambda^{-2}f\|_{L^{\infty}}\leq\|f\|_{L^{\infty}}\quad\Rightarrow\quad\|\partial^{2}_{x}\Lambda^{-2}f\|_{L^{\infty}}\leq
2\|f\|_{L^{\infty}},$
then we have
$\displaystyle
2^{j}\|\Delta_{j}\mathrm{J}\|_{L^{\infty}}\approx\|\partial_{x}\mathrm{J}\|_{L^{\infty}}\leq
C\|u\|^{5}_{C^{0,1}}\leq Cn^{-\frac{5}{3}}(\log n)^{5}.$
Similarly,
$\displaystyle 2^{j}\|\Delta_{j}\mathrm{K}\|_{L^{\infty}}\leq
C\|u\|^{5}_{C^{0,1}}\leq Cn^{-\frac{5}{3}}(\log n)^{5}.$
Then, we deduce that
$\displaystyle 2^{j}\|\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\|_{L^{\infty}}\leq
C\|u\|^{5}_{C^{0,1}}\leq Cn^{-\frac{5}{3}}(\log n)^{5},$
which leads to
$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\|_{L^{\infty}}\leq
Cn^{-\frac{2}{3}}(\log n)^{5}.$
Combining Step 1 and Step 2, then for $t=\frac{1}{\log n}$, we obtain for
$n\gg 1$
$\displaystyle\|u(t)\|_{B^{1}_{\infty,1}}$
$\displaystyle\geq\|u(t)\|_{B^{1}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle\geq
C\sum_{j\in\mathbb{N}(n)}2^{j}\|(\Delta_{j}u)\circ\phi\|_{L^{\infty}}$
$\displaystyle\geq ct(\log n)^{2}-Cn^{-\frac{2}{3}}(\log
n)^{5}-Cn^{-\frac{1}{3}}(\log n)^{3}$ $\displaystyle\geq\log\log n,$
which contradicts the hypothesis (3.19). Thus, Theorem 1.1 is proved.
$\square$
## 4 Discussion
By the clever Lagrangian coordinate transformation used in [15] and
constructing a new initial data, we prove that the Novikov equation is ill-
posed in critical Besov spaces $B^{1}_{\infty,1}(\mathbb{R})$. Thus our
results (Theorem 1.1 and Corollary 1.1) indicate that the local well-posedness
and ill-posedness for the Novikov equation in all critical Besov spaces
$B^{1+1/p}_{p,r}(\mathbb{R})$ have been solved completely. Since the Novikov
equation has cubic nonlinear term, we expect that norm inflation is stemmed
from the worst term $u(\partial_{x}u)^{2}$, which is different from the
quadratic term $(\partial_{x}u)^{2}$ for the Camassa-Holm equation. Our new
idea is to construct a initial data which includes two parts, one of whose
Fourier transform is supported at high frequencies and the other is supported
at low frequencies. Then the cubic nonlinear term $u(\partial_{x}u)^{2}$ will
generate the low-high-high frequency interaction, which contributes a large
quantity lead to the norm inflation. Lastly, we should mention that, by
dropping the low frequency term, the initial data
$u_{0}=2^{-\frac{n}{2}}u_{0}^{\mathrm{H}}$ can be as an example which leads to
the norm inflation for the Camassa-Holm equation (see Remark 3.1) .
## Acknowledgments
J. Li is supported by the National Natural Science Foundation of China
(11801090 and 12161004) and Jiangxi Provincial Natural Science Foundation
(20212BAB211004). Y. Yu is supported by the National Natural Science
Foundation of China (12101011) and Natural Science Foundation of Anhui
Province (1908085QA05). W. Zhu is supported by the National Natural Science
Foundation of China (12201118) and Guangdong Basic and Applied Basic Research
Foundation (2021A1515111018).
## Data Availability
No data was used for the research described in the article.
## Conflict of interest
The authors declare that they have no conflict of interest.
## References
* [1] H. Bahouri, J. Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Springer, Heidelberg, 2011.
* [2] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
* [3] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier, 50 (2000), 321-362.
* [4] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85.
* [5] A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970.
* [6] A. Constantin, J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
* [7] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
* [8] A. Constantin, W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
* [9] R. Danchin, A few remarks on the Camassa-Holm equation, Differ. Integral Equ., 14 (2001), 953-988.
* [10] A. Degasperis, D. Holm, A. Hone, A new integral equation with peakon solutions. Theoret. Math. Phys. 133 (2002), 1463-1474.
* [11] J. Escher, Y. Liu, Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J. 56 (2007), 87-177.
* [12] A. Fokas, B. Fuchssteiner, Symplectic structures, their bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981/1982) 47-66.
* [13] K. Grayshan, A. Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232.
* [14] Z. Guo, X. Liu, L. Molinet, Z. Yin, Ill-posedness of the Camassa-Holm and related equations in the critical space, J. Differ. Equ., 266 (2019), 1698-1707.
* [15] Y. Guo, W. Ye, Z. Yin, Ill-posedness for the Cauchy problem of the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$, J. Differ. Equ., 327 (2022), 127-144.
* [16] A. A. Himonas, C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449–479.
* [17] A. Himonas, C. Holliman, The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differ. Equ., 19 (2014) 161-200.
* [18] A. Himonas, D. Matzavinos, An ab-family ofequations with peakon traveling waves, Proc. Amer. Math. Soc., 144 (2016), 3797–3811.
* [19] A. Hone, H. Lundmark, J. Szmigielski, Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289.
* [20] A. Hone, J. Wang, Integrable peakon equations with cubic nonlinearity, J Phys A., 41 (2008) 372002: 1-11.
* [21] J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl. 306 (2005), 72-82.
* [22] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198.
* [23] Y. Li, P.J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 162:1 (2000) 27-63.
* [24] J. Li, Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differ. Equ., 261 (2016), 6125-6143.
* [25] J. Li, Y. Yu, W. Zhu, Well-posedness and continuity properties of the Degasperis-Procesi equation in critical Besov space. Monatsh. Math., (2022) doi.org/10.1007/s00605-022-01691-4
* [26] J. Li, Y. Yu, W. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov spaces, J. Differ. Equ., 306 (2022), 403-417.
* [27] J. Li, Y. Yu, W. Zhu, Sharp ill-posedness for the generalized Camassa-Holm equation in Besov spaces, J. Evol. Equ., 22:29 (2022) 11pp.
* [28] L. Ni, Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differ. Equ., 250 (2011), 3002-3021.
* [29] V. Novikov, Generalization of the Camassa-Holm equation, J. Phys. A., 42 (2009), 342002.
* [30] G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal. 46 (3) (2001), 309-327.
* [31] X. Wu, Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Classe Sci. Ser. V 11 (2012), 707-727.
* [32] X. Wu, Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137.
* [33] W. Yan, Y. Li, Y. Zhang, The Cauchy problem for the integrable Novikov equation, J. Differ. Equ., 253 (2012), 298-318.
* [34] W. Yan, Y. Li, Y. Zhang, The Cauchy problem for the Novikov equation, NoDEA-Nonlinear Differ. Equ. Appl., 20 (2013), 1157–1169.
* [35] W. Ye, Z. Yin, Y. Guo, A new result for the local well-posedness of the Camassa-Holm type equations in critial Besov spaces $B^{1+1/p}_{p,1}$, $1\leq p<\infty$, arXiv, 2021.
* [36] Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal. 212 (2004), 182-194.
|
# Bi-Lipschitz invariance of
planar $BV$\- and $W^{1,1}$-extension domains
Miguel García-Bravo , Tapio Rajala and Zheng Zhu University of Jyvaskyla
Department of Mathematics and Statistics
P.O. Box 35 (MaD)
FI-40014 University of Jyvaskyla
Finland<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.
We prove that a bi-Lipschitz image of a planar $BV$-extension domain is also a
$BV$-extension domain, and that a bi-Lipschitz image of a planar
$W^{1,1}$-extension domain is again a $W^{1,1}$-extension domain.
###### Key words and phrases:
Sobolev extension, BV-extension
###### 2000 Mathematics Subject Classification:
Primary 46E35.
The first two authors acknowledge the support from the Academy of Finland,
grant no. 314789. The third author was supported by the Academy of Finland via
the Centre of Excellence in Analysis and Dynamics Research, grant no. 323960.
## 1\. Introduction
Let ${\Omega}\subset{\mathbb{R}}^{n}$ be a domain. For $1\leq p\leq\infty$, we
define the Sobolev space $W^{1,p}({\Omega})$ by setting
$W^{1,p}:=\\{u\in L^{p}({\Omega})\,:\,\nabla u\in
L^{p}({\Omega};{{{\mathbb{R}}}^{n}})\\},$
where $\nabla u$ means the weak (distributional) derivative of $u$. The
Sobolev space $W^{1,p}({\Omega})$ is equipped with the norm
$\|u\|_{W^{1,p}({\Omega})}:=\|u\|_{L^{p}({\Omega})}+\||\nabla
u|\|_{L^{p}({\Omega})}.$
We say that ${\Omega}$ is a $W^{1,p}$-extension domain if there exists a
bounded extension operator $T\colon W^{1,p}({\Omega})\to
W^{1,p}({{{\mathbb{R}}}^{n}})$, meaning that for every $u\in
W^{1,p}({\Omega})$ we have $T(u)\big{|}_{\Omega}\equiv u$ and
$\|T(u)\|_{W^{1,p}({{{\mathbb{R}}}^{n}})}\leq C\|u\|_{W^{1,p}({\Omega})},$
where the constant $C$ is independent of $u$. The minimal possible constant
$C$ above is denoted by $\|T\|$. Based on results in [7, 8], for
$1<p\leq\infty$, whenever ${\Omega}$ is a $W^{1,p}$-extension domain, we can
construct a bounded linear extension operator $T\colon W^{1,p}({\Omega})\to
W^{1,p}({{{\mathbb{R}}}^{n}})$. By [12], for planar bounded simply connected
$W^{1,1}$-extension domains ${\Omega}$, we can also construct a bounded linear
extension operator $T\colon W^{1,1}({\Omega})\to W^{1,1}({\mathbb{R}}^{2})$.
The classical results due to Calderón and Stein [3, 15] tell us that Lipschitz
domains are $W^{1,p}$-extension domains, for every $1\leq p\leq\infty$. Later,
Jones [9] defined a class of so-called $(\epsilon,\delta)$-domains which are a
generalization of Lipschitz domains. He also proved that these domains are
$W^{1,p}$-extension domains for every $1\leq p\leq\infty$. Moreover, in the
works [11, 12, 14], a geometric characterization of planar bounded simply
connected $W^{1,p}$-extension domains was established.
For arbitrary $u\in W^{1,p}({{{\mathbb{R}}}^{n}})$ with $1\leq p\leq\infty$,
the inequality
$|u(x)-u(y)|\leq|x-y|\left(CM[|\nabla u|](x)+CM[|\nabla u|](y)\right)$
holds on every Lebesgue point of $u$, where the constant $C$ is independent of
$u$ and $M[|\nabla u|]$ denotes the Hardy-Littlewood maximal function of
$|\nabla u|$. Motivated by this estimate, Hajłasz defined the so-called
Hajłasz-Sobolev space $M^{1,p}({\Omega})$ which consists of all functions
$u\in L^{p}({\Omega})$ such that there exists a function $0\leq g\in
L^{p}({\Omega})$ satisfying the inequality
$|u(x)-u(y)|\leq|x-y|(g(x)+g(y))$ (1.1)
for every $x,y\in{\Omega}\setminus F$ where the exceptional set $F$ satisfies
$|F|=0$. We use $\mathcal{D}_{p}(u)$ to denote the class of all nonnegative
functions $g\in L^{p}({\Omega})$ which satisfy the inequality (1.1). The
Hajłasz-Sobolev space $M^{1,p}({\Omega})$ is then equipped with the norm
$\|u\|_{M^{1,p}({\Omega})}:=\|u\|_{L^{p}({\Omega})}+\inf_{g\in\mathcal{D}_{p}(u)}\|g\|_{L^{p}({\Omega})}.$
For $1\leq p\leq\infty$, one always has $M^{1,p}({\Omega})\subset
W^{1,p}({\Omega})$ and the inclusion is strict for $p=1$, see [13]. By [6],
for $1<p\leq\infty$, the equality $M^{1,p}({\Omega})=W^{1,p}({\Omega})$ holds
for a bounded $W^{1,p}$-extension domain ${\Omega}$. Similarly to
$W^{1,p}$-extension domains, a domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is
said to be an $M^{1,p}$-extension domain, if there exists a bounded extension
operator $T\colon M^{1,p}({\Omega})\to M^{1,p}({{{\mathbb{R}}}^{n}})$ (the
existence of such an operator implies the existence of a linear one for the
cases $1\leq p<\infty$ by [8]). Observe that since $M^{1,\infty}({\Omega})$
consists of Lipschitz functions, by Kirszbraun theorem, every domain is a
$M^{1,\infty}$-extension domain. Furthermore, in [8], it is proved that for
$1\leq p<\infty$ we have that ${\Omega}$ is an $M^{1,p}$-extension domain if
and only if ${\Omega}$ is Ahlfors $n$-regular. We say that a domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is Ahlfors $n$-regular, if for every
$x\in{\Omega}$ and $0<r<1$, we have
$|B(x,r)\cap{\Omega}|\geq c|B(x,r)|$
with a constant $0<c<1$ independent of $x$ and $r$. Combining these two
results, one can prove that for $1<p<\infty$, a domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is a $W^{1,p}$-extension domain if and
only if $W^{1,p}({\Omega})=M^{1,p}({\Omega})$ and ${\Omega}$ is Ahlfors
$n$-regular (see [7, Theorem 5]).
In the case that ${\Omega}$ and ${\Omega}^{\prime}$ are bi-Lipschitz
equivalent, one can easily check that
$M^{1,p}({\Omega})=M^{1,p}({\Omega}^{\prime})$ isomorphically, and also the
fact that ${\Omega}$ is Ahlfors $n$-regular if and only if ${\Omega}^{\prime}$
is Ahlfors $n$-regular. Moreover, a domain is a $W^{1,\infty}$-extension
domain if and only if it is locally quasiconvex and local quasiconvexity is
bi-Lipschitz invariant. We say that a set $E\subset{\mathbb{R}}^{n}$ is
quasiconvex, if there exists a constant $c\geq 1$ such that for every $x,y\in
E$, there exists a rectifiable curve $\gamma\subset E$ connecting $x$ and $y$
with the length controlled from above by $c|x-y|$. In such case $E$ is called
$c$-quasiconvex. These last observations lead to the following theorem,
presented by Hajłasz, Koskela and Tuominen in [7].
###### Theorem 1.1.
If ${\Omega}$ and ${\Omega}^{\prime}$ are bi-Lipschitz equivalent, for
$1<p\leq\infty$, ${\Omega}$ is a $W^{1,p}$-extension domain if and only if
${\Omega}^{\prime}$ is a $W^{1,p}$-extension domain.
Since $M^{1,1}({\Omega})$ is strictly included in $W^{1,1}({\Omega})$ for
arbitrary domains ${\Omega}$, the above method does not work for the case
$p=1$. In the same paper, Hajłasz, Koskela and Tuominen raised the question.
###### Question 1.2.
Is Theorem 1.1 true for $p=1$?
A partial affirmative answer to this question was provided in [10, Corollary
1.3]. There, it was shown that Theorem 1.1 holds for bounded simply connected
planar domains also in the case $p=1$, and also for $BV$-functions. In [10] it
was also conjectured that the hypothesis of simply connectivity was
superfluous. In this paper we will show that they were right. We will extend
these previous results and answer Question 1.2 positively for general bounded
planar domains. We will do this by first resolving the question for
$BV$-functions via decomposition of sets of finite perimeter into Jordan
domains, and then employing a recent result from [4] to pass to
$W^{1,1}$-functions. Both, the proof in [10] and our proof, rely on the
results of Väisälä [16] and on the quasiconvexity of the connected open
components of the complement of planar $BV$-extension domains. The difference
is that in [10] the bi-Lipschitz function was extended to the complement by
using the fact that the bi-Lipschitz map can be extended to a small
neighbourhood. Here we use a decomposition of sets of finite perimeter and the
bi-Lipschitz invariance of the quasiconvexity of the holes, see Section 2 for
the definitions and results needed for this approach.
Recall that the space of functions of bounded variation $BV({\Omega})$ is
defined by setting
$BV({\Omega}):=\\{u\in L^{1}({\Omega})\ :\|Du\|({\Omega})<\infty\\}$
where
$\|Du\|({\Omega})=\sup\left\\{\int_{\Omega}u{\rm div}(v)dx\,:\,v\in
C_{0}^{\infty}({\Omega};{{{\mathbb{R}}}^{n}}),|v|\leq 1\right\\}$
means the total variation of $u$ on ${\Omega}$. The function space
$BV({\Omega})$ is equipped with the norm
$\|u\|_{BV({\Omega})}:=\|u\|_{L^{1}({\Omega})}+\|Du\|({\Omega}).$
Note that $\|Du\|$ is a Radon measure on ${\Omega}$ that is defined for every
set $F\subset{\Omega}$ as
$\|Du\|(F):=\inf\\{\|Du\|(U)\,:\,F\subset U,U\ {\rm is\ open}\\}.$
A domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is said to be a $BV$-extension
domain, if there exists a bounded extension operator $T\colon BV({\Omega})\to
BV({{{\mathbb{R}}}^{n}})$ with $T(u)\big{|}_{\Omega}\equiv u$ and an absolute
constant $C>0$ so that
$\|T(u)\|_{BV({{{\mathbb{R}}}^{n}})}\leq C\|u\|_{BV({\Omega})}$
for every $u\in BV({\Omega})$. By a result in [10], a $W^{1,1}$-extension
domain is also a $BV$-extension domain. A typical example showing that the
converse is not true is the slit disk in the plane.
In Section 2 we will prove the following result.
###### Theorem 1.3.
Let $\Omega\subset\mathbb{R}^{2}$ be a bounded $BV$-extension domain and
$f\colon\Omega\to\Omega^{\prime}$ a bi-Lipschitz map. Then $\Omega^{\prime}$
is also a $BV$-extension domain.
Let us show how Theorem 1.3 implies the same result for $W^{1,1}$-extension
domains. We use the recent characterization of $W^{1,1}$-extension domains
among bounded $BV$-extension domains that was proven by the first and second
named authors in [4].
###### Theorem 1.4.
Let $\Omega\subset\mathbb{R}^{2}$ be a bounded $BV$-extension domain. Then
$\Omega$ is a $W^{1,1}$-extension domain if and only if the set
$\partial\Omega\setminus\bigcup_{i\in I}\overline{\Omega_{i}}$
is purely $1$-unrectifiable, where $\\{\Omega_{i}\\}_{i\in I}$ are the
connected components of $\mathbb{R}^{2}\setminus\overline{\Omega}$.
Recall that a set $H\subset\mathbb{R}^{2}$ is called purely $1$-unrectifiable
if for every Lipschitz map $f\colon\mathbb{R}\to\mathbb{R}^{2}$ we have
$\mathcal{H}^{1}(H\cap f(\mathbb{R}))=0$.
Suppose that $\Omega\subset\mathbb{R}^{2}$ is a bounded $W^{1,1}$-extension
domain. By Theorem 1.4 the set
$H=\partial\Omega\setminus\bigcup_{i\in I}\overline{\Omega_{i}}$
is purely $1$-unrectifiable and so is the image $H^{\prime}=f(H)$ under a bi-
Lipschitz map $f\colon\Omega\to\Omega^{\prime}$ that is extended to the
closures $\overline{\Omega}$ and $\overline{\Omega}^{\prime}$ as a bi-
Lipschitz map. Hence, recalling that as a $W^{1,1}$-extension domain $\Omega$
is also a $BV$-extension domain, Theorem 1.3 implies that
$\Omega^{\prime}=f(\Omega)$ is also a $BV$-extension domain. Now, from Theorem
1.4 we conclude that $\Omega^{\prime}$ is a $W^{1,1}$-extension domain. We
have then established the following.
###### Corollary 1.5.
Let $\Omega\subset\mathbb{R}^{2}$ be a bounded $W^{1,1}$-extension domain and
$f\colon\Omega\to\Omega^{\prime}$ a bi-Lipschitz map. Then $\Omega^{\prime}$
is a $W^{1,1}$-extension domain.
Let us remark that one could also prove Corollary 1.5 with a similar proof as
we provide for Theorem 1.3 in Section 2; via the invariance of quasiconvexity
of the components of the complement, a characterization of $W^{1,1}$-extension
domains as the domains with the strong extension property for sets of finite
perimeter and by slightly pushing the boundary of the extension of a Jordan
domain away from the boundary of $\partial\Omega$, see [4] for more details on
these tools. This alternative approach indicates that if we were able to prove
the bi-Lipschitz invariance of $BV$-extension domains in higher dimensions,
and were able to push the boundaries of sets of finite perimeter away from
$\partial\Omega$ in a controlled manner, the bi-Lipschitz invariance of
$W^{1,1}$-extension domains would follow. However, at the moment we are not
able to complete such proof. An alternative approach for trying to solve the
higher dimensional case could be to use the characterization of
$W^{1,1}$-functions from [5], similar to $M^{1,1}$.
## 2\. Proof of Theorem 1.3
We will prove Theorem 1.3 by using the bi-Lipschitz invariance of the
quasiconvexity of the connected components of
$\mathbb{R}^{2}\setminus\overline{\Omega}$, which are referred to as the
_holes_ of $\overline{\Omega}$. We recall the following result of Väisälä [16,
Corollary 4.11].
###### Theorem 2.1.
Let $G\subset\mathbb{R}^{2}$ be a bounded continuum such that each hole of $G$
is $c$-quasiconvex and let $f\colon G\to\mathbb{R}^{2}$ be $L$-bilipschitz.
Then each hole of $G^{\prime}=f(G)$ is $c^{\prime}$-quasiconvex with
$c^{\prime}(c,L)$.
In order to use the Theorem 2.1 we need to observe that the holes of a bounded
planar $BV$-extension domain are quasiconvex. This was established for simply
connected domains in [10, Theorem 1.1], and the proof works with minor
modifications in the more general case considered here, see [4, Lemma 5.2].
###### Lemma 2.2.
Suppose that $\Omega\subset\mathbb{R}^{2}$ is a bounded $BV$-extension domain.
Then there exists a constant $c>0$ so that each hole of $\overline{\Omega}$ is
$c$-quasiconvex.
The invariance of quasiconvexity of the holes is easier to use for the
boundaries of sets of finite perimeter, rather than for $BV$-functions. The
passage from $BV$-functions to sets of finite perimeter is provided by Lemma
2.3, which is a combination of the works [2, 10]. Before stating it we need to
recall some definitions.
A Lebesgue measurable subset $E\subset\mathbb{R}^{n}$ has finite perimeter in
$\Omega$ if $\chi_{E}\in BV(\Omega)$, where $\chi_{E}$ denotes the
characteristic function. We set $P(E,\Omega)=\|D\chi_{F}\|(\Omega)$ and call
it the perimeter of $E$ in $\Omega$. We will say that $\Omega$ has the
extension property for sets of finite perimeter if there exists $C>0$ so that
for every set $E\subset\Omega$ of finite perimeter in $\Omega$ one may find
$\widetilde{E}\subset\mathbb{R}^{n}$ of finite perimeter in $\mathbb{R}^{n}$
such that $\widetilde{E}\cap\Omega=E$, modulo a measure zero set, and
$P(\widetilde{E},\mathbb{R}^{n})\leq CP(E,\Omega)$.
###### Lemma 2.3.
Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain. Then the following are
equivalent:
1. (1)
$\Omega$ is a $BV$-extension domain.
2. (2)
$\Omega$ has the extension property for sets of finite perimeter.
One more tool that we use is the decomposition of planar sets of finite
perimeter into Jordan domains.
We say that $\Gamma\subset\mathbb{R}^{2}$ is a Jordan curve if
$\Gamma=\gamma([a,b])$ for some $a,b\in\mathbb{R}$, $a<b$, and some continuous
map $\gamma$, injective on $[a,b)$ and such that $\gamma(a)=\gamma(b)$. The
Jordan curve theorem assures that $\Gamma$ splits
$\mathbb{R}^{2}\setminus\Gamma$ into exactly two connected components, a
bounded one and an unbounded one that we denote by $\text{int}(\Gamma)$ and
$\text{ext}(\Gamma)$ respectively. A set $U$ whose boundary $\partial U$ is a
Jordan curve is called a Jordan domain.
For technical reasons we also add to the class of Jordan curves the formal
“Jordan” curves $J_{0}$ and $J_{\infty}$, whose interiors are $\mathbb{R}^{2}$
and the empty set respectively and for which we set
$\mathcal{H}^{1}(J_{0})=\mathcal{H}^{1}(J_{\infty})=0$.
For a measurable set $E\subset{{{\mathbb{R}}}^{n}}$, we denote by
$\partial^{M}E$ its essential boundary, which consists of points such that
both $E$ and ${{{\mathbb{R}}}^{n}}\setminus E$ have positive upper density on
them, that is
$\partial^{M}E=\left\\{x\in\mathbb{R}^{n}\,:\,\limsup_{r\searrow
0}\frac{|E\cap B(x,r)|}{|B(x,r)|}>0\;\text{and}\;\limsup_{r\searrow
0}\frac{|(\mathbb{R}^{n}\setminus E)\cap B(x,r)|}{|B(x,r)|}>0\right\\}.$
A set $E\subset\mathbb{R}^{n}$ of finite perimeter is called decomposable, if
there exist sets $A,B\subset\mathbb{R}^{n}$ of positive Lebesgue measure such
that $E=A\cup B$, $A\cap B=\emptyset$, and
$P(E,\mathbb{R}^{n})=P(A,\mathbb{R}^{n})+P(B,\mathbb{R}^{n})$. A set is called
indecomposable, if it is not decomposable.
The following was proven in [1, Corollary 1].
###### Theorem 2.4.
Let $E\subset\mathbb{R}^{2}$ have finite perimeter. Then, there exists a
unique decomposition of $\partial^{M}E$ into rectifiable Jordan curves
$\\{C_{i}^{+},C_{k}^{-}\,:\,i,k\in\mathbb{N}\\}$, modulo
$\mathcal{H}^{1}$-measure zero sets, such that
1. (1)
Given $\text{int}(C_{i}^{+})$, $\text{int}(C_{k}^{+})$, $i\neq k$, they are
either disjoint or one is contained in the other; given
$\text{int}(C_{i}^{-})$, $\text{int}(C_{k}^{-})$, $i\neq k$, they are either
disjoint or one is contained in the other. Each $\text{int}(C_{i}^{-})$ is
contained in one of the $\text{int}(C_{k}^{+})$.
2. (2)
$P(E,\mathbb{R}^{2})=\sum_{i}\mathcal{H}^{1}(C_{i}^{+})+\sum_{k}\mathcal{H}^{1}(C_{k}^{-})$.
3. (3)
If $\text{int}(C_{i}^{+})\subset\text{int}(C_{j}^{+})$, $i\neq j$, then there
is some rectifiable Jordan curve $C_{k}^{-}$ such that
$\text{int}(C_{i}^{+})\subset\text{int}(C_{k}^{-})\subset\text{int}(C_{j}^{+})$.
Similarly, if $\text{int}(C_{i}^{-})\subset\text{int}(C_{j}^{-})$, $i\neq j$,
then there is some rectifiable Jordan curve $C_{k}^{+}$ such that
$\text{int}(C_{i}^{-})\subset\text{int}(C_{k}^{+})\subset\text{int}(C_{j}^{-})$.
4. (4)
Setting $L_{j}=\\{i\,:\,\text{int}(C_{i}^{-})\subset\text{int}(C_{j}^{+})\\}$
the sets $Y_{j}=\text{int}(C_{j}^{+})\setminus\bigcup_{i\in
L_{j}}\text{int}(C_{i}^{-})$ are pairwise disjoint , indecomposable and
$E=\bigcup_{j}Y_{j}$.
Since sets of finite perimeter are defined via the total variation seminorm of
$BV$-functions, they are understood modulo $2$-dimensional measure zero sets.
In particular, the last equality in (4) of Theorem 2.4 is modulo measure zero
sets. To make precise the change of representatives, we use below the notation
$A\Delta B:=(A\setminus B)\cup(B\setminus A)$ for the symmetric difference
between subsets $A,B\subset\mathbb{R}^{2}$.
With the auxiliary tools now recalled, we are ready to prove the main result
of this paper.
###### Proof of Theorem 1.3.
Suppose $f\colon\Omega\to\Omega^{\prime}$ is $L$-bi-Lipschitz. First notice
that $f$ extends to $\overline{\Omega}\to\overline{\Omega^{\prime}}$ as a bi-
Lipschitz map. Let $\Omega_{i}^{\prime}$ be a connected component of
$\mathbb{R}^{2}\setminus\overline{\Omega^{\prime}}$ and $\Omega_{i}$ the
connected component of $\mathbb{R}^{2}\setminus\overline{\Omega}$ for which
$f(\partial\Omega_{i})=\partial\Omega_{i}^{\prime}$. Then by Lemma 2.2 there
exists a constant $c>0$ so that each hole ${\Omega_{i}}$ is $c$-quasiconvex.
Therefore, by Theorem 2.1 also each ${\Omega_{i}^{\prime}}$ is
$c^{\prime}$-quasiconvex, where the quasiconvexity constant $c^{\prime}$ does
not depend on $i$. Obviously, each $\overline{\Omega_{i}^{\prime}}$ is also
$c^{\prime}$-quasiconvex.
Suppose that $\Omega$ is a bounded $BV$-extension domain. By Lemma 2.3, we
only need to prove that having the extension property for sets of finite
perimeter is invariant under the bi-Lipschitz map $f$. Let
$E^{\prime}\subset\Omega^{\prime}$ be a set of finite perimeter. Then
$E=f^{-1}(E^{\prime})$ is also a set of finite perimeter with
$P(E,\Omega)\leq LP(E^{\prime},\Omega^{\prime}).$ (2.1)
Let $\widetilde{E}$ be the perimeter extension of $E$ to the whole
$\mathbb{R}^{2}$ with
$P(\widetilde{E},{\mathbb{R}}^{2})\leq CP(E,{\Omega}).$ (2.2)
By Theorem 2.4, there exists a class of Jordan curves
$\\{\widetilde{C}_{i}^{+},\widetilde{C}_{k}^{-}\\}_{i,k\in\mathbb{N}}$ with
$\mathcal{H}^{2}(F)=0$
for the symmetric difference
$F:=\widetilde{E}\Delta\left(\bigcup_{i}\text{int}(\widetilde{C}_{i}^{+})\setminus\bigcup_{k}\text{int}(\widetilde{C}_{k}^{-})\right),$
and
$P(\widetilde{E},\mathbb{R}^{2})=\sum_{i}\mathcal{H}^{1}(\widetilde{C}_{i}^{+})+\sum_{k}\mathcal{H}^{1}(\widetilde{C}^{-}_{k})$
(2.3)
Take
$J\in\\{\text{int}(\widetilde{C}_{i}^{+})\\}_{i}\cup\\{\text{int}(\widetilde{C}_{k}^{-})\\}_{k}.$
We will extend each $J^{\prime}=f(J\cap\Omega)\subset\Omega^{\prime}$ to the
whole $\mathbb{R}^{2}$ in order to define the final extension set
$\widetilde{E}^{\prime}$ of $E^{\prime}$. Consider a homeomorphism
$\gamma\colon\mathbb{S}^{1}\to\partial J$, given by the fact that $J$ is a
Jordan domain. The set $\gamma\setminus\overline{\Omega}$ consists of (at
most) countably many open arcs $\gamma_{i}$ with endpoints
$x_{i},y_{i}\in\partial\Omega_{j(i)}$ for some $j(i)$. Observe that since
$|x_{i}-y_{i}|\leq\mathcal{H}^{1}(\gamma_{i})$, we always have
$\mathcal{H}^{1}(\gamma\cap\overline{\Omega})+\sum_{i}|x_{i}-y_{i}|\leq\mathcal{H}^{1}(\gamma\cap\overline{\Omega})+\mathcal{H}^{1}(\gamma\setminus\overline{\Omega})\leq\mathcal{H}^{1}(\gamma).$
(2.4)
Let us write $I_{i}=\gamma^{-1}(\gamma_{i})$ for each $i$ and use the
$c^{\prime}$-quasiconvexity of $\overline{\Omega_{j(i)}^{\prime}}$ to find a
curve $\gamma_{i}^{\prime}\subset\overline{\Omega_{j(i)}^{\prime}}$ joining
$f(x_{i})$ to $f(y_{i})$ with
$\ell(\gamma_{i}^{\prime})\leq c^{\prime}|f(x_{i})-f(y_{i})|.$ (2.5)
For convenience, we use the parametrization $\gamma_{i}^{\prime}\colon
I_{i}\to\mathbb{R}^{2}$ so that
$\gamma_{i}^{\prime-1}(x_{i})=\gamma^{-1}(x_{i})$ and
$\gamma_{i}^{\prime-1}(y_{i})=\gamma^{-1}(y_{i})$. The combination of
$f(\gamma\cap\overline{\Omega})$ with the curves $\gamma_{i}^{\prime}$ results
in a continuous curve $\gamma^{\prime}\colon\mathbb{S}^{1}\to\mathbb{R}^{2}$
defined as
$\gamma^{\prime}(t)=\begin{cases}f(\gamma(t)),&\text{if
}\gamma(t)\in\overline{\Omega},\\\ \gamma_{i}^{\prime}(t),&\text{if }t\in
I_{i}.\end{cases}$
Now, we define the extension domain $\widetilde{J}^{\prime}$ of
$J^{\prime}=f(\Omega\cap J)$ as the union of all the connected components of
$\mathbb{R}^{2}\setminus\gamma^{\prime}$ that intersect $J^{\prime}$. Then,
combining (2.5) with (2.4) and the fact that $f$ is $L$-bi-Lipschitz, gives
$\begin{split}P(\widetilde{J}^{\prime},\mathbb{R}^{2})&\leq\mathcal{H}^{1}(\gamma^{\prime})\\\
&=\mathcal{H}^{1}(f(\gamma\cap\overline{\Omega}))+\sum_{i}\mathcal{H}^{1}(\gamma_{i}^{\prime})\\\
&\leq\mathcal{H}^{1}(f(\gamma\cap\overline{\Omega}))+\sum_{i}c^{\prime}|f(x_{i})-f(y_{i})|\\\
&\leq
L\mathcal{H}^{1}(\gamma\cap\overline{\Omega})+c^{\prime}L\sum_{i}\mathcal{|}x_{i}-y_{i}|\\\
&\leq c^{\prime}L\mathcal{H}^{1}(\gamma)\\\
&=c^{\prime}LP(J,\mathbb{R}^{2}).\end{split}$ (2.6)
We finally set our extension of $E^{\prime}$ to be
$\widetilde{E}^{\prime}=\bigcup_{J\in\\{\text{int}(\widetilde{C}_{i}^{+})\\}_{i}}\widetilde{J}^{\prime}\setminus\bigcup_{J\in\\{\text{int}(\widetilde{C}_{k}^{-})\\}_{k}}\widetilde{J}^{\prime}.$
Since the decomposition of $\widetilde{E}$ was left unchanged inside $\Omega$,
we have that
$(\widetilde{E}^{\prime}\cap\Omega^{\prime})\Delta E^{\prime}=f(F\cap\Omega)$
has zero $2$-dimensional measure, since $F$ is measure-zero and $f$ is bi-
Lipschitz. Hence, $\widetilde{E^{\prime}}$ is indeed an extension of
$E^{\prime}$
Now, summing the estimate (2.6) over all the Jordan curves and using (2.3),
(2.2), and (2.1), we get
$\displaystyle P(\widetilde{E}^{\prime},\mathbb{R}^{2})$
$\displaystyle\leq\sum_{J\in\\{\text{int}(\widetilde{C}_{i}^{+}),\text{int}(\widetilde{C}_{k}^{-})\\}_{i,k}}P(\widetilde{J^{\prime}},\mathbb{R}^{2})$
$\displaystyle\leq
c^{\prime}L\sum_{J\in\\{\text{int}(\widetilde{C}_{i}^{+}),\text{int}(\widetilde{C}_{k}^{-})\\}_{i,k}}P(J,\mathbb{R}^{2})$
$\displaystyle=c^{\prime}LP(\widetilde{E},\mathbb{R}^{2})$ $\displaystyle\leq
Cc^{\prime}LP(E,\Omega)$ $\displaystyle\leq
Cc^{\prime}L^{2}P(E^{\prime},\Omega^{\prime}).$
This shows that $\widetilde{E}^{\prime}$ is a perimeter extension of $E$ as
required. Thus, by Lemma 2.3 we conclude that the domain $\Omega^{\prime}$ is
a $BV$-extension domain. ∎
## References
* [1] L. Ambrosio, V. Caselles, S. Masnou and J.-M. Morel, _Connected components of sets of finite perimeter and applications to image processing._ J. Eur. Math. Soc. 3, 39–92 (2001).
* [2] Y. D. Burago and V. G. Maz’ya, _Certain questions of potential theory and function theory for regions with irregular boundaries_ (Russian), Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 3 (1967) 152pp.
* [3] A. P. Calderón, _Lebesgue spaces of differentiable functions and distributions_ , in Proc. Symp. Pure Math., Vol $IV$, 1961, 33-49.
* [4] M. García-Bravo and T. Rajala, _Strong $BV$-extension and $W^{1,1}$-extension domains_, preprint 2021.
* [5] P. Hajłasz, _A new characterization of the Sobolev space_ , Stud. Math. 159 (2003), no. 2, 263–275.
* [6] P. Hajłasz, _Sobolev spaces on an arbitrary metric space_ , Potential Anal. 5 (1996), no. 4, 403-415.
* [7] P. Hajłasz, P. Koskela and H. Tuominen, _Sobolev embeddings, extensions and measure density condition_ , J. Funct. Anal. 254 (2008), no. 5, 1217–1234.
* [8] P. Hajłasz, P. Koskela and H. Tuominen, _Measure density and extendability of Sobolev functions_ , Rev. Mat. Iberoam. 24 (2008), no. 2, 645–669.
* [9] P. W. Jones, _Quasiconformal mappings and extendability of Sobolev functions_ , Acta. Math. 47 (1981), 71-88.
* [10] P. Koskela, M. Jr. Miranda and N. Shanmugalingam, _Geometric properties of planar $BV$-extension domains._ Around the research of Vladimir Maz’ya. $I$, 255–272, Int.Math.Ser.(N.Y.), 11, Springer, New York, 2010.
* [11] P. Koskela, T. Rajala and Y. Zhang, _A geometric characterization of planar Sobolev extension domains._ arXiv:1502.04139v6.
* [12] P. Koskela, T. Rajala and Y. Zhang, _Planar $W^{1,1}$-extension domains_, preprint.
* [13] P. Koskela and E. Saksman _Pointwise characterizations of Hardy-Sobolev functions_ , Math. Res. Lett. 15 (2008), 727–744.
* [14] P. Shvartsman, _On Sobolev extension domains in ${{{\mathbb{R}}}^{n}}$_, J. Funct. Anal. 258 (2010), no. 7, 2205-2245.
* [15] E. M. Stein, _Singular integrals and differentiability properties of functions_ , Princeton University Press, Princeton, New Jersey, 1970.
* [16] J. Väisälä, _Holes and maps of Euclidean spaces_ , Conform. Geom. Dyn. 12 (2008) 58–66.
|
# Projective quantum Monte Carlo simulations guided by unrestricted neural
network states
E. M. Inack The Abdus Salam International Centre for Theoretical Physics,
34151 Trieste, Italy SISSA - International School for Advanced Studies, 34136
Trieste, Italy INFN, Sezione di Trieste, 34136 Trieste, Italy G. E. Santoro
The Abdus Salam International Centre for Theoretical Physics, 34151 Trieste,
Italy SISSA - International School for Advanced Studies, 34136 Trieste, Italy
CNR-IOM Democritos National Simulation Center, Via Bonomea 265, 34136 Trieste,
Italy L. Dell’Anna Dipartimento di Fisica e Astronomia “Galileo Galilei”,
Università di Padova, 35131 Padova, Italy CNISM, Sezione di Padova, Italy S.
Pilati School of Science and Technology, Physics Division, Università di
Camerino, 62032 Camerino (MC), Italy
###### Abstract
We investigate the use of variational wave-functions that mimic stochastic
recurrent neural networks, specifically, unrestricted Boltzmann machines, as
guiding functions in projective quantum Monte Carlo (PQMC) simulations of
quantum spin models. As a preliminary step, we investigate the accuracy of
such unrestricted neural network states as variational Ansätze for the ground
state of the ferromagnetic quantum Ising chain. We find that by optimizing
just three variational parameters, independently on the system size, accurate
ground-state energies are obtained, comparable to those previously obtained
using restricted Boltzmann machines with few variational parameters per spin.
Chiefly, we show that if one uses optimized unrestricted neural network states
as guiding functions for importance sampling the efficiency of the PQMC
algorithms is greatly enhanced, drastically reducing the most relevant
systematic bias, namely that due to the finite random-walker population. The
scaling of the computational cost with the system size changes from the
exponential scaling characteristic of PQMC simulations performed without
importance sampling, to a polynomial scaling, even at the ferromagnetic
quantum critical point. The important role of the protocol chosen to sample
hidden-spins configurations, in particular at the critical point, is analyzed.
We discuss the implications of these findings for what concerns the problem of
simulating adiabatic quantum optimization using stochastic algorithms on
classical computers.
## I Introduction
Quantum Monte Carlo (QMC) algorithms are generally believed to be capable of
predicting equilibrium properties of quantum many-body systems at an
affordable computational cost, even for relatively large system sizes, at
least when the sign problem does not occur. However, it has recently been
shown that the computational cost to simulate the ground state of a quantum
Ising model with a simple projective QMC (PQMC) algorithm that does not
exploit importance sampling techniques scales exponentially with the system
size, making large-scale simulations unfeasible Inack _et al._ (2018). This
happens in spite of the fact that the Hamiltonian is sign-problem free. PQMC
methods have found vast use in condensed matter physics, in chemistry, and
beyond (see, e.g., Refs. Ceperley and Alder (1986); Hammond _et al._ (1994);
Foulkes _et al._ (2001); Carlson _et al._ (2015)). Shedding light on their
computational complexity, and possibly improving it by using importance
sampling techniques based on novel variational wave-functions, are therefore
very important tasks. We address them in this Article.
PQMC algorithms have recently emerged as useful computational tools also to
investigate the potential efficiency of adiabatic quantum computers in solving
large-scale optimization problems via quantum annealing Finnila _et al._
(1994); Santoro _et al._ (2002); Boixo _et al._ (2014); Inack and Pilati
(2015); Heim _et al._ (2015). In particular, it has been shown that the
(stochastic) dynamics of simple PQMC simulations allows to tunnel through tall
barriers of (effectively) double-well models even more efficiently than an
adiabatic quantum computer which exploits incoherent quantum tunneling Isakov
_et al._ (2016); Jiang _et al._ (2017); Mazzola _et al._ (2017); Inack _et
al._ (2018). This result seems to suggest that there might be no systematic
quantum speed-up in using a quantum annealing device to solve an optimization
problem, compared to a stochastic QMC simulation performed on a classical
computer Isakov _et al._ (2016). Remarkably, this computational advantage of
the PQMC simulations with respect to the expected behavior of a quantum
annealing device occurs also in more challenging models with frustrated
couplings Inack _et al._ (2018), as in the recently introduced Shamrock
model, where QMC algorithms based on the (finite temperature) path-integral
formalism display instead an exponential slowdown of the tunneling dynamics
Andriyash and Amin (2017). This result further stresses the importance of
shedding light on the computational complexity of PQMC algorithms: if these
computational techniques allowed one to simulate, with a polynomially scaling
computational cost, both the ground-state properties of a model Hamiltonian,
and also the tunneling dynamics of a quantum annealing device described by
such Hamiltonian Inack _et al._ (2018), then the quantum speedup mentioned
above would be very unlikely to be achieved. We focus in this paper on the
first of the two aspects, specifically, on analyzing and improving the scaling
of the computational cost to simulate ground-state properties of quantum Ising
models.
It is well known that the efficiency of PQMC algorithms can be enhanced by
implementing importance sampling techniques using as guiding functions
accurate variational Ansätze Foulkes _et al._ (2001). However, building
accurate variational wave-functions for generic many-body systems is a highly
non trivial task. Recently, variational wave-functions that mimic the
structure of neural networks have been shown to accurately describe ground-
state properties of quantum spin and lattice models Carleo and Troyer (2017);
Saito (2017); Saito and Kato (2018). The representational power and the
entanglement content of such variational states, now referred to as neural
network states, have been investigated Deng _et al._ (2017); Chen _et al._
(2018); Glasser _et al._ (2018); Gao and Duan (2017); Freitas _et al._
(2018), showing, among other properties, that they are capable of describing
volume-law entanglement. The authors of Ref. Carleo and Troyer (2017)
considered neural network states that mimic restricted Boltzmann machines
(RBM), i.e. such that no interaction among hidden spins is allowed. One very
appealing feature of such restricted neural network states is that the role of
the hidden spins can be accounted for analytically, without the need of Monte
Carlo sampling over hidden variables. Furthermore, such states provide very
accurate ground-state energy predictions, which can be systematically improved
by increasing the number of hidden spins per visible spin (later on referred
to as hidden-spin density). However, this high accuracy is obtained at the
cost of optimizing a number of variational parameters that increases with the
system size. This optimization task can be tackled using powerful optimization
algorithms such as the stochastic reconfiguration method (see, e.g, Ref.
Sorella _et al._ (2007)). Yet, having to optimize a large number of
variational parameters is not desirable in the context of quantum annealing
simulations, since one would be dealing with a variational optimization
problem, potentially even more difficult than the original classical
optimization problem.
In this Article, we consider instead neural network states that mimic
unrestricted Boltzmann machines (uRBMs), allowing intra-layer correlations
among hidden spins, beyond the inter-layer hidden-visible correlations and the
intra-layer visible-visible correlations (see Fig. 1). The structure of these
states resembles the one of the shadow wave functions originally introduced to
describe quantum fluids and solids Vitiello _et al._ (1988); Reatto and
Masserini (1988). We test their representational power considering as a
testbed the ferromagnetic quantum Ising chain. We find that by optimizing just
three variational parameters, independently on the system size, very accurate
ground-state energies are obtained, comparable to the case of restricted
neural network states with one hidden spin per visible spin. Such a small
number of variational parameters is a particularly appealing feature in the
context of quantum annealing problems. However, it comes at the prize of
having to perform Monte Carlo sampling over hidden-spin configurations.
The main goal of this Article is to show that the above-mentioned unrestricted
neural network states can be used as a guide for importance sampling in PQMC
simulations. This also implies that the development of neural network states
can be limited to obtaining reasonably accurate, but not necessarily exact,
variational Ansätze, since the residual error can be eliminated within the
PQMC simulation. In particular, we provide numerical evidence that the major
source of systematic bias of the PQMC algorithms, namely the bias originating
from the finite size of the random-walker population which has to be
stochastically evolved in any PQMC simulation, can be drastically reduced
using optimized unrestricted neural network states, even at the point of
changing the scaling of the required population size from exponential
(corresponding to the case without importance sampling) to polynomial in the
system size. This also implies a change of computational complexity from
exponential to polynomial. For comparison, we show that a conventional
variational wave-function of the Boltzmann type (with no hidden spins),
instead, does not determine a comparable efficiency improvement.
The rest of the Article is organized as follows: in Section II we define the
conventional Boltzmann-type variational wave functions and the unrestricted
neural network states, and we then analyze how accurately they predict the
ground-state energy of the quantum Ising chain via optimization of,
respectively one and three, variational parameters. Section III deals with the
continuous-time PQMC algorithm and with the implementation of importance
sampling using both Boltzmann-type wave functions and, chiefly, unrestricted
neural network states, showing how the systematic bias due to the finite
random-walker population is affected, both at and away from the quantum
critical point. The important effect of choosing different sampling protocols
for the hidden spins is also analyzed. Our conclusions and the outlook are
reported in Section IV.
## II Unrestricted neural network states for quantum Ising models
In this article, we consider as a test bed the one-dimensional ferromagnetic
quantum Ising Hamiltonian:
$\hat{H}=\hat{H}_{\mathrm{cl}}+\hat{H}_{\mathrm{kin}},$ (1)
where
$\hat{H}_{\mathrm{cl}}=-J\sum_{i=1}^{N}{\sigma}^{z}_{i}{\sigma}^{z}_{i+1}$ and
$\hat{H}_{\mathrm{kin}}=-\Gamma\sum_{i=1}^{N}{\sigma}^{x}_{i}$.
$\sigma^{x}_{i}$, $\sigma^{y}_{i}$, and $\sigma^{z}_{i}$ indicate Pauli
matrices acting on spins at the lattice site $i$. $N$ is the total number of
spins, and we adopt periodic boundary conditions, i.e.
${\sigma}^{\alpha}_{N+1}={\sigma}^{\alpha}_{1}$, with $\alpha=x,y,z$. The
parameter $J>0$ fixes the strength of the ferromagnetic interactions among
nearest-neighbor spins. In the following, we set $J=1$. All energy scales are
henceforth expressed in units of $J$. The parameter $\Gamma$ fixes the
intensity of a transverse magnetic field. Given $\left|x_{i}\right>$ an
eigenstate of the Pauli matrix ${\sigma^{z}_{i}}$ with eigenvalue $x_{i}=1$
when $\left|x\right>=\left|\uparrow\right>$ and $x_{i}=-1$ when
$\left|x\right>=\left|\downarrow\right>$, the quantum state of $N$ spins is
indicated by
$\left|\mathit{\boldsymbol{x}}\right>=\left|x_{1}x_{2}...x_{N}\right>$. Notice
that the function
$E_{\mathrm{cl}}(\mathit{\boldsymbol{x}})=\langle\mathit{\boldsymbol{x}}|\hat{H}_{\mathrm{cl}}|\mathit{\boldsymbol{x}}\rangle$
(with $\mathit{\boldsymbol{x}}=\left(x_{1},x_{2},\dots,x_{N}\right)$)
corresponds to the Hamiltonian function of a classical Ising model, while the
operator $\hat{H}_{\mathrm{kin}}$ introduces quantum (kinetic) fluctuations.
Our first goal is to develop trial wave functions that closely approximate the
ground state wave function
$\Psi_{0}{(\mathit{\boldsymbol{x}})}=\left<\mathit{\boldsymbol{x}}|\Psi_{0}\right>$
of the Hamiltonian (1). A simple Ansatz can be defined as
$\Psi_{\boldsymbol{\kappa}}{(\mathit{\boldsymbol{x}})}=e^{-\beta
E_{\mathrm{cl}}({\mathit{\boldsymbol{x}}})}=e^{-K_{1}\sum_{i=1}^{N}x_{i}x_{i+1}}\;.$
(2)
$\boldsymbol{\kappa}$ is here a set of real variational parameters to be
optimized. Their values are obtained by minimizing the average of the energy,
as in standard variational quantum Monte Carlo approaches. In this case, only
one parameter $K_{1}=\beta$ is present, $\boldsymbol{\kappa}=\\{K_{1}\\}$.
This choice is inspired by the classical Boltzmann distribution where $\beta$
would play the role of a fictitious inverse temperature. The above Ansatz will
be referred to as Boltzmann-type wave function.
A more sophisticated Ansatz can be constructed by using a generative
stochastic artificial neural network, namely an uRBM (see Fig. 1). Beyond the
visible spin variables
$\mathit{\boldsymbol{x}}=\left(x_{1},x_{2},\dots,x_{N}\right)$, one introduces
$N$ hidden spin variables
$\mathit{\boldsymbol{h}}=\left(h_{1},h_{2},\dots,h_{N}\right)$, taking values
$h_{i}=\pm 1$ (with $i=1,\dots,N$). Periodic boundary conditions within the
layers are also incorporated, i.e $x_{N+1}=x_{1}$ and $h_{N+1}=h_{1}$. The
trial wave function is thus written in the following integral form:
$\Psi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}})=\sum_{\mathit{\boldsymbol{h}}}\phi_{\boldsymbol{\kappa}}\left(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}\right)\;,$
(3)
where,
$\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})=e^{-\sum_{i=1}^{N}\left(K_{1}x_{i}x_{i+1}+K_{2}h_{i}h_{i+1}+K_{3}x_{i}h_{i}\right)}\;.$
(4)
Notice that the architecture of this uRBM includes correlations between
nearest-neighbor visible spins, between nearest-neighbor hidden spins, as well
as between pairs of visible and hidden spins with the same index $i$. These
three correlations are parametrized by the three constants $K_{1}$, $K_{2}$,
and $K_{3}$, respectively. With this uRBM trial Ansatz, the set of variational
parameters is $\boldsymbol{\kappa}=\\{K_{1},K_{2},K_{3}\\}$. It is
straightforward to generalize the uRBM Ansatz including more layers of hidden
spins. Every additional hidden-spin layer adds two more variational
parameters, and it effectively represents the application of an imaginary-time
Suzuki-Trotter step $e^{-\Delta\tau\hat{H}}$ for a certain time step
$\Delta\tau$. Thus, a deep neural network state with many hidden layers can
represent a long imaginary-time dynamics, which projects out the ground state
provided that the initial state is not orthogonal to it. In fact, the mapping
between deep neural networks and the imaginary time projection has been
exploited in Refs. Carleo _et al._ (2018); Freitas _et al._ (2018) to
construct more complex neural network states. In this article we consider only
the single hidden-spin layer uRBM, since this Ansatz turns out to be adequate
for the ferromagnetic quantum Ising chain. The multi hidden-spin layer Ansatz
might be useful to address more complex models as, e.g, frustrated Ising spin
glasses. Extensions along these lines are left as future work.
In a recent work Carleo and Troyer (2017), Carleo and Troyer considered a
restricted Boltzmann machine (RBM), where direct correlations among hidden
spins were not allowed. Their Ansatz included a larger number of hidden spins,
as well as more connections between visible and hidden spins, leading to an
extensive number of variational parameter proportional to $\alpha N$, where
$\alpha=1,2,\dots$. One advantage of the RBM, due to the absence of hidden-
hidden correlations, is that the role of hidden spins can be analytically
traced out. The uRBM we employ, which is analogous to the shadow wave
functions used to describe quantum fluid and solids, includes only three
variational parameters, independently of the system size. However, their
effect has to be addressed by performing sampling of hidden spins
configurations, as described below. It is worth pointing out that correlations
beyond nearest-neighbor spins could also be included in the uRBM Ansatz, with
straightforward modifications in the sampling algorithms described below. We
mention here also that, as shown in Ref. Gao and Duan (2017), neural network
states with intra-layer correlations can be mapped to deep neural networks
with more hidden layers, but no intra-layer correlations.
Figure 1: (color online). Structure of the unrestricted Boltzmann machine. The
lower (yellow) nodes depict visible spins, the upper (magenta) nodes depict
the hidden spins. The horizontal segments indicate intralayer visible-visible
and hidden-hidden correlations. The vertical (blue) segments represent the
interlayer correlations between the corresponding visible and hidden spins.
The green lines allude to a possible extension to deep layers architectures.
In the case of an uRBM variational wave function, the average value of the
energy $E={\langle\hat{H}\rangle}$ is computed as follows
$\displaystyle{\langle\hat{H}\rangle}$ $\displaystyle=$
$\displaystyle\frac{\langle\Psi_{\boldsymbol{\kappa}}|\hat{H}|\Psi_{\boldsymbol{\kappa}}\rangle}{\langle\Psi_{\boldsymbol{\kappa}}|\Psi_{\boldsymbol{\kappa}}\rangle}=\frac{\sum_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}\Psi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}})H_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}\Psi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}}^{\prime})}{\sum_{\mathit{\boldsymbol{x}}}\Psi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}})\Psi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}})}$
(5) $\displaystyle=$
$\displaystyle\frac{\sum_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}\left[\sum_{\mathit{\boldsymbol{h}}_{\bf
a}}\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
a})\right]H_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}\left[\sum_{\mathit{\boldsymbol{h}}_{\bf
b}}\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}}^{\prime},\mathit{\boldsymbol{h}}_{\bf
b})\right]}{\sum_{\mathit{\boldsymbol{x}}}\left[\sum_{\mathit{\boldsymbol{h}}_{\bf
a}}\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
a})\right]\left[\sum_{\mathit{\boldsymbol{h}}_{\bf
b}}\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
b})\right]}$ $\displaystyle=$ $\displaystyle\ll
E_{\mathrm{loc}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
b})\gg\;,$
where the local energy
$E_{\mathrm{loc}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})$ is defined
as
$E_{\mathrm{loc}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})=\frac{\sum_{\mathit{\boldsymbol{x}}^{\prime}}H_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}}^{\prime},\mathit{\boldsymbol{h}})}{\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})}\;,$
(6)
with
$H_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}=\langle\mathit{\boldsymbol{x}}|\hat{H}|\mathit{\boldsymbol{x}}^{\prime}\rangle$.
$\mathit{\boldsymbol{h}}_{\bf a}$ and $\mathit{\boldsymbol{h}}_{\bf b}$
indicate two hidden spin configurations. Notice that the formula for the local
energy can be symmetrized with respect to the two sets of hidden spins
$\mathit{\boldsymbol{h}}_{\bf a}$ and $\mathit{\boldsymbol{h}}_{\bf b}$,
providing results with slightly reduced statistical fluctuations. The double
brackets $\ll\cdots\gg$ indicate the expectation value over the visible-spin
configurations $\mathit{\boldsymbol{x}}$ and two sets of hidden spins
configurations $\mathit{\boldsymbol{h}}_{\bf a}$ and
$\mathit{\boldsymbol{h}}_{\bf b}$, sampled from the following normalized
probability distribution:
$p(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
a},\mathit{\boldsymbol{h}}_{\bf
b})=\frac{\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
a})\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
b})}{\sum_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
a},\mathit{\boldsymbol{h}}_{\bf
b}}\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
a})\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
b})}\;.$ (7)
As in standard Monte Carlo approaches, this expectation value is estimated as
the average of
$E_{\mathrm{loc}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})$ over a
(large) set of uncorrelated configurations, sampled according to
$p(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}_{\bf
a},\mathit{\boldsymbol{h}}_{\bf b})$. The statistical uncertainty can be
reduced at will by increasing the number of sampled configurations. The
optimal variational parameters $\boldsymbol{\kappa}_{\mathrm{opt}}$ that
minimize the energy expectation value can be found using a stochastic
optimization method. We adopt a relatively simple yet quite efficient one,
namely the stochastic gradient descent algorithm (see, e.g., Becca and Sorella
(2017)). While more sophisticated algorithms exist as, e.g., the stochastic
reconfiguration method Sorella _et al._ (2007), such methods are not
necessary here since the Ansätze that we consider include a very small number
of variational parameters, one or three. In fact, in these cases the optimal
variational parameters can be obtained also by performing a scan on a fine
grid. By doing so, we obtain essentially the same results provided by the
stochastic gradient descent algorithm.
We assess the accuracy of the optimized variational wave functions by
calculating the relative error
$e_{\mathrm{rel}}=\frac{\left|E-E_{\mathrm{JW}}\right|}{\left|E_{\mathrm{JW}}\right|}\;,$
(8)
in the obtained variational estimate $E$ of the ground state energy of the
Hamiltonian in Eq. (1). $E_{\mathrm{JW}}$ is the exact finite size ground
state energy of the quantum Ising chain. It is obtained by performing the
Jordan–Wigner transformation, followed by a Fourier and the Bogoliubov
transformations.
Figure 2 displays the relative error $e_{\mathrm{rel}}$ in Eq. (8)
corresponding to the variational wave functions introduced above, as a
function of the transverse field $\Gamma$. The system size is $N=80$, which is
here representative of the thermodynamic limit. The Boltzmann-type Ansatz does
not provide particularly accurate predictions. In the ferromagnetic phase
$\Gamma<1$, the relative error is up to $10\%$. The uRBM, instead, provides
very accurate predictions. The relative error is always below $0.1\%$. The
largest discrepancy occurs at the quantum critical point $\Gamma=1$. Such high
accuracy is remarkable, considering that the uRBM Ansatz involves only $3$
variational parameters. It is also worth mentioning that very similar
accuracies are obtained also for different system sizes. Therefore, the uRBM
Ansatz represents a promising guiding function for simulations of quantum
annealing optimization of disordered models. As a term of comparison, we show
in Fig. 2 the results obtained in Ref. Carleo and Troyer, 2017 using the RBM
Ansatz. The relative errors corresponding to the RBM with hidden-unit density
$\alpha=1$ are larger than those corresponding to the uRBM, despite the fact
that the RBM Ansatz involves a larger number of variational parameters.
However, it is worth stressing that the RBM results can be systematically
improved by increasing $\alpha$. For example, with $\alpha=2$ the RBM relative
errors are approximately an order of magnitude smaller than those
corresponding to the uRBM Ansatz.
Figure 2: (color online). Relative error $e_{\mathrm{rel}}$ in the variational
estimates of the ground-state energy, see Eq. (8), as a function of the
transverse field $\Gamma$, obtained using the simple Boltzmann wave function
and for the unrestricted Boltzmann machine (uRBM) Ansatz. The system size is
$N=80$. For comparison, we also show the data corresponding to the restricted
Boltzmann machine (RBM) from Ref. Carleo and Troyer, 2017, where $\alpha$
indicates the hidden-spin density. The thin lines are guides to the eyes.
## III Importance sampling guided by unrestricted neural network states
In this section we discuss how optimized variational wave functions can be
utilized to boost the performance of PQMC simulations. First, we consider the
implementation of the PQMC algorithm without guiding functions. PQMC methods
allow one to extract ground-state properties of quantum many-body systems
Anderson (1975); Kalos and Whitlock (2008) by stochastically simulating the
Schrödinger equation in imaginary-time $\tau=it$. In the Dirac notation, this
equation is written as:
$\displaystyle-\frac{\partial}{\partial\tau}|\Psi(\tau)\rangle=(\hat{H}-E_{\mathrm{ref}})|\Psi(\tau)\rangle.$
(9)
The reduced Planck constant is set to $\hbar=1$ throughout this Article.
$E_{\mathrm{ref}}$ is a reference energy introduced to stabilize the
simulation, as discussed later. Eq. (9) is simulated by iteratively applying
the equation
$\Psi(\mathit{\boldsymbol{x}},\tau+\Delta\tau)=\sum_{\mathit{\boldsymbol{x}}^{\prime}}G(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime},\Delta\tau)\Psi(\mathit{\boldsymbol{x}}^{\prime},\tau)$.
$\Delta\tau$ is a (short) time step and
$G({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)=\langle\mathit{\boldsymbol{x}}|e^{-\Delta\tau(\hat{H}-E_{\mathrm{ref}})}|\mathit{\boldsymbol{x}}^{\prime}\rangle$
is the Green’s function of Eq. (9). Below it is discussed how one can write a
suitable explicit expression. Long propagation times $\tau$ are achieved by
iterating many (small) time steps $\Delta\tau$, allowing one to sample, in the
$\tau\rightarrow\infty$ limit, spin configurations with a probability density
proportional to the ground state wave function
$\Psi_{0}(\mathit{\boldsymbol{x}})$ (assumed to be real and non negative). One
should notice that the Green’s function
$G({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)$
does not define a stochastic matrix; while its elements are nonnegative, one
has
$\sum_{\mathit{\boldsymbol{x}}}G({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)\neq
1$, in general. Therefore, it cannot be utilized to define the transition
matrix of a conventional Markov chain Monte Carlo simulation. This problem can
be circumvented by rewriting the Green’s function as
$G({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)=G_{\mathrm{T}}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)b_{\mathit{\boldsymbol{x}}^{\prime}}$,
where
$G_{\mathrm{T}}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)$
is by definition stochastic, and the normalization factor is
$b_{\mathit{\boldsymbol{x}}^{\prime}}=\sum_{\mathit{\boldsymbol{x}}}G({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)$.
A stochastic process can then be implemented, where a large population of
equivalent copies of the system, in jargon called walkers, is evolved. Each
walker represents one possible spin configuration
${\mathit{\boldsymbol{x}}}_{n}^{\prime}$ (the index $n$ labels different
walkers), and is gradually modified by performing spin-configuration updates
according to
$G_{\mathrm{T}}({\mathit{\boldsymbol{x}}}_{n},{\mathit{\boldsymbol{x}}}_{n}^{\prime},\Delta\tau)$.
Thereafter, their (relative) weights $w_{n}$ are accumulated according to the
rule $w_{n}\rightarrow w_{n}b_{\mathit{\boldsymbol{x}}_{n}^{\prime}}$,
starting with equal initial weights $w_{n}=1$ for all the walkers in the
initial population. While this implementation is in principle correct, it is
known to lead to an exponentially fast signal loss as the number of Monte
Carlo steps increases. This is due to the fact that the relative weight of few
walkers quickly becomes dominant, while most other walkers give a negligible
contribution to the signal. An effective remedy consists in introducing a
branching process, where each walker is replicated (or annihilated) a number
of times corresponding, on average, to the weight $w_{n}$. The simplest
correct rule consists in generating, for each walker in the population at a
certain imaginary time $\tau$, a number of descendants $n_{\mathrm{d}}$ in the
population at imaginary time $\tau+\Delta\tau$. $n_{\mathrm{d}}$ is defined as
$\mathrm{int}\left[w_{n}+\eta\right]$, where $\eta\in\left[0,1\right]$ is a
uniform random number, and the function $\mathrm{int}\left[\right]$ gives the
integer part of the argument Thijssen (2007). Clearly, after branching has
been performed, all walkers have the same weight $w_{n}=1$. Therefore, the
number of walkers in the population fluctuates at each PQMC iteration and can
be kept close to a target value by adjusting the reference energy
$E_{\mathrm{ref}}$. Introducing the branching process provides one with a
feasible, possibly efficient algorithm. However, such as process might
actually introduce a systematic bias if the average population size $N_{w}$ is
not large enough. The bias originates from the spurious correlations among
walkers generated from the same ancestor Becca and Sorella (2017). This effect
becomes negligible in the $N_{w}\rightarrow\infty$ limit, but might be sizable
for finite $N_{w}$. It is known to be the most relevant and subtle possible
source of systematic errors in PQMC algorithms Nemec (2010); Boninsegni and
Moroni (2012); Pollet _et al._ (2018). In fact, it was shown in Ref. Inack
_et al._ (2018) that in order to determine with a fixed target relative error,
the ground state energy of the ferromagnetic quantum Ising chain with the
(simple) diffusion Monte Carlo algorithm (which belongs to the category of
PQMC methods), the walker-population size $N_{w}$ has to exponentially
increase with the system size $N$. This implies an exponentially scaling
computational cost.
Figure 3: (color online). Number of random walkers $N_{w}$ required to
determine, using the PQMC algorithm without importance sampling, the ground-
state energy with a relative error $e_{\mathrm{rel}}$, see Eq. (8), as a
function of the system size $N$. Different datasets correspond to different
transverse field intensities $\Gamma$ and different relative errors. The lines
represent exponential fitting functions.
A promising strategy to circumvent the aforementioned problem is to introduce
the so-called importance sampling technique. This is indeed a well established
approach to boost the efficiency of PQMC simulations (see, e.g, Ref. Foulkes
_et al._ (2001)) because it has the potential to reduce the number of walkers
needed to attain a given accuracy Becca and Sorella (2017). It consists in
evolving a function
$f({\mathit{\boldsymbol{x}}},\tau)=\Psi({\mathit{\boldsymbol{x}}},\tau)\psi_{T}({\mathit{\boldsymbol{x}}})$
via a modified imaginary-time Schrödinger equation.
$\psi_{T}({\mathit{\boldsymbol{x}}})$ is a guiding function designed to
accurately approximate the ground-state wave function. Its role is to favor
the sampling of configurations with high probability amplitude. The obtained
modified imaginary-time Schrödinger equation is solved via a Markov process
defined by the following equation:
$f(\mathit{\boldsymbol{x}},\tau+\Delta\tau)=\sum_{\mathit{\boldsymbol{x}}^{\prime}}\tilde{G}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime},\Delta\tau)f(\mathit{\boldsymbol{x}}^{\prime},\tau),$
(10)
where the modified Green’s function is given by
$\tilde{G}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)=G(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime},\Delta\tau)\frac{\psi_{T}({\mathit{\boldsymbol{x}}})}{\psi_{T}({\mathit{\boldsymbol{x}}^{\prime}})}$.
A suitable approximation for the modified Green’s function can be obtained by
dividing the time step $\Delta\tau$ into $M$ shorter time steps
$\delta\tau=\Delta\tau/M$. If $\delta\tau$ is sufficiently short, one can
employ a Taylor expansion truncated at the linear term,
$\tilde{G}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)\cong\left[\tilde{g}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\delta\tau)\right]^{M}$,
where:
$\tilde{g}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\delta\tau)=\big{[}\delta_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}-\delta\tau(H_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}-E_{\mathrm{ref}}\delta_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}})\big{]}\frac{\psi_{T}({\mathit{\boldsymbol{x}}})}{\psi_{T}({\mathit{\boldsymbol{x}}^{\prime}})}\;.$
(11)
With this approximation, Eq. (10) defines a stochastic implementation of the
power method of linear algebra. Convergence to the exact ground state is
guaranteed as long as $\delta\tau$ is smaller than a finite value,
sufficiently small to ensure that all matrix elements of
$\tilde{g}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\delta\tau)$
are not negative Schmidt _et al._ (2005). As the system size increases,
shorter and shorter time steps $\delta\tau$ are required. This leads to
pathologically inefficient simulations, since in this regime the identity
operator dominates, resulting in extremely long autocorrelation times. This
problem can be solved by adopting the continuous-time Green’s function Monte
Carlo (CTGFMC) algorithm. The derivation and the details of this algorithm are
given in Ref. Becca and Sorella (2017); Sorella and Capriotti (2000), and so
we only sketch it here. The idea is to formally take the $M\rightarrow\infty$
limit, and determine the (stochastic) time interval $\delta\tau^{\prime}$ that
passes before the next configuration update occurs. It is convenient to
bookkeep the remaining time $\delta\tau_{t}$ left to complete a total interval
of time $\Delta\tau$. This is to ensure that each iteration of the PQMC
simulation corresponds to a time step of duration $\Delta\tau$. The time
interval $\delta\tau^{\prime}$ is sampled using the formula
$\delta\tau^{\prime}=\mathrm{Min}\big{(}\delta\tau_{t},\frac{\ln(1-\xi)}{E_{\mathrm{loc}}(\mathit{\boldsymbol{x}}^{\prime})-E_{\mathrm{cl}}(\mathit{\boldsymbol{x}}^{\prime})}\big{)}$
with $\xi\in(0,1)$ being a uniform random number. The spin-configuration
update $\mathit{\boldsymbol{x}}^{\prime}\rightarrow\mathit{\boldsymbol{x}}$
(with $\mathit{\boldsymbol{x}}^{\prime}\neq\mathit{\boldsymbol{x}}$) is
randomly selected from the probability distribution
$\begin{array}[]{ll}t_{{\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime}}&=\displaystyle\frac{p_{{\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime}}}{\sum_{\mathit{\boldsymbol{x}}\neq\mathit{\boldsymbol{x}}^{\prime}}p_{{\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime}}}\vspace{2mm}\\\
p_{{\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime}}&=\displaystyle\frac{\tilde{g}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\delta\tau^{\prime})}{\sum_{\mathit{\boldsymbol{x}}}\tilde{g}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\delta\tau^{\prime})}\end{array}\;.$
(12)
Notice that, with the Hamiltonian (1), $\mathit{\boldsymbol{x}}$ differs from
$\mathit{\boldsymbol{x}}^{\prime}$ only for one spin flip. The weight-update
factor for the branching process takes the exponential form
$b_{\mathit{\boldsymbol{x}}^{\prime}}=e^{-\delta\tau^{\prime}[E_{\mathrm{loc}}(\mathit{\boldsymbol{x}}^{\prime})-E_{\mathrm{ref}}]}$,
where the local energy is now
$E_{\mathrm{loc}}(\mathit{\boldsymbol{x}}^{\prime})=\sum_{\mathit{\boldsymbol{x}}}H_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime}}\frac{\psi_{T}({\mathit{\boldsymbol{x}}})}{\psi_{T}({\mathit{\boldsymbol{x}}^{\prime}})}$.
In summary, the CTGFMC algorithm requires to perform, for each walker $n$ in
the population, the following steps:
i)
initialize the time interval $\delta\tau_{t}=\Delta\tau$, and the weight
factor $w_{n}=1$;
ii)
sample the time $\delta\tau^{\prime}$ at which the the configuration update
$\mathit{\boldsymbol{x}}^{\prime}\rightarrow\mathit{\boldsymbol{x}}$ might
occur;
iii)
if $\delta\tau^{\prime}<\delta\tau_{t}$, update
$\mathit{\boldsymbol{x}}^{\prime}$ with a transition probability
$t_{{\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime}}$ in Eq.
(12), else set $\delta\tau^{\prime}=\delta\tau_{t}$;
iv)
accumulate the weight factor according to the rule $w_{n}\rightarrow
w_{n}b_{\mathit{\boldsymbol{x}}^{\prime}}$ and set
$\delta\tau_{t}\rightarrow\delta\tau_{t}-\delta\tau^{\prime}$;
v)
Go back to step ii) until $\delta\tau_{t}=0$;
vi)
finally, perform branching according to the total accumulated weight factor
$w_{n}$.
This continuous-time algorithm implicitly implements the exact imaginary-time
modified Green’s function
$\tilde{G}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau)$.
Figure 4: (color online). Number of random walkers $N_{w}$ required to
determine, using the optimized Boltzmann-type wave function to guide
importance sampling in the PQMC simulation, the ground-state energy with a
relative error $e_{\mathrm{rel}}$, see Eq. (8), as a function of the system
size $N$. Different datasets correspond to different transverse field
intensities $\Gamma$. The (red) dotted and (blue) dot-dashed lines represent
exponential fitting functions, while the (green) dashed line represents a
power-law fit with power $b=0.54(5)$. Figure 5: (color online). Number of
random walkers $N_{w}$ required to determine, using the optimized uRBM Ansatz
to guide importance sampling in the PQMC simulation, the ground-state energy
with a relative error $e_{\mathrm{rel}}$, see Eq. (8), as a function of the
system size $N$. The number of single-spin Metropolis updates of the hidden
spins per CTGFMC hidden-spin update is $k=0.1N$. The (red) dotted line
represents and exponential fit, while the (blue) dot-dashed line represents a
linear fit.
In the long imaginary-time limit, the walkers sample spin configurations with
a probability distribution proportional to
$f(\mathit{\boldsymbol{x}},\tau\rightarrow\infty)=\Psi_{0}({\mathit{\boldsymbol{x}}})\psi_{T}({\mathit{\boldsymbol{x}}})$.
If $\psi_{T}({\mathit{\boldsymbol{x}}})$ is a good approximation of the
ground-state wave function, this distribution closely approximates the
quantum-mechanical probability of finding the system in the spin configuration
$\mathit{\boldsymbol{x}}$. It is important to notice that if our guiding wave
function was exact, i.e. if
$\psi_{T}({\mathit{\boldsymbol{x}}})=\Psi_{0}({\mathit{\boldsymbol{x}}})$,
then the local energy $E_{\mathrm{loc}}(\mathit{\boldsymbol{x}})$ would be a
constant function. This would completely suppress the fluctuations of the
number of walkers, therefore eliminating the bias due to the finite walkers
population $N_{w}$. If $\psi_{T}({\mathit{\boldsymbol{x}}})$ is, albeit not
exact, a good approximation of $\Psi_{0}({\mathit{\boldsymbol{x}}})$, the
fluctuations of the number of walkers are still reduced compared to the case
of the simple CTGFMC algorithm (which corresponds to setting
$\psi_{T}(\mathit{\boldsymbol{x}})=1$) giving a faster convergence to the
exact $N_{w}\rightarrow\infty$ limit. Below we consider the use of the
variational wave-functions
$\Psi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}})$ described in Sec. II as
guiding wave-functions for the PQMC algorithm, setting the variational
parameters $\boldsymbol{\kappa}$ at their optimal values.
In order to employ the unrestricted neural-network states as guiding
functions, the PQMC algorithm has to be modified. One has to implement a
combined dynamics of the visible-spin configurations $\mathit{\boldsymbol{x}}$
and of the hidden-spin configurations $\mathit{\boldsymbol{h}}$. We will
indicate the global configuration as
$\mathit{\boldsymbol{y}}=(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})$.
The goal is to sample global configurations with the (normalized) probability
distribution
$p(\mathit{\boldsymbol{y}})=\frac{\Psi_{0}(\mathit{\boldsymbol{x}})\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})}{\sum_{\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}}\Psi_{0}(\mathit{\boldsymbol{x}})\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})}\;.$
(13)
This allows one to compute the ground state energy as
$E=\lim_{N_{c}\rightarrow\infty}\sum_{i=1}^{N_{c}}E_{\mathrm{loc}}(\mathit{\boldsymbol{x}}_{i},\mathit{\boldsymbol{h}}_{i})/N_{c}$,
where $N_{c}$ is a number of uncorrelated configurations
$\left\\{\mathit{\boldsymbol{y}}_{i}\right\\}$ sampled from
$p(\mathit{\boldsymbol{y}})$. The local energy
$E_{\mathrm{loc}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})$ is defined
as in Eq. (6). A suitable algorithm was implemented in Ref. Vitiello and
Whitlock (1991) in the case of the continuous-space Green’s function Monte
Carlo algorithm, where importance sampling was implemented using shadow wave
functions. Here we modify the approach of Ref. Vitiello and Whitlock (1991) to
address quantum spin models. The visible-spins configurations
$\mathit{\boldsymbol{x}}$ are evolved according to the CTGFMC described above,
keeping the hidden-spin configuration $\mathit{\boldsymbol{h}}^{\prime}$
fixed. The modified imaginary-time Green’s function is now
$\tilde{G}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau|\mathit{\boldsymbol{h}}^{\prime})=G(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}^{\prime},\Delta\tau)\frac{\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}^{\prime})}{\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}}^{\prime},\mathit{\boldsymbol{h}}^{\prime})}$.
As discussed above, this has to be rewritten as the product of a stochastic
matrix, which defines how the visible-spin configurations updates are
selected, and a weight term, which is taken into account with the branching
process. The weight-update factor is
$b_{\mathit{\boldsymbol{y}}^{\prime}}=\sum_{\mathit{\boldsymbol{x}}}\tilde{G}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau|\mathit{\boldsymbol{h}}^{\prime})$.
The dynamics of the hidden-spins configurations is dictated by a (classical)
Markov chain Monte Carlo algorithm. Considering
$\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})$
as an unnormalized probability distribution allows one to write — for any
fixed visible-spin configuration $\mathit{\boldsymbol{x}}$ — the Master
equation:
$\displaystyle\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})=\sum_{\mathit{\boldsymbol{h}}^{\prime}}T(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{h}}^{\prime}|\mathit{\boldsymbol{x}})\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}^{\prime}),$
(14)
where
$T(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{h}}^{\prime}|\mathit{\boldsymbol{x}})$
is the transition matrix that defines the Markov process. Clearly, the
following condition must be fulfilled
$\sum_{\mathit{\boldsymbol{h}}}T(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{h}}^{\prime}|\mathit{\boldsymbol{x}})=1$,
for any $\mathit{\boldsymbol{x}}$.
Our choice is a single spin flip Metropolis algorithm, where the flip of a
randomly selected spin is proposed, and accepted with the probability
$A(\mathit{\boldsymbol{h}}^{\prime}\rightarrow\mathit{\boldsymbol{h}}|\mathit{\boldsymbol{x}})=\mathrm{Min}\left\\{1,\frac{\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}})}{\phi_{\boldsymbol{\kappa}}(\mathit{\boldsymbol{x}},\mathit{\boldsymbol{h}}^{\prime})}\right\\}.$
(15)
Here, $\mathit{\boldsymbol{h}}$ differs from
$\mathit{\boldsymbol{h}}^{\prime}$ only for the (randomly selected) flipped
spin. One could perform a certain number, call it $k$, of Metropolis updates,
without modifying the formalism. In fact, this turns out to be useful, as
discussed below. The combined dynamics of the visible and the hidden spins is
driven by the following equation:
$f(\mathit{\boldsymbol{y}},\tau+\Delta\tau)=\sum_{\mathit{\boldsymbol{y}}^{\prime}}G(\mathit{\boldsymbol{y}},\mathit{\boldsymbol{y}}^{\prime},\Delta\tau)f(\mathit{\boldsymbol{y}}^{\prime},\tau),$
(16)
with
$G(\mathit{\boldsymbol{y}},\mathit{\boldsymbol{y}}^{\prime},\Delta\tau)=T(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{h}}^{\prime}|\mathit{\boldsymbol{x}})\tilde{G}({\mathit{\boldsymbol{x}}},{\mathit{\boldsymbol{x}}}^{\prime},\Delta\tau|\mathit{\boldsymbol{h}}^{\prime})$.
It can be shown Vitiello and Whitlock (1991) that the equilibrium probability
distribution of this equation is the desired joint probability distribution
$p(\mathit{\boldsymbol{y}})$ in Eq. (13). The stochastic process corresponding
to this equation can be implemented with the following steps:
i)
perform the visible-spin configuration update
$\mathit{\boldsymbol{x}}^{\prime}\rightarrow\mathit{\boldsymbol{x}}$, keeping
$\mathit{\boldsymbol{h}}^{\prime}$ fixed, according to the CTGFMC algorithm
described above (including accumulation of the weight factor);
ii)
perform $k$ single-spin Metropolis updates of the hidden-spin configuration
$\mathit{\boldsymbol{h}}^{\prime}$, keeping $\mathit{\boldsymbol{x}}$ fixed;
iii)
perform branching of the global configuration.
It is easily shown that the hidden-spin dynamics does not directly affect the
weight factor since the normalization of the Green function of the combined
dynamics is set by $b_{\mathit{\boldsymbol{y}}^{\prime}}$.
Since the optimized uRBM describes the ground state wave function with high
accuracy, one expects that its use as guiding function leads to a drastic
reduction of the systematic errors due to the finite random walker population.
However, one should take into account that there might be statistical
correlations among subsequent hidden-spin configurations along the Markov
chain. This might in turn affect the systematic error. Clearly, increasing the
number of Metropolis steps $k$ per CTGFMC visible-spin configuration update
allows one to suppress such correlations, possibly reducing the systematic
error. This will indeed turn out to be important, in particular at the quantum
critical point where statistical correlations along the Markov chain are more
significant.
Figure 6: (color online). Number of random walkers $N_{w}$ required to
determine, using the optimized uRBM to guide importance sampling in the PQMC
simulation, the ground-state energy with a relative error $e_{\mathrm{rel}}$,
see Eq. (8), as a function of the system size $N$. The transverse field
intensity is set at the ferromagnetic quantum critical point $\Gamma=1$.
Different datasets correspond to different values of the the number of single-
spin Metropolis updates $k$. The (red) dotted line represents an exponential
fit, while the (black) dot-dashed line represents a power-law fit, with power
$b=0.55(1)$.
Following Ref. Inack _et al._ , 2018, we analyze the computational complexity
of the PQMC algorithm by determining the number of walkers $N_{w}$ needed to
determine the ground state energy of the Hamiltonian (1) with a prescribed
accuracy. All data described below have been obtained with a time step
$\Delta\tau=0.1$, and all simulations have been run for a long enough total
imaginary time to ensure equilibration.
First, we consider the simple PQMC algorithm i.e., performed without
importance sampling. Fig. 3 displays the scaling with the system size $N$ of
the number of walkers $N_{w}$ required to keep the relative error
$e_{\mathrm{rel}}$, defined in Eq. (8), at the chosen threshold. This scaling
is evidently exponential, below, above, and also at the quantum critical
point. The most severe scaling comes from the ordered phase and could be
attributed to the fact that the simple PQMC is formally equivalent to PQMC
with a constant $\psi_{T}(\mathit{\boldsymbol{x}})$ for importance sampling.
This turns out to be a very poor choice of the guiding function in the ordered
regime given that it treats all configurations on an equal footing. Analogous
results have been obtained in Ref. Inack _et al._ , 2018 using the diffusion
Monte Carlo algorithm. This is another PQMC method — in fact very similar to
the CTGFMC algorithm employed here — whose transition matrix is defined from
the imaginary time Green’s function derived within the symmetrized Trotter
decomposition. Introducing importance sampling using the optimized Boltzmann-
type Ansatz as guiding function significantly reduces the systematic error due
to the finite random walker population, allowing one to reach quite small
relative errors. In particular, in the paramagnetic phase at $\Gamma=1.4$, the
scaling of $N_{w}$ versus $N$ is quite flat (see Fig. 4); it appears to be
well described by the power-law $N_{w}\sim N^{b}$ with the small power
$b=0.54(5)$, rather than by an exponential. However, in the ferromagnetic
phase at $\Gamma=0.6$ and at the quantum critical point $\Gamma=1$ the scaling
is still clearly exponential. This means that the simple Boltzmann-type Ansatz
is, in general, insufficient to ameliorate the exponentially scaling
computational cost of the PQMC algorithm. Fig. 5 shows the scaling of $N_{w}$
obtained using the optimized uRBM Ansatz as the guiding function. The number
of hidden-spin Metropolis steps per visible-spin update is set to a (small)
fraction of the system size $N$, namely to $k=0.1N$. At $\Gamma=0.6$, the
required walker population size $N_{w}$ turns out to be essentially
independent on the system size $N$. It is worth noticing that the prescribed
relative error is here as small as $e_{\mathrm{rel}}=10^{-6}$, and that this
high accuracy is achieved with a rather small walkers population
$N_{w}\lesssim 1000$. However, at the quantum critical point, $N_{w}$ still
displays an exponential scaling with system size. This effect can be traced
back to the diverging statistical correlations among subsequent hidden-spin
configurations along the Markov chain, due to quantum criticality. As
anticipated above, these statistical correlations can be suppressed by
increasing the number of hidden-spin updates $k$. Fig. 6 displays the scaling
of $N_{w}$, at the quantum critical point, for different $k$ values. One
observes that the scaling substantially improves already for moderately larger
$k$ values, leading to a crossover from the exponential scaling obtained with
$k=0.1N$, to a square-root like scaling $N_{w}\sim N^{0.55(1)}$ when $k=10N$.
It is important to point out that increasing $k$ implies a correspondingly
increasing contribution to the global computational cost of the PQMC
algorithm. However, since $k$ is here linear in the system size, this
contribution does not modify, to leading order, the scaling of the global
computational cost. Therefore, one can conclude that the uRBM Ansatz is
sufficient to change the scaling of the computational cost of the PQMC
algorithm from exponential in the system size, to an amenable polynomial
scaling. In the simulations presented here, single-spin flip Metropolis
updates are employed for the hidden variables. It is possible that cluster
spin updates would lead to an even faster convergence to the exact
$N_{w}\rightarrow\infty$ limit, due to the more efficient sampling of the
hidden-spin configurations. However, such cluster updates cannot always be
implemented, in particular for frustrated disordered Hamiltonians relevant for
optimization problems; therefore, we do not consider them here.
## IV Conclusions
The accuracy of variational wave-functions that mimic unrestricted Boltzmann
machines, which we refer to as unrestricted neural network states, has been
analyzed using the one-dimensional ferromagnetic Ising model as a testbed. By
optimizing just three variational parameters, ground-state energies with a
relative error smaller than $10^{-3}$ have been obtained. The ferromagnetic
quantum phase transition turns out to be the point where the relative error is
the largest. This accuracy is comparable to the one previously obtained using
restricted neural network states with few hidden variables per visible spin
Carleo and Troyer (2017). These restricted neural network states involve a
number of variational parameters proportional to the system size, as opposed
to the unrestricted neural network states considered here, where the (small)
number of variational parameters is fixed. This feature of the unrestricted
states makes them very suitable in the context of quantum annealing
simulations for Ising-type models (which are sign-problem free). However,
since one has to integrate over hidden-spins configurations via Monte Carlo
sampling, as opposed to the case of the restricted neural network states
Carleo and Troyer (2017) — for which the hidden-spin configurations can be
integrated out — they represent a less promising approach to model ground-
states of Hamiltonian where the negative sign-problem occurs. Indeed, in such
case an accurate variational Ansatz might have to include also hidden-spins
configurations with negative wave-function amplitude, making Monte Carlo
integration via random sampling inapplicable.
The variational study summarized here represented a necessary preliminary step
to investigate the use of optimized unrestricted neural network states as
guiding functions for importance sampling in PQMC simulations. We have found
that unrestricted neural network states allow one to drastically reduce the
systematic bias of the PQMC algorithm originating from the finite size of the
random-walker population. Specifically, the scaling of the population size
required to keep a fixed relative error as the system size increases changes
from the exponential scaling characteristic of simple PQMC simulations
performed without guiding functions, to a polynomial scaling. This also
implies a corresponding change in the scaling of the computational cost. This
qualitative scaling change occurs above, below, and also at the ferromagnetic
quantum phase transition. Instead, a conventional variational Ansatz of the
Boltzmann type was found to provide a significant improvement of the
computational cost only above the critical point (in the paramagnetic phase),
but to provide only a marginal improvement at and below the transition. It is
worth emphasizing that the use of unrestricted neural network states as
guiding functions in PQMC simulations requires the sampling of both the
visible and the hidden spins, using the combined algorithm described in Sec.
III (more efficient variants might be possible). The role of the statistical
correlations among hidden-spin configurations shows up in particular at the
ferromagnetic quantum critical point. We found that these correlations can be
eliminated by performing several single-spin updates, still without affecting,
to leading order, the global computational complexity of the simulation.
In Ref. Bravyi and Gosset (2017) it was proven that it is possible to devise
polynomially-scaling numerical algorithms to determine the ground-state
energy, with a small additive error, of various ferromagnetic spin models,
including the ferromagnetic Ising chain considered here. However, practical
implementations have not been provided. The numerical data we have reported in
this manuscript indicate that the PQMC algorithm guided by an optimized
unrestricted neural network state represents a practical algorithm with
polynomial computational complexity for the ferromagnetic quantum Ising chain.
More in general, it was shown in Ref. Bravyi (2015) that the problem of
estimating the ground-state energy of a generic sign-problem free Hamiltonian
with a small additive error is at least NP-hard. Indeed, this task encompasses
hard optimization problems such as $k-$SAT and MAX-CUT. This suggest that
there might be relevant models where the unrestricted neural network states
discussed here are not sufficient to make the computational cost of the PQMC
simulations affordable. Relevant candidates are Ising spin-glass models with
frustrated couplings. Such systems might require more sophisticated guiding
functions obtained, e.g., including more hidden-spin layers in the
unrestricted neural network state, as discussed in Sec. II. In future work we
plan to search for models that make PQMC simulation problematic. We argue that
this will help us in understanding if and for which models a systematic
quantum speed-up in solving optimization problems using quantum annealing
devices, instead of PQMC simulations performed on classical computer, could be
achieved.
We acknowledge insightful discussions with Giuseppe Carleo, Rosario Fazio,
Guglielmo Mazzola, Francesco Pederiva, Sandro Sorella, and Matteo Wauters. S.
P. and L. D. acknowledge financial support from the BIRD2016 project
“Superfluid properties of Fermi gases in optical potentials” of the University
of Padova. GES acknowledges support by the EU FP7 under ERC-MODPHYSFRICT,
Grant Agreement No. 320796.
## References
* Inack _et al._ (2018) E. M. Inack, G. Giudici, T. Parolini, G. Santoro, and S. Pilati, Phys. Rev. A 97, 032307 (2018).
* Ceperley and Alder (1986) D. Ceperley and B. Alder, Science 231, 555 (1986).
* Hammond _et al._ (1994) B. L. Hammond, W. A. Lester, and P. J. Reynolds, _Monte Carlo methods in ab initio quantum chemistry_, Vol. 1 (World Scientific, 1994).
* Foulkes _et al._ (2001) W. Foulkes, L. Mitas, R. Needs, and G. Rajagopal, Rev. Mod. Phys. 73, 33 (2001).
* Carlson _et al._ (2015) J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. Schmidt, and R. B. Wiringa, Rev. Mod. Phys. 87, 1067 (2015).
* Finnila _et al._ (1994) A. B. Finnila, M. A. Gomez, C. Sebenik, C. Stenson, and J. D. Doll, Chem. Phys. Lett. 219, 343 (1994).
* Santoro _et al._ (2002) G. E. Santoro, R. Martoňák, E. Tosatti, and R. Car, Science 295, 2427 (2002).
* Boixo _et al._ (2014) S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Nat. Phys. 10, 218 (2014).
* Inack and Pilati (2015) E. M. Inack and S. Pilati, Phys. Rev. E 92, 053304 (2015).
* Heim _et al._ (2015) B. Heim, T. F. Rønnow, S. V. Isakov, and M. Troyer, Science 348, 215 (2015).
* Isakov _et al._ (2016) S. V. Isakov, G. Mazzola, V. N. Smelyanskiy, Z. Jiang, S. Boixo, H. Neven, and M. Troyer, Phys. Rev. Lett. 117, 180402 (2016).
* Jiang _et al._ (2017) Z. Jiang, V. N. Smelyanskiy, S. V. Isakov, S. Boixo, G. Mazzola, M. Troyer, and H. Neven, Phys. Rev. A 95, 012322 (2017).
* Mazzola _et al._ (2017) G. Mazzola, V. N. Smelyanskiy, and M. Troyer, Phys. Rev. B 96, 134305 (2017).
* Andriyash and Amin (2017) E. Andriyash and M. H. Amin, arXiv preprint arXiv:1703.09277 (2017).
* Carleo and Troyer (2017) G. Carleo and M. Troyer, Science 355, 602 (2017).
* Saito (2017) H. Saito, J. Phys. Soc. Jpn. 86, 093001 (2017).
* Saito and Kato (2018) H. Saito and M. Kato, J. Phys. Soc. Jpn. 87, 014001 (2018).
* Deng _et al._ (2017) D.-L. Deng, X. Li, and S. D. Sarma, Phys. Rev. X 7, 021021 (2017).
* Chen _et al._ (2018) J. Chen, S. Cheng, H. Xie, L. Wang, and T. Xiang, Phys. Rev. B 97, 085104 (2018).
* Glasser _et al._ (2018) I. Glasser, N. Pancotti, M. August, I. D. Rodriguez, and J. I. Cirac, Phys. Rev. X 8, 011006 (2018).
* Gao and Duan (2017) X. Gao and L.-M. Duan, Nat. Commun. 8, 662 (2017).
* Freitas _et al._ (2018) N. Freitas, G. Morigi, and V. Dunjko, arXiv preprint arXiv:1803.02118 (2018).
* Sorella _et al._ (2007) S. Sorella, M. Casula, and D. Rocca, J. Chem. Phys. 127, 014105 (2007).
* Vitiello _et al._ (1988) S. Vitiello, K. Runge, and M. H. Kalos, Phys. Rev. Lett. 60, 1970 (1988).
* Reatto and Masserini (1988) L. Reatto and G. Masserini, Phys. Rev. B 38, 4516 (1988).
* Carleo _et al._ (2018) G. Carleo, Y. Nomura, and M. Imada, arXiv preprint arXiv:1802.09558 (2018).
* Becca and Sorella (2017) F. Becca and S. Sorella, _Quantum Monte Carlo Approaches for Correlated Systems_ (Cambridge University Press, 2017).
* Anderson (1975) J. B. Anderson, J. Chem. Phys. 63, 1499 (1975).
* Kalos and Whitlock (2008) M. H. Kalos and P. A. Whitlock, _Monte carlo methods_ (John Wiley & Sons, 2008).
* Thijssen (2007) J. Thijssen, _Computational physics_ (Cambridge University Press, 2007).
* Nemec (2010) N. Nemec, Phys. Rev. B 81, 035119 (2010).
* Boninsegni and Moroni (2012) M. Boninsegni and S. Moroni, Phys. Rev. E 86, 056712 (2012).
* Pollet _et al._ (2018) L. Pollet, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. B 98, 085102 (2018).
* Schmidt _et al._ (2005) K. E. Schmidt, P. Niyaz, A. Vaught, and M. A. Lee, Phys. Rev. E 71, 016707 (2005).
* Sorella and Capriotti (2000) S. Sorella and L. Capriotti, Phys. Rev. B 61, 2599 (2000).
* Vitiello and Whitlock (1991) S. A. Vitiello and P. A. Whitlock, Phys. Rev. B 44, 7373 (1991).
* Bravyi and Gosset (2017) S. Bravyi and D. Gosset, Phys. Rev. Lett. 119, 100503 (2017).
* Bravyi (2015) S. Bravyi, Quantum Inf. Comput. 15, 1122 (2015).
|
# Exponentially convergent trapezoidal rules to approximate fractional powers
of operators
Lidia Aceto Lidia Aceto
Università del Piemonte Orientale
Dipartimento di Scienze e Innovazione Tecnologica, viale Teresa Michel 11,
15121 Alessandria
Italy<EMAIL_ADDRESS>and Paolo Novati Paolo Novati
Università di Trieste
Dipartimento di Matematica e Geoscienze, via Valerio 12/1, 34127 Trieste
Italy<EMAIL_ADDRESS>
###### Abstract.
In this paper we are interested in the approximation of fractional powers of
self-adjoint positive operators. Starting from the integral representation of
the operators, we apply the trapezoidal rule combined with a single-
exponential and a double-exponential transform of the integrand function. For
the first approach our aim is only to review some theoretical aspects in order
to refine the choice of the parameters that allow a faster convergence. As for
the double exponential transform, in this work we show how to improve the
existing error estimates for the scalar case and also extend the analysis to
operators. We report some numerical experiments to show the reliability of the
estimates obtained.
###### Key words and phrases:
Matrix functions, Single-exponential transform, Double-exponential transform,
Trapezoidal rule, Fractional Laplacian
This work was partially supported by GNCS-INdAM and FRA-University of Trieste.
The authors are members of the INdAM research group GNCS
## 1\. Introduction
In this work we are interested in the numerical approximation of
${\mathcal{L}}^{-\alpha}$, $\alpha\in(0,1).$ Here ${\mathcal{L}}$ is a self-
adjoint positive operator acting in an Hilbert space ${\mathcal{H}}$ in which
the eigenfunctions of ${\mathcal{L}}$ form an orthonormal basis of
${\mathcal{H}},$ so that ${\mathcal{L}}^{-\alpha}$ can be written through the
spectral decomposition of ${\mathcal{L}}$. In other words, for a given
$g\in{\mathcal{H}}$, we have
(1) ${\mathcal{L}}^{-\alpha}g=\sum_{j=1}^{+\infty}\mu_{j}^{-\alpha}\langle
g,\varphi_{j}\rangle\varphi_{j}$
where $\mu_{j}$ and $\varphi_{j}$ are the eigenvalues and the eigenfunctions
of ${\mathcal{L}},$ respectively, and $\langle\cdot,\cdot\rangle$ denotes the
${\mathcal{H}}$-inner product. Throughout the paper we also assume
$\sigma(\mathcal{L})\subseteq[1,+\infty)$, where $\sigma(\mathcal{L})$ denotes
the spectrum of ${\mathcal{L}}$.
Applications of (1) include the numerical solution of fractional equations
involving the anomalous diffusion, in which ${\mathcal{L}}$ is related to the
Laplacian operator, and this is the main reason for which in recent years a
lot of attention has been placed on the efficient approximation of fractional
powers. Among the approaches recently introduced we quote here the methods
based on the best uniform rational approximations of functions closely related
to $\lambda^{-\alpha}$ that have been studied in [6, 7, 8, 9]. Another class
of methods relies on quadrature rules arising from the Dunford-Taylor integral
representation of $\lambda^{-\alpha}$ [1, 2, 3, 4, 5, 17, 18]. Very recently,
time stepping methods for a parabolic reformulation of fractional diffusion
equations, proposed in [19], have been interpreted by Hofreither in [10] as
rational approximations of $\lambda^{-\alpha}.$
In this work, starting from the integral representation
(2)
$\mathcal{L}^{-\alpha}=\frac{2\sin(\alpha\pi)}{\pi}\int_{0}^{+\infty}t^{2\alpha-1}({\mathcal{I}}+t^{2}\mathcal{L})^{-1}dt,\qquad\alpha\in(0,1),$
where ${\mathcal{I}}$ is the identity operator in ${\mathcal{H}}$, we consider
the trapezoidal rule applied to the single and the double-exponential
transform of the integrand function. The former approach has been extensively
studied in [5], where the authors also provide reliable error estimates. The
rate of convergence has been shown to be of type
(3) $\exp(-c\sqrt{n}),$
where $n$ is closely related to the number of nodes. Our aim here is just to
review some theoretical aspects in order to refine the choice of the
parameters that allow faster convergence, even if, still of type (3). As for
the double-exponential transform, widely investigated in [12, 13, 14, 15, 16]
for general scalar functions, in this work we show how to improve the existing
error estimates for the function $\lambda^{-\alpha}$. We also extend the
analysis to operators, showing that it is possible to reach a convergence rate
of type
$\exp\left(-c\sqrt{\frac{n}{\ln n}}\right).$
While theoretically disadvantageous with respect to the single-exponential
approach, we show that the double-exponential approach is actually faster at
least for $\alpha\in[1/2,1)$.
The paper is organized as follows. In Section 2 we make a short background
concerning the trapezoidal rule with particular attention to functions that
decay exponentially at infinity. In Section 3 we review the existing
convergence analysis for the trapezoidal rule combined with a single-
exponential transform and we refine the choice of the parameters that allow a
faster convergence. Section 4 is devoted to the trapezoidal rule combined with
a double-exponential transform. Here the convergence analysis is derived for
the approximation of the scalar function $\lambda^{-\alpha}$ and is then
extended to the case of the operator ${\mathcal{L}}^{-\alpha}.$ Some
concluding remarks are finally reported in Section 5.
## 2\. A general convergence result for the trapezoidal rule
Given a generic continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$, in
this section we make a short background concerning the trapezoidal
approximation
$I(f)=\int_{-\infty}^{+\infty}f(x)dx\approx
h\sum_{\ell=-\infty}^{+\infty}f(\ell h),$
where $h$ is a suitable positive value. Given $M$ and $N$ positive integers,
we denote the truncated trapezoidal rule by
$T_{M,N,h}(f)=h\sum_{\ell=-M}^{N}f(\ell h).$
For the error we have
$\mathcal{E}_{M,N,h}(f):=\left|I(f)-T_{M,N,h}(f)\right|\leq\mathcal{E}_{D}+\mathcal{E}_{T_{L}}+\mathcal{E}_{T_{R}},$
where
$\displaystyle\mathcal{E}_{D}$
$\displaystyle=\left|\int_{-\infty}^{+\infty}f(x)dx-h\sum_{\ell=-\infty}^{+\infty}f(\ell
h)\right|,$ $\displaystyle\mathcal{E}_{T_{L}}$
$\displaystyle=h\sum_{\ell=-\infty}^{-M-1}\left|f(\ell
h)\right|,\qquad\mathcal{E}_{T_{R}}=h\sum_{\ell=N+1}^{+\infty}\left|f(\ell
h)\right|.$
The quantities $\mathcal{E}_{D}$ and
$\mathcal{E}_{T}:=\mathcal{E}_{T_{L}}+\mathcal{E}_{T_{R}}$ are referred to as
the discretization error and the truncation error, respectively.
###### Definition 1.
[11, Definition 2.12] For $d>0,$ let $\mathcal{D}_{d}$ be the infinite strip
domain of width $2d$ given by
$\mathcal{D}_{d}=\\{\zeta\in\mathbb{C}:\,|\operatorname{Im}(\zeta)|<d\\}.$
Let $B(\mathcal{D}_{d})$ be the set of functions analytic in $\mathcal{D}_{d}$
that satisfy
$\int_{-d}^{d}|f(x+i\eta)|d\eta=\mathcal{O}(|x|^{a}),\quad
x\rightarrow\pm\infty,\quad 0\leq a<1,$
and
$\mathcal{N}(f,d)=\lim_{\eta\rightarrow
d^{-}}\left\\{\int_{-\infty}^{+\infty}|f(x+i\eta)|dx+\int_{-\infty}^{+\infty}|f(x-i\eta)|dx\right\\}<+\infty.$
The next theorem gives an estimate for the discretization error of the
trapezoidal rule when applied to functions in $B(\mathcal{D}_{d})$.
###### Theorem 1.
[11, Theorem 2.20] Assume $f\in B(\mathcal{D}_{d})$. Then
(4) $\mathcal{E}_{D}\leq\frac{\mathcal{N}(f,d)}{2\sinh(\pi d/h)}e^{-\pi d/h}.$
###### Theorem 2.
Assume $f\in B(\mathcal{D}_{d})$ and that there are positive constants
$\beta,\gamma$ and $C$ such that
(5) $|f(x)|\leq C\left\\{\begin{array}[]{cc}\exp(-\beta|x|),&x<0,\\\
\exp(-\gamma|x|),&x\geq 0.\end{array}\right.$
Then,
(6) $\mathcal{E}_{M,N,h}(f)\leq\frac{N(f,d)}{2\sinh(\pi d/h)}e^{-\pi
d/h}+\frac{C}{\beta}e^{-\beta Mh}+\frac{C}{\gamma}e^{-\gamma Nh}.$
###### Proof.
By (5) we immediately have
$\mathcal{E}_{T_{L}}\leq\frac{C}{\beta}e^{-\beta
Mh},\quad\mathcal{E}_{T_{R}}\leq\frac{C}{\gamma}e^{-\gamma Nh}.$
Using Theorem 1 we obtain (6). ∎
The above result states that for functions that decay exponentially for
$x\rightarrow\pm\infty$ it may be possible to have exponential convergence
after a proper selection of $h$. When working with the more general situation
(7) $I(g):=\int_{a}^{b}g(t)dt,$
one can consider a suitable conformal map
$\psi:(-\infty,+\infty)\rightarrow(a,b),$
and, through the change of variable $t=\psi(x)$, transform (7) to
$I(g_{\psi}):=\int_{-\infty}^{+\infty}g_{\psi}(x)dx,\qquad
g_{\psi}(x)=g(\psi(x))\psi^{\prime}(x).$
A suitable choice of the mapping $\psi$ may allow to work with a function
$g_{\psi}$ that fulfills the hypothesis of Theorem 2 so that $I(g)$ can be
evaluated with an error that decays exponentially.
Since the aim of the paper is the computation of $\mathcal{L}^{-\alpha}$ with
$\sigma(\mathcal{L})\subseteq[1,+\infty),$ for $\lambda\geq 1$ we consider now
the integral representation (2)
(8)
$\lambda^{-\alpha}=\frac{2\sin(\alpha\pi)}{\pi}\int_{0}^{+\infty}t^{2\alpha-1}(1+t^{2}\lambda)^{-1}dt,\qquad\alpha\in(0,1).$
Defining
(9) $g_{\lambda}(t):=t^{2\alpha-1}(1+t^{2}\lambda)^{-1},$
and a change of variable $t=\psi(x)$,
$\psi:(-\infty,+\infty)\rightarrow(0,+\infty)$, let
(10) $g_{\lambda,\psi}(x)=g_{\lambda}(\psi(x))\psi^{\prime}(x).$
Let moreover
$\mathcal{Q}_{M,N,h}^{\alpha}(g_{\lambda,\psi})=\frac{2\sin(\alpha\pi)}{\pi}h\sum_{\ell=-M}^{N}g_{\lambda,\psi}(\ell
h)$
be the truncated trapezoidal rule for the computation of $\lambda^{-\alpha}$,
that is, for the computation of
$\frac{2\sin(\alpha\pi)}{\pi}\int_{0}^{+\infty}g_{\lambda}(t)dt=\frac{2\sin(\alpha\pi)}{\pi}\int_{-\infty}^{+\infty}g_{\lambda,\psi}(x)dx.$
We denote the error by
$\displaystyle E_{M,N,h}(\lambda)$
$\displaystyle=\left|\lambda^{-\alpha}-\mathcal{Q}_{M,N,h}^{\alpha}(g_{\lambda,\psi})\right|$
(11)
$\displaystyle=\frac{2\sin(\alpha\pi)}{\pi}\mathcal{E}_{M,N,h}(g_{\lambda,\psi}),$
and for operator argument
(12)
$E_{M,N,h}(\mathcal{L})=\left\|{\mathcal{L}}^{-\alpha}-\mathcal{Q}_{M,N,h}^{\alpha}(g_{{\mathcal{L}},\psi})\right\|_{\mathcal{H}\rightarrow\mathcal{H}}.$
The remainder of the paper is devoted to the analysis of two special choices
for $\psi$, the single-exponential (SE) and the double-exponential (DE)
transforms.
## 3\. The Single-Exponential transform
The SE transform is defined by
(13) $\psi_{SE}(x)=\exp(x),$
so that from (9) and (10) we get
$g_{\lambda,\psi_{SE}}(x)=e^{2\alpha x}(1+e^{2x}\lambda)^{-1}.$
Since the poles of this function are given by
$x_{k}=-\frac{1}{2}\log\lambda-i(2k+1)\frac{\pi}{2},\qquad k\in\mathbb{Z},$
we have that $g_{\lambda,\psi_{SE}}$ is analytic in
$\mathcal{D}_{\psi_{SE}}=\left\\{\zeta\in\mathbb{C}:\,|\operatorname{Im}(\zeta)|<\frac{\pi}{2}\right\\},$
that is, the strip domain with $d=\pi/2,$ independently of $\alpha$ and
$\lambda.$ Now, in order to prove that $g_{\lambda,\psi_{SE}}$ belongs to
$B(\mathcal{D}_{\psi_{SE}})$ (see Definition 1), following the analysis given
in [5] we first note that for $\eta\in\mathbb{R},$ $|\eta|<\pi/2$ and
$\lambda\geq 1,$
$\left|(1+e^{2(x+i\eta)}\lambda)^{-1}\right|\leq\left\\{\begin{array}[]{cl}1,&x<0,\\\
e^{-2x},&x\geq 0.\end{array}\right.$
Therefore,
(14)
$\left|g_{\lambda,\psi_{SE}}(x+i\eta)\right|\leq\left\\{\begin{array}[]{ll}e^{2\alpha
x},&\mbox{for }x<0,\\\ e^{-2(1-\alpha)x},&\mbox{for }x\geq
0.\end{array}\right.$
This implies that
$\displaystyle\mathcal{N}\left(g_{\lambda,\psi_{SE}},\pi/2\right)$
$\displaystyle=\lim_{\eta\rightarrow(\pi/2)^{-}}\left\\{\int_{-\infty}^{+\infty}\left|g_{\lambda,\psi_{SE}}(x+i\eta)\right|dx+\int_{-\infty}^{+\infty}\left|g_{\lambda,\psi_{SE}}(x-i\eta)\right|dx\right\\}$
$\displaystyle\leq\frac{1}{\alpha(1-\alpha)},$
and also that
$\int_{-\pi/2}^{\pi/2}|g_{\lambda,\psi_{SE}}(x+i\eta)|d\eta=\mathcal{O}(1),\quad
x\rightarrow+\infty.$
Therefore, by (1) we can conclude that $g_{\lambda,\psi_{SE}}$ belongs to
$B(\mathcal{D}_{\psi_{SE}}).$ By (4), for the discretization error we have
$\left|\int_{-\infty}^{+\infty}g_{\lambda,\psi_{SE}}(x)dx-h\sum_{\ell=-\infty}^{+\infty}g_{\lambda,\psi_{SE}}(\ell
h)\right|\leq\frac{1}{\alpha(1-\alpha)}\frac{e^{-\pi d/h}}{2\sinh(\pi
d/h)},\quad d=\pi/2.$
Since
$\frac{e^{-t}}{2\sinh(t)}=e^{-2t}\left(1+\mathcal{O}(e^{-2t})\right)\quad\text{as
}t\rightarrow+\infty,$
by (2), for $h\leq 2\pi d$ we obtain
$\displaystyle\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{SE}})$
$\displaystyle=\left|\int_{-\infty}^{+\infty}g_{\lambda,\psi_{SE}}(x)dx-h\sum_{\ell=-M}^{N}g_{\lambda,\psi_{SE}}(\ell
h)\right|$ (15) $\displaystyle\leq\frac{1}{\alpha(1-\alpha)}e^{-2\pi
d/h}+\frac{1}{2\alpha}e^{-2\alpha
Mh}+\frac{1}{2(1-\alpha)}e^{-2(1-\alpha)Nh}.$
After choosing $h$, the contribute of the three exponentials can be equalized
by taking $M$ and $N$ such that
$\pi d/h\approx\alpha Mh\approx(1-\alpha)Nh,$
that is
$M=\left\lceil\frac{\pi d}{\alpha h^{2}}\right\rceil,\quad
N=\left\lceil\frac{\pi d}{(1-\alpha)h^{2}}\right\rceil,$
where $\left\lceil\cdot\right\rceil$ is the ceiling function. Denoting by
$n=M+N+1$ the total number of inversions we have that
$n\approx\frac{\pi d}{h^{2}}\frac{1}{\alpha(1-\alpha)},$
and therefore by (15) we obtain
$\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{SE}})\leq\frac{3}{2}\frac{1}{\alpha(1-\alpha)}\exp\left(-2\sqrt{\pi
d\alpha(1-\alpha)}\sqrt{n}\right).$
By (12), since $\mathcal{L}$ is assumed to be self-adjoint and
$\sigma(\mathcal{L})\subseteq[1,+\infty)$, we have that
$E_{M,N,h}(\mathcal{L})\leq\frac{2\sin(\alpha\pi)}{\pi}\max_{\lambda\geq
1}\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{SE}}),$
and then, finally, taking $d=\pi/2$ we obtain
(16)
$E_{M,N,h}(\mathcal{L})\leq\frac{\sin(\alpha\pi)}{\pi}\frac{3}{\alpha(1-\alpha)}\exp\left(-\pi\sqrt{2\alpha(1-\alpha)}\sqrt{n}\right),\quad
n=M+N+1.$
The analysis just presented is almost identical to the one given in [5].
Nevertheless in that paper the authors define $d=\pi/4$, while here we have
shown that one can take $d=\pi/2$. This choice produces a remarkable speedup,
as shown in Figure 1, where for different values of $\alpha$ we have
considered the error in the computation of $\mathcal{L}^{-\alpha}$ for the
artificial operator
(17)
$\mathcal{L}=\left[\operatorname{diag}(1,2,\dots,100)\right]^{8},\quad\sigma(\mathcal{L})\subseteq[1,10^{16}]\text{.}$
Figure 1. Error for the trapezoidal rule applied with the single-exponential
transform with $d=\pi/4$ and $d=\pi/2$, and error estimate given by (16).
## 4\. Double-Exponential transformation
The DE transform we use here is given by
(18) $\psi_{DE}(x)=\exp\left(\frac{\pi}{2}\sinh(x)\right).$
We consider in (8) the change of variable
$\tau t^{2}=\left(\psi_{DE}(x)\right)^{2}=\exp(\pi\sinh(x)),\qquad\tau>0.$
The function involved in this case is
$\displaystyle g_{\lambda,\psi_{DE}}(x)$
$\displaystyle=\frac{\pi}{2}\tau^{1-\alpha}\frac{\exp(\alpha\pi\sinh(x))}{\tau+\lambda\exp(\pi\sinh(x))}\cosh(x)$
(19)
$\displaystyle=\frac{\pi}{2}\lambda^{-\alpha}\frac{\left(\lambda/\tau\exp(\pi\sinh(x))\right)^{\alpha}}{1+\lambda/\tau\exp(\pi\sinh(x))}\cosh(x),$
and we employ the trapezoidal rule to compute
$\lambda^{-\alpha}=\frac{2\sin(\alpha\pi)}{\pi}\int_{-\infty}^{+\infty}g_{\lambda,\psi_{DE}}(x)dx.$
The parameter $\tau$ needs to be selected in some way and the analysis is
provided in Section 4.5. Its introduction is motivated by the fact that, when
moving from $\lambda$ to $\mathcal{L}$, the method (the choice of $M$, $N$ and
$h$) and the error estimates have to be derived by working uniformly in the
interval $[1,+\infty)$ containing $\sigma(\mathcal{L})$. As in the SE case,
the function $g_{\lambda,\psi_{DE}}(x)$ exhibits a fast decay for
$x\rightarrow\pm\infty$, but the definition of the strip of analyticity is now
much more difficult to handle since everything now depends on $\lambda$ and
$\tau$.
### 4.1. Asymptotic behavior of the integrand function
In order to apply Theorem 2 we need to study
$|g_{\lambda,\psi_{DE}}(x+i\eta)|$. From (19) we have
$\left|g_{\lambda,\psi_{DE}}(x+i\eta)\right|=\frac{\pi}{2}\lambda^{-\alpha}\left|\frac{\left(\lambda/\tau\exp(\pi\sinh(x+i\eta))\right)^{\alpha}}{1+\lambda/\tau\exp(\pi\sinh(x+i\eta))}\right||\cosh(x+i\eta)|.$
After simple manipulations based on standard relations we find
$\left|\cosh(x+i\eta)\right|=\sqrt{\cosh^{2}x-\sin^{2}\eta},$
and therefore
$\left|\cosh(x+i\eta)\right|\leq\cosh x.$
Moreover
$\left|\left(\lambda/\tau\exp(\pi\sinh(x+i\eta))\right)^{\alpha}\right|=\left(\frac{\lambda}{\tau}\right)^{\alpha}\left|\exp(\alpha\pi\sinh
x\cos\eta))\right|.$
In addition, we can bound the denominator using the results given in [13, p.
388], that is,
$\left|\frac{1}{1+\lambda/\tau\exp(\pi\sinh(x+i\eta))}\right|\leq\frac{1}{1+\lambda/\tau\exp(\pi\sinh
x\cos\eta)\cos(\pi/2\sin\eta)}.$
From the above relations we find
$\left|g_{\lambda,\psi_{DE}}(x+i\eta)\right|\leq\frac{\pi}{2}\lambda^{-\alpha}\frac{\cosh
x}{\cos(\pi/2\sin\eta)}G_{\alpha}(x,\eta),$
where
$G_{\alpha}(x,\eta)=\frac{\left(\lambda/\tau\exp(\pi\sinh
x\cos\eta)\right)^{\alpha}}{1+\lambda/\tau\exp(\pi\sinh x\cos\eta)}.$
Let $x^{\ast}$ be such that
$\pi\sinh x^{\ast}\cos\eta=\ln(\tau/\lambda);$
we have
(20)
$G_{\alpha}(x,\eta)\leq\left\\{\begin{array}[]{ll}(\lambda/\tau)^{\alpha}\exp(\alpha\pi\cos\eta\sinh
x),&x\leq x^{\ast},\\\
(\lambda/\tau)^{\alpha-1}\exp(-(1-\alpha)\pi\cos\eta\sinh
x),&x>x^{\ast}.\end{array}\right.$
### 4.2. Error estimate for the scalar case
The bound (20) implies that
$\displaystyle\mathcal{N}\left(g_{\lambda,\psi_{DE}},d\right)$
$\displaystyle=\lim_{\eta\rightarrow
d^{-}}\left\\{\int_{-\infty}^{+\infty}\left|g_{\lambda,\psi_{DE}}(x+i\eta)\right|dx+\int_{-\infty}^{+\infty}\left|g_{\lambda,\psi_{DE}}(x-i\eta)\right|dx\right\\}$
$\displaystyle\leq\lim_{\eta\rightarrow
d^{-}}\pi\lambda^{-\alpha}\left\\{\frac{1}{\cos(\pi/2\sin\eta)}\int_{-\infty}^{+\infty}G_{\alpha}(x,\eta)\cosh
xdx\right\\}$ $\displaystyle\leq\lim_{\eta\rightarrow
d^{-}}\frac{\pi\lambda^{-\alpha}}{\cos(\pi/2\sin\eta)}\left\\{(\lambda/\tau)^{\alpha}\int_{-\infty}^{x^{\ast}}\exp(\alpha\pi\cos\eta\sinh
x)\cosh x\,dx\right.+$
$\displaystyle+\left.(\lambda/\tau)^{\alpha-1}\int_{x^{\ast}}^{+\infty}\exp(-(1-\alpha)\pi\cos\eta\sinh
x)\cosh x\,dx\right\\}$
$\displaystyle\leq\frac{1}{\alpha(1-\alpha)}\frac{2}{\cos d\cos(\pi/2\sin
d)}\lambda^{-\alpha}.$
In addition, assuming $d=d(\lambda,\tau)<\pi/2$, it can be observed that
$\displaystyle\int_{-d(\lambda,\tau)}^{d(\lambda,\tau)}|g_{\lambda,\psi_{DE}}(x+i\eta)|d\eta$
$\displaystyle\leq\frac{\pi}{2}\lambda^{-\alpha}\int_{-d(\lambda,\tau)}^{d(\lambda,\tau)}\frac{G_{\alpha}(x,\eta)\cosh
x}{\cos(\pi/2\sin\eta)}d\eta$ $\displaystyle=\mathcal{O}(1)\quad\text{for
}x\rightarrow\pm\infty.$
Using (1), for the discretization error we have
$\left|\int_{-\infty}^{+\infty}g_{\lambda,\psi_{DE}}(x)dx-h\sum_{\ell=-\infty}^{+\infty}g_{\lambda,\psi_{DE}}(\ell
h)\right|\leq\xi(d)\frac{1}{\alpha(1-\alpha)}\lambda^{-\alpha}\frac{e^{-\pi
d/h}}{2\sinh(\pi d/h)},$
where
(22) $\xi(d)=\frac{2}{{\cos d\cos(\pi/2\sin d)}}.$
The remaining task is to estimate the truncation error. Using (20) we obtain
$\displaystyle
h\sum\nolimits_{\ell=-\infty}^{-M-1}\left|g_{\lambda,\psi_{DE}}(\ell
h)\right|$
$\displaystyle\leq\frac{\pi}{2}\tau^{-\alpha}h\sum\nolimits_{\ell=-\infty}^{-M-1}\exp(\alpha\pi\sinh(\ell
h))\cosh(\ell h)$
$\displaystyle\leq\frac{\pi}{2}\tau^{-\alpha}\int_{-\infty}^{-Mh}\exp(\alpha\pi\sinh
x)\cosh(x)dx$
$\displaystyle\leq\frac{\tau^{-\alpha}}{2\alpha}\exp\left(-\alpha\pi\sinh(Mh)\right)$
$\displaystyle\leq\frac{\tau^{-\alpha}}{2\alpha}\exp\left(\frac{\alpha\pi}{2}\right)\exp\left(-\frac{\alpha\pi}{2}\exp(Mh)\right).$
Similarly,
$\displaystyle
h\sum\nolimits_{\ell=N+1}^{+\infty}\left|g_{\lambda,\psi_{DE}}(\ell h)\right|$
$\displaystyle\leq\frac{\pi}{2}\lambda^{-1}\tau^{1-\alpha}h\sum\nolimits_{\ell=N+1}^{+\infty}\exp(-(1-\alpha)\pi\sinh(\ell
h))\cosh(\ell h)$
$\displaystyle\leq\frac{\pi}{2}\lambda^{-1}\tau^{1-\alpha}h\int_{Nh}^{+\infty}\exp(-(1-\alpha)\pi\sinh
x)\cosh(x)dx$
$\displaystyle\leq\frac{\lambda^{-1}\tau^{1-\alpha}}{2(1-\alpha)}\exp\left(\frac{(1-\alpha)\pi}{2}\right)\exp\left(-\frac{(1-\alpha)\pi}{2}\exp(Nh)\right).$
The above results are summarized as follows.
###### Proposition 1.
Using the double-exponential transform, for the quadrature error it holds
(23) $\displaystyle\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{DE}})$
$\displaystyle\leq\frac{1}{\alpha(1-\alpha)}\xi(d)\lambda^{-\alpha}\frac{e^{-\pi
d/h}}{2\sinh(\pi d/h)}+$ (24)
$\displaystyle\frac{\tau^{-\alpha}}{2\alpha}\exp\left(\frac{\alpha\pi}{2}\right)\exp\left(-\frac{\alpha\pi}{2}\exp(Mh)\right)+$
(25)
$\displaystyle\frac{\lambda^{-1}\tau^{1-\alpha}}{2(1-\alpha)}\exp\left(\frac{(1-\alpha)\pi}{2}\right)\exp\left(-\frac{(1-\alpha)\pi}{2}\exp(Nh)\right),$
where $\xi(d)$ is defined by (22).
Defining
(26)
$h=\ln\left(\frac{4dn}{\mu}\right)\frac{1}{n},\quad\text{for\quad}n\geq\frac{\mu
e}{4d},\quad\mu=\min(\alpha,1-\alpha)$
as in [13, Theorem 2.14], we first observe that (see (23))
(27) $\frac{\exp\left(-\frac{\pi d}{h}\right)}{2\sinh\left(\frac{\pi
d}{h}\right)}\leq\frac{1}{1-e^{-\frac{\pi}{2}\mu e}}\exp\left(\frac{-2\pi
dn}{\ln\left(\frac{4dn}{\mu}\right)}\right).$
Setting $M=N=n$, the choice of $h$ as in (26) leads to a truncation error that
decays faster than the discretization one, because for an arbitrary constant
$c$ (see (24)-(25))
$\exp\left(-c\exp\left(nh\right)\right)=\exp\left(-\frac{4cdn}{\mu}\right).$
As consequence the idea is to assume the discretization error as estimator for
the global quadrature error, that is, using (23) and (27),
(28)
$\mathcal{E}_{n,h}(g_{\lambda,\psi_{DE}})=\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{DE}})\approx
K_{\alpha}\xi(d)\lambda^{-\alpha}\exp\left(\frac{-2\pi
dn}{\ln\left(\frac{4dn}{\mu}\right)}\right),$
where
(29) $K_{\alpha}=\frac{1}{\alpha(1-\alpha)}\frac{1}{1-e^{-\frac{\pi}{2}\mu
e}}.$
Formula (28) is very similar to the one given in [13, Theorem 2.14], that
reads
(30)
$\mathcal{\hat{E}}_{M,N,h}(g_{\lambda,\psi_{DE}})\approx\frac{\tau^{-\alpha}}{\mu}\alpha(1-\alpha)\left(K_{\alpha}\xi(d)+e^{\frac{\pi}{2}\nu}\right)\exp\left(\frac{-2\pi
dn}{\ln\left(\frac{4dn}{\mu}\right)}\right)$
where $\nu=\max\left(\alpha,1-\alpha\right)$ and $M=n$, $N=n-\chi$ (or
viceversa depending on $\alpha$), where $\chi>0$ is defined in order to
equalize the contribute of the truncation errors [13, Theorem 2.11]. The
important difference is given by the factor $\lambda^{-\alpha}$ that replaces
$\tau^{-\alpha}$, and this is crucial to correctly handle the case of
$\lambda\rightarrow+\infty$. In this situation the error of the trapezoidal
rule goes 0 because $g_{\lambda,\psi_{DE}}(x)\rightarrow 0$ as
$\lambda\rightarrow+\infty$ (see (19)). Anyway, as we shall see, $d\rightarrow
0$ as $\lambda\rightarrow+\infty$, so that the exponential term itself is not
able to reproduce this situation. An example is given in Figure 2 in which we
consider $\lambda=10^{12}$ and $\tau=100.$
Figure 2. Error for the trapezoidal rule applied with the double-exponential
transform (error DE), estimates (28) and (30) vs the number of inversions, for
the computation of $\lambda^{-\alpha}$ with $\lambda=10^{12}$ and $\tau=100$.
### 4.3. The poles of the integrand function
All the analysis presented so far is based on the assumption that the
integrand function
$g_{\lambda,\psi_{DE}}(x)=\frac{\pi}{2}\tau^{1-\alpha}\frac{\exp\left(\alpha\pi\sinh
x\right)}{\tau+\lambda\exp\left(\pi\sinh x\right)}\cosh x$
is analytic in the strip $\mathcal{D}_{d}$, for a certain $d=d(\lambda,\tau)$.
Therefore we have to study the poles of this function, that is, we have to
study the equation
$\tau+\lambda\exp\left(\pi\sinh x\right)=0.$
We have
$\displaystyle\exp\left(\pi\sinh x\right)$
$\displaystyle=\frac{\tau}{\lambda}e^{i\pi},$ $\displaystyle\sinh x$
$\displaystyle=\frac{1}{\pi}\ln\frac{\tau}{\lambda}+i(2k+1),\quad
k\in\mathbb{Z}.$
By solving the above equation for each $k$, we obtain the complete set of
poles. Assuming to work with the principal value of the logarithm and taking
$k=0$, we obtain the poles closest to the real axis $x_{0}$ and its conjugate
$\overline{x_{0}}$, where
$\displaystyle x_{0}$
$\displaystyle=\sinh^{-1}\left(\frac{1}{\pi}\ln\frac{\tau}{\lambda}+i\right)$
(31)
$\displaystyle=\ln\left(\frac{1}{\pi}\ln\frac{\tau}{\lambda}+i+\sqrt{\left(\frac{1}{\pi}\ln\frac{\tau}{\lambda}\right)^{2}+2i\frac{1}{\pi}\ln\frac{\tau}{\lambda}}\right).$
In order to apply the bound on the strip we have to define
(32) $d=d(\lambda,\tau)=r\operatorname{Im}x_{0},\quad 0<r<1.$
The introduction of the factor $r$ is necessary to avoid
${\xi}{(d)\rightarrow+\infty}$ as $\operatorname{Im}x_{0}\rightarrow\pi/2$,
which verifies for $\lambda\rightarrow\tau$ (see (22)).
#### 4.3.1. Asymptotic behaviors
Setting
$s=\frac{1}{\pi}\ln\frac{\lambda}{\tau},$
we have
$\frac{1}{\pi}\ln\frac{\tau}{\lambda}=-s,$
and therefore we can write (31) as
$x_{0}=\ln\left(s\left(-1+\frac{i}{s}+\sqrt{1-\frac{2i}{s}}\right)\right).$
Assuming $\lambda\gg\tau$, that is, $s\gg 1$, and using
(33) $\sqrt{1-x}\approx
1-\frac{1}{2}x-\frac{1}{8}x^{2}-\frac{1}{16}x^{3},\quad x\approx 0,$
we obtain
$\sqrt{1-\frac{2i}{s}}\approx
1-\frac{i}{s}+\frac{1}{2s^{2}}+\frac{i}{2s^{3}}.$
Using also $\ln(1+x)\approx x$,
$\displaystyle x_{0}$
$\displaystyle\approx\ln\left(s\left(-1+\frac{i}{s}+1-\frac{i}{s}+\frac{1}{2s^{2}}+\frac{i}{2s^{3}}\right)\right)$
$\displaystyle=\ln\left(s\left(\frac{1}{2s^{2}}+\frac{i}{2s^{3}}\right)\right)$
$\displaystyle=\ln\left(\frac{1}{2s}\right)+\ln\left(1+\frac{i}{s}\right)$
$\displaystyle\approx\ln\left(\frac{1}{2s}\right)+\frac{i}{s}.$
Therefore, for $\lambda\gg\tau$,
(34)
$\operatorname{Im}x_{0}\approx\frac{1}{s}=\frac{\pi}{\ln\frac{\lambda}{\tau}}.$
Assume now $\lambda=1$ and $\tau\gg 1$. By (31) we have
$x_{0}=\ln\left(\frac{1}{\pi}\ln\tau+i+\sqrt{\left(\frac{1}{\pi}\ln\tau\right)^{2}+2i\frac{1}{\pi}\ln\tau}\right).$
Setting
$s=\frac{1}{\pi}\ln\tau,$
we have
$\displaystyle x_{0}$
$\displaystyle=\ln\left(s\left(1+\frac{i}{s}+\sqrt{1+\frac{2i}{s}}\right)\right)$
$\displaystyle\approx\ln\left(s\left(1+\frac{i}{s}+1+\frac{i}{s}\right)\right)$
$\displaystyle=\ln\left(2s\left(1+\frac{i}{s}\right)\right)$
$\displaystyle\approx\ln\left(2s\right)+\frac{i}{s},$
that finally leads to
(35) $\operatorname{Im}x_{0}\approx\frac{1}{s}=\frac{\pi}{\ln\tau}.$
### 4.4. The minimax problem
Let us define the function
$\varphi(\lambda,\tau)=\xi(d)\lambda^{-\alpha}\exp\left(\frac{-2\pi
dn}{\ln\left(\frac{4dn}{\mu}\right)}\right),\quad d=d(\lambda,\tau),$
representing the $(\lambda,\tau)$-dependent factor of the error estimate given
by (28), that is,
$\mathcal{E}_{n,h}(g_{\lambda,\psi_{DE}})\approx
K_{\alpha}\varphi(\lambda,\tau),$
where $K_{\alpha}$ is defined by (29). Since our aim is to work with a self-
adjoint operator with spectrum contained in $[1,+\infty)$ the problem consists
in defining properly the parameter $\tau.$ This can be done by solving
(36) $\min_{\tau\geq 1}\max_{\lambda\geq 1}\varphi(\lambda,\tau).$
As for the true error, experimentally one observes that $\tau$ must be taken
much greater than 1, independently of $\alpha$. Therefore, from now on the
analysis will be based on the assumption $\tau\gg 1$. Regarding the function
$\varphi(\lambda,\tau)$, by taking $d=d(\lambda,\tau)$ as in (32) and $n$
sufficiently large, again, one experimentally observes that with respect to
$\lambda$ the function initially decreases, reaches a local minimum (for
$\lambda=\tau$ in which $d=r\pi/2$), then a local maximum (much greater than
$\tau$), and finally goes to 0 for $\lambda\rightarrow+\infty$ (see Figure 3).
In this view, denoting by $\overline{\lambda}$ the local maximum, for $n$
sufficiently large the problem (36) reduces to the solution of
(37) $\varphi(1,\tau)=\varphi(\overline{\lambda},\tau).$
#### 4.4.1. Evaluating the local maximum
Since $0<d\leq r\pi/2$, $0<r<1$, we have
$0<C\leq\cos d\cos\left(\frac{\pi}{2}\sin d\right)<1,$
where $C$ is a constant depending on $r$. Therefore by (22),
$2<\xi(d)\leq\frac{2}{C},$
so that we neglect the contribute of this function in what follows.
Since the maximum is seen to be much larger than $\tau$, we consider the
approximation (34). Therefore we have to solve
$\frac{d}{d\lambda}\lambda^{-\alpha}\exp\left(-\frac{2\pi
r\frac{\pi}{\ln\frac{\lambda}{\tau}}n}{\ln\left(\frac{4}{\mu}nr\frac{\pi}{\ln\frac{\lambda}{\tau}}\right)}\right)=0,$
that, after some manipulation leads to
$\frac{d}{d\lambda}\lambda^{-\alpha}\exp\left(-\frac{c_{1}n}{\ln\frac{\lambda}{\tau}\,q(\lambda)}\right)=0,$
where
(38) $c_{1}=2\pi^{2}r,\quad
q(\lambda)=\ln\left(c_{2}n\right)-\ln\left(\ln\frac{\lambda}{\tau}\right),\quad
c_{2}=\frac{4}{\mu}\pi r.$
We find the equation
$-\alpha\lambda^{-1}-\frac{d}{d\lambda}\left(\frac{c_{1}n}{\ln\frac{\lambda}{\tau}\,q(\lambda)}\right)=0,$
and since
$\displaystyle\frac{d}{d\lambda}\left(\frac{c_{1}n}{\ln\frac{\lambda}{\tau}\,q(\lambda)}\right)$
$\displaystyle=\frac{c_{1}n}{\lambda}\frac{1-q(\lambda)}{\left(\ln\frac{\lambda}{\tau}\right)^{2}q(\lambda)^{2}},$
we finally have to solve
(39)
$\alpha+c_{1}n\frac{1-q(\lambda)}{\left(\ln\frac{\lambda}{\tau}\right)^{2}q(\lambda)^{2}}=0.$
For large $n$ we have
$\displaystyle q(\lambda)$ $\displaystyle\approx\ln\left(c_{2}n\right),$
$\displaystyle\frac{q(\lambda)-1}{q(\lambda)^{2}}$
$\displaystyle\approx\frac{1}{q(\lambda)}\approx\frac{1}{\ln\left(c_{2}n\right)},$
so that the solution of (39) can be approximated by
(40)
$\lambda^{\ast}=\tau\exp\left(\sqrt{\frac{c_{1}n}{\alpha\ln\left(c_{2}n\right)}}\right).$
For any given $\tau\geq 1,$ it can be observed experimentally that
$\lambda^{\ast}$ is a very good approximation of the local maximum (see Figure
3).
We also remark that the assumption on $n$ stated in (26), that leads to the
error estimate (28), is automatically fulfilled for $\lambda=\lambda^{\ast}$,
at least for $\alpha$ not too small. Indeed, using (32) and (34) we first
observe that
(41)
$d(\lambda^{\ast},\tau)\approx\frac{r\pi}{\ln\frac{\lambda^{\ast}}{\tau}}=r\pi\sqrt{\frac{\alpha\ln\left(c_{2}n\right)}{c_{1}n}}.$
Then by (38), using $\mu\leq 1/2$ and assuming for instance $0.9<r<1,$ after
some simple computation we find
$\displaystyle\frac{\mu e}{4d(\lambda^{\ast},\tau)}$
$\displaystyle\approx\frac{\mu e}{4\pi
r}\sqrt{\frac{c_{1}n}{\alpha\ln\left(c_{2}n\right)}}$
$\displaystyle\leq\frac{1}{3}\sqrt{\frac{n}{\alpha}}.$
Therefore the condition (26) holds true for $n\geq 1/(9\alpha).$
#### 4.4.2. The error at the local maximum
By (41) clearly $d(\lambda^{\ast},\tau)\rightarrow 0$ for
$n\rightarrow+\infty$, and therefore from (22) we deduce that
$\xi(d(\lambda^{\ast},\tau))\rightarrow 2$ for $n\rightarrow+\infty$. As
consequence
$\varphi(\lambda^{\ast},\tau)\approx
2\left(\lambda^{\ast}\right)^{-\alpha}\exp\left(\frac{-2\pi
d(\lambda^{\ast},\tau)n}{\ln\left(\frac{4d(\lambda^{\ast},\tau)n}{\mu}\right)}\right).$
By defining
(42) $s_{n}=\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}},$
from (40) and (41) we have
$\displaystyle\lambda^{\ast}$
$\displaystyle=\tau\exp\left(\frac{s_{n}}{\sqrt{\alpha}}\right),$
$\displaystyle d(\lambda^{\ast},\tau)$
$\displaystyle\approx\frac{\sqrt{\alpha}r\pi}{s_{n}},$
and hence, after some computation
$\displaystyle\left(\lambda^{\ast}\right)^{-\alpha}\exp\left(\frac{-2\pi
d(\lambda^{\ast},\tau)n}{\ln\left(\frac{4d(\lambda^{\ast},\tau)n}{\mu}\right)}\right)$
$\displaystyle\approx\tau^{-\alpha}\exp\left(-\sqrt{\alpha}s_{n}\right)\exp\left(\frac{-2\pi\frac{\sqrt{\alpha}r\pi}{s_{n}}n}{\ln\left(\frac{4\frac{\sqrt{\alpha}r\pi}{s_{n}}n}{\mu}\right)}\right)$
$\displaystyle=\tau^{-\alpha}\exp\left(-\sqrt{\alpha}\left(s_{n}+\frac{c_{1}n}{s_{n}\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}\right)\right).$
By (42) we have
$\displaystyle
s_{n}+\frac{c_{1}n}{s_{n}\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}$
$\displaystyle=\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}+\frac{c_{1}n}{\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}$
$\displaystyle=\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\left(1+\frac{\ln\left(c_{2}n\right)}{\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}\right)$
$\displaystyle\approx 3\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}},$
because
$\frac{\ln\left(c_{2}n\right)}{\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}\rightarrow
2\text{\quad for\quad}n\rightarrow+\infty.$
Joining the above approximations we finally obtain
$\displaystyle\varphi(\lambda^{\ast},\tau)$ $\displaystyle\approx
2\tau^{-\alpha}\exp\left(-3\sqrt{\alpha}\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\right)$
(43) $\displaystyle=2\tau^{-\alpha}\exp\left(-3\sqrt{\alpha}s_{n}\right).$
#### 4.4.3. Error at $\lambda=1$
By (35), that is,
$d(1,\tau)\approx r\frac{\pi}{\ln\tau},\quad\tau\gg 1,$
we have again $\xi(d(1,\tau))\approx 2$ and therefore
$\varphi(1,\tau)\approx 2\exp\left(\frac{-2\pi
d(1,\tau)n}{\ln\left(\frac{4d(1,\tau)n}{\mu}\right)}\right).$
Using (38) we find
$\displaystyle\varphi(1,\tau)$ $\displaystyle\approx 2\exp\left(-\frac{2\pi
r\frac{\pi}{\ln\tau}n}{\ln\left(\frac{4}{\mu}nr\frac{\pi}{\ln\tau}\right)}\right)$
$\displaystyle=2\exp\left(-\frac{c_{1}n}{\ln\tau\left(\ln\left(c_{2}n\right)-\ln\left(\ln\tau\right)\right)}\right)$
$\displaystyle\approx
2\exp\left(-\frac{c_{1}n}{\ln\tau\ln\left(c_{2}n\right)}\right)$ (44)
$\displaystyle=2\exp\left(-\frac{s_{n}^{2}}{\ln\tau}\right).$
### 4.5. Approximating the optimal value for $\tau$
We need to solve (37). Using the approximations (44) and (43) we impose
$\displaystyle\exp\left(-\frac{s_{n}^{2}}{\ln\tau}\right)$
$\displaystyle=\tau^{-\alpha}\exp\left(-3\sqrt{\alpha}s_{n}\right)$
$\displaystyle=\exp\left(-3\sqrt{\alpha}s_{n}-\alpha\ln\tau\right),$
that is,
$-\frac{s_{n}^{2}}{\ln\tau}=-3\sqrt{\alpha}s_{n}-\alpha\ln\tau.$
Solving the above equation we find
$\displaystyle\ln\tau$
$\displaystyle=\frac{\left(-3+\sqrt{13}\right)\sqrt{\alpha}s_{n}}{2\alpha}$
$\displaystyle\approx 0.3\frac{s_{n}}{\sqrt{\alpha}},$
so that
(45) $\tau^{\ast}=\exp\left(0.3\frac{s_{n}}{\sqrt{\alpha}}\right)$
represents an approximate solution of (37).
Figure 3. Plot of the function $\varphi(\lambda,\tau^{\ast})$ for $n=40$ and
$\alpha=1/2$. The asterisk represents the approximation of the local maximum
given by (40), that is, the point
$\left(\lambda^{\ast},\varphi(\lambda^{\ast},\tau^{\ast})\right)$. The diamond
represents the approximation of $\varphi(\lambda^{\ast},\tau^{\ast})$ stated
in (43). Finally the circle is the approximation of $\varphi(1,\tau^{\ast})$
given in (44).
In Figure 3 we plot the function $\varphi(\lambda,\tau^{\ast})$ for
$\lambda\in[1,10^{20}]$, in an example in which $n=40$, $\alpha=1/2$, and
$\tau^{\ast}\cong 84.4$ defined by (45). Moreover we show the results of the
approximations (40), (43) and (44), for $\tau=\tau^{\ast}$. Clearly the ideal
situation would be to have $\tau^{\ast}$ such that
$\varphi(1,\tau^{\ast})=\varphi(\overline{\lambda},\tau^{\ast})$, but
notwithstanding all the approximations used, the results are fairly good and
allow to have a simple expression for $\tau^{\ast}$.
By using (45) in (43) we obtain
(46) $\varphi(\lambda^{\ast},\tau^{\ast})\approx
2\exp\left(-3.3\sqrt{\alpha}\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\right).$
Remembering that
$E_{n,h}(\mathcal{L})\leq
2\frac{\sin\left(\alpha\pi\right)}{\pi}\max_{\lambda\geq
1}\mathcal{E}_{n,h}(g_{\mathcal{\lambda},\psi_{DE}}),$
using (46) we finally obtain the error estimate
(47)
$E_{n,h}(\mathcal{L})\approx\overline{K}_{\alpha}\exp\left(-3.3\sqrt{\alpha}\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\right),$
where
$\displaystyle\overline{K}_{\alpha}$
$\displaystyle=4\frac{\sin\left(\alpha\pi\right)}{\pi}K_{\alpha}$
$\displaystyle=4\frac{\sin\left(\alpha\pi\right)}{\pi}\frac{1}{\alpha(1-\alpha)}\frac{1}{1-e^{-\frac{\pi}{2}\mu
e}}.$
In Figure 4 we show the behavior of the method for the computation of
$\mathcal{L}^{-\alpha}$, with $\mathcal{L}$ defined in (17), together with the
estimate (47). For comparison, in the same pictures we also plot the error of
the SE approach. As mentioned in the introduction, the DE approach appears to
be faster for $1/2\leq\alpha<1.$
Figure 4. Error for the trapezoidal rule applied with the double-exponential
transform (error DE), with the single-exponential transform (error SE) and
error estimate given by (47).
## 5\. Conclusions
In this work we have analyzed the behavior of the trapezoidal rule for the
computation of $\mathcal{L}^{-\alpha}$, in connection with the single and the
double-exponential transformations. All the analysis has been based on the
assumption of $\mathcal{L}$ unbounded, so that the results can be applied even
to discrete operators, with spectrum arbitrarily large, without the need to
know its amplitude, that is, the largest eigenvalue. In particular we have
revised the analysis for the single-exponential transform and we have
introduced new error estimates for the scalar and the operator case for the
double-exponential transform. The sharp estimate obtained for the scalar case
has been fundamental for the proper selection of the parameter $\tau$ that is
necessary to obtain good results also for the operator case.
## References
* [1] L. Aceto, D. Bertaccini, F. Durastante, P. Novati. Rational Krylov methods for functions of matrices with applications to fractional partial differential equations. J. Comput. Phys. 396, 470-482 (2019)
* [2] L. Aceto, P. Novati. Padé-type approximations to the resolvent of fractional powers of operators. J. Sci. Comput. 83 (2020): 13, DOI:10.1007/s10915-020-01198-w
* [3] L. Aceto, P. Novati. Rational approximations to fractional powers of self-adjoint positive operators. Numer. Math. 143, 1-16 (2019)
* [4] L. Aceto, P. Novati. Fast and accurate approximations to fractional powers of operators. IMA J. Numer. Anal. (2021), drab002, https://doi.org/10.1093/imanum/drab002
* [5] A. Bonito, J.E. Pasciak. Numerical approximation of fractional powers of elliptic operators. Math. Comp. 84, 2083-2110 (2015)
* [6] S. Harizanov, R. Lazarov, S. Margenov, P. Marinov, Y. Vutov. Optimal solvers for linear systems with fractional powers of sparse SPD matrices. Numer. Linear Algebra Appl. 25(5), e2167 (2018)
* [7] S. Harizanov, R. Lazarov, P. Marinov, S. Margenov, J.E. Pasciak. Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation. arXiv:1905.08155v2 (2019)
* [8] S. Harizanov, R. Lazarov, P. Marinov, S. Margenov, J.E. Pasciak. Comparison analysis of two numerical methods for fractional diffusion problems based on the best rational approximations of $t^{\gamma}$ on $[0,1].$ In: Apel T., Langer U., Meyer A., Steinbach O. (eds) Advanced Finite Element Methods with Applications. FEM 2017. Lecture Notes in Computational Science and Engineering, vol 128. Springer, Cham, 2019
* [9] S. Harizanov, S. Margenov. Positive approximations of the inverse of fractional powers of SPD M-Matrices. In: Feichtinger G., Kovacevic R., Tragler G. (eds) Control Systems and Mathematical Methods in Economics. Lecture Notes in Economics and Mathematical Systems, vol 687. Springer, Cham, 2018
* [10] C. Hofreither. A unified view of some numerical methods for fractional diffusion. Comput. Math. Appl. 80(2), 351-366 (2020)
* [11] J. Lund, K.L. Bowers. Sinc Methods for Quadrature and Differential Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR1171217 (93i:65004)
* [12] T. Okayama, K. Tanaka, T. Matsuo, M. Sugihara. DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods Part I: definite integration and function approximation Numer. Math. 125, 511-543 (2013)
* [13] T. Okayama, T. Matsuo, M. Sugihara. Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration. Numer. Math. 124, 361-394 (2013)
* [14] M. Mori. Discovery of the double exponential transformation and its developments. Publ. RIMS, Kyoto Univ. 41, 897-935 (2005)
* [15] K. Tanaka, M. Sugihara, K. Murota, M. Mori. Function classes for double exponential integration formulas. Numer. Math. 111, 631-655 (2009)
* [16] L.N. Trefethen, J.A.C. Weideman, The exponentially convergent trapezoidal rule. SIAM Review 56(3), 385-458 (2014)
* [17] P.N. Vabishchevich. Approximation of a fractional power of an elliptic operator. CoRR abs/1905.10838 (2019)
* [18] P.N. Vabishchevich. Numerical solution of time-dependent problems with fractional power elliptic operator. Comput. Meth. in Appl. Math. 18(1), 111-128 (2018)
* [19] P.N. Vabishchevich. Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys. 282, 289-302 (2015)
|
* [98] L.-y. Dong, S.-j. Ji, C.-j. Zhang, Q. Zhang, D. W. Chiu, L.-q. Qiu, and D. Li, “An unsupervised topic-sentiment joint probabilistic model for detecting deceptive reviews,” _Expert Systems with Applications_ , vol. 114, pp. 210–223, 2018.
* [99] G. Xu, M. Hu, C. Ma, and M. Daneshmand, “Gscpm: Cpm-based group spamming detection in online product reviews,” in _ICC 2019 - 2019 IEEE International Conference on Communications (ICC)_ , 2019, pp. 1–6.
* [100] C. Zhu, W. Zhao, Q. Li, P. Li, and Q. Da, “Network embedding-based anomalous density searching for multi-group collaborative fraudsters detection in social media,” _Computers, Materials and Continua_ , vol. 60, no. 1, pp. 317–333, 2019. [Online]. Available: http://dx.doi.org/10.32604/cmc.2019.05677
* [101] R. Wen, J. Wang, C. Wu, and J. Xiong, “Asa: Adversary situation awareness via heterogeneous graph convolutional networks,” in _Companion Proceedings of the Web Conference 2020_ , ser. WWW ’20. New York, NY, USA: Association for Computing Machinery, 2020, p. 674–678. [Online]. Available: https://doi.org/10.1145/3366424.3391266
* [102] R. Narayan, J. K. Rout, and S. K. Jena, “Review spam detection using semi-supervised technique,” in _Progress in Intelligent Computing Techniques: Theory, Practice, and Applications_. Springer, 2018, pp. 281–286.
* [103] R. Hassan and M. R. Islam, “Detection of fake online reviews using semi-supervised and supervised learning,” in _2019 International Conference on Electrical, Computer and Communication Engineering (ECCE)_ , 2019, pp. 1–5.
* [104] A. Pathak and R. K. Srihari, “Breaking! presenting fake news corpus for automated fact checking,” in _Proceedings of the 57th annual meeting of the association for computational linguistics: student research workshop_ , 2019, pp. 357–362.
* [105] H. Rashkin, E. Choi, J. Y. Jang, S. Volkova, and Y. Choi, “Truth of varying shades: Analyzing language in fake news and political fact-checking,” in _Proceedings of the 2017 conference on empirical methods in natural language processing_ , 2017, pp. 2931–2937.
* [106] R. Vijjali, P. Potluri, S. Kumar, and S. Teki, “Two stage transformer model for covid-19 fake news detection and fact checking,” _arXiv preprint arXiv:2011.13253_ , 2020.
* [107] S. Wang, W. Mao, P. Wei, and D. D. Zeng, “Knowledge structure driven prototype learning and verification for fact checking,” _Knowledge-Based Systems_ , vol. 238, p. 107910, 2022. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0950705121010650
* [108] A. Porshnev, I. Redkin, and A. Shevchenko, “Machine learning in prediction of stock market indicators based on historical data and data from twitter sentiment analysis,” in _2013 IEEE 13th International Conference on Data Mining Workshops_. IEEE, 2013, pp. 440–444.
* [109] S. Li, Y. Wang, J. Xue, N. Zhao, and T. Zhu, “The impact of covid-19 epidemic declaration on psychological consequences: a study on active weibo users,” _International journal of environmental research and public health_ , vol. 17, no. 6, p. 2032, 2020.
* [110] J. Marciano, “Fake online reviews cost $152 billion a year. Here’s how e-commerce sites can stop them,” https://www.weforum.org/agenda/2021/08/fake-online-reviews-are-a-152-billion-problem-heres-how-to-silence-them/, Aug 10, 2021, [Online; accessed 04-Aug-2022].
* [111] S. Dickens, “Introducing New Machine Learning Techniques to Help Stop Scams,” https://www.facebook.com/notes/10157814548136886/, March 26, 2018, [Online; accessed 04-Aug-2022].
* [112] J. D. Day and H. Zimmermann, “The osi reference model,” _Proceedings of the IEEE_ , vol. 71, no. 12, pp. 1334–1340, 1983.
* [113] P. Gupta, S. Gandhi, and B. R. Chakravarthi, “Leveraging transfer learning techniques-bert, roberta, albert and distilbert for fake review detection,” in _Forum for Information Retrieval Evaluation_ , 2021, pp. 75–82.
* [114] M. Matassoni, R. Gretter, D. Falavigna, and D. Giuliani, “Non-native children speech recognition through transfer learning,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 6229–6233.
* [115] P. G. Shivakumar and P. Georgiou, “Transfer learning from adult to children for speech recognition: Evaluation, analysis and recommendations,” _Computer speech & language_, vol. 63, p. 101077, 2020.
* [116] S. Shen, M. Sadoughi, M. Li, Z. Wang, and C. Hu, “Deep convolutional neural networks with ensemble learning and transfer learning for capacity estimation of lithium-ion batteries,” _Applied Energy_ , vol. 260, p. 114296, 2020.
* [117] Y. Yao, B. Viswanath, J. Cryan, H. Zheng, and B. Y. Zhao, “Automated crowdturfing attacks and defenses in online review systems,” in _Proceedings of the 2017 ACM SIGSAC conference on computer and communications security_ , 2017, pp. 1143–1158.
* [118] M. A. Al-Garadi, K. D. Varathan, S. D. Ravana, E. Ahmed, and V. Chang, “Identifying the influential spreaders in multilayer interactions of online social networks,” _Journal of Intelligent & Fuzzy Systems_, vol. 31, no. 5, pp. 2721–2735, 2016.
* [119] B. Oselio, A. Kulesza, and A. O. Hero, “Multi-layer graph analysis for dynamic social networks,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 8, no. 4, pp. 514–523, 2014.
* [120] S. Sivasankari and G. Vadivu, “Tracing the fake news propagation path using social network analysis,” _Soft Computing_ , pp. 1–9, 2021.
* [121] W. L. Hamilton, R. Ying, and J. Leskovec, “Inductive representation learning on large graphs,” in _Proceedings of the 31st International Conference on Neural Information Processing Systems_ , ser. NIPS’17. Red Hook, NY, USA: Curran Associates Inc., 2017, p. 1025–1035. |
# Boosting Model Resilience via Implicit Adversarial Data Augmentation
Xiaoling Zhou1 Wei Ye1111Corresponding authors. Zhemg Lee2 Rui Xie1&Shikun
Zhang1111Corresponding authors.
1National Engineering Research Center for Software Engineering, Peking
University, China
2Tianjin University, Tianjin, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Data augmentation plays a pivotal role in enhancing and diversifying training
data. Nonetheless, consistently improving model performance in varied learning
scenarios, especially those with inherent data biases, remains challenging. To
address this, we propose to augment the deep features of samples by
incorporating their adversarial and anti-adversarial perturbation
distributions, enabling adaptive adjustment in the learning difficulty
tailored to each sample’s specific characteristics. We then theoretically
reveal that our augmentation process approximates the optimization of a
surrogate loss function as the number of augmented copies increases
indefinitely. This insight leads us to develop a meta-learning-based framework
for optimizing classifiers with this novel loss, introducing the effects of
augmentation while bypassing the explicit augmentation process. We conduct
extensive experiments across four common biased learning scenarios: long-tail
learning, generalized long-tail learning, noisy label learning, and
subpopulation shift learning. The empirical results demonstrate that our
method consistently achieves state-of-the-art performance, highlighting its
broad adaptability.
## 1 Introduction
Data augmentation techniques, designed to enrich the quantity and diversity of
training samples, have demonstrated their effectiveness in improving the
performance of deep neural networks (DNNs) Maharana et al. (2022). Existing
methods can be divided into two categories. The first, explicit data
augmentation Cubuk et al. (2020); Xu and Zhao (2023), applies geometric
transformations to samples and directly integrates augmented instances into
the training process, leading to reduced training efficiency. The second, a
recent addition to this field, is the implicit data augmentation technique
Wang et al. (2019), which is inspired by the existence of numerous semantic
vectors within the deep feature space of DNNs. This method emphasizes
enhancing the deep features of samples and is achieved by optimizing robust
losses instead of explicitly conducting the augmentation process, resulting in
a more efficient and effective approach. Subsequent studies in imbalanced
learning have extended this approach. For example, MetaSAug Li et al. (2021)
refines the accuracy of covariance matrices for minor classes by minimizing
losses on a balanced validation set. RISDA Chen et al. (2022) generates
diverse instances for minor classes by extracting semantic vectors from the
deep feature space of both the current class and analogous classes.
Figure 1: (a) Illustration for our augmentation strategy, which augments
samples within their adversarial and anti-adversarial perturbation
distributions. (b) Illustration of an imbalanced learning scenario. Our method
employs adversarial and anti-adversarial augmentations for the minor and major
classes, respectively.
Despite these promising efforts, current augmentation strategies still exhibit
notable limitations. Firstly, these approaches mainly enhance samples within
the original training data space Xu and Zhao (2023); Wang et al. (2019),
falling short in effectively mitigating the distributional discrepancies
between training and test data, such as noise Zheng et al. (2021) and
subpopulation shifts Yao et al. (2022). Secondly, previous algorithms Li et
al. (2021); Chen et al. (2022) mostly operate at the category level, causing
different samples within the same class to share identical augmentation
distributions and strengths, which is potentially unreasonable and inaccurate.
For instance, in noisy learning scenarios, it is advisable to treat noisy
samples separately to mitigate their negative impact on model training Zhou et
al. (2023).
This study introduces a novel Implicit Adversarial Data Augmentation (IADA)
approach, which conceptually embodies two main characteristics to address the
two bottlenecks above. Firstly, as illustrated in Fig. 1(a), IADA enriches the
deep features of samples by randomly sampling perturbation vectors from their
adversarial and anti-adversarial perturbation distributions, surpassing the
limitation of augmenting within the original training distribution. With this
strategy, the classifier is anticipated to be dynamically adjusted by
modifying the learning difficulty of samples. Secondly, the augmentation
distribution for each sample is tailored based on its unique training
characteristics, granting it the ability to address data biases beyond the
category level. Specifically, these distributions are modeled as multivariate
normal distributions, characterized by sample-wise perturbations and class-
specific covariance matrices. Fig. 1(b) demonstrates the two features of our
method using imbalanced learning as an example scenario, where the classifier
typically favors major classes while underperforming on the minor ones, as
well as favors easy samples (e.g., dogs on grass) but commonly mispredicting
hard samples (e.g., dogs in the water)111This example stems from the
observation that the majority of dogs in the training set are on grass and
seldom in the water.. In this scenario, our method adopts anti-adversarial
augmentation for major classes and adversarial augmentation (with higher
strength) for minor ones. Meanwhile, adversarial augmentation is also applied
to hard samples within major classes, ultimately facilitating better learning
of class boundaries.
By exploring an infinite number of augmentations, we theoretically derive a
surrogate loss for our augmentation strategy, thereby eliminating the need for
explicit augmentation. Subsequently, to determine the perturbation strategies
for samples in this loss, we construct a meta-learning-based framework, in
which a perturbation network is tasked with computing perturbation strategies
by leveraging diverse training characteristics extracted from the classifier
as inputs. The training of all parameters in the perturbation network is
guided by a small, unbiased meta dataset, enabling the generation of well-
founded perturbation strategies for samples. Extensive experiments show that
our method adeptly addresses various data biases, such as noise, imbalance,
and subpopulation shifts, consistently achieving state-of-the-art (SOTA)
performance among all compared methods.
In summary, our contributions in this paper are threefold.
* •
We propose a novel perspective on data augmentation, wherein samples undergo
augmentation within their adversarial and anti-adversarial perturbation
distributions, to facilitate model training across diversified learning
scenarios, particularly those with data biases.
* •
In accordance with our augmentation perspective, we derive a new logit-
adjusted loss and incorporate it into a well-designed meta-learning framework
to optimize the classifier, unlocking the potential of data augmentation
without an explicit augmentation procedure.
* •
We conduct extensive experiments across four typical biased learning
scenarios, encompassing long-tail (LT) learning, generalized long-tail (GLT)
learning, noisy label learning, and subpopulation shift learning. The results
conclusively demonstrate the effectiveness and broad applicability of our
approach.
## 2 Related Work
#### Data Augmentation
methods have showcased their capacity to improve DNNs’ performance by
expanding and diversifying training data Maharana et al. (2022). Explicit
augmentation directly incorporates augmented data into the training process,
albeit at the expense of reduced training efficiency Cubuk et al. (2020);
Taylor and Nitschke (2018); Xu and Zhao (2023). Recently, Wang et al. Wang et
al. (2019) introduced an implicit semantic data augmentation approach, named
ISDA, which transforms the deep features of samples within the semantic space
of DNNs and boils down to the optimization of a robust loss. Subsequent
studies Li et al. (2021); Chen et al. (2022) in image classification tasks
have extended this approach. However, these methods still struggle with
effectively improving model performance when dealing with data biases that go
beyond the category level.
Figure 2: The overview of our method pipeline. We initiate with a sample-wise
adversarial data augmentation strategy (Box 1), enriching the deep features of
samples using perturbation vectors extracted from their adversarial and anti-
adversarial perturbation distributions. Subsequently, by considering an
infinite number of augmented instances, we derive a novel robust loss, termed
IADA (Box 2). Regularization analysis reveals the efficacy of IADA in
improving model generalization, robustness, and inter-class fairness. To
facilitate optimization with IADA, we then establish a meta-learning-based
framework called Meta-IADA (Box 3). Within it, a perturbation network is
tasked with generating perturbation strategies for samples (denoted as
$\epsilon_{\boldsymbol{x}}$) in the IADA loss, leveraging a set of
($K\\!=\\!15$) training characteristics as inputs.
#### Adversarial and Anti-Adversarial Perturbations
transform samples in directions that respectively move towards and away from
the decision boundary, thereby modifying samples’ learning difficulty Lee et
al. (2023); Zhou et al. (2023). Consequently, models allocate varying levels
of attention to samples subjected to their perturbations. Research has
confirmed that incorporating adversarial and anti-adversarial samples during
training assists models in achieving a better tradeoff between robustness and
generalization Zhou et al. (2023); Zhu et al. (2021). However, existing
adversarial training methods primarily focus on two specific types of
perturbations that maximize and minimize losses Xu et al. (2021); Zhou et al.
(2023), posing limitations. Moreover, generating adversarial perturbations
within the input space is time-consuming Madry et al. (2018). Different from
prior studies, our approach randomly selects perturbation vectors from both
adversarial and anti-adversarial perturbation distributions, enabling the
generation of multiple distinct adversarial and anti-adversarial samples.
Furthermore, the perturbations are generated within the deep feature space,
enhancing efficiency and ensuring universality across various data types.
## 3 Implicit Adversarial Data Augmentation
We initially introduce a sample-wise adversarial data augmentation strategy to
facilitate model training across various learning scenarios. By considering
infinite augmentations, we then derive a surrogate loss for our augmentation
strategy.
### 3.1 Adversarial Data Augmentation
Consider training a deep classifier $\mathcal{F}$, with weights
$\boldsymbol{\Phi}$ on a training set, denoted as
$\mathcal{D}^{{tr}}=\\{(\boldsymbol{x}_{i},y_{i})\\}_{i=1}^{N}$, where $N$
refers to the number of training samples, and
$y_{i}\in\\{{1},\cdots,{\mathcal{C}}\\}$ represents the label of sample
$\boldsymbol{x}_{i}$. The deep feature (before logit) learned by $\mathcal{F}$
for $\boldsymbol{x}_{i}$ is represented as a $\mathcal{H}$-dimensional vector
$\boldsymbol{h}_{i}={\mathcal{F}}_{\boldsymbol{\Phi}}(\boldsymbol{x}_{i})\in\mathbb{R}^{\mathcal{H}}$.
Our augmentation strategy enhances samples within the deep feature space of
DNNs. The perturbation vectors for the deep feature of each sample are
randomly extracted from either its adversarial or anti-adversarial
perturbation distributions. These distributions are modeled as multivariate
normal distributions,
$\mathcal{N}(\boldsymbol{\delta}_{i},{\boldsymbol{\Sigma}}_{y_{i}})$, where
$\boldsymbol{\delta}_{i}$ refers to the sample perturbation, and
${\boldsymbol{\Sigma}}_{y_{i}}$ represents the class-specific covariance
matrix estimated from the features of all training samples in class $y_{i}$.
As samples undergo augmentation within the deep feature space, perturbations
should also be generated within this space, facilitating semantic alterations
for training samples. Consequently, the perturbation vector
$\boldsymbol{\delta}_{i}$ for sample $\boldsymbol{x}_{i}$ is calculated as
$\epsilon_{i}\\!\cdot\\!{sign}(\nabla_{\boldsymbol{h}_{i}}\ell_{i}^{CE})$,
where ${sign}(\nabla_{\boldsymbol{h}_{i}}\ell_{i}^{CE})$ signifies the
gradient sign of the CE loss $\ell_{i}^{CE}$ with respect to
$\boldsymbol{h}_{i}$. The parameter $\epsilon_{i}$ plays a pivotal role in
determining the perturbation strategy applied to $\boldsymbol{x}_{i}$,
encompassing both the perturbation direction and bound. Its positive or
negative sign signifies adversarial or anti-adversarial perturbations,
respectively. Furthermore, the absolute value $|\epsilon_{i}|$ governs the
perturbation bound. In practical applications, the value of $\epsilon_{i}$ is
dynamically computed through a perturbation network based on the training
characteristics of $\boldsymbol{x}_{i}$, which will be elaborated in Section
4. Additionally, the class-specific covariance matrix
$\boldsymbol{\Sigma}_{y_{i}}$ within this distribution aids in preserving the
covariance structure of each class. Its value is estimated in real-time by
aggregating statistics from all mini-batches, as detailed in Section I of the
Appendix. Regarding the augmentation strength quantified by the number of
augmented instances $\mathcal{M}_{i}$ for $\boldsymbol{x}_{i}$, we define
$\mathcal{M}_{i}$ as $\mathcal{M}/\pi_{y_{i}}$, where $\mathcal{M}$ is a
constant and $\pi_{y_{i}}$ represents the proportion of class $y_{i}$ in the
training data. Accordingly, a smaller proportion results in a larger number of
augmented instances, ensuring class balance.
To compute the augmented features $\tilde{\boldsymbol{h}}_{i}$ from
${\boldsymbol{h}}_{i}$, we transform ${\boldsymbol{h}}_{i}$ along random
directions sampled from
$\mathcal{N}(\boldsymbol{\delta}_{i},{\boldsymbol{\Sigma}}_{y_{i}})$. This
transformation yields
$\tilde{\boldsymbol{h}}_{i}\sim\mathcal{N}({\boldsymbol{h}}_{i}+\boldsymbol{\delta}_{i},\alpha{\boldsymbol{\Sigma}}_{y_{i}})$,
where the parameter $\alpha$ controls the extent of dispersion for augmented
samples. In summary, our adversarial data augmentation strategy offers the
following advantages:
* •
Instead of augmenting samples within the original data space, our approach
enhances them within their adversarial and anti-adversarial perturbation
distributions. This method effectively adjusts the learning difficulty
distribution of training samples, fostering improved generalization and
robustness in DNNs.
* •
Our sample-wise augmentation distribution customizes the mean vector based on
the unique training characteristics of each sample. This personalized strategy
significantly enhances models’ ability to address data biases, encompassing
those beyond the category level.
### 3.2 IADA Loss
With our augmentation strategy, a straightforward way to train a classifier
involves augmenting each ${\boldsymbol{h}}_{i}$ for $\mathcal{M}_{i}$ times.
This procedure generates an augmented feature set for each sample,
$\\{\tilde{\boldsymbol{h}}_{i}^{1},\cdots,\tilde{\boldsymbol{h}}_{i}^{\mathcal{M}_{i}}\\}$.
Subsequently, the CE loss for all augmented samples is as follows:
$\displaystyle\mathcal{L}^{\mathcal{M}}(\boldsymbol{W},\boldsymbol{b})=\frac{1}{\hat{\mathcal{M}}}\sum_{i=1}^{N}\sum_{k=1}^{\mathcal{M}_{i}}-\log\frac{e^{{\boldsymbol{w}_{y_{i}}^{T}\tilde{\boldsymbol{h}}_{i}^{k}+b_{y_{i}}}}}{\sum_{j=1}^{\mathcal{C}}e^{{\boldsymbol{w}_{j}^{T}\tilde{\boldsymbol{h}}_{i}^{k}+b_{j}}}},$
(1)
where $\hat{\mathcal{M}}\\!=\\!\sum_{i=1}^{N}\mathcal{M}_{i}$. Additionally,
$\boldsymbol{W}\\!=\\![\boldsymbol{w}_{1},\\!\cdots\\!,\boldsymbol{w}_{{\mathcal{C}}}]^{T}\\!\in\\!\mathbb{R}^{\mathcal{C}\times\mathcal{H}}$
and
$\boldsymbol{b}=[b_{1},\cdots,b_{{{\mathcal{C}}}}]^{T}\in\mathbb{R}^{\mathcal{C}}$,
in which $\boldsymbol{w}_{j}$ and $b_{j}$ refer to the weight vector and bias
corresponding to the last fully connected layer for class $j$. Considering
augmenting more data while enhancing training efficiency, we explore
augmenting an infinite number of times for the deep feature of each training
sample. As $\mathcal{M}$ in $\mathcal{M}_{i}$ approaches infinity, the
expected CE loss is expressed as:
$\displaystyle\mathcal{L}^{\mathcal{M}\rightarrow\infty}(\boldsymbol{W},\boldsymbol{b})$
$\displaystyle=\sum_{i=1}^{N}\frac{1}{\pi_{y_{i}}}\mathrm{E}_{\tilde{\boldsymbol{h}}_{i}}\big{[}-\log\frac{e^{{\boldsymbol{w}_{y_{i}}^{T}\tilde{\boldsymbol{h}}_{i}+b_{y_{i}}}}}{\sum_{j=1}^{\mathcal{C}}e^{{\boldsymbol{w}_{j}^{T}\tilde{\boldsymbol{h}}_{i}+b_{j}}}}\big{]}.$
(2)
However, accurately calculating Eq. (2) poses a challenge. Hence, we proceed
to derive a computationally efficient surrogate loss for it. Given the
concavity of the logarithmic function $\log\left(\cdot\right)$, using Jensen’s
inequality, $\mathrm{E}[\log X]\leq\log\mathrm{E}[X]$, we derive an upper
bound of Eq. (2) as follows:
$\begin{aligned}
\mathcal{L}^{\mathcal{M}\rightarrow\infty}(\boldsymbol{W},\boldsymbol{b})\leq\sum_{i=1}^{N}\frac{1}{{\pi_{y_{i}}}}\log\bigg{(}\sum_{j=1}^{\mathcal{C}}\mathrm{E}_{\tilde{{\boldsymbol{h}}}_{i}}\big{[}e^{\Delta\boldsymbol{w}_{j,y_{i}}\tilde{\boldsymbol{h}}_{i}+\Delta
b_{j,y_{i}}}\big{]}\bigg{)},\end{aligned}$
(3)
where
$\Delta\boldsymbol{w}_{j,y_{i}}=\boldsymbol{w}_{j}^{T}-\boldsymbol{w}_{y_{i}}^{T}$
and $\Delta b_{j,y_{i}}=b_{j}-b_{y_{i}}$. As $\tilde{\boldsymbol{h}}_{i}$ is a
Gaussian random variable adhering to
$\mathcal{N}({\boldsymbol{{h}}}_{i}+\boldsymbol{\delta}_{i},\alpha{\boldsymbol{\Sigma}}_{y_{i}})$,
we know that
$\Delta\boldsymbol{w}_{j,y_{i}}\tilde{\boldsymbol{h}}_{i}+\Delta{b}_{j,y_{i}}\\!\sim\\!\mathcal{N}\big{(}\Delta\boldsymbol{w}_{j,y_{i}}\left({\boldsymbol{h}}_{i}+\boldsymbol{\delta}_{i}\right)+\Delta{b}_{j,y_{i}},\alpha\Delta\boldsymbol{w}_{j,y_{i}}{\boldsymbol{\Sigma}}_{y_{i}}\Delta\boldsymbol{w}_{j,y_{i}}^{T}\big{)}$.
Subsequently, utilizing the moment-generating function,
$\mathrm{E}\left[e^{tX}\right]=e^{t\mu+\frac{1}{2}\sigma^{2}t^{2}},\quad
X\sim\mathcal{N}(\mu,\sigma^{2}),$ (4)
the upper bound of Eq. (3) can be represented as
$\displaystyle\mathcal{L}^{\mathcal{M}\rightarrow\infty}(\boldsymbol{W},\boldsymbol{b})\leq\sum_{i=1}^{N}-\frac{1}{\pi_{y_{i}}}\log\frac{e^{\mathcal{Z}_{i}^{y_{i}}}}{\sum_{j=1}^{\mathcal{C}}e^{\mathcal{Z}_{i}^{j}}},$
(5)
where
$\mathcal{Z}_{i}^{j}=\boldsymbol{w}_{j}(\boldsymbol{h}_{i}+\boldsymbol{\delta}_{i})+b_{j}+\alpha\rho_{i}^{j}$
and
$\rho_{i}^{j}\\!=\\!\frac{1}{2}\Delta\boldsymbol{w}_{j,y_{i}}{\boldsymbol{\Sigma}}_{y_{i}}\Delta\boldsymbol{w}_{j,y_{i}}^{T}$.
Name | Term | Function
---|---|---
$\mathcal{G}$ | $\sum_{i=1}^{N}\sum_{j\neq y_{i}}{q}_{i,j}\rho_{i}^{j}$ | Generalization
$\mathcal{R}$ | $\sum_{i=1}^{N}\sum_{j\neq y_{i}}{q}_{i,j}\Delta\boldsymbol{w}_{j,y_{i}}\boldsymbol{\delta}_{i}$ | Robustness
$\mathscr{F}$ | $\sum_{i=1}^{N}\sum_{j\neq y_{i}}{q}_{i,j}\log(\pi_{j}/\pi_{y_{i}})$ | Inter-class fairness
Table 1: The regularization terms incorporated in the IADA loss and their
corresponding functions. $\boldsymbol{q}_{i}$ denotes the Softmax output
linked to sample $\boldsymbol{x}_{i}$. The derivation process is presented in
the Appendix.
Drawing inspiration from the Logit Adjustment (LA) approach Menon et al.
(2021), we introduce its logit adjustment term in place of the class-wise
weight $1/\pi_{y_{i}}$, providing a more effective solution for imbalanced
class distributions. Consequently, the final IADA loss is formulated as
follows:
$\displaystyle\mathcal{L}^{{IADA}}(\boldsymbol{W},\boldsymbol{b})\coloneq\sum_{i=1}^{N}-\log\frac{e^{{\tilde{\mathcal{Z}}}_{i}^{y_{i}}}}{\sum_{j=1}^{\mathcal{C}}e^{{\tilde{\mathcal{Z}}}_{i}^{j}}},$
(6)
where
${\tilde{\mathcal{Z}}}_{i}^{j}=\boldsymbol{w}_{j}(\boldsymbol{h}_{i}+\boldsymbol{\delta}_{i})+b_{j}+\alpha\mathcal{\rho}_{i}^{j}+\beta\log\pi_{j}$.
The symbols $\alpha$ and $\beta$ serve as two hyperparameters in the IADA
loss, frequently set to 0.5 and 1, respectively, in practical applications.
The IADA loss, essentially a logit-adjusted variant of the CE loss, operates
as a surrogacy for our proposed adversarial data augmentation strategy.
Therefore, instead of explicitly executing the augmentation process, we can
directly optimize this loss, thereby improving efficiency.
We further explain the IADA loss, defined in Eq. (6), from a regularization
perspective using the Taylor expansion. This process reveals three
regularization terms stemming from our adversarial data augmentation strategy,
as outlined in Table 1. A detailed analysis of these terms is presented in
Section III of the Appendix. Through our analysis, the first term
$\mathcal{G}$ diminishes the mapped variances of deep features within each
class, thereby enhancing intra-class compactness and improving the
generalization ability of models. The second term $\mathcal{R}$ reinforces
model robustness by increasing the cosine similarity between the
classification boundary and the gradient vectors of adversarial samples.
Furthermore, the third term $\mathscr{F}$ promotes fairness among classes by
favoring less-represented categories. Collectively, the regularization terms
incorporated in the IADA loss significantly contribute to strengthening the
generalization, robustness, and inter-class fairness of DNNs, as depicted in
Fig. 2 (Box 2).
## 4 Optimization Using IADA Loss
Due to the necessity of pre-determining the value of $\epsilon_{i}$ in
$\boldsymbol{\delta}_{i}$, which governs the perturbation strategies, when
utilizing the IADA loss, we construct a meta-learning-based framework named
Meta-IADA. This framework is designed to optimize classifiers using the IADA
loss. As illustrated in Fig. 2 (Box 3), Meta-IADA comprises four main
components: the classifier, the characteristics extraction module, the
perturbation network, and the meta-learning-based learning strategy.
Considering that determining perturbation strategies for samples involves
factors including their learning difficulty, class distribution, and noise
levels Zhou et al. (2023); Xu et al. (2021), we extract fifteen training
characteristics (e.g., sample loss and margin) from the classifier to
encompass these aspects. All characteristics are comprehensively introduced in
Section IV of the Appendix. These extracted characteristics then serve as
inputs to the perturbation network, assisting in computing the values of
$\epsilon_{i}$. Within our framework, the perturbation network employs a two-
layer MLP222A comparative analysis of alternative architectures for the
perturbation network is detailed in Section V of the Appendix., known
theoretically as a universal approximator for nearly any continuous function.
Its output passes through a Tanh function to constrain the range of
$\epsilon_{i}$ within $(-1,1)$.
Input: Training data $\mathcal{D}^{tr}$, metadata ${\mathcal{D}}^{meta}$,
batch sizes $n$ and $m$, ending iterations $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$.
Output: Learned classifier parameter ${\boldsymbol{\Phi}}$.
1Initialize classifier parameter ${\boldsymbol{\Phi}}^{1}$ and perturbation
network parameter $\boldsymbol{\Omega}^{1}$;
2for _$t\leq\mathcal{T}_{1}$_ do
3 Sample $\\{(\boldsymbol{x}_{i},{y}_{i})\\}_{i=1}^{n}$ from
$\mathcal{D}^{tr}$;
4 Update
${\boldsymbol{\Phi}}^{t+1}\leftarrow{\boldsymbol{\Phi}}^{t}-\eta_{1}\frac{1}{n}\sum\nolimits_{i=1}^{n}\nabla_{{{\boldsymbol{\Phi}}}}\ell^{{CE}}_{i}$;
5
6 end for
7for _$\mathcal{T}_{1} <t\leq\mathcal{T}_{2}$_ do
8 Sample $\\{(\boldsymbol{x}_{i},{y}_{i})\\}_{i=1}^{n}$ from
$\mathcal{D}^{tr}$;
9 Sample $\\{(\boldsymbol{x}^{meta}_{i},{y}^{meta}_{i})\\}_{i=1}^{m}$ form
$\mathcal{D}^{meta}$;
10
11 Obtain current covariance matrices ${\boldsymbol{\Sigma}}$;
12 Formulate $\overline{{\boldsymbol{\Phi}}}^{t+1}$ by Eq. (7);
13 Update $\boldsymbol{\Omega}^{t+1}$ by Eq. (8);
14 Update ${\boldsymbol{\Sigma}}^{t+1}$ by Eq. (9);
15 Update ${\boldsymbol{\Phi}}^{t+1}$ by Eq. (10);
16
17 end for
Algorithm 1 Algorithm of Meta-IADA
Meta-IADA employs a meta-learning-based strategy to iteratively update the
classifier and the perturbation network. This involves leveraging a high-
quality (clean and balanced) yet small meta dataset,
$\mathcal{D}^{{meta}}\\!=\\!\\{(\boldsymbol{x}_{i}^{{meta}},y_{i}^{{meta}})\\}_{i=1}^{M}$.
Additionally, to mitigate the accuracy compromise within
${\boldsymbol{\Sigma}}_{y_{i}}$ for minor classes due to the constraints of
limited training data, their values are further updated on the metadata. Let
$\boldsymbol{\Omega}$ represent the parameters of the perturbation network.
The optimization process within Meta-IADA is detailed as follows:
Initially, the parameters of the classifier, $\boldsymbol{\Phi}$, are updated
using stochastic gradient descent (SGD) on a mini-batch of training samples
$\\{(\boldsymbol{x}_{i},{y}_{i})\\}_{i=1}^{n}$ with the following objective:
$\begin{aligned}
\overline{{\boldsymbol{\Phi}}}^{t+1}\leftarrow{\boldsymbol{\Phi}}^{t}-\eta_{1}\frac{1}{n}\sum\nolimits_{i=1}^{n}\nabla_{{{\boldsymbol{\Phi}}}}\ell^{{IADA}}\big{(}{\mathcal{F}}(\boldsymbol{x}_{i};{\boldsymbol{\Phi}}^{t}),y_{i};{\boldsymbol{\Sigma}}^{t}_{y_{i}},\epsilon({\boldsymbol{f}}_{i}^{t},{\boldsymbol{\Omega}}^{t})\big{)},\end{aligned}$
(7)
where $\eta_{1}$ is the step size. Additionally, $\boldsymbol{f}_{i}^{t}$
denotes the concatenated vector containing the extracted training
characteristics for $\boldsymbol{x}_{i}$ at the $t$th iteration. Subsequently,
leveraging the optimized $\overline{\boldsymbol{\Phi}}^{t+1}$, the parameter
update in the perturbation network, $\boldsymbol{\Omega}$, entails utilizing a
mini-batch of metadata
$\\{(\boldsymbol{x}_{i}^{{meta}},{y}_{i}^{{meta}})\\}_{i=1}^{m}$:
$\begin{aligned}
\boldsymbol{\Omega}^{t+1}&\leftarrow\boldsymbol{\Omega}^{t}-\eta_{2}\frac{1}{m}\sum\nolimits_{i=1}^{m}\nabla_{\boldsymbol{\Omega}}\ell^{{CE}}\big{(}{\mathcal{F}}(\boldsymbol{x}_{i}^{{meta}};\overline{\boldsymbol{\Phi}}^{t+1}),y_{i}^{{meta}}\big{)},\end{aligned}$
(8)
where $\eta_{2}$ denotes the step size. Simultaneously, the optimization of
covariance matrices is conducted using the metadata, outlined as follows:
$\begin{aligned}
{\boldsymbol{\Sigma}}^{t+1}&\leftarrow{\boldsymbol{\Sigma}}^{t}-\eta_{2}\frac{1}{m}\sum\nolimits_{i=1}^{m}\nabla_{\boldsymbol{\Sigma}}\ell^{{CE}}\big{(}{\mathcal{F}}(\boldsymbol{x}_{i}^{{meta}};\overline{\boldsymbol{\Phi}}^{t+1}),y_{i}^{{meta}}\big{)}.\end{aligned}$
(9)
Finally, leveraging the computed perturbations and updated covariance
matrices, we proceed to update the parameters $\boldsymbol{\Phi}$ of the
classifier in the following manner:
$\begin{aligned}
{\boldsymbol{\Phi}}^{t+1}&\leftarrow{\boldsymbol{\Phi}}^{t}-\eta_{1}\frac{1}{n}\sum\nolimits_{i=1}^{n}\nabla_{{{\boldsymbol{\Phi}}}}\ell^{{IADA}}\big{(}{\mathcal{F}}(\boldsymbol{x}_{i};\boldsymbol{\Phi}^{t}),y_{i};{\boldsymbol{\Sigma}}^{t+1}_{y_{i}},\epsilon({\boldsymbol{f}}_{i}^{t},{\boldsymbol{\Omega}}^{t+1})\big{)}.\end{aligned}$
(10)
Similar to MetaAug Li et al. (2021), to acquire better-generalized
representations, the classifier is initially trained using vanilla CE loss,
followed by trained with Meta-IADA. The algorithm for Meta-IADA is delineated
in Algorithm 1.
Dataset | CIFAR10 | CIFAR100
---|---|---
Imbalance ratio | 100:1 | 10:1 | 100:1 | 10:1
Class-Balanced CE† Cui et al. (2019) | 72.68 | 86.90 | 38.77 | 57.57
Class-Balanced Focal† Cui et al. (2019) | 74.57 | 87.48 | 39.60 | 57.99
LDAM-DRW† Cao et al. (2019) | 78.12 | 88.37 | 42.89 | 58.78
Meta-Weight-Net† Shu et al. (2019) | 73.57 | 87.55 | 41.61 | 58.91
De-confound-TDE∗ Tang et al. (2020) | 80.60 | 88.50 | 44.10 | 59.60
LA Menon et al. (2021) | 77.67 | 88.93 | 43.89 | 58.34
MiSLAS∗ Zhong et al. (2021) | 82.10 | 90.00 | 47.00 | 63.20
LADE Hong et al. (2021) | 81.17 | 89.15 | 45.42 | 61.69
MetaSAug† Li et al. (2021) | 80.54 | 89.44 | 46.87 | 61.73
LPL∗ Li et al. (2022) | 77.95 | 89.41 | 44.25 | 60.97
RISDA∗ Chen et al. (2022) | 79.89 | 89.36 | 50.16 | 62.38
LDAM-DRW-SAFA∗ Hong et al. (2022) | 80.48 | 88.94 | 46.04 | 59.11
BKD∗ Zhang et al. (2023) | 82.50 | 89.50 | 46.50 | 62.00
Meta-IADA (Ours) | 84.01 | 91.73 | 52.18 | 64.72
Table 2: Accuracy (%) comparison on CIFAR-LT benchmark. The best results are
highlighted in bold. “$\ast$” represents the results in the original paper,
while “$\dagger$” denotes the results in Li et al. (2021). Figure 3: Ratio of
adversarial samples in each class during the last forty epochs on CIFAR10
under imbalance ratios of 10:1 and 100:1. From “C1” to “C10”, the class
proportions progressively decrease.
## 5 Experiments
We experiment across four typical biased learning scenarios, including LT
learning, GLT learning, noisy label learning, and subpopulation shift
learning, involving image and text datasets. The excluded settings and results
(including those on standard datasets) are detailed in the Appendix. The code
for Meta-IADA is available in the supplementary materials.
Method | Accuracy$\uparrow$
---|---
Class-Balanced CE† Cui et al. (2019) | 66.43
Class-Balanced Focal† Cui et al. (2019) | 61.12
LDAM-DRW† Cao et al. (2019) | 68.00
BBN† Zhou et al. (2020) | 66.29
Meta-Class-Weight† Jamal et al. (2020) | 67.55
MiSLAS∗ Zhong et al. (2021) | 71.60
LADE∗ Hong et al. (2021) | 70.00
MetaSAug† Li et al. (2021) | 68.75
LDAM-DRS-SAFA∗ Hong et al. (2022) | 69.78
RISDA∗ Chen et al. (2022) | 69.15
BKD∗ Zhang et al. (2023) | 71.20
Meta-IADA (Ours) | 72.55
Table 3: Accuracy (%) comparison on iNat 2018 dataset.
### 5.1 Long-Tail Learning
Four LT image classification benchmarks, CIFAR-LT Cui et al. (2019), ImageNet-
LT Liu et al. (2019), Places-LT Liu et al. (2019), and iNaturalist (iNat) 2018
Jamal et al. (2020), are evaluated. Additionally, two imbalanced text
classification datasets are included. Due to space limitations, we only
present experiments for CIFAR-LT and iNat in the main text.
#### Experiments on CIFAR-LT Datasets.
We employ the ResNet-32 model He et al. (2016) with an initial learning rate
of 0.1. The training employs the SGD optimizer with a momentum of 0.9 and a
weight decay of $5\\!\times\\!10^{-4}$ on a single GPU, spanning 200 epochs.
The learning rate is decayed by 0.01 at the $160$th and $180$th epochs.
Additionally, the perturbation network is optimized using Adam, with an
initial learning rate of $1\\!\times\\!10^{-3}$. To construct metadata, we
randomly select ten images per class from the validation data. For the
hyperparameters in the IADA loss, $\alpha$ is selected from {0.1, 0.25, 0.5,
0.75, 1}, while keeping $\beta$ fixed at 1.
From the results reported in Table 2, Meta-IADA displays remarkable
superiority over other LT baselines, underscoring its effectiveness in
managing imbalanced class distributions. Additionally, it outperforms previous
implicit augmentation methods, providing evidence for the efficacy of
augmenting samples within their adversarial and anti-adversarial perturbation
distributions. Furthermore, as depicted in Fig. 3, Meta-IADA induces a lower
proportion of adversarial samples in major classes (“C1” to “C5”), whereas
minor classes (“C6” to “C10”) exhibit a higher proportion. This manifests the
model’s increased attention on samples in tail classes. Comparative analyses
of confusion matrices involving CE loss, MetaSAug, and Meta-IADA are presented
in Section V of the Appendix. The results unveil that Meta-IADA remarkably
increases the accuracy of both major and minor classes. Conversely, while
MetaSAug improves the performance of minor categories, it compromises the
accuracy of major classes.
#### Experiments on iNat 2018 Dataset.
The ResNet-50 model serves as the backbone classifier, pre-trained on ImageNet
Russakovsky et al. (2015) and iNat 2017 Horn et al. (2018) datasets. The
perturbation network is optimized using Adam, with an initial learning rate of
$1\\!\times\\!10^{-4}$. Other settings follow those outlined in the MetaSAug
Li et al. (2021) paper. As reported in Table 3, Meta-IADA outperforms other
comparative approaches tailored for LT learning. This suggests that, even in
situations with imbalanced class distributions, employing a sample-level
strategy proves more effective, as finer-grained imbalances within each class
may also exist.
Protocol | CLT | GLT | ALT
---|---|---|---
Metric | Acc.$\uparrow$ | Prec.$\uparrow$ | Acc.$\uparrow$ | Prec.$\uparrow$ | Acc.$\uparrow$ | Prec.$\uparrow$
CE loss‡ | 42.52 | 47.92 | 34.75 | 40.65 | 41.73 | 41.74
MixUp‡ Zhang et al. (2018) | 38.81 | 45.41 | 31.55 | 37.44 | 42.11 | 42.42
LDAM‡ Cao et al. (2019) | 46.74 | 46.86 | 38.54 | 39.08 | 42.66 | 41.80
ISDA Wang et al. (2019) | 42.66 | 44.98 | 36.44 | 37.26 | 43.34 | 43.56
cRT‡ Kang et al. (2020) | 45.92 | 45.34 | 37.57 | 37.51 | 41.59 | 41.43
LWS‡ Kang et al. (2020) | 46.43 | 45.90 | 37.94 | 38.01 | 41.70 | 41.71
De-confound-TDE‡ Tang et al. (2020) | 45.70 | 44.48 | 37.56 | 37.00 | 41.40 | 42.36
BLSoftmax‡ Ren et al. (2020) | 45.79 | 46.27 | 37.09 | 38.08 | 41.32 | 41.37
BBN‡ Zhou et al. (2020) | 46.46 | 49.86 | 37.91 | 41.77 | 43.26 | 43.86
RandAug‡ Cubuk et al. (2020) | 46.40 | 52.13 | 38.24 | 44.74 | 46.29 | 46.32
LA‡ Menon et al. (2021) | 46.53 | 45.56 | 37.80 | 37.56 | - | -
MetaSAug Li et al. (2021) | 48.53 | 54.21 | 40.27 | 44.38 | 47.62 | 48.26
IFL‡ Tang et al. (2022) | 45.97 | 52.06 | 37.96 | 44.47 | 45.89 | 46.42
RISDA Chen et al. (2022) | 46.31 | 51.24 | 38.45 | 42.77 | 43.65 | 43.23
BKD Zhang et al. (2023) | 46.51 | 50.15 | 37.93 | 41.50 | 42.17 | 41.83
Meta-IADA (Ours) | 53.45 | 58.05 | 44.36 | 50.07 | 52.54 | 53.23
Table 4: Accuracy and precision (%) of CLT, GLT, and ALT protocols on the
ImageNet-GLT benchmark. ${\ddagger}$ indicates the results reported in Tang et
al. (2022). Figure 4: Ratio of adversarial samples for noisy and clean samples
in the final forty epochs on CIFAR10 with 20% and 40% flip noise.
### 5.2 Generalized Long-Tail Learning
GLT learning considers both long-tailed class and attribute distributions in
the training data. We employ two GLT benchmarks, ImageNet-GLT, and MSCOCO-GLT
Tang et al. (2022). Each benchmark comprises three protocols, CLT, ALT, and
GLT, showcasing variations in class distribution, attribute distribution, and
combinations of both between training and testing datasets. The ResNeXt-50 Xie
et al. (2017) model is utilized as the backbone network. We train models with
a batch size of 256 and an initial learning rate of 0.1, using SGD with a
weight decay of $5\\!\times\\!10^{-4}$ and a momentum of 0.9. The perturbation
network is optimized using the Adam optimizer, initialized with a learning
rate of $1\\!\times\\!10^{-3}$. The metadata utilized in this experiment is
balanced in both classes and attributes. We adopt the construction method of
Tang et al. Tang et al. (2022), which involves clustering images in each class
into six groups using KMeans with a pre-trained ResNet-50 model. Two images
are then randomly selected from each group and class within the validation
data.
The comparative results for ImageNet-GLT are outlined in Table 4, while those
for MSCOCO-GLT are presented in the Appendix. Meta-IADA demonstrates
substantial enhancements in accuracy and precision across all three protocols,
emphasizing its ability to address distributional skewness in three scenarios,
including attribute imbalance, class imbalance, and their combination. The
efficacy of Meta-IADA stems from its ability to apply adversarial augmentation
to samples from tail classes and those with rare attributes. This
significantly magnifies the impact of these samples on model training,
resulting in improved prediction accuracy. However, methods specifically
designed for LT learning demonstrate subpar performance on the ALT protocol,
primarily due to their class-level characteristics.
Dataset | CIFAR10 | CIFAR100
---|---|---
Noise ratio | 20% | 40% | 20% | 40%
CE loss | 76.85 | 70.78 | 50.90 | 43.02
D2L Ma et al. (2018) | 87.64 | 83.90 | 63.39 | 51.85
Co-teaching Han et al. (2018) | 82.85 | 75.43 | 54.19 | 44.92
GLC Hendrycks et al. (2018) | 89.77 | 88.93 | 63.15 | 62.24
MentorNet Jiang et al. (2018) | 86.41 | 81.78 | 62.00 | 52.71
L2RW Ren et al. (2018) | 87.88 | 85.70 | 57.51 | 51.00
DMI Xu et al. (2019) | 88.43 | 84.00 | 58.87 | 42.95
Meta-Weight-Net Shu et al. (2019) | 90.35 | 87.65 | 64.31 | 58.67
APL Ma et al. (2020) | 87.45 | 81.08 | 59.86 | 53.31
JoCoR Wei et al. (2020) | 90.78 | 83.67 | 65.21 | 46.44
MLC Zheng et al. (2021) | 91.55 | 89.53 | 66.32 | 62.29
MFRW-MES Ricci et al. (2023) | 91.45 | 90.72 | 65.27 | 62.35
Meta-IADA (Ours) | 93.44 | 91.99 | 69.16 | 64.37
Table 5: Accuracy (%) comparison on CIFAR datasets with 20% and 40% flip
noise.
Dataset | CelebA | CMNIST | Waterbirds | CivilComments
---|---|---|---|---
Metric | Avg.$\uparrow$ | Worst$\uparrow$ | Avg.$\uparrow$ | Worst$\uparrow$ | Avg.$\uparrow$ | Worst$\uparrow$ | Avg.$\uparrow$ | Worst$\uparrow$
CORAL∔ Li et al. (2018) | 93.8 | 76.9 | 71.8 | 69.5 | 90.3 | 79.8 | 88.7 | 65.6
IRM∔ Arjovsky et al. (2019) | 94.0 | 77.8 | 72.1 | 70.3 | 87.5 | 75.6 | 88.8 | 66.3
GroupDRO∔ Sagawa et al. (2020) | 92.1 | 87.2 | 72.3 | 68.6 | 91.8 | 90.6 | 89.9 | 70.0
DomainMix∔ Xu et al. (2020) | 93.4 | 65.6 | 51.4 | 48.0 | 76.4 | 53.0 | 90.9 | 63.6
IB-IRM∔ Ahuja et al. (2021) | 93.6 | 85.0 | 72.2 | 70.7 | 88.5 | 76.5 | 89.1 | 65.3
V-REx∔ Krueger et al. (2021) | 92.2 | 86.7 | 71.7 | 70.2 | 88.0 | 73.6 | 90.2 | 64.9
UW∔ Yao et al. (2022) | 92.9 | 83.3 | 72.2 | 66.0 | 95.1 | 88.0 | 89.8 | 69.2
Fish∔ Shi et al. (2022) | 93.1 | 61.2 | 46.9 | 35.6 | 85.6 | 64.0 | 89.8 | 71.1
LISA∔ Yao et al. (2022) | 92.4 | 89.3 | 74.0 | 73.3 | 91.8 | 89.2 | 89.2 | 72.6
Meta-IADA (Ours) | 94.5 | 91.2 | 78.0 | 75.9 | 94.5 | 92.5 | 92.1 | 74.8
Table 6: Comparison of the average and worst-group accuracy (%) on four
subpopulation shift datasets. ${\dotplus}$ indicates the results reported in
Yao et al. (2022).
### 5.3 Noisy Label Learning
We examine two types of label corruptions: uniform and pair-flip noise Shu et
al. (2019), using CIFAR Krizhevsky and Hinton (2009) datasets. The Wide
ResNet-28-10 (WRN-28-10) Zagoruyko and Komodakis (2016) and ResNet-32 models
are utilized for uniform and flip noises, respectively. 1,000 images with
clean labels are selected from the validation set to compile the metadata. The
ResNet settings match those used for CIFAR-LT. WRN-28-10 is trained using SGD
with an initial learning rate of 0.1, a momentum of 0.9, and a weight decay of
$5\\!\times\\!10^{-4}$; the learning rate is decayed by 0.1 at the $50$th and
$55$th epochs during the total 60 epochs. The Adam optimizer, initialized with
a learning rate of $1\\!\times\\!10^{-3}$, is utilized for optimizing the
perturbation network.
Table 5 presents the comparison results for flip noise, while those for
uniform noise are detailed in the Appendix. Meta-IADA consistently achieves
SOTA performance when compared to all other methods, surpassing the highest
accuracy among comparative methods by an average of 2%. These outcomes
underscore its capacity to bolster DNNs’ robustness against noise. An analysis
of Fig. 4 indicates that nearly all noisy samples undergo anti-adversarial
augmentations in Meta-IADA, effectively mitigating the negative impact of
noisy samples on the overall performance of DNNs.
Figure 5: Visualization of instances corresponding to deep features augmented by Meta-IADA. Regularization terms | CIFAR10 | CIFAR100
---|---|---
$\mathcal{G}$ | $\mathcal{R}$ | $\mathscr{F}$ | 10:1 | 100:1 | 10:1 | 100:1
✓ | ✓ | ✓ | 8.17 | 15.99 | 35.28 | 47.82
✗ | ✓ | ✓ | 9.48 | 18.20 | 36.95 | 51.01
✓ | ✗ | ✓ | 10.06 | 19.11 | 37.62 | 52.47
✓ | ✓ | ✗ | 9.77 | 17.36 | 37.18 | 49.87
Table 7: Results of ablation studies using the ResNet-32 model on CIFAR-LT
benchmark. Top-1 error rates are reported.
### 5.4 Subpopulation Shift Learning
We conduct evaluations on four binary classification datasets characterized by
subpopulation shifts: Colored MNIST (CMNIST) Yao et al. (2022), Waterbirds
Sagawa et al. (2020), CelebA Liu et al. (2016), and CivilComments Borkan et
al. (2019). For three image datasets (i.e., CMNIST, Waterbirds, and CelebA),
we utilize ResNet-50 He et al. (2016) as the backbone network, while for the
text dataset, CivilComments, we employ DistilBert Sanh et al. (2019). To
provide a comprehensive assessment, we report both average and worst-group
accuracy. Detailed experimental settings and dataset introductions are
available in Section V of the Appendix.
Based on the results in Table 6, Meta-IADA achieves the highest worst-group
accuracy across the four datasets. This emphasizes its effectiveness in
enhancing performance for underrepresented groups, such as samples in the
landbird class with a water background. Typically, these samples benefit from
adversarial augmentation, enhancing their influence on model training.
Furthermore, except for the Waterbirds dataset, Meta-IADA surpasses other
methods in terms of average accuracy. These findings illustrate Meta-IADA’s
capability to bolster model resilience against subpopulation shifts.
### 5.5 Visualization Results
We utilize the visualization method introduced in ISDA Wang et al. (2019),
projecting the features augmented by Meta-IADA back into the pixel space. The
results, depicted in Fig. 5, highlight the diversity encapsulated within the
generated adversarial and anti-adversarial samples. Additionally, samples
created through adversarial and anti-adversarial augmentation commonly
showcase varying levels of difficulty compared to the original samples. These
modifications are attributed to transformations within the deep feature space,
influenced by semantic vectors associated with attributes such as angles,
colors, and backgrounds.
### 5.6 Ablation and Sensitivity Studies
Ablation studies are conducted to analyze the impact of the three
regularization terms within the IADA loss. The results reported in Table 7,
emphasize the crucial roles played by both the generalization term
$\mathcal{G}$ and the robustness term $\mathcal{R}$. Additionally, in the
context of an imbalanced scenario, the fairness term $\mathscr{F}$ also
demonstrates its significance. Furthermore, sensitivity tests are performed on
the hyperparameters $\alpha$ and $\beta$, which respectively control the
effects of $\mathcal{G}$ and $\mathscr{F}$. As shown in Fig. 6, Meta-IADA
achieves optimal performance when $\alpha$ is around 0.5 and $\beta$
approaches 1. Besides, the stable ranges for $\alpha$ and $\beta$ lie within
$[0.25,0.75]$ and $[0.75,1.25]$, respectively. Based on these findings, we
recommend setting $\alpha\\!=\\!0.5$ and $\beta\\!=\\!1$ for practical
applications.
Figure 6: Results of sensitivity tests on CIFAR datasets with an imbalance
ratio of 10:1, using the ResNet-32 model.
## 6 Conclusion
This paper presents a novel adversarial data augmentation strategy to
facilitate model training across diverse learning scenarios, particularly
those with data biases. This strategy enriches the deep features of samples by
incorporating their adversarial and anti-adversarial perturbation
distributions, dynamically adjusting the learning difficulty of training
samples. Subsequently, we formulate a surrogate loss for our augmentation
strategy and establish a meta-learning framework to optimize classifiers using
this loss. Extensive experiments are conducted across various biased learning
scenarios involving different networks and datasets, showcasing the
effectiveness and broad applicability of our approach.
## References
* Ahuja et al. [2021] Kartik Ahuja, Ethan Caballero, Dinghuai Zhang, Yoshua Bengio, Ioannis Mitliagkas, and Irina Rish. Invariance principle meets information bottleneck for out-of-distribution generalization. In NeurIPS, pages 3438–3450, 2021.
* Arjovsky et al. [2019] Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019.
* Borkan et al. [2019] Daniel Borkan, Lucas Dixon, Jeffrey Sorensen, Nithum Thain, and Lucy Vasserman. Nuanced metrics for measuring unintended bias with real data for text classification. In WWW, pages 491–500, 2019.
* Cao et al. [2019] Kaidi Cao, Colin Wei, Adrien Gaidon, Nikos Arechiga, and Tengyu Ma. Learning imbalanced datasets with label-distribution-aware margin loss. In NeurIPS, pages 1567–1578, 2019.
* Chen et al. [2022] Xiaohua Chen, Yucan Zhou, Dayan Wu, Wanqian Zhang, Yu Zhou, Bo Li, and Weiping Wang. Imagine by reasoning: A reasoning-based implicit semantic data augmentation for long-tailed classification. In AAAI, pages 356–364, 2022.
* Cubuk et al. [2020] Ekin D. Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V. Le. Randaugment: Practical automated data augmentation with a reduced search space. In CVPR Workshops, pages 3008–3017, 2020.
* Cui et al. [2019] Yin Cui, Menglin Jia, Tsung-Yi Lin, Yang Song, and Serge Belongie. Class-balanced loss based on effective number of samples. In CVPR, pages 9268–9277, 2019.
* Han et al. [2018] Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor W. Tsang, and Masashi Sugiyama. Co-teaching: Robust training of deep neural networks with extremely noisy labels. In NeurIPS, pages 8536–8546, 2018.
* He et al. [2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, pages 770–778, 2016.
* Hendrycks et al. [2018] Dan Hendrycks, Mantas Mazeika, Duncan Wilson, and Kevin Gimpel. Using trusted data to train deep networks on labels corrupted by severe noise. In NeurIPS, pages 10477–10486, 2018.
* Hong et al. [2021] Youngkyu Hong, Seungju Han, Kwanghee Choi, Seokjun Seo, Beomsu Kim, and Buru Chang. Disentangling label distribution for long-tailed visual recognition. In CVPR, pages 6626–6636, 2021.
* Hong et al. [2022] Yan Hong, Jianfu Zhang, Zhongyi Sun, and Ke Yan. Safa: Sample-adaptive feature augmentation for long-tailed image classification. In ECCV, pages 587–603, 2022.
* Horn et al. [2018] Grant Van Horn, Oisin Mac Aodha, Yang Song, Yin Cui, Chen Sun, Alex Shepard, Hartwig Adam, Pietro Perona, and Serge Belongie. The inaturalist species classification and detection dataset. In CVPR, pages 8769–8778, 2018.
* Jamal et al. [2020] Muhammad Abdullah Jamal, Matthew Brown, Ming-Hsuan Yang, Liqiang Wang, and Boqing Gong. Rethinking class-balanced methods for long-tailed visual recognition from a domain adaptation perspective. In CVPR, pages 7610–7619, 2020.
* Jiang et al. [2018] Lu Jiang, Zhengyuan Zhou, Thomas Leung, Li-Jia Li, and Fei-Fei Li. Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels. In ICML, pages 2304–2313, 2018.
* Kang et al. [2020] Bingyi Kang, Saining Xie, Marcus Rohrbach, Zhicheng Yan, Albert Gordo, Jiashi Feng, and Yannis Kalantidis. Decoupling representation and classifier for long-tailed recognition. In ICLR, 2020.
* Krizhevsky and Hinton [2009] Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, 2009.
* Krueger et al. [2021] David Krueger, Ethan Caballero, Joern-Henrik Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi Le Priol, and Aaron Courville. Out-of-distribution generalization via risk extrapolation (rex). In ICML, pages 5815–5826, 2021.
* Lee et al. [2023] Sungyoon Lee, Hoki Kim, and Jaewook Lee. Graddiv: Adversarial robustness of randomized neural networks via gradient diversity regularization. IEEE TPAMI, 45(2):2645–2651, 2023.
* Li et al. [2018] Haoliang Li, Sinno Jialin Pan, Shiqi Wang, and Alex C. Kot. Domain generalization with adversarial feature learning. In CVPR, pages 5400–5409, 2018.
* Li et al. [2021] Shuang Li, Kaixiong Gong, Chi-Harold Liu, Yulin Wang, Feng Qiao, and Xinjing Cheng. Metasaug: Meta semantic augmentation for long-tailed visual recognition. In CVPR, pages 5208–5217, 2021.
* Li et al. [2022] Mengyang Li, Fengguang Su, Ou Wu, et al. Logit perturbation. In AAAI, pages 1359–1366, 2022.
* Liu et al. [2016] Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In ICCV, pages 3730–3738, 2016.
* Liu et al. [2019] Ziwei Liu, Zhongqi Miao, Xiaohang Zhan, Jiayun Wang, Boqing Gong, and Stella X. Yu. Large-scale long-tailed recognition in an open world. In CVPR, pages 2537–2546, 2019.
* Ma et al. [2018] Xingjun Ma, Yisen Wang, Michael E. Houle, Shuo Zhou, Sarah Erfani, Shutao Xia, Sudanthi Wijewickrema, et al. Dimensionality-driven learning with noisy labels. In ICML, pages 3355–3364, 2018.
* Ma et al. [2020] Xingjun Ma, Hanxun Huang, Yisen Wang, Simone Romano, Sarah Erfani, and James Bailey. Normalized loss functions for deep learning with noisy labels. In ICML, pages 6543–6553, 2020.
* Madry et al. [2018] Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In ICLR, 2018.
* Maharana et al. [2022] Kiran Maharana, Surajit Mondal, and Bhushankumar Nemade. A review: Data pre-processing and data augmentation techniques. Global Transitions Proceedings, 3(1):91–99, 2022.
* Menon et al. [2021] Aditya Krishna Menon, Sadeep Jayasumana, Ankit Singh Rawat, Himanshu Jain, Andreas Veit, and Sanjiv Kumar. Long-tail learning via logit adjustment. In ICLR, 2021.
* Ren et al. [2018] Mengye Ren, Wenyuan Zeng, Bin Yang, and Raquel Urtasun. Learning to reweight examples for robust deep learning. In ICML, pages 4334–4343, 2018.
* Ren et al. [2020] Jiawei Ren, Cunjun Yu, Shunan Sheng, Xiao Ma, Haiyu Zhao, Shuai Yi, and Hongsheng Li. Balanced meta-softmax for long-tailed visual recognition. In NeurIPS, pages 4175–4186, 2020.
* Ricci et al. [2023] Simone Ricci, Tiberio Uricchio, and Alberto Del Bimbo. Meta-learning advisor networks for long-tail and noisy labels in social image classification. ACM TOMM, 19(5s):1–23, 2023.
* Russakovsky et al. [2015] Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, and Michael Bernstein. Imagenet large scale visual recognition challenge. IJCV, 115(3):211–252, 2015.
* Sagawa et al. [2020] Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, et al. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. In ICLR, 2020.
* Sanh et al. [2019] Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf. Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter. arXiv preprint arXiv:1910.01108, 2019.
* Shi et al. [2022] Yuge Shi, Jeffrey Seely, Philip H.S. Torr, N. Siddharth, Awni Hannun, Nicolas Usunier, and Gabriel Synnaeve. Gradient matching for domain generalization. In ICLR, 2022.
* Shu et al. [2019] Jun Shu, Qi Xie, Lixuan Yi, Qian Zhao, Sanping Zhou, Zongben Xu, and Deyu Meng. Meta-weight-net: Learning an explicit mapping for sample weighting. In NeurIPS, pages 1919–1930, 2019.
* Tang et al. [2020] Kaihua Tang, Jianqiang Huang, and Hanwang Zhang. Long-tailed classification by keeping the good and removing the bad momentum causal effect. In NeurIPS, pages 1513–1524, 2020.
* Tang et al. [2022] Kaihua Tang, Mingyuan Tao, Jiaxin Qi, Zhenguang Liu, and Hanwang Zhang. Invariant feature learning for generalized long-tailed classification. In ECCV, pages 709–726, 2022.
* Taylor and Nitschke [2018] Luke Taylor and Geoff Nitschke. Improving deep learning with generic data augmentation. In SSCI, pages 1542–1547, 2018.
* Wang et al. [2019] Yulin Wang, Xuran Pan, Shiji Song, Hong Zhang, Cheng Wu, and Gao Huang. Implicit semantic data augmentation for deep networks. In NeurIPS, pages 12635–12644, 2019.
* Wei et al. [2020] Hongxin Wei, Lei Feng, Xiangyu Chen, and Bo An. Combating noisy labels by agreement: A joint training method with co-regularization. In CVPR, pages 13723–13732, 2020.
* Xie et al. [2017] Saining Xie, Ross Girshick, Piotr Dollár, Zhuowen Tu, and Kaiming He. Aggregated residual transformations for deep neural networks. In CVPR, pages 5987–5995, 2017.
* Xu and Zhao [2023] Xiaogang Xu and Hengshuang Zhao. Universal adaptive data augmentation. In IJCAI, pages 1596–1603, 2023.
* Xu et al. [2019] Yilun Xu, Peng Cao, Yuqing Kong, and Yizhou Wang. L_dmi: A novel information-theoretic loss function for training deep nets robust to label noise. In NeurIPS, pages 6225–6236, 2019.
* Xu et al. [2020] Minghao Xu, Jian Zhang, Bingbing Ni, Teng Li, Chengjie Wang, Qi Tian, and Wenjun Zhang. Adversarial domain adaptation with domain mixup. In AAAI, pages 6502–6509, 2020.
* Xu et al. [2021] Han Xu, Xiaorui Liu, Yaxin Li, Anil Jain, and Jiliang Tang. To be robust or to be fair: Towards fairness in adversarial training. In ICML, pages 11492–11501, 2021.
* Yao et al. [2022] Huaxiu Yao, Yu Wang, Sai Li, Linjun Zhang, Weixin Liang, James Zou, and Chelsea Finn. Improving out-of-distribution robustness via selective augmentation. In ICML, pages 25407–25437, 2022.
* Zagoruyko and Komodakis [2016] Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.
* Zhang et al. [2018] Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz. Mixup: Beyond empirical risk minimization. In ICLR, 2018.
* Zhang et al. [2023] Shaoyu Zhang, Chen Chen, Xiyuan Hu, and Silong Peng. Balanced knowledge distillation for long-tailed learning. Neurocomputing, 527:36–46, 2023.
* Zheng et al. [2021] Guoqing Zheng, Ahmed Hassan Awadallah, and Susan Dumais. Meta label correction for noisy label learning. In AAAI, pages 11053–11061, 2021.
* Zhong et al. [2021] Zhisheng Zhong, Jiequan Cui, Shu Liu, and Jiaya Jia. Improving calibration for long-tailed recognition. In CVPR, pages 16489–16498, 2021.
* Zhou et al. [2020] Boyan Zhou, Quan Cui, Xiu-Shen Wei, and Zhao-Min Chen. Bbn: Bilateral-branch network with cumulative learning for long-tailed visual recognition. In CVPR, pages 9719–9728, 2020.
* Zhou et al. [2023] Xiaoling Zhou, Nan Yang, and Ou Wu. Combining adversaries with anti-adversaries in training. In AAAI, pages 11435–11442, 2023.
* Zhu et al. [2021] Jianing Zhu, Jingfeng Zhang, Bo Han, Tongliang Liu, Gang Niu, Hongxia Yang, Mohan Kankanhalli, and Masashi Sugiyama. Understanding the interaction of adversarial training with noisy labels. arXiv preprint arXiv:2102.03482, 2021.
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# Dyn-ASR: Compact, Multilingual Speech Recognition via Spoken Language and
Accent Identification
###### Abstract
Running automatic speech recognition (ASR) on edge devices is non-trivial due
to resource constraints, especially in scenarios that require supporting
multiple languages. We propose a new approach to enable multilingual speech
recognition on edge devices. This approach uses both language identification
and accent identification to select one of multiple monolingual ASR models on-
the-fly, each fine-tuned for a particular accent. Initial results for both
recognition performance and resource usage are promising with our approach
using less than 1/12th of the memory consumed by other solutions.
Index Terms— language identification, accent identification, speech
recognition, multilingual methods
## 1 Introduction
Speech-driven interfaces have become increasingly common, and COVID-19 has
accelerated interest in speech as one of the primary modes of interaction with
a variety of systems [1]. In the case of kiosks used for ticketing, banking or
informational purposes, developers of speech interfaces need to minimize
latency and resource consumption while maintaining a high level of performance
in speech-related tasks, like automatic speech recognition (ASR). At the same
time, kiosks placed in transportation or business hubs (e.g., airports or
international hotels) need to be able to support multiple languages.
Running any speech or language model on an edge system is non-trivial due to
the size of parameter sets in modern speech and language models and the
accelerated hardware needed to run many of these neural network based models.
If multiple models are needed for multiple languages, resource consumption [2]
and increased inference times could very easily prevent developers from
deploying applications in any environment, much less at the edge. In fact,
common enterprise ASR systems require developers to deploy separate,
dissimilar instances of model servers for each supported language, which
complicates infrastructure and could result in reliability or maintenance
issues.
Beyond language, accent and other demographic factors have been shown to
dramatically impact ASR performance [3, 4]. Those demographic factors need to
be addressed via larger, more computationally expensive speech models or via
models fine-tuned for particular demographic groups.
In this paper, we implement and test an approach that allows for dynamic usage
of monolingual, accent-specific speech recognition models in a multilingual
context. This dynamic usage of monolingual models reduces the need for higher
end compute and memory, preserves the recognition performance of monolingual
ASR models, and ultimately enables multilingual contactless interactions on
edge devices. In terms of user experience, the proposed methodology also
removes the necessity for an awkward, touch-driven user selection of a
language within an application.
## 2 Related Work
Many speech-driven systems at the edge detect a wake word and transfer audio
to the cloud or another remote endpoint for processing. Such a system may not
be desirable in cases where latency and privacy are a concern. For example,
per Google Cloud Speech-to-Text best practice recommendations [5], 16-bit,
16kHz mono PCM audio data needs to be sent to API endpoints with a frame size
of 100ms. Thus, round trip latency would be around 4s for an upload of around
3.2KB/second, which is beyond acceptable levels for many applications.
In terms of language support, developers often leverage slot-filling. For
example, developers might require a user to select a preferred language via a
touchscreen or speak the name of their preferred language. Both of these
interactions could be awkward as they represent interactions that do not
naturally occur in conversation. Moreover, the former implementation prevents
truly contactless interactions, and the latter presents challenges in dealing
with alternate language names, accents, and demographic factors.
Researchers have tried to extend ASR systems that rely on phonetically
inspired acoustic models to support multiple languages. Certain of these
extensions pool phonemes from all languages into a single set and others
manage separate sets (see [6] and references therein). Either approach
requires the management of expertly crafted linguistic information, and the
burden of curating that information grows with the number of languages.
There have also been an increasing number of attempts to create end-to-end
(E2E) models that recognize speech in multiple languages. For example, [7]
utilizes a single model which is trained while sharing parameters across 51
languages, and the model ends up containing over a billion parameters. The
scale of this type of model would make it a challenge to utilize on edge
devices because of resource consumption and/or the need for specialized
hardware accelerators. Other attempts at E2E multilingual ASR exhibit similar
characteristics [8].
In [9], the authors use an E2E architecture called RNN Transducer (RNN-T) that
shows promise in edge applications. A larger, adapted version of RNN-T takes
as input a vector representing a particular one of 9 supported languages. The
authors assume that the language is either specified manually by a user of the
model or determined automatically from a language identification system, but
they do not integrate any specific spoken language identification system.
In still another vein of research [10, 11, 12, 13, 14, 15], researchers have
tried to integrate spoken language identification directly into a joint speech
recognition and language ID model. Generally, this work adds the prediction of
one or more language identifications into the prediction of other outputs,
such as decoded characters. While these models can reduce the overall number
of models needed to support multilingual ASR, they can exhibit degradation in
performance for some or all of the supported languages in comparison to the
performance of corresponding
monolinguhttps://www.overleaf.com/project/6020b5d855cbeaf4a4bfea22al ASR
models. Moreover, the authors are not aware of any of these studies that
integrates identifications of accent
intohttps://www.overleaf.com/project/6020b5d855cbeaf4a4bfea22f speech
recognition models.
While the present study is related to recent work in spoken language
identification [16] and E2E ASR, it capitalizes on a new, simultaneous
combination of spoken language identification, spoken accent identification,
and fine-tuned E2E ASR, which was not considered in these earlier studies. In
particular, we investigate the efficacy of utilizing such a combination on
edge devices for high quality multilingual ASR.
## 3 Approach
An overview of the proposed Dynamic ASR (or Dyn-ASR, which we pronounce
”dinosaur”) to processing multilingual speech is presented in Fig. 1. In the
approach, we assume WAV file inputs which are first pre-processed to
normalize, trim silence from, and format the audio. In the case of audio fed
to speech recognition models, the audio is formatted to 16-bits and 16kHz. In
case of audio fed to language identification and accent identification models,
the audio is formatted to 16-bits and 8kHz, and we artificially repeat the
input audio to fill at least 10 seconds.
Fig. 1: The proposed processing steps and flow for recognizing multiple
languages.
After pre-processing, language and accent identification is performed. For
both language and accent identification we utilize a model with two LSTM
layers, each having 200 units and each followed by batch normalization. One
such model is used to classify the input audio into a language class. Then, we
utilize a separate accent identification model (corresponding to the
identified language) to further classify the input audio into an accent class.
An ASR model is trained for each language and accent pair that is to be
supported by the system. We fine-tune these language and accent specific ASR
models from general (i.e., not accent specific) ASR models. Because we are
targeting edge applications, we experimented with several different
phonetically inspired and E2E model architectures that were optimized for edge
devices using OpenVINO [17] and/or compact by nature. These models included
Deep Neural Network (DNN) acoustic models and RNN-T, Conformer [18],
DeepSpeech [19], and QuartzNet [20] E2E models. In the end, we found that the
Conformer and/or QuartzNet E2E models fulfilled our constraints in terms of
ASR performance and system resource consumption.
Depending on the size of the models and system constraints, each of the ASR
models can be loaded into memory when an application implementing the Fig. 1
process starts, or each ASR model could be loaded into memory on-the-fly. In
any event, the model corresponding to the identified language and accent pair
is dynamically chosen or loaded into memory after the language and accent is
identified. In this way, multiple compact monolingual models can be utilized
dynamically to recognize speech in multiple languages without significantly
sacrificing the performance of speech recognition or exceeding edge device
memory or processor constraints.
## 4 Experiments
To test the Fig. 1 process for multilingual speech recognition on edge
devices, we evaluated (i) the performance of our language and accent
identification models; (ii) the performance of our language and accent
specific ASR models; and (iii) the performance of an implementation of the
full Fig. 1 process with respect to resource consumption.
### 4.1 Data
In the following, we trained and tested our models/methods on English, Tamil,
and Mandarin speech data. The English data was segmented into 8 accents
(Scotland, Australia, England, India, USA, China, Malaysia, other) and the
Mandarin data was segmented into 3 accents (Mainland, Taiwan, Hong Kong).
For transcribed speech data with corresponding language and accent labels, we
relied on Mozilla’s Common Voice data, the Speech Accent Archive from George
Mason University (SAA), and the Singapore National Speech Corpus (NSC). For
additional Tamil speech data, we used Microsoft’s Indian Language Speech
Corpus. We used the SoX utility to normalize the speech files to 16kHz, 16-bit
WAV files for training and testing ASR tasks and 8kHz, 16-bit WAV files for
training and testing language and accent identification tasks.
### 4.2 LID and accent identification
As mentioned in Section 3, we utilize one LSTM-based model for language
identification and one LSTM-based model per language for accent
identification. For our combination of English, Tamil, and Mandarin, that
means that we have 1 spoken language identification model and 2 spoken accent
identification models (one for English and one for Mandarin). We sampled
38,400 samples per language to train the models. To train the accent
identification models we utilized rejection sampling due to the unbalanced
nature of the accent data.
Our language identification model gives 84.99 % accuracy on the 3 language
classes. On English accents, we achieve 74.41 % accuracy across the 8 accents,
and we achieve 79.83 % accuracy across the 3 Mandarin accents. The more
crucial step in the Fig. 1 approach is language identification, because
language identification determines if the ASR model used will correspond to
the spoken language or another language entirely. Correct accent
identification will further improve recognition accuracy, but to a lesser
degree. Our results here show that executing a spoken language identification
model prior to selection of ASR model could result in choosing a proper model
for at least 8-9 out of 10 inferences. Additionally, we found that using a
single model for LID and accent identification would not achieve comparable
accuracy on a similarly-sized model. Using a larger model for combined LID and
accent identification would also slow down the time-to-ASR for the combined
system.
### 4.3 Fine-tuned ASR
Assuming a proper language identification, we also wanted to validate the idea
that switching between monolingual ASR models (each fine-tuned for a
particular accent) could both: (i) outperform individual models trained on
data corresponding to multiple accents; and (ii) allow us to avoid more
complicated and/or larger multi-accent data and models. We created a test set
of Indian, Chinese, and Malaysian accented English by selecting these accents
out of the SAA. We then evaluated ASR models fine-tuned on each of these
accents alongside publicly available pre-trained models. For this evaluation,
we chose English because of the availability of multiple pre-trained models
for comparison and because it is one of the languages considered in our other
experiments.
The ASR models we fine-tuned were based on the QuartzNet architecture and
fine-tuned on Indian, Chinese, and Malaysian accented English data from the
Singapore National Speech Corpus. When evaluating these models (collectively
referred to below as the models of the Dynamic ASR system, or Dyn-ASR), we
loaded and utilized each of the models for the corresponding annotated accent.
This simulates the best case scenario when loading language and accent
specific models in the process illustrated in Fig. 1. Of course in any
implementation of the Fig. 1, the performance of the Dyn-ASR models will
depend on the performance of the language and accent identification models,
but this evaluation gives us a baseline for evaluating the set of ASR models
themselves.
The pre-trained models that we used as a reference are DeepSpeech trained on
US English (DS), QuartzNet trained on LibriSpeech (QN-LS), and QuartzNet
trained on multiple accents (QN-Multi). The results of this comparison are
presented in Table 1.
Table 1: Accent Specific ASR Results. All results are word error rates (WERs) per accent and per model or set of models. Model(s): | DS | QN-LS | QN-Multi | Dyn-ASR
---|---|---|---|---
India | 45.93 | 32.15 | 16.02 | 15.89
China | 57.10 | 44.01 | 27.45 | 26.05
Malaysia | 42.75 | 19.75 | 13.59 | 11.59
### 4.4 Resource consumption
(a) CPU Usage
(b) Memory Usage
(c) Inference Time
Fig. 2: Resource usage results for Dyn-ASR, VOSK-Server (VS-EN, VS-CMN) and
Deepspech (MDS-EN, PDS-CMN). The Dyn-ASR solution uses minimal memory and has
lowest inference time
To evaluate resource consumption, we created an implementation of the Fig. 1
approach for English and Mandarin. We compare the resource consumption of this
implementation (Dyn-ASR below) with the Vosk speech recognition toolkit server
[21] (both an English instance, VS-EN, and a Mandarin instance, VS-CMN),
Mozilla’s DeepSpeech implementation trained on US English (MDS-EN) [19], and
PaddlePaddle’s DeepSpeech implementation trained on Mandarin (PDS-CMN). Note,
the authors had difficulty in finding any practical, publicly available system
natively supporting multilingual ASR models or integrating spoken language
identification. As such, multiple instances and versions of these systems had
to be deployed, which demonstrates the operational barriers to practically
deploying a multilingual ASR system.
While there are portable versions of the Vosk servers for each language, we
picked the server version that would give the best quality speech recognition
results. We utilized two languages (English and Mandarin), each with two
accents (US and Chinese accented English and Mainland and Taiwanese accented
Mandarin) respectively for the input audio data. Table 2 includes the system
resource consumption for each solution.
Table 2: Storage and Memory in MB used by each solution. Dyn-ASR utilizes only 10 % and 25 % of highest storage and memory consumed by the other solutions. DeepSpeech models are not loaded in memory at installation. System | Dyn-ASR | VS-EN | VS-CMN | MDS- EN | PDS- CMN
---|---|---|---|---|---
Storage | 317 | 3200 | 426 | 1000 | 2800
Memory | 775 | 3049 | 469 | NA | 3130
To ensure that we could provide each solution with whatever resources it could
consume, we ran all of the ASR solutions on a Core i9 System (i9-7920X CPU)
which has 12 cores and 24 threads with 64GB of system memory and 500GB of
storage. All the audio file inputs were of type 16kHz, 16-bit PCM mono.
Results were captured in terms of the number of CPU cores utilized, memory
usage and total inference time and are included in Fig. 2
As shown in Fig. 2 part a, the Dyn-ASR container has not been pinned to a CPU
core, and thus it ends up using as many cores as needed to complete inference
at a constant time of less than 1 second (see Fig. 2 part c). It also uses
minimal incremental memory as shown in Fig. 2 part b (around 10MB). A
combination of the VS-EN + VS-CMN systems or the MDS-EN + PDS-CMN systems
would need to be assembled to match the multilingual ASR capabilities of Dyn-
ASR, yet any of these combinations would exceed the memory usage of the the
example Dyn-ASR system and increase the complexity of deployed infrastructure.
Further, neither of these combinations (VS-EN + VS-CMN or MDS-EN + PDS-CMN)
would solve the problem of selecting the correct ASR model corresponding to
the input language, which functionality is natively rolled into the Dyn-ASR
system. These characteristics together make the Dyn-ASR approach appealing for
edge deployments.
## 5 Conclusions and Future Work
We introduced a new approach to multilingual speech recognition that
selectively uses monolingual ASR models fine-tuned for particular accents. The
particular recognition models used for each inference is determined on-the-fly
using a language identification model and an accent identification model. An
implementation of this approach for English and Mandarin behaved favorably in
terms of resource consumption as compared to other publicly available ASR
solutions and also shows promise in terms of recognition performance. This
work explored certain model architectures, but we are exploring still other
architectures along with further optimization using Intel’s OpenVINO toolkit.
The authors would also like to integrate a step in the processing that uses a
text-based model and/or probabilities from the language/ accent identification
to deal with misidentified languages.
## References
* [1] K. Scates, “The growth of contactless and voice interfaces in a post-pandemic world,” Available at https://blog.soundhound.com/the-growth-of-voice-user-interfaces-in-a-post-pandemic-world-8bded452520b (2020/04/09).
* [2] R. Krishna, S. Mahlke, and T. Austin, “Insights into the memory demands of speech recognition algorithms,” in _Proc. of the 2nd Annual Workshop on Memory Performance Issues_ , 2002.
* [3] M. K. Audrey Mbogho, “The impact of accents on automatic recognition of south african english speech: A preliminary investigation,” _SAICSIT ’10_ , vol. 10, pp. 187–192, 2010.
* [4] J. Meyer, L. Rauchenstein, J. D. Eisenberg, and N. Howell, “Artie bias corpus: An open dataset for detecting demographic bias in speech applications,” in _Proceedings of the 12th Language Resources and Evaluation Conference_. Marseille, France: European Language Resources Association, May 2020, pp. 6462–6468. [Online]. Available: https://www.aclweb.org/anthology/2020.lrec-1.796
* [5] G. Cloud, “Ai and machine learning best practices guide,” Available at https://cloud.google.com/speech-to-text/docs/best-practices#frame_size (2020/09/18).
* [6] X. Li, S. Dalmia, J. Li, M. R. Lee, P. Littell, J. Yao, A. Anastasopoulos, D. R. Mortensen, G. Neubig, A. Black, and F. Metze, “Universal phone recognition with a multilingual allophone system,” _ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pp. 8249–8253, 2020.
* [7] V. Pratap, A. Sriram, P. Tomasello, A. Hannun, V. Liptchinsky, G. Synnaeve, and R. Collobert, “Massively multilingual asr: 50 languages, 1 model, 1 billion parameters,” _ArXiv_ , vol. abs/2007.03001, 2020.
* [8] S. Watanabe, T. Hori, and J. Hershey, “Language independent end-to-end architecture for joint language identification and speech recognition,” _2017 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU)_ , pp. 265–271, 2017.
* [9] A. Kannan, A. Datta, T. Sainath, E. Weinstein, B. Ramabhadran, Y. Wu, A. Bapna, Z. Chen, and S. Lee, “Large-scale multilingual speech recognition with a streaming end-to-end model,” in _INTERSPEECH_ , 2019.
* [10] H. Seki, S. Watanabe, T. Hori, J. L. Roux, and J. Hershey, “An end-to-end language-tracking speech recognizer for mixed-language speech,” _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pp. 4919–4923, 2018.
* [11] S. Punjabi, H. Arsikere, Z. Raeesy, C. Chandak, N. A. Bhave, A. Bansal, M. Müller, S. Murillo, A. Rastrow, S. Garimella, R. Maas, M. Hans, A. Mouchtaris, and S. Kunzmann, “Streaming end-to-end bilingual asr systems with joint language identification,” _ArXiv_ , vol. abs/2007.03900, 2020\.
* [12] J. Weiner, N. T. Vu, D. Telaar, F. Metze, T. Schultz, D. Lyu, C. E. Siong, and H. Li, “Integration of language identification into a recognition system for spoken conversations containing code-switches,” in _SLTU_ , 2012.
* [13] K. Li, J. Li, G. Ye, R. Zhao, and Y. Gong, “Towards code-switching asr for end-to-end ctc models,” _ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pp. 6076–6080, 2019.
* [14] H. Caesar, “Integrating language identification to improve multilingual speech recognition,” 2012.
* [15] J. Gonzalez-Dominguez, D. Eustis, I. Lopez-Moreno, A. Senior, F. Beaufays, and P. Moreno, “A real-time end-to-end multilingual speech recognition architecture,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 9, pp. 749–759, 2015.
* [16] M. Lindgren, “Deep learning for spoken language identification; syväoppiminen puhutun kielen tunnistamisessa,” G2 Pro gradu, diplomityö, Aalto University, 2020-05-19. [Online]. Available: http://urn.fi/URN:NBN:fi:aalto-202006213891
* [17] Y. Gorbachev, M. Fedorov, I. Slavutin, A. Tugarev, M. Fatekhov, and Y. Tarkan, “Openvino deep learning workbench: Comprehensive analysis and tuning of neural networks inference,” _2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW)_ , pp. 783–787, 2019.
* [18] A. Gulati, J. Qin, C.-C. Chiu, N. Parmar, Y. Zhang, J. Yu, W. Han, S. Wang, Z. Zhang, Y. Wu, and R. Pang, “Conformer: Convolution-augmented transformer for speech recognition,” _ArXiv_ , vol. abs/2005.08100, 2020.
* [19] A. Y. Hannun, C. Case, J. Casper, B. Catanzaro, G. Diamos, E. Elsen, R. Prenger, S. Satheesh, S. Sengupta, A. Coates, and A. Ng, “Deep speech: Scaling up end-to-end speech recognition,” _ArXiv_ , vol. abs/1412.5567, 2014\.
* [20] S. Kriman, S. Beliaev, B. Ginsburg, J. Huang, O. Kuchaiev, V. Lavrukhin, R. Leary, J. Li, and Y. Zhang, “Quartznet: Deep automatic speech recognition with 1d time-channel separable convolutions,” _ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pp. 6124–6128, 2020.
* [21] A. VOSK, “Vosk server github project,” Available at https://alphacephei.com/vosk/server (2020/06/21).
|
# Robust density estimation with the ${\mathbb{L}}_{1}$-loss. Applications to
the estimation of a density on the line satisfying a shape constraint
Yannick BARAUD , Hélène HALCONRUY and Guillaume MAILLARD Department of
Mathematics, University of Luxembourg Maison du nombre 6 avenue de la Fonte
L-4364 Esch-sur-Alzette Grand Duchy of Luxembourg<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: June 24, 2022)
###### Abstract.
We solve the problem of estimating the distribution of presumed i.i.d.
observations for the total variation loss. Our approach is based on density
models and is versatile enough to cope with many different ones, including
some density models for which the Maximum Likelihood Estimator (MLE for short)
does not exist. We mainly illustrate the properties of our estimator on models
of densities on the line that satisfy a shape constraint. We show that it
possesses some similar optimality properties, with regard to some global rates
of convergence, as the MLE does when it exists. It also enjoys some adaptation
properties with respect to some specific target densities in the model for
which our estimator is proven to converge at parametric rate. More important
is the fact that our estimator is robust, not only with respect to model
misspecification, but also to contamination, the presence of outliers among
the dataset and the equidistribution assumption. This means that the estimator
performs almost as well as if the data were i.i.d. with density $p$ in a
situation where these data are only independent and most of their marginals
are close enough in total variation to a distribution with density $p$. We
also show that our estimator converges to the average density of the data,
when this density belongs to the model, even when none of the marginal
densities belongs to it. Our main result on the risk of the estimator takes
the form of an exponential deviation inequality which is non-asymptotic and
involves explicit numerical constants. We deduce from it several global rates
of convergence, including some bounds for the minimax ${\mathbb{L}}_{1}$-risks
over the sets of concave and log-concave densities. These bounds derive from
some specific results on the approximation of densities which are monotone,
convex, concave and log-concave. Such results may be of independent interest.
###### Key words and phrases:
Density estimation, robust estimation, shape constraint, total variation loss,
minimax theory
###### 1991 Mathematics Subject Classification:
Primary 62G07, 62G35; Secondary 62C20
This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No 811017
## 1\. Introduction
Estimating a density under a shape constraint has been addressed by many
authors since the pioneer papers by Grenander (1956; 1981), Rao (1969),
Groeneboom (1985) and Birgé (1989) for estimating a nonincreasing density on
$(0,+\infty)$. It is well-known that this problem can elegantly be solved by
the Grenander estimator – see Grenander (1956)– which is the maximum
likelihood estimator (MLE for short) over the set of densities that satisfy
this monotonicity constraint on $(0,+\infty)$. Rao (1969) and Grenander (1956;
1981) established the local asymptotic properties of the Grenander estimator
while Groeneboom (1985) and Birgé (1989) studied its global estimation errors
for the ${\mathbb{L}}_{1}$-loss on some functional classes of interest. Birgé
proved that the uniform risk of the Grenander estimator over the set
${\mathscr{F}}(H,L)$, that consists of nonincreasing densities bounded by
$H>0$ and supported on $[0,L]$ with $L>0$, is of order $(\log(1+HL)/n)^{1/3}$.
Since this rate is optimal, the Grenander estimator performs almost as well
(apart maybe from numerical constants) as a minimax estimator over
${\mathscr{F}}(H,L)$ that would know the values of $H$ and $L$ in advance.
Even more surprising is the fact that the Grenander estimator can converge at
a parametric rate ($1/\sqrt{n}$) when the density is piecewise constant on the
elements of partition of $(0,+\infty)$ into a finite number of intervals – see
Grenander (1981), Groeneboom (1985) and Birgé (1989). As a consequence, the
Grenander estimator can nicely adapt to the specific features of the target
density even though these features are a priori unknown.
Because of these adaptation properties for estimating a monotone density, the
MLE has widely and almost exclusively been used to solve many other density
estimation problems under shape constraints. We refer to Groeneboom et al
(2001) for convex densities and to Balabdaoui and Wellner (2007) and Gao and
Wellner () for $k$-monotone ones. Although the construction of the MLE is not
based on any smoothness assumption in these cases, it still needs to have some
pieces of information on the support of the target density. It was already the
case for the monotonicity constraint since the left-endpoint of the support of
the density needed to be known to build the Grenander estimator. This kind of
prior information might, however, not be available in practice and a more
reasonable assumption would be that only an interval containing this support
be known. Unfortunately, under this weaker assumption the MLE does not exist
and the search of alternative estimators becomes necessary to solve this
issue.
To our knowledge, the first attempt to solve it dates back to Wegman, (1970).
He designed a MLE-type estimator restricted to a class of unimodal densities
that attain their modes on an interval of length not smaller than some
parameter ${\varepsilon}>0$. This parameter needs to be tuned by the
statistician and its choice influences the performance of the resulting
estimator. Wegman and Grenander estimators both converge at the same rate
except on an interval of length $\varepsilon$ around the mode. Birgé, (1997)
proposed a different approach based on data-driven choice of a Grenander
estimator among the collection of those associated to all possible modes. He
proved that the ${\mathbb{L}}_{1}$-risk of the selected estimator is the same
as that of the Grenander estimator that would know the value of the mode in
advance, up to an additional term of order $1/\sqrt{n}$.
The situation is different when the density is assumed to be log-concave on
${\mathbb{R}}$, or more generally on ${\mathbb{R}}^{d}$ with $d>1$. The
construction of the MLE is then free of any assumption on the support of the
density. The study the MLE on the set of log-concave densities has led to an
intensive work. We refer the reader to Dümbgen and K. Rufibach (2009), Doss
and Wellner (2016) , Cule and Samworth (2010), Kim and Samworth (2016) and
Feng et al (2021) as well as the references therein. Kim and Samworth (2016)
described the uniform rates of convergence of the MLE for the squared
Hellinger loss over the class of log-concave densities in dimension
$d\in\\{1,2,3\\}$ and they proved these rates to be minimax (up to a possible
logarithmic factor). Besides, as for the monotonicity constraint in dimension
one, the MLE also possesses for log-concave densities some adaptation
properties: it converges at parametric rate (for the Hellinger loss and up to
a possible logarithmic factor) when the logarithm of the target density is
piecewise affine on a suitable convex subset of ${\mathbb{R}}^{d}$ with
$d\in\\{1,2,3\\}$. This result was established by Kim et al (2018) when $d=1$
and extended to the dimensions $d\in\\{2,3\\}$ by Feng et al (2021). In
dimension $d\geqslant 4$, Kur et al. Kur et al., (2019) showed that the MLE
converges at a minimax rate - up to a logarithm factor - for the Hellinger
loss.
In the one dimensional case, our aim is to design a versatile estimation
strategy that can be applied to a wide variety of density models, including
some for which the MLE does not exist, and that automatically results in
estimators with good estimation properties. In particular, these estimators
should keep the nice minimax and adaptation properties of the MLE, when it
exists, for estimating a density under a shape constraint. They should also
remain stable with respect to a slight departure from the ideal situation
where the data are truly i.i.d. and their density satisfies the required
shape. In particular, the estimator should still perform well when the
equidistribution assumption is slightly violated and the data set contains a
small portion of outliers. It should also perform well when the shape of the
density is slightly different from what was originally expected, that is, when
the true density of the data does not satisfy the shape constraint but is
close enough (with respect to the ${\mathbb{L}}_{1}$-loss) to a density that
does satisfy it. In a nutshell, our aim is to build estimators that are
robust. Except for the Grenander estimator (which is a particular case of a
$\rho$-estimator — see Baraud and Birgé (2018)[Section 6]—, we are not aware
of any result that establishes such robustness properties for the MLE.
Actually, we are not aware of many robust strategies for estimating a density
under a shape constraint. For estimating concave and log-concave densities,
Chan et al (2014) proposed a piecewise linear estimator on a data-driven
partition of ${\mathbb{R}}$ into intervals. Their estimator is minimax optimal
on the sets of concave and log-concave densities and it enjoys some robustness
properties with respect to a departure (in ${\mathbb{L}}_{1}$-distance) of the
true density from the model. Their approach is based on the estimation
procedure described in Devroye and Lugosi (2001) and uses the fact that the
Yatracos class associated to the set of the densities that are piecewise
linear on a partition of the line into a fixed number of intervals is VC.
Despite the desirable properties described above, this estimator does not
possess some of the nice ones that makes the MLE so popular. For estimating a
log-concave density, the MLE converges at global rate of order $n^{-2/5}$ (for
the ${\mathbb{L}}_{1}$ and Hellinger distances) but, as already mentioned, it
also possesses some adaptation properties with respect to these densities the
logarithms of which are piecewise linear. The estimator proposed by Chan et al
does not possess such a property. Besides, their approach provides competitors
to the MLE for some specific density models only. Chan et al’s approach cannot
deal with the estimation of a monotone density on a half-line for example and
therefore cannot be used to provide a surrogate to the Grenander estimator.
In dimension one, Baraud and Birgé (2016)[Section 7] proposed to solve the
problem of robust estimation of a density under a shape constraint by using
$\rho$-estimation. Their results hold for the Hellinger loss while ours is for
the total variation one (TV-loss for short). The estimator we propose is more
specifically designed for this loss and quite surprisingly the risk bounds we
get for the TV-loss are slightly different from those obtained by Baraud and
Birgé for the Hellinger one. We do not know if $\rho$-estimators would satisfy
the same ${\mathbb{L}}_{1}$-risk bounds as those we establish here.
Our procedure shares some similarities with that proposed by Devroye and
Lugosi (2001). When the Yatracos class associated to density model is VC, the
risk bound we establish is similar to theirs except from the fact that we
provide explicit numerical constants. However, unlike them, we also consider
density models for which the Yatracos class is not VC, which is typically the
case for these models of densities that satisfy a shape constraint.
Nevertheless, it is likely that with the same techniques of proofs, we could
establish for Devroye and Lugosi’s estimators the similar results as those we
establish here for ours.
The theory of $\ell$-estimation introduced in Baraud (2021) provides a generic
way of building estimators that possess the robustness properties we are
looking for. Even though the present paper is in the same line, we modify
Baraud’s procedure and establish, for the modified $\ell$-estimator, risk
bounds with numerical constants that are essentially divided by a factor 2 as
compared to his. Another important difference with Baraud’s result lies in the
following fact. When the data are only independent with marginal densities
$p_{1}^{\star},\ldots,p_{n}^{\star}$, we measure the performance of our
density estimator $\widehat{p}$ in terms of its ${\mathbb{L}}_{1}$-distance
$\|p^{\star}-\widehat{p}\|$ between $\widehat{p}$ and the average of the
marginal densities $p^{\star}=n^{-1}\sum_{i=1}^{n}p_{i}^{\star}$. In contrast,
Baraud considered, as a loss function, the average of the
${\mathbb{L}}_{1}$-distances of $\widehat{p}$ to the $p_{i}^{\star}$, i.e. the
quantity $n^{-1}\sum_{i=1}^{n}\left\|{\widehat{p}-p_{i}^{\star}}\right\|$. As
a consequence, unlike Baraud, we can establish the convergence of our
estimator to $p^{\star}$, as soon as its belongs to the model, even in the
unfavourable situation where none of the marginals $p_{i}^{\star}$ belongs to
it.
The risk bounds we obtain hold for very general density models but our
applications focus on the estimation of a density on the line that satisfies a
shape constraint. In a nutshell, we establish the following results which are
to our knowledge new in the literature.
* •
The procedure applies to a large variety of density model including some for
which the MLE does not exist (the set of all monotone densities on a half-
line, the set of all unimodal densities on ${\mathbb{R}}$, the set of all
convex densities on an interval, etc).
* •
The global rates of convergence that we establish for our estimator are
optimal in all the models we consider.
* •
The estimator possesses some adaptation properties: it converges at parametric
rate when the data are i.i.d. with a density that belongs to model and
satisfies some special properties. In particular, our estimator shares similar
adaptation properties as those established for the MLE under a monotonicity or
a log-concavity constraint. We also establish some adaptation properties on
density models on which the MLE does not even exist.
* •
The estimator is robust with respect to model misspecification, contamination,
the presence of outliers and is robust with respect to a departure from the
equidistribution assumption we started from.
The paper is organized as follows. The statistical framework is described in
Section 2 and the construction of the estimator as well as its properties are
presented in Section 3. The more specific properties of our estimator for
estimating a mixture of densities that are monotone, convex or concave can be
found in Sections 4 and 5 respectively while the case of a log-concave density
is tackled in 6 respectively. These sections also contain some approximation
results which may be of independent interest and are central to our approach.
The proofs are postponed to Section 7.
## 2\. The statistical framework and main notations
Let $X_{1},\ldots,X_{n}$ be $n$ independent random variables and
$P_{1}^{\star},\ldots,P_{n}^{\star}$ their marginals on a measurable space
$({\mathscr{X}},{\mathscr{A}})$. Our aim is to estimate the $n$-tuple
${\mathbf{P}}^{\star}=(P_{1}^{\star},\ldots,P_{n}^{\star})$ from the
observation of ${\boldsymbol{X}}=(X_{1},\ldots,X_{n})$ on the basis of a
suitable model for ${\mathbf{P}}^{\star}$. More precisely, given a
$\sigma$-finite measure $\mu$ on $({\mathscr{X}},{\mathscr{A}})$ and a family
$\overline{\mathcal{M}}$ of densities with respect to $\mu$, we shall do as if
the $X_{i}$ were i.i.d. with a density that belongs to
$\overline{\mathcal{M}}$, even though this might not be true, and estimate
${\mathbf{P}}^{\star}$ by a $n$-tuple of the form
$(\widehat{P},\ldots,\widehat{P})$ where
$\widehat{P}=\widehat{P}({\boldsymbol{X}})=\widehat{p}\cdot\mu$ is a random
element of the set
$\overline{\mathscr{M}}=\\{P=p\cdot\mu,p\in\overline{\mathcal{M}}\\}$. We
refer to $\overline{\mathscr{M}}$ and $\overline{\mathcal{M}}$ as our
probability and density models respectively. For the sake of simplicity, we
abusively identify ${\mathbf{P}}^{\star}$ with the distribution $\bigotimes
P_{i}^{\star}$ of the observation ${\boldsymbol{X}}$.
The density models we have in mind are nonparametric and gather densities that
satisfy a given shape constraint: monotonicity on a half line, convexity on an
interval, log-concavity on the line, among other examples.
In order to evaluate the accuracy of our estimator, we use the TV-loss $d$ on
the set ${\mathscr{P}}$ of all probability measures on
$({\mathscr{X}},{\mathscr{A}})$. We denote by $\left\|{\cdot}\right\|$ the
${\mathbb{L}}_{1}$-norm on the set
${\mathbb{L}}_{1}({\mathscr{X}},{\mathscr{A}},\mu)$ that consists of the
equivalence classes of integrable functions on
$({\mathscr{X}},{\mathscr{A}},\mu)$. We recall that the TV-loss is a distance
defined for $P,Q\in{\mathscr{P}}$ by
(1)
$d\\!\left(P,Q\right)=\underset{A\in{\mathscr{A}}}{\sup}\left[{P(A)-Q(A)}\right]$
and if $P$ and $Q$ are absolutely continuous with respect to our dominating
measure $\mu$,
$d\\!\left(P,Q\right)=\frac{1}{2}\left\|{\frac{dP}{d\mu}-\frac{dQ}{d\mu}}\right\|.$
In general, whatever ${\mathbf{P}}^{\star}$, denote by $P^{\star}$ the uniform
mixture of the marginals:
$P^{\star}=\frac{1}{n}\sum_{i=1}^{n}P_{i}^{\star}.$
In particular, when the data are i.i.d., their common distribution is
$P^{\star}\in{\mathscr{P}}$.
The quantity $d\\!\left(P^{\star},P\right)$ is small as compared to 1 when
only a small portion of the marginals $P_{1}^{\star},\ldots,P_{n}^{\star}$ are
far away from $P$. Note that this quantity can be small even if none of the
$P_{i}^{\star}$ equals $P$.
Throughout this paper, we assume the following.
###### Assumption 1.
There exists a countable subset ${\mathcal{M}}$ of $\overline{\mathcal{M}}$
that is dense in $\overline{\mathcal{M}}$ for the ${\mathbb{L}}_{1}$-norm.
We recall that a subset of a separable metric space is separable. In
particular, when the space ${\mathbb{L}}_{1}({\mathscr{X}},{\mathscr{A}},\mu)$
is separable for the ${\mathbb{L}}_{1}$-norm, so is any subset
$\overline{\mathcal{M}}$ of densities on $({\mathscr{X}},{\mathscr{A}},\mu)$
and Assumption 1 is automatically satisfied. This is in particular the case
when
$({\mathscr{X}},{\mathscr{A}})=({\mathbb{R}}^{k},{\mathscr{B}}({\mathbb{R}}^{k}))$,
$k\geqslant 1$ and $\mu$ is the Lebesgue measure. If a family
$\overline{\mathcal{M}}$ of densities satisfies our Assumption 1, so does any
subset $\overline{\mathcal{D}}$ of $\overline{\mathcal{M}}$. The set
$\overline{\mathcal{D}}$ may in turn be associated to a subset ${\mathcal{D}}$
and a probability set ${\mathscr{D}}=\\{P=p\cdot\mu,\;p\in{\mathcal{D}}\\}$
that are both countable and respectively dense in
$(\overline{\mathcal{D}},\left\|{\cdot}\right\|)$ and
$\overline{\mathscr{D}}=\\{P=p\cdot\mu,\;p\in\overline{\mathcal{D}}\\}$ for
the total variation distance $d$. We may therefore write
$\inf_{P\in{\mathscr{D}}}d\\!\left(P^{\star},P\right)=\inf_{P\in\overline{\mathscr{D}}}d\\!\left(P^{\star},P\right).$
We shall repeatedly apply this equality to sets $\overline{\mathscr{D}}$ of
interest without any further notice. As a consequence, replacing a density
model $\overline{\mathcal{D}}$ by a countable and dense subset ${\mathcal{D}}$
changes nothing from the approximation point of view. Nevertheless, we prefer
to work with ${\mathcal{D}}$ rather than $\overline{\mathcal{D}}$ in order to
avoid some measurability issues that may result from the calculation of the
supremum of an empirical process indexed by $\overline{\mathcal{D}}$.
Throughout the present paper, we use the same kind of notations as
$\overline{\mathcal{D}},{\mathcal{D}},{\mathscr{D}},\overline{\mathscr{D}}$ in
order to distinguish between the density model, a countable and dense subset
of it and their corresponding probability models. Following these notations
${\mathscr{M}}=\\{P=p\cdot\mu,\;p\in{\mathcal{M}}\\}$. An interval $I$ of
${\mathbb{R}}$ is said to be nontrivial if its interior $\mathring{I}$ is not
empty or equivalently if its length is positive. Given
$a\in{\mathbb{R}}\cup\\{-\infty\\}$ and $b\in{\mathbb{R}}$, $(a,b|$ denotes
any of the intervals $(a,b)$ and $(a,b]$. When we say that $p$ is a density on
a (nontrivial) interval $I$, we mean that $p$ is a density that vanishes
outside $I$. The set of positive integers is denoted ${\mathbb{N}}^{\star}$
and $|A|$ is the cardinality of a set $A$. By convention,
$\sum_{\varnothing}=0$. For an integrable function $f$ on
$({\mathscr{X}}^{\otimes n},{\mathscr{A}}^{\otimes n})$,
${\mathbb{E}}[f({\boldsymbol{X}})]$ is the integral of $f$ with respect to the
probability measure $\bigotimes_{i=1}^{n}P_{i}^{\star}={\mathbf{P}}^{\star}$
while for $f$ on $({\mathscr{X}},{\mathscr{A}})$ and $S\in{\mathscr{P}}$,
${\mathbb{E}}_{S}[f(X)]$ is the integral of $f$ with respect to $S$. We use
the same conventions for $\mathop{\rm
Var}\nolimits\left(f({\boldsymbol{X}})\right)$ and $\mathop{\rm
Var}\nolimits_{S}\left(f(X)\right)$.
## 3\. An $\ell$-type estimator for the TV-loss
Let $\overline{\mathcal{M}}$ be a density model that satisfies our Assumption
1 for some ${\mathcal{M}}\subset\overline{\mathcal{M}}$. Given $P=p\cdot\mu$
and $Q=q\cdot\mu$ in ${\mathscr{M}}$, we define
(2) $t_{(P,Q)}=1\hskip-2.6pt{\rm l}_{q>p}-P(q>p)=P(p\geqslant
q)-1\hskip-2.6pt{\rm l}_{p\geqslant q}.$
Given the family ${\mathscr{T}}=\\{t_{(P,Q)},\,(P,Q)\in{\mathscr{M}}^{2}\\}$,
we define for $P,Q\in{\mathscr{M}}$ and ${\mathbf{x}}\in{\mathscr{X}}^{n}$
(3)
${\mathbf{T}}({\mathbf{x}},P,Q)=\sum_{i=1}^{n}t_{(P,Q)}(x_{i})=\sum_{i=1}^{n}\left[{1\hskip-2.6pt{\rm
l}_{q>p}(x_{i})-P(q>p)}\right]$
and
${\mathbf{T}}({\mathbf{x}},P)=\underset{Q\in{\mathscr{M}}}{\sup}{\mathbf{T}}({\mathbf{x}},P,Q).$
For ${\varepsilon}>0$, we finally define our estimator as any (measurable)
element $\widehat{P}=\widehat{p}\cdot\mu$ that belongs to the set
(4)
${\mathscr{E}}({\boldsymbol{X}})=\left\\{{P\in{\mathscr{M}},\,{\mathbf{T}}({\boldsymbol{X}},P)\leqslant\underset{P^{\prime}\in{\mathscr{M}}}{\inf}{\mathbf{T}}({\boldsymbol{X}},P^{\prime})+\varepsilon}\right\\}.$
We call $\widehat{P}$ and $\widehat{p}$ a TV-estimator on ${\mathscr{M}}$ and
${\mathcal{M}}$ respectively. The parameter ${\varepsilon}$ is introduced in
case a minimizer of $P\mapsto{\mathbf{T}}({\boldsymbol{X}},P)$ does not exist
on ${\mathscr{M}}$. Any ${\varepsilon}$-minimizer would do provided that
${\varepsilon}$ is not too large.
The construction of estimators from an appropriate family of test statistics
$t_{(P,Q)}$ is described in Baraud (2021) and our approach is in the same
line. In particular, we use the following key property on the family
${\mathscr{T}}$ (which can be compared to Assumption 1 in Baraud (2021)).
###### Lemma 1.
For all probabilities $P,Q\in{\mathscr{M}}$ and $S\in{\mathscr{P}}$,
(5)
$d\\!\left(P,Q\right)-d\\!\left(S,Q\right)\leqslant{\mathbb{E}}_{S}\left[{t_{(P,Q)}(X)}\right]\leqslant
d\\!\left(S,P\right).$
In particular,
(6)
$d\\!\left(P,Q\right)-d\\!\left(P^{\star},Q\right)\leqslant\frac{1}{n}\sum_{i=1}^{n}{\mathbb{E}}\left[{t_{(P,Q)}(X_{i})}\right]\leqslant
d\\!\left(P^{\star},P\right),$
where $P^{\star}=n^{-1}\sum_{i=1}^{n}P_{i}^{\star}$.
However, our family ${\mathscr{T}}$ does not satisfy the anti-symmetry
assumption, namely $t_{(P,Q)}=-t_{(Q,P)}$, which is required for Baraud’s
construction. The risk bound that we establish below cannot therefore be
deduced from Baraud (2021). In fact, for the specific problem we want to solve
here the anti-symmetry assumption can be relaxed which leads to an improvement
on the numerical constants that are involved in the risk bounds.
Our construction also shares some similarities with that proposed by Devroye
and Lugosi (2001)[Section 6.8 p.55]. However, a careful look at their
selection criterion shows that it is slightly different from ours. They
replace our function ${\mathbf{T}}(\cdot,P,Q)$ given by (3) by
${\mathbf{T}}_{\text{DL}}(\cdot,P,Q):{\mathbf{x}}\mapsto\left|{\sum_{i=1}^{n}\left[{1\hskip-2.6pt{\rm
l}_{q\geqslant p}(x_{i})-P(q\geqslant p)}\right]}\right|.$
Their approach leads to a set of estimators
${\mathscr{E}}_{\text{DL}}({\boldsymbol{X}})$ defined in the same way as (4)
for ${\mathbf{T}}_{\text{DL}}$ in place of ${\mathbf{T}}$ (with
${\varepsilon}=1$).
###### Proof of Lemma 1.
Let $P,Q\in{\mathscr{M}}$. Using the definition (2) of $t_{(P,Q)}$ and that of
the TV-loss given by (1), we obtain that for all $S\in{\mathscr{P}}$,
${\mathbb{E}}_{S}\left[{t_{(P,Q)}(X)}\right]=S(q>p)-P(q>p)\leqslant
d\\!\left(S,P\right),$
which is exactly the second inequality in (5). To establish the first one, we
use the fact that $d\\!\left(P,Q\right)=Q(q>p)-P(q>p)$. This leads to
$\displaystyle{\mathbb{E}}_{S}\left[{t_{(P,Q)}(X)}\right]$
$\displaystyle=S(q>p)-Q(q>p)+\left[{Q(q>p)-P(q>p)}\right]$
$\displaystyle\geqslant-d\\!\left(S,Q\right)+d\\!\left(P,Q\right).$
Finally, (6) results from the observation that
$\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[t_{(P,Q)}(X_{i})\right]=\mathbb{E}_{P^{\star}}\left[t_{(P,Q)}(X)\right].$
∎
### 3.1. Properties of the estimator
Our main result is based on the key notion of extremal point in a model.
###### Definition 1.
Let ${\mathcal{F}}$ be a class of real-valued functions on a set
${\mathscr{X}}$ with values in ${\mathbb{R}}$. We say that an element
$\overline{f}\in{\mathcal{F}}$ is extremal in ${\mathcal{F}}$ (or is an
extremal point of ${\mathcal{F}}$) with degree not larger than $D\geqslant 1$
if the classes of subsets
$\displaystyle{\mathscr{C}}^{>}({\mathcal{F}},\overline{f})$
$\displaystyle=\big{\\{}\\{x\in{\mathscr{X}}\,\big{|}\,q(x)>\overline{f}(x)\\},\;q\in{\mathcal{F}}\setminus\\{\overline{f}\\}\big{\\}}$
and
$\displaystyle{\mathscr{C}}^{<}({\mathcal{F}},\overline{f})$
$\displaystyle=\big{\\{}\\{x\in{\mathscr{X}}\,\big{|}\,q(x)<\overline{f}(x)\\},\;q\in{\mathcal{F}}\setminus\\{\overline{f}\\}\big{\\}}$
are both VC with dimension not larger than $D$.
Additionally, we say that $\overline{P}$ is an extremal point of
$\overline{\mathscr{M}}$ with degree not larger than $D\geqslant 1$ if there
exists $\overline{p}\in\overline{\mathcal{M}}$ such that
$\overline{P}=\overline{p}\cdot\mu$ and $\overline{p}$ is extremal in
$\overline{\mathcal{M}}$ with degree not larger than $D$. For each $D\geqslant
1$, we denote by $\overline{\mathcal{O}}(D)$ the set of extremal points
$\overline{p}$ in $\overline{\mathcal{M}}$ with degree not larger than $D$,
${\mathcal{O}}(D)$ a countable and dense subset of it, ${\mathscr{O}}(D)$ the
corresponding set of probability measures and
$\overline{\mathcal{O}}=\bigcup_{D\geqslant 1}\overline{\mathcal{O}}(D)$ the
set of all extremal points in $\overline{\mathcal{M}}$. Finally, let
${\mathcal{M}}$ be a countable and dense subset of $\overline{\mathcal{M}}$
containing $\bigcup_{D\geqslant 1}{\mathcal{O}}(D)$.
###### Theorem 1.
Let $\overline{\mathcal{M}}$ be a density model satisfying our Assumption 1
for which $\overline{\mathcal{O}}$ is nonempty. Any TV-estimator $\widehat{P}$
on ${\mathscr{M}}$ satisfies for all $\xi>0$ and all product distribution
${\mathbf{P}}^{\star}$, with a probability at least $1-e^{-\xi}$, for all
$D\geqslant 1$ and all $P\in\overline{\mathscr{O}}(D)$,
(7) $d\\!\left(P,\widehat{P}\right)\leqslant
2d\\!\left(P^{\star},P\right)+20\sqrt{\frac{5D}{n}}+\sqrt{\frac{2(\xi+\log
2)}{n}}+\frac{\varepsilon}{n}.$
In particular,
(8) $\displaystyle
d\\!\left(P^{\star},\widehat{P}\right)\leqslant\inf_{D\geqslant
1}\left[{3\inf_{P\in\overline{\mathscr{O}}(D)}d\\!\left(P^{\star},P\right)+20\sqrt{\frac{5D}{n}}}\right]+\sqrt{\frac{2(\log
2+\xi)}{n}}+\frac{\varepsilon}{n},$
with the convention $\inf_{\varnothing}=+\infty$. As a consequence of (7),
(9) $\mathbb{E}\left[d\\!\left(P,\widehat{P}\right)\right]\leqslant
2d\\!\left(P^{\star},P\right)+48\sqrt{\frac{D}{n}}+\frac{\varepsilon}{n}$
for all $D\geqslant 1$ and all $P\in\overline{\mathscr{O}}(D)$, moreover by
(8),
(10)
$\mathbb{E}\left[{d\\!\left(P^{\star},\widehat{P}\right)}\right]\leqslant\inf_{D\geqslant
1}\left\\{3\inf_{P\in\overline{\mathscr{O}}(D)}d\\!\left(P^{\star},P\right)+48\sqrt{\frac{D}{n}}\right\\}+\frac{\varepsilon}{n}.$
###### Proof.
The proof is postponed to Subsection 7.2. ∎
Let us now comment on this result.
In the favourable situation where the $X_{i}$ are i.i.d. with distribution
$P^{\star}$ in $\overline{\mathscr{O}}$, $\inf_{D\geqslant
1}\inf_{P\in\overline{\mathscr{O}}(D)}d\\!\left(P^{\star},P\right)=0$ and we
deduce from (10) that the estimator $\widehat{P}$ converges toward $P^{\star}$
at rate $1/\sqrt{n}$ for the total variation distance. More precisely, the
risk of the estimator is not larger than $48\sqrt{D/n}+{\varepsilon}/n$ when
$P^{\star}$ belongs to $\overline{\mathscr{O}}(D)$ for some $D\geqslant 1$.
Note that the result also holds when the data are independent only, provided
that $P^{\star}=n^{-1}\sum_{i=1}^{n}P_{i}^{\star}$ is extremal. This situation
may occur even when none of the marginals $P_{i}^{\star}$ is extremal or even
belongs to the model ${\mathscr{M}}$.
In the general case where the data are independent only and their marginals
write for all $i\in\\{1,\ldots,n\\}$ as
(11)
$P_{i}^{\star}=(1-\alpha_{i})\overline{P}+\alpha_{i}R_{i}=\overline{P}+\alpha_{i}\left({R_{i}-\overline{P}}\right)$
for some $\overline{P}\in\overline{\mathscr{O}}(D)$ with $D\geqslant 1$,
$\alpha_{1},\ldots,\alpha_{n}$ in $[0,1]$ and distributions
$R_{1},\ldots,R_{n}$ in ${\mathscr{P}}$, we deduce from (9) that
$\displaystyle\mathbb{E}\left[{d\\!\left(\overline{P},\widehat{P}\right)}\right]$
$\displaystyle\leqslant
2d\\!\left(P^{\star},\overline{P}\right)+48\sqrt{\frac{D}{n}}+\frac{\varepsilon}{n}$
$\displaystyle\leqslant\frac{2}{n}\sum_{i=1}^{n}\alpha_{i}+48\sqrt{\frac{D}{n}}+\frac{\varepsilon}{n}.$
As compared to the previous situation where
$P_{i}^{\star}=\overline{P}\in\overline{\mathscr{O}}(D)$, hence $\alpha_{i}=0$
for all $i$, we see that the risk bound we get only inflates by the additional
term $2\overline{\alpha}=(2/n)\sum_{i=1}^{n}\alpha_{i}$ and it remains thus of
the same order when $\overline{\alpha}$ is small enough as compared to
$\sqrt{D/n}$. Note that this situation may occur even when $\alpha_{i}>0$ for
all $i$, i.e. when none of the marginals $P_{i}^{\star}$ belongs to
$\overline{\mathscr{O}}(D)$. In order to be more specific, we may consider the
two following situations. In the first one, there exists some subset of the
data which are i.i.d. with distribution
$\overline{P}\in\overline{\mathscr{O}}(D)$ while the other part, corresponding
to what we shall call outliers, are independently drawn according to some
arbitrary distributions. In this case, there exists a subset
$S\subset\\{1,\ldots,n\\}$ such that $\alpha_{i}=1$ for $i\in S$ and
$\alpha_{i}=0$ otherwise in (11). Our procedure is stable with respect to the
presence of such outliers as soon as $\overline{\alpha}=|S|/n$ remains small
as compared to $\sqrt{D/n}$. In the other situation, which is called the
contamination case, the data are i.i.d., a portion $\alpha\in(0,1]$ of them
are drawn according to an arbitrary distribution $R$ while the other part
follows the distribution $\overline{P}\in\overline{\mathscr{O}}(D)$. Then (11)
holds with $\alpha_{i}=\alpha$ and $R_{i}=R$ for all $i\in\\{1,\ldots,n\\}$.
The risk bound we get remains of the stable under contamination as long as the
level $\overline{\alpha}=\alpha$ of contamination remains small as compared to
$\sqrt{D/n}$.
A bound similar to (8) has been established in Baraud (2021) for his
$\ell$-estimators. His inequality (48) can be reformulated in our context as
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}d\\!\left(P_{i}^{\star},\widehat{P}\right)\leqslant$
$\displaystyle\inf_{D\geqslant
1}\left[{6\inf_{P\in\overline{\mathscr{O}}(D)}\left[{\frac{1}{n}\sum_{i=1}^{n}d\\!\left(P_{i}^{\star},P\right)}\right]+40\sqrt{\frac{5D}{n}}}\right]+2\sqrt{\frac{2\xi}{n}}+\frac{2\varepsilon}{n}$
$\displaystyle\quad-\inf_{P\in\overline{\mathscr{M}}}\left[{\frac{1}{n}\sum_{i=1}^{n}d\\!\left(P_{i}^{\star},P\right)}\right].$
In comparison, equation (9) and the triangle inequality imply that
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}d\\!\left(P_{i}^{\star},\widehat{P}\right)$
$\displaystyle\leqslant\inf_{D\geqslant
1}\left[{3\inf_{P\in\overline{\mathscr{O}}(D)}\left[{\frac{1}{n}\sum_{i=1}^{n}d\\!\left(P_{i}^{\star},P\right)}\right]+20\sqrt{\frac{5D}{n}}}\right]$
$\displaystyle\quad+\sqrt{\frac{2(\log 2+\xi)}{n}}+\frac{\varepsilon}{n}$
If we omit the term
$\inf_{P\in\overline{\mathscr{M}}}\left[{\frac{1}{n}\sum_{i=1}^{n}d\\!\left(P_{i}^{\star},P\right)}\right]$
that appears in his inequality and $\log 2$ that appears in ours, all the
constants we get are divided by a factor 2 as compared to his.
When the Yatracos class $\\{\\{p>q\\},p,q\in\overline{\mathcal{M}}\\}$ is VC
with dimension not larger than $D\geqslant 1$, all the elements of
$\overline{\mathcal{M}}$ are extremal with degree not larger than $D$ and (10)
becomes
(12)
$\displaystyle{\mathbb{E}}\left[{d\\!\left(P^{\star},\widehat{P}\right)}\right]\leqslant
3\inf_{P\in\overline{\mathscr{M}}}d\\!\left(P^{\star},P\right)+48\sqrt{\frac{D}{n}}+\frac{\varepsilon}{n}.$
In the particular case of i.i.d. data with common distribution $P^{\star}$, an
inequality of the same flavour was established by Devroye and Lugosi
(2001)[Section 8.2] for their minimum distance estimate. Both inequalities
involve a constant 3 in front of the approximation term
$\inf_{P\in\overline{\mathscr{M}}}d\\!\left(P^{\star},P\right)$. In our
inequality the numerical constants are explicit.
In the next sections, we take advantage of the stronger inequality (8) to
consider density models $\overline{\mathcal{M}}$ for which the Yatracos
classes $\\{\\{p>q\\},p,q\in\overline{\mathcal{M}}\\}$ are not VC.
In the remaining part of this paper,
$({\mathscr{X}},{\mathscr{A}})=({\mathbb{R}},{\mathscr{B}}({\mathbb{R}}))$ and
$\mu$ is the Lebesgue measure on ${\mathbb{R}}$. Since
${\mathbb{L}}_{1}({\mathbb{R}},{\mathscr{B}}({\mathbb{R}}),\mu)$ is separable,
Assumption 1 is automatically satisfied. Each of the forthcoming sections is
devoted to the problem of estimating densities or a mixture of densities with
respect to $\mu$ under the assumption that they satisfy one of the following
shape constraint: monotonicity, concavity, convexity or $\log$-concavity.
## 4\. Estimating a piecewise monotone density
We denote by ${\mathcal{A}}(k)$ the class of nonempty subsets
$A\subset{\mathbb{R}}$ with cardinality not larger than $k\geqslant 1$. The
elements of $A$ provide a partition of ${\mathbb{R}}$ into $l=|A|+1\leqslant
k+1$ intervals $I_{1},\ldots,I_{l}$ the endpoints of which belong to $A$. We
denote by ${\mathbf{I}}(A)$ the set
$\\{\mathring{I}_{1},\ldots,\mathring{I}_{l}\\}$ of their interiors. Although
there exist several ways of partitioning ${\mathbb{R}}$ into intervals with
endpoints in $A$, the set ${\mathbf{I}}(A)$ is uniquely defined.
### 4.1. Piecewise monotone densities
###### Definition 2.
Let $k\geqslant 2$. A function $g$ on ${\mathbb{R}}$ is said to be
$k$-piecewise monotone if there exists $A\in{\mathcal{A}}(k-1)$ such that the
restriction of $g$ to each interval $I\in{\mathbf{I}}(A)$ is monotone. In
particular, there exist at most $k$ monotone functions $g_{I}$ on
$I\in{\mathbf{I}}(A)$ such that
$g(x)=\sum_{I\in{\mathbf{I}}(A)}g_{I}(x)1\hskip-2.6pt{\rm
l}_{I}(x)\quad\text{for all $x\in{\mathbb{R}}\setminus A$.}$
A $k$-piecewise monotone function $g$ associated to $A\in{\mathcal{A}}(k-1)$
may not be monotone on each element of a partition based on $A$. We only
require that $g$ be monotone on the interiors of these elements. The function
$\begin{array}[]{l|rcl}g:&{\mathbb{R}}&\longrightarrow&{\mathbb{R}}_{+}\\\
&x&\longmapsto&{\frac{1}{\sqrt{|x|}}1\hskip-2.6pt{\rm l}_{|x|>0}}\end{array}$
is 2-piecewise monotone, associated to $A=\\{0\\}\in{\mathcal{A}}(1)$, but $g$
is neither monotone on $(-\infty,0]$ nor on $[0,+\infty)$.
We denote by $\overline{\mathcal{M}}_{k}$ the set of $k$-piecewise monotone
densities. The sets $\overline{\mathcal{M}}_{k}$ are obviously increasing with
$k$ for the inclusion. The set $\overline{\mathcal{M}}_{2}$ contains the
unimodal densities on the line and in particular all the densities that are
monotone on a half-line and vanish elsewhere.
Of special interest are those densities in $\overline{\mathcal{M}}_{k}$ which
are also piecewise constant on a finite partition of ${\mathbb{R}}$ into
intervals. More precisely, for $D\geqslant 1$ let
$\overline{\mathcal{O}}_{D,k}$ be the subset of $\overline{\mathcal{M}}_{k}$
that consists of those densities that are constant on each element of a class
${\mathbf{I}}(A)$ with $A\in{\mathcal{A}}(D+1)$. The number $D$ is a bound on
the number of bounded intervals in the class, hence on the number of positive
values that a density in $\overline{\mathcal{O}}_{D,k}$ may take. The uniform
distribution on a nontrivial interval has a density that belongs to
$\overline{\mathcal{O}}_{1,2}$, and also to $\overline{\mathcal{O}}_{D,k}$ for
all $D\geqslant 1$ and $k\geqslant 2$. The sets $\overline{\mathcal{O}}_{D,k}$
with $D\geqslant 1$ and $k\geqslant 2$ are therefore nonempty. They satisfy
the following property which is a consequence of Proposition 3 of Baraud and
Birgé (2016).
###### Proposition 1.
For all $D\geqslant 1$ and $k\geqslant 2$, all the elements of
$\overline{\mathcal{O}}_{D,k}$ are extremal in $\overline{\mathcal{M}}_{k}$
with degree not larger than $3(k+D+1)$.
###### Proof.
The proof is postponed to Subsection 7.3. ∎
For all $D\geqslant 1$ and $k\geqslant 2$, let ${\mathcal{O}}_{D,k}$ be a
countable and dense subset of $\overline{\mathcal{O}}_{D,k}$ (for the
${\mathbb{L}}_{1}$-norm) and ${\mathcal{M}}_{k}$ a countable and dense subset
of $\overline{\mathcal{M}}_{k}$ that contains $\bigcup_{D\geqslant
1}{\mathcal{O}}_{D,k}$. It follows from Proposition 1 that the elements of
${\mathcal{O}}_{D,k}$ are also extremal in ${\mathcal{M}}_{k}$ with degree not
larger than $3(k+D+1)$ for all $D\geqslant 1$. We immediately deduce from
Theorem 1 the following result.
###### Theorem 2.
Let $k\geqslant 2$. Whatever the product distribution ${\mathbf{P}}^{\star}$
of the data, any TV-estimator $\widehat{P}$ on ${\mathscr{M}}_{k}$ satisfies
(13)
$\mathbb{E}\left[{d\\!\left(P^{\star},\widehat{P}\right)}\right]\leqslant\inf_{D\geqslant
1}\left[{3\inf_{P\in\overline{\mathscr{O}}_{D,k}}d\\!\left(P^{\star},P\right)+83.2\sqrt{\frac{D+k+1}{n}}}\right]+\frac{\varepsilon}{n}.$
Our approach solves the problem of estimating a nonincreasing density on a
half-line $I$ by taking $k=2$. For this specific problem, a bound of the same
flavour was established by Baraud (2021) (see Proposition 6) for his
$\ell$-estimator. When the data are i.i.d. with distribution $P^{\star}$, our
result shows better constants in the risk bound (13) as compared to Baraud’s.
In particular, the constant 3 in front of the approximation term
$\inf_{P\in\overline{\mathscr{O}}_{D,k}}d\\!\left(P^{\star},P\right)$ improves
on his constant 5. Our estimator (as well as Baraud’s) also improves on the
Grenander estimator since our construction does not require the prior
knowledge of the half-line $I$.
For general values of $k\geqslant 2$, the problem of estimating a
$k$-piecewise monotone density was also considered in Baraud and Birgé (2016).
The authors used a $\rho$-estimator and the Hellinger loss in place of the
total variation one to evaluate its risk— see their Corollary 2. Their bound
is similar to ours except from the fact that the quantity $\sqrt{(D+k+1)/n}$
appears there multiplied by a logarithmic factor. This logarithmic factor
turns out to be necessary when one deals with the Hellinger loss while it
disappears with the total variation one.
In order to specify further the risk bound given by (13), and see what
properties can be established on our estimator, let us consider different
assumptions on the distribution of the data. These assumptions can be done a
posteriori, since the estimator is solely based on the model
$\overline{\mathscr{M}}_{k}$ (or a dense subset of it). Such assumptions
enable us to bound the quantity
$A=\inf_{D\geqslant
1}\left[{3\inf_{P\in\overline{\mathscr{O}}_{D,k}}d\\!\left(P^{\star},P\right)+83.2\sqrt{\frac{D+k+1}{n}}}\right]$
that appear in the right-hand side of (13).
If the $X_{i}$ are not i.i.d. but only independent and their marginals are
close enough to a distribution of the form
$\overline{P}=\overline{p}\cdot\mu$, by the triangle inequality we may bound
$A$ by
$3d\\!\left(P^{\star},\overline{P}\right)+{\mathbb{B}}_{k,n}(\overline{p})$
where ${\mathbb{B}}_{k,n}(p)$ is defined for a density $p$ on the line by the
formulas
$\displaystyle{\mathbb{B}}_{k,n}(p)$ $\displaystyle=\inf_{D\geqslant
1}\left[{3\inf_{Q\in\overline{\mathscr{O}}_{D,k}}d\\!\left(P,Q\right)+83.2\sqrt{\frac{D+k+1}{n}}}\right]$
(14) $\displaystyle=\inf_{D\geqslant
1}\left[{\frac{3}{2}\inf_{q\in\overline{\mathcal{O}}_{D,k}}\left\|{p-q}\right\|+83.2\sqrt{\frac{D+k+1}{n}}}\right].$
This means that as long as $3d\\!\left(P^{\star},\overline{P}\right)$ remains
small as compared to ${\mathbb{B}}_{k,n}(\overline{p})$, the bound on
${\mathbb{E}}[d(P^{\star},\widehat{P})]$ would be almost the same as if the
$X_{i}$ were truly i.i.d. with distribution
$\overline{P}=\overline{p}\cdot\mu$. This property accounts for the robustness
of our approach.
If we apply a location-scale transformation to the data, that is, if in place
of the original data $X_{1},\ldots,X_{n}$ with density $\overline{p}$ we
observe the random variables $Y_{i}=\sigma X_{i}+m$ for $i\in\\{1,\ldots,n\\}$
with $(m,\sigma)\in{\mathbb{R}}\times(0,+\infty)$, the density
$\overline{p}_{m,\sigma}$ of the new data would satisfy
${\mathbb{B}}_{k,n}(\overline{p}_{m,\sigma})={\mathbb{B}}_{k,n}(\overline{p})$
since the set $\overline{\mathcal{O}}_{D,k}$ and the total variation distance
remain invariant under such a transformation. This means that the performance
of the TV-estimator is independent of the unit that is used to measure the
data.
In the remaining part this section, we provide upper bounds on the quantity
${\mathbb{B}}_{k,n}(p)$ for some densities $p\in\overline{\mathcal{M}}_{k}$ of
special interest.
### 4.2. Estimation of bounded and compactly supported $k$-piecewise monotone
densities
For $k\geqslant 3$, let $\overline{\mathcal{M}}_{k}^{\infty}$ be the subset of
$\overline{\mathcal{M}}_{k}$ that consists of the densities on ${\mathbb{R}}$
which coincide almost everywhere with a density of the form
(15) $p=\sum_{i=1}^{k-2}w_{i}p_{i}1\hskip-2.6pt{\rm l}_{(x_{i-1},x_{i})}$
where
1. (i)
$(x_{i})_{i\in\\{0,\ldots,k-2\\}}$ is an increasing sequence of real numbers;
2. (ii)
$w_{1},\ldots,w_{k-2}$ are nonnegative numbers such that
$\sum_{i=1}^{k-2}w_{i}=1$;
3. (iii)
for $i\in\\{1,\ldots,k-2\\}$, $p_{i}$ is a monotone density on the interval
$I_{i}=(x_{i-1},x_{i})$ of length $L_{i}>0$ with variation
$V_{i}=\sup_{x\in I_{i}}p_{i}(x)-\inf_{x\in I_{i}}p_{i}(x)<+\infty.$
A density $p$ in $\overline{\mathcal{M}}_{k}^{\infty}$ is necessarily bounded
and compactly supported. A monotone density on ${\mathbb{R}}_{+}$ which is
bounded and compactly supported belongs to
$\overline{\mathcal{M}}_{3}^{\infty}$. A bounded unimodal density supported on
a compact interval belongs to $\overline{\mathcal{M}}_{4}^{\infty}$.
For $p\in\overline{\mathcal{M}}_{k}^{\infty}$, we set
(16)
$\boldsymbol{R}_{k,0}(p)=\inf\left[{\sum_{i=1}^{k-2}\sqrt{w_{i}\log\left({1+L_{i}V_{i}}\right)}}\right]^{2}$
where the infimum runs among all ways of writing $p$ under the form (15) a.e.
Note that we allow some of the $w_{i}$ to be zero in which case the
corresponding densities $p_{i}$ may be chosen arbitrarily and their choices do
not contribute to the value of $\boldsymbol{R}_{k,0}(p)$. For $k\geqslant 3$
and $R>0$, let $\overline{\mathcal{M}}_{k}^{\infty}(R)$ be the subset of
$\overline{\mathcal{M}}_{k}^{\infty}$ that gathers these densities $p$ for
which $\boldsymbol{R}_{k,0}(p)<R$. When a density $p$ belongs to
$\overline{\mathcal{M}}_{k}^{\infty}(R)$, we may therefore write $p$ under the
form (15) a.e. with $L_{i}$ and $V_{i}$ such that
$\left[{\sum_{i=1}^{k-2}\sqrt{w_{i}\log\left({1+L_{i}V_{i}}\right)}}\right]^{2}<R$.
The classes of sets $(\overline{\mathcal{M}}_{k}^{\infty})_{k\geqslant 3}$ and
$(\overline{\mathcal{M}}_{k}^{\infty}(R))_{k\geqslant 3}$ are both increasing
(for the inclusion): if $p\in\overline{\mathcal{M}}_{l}^{\infty}(R)$ with
$l<k$, we may write
$p=\sum_{i=1}^{l-2}w_{i}p_{i}1\hskip-2.6pt{\rm
l}_{(x_{i-1},x_{i})}\quad\text{a.e.\
with}\quad\left[{\sum_{i=1}^{l-2}\sqrt{w_{i}\log\left({1+L_{i}V_{i}}\right)}}\right]^{2}<R$
and alternatively
$p=\sum_{i=1}^{l-2}w_{i}p_{i}1\hskip-2.6pt{\rm
l}_{(x_{i-1},x_{i})}+\sum_{j=l-1}^{k-2}0\times 1\hskip-2.6pt{\rm
l}_{(x_{i}+j-l+1),x_{i}+j-l+2)}\ \text{a.e.}$
hence, $p\in\overline{\mathcal{M}}_{k}^{\infty}$ and
$\boldsymbol{R}_{k,0}(p)\leqslant\left[{\sum_{i=1}^{l-2}\sqrt{w_{i}\log\left({1+L_{i}V_{i}}\right)}+0}\right]^{2}<R.$
It is not difficult to check that the set
$\overline{\mathcal{M}}_{k}^{\infty}(R)$ is invariant under a location-scale
transformation.
###### Theorem 3.
Let $k\geqslant 3$ and $R>0$. If $p\in\overline{\mathcal{M}}_{k}^{\infty}(R)$,
(17) ${\mathbb{B}}_{k,n}(p)\leqslant
41.3\left({\frac{R}{n}}\right)^{1/3}+83.2\sqrt{\frac{2k}{n}}.$
This result is to our knowledge new in the literature. We deduce that the
minimax risk for the ${\mathbb{L}}_{1}$-norm over
$\overline{\mathcal{M}}_{k}^{\infty}(R)$ is not larger than $(R/n)^{1/3}\vee
kn^{-1/2}$ up to a positive multiplicative constant. For large enough values
of $n$, the bound is of order $(R/n)^{1/3}$ while for moderate ones and values
of $R$ which are close enough to 0, which means that the densities in
$\overline{\mathcal{M}}_{k}^{\infty}(R)$ are close to a mixture of $k-2$
uniform distributions, the bound is of order $\sqrt{k/n}$.
It is interesting to compare this result to that established in Baraud and
Birgé (2016)[page 3900] for their $\rho$-estimators. For the problem of
estimating a bounded unimodal density $p$ supported on an interval of length
$L$, which is an element of $\overline{\mathcal{M}}_{4}$, Baraud and Birgé
show that the Hellinger risk of the $\rho$-estimator is not larger than
$(\sqrt{L\left\|{p}\right\|_{\infty}}/n)^{1/3}\log n$ up to some numerical
constant. With our $\ell$-estimator, the bound we get is of order
$(\log(1+L\left\|{p}\right\|_{\infty})/n)^{1/3}$ and only depends
logarithmically on the quantity $L\left\|{p}\right\|_{\infty}$ (which is not
smaller than 1 since $p$ is a density).
###### Proof.
The proof of Theorem 3 is based on (14) and the following approximation
result. The complete proof is postponed to subsection 7.3. ∎
###### Proposition 2.
Let $V\geqslant 0$ and $I$ be a bounded interval of length $L>0$. For all
$D\geqslant 1$, there exists a partition ${\mathcal{J}}={\mathcal{J}}(D,L,V)$
of $I$ into $D\geqslant 1$ nontrivial intervals with the following properties.
For any monotone density $p$ on $I$ for which
$V_{I}(p)=\sup_{x\in\mathring{I}}p(x)-\inf_{x\in\mathring{I}}p(x)\leqslant V,$
the function $\overline{p}=\overline{p}({\mathcal{J}})$ defined by
$\overline{p}=\sum_{J\in{\mathcal{J}}}\overline{p}_{J}1\hskip-2.6pt{\rm
l}_{J}\quad\text{with}\quad\overline{p}_{J}=\frac{1}{\mu(J)}\int_{J}p\;d\mu,$
is a monotone density on $I$ that satisfies
(18)
$\int_{I}\left|{p-\overline{p}}\right|d\mu\leqslant\left[{(1+VL)^{1/D}-1}\right]\wedge
2\leqslant\frac{2\log\left({1+VL}\right)}{D}.$
###### Proof.
The proof is postponed to subsection 7.3. ∎
Although the result is hidden in his calculations, Birgé (1987) has
established a bound of the same flavour except from the fact that the
variation $V$ is replaced by a uniform bound on $p$. Unlike his, our bound
(18) allows to recover the fact that when $V=0$, i.e. when $p$ is constant on
$I$, the left-hand side equals 0 as expected. The combination of Theorem 2 and
Theorem 3 immediately leads to the following corollary:
###### Corollary 1.
Let $k\geqslant 3$. If $X_{1},\dots,X_{n}$ is a $n$ sample which density
$p\in\overline{\mathcal{M}}_{k}^{\infty}(R)$, then the TV-estimator
$\widehat{p}$ on ${\mathcal{M}}_{k}$ satisfies
(19) $\mathbb{E}\left[{\|p-\widehat{p}\|}\right]\leqslant
82.6\left({\frac{R}{n}}\right)^{1/3}+166.4\sqrt{\frac{2k}{n}}+\frac{2\varepsilon}{n}.$
### 4.3. Estimation of other $k$-piecewise monotone densities
Corollary 1 provides an upper bound on the ${\mathbb{L}}_{1}$-risk of the TV-
estimator for estimating a density $p\in\overline{\mathcal{M}}_{k}^{\infty}$.
A natural question is how the estimator performs when the density $p$ is
neither bounded nor supported on a compact interval. Since for such densities
we may write
(20)
${\mathbb{B}}_{k,n}(p)\leqslant\inf_{\overline{p}\in\overline{\mathcal{M}}_{k}^{\infty}}\left[{\frac{3}{2}\left\|{p-\overline{p}}\right\|+{\mathbb{B}}_{k,n}(\overline{p})}\right],$
an upper bound on ${\mathbb{B}}_{k,n}(p)$ can be obtained by combining Theorem
3 with an approximationt result showing how general densities in
$\overline{\mathcal{M}}_{k}$ can be approximated by elements of
$\overline{\mathcal{M}}_{k}^{\infty}$. In this section, we therefore study the
approximation properties of the set $\overline{\mathcal{M}}_{k}^{\infty}$ with
respect to possibly unbounded and non-compactly supported densities. We start
with the case of a monotone density on a half-line and introduce the following
definitions.
###### Definition 3.
Given a nonincreasing density $p$ on $(a,+\infty)$ with $a\in{\mathbb{R}}$, we
define $\widetilde{p}$ as the mapping on $(0,+\infty)$ given by
(21) $\widetilde{p}(y)=\inf\left\\{{x>0,\;p(a+x)<y}\right\\}\geqslant 0.$
We define the $x$-tail function $\tau_{x}(p,\cdot)$ associated to $p$ as
$\begin{array}[]{l|rcl}\tau_{x}(p,\cdot):&[0,+\infty)&\longrightarrow&{\mathbb{R}}_{+}\\\
&t&\longmapsto&\displaystyle{\int_{t}^{+\infty}p(a+x)d\mu(x)}\end{array},$
the $y$-tail function $\tau_{y}(p,\cdot)$ as
$\begin{array}[]{l|rcl}\tau_{y}(p,\cdot):&[0,+\infty)&\longrightarrow&{\mathbb{R}}_{+}\\\
&t&\longmapsto&\displaystyle{\int_{t}^{+\infty}\widetilde{p}(y)d\mu(y)}\end{array}$
and the tail function $\tau(p,\cdot)$ as
(22)
$\tau(p,t)=\inf_{s>0}\left[{\tau_{x}(p,st)+\tau_{y}(p,p(a+s))}\right]\quad\text{for
all $t\geqslant 1$}.$
When $p$ is a nondecreasing density on $(-\infty,-a)$ with $a\in{\mathbb{R}}$,
we define $\tau_{x}(p,\cdot)$, $\tau_{y}(p,\cdot)$ and $\tau(p,\cdot)$ as
respectively the $x$-tail, $y$-tail and tail functions of the nonincreasing
density $x\mapsto p(-x)$ .
Let us comment these definitions. When $p$ is a continuous decreasing density
from $(0,+\infty)$ onto $(0,+\infty)$, $\widetilde{p}$ is the reciprocal
function $p^{-1}$. By taking the symmetric of the graph $x\mapsto p(x)$ with
respect to the first diagonal, we easily see that $\widetilde{p}=p^{-1}$ is a
nonincreasing density on $(0,+\infty)$. This property remains true in the
general case as shown by the lemma below with the special value $B=0$. As a
consequence, $\tau_{y}(p,\cdot)$ can be interpreted as the tail of the
distribution function associated to the density $\widetilde{p}$ while
$\tau_{x}(p,\cdot)$ is that of $p(a+\cdot)$.
###### Lemma 2.
Let $p$ be a nonincreasing density on $(a,+\infty)$ with $a\in{\mathbb{R}}$
and $\widetilde{p}$ the mapping defined by (21). For all $B\geqslant 0$,
(23)
$\int_{0}^{+\infty}\left[{p(a+x)-B}\right]_{+}d\mu(x)=\int_{B}^{+\infty}\widetilde{p}(y)d\mu(y)=\tau_{y}(p,B).$
By changing $p$ into $x\mapsto p(-x)$, we also obtain that when $p$ is
nondecreasing on $(-\infty,-a)$,
$\int_{-\infty}^{0}\left[{p(-a+x)-B}\right]_{+}d\mu(x)=\int_{B}^{+\infty}\widetilde{p}(y)d\mu(y)=\tau_{y}(p,B)\quad\text{for
all $B\geqslant 0$.}$
###### Proof.
Let $y>0$. Since $p$ is nonincreasing density on $(a,+\infty)$, it necessarily
tends to 0 at $+\infty$. The set $I(y)=\left\\{{x>0,\;p(a+x)<y}\right\\}$ is
therefore a nonempty unbounded interval with endpoint
$\widetilde{p}(y)<+\infty$ by definition of $\widetilde{p}(y)$. In particular,
$(\widetilde{p}(y),+\infty)\subset I(y)\subset[\widetilde{p}(y),+\infty),$
and by taking the complementary of those sets we obtain that for all
$(x,y)\in(0,+\infty)\times(0,+\infty)$
$1\hskip-2.6pt{\rm l}_{x<\widetilde{p}(y)}\leqslant 1\hskip-2.6pt{\rm
l}_{p(a+x)\geqslant y}\leqslant 1\hskip-2.6pt{\rm
l}_{x\leqslant\widetilde{p}(y)}.$
Integrating these inequalities on $(0,+\infty)\times(B,+\infty)$ with respect
to $\mu\otimes\mu$ and using Fubini’s theorem, we obtain that
$\displaystyle\int_{B}^{+\infty}\widetilde{p}(y)d\mu(y)$
$\displaystyle=\int_{B}^{+\infty}\left[{\int_{0}^{+\infty}1\hskip-2.6pt{\rm
l}_{x<\widetilde{p}(y)}d\mu(x)}\right]d\mu(y)\leqslant\int_{B}^{+\infty}\left[{\int_{0}^{+\infty}1\hskip-2.6pt{\rm
l}_{p(a+x)\geqslant y}d\mu(x)}\right]d\mu(y)$
$\displaystyle=\int_{0}^{+\infty}\left[{\int_{B}^{+\infty}1\hskip-2.6pt{\rm
l}_{p(a+x)\geqslant
y}d\mu(y)}\right]d\mu(x)=\int_{a}^{+\infty}\left[{p(a+x)-B}\right]_{+}d\mu(x)$
$\displaystyle\leqslant\int_{B}^{+\infty}\left[{\int_{0}^{+\infty}1\hskip-2.6pt{\rm
l}_{x\leqslant\widetilde{p}(y)}d\mu(x)}\right]d\mu(y)=\int_{B}^{+\infty}\widetilde{p}(y)d\mu(y),$
which proves (23). ∎
It follows from (23) that if $p$ is a nonincreasing density on $(a,+\infty)$,
$\tau(p,t)$ also writes for all $t\geqslant 1$ as
$\tau(p,t)=\inf_{s>0}\left[{\int_{st}^{+\infty}p(a+x)d\mu(x)+\int_{0}^{s}\left[{p(a+x)-p(a+s)}\right]d\mu(x)}\right].$
It is not difficult to check that the mapping $\tau(p,\cdot)$ is nonincreasing
on $[1,+\infty)$, tends to 0 at $+\infty$ and is invariant under a location-
scale transformation, i.e. by changing $p$ into the density
$\sigma^{-1}p[\sigma^{-1}(\cdot-m)]$ on $(\sigma a+m,+\infty)$ with
$m\in{\mathbb{R}}$ and $\sigma>0$.
We consider the general situation where $p$ is an arbitrary element of
$\overline{\mathcal{M}}_{\ell}$ with $\ell\geqslant 2$. Changing the values of
$p$, if ever necessary, on a negligible set, which will not change the way it
can be approximated in ${\mathbb{L}}_{1}$-norm, we may assume with no loss of
generality that it writes as
(24) $p=w_{1}p_{1}1\hskip-2.6pt{\rm
l}_{(-\infty,x_{1})}+\sum_{i=2}^{\ell-1}w_{i}p_{i}1\hskip-2.6pt{\rm
l}_{(x_{i-1},x_{i})}+w_{\ell}p_{\ell}1\hskip-2.6pt{\rm
l}_{(x_{\ell-1},+\infty)}$
where $p_{1}$ and $p_{\ell}$ are monotone densities on $(-\infty,x_{1})$ and
$(x_{\ell-1},+\infty)$ respectively, $w_{1}=\int_{-\infty}^{x_{1}}pd\mu$,
$w_{\ell}=\int_{x_{\ell}}^{+\infty}pd\mu$ and when $\ell>2$,
$x_{1}<x_{2}<\ldots<x_{\ell-1}$ is an increasing sequence of real numbers, the
$p_{i}$ are monotone densities on $(x_{i-1},x_{i})$ and
$w_{i}=\int_{x_{i-1}}^{x_{i}}pd\mu$ for all $i\in\\{2,\ldots,\ell-1\\}$. For
$p$ written under the form (24), we set
(25)
$\tau_{\infty}(p,t)=\max_{i\in\\{1,\ldots,\ell\\}}\tau\left({p_{i},t}\right)\quad\text{for
all $t\geqslant 1$.}$
The mapping $t\mapsto\tau(p,t)$ is nonincreasing on ${\mathbb{R}}_{+}$ and
tends to 0 at $+\infty$.
###### Theorem 4.
Let $\ell\geqslant 2$, $k\geqslant 2\ell$ and $R\geqslant\ell\log 2$. If $p$
is a density of the form (24) a.e.,
(26)
$\inf_{\overline{p}\in\overline{\mathcal{M}}_{k}^{\infty}(R)}\left\|{p-\overline{p}}\right\|\leqslant
2\tau_{\infty}\left({p,\exp\left({\frac{R}{\ell}}\right)-1}\right).$
###### Proof.
The proof is postponed to Subsection 7.3. ∎
By combining Theorem 3 and Theorem 4 we obtain the following corollary.
###### Corollary 2.
Let $\ell\geqslant 2$ and $k\geqslant 2\ell$. If $p$ is a density of the form
(24) a.e.,
(27) ${\mathbb{B}}_{k,n}(p)\leqslant
44.3\left({\frac{\ell\log(1+r_{n})}{n}}\right)^{1/3}+83.2\sqrt{\frac{2k}{n}}$
where
(28) $r_{n}=\inf\left\\{{t\geqslant
1,\;\tau_{\infty}(p,t)\leqslant\left({\frac{\ell\log(1+t)}{n}}\right)^{1/3}}\right\\}$
and $\tau_{\infty}(p,\cdot)$ is defined by (25). Then, the TV-estimator
$\widehat{p}$ on ${\mathcal{M}}_{k}$ satisfies
(29) $\mathbb{E}\left[{\|p-\widehat{p}\|}\right]\leqslant
88.6\left({\frac{\ell\log(1+r_{n})}{n}}\right)^{1/3}+166.4\sqrt{\frac{2k}{n}}+\frac{2\varepsilon}{n}.$
###### Proof.
Since $\tau_{\infty}(p,\cdot)$ tends to 0 at $+\infty$, the set
${\mathcal{R}}=\left\\{{t\geqslant
1,\;\tau_{\infty}(p,t)\leqslant\left({\frac{\ell\log(1+t)}{n}}\right)^{1/3}}\right\\}$
is nonempty, $r_{n}$ is well-defined and for all $t>r_{n}\geqslant 1$,
$\tau_{\infty}(p,t)\leqslant\left({\frac{\ell\log(1+t)}{n}}\right)^{1/3}.$
Using (20), Theorems 3 and 4 with $R=\ell\log(1+t)>\ell\log 2$ we obtain that
$\displaystyle{\mathbb{B}}_{k,n}(p)$
$\displaystyle\leqslant\frac{3}{2}\inf_{\overline{p}\in\overline{\mathcal{M}}_{k}^{\infty}(R)}\left\|{p-\overline{p}}\right\|+\sup_{\overline{p}\in\overline{\mathcal{M}}_{k}^{\infty}(R)}{\mathbb{B}}_{k,n}(\overline{p})$
$\displaystyle\leqslant
3\tau_{\infty}\left({p,t}\right)+41.3\left({\frac{\ell\log(1+t)}{n}}\right)^{1/3}+83.2\sqrt{\frac{2k}{n}}$
$\displaystyle\leqslant
44.3\left({\frac{\ell\log(1+t)}{n}}\right)^{1/3}+83.2\sqrt{\frac{2k}{n}},$
and the result follows from the fact that $t$ is arbitrary in
$(r_{n},+\infty)$. ∎
###### Example 1.
Let $n\geqslant 2$, $\alpha\geqslant 0$, $\beta\geqslant-1$, $\gamma\in(0,1)$
and $q$ be the mapping defined by
$q(x)=\frac{2^{1-\gamma}}{x^{1-\gamma}}1\hskip-2.6pt{\rm
l}_{(0,2)}+\frac{2^{1+\alpha}\left({\log
2}\right)^{1+\beta}}{x^{1+\alpha}\left({\log
x}\right)^{1+\beta}}1\hskip-2.6pt{\rm l}_{[2,+\infty)}.$
When $(\alpha,\beta)\in(0,+\infty)\times[-1,+\infty)$ and when $\alpha=0$ and
$\beta>0$, $q$ is positive, integrable, nonincreasing function on
$(0,+\infty)$ and we may denote by $p$ the corresponding density, i.e. $p=cq$
for some $c>0$ depending on $\alpha,\beta$ and $\gamma$. The density $p$ may
be written is under the form (24) with $\ell=2$, $w_{1}=0$, $w_{2}=1$,
$a=x_{1}=0$ and $p_{2}=p$. Throughout this example, $C$ denotes a positive
number depending on $\alpha,\beta$ and $\gamma$ that may vary from line to
line.
It follows from Definition 3 that when $\alpha>0$, for all $t\geqslant 2$
$\displaystyle\frac{\tau_{x}(p,t)}{c2^{1+\alpha}\left({\log
2}\right)^{1+\beta}}$
$\displaystyle=\int_{t}^{+\infty}\frac{dx}{x^{1+\alpha}\left({\log
x}\right)^{1+\beta}}=\int_{\log t}^{+\infty}\frac{e^{-\alpha
s}}{s^{1+\beta}}ds\leqslant\frac{1}{\alpha t^{\alpha}\left({\log
t}\right)^{1+\beta}},$
and when $\alpha=0$ and $\beta>0$
$\displaystyle\frac{\tau_{x}(p,t)}{c2^{1+\alpha}\left({\log
2}\right)^{1+\beta}}$ $\displaystyle=\int_{\log
t}^{+\infty}\frac{1}{s^{1+\beta}}ds=\frac{1}{\beta\left({\log
t}\right)^{\beta}},$
For $y>c$, $\widetilde{p}:y\mapsto 2(c/y)^{1/(1-\gamma)}$, hence
$\tau_{y}(p,t)=\int_{t}^{+\infty}\widetilde{p}(y)d\mu(y)=\frac{2(1-\gamma)c^{1/(1-\gamma)}}{\gamma
t^{\gamma/(1-\gamma)}}\quad\text{for all $t\geqslant c$.}$
We deduce that for all $t\geqslant 1$ and $s\in[2/t,2]$, $p(s)\geqslant
p(2)=c$ and
$\displaystyle C^{-1}\tau_{\infty}(p,t)$
$\displaystyle\leqslant\begin{cases}\left[{{(st)^{\alpha}\left({\log(st)}\right)^{1+\beta}}}\right]^{-1}+s^{\gamma}&\text{when
$\alpha>0$ and $\beta\geqslant-1$}\\\
\left({\log(st)}\right)^{-\beta}+s^{\gamma}&\text{when $\alpha=0$ and
$\beta>0$}.\end{cases}$
Taking
$s=\begin{cases}2\left[{\left({t^{\alpha}\left({\frac{\log(1+t)}{\log
2}}\right)^{1+\beta}}\right)^{-\frac{1}{\alpha+\gamma}}\vee
t^{-1}}\right]&\text{when $\alpha>0$ and $\beta\geqslant-1$}\\\
2\left[{\left({\frac{\log(1+t)}{\log 2}}\right)^{-\frac{\beta}{\gamma}}\vee
t^{-1}}\right]&\text{when $\alpha=0$ and $\beta>0$}\end{cases}$
the value of which belongs to $[2/t,2]$, we obtain that for all $t\geqslant 1$
$\displaystyle C^{-1}\tau_{\infty}(p,t)$
$\displaystyle\leqslant\begin{cases}\left[{t^{\alpha}\left({\log(1+t)}\right)^{1+\beta}}\right]^{-\frac{\gamma}{\alpha+\gamma}}\vee
t^{-\gamma}&\text{when $\alpha>0$ and $\beta\geqslant-1$}\\\
\left({\log(1+t)}\right)^{-\beta}\vee t^{-\gamma}&\text{when $\alpha=0$ and
$\beta>0$}.\end{cases}$
If $\alpha>0$ and $\beta\geqslant-1$, by taking
$t=t_{n}=C^{\prime}n^{\frac{1}{3}\left({\frac{1}{\alpha}+\frac{1}{\gamma}}\right)}\log^{-\kappa}n\quad\text{with}\quad\kappa=\frac{1}{3}\left({\frac{1}{\alpha}+\frac{1}{\gamma}}\right)+\frac{1+\beta}{\alpha}$
for some constant $C^{\prime}>0$ large enough, we obtain that
$\tau_{\infty}(p,t_{n})\leqslant\left({2\log(1+t_{n})/n}\right)^{1/3}$ and
consequently, $r_{n}$ defined by (28) satisfies $r_{n}\leqslant t_{n}$.
Applying Corollary 2, we conclude that for all $k\geqslant 4$
${\mathbb{B}}_{k,n}(p)\leqslant C\left[{\left({\frac{\log
n}{n}}\right)^{1/3}+\sqrt{\frac{2k}{n}}}\right].$
If $\alpha=0$ and $\beta>1$, we take $t=t_{n}$ such that
$\log(1+t_{n})=C^{\prime}n^{1/(1+3\beta)}$ for some constant $C^{\prime}>0$
large enough, we obtain that
$\displaystyle\tau_{\infty}(p,t_{n})\leqslant
Cn^{\frac{-\beta}{1+3\beta}}\leqslant\left({\frac{2\log(1+t_{n})}{n}}\right)^{1/3},$
hence $r_{n}\leqslant t_{n}=C^{\prime}n^{1/(1+3\beta)}$ and we get that for
all $k\geqslant 4$
${\mathbb{B}}_{k,n}(p)\leqslant
C\left[{n^{\frac{-\beta}{1+3\beta}}+\sqrt{\frac{2k}{n}}}\right].$
## 5\. Convex-concave densities
### 5.1. Piecewise monotone convex-concave densities
In this section, our aim is to estimate a density on the line which is
piecewise monotone convex-concave in the sense defined below.
###### Definition 4.
A function $g$ is said to be convex-concave on an interval $I$ if it is either
convex or concave on $I$. For $k\geqslant 2$, a function $g$ on $\mathbb{R}$
is said to be $k$-piecewise monotone convex-concave if there exists
$A\in{\mathcal{A}}(k-1)$ such that the restriction of $g$ to the each interval
$I\in{\mathbf{I}}(A)$ is monotone and convex-concave. In particular, there
exist at most $k$ functions $\\{g_{I},\;I\in{\mathbf{I}}(A)\\}$, where $g_{I}$
is monotone and convex-concave on $I$ such that
$g(x)=\sum_{I\in{\mathbf{I}}(A)}g_{I}(x)1\hskip-2.6pt{\rm
l}_{I}(x)\quad\text{for all $x\in{\mathbb{R}}\setminus A$}.$
We denote by $\overline{{\mathcal{M}}}_{k}^{1}$ the set of $k$-piecewise
monotone convex-concave densities. The Laplace density $x\mapsto(1/2)e^{-|x|}$
belongs to $\overline{{\mathcal{M}}}_{2}^{1}$, the uniform density on a
(nontrivial) interval belongs to $\overline{{\mathcal{M}}}_{3}^{1}$, all
convex-concave densities on an interval belong to
$\overline{{\mathcal{M}}}_{4}^{1}$. A function
$g\in\overline{{\mathcal{M}}}_{k}^{1}$ associated to $A\in{\mathcal{A}}(k-1)$
admits left and right derivatives at any point $x\in{\mathbb{R}}\setminus A$.
These derivatives are denoted by $g_{l}^{\prime},g_{r}^{\prime}$ respectively.
More generally, when a function $f$ is continuous and convex-concave on a
nontrivial bounded interval $[a,b]$, we define
$f^{\prime}_{r}(z)=\lim_{x\downarrow z}\frac{f(x)-f(z)}{x-z}\quad\text{for all
$z\in[a,b)$}$
and
$f^{\prime}_{l}(z)=\lim_{x\uparrow z}\frac{f(x)-f(z)}{x-z}\quad\text{for all
$z\in(a,b]$}.$
These quantities are finite for all $z\in(a,b)$ and belong to
$[-\infty,+\infty]$ when $z\in\\{a,b\\}$. We say that $f$ admits a right
derivative at $a$ and a left derivative at $b$ when $f^{\prime}_{r}(a)$ and
$f^{\prime}_{l}(b)$ are finite respectively.
The role played by piecewise constant functions in the previous section is
here played by piecewise linear functions. For $D\geqslant 1$, let
$\overline{{\mathcal{O}}}_{D,k}^{1}$ be the subset of
$\overline{{\mathcal{M}}}_{k}^{1}$ that consists of those densities that are
left-continuous and affine on each interval of a class ${\mathbf{I}}(A)$ with
$A\in{\mathcal{A}}(D+1)$. For example, the left-continuous version of the
density of a uniform distribution on a nontrivial interval belongs to
$\overline{{\mathcal{O}}}_{1,3}^{1}$. The proposition below shows that the
elements of $\overline{{\mathcal{O}}}_{D,k}^{1}$ are extremal in
$\overline{{\mathcal{M}}}_{k}^{1}$.
###### Proposition 3.
Let $k\geqslant 2$, $D\geqslant 1$. If $p\in\overline{\mathcal{M}}_{k}^{1}$
and $q\in\overline{{\mathcal{O}}}_{D,k}^{1}$, the sets
$\\{x\in{\mathbb{R}},\,p(x)-q(x)>0\\}$ and
$\\{x\in{\mathbb{R}},\,p(x)-q(x)<0\\}$ are unions of at most $D+2k-1$
intervals. In particular, the elements of $\overline{{\mathcal{O}}}_{D,k}^{1}$
are extremal in $\overline{{\mathcal{M}}}_{k}^{1}$ with degree not larger than
$2(D+2k-1)$.
###### Proof.
The proof is postponed to Subsection 7.4. ∎
For all $D\geqslant 1$, $k\geqslant 2$, let ${\mathcal{O}}_{D,k}^{1}$ be a
countable and dense subset of $\overline{\mathcal{O}}_{D,k}^{1}$ (for the
${\mathbb{L}}_{1}$-norm) and ${\mathcal{M}}_{k}^{1}$ a countable and dense
subset of $\overline{{\mathcal{M}}}_{k}^{1}$ that contains
$\bigcup_{D\geqslant 1}\mathcal{O}_{D,k}^{1}$. By proposition 1, the elements
of ${\mathcal{O}}_{D,k}^{1}$ are also extremal in ${\mathcal{M}}_{k}^{1}$ with
degree no larger than $2(D+2k-1)$ for all $D\geqslant 1$. We deduce from
Theorem 1 the following result.
###### Theorem 5.
Let $k\geqslant 2$. Whatever the product distribution ${\mathbf{P}}^{\star}$
of the data, any TV-estimator $\widehat{P}$ on ${\mathscr{M}}_{k}^{1}$
satisfies
(30)
${\mathbb{E}}\left[d\\!\left(P^{\star},\hat{P}\right)\right]\leqslant\inf_{D\geqslant
1}\left\\{3\inf_{P\in\overline{{\mathscr{O}}}_{D,k}^{1}}d\\!\left(P^{\star},P\right)+68\sqrt{\frac{D+2k-1}{n}}\right\\}+\frac{\varepsilon}{n}.$
In the remaining part of this section we assume that the $X_{i}$ are i.i.d.
with a density $p\in\overline{\mathcal{M}}_{k}^{1}$, in which case, the right-
hand side of (30) writes as ${\mathbb{B}}_{k,n}^{1}(p)+{\varepsilon}/n$ with
(31) ${\mathbb{B}}_{k,n}^{1}(p)=\inf_{D\geqslant
1}\left[\frac{3}{2}\inf_{q\in\overline{{\mathcal{O}}}_{D,k}^{1}}\left\|{p-q}\right\|+68\sqrt{\frac{D+2k-1}{n}}\right].$
As we did in Section 4, our aim is to bound the quantity
${\mathbb{B}}_{k,n}^{1}(p)$ under some suitable additional assumptions on the
density $p$.
### 5.2. Approximation of a monotone convex-concave density by a piecewise
linear function
Let us now turn to the approximation of a monotone convex-concave density by a
convex-concave piecewise linear function. The approximation result that we
establish is actually true for a sub-density on an interval $[a,b]$, i.e. a
nonnegative function on $[a,b]$ the integral of which is not larger than 1. In
the remaining part of this chapter, we use the following convenient
definition.
###### Definition 5.
Let $D\geqslant 1$ and $f$ be a continuous function on a compact nontrivial
interval $[a,b]$. We say that $\overline{f}$ is a $D$-linear interpolation of
$f$ on $[a,b]$ if there exists a subdivision $a=x_{0}<\ldots<x_{D}=b$ such
that $\overline{f}(x_{i})=f(x_{i})$ and $\overline{f}$ is affine on
$[x_{i-1},x_{i}]$ for all $i\in\\{1,\ldots,D\\}$.
This definition automatically determines the values of $\overline{f}$ on
$[a,b]$ since $\overline{f}$ corresponds on $[x_{i-1},x_{i}]$ to the chord
that connects $(x_{i-1},f(x_{i-1}))$ to $(x_{i},f(x_{i}))$ for all
$i\in\\{1,\ldots,D\\}$. The function $\overline{f}$ is therefore continuous
and piecewise linear on a partition of $[a,b]$ into $D$ intervals and it
inherits of some of the features of the function $f$. For example, if $f$ is
nonnegative, increasing, decreasing, convex or concave, so is $\overline{f}$.
If $f$ is convex (respectively concave), $\overline{f}\geqslant f$
(respectively $\overline{f}\leqslant f$).
Given a continuous monotone convex-concave function $f$ with increment
$\Delta=(f(b)-f(a))/(b-a)$ on a bounded nontrivial interval $[a,b]$, we define
its linear index $\Gamma=\Gamma(f)$ as
$\Gamma=1-\frac{1}{2}\left({\frac{|p_{r}^{\prime}(a)|\wedge|p_{l}^{\prime}(b)|}{|\Delta|}+\frac{|\Delta|}{|p_{r}^{\prime}(a)|\vee|p_{l}^{\prime}(b)|}}\right),$
with the conventions $0/0=1$ and $1/(+\infty)=0$. Since $f$ is convex-concave,
monotone and continuous
$|p_{r}^{\prime}(a)|\wedge|p_{l}^{\prime}(b)|\leqslant|\Delta|\leqslant|p_{r}^{\prime}(a)|\vee|p_{l}^{\prime}(b)|\quad\text{and}\quad$
and
$\Delta=0\implies|p_{r}^{\prime}(a)|=|p_{l}^{\prime}(b)|=0.$
With our conventions, $\Gamma$ is well-defined and belongs to $[0,1]$. When
$f$ is affine, $\Delta=p_{r}^{\prime}(a)=p_{l}^{\prime}(b)$ and its linear
index is 0. In the opposite direction when $f$ is far from being affine, say
when for some $c\in(a,b)$ and $v>0$
$f(x)=\frac{v}{b-c}(x-c)_{+}\quad\text{for all $x\in[a,b]$}$
its linear index $\Gamma=1-(b-c)/[2(b-a)]$ increases to 1 as $c$ approaches
$b$.
###### Theorem 6.
Let $p$ be a monotone, continuous, convex-concave density on a bounded
interval $[a,b]$ of length $L>0$ with variation $V=|p(b)-p(a)|$ and linear
index $\Gamma\in[0,1]$. For all $D\geqslant 1$, there exists a $2D$-linear
interpolation $\overline{p}$ of $p$ such that
(32)
$\int_{a}^{b}\left|{p-\overline{p}}\right|d\mu\leqslant\frac{4}{3}\left[{\left({1+\sqrt{2LV\Gamma}}\right)^{1/D}-1}\right]^{2}.$
In particular, there exists a continuous convex-concave piecewise linear
density $q$ based on a partition of $[a,b]$ into $2D$ intervals that satisfies
(33) $\int_{a}^{b}\left|{p-q}\right|d\mu\leqslant
5.14\frac{\log^{2}\left({1+\sqrt{2LV\Gamma}}\right)}{D^{2}}.$
Since $\Gamma\in[0,1]$, (33) implies that
$\int_{a}^{b}\left|{p-q}\right|d\mu\leqslant
5.14\frac{\log^{2}\left({1+\sqrt{2LV}}\right)}{D^{2}}.$
Nevertheless, when $p$ is affine on $[a,b]$, $\Gamma=0$ and we recover the
fact that we may choose $\overline{p}$ and $q$ on $[a,b]$ such that
$\int_{a}^{b}\left|{p-\overline{p}}\right|d\mu=\int_{a}^{b}\left|{p-q}\right|d\mu=0$.
The fact that a bounded convex (or concave) function on a compact interval can
be approximated in ${\mathbb{L}}_{1}$ by piecewise affine functions at rate
$\mathcal{O}(1/D^{2})$ had already been established by Guérin et al., (2006).
What is novel in Theorem 6 is the fact that for probability densities the
approximation error depends logarithmically on the product $LV$.
###### Proof.
The proof is based on several preliminary approximation results whose
statements and proofs are postponed to Subsection 7.4. ∎
### 5.3. Estimation of $k$-piecewise monotone convex-concave bounded and
compactly supported densities
In this section, we consider a density $p\in\overline{\mathcal{M}}_{k}^{1}$,
with $k\geqslant 3$, that is of the form
(34) $p=\sum_{i=1}^{k-2}w_{i}p_{i}1\hskip-2.6pt{\rm l}_{(x_{i-1},x_{i})}$
where
1. (i)
$(x_{i})_{i\in\\{0,\ldots,k-2\\}}$ is an increasing sequence of real numbers;
2. (ii)
$w_{1},\ldots,w_{k-2}$ are nonnegative numbers such that
$\sum_{i=1}^{k-2}w_{i}=1$;
3. (iii)
for $i\in\\{1,\ldots,k-2\\}$, $p_{i}$ is a monotone continuous convex-concave
density on the interval $[x_{i-1},x_{i}]$ of length $L_{i}>0$, with variation
$V_{i}=\left|{p_{i}(x_{i-1})-p(x_{i})}\right|<+\infty$ and linear index
$\Gamma_{i}\in[0,1]$.
For a density $p$ satisfying (34) we define
(35)
$\boldsymbol{R}_{k,1}(p)=\inf\left[{\sum_{i=1}^{k-2}\left({w_{i}\log^{2}\left({1+\sqrt{2L_{i}V_{i}\Gamma_{i}}}\right)}\right)^{1/3}}\right]^{3/2},$
where the infimum runs among all ways of writing $p$ under the form (34). We
denote by $\overline{\mathcal{M}}_{k,1}^{\infty}$ the class of all densities
of the form (34) a.e. and for $R>0$,
$\overline{\mathcal{M}}_{k,1}^{\infty}(R)$ the subset of those which satisfy
$\boldsymbol{R}_{k,1}(p)<R$.
Note that a concave (or convex) density on a (necessarily) compact interval
belongs to $\overline{\mathcal{M}}_{4}^{1}$.
The following result holds.
###### Theorem 7.
Let $k\geqslant 3$ and $R>0$. For all
$p\in\overline{\mathcal{M}}_{k,1}^{\infty}(R)$
(36) ${\mathbb{B}}_{k,n}^{1}(p)\leqslant
7.71\left[{15.06\left({\frac{R}{n}}\right)^{2/5}+8.82\sqrt{\frac{4k-5}{n}}}\right],$
where ${\mathbb{B}}_{k,n}^{1}(p)$ is defined by (31).
Theorem 7 implies, together with Theorem 5, that the TV-estimator converges at
rate $n^{-2/5}$ in total variation distance, whenever the underlying density
$p^{\star}$ is bounded, compactly supported and belongs to the class
$\overline{{\mathcal{M}}}_{k,1}^{\infty}$ of $k-$piecewise monotone convex-
concave densities. The rate $n^{-2/5}$ matches the minimax lower bound
established in Devroye and Lugosi (2001)[Section 15.5] for bounded convex
densities. This rate is therefore optimal.
To the best of our knowledge, Theorem 8 provides the sharpest known minimax
upper bound in this setting, including the case of a monotone, convex or
concave density on a compact interval. In comparison, Gao and Wellner, 2009b
proved that the MLE on the set of convex non-increasing densities on a given
interval achieves the rate $n^{-2/5}$ (for the Hellinger distance). Note that
the construction of the MLE requires that the support of the target density is
known while our TV-estimator assumes nothing.
Consider now the special case of a continuous, concave density on an interval
$[a,b]$ (which is necessarily bounded). A monotone continuous concave density
$p$ on a bounded interval $[a,b]$ with length $L>0$ belongs to
$\overline{\mathcal{M}}_{3}^{1}$. It necessarily satisfies
$L|p(a)-p(b)|/2\leqslant 1$, hence
$\boldsymbol{R}_{3,1}(p)\leqslant\log(1+\sqrt{2L|p(a)-p(b)|})\leqslant\log 3$.
If $p$ is not monotone but only continuous and concave on $[a,b]$, we may
write $p$ as $w_{1}p_{1}+w_{2}p_{2}$ where $p_{1}$ and $p_{2}$ are monotone
and concave densities on the intervals $[a,c]$ and $[c,a]$ respectively where
$c$ is a maximizer of $p$ in $(a,b)$. The density $p$ belongs to
$\overline{\mathcal{M}}_{4}^{1}$ and by applying the previous inequality to
the densities $p_{1}$ and $p_{2}$ successively and the inequality
$z^{1/3}+(1-z)^{1/3}\leqslant 2^{2/3}$ which holds for all $z\in[0,1]$, we
obtain that
$\displaystyle\boldsymbol{R}_{4,1}^{2/3}(p)$
$\displaystyle\leqslant\left({w_{1}\log^{2}(1+\sqrt{2(c-a)\left({p_{1}(c)-p_{1}(a)}\right)}}\right)^{1/3}$
$\displaystyle\quad+\left({w_{2}\log^{2}(1+\sqrt{2(b-c)\left({p_{2}(c)-p_{2}(b)}\right)}}\right)^{1/3}$
$\displaystyle\leqslant\left({w_{1}^{1/3}+w_{2}^{1/3}}\right)(\log
3)^{2/3}\leqslant(2\log 3)^{2/3}.$
We immediately deduce from Theorems 5 and 7 the following corollary.
###### Corollary 3.
If $X_{1},\ldots,X_{n}$ is a $n$-sample which density is concave on an
interval of ${\mathbb{R}}$, then the TV-estimator $\widehat{p}$ on
${\mathcal{M}}_{4}^{1}$ satisfies
${\mathbb{E}}\left[{\left\|{p-\widehat{p}}\right\|}\right]\leqslant\frac{320}{n^{2/5}}+\frac{451}{\sqrt{n}}+\frac{2{\varepsilon}}{n}.$
In particular,
$\adjustlimits{\inf}_{\widetilde{p}}{\sup}_{p}{\mathbb{E}}\left[{\left\|{p-\widetilde{p}}\right\|}\right]\leqslant\frac{320}{n^{2/5}}+\frac{451}{\sqrt{n}},$
where the supremum runs among all concave densities $p$ on an interval of
${\mathbb{R}}$ and the infimum over all density estimators $\widetilde{p}$
based on a $n$-sample with density $p$.
###### Proof of Theorem 7.
Let $p\in\overline{\mathcal{M}}_{k,1}^{\infty}(R)$. With no loss of generality
we may assume that $p$ is of the form (34) everywhere and choose a subdivision
$(x_{i})_{i\in\\{0,\ldots,k-2\\}}$ in such a way that
$\left[{\sum_{i=1}^{k-2}\left({w_{i}\log^{2}\left({1+\sqrt{2L_{i}V_{i}\Gamma_{i}}}\right)}\right)^{1/3}}\right]^{3/2}\leqslant
R.$
Let $D\geqslant k-2$ and $D_{1},\ldots,D_{k-2}$ be some positive integers to
be chosen later on that satisfy the constraint $\sum_{i=1}^{k-2}D_{i}\leqslant
D$. By Theorem 6, we may find for all $i\in\\{1,\ldots,k-2\\}$ a density
$q_{i}$ that is continuous and supported on $[x_{i-1},x_{i}]$, piecewise
linear on a partition of $[x_{i-1},x_{i}]$ into $2D_{i}$ intervals, that
satisfies
$\int_{x_{i-1}}^{x_{i}}\left|{p_{i}-q_{i}}\right|d\mu\leqslant
5.14\frac{\log^{2}\left({1+\sqrt{2L_{i}V_{i}\Gamma_{i}}}\right)}{D_{i}^{2}}.$
The function $q=\sum_{i=1}^{k-2}w_{i}q_{i}1\hskip-2.6pt{\rm
l}_{(x_{i-1},x_{i}]}$ is a density, that is left-continuous, convex-concave on
each interval $I\in{\mathbf{I}}(\\{x_{0},\ldots,x_{k-2}\\})$ and affine on
each interval of a partition
$(x_{0},x_{k-2}]=\bigcup_{i=1}^{k-2}(x_{i-1},x_{i}]$ into
$\sum_{i=1}^{k-2}2D_{i}\leqslant 2D$ intervals. It therefore belongs to
$\overline{\mathcal{O}}_{2D,k}$. Besides,
$\displaystyle\left\|{p-q}\right\|$
$\displaystyle\leqslant\sum_{i=1}^{k-2}w_{i}\int_{x_{i-1}}^{x_{i}}\left|{p_{i}-q_{i}}\right|d\mu\leqslant
5.14\sum_{i=1}^{k-2}\frac{w_{i}\log^{2}\left({1+\sqrt{2L_{i}V_{i}\Gamma_{i}}}\right)}{D_{i}^{2}}$
and it follows from (31) that
$\displaystyle{\mathbb{B}}_{k,n}^{1}(p)$
$\displaystyle\leqslant\frac{3}{2}\inf_{q\in\overline{{\mathcal{O}}}_{2D,k}^{1}}\left\|{p-q}\right\|+68\sqrt{\frac{2D+2k-1}{n}}$
$\displaystyle\leqslant
7.71\sum_{i=1}^{k-2}\frac{w_{i}\log^{2}\left({1+\sqrt{2L_{i}V_{i}\Gamma_{i}}}\right)}{D_{i}^{2}}+68\sqrt{\frac{2D+2k-1}{n}}.$
Let us set $c=68/7.71$, $s_{0}=[nR^{4}/(2c^{2})]^{1/5}$,
$s_{i}=\left[{w_{i}\log^{2}\left({1+\sqrt{2L_{i}V_{i}\Gamma_{i}}}\right)}\right]^{1/3}\quad\text{for
all $i\in\\{1,\ldots,k-2\\}$}$
and choose $D=s_{0}+k-2$ and
$D_{i}=\left\lceil\frac{s_{0}s_{i}}{\sum_{j=1}^{k-2}s_{j}}\right\rceil\geqslant\frac{s_{0}s_{i}}{\sum_{j=1}^{k-2}s_{j}}\vee
1\quad\text{for all $i\in\\{1,\ldots,k-2\\}$},$
so that $\sum_{i=1}^{k-2}D_{i}\leqslant D$. Then,
$\displaystyle{\mathbb{B}}_{k,n}^{1}(p)$ $\displaystyle\leqslant
7.71\left[{\sum_{i=1}^{k-2}\frac{s_{i}^{3}}{D_{i}^{2}}+c\sqrt{\frac{2D+2k-1}{n}}}\right]$
$\displaystyle\leqslant
7.71\left[{\frac{\left({\sum_{j=1}^{k-2}s_{j}}\right)^{3}}{s_{0}^{2}}+c\sqrt{\frac{2s_{0}+4k-5}{n}}}\right]$
$\displaystyle\leqslant
7.71\left[{\frac{R^{2}}{s_{0}^{2}}+c\sqrt{\frac{2s_{0}}{n}}+c\frac{\sqrt{4k-5}}{n}}\right]$
$\displaystyle=7.71\left[{2(2c^{2})^{2/5}\frac{R^{2/5}}{n^{2/5}}+c\frac{\sqrt{4k-5}}{n}}\right]$
which gives (36). ∎
## 6\. Log-concave densities
In this section, we consider the set $\overline{\mathcal{M}}^{\text{LC}}$ of
log-concave densities. With the convention $\exp(-\infty)=0$,
$\overline{\mathcal{M}}^{\text{LC}}$ is the set of densities of the form
$p=\exp\phi$ where the set $J=J(p)=\\{x\in{\mathbb{R}},\;p(x)>0\\}$ is an open
interval and the mapping $\phi:{\mathbb{R}}\to{\mathbb{R}}\cup\\{-\infty\\}$
is a continuous and concave function on $J$. Given some $D\geqslant 1$, a
subset of $\overline{\mathcal{M}}^{\text{LC}}$ of special interest is the set
$\overline{\mathcal{O}}_{D}^{\text{LC}}$ of those densities the logarithm of
which is either affine or takes the value $-\infty$ on the elements of a set
${\mathbf{I}}(A)$ with $A\in{\mathcal{A}}(D)$. Since a density
$p\in\overline{\mathcal{O}}_{D}^{\text{LC}}$ is log-concave, the logarithm of
$p$ may take the value $-\infty$ on the two unbounded elements of
${\mathbf{I}}(A)$ only. For example, the densities $x\mapsto 1\hskip-2.6pt{\rm
l}_{x>0}\exp(-x)$ and $x\mapsto(1/2)\exp(-|x|)$ of the exponential and Laplace
distributions belong to $\overline{\mathcal{O}}_{1}^{\text{LC}}$ while the
standard Gaussian density belongs to $\overline{\mathcal{M}}^{\text{LC}}$.
###### Proposition 4.
Let $D\geqslant 1$. The elements of $\overline{{\mathcal{O}}}_{D}^{\text{LC}}$
are extremal in $\overline{\mathcal{M}}^{\text{LC}}$ with degree not larger
than $2(D+2)$.
###### Proof.
The proof is postponed to Subsection 7.5. ∎
For all $D\geqslant 1$, let ${\mathcal{O}}_{D}^{\text{LC}}$ be a countable and
dense subset of $\overline{{\mathcal{O}}}_{D}^{\text{LC}}$ (for the
${\mathbb{L}}^{1}$-norm) and ${\mathcal{M}}^{\text{LC}}$ a countable and dense
subset of $\overline{{\mathcal{M}}}^{\text{LC}}$ that contains
$\bigcup_{D\geqslant 1}{\mathcal{O}}_{D}^{\text{LC}}$. We immediately deduce
from Theorem 1 together with Proposition 4 the following result.
###### Theorem 8.
Whatever the product distribution ${\mathbf{P}}^{\star}$ of the data, any TV-
estimator $\widehat{P}$ on ${\mathscr{M}}^{\text{LC}}$ satisfies
(37)
$\mathbb{E}\left[{d\\!\left(P^{\star},\widehat{P}\right)}\right]\leqslant\inf_{D\geqslant
1}\left[{3\inf_{P\in\overline{\mathscr{O}}_{D}^{\text{LC}}}d\\!\left(P^{\star},P\right)+48\sqrt{2}\sqrt{\frac{D+2}{n}}}\right]+\frac{\varepsilon}{n}.$
In the remaining part of this section we assume that the $X_{i}$ are i.i.d.
with a density $p\in\overline{\mathcal{M}}_{\text{LC}}$, in which case, the
right-hand side of (37) writes as ${\mathbb{B}}_{n}(p)+{\varepsilon}/n$ with
(38) $\displaystyle{\mathbb{B}}_{n}(p)$ $\displaystyle=\inf_{D\geqslant
1}\left[{\frac{3}{2}\inf_{q\in\overline{\mathcal{O}}_{D}^{\text{LC}}}\left\|{p-q}\right\|+48\sqrt{2}\sqrt{\frac{D+2}{n}}}\right]$
In particular, if $p$ belongs to $\overline{\mathcal{O}}_{D}^{\text{LC}}$ for
some $D\geqslant 1$, then ${\mathbb{B}}_{n}(p)\leqslant\sqrt{D/n}$, so that
$\hat{p}$ converges at the parametric rate. Thus, the TV-estimator shares the
_adaptivity_ property established for the MLE by Kim et al (2018) (and Feng et
al., (2021)), and by Baraud and Birgé (2016) for their $\rho-$estimator. As
compared to theirs, our upper bound does not involve logarithmic factors. This
difference is due to the fact that Baraud and Birgé’s results are established
for the Hellinger distance and these logarithmic factors are sometimes
necessary for such a loss.
As we did in Sections 4 and 5 , our aim is now to bound the quantity
${\mathbb{B}}_{n}(p)$. This can be done by using the following approximation
result.
###### Proposition 5.
Let $D\geqslant 6$. For all log-concave densities $p$,
$\inf_{q\in\overline{\mathcal{O}}^{\text{LC}}_{6D}}\left\|{p-q}\right\|\leqslant\frac{2}{D^{2}}.$
###### Proof.
The proof is postponed to Subsection 7.5. ∎
Combining (38) with Proposition 5, we deduce that for all $D\geqslant 6$
$\displaystyle{\mathbb{B}}_{n}(p)$ $\displaystyle=\inf_{D\geqslant
6}\left[{\frac{3}{2}\inf_{q\in\overline{\mathcal{O}}_{6D}^{\text{LC}}}\left\|{p-q}\right\|+48\sqrt{2}\sqrt{\frac{6D+2}{n}}}\right]$
$\displaystyle\leqslant\inf_{D\geqslant
6}\left[{\frac{3}{D^{2}}+48\sqrt{2}\sqrt{\frac{6D+2}{n}}}\right].$
By choosing
$D=\left\lceil\left({\frac{3n}{96^{2}}}\right)^{1/5}\right\rceil\vee
6\leqslant\left({\frac{6n}{2\times 96^{2}}}\right)^{1/5}+6,$
we obtain the following estimation result for log-concave densities.
###### Corollary 4.
If $X_{1},\ldots,X_{n}$ is a $n$-sample which density $p$ is log-concave on
${\mathbb{R}}$, then the TV-estimator $\widehat{p}$ on ${\mathcal{M}}^{LC}$
satisfies
${\mathbb{E}}\left[{\left\|{p-\widehat{p}}\right\|}\right]\leqslant\frac{300}{n^{2/5}}+\frac{837}{\sqrt{n}}+\frac{2\varepsilon}{n}.$
The shape-constrained TV-estimator attains the global convergence rate of
$n^{-2/5}$ on the class of log-concave densities, i.e. the same global rate as
the MLE — see Kim and Samworth, (2016).
### Acknowledgement
One of the authors thanks Lutz Dümbgen for asking if we could establish a
result with $d(P^{\star},\widehat{P})$ in place of
$n^{-1}\sum_{i=1}^{n}d(P_{i},\widehat{P})$ at a conference where a preliminary
version of this work was presented. His question spurred the authors to
improve their result and obtain this new version of Theorem 1.
## 7\. Proofs
### 7.1. Technical result
First, we need a way to approximate $p$ not just by nonnegative functions, but
by densities. The following lemma shows that this can be achieved with a
simple renormalization.
###### Lemma 3.
Let $p$ be a density on a measured space $(E,{\mathcal{E}},\nu)$ and $f$ a
nonnegative integrable function on $(E,{\mathcal{E}},\nu)$ which is not
$\nu$-a.e. equal to 0 on $E$. Then
$\int_{E}\left|p-\frac{f}{\int_{E}fd\nu}\right|d\nu\leqslant
2\left[{1\wedge\int_{E}|p-f|d\nu}\right].$
This inequality cannot be improved in general since equality holds when
$f=p1\hskip-2.6pt{\rm l}_{I}$ and $I$ is a measurable subset of $E$ on which
$p$ is not $\nu$-a.e. equal to 0.
###### Proof.
The fact that
$\int_{E}\left|p-\frac{f}{\int_{E}fd\nu}\right|d\nu\leqslant 2$
comes from the triangle inequality. Let us now prove that
$\int_{E}\left|p-\frac{f}{\int_{E}fd\nu}\right|d\nu\leqslant
2\int_{E}|p-f|d\nu.$
We first assume that $c=\int_{E}fd\nu\in(0,1]$. Since $f/c$ and $p$ are two
densities,
$\displaystyle\int_{E}\left|\frac{f}{c}-p\right|d\nu$
$\displaystyle=2\int_{I}\left[p-\frac{f}{c}\right]1\hskip-2.6pt{\rm
l}_{\\{cp\geqslant f\\}}d\nu\leqslant 2\int_{E}[p-f]1\hskip-2.6pt{\rm
l}_{\\{cp\geqslant f\\}}d\nu$ $\displaystyle\leqslant 2\int_{E}|p-f|d\nu.$
This proves the lemma when $c\in(0,1]$. let us now assume that $c>1$. The
previous case of the lemma applies to the density $f/c$ and the nonnegative
function $p/c$ the integral of which is not larger than 1. This yields
$\displaystyle\int_{E}\left|\frac{f}{c}-p\right|d\nu$ $\displaystyle\leqslant
2\int_{E}\left|\frac{f}{c}-\frac{p}{c}\right|d\nu=\frac{2}{c}\int_{I}|f-p|d\nu\leqslant
2\int_{E}|f-p|d\nu.$
∎
### 7.2. Proofs of Section 3
###### Proof of Theorem 1.
Let $D\geqslant 1$ such that $\overline{\mathcal{O}}(D)$ is not empty. Such an
integer $D$ exists since $\overline{\mathcal{O}}$ is nonempty. Let
$\overline{P}=\overline{p}\cdot\mu$ be an arbitrary point in
${\mathscr{O}}(D)$ with $\overline{p}\in{\mathcal{O}}(D)$. For
$P,Q\in{\mathscr{M}}$ and $\zeta\geqslant 0$, we set
$\displaystyle{\mathbf{Z}}_{+}({\boldsymbol{X}},P)$
$\displaystyle=\sup_{Q\in{\mathscr{M}}}\left[{{\mathbf{T}}({\boldsymbol{X}},P,Q)-{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},P,Q)}\right]}\right]-\zeta$
$\displaystyle{\mathbf{Z}}_{-}({\boldsymbol{X}},P)$
$\displaystyle=\sup_{Q\in{\mathscr{M}}}\left[{{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},Q,P)}\right]-{\mathbf{T}}({\boldsymbol{X}},Q,P)}\right]-\zeta$
and
${\mathbf{Z}}({\boldsymbol{X}},P)={\mathbf{Z}}_{+}({\boldsymbol{X}},P)\vee{\mathbf{Z}}_{-}({\boldsymbol{X}},P).$
Applying the first inequality of (6) with $P=Q$ and $Q=\overline{P}$, we infer
that for all $Q\in{\mathscr{M}}$,
$\displaystyle nd\\!\left(\overline{P},Q\right)$ $\displaystyle\leqslant
nd\\!\left(P^{\star},\overline{P}\right)+{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},Q,\overline{P})}\right]$
$\displaystyle=nd\\!\left(P^{\star},\overline{P}\right)+{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},Q,\overline{P})}\right]-{\mathbf{T}}({\boldsymbol{X}},Q,\overline{P})+{\mathbf{T}}({\boldsymbol{X}},Q,\overline{P})$
$\displaystyle\leqslant
nd\\!\left(P^{\star},\overline{P}\right)+{\mathbf{Z}}({\boldsymbol{X}},\overline{P})+{\mathbf{T}}({\boldsymbol{X}},Q,\overline{P})+\zeta$
$\displaystyle\leqslant
nd\\!\left(P^{\star},\overline{P}\right)+{\mathbf{Z}}({\boldsymbol{X}},\overline{P})+{\mathbf{T}}({\boldsymbol{X}},Q)+\zeta.$
In particular, the inequality applies to
$Q=\widehat{P}\in{\mathscr{E}}({\boldsymbol{X}})$ and using the fact that
${\mathbf{T}}({\boldsymbol{X}},\widehat{P})\leqslant\inf_{P\in{\mathscr{M}}}{\mathbf{T}}({\boldsymbol{X}},P)+\varepsilon\leqslant{\mathbf{T}}({\boldsymbol{X}},\overline{P})+\varepsilon,$
we deduce that
(39) $\displaystyle nd\\!\left(\overline{P},\widehat{P}\right)$
$\displaystyle\leqslant
nd\\!\left(P^{\star},\overline{P}\right)+{\mathbf{Z}}({\boldsymbol{X}},\overline{P})+{\mathbf{T}}({\boldsymbol{X}},\overline{P})+\zeta+\varepsilon.$
Using now the second inequality of (6) with $P=\overline{P}$, we obtain that
$\displaystyle{\mathbf{T}}({\boldsymbol{X}},\overline{P})$
$\displaystyle=\sup_{Q\in{\mathscr{M}}}{\mathbf{T}}({\boldsymbol{X}},\overline{P},Q)$
$\displaystyle\leqslant\sup_{Q\in{\mathscr{M}}}\left[{{\mathbf{T}}({\boldsymbol{X}},\overline{P},Q)-{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},\overline{P},Q)}\right]-\zeta}\right]+\sup_{Q\in{\mathscr{M}}}{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},\overline{P},Q)}\right]+\zeta$
$\displaystyle\leqslant{\mathbf{Z}}({\boldsymbol{X}},\overline{P})+nd\\!\left(P^{\star},\overline{P}\right)+\zeta,$
which with (39) lead to
(40) $nd\\!\left(\overline{P},\widehat{P}\right)\leqslant
2nd\\!\left(P^{\star},\overline{P}\right)+2\zeta+\varepsilon+2{\mathbf{Z}}({\boldsymbol{X}},\overline{P}).$
Let us now bound from above ${\mathbf{Z}}({\boldsymbol{X}},\overline{P})$. We
set for $P\in{\mathscr{M}}$
$\displaystyle{\mathbf{w}}(P)$
$\displaystyle={\mathbb{E}}\left[{\sup_{Q\in{\mathscr{M}}}\left[{{\mathbf{T}}({\boldsymbol{X}},P,Q)-{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},P,Q)}\right]}\right]}\right]$
$\displaystyle\quad\vee{\mathbb{E}}\left[{\sup_{Q\in{\mathscr{M}}}\left[{{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},Q,P)}\right]-{\mathbf{T}}({\boldsymbol{X}},Q,P)}\right]}\right].$
The functions $t_{(\overline{P},Q)}$ satisfy
$|t_{(\overline{P},Q)}(x)-t_{(\overline{P},Q)}(x^{\prime})|\leqslant 1$ for
all $Q\in{\mathscr{M}}$ and $x,x^{\prime}\in E$, hence
$\left|{{\mathbf{Z}}_{+}((x_{1},\ldots,x_{i},\ldots,x_{n}),\overline{P})-{\mathbf{Z}}_{+}((x_{1},\ldots,x_{i}^{\prime},\ldots,x_{n}),\overline{P})}\right|\leqslant
1$
for all ${\mathbf{x}}\in{\mathbf{E}}$, $x_{i}^{\prime}\in E$ and
$i\in\\{1,\ldots,n\\}$. Following the same lines as in the proof of Lemma 2 of
Baraud (2021) (with $\xi+\log 2$ in place of $\xi$), we deduce that with a
probability at least $1-(1/2)e^{-\xi}$,
(41) $\displaystyle{\mathbf{Z}}_{+}({\boldsymbol{X}},\overline{P})$
$\displaystyle\leqslant{\mathbb{E}}\left[{{\mathbf{Z}}_{+}({\boldsymbol{X}},\overline{P})}\right]+\sqrt{\frac{n(\xi+\log
2)}{2}}$
$\displaystyle={\mathbb{E}}\left[{\sup_{Q\in{\mathscr{M}}}\left[{{\mathbf{T}}({\boldsymbol{X}},\overline{P},Q)-{\mathbb{E}}\left[{{\mathbf{T}}({\boldsymbol{X}},\overline{P},Q)}\right]}\right]}\right]+\sqrt{\frac{n(\xi+\log
2)}{2}}-\zeta$ (42)
$\displaystyle\leqslant{\mathbf{w}}(\overline{P})+\sqrt{\frac{n(\xi+\log
2)}{2}}-\zeta.$
Arguing similarly, with a probability at least $1-(1/2)e^{-\xi}$,
(43) $\displaystyle{\mathbf{Z}}_{-}({\boldsymbol{X}},\overline{P})$
$\displaystyle\leqslant{\mathbf{w}}(\overline{P})+\sqrt{\frac{n(\xi+\log
2)}{2}}-\zeta.$
Putting (42) and (43) together and choosing
$\zeta={\mathbf{w}}(\overline{P})+\sqrt{n(\xi+\log 2)/2}$, we obtain that with
a probability at least $1-e^{-\xi}$,
${\mathbf{Z}}({\boldsymbol{X}},\overline{P})={\mathbf{Z}}_{+}({\boldsymbol{X}},P)\vee{\mathbf{Z}}_{-}({\boldsymbol{X}},P)\leqslant{\mathbf{w}}(\overline{P})+\sqrt{\frac{n(\xi+\log
2)}{2}}-\zeta\leqslant 0$
which with (40) lead to the bound
(44) $d\\!\left(\overline{P},\widehat{P}\right)\leqslant
2d\\!\left(P^{\star},\overline{P}\right)+\frac{2{\mathbf{w}}(\overline{P})}{n}+\sqrt{\frac{2(\xi+\log
2)}{n}}+\frac{\varepsilon}{n}.$
It remains now to control ${\mathbf{w}}(\overline{P})$. Since
$\overline{p}\in{\mathcal{O}}(D)\subset\overline{\mathcal{O}}(D)$, it is
extremal in $\overline{\mathcal{M}}\supset{\mathcal{M}}$ with degree not
larger than $D$, the classes
$\left\\{{\\{q<\overline{p}\\},\;q\in{\mathcal{M}}\setminus\\{\overline{p}\\}}\right\\}$
and
$\left\\{{\\{q>\overline{p}\\},\;q\in{\mathcal{M}}\setminus\\{\overline{p}\\}}\right\\}$
are both VC with dimensions not larger than $D$. We may therefore apply
Proposition 3.1 in Baraud (2016) with $\sigma=1$ and get
(45)
$\displaystyle{\mathbb{E}}\left[{\sup_{q\in{\mathcal{M}}\setminus\\{\overline{p}\\}}\left|{\sum_{i=1}^{n}\left({1\hskip-2.6pt{\rm
l}_{\overline{p}>q}(X_{i})-P_{i}^{\star}(\overline{p}>q)}\right)}\right|}\right]$
$\displaystyle\leqslant 10\sqrt{5nD},$
and
(46)
$\displaystyle{\mathbb{E}}\left[{\sup_{q\in{\mathcal{M}}\setminus\\{\overline{p}\\}}\left|{\sum_{i=1}^{n}\left({1\hskip-2.6pt{\rm
l}_{\overline{p}<q}(X_{i})-P_{i}^{\star}(\overline{p}<q)}\right)}\right|}\right]$
$\displaystyle\leqslant 10\sqrt{5nD}.$
These inequalities entail ${\mathbf{w}}(\overline{P})\leqslant 10\sqrt{5nD}$,
and we infer from (44) that
(47) $d\\!\left(\overline{P},\widehat{P}\right)\leqslant
2d\\!\left(P^{\star},\overline{P}\right)+20\sqrt{\frac{5D}{n}}+\sqrt{\frac{2(\xi+\log
2)}{n}}+\frac{\varepsilon}{n}.$
Since $\overline{P}$ is arbitrary in the set ${\mathscr{O}}(D)$ which is dense
on $\overline{\mathscr{O}}(D)$, we infer that equation (47) holds for all
$\overline{P}\in\overline{\mathscr{O}}(D)$, which yields (7). Hence, by the
triangle inequality,
$\displaystyle d\\!\left(P^{\star},\widehat{P}\right)$
$\displaystyle\leqslant\inf_{P\in\overline{\mathscr{O}}(D)}\left\\{d\\!\left(P^{\star},P\right)+d\\!\left(P,\widehat{P}\right)\right\\}$
$\displaystyle\leqslant
3\inf_{P\in\overline{\mathscr{O}}(D)}d\\!\left(P^{\star},P\right)+20\sqrt{\frac{5D}{n}}+\sqrt{\frac{2(\xi+\log
2)}{n}}+\frac{\varepsilon}{n}.$
With our convention that $\inf_{\varnothing}=+\infty$, the inequality is also
true when $\overline{\mathscr{O}}(D)=\varnothing$, hence for all values of
$D$, which leads to (8). Inequality (10) follows by integrating this deviation
bound with respect to $\xi$. ∎
### 7.3. Proofs of Section 4
###### Proof of Proposition 2.
We restrict ourselves to the case where $p$ is nonincreasing on $I$, the proof
in the other case is similar. Let $q$ be the function that coincides with $p$
on $\mathring{I}=(a,b)$ and satisfies $q(a)=\sup_{x\in(a,b)}p(x)$ and
$q(b)=\inf_{x\in(a,b)}p(x)$. Clearly, $p=q$ a.e. and satisfies
$V_{I}(p)=q(a)-q(b)=V_{I}(q)$. With no loss of generality, we may therefore
assume that $I=[a,b]$ and that $V_{I}(p)=p(a)-p(b)$, what we shall do now.
Since $p$ is nonincreasing in $I$, for all intervals $J\subset I$ with
endpoints $u<v$,
(48)
$\int_{J}\left|{p-\overline{p}_{J}}\right|d\mu\leqslant\frac{(v-u)(p(u)-p(v))}{2}.$
In particular, when $D=1$ it suffices to take ${\mathcal{J}}=\\{I\\}$ and the
result follows from (48) with $u=a$ and $v=b$ and the trivial inequality
$\int_{I}|p-\overline{p}_{J}|d\mu\leqslant 2$. It remains to prove the result
for $D\geqslant 2$ and since (18) is trivially true for $V=0$ we may also
assume that $V>0$.
Let us set $\eta=(1+VL)^{1/D}-1>0$, $x_{0}=a$ and for all
$j\in\\{1,\ldots,D\\}$,
$x_{j}=x_{j-1}+L\frac{(1+\eta)^{j}}{\sum_{k=1}^{D}(1+\eta)^{k}}=x_{0}+L\frac{(1+\eta)^{j}-1}{(1+\eta)^{D}-1}=x_{0}+\frac{(1+\eta)^{j}-1}{V}.$
Then, we obtain an increasing sequence of points
$a=x_{0}<x_{1}<\ldots<x_{D}=b=a+L$ and a partition ${\mathcal{J}}$ of $I$ into
$D$ intervals based on $\\{x_{0},\ldots,x_{D}\\}$. Using (48) and the facts
that $x_{j+1}-x_{j}=(1+\eta)(x_{j}-x_{j-1})>x_{j}-x_{j-1}$ for
$j\in\\{1,\ldots,D-1\\}$, we obtain
$\displaystyle\int_{I}\left|{p-\overline{p}}\right|d\mu$
$\displaystyle=\sum_{J\in{\mathcal{J}}}\int_{J}\left|{p-\overline{p}_{J}}\right|d\mu\leqslant\frac{1}{2}\sum_{j=1}^{D}(x_{j}-x_{j-1})[p(x_{j-1})-p(x_{j})]$
$\displaystyle=\frac{1}{2}\left[{p(x_{0})(x_{1}-x_{0})+\sum_{j=1}^{D-1}p(x_{j})\left[{(x_{j+1}-x_{j})-(x_{j}-x_{j-1})}\right]}\right]$
$\displaystyle\quad-\frac{p(x_{D})(x_{D}-x_{D-1})}{2}$
$\displaystyle\leqslant\frac{1}{2}\left[{V_{I}(p)(x_{1}-x_{0})+\eta\sum_{j=1}^{D-1}p(x_{j})(x_{j}-x_{j-1})}\right]$
$\displaystyle\quad+\frac{p(x_{D})\left[{(x_{1}-x_{0})-(x_{D}-x_{D-1})}\right]}{2}$
$\displaystyle\leqslant\frac{1}{2}\left[{V(x_{1}-x_{0})+\eta\sum_{j=1}^{D-1}\int_{x_{j-1}}^{x_{j}}pd\mu}\right]\leqslant\frac{1}{2}\left[{V(x_{1}-x_{0})+\eta}\right]=\eta.$
Together with the trivial bound
$\int_{I}\left|{p-\overline{p}}\right|d\mu\leqslant 2$, this last inequality
leads to (18). The second inequality derives from the fact that
$(e^{x}-1)\wedge 2\leqslant 2x/\log 3\leqslant 2x$ for all $x\geqslant 0$. ∎
###### Proof of Theorem 3.
Let $D,D_{1},\ldots,D_{k-2}$ be positive integers and $p$ a density in
$\overline{\mathcal{M}}_{k}^{\infty}(R)$. We may therefore write $p$ under the
form (15) with
$\left[{\sum_{i=1}^{k-2}\sqrt{w_{i}\log\left({1+L_{i}V_{i}}\right)}}\right]^{2}\leqslant
R.$
Applying Proposition 2 to the density $p_{i}$, with $I=I_{i}=(x_{i-1},x_{i})$,
$L=L_{i}=(x_{i}-x_{i-1})$, $V=V_{i}$ and $D=D_{i}\geqslant 1$ for each
$i\in\\{1,\ldots,k-2\\}$, we build a monotone density $\overline{p}_{i}$ on
$I_{i}$ which is piecewise constant on partition of $I_{i}$ into $D_{i}$
nontrivial intervals and that satisfies
$\int_{I_{i}}\left|{p_{i}-\overline{p}_{i}}\right|d\mu\leqslant\frac{2S_{i}}{D_{i}}\quad\text{with}\quad
S_{i}=\log\left({1+V_{i}L_{i}}\right).$
Let us now take $D_{i}=\left\lceil
D\sqrt{w_{i}S_{i}}/(\sum_{i=1}^{k-2}\sqrt{w_{i}S_{i}})\right\rceil\vee 1$ for
all $i\in\\{1,\ldots,k-2\\}$. Since
$\frac{D\sqrt{w_{i}S_{i}}}{\sum_{i=1}^{k-2}\sqrt{w_{i}S_{i}}}\vee 1\leqslant
D_{i}\leqslant\frac{D\sqrt{w_{i}S_{i}}}{\sum_{i=1}^{k-2}\sqrt{w_{i}S_{i}}}+1,$
the density $\overline{p}=\sum_{i=1}^{k-2}w_{i}\overline{p}_{i}$ satisfies
$\displaystyle\left\|{p-\overline{p}}\right\|$
$\displaystyle\leqslant\sum_{i=1}^{k-2}w_{i}\int_{I_{i}}\left|{p_{i}-\overline{p}_{i}}\right|d\mu\leqslant\sum_{i=1}^{k-2}\frac{2w_{i}S_{i}}{D_{i}}\leqslant\frac{2}{D}\left({\sum_{i=1}^{k-2}\sqrt{w_{i}S_{i}}}\right)^{2}\leqslant\frac{2R}{D}.$
Besides, the density $\overline{p}$ is $k$-piecewise monotone, supported on
$[x_{1},x_{k-2}]$ and piecewise constant on a partition of ${\mathbb{R}}$
consisting of at most $\sum_{i=1}^{k-2}D_{i}\leqslant D+k-2$ bounded
intervals. It therefore belongs to $\overline{\mathcal{O}}_{D+k-2,k}$.
Finally, let us choose
$D=\left\lceil\left({\frac{9R^{2}n}{83.2^{2}}}\right)^{1/3}\right\rceil\leqslant\left({\frac{9R^{2}n}{83.2^{2}}}\right)^{1/3}+1.$
Using the sub-additivity property of the square root, we deduce from (14) that
$\displaystyle{\mathbb{B}}_{k,n}(p)$
$\displaystyle\leqslant\frac{3R}{2D}+83.2\sqrt{\frac{D-1+2k}{n}}$
$\displaystyle\leqslant\frac{3^{1/3}\times
83.2^{2/3}}{2}\left({\frac{R}{n}}\right)^{1/3}+\frac{83.2}{\sqrt{n}}\sqrt{\left({\frac{9nR^{2}}{83.2^{2}}}\right)^{1/3}+2k}$
$\displaystyle\leqslant\left[{\frac{3^{1/3}\times 83.2^{2/3}}{2}+3^{1/3}\times
83.2^{2/3}}\right]\left({\frac{R}{n}}\right)^{1/3}+83.2\sqrt{\frac{2k}{n}},$
which is (17). ∎
###### Proof of Theorem 4.
Let us start with the following lemma where we show that the mapping
$\tau(p,\cdot)$ controls the ${\mathbb{L}}_{1}$-approximation error of a
monotone density $p$ by the elements of the class
$\overline{\mathcal{M}}_{3}^{\infty}(R)$.
###### Lemma 4.
Let $p$ be a density on ${\mathbb{R}}$, $B$ some positive number and $I$ a
subset of ${\mathbb{R}}$ on which the density $p$ is not a.e. equal to 0. The
density $p_{|I}^{\wedge B}=(p\wedge B)1\hskip-2.6pt{\rm
l}_{I}/\int_{I}(p\wedge B)d\mu$ satisfies,
(49) $\left\|{p-p_{|I}^{\wedge B}}\right\|\leqslant
2\left[{\int_{I}\left({p-B}\right)_{+}d\mu+\int_{I^{c}}pd\mu}\right].$
If $p$ is a monotone density on a half-line
(50)
$\inf_{\overline{p}\in\overline{\mathcal{M}}_{3}^{\infty}(R)}\left\|{p-\overline{p}}\right\|\leqslant
2\tau\left({p,\exp(R)-1}\right)\quad\text{for all $R\geqslant\log 2$.}$
Besides, if $p$ is a nonincreasing density on $(a,a+l)$ (respectively a
nondecreasing density on $(a-l,a)$) with $a\in{\mathbb{R}}$ and
$l\in(0,+\infty]$, we may restrict the infimum to the nonincreasing densities
on $(a,a+l)$ (respectively the nondecreasing densities on $(a-l,a)$) that
belong to $\overline{\mathcal{M}}_{3}^{\infty}(R)$.
###### Proof.
Since $p$ is not equal to 0 a.e. on $I$, $\int_{I}(p\wedge B)d\mu>0$ and
$p_{|I}^{\wedge B}$ is therefore a well-defined density on $I$. By Lemma 3,
$\displaystyle\int_{{\mathbb{R}}}\left|{p-p_{|I}^{\wedge B}}\right|d\mu$
$\displaystyle\leqslant 2\int_{{\mathbb{R}}}\left|{p-(p\wedge
B)1\hskip-2.6pt{\rm l}_{I}}\right|d\mu$
$\displaystyle=2\int_{I^{c}}pd\mu+2\int_{I}\left({p-B}\right)_{+}d\mu.$
Changing $p$ into $x\mapsto p(-x)$ if necessary and possibly changing the
value of $p$ at the endpoint of the half-line, we may assume with no loss of
generality that $p$ is a nonincreasing density on a half-line of the form
$(a,+\infty)$ with $a\in{\mathbb{R}}$. Let us now set
$l=\sup\\{z>0,\;p(a+z)>0\\}\in(0,+\infty]\quad\text{and}\quad
t=\exp(R)-1\geqslant 1.$
We first consider the case where $l=+\infty$. Given $s>0$, let us take
$B=p(a+s)>0$ and $I=(a,a+st)$. Since $p$ is nonincreasing on $(a,+\infty)$,
$p(x)\geqslant p(a+s)=B$ for all $x\in(a,a+s)\subset I$ and consequently
$\int_{I}(p\wedge B)d\mu\geqslant\int_{a}^{a+s}(p\wedge B)d\mu=sB.$
The density $\overline{p}_{s}=p_{|I}^{\wedge B}$ belongs to
$\overline{\mathcal{M}}_{3}^{\infty}$, is supported on an interval of length
not larger than $st$ and its variation on $I$ is not larger that
$\frac{B}{\int_{I}(p\wedge B)d\mu}-p(a+st)<\frac{B}{\int_{I}(p\wedge B)d\mu}.$
Hence, it follows from (16) that
$\displaystyle\boldsymbol{R}_{k,0}(\overline{p}_{s})<\log\left({1+\frac{tsB}{\int_{I}(p\wedge
B)d\mu}}\right)\leqslant\log(1+t)=R$
and consequently, $\overline{p}_{s}\in\overline{\mathcal{M}}_{3}^{\infty}(R)$.
Applying (49) and Lemma 2, we obtain that for all $s>0$
(51)
$\inf_{\overline{p}\in\overline{\mathcal{M}}_{3}^{\infty}(R)}\left\|{p-\overline{p}}\right\|\leqslant\left\|{p-\overline{p}_{s}}\right\|\leqslant
2\left[{\tau_{x}(p,st)+\tau_{y}\left({p,p(a+s)}\right)}\right]$
and we derive (50) from (22). Since for all $s>0$, $\overline{p}_{s}$ is a
density on $(a,+\infty)$, we may restrict the infimum to these densities in
$\overline{\mathcal{M}}_{3}^{\infty}(R)$ that satisfy this property.
Let us now turn to the case where $l<+\infty$ and define $s_{0}=l/t\leqslant
l$. Given $s\in(0,s_{0})$, we take $B=p(a+s)>0$ and $I=(a,a+st)$. Since
$st<l$, $p(a+st)>0$ and by arguing as before, we obtain that
$\inf_{\overline{p}\in\overline{\mathcal{M}}_{3}^{\infty}(R)}\left\|{p-\overline{p}}\right\|\leqslant
2\left[{\tau_{x}(p,st)+\tau_{y}\left({p,p(a+s)}\right)}\right]\quad\text{for
all $s\in(0,s_{0})$.}$
It follows from the monotonicity of $\tau_{y}(p,\cdot)$ that for all
$0<s<s_{0}\leqslant s^{\prime}$,
$\tau_{y}\left({p,p(a+s)}\right)\leqslant\tau_{y}\left({p,p(a+s_{0})}\right)\leqslant\tau_{y}\left({p,p(a+s^{\prime})}\right),$
and since the mapping $u\mapsto\tau_{x}(p,u)$ is continuous and nonincreasing,
for all $s^{\prime}\geqslant s_{0}$
$\displaystyle\inf_{s\in(0,s_{0})}\left[{\tau_{x}(p,st)+\tau_{y}\left({p,p(a+s)}\right)}\right]$
$\displaystyle\leqslant\inf_{s\in(0,s_{0})}\tau_{x}(p,st)+\tau_{y}\left({p,p(a+s^{\prime})}\right)=\tau_{x}(p,s_{0}t)+\tau_{y}\left({p,p(a+s^{\prime})}\right)$
$\displaystyle=0+\tau_{y}\left({p,p(a+s^{\prime})}\right)=\tau_{x}(p,s^{\prime}t)+\tau_{y}\left({p,p(a+s^{\prime})}\right).$
Consequently, (51) remains satisfied for all $s>0$. Since it is actually
enough to restrict the infimum to those $s\in(0,s_{0})$ and since for such
values of $s$ the density $\overline{p}_{s}=p_{|I}^{\wedge B}$ vanishes
outside $(a,a+s)\subset(a,a+l)$, we may restrict the infimum in (50) to those
densities in $\overline{\mathcal{M}}_{3}^{\infty}(R)$ that vanish outside
$(a,a+l)$. ∎
Let us set $\eta=\tau_{\infty}\left({p,\exp\left({R/\ell}\right)-1}\right)$.
Since $R/\ell\geqslant\log 2$, by applying Lemma 4 to the densities $p_{i}$,
we may find for all $i\in\\{1,\ldots,\ell\\}$ a density
$\overline{p}_{i}\in\overline{\mathcal{M}}_{3}^{\infty}(R/\ell)$ such that
$\|p_{i}-\overline{p}_{i}\|\leqslant
2\tau\left({p_{i},\exp(R/\ell)-1}\right)$. In particular, the density
$\overline{p}=\sum_{i=1}^{\ell}w_{i}\overline{p}_{i}$ satisfies
(52)
$\left\|{p-\overline{p}}\right\|\leqslant\sum_{i=1}^{\ell}w_{i}\left\|{p_{i}-\overline{p}_{i}}\right\|\leqslant\max_{i\in\\{1,\ldots,\ell\\}}\left\|{p_{i}-\overline{p}_{i}}\right\|\leqslant
2\eta.$
When $\ell>2$ and $i\in\\{2,\ldots,\ell-1\\}$, it follows from the definition
of $\overline{\mathcal{M}}_{3}^{\infty}(R/\ell)$ and Lemma 4 that we may
choose $\overline{p}_{i}$ in such a way that it vanishes outside an interval
$I_{i}=(x_{i,0},x_{i,1})\subset(x_{i-1},x_{i})$ with
$\log\left[{1+\left({x_{i,0}-x_{i,1}}\right)\left({\sup_{x\in
I_{i}}\overline{p}_{i}(x)-\inf_{x\in
I_{i}}\overline{p}_{i}(x)}\right)}\right]<\frac{R}{\ell}$
and $x_{i,0}=x_{i-1}$ when $p_{i}$ is nonincreasing and $x_{i,1}=x_{i}$ when
$p_{i}$ is nondecreasing. The mapping $\overline{p}_{1}$ is a nondecreasing
density on an interval of the form $I_{1}=(x_{1,0},x_{1,1})$ with
$x_{1,0}<x_{1,1}=x_{1}$ and
$\log\left[{1+\left({x_{1,1}-x_{1,0}}\right)\left({\sup_{x\in
I_{1}}\overline{p}_{1}(x)-\inf_{x\in
I_{1}}\overline{p}_{1}(x)}\right)}\right]<\frac{R}{\ell}.$
Similarly, $\overline{p}_{\ell}$ is a nonincreasing density on an interval of
the form $I_{\ell}=(x_{\ell,0},x_{\ell,1})$ with
$x_{\ell,0}=x_{\ell-1}<x_{\ell,1}$ and
$\log\left[{1+\left({x_{\ell,1}-x_{\ell,0}}\right)\left({\sup_{x\in
I_{\ell}}\overline{p}_{\ell}(x)-\inf_{x\in
I_{\ell}}\overline{p}_{\ell}(x)}\right)}\right]<\frac{R}{\ell}.$
The density $\overline{p}=\sum_{i=1}^{\ell}w_{i}\overline{p}_{i}$ also writes
as
$\overline{p}_{1}1\hskip-2.6pt{\rm
l}_{(x_{1,0},x_{1})}+\sum_{i=2}^{\ell-2}w_{i}\left[{\overline{p}_{i}1\hskip-2.6pt{\rm
l}_{I_{i}}+01\hskip-2.6pt{\rm l}_{(x_{i-1},x_{i})\setminus
I_{i}}}\right]+\overline{p}_{\ell}1\hskip-2.6pt{\rm
l}_{(x_{\ell-1},x_{\ell,1})}$
and may therefore be written under the form (15) when $k\geqslant 2\ell$.
Moreover, by Cauchy-Schwarz inequality
$\displaystyle\boldsymbol{R}_{k,0}(\overline{p})<\left[{\sum_{i=1}^{\ell}\sqrt{w_{i}\left({\frac{R}{\ell}}\right)}+0}\right]^{2}=\frac{R}{\ell}\left[{\sum_{i=1}^{\ell}\sqrt{w_{i}}}\right]^{2}\leqslant
R$
and consequently, $\overline{p}\in\overline{\mathcal{M}}_{k}^{\infty}(R)$. We
deduce from (52) that
$\displaystyle\inf_{q\in\overline{\mathcal{M}}_{k}^{\infty}(R)}\left\|{p-q}\right\|$
$\displaystyle\leqslant\left\|{p-\overline{p}}\right\|\leqslant 2\eta$
which is (26). ∎
### 7.4. Proofs of Section 5
###### Proof of Proposition 3.
The proof relies on the following lemma the proof of which is a direct
consequence of convexity and is therefore omitted.
###### Lemma 5.
Let $g$ be a convex and continuous function on a nontrivial interval $J$. The
set $\\{x\in J,\;g(x)<0\\}$ is an interval (possibly empty). The set $\\{x\in
J,\;g(x)>0\\}$ has one of the following forms: $\varnothing,J$,
$J\cap(-\infty,c_{0})$, $J\cap(c_{1},+\infty)$,
$[J\cap(-\infty,c_{0})]\cup[J\cap(c_{1},+\infty)]$ with $c_{0},c_{1}\in J$ and
$c_{0}<c_{1}$. In particular $\\{x\in J,\;g(x)>0\\}$ is the union of at most
two intervals.
When $g$ is continuous and concave on $J$, the same conclusion holds with
$\\{x\in J,\;g(x)>0\\}$ in place of $\\{x\in J,\;g(x)<0\\}$ and vice-versa.
Since $p$ belongs to $\overline{\mathcal{M}}_{k}^{1}$ and $q$ belongs to
$\overline{{\mathcal{O}}}_{D,k}^{1}$, there exists
$A=\\{a_{1},\ldots,a_{l}\\}$ with $l\in\\{1,\ldots,k-1\\}$ such that $p$ is
convex-concave on each element of ${\mathbf{I}}(A)$ and there exists a subset
$B\subset{\mathbb{R}}$ with cardinality not larger than $D+2$ such that $q$ is
left-continuous on ${\mathbb{R}}$ and affine on each element of
${\mathbf{I}}(B)$. Since $p$ and $q$ are densities, $p$ is necessarily convex
on the two unbounded intervals of ${\mathbf{I}}(A)$ and $q$ vanishes on the
two unbounded intervals of ${\mathbf{I}}(B)$.
We define $J_{1}=(-\infty,a_{1}]$, $J_{l+1}=(a_{l},+\infty)$ and when
$l\geqslant 2$, $J_{i}=(a_{i-1},a_{i}]$ for all $i\in\\{2,\ldots,l\\}$.
Besides, we set $I_{i}=\mathring{J}_{i}$ for all $i\in\\{1,\ldots,l+1\\}$. We
shall repeatedly use Lemma 5 with $g=p-q$ throughout the proof. Given
$\epsilon\in\\{\pm 1\\}$, we set
(53) $C_{\epsilon}=\left\\{{x\in{\mathbb{R}},\;\epsilon
g(x)>0}\right\\}=\bigcup_{i=1}^{l+1}\left\\{{x\in J_{i},\;\epsilon
g(x)>0}\right\\}.$
Our aim is to show that $C_{\epsilon}$ is the union of at most $D+2k-1$
intervals. The second part of the proposition is a consequence of Lemma 1 of
Baraud and Birgé (2016).
If $m_{1}=|B\cap I_{1}|=0$ then $q=0$ on $J_{1}$ (since it is left-
continuous), $g=p-0$ is continuous, monotone (nondecreasing) and convex on
$I_{1}$, $\\{x\in J_{1},\;g(x)<0\\}=\varnothing$ and $\\{x\in
J_{1},\;g(x)>0\\}$ is an interval.
If $m_{1}=|B\cap I_{1}|\geqslant 1$, we may partition $J_{1}$ into
$s=m_{1}+1\geqslant 2$ consecutive intervals $K_{1},\ldots,K_{s}$ that we may
choose to be of the form $(a,b]$, $a<b$, $a\in{\mathbb{R}}\cup\\{-\infty\\}$,
$b\in{\mathbb{R}}$. Since $p$ is continuous on $I_{1}$ and $q$ is left-
continuous, $g$ is continuous on $K_{1},\ldots,K_{s-1}$ and on
$\mathring{K}_{s}$. On $K_{1}$, $\\{x\in K_{1},\;g(x)<0\\}=\varnothing$ and
the set $\Lambda_{1,]}^{+}=\\{x\in K_{1},\;g(x)>0\\}$ is an interval which is
either empty or contains the right endpoint of $K_{1}$. When $s\geqslant 2$,
we may apply Lemma 5 to $g$ and the intervals $K_{i}$ with
$i\in\\{2,\ldots,s-1\\}$. We obtain that $\Lambda_{i}^{-}=\\{x\in
K_{i},\;g(x)<0\\}$ is an interval and $\\{x\in K_{i},\;g(x)>0\\}$ is of the
form $\Lambda_{i,(}^{+}\cup\Lambda_{i,]}^{+}$ where
$\Lambda_{i,(}^{+},\Lambda_{i,]}^{+}$ are two (possibly empty) intervals and
when they are not, $\inf\Lambda_{i,(}^{+}=\inf K_{i}$ and the right endpoint
of $K_{i}$ belongs to $\Lambda_{i,]}^{+}$. The set $\\{x\in K_{s},\;g(x)<0\\}$
writes as $\Lambda_{s}^{-}\cup\Lambda_{s,]}^{-}$ where
$\Lambda_{s}^{-},\Lambda_{s,]}^{-}$ are two possibly empty intervals and when
$\Lambda_{s,]}^{-}$ is not empty, it reduces to $\\{a_{1}\\}$. The set
$\\{x\in K_{s},\;g(x)>0\\}$ writes $\Lambda_{s,(}^{+}\cup\Lambda_{s}^{+}$
where $\Lambda_{s,(}^{+},\Lambda_{s}^{+}$ are two possibly empty intervals and
when they are not $\inf\Lambda_{s,(}^{+}=\inf K_{s}$ and
$\sup\Lambda_{s}^{+}=\sup K_{s}$. We conclude that
$\displaystyle\left\\{{x\in
J_{1},\;g(x)<0}\right\\}=\left[{\bigcup_{i=2}^{s-1}\Lambda_{i}^{-}}\right]\cup\left[{\Lambda_{s}^{-}\cup\Lambda_{s,]}^{-}}\right]$
and
$\displaystyle\left\\{{x\in J_{1},\;g(x)>0}\right\\}$
$\displaystyle=\Lambda_{1,]}^{+}\cup\left[{\bigcup_{i=2}^{s-1}\left({\Lambda_{i,(}^{+}\cup\Lambda_{i,]}^{+}}\right)}\right]\cup\left[{\Lambda_{s,(}^{+}\cup\Lambda_{s}^{+}}\right]$
are both the unions of at most $s=m_{1}+1$ intervals.
By arguing similarly, we obtain that on the interval $J_{l+1}$: the sets
$\\{x\in J_{l+1},\;\epsilon g(x)>0\\}$ with $\epsilon\in\\{\pm 1\\}$ are the
unions of at most $m_{l+1}+1$ intervals where $m_{l+1}=|B\cap J_{l+1}|$.
When $l\geqslant 2$, let us now consider an interval of the form
$J_{i}=(a_{i-1},a_{i}]$ with $i\in\\{2,\ldots,l\\}$ and set $m_{i}=|B\cap
I_{i}|$. If $m_{i}=0$, $g$ is continuous and convex-concave on $I_{i}$ and by
arguing as for $K_{s}$, we obtain that $\\{x\in J_{i},\;\epsilon g(x)>0\\}$ is
the union of at most 2 intervals whatever $\epsilon\in\\{\pm 1\\}$. If
$m_{i}\geqslant 1$, we may partition $I_{i}$ with $m_{i}+1$ intervals of the
form $(a,b]$ with $a<b$, $a,b\in{\mathbb{R}}$. On each of these intervals, $g$
is continuous and convex-concave and by applying Lemma 5 and arguing as
previously, we obtain that $\\{x\in J_{i},\;\epsilon g(x)>0\\}$ is a union of
at most $m_{i}+2$ intervals.
Using (53) and the facts that $\sum_{i=1}^{l+1}m_{i}\leqslant|B|\leqslant D+1$
and $l\leqslant k-1$, we conclude that the sets $C_{{\varepsilon}}$ are unions
of at most
$\displaystyle
m_{1}+1+m_{l+1}+1+\sum_{i=2}^{l}\left({m_{i}+2}\right)\leqslant|B|+2l\leqslant
D+2k-1$
intervals. ∎
###### Proof Theorem 6.
The proof is based on Lemma 6 and three preliminary approximation results
given in Proposition 6, Proposition 7 and Proposition 8, whose proofs follow
that one.
###### Proposition 6.
Let $f$ be a convex-concave continuous function on a bounded nontrivial
interval $[a,b]$ and $\ell_{f}$ be the linear function
(54) $\ell_{f}:x\mapsto f(a)+\frac{f(b)-f(a)}{b-a}\left({x-a}\right).$
The following results hold.
1. (i)
If $f$ admits a right derivative $f^{\prime}_{r}(a)$ at $a$ and a left
derivative $f^{\prime}_{l}(b)$ at $b$,
(55)
$\int_{a}^{b}\left|{f-\ell_{f}}\right|d\mu\leqslant\frac{(b-a)^{2}}{8}\left|{f^{\prime}_{r}(a)-f^{\prime}_{l}(b)}\right|.$
2. (ii)
If $f$ is strictly monotone on $[a,b]$ with $f^{\prime}_{r}(a)\neq 0$ and
$f^{\prime}_{l}(b)\neq 0$
(56)
$\int_{a}^{b}\left|{f-\ell_{f}}\right|d\mu\leqslant\frac{(f(b)-f(a))^{2}}{8}\left|{\frac{1}{f^{\prime}_{r}(a)}-\frac{1}{f^{\prime}_{l}(b)}}\right|,$
with the convention $1/(\pm\infty)=0$.
Since $\ell_{f}$ is the equation of the chord connecting $(a,f(a))$ to
$(b,f(b))$, it is clear that $\ell_{f}(x)=f(x)$ for all $x\in\\{a,b\\}$,
$\ell_{f}\geqslant f$ on $[a,b]$ when $f$ is convex on $[a,b]$ and
$\ell_{f}\leqslant f$ when $f$ is concave.
###### Proposition 7.
Let $p$ be a monotone, continuous and convex-concave sub-density on a bounded
interval $[a,b]$ of length $L>0$ with a right derivative $p^{\prime}_{r}(a)$
at $a$ and a left derivative $p^{\prime}_{l}(b)$ at $b$. For all $D\geqslant
1$, there exists a $D$-linear interpolation $\overline{p}$ of $p$ such that
(57)
$\int_{a}^{b}\left|{p-\overline{p}}\right|d\mu\leqslant\frac{4}{3}\left[{\left({1+L\sqrt{\left|{p^{\prime}_{l}(b)-p^{\prime}_{r}(a)}\right|}}\right)^{1/D}-1}\right]^{2}.$
###### Proposition 8.
Let $p$ be a strictly monotone, continuous, convex-concave sub-density with
variation $V=|p(a)-p(b)|$ on a nontrivial bounded interval $[a,b]$. Assume
furthermore that $p^{\prime}_{r}(a)$ and $p^{\prime}_{l}(b)$ are nonzero.
Then, for all $D\geqslant 1$ there exists a $D$-linear interpolation
$\overline{p}$ of $p$ on $[a,b]$ such that
(58)
$\int_{a}^{b}\left|{p-\overline{p}}\right|d\mu\leqslant\frac{4}{3}\left[{\left({1+V\sqrt{\left|{\frac{1}{p^{\prime}_{r}(a)}-\frac{1}{p^{\prime}_{l}(b)}}\right|}}\right)^{1/D}-1}\right]^{2}.$
###### Lemma 6.
If $F$ is a nondecreasing, differentiable, concave function on
${\mathbb{R}}_{+}$, the mapping
$\phi:u\mapsto\left[F\left({\sqrt{u}}\right)-F(0)\right]^{2}$
is concave on $\mathbb{R}_{+}$. In particular for all $D\geqslant 1$,
$F_{D}:u\mapsto\left[{\left({1+\sqrt{u}}\right)^{1/D}-1}\right]^{2}$
is concave on ${\mathbb{R}}_{+}$.
###### Proof.
For all $u>0$,
$\phi^{\prime}(u)=\frac{F\left({\sqrt{u}}\right)-F(0)}{\sqrt{u}}F^{\prime}(\sqrt{u})$
is the product of $u\mapsto[F\left({\sqrt{u}}\right)-F(0)]/\sqrt{u}$ and
$u\mapsto F^{\prime}(\sqrt{u})$ which are both nonnegative and nonincreasing
since $F$ is nondecreasing and concave. The function $\phi^{\prime}$ is
therefore nonincreasing and $\phi$ concave. ∎
Let’s turn to the proof of Theorem 6. We first introduce some mappings of
interest we will use in the proofs of of Theorem 6 and Proposition 7. Let
${\mathcal{V}}$ be the linear space of continuous functions $f$ on $[a,b]$
that admits a right derivative $f^{\prime}_{r}(a)$ at $a$ and a left
derivative $f^{\prime}_{l}(b)$ at $b$. Given $m\in{\mathbb{R}}$, we define
${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2}$ as the mappings defined on
${\mathcal{V}}$ by
(59) ${\mathcal{T}}_{1}:f\mapsto[x\mapsto
f(a+b-x)],\quad{\mathcal{T}}_{2}:f\mapsto[x\mapsto m-f(x)].$
Although ${\mathcal{T}}_{2}$ depends on the choice of $m$, we drop this
dependency in the notation for the sake of convenience. The mappings
${\mathcal{T}}_{j}$ are one-to-one from ${\mathcal{V}}$ onto itself, isometric
with respect to the ${\mathbb{L}}_{1}$-norm on ${\mathcal{V}}$ and they
satisfy for all $f\in{\mathcal{V}}$
(60) $\displaystyle\left|{f(a)-f(b)}\right|$
$\displaystyle=\left|{{\mathcal{T}}_{j}(f)(a)-{\mathcal{T}}_{j}(f)(b)}\right|$
(61) $\displaystyle\left|{f^{\prime}_{r}(a)-f^{\prime}_{l}(b)}\right|$
$\displaystyle=\left|{({\mathcal{T}}_{j}(f))^{\prime}_{r}(a)-({\mathcal{T}}_{j}(f))^{\prime}_{l}(b)}\right|$
and ${\mathcal{T}}_{j}^{-1}={\mathcal{T}}_{j}$ for all $j\in\\{1,2\\}$.
Let us now turn to the proof of Theorem 6 and assume first that $p$ is
nondecreasing, continuous and convex on $[a,b]$ so that
$p_{r}^{\prime}(a)\geqslant 0$. If $p$ is constant, the result is clear by
taking $\overline{p}=p$. We may therefore assume that $p(a)<p(b)$ and choose a
point $c\in(a,b)$ such that $p(c)>p(a)$. In particular,
$w_{1}=\int_{a}^{c}pd\mu>0$ and
$0<\frac{p(c)-p(a)}{c-a}\leqslant
p^{\prime}_{l}(c)\leqslant\frac{p(b)-p(c)}{b-c}<+\infty.$
The restriction $p_{1}$ of $p$ on the interval $[a,c]$ is nondecreasing,
continuous and convex and so is the density $p_{1}/w_{1}$. We may therefore
apply Proposition 7 to $p_{1}/w_{1}$ and find a $D$-linear interpolation
$\overline{p}_{1}$ of $p$ on $[a,c]$ that satisfies
(62) $\displaystyle\int_{a}^{c}\left|{p-\overline{p}_{1}}\right|d\mu$
$\displaystyle\leqslant\frac{4w_{1}}{3}\left[{\left({1+(c-a)\sqrt{\frac{p^{\prime}_{l}(c)-p^{\prime}_{r}(a)}{w_{1}}}}\right)^{1/D}-1}\right]^{2}.$
The restriction $p_{2}$ of $p$ to $[c,b]$ is increasing, continuous and convex
with nonzero right and left derivatives at $c$ and $b$ respectively. We may
therefore apply Proposition 8 to the density $p_{2}/w_{2}$ with
$w_{2}=\int_{c}^{b}pd\mu=1-w_{1}>0$ and find a $D$-linear interpolation
$\overline{p}_{2}$ of $p$ on $[c,b]$ that satisfies
(63)
$\displaystyle\int_{c}^{b}\left|{p-\overline{p}_{2}}\right|d\mu\leqslant\frac{4w_{2}}{3}\left[{\left({1+\frac{p(b)-p(c)}{\sqrt{w_{2}}}\sqrt{\frac{1}{p^{\prime}_{r}(c)}-\frac{1}{p^{\prime}_{l}(b)}}}\right)^{1/D}-1}\right]^{2}.$
We may choose $c\in(a,b)$ such that
$p^{\prime}_{l}(c)\leqslant\Delta\quad\text{and}\quad
p^{\prime}_{r}(c)\geqslant\Delta\quad\text{with}\quad\Delta=\frac{p(b)-p(a)}{b-a}.$
Then
$\displaystyle A$
$\displaystyle=(c-a)^{2}\left({p^{\prime}_{l}(c)-p^{\prime}_{r}(a)}\right)+(p(b)-p(c))^{2}\left({\frac{1}{p^{\prime}_{r}(c)}-\frac{1}{p^{\prime}_{l}(b)}}\right)$
$\displaystyle\leqslant(b-a)^{2}\left({\Delta-p^{\prime}_{r}(a)}\right)+(p(b)-p(a))^{2}\left({\frac{1}{\Delta}-\frac{1}{p^{\prime}_{l}(b)}}\right)$
(64)
$\displaystyle=2(b-a)(p(b)-p(a))\left[{1-\frac{1}{2}\left({\frac{p^{\prime}_{r}(a)}{\Delta}+\frac{\Delta}{p^{\prime}_{l}(b)}}\right)}\right]=2LV\Gamma.$
The function $\overline{p}=\overline{p}_{1}1\hskip-2.6pt{\rm
l}_{[a,c)}+\overline{p}_{2}1\hskip-2.6pt{\rm l}_{[c,b]}$ is a $2D$-linear
interpolation of $p$ on $[a,b]$. Since by Lemma 6 the function $F_{D}$ is
concave and increasing, we deduce from (64), (62) and (63) that
$\displaystyle\frac{3}{4}\left\|{p-\overline{p}}\right\|$
$\displaystyle\leqslant
w_{1}\int_{a}^{c}\left|{p-\overline{p}_{1}}\right|d\mu+w_{2}\int_{c}^{b}\left|{p-\overline{p}_{2}}\right|d\mu$
$\displaystyle\leqslant
w_{1}F_{D}\left({\frac{(c-a)^{2}\left({p^{\prime}_{l}(c)-p_{r}(a)}\right)}{w_{1}}}\right)$
$\displaystyle\quad+w_{2}F_{D}\left({\frac{(p(b)-p(c))^{2}}{w_{2}}\left({\frac{1}{p^{\prime}_{r}(c)}-\frac{1}{p^{\prime}_{l}(b)}}\right)}\right)$
$\displaystyle\leqslant F_{D}(A)\leqslant F_{D}(2LV\Gamma),$
which is (32).
The density $q=\overline{p}/\int_{a}^{b}\overline{p}d\mu$ is continuous
convex-concave on $[a,b]$ and piecewise linear on a partition of $[a,b]$ into
$2D$ intervals and it follows from Lemma 3 that
(65) $\int_{a}^{b}\left|{p-q}\right|d\mu\leqslant
2\left\\{{1\wedge\left[{\frac{4}{3}\left({\left({1+\sqrt{2LV\Gamma}}\right)^{1/D}-1}\right)^{2}}\right]}\right\\}.$
The mapping $z\mapsto(4/3)(e^{z}-1)^{2}$ is not larger than 1 if and only if
$z\leqslant z_{0}=\log(1+\sqrt{3/4})$ and for all $z\in[0,z_{0}]$,
$e^{z}-1\leqslant(e^{z_{0}}-1)z/z_{0}$. Consequently, for all $z\geqslant 0$
$\displaystyle
2\left\\{{1\wedge\left[{\frac{4}{3}\left({e^{z}-1}\right)^{2}}\right]}\right\\}$
$\displaystyle=\frac{8}{3}\left({e^{z_{0}\wedge
z}-1}\right)^{2}\leqslant\frac{8}{3}\left({\frac{e^{z_{0}}-1}{z_{0}}z}\right)^{2}\leqslant
5.14z^{2}.$
Applying this inequality with $z=D^{-1}\log\left({1+\sqrt{2LV\Gamma}}\right)$,
we deduce (33) from (65).
Theorem 6 is therefore proven for a nondecreasing, continuous convex density
$p$. In order to prove the result in the other cases, we use the
transformations ${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2}$ introduced above
and defined by (59). These transformations are isometric with respect to the
${\mathbb{L}}_{1}$-norm and they preserve the variation of a monotone
function. Note that they also preserve the linear index, i.e. for all
continuous convex-concave function $f$ on $[a,b]$ and $j\in\\{1,2\\}$,
$\Gamma(f)=\Gamma\left({{\mathcal{T}}_{j}(f)}\right)$. Applying
${\mathcal{T}}_{1}$ to nonincreasing continuous convex densities we establish
(33) and we extend it to all monotone concave continuous densities by applying
${\mathcal{T}}_{2}$. ∎
###### Proof of Proposition 6.
Let us first assume that $f$ is concave on $[0,1]$, admits a right derivative
at 0, a left derivative at 1 and satisfies $f(0)=0$ and $f(1)$=1. Let us show
that
(66)
$\int_{0}^{1}\left|{f-\ell_{f}}\right|d\mu\leqslant\frac{1}{2}\frac{\left({f_{r}^{\prime}(0)-1}\right)\left({1-f^{\prime}_{l}(1)}\right)}{f^{\prime}_{r}(0)-f^{\prime}_{l}(1)},$
with the convention $0/0=1$. If $f$ is linear on $[0,1]$, then
$f_{r}^{\prime}(0)=f^{\prime}_{l}(1)=1$ and the inequality is satisfied with
our convention. Otherwise, $f_{l}^{\prime}(1)<1=f(1)-f(0)<f_{r}^{\prime}(0)$
and since $f$ lies above its chord and under its tangents at 0 and 1,
$\ell_{f}(x)\leqslant
f(x)\leqslant\min\left\\{{f^{\prime}_{r}(0)x,1+f_{l}^{\prime}(1)(x-1)}\right\\}\quad\text{for
all $x\in[0,1]$}.$
Since $\ell_{f}(x)=x$, we deduce that for all $c\in[0,1]$
$\displaystyle\int_{0}^{1}\left|{f-\ell_{f}}\right|d\mu$
$\displaystyle\leqslant\int_{0}^{c}(f^{\prime}_{r}(0)-1)xd\mu+\int_{c}^{1}(1-f_{l}^{\prime}(1))(1-x)d\mu$
$\displaystyle=\frac{1}{2}\left[{(f^{\prime}_{r}(0)-1)c^{2}+(1-f_{l}^{\prime}(1))(1-c)^{2}}\right].$
The result follows by minimizing with respect to $c$, i.e. by taking
$c=(1-f_{l}^{\prime}(1))/(f_{r}^{\prime}(0)-f_{l}^{\prime}(1))\in(0,1)$. In
particular, we deduce from (67) that
$\displaystyle\int_{0}^{1}\left|{f-\ell_{f}}\right|d\mu\leqslant\frac{c(1-c)}{2}\left({f^{\prime}_{r}(0)-f^{\prime}_{l}(1)}\right)$
and since $c(1-c)\leqslant 1/4$, we obtain that
(67)
$\int_{0}^{1}\left|{f-\ell_{f}}\right|d\mu\leqslant\frac{1}{8}\left({f^{\prime}_{r}(0)-f^{\prime}_{l}(1)}\right).$
Note that the inequality also holds when
$f^{\prime}_{r}(0)=f^{\prime}_{l}(1)=1$.
When $f$ is increasing on $[0,1]$ and satisfies $0<1-f_{l}^{\prime}(1)<1$,
i.e. $f_{l}^{\prime}(1)\neq 0$, we also deduce from (67) and the convexity of
$z\mapsto 1/z$ on $(0,+\infty)$ that
$\displaystyle\int_{0}^{1}\left|{f-\ell_{f}}\right|d\mu$
$\displaystyle\leqslant\frac{1}{4}\frac{1}{\frac{1}{2}\left[{\left({\frac{1}{1-f^{\prime}_{l}(1)}-1}\right)+\left({1+\frac{1}{f^{\prime}_{r}(0)-1}}\right)}\right]}$
$\displaystyle\leqslant\frac{1}{4}\left[{\frac{1}{2}\left({\frac{1}{\frac{1}{1-f^{\prime}_{l}(1)}-1}+\frac{1}{1+\frac{1}{f^{\prime}_{r}(0)-1}}}\right)}\right]$
$\displaystyle=\frac{1}{8}\left[{\frac{1-f^{\prime}_{l}(1)}{f^{\prime}_{l}(1)}+\frac{f^{\prime}_{r}(0)-1}{f^{\prime}_{r}(0)}}\right]$
which leads to
(68)
$\int_{0}^{1}\left|{f-\ell_{f}}\right|d\mu\leqslant\frac{1}{8}\left({\frac{1}{f^{\prime}_{l}(1)}-\frac{1}{f^{\prime}_{r}(0)}}\right).$
Note that the inequality is still satisfied when $f^{\prime}_{r}(0)=+\infty$
with the convention $1/(+\infty)=0$.
Let us now turn to the proofs of (55) and (56). Note that (55) is clearly true
when $f$ is constant on $[a,b]$ and we may therefore assume that $f(a)\neq
f(b)$. We obtain (55) and (56) by applying (67) and (68) respectively to the
function
$g(x)=\frac{f(a+x(b-a))-f(a)}{f(b)-f(a)}$
when $f$ is concave and satisfies $f(b)>f(a)$ or when $f$ is convex and
satisfies $f(a)>f(b)$. In the other cases, one may use the function
$g(x)=\frac{f(b-x(b-a))-f(b)}{f(a)-f(b)}.$
∎
###### Proof of Proposition 7.
Let us first consider a function $g$ that is monotone, continuous and convex
on $[a,b]$ and that satisfies $g(a)=g_{r}^{\prime}(a)=0$ and
$0<\int_{a}^{b}gd\mu\leqslant 1$. Then $g$ is nonnegative and nondecreasing on
$[a,b]$. Since $\int_{a}^{b}gd\mu>0$, $g$ is not identically equal to 0 on
$[a,b]$ and consequently,
$g_{l}^{\prime}(b)\geqslant\frac{g(b)-g(a)}{b-a}>0.$
Let $q>1$ and
$x_{0}=a\quad\text{and}\quad
x_{i}=x_{i-1}+L\frac{q^{-i}}{\sum_{j=1}^{D}q^{-j}}\quad\text{for all
$i\in\\{1,\ldots,D\\}$},$
so that $x_{D}=b$ and
$\Delta_{i+1}=x_{i+1}-x_{i}=q^{-1}(x_{i}-x_{i-1})=q^{-1}\Delta_{i}$ for all
$i\in\\{1,\ldots,D-1\\}$ and
$\Delta_{D}=L\frac{q^{-D}(1-q^{-1})}{q^{-1}-q^{-(D+1)}}=L\frac{q-1}{q^{D}-1}.$
Let $g^{\prime}$ be any nondecreasing function on $[a,b]$ satisfying
$0=g_{r}^{\prime}(a)=g^{\prime}(a)$, $g_{l}^{\prime}(b)=g^{\prime}(b)$ and
$g_{l}^{\prime}(x)\leqslant g^{\prime}(x)\leqslant g_{r}^{\prime}(x)$ for all
$x\in(a,b)$. Since $g$ is convex, we may write
(69) $g(x)\geqslant g(x_{i})+g^{\prime}(x_{i})(x-x_{i})\quad\text{for all
$i\in\\{1,\ldots,D\\}$ and $x\in[a,b]$}.$
In particular, for all $i\in\\{1,\ldots,D-1\\}$
$\displaystyle\int_{x_{i}}^{x_{i}+1}\left({g(x_{i})+g^{\prime}(x_{i})(x-x_{i})}\right)d\mu(x)$
$\displaystyle=\left[{g(x_{i})+\frac{g^{\prime}(x_{i})\Delta_{i+1}}{2}}\right]\Delta_{i+1}$
$\displaystyle\leqslant\int_{x_{i}}^{x_{i}+1}gd\mu,$
hence,
(70) $g^{\prime}(x_{i})\Delta_{i+1}^{2}\leqslant
2\int_{x_{i}}^{x_{i}+1}(g-g(x_{i}))d\mu.$
Moreover, applying (69) with $x_{i}=x_{D}=b$ and using the facts that $g$ is
nonnegative and nonincreasing, we get
$\displaystyle 1\geqslant\int_{a}^{b}gd\mu$
$\displaystyle\geqslant\int_{a}^{b}\left({g(b)+g^{\prime}(b)(x-b)}\right)_{+}d\mu(x)$
$\displaystyle=\int_{b-g(b)/g^{\prime}(b)}^{b}\left({g(b)+g^{\prime}(b)(x-b)}\right)d\mu(x)=\frac{g^{2}(b)}{2g^{\prime}(b)},$
consequently
(71) $g(b)\leqslant\sqrt{2g^{\prime}(b)}.$
Let $g_{i}$ be the restriction of $g$ on the interval $[x_{i-1},x_{i}]$ and
$\overline{g}$ the function on $[a,b]$ that coincides on $[x_{i-1},x_{i}]$
with $\ell_{g_{i}}$ defined by (54) with $f=g_{i}$ for all
$i\in\\{1,\ldots,D\\}$. The function $\overline{g}$ is a $D$-linear
interpolation of $g$ on $[a,b]$ that satisfies
$\displaystyle\int_{a}^{b}\left|{g-\overline{g}}\right|d\mu$
$\displaystyle=\sum_{i=1}^{D}\int_{x_{i-1}}^{x_{i}}\left({\overline{g}-g}\right)d\mu$
$\displaystyle\leqslant\frac{1}{8}\sum_{i=1}^{D}\Delta_{i}^{2}\left({g_{l}^{\prime}(x_{i})-g_{r}^{\prime}(x_{i-1})}\right)$
$\displaystyle\leqslant\frac{1}{8}\sum_{i=1}^{D}\Delta_{i}^{2}\left({g^{\prime}(x_{i})-g^{\prime}(x_{i-1})}\right)$
$\displaystyle=\frac{1}{8}\left[{-\Delta_{1}^{2}g^{\prime}(a)+\Delta_{D}^{2}g^{\prime}(b)+\sum_{i=1}^{D-1}g^{\prime}(x_{i})\left({\Delta_{i}^{2}-\Delta_{i+1}^{2}}\right)}\right].$
Since $g^{\prime}(a)=0$ and $\Delta_{i}=q\Delta_{i+1}$ for all
$i\in\\{1,\ldots,D-1\\}$ we deduce that
(72)
$\int_{a}^{b}\left|{g-\overline{g}}\right|d\mu\leqslant\frac{1}{8}\left[{\Delta_{D}^{2}g^{\prime}(b)+(q^{2}-1)\sum_{i=1}^{D-1}g^{\prime}(x_{i})\Delta_{i+1}^{2}}\right].$
Using (70), (71) and the fact that $g$ is nondecreasing,
$\displaystyle\frac{1}{2}\sum_{i=1}^{D-1}g^{\prime}(x_{i})\Delta_{i+1}^{2}$
$\displaystyle\leqslant\sum_{i=1}^{D-1}\int_{x_{i}}^{x_{i+1}}(g-g(x_{i}))d\mu$
$\displaystyle=\int_{x_{1}}^{x_{D}}gd\mu-\sum_{i=1}^{D-1}\Delta_{i+1}g(x_{i})=\int_{x_{1}}^{x_{D}}gd\mu-q^{-1}\sum_{i=1}^{D-1}\Delta_{i}g(x_{i})$
$\displaystyle\leqslant\int_{x_{1}}^{x_{D}}gd\mu-q^{-1}\sum_{i=1}^{D-1}\int_{x_{i-1}}^{x_{i}}gd\mu=\int_{x_{1}}^{x_{D}}gd\mu-q^{-1}\int_{x_{0}}^{x_{D-1}}gd\mu$
$\displaystyle=(1-q^{-1})\int_{x_{1}}^{x_{D}}gd\mu+q^{-1}\int_{x_{D-1}}^{x_{D}}gd\mu-q^{-1}\int_{x_{0}}^{x_{1}}gd\mu$
$\displaystyle\leqslant(1-q^{-1})\times 1+q^{-1}\Delta_{D}g(b)\leqslant
1-q^{-1}+q^{-1}\sqrt{2\Delta_{D}^{2}g^{\prime}(b)}.$
It follows from (72) that
$\displaystyle\int_{a}^{b}\left|{g-\overline{g}}\right|d\mu$
$\displaystyle\leqslant\frac{1}{8}\left[{\Delta_{D}^{2}g^{\prime}(b)+2(q^{2}-1)\left({1-q^{-1}+q^{-1}\sqrt{2\Delta_{D}^{2}g^{\prime}(b)}}\right)}\right]$
$\displaystyle\leqslant\frac{(q-1)^{2}}{8}\left[{\frac{L^{2}g^{\prime}(b)}{(q^{D}-1)^{2}}+2\left({1+\frac{1}{q}}\right)\left({1+\frac{L\sqrt{2g^{\prime}(b)}}{q^{D}-1}}\right)}\right].$
Finally, choosing $q$ such that
$q^{D}-1=L\sqrt{g^{\prime}(b)}\quad\text{i.e.}\quad
q=\left({1+L\sqrt{g^{\prime}(b)}}\right)^{1/D}>1,$
we get that
$\displaystyle\int_{a}^{b}\left|{g-\overline{g}}\right|d\mu$
$\displaystyle\leqslant\frac{1+4(1+\sqrt{2})}{8}\left[{\left({1+L\sqrt{g^{\prime}(b)}}\right)^{1/D}-1}\right]^{2}$
(73)
$\displaystyle\leqslant\frac{4}{3}\left[{\left({1+L\sqrt{g^{\prime}(b)}}\right)^{1/D}-1}\right]^{2}.$
Let us now prove Proposition 7 in the case where $p$ is convex and
nondecreasing. Then $p^{\prime}_{r}(a)\geqslant 0$ and we may set
$\ell:x\mapsto p(a)+p_{r}^{\prime}(a)(x-a)$ and $g:x\mapsto p(x)-\ell(x)$. The
function $g$ is nonnegative, nondecreasing, continuous and convex on $[a,b]$ |
# Photorealistic Video Generation with Diffusion Models
Agrim Gupta1,2,∗ Lijun Yu2 Kihyuk Sohn2 Xiuye Gu2 Meera Hahn2 Li Fei-Fei1
Irfan Essa2,3 Lu Jiang2 José Lezama2
1 Stanford University 2 Google Research 3 Georgia Institute of Technology
###### Abstract
We present W.A.L.T, a transformer-based approach for photorealistic video
generation via diffusion modeling. Our approach has two key design decisions.
First, we use a causal encoder to jointly compress images and videos within a
unified latent space, enabling training and generation across modalities.
Second, for memory and training efficiency, we use a window attention
architecture tailored for joint spatial and spatiotemporal generative
modeling. Taken together these design decisions enable us to achieve state-of-
the-art performance on established video (UCF-101 and Kinetics-600) and image
(ImageNet) generation benchmarks without using classifier free guidance.
Finally, we also train a cascade of three models for the task of text-to-video
generation consisting of a base latent video diffusion model, and two video
super-resolution diffusion models to generate videos of $512\times 896$
resolution at $8$ frames per second.
Figure 1: W.A.L.T samples for text-to-video generation. Our approach can
generate high-resolution, temporally consistent photorealistic videos from
text prompts. The samples shown are $512\times 896$ resolution over $3.6$
seconds duration at $8$ frames per second.
00footnotetext: ∗Work partially done during an internship at Google.
## 1 Introduction
Transformers [73] are highly scalable and parallelizable neural network
architectures designed to win the hardware lottery [39]. This desirable
property has encouraged the research community to increasingly favor
transformers over domain-specific architectures in diverse fields such as
language [55, 56, 57, 26], audio [1], speech [58], vision [18, 30], and
robotics [7, 89, 5]. Such a trend towards unification allows researchers to
share and build upon advancements in traditionally disparate domains. Thus,
leading to a virtuous cycle of innovation and improvement in model design
favoring transformers.
A notable exception to this trend is generative modelling of videos. Diffusion
models [67, 69] have emerged as a leading paradigm for generative modelling of
images [33, 16] and videos [36]. However, the U-Net architecture [62, 33],
consisting of a series of convolutional [46] and self-attention [73] layers,
has been the predominant backbone in all video diffusion approaches [33, 16,
36]. This preference stems from the fact that the memory demands of full
attention mechanisms in transformers scale quadratically with input sequence
length. Such scaling leads to prohibitively high costs when processing high-
dimensional signals like video.
Latent diffusion models (LDMs) [61] reduce computational requirements by
operating in a lower-dimensional latent space derived from an autoencoder [75,
72, 20]. A critical design choice in this context is the type of latent space
employed: spatial compression (per frame latents) versus spatiotemporal
compression. Spatial compression is often preferred because it enables
leveraging pre-trained image autoencoders and LDMs, which are trained on large
paired image-text datasets. However, this choice increases network complexity
and limits the use of transformers as backbones, especially in generating
high-resolution videos due to memory constraints. On the other hand, while
spatiotemporal compression can mitigate these issues, it precludes the use of
paired image-text datasets, which are much larger and diverse than their video
counterparts.
We present Window Attention Latent Transformer (W.A.L.T): a transformer-based
method for latent video diffusion models (LVDMs). Our method consists of two
stages. First, an autoencoder maps both videos and images into a unified,
lower-dimensional latent space. This design choice enables training a single
generative model jointly on image and video datasets and significantly reduces
the computational burden for generating high resolution videos. Subsequently,
we propose a new design of transformer blocks for latent video diffusion
modeling which is composed of self-attention layers that alternate between
non-overlapping, window-restricted spatial and spatiotemporal attention. This
design offers two primary benefits: firstly, the use of local window attention
significantly lowers computational demands. Secondly, it facilitates joint
training, where the spatial layers independently process images and video
frames, while the spatiotemporal layers are dedicated to modeling the temporal
relationships in videos.
While conceptually simple, our method provides the first empirical evidence of
transformers’ superior generation quality and parameter efficiency in latent
video diffusion on public benchmarks. Specifically, we report state-of-the-art
results on class-conditional video generation (UCF-101 [70]), frame prediction
(Kinetics-600 [9]) and class conditional image generation (ImageNet [15])
without using classifier free guidance. Finally, to showcase the scalability
and efficiency of our method we also demonstrate results on the challenging
task of photorealistic text-to-video generation. We train a cascade of three
models consisting of a base latent video diffusion model, and two video super-
resolution diffusion models to generate videos of $512\times 896$ resolution
at $8$ frames per second and report state-of-the-art zero-shot FVD score on
the UCF-101 benchmark.
Figure 2: W.A.L.T. We encode images and videos into a shared latent space. The
transformer backbone processes these latents with blocks having two layers of
window-restricted attention: spatial layers capture spatial relations in both
images and video, while spatiotemporal layers model temporal dynamics in
videos and passthrough images via identity attention mask. Text conditioning
is done via spatial cross-attention.
## 2 Related Work
##### Video Diffusion Models.
Diffusion models have shown impressive results in image [67, 68, 52, 33, 61,
38] and video generation [36, 34, 66, 29, 4, 24]. Video diffusion models can
be categorized into pixel-space [36, 34, 66] and latent-space [31, 83, 4, 24]
approaches, the later bringing important efficiency advantages when modeling
videos. Ho et al. [36] demonstrated that the quality of text conditioned video
generation can be significantly improved by jointly training on image and
video data. Similarly, to leverage image datasets, latent video diffusion
models inflate a pre-trained image model, typically a U-Net [62], into a video
model by adding temporal layers, and initializing them as the identity
function [34, 66, 4]. Although computationally efficient, this approach
couples the design of video and image models, and precludes spatiotemporal
compression. In this work, we operate on a unified latent space for images and
videos, allowing us to leverage large scale image and video datasets while
enjoying computational efficiency gains from spatiotemporal compression of
videos.
##### Transformers for Generative Modeling.
Multiple classes of generative models have utilized Transformers [73] as
backbone, such as, Generative adversarial networks [47, 42, 85],
autoregressive [59, 20, 10, 80, 77, 21, 11, 74, 81, 27, 78] and diffusion [2,
53, 22, 87, 41, 50] models. Inspired by the success of autoregressive
pretraining of large language models [55, 56, 57], Ramesh et al. [59] trained
a text-to-image generation model by predicting the next visual token obtained
from an image tokenizer. Subsequently, this approach was applied to multiple
applications including class-conditional image generation [20, 79], text-to-
image [59, 80, 17, 76] or image-to-image translation [21, 77]. Similarly, for
video generation, transformer-based models were proposed to predict next
tokens using 3D extensions of VQGAN [78, 23, 81, 37] or using per frame image
latents [27]. Autoregressive sampling of videos is typically impractical given
the very long sequences involved. To alleviate this issue, non-autoregressive
sampling [10, 11], _i.e_. parallel token prediction, has been adopted as a
more efficient solution for transformer-based video generation [27, 74, 81].
Recently, the community has started adopting transformers as the denoising
backbone for diffusion models in place of U-Net [38, 53, 87, 12, 50]. To the
best of our knowledge, our work is the first successful empirical
demonstration (§ 5.1) of a transformer-based backbone for jointly training
image and video latent diffusion models.
## 3 Background
Diffusion formulation. Diffusion models [67, 33, 69] are a class of generative
models which learn to generate data by iteratively denoising samples drawn
from a noise distribution. Gaussian diffusion models assume a forward noising
process which gradually applies noise ($\boldsymbol{\epsilon}$) to real data
($\boldsymbol{x_{0}}\sim p_{\text{data}}$). Concretely,
$\boldsymbol{x_{t}}=\sqrt{\gamma(t)}\ \boldsymbol{x_{0}}+\sqrt{1-\gamma(t)}\
\boldsymbol{\epsilon},$ (1)
where
$\boldsymbol{\epsilon}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{I}),t\in\left[0,1\right]$,
and $\gamma(t)$ is a monotonically decreasing function (noise schedule) from
$1$ to $0$. Diffusion models are trained to learn the reverse process that
inverts the forward corruptions:
$\mathbb{E}_{x\sim
p_{\text{data}},t\sim\mathcal{U}(0,1),\boldsymbol{\epsilon}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{I})}\left[\left\|\boldsymbol{y}-f_{\theta}(\boldsymbol{x_{t}};\boldsymbol{c},t)\right\|^{2}\right],$
(2)
where $f_{\theta}$ is the denoiser model parameterized by a neural network,
$\boldsymbol{c}$ is conditioning information e.g., class labels or text
prompts, and the target $\boldsymbol{y}$ can be random noise
$\boldsymbol{\epsilon}$, denoised input $\boldsymbol{x_{0}}$ or
$\boldsymbol{v}=\sqrt{1-\gamma(t)}\ \boldsymbol{\epsilon}-\sqrt{\gamma(t)}\
\boldsymbol{x_{0}}$. Following [63, 34], we use $\boldsymbol{v}$-prediction in
all our experiments.
Latent diffusion models (LDMs). Processing high-resolution images and videos
using raw pixels requires considerable computational resources. To address
this, LDMs operate on the low dimensional latent space of a VQ-VAE [72, 20].
VQ-VAE consists of an encoder $E(x)$ that encodes an input video
$x\in\mathbb{R}^{T\times H\times W\times 3}$ into a latent representation
$z\in\mathbb{R}^{t\times h\times w\times c}$. The encoder downsamples the
video by a factor of $f_{s}=H/h=W/w$ and $f_{t}=T/t$, where $T=t=1$
corresponds to using an image autoencoder. An important distinction from the
original VQ-VAE is the absence of a codebook of quantized embeddings as
diffusion models can operate on continous latent spaces. A decoder $D$ is
trained to predict a reconstruction of the video, $\hat{x}$, from $z$.
Following VQ-GAN [20], reconstruction quality can be further improved by
adding adversarial [25] and perceptual losses [43, 86].
## 4 W.A.L.T
### 4.1 Learning Visual Tokens
A key design decision in video generative modeling is the choice of latent
space representation. Ideally, we want a shared and unified compressed visual
representation that can be used for generative modeling of both images and
videos [74, 82]. The unified representation is important because joint image-
video learning is preferable due to a scarcity of labeled video data [34],
such as text-video pairs. Concretely, given a video sequence
$x\in\mathbb{R}^{(1+T)\times H\times W\times C}$, we aim to learn a low-
dimensional representation $z\in\mathbb{R}^{(1+t)\times h\times w\times c}$
that performs spatial-temporal compression by a factor of $f_{s}=H/h=W/w$ in
space and a factor of $f_{t}=T/t$ in time. To enable a unified representation
for both videos and static images, the first frame is always encoded
independently from the rest of the video. This allows static images
$x\in\mathbb{R}^{1\times H\times W\times C}$ to be treated as videos with a
single frame, _i.e_. $z\in\mathbb{R}^{1\times h\times w\times c}$.
We instantiate this design with the causal 3D CNN encoder-decoder architecture
of the MAGVIT-v2 tokenizer [82]. Typically the encoder-decoder consists of
regular 3D convolution layers which cannot process the first frame
independently [81, 23]. This limitation stems from the fact that a regular
convolutional kernel of size $(k_{t},k_{h},k_{w})$ will operate on
$\left\lfloor\frac{k_{t}-1}{2}\right\rfloor$ frames before and
$\left\lfloor\frac{k_{t}}{2}\right\rfloor$ frames after the input frames.
Causal 3D convolution layers solve this issue as the convolutional kernel
operates on only the past $k_{t}-1$ frames. This ensures that the output for
each frame is influenced solely by the preceding frames, enabling the model to
tokenize the first frame independently.
After this stage, the input to our model is a batch of latent tensors
$z\in\mathbb{R}^{(1+t)\times h\times w\times c}$ representing a single video
or a stack of $1+t$ independent images (Fig. 2). Different from [82], our
latent representation is real-valued and quantization-free. In the section
below we describe how our model jointly processes a mixed batch of images and
videos.
### 4.2 Learning to Generate Images and Videos
Patchify. Following the original ViT [18], we “patchify” each latent frame
independently by converting it into a sequence of non-overlapping $h_{p}\times
w_{p}$ patches where $h_{p}=h/p$, $w_{p}=w/p$ and $p$ is the patch size. We
use learnable positional embeddings [73], which are the sum of space and time
positional embeddings. Position embeddings are added to the linear projections
[18] of the patches. Note that for images, we simply add the temporal position
embedding corresponding to the first latent frame.
Window attention. Transformer models composed entirely of global self-
attention modules incur significant compute and memory costs, especially for
video tasks. For efficiency and for processing images and videos jointly we
compute self-attention in windows [73, 27], based on two types of non-
overlapping configurations: spatial (S) and spatiotemporal (ST), _cf_. Fig. 2.
Spatial Window (SW) attention is restricted to all the tokens within a latent
frame of size $1\times h_{p}\times w_{p}$ (the first dimension is time). SW
models the spatial relations in images and videos. Spatiotemporal Window (STW)
attention is restricted within a 3D window of size $(1+t)\times
h_{p}^{\prime}\times h_{w}^{\prime}$, modeling the temporal relationships
among video latent frames. For images, we simply use an identity attention
mask ensuring that the value embeddings corresponding to the image frame
latents are passed through the layer as is. Finally, in addition to absolute
position embeddings we also use relative position embeddings [49].
Our design, while conceptually straightforward, achieves computational
efficiency and enables joint training on image and video datasets. In contrast
to methods based on frame-level autoencoders [24, 4, 27], our approach does
not suffer from flickering artifacts, which often result from encoding and
decoding video frames independently. However, similar to Blattmann et al. [4],
we can also potentially leverage pre-trained image LDMs with transformer
backbones by simply interleaving STW layers.
### 4.3 Conditional Generation
To enable controllable video generation, in addition to conditioning on
timestep $t$, diffusion models are often conditioned on additional conditional
information $\boldsymbol{c}$ such as class labels, natural language, past
frames or low resolution videos. In our transformer backbone, we incorporate
three types of conditioning mechanisms as described in what follows:
Cross-attention. In addition to self-attention layers in our window
transformer blocks, we add a cross-attention layer for text conditioned
generation. When training models on just videos, the cross-attention layer
employs the same window-restricted attention as the self-attention layer,
meaning S/ST blocks will have SW/STW cross-attention layers (Fig. 2). However,
for joint training, we only use SW cross-attention layers. For cross-attention
we concatenate the input signal (query) with the conditioning signal (key,
value) as our early experiments showed this improves performance.
AdaLN-LoRA. Adaptive normalization layers are an important component in a
broad range of generative and visual synthesis models [52, 16, 54, 44, 19,
53]. A simple way to incorporate adaptive layer normalization is to include
for each layer $i$, an MLP layer to regress a vector of conditioning
parameters $A^{i}=\texttt{MLP}(\boldsymbol{c}+\boldsymbol{t})$, where
$A^{i}=\texttt{concat}(\gamma_{1},\gamma_{2},\beta_{1},\beta_{2},\alpha_{1},\alpha_{2})$,
$A^{i}\in\mathbb{R}^{6\times d_{\texttt{model}}}$, and
$\boldsymbol{c}\in\mathbb{R}^{d_{\texttt{model}}}$,
$\boldsymbol{t}\in\mathbb{R}^{d_{\texttt{model}}}$ are the condition and
timestep embeddings. In the transformer block, $\gamma$ and $\beta$ scale and
shift the inputs of the multi-head attention and MLP layers, respectively,
while $\alpha$ scales the output of both the multi-head attention and MLP
layers. The parameter count of these additional MLP layers scales linearly
with the number of layers and quadratically with the model’s dimensional size
($\texttt{num\\_blocks}\times d_{\texttt{model}}\times 6\times
d_{\texttt{model}}$). For instance, in a ViT-g model with $1$B parameters, the
MLP layers contribute an additional $475$M parameters. Inspired by [40], we
propose a simple solution dubbed AdaLN-LoRA, to reduce the model parameters.
For each layer, we regress conditioning parameters as
$\displaystyle\small A^{1}$
$\displaystyle=\small\texttt{MLP}(\boldsymbol{c}+\boldsymbol{t}),$
$\displaystyle\small A^{i}$ $\displaystyle=\small
A^{1}+W_{b}^{i}W_{a}^{i}(\boldsymbol{c}+\boldsymbol{t})\quad\forall i\neq 1,$
(3)
where $W_{b}^{i}\,{\in}\,\mathbb{R}^{d_{\texttt{model}}\times r}$,
$W_{a}^{i}\,{\in}\,\mathbb{R}^{r\times(6\times d_{\texttt{model}})}$. This
reduces the number of trainable model parameters significantly when
$r\,{\ll}\,d_{\texttt{model}}$. For example, a ViT-g model with $r\,{=}\,2$
reduces the MLP parameters from $475$M to $12$M.
Self-conditioning. In addition to being conditioned on external inputs,
iterative generative algorithms can also be conditioned on their own
previously generated samples during inference [3, 65, 13]. Specifically, Chen
et al. [13] modify the training process for diffusion models, such that with
some probability $p_{\text{sc}}$ the model first generates a sample
$\boldsymbol{\tilde{z}_{0}}=f_{\theta}(\boldsymbol{z_{t}};\boldsymbol{0},\boldsymbol{c},t)$
and then refines this estimate using another forward pass conditioned on this
initial sample:
$f_{\theta}(\boldsymbol{z_{t}};\texttt{stopgrad}(\boldsymbol{\tilde{z}_{0}}),\boldsymbol{c},t)$.
With probability $1-p_{\text{sc}}$, only a single forward pass is done. We
concatenate the model estimate with the input along the channel dimension and
found this simple technique to work well when used in conjunction with
$\boldsymbol{v}$-prediction.
Method | K600 FVD$\downarrow$ | UCF FVD$\downarrow$ | params. | steps
---|---|---|---|---
TrIVD-GAN-FP [51] | 25.7±0.7 | – | – | 1
Video Diffusion [36] | 16.2±0.3 | – | 1.1B | 256
RIN [41] | 10.8 | – | 411M | 1000
TATS [23] | – | 332±18 | 321M | 1024
Phenaki [74] | 36.4±0.2 | – | 227M | 48
MAGVIT [81] | 9.9±0.3 | 76±2 | 306M | 12
MAGVITv2 [82] | 4.3±0.1 | 58±2 | 307M | 24
W.A.L.T-L _Ours_ | 3.3±0.0 | 46±2 | 313M | 50
W.A.L.T-XL _Ours_ | – | 36±2 | 460M | 50
Table 1: Video generation evaluation on frame prediction on Kinetics-600 and
class-conditional generation on UCF-101.
### 4.4 Autoregressive Generation
For generating long videos via autoregressive prediction we also train our
model jointly on the task of frame prediction. This is achieved by
conditioning the model on past frames with a probability of $p_{\text{fp}}$
during training. Specifically, the model is conditioned using
$c_{\text{fp}}=\texttt{concat}(m_{\text{fp}}\circ\boldsymbol{z_{t}},m_{\text{fp}})$,
where $m_{\text{fp}}$ is a binary mask. The binary mask indicates the number
of past frames used for conditioning. We condition on either $1$ latent frame
(image to video generation) or $2$ latent frames (video prediction). This
conditioning is integrated into the model through concatenation along the
channel dimension of the noisy latent input. During inference, we use standard
classifier-free guidance with $c_{\text{fp}}$ as the conditioning signal.
### 4.5 Video Super Resolution
Generating high-resolution videos with a single model is computationally
prohibitive. Following [35], we use a cascaded approach with three models
operating at increasing resolutions. Our base model generates videos at
$128\times 128$ resolution which are subsequently upsampled twice via two
super resolution stages. We first spatially upscale the low resolution input
$\boldsymbol{z^{\text{lr}}}$ (video or image) using a depth-to-space
convolution operation. Note that, unlike training where ground truth low-
resolution inputs are available, inference relies on latents produced by
preceding stages (_cf_. teaching-forcing). To reduce this discrepancy and
improve the robustness of the super-resolution stages in handling artifacts
generated by lower resolution stages, we use noise conditioning augmentation
[35]. Concretely, noise is added in accordance with $\gamma(t)$, by sampling a
noise level as $t_{\text{sr}}\sim\mathcal{U}(0,t_{\text{max\\_noise}})$ and is
provided as input to our AdaLN-LoRA layers.
Method | Cost (Iter$\times$BS) | FID$\downarrow$ | IS$\uparrow$ | params. | steps
---|---|---|---|---|---
BigGAN-deep [6] | - | 6.95 | 171.4 | 160M | 1
LDM-4 [61] | 178k$\times$1200 | 10.56 | 103.5 | 400M | 250
DiT-XL/2 [53] | 7000k$\times$256 | 9.62 | 121.5 | 675M | 250
ADM [16] | - | 7.49 | 127.5 | 608M | 2000
MDT [22] | 6500k$\times$256 | 6.23 | 143.0 | 676M | 250
MaskDiT [87] | 1200k$\times$1024 | 5.69 | 178.0 | 736M | 40
RIN [41] | 600k$\times$1024 | 3.42 | 182.0 | 410M | 1000
simple diffusion [38] | 500K$\times$2048 | 2.77 | 211.8 | 2B | 512
VDM++ [45] | - | 2.40 | 225.3 | 2B | 512
W.A.L.T-L _Ours_ | 437k$\times$1024 | 2.56 | 215.1 | 460M | 50
Table 2: Class-conditional image generation on ImageNet 256$\times$256. We
adopt the evaluation protocol and implementation of ADM [16] and report
results without classifier free guidance.
Aspect-ratio finetuning. To simplify training and leverage broad data sources
with different aspect ratios, we train our base stage using a square aspect
ratio. We fine-tune the base stage on a subset of data to generate videos with
a $9:16$ aspect ratio by interpolating position embeddings.
## 5 Experiments
In this section, we evaluate our method on multiple tasks: class-conditional
image and video generation, frame prediction and text conditioned video
generation and perform extensive ablation studies of different design choices.
For qualitative results, see Fig. 1, Fig. 3, Fig. 4 and videos on our project
website. See appendix for additional details.
### 5.1 Visual Generation
Video generation. We consider two standard video benchmarks, UCF-101 [70] for
class-conditional generation and Kinetics-600 [9] for video prediction with
$5$ conditioning frames. We use FVD [71] as our primary evaluation metric.
Across both datasets, W.A.L.T significantly outperforms all prior works (Tab.
1). Compared to prior video diffusion models, we achieve state-of-the-art
performance with less model parameters, and require $50$ DDIM [68] inference
steps.
patch size $p$ | FVD$\downarrow$ | IS$\uparrow$
---|---|---
1 | 60.7 | 87.2
2 | 134.4 | 82.2
4 | 461.8 | 63.9
(a)
st window | FVD$\downarrow$ | IS$\uparrow$ | sps
---|---|---|---
$5\times 4\times 4$ | 56.9 | 87.3 | 2.24
$5\times 8\times 8$ | 59.6 | 87.4 | 2.00
$5\times 16\times 16$ | 55.3 | 87.4 | 1.75
full self attn. | 59.9 | 87.8 | 1.20
(b)
$p_{\text{sc}}$ | FVD$\downarrow$ | IS$\uparrow$
---|---|---
0.0 | 109.9 | 82.6
0.3 | 76.0 | 86.5
0.6 | 60.0 | 86.8
0.9 | 61.4 | 87.1
(c)
$r$ | FVD$\downarrow$ | IS$\uparrow$ | params
---|---|---|---
2 | 60.7 | 87.2 | 313 M
4 | 56.6 | 87.3 | 314 M
16 | 55.5 | 88.0 | 316 M
64 | 54.4 | 87.9 | 324 M
256 | 52.5 | 88.5 | 357 M
(d)
| FVD$\downarrow$ | IS$\uparrow$
---|---|---
w/o qk norm [14] | 59.0 | 86.8
w/o latent norm | 67.9 | 87.1
w/o zero snr [48] | 91.0 | 84.2
full method | 60.7 | 87.2
(e)
$c$ | rFVD$\downarrow$ | FVD$\downarrow$ | IS$\uparrow$
---|---|---|---
4 | 37.7 | 86.4 | 84.9
8 | 17.1 | 75.4 | 86.3
16 | 8.2 | 67.0 | 86.0
32 | 3.5 | 83.4 | 82.9
(f)
Table 3: Ablation experiments on UCF-101 [70]. We compare FVD and inception
scores to ablate important design decisions with the default setting: L model,
$1\times 16\times 16$ spatial window, $5\times 8\times 8$ saptiotemporal (st)
window, $p_{\text{sc}}=0.9$, $c=8$ and $r=2$.
Image generation. To verify the modeling capabilities of W.A.L.T on the image
domain, we train a version of W.A.L.T for the standard ImageNet class-
conditional setting. For evaluation, we follow ADM [16] and report the FID
[32] and Inception [64] scores calculated on $50$K samples generated in $50$
DDIM steps. We compare (Table 2) W.A.L.T with state-of-the-art image
generation methods for $256\times 256$ resolution. Our model outperforms prior
works without requiring specialized schedules, convolution inductive bias,
improved diffusion losses, and classifier free guidance. Although VDM++ [45]
has slightly better FID score, the model has significantly more parameters
(2B).
### 5.2 Ablation Studies
We ablate W.A.L.T to understand the contribution of various design decisions
with the default settings: model L, patch size 1, $1\times 16\times 16$
spatial window, $5\times 8\times 8$ spatiotemporal window,
$p_{\text{sc}}=0.9$, $c=8$ and $r=2$.
Patch size. In various computer vision tasks utilizing ViT[18]-based models, a
smaller patch size $p$ has been shown to consistently enhance performance [18,
84, 8, 28]. Similarly, our findings also indicate that a reduced patch size
improves performance (Table LABEL:tab:patch_size).
Window attention. We compare three different STW window configurations with
full self-attention (Table LABEL:tab:window). We find that local self-
attention can achieve competitive (or better) performance while being
significantly faster (up to $2\times$) and requiring less accelerator memory.
Self-conditioning. In Table LABEL:tab:self_cond we study the influence of
varying the self-conditioning rate $p_{\text{sc}}$ on generation quality. We
notice a clear trend: increasing the self conditioning rate from $0.0$ (no
self-conditioning) to $0.9$ improves the FVD score substantially ($44\%$).
AdaLN-LoRA. An important design decision in diffusion models is the
conditioning mechanism. We investigate the effect of increasing the bottleneck
dimension $r$ in our proposed AdaLN-LoRA layers (Table LABEL:tab:adaln_lora).
This hyperparameter provides a flexible way to trade off between number of
model parameters and generation performance. As shown in Table
LABEL:tab:adaln_lora, increasing $r$ improves performance but also increases
model parameters. This highlights an important model design question: given a
fixed parameter budget, how should we allocate parameters - either by using
separate AdaLN layers, or by increasing base model parameters while using
shared AdaLN-LoRA layers? We explore this in Table 4 by comparing two model
configurations: W.A.L.T-L with separate AdaLN layers and W.A.L.T-XL with
AdaLN-LoRA and $r=2$. While both configurations yield similar FVD and
Inception scores, W.A.L.T-XL achieves a lower final loss value, suggesting the
advantage of allocating more parameters to the base model and choosing an
appropriate $r$ value within accelerator memory limits.
Model | AdaLN | FVD$\downarrow$ | IS$\uparrow$ | params. | final loss
---|---|---|---|---|---
L | separate | 34.6 | 90.2 | 458M | 0.274
XL | LoRA-2 | 36.7 | 89.4 | 460M | 0.268
Table 4: Parameter matched comparison between AdaLN-LoRA and per layer adaln
layers. See text for details. Figure 3: Qualitative evaluation. Example videos
generated by W.A.L.T from natural language prompts at $512\times 896$
resolution over $3.6$ seconds duration at $8$ frames per second. The W.A.L.T
model is able to generate temporally consistent photorealistic videos that
align with the textual prompt.
Noise schedule. Common latent diffusion noise schedules [61] typically do not
ensure a zero signal-to-noise ratio (SNR) at the final timestep, i.e., at
$t=1,\gamma(t)>0$. This leads to a mismatch between training and inference
phases. During inference, models are expected to start from purely Gaussian
noise, whereas during training, at $t=1$, a small amount of signal information
remains accessible to the model. This is especially harmful for video
generation as videos have high temporal redundancy. Even minimal information
leakage at $t=1$ can reveal substantial information to the model. Addressing
this mismatch by enforcing a zero terminal SNR [48] significantly improves
performance (Table LABEL:tab:misc_improvements). Note that this approach was
originally proposed to fix over-exposure problems in image generation, but we
find it effective for video generation as well.
Autoencoder. Finally, we investigate one critical but often overlooked
hyperparameter in the first stage of our model: the channel dimension $c$ of
the autoencoder latent $z$. As shown in Table LABEL:tab:tokenizer, increasing
$c$ significantly improves the reconstruction quality (lower rFVD) while
keeping the same spatial $f_{s}$ and temporal compression $f_{t}$ ratios.
Empirically, we found that both lower and higher values of $c$ lead to poor
FVD scores in generation, with a sweet spot of $c=8$ working well across most
datasets and tasks we evaluated. We also normalize the latents before
processing them via transformer which further improves performance.
In our transformer models, we use query-key normalization [14] as it helps
stabilize training for larger models. Finally, we note that some of our
default settings are not optimal, as indicated by ablation studies. These
defaults were chosen early on for their robustness across datasets, though
further tuning may improve performance.
Figure 4: Examples of consistent 3D camera motion (5.1 secs). Prompts: _camera
turns around a{blue jay, bunny}, studio lighting, $360^{\circ}$ rotation_.
Best viewed in video format.
### 5.3 Text-to-video
We train W.A.L.T for text-to-video jointly on text-image and text-video pairs
(Sec. 4.2). We used a dataset of $\sim$970M text-image pairs and $\sim$89M
text-video pairs from the public internet and internal sources. We train our
base model at resolution $17\times 128\times 128$ (3B parameters), and two
$2\times$ cascaded super-resolution models for $17\times 128\times
224\rightarrow 17\times 256\times 448$ (L, 1.3B, $p=2$) and $17\times
256\times 448\rightarrow 17\times 512\times 896$ (L, 419M, $p=2$)
respectively. We fine-tune the base stage for the $9:16$ aspect ratio to
generate videos at resolution $128\times 224$. We use classifier free guidance
for all our text-to-video results.
#### 5.3.1 Quantitative Evaluation
Evaluating text-conditioned video generation systems scientifically remains a
significant challenge, in part due to the absence of standardized training
datasets and benchmarks. So far we have focused our experiments and analyses
on the standard academic benchmarks, which use the same training data to
ensure controlled and fair comparisons. Nevertheless, to compare with prior
work on text-to-video, we also report results on the UCF-101 dataset in the
zero-shot evaluation protocol in Table 5 [24, 66, 37]. Also see supplement.
Joint training. A primary strength of our framework is its ability to train
simultaneously on both image and video datasets. In Table 5 we ablate the
impact of this joint training approach. Specifically, we trained two versions
of W.A.L.T-L (each with $419$M params.) models using the default settings
specified in § 5.2. We find that joint training leads to a notable improvement
across both metrics. Our results align with the findings of Ho et al. [36],
who demonstrated the benefits of joint training for pixel-based video
diffusion models with U-Net backbones.
Scaling. Transformers are known for their ability to scale effectively in many
tasks [55, 14, 5]. In Table 5 we show the benefits of scaling our transformer
model for video diffusion. Scaling our base model size leads to significant
improvements on both the metrics. It is important to note, however, that our
base model is considerably smaller than leading text-to-video systems. For
instance, Ho et al. [34] trained base model of $5.7$B parameters. Hence, we
believe scaling our models further is an important direction of future work.
Method | IS ($\uparrow$) | FVD ($\downarrow$)
---|---|---
CogVideo (Chinese) [37] | 23.6 | 751.3
CogVideo (English) [37] | 25.3 | 701.6
MagicVideo [88] | - | 699.0
Make-A-Video [66] | 33.0 | 367.2
Video LDM [4] | 33.5 | 550.6
PYoCo [24] | 47.8 | 355.2
W.A.L.T _(Ours)_ 419M (video only) | 26.8 | 598.8
W.A.L.T _(Ours)_ 419M (video + image) | 31.7 | 344.5
W.A.L.T _(Ours)_ 3B (video + image) | 35.1 | 258.1
Table 5: UCF-101 text-to-video generation. Joint training on image and video
datasets in conjunction with scaling the model parameters is essential for
high quality video generation.
Comparison with prior work. In Table 5, we present a system-level comparison
of various text-to-video generation methods. Our results are promising; we
surpass all previous work in the FVD metric. In terms of the IS, our
performance is competitive, outperforming all but PYoCo [24]. A possible
explanation for this discrepancy might be PYoCo’s use of stronger text
embeddings. Specifically, they utilize both CLIP [57] and T5-XXL [60]
encoders, whereas we employ a T5-XL [60] text encoder only.
#### 5.3.2 Qualitative Results
As mentioned in § 4.4, we jointly train our model on the task of frame
prediction conditioned on $1$ or $2$ latent frames. Hence, our model can be
used for animating images (image-to-video) and generating longer videos with
consistent camera motion (Fig. 4). See videos on our project website.
## 6 Conclusion
In this work, we introduce W.A.L.T, a simple, scalable, and efficient
transformer-based framework for latent video diffusion models. We demonstrate
state-of-the-art results for image and video generation using a transformer
backbone with windowed attention. We also train a cascade of three W.A.L.T
models jointly on image and video datasets, to synthesize high-resolution,
temporally consistent photorealistic videos from natural language
descriptions. While generative modeling has seen tremendous recent advances
for images, progress on video generation has lagged behind. We hope that
scaling our unified framework for image and video generation will help close
this gap.
## Acknowledgements
We thank Bryan Seybold, Dan Kondratyuk, David Ross, Hartwig Adam, Huisheng
Wang, Jason Baldridge, Mauricio Delbracio and Orly Liba for helpful
discussions and feedback.
## References
* Agostinelli et al. [2023] Andrea Agostinelli, Timo I Denk, Zalán Borsos, Jesse Engel, Mauro Verzetti, Antoine Caillon, Qingqing Huang, Aren Jansen, Adam Roberts, Marco Tagliasacchi, et al. MusicLM: Generating music from text. _arXiv:2301.11325_ , 2023.
* Bao et al. [2022] Fan Bao, Chongxuan Li, Yue Cao, and Jun Zhu. All are worth words: a vit backbone for score-based diffusion models. In _NeurIPS 2022 Workshop on Score-Based Methods_ , 2022.
* Bengio et al. [2015] Samy Bengio, Oriol Vinyals, Navdeep Jaitly, and Noam Shazeer. Scheduled sampling for sequence prediction with recurrent neural networks. _Advances in neural information processing systems_ , 28, 2015.
* Blattmann et al. [2023] Andreas Blattmann, Robin Rombach, Huan Ling, Tim Dockhorn, Seung Wook Kim, Sanja Fidler, and Karsten Kreis. Align your latents: High-resolution video synthesis with latent diffusion models. In _CVPR_ , 2023.
* Bousmalis et al. [2023] Konstantinos Bousmalis, Giulia Vezzani, Dushyant Rao, Coline Devin, Alex X Lee, Maria Bauza, Todor Davchev, Yuxiang Zhou, Agrim Gupta, Akhil Raju, et al. Robocat: A self-improving foundation agent for robotic manipulation. _arXiv preprint arXiv:2306.11706_ , 2023.
* Brock et al. [2018] Andrew Brock, Jeff Donahue, and Karen Simonyan. Large scale GAN training for high fidelity natural image synthesis. In _ICLR_ , 2018.
* Brohan et al. [2022] Anthony Brohan, Noah Brown, Justice Carbajal, Yevgen Chebotar, Joseph Dabis, Chelsea Finn, Keerthana Gopalakrishnan, Karol Hausman, Alex Herzog, Jasmine Hsu, et al. Rt-1: Robotics transformer for real-world control at scale. _arXiv preprint arXiv:2212.06817_ , 2022.
* Caron et al. [2021] Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers. In _Proceedings of the IEEE/CVF international conference on computer vision_ , pages 9650–9660, 2021.
* Carreira et al. [2018] Joao Carreira, Eric Noland, Andras Banki-Horvath, Chloe Hillier, and Andrew Zisserman. A short note about Kinetics-600. _arXiv:1808.01340_ , 2018.
* Chang et al. [2022] Huiwen Chang, Han Zhang, Lu Jiang, Ce Liu, and William T Freeman. MaskGIT: Masked generative image transformer. In _CVPR_ , 2022.
* Chang et al. [2023] Huiwen Chang, Han Zhang, Jarred Barber, AJ Maschinot, José Lezama, Lu Jiang, Ming-Hsuan Yang, Kevin Murphy, William T Freeman, Michael Rubinstein, et al. Muse: Text-to-image generation via masked generative transformers. In _ICML_ , 2023.
* Chen et al. [2023] Junsong Chen, Jincheng Yu, Chongjian Ge, Lewei Yao, Enze Xie, Yue Wu, Zhongdao Wang, James Kwok, Ping Luo, Huchuan Lu, et al. Pixart-$\backslash alpha$: Fast training of diffusion transformer for photorealistic text-to-image synthesis. _arXiv preprint arXiv:2310.00426_ , 2023.
* Chen et al. [2022] Ting Chen, Ruixiang Zhang, and Geoffrey Hinton. Analog bits: Generating discrete data using diffusion models with self-conditioning. _arXiv preprint arXiv:2208.04202_ , 2022.
* Dehghani et al. [2023] Mostafa Dehghani, Josip Djolonga, Basil Mustafa, Piotr Padlewski, Jonathan Heek, Justin Gilmer, Andreas Peter Steiner, Mathilde Caron, Robert Geirhos, Ibrahim Alabdulmohsin, et al. Scaling vision transformers to 22 billion parameters. In _International Conference on Machine Learning_ , pages 7480–7512. PMLR, 2023.
* Deng et al. [2009] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. ImageNet: A large-scale hierarchical image database. In _CVPR_ , 2009.
* Dhariwal and Nichol [2021] Prafulla Dhariwal and Alexander Nichol. Diffusion models beat GANs on image synthesis. In _NeurIPS_ , 2021.
* Ding et al. [2021] Ming Ding, Zhuoyi Yang, Wenyi Hong, Wendi Zheng, Chang Zhou, Da Yin, Junyang Lin, Xu Zou, Zhou Shao, Hongxia Yang, et al. CogView: Mastering text-to-image generation via transformers. In _NeurIPS_ , 2021.
* Dosovitskiy et al. [2020] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In _ICLR_ , 2020.
* Dumoulin et al. [2016] Vincent Dumoulin, Jonathon Shlens, and Manjunath Kudlur. A learned representation for artistic style. _arXiv preprint arXiv:1610.07629_ , 2016.
* Esser et al. [2021] Patrick Esser, Robin Rombach, and Bjorn Ommer. Taming transformers for high-resolution image synthesis. In _CVPR_ , 2021.
* Gafni et al. [2022] Oran Gafni, Adam Polyak, Oron Ashual, Shelly Sheynin, Devi Parikh, and Yaniv Taigman. Make-a-scene: Scene-based text-to-image generation with human priors. In _European Conference on Computer Vision_ , pages 89–106. Springer, 2022.
* Gao et al. [2023] Shanghua Gao, Pan Zhou, Ming-Ming Cheng, and Shuicheng Yan. Masked diffusion transformer is a strong image synthesizer. _arXiv:2303.14389_ , 2023.
* Ge et al. [2022] Songwei Ge, Thomas Hayes, Harry Yang, Xi Yin, Guan Pang, David Jacobs, Jia-Bin Huang, and Devi Parikh. Long video generation with time-agnostic VQGAN and time-sensitive transformer. In _ECCV_ , 2022.
* Ge et al. [2023] Songwei Ge, Seungjun Nah, Guilin Liu, Tyler Poon, Andrew Tao, Bryan Catanzaro, David Jacobs, Jia-Bin Huang, Ming-Yu Liu, and Yogesh Balaji. Preserve your own correlation: A noise prior for video diffusion models. _arXiv preprint arXiv:2305.10474_ , 2023.
* Goodfellow et al. [2014] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. _Advances in neural information processing systems_ , 27, 2014.
* Google [2023] Google. PaLM 2 technical report. _arXiv:2305.10403_ , 2023.
* Gupta et al. [2022] Agrim Gupta, Stephen Tian, Yunzhi Zhang, Jiajun Wu, Roberto Martín-Martín, and Li Fei-Fei. MaskViT: Masked visual pre-training for video prediction. In _ICLR_ , 2022.
* Gupta et al. [2023] Agrim Gupta, Jiajun Wu, Jia Deng, and Li Fei-Fei. Siamese masked autoencoders. _arXiv preprint arXiv:2305.14344_ , 2023.
* Harvey et al. [2022] William Harvey, Saeid Naderiparizi, Vaden Masrani, Christian Weilbach, and Frank Wood. Flexible diffusion modeling of long videos. _Advances in Neural Information Processing Systems_ , 35:27953–27965, 2022.
* He et al. [2022] Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Dollár, and Ross Girshick. Masked autoencoders are scalable vision learners. In _CVPR_ , 2022.
* He et al. [2023] Yingqing He, Tianyu Yang, Yong Zhang, Ying Shan, and Qifeng Chen. Latent video diffusion models for high-fidelity long video generation. _arXiv preprint arXiv:2211.13221_ , 2023.
* Heusel et al. [2017] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs trained by a two time-scale update rule converge to a local nash equilibrium. In _NeurIPS_ , 2017.
* Ho et al. [2020] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In _NeurIPS_ , 2020.
* Ho et al. [2022a] Jonathan Ho, William Chan, Chitwan Saharia, Jay Whang, Ruiqi Gao, Alexey Gritsenko, Diederik P Kingma, Ben Poole, Mohammad Norouzi, David J Fleet, et al. Imagen video: High definition video generation with diffusion models. _arXiv:2210.02303_ , 2022a.
* Ho et al. [2022b] Jonathan Ho, Chitwan Saharia, William Chan, David J Fleet, Mohammad Norouzi, and Tim Salimans. Cascaded diffusion models for high fidelity image generation. _JMLR_ , 23(1):2249–2281, 2022b.
* Ho et al. [2022c] Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J Fleet. Video diffusion models. In _ICLR Workshops_ , 2022c.
* Hong et al. [2022] Wenyi Hong, Ming Ding, Wendi Zheng, Xinghan Liu, and Jie Tang. CogVideo: Large-scale pretraining for text-to-video generation via transformers. _arXiv:2205.15868_ , 2022.
* Hoogeboom et al. [2023] Emiel Hoogeboom, Jonathan Heek, and Tim Salimans. simple diffusion: End-to-end diffusion for high resolution images. In _ICML_ , 2023.
* Hooker [2021] Sara Hooker. The hardware lottery. _Communications of the ACM_ , 64(12):58–65, 2021\.
* Hu et al. [2021] Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. Lora: Low-rank adaptation of large language models. In _ICLR_ , 2021.
* Jabri et al. [2023] Allan Jabri, David J Fleet, and Ting Chen. Scalable adaptive computation for iterative generation. In _ICML_ , 2023.
* Jiang et al. [2021] Yifan Jiang, Shiyu Chang, and Zhangyang Wang. Transgan: Two pure transformers can make one strong gan, and that can scale up. _Advances in Neural Information Processing Systems_ , 34:14745–14758, 2021.
* Johnson et al. [2016] Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In _ECCV_ , 2016.
* Karras et al. [2019] Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In _CVPR_ , 2019.
* Kingma and Gao [2023] Diederik P Kingma and Ruiqi Gao. Understanding the diffusion objective as a weighted integral of elbos. _arXiv:2303.00848_ , 2023.
* LeCun et al. [1998] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. _Proceedings of the IEEE_ , 86(11):2278–2324, 1998.
* Lee et al. [2021] Kwonjoon Lee, Huiwen Chang, Lu Jiang, Han Zhang, Zhuowen Tu, and Ce Liu. Vitgan: Training gans with vision transformers. _arXiv preprint arXiv:2107.04589_ , 2021.
* Lin et al. [2023] Shanchuan Lin, Bingchen Liu, Jiashi Li, and Xiao Yang. Common diffusion noise schedules and sample steps are flawed. _arXiv preprint arXiv:2305.08891_ , 2023.
* Liu et al. [2021] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. Swin transformer: Hierarchical vision transformer using shifted windows. In _Proceedings of the IEEE/CVF international conference on computer vision_ , pages 10012–10022, 2021.
* Lu et al. [2023] Haoyu Lu, Guoxing Yang, Nanyi Fei, Yuqi Huo, Zhiwu Lu, Ping Luo, and Mingyu Ding. Vdt: General-purpose video diffusion transformers via mask modeling. _arXiv preprint arXiv:2305.13311_ , 2023.
* Luc et al. [2020] Pauline Luc, Aidan Clark, Sander Dieleman, Diego de Las Casas, Yotam Doron, Albin Cassirer, and Karen Simonyan. Transformation-based adversarial video prediction on large-scale data. _arXiv:2003.04035_ , 2020.
* Nichol and Dhariwal [2021] Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In _International Conference on Machine Learning_ , pages 8162–8171. PMLR, 2021.
* Peebles and Xie [2022] William Peebles and Saining Xie. Scalable diffusion models with transformers. _arXiv:2212.09748_ , 2022.
* Perez et al. [2018] Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville. Film: Visual reasoning with a general conditioning layer. In _Proceedings of the AAAI conference on artificial intelligence_ , 2018.
* Radford et al. [2018] Alec Radford, Karthik Narasimhan, Tim Salimans, Ilya Sutskever, et al. Improving language understanding by generative pre-training. 2018\.
* Radford et al. [2019] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. Language models are unsupervised multitask learners. _OpenAI blog_ , 1(8):9, 2019.
* Radford et al. [2021] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In _ICML_ , 2021.
* Radford et al. [2023] Alec Radford, Jong Wook Kim, Tao Xu, Greg Brockman, Christine McLeavey, and Ilya Sutskever. Robust speech recognition via large-scale weak supervision. In _International Conference on Machine Learning_ , pages 28492–28518. PMLR, 2023.
* Ramesh et al. [2021] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. In _ICML_ , 2021.
* Roberts et al. [2019] Adam Roberts, Colin Raffel, Katherine Lee, Michael Matena, Noam Shazeer, Peter J Liu, Sharan Narang, Wei Li, and Yanqi Zhou. Exploring the limits of transfer learning with a unified text-to-text transformer. 2019\.
* Rombach et al. [2022] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models. In _CVPR_ , 2022.
* Ronneberger et al. [2015] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-Net: Convolutional networks for biomedical image segmentation. In _MICCAI_ , 2015.
* Salimans and Ho [2022] Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. _arXiv preprint arXiv:2202.00512_ , 2022.
* Salimans et al. [2016] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training GANs. In _NeurIPS_ , 2016.
* Savinov et al. [2021] Nikolay Savinov, Junyoung Chung, Mikolaj Binkowski, Erich Elsen, and Aaron van den Oord. Step-unrolled denoising autoencoders for text generation. _arXiv preprint arXiv:2112.06749_ , 2021.
* Singer et al. [2022] Uriel Singer, Adam Polyak, Thomas Hayes, Xi Yin, Jie An, Songyang Zhang, Qiyuan Hu, Harry Yang, Oron Ashual, Oran Gafni, et al. Make-a-video: Text-to-video generation without text-video data. _arXiv:2209.14792_ , 2022.
* Sohl-Dickstein et al. [2015] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In _ICML_ , 2015.
* Song et al. [2020] Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. _arXiv preprint arXiv:2010.02502_ , 2020.
* Song and Ermon [2019] Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. In _NeurIPS_ , 2019.
* Soomro et al. [2012] Khurram Soomro, Amir Roshan Zamir, and Mubarak Shah. UCF101: A dataset of 101 human actions classes from videos in the wild. _arXiv:1212.0402_ , 2012.
* Unterthiner et al. [2018] Thomas Unterthiner, Sjoerd van Steenkiste, Karol Kurach, Raphael Marinier, Marcin Michalski, and Sylvain Gelly. Towards accurate generative models of video: A new metric & challenges. _arXiv:1812.01717_ , 2018.
* Van Den Oord et al. [2017] Aaron Van Den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In _NeurIPS_ , 2017.
* Vaswani et al. [2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In _NeurIPS_ , 2017.
* Villegas et al. [2022] Ruben Villegas, Mohammad Babaeizadeh, Pieter-Jan Kindermans, Hernan Moraldo, Han Zhang, Mohammad Taghi Saffar, Santiago Castro, Julius Kunze, and Dumitru Erhan. Phenaki: Variable length video generation from open domain textual description. _arXiv:2210.02399_ , 2022.
* Vincent et al. [2008] Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. Extracting and composing robust features with denoising autoencoders. In _Proceedings of the 25th international conference on Machine learning_ , pages 1096–1103, 2008.
* Wu et al. [2021] Chenfei Wu, Lun Huang, Qianxi Zhang, Binyang Li, Lei Ji, Fan Yang, Guillermo Sapiro, and Nan Duan. Godiva: Generating open-domain videos from natural descriptions. _arXiv preprint arXiv:2104.14806_ , 2021.
* Wu et al. [2022] Chenfei Wu, Jian Liang, Lei Ji, Fan Yang, Yuejian Fang, Daxin Jiang, and Nan Duan. Nüwa: Visual synthesis pre-training for neural visual world creation. In _European conference on computer vision_ , pages 720–736. Springer, 2022.
* Yan et al. [2021] Wilson Yan, Yunzhi Zhang, Pieter Abbeel, and Aravind Srinivas. VideoGPT: Video generation using VQ-VAE and transformers. _arXiv:2104.10157_ , 2021.
* Yu et al. [2022a] Jiahui Yu, Xin Li, Jing Yu Koh, Han Zhang, Ruoming Pang, James Qin, Alexander Ku, Yuanzhong Xu, Jason Baldridge, and Yonghui Wu. Vector-quantized image modeling with improved VQGAN. In _ICLR_ , 2022a.
* Yu et al. [2022b] Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gunjan Baid, Zirui Wang, Vijay Vasudevan, Alexander Ku, Yinfei Yang, Burcu Karagol Ayan, et al. Scaling autoregressive models for content-rich text-to-image generation. _arXiv:2206.10789_ , 2022b.
* Yu et al. [2023a] Lijun Yu, Yong Cheng, Kihyuk Sohn, José Lezama, Han Zhang, Huiwen Chang, Alexander G Hauptmann, Ming-Hsuan Yang, Yuan Hao, Irfan Essa, et al. MAGVIT: Masked generative video transformer. In _CVPR_ , 2023a.
* Yu et al. [2023b] Lijun Yu, José Lezama, Nitesh B Gundavarapu, Luca Versari, Kihyuk Sohn, David Minnen, Yong Cheng, Agrim Gupta, Xiuye Gu, Alexander G Hauptmann, et al. Language model beats diffusion–tokenizer is key to visual generation. _arXiv preprint arXiv:2310.05737_ , 2023b.
* Yu et al. [2023c] Sihyun Yu, Kihyuk Sohn, Subin Kim, and Jinwoo Shin. Video probabilistic diffusion models in projected latent space. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 18456–18466, 2023c.
* Zhai et al. [2022] Xiaohua Zhai, Alexander Kolesnikov, Neil Houlsby, and Lucas Beyer. Scaling vision transformers. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 12104–12113, 2022.
* Zhang et al. [2022] Bowen Zhang, Shuyang Gu, Bo Zhang, Jianmin Bao, Dong Chen, Fang Wen, Yong Wang, and Baining Guo. Styleswin: Transformer-based gan for high-resolution image generation. In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 11304–11314, 2022.
* Zhang et al. [2018] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In _CVPR_ , 2018.
* Zheng et al. [2023] Hongkai Zheng, Weili Nie, Arash Vahdat, and Anima Anandkumar. Fast training of diffusion models with masked transformers. _arXiv:2306.09305_ , 2023.
* Zhou et al. [2022] Daquan Zhou, Weimin Wang, Hanshu Yan, Weiwei Lv, Yizhe Zhu, and Jiashi Feng. Magicvideo: Efficient video generation with latent diffusion models. _arXiv preprint arXiv:2211.11018_ , 2022.
* Zitkovich et al. [2023] Brianna Zitkovich, Tianhe Yu, Sichun Xu, Peng Xu, Ted Xiao, Fei Xia, Jialin Wu, Paul Wohlhart, Stefan Welker, Ayzaan Wahid, et al. RT-2: Vision-language-action models transfer web knowledge to robotic control. In _CoRL_ , 2023.
| T2V (base)
---|---
Input | $5\times 16\times 28$
Spatial window | $1\times 16\times 28$
Spatiotermporal window | $5\times 8\times 14$
Training steps | 250000
Batch size | 512
lr schedule | Constant
Optimizer | Adafactor
lr | 0.00005
Table 6: Hyperparameters for aspect-ratio finetuning.
## Appendix A Implementation Details
For the first stage, we follow the architecture and hyperparameters from Yu et
al. [82]. We report hyperparameters specific for training our model in Table
8. To train the second stage transformer model, we use the default settings of
$1\times 16\times 16$ spatial window, $5\times 8\times 8$ spatiotemporal
window, $p_{\text{sc}}=0.9$, $c=8$ and $r=2$. We summarize additional training
and inference hyperparameters for all tasks in Table 8. The UCF-101 model
results reported in Tables 1 and 4 are trained for $60,000$ steps. We perform
all ablations on UCF-101 with $35,000$ training steps.
Aspect-ratio finetuning. To simplify training and leverage broad data sources
with different aspect ratios, we train the base stage using a square aspect
ratio. We fine-tune the base the stage on a subset of data to generate videos
with a $9:16$ aspect ratio. We interpolate the absolute and relative position
embeddings and scale the window sizes. We summarize the finetuning
hyperparameters in Table 6.
Long video generation. As described in § 4.4, we train our model jointly on
the task of frame prediction. During inference, we generate videos as follows:
Given a natural language description of a video, we first generate the initial
$17$ frames using our base model. Next, we encode the last $5$ frames into $2$
latent frames using our causal 3D encoder. Providing $2$ latent frames as
input for subsequent autoregressive generation helps ensure that our model can
maintain continuity of motion and produce temporally consistent videos.
UCF-101 Text-to-Video. We follow the evaluation protocol of prior work [24],
and adapt their prompts to better describe the UCF-101 classes.
## Appendix B Additional Results
### B.1 Image Generation
We compare (Table 7) W.A.L.T with state-of-the-art image generation methods
for $256\times 256$ resolution with classifier free guidance. Unlike, prior
work [53, 22, 87] using Transformer for diffusion modelling, we did not
observe significant benefit of using vanilla classifier free guidance. Hence,
we report results using the power cosine schedule proposed by Gao et al. [22].
Our model performs better than prior works on the Inception Score metric, and
achieves competitive FID scores. Fig. 5 shows qualitative samples.
Method | Cost (Iter$\times$BS) | FID$\downarrow$ | IS$\uparrow$ | Params. | Steps
---|---|---|---|---|---
LDM-4 [61] | 178k$\times$1200 | 3.60 | 247.7 | 400M | 250
DiT-XL/2 [53] | 7000k$\times$256 | 2.27 | 278.2 | 675M | 250
ADM [16] | - | 3.94 | 215.8 | 608M | 2000
MDT [22] | 6500k$\times$256 | 1.79 | 283.0 | 676M | 250
MaskDiT [87] | 1200k$\times$1024 | 2.28 | 276.6 | 736M | 40
simple diffusion [38] | 500K$\times$2048 | 2.44 | 256.3 | 2B | 512
VDM++ [45] | - | 2.12 | 267.7 | 2B | 512
W.A.L.T-L _Ours_ | 437k$\times$1024 | 2.40 | 290.5 | 460M | 50
Table 7: Class-conditional image generation on ImageNet 256$\times$256. We
adopt the evaluation protocol and implementation of ADM [16] and report
results with classifier free guidance.
| ImageNet | UCF-101 | K600 | T2V (base) | T2V (2$\times$) | T2V (2$\times$2$\times$)
---|---|---|---|---|---|---
First Stage | | | | | |
Input | $1\times 256\times 256$ | $17\times 128\times 128$ | $17\times 128\times 128$ | $17\times 128\times 128$ | $17\times 256\times 448$ | $17\times 512\times 896$
$f_{s},f_{t}$ | 8, - | 8, 4 | 8, 4 | 8, 4 | 8, 4 | 8, 4
Channels | 128 | 128 | 128 | 128 | 128 | 128
Channel multiplier | 1,1,2,4 | 1, 2, 2, 4 | 1, 2, 2, 4 | 1, 2, 2, 4 | 1, 2, 2, 4 | 1, 2, 2, 4
Training duration | 270 epochs | 2000 epochs | 270000 steps | 1000000 steps | 1000000 steps | 1000000 steps
Batch size | 256 | 256 | 256 | 256 | 256 | 256
lr schedule | Cosine | Cosine | Cosine | Cosine | Cosine | Cosine
Optimizer | Adam | Adam | Adam | Adam | Adam | Adam
Second Stage | | | | | |
Input | $1\times 32\times 32$ | $5\times 16\times 16$ | $5\times 16\times 16$ | $5\times 16\times 16$ | $5\times 32\times 56$ | $5\times 64\times 112$
Layers | 24 | 28 | 24 | 52 | 40 | 24
Hidden size | 1024 | 1152 | 1024 | 9216 | 1408 | 1024
Heads | 16 | 16 | 16 | 16 | 16 | 16
Training duration | 350 epochs | 60000 steps | 360 epochs | 550000 steps | 675000 steps | 275000 steps
Batch size | 1024 | 256 | 512 | 512 | 512 | 512
lr schedule | Cosine | Cosine | Cosine | Cosine | Cosine | Cosine
Optimizer | AdamW | AdamW | AdamW | Adafactor | Adafactor | Adafactor
lr | 0.0005 | 0.0005 | 0.0005 | 0.0002 | 0.0005 | 0.0005
EMA | $\checkmark$ | $\checkmark$ | $\checkmark$ | ✗ | ✗ | ✗
Patch size | 1 | 1 | 1 | 1 | 2 | 4
AdaLN-LoRA | ✗ | 2 | 2 | 2 | 2 | 2
Diffusion | | | | | |
Diffusion Steps | 1000 | 1000 | 1000 | 1000 | 1000 | 1000
Noise schedule | Linear | Linear | Linear | Linear | Linear | Linear
$\beta_{0}$ | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001
$\beta_{1000}$ | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02
Sampler | DDIM | DDIM | DDIM | DDIM | DDIM | DDIM
Sampling steps | 50 | 50 | 50 | 50 | 50 | 50
Guidance | ✗ | ✗ | ✗ | $\checkmark$ | $\checkmark$ | $\checkmark$
Table 8: Training and evaluation hyperparameters.
### B.2 Video Generation
We show samples for Kinetics-600 frame prediction in Fig. 6.
Figure 5: ImageNet class-conditional generation samples.
Figure 6: Frame prediction samples on Kinetics-600. Top: ground-truth, where
unobserved frames are shaded. Bottom: generation.
### B.3 Image-to-Video
As noted in Section 4.4, we train our model jointly on the task of frame
prediction, where we condition on $1$ latent frame. This allows us to leverage
the high quality first frame from the image generator as context for
predicting subsequent frames. For qualitative results see videos on our
project website.
|
# Optimal Generation of Strictly
Increasing Binary Trees and Beyond
Olivier Bodini LIPN
Villetaneuse, FranceInstitut Galilé
Université Sorbonne Paris-Nord 0000-0002-1867-667X<EMAIL_ADDRESS>paris13.fr LIPN
Villetaneuse, FranceInstitut Galilé
Université Sorbonne Paris-NordLAGA
Villetaneuse, FranceInstitut Galilé
Université Sorbonne Paris-Nord Francis Durand LIPN
Villetaneuse, FranceInstitut Galilé
Université Sorbonne Paris-Nord 0009-0004-4146-0289 francis.durand@ens-paris-
saclay.fr LAGA
Villetaneuse, FranceInstitut Galilé
Université Sorbonne Paris-Nord Philippe Marchal LAGA
Villetaneuse, FranceInstitut Galilé
Université Sorbonne Paris-Nord 0000-0001-8236-5713<EMAIL_ADDRESS>paris13.fr
## 1 Introduction
Tree-like data structures are fundamental in computer science, serving as
critical tools for modeling phenomena, testing, and creating diverse and
representative datasets that enable effective training of machine learning
frameworks by exposing models to a wide range of possible input scenarios.
Recognizing the necessity for random samplers of these structures, a
comprehensive body of work has emerged offering algorithms for generating
simple tree families. These algorithms, ranging from general approaches like
the Recursive method and Boltzmann sampler [3] to specific ones like the BBJ
algorithm for binary trees [2], excel in efficiency and minimal randomness
usage. Despite the abundance of techniques for simple trees, the random
generation of increasing trees—a vital component in priority queue
management—remains underexplored. This gap is attributed to their non-uniform
internal structure, challenging the creation of homogeneous algorithms. Our
paper introduces a groundbreaking algorithm for the optimal generation of
strictly increasing binary trees, leveraging a novel approach pioneered by Ph.
Marchal [4] that ensures both entropy and time efficiency and we prove here
random bit complexity. Additionally, we present an enhanced algorithm that
adopts an innovative approximation schema for the recursive method. This
method is tailored for all weighted unary-binary increasing trees,
guaranteeing minimal randomness consumption. This dual approach not only
advances the field of random increasing tree generation but also sets a new
standard for algorithmic efficiency and randomness optimization. The next
section is devoted to presenting the first algorithm and evaluating its cost.
Section 3 addresses the second algorithm, which represents an improvement of
the recursive method through the use of a Monte Carlo process. The final
section demonstrates how the second method can be adapted to operate within a
more general framework.
## 2 An Ad Hoc Approach for Sampling Strictly Increasing Trees
Before introducing the first algorithm, let us make a first easy but important
observation. Strictly increasing binary trees are intimately related to
permutations (this has been known for a long time). Given a sequence of
distinct real numbers in the interval [0,1], we can derive a permutation based
on the relative order of the sequence, and thus an increasing tree in the
following way. Starting with the position of the smallest number in the
sequence, we place a 1 at this position in the permutation (and at the same
time, we create a root for the tree), and continue this process until all the
numbers have been placed (in the permutation and in the binary tree). If the
sequence alternates between increasing and decreasing values, the resulting
permutation will also alternate and the tree will be a strictly increasing
binary tree. Furthermore, suppose the sequence is chosen randomly and
uniformly. In that case, the permutation reflects a uniform distribution of
all possible permutations and the binary tree constructed is uniform among
binary trees of size $n$. This leads us to look for an algorithm that
generates a random, uniform, and alternating sequence, which in turn allows us
to construct a uniform alternating permutation and hence a uniform strictly
increasing binary tree.
Algorithm 1 Generate an Alternating Permutation of Size $n$
1:$n>1$
2:$D,Y\leftarrow\text{arrays of size }n$
3:$p\leftarrow\lfloor\log_{2}(n)\rfloor+1$
4:while precision on $Y$ is not sufficient do
5: $p\leftarrow p+1$
6: while $Y$ is empty do
7: $r\leftarrow U_{0}\text{with }p\text{ digits.}$
8: $D[0]\leftarrow 1-\sin^{2}\left(\frac{\pi}{2}r\right)$
9: for $i\leftarrow 0$ to $n-2$ do
10: $r\leftarrow U_{i+1}\text{with }p\text{ digits.}$
11: $D[i+1]=r^{2}(1-D[i])$
12: $\alpha\leftarrow\sqrt{\frac{1-D[n-1]}{1-D[0]}}$
13: $threshold\leftarrow\frac{1}{\alpha+\alpha^{-1}}$
14: $proba\leftarrow\text{rand()}$
15: if $proba\leq threshold$ then
16: $Y\leftarrow(1-D[0],D[1],1-D[2],\ldots)$
17: else if $threshold<proba\leq 2\cdot threshold$ then
18: $Y\leftarrow(1-D[n-1],D[n-2],1-D[n-3],\ldots)$
19: else
20: restart
21: Sort $Y$ to determine if its elements are strictly ordered
22:return the alternating sequence associated with $Y$
This algorithm is essentially a rewritten version of Marchal’s algorithm,
modified to bypass inefficient trigonometric calculations. More precisely, by
making the following change of variable
$C_{n}=\sin^{2}\left(\frac{\pi}{2}X_{n}\right)$ where $X_{n}$ is the sequence
described in [4], we do not alter the properties of the sequence and we
significantly improve the computation. We refer the reader to the initial note
by Marchal [4] for the proof of his algorithm’s validity. A key point is that
the algorithm uses a rejection procedure but it is proved in [4] that the
rejection probability is bounded above by $1-\frac{2}{3\pi}$ (which is
independent of $n$).
The remaining issue concerns the random-bit complexity. Naturally, for each
uniform i.i.d. random variable $U_{i}$, we aim to generate only enough digits
to ensure sufficient precision, thus avoiding any order ambiguity in the final
sequence.
As a preliminary, the number of strictly increasing binary trees of size $n$
is asymptotically equivalent to $2\left(\frac{2}{\pi}\right)^{n+1}n!$. Thus,
the entropy is $n\log_{2}(n)+\frac{2n}{\pi}+o(n)$. As we draw $n$ uniform
variables, we need at least $\log_{2}(n)$ bits of precision to be coherent
with the entropy.
Now, note that $D[n]$ converges when $n$ goes to the infinity to the
distribution
$Q:=\sum_{n=0}^{\infty}(-1)^{n}\prod_{i=0}^{n}U_{i}^{2}$
Indeed, we can derive from the algorithm the following induction:
$Q=U^{2}(1-V^{2}Q)$ where $U$ and $V$ are i.i.d. uniform distributions on
$[0,1]$. Consequently, the density function $d(z)$ of $Q$ follows the
functional equation:
$d(z)=\int_{z}^{1}\frac{1}{2\sqrt{t}}\cdot\frac{d(1-z/t)}{t}\,dt.$
To our knowledge, solving this equation is not straightforward. For that, let
us start from the density of the random variable $Q_{0}=U^{2}(1-V^{2})$, which
is:
$d(z):=\frac{\arctan\left(\frac{\sqrt{1-z}}{\sqrt{z}}\right)}{2\sqrt{z}},$
from which we can observe that the iteration $Q_{n+1}=U^{2}(1-V^{2}Q_{n})$
generates random variables whose density is alternatively a polynomial in
$A:=\arctan\left(\frac{\sqrt{1-z}}{\sqrt{z}}\right)$ and in
$\bar{A}:=\arctan\left(\frac{\sqrt{z}}{\sqrt{1-z}}\right)$, divided by
$\sqrt{z}$. Moreover, the polynomials follow a quite simple recurrence which
alternates
$P_{n+1}(\bar{A})=\int_{\bar{A}}^{\frac{\pi}{2}}P_{n}(A)\,dA$
and
$P_{n+1}(A)=\int_{0}^{A}P_{n}(\bar{A})\,d\bar{A}.$
So, the limiting polynomial follows the functional equation
$\int_{0}^{x}\left(\int_{x}^{\frac{\pi}{2}}P(t)\,dt\right)dx=P(x),$
from which we derive that the limiting distribution admits two solutions
depending on the parity on $n$: $\frac{2\sqrt{1-z}}{\pi\sqrt{z}}$ and
$\frac{2\sqrt{z}}{\pi\sqrt{1-z}}$. Therefore, we can conclude that the density
of the limiting random variable is
$ds(z)=\frac{\left(\frac{\sqrt{1-z}}{\sqrt{z}}\right)+\left(\frac{\sqrt{z}}{\sqrt{1-z}}\right)}{\pi}.$
In order to evaluate the required precision, let us assume that all the $D[i]$
are independent and identically distributed and follow the symmetrized
distribution $Q$ of density $ds(z)$ This statement is not strictly true;
however, it is asymptotically acceptable. This acceptability arises because
the dependency between $D[i]$ and $D[j]$ decreases exponentially fast as the
distance $|j-i|$ increases. This property follows directly from
$Q:=\sum_{n=0}^{\infty}(-1)^{n}\prod_{i=0}^{n}U_{i}^{2}$
and is really important because it implies that the errors do not accumulate.
Let $\delta$ denote the number of digits of precision (in base 2). Then, the
probability $P_{n,\delta}$ that there is no ambiguity in a calculation of the
alternating sequence of size $n$ with $\delta$ digits of precision is given
by:
$P_{n,\delta}:=n![z^{n}]\prod_{i=1}^{2^{\delta}-1}\left(1+z\frac{ds\left(\frac{i}{2^{\delta}}\right)}{\sum_{i=1}^{2^{\delta}-1}ds\left(\frac{i}{2^{\delta}}\right)}\right),$
Now, putting $N:=2^{\delta}$ and
$p_{i,n}:=\frac{ds\left(\frac{i}{N}\right)}{\sum_{i=1}^{N-1}ds\left(\frac{i}{N}\right)}$,
when $n$ is large, $\sum_{i=1}^{N-1}ds\left(\frac{i}{N}\right)/N$ tends to
$1$, and $\sum_{i=1}^{N-1}p_{i,n}^{2}$ is asymptotically equivalent to
$\frac{1}{n}\int_{1/n}^{1-1/n}ds(x)dx$, which simplifies to
$\sum_{i=1}^{N-1}p_{i,n}^{2}\sim\frac{2\ln(n)}{\pi^{2}n}$. Consequently,
$P_{n,\delta}$ is asymptotically equivalent to
$1-\frac{N(N-1)}{2}\frac{2\ln(n)}{\pi^{2}n}$. Indeed, the extraction $[z^{n}]$
is nothing more than the $PSet_{n}$ of the $B(\boldsymbol{p})=\sum p_{i,n}$
where the $p_{i,n}$ are seen as $n-1$ different atoms and it is well known
that $PSet_{n}$ can be expressed as a multivariate polynomial in the
$B(\boldsymbol{p}^{i})$. Moreover the two dominant contributions are
$\frac{B(\boldsymbol{p})^{n}}{n!}-\frac{B(\boldsymbol{p}^{2})B(\boldsymbol{p})^{n-2}}{2(n-2)!}$.
Finally, to achieve a probability of rejection $\varepsilon$, we require
$\delta=2\log_{2}(n)+\log_{2}(\ln(n))+\kappa+\log_{2}{\varepsilon}$. Thus, the
required precision is proportional (factor 2) to the entropy. It is worth
noting that improving the precision by one unit halves the risk of ambiguity.
This algorithm is highly efficient, but it cannot be extended to other types
of increasing trees. The following section introduces a more flexible
approach.
## 3 Toward an Efficient Recursive Methods Without Preliminary Calculations
We aim at generating a random strictly increasing binary tree with an exact
size of $n$. The generating function for our combinatorial family satisfies
the non-linear differential equation $T^{\prime}(z)=T^{2}(z)+1$ and let us
denote by $t_{n}$ the number of trees of size $n$. Utilizing the symbolic
method for combinatorial structures, we derive a recursive method algorithm
for our purpose. Our approach is now to avoid the exact and costly computation
of $t_{n}$ by a well-controlled approximation process, which paves the way for
a systematic and efficient algorithm to generate these structures, as shown in
Algorithm 2.
Algorithm 2 Generating $T$ of Size $n$ with the recursive method
1:If $n=1$ : Return the tree with one leaf labeled $1$
2:_Intelligently_ generate $M\in\\{0,1,..,n-1\\}$ the size of the left son of
$T$. Note that $P(M=m)$ has to be proportional to
$\binom{n-1}{m}t_{m}t_{n-1-m}$.
3:Generate recursively $T^{\prime}$ and $T^{\prime\prime}$ of size $m$ and
$n-1-m$ and return $T$ with $T^{\prime}$ as a left son and $T^{\prime\prime}$
as a right son with the indexes ”shuffled” among the $n-1$ remaining atoms.
There are a few problems with this generation scheme: How do we compute the
$t_{n}$’s? The classical way to do so is to pre-compute all of them but doing
so requires at least $\Omega(n^{2})$ in space and even more in time.
Generating $M$ can also be an issue, the classical way to generate $M$ would
require computing all the $t_{n}$’s and inverting the probability function
which can be ineffective.
We describe a way to compute only the necessary bits of precision of $t_{n}$
and a way to generate $M$ without the exact knowledge of $t_{n}$, solving the
two issues.
## 4 Generating $M$, the size of the left son
### 4.1 Having a direct expression for $t_{n}$
The differential equation $T^{\prime}(z)=T^{2}(z)+1$ can be integrated in
$T(z)=\tan(z)$ because $T(0)=0$ (there is no strictly increasing binary tree
of size $0$).
We can hence use the tangent development: $\tan(z)=\sum_{k\geq
1}\frac{4^{k}(4^{k}-1)|B_{2k}|}{(2k)!}z^{2k-1}$ with
$|B_{2k}|=\frac{2(2k)!\zeta(2k)}{(2\pi)^{2k}}$ being the absolute value of the
Bernoulli numbers, and $\zeta$ being the Riemann’s zeta function. So we can
extract the number of strictly increasing binary trees of size $n$. If $n$ is
even, $t_{n}=0$ because $\tan$ is odd, and for $n=2k-1$,
$t_{2k-1}=(2k-1)!\frac{2(4^{k}-1)\zeta(2k)}{\pi^{2k}}$.
In the following, we assume that $n=2l-1$ is odd since there is nothing to
generate if $n$ is even.
Since $P(M=m)$ is proportional to $\binom{n-1}{m}t_{m}t_{n-m-1}$, so
$P(M=m)=0$ when $m$ is even. So we can write, for $m=2k-1$, that $P(M=m)$ is
proportional to
$\binom{2l}{2k-1}\frac{(2k-1)!2(4^{k}-1)4^{k}\zeta(2k)}{(2\pi)^{2k}}\frac{(2(l-k)-1)!(4^{l-k}-1)4^{l-k}\zeta(2(l-k))}{(2\pi)^{2(l-k)}}$.
We can simplify this expression and get rid of the constant terms, we get that
$P(M=2k-1)$ is proportional to
$f_{k}:=(4^{k}-1)\zeta(2k)(4^{l-k}-1)\zeta(2(l-k))$, with
$k\in\\{1,2,..,l-1\\}$.
Since $\zeta(2k)=\sum_{i\geq 1}\frac{1}{i^{2k}}$, by truncating the sum,
$\zeta(2k)\geq 1+\frac{1}{4^{k}}$. This leads to the following bound on
$f_{k}$:
$f_{k}\geq(4^{k}-1)(1+\frac{1}{4^{k}})(4^{n-k}-1)(1+\frac{1}{4^{n-k}})$, i.e.
$f_{k}\geq(4^{k}-\frac{1}{4^{k}})(4^{n-k}-\frac{1}{4^{n-k}})$. With this lower
bound and the fact that $(f_{k})$ are symmetric decreasing from $1$ to
$\lfloor\frac{l}{2}\rfloor$, we can bound $f_{k}$.
###### Lemma 1.
For $k\in\\{1,2,..,l-1\\}$, $f_{1}\geq f_{k}\geq
f_{\lfloor\frac{l}{2}\rfloor}$ and
$\frac{f_{1}}{f_{\lfloor\frac{l}{2}\rfloor}}\leq\frac{4\pi^{4}}{45}\leq 9$
This lemma implies a Monte-Carlo method for sampling $M$.
Algorithm 3 Generate $M$ such that $P(M=2k-1)$ is proportional to $f_{k}$
1:Generate $X\in\\{1,2,..,l-1\\}$.
2:Test $Uf_{1}\leq f_{X}$ where $U$ is a uniform random number in $[0,1]$, if
true return $M=X$, else generate another $X$ and start again.
Lemma 1 implies that Algorithm 3 has a constant reject.
###### Lemma 2.
Algorithm 3 has a running time of $O(\log(n))$
This lemma is not only due to the fact that the reject is constant but also to
the fact that it is possible to compute only the first bits of $f_{1}$,
$f_{k}$, and $U$ and check whether we can conclude that $Uf_{1}\leq f_{X}$ or
$Uf_{1}>f_{X}$. If we cannot conclude, we can add the next bits and recheck
until we can accept or reject $X$ as $M$.
It is possible to compute only the first bits of
$f_{k}=(4^{k}-1)\zeta(2k)(4^{l-k}-1)\zeta(2(l-k))$ because we can truncate
$\zeta(2k)=\sum_{i\geq 1}\frac{1}{i^{2k}}$ to a finite order.
With this method of generating $M$, we can state the complexity of Algorithm
2.
###### Theorem 1.
Algorithm 2 has a running time of $O(n\log(n))$, which is optimal.
The algorithm is optimal because of the Shannon entropic principle: It is not
possible to generate a strictly increasing binary tree of size $n$ ($n$ odd)
with less than $\log_{2}(t_{n})$ random bits, so the best sampling algorithm
will run in at least $\log_{2}(t_{n})=\Omega(n\log(n))$
## 5 Adapting the Algorithm for Different Families of Binary-Increasing Trees
### 5.1 Binary Trees
Consider various families of increasing binary trees, each characterized by
distinct generating functions; for instance, non-plane binary trees have
$T^{\prime}(z)=\frac{T^{2}(z)}{2}+1$, leading to
$T(z)=\frac{\tan(z/\sqrt{2})}{\sqrt{2}}$. For all families of binary-
increasing trees with generating functions of the form $T^{\prime}=aT^{2}+c$
as described in Algorithm 2, the algorithm remains applicable. This is due to
the consistent structure shared by these trees, characterized by having one
more leaf than internal nodes.
### 5.2 Binary-Unary Trees
For a family with generating function
$T^{\prime}(z)=a(T(z)+\alpha)^{2}+a\gamma^{2}$, where $a,c>0$ and $b\geq 0$,
we focus on those written as $T^{\prime}(z)=a(T(z)+\alpha)^{2}+a\gamma^{2}$,
the case $T^{\prime}(z)=a(T(z)+\alpha)^{2}-a\gamma^{2}$ being similar. The
algorithm generates trees of size $n$ with
$t_{n}\sim\frac{n!a^{n}\gamma^{n+1}}{(\frac{\pi}{2}-\tan^{-1}(\frac{\alpha}{\gamma}))^{n+1}}$,
allowing effective recursive generation. And
$t_{n}=n!a^{n}\gamma^{n+1}\left(\sum_{k\geq
0}\frac{(-1)^{n+1}}{(\tan^{-1}(\frac{\alpha}{\gamma})-\pi(k+\frac{1}{2}))^{n+1}}+\frac{(-1)^{n+1}}{(\tan^{-1}(\frac{\alpha}{\gamma})+\pi(k+\frac{1}{2}))^{n+1}}\right)$
is still computable by truncating its expression.
Algorithm 4 Generating $T$ of Size $n$ for Binary-Unary Trees
1:If $n=1$ : Return the tree with one leaf colored in one of the $c$ colors.
2:For $n>1$ : The tree can have a unary root with probability
$\frac{bt_{n-1}}{t_{n}}$ or a binary root.
3:Let $X$ be a Bernoulli variable with parameter $\frac{bt_{n-1}}{t_{n}}$; if
$X=1$, call the algorithm recursively with parameter $n-1$ and return the tree
consisting of a unary root and labeled $1$ with $T^{\prime}$ as its only son.
4:Generate $M\in\\{0,1,..,n-1\\}$ with $P(M=m)$ proportional to
$\binom{n-1}{m}t_{m}t_{n-1-m}$.
5:Generate recursively $T^{\prime}$ and $T^{\prime\prime}$ of size $m$ and
$n-1-m$ and return $T$ with $T^{\prime}$ as a left son and $T^{\prime\prime}$
as a right son with the indexes ”shuffled” among the $n-1$ remaining axioms.
## References
* [1]
* [2] Axel Bacher, Olivier Bodini & Alice Jacquot (2017): _Efficient random sampling of binary and unary-binary trees via holonomic equations_. Theor. Comput. Sci. 695, pp. 42–53, 10.1016/J.TCS.2017.07.009.
* [3] Philippe Duchon, Philippe Flajolet, Guy Louchard & Gilles Schaeffer (2004): _Boltzmann Samplers for the Random Generation of Combinatorial Structures_. Combinatorics, Probability and Computing 13(4–5), p. 577–625, 10.1017/S0963548304006315.
* [4] Ph. Marchal (2012): _Generating random alternating permutations in time $n\log n$_. Available at https://www.math.univ-paris13.fr/~marchal/altperm1.pdf. Unpublished note.
|
An Adaptive Dual-level Reinforcement Learning Approach for Optimal Trade
Execution
Soohan Kimb (zeanvszed@g.skku.edu), Jimyeong Kimb(jimkim@g.skku.edu), Hong Kee
Sula (hksul@cau.ac.kr), Youngjoon Hongb<EMAIL_ADDRESS>
a Department of Finance, Chung-Ang University, Seoul, Republic of Korea.
b Department of Mathematics, Sungkyunkwan University, Suwon, Republic of Korea
Corresponding Author:
Youngjoon Hong
Department of Mathematics, Sungkyunkwan University, Suwon, Republic of Korea
Email<EMAIL_ADDRESS>
Hong Kee Sul
Department of Finance, Chung-Ang University, Seoul, Republic of Korea.
Email<EMAIL_ADDRESS>
###### Abstract
The purpose of this research is to devise a tactic that can closely track the
daily cumulative volume-weighted average price (VWAP) using reinforcement
learning. Previous studies often choose a relatively short trading horizon to
implement their models, making it difficult to accurately track the daily
cumulative VWAP since the variations of financial data are often insignificant
within the short trading horizon. In this paper, we aim to develop a strategy
that can accurately track the daily cumulative VWAP while minimizing the
deviation from the VWAP. We propose a method that leverages the U-shaped
pattern of intraday stock trade volumes and use Proximal Policy Optimization
(PPO) as the learning algorithm. Our method follows a dual-level approach: a
Transformer model that captures the overall(global) distribution of daily
volumes in a U-shape, and a LSTM model that handles the distribution of orders
within smaller(local) time intervals. The results from our experiments suggest
that this dual-level architecture improves the accuracy of approximating the
cumulative VWAP, when compared to previous reinforcement learning-based
models.
###### keywords:
Volume-Weighted Average Price , Reinforcement Learning , Optimal Trading
Execution , Proximal Policy Optimization , Markov Decision Process
††journal: Expert Systems with Applications
## 1 Introduction
The optimal trade execution problem aims to find a strategy to optimally trade
large orders within a given period of time. One of the most common and
practical methods that practitioners frequently use is known as Volume
Weighted Average Price (VWAP) trading. VWAP is calculated by adding up the
dollars traded for every transaction (price multiplied by the number of shares
traded) and then dividing by the total shares traded for the day. This gives
an average price that takes into account both the price and the volume of
shares traded. Funds and traders often use VWAP as a benchmark to compare the
price at which they executed trades to the overall market price for the
security, in order to evaluate the performance of their trades.
Recently, the use of Reinforcement Learning (RL) in optimal trade execution
has become popular as it is a type of stochastic decision-making process that
automates the practitioner’s task of using past data to make decisions on when
to execute orders. Since RL approaches have many advantages, e.g. capable of
capturing the market’s microstructure, the trading results conducted by RL
solutions often outperform traditional VWAP tracking approaches such as
Bialkowski et al. (2008) and Podobnik et al. (2009).
This paper aims to develop a strategy that can consistently track the daily
cumulative VWAP. To achieve this goal, we start with the observation of an
important stock trading characteristic, the U-shaped intraday trading pattern,
documented in Jain & Joh (1988) and Goodhart & O’Hara (1997). This is a well-
known stylized fact stating that a stock’s trading volume often follows a
U-shaped pattern throughout the day. This means that volume is highest at the
opening, falls rapidly to lower levels, and then rises again towards the close
of the market, but is relatively low in the afternoon. (See Figure 1).
Figure 1: Trade volume ratio over each 20-minute period throughout the day.
The lines represent the 1-year averages of the ratios and the shaded regions
are drawn from daily deviations from the averages.
In view of this observation, we propose a novel dual-level approach to
minimize the market impact and track the daily cumulative VWAP accurately and
consistently. As the name suggests, our model consists of two stages. In the
first stage, the model decides how much trade volume will be executed in each
interval, using the U-shape property as a guideline. In the second stage, the
RL model decides how the orders will be executed in each interval. We propose
two methods for the first stage, the statistical U-shape method and the
U-shape Transformer method. The statistical U-shape method allocates the
volume based on the historical average trade volumes. However, this approach
cannot fully capture the day-to-day variations (See Figure 1). Alternatively,
we propose the U-shape Transformer for figuring out these variations. In the
next stage, we apply LSTMs and PPO to distribute orders efficiently within
each interval.
We expect that this dual-level approach improves the accuracy of approximating
the cumulative VWAP by properly distributing the total order. Our
contributions are as follows:
1. 1.
We propose a novel approach for optimal trade execution that utilizes a dual-
level strategy. Orders are allocated through two stages: in the first stage we
utilize the U-shaped pattern of intraday volumes and in the second stage we
implement deep reinforcement learning.
2. 2.
We further propose the U-shape Transformer model to capture the day-to-day
oscillations of the intraday U-shape distribution.
3. 3.
We conduct ablation studies to show the strengths of our methods in
approximating the daily cumulative VWAP and reveal that utilizing deep
reinforcement learning for short trading horizons is ineffective.
## 2 Backgrounds
In this section, we discuss the basics of the limit order book (LOB) and the
optimal trade execution process. Most modern financial exchanges, including
the Korea Stock Exchange(KRX), offer electronic trading platforms with access
to limit order books. Traders often rely on the information provided by the
limit order book to develop successful trading strategies.
### 2.1 Limit Order Book
In financial markets, a limit order is an order to buy or sell a fixed number
of shares at a specified price. This order becomes part of the limit order
book, which records all limit orders for a specific stock. The bid price is
the highest price a buyer is willing to pay for the security, and the ask
price is the lowest price in which the seller is willing to sell. A market
order, by contrast, is an order to buy or sell a fixed number of shares at the
current market price without stating the price. When a market order is placed,
it is matched by the best available limit orders in the LOB. The LOB is
important for optimal trade execution and helps traders track supply and
demand for a stock, making it easier to identify the best time and price to
buy or sell shares.
### 2.2 Optimal Trade Execution
Optimal trade execution refers to the process of buying or selling a financial
asset while achieving the desired objective. Generally, this problem
formulates as follows: Within a timeframe of $T$ timesteps,
${0,1,\cdots,T-1}$, a trader who possesses an inventory of $O$ shares must buy
or sell the entirety of the inventory. For each timestep
$t\in\\{0,1,\cdots,T-1\\}$, the trader determines the number of shares to
order $O_{t}$ based on the information from LOB, and carries out the trade at
an execution price $p_{t}$. For the group of traders seeking to maximize
trading profit, their objective is to find a strategy that maximizes or
minimizes the average execution price $\bar{P}$, which is calculated as
$\bar{P}=\sum^{T-1}_{t=0}\frac{O_{t}}{O}p_{t}.$
While many financial firms pursue profits through trading, there are also
numerous firms within the financial industry that prioritize consistently
tracking the VWAP over maximizing trading profit. The focus of our paper is
tailored for this second group of traders. The VWAP is calculated as
$\text{VWAP}=\frac{\sum^{T-1}_{i=0}p_{t}q_{t}}{\sum^{T-1}_{i=0}q_{t}},$
where $q_{t}$ is the execution volume determined by the market. Therefore, the
primary objective of this paper is to minimize the difference between the VWAP
and $\bar{P}$, using a dual-level approach.
## 3 Related Works
### 3.1 Optimal Execution without RL
In Bertsimas & Lo (1998), a fundamental work for optimal trade execution is
proposed, where the authors assume that market prices follow an arithmetic
random walk. They use a dynamic programming principle to find an explicit
closed-form solution. Building on this work, Huberman & Stanzl (2005) and
Almgren & Chriss (2001) extended the result in Bertsimas & Lo (1998) by
incorporating transaction costs, more complex price impact functions, and risk
aversion parameters, under the assumption that market prices follow a Brownian
motion. These dynamical approaches, however, are difficult to apply directly
in the real world due to the discrepancy between their strong market
assumptions and the complexity of actual market conditions.
On the other hand, both practitioners and researchers have widely used the
time-weighted average price (TWAP) strategy and the volume-weighted average
price (VWAP) strategy (Berkowitz et al. (1988); Kakade et al. (2004)), which
are based on either pure rules or statistical rules. Especially in Kakade et
al. (2004), the authors used historical data to estimate the average volume
traded for each time interval and split the order accordingly. However, this
strategy is not well-suited for capturing unexpected volatility.
### 3.2 Optimal Execution with RL
From the perspective that optimal execution problems are a type of sequential
decision-making task, Reinforcement Learning (RL) has been commonly applied in
this field. To the best of our knowledge, Nevmyvaka et al. (2006) is the first
work to leverage RL frameworks such as Q-learning Watkins & Dayan (1992) to
optimal trade execution problems. It is important to note that the curse of
dimensionality in Q-learning makes it challenging to handle high-dimensional
data. While Hendricks & Wilcox (2014) attempts to combine RL with the Almgren-
Chriss model, it is a challenging task as the model depends on certain
assumptions about the market dynamics.
Thanks to the advancements in deep RL, recent studies such as Ning et al.
(2021) and Lin & Beling (2019) have utilized Deep Q-Networks (DQNs) Mnih et
al. (2013) for optimal trade execution, addressing the challenges of high-
dimensional data and the complexity of the financial market without relying on
any market assumptions. However, these methods require the design of specific
attributes, which can be labor-intensive. Recently, several studies have
investigated the use of proximal policy optimization (PPO) Schulman et al.
(2017) based optimal execution frameworks, which do not require manually
designed feature engineering. For instance, Lin & Beling (2020), Fang et al.
(2021) and Pan et al. (2022) have explored this approach in different
scenarios. Specifically, Pan et al. (2022) focused on optimal execution with
limit orders, while Lin & Beling (2020) and Fang et al. (2021) used market
orders.
Prior literature often considers a relatively short trading horizon or the
entire day for RL implementation. However, VWAP approximation becomes
challenging when a short time frame is used since the variations of financial
data are often insignificant within short time intervals. Moreover, the naive
approach of considering the entire day can put a strain on the network
architecture due to the long sequence length. To address these issues, we
propose a dual-level approach method to enhance the accuracy of approximating
the daily cumulative VWAP. In our approach, we make use of the fact that
practitioners usually utilize the statistical U-shape property to divide the
total order for the day in order to track the daily cumulative VWAP.
## 4 Method Description
In this section, we describe our dual-level approach and formulation of the
optimal trade execution problem under the lens of reinforcement learning. We
also provide details on the architecture of the Transformer and LSTM neural
networks and how they are integrated into our proposed method.
Figure 2: A schematic illustration of our dual-level approach.
### 4.1 Dual-level Approach
Our approach involves two stages of distributing orders. In the first stage,
we allocate the total daily orders $O$ into $L$ intervals. In the second
stage, we further divide the orders from each interval into smaller executable
orders using RL. Our approach begins by dividing the market hours of a day
into $L$ intervals. For each interval $l$, we let $O^{l}$ as the total order
to be executed, ensuring that the sum of all interval orders equals the total
daily order, or $\sum^{L-1}_{l=0}O^{l}=O$. We suggest two methods for
implementing the first stage. The naive approach is to adhere to the
statistical U-shape trade volume distribution for each interval obtained from
past data (e.g. the historical average of the past year). A more advanced
method would be to use the output (predicted U-shape distribution) generated
by a U-shape Transformer model. In the latter case, $O^{l}$ is calculated
progressively by referring only to market information from past days and
previous intervals within the day. For such an $l^{th}$ interval, which is
comprised of $T$ timesteps, the LSTM model outputs $O^{l}_{t}$, which is the
number of orders to be executed at timestep $t$ such that
$\sum^{T-1}_{t=0}O^{l}_{t}=O^{l}$. Once $O^{l}_{t}$ is decided, it is averaged
over $I$ execution steps in the corresponding $t^{th}$ subinterval. In other
words, for each execution step, which is set as every 5 seconds, the final
number of orders executed is $O^{l}_{t}/I$. The choice of averaging is
supported by our findings that there is no significant difference in
approximating the VWAP, regardless of how orders are distributed within the
subintervals, as detailed in Section 5.2. A schematic illustration of this
design is provided in Figure 2.
### 4.2 MDP Formulation for Optimal Execution
In this section, we explain our Markov Decision Process (MDP) formulation of
optimal trade execution. A MDP is typically represented using the tuple
($\mathcal{S}$, $\mathcal{A}$, $\mathcal{P}$, $r$, $\gamma$), where
$\mathcal{S}$ is the state space, $\mathcal{A}$ is the action space,
$\mathcal{P}$ is the transition probability, $r$ is the reward function, and
$\gamma$ is the discount factor. Our configurations of the state space
$\mathcal{S}$ and action space $\mathcal{A}$ are similar to that of Lin &
Beling (2020). It’s worth noting that the MDP implementation in this paper is
applied on a per $l^{th}$ interval basis for a given day.
#### 4.2.1 State
The state $s^{l}_{t}\in\mathcal{S}$ consists of both public and private
information. The public state is made up of the top 5 bid and ask prices,
along with their associated volumes, while the private state includes the
elapsed time and the current remaining volume to be executed. The elapsed time
ranges from $0$ to $T-1$ and the current remaining volume at timestep $t$ is
equal to $O^{l}-\sum^{t-1}_{j=0}O^{l}_{j}$.
#### 4.2.2 Action
Our LSTM model outputs a policy $\pi(\cdot|s^{l}_{t})\in\mathbb{R}^{21}$,
which is a discrete categorical probability distribution. From this, an action
$a^{l}_{t}\in\mathcal{A}:=\\{0,0.1,0.2,\cdots,2\\}$ is sampled to determine
the number of orders to execute, with $O^{l}_{t}=a^{l}_{t}\frac{O^{l}}{T}$. To
ensure that the total number of orders executed over the entire period is
equal to $O^{l}$, that is $\sum^{T-1}_{t=0}O^{l}_{t}=O^{l}$, we impose the
following two restrictions.
* 1.
If $\sum^{j}_{t=0}O^{l}_{t}>O^{l}$ for some $j\in\\{0,1,2,\cdots,T-1\\}$, then
$O^{l}_{j}=O^{l}-\sum^{j-1}_{t=0}O^{l}_{t}$
* 2.
$O^{l}_{T-1}$ is equal to the remaining volume to be executed at the last
timestep for each interval (i.e. $0\leq l\leq L-1$)
These restrictions are in line with the fact that it is essential for brokers
to promptly acquire or liquidate all shares requested by their customers.
#### 4.2.3 Reward
The reward function is designed to encourage the agent (LSTM model) to
generate actions that result in orders that are close to the target volume
$O_{t}^{l^{*}}$, in order to track the VWAP. This is achieved by using the
price calculated from the executed orders. Here, $O_{t}^{l^{*}}$ denotes the
desired VWAP order at timestep $t$ in the $l^{th}$ interval such that
$O^{l}=\sum^{T-1}_{t=0}O^{l^{*}}_{t}$. The daily VWAP can be obtained by
$VWAP_{day}=\sum^{L-1}_{l=0}\sum^{T-1}_{t=0}\frac{O^{l^{*}}_{t}}{O}p^{l}_{t},$
(1)
where $p^{l}_{t}$ is the associated average market traded price of the
corresponding $t^{th}$ subinterval.
Our reward function compares $O_{t}^{l^{*}}$ and $O^{l}_{t}$ directly for a
given $a^{l}_{t}$ as follows:
$r(a^{l}_{t}):=\left\\{\begin{array}[]{ll}1&M^{l}_{t}<0.01\\\ 0&0.01\leq
M^{l}_{t}<0.05\\\ -1&\text{otherwise}\end{array}\right.$ (2)
where $M^{l}_{t}:=\frac{|O^{l}_{t}-O^{l^{*}}_{t}|}{O^{l^{*}}_{t}}$.
Once MDP is determined, the goal of policy-based RL is to find an optimal
policy parameter $\theta^{*}$, i.e.
$\theta^{*}=\arg\max\mathbb{E}\left[\sum^{T}_{t=0}\gamma^{t}r_{t}|a_{t}\sim\pi_{\theta}(\cdot|s_{t}),s_{t+1}\sim\mathcal{P}(\cdot|s_{t},a_{t})\right],$
(3)
where $r_{t}=r(a_{t})$.
### 4.3 Neural Network Architecture and Training
We now turn to the architectural details of the Transformer and LSTM models
and elaborate on how they are used in our dual-level approach.
#### 4.3.1 Level 1: Transformer for Global U-shape Approximation
The objective of the Transformer model is to progressively predict the U-shape
ratio for each interval in a given day. We use the structure of the
Transformer Encoder and Decoder as proposed in Vaswani et al. (2017). The
overall architecture of the Transformer model that we use is depicted in
Figure 3.
U-shape Encoder. The U-shape Encoder $\mathcal{E}$ is composed of a
Transformer Encoder and $L$ linear layers, all of which map to the same
dimension. This module is responsible for learning the general dynamics of the
U-shape distribution. The input $E_{in}$ is a sequence of length $L$ of
vectors each containing $N$ historical daily volume ratios of the
corresponding interval in the sequence. The days from which these ratios are
drawn are randomly selected. Each vector in the sequence is processed by a
different linear layer that serves as an embedding to learn the unique
features of each interval. After passing through the embedding layer, the
input sequence is then processed by the Transformer Encoder, resulting in the
output $E_{out}$.
Figure 3: The architecture of the U-shape Transformer. The U-shape Decoder in
this figure represents the $l^{th}$ decoding step.
U-shape Decoder. While $\mathcal{E}$ learns the general dynamics of the
U-shape distribution, the U-shape Decoder $\mathcal{D}$ focuses on handling
the day-to-day variations of the distribution. The U-shape Decoder is composed
of a Transformer Decoder and $L$ linear layers, each mapping to different
dimensions. In contrast to $\mathcal{E}$, these linear layers are applied to
the output of the Transformer Decoder.
For every $l^{th}$ $(l>0)$ decoding step, both $E_{out}$ and the cumulative
input sequence are fed as input to the Transformer Decoder, $\mathcal{D}$. We
denote the cumulative input sequence as $D^{l}_{in}=(D^{l}_{in,j})_{0\leq
j\leq l-1}$, where $D^{l}_{in,j}:=\text{CONCAT}(D^{j}_{out},h^{j}_{T-1})$.
Here, $D^{j}_{out}$ represents the output of $\mathcal{D}$ at the $j^{th}$
step, and $h^{j}_{T-1}$ is the last LSTM hidden vector from the $j^{th}$
interval. The output from the Transformer Decoder at the $l^{th}$ decoding
step is a vector of dimension $L+H$, which is then passed through a linear
layer that reduces its dimension to $L-l$. The softmax function is applied to
predict the ratios of the remaining $L-l$ intervals, and the first $l$
components are zero-padded. Note that we use $h^{j}_{T-1}$ to capture the day-
to-day variations in the U-shape distribution, as it holds interval-specific
information for that particular day. To ensure
$\sum^{L-1}_{l=0}u^{l}_{pred}=1$, the final $l^{th}$ interval volume ratio
prediction is calculated as
$u^{l}_{pred}=\left(1-\sum^{l-1}_{j=0}u^{j}_{pred}\right)\cdot
D^{l}_{out}[l].$ (4)
It’s worth noting that $D^{0}_{in}$ is constructed by applying an additional
linear layer that transforms the dimension of the vector holding the top 5
bid/ask volume averages of pre-market data to $L+H$.
Figure 4: The architecture of the LSTM model.
#### 4.3.2 Level 2: LSTM for Local Order Distribution
The objective of the LSTM model is to optimally allocate orders within the
intervals of a given day. The structure of our LSTM model, which is outlined
in Figure 4, is similar to that of Lin & Beling (2020). Given $s^{l}_{t}$ as
input, it first passes through two linear layers with hidden dimension $128$
and is then concatenated with $a^{l}_{t-1}$ and $r^{l}_{t-1}:=r(a^{l}_{t-1})$.
The resulting vector is then processed by an LSTM Cell with hidden dimension
$H$ to obtain $h^{l}_{t}$, which is then passed separately through two linear
layers to produce the policy $\pi(\cdot|s^{l}_{t})$ and the value
$V(s^{l}_{t})$ respectively. We note that the vector $h^{l}_{T-1}$ which is
obtained at the last timestep is used as input to the Transformer Decoder. The
action $a^{l}_{t}$, reward $r^{l}_{t}$, and the number of orders to execute
$O^{l}_{t}$ is calculated as is described in Section 4.2. The LSTM Cell also
receives $h^{l}_{t-1}$ and $c^{l}_{t-1}$ as input and its output includes
$c^{l}_{t}$ as well.
#### 4.3.3 Training Procedure
U-shape Transformer. Since the goal of the first level of order allocation is
to approximate the U-shape distribution while also tracking the daily
cumulative VWAP, the training objective of the Transformer model is to
minimize
$J_{TF}:=\mathbb{E}_{days}\left[\frac{c_{1}}{L}\sum^{L-1}_{l=0}\left(u^{l}_{true}-u^{l}_{pred}\right)^{2}+c_{2}VAA\right]$
(5)
where $c_{1}$ and $c_{2}$ are coefficients and
$\text{VAA}:=\left|\frac{MP_{day}-VWAP_{day}}{VWAP_{day}}\right|$
denotes the VWAP Approximation Accuracy (VAA), which we will further use as a
general metric for performance evaluation in experiments, and
$MP_{day}:=\sum^{L-1}_{l=0}\sum^{T-1}_{t=0}\frac{O^{l}_{t}}{O}p^{l}_{t}$
is the price yielded by our model. This is one of the measures used in
practice to gauge how close a trader has traded closer to their benchmark, the
VWAP. If a trader has traded at a price that is exactly equal to the VWAP,
then $VAA=0$. Thus lower VAA would be better. The first term inside the
expectation of (5) learns to make ratio predictions closer to ground-truth
values. The second term penalizes the Transformer model if the final daily
acquisition price generated by the combination of the Transformer and LSTM
models deviates from the daily VWAP. The primary objective of the U-shaped
transformer in the first level is to distribute the daily total order with
precision. However, relying solely on the first term of the loss function may
not be enough to accomplish this objective. This is because $u^{l}_{true}$
does not consider the local information, such as the LOB, which can pose
difficulties in tracking the daily VWAP accurately. Furthermore, if only the
first term in (5) is taken into account, the reinforcement learning framework
would not influence the first level, leading to independent operation of the
first and second levels. To ensure that the micro information has an
appropriate impact on the first level, we incorporate the second term into the
loss function.
LSTM. To find the optimal parameter $\theta$ for the LSTM model, we use an
actor-critic style PPO algorithm Schulman et al. (2017), which is one of the
most popular on-policy RL algorithms, as our base learner. One of the major
advantages of the PPO in this task is its ability to adapt to changing market
conditions and effectively learn from noisy and high-dimensional data. This
algorithm uses the actor loss $J_{\text{PPO}}^{CLIP}(\theta)$ and the critic
loss function $J_{\text{PPO}}^{VF}(\theta)$, which are defined as follows:
$\begin{split}&J_{\text{PPO}}^{CLIP}(\theta)\\\
&:=\mathbb{E}_{t}\left[\min\left(q_{t}(\theta)\hat{A}_{t},\text{clip}(q_{t}(\theta),1-\varepsilon,1+\varepsilon)\hat{A}_{t}\right)\right],\end{split}$
(6)
and
$J_{\text{PPO}}^{VF}(\theta)=\mathbb{E}_{t}\left[(V^{\text{targ}}_{t}-V_{\theta}(s_{t}))^{2}\right],$
(7)
where
$q_{t}(\theta)=\frac{\pi_{\theta}(a_{t}|s_{t})}{\pi_{\theta_{\text{old}}}(a_{t}|s_{t})}$,
$V^{\text{targ}}_{t}=\sum_{k=t}^{T-1}\gamma^{k-t}r_{k}\ \text{and}\
\hat{A}_{t}=V^{\text{targ}}_{t}-V_{\theta}(s_{t}).$ (8)
PPO’s goal is to find a parameter $\theta$ that maximizes the main objective
function $J_{\text{PPO}}(\theta)$, which is defined by
$J_{\text{PPO}}(\theta)=J_{\text{PPO}}^{CLIP}(\theta)-c_{3}J_{\text{PPO}}^{VF}(\theta)+c_{4}\mathbb{E}_{t}[S[\pi_{\theta}](s_{t})],$
(9)
where $c_{3},c_{4}$ are coefficients, and
$\mathbb{E}_{t}[S[\pi_{\theta}](s_{t})]$ denotes an entropy loss term, where
$S[\pi_{\theta}](s_{t})$ is defined by
$S[\pi_{\theta}](s_{t}):=-\sum_{a\in\mathcal{A}}\pi_{\theta}(a|s_{t})\log{\pi_{\theta}(a|s_{t})}.$
(10)
By incorporating an entropy term, PPO can incentivize the agent to explore
alternative actions, prevent getting trapped in suboptimal policies, and
mitigate overfitting to the training data (Williams, 1992; Mnih et al., 2016).
To find the parameter $\theta$, PPO iteratively gathers episodes and updates
the parameter $\theta$ using the following scheme:
$\theta_{k+1}=\arg\max_{\theta}\mathbb{E}_{s}\mathbb{E}_{a\sim\pi_{\theta_{k}}(\cdot|s)}[J_{\text{PPO}}(\theta)],$
(11)
where $k$ stands for the $k^{\text{th}}$ step. The “clip” operator in (6)
allows PPO to learn from previous experiences without becoming overly
dependent on them. Moreover, PPO can effectively retain knowledge from past
experiences while also acquiring insights from new experiences. This key
attribute of PPO is one of its significant advantages.
## 5 Experiments
Our dual-level approach significantly enhances the accuracy and consistency of
tracking the daily cumulative VWAP, despite the challenges in improving the
performance of RL frameworks for optimal trade execution in relatively short
trading horizons. Furthermore, the U-shape Transformer proves efficient in
capturing the daily variations of the U-shape distribution. In the following
section, we conduct a series of experiments to verify these assertions.
### 5.1 Experiment Settings
#### 5.1.1 Implementation
In all of our experiments, the hyperparameters for the loss functions, the
length of the intervals, the number of total orders, and the trading horizon
are fixed and set according to the following specifications:
1. -
The coefficients for $J_{\text{TF}}$: $c_{1}=0.5,c_{2}=0.5$
2. -
The coefficients for $J_{\text{PPO}}$: $c_{3}=1,c_{4}=0.01$
3. -
The length of the intervals: $L=19$, $T=20$
4. -
The number of total orders: $O$ $\sim$ $\mathcal{N}(2.5\times
10^{-3}\mu,6.25\times 10^{-6}\sigma^{2}$), where $\mu$ and $\sigma$ is the
average and standard deviation of the day total volume for the previous sixty
days, respectively.
5. -
Trading horizon of a day: $380$ minutes (From when the market opens at
09:00:00 to when it closes at 15:20:00).
Since a typical day allows for 380 minutes of trading, our model first takes
the daily total order, $O$, and breaks it down to $O^{l}$, 19 intervals of 20
minutes each. ($l\in[0,18]$) We either apply the statistical U-shape method or
the Transformer model for this first stage allocation process. Then, within
each interval, the LSTM model further subdivides $O^{l}$ into 20 sets of
1-minute subintervals, $O^{l}_{t}$. ($t\in[0,19]$)
In each of these subintervals, $O^{l}_{t}$ is divided into equal portions and
executed over a span of 12 steps, assuming that orders are executed every 5
seconds. The method and hyperparameters used to train our agents for the
experiment are summarized in Algorithms 1, 2 and Table 3, which are located at
the end of the paper.
To conduct realistic simulations, we determine $O$ in a way that takes into
account the volatility of daily volume, which differs from most previous works
where it is fixed to a certain value. This choice aims to simulate the
fluctuations of total orders that financial firms have to execute in a day,
taking into consideration recent trade volume statistics for each stock. By
considering the volume statistics for each stock, we account for the
differences in total orders for various stocks, which is often encountered in
real-world scenarios. Additionally, during testing, we only use the daily
ground truth U-shape volume ratios of dates in the training data for our
Transformer Encoder to prevent looking ahead into the future. As in previous
works Nevmyvaka et al. (2006), Hendricks & Wilcox (2014), Ning et al. (2021),
and Lin & Beling (2020), we make the following assumptions in our experiments.
1. 1.
We assume that the actions taken by our model only affect a temporary market,
and that the market will recover to the equilibrium level at the next time
step.
2. 2.
Commissions and exchange fees are ignored.
3. 3.
Our model’s orders are traded immediately without order arrival delays.
We believe that the above assumptions are reasonable as we consider relatively
small total orders in comparison to the daily total market volumes of the
stocks.
#### 5.1.2 Datasets
Our millisecond trade and limit order book (LOB) tick data is from the Korea
Exchange (KRX). We use the daily trade and LOB data of Samsung Electronics
(SE), SK Hynix (SH), Kia Corporation (KC) and POSCO Holdings Inc (PH) from
January 1st, 2021 to December 31st, 2021. Our choice of the four firms in the
KOSPI index is based on their liquidity and trade volume and variety in market
capitalization and industry. In order to evaluate the effectiveness of our
approach in relation to trading volume, we have opted to select stocks with a
range of trade volumes (see Table 1).
Stocks | SE | SH | KC | PH
---|---|---|---|---
Volume | 17,000K | 3,800K | 3,900K | 430K
Table 1: Average volume for each stock.
We divide the data into two sets, using January 1st to September 30th as
training data and October 1st to December 31st as test data.
The millisecond raw data is preprocessed by first dividing them into groups of
5 seconds, and extracting data that represents each 5-second interval. We take
the last LOB data of the 5-second interval to construct the MDP state and the
5-second VWAP and total traded volume for calculating the daily VWAP. The
statistical U-shape statistics are constructed by taking the year average of
the U-shape ratios of the 20-minute intervals in the day. The data utilized in
generating $D^{0}_{in}$ shown in Section 4.3 are the top 5 bid/ask volume
averages of the raw pre-market data (from 08:30:00 to 09:00:00).
#### 5.1.3 Ablation Study
We provide a brief explanation of all methods used for performance comparison
during our experiments here. The hyperparameters for each method can be found
in Table 3.
Stocks | Metric | OPD | DQN | PPO | HUL | TUL
---|---|---|---|---|---|---
SE | Mean | 26.27 | 11.53 | 12.00 | 8.77 | 7.13
Standard Deviation | 18.50 | 8.84 | 8.88 | 7.17 | 2.44
% in 10bps | 25.00 | 57.38 | 49.18 | 65.57 | 88.52
SH | Mean | 38.55 | 21.62 | 22.57 | 7.13 | 7.65
Standard Deviation | 24.90 | 17.71 | 16.92 | 2.81 | 3.63
% in 10bps | 10.42 | 30.00 | 25.00 | 91.67 | 88.33
KA | Mean | 27.79 | 13.16 | 12.47 | 6.59 | 6.03
Standard Deviation | 18.76 | 10.25 | 11.55 | 8.36 | 3.30
% in 10bps | 18.75 | 46.67 | 56.67 | 90.00 | 86.67
PH | Mean | 32.44 | 14.84 | 14.64 | 22.52 | 12.27
Standard Deviation | 20.89 | 13.47 | 12.63 | 47.11 | 7.90
% in 10bps | 18.75 | 40.98 | 40.98 | 32.79 | 44.26
Table 2: Table 2 summarizes the mean, standard deviation, and percentage of
VAAs within the range [0, 10bps]. The Mean and Standard Deviation are
represented in units of basis points (bps), and the $\%$ in 10 bps are
represented in units of percentages. For OPD, we excluded VAAs above the 80th
percentile for fair comparisons since our training dataset was smaller than
that used in Fang et al. (2021).
1. -
DQN is the method proposed in Ning et al. (2021). To apply this model, we
begin by dividing a given day into six 1-hour intervals, excluding the final
20 minutes of market hours to replicate the experimental setup in Ning et al.
(2021). The total order for the day is then equally distributed amongst each
hour, and DQN further allocates the order for five sub-intervals of 12 minutes
each. The orders are executed equally for each 5-second time step within the
12-minute sub-intervals. To ensure fair comparisons, we evaluate the daily
price generated by DQN against the daily cumulative VWAP calculated by
excluding the final 20 minutes.
2. -
OPD is the method proposed in Fang et al. (2021). To implement this model, we
first divide a given day into ten intervals, each spanning 38 minutes, and
then use OPD to allocate orders for each interval. Within each interval, the
allocated order is equally distributed amongst each minute.
3. -
PPO is the method proposed in Lin & Beling (2020). To apply this model, we
equally distribute the day total order to each minute in the day, i.e.
$O_{j}=O/380$ for all $j\in\\{1,\cdots,380\\}$ as in Lin & Beling (2020).
4. -
(HU-)PPO utilizes the statistical U-shape to determine $O^{l}$, and within
each interval, the allocated order is equally divided among every minute, i.e.
$O^{l}_{t}=O^{l}/20$ for all $t\in\\{0,\cdots,19\\}$, where PPO is finally
used to distribute orders within the minute.
5. -
HUL is our proposed dual-level approach which uses the statistical U-shape to
determine $O^{l}$.
6. -
TUL is our proposed dual-level approach that employs the Transformer model to
determine $O^{l}$.
Figure 5: VAA’s of HU-PPO_LSTM, PPO_LSTM (in dark blue), and their alternative
versions (in brown). In these alternative versions, orders are executed in the
first step for every 1-minute trading horizon rather than using deep
reinforcement learning.
Figure 6: VAAs for the three best-performing models on the test data for each stock. SE |
---|---
SH |
KC |
PH |
Figure 7: The histograms of the VAA distributions for each model and stock on
the test dates.
### 5.2 Experiment Results
We test our models, which are trained for 100,000 iterations, by conducting
trading simulations in the buying direction. We first verify that using a
short trading horizon for allocating orders degrades performance in
approximating the daily cumulative VWAP. To do this, we investigate whether
the decisions made by deep RL for the PPO and (HU-)PPO models actually
contribute to lowering the VAA’s produced during testing. Note that for both
models, deep RL is using a 1-minute trading horizon. Thus, we intentionally
construct a naive alternative version, in which the orders are simply executed
in the first 5 seconds in the 1 minute trading horizon. The procedures taken
to obtain the total order for the 1-minute interval for these versions are the
same as their original counterparts. Figure 5 displays the comparison of the
VAA’s on the test dates given by (HU-)PPO, PPO, and their alternative
versions.
Figure 8: U-shape averages on test days for ground truths and Transformer-
predicted values.
Figure 9: The ground truth and Transformer-predicted U-shape plots for two
selected test dates for PH.
Both (HU-)PPO and PPO exhibit almost identical performance compared to their
alternative versions, which suggests that in the context of VWAP the use of
deep reinforcement learning does not significantly improve the approximation
capability if the trading horizon is too short. This implies that in order to
effectively leverage the benefits of deep RL in sequential decision-making, it
should be implemented on a longer trading horizon.
Secondly, we present Table 2 and Figure 6 to demonstrate how our dual-level
approach significantly improves performance. Table 2 reports the mean and
standard deviation values of VAA’s produced by DQN, OPD, PPO, HUL and TUL
across the test dates, and the percentage of them that fall under 10 bps.
These methods are used by practitioners, to measure how the traders were
performing relative to the VWAP. Our proposed dual-level approach demonstrates
significant improvements compared to using short trading horizons and single-
level neural networks. Although HUL occasionally yields better accuracy, TUL
generally has much less deviation from the daily cumulative VWAP than all
other methods. Especially for the stock PH, which is relatively less liquid
compared to the other stocks, DQN and PPO show better mean and standard
deviation values than HUL. However, TUL outperforms both models and
demonstrates the best mean and standard deviation values. The histograms for
VAA distributions given by DQN, OPD, PPO, HUL and TUL on each stock test data
are drawn in Figure 7, which depicts the clustering tendency of accuracies
produced by TUL.
Lastly, we demonstrate the effectiveness of the proposed U-shape Transformer
using Figures 8 and 9. Figure 8 shows the plots of the ground truth U-shape
and Transformer-predicted U-shape values, where the U-shape ratio values for
each 20-minute interval were averaged across all test dates. We observe that
our U-shape Transformer is able to follow the U-shape pattern for the test
dates on average for all stocks, which is in line with the comparable (and
sometimes better) performance of TUL against HUL. This is further supported by
the fact that HUL only utilizes the average U-shape ratio values of the past,
while TUL approximates the U-shape distribution for the test dates accurately
and uses predicted values for order allocation. Thus, TUL becomes more
adaptive in tracking the daily cumulative VWAP as it incorporates predictions
of fluctuations that may appear in the U-shape distribution in the future.
We also validate the performance improvements of TUL compared to other models,
particularly HUL, by examining TUL’s ability to capture daily fluctuations in
the U-shape. Figure 9 displays the ground truth U-shape and Transformer-
predicted distributions for two sampled test dates for the stock PH. We
deliberately choose PH since the performance gains of TUL when compared to HUL
were the largest for this particular stock, which has daily volume
distributions that deviates the most from the average due to relatively low
liquidity. While the U-shape Transformer may occasionally make erroneous
predictions, it quickly corrects itself and adapts to changes in the daily
volume distribution, attempting to accurately follow ratio changes throughout
the day.
## 6 Conclusion
In this study, we propose a dual-level approach to address the problem of
tracking the VWAP with accuracy and consistency. Our model is unique in that
it takes into account the well-known U-shaped intraday trading pattern. In the
first stage, we allocate the number of orders to execute in each interval,
taking the U-shape pattern into account. We consider two methods, statistical
U-shape and U-shape Transformer, for implementing the first stage. The naive
approach of using the statistical U-shape distribution improves performance.
However, our experiments demonstrate that the U-shape Transformer performs
even better. After the total daily orders are allocated to each interval in
the first stage, the LSTM model is used in the second stage to determine how
the orders in each interval should be executed Our simulation results show
that the dual-level approach consistently and accurately tracks the daily
cumulative VWAP.
Hyperparameter | DQN | OPD | PPO | HUL | TUL
---|---|---|---|---|---
Outer iterations | 10000 | 10000 | 10000 | 10000 | 10000
Inner iterations | 20 | 10 | 10 | 10 | 10
Batch size | 50 | 10 | 20 | 10 | 10
PPO CLIP $\varepsilon$ | - | 0.2 | 0.2 | 0.2 | 0.2
Discount factor $\gamma$ | - | 1 | 1 | 1 | 1
Number of trajectories per each outer iteration | 1000 | 160 | 200 | 152 | 228
LSTM hidden dimension $H$ | - | - | 128 | 128 | 129
LSTM model learning rate | - | - | 5e-5$\searrow$1e-5 | 5e-5$\searrow$1e-5 | 5e-5$\searrow$1e-5
DQN or OPD model learning rate | 1e-4 | 1e-4 | - | - | -
U-shape Encoder input vector dimension $N$ | - | - | - | - | 20
U-shape Encoder embedding dimension | - | - | - | - | 148
Transformer Encoder & Decoder number of heads | - | - | - | - | 4
Transformer Encoder & Decoder number of layers | - | - | - | - | 1
Transformer Encoder & Decoder PFFN dimension | - | - | - | - | 128
Transformer model learning rate | - | - | - | - | 1e-3$\searrow$2e-4
Table 3: Hyperparameters for DQN, OPD, PPO, HUL and TUL. Note that $\searrow$
indicates a linearly annealing learning rate schedule. Network configurations
of DQN and OPD follow Ning et al. (2021) and Fang et al. (2021), respectively.
Algorithm 1 Dual-level Neural Network
Require: Number of $outer$ $iterations$ and $num$ $days$
Randomly initialize learnable parameters $\theta^{\mathcal{E}}$,
$\theta^{\mathcal{D}}$, and $\theta^{\mathcal{L}}$
Initialize trajectory buffer $\mathcal{T}$
for $j=1$ to $outer$ $iterations$ do
for $d=1$ to $num$ $days$ do
Randomly choose a date in the training set
$\mu\leftarrow$ the average of the daily volume for the last 60 days
$\sigma\leftarrow$ the standard deviation of the daily volume for the last 60
days
$\mathcal{T}\leftarrow\mathcal{T}\bigcup$ Gathering Episodes(date, $\mu$,
$\sigma$)
end for
Using $\mathcal{T}$, update $\mathcal{E}$, $\mathcal{D}$ by optimizing the
loss function $J_{TF}$ in (5) with respect to $\theta^{\mathcal{E}}$ and
$\theta^{\mathcal{D}}$
Optimize the loss function $J_{\text{PPO}}$ in (9) with respect to
$\theta^{\mathcal{L}}$
$\mathcal{T}\leftarrow\emptyset$
end for
Algorithm 2 Gathering Episodes
Require: The number of intervals in a day $L$
Input: Date, $\mu$, $\sigma$
Output: Trajectory buffer of $i^{th}$ day $\mathcal{T}_{i}$
Initialize $\mathcal{T}_{i}$
$O\sim\mathcal{N}$($2.5$ $\times$ $10^{-3}\mu$, $6.25$ $\times$
$10^{-6}\sigma^{2}$)
$E_{out}=\mathcal{E}(E_{in})$
for $l=0$ to $L-1$ do
if $l=0$ then
Construct $D^{0}_{in}$ from raw pre-market data
else
$D^{l}_{in}\leftarrow D^{l-1}_{in}\bigcup Concat(D^{l-1}_{out},h^{l-1}_{T-1})$
end if
Obtain $u^{l}_{pred}$ and $D^{l}_{out}$ by procedure depicted in Section 4.3.1
$O^{l}\leftarrow O\cdot u^{l}_{pred}$
Obtain
$\tau_{l}=\\{s^{l}_{t},a^{l}_{t},r(a^{l}_{t}),V(s^{l}_{t}),\pi(a^{l}_{t}|s^{l}_{t}),O^{l}_{t}\\}_{t\in\\{0,1,2,\cdots,T-1\\}}$
and $h^{l}_{T-1}$ by procedure depicted in Section 4.3.2
$\mathcal{T}_{i}\leftarrow\mathcal{T}_{i}\bigcup\tau_{l}$
end for
## Acknowledgments
The work of Y. Hong was supported by Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded by the Ministry of
Education (NRF-2021R1A2C1093579) and the Korea government (MSIT)(No.
2022R1A4A3033571).
## References
* Almgren & Chriss (2001) Almgren, R., & Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3, 5–40.
* Berkowitz et al. (1988) Berkowitz, S. A., Logue, D. E., & Noser Jr, E. A. (1988). The total cost of transactions on the nyse. The Journal of Finance, 43, 97–112.
* Bertsimas & Lo (1998) Bertsimas, D., & Lo, A. W. (1998). Optimal control of execution costs. Journal of financial markets, 1, 1–50.
* Bialkowski et al. (2008) Bialkowski, J., Darolles, S., & Fol, G. L. (2008). Improving vwap strategies: A dynamic volume approach. Journal of Banking $\&$ Finance, 32, 1709–1722.
* Fang et al. (2021) Fang, Y., Ren, K., Liu, W., Zhou, D., Zhang, W., Bian, J., Yu, Y., & Liu, T.-Y. (2021). Universal trading for order execution with oracle policy distillation. In Proceedings of the AAAI Conference on Artificial Intelligence (pp. 107–115). volume 35.
* Goodhart & O’Hara (1997) Goodhart, C. A., & O’Hara, M. (1997). High frequency data in financial markets: Issues and applications. Journal of Empirical Finance, 4, 73–114.
* Hendricks & Wilcox (2014) Hendricks, D., & Wilcox, D. (2014). A reinforcement learning extension to the almgren-chriss framework for optimal trade execution. In 2014 IEEE Conference on Computational Intelligence for Financial Engineering & Economics (CIFEr) (pp. 457–464). IEEE.
* Huberman & Stanzl (2005) Huberman, G., & Stanzl, W. (2005). Optimal liquidity trading. Review of Finance, (pp. 165–200).
* Jain & Joh (1988) Jain, P. C., & Joh, G.-H. (1988). The dependence between hourly prices and trading volume. The Journal of Financial and Quantitative Analysis, 23, 269–283.
* Kakade et al. (2004) Kakade, S. M., Kearns, M., Mansour, Y., & Ortiz, L. E. (2004). Competitive algorithms for vwap and limit order trading. In Proceedings of the 5th ACM conference on Electronic commerce (pp. 189–198).
* Lin & Beling (2019) Lin, S., & Beling, P. A. (2019). Optimal liquidation with deep reinforcement learning. In Proceedings of the 33rd Conference on Neural Information Processing Systems, Deep Reinforcement Learning Workshop, Vancouver, Canada.
* Lin & Beling (2020) Lin, S., & Beling, P. A. (2020). An end-to-end optimal trade execution framework based on proximal policy optimization. In IJCAI (pp. 4548–4554).
* Mnih et al. (2016) Mnih, V., Badia, A. P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., & Kavukcuoglu, K. (2016). Asynchronous methods for deep reinforcement learning. In International conference on machine learning (pp. 1928–1937). PMLR.
* Mnih et al. (2013) Mnih, V., Kavukcuoglu, K., Silver, D., Graves, A., Antonoglou, I., Wierstra, D., & Riedmiller, M. (2013). Playing atari with deep reinforcement learning, . URL: http://arxiv.org/abs/1312.5602. Cite arxiv:1312.5602Comment: NIPS Deep Learning Workshop 2013.
* Nevmyvaka et al. (2006) Nevmyvaka, Y., Feng, Y., & Kearns, M. (2006). Reinforcement learning for optimized trade execution. In Proceedings of the 23rd international conference on Machine learning (pp. 673–680).
* Ning et al. (2021) Ning, B., Lin, F. H. T., & Jaimungal, S. (2021). Double deep q-learning for optimal execution. Applied Mathematical Finance, 28, 361–380.
* Pan et al. (2022) Pan, F., Zhang, T., Luo, L., He, J., & Liu, S. (2022). Learn continuously, act discretely: Hybrid action-space reinforcement learning for optimal execution. In IJCAI (pp. 3912–3918).
* Podobnik et al. (2009) Podobnik, B., Horvatic, D., Petersen, A. M., & Stanley, H. E. (2009). Cross-correlations between volume change and price change. Proceedings of the National Academy of Sciences, 106, 22079–22084.
* Schulman et al. (2017) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., & Klimov, O. (2017). Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, .
* Vaswani et al. (2017) Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, Ł., & Polosukhin, I. (2017). Attention is all you need. Advances in neural information processing systems, 30.
* Watkins & Dayan (1992) Watkins, C. J., & Dayan, P. (1992). Q-learning. Machine learning, 8, 279–292.
* Williams (1992) Williams, R. J. (1992). Simple statistical gradient-following algorithms for connectionist reinforcement learning. Reinforcement learning, (pp. 5–32).
|
# Coordinated Followup Could Have Enabled the Discovery of the GW190425
Kilonova
Ido Keinan The School of Physics and Astronomy, Tel Aviv University, Tel Aviv
69978, Israel Iair Arcavi The School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 69978, Israel Ido Keinan<EMAIL_ADDRESS>
###### Abstract
The discovery of a kilonova associated with the GW170817 binary neutron star
merger had far-reaching implications for our understanding of several open
questions in physics and astrophysics. Unfortunately, since then, only one
robust binary neutron star merger was detected through gravitational waves,
GW190425, and no electromagnetic counterpart was identified for it. We analyze
all reported electromagnetic followup observations of GW190425 and find that
while the gravitational-wave localization uncertainty was large, most of the
90% probability region could have been covered within hours had the search
been coordinated. Instead, more than 5 days after the merger, the
uncoordinated search covered only 50% of the probability, with some areas
observed over 100 times, and some never observed. We further show that,
according to some models, it would have been possible to detect the GW190425
kilonova, despite the larger distance and higher component masses compared to
GW170817. These results emphasize the importance of coordinating followup of
gravitational-wave events, not only to avoid missing future kilonovae, but
also to discover them early. Such coordination, which is especially important
given the rarity of these events, can be achieved with the Treasure Map, a
tool developed specifically for this purpose.
††software: astropy (Astropy Collaboration et al., 2022), healpy (Górski et
al., 2005), matplotlib (Hunter, 2007), MOSFiT, numpy (van der Walt et al.,
2011), pandas, pyphot.
## 1 Introduction
Binary neutron star (BNS) mergers were predicted to produce both gravitational
waves (GWs; Clark & Eardley, 1977) and electromagnetic (EM) radiation. The
latter was theorized to be emitted in the form of two types of transients: a
short gamma-ray burst (GRB; Eichler et al., 1989), and longer-duration
emission from the radioactive decay of heavy elements synthesized via the
$r$-process in the ejecta (Li & Paczyński, 1998). This transient has been
nicknamed “macronova” (Kulkarni, 2005) or “kilonova” (Metzger et al.,
2010)111Hereafter we use the term “kilonova” rather than “macronova” because
it has been more widely adopted by the community..
All three predictions were confirmed with the first discovery of a GW signal
from a BNS merger, GW170817, detected and localized by the Advanced Large
Interferometer Gravitational-wave Observatory (LIGO; LIGO Scientific
Collaboration et al., 2015), and the Advanced Virgo detector (Acernese et al.,
2014), during their second observation run (O2), on August 17, 2017 (Abbott et
al., 2017a). The GW source was localized to a region of $\sim 30\ deg^{2}$ at
a distance of $\sim 40\ Mpc$ and had a total binary mass of
$2.74^{+0.04}_{-0.01}\ M_{\odot}$ (Abbott et al., 2017a). A short GRB,
GRB170817A, consistent with the GW localization, was detected $\sim$2 seconds
after the GW-determined merger time (Abbott et al., 2017b) by the Fermi Gamma-
ray Burst Monitor (Fermi-GBM; Meegan et al., 2009), and the International
Gamma-Ray Astrophysics Laboratory (INTEGRAL; Winkler et al., 2003). An optical
transient, AT 2017gfo, also consistent with the GW localization, was detected
11 hours later (Coulter et al., 2017a). Both EM transients were later
confirmed to be physically associated with the BNS merger (Abbott et al.,
2017).
Specifically, AT 2017gfo was consistent with kilonova predictions whereby at
the coalescence of a BNS system, $10^{-4}-10^{-2}M_{\odot}$ of neutron-rich
material are ejected at velocities of $0.1-0.3c$ in the equatorial plane due
to tidal effects (e.g. Rosswog et al., 1999; Hotokezaka et al., 2013).
Additional mass was predicted to be ejected in the polar direction from the
contact region at the time of the merger (e.g. Bauswein et al., 2013;
Hotokezaka et al., 2013). Lanthanides formed in low electron-fraction
material, such as the equatorial tidal ejecta, have a high opacity (Kasen et
al., 2013; Tanaka & Hotokezaka, 2013), making the light curve redder, fainter,
and longer lived compared to low lanthanide ejecta (Barnes & Kasen, 2013;
Grossman et al., 2014), such as that from to polar regions. The result, seen
in AT 2017gfo, is a kilonova of at least two components: one blue and short-
lived from the lanthanide-poor ejecta, and one red and longer-lived from the
lanthanide-rich ejecta. For a review of this event, see for e.g. Nakar (2020)
and Margutti & Chornock (2021).
GW170817 and its EM counterparts confirmed that BNS mergers are sites of
$r$-process nucleosynthesis (e.g. Kasen et al., 2017), and confirmed the
connection between short GRBs and BNS mergers. This event also demonstrated a
novel method to constrain the Hubble constant (Abbott et al., 2017). However,
many open questions regarding the properties of the kilonova emission remain,
such as whether the source of the early blue emission is indeed a distinct
lanthanide-poor ejecta component (other physical mechanisms were proposed,
e.g. Kasliwal et al., 2017; Piro & Kollmeier, 2018; Waxman et al., 2018). More
BNS detections and multi-wavelength observations of their EM counterparts,
especially during the first few hours (e.g. Arcavi, 2018) are needed to answer
these questions.
The second (and so far, last) robust GW detection of a BNS merger occurred at
the start of the third GW-detector observing run (O3). GW190425 was discovered
on April 25, 2019, at 08:18:05 UTC, at a distance of $159^{+69}_{-72}\ Mpc$
and with a total binary mass of $3.4^{+0.3}_{-0.1}\ M_{\odot}$ (Abbott et al.,
2020). The event was detected only by a single LIGO detector, making its
localization poorly constrained, with a $10183\ deg^{2}$ 90% uncertainty
region initially (Ligo Scientific Collaboration & VIRGO Collaboration, 2019a).
This region shrank about one day later to $7461\ deg^{2}$ (Ligo Scientific
Collaboration & VIRGO Collaboration, 2019b), but grew back to $9881\ deg^{2}$
in the final localization published in the second GW Transient Catalog
(GWTC-2; Abbott et al., 2021). The localization parameters are summarized in
Table 1, and the localization maps are shown in Figure 1.
No coincident GRB was detected by the Fermi Large Area Telescope (Fermi-LAT;
Atwood et al., 2009; Axelsson et al., 2019), the Swift Burst Alert Telescope
(Swift/BAT; Gehrels et al., 2004; Sakamoto et al., 2019) nor by the Fermi-GBM
(Fletcher et al., 2019). The INTErnational Gamma-Ray Astrophysics Laboratory
(INTEGRAL; Winkler et al., 2003) reported a detection of a low significance
signal, associated with a BNS merger (Pozanenko et al., 2020), but with no
localization information to robustly tie it to GW190425 (Savchenko et al.,
2019).
Table 1: GW190425 localization areas and distances.
Epoch | Date | 90% Area | 50% Area | Distance
---|---|---|---|---
| [UTC] | [$deg^{2}$] | [$deg^{2}$] | [Mpc]
Initial | 2019-04-25 | 10183 | 2806 | $154\pm 45$
Alert | 08:19:27 | | |
Update | 2019-04-26 | 7461 | 1378 | $156\pm 41$
Alert | 10:48:17 | | |
GWTC-2 | 2020-07-27 | 9881 | 2400 | $157\pm 43$
| 16:41:49 | | |
Note. — “90% (50%) Area” refers to the sky area that contains 90% (50%) of the
probability for the location of the source.
Figure 1: GW190425 localizations released in the initial (top) and update
(middle) alerts, and in the final GWTC-2 publication (bottom; Abbott et al.,
2021). White and grey contours denote the regions containing 50% and 90% of
the probability of the source location, respectively. The Swift/Bat and Fermi-
LAT footprints at the time of the merger are presented in red solid and dashed
lines, covering areas of $4596\ deg^{2}$ and $8602\ deg^{2}$, respectively.
The position of the Sun and Moon at the time of the merger are denoted in
their respective symbols.
An extensive EM followup effort to find the kilonova associated with GW190425
began shortly after the initial alert. However, no significant kilonova
candidate was identified. This was attributed to the larger distance, higher
component masses, more equal mass ratio, and to the larger localization
compared to GW170817 (e.g. Hosseinzadeh et al., 2019a; Lundquist et al., 2019;
Abbott et al., 2020; Foley et al., 2020; Gompertz et al., 2020; Kyutoku et
al., 2020; Sagués Carracedo et al., 2021; Camilletti et al., 2022; Dudi et
al., 2022; Radice et al., 2024). However, here we show that, according to some
models, the emission from the GW190425 kilonova could have been detected by
existing facilities, and that most of the localization could have been covered
within hours, had the EM followup effort been coordinated.
Coordinating EM followup of GW events is challenging. Reports of observations
and possible kilonova candidates during O2 and O3 were primarily communicated
through GRB Coordinates Network222https://gcn.nasa.gov/ (GCN; now the General
Coordinates Network) circulars. GCN circulars were designed to be human
readable, with no unified text format. As such they are not ideal when
reporting lists of coordinates, bands, and sensitivities observed. The non
machine-readable format makes it very difficult to parse, store and interpret
such information in real time.
One tool designed to address the challenge of coordinating EM followup of GW
events is the Treasure Map333https://treasuremap.space/ (Wyatt et al., 2020).
Treasure Map is a web application that collects, stores and distributes
followup reports through a machine-friendly Application Programming Interface
(API). The API allows observers to report their planned and executed
observations and to retrieve those of other facilities in order to inform
their own counterpart search plan in real time. In addition, an interactive
visualization allows observers to see their observations and those of other
facilities on a map, together with GW localization in an intuitive way. All of
the reports are also available to download at a later date for analyzing
followup efforts and non-detection statistics. Unfortunately, the Treasure Map
did not exist during GW190425, leading to very inefficient followup, as we
will show here. Our goal is to encourage the community to use tools such as
the Treasure Map during the next BNS merger, in order to increase the chances
of finding the EM counterpart, and to do so as quickly as possible. We do so
by demonstrating the impact of uncoordinated observations on missing the
counterpart to GW190425.
In this work, we analyze all reported infrared, optical and ultraviolet
followup efforts of GW190425. In Section 2 we describe our data sources and
collection process; in Section 3, we analyze the sky coverage and depth
achieved in the context of the expected kilonova emission from this event;
and, finally, in Section 4, we discuss our conclusions.
## 2 Data
In order to analyze the efficiency of the search for the GW190425 EM
counterpart we collected pointing and photometric data of infrared, optical
and ultraviolet observations obtained as followup of GW190425 up to 2019 April
30, 23:57:21 UTC, which were reported in one of the following sources
(removing duplicate reports of the same observations): The Treasure Map (1419
observations reported retroactively), 122 GCN circulars (1339 observations),
the MASTER global robotic net (MASTER-Net; Lipunov et al., 2010)
website444http://observ.pereplet.ru/ (4491 observations), the Coulter et al.
(2024) report on the One-meter Two-hemispheres (1M2H) followup (345
observations), and the Smartt et al. (2024) report of the Panoramic Survey
Telescope and Rapid Response System (PAN-STARRS) and the Asteroid Terrestrial-
impact Last Alert System (ATLAS) followup (6631 observations). In total, 14255
unique observations were collected, including time stamps, limiting
magnitudes, and bands used. Instrument footprints and fields of view (FOVs)
were taken from the Treasure Map when available, from GCN reports when
provided there, and from instrument-specific websites and publications
otherwise (all references are provided in Table 2). Some galaxy-targeted
searches provided target galaxy names rather than exact coordinates pointed
to. In such cases, we obtained the galaxy coordinates from the Galaxy List for
the Advanced Detector Era (GLADE) catalog (Dálya et al., 2018) or the SIMBAD
Astronomical Database (Wenger et al., 2000) and assumed the target galaxies
were positioned at the center of the observed footprint. A summary of the
observations and instruments used in this analysis is presented in Table 2,
and the full list of all pointings gathered is presented in Table 3.
Table 2: GW190425 EM followup observations reported in circulars, the Treasure
Map, by MASTER-Net on their website, and in Coulter et al. (2024) and Smartt
et al. (2024), out to 5.65 days after merger.
Group/Facility/Instrument | No. of | Bands | Field of View | Median $5\sigma$ Depth | Pointings Reference
---|---|---|---|---|---
| Pointings | | [$deg^{2}$] | [AB Mag] |
Pan-STARRSa | 6209 | w,i,z | 7.07 | 21.58 | 1
MASTER-Netb | 4491 | Clear | 8.00 | 19.10 | 2
ZTFc | 596 | g,r,i | 46.73 | 21.05 | 3
ATLASd | 422 | o,c | 28.89 | 19.26 | 1
Swift/UVOTe | 392 | u | 8.0e-02 | 19.40 | 3
GOTOf | 303 | V,g | 18.85 | 19.85 | 3
J-GEM/Subarug | 154 | r | 2.0e-03 | 24.00 | 4
1M2H/Swopeh | 151 | B,V,u,g,r,i | 0.25 | 21.55 | 5
COATLIi | 128 | w | 3.3e-02 | n/a | 6
KMTNet 1.6mj | 120 | R | 4.00 | n/a | 7
KAITk | 101 | Clear | 1.2e-02 | 19.00 | 8
Harold Johnson Telescope/RATIRl | 99 | J,H,Y,Z,g,r,i | 8.1e-03 | 20.74 | 9, 10
GROWTH/LOTm | 93 | R,g,r,i | 3.6e-02 | 19.79 | 11, 12, 13
1M2H/Nickeln | 93 | r,i | 1.1e-02 | 20.24 | 5
MMTO/MMTCamo | 81 | g,i | 2.0e-03 | 21.48 | 3, 14
GWAC-F60Ap | 80 | Clear | 9.0e-02 | 18.97 | 15
CNEOSTq | 75 | R | 9.00 | 19.86 | 16
BOOTES-5r | 63 | Clear | 2.78 | 20.50 | 17
SAGUARO/CSS 1.5ms | 61 | Clear | 5.00 | 21.30 | 3
LCO 1mt | 58 | g,r,i | 0.20 | 21.30 | 18, 19, 20, 21
1M2H/Thacheru | 53 | g,r,z | 0.12 | 19.55 | 5
GRANDMA/LesMakes T60v | 52 | Clear | 17.64 | 19.20 | 22
GRANDMA/AZT$-$8w | 52 | B,R | 0.25 | 19.00 | 22
GRANDMA/Abastumani T70x | 40 | R | 0.25 | 16.29 | 22
Xinglong 60/90cmy | 35 | Clear | 2.25 | 18.00 | 23
McDonald Observatory 2.1m/CQUEANz | 30 | i | 6.1e-03 | 20.44 | 24
1M2H/ANDICAM-CCDaa | 27 | I | 1.1e-02 | 20.70 | 5
GROWTH/Kitt Peak 2.1m/KPEDab | 25 | I,g,r | 7.0e-03 | 20.40 | 25, 26, 27
YAHPTac | 23 | R | 3.4e-02 | 18.18 | 28
1M2H/ANDICAM-IRad | 21 | J,H,K | 1.5e-03 | 13.45 | 5
Liverpool Telescope/IO:Oae | 19 | g,r,i,z | 2.8e-02 | 22.24 | 29, 30
LOAO 1maf | 13 | R | 0.22 | 19.23 | 31
GROWTH-Indiaag | 10 | g,r,i | 0.49 | 18.41 | 32, 33
Konkoly 0.8mah | 10 | g,r | 9.0e-02 | 20.40 | 34
IRSF 1.4m/SIRIUSai | 7 | J,H,K | 1.6e-02 | 15.31 | 35, 36
TAROT TREaj | 7 | R | 17.64 | 17.23 | 37
Konkoly 0.6/0.9mak | 5 | Clear | 1.36 | 21.50 | 34
Lijiang 2.4mal | 5 | g,r | 2.56 | 19.04 | 38
GRANDMA/CAHA 2.2mam | 5 | U,B,V,R,I | 4.5e-02 | 22.30 | 39
DECaman | 4 | g,r,i,z | 3.35 | 23.85 | 40
MPG/GRONDao | 3 | r,i,z | 8.1e-03 | 20.00 | 41
NOT/ALFOSCaq | 3 | g,r,i | 1.1e-02 | 20.39 | 42
OSN 1.5mar | 2 | R | 6.8e-02 | 21.30 | 43, 44
ANU/SkyMapperas | 2 | i | 5.62 | 20.00 | 45
GTC/OSIRISat | 1 | r | 1.8e-02 | 22.80 | 44
VISTA/VIRCAMau | 1 | K | 2.14 | 19.55 | 46
††footnotetext: Instrument references: aPanoramic Survey Telescope and Rapid
Response System (Chambers et al., 2016) bMobile Astronomical System of
TElescope Robots (Lipunov et al., 2010); cZwicky Transient Facility (Masci et
al., 2018); dAsteroid Terrestrial-impact Last Alert System Tonry et al. (2018)
eSwift Ultraviolet/Optical Telescope (Roming et al., 2005); fGravitational-
wave Optical Transient Observer (Dyer et al., 2022); gJapanese Collaboration
for Gravitational-wave ElectroMagnetic followup (Yoshida et al., 2000;
Kashikawa et al., 2002); hOne-Meter Two-Hemispheres/Swope (Bowen & Vaughan,
1973); iCOATLI (Watson et al., 2016); jKorea Microlensing Telescope Network
(Kim et al., 2016); kKatzman Automatic Imaging Telescope (Filippenko et al.,
2001); lReionization and Transients Infrared Camera (Farah et al., 2010);
mGlobal Relay of Observatories Watching Transients Happen/Lulin One-meter
Telescope (Kasliwal et al., 2019); nNickel Direct Imaging Camera
(https://mthamilton.ucolick.org/techdocs/instruments/nickel_direct/intro/);
oMMT Observatory (Fabricant et al., 2019); pGround Wide Angle Cameras Array
(Han et al., 2021); qChinese Near Earth Object Survey Telescope; rBurst
Observer and Optical Transient Exploring System 5; sSearches After
Gravitational-waves Using ARizona’s Observatories/Catalina Sky Survey (Drake
et al., 2009); tLas Cumbres Observatory (Brown et al., 2013); uThacher ACP
Camera Swift et al. (2022); vGlobal Rapid Advanced Network Devoted to the
Multi-messenger Addicts (Antier et al., 2020); wAstronomical Reflecting
Telescope 8 (Antier et al., 2020); xAbastumani(Antier et al., 2020); yXinglong
60/90cm, Chinese Academy of Sciences; zCamera for QUasars in EArly uNiverse
(Park et al., 2012); aaA Novel Dual Imaging CAMera (ANDICAM) CCD (DePoy et
al., 2003); abKitt Peak EMCCD Demonstrator; acYAoan High Precision Telescope
(Guo et al., 2022); adANDICAM-IR (DePoy et al., 2003); aeInfrared-
Optical:Optical (Steele et al., 2004); afLemmonsan Optical Astronomy
Observatory (Han et al., 2005); agKasliwal et al. (2019); ahKonkoly 0.8m
Telescope, Research Centre for Astronomy and Earth Sciences; aiInfraRed Survey
Facility/Simultaneous InfraRed Imager for Unbiased Survey (Nagayama et al.,
2003); ajTélescope à Action Rapide pour les Objets Transitoires (Boër et al.,
1999); ak Konkoly 0.6/0.9m (https://old.konkoly.hu/konkoly/telescopes.html);
al(Wang et al., 2019); amCentro Astronómico Hispano en Andalucía (Antier et
al., 2020); anDark Energy Camera (Flaugher et al., 2015); aoMax Planck
Gesellschaft/Gamma-Ray burst Optical and Near-infrared Detector (Greiner et
al., 2008); aqNordic Optical Telescope/ALhambra Faint Object Spectrograph and
Camera (Djupvik & Andersen, 2010); arObservatorio de Sierra Nevada
(https://www.osn.iaa.csic.es/en/page/15-m-telescope); asAustralian National
University (Keller et al., 2007); atGran Telescopio Canarias/Optical System
for Imaging and low-intermediate Resolution Integrated Spectroscopy (Cobos et
al., 2002); auVisible and Infrared Survey Telescope for Astronomy/VISTA
InfraRed CAMera (Emerson et al., 2006; Dalton et al., 2006).
††footnotetext: Pointings references: 1Smartt et al. (2024)
2https://observ.pereplet.ru/ (Lipunov et al., 2010);
3https://treasuremap.space/ (Wyatt et al., 2020); 4Sasada et al. (2019);
5Coulter et al. (2024); 6Watson et al. (2019); 7Kim et al. (2019); 8Zheng et
al. (2019); 9Butler et al. (2019); 10Troja et al. (2019); 11Tan et al.
(2019a); 12Tan et al. (2019b); 13Kong (2019); 14Hosseinzadeh et al. (2019b);
15Xin et al. (2019); 16Li et al. (2019a); 17Hu et al. (2019a); 18Burke et al.
(2019); 19Hiramatsu et al. (2019a); 20Hiramatsu et al. (2019b); 21Arcavi et
al. (2019); 22Howell et al. (2019); 23Xu et al. (2019); 24Im et al. (2019);
25Ahumada et al. (2019b); 26Ahumada et al. (2019a); 27Ahumada et al. (2019c);
28Sun et al. (2019); 29Perley & Copperwheat (2019); 30Perley et al. (2019);
31Paek et al. (2019); 32Bhalerao et al. (2019); 33Waratkar et al. (2019);
34Vinko et al. (2019); 35Morihana et al. (2019a); 36Morihana et al. (2019b);
37Blazek et al. (2019); 38Li et al. (2019b); 39Kann et al. (2019); 40Bloom et
al. (2019); 41Schady et al. (2019); 42Izzo et al. (2019); 43Castro-Tirado et
al. (2019); 44Hu et al. (2019b); 45Chang et al. (2019); 46Tanvir et al.
(2019).
Note. — Some sources did not provide their observation depths. These are
marked with “n/a” in the depth column.
Table 3: List of the observations used in this work.
Facility/Instrument | MJD | R.A. [deg] | Dec. [deg] | Band | $5\sigma$ Limiting Mag. | Source
---|---|---|---|---|---|---
ZTF | 58598.3460 | 180.0000 | 62.1500 | r | 20.46 | Treasure Map
ZTF | 58598.3465 | -168.6100 | 54.9500 | r | 20.95 | Treasure Map
ZTF | 58598.3470 | -165.0502 | 47.7500 | r | 21.05 | Treasure Map
ZTF | 58598.3474 | -167.7500 | 40.5500 | r | 20.80 | Treasure Map
Note. — This table is published in its entirety in the machine readable
format. A portion is shown here for guidance regarding its form and content.
## 3 Analysis and Results
In order to measure the efficiency of the search for the GW190425 EM
counterpart, we first analyze the observability of the GW localization region
at the time of the merger (Section 3.1), followed by the amount of the GW
localization that was covered as a function of time, in terms of the total
probability, area, and galaxy luminosity (Sections 3.2 and 3.3). We then
examine the reported non-detection limiting magnitudes and compare them to
light curves of the GW170817 kilonova and of kilonova models tuned to the
parameters of GW190425 (Section 3.4).
GW localizations are provided as Hierarchical Equal Area isoLatitude
Pixelation555https://HEALPix.sourceforge.io/ (HEALPix; Górski et al., 2005)
maps. These maps divide the sky into pixels of equal area and assign a
probability value for the location of the GW source to each pixel. Each pixel
is also assigned a distance to the source (assuming it is at that position)
and a distance error estimate. The number of pixels $N_{pixels}$ in the map is
$12\cdot(nside)^{2}$ with $nside=2^{n}$ for some integer $n\geq 0$. The
localization maps provided for GW190425 have a resolution of $nside=256$, or
$N_{pixels}=786432$. This translates into pixels with an area of $0.052\
deg^{2}$ each. We perform all of our analysis using the maps in this
resolution, as they were provided, except for the observability analysis
(Section 3.1), for which we downsample the maps to $nside=32$ (i.e.
$N_{pixels}=12288$ and a pixel area of $3.36\ deg^{2}$) for computational
ease.
### 3.1 Observability
We define that a region on the sky is “visible” at a point in time from a
certain location on Earth when the Sun is a least $12^{\circ}$ below the
horizon, the Moon separation is at least $20^{\circ}$ (the moon illumination
was 66% at the time of the GW190425 merger) and the airmass is less than 2.5.
According to this definition, of the $9881\ deg^{2}$ ($2400\ deg^{2}$) of the
90% (50%) final localization, $8182\ deg^{2}$ ($2182\ deg^{2}$), or roughly
83% (90%), were visible during the 24 hours following the merger from the
combined locations of all ground-based observatories listed in Table 2.
Next, we define the “accessibility” of each area of the sky to each instrument
as the amount of time per day that area is visible to that instrument,
$t_{visible}$, weighted by the instrument FOV (to take into account that
larger FOV instruments can cover more area simultaneously) and divided by the
pixel area of the map ($3.36\ deg^{2}$ in this case):
$t_{accessibility}=t_{visible}\left(\frac{FOV}{3.36deg^{2}}\right)$ (1)
We define an inaccessible area as one with total accessibility (i.e. summed
over all ground-based instruments considered here) $t_{accessibility}<5\ min$
(corresponding to a typical single exposure time). Using this definition, 6284
$deg^{2}$ of the sky were inaccessible due to sun and moon constraints, of
which 2098 $deg^{2}$ were in the GWTC-2 90% localization region. The rest of
the localization region, encompassing 75% of the probability, was visible with
a median total accessibility time of 41.5 hours. The total accessibility of
the 90% localization region during the first day since merger is presented in
Figures 2 and 3, and in Table 4.
Figure 2: Accessible hours during the first day after the merger summed over all ground-based observatories listed in Table 2. An accessible hour is one that is over an airmass of 2.5 and not sun or moon-constrained, normalized to the FOV of each instrument (see text for details). Grey regions indicate inaccessible areas $(t_{accessibility}<5\ min)$. Black contours are of the GWTC-2 localization. The sun and moon locations in the figure at the time of the merger. Table 4: GW190425 accessible localization areas and probabilities, and median accessibility. Epoch | 90% Area | Accessible | Accessible | Median
---|---|---|---|---
| | 90% Area | Probability | Accessibility
| [$deg^{2}$] | [$deg^{2}$] | | [Hr]
Initial | 10183 | 8216 | 72% | 80.0
Update | 7461 | 5833 | 76% | 32.0
GWTC-2 | 9881 | 7783 | 75% | 41.5
Figure 3: Accessible hours of the GWTC-2 90% localization during the first day
after merger. Out of the 90% localization area, 2098 $deg^{2}$ were
inaccessible ($t_{accessibility}<5\ min$, leftmost bin), leaving 75% of the
probability accessible, with a median total accessibility of 41.5 hours.
### 3.2 Area and Probability Coverage
Figure 4: Total number of infrared, optical and ultraviolet followup
observations up to 5.65 days post merger for GW190425. Grey areas are where no
observations were reported. Lines mark the 50% and 90% probability regions of
the final GWTC-2 localization. While some parts of the localization were
observed over 100 times, some were not observed at all, even though they were
accessible (Figure 2).
We count the sky area and the probability observed up to 5.65 days after the
time of the merger, both with and without overlap between observations, in
order to measure how much of the localization was covered and how much of it
was observed multiple times. Measuring total area and probability on a HEALPix
map requires counting the pixels that overlap with an observed footprint. Due
to the resolution of the map, the minimal area that can be counted per
footprint is $0.052\ deg^{2}$. However, there are footprints with a smaller
FOV (as seen in Table 2). Here we use the
healpy666https://healpy.readthedocs.io/ python package, which includes two
methods for counting (“querying”) pixels inside an observed footprint. The
“inclusive” method counts every pixel that overlaps with the footprint, while
the “exclusive” method only counts pixels that have their center fall inside
the footprint. An illustration of these two methods is presented in Figure 8
in Appendix A. We choose the exclusive method for our analysis, as it is a
more conservative accounting of the actual area and probability covered, while
the inclusive method over counts pixels, causing an overestimation of the area
and probability covered.
Figure 4 shows the total coverage of all reported followup observations up to
5.65 days post-merger outlined in Section 2. Some regions, in the northern
part of the localization, were observed more than 100 times while most of the
accessible southern part of the localization was not observed at all.
The GW localization was updated 26 hours after the initial localization was
released. At the time of the update, earlier footprints had their probability
coverage changed. We take that into account by retroactively recounting the
probability covered by all of the observations using the updated localization
from the time it was released. The area covered does not depend on the
localization map and thus, no recounting of the area covered is needed at the
time of the update. The sum of area and probability covered is performed in
two different ways, first without regard to whether observations overlap or
not, and second with overlapping regions counted only once. We compare the
probability covered to the 100% probability of the entire sky and the area
covered to the initial and updated 90% probability areas.
The probability and area covered as a function of time are shown in Figure 5.
Within 5.65 days following the merger, 11486 $deg^{2}$ were covered (once or
more). This is more than the area of the GWTC-2 90% probability region, but
not all of the area covered is in that region (Fig. 4). In fact, at 5.65 days
post-merger, only 54% of the probability (of the GWTC-2 localization) was
covered, while enough observations were made to cover over 120000 $deg^{2}$ by
that time. The 90% probability region could have theoretically been covered
within three hours from merger777Here we provide the theoretical time period
it would have taken ideally coordinated observations to cover the
localization, without simulating a detailed coordinated observing strategy in
order to provide a simple order-of-magnitude estimate of the time it would
have taken to cover the region. Instrument-specific observability constraints
could prolong this time estimate..
Figure 5: Cumulative area (top) and probability (bottom) observed during the
5.65 days following the GW190425 merger. Colored dashed lines are for all
observations (including overlapping ones) while solid lines are for unique
positions on the sky (without overlap). Horizontal grey dotted lines denote
the total area inside the 90% localization region and 100% probability
reported at the time, for the top and bottom plots, respectively, while the
grey dash-dotted lines are for the accessible area and probability (see
Section 3.1). The vertical grey dotted line denotes the time of the updated
localization.
### 3.3 Galaxy Luminosity Coverage
We define the galaxy luminosity coverage as the fraction of $B$-band
luminosity of galaxies observed out of the total $B$-band luminosity of all
galaxies with $L>L_{B}^{*}$ in the 90% localization volume within 3$\sigma$ of
the mean localization distance. Here, $L_{B}^{*}$ is the $B$-band luminosity
corresponding to $L^{*}$, the charactesitic luminosity of the Schechter
Function (Schechter, 1976). We adopt a corresponding absolute magnitude in the
$B$ band of $M_{B}^{*}=-20.7$, following Arcavi et al. (2017a). The GLADE
catalog is less complete to galaxies fainter than $L^{*}$ at large distances
(see Dálya et al., 2018, for more details).
We count only galaxies that have a $B$-band apparent magnitude and a distance
estimate in Version 2.3 of the GLADE catalog. There are a total of 47664
(34302) such galaxies in the catalog inside the initial (updated) 90%
localization.
We count galaxies that are inside observed footprints and the localization
volume, once with recounting galaxies and once without. Figure 6 shows the
galaxy luminosity coverage as a function of time. After 5.65 days from the
merger, only 52% of the $B$-band luminosity in galaxies inside the updated
localization region was observed, while 11.88 times the localization
luminosity was observed in total due to overlapping observations.
Figure 6: $B$-band Galaxy luminosity fraction observed out of all galaxies
with luminosity larger than $L_{B}^{*}$ from the GLADE 2.3 catalog that are
inside the 90% localization region and up to a distance of 3$\sigma$ of the
mean estimated distance. The initial and updated localizations are separated
by the vertical dotted grey line. The dashed line shows the total luminosity
observed including overlapping galaxies, while the solid line shows the total
luminosity observed in unique galaxies.
### 3.4 Kilonova Detection
Table 5: Parameters and distributions used for generating the ensemble of
light curves from the Nicholl et al. (2021) BNS merger kilonova model.
Parameter | Description | Type | Values
---|---|---|---
$\mathcal{M}^{a}$ | Chirp mass | Gaussian | $\mu=1.44$, $\sigma=0.02$
$q^{a}$ | Mass ratio | uniform | $\min=0.4$, $\max=1.0$
s | Shock density profile power law index | fixed | 1.0
$M_{TOV}$ | NS maximum theoretical mass | fixed | 2.2
$\alpha^{b}$ | Enhancement of blue ejecta | uniform | $\min=0.1$, $\max=1.0$
$disk_{frac}^{b}$ | Fraction of disk ejected | uniform | $\min=0.05$, $\max=0.5$
$\cos\theta^{b}$ | Cosine of viewing angle | uniform | $\min=0.0$, $\max=1.0$
$\cos\theta_{cocoon}^{b}$ | Cosine of cocoon opening angle | uniform | $\min=0.707$, $\max=1.0$
$\cos\theta_{open}^{b}$ | Cosine of squeezed ejecta opening angle | fixed | 0.707
$\log N_{H}^{b}$ | Hydrogen column density | uniform | $\min=19$, $\max=23$
aafootnotetext: Based on the inferred values in Abbott et al. (2020).
bbfootnotetext: Based on the priors used in Nicholl et al. (2021).
Even if the localization would have been covered as efficiently as possible,
there is still a possibility that the EM counterpart was too faint to have
been detected. In order to check this, we compare the non-detection limiting
magnitudes reported for each of the followup observations to the light curve
of GW170817 and to light curves generated using the Kasen et al. (2017) and
Nicholl et al. (2021) kilonova models.
We use the InfraRed Survey Archive (IRSA) dust extinction service
queries888https://astroquery.readthedocs.io/en/latest/ipac/irsa/irsa_dust/irsa_dust.html
to correct all reported magnitude limits for Milky Way dust extinction at the
center of their footprint by using the Schlafly & Finkbeiner (2011) galactic
dust extinction estimates. For unfiltered observations, we assume $r$-band
extinction. We then translate all limits to absolute magnitudes using the mean
distance of the pixel that overlaps with the footprint center.
The GW170817 data is taken from the Arcavi (2018) compilation of observations
by Andreoni et al. (2017); Arcavi et al. (2017b); Coulter et al. (2017b);
Cowperthwaite et al. (2017); Díaz et al. (2017); Drout et al. (2017); Evans et
al. (2017); Hu et al. (2017); Kasliwal et al. (2017); Lipunov et al. (2017);
Pian et al. (2017); Shappee et al. (2017); Smartt et al. (2017); Tanvir et al.
(2017); Troja et al. (2017); Utsumi et al. (2017); Valenti et al. (2017) and
Pozanenko et al. (2018).
The Kasen et al. (2017) model is based on a 1D Monte Carlo simulation of the
radiative transfer of photons from the radioactive decay of $r$-process
elements through kilonova ejecta. The ejecta is modeled using three
parameters: ejecta mass $M_{ej}$, characteristic expansion velocity $v_{k}$,
and lanthanide fraction power index $X_{lan}$ (where the lanthanide fraction
is $10^{X_{lan}}$). The calculation produces a spectral energy distribution of
the emission. We use pyphot999https://mfouesneau.github.io/pyphot/index.html
to create synthetic light curves in the relevant bands from these model
spectra. We use the ejecta parameters assumed by Foley et al. (2020) for the
GW190415 kilonova: $M_{ej}=0.04M_{\odot}$, $v_{k}=0.15c$ and $X_{lan}=-2$,
which they propose assuming a BNS system with masses $1.4M_{\odot}$ and
$2.0M_{\odot}$. This lanthanide fraction results in a high opacity of
${\kappa_{ej}}\approx 10\ cm^{2}/g$ and a transient that is observable mostly
in the infrared. This led Foley et al. (2020) to conclude that the GW190425
kilonova would not have likely been observable to most facilities that
participated in the search.
Nicholl et al. (2021) provide an analytic 2D kilonova model that assumes a
three-component ejecta: one component is the “tidal ejecta” in the equatorial
region and a second is the “polar ejecta” concentrated in the polar region.
For both components, a blackbody is assumed. The third component is from GRB-
shocked material in a “cocoon” around the polar region (Nakar & Piran, 2016).
The model parameters relate directly to the GW-measured binary system chirp
mass $\mathcal{M}$ and mass ratio $q$, and to neutron star (NS) equation of
state parameters, such as the NS maximum theoretical mass $M_{TOV}$. These are
used to calculate the mass $M_{ej}$, velocity $v_{k}$, and the opacity
${\kappa_{ej}}$ of each ejecta component corresponding to three opacity
regimes: the “red” high opacity (low $Y_{e}$) tidal ejecta, the “blue” low
opacity (high $Y_{e}$) polar ejecta, and the intermediate-opacity “purple”
ejecta. The blue and red opacities are assumed to be $\kappa_{ej,blue}=0.5\
cm^{2}/g$ and $\kappa_{ej,red}=10\ cm^{2}/g$, respectively. The asymmetry of
the ejecta is parameterized by the opening angle around the poles
${\theta_{open}}$, which separates the polar ejecta and the tidal ejecta, and
the opening angle of the shocked cocoon ${\theta_{cocoon}}$. Finally, the
observer’s viewing angle ${\theta}$ and the host hydrogen column density
$N_{H}$ are taken as parameters as well. The model is implemented in the
Modular Open Source Fitter for Transients (MOSFiT; Guillochon et al. 2018),
which can be used to generate light curves in various bands. We generate an
ensemble of models, using the binary parameters $\mathcal{M}$ and $q$ from
Abbott et al. (2020). For the rest of the parameters, we sample 100
realizations of the ensemble from the prior distributions that Nicholl et al.
(2021) used for their GW170817 kilonova fit (which included three fixed
parameters, as we repeat here). The full set of parameters and the values used
are described in Table 5. These parameters produce a blue kilonova ejecta
component with $M_{ej,blue}=0.00154\pm 0.00036M_{\odot}$, and
$v_{k,blue}=0.1513\pm 0.0024c$, a red ejecta component with
$M_{ej,red}=0.0426\pm 0.0047M_{\odot}$, and $v_{k,red}=0.2586\pm 0.0024c$, and
a purple ejecta component with $M_{ej,purple}=0.0235\pm 0.00036M_{\odot}$,
$v_{k,purple}=0.0485\pm 0.0011c$, and $\kappa_{ej,purple}\simeq 5.6\
cm^{2}/g$. The total mass ejected of $M_{ej}=0.0676\pm 0.0045M_{\odot}$ has a
mean opacity of $\kappa_{ej}=7.95\pm 0.21\ cm^{2}/g$, making the light curves
bluer and more luminous overall than the Kasen et al. (2017) ones.
The comparison of reported non-detection limiting magnitudes to the GW170817
kilonova and the model light curves described above can be seen in Figure 7.
Figure 7: 5$\sigma$ non-detection limits of the GW190425 kilonova (triangles)
compared to a single kilonova model from Kasen et al. (2017) (dashed lines),
an ensemble of models from Nicholl et al. (2021) (semi-transparent solid grey
lines; see text for details), and the GW170817 kilonova (circles; see text for
data sources). The upper limit triangles are displayed fully opaque for
observations within the 50% GWTC-2 localization region, partially opaque for
observations outside the 50% GWTC-2 localization region but within the 90%
GWTC-2 localization region, and almost transparent for observations outside
the 90% GWTC-2 localization region. The limits are converted to absolute
magnitude after Milky Way extinction correction and using the distance
estimate at the HEALPix closest to the center of the pointing for each limit.
Data in the ‘clear’ band is unfiltered and is compared to models in the
$r$-band. Limits with magnitudes brighter than -20 are not shown.
We find that, had the GW190425 kilonova been similar to that of GW170817, it
should have been easily detected if the entire localization region had been
observed, even given the larger distance of GW190425 compared to that of
GW170817. In fact, limits in the $g$\- and $r$-bands are 3–4 magnitudes deeper
than what would have been required to observe a GW170817-like kilonova at
peak. However, given the different merger parameters, it is not clear whether
the GW190425 kilonova would have been as luminous as that of GW170817. Indeed,
the Kasen et al. (2017) models predict much fainter optical emission.
According to that model, the kilonova would just have barely been detected in
a few of the bands used in the search. The Nicholl et al. (2021) models, on
the other hand, predict a kilonova that would have been detected easily in the
optical bands as well. The two model predictions differ by as much as $6$
magnitudes in the optical bands.
## 4 Summary and Conclusions
We analyzed all reported ultraviolet, optical and infrared followup
observations of GW190425, the only robust BNS merger detected since GW170817,
and showed that:
1. 1.
Other than some sun and moon constrained regions, most of the localization
region of GW190425 was observable at the time of the trigger.
2. 2.
Enough observational resources were invested in the followup of GW190425 to
allow the accessible part of the 90% probability region to have been covered
potentially in a few hours. Instead, several regions were observed over 100
times, while others were never observed, and more than 5 days after the
merger, only 50% of the probability was covered.
3. 3.
Even if the GW190425 kilonova were 3–4 magnitudes fainter than the GW170817
kilonova in its optical peak, it could still have been detected. According to
more conservative models without a blue emission component, the kilonova might
have been only marginally detected around peak.
These results only take into account observations undertaken by facilities
which decided to followup GW190425. It is not possible to know which
additional facilities prefered not to trigger given the large localization
region, but which might have participated in the search, had they known that
roughly half of the probability was already being covered by others.
We conclude that even for relatively distant BNS mergers with large
localizations, coordinated followup can significantly increase the chances of
discovering an EM counterpart. The lack of coordination during the followup of
GW190425 strongly contributed to the failure to identify its EM counterpart.
In addition, lack of coordination weakens our ability to constrain models of
kilonova emission, given the amount of localization probability that was not
covered, and for which no limits are available. Even for smaller localiztions,
coordination can help find the kilonova sooner. Observations on few-hour time
scales are critical for constraining emission models (e.g. Arcavi, 2018).
Real-time coordination in such a competitive field, requiring very rapid
response, is challenging. Tools like the Treasure Map have been built exactly
to overcome this challenge. We encourage the community to report their
pointings to the Treasure Map, and to use the information there to guide their
search, even if it means searching lower probability areas (instead of
contributing the 100th observation to a higher probability region).
With the lack of additional significant BNS merger discoveries since GW190425,
it is possible that the BNS merger rate is on the lower end of its large
uncertainty range, and events as nearby and well localized as GW170817 are
likely very rare. If we wish to unleash the potential of GW-EM multi-messenger
astronomy for nuclear physics, astrophysics, and cosmology, we must better
coordinate our GW followup searches.
We thank A. Tohuvavohu for assisting with querying Treasure Map. We also thank
M. Nicholl for help using the kilonova model implemented in MOSFiT. I.A.
acknowledges support from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (grant agreement
number 852097), and from the Pazy foundation (grant number 216312). I.K. and
I.A. acknowledge support from the Israel Science Foundation (grant number
2752/19), from the United States - Israel Binational Science Foundation (BSF;
grant number 2018166). This paper has made use of version 2.3 of the GLADE
galaxy catalog (Dálya et al., 2018), the NASA/IPAC Infrared Science Archive,
which is funded by the National Aeronautics and Space Administration and
operated by the California Institute of Technology (Schlafly & Finkbeiner,
2011) and the SIMBAD database, operated at CDS, Strasbourg, France.
## References
* Abbott et al. (2017a) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a, Phys. Rev. Lett., 119, 161101, doi: 10.1103/PhysRevLett.119.161101
* Abbott et al. (2017b) —. 2017b, ApJ, 848, L13, doi: 10.3847/2041-8213/aa920c
* Abbott et al. (2017) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, The Astrophysical Journal Letters, 848, L12, doi: 10.3847/2041-8213/aa91c9
* Abbott et al. (2017) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, Nature, 551, 85, doi: 10.1038/nature24471
* Abbott et al. (2020) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2020, The Astrophysical Journal Letters, 892, L3, doi: 10.3847/2041-8213/ab75f5
* Abbott et al. (2021) Abbott, R., Abbott, T. D., Abraham, S., & et al. 2021, Physical Review X, 11, 021053, doi: 10.1103/PhysRevX.11.021053
* Acernese et al. (2014) Acernese, F., Agathos, M., Agatsuma, K., et al. 2014, Classical and Quantum Gravity, 32, 024001, doi: 10.1088/0264-9381/32/2/024001
* Ahumada et al. (2019a) Ahumada, T., Coughlin, M. W., Staats, K., Burdge, K., & Dekany, R. G. 2019a, GRB Coordinates Network, 24320, 1
* Ahumada et al. (2019b) Ahumada, T., Coughlin, M. W., Staats, K., & Dekany, R. G. 2019b, GRB Coordinates Network, 24198, 1
* Ahumada et al. (2019c) Ahumada, T., Coughlin, M. W., Staats, K., et al. 2019c, GRB Coordinates Network, 24343, 1
* Andreoni et al. (2017) Andreoni, I., Ackley, K., Cooke, J., et al. 2017, PASA, 34, e069, doi: 10.1017/pasa.2017.65
* Antier et al. (2020) Antier, S., Agayeva, S., Aivazyan, V., et al. 2020, MNRAS, 492, 3904, doi: 10.1093/mnras/stz3142
* Arcavi (2018) Arcavi, I. 2018, The Astrophysical Journal Letters, 855, L23, doi: 10.3847/2041-8213/aab267
* Arcavi et al. (2019) Arcavi, I., Howell, D. A., McCully, C., et al. 2019, GRB Coordinates Network, 24307, 1
* Arcavi et al. (2017a) Arcavi, I., McCully, C., Hosseinzadeh, G., et al. 2017a, ApJ, 848, L33, doi: 10.3847/2041-8213/aa910f
* Arcavi et al. (2017b) Arcavi, I., Hosseinzadeh, G., Howell, D. A., et al. 2017b, Nature, 551, 64, doi: 10.1038/nature24291
* Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
* Atwood et al. (2009) Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, ApJ, 697, 1071, doi: 10.1088/0004-637X/697/2/1071
* Axelsson et al. (2019) Axelsson, M., Omodei, N., Kocevski, D., & Longo, F. 2019, GRB Coordinates Network, 24174, 1
* Barnes & Kasen (2013) Barnes, J., & Kasen, D. 2013, The Astrophysical Journal, 775, 18, doi: 10.1088/0004-637X/775/1/18
* Bauswein et al. (2013) Bauswein, A., Goriely, S., & Janka, H. T. 2013, ApJ, 773, 78, doi: 10.1088/0004-637X/773/1/78
* Bhalerao et al. (2019) Bhalerao, V., Kumar, H., Karambelkar, V., et al. 2019, GRB Coordinates Network, 24201, 1
* Blazek et al. (2019) Blazek, M., Christensen, N., Howell, E., et al. 2019, GRB Coordinates Network, 24227, 1
* Bloom et al. (2019) Bloom, J. S., Zucker, C., Schlafly, E., et al. 2019, GRB Coordinates Network, 24337, 1
* Boër et al. (1999) Boër, M., Bringer, M., Klotz, A., et al. 1999, A&AS, 138, 579, doi: 10.1051/aas:1999356
* Bowen & Vaughan (1973) Bowen, I. S., & Vaughan, A. H., J. 1973, Appl. Opt., 12, 1430, doi: 10.1364/AO.12.001430
* Brown et al. (2013) Brown, T. M., Baliber, N., Bianco, F. B., et al. 2013, PASP, 125, 1031, doi: 10.1086/673168
* Burke et al. (2019) Burke, J., Hiramatsu, D., Arcavi, I., et al. 2019, GRB Coordinates Network, 24206, 1
* Butler et al. (2019) Butler, N., Watson, A. M., Troja, E., et al. 2019, GRB Coordinates Network, 24238, 1
* Camilletti et al. (2022) Camilletti, A., Chiesa, L., Ricigliano, G., et al. 2022, Monthly Notices of the Royal Astronomical Society, 516, 4760, doi: 10.1093/mnras/stac2333
* Castro-Tirado et al. (2019) Castro-Tirado, A. J., Hu, Y. D., Li, X. Y., et al. 2019, GRB Coordinates Network, 24214, 1
* Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560, doi: 10.48550/arXiv.1612.05560
* Chang et al. (2019) Chang, S. W., Wolf, C., Onken, C. A., Luvaul, L., & Scott, S. 2019, GRB Coordinates Network, 24325, 1
* Clark & Eardley (1977) Clark, J. P. A., & Eardley, D. M. 1977, ApJ, 215, 311, doi: 10.1086/155360
* Cobos et al. (2002) Cobos, F. J., Gonzalez, J. J., Tejada, C., Cepa, J., & Rasilla, J. L. 2002, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4832, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 249–257, doi: 10.1117/12.486467
* Coulter et al. (2017a) Coulter, D. A., Kilpatrick, C. D., Siebert, M. R., et al. 2017a, GRB Coordinates Network, 21529, 1
* Coulter et al. (2017b) Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017b, Science, 358, 1556, doi: 10.1126/science.aap9811
* Coulter et al. (2024) Coulter, D. A., Kilpatrick, C. D., Jones, D. O., et al. 2024, arXiv e-prints, arXiv:2404.15441, doi: 10.48550/arXiv.2404.15441
* Cowperthwaite et al. (2017) Cowperthwaite, P. S., Berger, E., Villar, V. A., et al. 2017, ApJ, 848, L17, doi: 10.3847/2041-8213/aa8fc7
* Dalton et al. (2006) Dalton, G. B., Caldwell, M., Ward, A. K., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6269, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. I. S. McLean & M. Iye, 62690X, doi: 10.1117/12.670018
* Dálya et al. (2018) Dálya, G., Galgóczi, G., Dobos, L., et al. 2018, MNRAS, 479, 2374, doi: 10.1093/mnras/sty1703
* DePoy et al. (2003) DePoy, D. L., Atwood, B., Belville, S. R., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Instrument Design and Performance for Optical/Infrared Ground-based Telescopes, ed. M. Iye & A. F. M. Moorwood, 827–838, doi: 10.1117/12.459907
* Díaz et al. (2017) Díaz, M. C., Macri, L. M., Garcia Lambas, D., et al. 2017, ApJ, 848, L29, doi: 10.3847/2041-8213/aa9060
* Djupvik & Andersen (2010) Djupvik, A. A., & Andersen, J. 2010, in Astrophysics and Space Science Proceedings, Vol. 14, Highlights of Spanish Astrophysics V, 211, doi: 10.1007/978-3-642-11250-8_21
* Drake et al. (2009) Drake, A. J., Djorgovski, S. G., Mahabal, A., et al. 2009, ApJ, 696, 870, doi: 10.1088/0004-637X/696/1/870
* Drout et al. (2017) Drout, M. R., Piro, A. L., Shappee, B. J., et al. 2017, Science, 358, 1570, doi: 10.1126/science.aaq0049
* Dudi et al. (2022) Dudi, R., Adhikari, A., Brügmann, B., et al. 2022, Phys. Rev. D, 106, 084039, doi: 10.1103/PhysRevD.106.084039
* Dyer et al. (2022) Dyer, M. J., Ackley, K., Lyman, J., et al. 2022, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 12182, Ground-based and Airborne Telescopes IX, ed. H. K. Marshall, J. Spyromilio, & T. Usuda, 121821Y, doi: 10.1117/12.2629369
* Eichler et al. (1989) Eichler, D., Livio, M., Piran, T., & Schramm, D. N. 1989, Nature, 340, 126, doi: 10.1038/340126a0
* Emerson et al. (2006) Emerson, J., McPherson, A., & Sutherland, W. 2006, The Messenger, 126, 41
* Evans et al. (2017) Evans, P. A., Cenko, S. B., Kennea, J. A., et al. 2017, Science, 358, 1565, doi: 10.1126/science.aap9580
* Fabricant et al. (2019) Fabricant, D., Fata, R., Epps, H., et al. 2019, PASP, 131, 075004, doi: 10.1088/1538-3873/ab1d78
* Farah et al. (2010) Farah, A., Barojas, E., Butler, N. R., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 77357Z, doi: 10.1117/12.857828
* Filippenko et al. (2001) Filippenko, A. V., Li, W. D., Treffers, R. R., & Modjaz, M. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 246, IAU Colloq. 183: Small Telescope Astronomy on Global Scales, ed. B. Paczynski, W.-P. Chen, & C. Lemme, 121
* Flaugher et al. (2015) Flaugher, B., Diehl, H. T., Honscheid, K., et al. 2015, AJ, 150, 150, doi: 10.1088/0004-6256/150/5/150
* Fletcher et al. (2019) Fletcher, C., Fermi-GBM Team, & GBM-LIGO/Virgo Group. 2019, GRB Coordinates Network, 24185, 1
* Foley et al. (2020) Foley, R. J., Coulter, D. A., Kilpatrick, C. D., et al. 2020, Monthly Notices of the Royal Astronomical Society, 494, 190, doi: 10.1093/mnras/staa725
* Foley et al. (2020) Foley, R. J., Coulter, D. A., Kilpatrick, C. D., et al. 2020, MNRAS, 494, 190, doi: 10.1093/mnras/staa725
* Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091
* Gompertz et al. (2020) Gompertz, B. P., Cutter, R., Steeghs, D., et al. 2020, MNRAS, 497, 726, doi: 10.1093/mnras/staa1845
* Górski et al. (2005) Górski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759, doi: 10.1086/427976
* Greiner et al. (2008) Greiner, J., Bornemann, W., Clemens, C., et al. 2008, PASP, 120, 405, doi: 10.1086/587032
* Grossman et al. (2014) Grossman, D., Korobkin, O., Rosswog, S., & Piran, T. 2014, Monthly Notices of the Royal Astronomical Society, 439, 757, doi: 10.1093/mnras/stt2503
* Guillochon et al. (2018) Guillochon, J., Nicholl, M., Villar, V. A., et al. 2018, ApJS, 236, 6, doi: 10.3847/1538-4365/aab761
* Guo et al. (2022) Guo, B., Peng, Q., Chen, Y., et al. 2022, Research in Astronomy and Astrophysics, 22, 055007, doi: 10.1088/1674-4527/ac5959
* Han et al. (2005) Han, W., Mack, P., Lee, C.-U., et al. 2005, PASJ, 57, 821, doi: 10.1093/pasj/57.5.821
* Han et al. (2021) Han, X., Xiao, Y., Zhang, P., et al. 2021, Publications of the Astronomical Society of the Pacific, 133, 065001, doi: 10.1088/1538-3873/abfb4e
* Hiramatsu et al. (2019a) Hiramatsu, D., Arcavi, I., Burke, J., et al. 2019a, GRB Coordinates Network, 24194, 1
* Hiramatsu et al. (2019b) Hiramatsu, D., Pellegrino, C., Burke, J., et al. 2019b, GRB Coordinates Network, 24225, 1
* Hosseinzadeh et al. (2019a) Hosseinzadeh, G., Cowperthwaite, P. S., Gomez, S., et al. 2019a, ApJ, 880, L4, doi: 10.3847/2041-8213/ab271c
* Hosseinzadeh et al. (2019b) Hosseinzadeh, G., Berger, E., Blanchard, P. K., et al. 2019b, GRB Coordinates Network, 24182, 1
* Hotokezaka et al. (2013) Hotokezaka, K., Kiuchi, K., Kyutoku, K., et al. 2013, Phys. Rev. D, 87, 024001, doi: 10.1103/PhysRevD.87.024001
* Howell et al. (2019) Howell, E., Christensen, N., Blazek, M., et al. 2019, GRB Coordinates Network, 24256, 1
* Hu et al. (2017) Hu, L., Wu, X., Andreoni, I., et al. 2017, Science Bulletin, 62, 1433, doi: 10.1016/j.scib.2017.10.006
* Hu et al. (2019a) Hu, Y. D., Li, X. Y., Carrasco, I., et al. 2019a, GRB Coordinates Network, 24270, 1
* Hu et al. (2019b) Hu, Y. D., Castro-Tirado, A. J., Li, X. Y., et al. 2019b, GRB Coordinates Network, 24324, 1
* Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
* Im et al. (2019) Im, M., Paek, G. S. H., Lim, G., et al. 2019, GRB Coordinates Network, 24183, 1
* Izzo et al. (2019) Izzo, L., Malesani, D. B., & Carini, R. 2019, GRB Coordinates Network, 24319, 1
* Kann et al. (2019) Kann, D. A., Thoene, C., Stachie, C., et al. 2019, GRB Coordinates Network, 24459, 1
* Kasen et al. (2013) Kasen, D., Badnell, N. R., & Barnes, J. 2013, The Astrophysical Journal, 774, 25, doi: 10.1088/0004-637X/774/1/25
* Kasen et al. (2017) Kasen, D., Metzger, B., Barnes, J., Quataert, E., & Ramirez-Ruiz, E. 2017, Nature, 551, 80, doi: 10.1038/nature24453
* Kashikawa et al. (2002) Kashikawa, N., Aoki, K., Asai, R., et al. 2002, PASJ, 54, 819, doi: 10.1093/pasj/54.6.819
* Kasliwal et al. (2017) Kasliwal, M. M., Nakar, E., Singer, L. P., et al. 2017, Science, 358, 1559, doi: 10.1126/science.aap9455
* Kasliwal et al. (2019) Kasliwal, M. M., Cannella, C., Bagdasaryan, A., et al. 2019, PASP, 131, 038003, doi: 10.1088/1538-3873/aafbc2
* Keller et al. (2007) Keller, S. C., Schmidt, B. P., Bessell, M. S., et al. 2007, PASA, 24, 1, doi: 10.1071/AS07001
* Kim et al. (2019) Kim, J., Im, M., Lee, C. U., et al. 2019, GRB Coordinates Network, 24216, 1
* Kim et al. (2016) Kim, S.-L., Lee, C.-U., Park, B.-G., et al. 2016, Journal of Korean Astronomical Society, 49, 37, doi: 10.5303/JKAS.2016.49.1.37
* Kong (2019) Kong, A. 2019, GRB Coordinates Network, 24303, 1
* Kulkarni (2005) Kulkarni, S. R. 2005, arXiv e-prints, astro, doi: 10.48550/arXiv.astro-ph/0510256
* Kyutoku et al. (2020) Kyutoku, K., Fujibayashi, S., Hayashi, K., et al. 2020, ApJ, 890, L4, doi: 10.3847/2041-8213/ab6e70
* Li et al. (2019a) Li, B., Xu, D., Zhou, X., & Lu, H. 2019a, GRB Coordinates Network, 24285, 1
* Li & Paczyński (1998) Li, L.-X., & Paczyński, B. 1998, The Astrophysical Journal, 507, L59, doi: 10.1086/311680
* Li et al. (2019b) Li, W. X., Chen, J. C., Xiaofeng, W. X., et al. 2019b, GRB Coordinates Network, 24267, 1
* Ligo Scientific Collaboration & VIRGO Collaboration (2019a) Ligo Scientific Collaboration, & VIRGO Collaboration. 2019a, GRB Coordinates Network, 24168, 1
* Ligo Scientific Collaboration & VIRGO Collaboration (2019b) —. 2019b, GRB Coordinates Network, 24228, 1
* LIGO Scientific Collaboration et al. (2015) LIGO Scientific Collaboration, T., Aasi, J., Abbott, B. P., et al. 2015, Classical and Quantum Gravity, 32, 074001, doi: 10.1088/0264-9381/32/7/074001
* Lipunov et al. (2010) Lipunov, V., Kornilov, V., Gorbovskoy, E., et al. 2010, Advances in Astronomy, 2010, 349171, doi: 10.1155/2010/349171
* Lipunov et al. (2017) Lipunov, V. M., Gorbovskoy, E., Kornilov, V. G., et al. 2017, ApJ, 850, L1, doi: 10.3847/2041-8213/aa92c0
* Lundquist et al. (2019) Lundquist, M. J., Paterson, K., Fong, W., et al. 2019, ApJ, 881, L26, doi: 10.3847/2041-8213/ab32f2
* Margutti & Chornock (2021) Margutti, R., & Chornock, R. 2021, ARA&A, 59, 155, doi: 10.1146/annurev-astro-112420-030742
* Masci et al. (2018) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2018, Publications of the Astronomical Society of the Pacific, 131, 018003, doi: 10.1088/1538-3873/aae8ac
* Meegan et al. (2009) Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, The Astrophysical Journal, 702, 791, doi: 10.1088/0004-637X/702/1/791
* Metzger et al. (2010) Metzger, B. D., Martínez-Pinedo, G., Darbha, S., et al. 2010, Monthly Notices of the Royal Astronomical Society, 406, 2650, doi: 10.1111/j.1365-2966.2010.16864.x
* Morihana et al. (2019a) Morihana, K., Jian, M., & Nagayama, T. 2019a, GRB Coordinates Network, 24219, 1
* Morihana et al. (2019b) —. 2019b, GRB Coordinates Network, 24328, 1
* Nagayama et al. (2003) Nagayama, T., Nagashima, C., Nakajima, Y., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Instrument Design and Performance for Optical/Infrared Ground-based Telescopes, ed. M. Iye & A. F. M. Moorwood, 459–464, doi: 10.1117/12.460770
* Nakar (2020) Nakar, E. 2020, Phys. Rep., 886, 1, doi: 10.1016/j.physrep.2020.08.008
* Nakar & Piran (2016) Nakar, E., & Piran, T. 2016, The Astrophysical Journal, 834, 28, doi: 10.3847/1538-4357/834/1/28
* Nicholl et al. (2021) Nicholl, M., Margalit, B., Schmidt, P., et al. 2021, MNRAS, 505, 3016, doi: 10.1093/mnras/stab1523
* Nicholl et al. (2021) Nicholl, M., Margalit, B., Schmidt, P., et al. 2021, Monthly Notices of the Royal Astronomical Society, 505, 3016, doi: 10.1093/mnras/stab1523
* Paek et al. (2019) Paek, G. S. H., Im, M., Lim, G., et al. 2019, GRB Coordinates Network, 24188, 1
* Park et al. (2012) Park, W.-K., Pak, S., Im, M., et al. 2012, PASP, 124, 839, doi: 10.1086/667390
* Perley & Copperwheat (2019) Perley, D. A., & Copperwheat, C. M. 2019, GRB Coordinates Network, 24202, 1
* Perley et al. (2019) Perley, D. A., Copperwheat, C. M., & Taggart, K. L. 2019, GRB Coordinates Network, 24314, 1
* Pian et al. (2017) Pian, E., D’Avanzo, P., Benetti, S., et al. 2017, Nature, 551, 67, doi: 10.1038/nature24298
* Piro & Kollmeier (2018) Piro, A. L., & Kollmeier, J. A. 2018, ApJ, 855, 103, doi: 10.3847/1538-4357/aaaab3
* Pozanenko et al. (2020) Pozanenko, A. S., Minaev, P. Y., Grebenev, S. A., & Chelovekov, I. V. 2020, Astronomy Letters, 45, 710, doi: 10.1134/S1063773719110057
* Pozanenko et al. (2018) Pozanenko, A. S., Barkov, M. V., Minaev, P. Y., et al. 2018, ApJ, 852, L30, doi: 10.3847/2041-8213/aaa2f6
* Radice et al. (2024) Radice, D., Ricigliano, G., Bhattacharya, M., et al. 2024, Monthly Notices of the Royal Astronomical Society, 528, 5836, doi: 10.1093/mnras/stae400
* Roming et al. (2005) Roming, P. W. A., Kennedy, T. E., Mason, K. O., et al. 2005, Space Sci. Rev., 120, 95, doi: 10.1007/s11214-005-5095-4
* Rosswog et al. (1999) Rosswog, S., Liebendörfer, M., Thielemann, F. K., et al. 1999, A&A, 341, 499, doi: 10.48550/arXiv.astro-ph/9811367
* Sagués Carracedo et al. (2021) Sagués Carracedo, A., Bulla, M., Feindt, U., & Goobar, A. 2021, MNRAS, 504, 1294, doi: 10.1093/mnras/stab872
* Sakamoto et al. (2019) Sakamoto, T., Barthelmy, S. D., Lien, A. Y., et al. 2019, GRB Coordinates Network, 24184, 1
* Sasada et al. (2019) Sasada, M., Akitaya, H., Nakaoka, T., et al. 2019, GRB Coordinates Network, 24192, 1
* Savchenko et al. (2019) Savchenko, V., Ferrigno, C., Martin-Carillo, A., et al. 2019, GRB Coordinates Network, 24178, 1
* Schady et al. (2019) Schady, P., Chen, T. W., Schweyer, T., Malesani, D. B., & Bolmer, J. 2019, GRB Coordinates Network, 24229, 1
* Schechter (1976) Schechter, P. 1976, ApJ, 203, 297, doi: 10.1086/154079
* Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103, doi: 10.1088/0004-637X/737/2/103
* Shappee et al. (2017) Shappee, B. J., Simon, J. D., Drout, M. R., et al. 2017, Science, 358, 1574, doi: 10.1126/science.aaq0186
* Smartt et al. (2017) Smartt, S. J., Chen, T. W., Jerkstrand, A., et al. 2017, Nature, 551, 75, doi: 10.1038/nature24303
* Smartt et al. (2024) Smartt, S. J., Nicholl, M., Srivastav, S., et al. 2024, MNRAS, 528, 2299, doi: 10.1093/mnras/stae100
* Steele et al. (2004) Steele, I. A., Smith, R. J., Rees, P. C., et al. 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5489, Ground-based Telescopes, ed. J. Oschmann, Jacobus M., 679–692, doi: 10.1117/12.551456
* Sun et al. (2019) Sun, T., Chen, J., Hu, L., et al. 2019, GRB Coordinates Network, 24234, 1
* Swift et al. (2022) Swift, J. J., Andersen, K., Arculli, T., et al. 2022, Publications of the Astronomical Society of the Pacific, 134, 035005, doi: 10.1088/1538-3873/ac5aca
* Tan et al. (2019a) Tan, H. J., Yu, P. C., Ngeow, C. C., & Ip, W. H. 2019a, GRB Coordinates Network, 24193, 1
* Tan et al. (2019b) Tan, H. J., Yu, P. C., Patil, A. S., et al. 2019b, GRB Coordinates Network, 24274, 1
* Tanaka & Hotokezaka (2013) Tanaka, M., & Hotokezaka, K. 2013, The Astrophysical Journal, 775, 113, doi: 10.1088/0004-637X/775/2/113
* Tanvir et al. (2019) Tanvir, N. R., Gonzalez-Fernandez, C., Levan, A. J., Malesani, D. B., & Evans, P. A. 2019, GRB Coordinates Network, 24334, 1
* Tanvir et al. (2017) Tanvir, N. R., Levan, A. J., González-Fernández, C., et al. 2017, ApJ, 848, L27, doi: 10.3847/2041-8213/aa90b6
* Tonry et al. (2018) Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505, doi: 10.1088/1538-3873/aabadf
* Troja et al. (2017) Troja, E., Piro, L., van Eerten, H., et al. 2017, Nature, 551, 71, doi: 10.1038/nature24290
* Troja et al. (2019) Troja, E., Watson, A. M., Becerra, R. L., et al. 2019, GRB Coordinates Network, 24335, 1
* Utsumi et al. (2017) Utsumi, Y., Tanaka, M., Tominaga, N., et al. 2017, PASJ, 69, 101, doi: 10.1093/pasj/psx118
* Valenti et al. (2017) Valenti, S., Sand, D. J., Yang, S., et al. 2017, ApJ, 848, L24, doi: 10.3847/2041-8213/aa8edf
* van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22, doi: 10.1109/MCSE.2011.37
* Vinko et al. (2019) Vinko, A., Bodi, L., Kriskovics, K., Sarneczky, K., & Pal, A. 2019, GRB Coordinates Network, 24367, 1
* Wang et al. (2019) Wang, C.-J., Bai, J.-M., Fan, Y.-F., et al. 2019, Research in Astronomy and Astrophysics, 19, 149, doi: 10.1088/1674-4527/19/10/149
* Waratkar et al. (2019) Waratkar, G., Kumar, H., Bhalerao, V., Stanzin, J., & Anupama, G. C. 2019, GRB Coordinates Network, 24304, 1
* Watson et al. (2016) Watson, A. M., Cuevas Cardona, S., Alvarez Nuñez, L. C., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, Ground-based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, L. Simard, & H. Takami, 99085O, doi: 10.1117/12.2233000
* Watson et al. (2019) Watson, A. M., Butler, N., Troja, E., et al. 2019, GRB Coordinates Network, 24239, 1
* Waxman et al. (2018) Waxman, E., Ofek, E. O., Kushnir, D., & Gal-Yam, A. 2018, MNRAS, 481, 3423, doi: 10.1093/mnras/sty2441
* Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9, doi: 10.1051/aas:2000332
* Winkler et al. (2003) Winkler, C., Courvoisier, T. J. L., Di Cocco, G., et al. 2003, A&A, 411, L1, doi: 10.1051/0004-6361:20031288
* Wyatt et al. (2020) Wyatt, S. D., Tohuvavohu, A., Arcavi, I., et al. 2020, ApJ, 894, 127, doi: 10.3847/1538-4357/ab855e
* Xin et al. (2019) Xin, L. P., Han, X. H., Wei, J. Y., et al. 2019, GRB Coordinates Network, 24315, 1
* Xu et al. (2019) Xu, D., Zhu, Z. P., Yu, B. Y., et al. 2019, GRB Coordinates Network, 24190, 1
* Yoshida et al. (2000) Yoshida, M., Shimizu, Y., Sasaki, T., et al. 2000, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4009, Advanced Telescope and Instrumentation Control Software, ed. H. Lewis, 240–249, doi: 10.1117/12.388394
* Zheng et al. (2019) Zheng, W., Zhang, K., Vasylyev, S., & Filippenko, A. V. 2019, GRB Coordinates Network, 24179, 1
## Appendix A Healpy Exclusive vs. Inclusive Methods
We present an illustration of the inclusive vs. the exclusive methods of
healpy pixel queries in Figure 8. The inclusive method can significantly
overestimate the area covered by a footprint with a field of view smaller than
the pixel size. The exclusive method is much more conservative, and hence we
chose it for our analysis.
Figure 8: An illustration of the exclusive and inclusive healpy pixel query
methods. Each black square represents a pixel centered around a black dot and
the yellow rectangles are hypothetical observational footprints. Grey pixels
are the ones returned according to each method and white pixels are ignored.
The inclusive method (left) returns every pixel that intersects with the
observed footprint, while the exclusive method (right) returns only pixels
with centers included in the observed footprint. For footprints with a field
of view (FOV) much smaller than the pixel area (bottom), the inclusive method
can significantly overestimate the area covered.
|
# Persistence of the Pattern in the Interior of 5d Moduli Spaces
Tom Rudelius<EMAIL_ADDRESS>Department of Mathematical
Sciences, Durham University, Durham DH1 3LE United Kingdom
###### Abstract
Castellano, Ruiz, and Valenzuela recently observed a remarkable “pattern” in
infinite-distance limits of moduli spaces in quantum gravity, which relates
the field space variation of the mass of the lightest tower of particles to
the field space variation of the species scale. In this work, we show how a
version of this pattern can be proven to hold for BPS particles and strings
throughout the vector multiplet moduli space of a 5d supergravity theory, even
in regions where the particle masses and string tensions are substantially
modified relative to their asymptotic behavior in the infinite-distance
limits. This suggests that a suitably defined version of the pattern may hold
not merely in the asymptotic limits of moduli space, but in the interior as
well.
###### pacs:
## I Introduction
Recent explorations of moduli spaces in quantum gravity have uncovered
evidence of universal structures within infinite-distance limits. As pointed
out long ago by Ooguri and Vafa [1], such limits seem to feature a tower of
light particles, whose masses decay exponentially with geodesic distance
$||\phi||$ as $m\sim\exp(-\alpha||\phi||)$. Such a tower of light particles
implies that quantum gravity becomes strongly coupled at an energy scale
$\Lambda_{\rm QG}$ (often referred to as the “species scale”) which is
parametrically smaller than the Planck scale, and which itself decays as
$\Lambda_{\rm QG}\sim\exp(-\lambda_{\rm QG}||\phi||)$.
It has long been understood that these exponential decay coefficients
$\alpha$, $\lambda_{\rm QG}$ should be order-one numbers in Planck units,111We
will set $8\pi G=M_{\textrm{Pl};d}^{2-d}=1$ throughout this letter. but
recently these values have been pinned down with greater precision. Reference
[2] presented strong evidence that the coefficient $\alpha$ of the lightest
tower in a given infinite-distance limit must satisfy
$\frac{1}{\sqrt{d-2}}\leq\alpha\leq\sqrt{\frac{d-1}{d-2}}$ in $d$ spacetime
dimensions, and several subsequent works [3, 4, 5] argued that the species
scale should decay with a coefficient
$\frac{1}{\sqrt{(d-1)(d-2)}}\leq\lambda_{\rm QG}\leq\frac{1}{\sqrt{d-2}}$.
Finally, [6, 7] provided strong evidence for not merely an inequality, but
actually an _equality_ , which relates the two decay coefficients mentioned
above. In the case of a 1-dimensional moduli space, this equality takes the
form $\alpha\lambda_{\rm QG}=1/(d-2)$. More generally, the equality takes the
form
$\frac{\vec{\nabla}m}{m}\cdot\frac{\vec{\nabla}\Lambda_{\rm QG}}{\Lambda_{\rm
QG}}=\frac{1}{d-2}\,,$ (1)
where $m$ is the characteristic mass scale of the lightest tower in the
infinite distance limit, the gradient $\vec{\nabla}$ involves the derivative
with respect to every massless modulus of the theory, and the inner product is
taken with respect to the metric $g_{ij}$ on moduli space, which is
parametrized by vacuum expectation values of the massless scalar fields.
Plugging in $m=m_{0}\exp(-\alpha\phi)$ and $\Lambda_{\rm
QG}=\Lambda_{0}\exp(-\lambda_{\rm QG}\phi)$, we see that (1) indeed reduces to
$\alpha\lambda_{\rm QG}=1/(d-2)$ for a 1-dimensional moduli space.
In [6, 7], equation (1)–which the authors referred to as the “pattern”–was
conjectured to apply in any infinite-distance limit in the moduli space.
Within such limits, the pattern may be viewed as a consequence of the Emergent
String Conjecture [8], which leads to tight constraints on the set of
exponentially light towers and the relationships between them [9]. The
Emergent String Conjecture holds that every infinite-distance limit is either
a decompactification limit or an emergent string limit in an appropriate
duality frame, which in turn implies that the lightest tower of particles is
either a tower of Kaluza-Klein modes or a tower of string oscillation modes.
In addition, it suggests that the scale $\Lambda_{\rm QG}$ should be (roughly)
identified with either a string scale or a higher-dimensional Planck scale,
which can be used to justify the pattern (1), as well as many of the other
bounds on light towers and the species scale mentioned above [9].
It is not so clear, however, how this whole story carries over to the interior
of moduli space, which is the region of greatest practical importance for
phenomenology and cosmology. As pointed out in [7], even the notion of the
“lightest tower” in moduli space is not generically well-defined outside the
asymptotic regimes of moduli space. One exception to this, however, comes from
towers of BPS particles, which remain well-defined throughout moduli space.
Similarly, although the definition of the species scale may be subtle, the
tension $T$ of a BPS string is well-defined throughout moduli space, and from
it we may define a string scale via $M_{\rm string}\equiv\sqrt{2\pi T}$.
One setting in which both BPS particles and BPS strings appear is supergravity
in five dimensions, which offers a rich and diverse landscape of quantum
gravity theories arising from M-theory compactifications on Calabi-Yau
threefolds. In this work, we explore this landscape, demonstrating that a
suitably refined version of (1) persists even into the interior of moduli
space, where other universal features of infinite-distance limits cease to be
valid.
In particular, we focus on vector multiplet moduli spaces of 5d supergravity
theories that feature emergent string limits, in which a BPS string becomes
tensionless in the asymptotic limit. In such theories, we find that the
pattern in (1) is satisfied throughout vector multiplet moduli space provided
that (a) $m$ is defined to be the mass scale associated with a tower of BPS
particles which become massless in some infinite-distance limit, and (b) the
quantum gravity scale $\Lambda_{\rm QG}$ is defined to be the string scale
$M_{\rm string}=\sqrt{2\pi T}$ of an emergent BPS string of tension $T$.
Remarkably, we find that (1) remains satisfied even when the lengths of the
individual vectors ${\vec{\nabla}m}/{m}$ and ${\vec{\nabla}\Lambda_{\rm
QG}}/{\Lambda_{\rm QG}}$ vary significantly throughout moduli space, and it
remains satisfied even after passing (via flop transitions of the Calabi-Yau
threefold) into distinct phases of the theory.
Our analysis suggests that the pattern is indeed pointing toward a universal
feature of quantum gravity, and it further suggests a possible refinement of
the pattern. Namely, our results suggest that one should set $m$ in (1) to be
the mass scale associated with the lightest tower of Kaluza-Klein modes or
string oscillation modes (which become massless in an appropriate infinite-
distance limit), and one should take the quantum gravity scale $\Lambda_{\rm
QG}$ to be a string scale or higher-dimensional Planck scale (each of which
also vanish in an infinite-distance limit). This choice for the quantum
gravity scale $\Lambda_{\rm QG}$ agrees with the species scale in asymptotic
limits, but in the interior of moduli space the two may differ in subtle but
important ways, as we discuss in §IV below. Although the full scope of this
refinement is unclear at present, it is encouraging that some features of the
asymptotic limits of moduli space seem to apply in the interior as well.
Clearly, further research is needed.
The remainder of this letter is structured as follows: in §II, we review
relevant aspects of 5d supergravity. In §III, we show how equation (1), when
properly defined, persists throughout vector multiplet moduli spaces of 5d
supergravity theories. In §IV, we end with a discussion of implications and
possible refinements of the pattern.
## II Basics of 5d Supergravity
In this section, we review relevant aspects of 5d supergravity. For further
details, see [10].
Many features of a 5d supergravity theory are captured by its prepotential, a
cubic function of the coordinates $Y^{I}$:
$\mathcal{F}=\frac{1}{6}C_{IJK}Y^{I}Y^{J}Y^{K}\,,$ (2)
where here and henceforth repeated indices are summed. In an M-theory
compactification to 5d on a Calabi-Yau threefold $X$, indices $I,J,K$ run from
$0$ to $h^{1,1}(X)-1$, the constants $C_{IJK}$ are the triple intersection
numbers of the manifold, and the moduli $Y^{I}$ are volumes of certain two-
cycles. The vector multiplet moduli space is given by the slice
$\mathcal{F}=1$, which means that this moduli space has dimension $n\equiv
h^{1,1}(X)-1$.
At a generic point in moduli space, the gauge group is $U(1)^{h^{1,1}(X)}$,
and the gauge kinetic matrix is given by
$a_{IJ}=\mathcal{F}_{I}\mathcal{F}_{J}-\mathcal{F}_{IJ},$ (3)
with
$\mathcal{F}_{I}=\partial_{I}\mathcal{F}=\frac{1}{2}C_{IJK}Y^{J}Y^{K},~{}~{}\mathcal{F}_{IJ}=\partial_{I}\partial_{J}\mathcal{F}=C_{IJK}Y^{K}.$
(4)
The metric on moduli space is given (up to a factor of $1/2$) by the pullback
of $a_{IJ}$ to the $\mathcal{F}=1$ slice:222Note that our definition of the
moduli space metric $g_{ij}$ differs from that of [10] by a factor of $2$:
$g_{ij}^{\rm here}=\frac{1}{2}\mathfrak{g}_{ij}^{\rm there}$.
$g_{ij}=\frac{1}{2}a_{IJ}\partial_{i}Y^{I}\partial_{j}Y^{J},$ (5)
where $\partial_{i}\equiv\frac{\partial}{\partial\phi^{i}}$, and we have
parametrized $Y^{I}=Y^{I}(\phi)$ in terms of the $n$ moduli $\phi^{i}$,
$i=1,...,n$.
Using the definitions above, it is possible to prove the following useful
relation between the inverse metric $g^{ij}$ and the inverse gauge kinetic
matrix $a^{IJ}$ [10]:
$\displaystyle
a^{IJ}=\frac{1}{2}g^{ij}\partial_{i}Y^{I}\partial_{j}Y^{J}+\frac{1}{3}Y^{I}Y^{J},$
(6)
Similarly, using the simple identities
$Y^{J}\mathcal{F}_{IJ}=2\mathcal{F}_{I}$,
$Y^{I}\mathcal{F}_{I}=3\mathcal{F}=1$, and $a_{IJ}a^{JK}=\delta_{I}^{K}$, it
is easy to see that
$a_{IJ}Y^{J}=\mathcal{F}_{I}~{}~{}\Rightarrow~{}~{}a^{IJ}\mathcal{F}_{I}=Y^{J}\,.$
(7)
A particle is labeled by a vector of electric charges, $q_{I}$. The mass of
such a particle is bounded from below by the BPS bound:
$\displaystyle
m(q_{I})\geq\left(\sqrt{2}\pi\right)^{1/3}|Z|=\left(\sqrt{2}\pi\right)^{1/3}|q_{I}Y^{I}|\,,$
(8)
Particles that saturate the BPS bound are called BPS particles.
Meanwhile, a string may carry magnetic charge under the gauge fields, which is
labeled by a magnetic charge vector $\tilde{q}^{I}$. The tension of such a
charged string is bounded from below by the BPS bound for strings:
$\displaystyle
T(\tilde{q}^{I})\geq\frac{1}{2}\left(\frac{1}{\sqrt{2}\pi}\right)^{1/3}|\tilde{Z}|=\frac{1}{2}\left(\frac{1}{\sqrt{2}\pi}\right)^{1/3}|\tilde{q}^{I}\mathcal{F}_{I}|\,.$
(9)
Strings that saturate this bound are called BPS strings.
## III The Pattern in 5d Supergravity
Suppose there exists an infinite-distance limit in vector multiplet moduli
space in which a tower of BPS particles of mass $m_{k}=km$ become light, and a
BPS string of tension $T$ becomes tensionless. The results of [2, 11, 9]
indicate that in the asymptotic limit, the tension of the BPS string and the
mass of the BPS particles will satisfy
$\displaystyle
g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{j}m}{m}=\frac{4}{3}\,,~{}~{}~{}~{}~{}g^{ij}\frac{\partial_{i}T}{T}\frac{\partial_{j}T}{T}=\frac{4}{3}\,,$
(10)
and
$\displaystyle
g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{i}T}{T}=\frac{2}{3}\,.$ (11)
Here, (11) matches the pattern equation (1) after setting $d=5$, $\Lambda_{\rm
QG}=M_{\rm string}\equiv\sqrt{2\pi T}$. In what follows, we will prove that
(11) remains true at _all_ points of the vector multiplet moduli space, even
points located in different phases of the theory. This is remarkable because
the relations in (10) do not, in general, hold beyond the asymptotic regimes
of moduli space.
To prove this, we must use one important fact: the tower of light particles
(of charge $kq_{I}$, for increasing $k$) and the asymptotically tensionless
BPS string (of magnetic charge $\tilde{q}^{I}$) have vanishing Dirac pairing,
i.e., $q_{I}\tilde{q}^{I}=0$. This fact follows from the discussion in §5 of
[10], which showed that all asymptotic limits of the moduli space feature a
tensionless BPS string and a tower of light BPS particles with vanishing Dirac
pairing.333At least, any such limit features a tensionless BPS string of
charge $\tilde{q}^{I}$ and a set of charges $kq_{I},k\in\mathbb{Z}$ with
$q_{I}\tilde{q}^{I}=0$ and $Z=q_{I}Y^{I}\rightarrow 0$. That these charges are
occupied by BPS particles remains to be proven in full generality, but this is
true in all known examples and follows from many well-supported quantum
gravity conjectures [10]. In contrast, boundaries of moduli space involving
light particles and tensionless strings of nontrivial Dirac pairing
($q_{I}\tilde{q}^{I}\neq 0$) lie at finite distance rather than infinite
distance.444Geometrically, such finite-distance boundaries correspond to the
collapse of a divisor to a point, whereas infinite-distance boundaries
correspond to the collapse of the entire Calabi-Yau threefold [12].
With this fact in hand, the relation (11) follows straightforwardly from the
definitions and identities of §II. To begin, we plug the formulas for the mass
(tension) of BPS particles (strings) into the left-hand side of (11):
$\displaystyle
g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{j}T}{T}=\frac{\partial_{i}(q_{I}Y^{I})}{q_{K}Y^{K}}\frac{\partial_{j}(\tilde{q}^{J}\mathcal{F}_{J})}{\tilde{q}^{L}\mathcal{F}_{L}}\,.$
(12)
From here, we use the definition of $\mathcal{F}_{IJ}$ in (4) to set
$\partial_{j}\mathcal{F}_{J}=\partial_{j}Y^{P}\mathcal{F}_{PJ}$, and we use
the identity of (6) to rewrite this as
$\displaystyle g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{j}T}{T}$
$\displaystyle=\frac{2q_{I}a^{IP}\mathcal{F}_{PJ}\tilde{q}^{J}}{(q_{K}Y^{K})(\tilde{q}^{L}\mathcal{F}_{L})}-\frac{2}{3}\frac{(q_{I}Y^{I})\mathcal{F}_{PJ}Y^{P}\tilde{q}^{J}}{(q_{K}Y^{K})(\tilde{q}^{L}\mathcal{F}_{L})}$
$\displaystyle=\frac{2q_{I}a^{IP}\mathcal{F}_{PJ}\tilde{q}^{J}}{(q_{K}Y^{K})(\tilde{q}^{L}\mathcal{F}_{L})}-\frac{4}{3}\,,$
(13)
where in the second line we have used the fact that
$\mathcal{F}_{PJ}Y^{P}=2\mathcal{F}_{J}$. Next, by the definition of $a_{IJ}$
in (3), we have
$\displaystyle\delta^{I}_{J}=a^{IP}a_{PJ}=a^{IP}(\mathcal{F}_{P}\mathcal{F}_{J}-\mathcal{F}_{PJ})\,,$
(14)
which implies
$\displaystyle
a^{IP}\mathcal{F}_{PJ}=-\delta^{I}_{J}+a^{IP}\mathcal{F}_{P}\mathcal{F}_{J}=-\delta^{I}_{J}+Y^{I}\mathcal{F}_{J}\,.$
(15)
where in the last equation we have used the identity in (7). Plugging this
into (13), we arrive finally at
$\displaystyle g^{ij}\frac{\partial_{i}m}{m}\frac{\partial_{j}T}{T}$
$\displaystyle=\frac{2}{3}-\frac{2q_{I}\tilde{q}^{I}}{(q_{K}Y^{K})(\tilde{q}^{L}\mathcal{F}_{L})}\,.$
(16)
Imposing the vanishing of the Dirac pairing between the BPS particles and the
BPS string, $q_{I}\tilde{q}^{I}=0$, we find the desired result (11).
## IV Discussion
Within 5d supergravity, we have shown that _any_ BPS particle of charge
$q_{I}$ and _any_ BPS string of charge $\tilde{q}^{I}$ with vanishing Dirac
pairing $q_{I}\tilde{q}^{I}=0$ satisfy the relation (11). This implies the
pattern of (1) after setting $\Lambda_{\rm QG}=M_{\rm string}=\sqrt{2\pi T}$.
It is relatively well-established that this relation is satisfied in the
asymptotic limits of moduli space; our result shows that it extends to the
_entire_ moduli space. This agrees with observations made in [7], which showed
that the pattern is satisfied exactly (i.e., not merely at leading order in an
expansion) in certain infinite-distance limits in vector multiplet moduli
spaces of 4d supergravity theories.
Our result is nontrivial for a couple of reasons. To begin, it is not hard to
find examples in which the closely related formulas of (10) are badly violated
in the interior of moduli space, even though they are indeed satisfied in the
asymptotic limits.555Type I’ string theory in 9d is another well-studied case
where formulas analogous to those of (10) are violated–see [13] for further
details. In some cases, the length of the vector $\partial_{i}m/m$ may even
diverge in certain limits of moduli space, yet even then the relation (11)
remains valid. (The $h^{1,1}=3$ geometry of [14], studied extensively in [10],
is one example where this occurs.) Secondly, the interior of a 5d vector
multiplet moduli space can feature complicated phase transitions
(geometrically realized by flop transitions or Weyl reflections [15]), each of
which involves a modification of the triple intersection numbers $C_{IJK}$ and
hence of the prepotential $\mathcal{F}$. Remarkably, our derivation of (11) is
unaffected by such modifications, and thus the relation persists even across
the distinct phases of vector multiplet moduli space. This suggests that,
whereas many features of infinite-distance limits of moduli spaces in quantum
gravity may be consequences of the weakly coupled lamppost, the pattern of
(1), suitably interpreted, may have a wider range of applicability.
In addition, our work suggests a possible modification of the pattern of (1),
in which the species scale is replaced by a string scale or higher-dimensional
Planck scale. One possibility takes the form
$\displaystyle\frac{\vec{\nabla}m}{m}\cdot\frac{\vec{\nabla}M}{M}$
$\displaystyle=\frac{1}{d-2}\,,$
$\displaystyle\text{with}~{}m=\min(\mathcal{T})~{}$ $\displaystyle\text{for
some}~{}M\in\mathcal{S}\cup\mathcal{P},$ (17)
where $\mathcal{T}$ is the set comprised of the mass scales of all of the
Kaluza-Klein towers or string oscillator towers which become light in the
various asymptotic limits of the theory, $\mathcal{S}$ is the set of all of
the string scales $M_{\rm string}=\sqrt{2\pi T}$ of the fundamental strings
which become light in the emergent string limits of the theory, and
$\mathcal{P}$ is the set of higher-dimensional Planck scales that arise in the
decompactification limits of the theory. In many cases, including in
asymptotic limits of moduli space, $M$ can in fact be chosen to be the
smallest mass scale in the set $\mathcal{S}\cup\mathcal{P}$.
This modified version of the pattern is more or less equivalent to (1) in
infinite-distance limits of moduli space. The advantage of the modification
can be seen, for instance, in Type IIB string theory in 10 dimensions. Here,
the species scale $\Lambda_{\rm sp}$ (defined as the coefficient of the
$R^{4}$ term in the action) varies smoothly over moduli space, reaching a
critical point with $\vec{\nabla}\Lambda_{\rm sp}=0$ at $\tau=i$ [16]. As a
result, (1) breaks down in a neighborhood of this point if the quantum gravity
scale is simply defined to be the species scale, $\Lambda_{\rm
QG}=\Lambda_{\rm sp}$, and $m$ is simply defined to be the mass scale of the
lightest tower. A similar breakdown may be expected to occur near critical
points of the species scale in 5d supergravity as well, which can occur at a
boundary between two phases of the theory. (The symmetric flop geometry of
[10] is one example where this occurs.)
In contrast, the choice of the scale $M$ used in (17) has the advantage that
it, like the mass scale of the lightest tower $m$, can have discontinuous
first derivative across the boundaries between different regions of moduli
space, as different string scales/higher-dimensional Planck scales may satisfy
the bound at different points in moduli space. As a result, (17) is satisfied
at all points in Type IIB moduli space and all points in the vector multiplet
moduli space of the aforementioned symmetric flop example of [10].
The full scope of the relation proposed in (17) remains to be understood.
Indeed, it is not even clear that it holds in 5d supergravity theories when
the lightest tower consists of non-BPS particles (such as the Kaluza-Klein
modes for a decompactification limit to 11d M-theory). Nonetheless, it is
exciting to think that the pattern first observed in [6, 7] may have a broader
range of applicability than previously realized, and it is refreshing to see
that some of the conjectured features of quantum gravity may extend beyond the
infinite-distance lamppost.
Acknowledgments. We thank Muldrow Etheredge, Ben Heidenreich, Damian van de
Heisteeg, Jacob McNamara, Ignacio Ruiz, Cumrun Vafa, Irene Valenzuela, and Max
Wiesner for useful discussions, and we further thank Ben Heidenreich, Ignacio
Ruiz, and Irene Valenzuela for comments on a draft. This work was supported in
part by STFC through grant ST/T000708/1.
## References
* Ooguri and Vafa [2007] H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl.Phys. B766, 21 (2007), arXiv:hep-th/0605264 [hep-th] .
* Etheredge _et al._ [2022] M. Etheredge, B. Heidenreich, S. Kaya, Y. Qiu, and T. Rudelius, Sharpening the Distance Conjecture in Diverse Dimensions, (2022), arXiv:2206.04063 [hep-th] .
* van de Heisteeg _et al._ [2023a] D. van de Heisteeg, C. Vafa, and M. Wiesner, Bounds on Species Scale and the Distance Conjecture, (2023a), arXiv:2303.13580 [hep-th] .
* Calderón-Infante _et al._ [2023] J. Calderón-Infante, A. Castellano, A. Herráez, and L. E. Ibáñez, Entropy Bounds and the Species Scale Distance Conjecture, (2023), arXiv:2306.16450 [hep-th] .
* van de Heisteeg _et al._ [2023b] D. van de Heisteeg, C. Vafa, M. Wiesner, and D. H. Wu, Bounds on Field Range for Slowly Varying Positive Potentials, (2023b), arXiv:2305.07701 [hep-th] .
* Castellano _et al._ [2023a] A. Castellano, I. Ruiz, and I. Valenzuela, A Universal Pattern in Quantum Gravity at Infinite Distance, (2023a), arXiv:2311.01501 [hep-th] .
* Castellano _et al._ [2023b] A. Castellano, I. Ruiz, and I. Valenzuela, Stringy Evidence for a Universal Pattern at Infinite Distance, (2023b), arXiv:2311.01536 [hep-th] .
* Lee _et al._ [2019] S.-J. Lee, W. Lerche, and T. Weigand, Emergent Strings from Infinite Distance Limits, (2019), arXiv:1910.01135 [hep-th] .
* [9] M. Etheredge, B. Heidenreich, T. Rudelius, I. Ruiz, and I. Valenzuela, To appear, .
* Alim _et al._ [2021] M. Alim, B. Heidenreich, and T. Rudelius, The Weak Gravity Conjecture and BPS Particles, (2021), arXiv:2108.08309 [hep-th] .
* Rudelius [2023] T. Rudelius, Gopakumar-Vafa Invariants and the Emergent String Conjecture, (2023), arXiv:2309.10024 [hep-th] .
* Witten [1996] E. Witten, Phase transitions in M theory and F theory, Nucl. Phys. B471, 195 (1996), arXiv:hep-th/9603150 [hep-th] .
* Etheredge _et al._ [2023] M. Etheredge, B. Heidenreich, J. McNamara, T. Rudelius, I. Ruiz, and I. Valenzuela, Running Decompactification, Sliding Towers, and the Distance Conjecture, (2023), arXiv:2306.16440 [hep-th] .
* Klemm _et al._ [1997] A. Klemm, P. Mayr, and C. Vafa, BPS states of exceptional noncritical strings, _Advanced quantum field theory. Proceedings, European Research Conference in the Memory of Claude Itzykson, La Londe les Maures, France, August 31-September 5, 1996_ , Nucl. Phys. Proc. Suppl. 58, 177 (1997), [,177(1996)], arXiv:hep-th/9607139 [hep-th] .
* Gendler _et al._ [2022] N. Gendler, B. Heidenreich, L. McAllister, J. Moritz, and T. Rudelius, Moduli Space Reconstruction and Weak Gravity, (2022), arXiv:2212.10573 [hep-th] .
* van de Heisteeg _et al._ [2023c] D. van de Heisteeg, C. Vafa, M. Wiesner, and D. H. Wu, Species Scale in Diverse Dimensions, (2023c), arXiv:2310.07213 [hep-th] .
|
# The relation between the symplectic group $Sp(4,\mathbb{R})$ and its Lie
algebra: its application in polymer quantum mechanics
Guillermo Chacón-Acosta<EMAIL_ADDRESS>Departamento de Matemáticas
Aplicadas y Sistemas, Universidad Autónoma Metropolitana Cuajimalpa, Vasco de
Quiroga 4871, Ciudad de México 05348, MEXICO Angel Garcia-Chung
<EMAIL_ADDRESS>Departamento de Física, Universidad Autónoma
Metropolitana - Iztapalapa,
San Rafael Atlixco 186, Ciudad de México 09340, México Universidad
Panamericana,
Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac, C.P. 01050 Ciudad de
México, México
###### Abstract
In this paper, we show the relation between $sp(4,\mathbb{R})$, the Lie
algebra of the symplectic group, and the elements of $Sp(4,\mathbb{R})$. We
use this result to obtain some special cases of symplectic matrices relevant
to the study of squeezed states. In this regard, we provide some applications
in quantum mechanics and analyze the squeezed polymer states obtained from the
polymer representation of the symplectic group. Remarkably, the polymer’s
dispersions are the same as those obtained for the squeezed states in the
usual representation.
###### Contents
1. I Introduction
2. II $Sp(4,\mathbb{R})$ group analysis and $sp(4,\mathbb{R})$ Lie algebra
3. III Quantum relations and examples
1. III.1 Relation between $sp(4,\mathbb{R})$ and ${\cal P}(2,\mathbb{R})$
2. III.2 Examples
1. III.2.1 Case ${\bf a},{\bf c}\neq{\bf 0}$ and ${\bf b}={\bf 0}$.
2. III.2.2 Case ${\bf c}={\bf a}=\mbox{diag}(a_{11},a_{22})$ and ${\bf b}=$diag$(b_{11},b_{22})$.
3. III.2.3 Case ${\bf a}={\bf c}={\bf 0}$ and ${\bf b}\neq{\bf 0}$.
4. IV Quantum representation and its applications
1. IV.1 Schrödinger representation of $Sp(2n,\mathbb{R})$
2. IV.2 Schrödinger representation of the squeeze operator for a bi-partite system.
3. IV.3 Covariance matrix for squeezed states
5. V Squeezed states in polymer quantum mechanics
6. VI Conclusions
7. VII Acknowledgments
8. VIII Appendix: calculation of ${\bf m}^{2n}$
9. IX Appendix: Series analysis
10. X Appendix: Covariance matrix coefficients
## I Introduction
Squeezed states are broadly used in many areas of physics walls1983squeezed ;
schnabel2017squeezed ; walls2007quantum ; adesso2014continuous ;
braunstein2005quantum . Of particular interest is the use of these squeezed
states in cosmology grishchuk1990squeezed ; polarski1996semiclassicality ;
lesgourgues1997quantum ; kiefer1998quantum ; guth1985quantum ;
martin2016quantum ; albrecht1994inflation ; gasperini1993quantum ;
grain2019squeezing , specifically when arguing for the emergence of semi-
classical behavior in the early universe. Loop Quantum Cosmology (LQC)
ashtekar2003mathematical ; bojowald2010canonical ; bojowald2011quantum ;
ashtekar2006quantum ; ashtekar2006quantum2 is another scenario in which
squeezed states are relevant. There, squeezed states for a single-mode show
some of the features of the quantum bounce and closely approximate solutions
to the classical Einstein equations taveras2008corrections ;
mielczarek2012gaussian ; gazeau2013quantum ; diener2014numerical ;
diener2014numerical2 ; corichi2011coherent . The squeezed states used in LQC
are constructed by hand, imposing the Gaussian form of the states to obtain
the squeezing nature of the dispersion relations. Moreover, the states
describe systems with only one degree of freedom, i.e., single-mode squeezed
states milburn1984multimode ; gerry2005introductory .
In quantum optics, squeezed states can be used to improve the sensitivity of
measurement devices beyond the usual quantum noise limits simon1988gaussian ;
braunstein2005quantum ; walls2007quantum ; adesso2014continuous ;
ma1990multimode ; schnabel2017squeezed ; pirandola2009correlation ;
weedbrook2012gaussian . They are defined by the squeeze operators’ action on
coherent states, or the vacuum state walls2007quantum ; adesso2014continuous ;
braunstein2005quantum . These operators are defined within the Fock
representation using the annihilation and creation operators, or in the Wigner
representation, using the Wigner functional. A particular squeezed state used
in quantum optics is the two-mode squeezed state which plays a prominent role
in the study of entanglement for bipartite systems. Particularly in the limit
when the amount of squeezing is infinitely large, the states become EPR-like
states EPRpaper .
Based on the relevance that squeezed states play in cosmology, LQC, and
quantum optics, one might ask whether there is a relation between them and
whether it is possible to obtain squeezed states in LQC the same way squeezed
states are defined in quantum optics. Recall that the construction used in LQC
for the squeezed states is somewhat artificial and does not correspond to any
mechanism in the cosmological events. Hence, exploring whether LQC formulation
admits an operator similar to the squeeze operator and whose action on some
state yields a squeezed state might pave the way to construct such a mechanism
in LQC.
To do so, one must consider that in the LQC, the representation of the
operators is not weakly continuous, hence the Fock representation is not
suitable for the physical description. Instead, the Schrödinger
representation, which is the scheme inherited from the quantization procedure,
seems to be the natural scheme to be considered ashtekar2003quantum ;
corichi2007polymer ; flores2013propagators ; velhinho2007quantum ;
pawlowski2014separable ; Garcia-Chung:2020cag . Despite the Schrödinger
representation of the squeeze operator might be obtained using the
representation of the infinitesimal squeeze operator via the exponential map
hall2018theory , in LQC this cannot be done. In addition to the mathematical
challenge that this operation requires in standard quantum mechanics, this is
not possible in LQC because there is no infinitesimal representation of the
squeeze operator. Therefore, in the present work, we will use the
representation of the symplectic group $Sp(2n,\mathbb{R})$ in polymer quantum
mechanics Garcia-Chung:2020cag . It is worth mentioning that Polymer Quantum
Mechanics (PQM) can be considered as a “toy model” of LQC because they share
Hilbert spaces with the same mathematical structures. As a result, the
representation of the symplectic group in PQM is mathematically the same as in
LQC.
To analyze the squeeze operator corresponding to the bipartite squeezed states
in LQC, we will provide the relation between the Lie algebra of the symplectic
group $sp(4,\mathbb{R})$ and the Lie group $Sp(4,\mathbb{R})$. As far as the
authors’ knowledge, this relation has not been reported before. With this
result, we show some specific cases and then move to the analysis within
polymer quantum mechanics. Also, this relation allows us to describe the
single-mode squeeze operator (specifically, the product of two single-mode
operators) as a particular case of a symplectic matrix in $Sp(4,\mathbb{R})$.
We will show that the squeezed states derived in this way for LQC share the
same features as those used in quantum optics. In particular, the
correlations’ structure is the same for both the single-mode and the two-mode
squeezed states. However, there is no need for a Gaussian-like structure for
the initial states upon which the polymer squeeze operators act and such
structure is absent in the polymer squeezed states.
This paper is organized as follows: in section (II) we calculate the relation
between $sp(4,\mathbb{R})$ and $Sp(4,\mathbb{R})$. In section (III) we discuss
the isomorphism between $sp(4,\mathbb{R})$ and the second-order polynomial
operators ${\cal P}(2,\mathbb{R})$ and provide some examples. In Section (IV),
we show some of the applications of the results given in section III; in
particular, we determine the covariance matrix for the squeezed states in
standard quantum mechanics. In Section (V), we analyze the squeeze operators’
representation in polymer quantum mechanics and construct the polymer squeezed
state. We also calculate the dispersion relations and show that they are equal
to those obtained for the standard squeezed states. We give our conclusion in
Section (VI).
## II $Sp(4,\mathbb{R})$ group analysis and $sp(4,\mathbb{R})$ Lie algebra
In this section we will detail the relation between an arbitrary element of
$sp(4,\mathbb{R})$ and its corresponding element in the group
$Sp(4,\mathbb{R})$. This relation is the main result of this section and has
not been reported as far as we know. First, let us introduce some preliminary
concepts and notation, which we will use throughout the paper, to make the
presentation self-contained.
Let us begin by considering the Poisson manifold $(\mathbb{R}^{2n},\\{,\\})$
with Poisson bracket for the coordinates $q_{j}$ and momenta $p_{j}$
($j=1,2,3,\dots,n$) given by
$\left\\{q_{j},q_{k}\right\\}=0,\qquad\left\\{p_{j},p_{k}\right\\}=0,\qquad\left\\{q_{j},p_{k}\right\\}=\delta_{jk}.$
(1)
These coordinates are collected using the array
$\vec{Y}^{T}=(q_{1},\,p_{1},\,q_{2},\,p_{2},\,\ldots,\,q_{n},\,p_{n})$ for
which the Poisson bracket (1) takes the form
$\left\\{\vec{Y},\vec{Y}^{T}\right\\}=\left(\begin{array}[]{cccc}{\bf J}&{\bf
0}&\cdots&{\bf 0}\\\ {\bf 0}&{\bf J}&\cdots&{\bf 0}\\\
\vdots&\vdots&\vdots&\vdots\\\ {\bf 0}&{\bf 0}&\cdots&{\bf
J}\end{array}\right)={\bf 1}_{n\times n}\otimes{\bf J},$ (2)
where the ${\bf 0}$ is the $2\times 2$ null matrix, ${\bf 1}_{n\times n}$ is
the identity matrix and the matrix ${\bf J}$ is given by
${\bf J}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right).$ (3)
The group action over the manifold $\mathbb{R}^{2n}$ is
$Sp(2n,\mathbb{R})\times\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n};\left({\bf
M},\vec{Y}\right)\mapsto\vec{Y}^{\prime T}={\bf M}\;\vec{Y}^{T},$ (4)
provided that the matrix ${\bf M}$ satisfies the condition
$\left({\bf 1}_{n\times n}\otimes{\bf J}\right)={\bf M}\left({\bf 1}_{n\times
n}\otimes{\bf J}\right){\bf M}^{T},$ (5)
where ${\bf M}^{T}$ is the transpose matrix. That is, the symplectic group
$Sp(2n,\mathbb{R})$ can be defined as the set of $2n\times 2n$ real matrices
satisfying (5) and, additionally, its group action on the Poisson manifold
$(\mathbb{R}^{2n},\\{,\\})$ is given by (4). Note that a “coordinatization” of
$({\mathbb{R}^{2n}},\\{,\\})$ different from $\vec{Y}$ yields a condition for
the symplectic group matrices different to that in (5). To show this, consider
now the array $\vec{X}^{T}=(\vec{q}^{\;T}\;,\;\vec{p}^{\;T})$ where
$\vec{q}^{\;T}=(q_{1},q_{2},\dots,q_{n})$ and
$\vec{p}^{\;T}=(p_{1},p_{2},\dots,p_{n})$ are the coordinates on the space
$\mathbb{R}^{2n}$. The Poisson bracket for this array is given by
$\left\\{\vec{X},\vec{X}^{T}\right\\}=\left(\begin{array}[]{cc}{\bf 0}&{\bf
1}_{n\times n}\\\ -{\bf 1}_{n\times n}&{\bf 0}\end{array}\right)={\bf
J}\otimes{\bf 1}_{n\times n}.$ (6)
The group action is now given by
$Sp(2n,\mathbb{R})\times\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n};\left(\widetilde{\bf
M},\vec{X}\right)\mapsto\vec{X}^{\prime}$ where $\vec{X}^{\prime}$ is
$\vec{X}^{\prime T}=\widetilde{\bf M}\,\vec{X}^{T},$ (7)
and the matrix $\widetilde{\bf M}$ satisfies
$\left({\bf J}\otimes{\bf 1}_{n\times n}\right)=\widetilde{\bf M}\left({\bf
J}\otimes{\bf 1}_{n\times n}\right)\widetilde{\bf M}^{T}.$ (8)
Hence, both conditions (5) and (8), can be considered as definitions for the
symplectic group in different “coordinatizations” of the phase space
$\mathbb{R}^{2n}$. Naturally, both group actions $\widetilde{\bf M}$ and ${\bf
M}$ are related via the similarity transformation ${\bf\Gamma}(n)$
adesso2014continuous as
$\widetilde{\bf M}={\bf\Gamma}(n)\;{\bf M}\;{\bf\Gamma}^{-1}(n),$ (9)
where ${\bf\Gamma}(n)$ is given by
$\vec{X}^{T}={\bf\Gamma}(n)\,\vec{Y}^{T},$ (10)
and is such that ${\bf\Gamma}^{T}(n)={\bf\Gamma}^{-1}(n)$. Since the present
work concerns the case where $n=2$, it is worth showing the explicit form of
${\bf\Gamma}(2)$ which is
${\bf\Gamma}(2)=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{array}\right).$ (11)
Having provided the two group actions over the manifold $\mathbb{R}^{2n}$
using different “coordinatizations” and their relation for arbitrary $n$, let
us now focus on the symplectic group $Sp(4,\mathbb{R})$. According to (5) this
group is given by $4\times 4$ real matrices ${\bf M}$ for which the following
condition holds
$\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf J}\end{array}\right)={\bf
M}\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf J}\end{array}\right){\bf
M}^{T},$ (12)
The matrix ${\bf M}$ can be written in block form as
${\bf M}:=\left(\begin{array}[]{cc}{\bf A}&{\bf B}\\\ {\bf C}&{\bf
D}\end{array}\right),$ (13)
where the $2\times 2$ block matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and ${\bf
D}$ satisfy the conditions
${\bf J}={\bf A}{\bf J}{\bf A}^{T}+{\bf B}{\bf J}{\bf B}^{T}={\bf C}{\bf
J}{\bf C}^{T}+{\bf D}{\bf J}{\bf D}^{T},\qquad{\bf 0}={\bf A}{\bf J}{\bf
C}^{T}+{\bf B}{\bf J}{\bf D}^{T},$ (14)
which result from (12).
The Lie algebra of $Sp(4,\mathbb{R})$, denoted as $sp(4,\mathbb{R})$, is given
by $4\times 4$ matrices ${\bf m}$ such that the exponential map hall2018theory
of the Lie algebra element ${\bf m}$ yields symplectic matrices ${\bf M}$
close to the identity, i.e.,
${\bf M}=e^{\bf m}:={\bf 1}+{\bf m}+\frac{1}{2}{\bf
m}^{2}+\dots+\frac{1}{n!}{\bf m}^{n}+\dots$ (15)
It can be shown that the matrices in $sp(4,\mathbb{R})$ can be written as the
product
${\bf m}=\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf J}\end{array}\right){\bf
L},$ (16)
where ${\bf L}$ is a real symmetric matrix written in block form as
${\bf L}=\left(\begin{array}[]{cc}{\bf a}&{\bf b}\\\ {\bf b}^{T}&{\bf
c}\end{array}\right),$ (17)
and where ${\bf b}$ is a $2\times 2$ real matrix, whereas ${\bf a}$ and ${\bf
c}$ are also real but $2\times 2$ symmetric matrices.
If a matrix ${\bf M}$ can be written as in (15), then its inverse ${\bf
M}^{-1}$, its transpose ${\bf M}^{T}$ and the $n$-power matrix $({\bf
M})^{n}$, can be written respectively as follows
$\displaystyle{\bf M}^{-1}$ $\displaystyle=$
$\displaystyle\exp{\left[-\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right){\bf L}\right]},$ (20) $\displaystyle{\bf M}^{T}$
$\displaystyle=$ $\displaystyle-\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right)\,{\bf M}^{-1}\,\left(\begin{array}[]{cc}{\bf J}&0\\\
0&{\bf J}\end{array}\right),$ (25) $\displaystyle({\bf M})^{n}$
$\displaystyle=$ $\displaystyle\exp{\left[\left(\begin{array}[]{cc}{\bf
J}&0\\\ 0&{\bf J}\end{array}\right)(n{\bf L})\right]}.$ (28)
Thus, the Lie algebra multiplication in $sp(4,\mathbb{R})$ is given by the
matrix commutator $[,]_{m}$. When this multiplication acts on two arbitrary
elements ${\bf m}_{1}$ and ${\bf m}_{2}$ gives the element ${\bf m}_{3}$
defined as
${\bf m}_{3}:=\left[{\bf m}_{1},{\bf
m}_{2}\right]_{m}=\left[\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right){\bf L}_{1},\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right){\bf L}_{2}\right]=\left(\begin{array}[]{cc}{\bf J}&0\\\
0&{\bf J}\end{array}\right){\bf L}_{3},$ (29)
where the matrix ${\bf L}_{3}$ is also a real symmetric matrix with components
of the form
${\bf L}_{3}=\left(\begin{array}[]{cc}{\bf a}_{1}{\bf J}{\bf a}_{2}+{\bf
b}_{1}{\bf J}{\bf b}^{T}_{2}-{\bf a}_{2}{\bf J}{\bf a}_{1}-{\bf b}_{2}{\bf
J}{\bf b}^{T}_{1}&{\bf a}_{1}{\bf J}{\bf b}_{2}+{\bf b}_{1}{\bf J}{\bf
c}_{2}-{\bf a}_{2}{\bf J}{\bf b}_{1}-{\bf b}_{2}{\bf J}{\bf c}_{1}\\\ {\bf
b}^{T}_{1}{\bf J}{\bf a}_{2}+{\bf c}_{1}{\bf J}{\bf b}^{T}_{2}-{\bf
b}^{T}_{2}{\bf J}{\bf a}_{1}-{\bf c}_{2}{\bf J}{\bf b}^{T}_{1}&{\bf c}_{1}{\bf
J}{\bf c}_{2}+{\bf b}^{T}_{1}{\bf J}{\bf b}_{2}-{\bf c}_{2}{\bf J}{\bf
c}_{1}-{\bf b}^{T}_{2}{\bf J}{\bf b}_{1}\end{array}\right),$ (30)
hence, ${\bf m}_{3}$ is clearly an element in $sp(4,\mathbb{R})$.
Up to this point, we introduced the main concepts and notations required to
derive the relation between $sp(4,\mathbb{R})$ and its corresponding Lie group
$Sp(4,\mathbb{R})$. Let us proceed then to obtain the explicit relation
between the block matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and ${\bf D}$ and
the Lie algebra element ${\bf L}$. It is worth noting that the following
procedure can be applied to higher-order symplectic groups $Sp(2n,\mathbb{R})$
for $n\geq 3$, being this the main reason for its exposition in this section.
Let us collect the even and odd terms of the expansion in (15) as follows
$\displaystyle{\bf M}$ $\displaystyle=$ $\displaystyle\left[{\bf
1}+\frac{1}{2!}{\bf m}^{2}+\dots+\frac{1}{(2n)!}{\bf m}^{2n}+\dots\right]+{\bf
m}\left[{\bf 1}+\frac{1}{3!}{\bf m}^{2}+\dots+\frac{1}{(2n+1)!}{\bf
m}^{2n}+\dots\right],$ (31)
where ${\bf m}^{2}$ takes the form
${\bf m}^{2}=\left(\begin{array}[]{cc}-(\det{\bf a}+\det{\bf b}){\bf
1}_{2\times 2}&{\bf J}{\bf d}\\\ -{\bf J}{\bf d}^{T}&-(\det{\bf b}+\det{\bf
c}){\bf 1}_{2\times 2}\end{array}\right),$ (32)
and the matrix ${\bf d}$ is defined as
${\bf d}={\bf a}{\bf J}{\bf b}+{\bf b}{\bf J}{\bf c}.$ (33)
As can be seen from the expansion (31), to obtain the expression for ${\bf M}$
we need first to determine ${\bf m}^{2n}$. In Appendix VIII we obtain the
expression for ${\bf m}^{2n}$ given in Eq. (119). Let us replace this result
in the series expansion (31), which, after collecting the even and odd terms,
gives the following
$\displaystyle{\bf A}$ $\displaystyle=$
$\displaystyle\alpha^{(e)}+(\alpha^{(o)}-\beta^{(o)}\det{\bf b})\,{\bf
J}\,{\bf a}+\beta^{(o)}{\bf J}\,{\bf b}\,{\bf J}\,{\bf c}\,{\bf J}\,{\bf
b}^{T},$ (34) $\displaystyle{\bf B}$ $\displaystyle=$
$\displaystyle(\gamma^{(o)}-\beta^{(o)}\det{\bf a})\,{\bf J}\,{\bf
b}+\beta^{(e)}({\bf J}\,{\bf a}\,{\bf J}\,{\bf b}+{\bf J}\,{\bf b}\,{\bf
J}\,{\bf c})+\beta^{(o)}{\bf J}\,{\bf a}\,{\bf J}\,{\bf b}\,{\bf J}\,{\bf c},$
(35) $\displaystyle{\bf C}$ $\displaystyle=$
$\displaystyle(\alpha^{(o)}-\beta^{(o)}\det{\bf c})\,{\bf J}\,{\bf
b}^{T}+\beta^{(e)}({\bf J}\,{\bf b}^{T}\,{\bf J}\,{\bf a}+{\bf J}\,{\bf
c}\,{\bf J}\,{\bf b}^{T})+\beta^{(o)}{\bf J}\,{\bf c}\,{\bf J}\,{\bf
b}^{T}\,{\bf J}\,{\bf a},$ (36) $\displaystyle{\bf D}$ $\displaystyle=$
$\displaystyle\gamma^{(e)}+(\gamma^{(o)}-\beta^{(o)}\det{\bf b})\,{\bf
J}\,{\bf c}+\beta^{(o)}{\bf J}\,{\bf b}^{T}\,{\bf J}\,{\bf a}\,{\bf J}\,{\bf
b}.$ (37)
The coefficients $\alpha^{(e)}$, $\alpha^{(o)}$, $\beta^{(e)}$, $\beta^{(o)}$,
$\gamma^{(e)}$ and $\gamma^{(o)}$ were defined in the appendix IX.
These expressions for the matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and ${\bf
D}$ link the components of the Lie algebra element ${\bf L}$ with the
corresponding symplectic matrix ${\bf M}$ and constitute the main result of
this section. Note also the “non-linear matrix relation” between the Lie
algebra elements and the group elements, particularly the role of the block
matrix ${\bf b}$.
A remarkable and direct application of this result is that it allows us to
compute the symplectic eigenvalues of the matrix ${\bf M}$. To do so, recall
that the characteristic polynomial for a $4\times 4$ matrix ${\bf M}$ with
$\det({\bf M})=1$ can be written in terms of the trace of its first three
powers by the expression
$\displaystyle\Lambda^{4}-(\mbox{Tr}({\bf
M}))\Lambda^{3}+\frac{\left[(\mbox{Tr}({\bf M}))^{2}-(\mbox{Tr}({\bf
M}^{2}))\right]}{2}\Lambda^{2}+\left[-\frac{(\mbox{Tr}({\bf
M}))^{3}}{6}+\frac{(\mbox{Tr}({\bf M}))(\mbox{Tr}({\bf M}^{2}))}{2}+\right.$
$\displaystyle\left.-\frac{(\mbox{Tr}({\bf M}^{3}))}{3}\right]\Lambda+1=0.$
(38)
where $\Lambda$ are the eigenvalues of the arbitrary matrix ${\bf M}$ and
$\mbox{Tr}({\bf M})$ is the trace of the matrix.
Using the relations (34)–(37) we obtain that the trace $\mbox{Tr}({\bf M})$ is
given by
$\mbox{Tr}({\bf
M})=2(\alpha^{(e)}+\gamma^{(e)})=\cosh\left(\sqrt{\lambda_{+}}\right)+\cosh\left(\sqrt{\lambda_{-}}\right),$
(39)
where $\alpha^{(e)}$ and $\gamma^{(e)}$ are given in (157) and (161),
respectively, and eigenvalues $\lambda_{\pm}$ are given in (142). Moreover,
due to the linearity of the trace and the relation (28), we can verify that
$\mbox{Tr}({\bf M}^{n})$ is given by
$\mbox{Tr}({\bf
M}^{n})=\cosh\left(n\sqrt{\lambda_{+}}\right)+\cosh\left(n\sqrt{\lambda_{-}}\right),$
(40)
from which we obtain the expressions for $\mbox{Tr}({\bf M}^{2})$ and
$\mbox{Tr}({\bf M}^{3})$.
Inserting all these expressions in (38) for a symplectic matrix of the form
(15), and when calculating the roots of the polynomial, we obtain that their
eigenvalues are given by
$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\cosh\left(\sqrt{\lambda_{+}}\right)-\sinh\left(\sqrt{\lambda_{+}}\right)=e^{-\sqrt{\lambda_{+}}},$
(41) $\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle\cosh\left(\sqrt{\lambda_{-}}\right)-\sinh\left(\sqrt{\lambda_{-}}\right)=e^{-\sqrt{\lambda_{-}}},$
(42) $\displaystyle\Lambda_{3}$ $\displaystyle=$
$\displaystyle\sinh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{-}}\right)=e^{\sqrt{\lambda_{-}}},$
(43) $\displaystyle\Lambda_{4}$ $\displaystyle=$
$\displaystyle\sinh\left(\sqrt{\lambda_{+}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)=e^{\sqrt{\lambda_{+}}},$
(44)
Note that if $\lambda_{+}>\lambda_{-}$, then the eigenvalues are ordered as
$\Lambda_{1}<\Lambda_{2}<\Lambda_{3}<\Lambda_{4}$ which coincides with the
results of Williamson’s theorem Arnold ; Gosson2006 .
In milburn1984multimode ; castanos1987matrix an alternative (and different)
formulation for each of the symplectic group generators was provided. Our
approach, however, not only reproduces to the same expressions for the
generators but also provides a direct relation with the Lie algebra matrix
generators ${\bf a}$, ${\bf b}$ and ${\bf c}$ a point that is absent in
milburn1984multimode ; castanos1987matrix . As a result, we can relate a
broader range of Lie algebra elements with their corresponding group elements.
Let us now show some of the relevant matrices and examples in the next section
in which this result can be applied.
## III Quantum relations and examples
This section provides three examples where the relation between the Lie
algebra element and the group element is explicit. However, before proceeding,
let us introduce additional concepts and notations (see hall2018theory ;
adesso2014continuous for more details), which will be relevant for the
quantum description.
### III.1 Relation between $sp(4,\mathbb{R})$ and ${\cal P}(2,\mathbb{R})$
Consider the Lie algebra formed by second-order (operator) polynomials,
denoted by ${\cal P}(2,\mathbb{R})$. An arbitrary element $\widehat{s}$ is
given as
$\displaystyle\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[\frac{a_{11}}{2}\widehat{q}^{2}_{1}+\frac{a_{12}}{2}(\widehat{q}_{1}\widehat{p}_{1}+\widehat{p}_{1}\widehat{q}_{1})+\frac{a_{22}}{2}\widehat{p}^{2}_{1}+b_{11}\widehat{q}_{1}\widehat{q}_{2}+b_{12}\widehat{q}_{1}\widehat{p}_{2}+b_{21}\widehat{p}_{1}\widehat{q}_{2}+b_{22}\widehat{p}_{1}\widehat{p}_{2}+\right.$
(45)
$\displaystyle\left.+\frac{c_{11}}{2}\widehat{q}^{2}_{2}+\frac{c_{12}}{2}(\widehat{q}_{2}\widehat{p}_{2}+\widehat{p}_{2}\widehat{q}_{2})+\frac{c_{22}}{2}\widehat{p}^{2}_{2}\right].$
Here, $\widehat{q}_{j}$ and $\widehat{p}_{j}$, with $j=1,2$, are the position
and momenta operators satisfying the canonical commutation relations
$[\widehat{q}_{j},\widehat{p}_{k}]=i\hbar\delta_{j,k}$, and $a_{ij},\;b_{ij}$
and $c_{ij}$ are all real coefficients. The reason for this notation is that
formally $i\hbar\widehat{s}$ is a self-adjoint operator to be represented in a
Hilbert space ${\cal H}$, hence the exponential map $e^{\widehat{s}}$ gives
rise to a unitary operator in ${\cal H}$. Thus, in this sense, this notation
smoothes the way to the quantum representation analysis in section IV.
It is easy to check that $\widehat{s}$ can be written in the following form
$\widehat{s}=-\frac{i}{2\hbar}\widehat{\bf
R}^{T}\left(\begin{array}[]{cccc}a_{11}&a_{12}&b_{11}&b_{12}\\\
a_{12}&a_{22}&b_{21}&b_{22}\\\ b_{11}&b_{21}&c_{11}&c_{12}\\\
b_{12}&b_{22}&c_{12}&c_{22}\end{array}\right)\widehat{\bf R},$ (46)
that is a symmetric matrix, and where
$\widehat{\bf
R}^{T}=\left(\begin{array}[]{cccc}\widehat{q}_{1}&\widehat{p}_{1}&\widehat{q}_{2}&\widehat{p}_{2}\end{array}\right).$
(47)
Instead of the matrix commutator, the Lie algebra multiplication in ${\cal
P}(2,\mathbb{R})$ is given by the operator commutator $[,]$. Therefore, the
Lie algebra multiplication of two elements ${\widehat{s}_{1}}$ and
${\widehat{s}_{2}}$ gives a third element $\widehat{s}_{3}$ of the form
$\widehat{s}_{3}=\left[\widehat{s}_{1},\widehat{s}_{2}\right]=-\frac{i}{2\hbar}\widehat{\bf
R}^{T}{\bf L}_{3}\widehat{\bf R},$ (48)
where the matrix ${\bf L}_{3}$ is given by (30). Due to ${\bf L}_{3}$ is a
symmetric matrix, the operator ${\widehat{s}}_{3}$ is clearly in ${\cal
P}(2,\mathbb{R})$. Naturally, this result provides the isomorphism between
$sp(4,\mathbb{R})$ and ${\cal P}(2,\mathbb{R})$, i.e., the map
$\iota:sp(4,\mathbb{R})\rightarrow{\cal P}(2,\mathbb{R});{\bf
m}\mapsto\widehat{s}=\iota({\bf m})=-\frac{i}{2\hbar}\widehat{\bf R}^{T}\,{\bf
L}\,\widehat{\bf R},$ (49)
and this map preserves the linear properties of both Lie algebras, i.e., it is
a Lie algebra isomorphism.
An implication of this isomorphism is that due to ${\cal P}(2,\mathbb{R})$ is
a Lie algebra isomorphic to $sp(4,\mathbb{R})$, then the exponential map of
its elements ($\widehat{s}\mapsto e^{\widehat{s}}$) gives a (quantum) unitary
operator ($e^{\widehat{s}}$) which can be seen as the (quantum) unitary
representation of $Sp(4,\mathbb{R})$ as showed in the following diagram
$\begin{array}[]{ccc}\boxed{sp(4,\mathbb{R})}&\xLongleftrightarrow{\iota}&\boxed{{\cal
P}(2,\mathbb{R})}\\\ \big{\downarrow}&&\big{\downarrow}\\\
\boxed{Sp(4,\mathbb{R})}&\longrightarrow&\boxed{\widehat{Sp}(4,\mathbb{R})}\end{array}$
(50)
We can expect that if a representation of ${\cal P}(2,\mathbb{R})$ in a
Hilbert space ${\cal H}$ is known, then there is also a representation of
$\widehat{Sp}(4,\mathbb{R})$ in ${\cal H}$. However, in some scenarios like in
polymer quantum mechanics and LQC, it is not possible to obtain the
representation of $\widehat{S}$ out of the representation of $\widehat{s}$ in
${\cal H}$. The reason is that some elements of ${\cal P}(2,\mathbb{R})$
cannot be represented in the corresponding Hilbert space. This difficulty can
be overcome if we can represent directly the exponential $e^{\widehat{s}}$
instead of its infinitesimal generator $\widehat{s}$. This approach was done
for the case of polymer quantum mechanics in Garcia-Chung:2020cag .
Consequently, due to the operators in (45) can be used to describe the
dynamics of many physical systems ranging from two decoupled quantum harmonic
oscillators to the bipartite squeeze operators, a polymer representation of
these operators is possible, as we will show in section V. More details about
these aspects will be provided in section V.
In the next subsection, we show some of the explicit forms of ${\bf M}$.
### III.2 Examples
#### III.2.1 Case ${\bf a},{\bf c}\neq{\bf 0}$ and ${\bf b}={\bf 0}$.
Let us consider the Lie algebra element with ${\bf b}={\bf 0}$ and ${\bf
a},{\bf c}\neq 0$, which, according to the expression (46), implies that there
is no interaction between the subsystems, that is, $\widehat{s}$ is of the
form
$\displaystyle\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{2\hbar}\left[a_{11}\widehat{q}^{2}_{1}+a_{12}(\widehat{q}_{1}\widehat{p}_{1}+\widehat{p}_{1}\widehat{q}_{1})+a_{22}\widehat{p}^{2}_{1}+c_{11}\widehat{q}^{2}_{2}+c_{12}(\widehat{q}_{2}\widehat{p}_{2}+\widehat{p}_{2}\widehat{q}_{2})+c_{22}\widehat{p}^{2}_{2}\right].$
(51)
In this case, ${\bf d}={\bf 0}$ and $\lambda_{+}=-\det{\bf a}$ and
$\lambda_{-}=-\det{\bf c}$. After inserting ${\bf b}=0$ and the expressions
for $\lambda_{\pm}$ in (157)-(162) we obtain the following symplectic matrix
${\bf M}_{1}=\left(\begin{array}[]{cc}\cosh\left(\sqrt{-\det{\bf
a}}\right)+\frac{\sinh\left(\sqrt{-\det{\bf a}}\right)}{\sqrt{-\det{\bf
a}}}\,{\bf J}\,{\bf a}&{\bf 0}\\\ {\bf 0}&\cosh\left(\sqrt{-\det{\bf
c}}\right)+\frac{\sinh\left(\sqrt{-\det{\bf c}}\right)}{\sqrt{-\det{\bf
c}}}\,{\bf J}\,{\bf c}\end{array}\right).$ (52)
As can be seen, both block matrices in (52) are elements of $Sp(2,\mathbb{R})$
hence, the Lie algebra elements given by the parameters ${\bf a}$ and ${\bf
c}$ can be considered as the Lie algebra generators of
$Sp(2,\mathbb{R})\otimes Sp(2,\mathbb{R})\subset Sp(4,\mathbb{R})$. Moreover,
the matrix ${\bf M}_{1}$ is diagonal if and only if ${\bf a}$ and ${\bf c}$
are anti-diagonal matrices, i.e., only when there are no squared terms in
(45).
An important symplectic matrix of this type is
$\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf J}\end{array}\right),$ (53)
which is often used to derive the transpose matrix as in (25). One can check
that this matrix can be obtained from (52) when ${\bf a}={\bf
c}=\mbox{diag}(\frac{\pi}{2},\frac{\pi}{2})$, i.e.,
$\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right)=\exp{\left[\frac{\pi}{2}\left(\begin{array}[]{cc}{\bf
J}&0\\\ 0&{\bf J}\end{array}\right)\right]}.$ (54)
#### III.2.2 Case ${\bf c}={\bf a}=\mbox{diag}(a_{11},a_{22})$ and ${\bf
b}=$diag$(b_{11},b_{22})$.
In this case the operator $\widehat{s}$ is of the form
$\displaystyle\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[a_{22}\left(\frac{1}{2}\widehat{p}^{2}_{1}+\frac{1}{2}\widehat{p}^{2}_{2}\right)+a_{11}\left(\frac{1}{2}\widehat{q}^{2}_{1}+\frac{1}{2}\widehat{q}^{2}_{2}\right)+b_{11}\widehat{q}_{1}\widehat{q}_{2}+b_{22}\widehat{p}_{1}\widehat{p}_{2}\right],$
(55)
i.e., the sub-systems interact via the matrix ${\bf b}$ but only with
couplings between coordinates $\widehat{q}_{1}\widehat{q}_{2}$ and momenta
operators $\widehat{p}_{1}\widehat{p}_{2}$. According to (142), the expression
for $\lambda_{\pm}$ for this case is
$\lambda_{\pm}=-(a_{11}\,a_{22}+b_{11}\,b_{22})\pm(a_{11}\,b_{22}+a_{22}\,b_{11})=-(a_{11}\mp
b_{11})(a_{22}\mp b_{22}).$ (56)
Note that when $b_{11}=\pm a_{11}$ or $b_{22}=\pm a_{22}$ the eigenvalues are
null and two particular systems emerge with their operators given by
$\displaystyle b_{11}=\pm a_{11}\rightarrow\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[a_{22}\left(\frac{1}{2}\widehat{p}^{2}_{1}+\frac{1}{2}\widehat{p}^{2}_{2}\right)+\frac{a_{11}}{2}\left(\widehat{q}_{1}\pm\widehat{q}_{2}\right)^{2}+b_{22}\widehat{p}_{1}\widehat{p}_{2}\right],$
(57) $\displaystyle b_{22}=\pm a_{22}\rightarrow\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[\frac{a_{22}}{2}\left(\widehat{p}_{1}+\widehat{p}_{2}\right)^{2}+a_{11}\left(\frac{1}{2}\widehat{q}^{2}_{1}+\frac{1}{2}\widehat{q}^{2}_{2}\right)+b_{11}\widehat{q}_{1}\widehat{q}_{2}\right].$
(58)
Both systems represent two interacting harmonic oscillators with a coupling
term in the momenta and the coordinates, respectively.
The symplectic matrix, denoted in this case as ${\bf M}_{2}$, is given by
${\bf
M}_{2}=\left(\begin{array}[]{cccc}\frac{\cosh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{22}+b_{22})S_{-}+(a_{22}-b_{22})S_{+}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)-\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{22}+b_{22})S_{-}+(b_{22}-a_{22})S_{+}}{2}\\\
\frac{(b_{11}-a_{11})S_{+}-(a_{11}+b_{11})S_{-}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{11}-b_{11})S_{+}-(a_{11}+b_{11})S_{-}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)-\cosh\left(\sqrt{\lambda_{+}}\right)}{2}\\\
\frac{\cosh\left(\sqrt{\lambda_{-}}\right)-\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{22}+b_{22})S_{-}+(b_{22}-a_{22})S_{+}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{22}+b_{22})S_{-}+(a_{22}-b_{22})S_{+}}{2}\\\
\frac{(a_{11}-b_{11})S_{+}-(a_{11}+b_{11})S_{-}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)-\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(b_{11}-a_{11})S_{+}-(a_{11}+b_{11})S_{-}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)}{2}\\\
\end{array}\right),$ (59)
where we introduce the parameters $S_{\pm}$ as
$S_{\pm}:=\frac{\sinh\left(\sqrt{\lambda_{\pm}}\right)}{\sqrt{\lambda_{\pm}}}.$
(60)
#### III.2.3 Case ${\bf a}={\bf c}={\bf 0}$ and ${\bf b}\neq{\bf 0}$.
In this case, the operator $\widehat{s}$ is of the form
$\displaystyle\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[b_{11}\widehat{q}_{1}\widehat{q}_{2}+b_{12}\widehat{q}_{1}\widehat{p}_{2}+b_{21}\widehat{p}_{1}\widehat{q}_{2}+b_{22}\widehat{p}_{1}\widehat{p}_{2}\right],$
(61)
and this system corresponds, as we will see in the next section, to the
general case of the squeeze operator for a bi-partite system
adesso2014continuous .
Note that in this case, not only the matrices ${\bf a}={\bf c}$ are null, but
also the matrix ${\bf d}$, which implies that
$\lambda_{+}=\lambda_{-}=-\det{\bf b}$. Once we replace these expressions in
(157)-(162) the symplectic matrix takes the form
${\bf M}_{3}=\left(\begin{array}[]{cc}\cosh\sqrt{-\det{\bf
b}}&\frac{\sinh\sqrt{-\det{\bf b}}}{\sqrt{-\det{\bf b}}}\,{\bf J}\,{\bf b}\\\
\frac{\sinh\sqrt{-\det{\bf b}}}{\sqrt{-\det{\bf b}}}\,{\bf J}\,{\bf
b}^{T}&\cosh\sqrt{-\det{\bf b}}\end{array}\right),$ (62)
where the block matrices ${\bf A}$ and ${\bf D}$ are diagonal matrices.
Clearly, when $\det{\bf b}<0$ the coefficients of ${\bf M}_{3}$ will be given
by hyperbolic functions. In case $\det{\bf b}>0$ the coefficients are
described by trigonometric functions instead.
After giving some examples of symplectic matrices obtained through the
exponential map, we are ready to show some of their quantum mechanics
applications, both in the standard representation and the so-called polymer or
loop representation.
## IV Quantum representation and its applications
The unitary representation of the group $Sp(2n,\mathbb{R})$ was given by
Moshinsky and Quesne in moshinsky1971linear . A review and a historical
analysis can be found in torre2005linear ; wolf2016development . However, to
be self-contained, we will show the main aspects of this group’s quantum
representation in standard quantum mechanics in the next subsection.
### IV.1 Schrödinger representation of $Sp(2n,\mathbb{R})$
The symplectic group is a non-compact group which implies an infinite-
dimensional Hilbert space for its unitary representation. Consider the Hilbert
space ${\cal H}=L^{2}(\mathbb{R}^{n},d\vec{x})$ where $d\vec{x}$ is the
standard Lebesgue measure. The unitary representation of $Sp(2n,\mathbb{R})$
is the map
$\widehat{C}:Sp(2n,\mathbb{R})\rightarrow{\cal U}({\cal H});\;\widetilde{\bf
M}\mapsto\widehat{C}_{\widetilde{\bf M}},$ (63)
where $\widehat{C}_{\widetilde{\bf M}}$ is a unitary operator over ${\cal H}$,
i.e., formally $\widehat{C}^{\dagger}_{\widetilde{\bf
M}}=\widehat{C}^{-1}_{\widetilde{\bf M}}$. Note that the group action
considered in this map is $\widetilde{\bf M}$ instead of ${\bf M}$, i.e., we
used the “coordinatization” given by $\vec{X}$ introduced in section II.
Hence, in order to obtain a quantum (unitary) representation of a given
symplectic matrix ${\bf M}$ we first have to transform it into the other group
action $\widetilde{\bf M}$ using Eq. (9) with the corresponding matrix
${\bf\Gamma}(n)$ given by (10) or ${\bf\Gamma}(2)$ for $Sp(4,\mathbb{R})$
given in (11).
The map $\widehat{C}$ is given by the integral operator
$\widehat{C}_{\widetilde{\bf M}}\Psi(\vec{x})=\int
d\vec{x}^{\prime}C_{\widetilde{\bf
M}}(\vec{x},\vec{x}^{\prime})\Psi(\vec{x}^{\prime}),\qquad\quad\Psi(\vec{x})\in{\cal
H},$ (64)
and the kernel $C_{\widetilde{\bf M}}(\vec{x},\vec{x}^{\prime})$ of this
integral is
$C_{\widetilde{\bf
M}}(\vec{x},\vec{x}^{\prime})=\frac{e^{\frac{i}{2\hbar}\left[\vec{x}^{T}{\widetilde{\bf
D}}{\widetilde{\bf B}}^{-1}\vec{x}-2\vec{x}^{\prime T}{\widetilde{\bf
B}}^{-1}\vec{x}+\vec{x}^{\prime T}{\widetilde{\bf B}}^{-1}{\widetilde{\bf
A}}\vec{x}^{\prime}\right]}}{\sqrt{(2\pi i\hbar)^{n}\det{\widetilde{\bf
B}}}}.$ (65)
According to moshinsky1971linear , this representation results from imposing
two conditions on the operators $\widehat{C}_{\widetilde{\bf M}}$. The first
one is given by
$\widehat{C}_{\widetilde{\bf
M}}\left(\begin{array}[]{c}\vec{\widehat{q}}^{T}\\\
\vec{\widehat{p}}^{T}\end{array}\right)\widehat{C}^{-1}_{\widetilde{\bf
M}}=\widetilde{\bf M}^{-1}\left(\begin{array}[]{c}\vec{\widehat{q}}^{T}\\\
\vec{\widehat{p}}^{T}\end{array}\right),$ (66)
and relates the symplectic group elements $\widetilde{\bf M}$ with the
operators $\widehat{C}_{\widetilde{\bf M}}$. Here,
$\vec{\widehat{q}}:=(\widehat{q}_{1},\widehat{q}_{2},\dots,\widehat{q}_{n})$
and
$\vec{\widehat{p}}:=(\widehat{p}_{1},\widehat{p}_{2},\dots,\widehat{p}_{n})$
are the coordinate and momenta operators associated to the Heisenberg Lie
algebra of the system. The second condition is that
$\widehat{C}_{\widetilde{\bf M}}\cdot\left(\widehat{C}_{\widetilde{\bf
M}}\right)^{\dagger}=\widehat{1},$ (67)
where $\widehat{1}$ is the identity operator and this results in the unitarity
of $\widehat{C}_{\widetilde{\bf M}}$.
The factor $\det{\bf B}$ in (65) gives rise to a well define operator even in
the case where the matrix ${\bf B}$ is singular (for more details see
moshinsky1971linear ; wolf2016development ). Finally, it is worth mentioning
that this representation (64) is valid for the entire symplectic group and not
just for those elements close to the group identity.
Since the fundamental operators are unbounded the condition (66) only holds in
a subspace given by the domain of the operators $\widehat{q}_{j}$ and
$\widehat{p}_{j}$ in ${\cal H}$. To obtain a condition valid in the full
Hilbert space, we are forced to introduce the exponentiated version of
$\widehat{q}_{j}$ and $\widehat{p}_{j}$, that is to say, the Weyl algebra.
Briefly, the Weyl algebra is a $C^{*}$-unital algebra whose generators,
denoted by $\widehat{W}(\vec{a},\vec{b})$, are related with $\widehat{q}_{j}$
and $\widehat{p}_{j}$ with the following relation
$\widehat{W}(\vec{a},\vec{b}):=e^{\frac{i}{\hbar}\left(\vec{a}\;\vec{\widehat{q}}^{\,T}+\vec{b}\;\vec{\widehat{p}}^{\;T}\right)},$
(68)
and such that the real arrays $\vec{a}=(a_{1},a_{2},\dots,a_{n})$ and
$\vec{b}=(b_{1},b_{2},\dots,b_{n})$, which have dimensions
$[a_{j}]=\mbox{momentum}$ and $[b_{j}]=\mbox{position}$, label the Weyl
algebra generators.
The standard Schrödinger representation of $\widehat{q}_{j}$ and
$\widehat{p}_{j}$ is now used to obtain a representation for the generators
$\widehat{W}(\vec{a},\vec{b})$ in ${\cal H}$ given by
$\widehat{W}(\vec{a},\vec{b})\Psi(\vec{x})=e^{\frac{i}{2\hbar}\vec{a}\;\vec{b}^{\,T}}e^{\frac{i}{\hbar}\vec{a}\;\vec{x}^{\,T}}\Psi(\vec{x}+\vec{b}),$
(69)
and such that the canonical commutation relations give rise to the Weyl
algebra mutiplication
$\widehat{W}(\vec{a}_{1},\vec{b}_{1})\widehat{W}(\vec{a}_{2},\vec{b}_{2})=e^{-\frac{i}{2\hbar}\left(\vec{a}_{1}\;\vec{b}^{T}_{2}-\vec{b}_{1}\;\vec{a}^{T}_{2}\right)}\widehat{W}(\vec{a}_{1}+\vec{a}_{2},\vec{b}_{1}+\vec{b}_{2}).$
(70)
Combining (66) and (68) to obtain the exponentiated version of (66) yields
$\widehat{C}_{\widetilde{\bf
M}}\,\widehat{W}(\vec{a},\vec{b})\,(\widehat{C}_{\widetilde{\bf
M}})^{-1}=\widehat{W}(\vec{a}\;\widetilde{\bf D}^{T}-\vec{b}\;\widetilde{\bf
C}^{T},-\vec{a}\;\widetilde{\bf B}^{T}+\vec{b}\;\widetilde{\bf A}^{T}),$ (71)
where $\widetilde{\bf A}$, $\widetilde{\bf B}$, $\widetilde{\bf C}$ and
$\widetilde{\bf D}$ are the block matrices in $\widetilde{\bf M}$. This
relation allows us to obtain a representation of the symplectic group in the
Hilbert space used in polymer quantum mechanics and in loop quantum cosmology
Garcia-Chung:2020cag .
We are now ready to show, in the next subsections, some of the applications of
the representation of $Sp(4,\mathbb{R})$ given by (64) and (65).
### IV.2 Schrödinger representation of the squeeze operator for a bi-partite
system.
The squeeze operator $\widehat{S}(\zeta)$ for a bi-partite system is given by
the exponential map
$\widehat{S}(\zeta)=e^{\widehat{s}_{\zeta}},$ (72)
where the operator $\widehat{s}_{\zeta}$, is given by
$\widehat{s}_{\zeta}:={\frac{1}{2}\left(\zeta^{*}\widehat{a}_{1}\widehat{a}_{2}-\zeta\widehat{a}^{\dagger}_{1}\widehat{a}^{\dagger}_{2}\right)}.$
(73)
Here, $\widehat{a}_{1}$ and $\widehat{a}_{2}$ are the annihilation operators
for each of the sub-systems, say, 1 and 2, of the bi-partite system,
$\widehat{a}^{\dagger}_{1}$ and $\widehat{a}^{\dagger}_{2}$ are their adjoint
operators respectively and $\zeta$ is a complex number labelling the amount of
squeezing. The operator $\widehat{S}(\zeta)$, when acting on the vacuum state
of the bi-partite quantum harmonic oscillators, gives a family of squeezed
states labelled by $\zeta$.
The operators in (73) are in the Fock representation, hence, let us transform
(73) to the Schrödinger representation described with operators
$\widehat{q}_{1}$, $\widehat{q}_{2}$, $\widehat{p}_{1}$ and $\widehat{p}_{2}$.
The relation between these representations is given by
$\displaystyle\widehat{a}_{j}=\frac{1}{\sqrt{2}}\frac{\widehat{q}_{j}}{l_{j}}+\frac{i}{\sqrt{2}}\frac{l_{j}\widehat{p}_{j}}{\hbar},\qquad\widehat{a}^{\dagger}_{j}=\frac{1}{\sqrt{2}}\frac{\widehat{q}_{j}}{l_{j}}-\frac{i}{\sqrt{2}}\frac{l_{j}\widehat{p}_{j}}{\hbar},$
(74)
for $j=1,2$ and $l_{j}:=\sqrt{\frac{\hbar}{m_{j}\omega_{j}}}$ where $m_{j}$
and $\omega_{j}$ stand for the masses and the frequencies of the oscillators.
Inserting these expressions for $\widehat{a}_{j}$ and
$\widehat{a}^{\dagger}_{j}$ in (73) the operator $\widehat{s}_{\zeta}$ takes
the following form
$\displaystyle\widehat{s}_{\zeta}$ $\displaystyle=$
$\displaystyle\frac{1}{2i\hbar}\left[\frac{\hbar\,\zeta_{y}}{l_{1}l_{2}}\;\widehat{q}_{1}\widehat{q}_{2}-\frac{l_{2}\,\zeta_{x}}{l_{1}}\;\widehat{q}_{1}\widehat{p}_{2}-\frac{l_{1}\,\zeta_{x}}{l_{2}}\;\widehat{p}_{1}\widehat{q}_{2}-\frac{l_{1}l_{2}\,\zeta_{y}}{\hbar}\widehat{p}_{1}\widehat{p}_{2}\right],$
(75)
where $\zeta_{x}$ and $\zeta_{y}$ are the real and imaginary parts of $\zeta$.
We now rewrite this operator in the form
$\widehat{s}_{\zeta}=-\frac{i}{4\hbar}(\vec{\widehat{R}}^{T}_{1},\vec{\widehat{R}}^{T}_{2})\left(\begin{array}[]{cc}0&{\bf
b}\\\ {\bf
b}^{T}&0\end{array}\right)\left(\begin{array}[]{c}\vec{\widehat{R}}_{1}\\\
\vec{\widehat{R}}_{2}\end{array}\right),$ (76)
where the matrix ${\bf b}$ is the following
${\bf
b}=\left(\begin{array}[]{cc}\frac{\hbar\,\zeta_{y}}{l_{1}l_{2}}&-\frac{l_{2}\,\zeta_{x}}{l_{1}}\\\
-\frac{l_{1}\,\zeta_{x}}{l_{2}}&-\frac{l_{1}l_{2}\,\zeta_{y}}{\hbar}\end{array}\right).$
(77)
Using the isomorphism $\iota^{-1}$ defined in (49) we obtain that the
corresponding Lie algebra element ${\bf
m}_{\zeta}=\iota^{-1}(\widehat{s}_{\zeta})$ is given by
${\bf m}_{\zeta}=\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right)\left(\begin{array}[]{cc}0&{\bf b}\\\ {\bf
b}^{T}&0\end{array}\right).$ (78)
Note that the Lie algebra matrix ${\bf m}_{\zeta}$ isomorphic to the squeeze
operator $\widehat{s}_{\zeta}$, is of the type given in the third case
(III.2.3).
To obtain the symplectic matrix associated to this Lie algebra element, we
insert ${\bf m}_{\zeta}$ and its expressions for ${\bf a}$, ${\bf b}$ and
${\bf c}$ in (62). This results in the following symplectic matrix
${\bf M}_{\bf
s}(r,\phi)=\left(\begin{array}[]{cccc}\cosh(r)&0&-\sinh(r)\cos(\phi)\frac{l_{1}}{l_{2}}&-\sinh(r)\sin(\phi)\frac{l_{1}l_{2}}{\hbar}\\\
0&\cosh(r)&-\sinh(r)\sin(\phi)\frac{\hbar}{l_{1}l_{2}}&\sinh(r)\cos(\phi)\frac{l_{2}}{l_{1}}\\\
-\sinh(r)\cos(\phi)\frac{l_{2}}{l_{1}}&-\sinh(r)\sin(\phi)\frac{l_{1}l_{2}}{\hbar}&\cosh(r)&0\\\
-\sinh(r)\sin(\phi)\frac{\hbar}{l_{1}l_{2}}&\sinh(r)\cos(\phi)\frac{l_{1}}{l_{2}}&0&\cosh(r)\\\
\end{array}\right),$ (79)
where $r$ and $\phi$ are defined as $\zeta=re^{i\phi}$. Matrix ${\bf M}_{\bf
s}(r,\phi)$ can be considered as the classical symplectic transformation such
that when represented in $L^{2}(\mathbb{R}^{2},d^{2}\vec{x})$, gives rise to
the quantum operator $\widehat{S}(\zeta)$. Naturally, this means also that the
unitary representation of $\widehat{S}(\zeta)$ in the Schrödinger
representation is given by $\widehat{C}_{{\bf M}_{\bf s}}$, i.e.,
$\widehat{C}_{{\bf M}_{\bf s}}=e^{\widehat{s}_{\zeta}}$.
It is worth to mention that although the expression (79) depends on the proper
lengths $l_{1}$ and $l_{2}$, the matrix ${\bf M}_{\bf s}(r,\phi)$ is
$\hbar-$independent, i.e., it is entirely a classical object. Also, matrix
${\bf M}_{\bf s}$ produces classical squeezing but of course, adapted to the
classical phase space, which in this case is $(\mathbb{R}^{4},\\{,\\})$. To
illustrate the squeezing and the rotation properties of the matrix ${\bf
M}_{\bf s}$ as a canonical transformation for different values of $r$ and
$\phi$ we consider its action on a circular trajectory
$(q_{1}(t),p_{1}(t),q_{2}(t),p_{2}(t))$ where,
$q_{j}(t)=\cos(t)\;q_{j}+\sin(t)\;p_{j}$ and
$p_{j}(t)=-\sin(t)\;q_{j}+\cos(t)\;p_{j}$, for $j=1,2$.
(a) Squeezing a circular trajectory.
(b) Rotating a squeezed trajectory with $r=0.6$.
Figure 1: In both figures, the solid, the dashed and the dotted lines
correspond to: (a) $r=0$, $r=0.3$ and $r=0.6$, respectively and (b) to
$\phi=0$, $\phi=\pi/4$ and $\phi=\pi/2$, respectively.
The action of ${\bf M}_{\bf s}$ on the trajectories is explicitly of the form
$\left(\begin{array}[]{c}q^{\prime}_{1}(t)\\\ p^{\prime}_{1}(t)\\\
q^{\prime}_{2}(t)\\\ p^{\prime}_{2}(t)\end{array}\right)={\bf M}_{\bf
s}\left(r,\phi\right)\left(\begin{array}[]{c}q_{1}(t)\\\ p_{1}(t)\\\
q_{2}(t)\\\ p_{2}(t)\end{array}\right).$ (80)
In figure 1 we showed the plot of $(q^{\prime}_{1}(t),p^{\prime}_{1}(t))$. As
expected, we note in Fig. (1(a)), that the amount of squeezing $r$ squeezes
the circular trajectory. Recall that symplectic transformations also preserve
the area, hence the trajectories are squeezed but the area is preserved. On
the other hand, the rotation angle $\phi$, as showed in (1(b)), rotates the
trajectories and also preserves the area.
Finally, observe that ${\bf M}_{\bf s}$ is given in the $\vec{Y}$
“coordinatization” which is not suitable for its quantum representation. To
make it suitable, let us provide the expression for the matrix $\widetilde{\bf
M}_{\bf s}$, which is given by
$\widetilde{\bf M}_{\bf
s}=\left(\begin{array}[]{cccc}\cosh(r)&-\frac{l_{1}\sinh(r)\cos(2\phi)}{l_{2}}&0&-\frac{l_{1}l_{2}\sinh(r)\sin(2\phi)}{\hbar}\\\
-\frac{l_{2}\sinh(r)\cos(2\phi)}{l_{1}}&\cosh(r)&-\frac{l_{1}l_{2}\sinh(r)\sin(2\phi)}{\hbar}&0\\\
0&-\frac{\hbar\sinh(r)\sin(2\phi)}{l_{1}l_{2}}&\cosh(r)&\frac{l_{2}\sinh(r)\cos(2\phi)}{l_{1}}\\\
-\frac{\hbar\sinh(r)\sin(2\phi)}{l_{1}l_{2}}&0&\frac{l_{1}\sinh(r)\cos(2\phi)}{l_{2}}&\cosh(r)\\\
\end{array}\right).$ (81)
This expression will be used to explore the analog of the bipartite squeeze
operator in polymer quantum mechanics in section V.
### IV.3 Covariance matrix for squeezed states
Now we will show the relation between the covariance matrix, denoted by ${\bf
V}^{(2)}$, and the symplectic matrix $\widetilde{\bf M}$. Let us consider the
state $|\Psi_{\widetilde{\bf M}}\rangle\in L^{2}(\mathbb{R}^{n},d\vec{x})$
related with the symplectic matrix $\widetilde{\bf M}$ as
$|\Psi_{\widetilde{\bf M}}\rangle=\widehat{C}_{\widetilde{\bf M}}|0\rangle,$
(82)
where $|0\rangle=\int d\vec{x}\;\Psi_{0}(\vec{x})\,|\vec{x}\rangle$ is the
state $|0\rangle=|0\rangle_{1}\otimes|0\rangle_{2}\dots|0\rangle_{n}$, and the
ket $|0\rangle_{j}$ is the vacuum state of the $j$-th quantum harmonic
oscillator. Note that this construction can be extended to other states in
$L^{2}(\mathbb{R}^{n},d\vec{x})$ and not only for $|0\rangle$. However, for
simplicity in our exposition, let us consider the simplest example of the
covariance matrix for $\widehat{C}_{\widetilde{\bf M}}|0\rangle$.
To obtain the covariance matrix we first calculate the following amplitude
$\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf M}}\rangle=\langle
0|\widehat{C}^{\dagger}_{\widetilde{\bf
M}}\;\widehat{W}(\vec{a},\vec{b})\;\widehat{C}_{\widetilde{\bf M}}|0\rangle,$
(83)
where $\widehat{W}(\vec{a},\vec{b})$ is the Weyl-algebra generator introduced
in (68). Combining (64), (69) and the Gaussian form of the vacuum state of the
system given by $n$-decoupled harmonic oscillators, we obtain the following
expression for the amplitude in (83)
$\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle=\exp\left\\{-\frac{1}{4}\left(\begin{array}[]{cc}\vec{a}&\vec{b}\end{array}\right)^{T}{\bf\Lambda}\left(\begin{array}[]{c}\vec{a}\\\
\vec{b}\end{array}\right)\right\\},$ (84)
where the matrix ${\bf\Lambda}$ is given by
${\bf\Lambda}:={\widetilde{\bf
M}}\left(\begin{array}[]{cc}\frac{1}{\hbar^{2}}{\bf L}^{2}&{\bf 0}\\\ {\bf
0}&{\bf L}^{-2}\end{array}\right){\widetilde{\bf M}}^{T},$ (85)
and ${\bf L}=\mbox{diag}(l_{1},l_{2},\dots,l_{n})$, where $l_{j}$ was defined
earlier (74).
The covariance matrix ${\bf V}^{(2)}$ has components given by
${\bf V}^{(2)}=\left(\begin{array}[]{cc}\langle\Psi_{\widetilde{\bf
M}}|\widehat{x}_{j}\;\widehat{x}_{k}|\Psi_{\widetilde{\bf
M}}\rangle&\frac{1}{2}\langle\Psi_{\widetilde{\bf
M}}|\left\\{\widehat{x}_{j},\;\widehat{p}_{k}\right\\}|\Psi_{\widetilde{\bf
M}}\rangle\\\ \frac{1}{2}\langle\Psi_{\widetilde{\bf
M}}|\left\\{\widehat{p}_{j},\;\widehat{x}_{k}\right\\}|\Psi_{\widetilde{\bf
M}}\rangle&\langle\Psi_{\widetilde{\bf
M}}|\widehat{p}_{j}\;\widehat{p}_{k}|\Psi_{\widetilde{\bf
M}}\rangle\end{array}\right),$ (86)
and these components can be obtained from (84) using the following relations
$\displaystyle\langle\Psi_{\widetilde{\bf
M}}|\widehat{x}_{j}\;\widehat{x}_{k}|\Psi_{\widetilde{\bf
M}}\rangle=-\hbar^{2}\partial^{2}_{a_{j}a_{k}}\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle|_{\vec{a},\vec{b}=0},$ (87)
$\displaystyle\frac{1}{2}\langle\Psi_{\widetilde{\bf
M}}|\left\\{\widehat{x}_{j},\;\widehat{p}_{k}\right\\}|\Psi_{\widetilde{\bf
M}}\rangle=-\hbar^{2}\partial^{2}_{a_{j}b_{k}}\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle|_{\vec{a},\vec{b}=0},$ (88)
$\displaystyle\frac{1}{2}\langle\Psi_{\widetilde{\bf
M}}|\left\\{\widehat{p}_{j},\;\widehat{x}_{k}\right\\}|\Psi_{\widetilde{\bf
M}}\rangle=-\hbar^{2}\partial^{2}_{b_{j}a_{k}}\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle|_{\vec{a},\vec{b}=0},$ (89)
$\displaystyle\langle\Psi_{\widetilde{\bf
M}}|\widehat{p}_{j}\;\widehat{p}_{k}|\Psi_{\widetilde{\bf
M}}\rangle=-\hbar^{2}\partial^{2}_{b_{j}b_{k}}\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle|_{\vec{a},\vec{b}=0}.$ (90)
Remarkably, the resulting expression for ${\bf V}^{(2)}$ in terms of the
symplectic matrix $\widetilde{\bf M}$ is
${\bf V}^{(2)}=\frac{1}{2}{\widetilde{\bf M}}\left(\begin{array}[]{cc}{\bf
L}^{2}&{\bf 0}\\\ {\bf 0}&\hbar^{2}{\bf
L}^{-2}\end{array}\right){\widetilde{\bf M}}^{T},$ (91)
and this shows the direct relation between the covariance matrix ${\bf
V}^{(2)}$ for the state $|\Psi_{\widetilde{\bf M}}\rangle$ and the symplectic
matrix ${\widetilde{\bf M}}$ associated with the unitary operator
$\widehat{C}_{\widetilde{\bf M}}$. Moreover, if we now consider the definition
(8), it can be shown that $\frac{2}{\hbar}{\bf V}^{(2)}$ is actually a
symplectic matrix. Let us apply this formula to some of the systems considered
before.
Consider the matrix ${\bf M}_{1}$ given in (52). Using (9) we obtain the
expression for ${\widetilde{\bf M}}_{1}$ which then is replaced in (91) giving
rise to the following covariance matrix
${\bf
V}^{(2)}_{1}=\frac{1}{2}\left(\begin{array}[]{cccc}{V}^{(2)}_{11}&0&{V}^{(2)}_{13}&0\\\
0&{V}^{(2)}_{22}&0&{V}^{(2)}_{24}\\\ {V}^{(2)}_{13}&0&{V}^{(2)}_{33}&0\\\
0&{V}^{(2)}_{24}&0&{V}^{(2)}_{44}\end{array}\right).$ (92)
Its components are given in the appendix (X) and in the particular case where
$a_{11}=a_{22}=c_{11}=c_{22}=0$, the covariance matrix ${\bf V}^{(2)}_{1}$
reduces to
${\bf
V}^{(2)}_{1}=\frac{1}{2}\left(\begin{array}[]{cccc}l^{2}_{1}e^{2a_{12}}&0&0&0\\\
0&l^{2}_{2}e^{2c_{12}}&0&0\\\ 0&0&\frac{\hbar^{2}}{l^{2}_{1}}e^{-2a_{12}}&0\\\
0&0&0&\frac{\hbar^{2}}{l^{2}_{2}}e^{-2c_{12}}\end{array}\right).$ (93)
We use this result to derive the uncertainties in the coordinates $\Delta
x_{j}$ for $j=1,2$ which according to (93) are given by
$\Delta x_{j}:=\sqrt{\langle\Psi_{\widetilde{\bf
M}}|\widehat{x}^{2}_{j}|\Psi_{\widetilde{\bf
M}}\rangle-\langle\Psi_{\widetilde{\bf
M}}|\widehat{x}_{j}|\Psi_{\widetilde{\bf
M}}\rangle^{2}}=\frac{l_{j}e^{\alpha_{j}}}{\sqrt{2}},$ (94)
where due to the symmetry of the vacuum wavefunction we have
$\langle\Psi_{\widetilde{\bf M}}|\widehat{x}_{j}|\Psi_{\widetilde{\bf
M}}\rangle=0$. This can be verified calculating the first derivatives in (84).
Here, for simplicity we make $\alpha_{1}=a_{12}$ and $\alpha_{2}=c_{12}$. Note
the remarkably property of the squeezed states like in (82) which is that
$\Delta x_{j}$ can be smaller than the proper length of the vacuum state
$l_{j}$ when $\alpha_{j}<0$.
Similarly, the uncertainties in the momenta $\Delta p_{j}$, are
$\Delta p_{j}:=\sqrt{\langle\Psi_{\widetilde{\bf
M}}|\widehat{p}^{2}_{j}|\Psi_{\widetilde{\bf
M}}\rangle-\langle\Psi_{\widetilde{\bf
M}}|\widehat{p}_{j}|\Psi_{\widetilde{\bf
M}}\rangle^{2}}=\frac{\hbar}{\sqrt{2}l_{j}e^{\alpha_{j}}},$ (95)
which can also be smaller than $\frac{\hbar}{l_{j}}$ when $\alpha_{j}>0$ and
also, $\langle\Psi_{\widetilde{\bf M}}|\widehat{p}_{j}|\Psi_{\widetilde{\bf
M}}\rangle=0$ as the previous case. Nevertheless, both uncertainties satisfy
Heisenberg’s uncertainty principle:
$\Delta x_{j}\,\Delta
p_{j}=\left(\frac{l_{j}e^{\alpha_{j}}}{\sqrt{2}}\right)\left(\frac{\hbar}{\sqrt{2}l_{j}e^{\alpha_{j}}}\right)=\hbar/2.$
(96)
Another interesting covariance matrix is the one related with the bipartite
squeeze operator (81) derived in the previous subsection. Inserting (81) in
(91) yields
${\bf
V}^{(2)}(r,\phi)=\left(\begin{array}[]{cccc}\frac{l_{1}^{2}\cosh(2r)}{2}&-\frac{l_{1}l_{2}\sinh(2r)\cos(2\phi)}{2}&0&-\frac{l_{1}\hbar\sinh(2r)\sin(2\phi)}{2l_{2}}\\\
-\frac{l_{1}l_{2}\sinh(2r)\cos(2\phi)}{2}&\frac{l_{2}^{2}\cosh(2r)}{2}&-\frac{l_{2}\hbar\sinh(2r)\sin(2\phi)}{2l_{1}}&0\\\
0&-\frac{l_{2}\hbar\sinh(2r)\sin(2\phi)}{2l_{1}}&\frac{\hbar^{2}\cosh(2r)}{2l_{1}^{2}}&\frac{\hbar^{2}\sinh(2r)\cos(2\phi)}{2l_{1}l_{2}}\\\
-\frac{l_{1}\hbar\sinh(2r)\sin(2\phi)}{2l_{2}}&0&\frac{\hbar^{2}\sinh(2r)\cos(2\phi)}{2l_{1}l_{2}}&\frac{\hbar^{2}\cosh(2r)}{2l_{2}^{2}}\end{array}\right),$
(97)
and this allows us to determine the correlation between the second moments of
the subsystem 1 and the subsystem 2
$\displaystyle\langle\Psi_{\widetilde{\bf
M}}|\left(\widehat{x}_{1}\pm\widehat{x}_{2}\right)^{2}|\Psi_{\widetilde{\bf
M}}\rangle$ $\displaystyle=$
$\displaystyle\frac{e^{2r}}{4}\left[l^{2}_{1}+l^{2}_{2}\mp
2l_{1}\,l_{2}\cos(2\phi)\right]+\frac{e^{-2r}}{4}\left[l^{2}_{1}+l^{2}_{2}\pm
2l_{1}\,l_{2}\cos(2\phi)\right],$ (98)
$\displaystyle\langle\Psi_{\widetilde{\bf
M}}|\left(\widehat{p}_{1}\pm\widehat{p}_{2}\right)^{2}|\Psi_{\widetilde{\bf
M}}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}e^{2r}}{4l^{2}_{1}\,l^{2}_{2}}\left[l^{2}_{1}+l^{2}_{2}\pm
2l_{1}\,l_{2}\cos(2\phi)\right]+\frac{\hbar^{2}e^{-2r}}{4l^{2}_{1}\,l^{2}_{2}}\left[l^{2}_{1}+l^{2}_{2}\mp
2l_{1}\,l_{2}\cos(2\phi)\right].$ (99)
In the particular case where $l_{1}=l_{2}=l$ and $\phi=\frac{\pi}{2}$, the
uncertainties $(\Delta x_{1})_{\widetilde{\Psi}}$ and $(\Delta
x_{2})_{\widetilde{\Psi}}$ for the state $\widetilde{\Psi}$ are correlated as
follows
$(\Delta x_{1})^{2}_{\widetilde{\Psi}}+(\Delta
x_{2})^{2}_{\widetilde{\Psi}}=l^{2}\cosh(2r).$ (100)
These are the main results, at the standard quantum mechanics level, which we
want to show regarding the representation of the symplectic group in quantum
mechanics. There are others applications like the analysis of the Bohmian
trajectories for bipartite squeezed states, the analysis of the fidelity for
bipartite or tripartite squeezed states, and others which are currently in
preparation. For now, let us move to the analysis of the squeezed states in
polymer quantum mechanics given in the next section.
## V Squeezed states in polymer quantum mechanics
Polymer quantum mechanics ashtekar2003quantum ; corichi2007polymer ;
velhinho2007quantum ; pawlowski2014separable , is a quantization scheme which
can be considered as a “toy model” looming from loop quantum cosmology. Hence,
exploring the nature and properties of squeezed states in polymer quantum
mechanics will help study those scenarios in loop quantum cosmology where such
states might play a significant role.
For this example, we will consider a system with two degrees of freedom and
both will be polymer quantized. Therefore, the Hilbert space of the entire
system is given by
${\cal H}_{poly}={\cal H}^{(1)}_{poly}\times{\cal H}^{(2)}_{poly},$ (101)
where the Hilbert spaces ${\cal H}^{(j)}_{poly}$ with $j=1,2$ are of the form
${\cal H}^{(j)}_{poly}=L^{2}(\overline{\mathbb{R}},dp^{(j)}_{Bohr}),$ (102)
where $\overline{\mathbb{R}}$ is the Bohr compactification of real line and
$dp^{(j)}_{Bohr}$ is the Bohr measure (see velhinho2007quantum for more
details). This Hilbert space resembles the momentum representation used in the
standard quantum mechanics.
An arbitrary state in this Hilbert space ${\cal H}_{poly}$ is given by
$\Psi(p_{1},p_{2})=\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x}_{j}}e^{\frac{i}{\hbar}\vec{x}^{T}_{j}\vec{p}},$
(103)
where $\\{\vec{x}_{j}\\}$ is a shorthand notation for the graph
$\\{(x^{(1)}_{j},x^{(2)}_{j})\\}^{j=n}_{j=1}$ associated with the state
$\Psi(\vec{p})$. In this notation, the array
$\vec{p}=\left(p_{1},p_{2}\right)$ denotes the momentum variables for the
system 1 and 2, respectively. The coefficients $\Psi_{\vec{x}_{j}}$ provide
the value for the norm of the state which is given by
$||\Psi(\vec{p})||=\sum_{\\{\vec{x}_{j}\\}}|\Psi_{\vec{x}_{j}}|^{2},$ (104)
hence these coefficients are different from zero and the sum converges (they
are non-null over countable number of points in the graph
$\\{\vec{x}_{j}\\}$). This norm arises from the inner product
$\langle\Psi|\Phi\rangle=\lim_{L_{1},L_{2}\rightarrow\infty}\frac{1}{4\,L_{1}L_{2}}\int^{L_{1}}_{-L_{1}}\int^{L_{2}}_{-L_{2}}\Psi^{*}(\vec{p})\,\Phi(\vec{p})dp_{1}dp_{2},$
(105)
which for the specific case of the plane waves takes the form of the Kronecker
delta
$\langle
e^{\frac{i}{\hbar}\vec{x}^{T}\vec{p}}|e^{\frac{i}{\hbar}\vec{x}^{\prime
T}\vec{p}}\rangle=\delta_{\vec{x},\vec{x}^{\prime}}.$ (106)
This inner product is the main signature of the polymer quantization as it
violates the Stone-von Neumann theorem. Consequently, polymer quantum
mechanics is not unitarily equivalent to the standard Schrödinger
representation. Moreover, in polymer quantum mechanics there is no momentum
operator hence infinitesimal spatial translations cannot be implemented.
Nevertheless, we can obtain a representation for the position operator, which
in the present case is given by
$\displaystyle\widehat{q}_{1}\Psi(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle i\hbar\frac{\partial}{\partial
p_{1}}\Psi(p_{1},p_{2})=-\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x}_{j}}x^{(1)}_{j}e^{\frac{i}{\hbar}\vec{x}^{T}_{j}\vec{p}},$
(107) $\displaystyle\widehat{q}_{2}\Psi(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle i\hbar\frac{\partial}{\partial
p_{2}}\Psi(p_{1},p_{2})=-\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x}_{j}}x^{(2)}_{j}e^{\frac{i}{\hbar}\vec{x}^{T}_{j}\vec{p}}.$
(108)
Despite these peculiarities with the non-regularity of the polymer
representation, the representation of the symplectic group $Sp(2n,\mathbb{R})$
on the Hilbert space of polymer quantum mechanics was provided recently by one
of the authors in Ref. Garcia-Chung:2020cag . There, the representation is
given by the map $\widehat{C}^{(poly)}:Sp(2n,\mathbb{R})\rightarrow L({\cal
H}_{poly}),\;\widetilde{\bf M}\mapsto\widehat{C}^{(poly)}_{\widetilde{\bf
M}}$, where the linear operator $\widehat{C}^{(poly)}_{\widetilde{\bf M}}$
acts on ${\cal H}_{poly}$ as
$\widehat{C}^{(poly)}_{\widetilde{\bf
M}}\Psi(\vec{p})=\lim_{L_{1},L_{2}\rightarrow\infty}\frac{1}{4\,L_{1}\,L_{2}}\int^{L_{1}}_{-L_{1}}\int^{L_{2}}_{-L_{2}}C^{(poly)}_{\widetilde{\bf
M}}(\vec{p},\vec{p}^{\prime})\Psi(\vec{p}^{\prime})d\vec{p}^{\prime}.$ (109)
The polymer kernel $C^{(poly)}_{\widetilde{\bf M}}(\vec{p},\vec{p}^{\prime})$
is given by
$C^{(poly)}_{\widetilde{\bf M}}(\vec{p},\vec{p}^{\prime})=\det(\widetilde{\bf
D}\widetilde{\bf
A}^{T})^{-\frac{1}{4}}e^{-\frac{i}{2\hbar}\vec{p}^{T}\widetilde{\bf
B}\widetilde{\bf
D}^{-1}\vec{p}}\sum_{\vec{x}}e^{\frac{i}{\hbar}\vec{p}^{T}\vec{x}-\frac{i}{\hbar}\vec{p}^{\prime
T}\widetilde{\bf D}^{T}\vec{x}+\frac{i}{2\hbar}\vec{x}^{T}{\widetilde{\bf
D}\widetilde{\bf C}^{T}}\vec{x}},$ (110)
and note that when the factor $\det(\widetilde{\bf D}\widetilde{\bf
A}^{T})^{-\frac{1}{4}}\neq 1$ it implies that this representation is not
unitary Garcia-Chung:2020cag .
Recall that one of the main features of loop quantum cosmology is its
intrinsic length scale given by the Planck length. As a toy model, polymer
quantum mechanics does not have an intrinsic length scale. However, it admits
a length scale that mimics some of the features of loop quantum cosmology.
This length scale is introduced at hand and is called polymer scale, usually
denoted by $\mu$. This polymer scale constitutes the analog of minimum length
for polymer quantum mechanics models, and therefore, it can be considered as a
lower bound for the uncertainties. In the present analysis, each system admits
a polymer scale $\mu_{1}$ and $\mu_{2}$ when the dynamics is considered.
Let us now consider the following questions: (1) is it possible to have
polymer states such that their uncertainties are lower than the polymer scale?
Furthermore, (2) do the correlations found in (98) have an analog in polymer
quantum mechanics?
To answer these questions let us consider the matrix $\widetilde{\bf M}_{1}$
with $a_{11}=a_{22}=c_{11}=c_{22}=0$, used to calculate the covariance matrix
${\bf V}^{(2)}_{1}$ in (93), but now with $a_{12}=-r_{1}$ and $c_{12}=-r_{2}$.
The explicit form is $\widetilde{\bf M}_{1}$
$\widetilde{\bf M}_{1}=\left(\begin{array}[]{cccc}e^{-r_{1}}&0&0&0\\\
0&e^{-r_{2}}&0&0\\\ 0&0&e^{r_{1}}&0\\\ 0&0&0&e^{r_{2}}\end{array}\right).$
(111)
The action of the group element $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{1}}$
on an arbitrary polymer state (103) gives the following state
$\widetilde{\Psi}_{\widetilde{\bf
M}_{1}}(p_{1},p_{2})=\widehat{C}^{(poly)}_{\widetilde{\bf
M}_{1}}\Psi(p_{1},p_{2})=\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x}_{j}}e^{\frac{i}{\hbar}\left(e^{-r_{1}}p_{1}x^{(1)}_{j}+e^{-r_{2}}p_{2}x^{(2)}_{j}\right)}.$
(112)
Using the representation of the position operators $\widehat{q}_{1}$ and
$\widehat{q}_{2}$ given in (107) and (108) we obtain the dispersion relations
$(\Delta x_{1})_{\widetilde{\Psi}}=e^{-r_{1}}(\Delta
x_{1})_{\Psi},\qquad(\Delta x_{2})_{\widetilde{\Psi}}=e^{-r_{2}}(\Delta
x_{2})_{\Psi},$ (113)
which show that the squeezed polymer state is indeed squeezed by a factor
$e^{-r_{1}}$ or $e^{-r_{2}}$. Consequently, if the initial dispersion of the
polymer state is given by $(\Delta x_{1})_{\Psi}$ or $(\Delta x_{2})_{\Psi}$,
then the squeeze operator $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{1}}$ gives
rise to a polymer state $\widetilde{\Psi}$ (112) whose dispersion is smaller
than that of the initial polymer state $\Psi$. Moreover, due to there is no
upper bound for the parameter $r$, these dispersion relations can be smaller
than the corresponding polymer scales $\mu_{1}$ and $\mu_{2}$.
Let us now consider the analog of the correlations (100) but for polymer
states. To do so, consider the polymer representation
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ of the symplectic matrix
(81) corresponding to a bi-partite system. The action of
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ on an arbitrary polymer
state (103) is given by
$\widetilde{\Psi}_{s}=\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf
s}}\Psi(p_{1},p_{2})=\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x_{j}}}e^{\frac{i}{\hbar}\left[\left(\cosh(r)\,x^{(1)}_{j}+\sinh(r)\,x^{(2)}_{j}\right)p_{1}+\left(\cosh(r)\,x^{(2)}_{j}+\sinh(r)\,x^{(1)}_{j}\right)p_{2}\right]},$
(114)
where again the representation in (109) was used.
We combine this result with the representation of the position operators in
(107) and (108) and obtain the following relations
$(\Delta x_{1})^{2}_{\widetilde{\Psi}_{s}}-(\Delta
x_{2})^{2}_{\widetilde{\Psi}_{s}}=(\Delta x_{1})^{2}_{\Psi}-(\Delta
x_{2})^{2}_{\Psi},\qquad\forall\quad\Psi\in{\cal H}_{poly},$ (115)
where the conditions $l_{1}=l_{2}$ and $\phi=\\{0,\frac{\pi}{2},\pi\\}$ were
imposed. Remarkably, this result not only is independent of the parameter $r$
(which labels the amount of squeezing) but also applies to any polymer state
$\Psi\in{\cal H}_{poly}$. As can be seen, the difference of dispersions
squared is conserved, regardless of the amount of squeezing. Also, note that
$l_{1}$ and $l_{2}$ are considered group parameters and have no relation to
the dynamics, i.e., we are considering general states in ${\cal H}_{poly}$.
The same applies for $\phi$.
Let us now consider pure and symmetric polymeric states. The pure states are
those that can be written as the following product
$\Psi^{(p)}(p_{1},p_{2})=\left(\sum_{\\{x^{(1)}_{j}\\}}\Psi^{(1)}_{x^{(1)}_{j}}e^{\frac{i}{\hbar}p_{1}x^{(1)}_{j}}\right)\left(\sum_{\\{x^{(2)}_{j}\\}}\Psi^{(2)}_{x^{(2)}_{j}}e^{\frac{i}{\hbar}p_{2}x^{(2)}_{j}}\right).$
(116)
Secondly, both lattices $\\{x^{(1)}_{j}\\}$ and $\\{x^{(2)}_{j}\\}$ are
symmetric, i.e., for every positive point $0<x^{(s)}_{j}\in\\{x^{(s)}_{j}\\}$
there exist a negative point $0>x^{(s)}_{j^{\prime}}\in\\{x^{(s)}_{j}\\}$,
such that $x^{(s)}_{j}+x^{(s)}_{j^{\prime}}=0$, and the states are also
symmetric which implies that
$\Psi^{(s)}_{x^{(s)}_{j}}=\Psi^{(s)}_{x^{(s)}_{j^{\prime}}}$. These states are
the analog of the states described with even functions in the standard quantum
mechanics.
The dispersion relation for squeezed pure symmetric polymer states is given by
$(\Delta x_{1})^{2}_{\widetilde{\Psi}_{s}}+(\Delta
x_{2})^{2}_{\widetilde{\Psi}_{s}}=\cosh(2r)\left((\Delta
x_{1})^{2}_{\Psi^{(p)}}+(\Delta x_{2})^{2}_{\Psi^{(p)}}\right),$ (117)
which takes the form
$(\Delta x_{1})^{2}_{\widetilde{\Psi}_{s}}+(\Delta
x_{2})^{2}_{\widetilde{\Psi}_{s}}=l^{2}\cosh(2r),$ (118)
when the dispersion of the pure states are $\left((\Delta
x_{1})_{\Psi^{(p)}}=(\Delta x_{2})_{\Psi^{(p)}}\right)=l/\sqrt{2}$. Notably,
Eq. (118) is the same as that obtained in (100) for the Schrödinger
representation. This shows that the correlations present in the standard
quantum mechanics using the $\widehat{C}_{\widetilde{\bf M}}$ operator for
both symplectic matrices $\widetilde{\bf M}_{1}$ and $\widetilde{\bf M}_{\bf
s}$ are the same to those obtained in polymer quantum mechanics using the
operator $\widehat{C}^{(poly)}_{\widetilde{\bf M}}$.
## VI Conclusions
In this paper we provided the direct relation between the Lie algebra
$sp(4,\mathbb{R})$ and the symplectic group $Sp(4,\mathbb{R})$. The expression
shows the link between the block matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and
${\bf D}$ with those of the Lie algebra ${\bf a}$, ${\bf b}$ and ${\bf c}$
given in the Eqs. (34)-(37). This result has not been reported before and
applies to the full Lie algebra $sp(4,\mathbb{R})$ of the symplectic group
$Sp(4,\mathbb{R})$.
Such relation allows us to obtain some important symplectic matrices that were
used in subsequent sections. In the first example for ${\bf a},{\bf c}\neq{\bf
0}$ and ${\bf b}=0$, we show that the corresponding symplectic matrix ${\bf
M}_{1}$, given in (52), can be written as $\left({\bf
M}^{\prime}_{1}\otimes{\bf 1}\right)\cdot\left({\bf 1}\otimes{\bf
M}^{\prime}_{2}\right)$, where ${\bf 1},{\bf M}^{\prime}_{1},{\bf
M}^{\prime}_{2}\in Sp(2,\mathbb{R})$. Here, ${\bf M}^{\prime}_{1}$ and ${\bf
M}^{\prime}_{2}$ are symplectic matrices acting over each of the sub-systems
with coordinates $(q_{1},p_{1})$ and $(q_{2},p_{2})$ respectively. In the case
${\bf a}={\bf c}=\mbox{diag}(a_{11},a_{22})$ and ${\bf
b}=\mbox{diag}(b_{11},b_{22})$ the symplectic matrix ${\bf M}_{2}$ in (59),
describes two coupled harmonic oscillators with interaction terms labeled by
the coefficients of the matrix ${\bf b}$. Finally, in the equation (62) we
showed the symplectic matrix ${\bf M}_{3}$ for the case in which ${\bf a}={\bf
c}={\bf 0}$ and a general form of the matrix ${\bf b}$.
In section (IV) we analyzed the classical description of squeeze operators. We
showed that the symplectic matrix ${\bf M}_{\bf s}$ is the classical analog of
the squeeze operator
${\widehat{S}(\zeta)=e^{\widehat{s}_{\zeta}}}=\widehat{C}_{\widetilde{\bf
M}_{\bf s}}$. Also, we remarked the isomorphism between the Lie algebra
$sp(4,\mathbb{R})$ and ${\cal P}(2,\mathbb{R})$. Additionally, the general
form of the covariance matrix ${\bf V}^{(2)}$ for the squeezed vacuum state
$|\Psi_{\widetilde{\bf M}}\rangle$ was derived using the Weyl algebra
representation and the symplectic matrix ${\bf M}_{1}$. The components of this
covariant matrix were used to calculate the dispersion relations (94) and (95)
for the particular case where $a_{11}=a_{22}=c_{11}=c_{22}=0$. As is already
known, these dispersions can be smaller than the vacuum characteristic length
for the harmonic oscillators. They also satisfy the Heisenberg uncertainty
principle as was shown in (96). We then calculated the covariance matrix for
the symplectic matrix ${\bf M}_{\bf s}$ corresponding to the classical analog
of the bipartite squeeze operator. With this matrix we determined the
correlation (100). We also provided the general expressions for these
correlations in equations (98) and (99).
Applying the previous results, it is also possible to represent operators in
non-regular Hilbert spaces that are non-unitarily equivalent to the Fock-
Schrödinger representation, so in section (V) we analyze polymer quantized
systems. We calculated the dispersion relation for an arbitrary polymer state
using the representation of the symplectic group in polymer quantum mechanics.
We obtained that the polymer representation of the squeeze operator given by
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{1}}$, yields a dispersion relation
Eq. (113), which can be smaller than those of the initial state. This implies
that $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{1}}$ is indeed a polymer
squeeze operator and (112) describes a polymer squeezed state.
On the other hand, the polymer representation of the bipartite squeeze
operator given by $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ was used
to derive the polymer correlations (115) and (118). The first correlation
(115) shows that the difference of the dispersions’ square is preserved and is
independent of the initial polymer state. Clearly, this result only holds for
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ so a symplectic matrix
different than the one used in (111) will produce a different result. In the
case of (118), the result has the same form as the standard correlation in Eq.
(100), hence, the polymer representation of
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ can be used to construct
correlated squeezed states for bipartite polymer systems. Naturally, this
brings some questions like whether there is any mechanism in nature, say loop
quantum cosmology or the interior of a black hole, from which a polymer
squeezed state can be created.
Moreover, establishing that squeezing is a property also present in non-
regular representations questions its role in the classicality of some
cosmological models. As we showed, it is possible to construct entangled
polymer states using $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$. Such
polymer entangled states correlations satisfy a relation identical to the one
obtained in the standard quantum mechanics. In this case, the states are
polymer bipartite squeezed states, similar to those used in the quantum
description of the inflaton field.
Finally, it is worth to mention that the polymer squeezed states obtained as a
result of the representation of $Sp(4,\mathbb{R})$ in ${\cal H}_{poly}$, given
in (112, 114), differ from those reported in the LQC literature
taveras2008corrections ; mielczarek2012gaussian ; gazeau2013quantum ;
diener2014numerical ; diener2014numerical2 ; corichi2011coherent . In these
references, a Gaussian form of the states is considered, whereas in our case,
the polymer state (112) is a general polymer state. In LQC, the squeezed
states are constructed by hand due after imposing some conditions to achieve
the squeezed nature of the dispersion relations. In our case, the squeezed
state results from the action of the squeeze operator.
The results of this work open the doors for studying the entangled states of
matter and geometry and the role that their correlations might play in some
physical scenarios.
## VII Acknowledgments
I thank Academia de Matemáticas and Colegio de Física, UP, for the support and
enthusiasm.
## VIII Appendix: calculation of ${\bf m}^{2n}$
In this appendix the expression for the matrix ${\bf m}^{2n}$ is obtained. To
do so, recall that the matrix ${\bf m}^{2}$ is formed by four $2\times 2$
block matrices where the upper left and the lower right matrices are multiples
of the identity matrix ${\bf 1}_{2\times 2}$. The upper right block is the
matrix ${\bf J}{\bf d}$ whereas the lower left is $-{\bf J}{\bf d}^{T}$.
Notably, we found that this block structure is preserved after exponentiating
the matrix ${\bf m}^{2}$ an integer number of times. That is to say, the
n-power of matrix ${\bf m}^{2}$ yields a new matrix $({\bf m}^{2})^{n}$ given
by
$({\bf m}^{2})^{n}=\left(\begin{array}[]{cc}\alpha_{n}{\bf 1}_{2\times
2}&\beta_{n}{\bf J}{\bf d}\\\ -\beta_{n}{\bf J}{\bf d}^{T}&\gamma_{n}{\bf
1}_{2\times 2}\end{array}\right).$ (119)
It is this pattern the one to be considered when this procedure is applied to
higher order symplectic groups.
The coefficients $\alpha_{n}$, $\beta_{n}$ and $\gamma_{n}$, are to be
determined and depend on the values of the matrices ${\bf a}$, ${\bf b}$,
${\bf c}$ and ${\bf d}$. For $n=1$, these coefficients are given by the
factors in the block matrices of ${\bf m}^{2}$ given in (32) and can be
directly defined as
$\alpha_{1}:=-(\det{\bf a}+\det{\bf
b}),\qquad\beta_{1}:=+1,\qquad\gamma_{1}:=-(\det{\bf c}+\det{\bf b}).$ (120)
To calculate $\alpha_{n}$, $\beta_{n}$ and $\gamma_{n}$ for arbitrary $n$,
first note that they can be generated with the $(n-1)$-power of the matrix
${\bf T}$ as
$\displaystyle\left(\begin{array}[]{c}\alpha_{n}\\\ \beta_{n}\\\
\gamma_{n}\end{array}\right)={\bf
T}^{n-1}\left(\begin{array}[]{c}\alpha_{1}\\\ \beta_{1}\\\
\gamma_{1}\end{array}\right),$ (127)
where the matrix ${\bf T}$ is given by
${\bf T}=\left(\begin{array}[]{ccc}\alpha_{1}&\beta_{1}\det{\bf d}&0\\\
\beta_{1}&\gamma_{1}&0\\\ 0&\beta_{1}\det{\bf
d}&\gamma_{1}\end{array}\right).$ (128)
The calculation shows that ${\bf T}^{n-1}$ is a matrix of the form
${\bf T}^{n-1}=\left(\begin{array}[]{cc}{\bf U}^{n-1}&\vec{0}^{T}\\\
\vec{u}^{T}\gamma_{1}^{n-2}\sum^{n-2}_{j=0}\gamma^{-j}_{1}{\bf
U}^{j}&\gamma^{n-1}_{1}\end{array}\right),$ (129)
where $\vec{0}=(0,0)$ and $\vec{u}=(0,\beta_{1}\det{\bf d})$ and matrix ${\bf
U}$ is given by
$\displaystyle{\bf U}=\left(\begin{array}[]{cc}\alpha_{1}&\beta_{1}\det{\bf
d}\\\ \beta_{1}&\gamma_{1}\end{array}\right).$ (132)
Then, using (127) we have the following relation for the coefficients
$\displaystyle\left(\begin{array}[]{c}\alpha_{n}\\\
\beta_{n}\end{array}\right)={\bf U}^{n-1}\left(\begin{array}[]{c}\alpha_{1}\\\
\beta_{1}\end{array}\right),\qquad\gamma_{n}=\gamma^{n}_{1}+\vec{u}^{T}\gamma^{n-2}_{1}\sum^{n-2}_{j=0}\gamma^{-j}_{1}{\bf
U}^{j}\left(\begin{array}[]{c}\alpha_{1}\\\ \beta_{1}\end{array}\right).$
(139)
In order to calculate ${\bf U}^{n-1}$ we need to diagonalize the matrix ${\bf
U}$. Let ${\bf P}$ be the matrix diagonalizing ${\bf U}$, then
${\bf U}={\bf P}\,{\bf D}_{0}\,{\bf P}^{-1},$ (140)
where the matrix ${\bf P}$ is
${\bf
P}=\left(\begin{array}[]{cc}\frac{(\lambda_{+}-\gamma_{1})}{\beta_{1}}k_{1}&\frac{(\lambda_{-}-\gamma_{1})}{\beta_{1}}k_{2}\\\
k_{1}&k_{2}\end{array}\right).$ (141)
The real arbitrary parameters $k_{1}$ and $k_{2}$ result from the
diagonalization procedure. Its values will be automatically cancelled as part
of the calculation of ${\bf U}^{n-1}$ further below. The eigenvalues of ${\bf
U}$, denoted by $\lambda_{\pm}$, have the following expression
$\displaystyle\lambda_{\pm}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}+\gamma_{1}}{2}\pm\frac{1}{2}\sqrt{(\alpha_{1}-\gamma_{1})^{2}+4\beta^{2}_{1}\det{\bf
d}},$ (142) $\displaystyle=$ $\displaystyle-\frac{\det{\bf a}+\det{\bf
c}+2\det{\bf b}}{2}\pm\frac{1}{2}\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}.$
and the diagonal matrix ${\bf D}_{0}$ is
${\bf D}_{0}=\left(\begin{array}[]{cc}\lambda_{+}&0\\\
0&\lambda_{-}\end{array}\right).$ (143)
We now take the $n-1$ power of ${\bf U}$ given in (140) to obtain the
following result
$\displaystyle{\bf
U}^{n-1}=\left(\begin{array}[]{cc}\frac{(\lambda_{+}-\gamma_{1})}{\beta_{1}}k_{1}&\frac{(\lambda_{-}-\gamma_{1})}{\beta_{1}}k_{2}\\\
k_{1}&k_{2}\end{array}\right)\left(\begin{array}[]{cc}\lambda^{n-1}_{+}&0\\\
0&\lambda^{n-1}_{-}\end{array}\right)\left(\begin{array}[]{cc}\frac{(\lambda_{+}-\gamma_{1})}{\beta_{1}}k_{1}&\frac{(\lambda_{-}-\gamma_{1})}{\beta_{1}}k_{2}\\\
k_{1}&k_{2}\end{array}\right)^{-1},$ (150)
which, when combined with the result in (139) together with the expression for
$\vec{u}$, gives
$\displaystyle\alpha_{n}$ $\displaystyle=$
$\displaystyle\frac{(\lambda_{+}-\det{\bf b}-\det{\bf
c})\lambda^{n}_{+}-(\lambda_{-}-\det{\bf b}-\det{\bf
c})\lambda^{n}_{-}}{\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\beta^{2}_{1}\det{\bf
d}}},$ (151) $\displaystyle\beta_{n}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{n}_{+}-\lambda^{n}_{-}}{\sqrt{(\det{\bf
a}-\det{\bf c})^{2}+4\beta^{2}_{1}\det{\bf d}}},$ (152)
$\displaystyle\gamma_{n}$ $\displaystyle=$
$\displaystyle\frac{\left[(\lambda_{+}-\det{\bf b}-\det{\bf
c})\lambda^{n}_{-}-(\lambda_{-}-\det{\bf b}-\det{\bf
c})\lambda^{n}_{+}\right]}{\sqrt{(\det{\bf a}-\det{\bf
c})^{2}+4\beta^{2}_{1}\det{\bf d}}}.$ (153)
These are the final expressions for the coefficients in $({\bf m}^{2})^{n}$.
## IX Appendix: Series analysis
In this appendix we calculate the series expansion terms. To do so, note that
once the expression for $({\bf m}^{2})^{n}$ is inserted the expansion (31) and
the even and odd terms are collected, the matrix ${\bf M}({\bf a},{\bf b},{\bf
c})$ takes the form
${\bf M}({\bf a},{\bf b},{\bf c})=\left(\begin{array}[]{cc}\alpha^{(e)}{\bf
1}_{2\times 2}&\beta^{(e)}{\bf J}{\bf d}\\\ -\beta^{(e)}{\bf J}{\bf
d}^{T}&\gamma^{(e)}{\bf 1}_{2\times 2}\end{array}\right)+{\bf
m}\left(\begin{array}[]{cc}\alpha^{(o)}{\bf 1}_{2\times 2}&\beta^{(o)}{\bf
J}{\bf d}\\\ -\beta^{(o)}{\bf J}{\bf d}^{T}&\gamma^{(o)}{\bf 1}_{2\times
2}\end{array}\right),$ (154)
where the following coefficients are given by
$\alpha^{(e)}:=1+\sum^{+\infty}_{n=1}\frac{1}{(2n)!}\alpha_{n},\qquad\beta^{(e)}:=\sum^{+\infty}_{n=1}\frac{1}{(2n)!}\beta_{n},\qquad\gamma^{(e)}:=1+\sum^{+\infty}_{n=1}\frac{1}{(2n)!}\gamma_{n},$
(155)
$\alpha^{(o)}:=1+\sum^{+\infty}_{n=1}\frac{1}{(2n+1)!}\alpha_{n},\qquad\beta^{(o)}:=\sum^{+\infty}_{n=1}\frac{1}{(2n+1)!}\beta_{n},\qquad\gamma^{(o)}:=1+\sum^{+\infty}_{n=1}\frac{1}{(2n+1)!}\gamma_{n}.$
(156)
We now insert (151), (152) and (153) in the relations (155) - (156) to obtain
$\alpha^{(e)}=\frac{1}{2}\left[\cosh\left(\sqrt{\lambda_{+}}\right)+\cosh\left(\sqrt{\lambda_{-}}\right)\right]+\frac{\left(\det{\bf
c}-\det{\bf
a}\right)\left[\cosh\left(\sqrt{\lambda_{+}}\right)-\cosh\left(\sqrt{\lambda_{-}}\right)\right]}{2\sqrt{(\det{\bf
a}-\det{\bf c})^{2}+4\det{\bf d}}},$ (157)
$\displaystyle\alpha^{(o)}=\frac{1}{2}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}+\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right]+\frac{\left(\det{\bf
c}-\det{\bf a}\right)}{2\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}-\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right],$
(158) $\beta^{(e)}=\frac{1}{\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}}\left[\cosh\left(\sqrt{\lambda_{+}}\right)-\cosh\left(\sqrt{\lambda_{-}}\right)\right],$
(159) $\beta^{(o)}=\frac{1}{\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}-\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right],$
(160)
$\gamma^{(e)}=\frac{1}{2}\left[\cosh\left(\sqrt{\lambda_{+}}\right)+\cosh\left(\sqrt{\lambda_{-}}\right)\right]-\frac{\left(\det{\bf
c}-\det{\bf
a}\right)\left[\cosh\left(\sqrt{\lambda_{+}}\right)-\cosh\left(\sqrt{\lambda_{-}}\right)\right]}{2\sqrt{(\det{\bf
a}-\det{\bf c})^{2}+4\det{\bf d}}},$ (161)
$\displaystyle\gamma^{(o)}=\frac{1}{2}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}+\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right]-\frac{\left(\det{\bf
c}-\det{\bf a}\right)}{2\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}-\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right],$
(162)
where we have to recall the expression for the eigenvalues $\lambda_{\pm}$ in
(142).
## X Appendix: Covariance matrix coefficients
In this appendix we show the explicit form of the coefficients of the
covariance matrix ${\bf V}^{(2)}$.
$\displaystyle{V}^{(2)}_{11}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(\sqrt{-\det{\bf
a}})\left(a_{12}^{2}l^{4}_{1}+a_{22}^{2}\hbar^{2}\right)}{l^{2}_{1}\det{\bf
a}}+\frac{a_{12}l^{2}_{1}\sinh(2\sqrt{-\det{\bf a}})}{\sqrt{-\det{\bf
a}}}+l^{2}_{1}\cosh^{2}(\sqrt{-\det{\bf a}}),$ (163)
$\displaystyle{V}^{(2)}_{22}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(\sqrt{-\det{\bf
c}})\left(c_{12}^{2}l^{4}_{2}+c_{22}^{2}\hbar^{2}\right)}{l^{2}_{2}\det{\bf
c}}+\frac{c_{12}l^{2}_{2}\sinh(2\sqrt{-\det{\bf c}})}{\sqrt{-\det{\bf
c}}}+l^{2}_{2}\cosh^{2}(\sqrt{-\det{\bf c}}),$ (164)
$\displaystyle{V}^{(2)}_{13}$ $\displaystyle=$
$\displaystyle\frac{\sinh(2\sqrt{-\det{\bf
a}})\left(a_{22}\hbar^{2}-a_{11}l^{4}_{1}\right)}{2l^{2}_{1}\sqrt{-\det{\bf
a}}}+\frac{a_{12}\sinh^{2}(\sqrt{-\det{\bf
a}})\left(a_{11}l^{4}_{1}+a_{22}\hbar^{2}\right)}{l^{2}_{1}\det{\bf a}},$
(165) $\displaystyle{V}^{(2)}_{24}$ $\displaystyle=$
$\displaystyle\frac{\sinh(2\sqrt{-\det{\bf
c}})\left(c_{22}\hbar^{2}-c_{11}l^{4}_{2}\right)}{2l^{2}_{2}\sqrt{-\det{\bf
c}}}+\frac{c_{12}\sinh^{2}(\sqrt{-\det{\bf
c}})\left(c_{11}l^{4}_{2}+c_{22}\hbar^{2}\right)}{l^{2}_{2}\det{\bf c}},$
(166) $\displaystyle{V}^{(2)}_{33}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(\sqrt{-\det{\bf
a}})\left(a_{11}^{2}l^{4}_{1}+a_{12}^{2}\hbar^{2}\right)}{l^{2}_{1}\det{\bf
a}}+\frac{a_{12}\hbar^{2}\sinh(2\sqrt{-\det{\bf a}})}{l^{2}_{1}\sqrt{-\det{\bf
a}}}+\frac{\hbar^{2}\cosh^{2}(\sqrt{-\det{\bf a}})}{l^{2}_{1}},$ (167)
$\displaystyle{V}^{(2)}_{44}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(\sqrt{-\det{\bf
c}})\left(c_{11}^{2}l^{4}_{2}+c_{12}^{2}\hbar^{2}\right)}{l^{2}_{2}\det{\bf
c}}+\frac{c_{12}\hbar^{2}\sinh(2\sqrt{-\det{\bf c}})}{l^{2}_{2}\sqrt{-\det{\bf
c}}}+\frac{\hbar^{2}\cosh^{2}(\sqrt{-\det{\bf c}})}{l^{2}_{2}}.$ (168)
## References
* (1) Walls, D. F., “Squeezed states of light”, _Nature_ , 306, 5939, 1983, Springer.
* (2) Braunstein, S. L. and Van Loock, P., “Quantum information with continuous variables”, _Reviews of Modern Physics_ , 77, 2, 2005.
* (3) Walls, D. F. and Milburn, G. J., “Quantum optics”, 2007, _Springer Science & Business Media_.
* (4) Adesso, G., Ragy, S. and Lee, A. R., “Continuous variable quantum information: Gaussian states and beyond”, _Open Systems & Information Dynamics_, 21, 2014, World Scientific.
* (5) Schnabel, R., “Squeezed states of light and their applications in laser interferometers”, _Physics Reports_ , 684, 2017, Elsevier.
* (6) Grishchuk, L. P. and Sidorov, Y. V., “Squeezed quantum states of relic gravitons and primordial density fluctuations”, _Physical Review D_ , (42), 1990.
* (7) Polarski, D. and Starobinsky, A. A., “Semiclassicality and decoherence of cosmological perturbations”, _Classical and Quantum Gravity_ , (13), 1996.
* (8) Lesgourgues, J., Polarski, D. and Starobinsky, A. A., “Quantum-to-classical transition of cosmological perturbations for non-vacuum initial states”, _Nuclear Physics B_ , 497, 1997.
* (9) Kiefer, C., Polarski, D. and Starobinsky, A. A., “Quantum-to-classical transition for fluctuations in the early universe”, _International Journal of Modern Physics D_ , 7, 1998.
* (10) Guth, A. H. and Pi, S., “Quantum mechanics of the scalar field in the new inflationary universe”, _Physical Review D_ , 32, 1985.
* (11) Martin, J. and Vennin, V., “Quantum discord of cosmic inflation: Can we show that CMB anisotropies are of quantum-mechanical origin?”, _Physical Review D_ , 93, 2016.
* (12) Albrecht, A., Ferreira, P., Joyce, M. and Prokopec, T., “Inflation and squeezed quantum states”, _Physical Review D_ , 50, 1994.
* (13) Gasperini, M. and Giovannini, M., “Quantum squeezing and cosmological entropy production”, _Classical and Quantum Gravity_ , 10, 9, 1993.
* (14) Grain, J. and Vennin, V., “Squeezing formalism and canonical transformations in cosmology”, _arXiv preprint arXiv:1910.01916_ , 2019.
* (15) Ashtekar, A., Bojowald, M. and Lewandowski, J., “Mathematical structure of loop quantum cosmology”, _Advances in Theoretical and Mathematical Physics_ , 7, 2, 2003.
* (16) Bojowald, M., “Canonical gravity and applications: cosmology, black holes, and quantum gravity”, _Cambridge University Press_ , 2010.
* (17) Bojowald, M., “Quantum cosmology: a fundamental description of the universe”, vol. 835, _Springer Science & Business Media_, 2011.
* (18) Ashtekar, A., Pawlowski, T. and Singh, P., “Quantum nature of the big bang: an analytical and numerical investigation”, _Physical Review D_ , 73, 12, 2006.
* (19) Ashtekar, A., Pawlowski, T. and Singh, P., “Quantum nature of the big bang: Improved dynamics”, _Physical Review D_ , 74, 8, 2006.
* (20) Mielczarek, J. and Piechocki, W., “Gaussian state for the bouncing quantum cosmology”, _Physical Review D_ , 86, 8, 2012.
* (21) Gazeau, J. P., Mielczarek, J. and Piechocki, W., “Quantum states of the bouncing universe”, _Physical Review D_ , 87, 12, 2013.
* (22) Diener, P., Gupt, B., Megevand, M. and Singh, P., “Numerical evolution of squeezed and non-Gaussian states in loop quantum cosmology”, _Classical and Quantum Gravity_ , 31, 16, 2014.
* (23) Diener, P., Gupt, B. and Singh, P., “Numerical simulations of a loop quantum cosmos: robustness of the quantum bounce and the validity of effective dynamics”, _Classical and Quantum Gravity_ , 31, 2014.
* (24) Taveras, V., “Corrections to the Friedmann equations from loop quantum gravity for a universe with a free scalar field”, _Physical Review D_ , 78, 2008.
* (25) Corichi, A. and Montoya, E., “Coherent semiclassical states for loop quantum cosmology”, _Physical Review D_ , 84, 2011.
* (26) Milburn, G. J., “Multimode minimum uncertainty squeezed states”, _Journal of Physics A: Mathematical and General_ , 17, 4, 1984.
* (27) Gerry, Ch., Knight, P. and Knight, P. L., _Introductory quantum optics_ , (2005), Cambridge university press.
* (28) Simon, R. Sudarshan, E. C. G. and Mukunda, N., “Gaussian pure states in quantum mechanics and the symplectic group”, _Physical Review A_ , 37, 8, 1988.
* (29) Ma, X. and Rhodes, W., “Multimode squeeze operators and squeezed states”, _Physical Review A_ , 41, 9, 1990.
* (30) Pirandola, S., Serafini, A. and Lloyd, S., “Correlation matrices of two-mode bosonic systems”, _Physical Review A_ , 79, 2009.
* (31) Weedbrook, C. et. al., “Gaussian quantum information”, _Reviews of Modern Physics_ , 84, 2, 2012.
* (32) Einstein, A., Podolsky, B. and Rosen, N. “Can quantum mechanical description of physical reality be considered complete?” _Phys. Rev._ 47, (1935).
* (33) Ashtekar, A., Fairhurst, S. and Willis, J. L., “Quantum gravity, shadow states and quantum mechanics”, _Classical and Quantum Gravity_ , 20, 6, 2003.
* (34) Corichi, A., Vukasinac, T. and Zapata, José A., “Polymer quantum mechanics and its continuum limit”, Phys. Review D, 76, 4, 2007.
* (35) Flores-González, E. and Morales-Técotl, H. A. and Reyes, J. D., “Propagators in polymer quantum mechanics”, _Annals of Physics_ , 336, 2013.
* (36) Velhinho, J. M., “The quantum configuration space of loop quantum cosmology”, _Classical and Quantum Gravity_ , 24, 14, 2007.
* (37) Pawlowski, T., Villaseñor, E. J. S. and Barbero, J. F., “Separable Hilbert space for loop quantization”, Phys. Review D, 90, 6, 2014.
* (38) Garcia-Chung, A.“The symplectic group in Polymer Quantum Mechanics,” _Physical Review D_ , 101, 10, 2020. arXiv:2003.00388 [gr-qc].
* (39) Hall, M., “The theory of groups”, 2018, _Courier Dover Publications_
* (40) Arnold, V. I, _Mathematical methods of classical mechanics_ , Springer, 1989.
* (41) De Gosson, M. A. _Symplectic geomtry and quantum mechanics_ , Birkhauser, 2006
* (42) Castaños, O. and Moshinsky, M. _Matrix representation of the generators of symplectic algebras. I. The case of sp (4, R)_ , Journal of Physics A: Mathematical and General, 20, 3, 1987.
* (43) Moshinsky, M. and Quesne, C., “Linear canonical transformations and their unitary representations”, _Journal of Mathematical Physics_ , 12, 8, 1971.
* (44) Wolf, K., “Development of linear canonical transforms: a historical sketch”, in _Linear Canonical Transforms_ , 3-28, 2016, Springer.
* (45) Torre, A., “Linear ray and wave optics in phase space: bridging ray and wave optics via the Wigner phase-space picture”, 2005, Elsevier.
|
# Shortest Beer Path Queries in Outerplanar Graphs††thanks: A preliminary
version was presented at the 32nd Annual International Symposium on Algorithms
and Computation (ISAAC 2021). JB was supported by an NSERC Undergraduate
Student Research Award. MS was suported by NSERC.
Joyce Bacic School of Computer Science, Carleton University, Ottawa, Canada.
Saeed Mehrabi UMass Lowell, USA Michiel Smid22footnotemark: 2
###### Abstract
A _beer graph_ is an undirected graph $G$, in which each edge has a positive
weight and some vertices have a beer store. A _beer path_ between two vertices
$u$ and $v$ in $G$ is any path in $G$ between $u$ and $v$ that visits at least
one beer store.
We show that any outerplanar beer graph $G$ with $n$ vertices can be
preprocessed in $O(n)$ time into a data structure of size $O(n)$, such that
for any two query vertices $u$ and $v$, (i) the weight of the shortest beer
path between $u$ and $v$ can be reported in $O(\alpha(n))$ time (where
$\alpha(n)$ is the inverse Ackermann function), and (ii) the shortest beer
path between $u$ and $v$ can be reported in $O(L)$ time, where $L$ is the
number of vertices on this path. Both results are optimal, even when $G$ is a
beer tree (i.e., a beer graph whose underlying graph is a tree).
## 1 Introduction
Imagine that you are going to visit a friend and, not wanting to show up empty
handed, you decide to pick up some beer along the way. In this paper we
determine the fastest way to go from your place to your friend’s place while
stopping at a beer store to buy some drinks.
A _beer graph_ is a undirected graph $G=(V,E)$, in which each edge $(u,v)$ has
a positive weight $\omega(u,v)$ and some of the vertices are beer stores. For
two vertices $u$ and $v$ of $G$, we define the _shortest beer path_ from $u$
to $v$ to be the shortest (potentially non-simple) path that starts at $u$,
ends at $v$, and visits at least one beer store. We denote this shortest path
by $\mathsf{SP}_{B}(u,v)$. The _beer distance_ $\mathsf{dist}_{B}(u,v)$
between $u$ and $v$ is the weight of the path $\mathsf{SP}_{B}(u,v)$, i.e.,
the sum of the edge weights on $\mathsf{SP}_{B}(u,v)$.
Observe that even though the shortest beer path from $u$ to $v$ may be a non-
simple path, it is always composed of two simple paths: the shortest path from
$u$ to a beer store and the shortest path from this same beer store to $v$.
Thus, when looking at the shortest beer path problem, we often need to
consider the shortest path between vertices. We denote the shortest path in
$G$ from $u$ to $v$ by $\mathsf{SP}(u,v)$ and we use $\mathsf{dist}(u,v)$ to
denote the weight of this path. We also say that $\mathsf{dist}(u,v)$ is the
_distance_ between $u$ and $v$ in $G$.
To the best of our knowledge, the problem of computing shortest beer paths has
not been considered before. Let $s$ be a fixed source vertex of $G$. Recall
that Dijkstra’s algorithm computes $\mathsf{dist}(s,v)$ for all vertices $v$,
by maintaining a “tentative distance” $\delta(v)$, which is the weight of the
shortest path from $s$ to $v$ computed so far. If we also maintain a
“tentative beer distance” $\delta_{B}(v)$ (which is the weight of the shortest
beer path from $s$ to $v$ that has been found so far), then a modification of
Dijkstra’s algorithm allows us to compute $\mathsf{dist}_{B}(s,v)$ for all
vertices $v$, in $O(|V|\log|V|+|E|)$ total time.
As far as we know, no non-trivial results are known for beer distance queries.
In this case, we want to preprocess the beer graph $G$ into a data structure,
such that, for any two query vertices $u$ and $v$, the shortest beer path
$\mathsf{SP}_{B}(u,v)$, or its weight $\mathsf{dist}_{B}(u,v)$, can be
reported.
### 1.1 Our Results
We present data structures that can answer shortest beer path queries in
outerplanar beer graphs. Recall that a graph $G$ is _outerplanar_ , if $G$ can
be embedded in the plane, such that all vertices are on the outer face, and no
two edges cross.
Our first result is stated in terms of the inverse Ackermann function. We use
the definition as given in [3]: Let $A_{0}(i)=i+1$ and, for $\ell\geq 0$,
$A_{\ell+1}(i)=A_{\ell}^{(i+1)}(i+8)$, where $A_{\ell}^{(i+1)}$ is the
function $A_{\ell}$ iterated $i+1$ times. We define $\alpha(m,n)$ to be the
smallest value of $\ell$ for which $A_{\ell}(\lfloor m/n\rfloor)>n$, and we
define $\alpha(n)=\alpha(n,n)$.
###### Theorem 1
Let $G$ be an outerplanar beer graph with $n$ vertices. For any integer $m\geq
n$, we can preprocess $G$ in $O(m)$ time into a data structure of size $O(m)$,
such that for any two query vertices $u$ and $v$, both $\mathsf{dist}(u,v)$
and $\mathsf{dist}_{B}(u,v)$ can be computed in $O(\alpha(m,n))$ time.
By taking $m=n$, both the preprocessing time and the space used are $O(n)$,
and for any two query vertices $u$ and $v$, both $\mathsf{dist}(u,v)$ and
$\mathsf{dist}_{B}(u,v)$ can be computed in $O(\alpha(n))$ time.
As another example, let $\log^{*}n$ be the number of times the function $\log$
must be applied, when starting with the value $n$, until the result is at most
$1$, and let $\log^{**}n$ be the number of times the function $\log^{*}$ must
be applied, again starting with $n$, until the result is at most $1$. Let
$m=n\log^{**}n$. Since $\alpha(m,n)=O(1)$, we obtain a data structure with
space and preprocessing time $O(n\log^{**}n)$ that can answer both distance
and beer distance queries in $O(1)$ time.
As we mentioned before, beer distance queries have not been considered for any
class of graphs. In fact, the only result on (non-beer) distance queries in
outerplanar graphs that we are aware of is by Djidjev _et al._ [5]. They show
that an outerplanar graph with $n$ vertices can be preprocessed in $O(n\log
n)$ time into a data structure of size $O(n\log n)$, such that any distance
query can be answered in $O(\log n)$ time. Our result in Theorem 1
significantly improves their result.
We also show that the result in Theorem 1 is optimal for beer distance
queries, even if $G$ is a _beer tree_ (i.e., a beer graph whose underlying
graph is a tree). We do not know if the query time is optimal for (non-beer)
distance queries.
Our second result is on reporting the shortest beer path between two query
vertices.
###### Theorem 2
Let $G$ be an outerplanar beer graph with $n$ vertices. We can preprocess $G$
in $O(n)$ time into a data structure of size $O(n)$, such that for any two
vertices $u$ and $v$, the shortest beer path from $u$ to $v$ can be reported
in $O(L)$ time, where $L$ is the number of vertices on this beer path.
Observe that the query time in Theorem 2 does not depend on the number $n$ of
vertices of the graph. Again, we are not aware of any previous work on
reporting shortest beer paths. Djidjev _et al._ [5] show that, after $O(n\log
n)$ preprocessing and using $O(n\log n)$ space, the shortest (non-beer) path
between two query vertices can be reported in $O(\log n+L)$ time, where $L$ is
the number of vertices on the path.
### 1.2 Preliminaries and Organization
Throughout this paper, we only consider outerplanar beer graphs $G$. The
number of vertices of $G$ is denoted by $n$. It is well known that $G$ has at
most $2n-3$ edges. As in [5], we say that $G$ satisfies the _generalized
triangle inequality_ , if for every edge $(u,v)$ in $G$,
$\mathsf{dist}(u,v)=\omega(u,v)$, i.e., the shortest path between $u$ and $v$
is the edge $(u,v)$.
The outerplanar graph $G$ is called _maximal_ , if adding an edge between any
two non-adjacent vertices of $G$ results in a graph that is not outerplanar.
In this case, the number of edges is equal to $2n-3$. A maximal outerplanar
graph $G$ is 2-connected, each internal face of $G$ is a triangle and the
outer face of $G$ forms a Hamiltonian cycle. In such a graph, edges on the
outer face will be referred to as _external_ edges, where all other edges will
be referred to as _internal_ edges.
The _weak dual_ of a maximal outerplanar graph $G$ is the graph $D(G)$ whose
node set is the set of all internal faces of $G$, and in which
$(F,F^{\prime})$ is an edge if and only if the faces $F$ and $F^{\prime}$
share an edge in $G$; see Figure 1. For simplicity, we will refer to $D(G)$ as
the dual of $G$. Observe that $D(G)$ is a tree with $n-2$ nodes, each of which
has degree at most three.
Figure 1: A maximal outerplanar graph shown in black. Its dual is shown in
red.
If $H$ is a subgraph of the beer graph $G$, and $u$ and $v$ are vertices of
$H$, then $\mathsf{dist}(u,v,H)$ and $\mathsf{dist}_{B}(u,v,H)$ denote the
distance and beer distance between $u$ and $v$ in $H$, respectively. The
shortest beer path in $H$ between $u$ and $v$ must be entirely within $H$.
Observe that we use the shorthand $\mathsf{dist}(u,v)$ for
$\mathsf{dist}(u,v,G)$, and $\mathsf{dist}_{B}(u,v)$ for
$\mathsf{dist}_{B}(u,v,G)$.
It will not be surprising that the algorithms for computing shortest beer
paths use the dual $D(G)$. Thus, our algorithms will need some basic data
structures on trees. These data structures will be presented in Section 2.
In Section 3, we will prove Theorem 1 for maximal outerplanar beer graphs. We
also prove that the result in Theorem 1 is optimal, even for beer trees. The
proof of Theorem 2, again for maximal outerplanar beer graphs, will be
presented in Section 4. Both Sections 3 and 4 will use the result in Lemma 4,
whose detailed proof will be given in Section 5.
The extensions of Theorems 1 and 2 to arbitrary outerplanar beer graphs will
be given in Section 6. Finally, Section 7 will present an $O(n)$-time
algorithm for computing the single-source shortest beer path tree for any
given source vertex.
## 2 Query Problems on Trees
Our algorithms for computing beer shortest paths in an outerplanar graph $G$
will use the dual of $G$, which is a tree. In order to obtain fast
implementations of these algorithms, we need to be able to solve several query
problems on this tree. In this section, we present all query problems that
will be used in later sections.
###### Lemma 1
Let $T$ be a tree with $n$ nodes that is rooted at an arbitrary node. We can
preprocess $T$ in $O(n)$ time, such that each of the following queries can be
answered in $O(1)$ time:
1. 1.
Given a node $u$ of $T$, return its level, denoted by $\mathord{\it
level}(u)$, which is the number of edges on the path from $u$ to the root.
2. 2.
Given two nodes $u$ and $v$ of $T$, report their lowest common ancestor,
denoted by $\mathsf{LCA}(u,v)$.
3. 3.
Given two nodes $u$ and $v$ of $T$, decide whether or not $u$ is in the
subtree rooted at $v$.
4. 4.
Given two distinct nodes $u$ and $v$ of $T$, report the second node on the
path from $u$ to $v$.
5. 5.
Given three nodes $u$, $v$, and $w$, decide whether or not $w$ is on the path
between $u$ and $v$.
Proof. The first claim follows from the fact that by performing an $O(n)$–time
pre-order traversal of $T$, we can compute $\mathord{\it level}(u)$ for each
node $u$. A proof of the second claim can be found in Harel and Tarjan [6] and
Bender and Farach-Colton [2]. The third claim follows from the fact that $u$
is in the subtree rooted at $v$ if and only $\mathsf{LCA}(u,v)=v$. A proof of
the fourth claim can be found in Chazelle [4, Lemma 15]. The fifth claim
follows from the following observations. Assume that $u$ is in the subtree
rooted at $v$. Then $w$ is on the path between $u$ and $v$ if and only if
$\mathsf{LCA}(u,w)=w$ and $w$ is in the subtree rooted at $v$. The case when
$v$ is in the subtree rooted at $u$ is symmetric. Assume that
$\mathsf{LCA}(u,v)\not\in\\{u,v\\}$. Then $w$ is on the path between $u$ and
$v$ if and only if $w$ is on the path between $u$ and $\mathsf{LCA}(u,v)$ or
$w$ is on the path between $v$ and $\mathsf{LCA}(u,v)$.
### 2.1 Closest-Colour Queries in Trees
Let $T$ be a tree with $n$ nodes and let $\mathcal{C}$ be a set of _colours_.
For each colour $c$ in $\mathcal{C}$, we are given a path $P_{c}$ in $T$. Even
though these paths may share nodes, each node of $T$ belongs to at most a
constant number of paths. This implies that the total size of all paths
$P_{c}$ is $O(n)$. We assume that each node $u$ of $T$ stores the set of all
colors $c$ such that $u$ is on the path $P_{c}$.
In a _closest-colour query_ , we are given two nodes $u$ and $v$ of $T$, and a
colour $c$, such that $u$ is on the path $P_{c}$. The answer to the query is
the node on $P_{c}$ that is closest to $v$. Refer to Figure 2 for an
illustration.
Figure 2: A tree $T$ and a collection of coloured paths. For a query with
nodes $u$ and $v$, and color “red”, the answer is the node $w$.
###### Lemma 2
After an $O(n)$–time preprocessing, we can answer any closest-colour query in
$O(1)$ time.
Proof. We take an arbitrary node of $T$ and make it the root. Then we
preprocess $T$ such that each of the queries in Lemma 1 can be answered in
$O(1)$ time.
For each colour $c$, let $c^{1}$ and $c^{2}$ be the end nodes of the path
$P_{c}$, and let $c^{h}$ be the highest node on $P_{c}$ in the tree (i.e., the
node on $P_{c}$ that is closest to the root). With each node of $P_{c}$, we
store pointers to $c^{1}$, $c^{2}$, and $c^{h}$.
Since each node of $T$ is in a constant number of coloured paths, we can
compute the pointers for all the coloured paths in $O(n)$ total time.
The query algorithm does the following. Let $u$ and $v$ be two nodes of $T$,
and let $c$ be a colour such that $u$ is on the $c$-coloured path $P_{c}$.
If $u=v$ or $v$ is also on $P_{c}$, then we return the node $v$. From now on,
assume that $u\neq v$ and $v$ is not on $P_{c}$. Below, we consider all
possible cases, which are illustrated in Figure 3.
Figure 3: Illustrating all possible cases in the proof of Lemma 2. The path
$P_{c}$ is red and the blue square indicates the node that is returned by the
closest-colour query.
1. 1.
If $\mathsf{LCA}(u,v)=v$, then $u$ is in the subtree rooted at $v$. In this
case, we return $c^{h}$, the highest $c$-coloured node.
2. 2.
Assume that $\mathsf{LCA}(u,v)=u$. Then $v$ is in the subtree rooted at $u$.
The closest $c$-coloured node to $v$ is either $\mathsf{LCA}(v,c^{1})$ or
$\mathsf{LCA}(v,c^{2})$. Since $v$ is lower than $u$ in the tree, we know that
the closest $c$-colored node to $v$ is at $\mathord{\it level}(u)$ or greater.
If $\mathord{\it level}(\mathsf{LCA}(v,c^{1}))>\mathord{\it
level}(\mathsf{LCA}(v,c^{2}))$, then $\mathsf{LCA}(v,c^{1})$ is lower in the
tree and closer to $v$, so we return $\mathsf{LCA}(v,c^{1})$ . Otherwise,
$\mathsf{LCA}(v,c^{2})$ is lower in $T$ or equal to both
$\mathsf{LCA}(v,c^{1})$ and $u$, so we return $\mathsf{LCA}(v,c^{2})$.
3. 3.
Assume that $\mathsf{LCA}(u,v)\neq u$ and $\mathsf{LCA}(u,v)\neq v$. Then $u$
and $v$ are in different subtrees of $\mathsf{LCA}(u,v)$.
1. (a)
If $\mathord{\it level}(c^{h})>\mathord{\it level}(\mathsf{LCA}(u,v))$, then
we return $c^{h}$.
2. (b)
If $\mathord{\it level}(c^{h})<\mathord{\it level}(\mathsf{LCA}(u,v))$, then
we return $\mathsf{LCA}(u,v)$.
3. (c)
Assume that $\mathord{\it level}(c^{h})=\mathord{\it
level}(\mathsf{LCA}(u,v))$. Observe that exactly one end node of the
$c$-coloured path is in the subtree rooted at $u$.
1. i.
If $c^{1}$ is in the subtree rooted at $u$, then we return
$\mathsf{LCA}(v,c^{2})$.
2. ii.
If $c^{2}$ is in the subtree rooted at $u$, then we return
$\mathsf{LCA}(v,c^{1})$.
Using Lemma 1, each of these case takes $O(1)$ time. Therefore, the entire
query algorithm takes $O(1)$ time.
### 2.2 Path-Sum Queries in Trees
Let $(W,\oplus)$ be a semigroup. Thus, $W$ is a set and $\oplus:W\times
W\rightarrow W$ is an associative binary operator. We assume that for any two
elements $s$ and $s^{\prime}$ in $W$, the value of $s\oplus s^{\prime}$ can be
computed in $O(1)$ time.
Let $T$ be a tree with $n$ nodes in which each edge $e$ stores a value $s(e)$,
which is an element of $W$. For any two distinct nodes $u$ and $v$ in $T$, we
define their _path-sum_ $\mathsf{PS}(u,v)$ as follows: Let
$e_{1},e_{2},\ldots,e_{k}$ be the edges on the path in $T$ between $u$ and
$v$. Then we define $\mathsf{PS}(u,v)=\oplus_{i=1}^{k}s(e_{i})$.
Chazelle [4] considers the problem of preprocessing the tree $T$, such that
for any two distinct query nodes $u$ and $v$, the value of $\mathsf{PS}(u,v)$
can be reported. (See also Alon and Schieber [1], Thorup [9], and Chan _et
al._ [3].) Chazelle’s result is stated in terms of the inverse Ackermann
function; see Section 1.1.
###### Lemma 3
Let $T$ be a tree with $n$ nodes in which each edge stores an element of the
semigroup $(W,\oplus)$. For any integer $m\geq n$, we can preprocess $T$ in
$O(m)$ time into a data structure of size $O(m)$, such that any path-sum query
can be answered in $O(\alpha(m,n))$ time.
###### Remark 1
_Assume that $(W,\oplus)$ is the semigroup, where $W$ is the set of all real
numbers and the operator $\oplus$ takes the minimum of its arguments. In this
case, we will refer to a query as a _path-minimum query_. For this semigroup,
the result of Lemma 3 is optimal: Any data structure that can be constructed
in $O(m)$ time has worst-case query time $\Omega(\alpha(m,n))$. To prove this,
assume that we can answer any query in $o(\alpha(m,n))$ time. Then the on-line
minimum spanning tree verification problem on a tree with $n$ vertices and
$m\geq n$ queries can be solved in $o(m\cdot\alpha(m,n))$ time, by performing
a path-maximum query for the endpoints of each edge $e$ and checking that the
weight of $e$ is larger than the path-maximum. This contradicts the lower
bound for this problem proved by Pettie [8]. _
## 3 Beer Distance Queries in Maximal Outerplanar Graphs
Let $G$ be a maximal outerplanar beer graph with $n$ vertices that satisfies
the generalized triangle inequality. We will show how to preprocess $G$, such
that for any two vertices $u$ and $v$, the weight, $\mathsf{dist}_{B}(u,v)$,
of a shortest beer path between $u$ and $v$ can be reported. Our approach will
be to define a special semigroup $(W,\oplus)$, such that each element of $W$
“contains” certain distances and beer distances. With each edge of the dual
$D(G)$, we will store one element of the set $W$. As we will see later, a beer
distance query can then be reduced to a path-sum query in $D(G)$. Thus, by
applying the results of Section 2.2, we will obtain a proof of Theorem 1.
We will need the first claim in the following lemma. The second claim will be
used in Section 4.
###### Lemma 4
Consider the beer graph $G$ as above.
1. 1.
In $O(n)$ total time, we can compute $\mathsf{dist}_{B}(u,u)$ for each vertex
$u$ of $G$, and $\mathsf{dist}_{B}(u,v)$ for each edge $(u,v)$ in $G$.
2. 2.
After an $O(n)$–time preprocessing of $G$, we can report,
1. (a)
for any query edge $(u,v)$ of $G$, the shortest beer path between $u$ and $v$
in $O(L)$ time, where $L$ is the number of vertices on this path,
2. (b)
for any query vertex $u$ of $G$, the shortest beer path from $u$ to itself in
$O(L)$ time, where $L$ is the number of vertices on this path.
Proof. We choose an arbitrary face $R$ of $G$ and make it the root of $D(G)$.
Let $(u,v)$ be any edge of $G$. This edge divides $G$ into two outerplanar
subgraphs, both of which contain $(u,v)$ as an edge. Let $G_{uv}^{R}$ be the
subgraph that contains the face $R$, and let $G_{uv}^{\neg R}$ denote the
other subgraph. Note that if $(u,v)$ is an external edge, then $G_{uv}^{R}=G$
and $G_{uv}^{\neg R}$ consists of the single edge $(u,v)$. By the generalized
triangle inequality, the shortest beer path between $u$ and $v$ is completely
in $G_{uv}^{R}$ or completely in $G_{uv}^{\neg R}$. The same is true for the
shortest beer path from $u$ to itself. Thus, for each edge $(u,v)$ of $G$,
$\mathsf{dist}_{B}(u,v)=\min\left(\mathsf{dist}_{B}(u,v,G_{uv}^{R}),\mathsf{dist}_{B}(u,v,G_{uv}^{\neg
R})\right),$
$\mathsf{dist}_{B}(u,u)=\min\left(\mathsf{dist}_{B}(u,u,G_{uv}^{R}),\mathsf{dist}_{B}(u,u,G_{uv}^{\neg
R})\right).$
By performing a post-order traversal of $D(G)$, we can compute
$\mathsf{dist}_{B}(u,v,G_{uv}^{\neg R})$ and
$\mathsf{dist}_{B}(u,u,G_{uv}^{\neg R})$ for all edges $(u,v)$, in $O(n)$
total time. After these values have been computed, we perform a pre-order
traversal of $D(G)$ and obtain $\mathsf{dist}_{B}(u,v,G_{uv}^{R})$ and
$\mathsf{dist}_{B}(u,u,G_{uv}^{R})$, again for all edges $(u,v)$, in $O(n)$
total time. The details will be given in Section 5.
In the rest of this section, we assume that all beer distances in the first
claim of Lemma 4 have been computed.
For any two distinct internal faces $F$ and $F^{\prime}$ of $G$, let
$Q_{F,F^{\prime}}$ be the union of the two sets
$\\{(u,v,\mathsf{dist}(u,v),\mathsf{D})\mid u\mbox{ is a vertex of }F,v\mbox{
is a vertex of }F^{\prime}\\}$
and
$\\{(u,v,\mathsf{dist}_{B}(u,v),\mathsf{BD})\mid u\mbox{ is a vertex of
}F,v\mbox{ is a vertex of }F^{\prime}\\},$
where the “bits” $\mathsf{D}$ and $\mathsf{BD}$ indicate whether the tuple
represents a distance or a beer distance. In words, $Q_{F,F^{\prime}}$ is the
set of all shortest path distances and all shortest beer distances between a
vertex in $F$ and a vertex in $F^{\prime}$. Since each internal face has three
vertices, the set $Q_{F,F^{\prime}}$ has exactly $18$ elements.
###### Observation 1
Let $u$ and $v$ be vertices of $G$, and let $F$ and $F^{\prime}$ be internal
faces that contain $u$ and $v$ as vertices, respectively.
1. 1.
If $F=F^{\prime}$, then we can determine both $\mathsf{dist}(u,v)$ and
$\mathsf{dist}_{B}(u,v)$ in $O(1)$ time.
2. 2.
If $F\neq F^{\prime}$ and we are given the set $Q_{F,F^{\prime}}$, then we can
determine both $\mathsf{dist}(u,v)$ and $\mathsf{dist}_{B}(u,v)$ in $O(1)$
time.
Proof. First assume that $F=F^{\prime}$. If $u=v$, then $\mathsf{dist}(u,v)=0$
and $\mathsf{dist}_{B}(u,v)$ has been precomputed. If $u\neq v$, then $(u,v)$
is an edge of $G$ and, thus, $\mathsf{dist}(u,v)=\omega(u,v)$ and
$\mathsf{dist}_{B}(u,v)$ has been precomputed.
Assume that $F\neq F^{\prime}$. If we know the set $Q_{F,F^{\prime}}$, then we
can find $\mathsf{dist}(u,v)$ and $\mathsf{dist}_{B}(u,v)$ in $O(1)$ time,
because these two distances are in $Q_{F,F^{\prime}}$.
In the rest of this section, we will show that Lemma 3 can be used to compute
the set $Q_{F,F^{\prime}}$ for any two distinct internal faces $F$ and
$F^{\prime}$.
###### Lemma 5
For any edge $(F,F^{\prime})$ of $D(G)$, the set $Q_{F,F^{\prime}}$ can be
computed in $O(1)$ time.
Proof. Let $u$ be a vertex of $F$ and let $v$ be a vertex of $F^{\prime}$.
Consider the subgraph $G[F,F^{\prime}]$ of $G$ that is induced by the four
vertices of $F$ and $F^{\prime}$; this subgraph has five edges. By the
generalized triangle inequality,
$\mathsf{dist}(u,v)=\mathsf{dist}(u,v,G[F,F^{\prime}])$. Thus,
$\mathsf{dist}(u,v)$ can be computed in $O(1)$ time.
We now show how $\mathsf{dist}_{B}(u,v)$ can be computed in $O(1)$ time. If
$u=v$ or $(u,v)$ is an edge of $G$, then $\mathsf{dist}_{B}(u,v)$ has been
precomputed. Assume that $u\neq v$ and $(u,v)$ is not an edge of $G$. Let $w$
and $w^{\prime}$ be the two vertices that are shared by $F$ and $F^{\prime}$.
Since any path in $G$ between $u$ and $v$ contains at least one of $w$ and
$w^{\prime}$, $\mathsf{dist}_{B}(u,v)$ is the minimum of
1. 1.
$\mathsf{dist}_{B}(u,w)+\omega(w,v)$,
2. 2.
$\omega(u,w)+\mathsf{dist}_{B}(w,v)$,
3. 3.
$\mathsf{dist}_{B}(u,w^{\prime})+\omega(w^{\prime},v)$,
4. 4.
$\omega(u,w^{\prime})+\mathsf{dist}_{B}(w^{\prime},v)$.
Since $(u,w)$, $(w,v)$, $(u,w^{\prime})$, and $(w^{\prime},v)$ are edges of
$G$, all terms in these four sums have been precomputed. Therefore,
$\mathsf{dist}_{B}(u,v)$ can be computed in $O(1)$ time.
We have shown that each of the $18$ elements of $Q_{F,F^{\prime}}$ can be
computed in $O(1)$ time. Therefore, this entire set can be computed in $O(1)$
time.
###### Lemma 6
Let $F$, $F^{\prime}$, and $F^{\prime\prime}$ be three pairwise distinct
internal faces of $G$, such that $F^{\prime}$ is on the path in $D(G)$ between
$F$ and $F^{\prime\prime}$. If we are given the sets $Q_{F,F^{\prime}}$ and
$Q_{F^{\prime},F^{\prime\prime}}$, then the set $Q_{F,F^{\prime\prime}}$ can
be computed in $O(1)$ time.
Proof. Let $u$ be a vertex of $F$ and let $v$ be a vertex of
$F^{\prime\prime}$. Since $G$ is an outerplanar graph, any path in $G$ between
$u$ and $v$ must contain at least one vertex of $F^{\prime}$. It follows that
$\mathsf{dist}(u,v)=\min\\{\mathsf{dist}(u,w)+\mathsf{dist}(w,v)\mid w\mbox{
is a vertex of }F^{\prime}\\}.$
Thus, since $(u,w,\mathsf{dist}(u,w),\mathsf{D})\in Q_{F,F^{\prime}}$ and
$(w,v,\mathsf{dist}(w,v),\mathsf{D})\in Q_{F^{\prime},F^{\prime\prime}}$, the
value of $\mathsf{dist}(u,v)$ can be computed in $O(1)$ time.
Figure 4: Any beer path from $u$ to $v$ contains at least one vertex of
$F^{\prime}$. In (a), we consider the shortest beer path from $u$ to $w$,
followed by the shortest path from $w$ to $v$. In (b), we consider the
shortest path from $u$ to $w$, followed by the shortest beer path from $w$ to
$v$.
By a similar argument, $\mathsf{dist}_{B}(u,v)$ is equal to (refer to Figure
4)
$\min\\{\min(\mathsf{dist}_{B}(u,w)+\mathsf{dist}(w,v),\mathsf{dist}(u,w)+\mathsf{dist}_{B}(w,v)):\mbox{
$w$ is a vertex of $F^{\prime}$}\\}.$
All values $\mathsf{dist}(u,w)$, $\mathsf{dist}(w,v)$,
$\mathsf{dist}_{B}(u,w)$, and $\mathsf{dist}_{B}(w,v)$ are encoded in the sets
$Q_{F,F^{\prime}}$ and $Q_{F^{\prime},F^{\prime\prime}}$. Therefore, we can
compute $\mathsf{dist}_{B}(u,v)$ in $O(1)$ time.
Thus, since each of the $18$ elements of $Q_{F,F^{\prime\prime}}$ can be
computed in $O(1)$ time, the entire set can be computed in $O(1)$ time.
We define
$W=\\{Q_{F,F^{\prime}}\mid\mbox{ $F$ and $F^{\prime}$ are distinct internal
faces of $G$}\\}\cup\\{\bot\\},$
where $\bot$ is a special symbol. We define the operator $\oplus:W\times
W\rightarrow W$ in the following way.
1. 1.
If $F$ and $F^{\prime}$ are distinct internal faces of $G$, then
$Q_{F,F^{\prime}}\oplus Q_{F,F^{\prime}}=Q_{F,F^{\prime}}$.
2. 2.
If $F$, $F^{\prime}$, and $F^{\prime\prime}$ are pairwise distinct internal
faces of $G$ such that $F^{\prime}$ is on the path in $D(G)$ between $F$ and
$F^{\prime\prime}$, then $Q_{F,F^{\prime}}\oplus
Q_{F^{\prime},F^{\prime\prime}}=Q_{F,F^{\prime\prime}}$.
3. 3.
In all other cases, the operator $\oplus$ returns $\bot$.
It is not difficult to verify that $\oplus$ is associative, implying that
$(W,\oplus)$ is a semigroup. By Lemma 5, we can compute $Q_{F,F^{\prime}}$ for
all edges $(F,F^{\prime})$ of $D(G)$, in $O(n)$ total time.
Recall from Lemma 1 that, after an $O(n)$–time preprocessing, we can decide in
$O(1)$ time, for any three internal faces $F$, $F^{\prime}$, and
$F^{\prime\prime}$ of $G$, whether $F^{\prime}$ is on the path in $D(G)$
between $F$ and $F^{\prime\prime}$. Therefore, using Lemma 6, the operator
$\oplus$ takes $O(1)$ time to evaluate for any two elements of $W$.
Finally, let $F$ and $F^{\prime}$ be two distinct internal faces of $G$, and
let $F=F_{0},F_{1},F_{2},\ldots,F_{k}=F^{\prime}$ be the path in $D(G)$
between $F$ and $F^{\prime}$. Then
$Q_{F,F^{\prime}}=\oplus_{i=0}^{k-1}Q_{F_{i},F_{i+1}}$. Thus, if we store with
each edge of the tree $D(G)$, the corresponding element of the semigroup, then
computing $Q_{F,F^{\prime}}$ becomes a path-sum query as in Section 2.2.
To summarize, all conditions to apply Lemma 3 are satisfied. As a result, we
have proved Theorem 1 for maximal outerplanar graphs that satisfy the
generalized triangle inequality.
### 3.1 The Result in Theorem 1 is Optimal
In Section 2.2, see also Remark 1, we have seen path-minimum queries in a
tree, in which each edge $e$ stores a real number $s(e)$. In such a query, we
are given two distinct nodes $u$ and $v$, and have to return the smallest
value $s(e)$ among all edges $e$ on the path between $u$ and $v$. Lemma 3
gives a trade-off between the preprocessing and query times when answering
such queries.
Let $D$ be an arbitrary data structure that answers beer distance queries in
any beer tree. Let $P(n)$, $S(n)$, and $Q(n)$ denote the preprocessing time,
space, and query time of $D$, respectively, when the beer tree has $n$ nodes.
We will show that $D$ can be used to answer path-minimum queries.
Consider an arbitrary tree $T$ with $n$ nodes, such that each edge $e$ stores
a real number $s(e)$. We may assume without loss of generality that $0<s(e)<1$
for each edge $e$ of $T$.
By making an arbitrary node the root of $T$, the number of edges on the path
in $T$ between two nodes $u$ and $v$ is equal to
$\mathord{\it level}(u)+\mathord{\it level}(v)-2\cdot\mathord{\it
level}(\mathsf{LCA}(u,v)).$
Thus, by Lemma 1, after an $O(n)$–time preprocessing, we can compute the
number of edges on this path in $O(1)$ time.
We create a beer tree $T^{\prime}$ as follows. Initially, $T^{\prime}$ is a
copy of $T$. For each edge $e=(u,v)$ of $T^{\prime}$, we introduce a new node
$x_{e}$ and replace $e$ by two edges $(u,x_{e})$ and $(v,x_{e})$; we assign a
weight of $1$ to each of these two edges. In the current tree $T^{\prime}$,
none of the nodes has a beer store. For every node $x_{e}$ in $T^{\prime}$, we
introduce a new node $x^{\prime}_{e}$, add the edge $(x_{e},x^{\prime}_{e})$,
assign a weight of $s(e)$ to this edge, and make $x^{\prime}_{e}$ a beer
store. Finally, we construct the data structure $D$ for the resulting beer
tree $T^{\prime}$. Since $T^{\prime}$ has $n+2(n-1)=3n-2$ nodes, it takes
$P(3n-2)+O(n)$ time to construct $D$ from the input tree $T$. Moreover, the
amount of space used is $S(3n-2)+O(n)$.
Let $u$ and $v$ be two distinct nodes in the original tree $T$, let $\pi$ be
the path in $T$ between $u$ and $v$, and let $\ell$ be the number of edges on
$\pi$. The corresponding path $\pi^{\prime}$ in $T^{\prime}$ between $u$ and
$v$ has weight $2\ell$.
For any edge $e$ of $T$, let $\pi^{\prime}_{e}$ be the beer path in
$T^{\prime}$ that starts at $u$, goes to $x_{e}$, then goes to
$x^{\prime}_{e}$ and back to $x_{e}$, and continues to $v$.
If $e$ is an edge of $\pi$, then the weight of $\pi^{\prime}_{e}$ is equal to
$2\ell+2\cdot s(e)$, which is less than $2\ell+2$. On the other hand, if $e$
is an edge of $T$ that is not on $\pi$, then the weight of $\pi^{\prime}_{e}$
is at least $2\ell+2+2\cdot s(e)$, which is larger than $2\ell+2$. It follows
that the shortest beer path in $T^{\prime}$ between $u$ and $v$ visits the
beer store $x^{\prime}_{e}$, where $e$ is the edge on $\pi$ for which $s(e)$
is minimum.
Thus, by computing $\ell$ and querying $D$ for the beer distance in
$T^{\prime}$ between $u$ and $v$, we obtain the smallest value $s(e)$ among
all edges $e$ on the path in $T$ between $u$ and $v$. The query time is
$Q(3n-2)+O(1)$.
By combining this reduction with Remark 1, it follows that the result of
Theorem 1 is optimal.
## 4 Reporting Shortest Beer Paths in Maximal Outerplanar Graphs
Let $G$ be a maximal outerplanar beer graph with $n$ vertices that satisfies
the generalized triangle inequality. In this section, we show that, after an
$O(n)$–time preprocessing, we can report, for any two query vertices $s$ and
$t$, the shortest beer path $\mathsf{SP}_{B}(s,t)$ from $s$ to $t$, in $O(L)$
time, where $L$ is the number of vertices on this path. As before, $D(G)$
denotes the dual of $G$.
###### Observation 2
Let $v$ be a vertex of $G$. The faces of $G$ containing $v$ form a path of
nodes in $D(G)$.
Define $P_{v}$ to be the path in $D(G)$ formed by the faces of $G$ containing
the vertex $v$. Let $G[P_{v}]$ be the subgraph of $G$ induced by the faces of
$G$ containing $v$. Note that $G[P_{v}]$ has a fan shape. Let $\mathsf{CW}(v)$
denote the clockwise neighbor of $v$ in $G[P_{v}]$ and let $\mathsf{CCW}(v)$
denote the counterclockwise neighbor of $v$ in $G[P_{v}]$. We will refer to
the clockwise path from $\mathsf{CW}(v)$ to $\mathsf{CCW}(v)$ in $G[P_{v}]$ as
the $v$-_chain_ and denote it by $\rho_{v}$. (Refer to Figure 5.)
Figure 5: A maximal outerplanar graph $G$. The subgraphs $G[P_{v}]$ and
$G[P_{u}]$ are shown in red and blue, respectively. Both the $v$-chain
$\rho_{v}$ and the $u$-chain $\rho_{u}$ are shown in bold. Both paths $P_{v}$
and $P_{u}$ are shown in black. Observe that $P_{u}$ is a single node.
###### Lemma 7
After an $O(n)$–time preprocessing, we can answer the following queries, for
any three query vertices $v$, $u$, and $w$, such that both $u$ and $w$ are on
the $v$-chain $\rho_{v}$:
1. 1.
Report the weight $\mathsf{dist}(u,w,\rho_{v})$ of the path from $u$ to $w$
along $\rho_{v}$ in $O(1)$ time.
2. 2.
Report the path $\mathsf{SP}(u,w,\rho_{v})$ from $u$ to $w$ along $\rho_{v}$
in $O(L)$ time, where $L$ is the number of vertices on this path.
Proof. For any vertex $v$ and any vertex $u$ on $\rho_{v}$, we store the
weight of the path from $u$ to $\mathsf{CW}(v)$ along $\rho_{v}$. Observe that
$\mathsf{dist}(u,w,\rho_{v})=|\mathsf{dist}(u,\mathsf{CW}(v),\rho_{v})-\mathsf{dist}(w,\mathsf{CW}(v),\rho_{v})|.$
Any exterior edge in $G$ is in exactly one chain and any interior edge in $G$
is in exactly two chains. Thus, the sum of the number of edges on each chain
is proportional to the number of edges of $G$, which is $O(n)$.
###### Lemma 8
After an $O(n)$–time preprocessing, we can answer the following query in
$O(1)$ time: Given three query vertices $v$, $u$, and $w$, such that both $u$
and $w$ are vertices of $G[P_{v}]$, report $\mathsf{dist}(u,w)$, i.e., the
distance between $u$ and $w$ in $G$.
Proof. We get the following cases; the correctness follows from the
generalized triangle inequality:
1. 1.
If $u=w$ then $\mathsf{dist}(u,w)=0$.
2. 2.
If $u=v$ then $(u,w)$ is an edge and we return $\omega(u,w)$. Similarly if
$w=v$, we return $\omega(u,w)$.
3. 3.
Otherwise $u$ and $w$ are both on $\rho_{v}$ and we return
$\min(\mathsf{dist}(u,w,\rho_{v}),\omega(u,v)+\omega(v,w))$.
###### Lemma 9
After an $O(n)$–time preprocessing, we can report, for any three vertices $v$,
$u$, and $w$, such that both $u$ and $w$ are vertices of $G[P_{v}]$,
$\mathsf{SP}(u,w)$ in $O(L)$ time, where $L$ is the number of vertices on the
path.
Proof. Using Lemma 8, we can determine in $O(1)$ if the shortest path from $u$
to $w$ goes through $v$ or follows the $v$-chain $\rho_{v}$. (Refer to Figure
6). If it goes through $v$, then $\mathsf{SP}(u,w)=(u,v,w)$. Otherwise,
$\mathsf{SP}(u,w)$ takes the path along $\rho_{v}$ and by Lemma 7, we can find
this path in $O(L)$ time.
Figure 6: Two possible cases for the shortest path between $u$ and $w$: (a) it
goes through vertex $v$ (shown in dashed red), or (b) it goes through the
vertices of the $v$-chain between $u$ and $w$ (shown in dashed blue).
###### Lemma 10
After an $O(n)$–time preprocessing, we can report, for any three vertices $v$,
$u$ and $w$, such that both $u$ and $w$ are vertices of $G[P_{v}]$, the beer
distance $\mathsf{dist}_{B}(u,w)$ in $O(1)$ time. The corresponding shortest
beer path $\mathsf{SP}_{B}(u,w)$ can be reported in $O(L)$ time, where $L$ is
the number of vertices on the path.
Proof. Recall from Lemma 4 that we can compute $\mathsf{dist}_{B}(u,v)$ for
every edge $(u,v)$ in $G$, and $\mathsf{dist}_{B}(v,v)$ for every vertex $v$
in $G$, in $O(n)$ time.
Let $\rho_{v}=(\mathsf{CW}(v)=u_{1},u_{2},\dots,u_{N}=\mathsf{CCW}(v))$. Let
$A_{v}[~{}]$ be an array of size $N-1$. For $i=1,\ldots,N-1$, we set
$A_{v}[i]=\mathsf{dist}_{B}(u_{i},u_{i+1})-\omega(u_{i},u_{i+1})$. Recall that
by the generalized triangle inequality,
$\omega(u_{i},u_{i+1})=\mathsf{dist}(u_{i},u_{i+1})$. Therefore, $A[i]$ holds
the difference between the weights of the shortest path from $u_{i}$ to
$u_{i+1}$ and the shortest beer path from $u_{i}$ to $u_{i+1}$. After
preprocessing the array $A_{v}[~{}]$ in $O(N)$ time, we can conduct range
minimum queries in $O(1)$ time. (Bender and Farach-Colton [2] show that these
queries are equivalent to $\mathsf{LCA}$-queries in the Cartesian tree of the
array.) Thus, for each $v$-chain of $N$ nodes, we spend $O(N)$ time processing
the $v$-chain. Since every edge is in at most two chains, processing all
$v$-chains takes $O(n)$ time and space.
Given two vertices $u$ and $w$ of $G[P_{v}]$, we determine the beer distance
$\mathsf{dist}_{B}(u,w)$ as follows:
1. 1.
If $u=w$ then $\mathsf{dist}_{B}(u,w)$ has already been computed by Lemma 4.
2. 2.
If $u=v$ or $w=v$, then there is an edge from $v$ to the other vertex. Thus,
$\mathsf{dist}_{B}(u,w)$ has already been computed by Lemma 4.
3. 3.
Otherwise, $u$, $w$ and $v$ are three distinct vertices. Assume without loss
of generality that $w$ is clockwise from $u$ on the $v$-chain. We take the
minimum of the following two cases:
1. (a)
The shortest beer path from $u$ to $w$ that goes through $v$. Since a beer
store must be visited before or after $v$, this beer path has a weight of
$\min(\mathsf{dist}_{B}(u,v)+\omega(v,w),\omega(u,v)+\mathsf{dist}_{B}(v,w))$.
2. (b)
The shortest beer path through the vertices of the $v$-chain. Note that this
beer path will visit each vertex on the $v$-chain between $u$ and $w$, but may
go off the $v$-chain to visit a beer store. On $\mathsf{SP}_{B}(u,w)$, there
is one pair of vertices, $u_{i}$ and $u_{i+1}$, such that a beer path is taken
between $u_{i}$ and $u_{i+1}$, and $u_{i}$ and $u_{i+1}$ are adjacent on the
$v$-chain; refer to Figure 7. The shortest path is taken between all other
pairs of adjacent vertices on the $v$-chain. From Lemma 7, we can compute
$\mathsf{dist}(u,w,\rho_{v})$ in $O(1)$ time. The shortest beer path through
the vertices of the $v$-chain has a weight of
$\mathsf{dist}(u,w,\rho_{v})+A_{v}[i]$, where $A_{v}[i]$ is the additional
distance needed to visit a beer store between $u_{i}$ and $u_{i+1}$. Let $u$
be the $j^{th}$ vertex on $\rho_{v}$ and let $w$ be the $k^{th}$ vertex in
$\rho_{v}$. Then $A_{v}[i]$ is the minimum value in $A_{v}[j,\dots,k-1]$. We
can determine $A_{v}[i]$ in constant time using a range minimum query.
Note that in case 1 and case 2, $\mathsf{SP}_{B}(u,w)$ can be constructed in
$O(L)$ time by Lemma 4. For case 3 (a) let $p=(u,v,w)$ and for case 3 (b) let
$p=\mathsf{SP}(u,w,\rho_{v})$. Let $u_{i}$, $u_{i+1}$ be the pair of adjacent
vertices on $p$ between which a beer path was taken. Using Lemma 4 we can find
$\mathsf{SP}_{B}(u_{i},u_{i+1})$ in $O(L)$ time. We obtain
$\mathsf{SP}_{B}(u,w)$ by replacing the edge $(u_{i},u_{i+1})$ in $p$ with
$\mathsf{SP}_{B}(u_{i},u_{i+1})$.
Figure 7: Both figures show a shortest beer path from $u$ to $w$ through the
vertices on the $v$-chain. Thicker edges on the blue beer path are edges that
are traversed twice; once in each direction.
### 4.1 Answering Shortest Beer Path Queries
Recall that, for any vertex $v$ of $G$, $P_{v}$ denotes the path in $D(G)$
formed by the faces of $G$ containing $v$. Moreover, $G[P_{v}]$ denotes the
subgraph of $G$ induced by these faces.
Consider two query vertices $s$ and $t$ of $G$. Our goal is to compute the
shortest beer path $\mathsf{SP}_{B}(s,t)$.
Let $F_{s}$ and $F_{t}$ be arbitrary faces containing $s$ and $t$,
respectively. If $t$ is in $G[P_{s}]$ then, by Lemma 10, we can construct
$\mathsf{SP}_{B}(s,t)$ in $O(L)$ time. For the remainder of this section, we
assume that $t$ is not in $G[P_{s}]$. To find $\mathsf{SP}_{B}(s,t)$, we start
by constructing a directed acyclic graph (DAG), $H$. In this DAG, vertices
will be arranged in columns of constant size, and all edges go from left to
right between vertices in adjacent columns. In $H$, each column will contain
one vertex that is on $\mathsf{SP}_{B}(s,t)$. First we will construct $H$ and
then we will show how we can use $H$ to construct $\mathsf{SP}_{B}(s,t)$. The
entire construction is illustrated in Figure 8.
###### Observation 3
Any interior edge $(a,b)$ of $G$ splits $G$ into two subgraphs such that if
$s$ is in one subgraph and $t$ is in the other, then any path in $G$ from $s$
to $t$ must visit at least one of $a$ and $b$.
Let $P$ be the unique path between $F_{s}$ and $F_{t}$ in $D(G)$. Consider
moving along $P$ from $F_{s}$ to $F_{t}$. Let $F_{1}$ be the node on $P_{s}$
that is closest to $F_{t}$, and let $F^{\prime}_{1}$ be the successor of
$F_{1}$ on $P$. Note that, by Lemmas 1 and 2, we can find $F_{1}$ and
$F^{\prime}_{1}$ in $O(1)$ time.111To apply Lemma 2, we consider each vertex
of $G$ to be a colour. For each vertex $v$ of $G$, the $v$-coloured path in
the tree $D(G)$ is the path $P_{v}$. The face $F_{1}$ is the answer to the
closest-colour query with nodes $F_{s}$ and $F_{t}$ and colour $s$. Let
$e_{1}=(a_{1},b_{1})$ be the edge in $G$ shared by the faces $F_{1}$ and
$F^{\prime}_{1}$. Since $\mathsf{SP}_{B}(s,t)$ must visit both of these faces,
by Observation 3, at least one of $a_{1}$ or $b_{1}$ is on the shortest beer
path.
We place $s$ in the first column of $H$ and $a_{1}$ and $b_{1}$ in the second
column of $H$. We then add two directed edges from $s$ to $a_{1}$, one with
weight $\mathsf{dist}(s,a_{1})$ and the other with weight
$\mathsf{dist}_{B}(s,a_{1})$. Similarly, we add two directed edges from $s$ to
$b_{1}$ with weights $\mathsf{dist}(s,b_{1})$ and
$\mathsf{dist}_{B}(s,b_{1})$.
Figure 8: An outerplanar graph $G$ (a) and the DAG $H$ constructed for the
shortest beer path query from $s$ to $t$ (b). The path $P$ from $F_{s}$ to
$F_{t}$ is shown in red. Each edge $e_{i}=(a_{i},b_{i})$ such that $e_{i}$ is
shared by $F_{i}$ and $F^{\prime}_{i}$ is shown in blue. The green edges of
$H$ represent the beer edges.
When $i\geq 2$ we construct the $(i+1)^{th}$ column of $H$ in the following
way. Let $e_{i-1}=(a_{i-1},b_{i-1})$ be the edge shared by the faces $F_{i-1}$
and $F^{\prime}_{i-1}$. The $i^{th}$ column of $H$ contains the vertices
$a_{i-1}$ and $b_{i-1}$. Note that $F^{\prime}_{i-1}$ is in both $P_{b_{i-1}}$
and $P_{a_{i-1}}$. Using Lemma 2, we find the node $F_{i}^{b}$ on
$P_{b_{i-1}}$ that is closest to $F_{t}$. If the vertex $a_{i-1}$ is not in
$F_{i}^{b}$, then we let $F_{i}=F_{i}^{b}$. Otherwise, we let $F_{i}$ be the
node on $P_{a_{i-1}}$ that is closest to $F_{t}$.
If $t$ is not a vertex of $F_{i}$, then let $F^{\prime}_{i}$ be the node that
follows $F_{i}$ on $P$; we find $F^{\prime}_{i}$ using Lemma 1. Let
$e_{i}=(a_{i},b_{i})$ be the edge of $G$ shared by the faces $F_{i}$ and
$F^{\prime}_{i}$. In the $(i+1)^{th}$ column, we place $a_{i}$ and $b_{i}$.
For each $u\in\\{a_{i-1},b_{i-1}\\}$ and each $v\in\\{a_{i},b_{i}\\}$ we add
two directed edges $(u,v)$ to the DAG, one with weight $\mathsf{dist}(u,v)$
and the other with weight $\mathsf{dist}_{B}(u,v)$. If $F_{i}$ is in
$P_{a_{i-1}}$, all these vertices are in $G[P_{a_{i-1}}]$; otherwise, $F_{i}$
is in $P_{b_{i-1}}$, and all these vertices are in $G[P_{b_{i-1}}]$. Thus, by
Lemmas 8 and 10, we can find the distances and beer distances to assign to
these edges in constant time.
If $t$ is in $F_{i}$, then in the $(i+1)^{th}$ column we only place the vertex
$t$. In this case, for each $u\in\\{a_{i-1},b_{i-1}\\}$, we add two directed
edges $(u,t)$ to the DAG with weights $\mathsf{dist}(u,t)$ and
$\mathsf{dist}_{B}(u,t)$. At this point we are done constructing $H$.
We define a _beer edge_ to be an edge of $H$ that was assigned a weight of a
beer path during the construction of $H$. We find the beer distance from $s$
to $t$ in $G$ using the following dynamic programming approach in $H$.
Let $M$ denote the number of columns in $H$. For $i=3,\dots,M$ and for all $u$
in the $i^{th}$ column of $H$, compute
$\mathsf{dist}_{B}(s,u)=\min\begin{cases}\mathsf{dist}_{B}(s,a_{i-2})+\mathsf{dist}(a_{i-2},u),\\\
\mathsf{dist}(s,a_{i-2})+\mathsf{dist}_{B}(a_{i-2},u),\\\
\mathsf{dist}_{B}(s,b_{i-2})+\mathsf{dist}(b_{i-2},u),\\\
\mathsf{dist}(s,b_{i-2})+\mathsf{dist}_{B}(b_{i-2},u)\end{cases}$
and
$\mathsf{dist}(s,u)=\min\begin{cases}\mathsf{dist}(s,a_{i-2})+\mathsf{dist}(a_{i-2},u),\\\
\mathsf{dist}(s,b_{i-2})+\mathsf{dist}(b_{i-2},u).\\\ \end{cases}$
The vertices $a_{i-2}$ and $b_{i-2}$ occur in the $(i-1)^{th}$ column. Thus,
$\mathsf{dist}_{B}(s,a_{i-2})$, $\mathsf{dist}_{B}(s,b_{i-2})$,
$\mathsf{dist}(s,a_{i-2})$, and $\mathsf{dist}(s,b_{i-2})$ will be computed
before computing the values for the $i^{th}$ column. We get
$\mathsf{dist}(a_{i-2},u)$, $\mathsf{dist}_{B}(a_{i-2},u)$,
$\mathsf{dist}(b_{i-2},u)$ and $\mathsf{dist}_{B}(b_{i-2},u)$ from the weights
of the DAG-edges between the $(i-1)^{th}$ and $i^{th}$ columns of $H$.
By keeping track of which expression produced $\mathsf{dist}_{B}(s,u)$ and
$\mathsf{dist}(s,u)$, we can backwards reconstruct the shortest beer path in
the DAG. Knowing the shortest beer path in the DAG enables us to construct the
corresponding beer path in $G$ as follows.
1. 1.
Define $P_{st}$ to be an empty path.
2. 2.
For each edge $(w,v)$ of the shortest beer path in the DAG.
1. (a)
If $(w,v)$ was a beer edge, let $P_{wv}=\mathsf{SP}_{B}(w,v)$, which can be
constructed in time proportional to its number of vertices via Lemma 10.
2. (b)
Otherwise, let $P_{wv}=\mathsf{SP}(w,v)$ which can be constructed in time
proportional to its number of vertices as seen in Lemma 9.
Let $P_{st}=P_{st}\cup P_{wv}$.
3. 3.
Return $P_{st}$, which is equal to $\mathsf{SP}_{B}(w,v)$.
Let $L$ denote the number of vertices on $\mathsf{SP}_{B}(s,t)$. In order for
the above query algorithm to take $O(L)$ time, the size of the DAG must be
$O(L)$. The following three lemmas will show this to be true.
###### Lemma 11
For $2\leq i<M-1$, $F_{i}$ contains either $a_{i-1}$ or $b_{i-1}$, but not
both.
Proof. Recall that we defined $F_{i}^{b}$ to be the last node on $P$ that is
also on $P_{b_{i-1}}$. We similarly define $F_{i}^{a}$ to be the last node on
$P$ that is also on $P_{a_{i-1}}$. From the way we choose $F_{i}$, $F_{i}$ is
either $F_{i}^{b}$ or $F_{i}^{a}$. We only choose $F_{i}=F_{i}^{b}$ after
having checked that $a_{i-1}$ is not in $F_{i}^{b}$; thus in this case we can
be sure that $F_{i}$ only contains $b_{i-1}$.
Assume for the purpose of contradiction that we choose $F_{i}=F_{i}^{a}$ and
$b_{i-1}$ is also in $F_{i}$. Let the third vertex of $F_{i}$ be $c$. Let the
face on $P$ immediately following $F_{i}$ be $F_{i}^{\prime}$. The edge shared
by $F_{i}$ and $F^{\prime}_{i}$ is either $(b_{i-1},c)$ or $(a_{i-1},c)$. If
$(b_{i-1},c)$ is the shared edge, then $F_{i}^{\prime}$ is a face closer to
$F_{t}$ that contains $b_{i-1}$ and not $a_{i-1}$, so we would have chosen
$F_{i}=F_{i}^{b}$, which is a contradiction. Otherwise, $(a_{i-1},c)$ is the
edge shared by $F_{i}$ and $F_{i}^{\prime}$, which implies that there is a
face containing $a_{i-1}$ closer to $F_{t}$ in $P$ than $F_{i}^{a}$, which
contradicts the definition of $F_{i}^{a}$.
###### Lemma 12
Every vertex of $G$ appears in at most one column of $H$.
Proof. Since $(a_{1},b_{1})$ is an edge shared by both the last face of $P$
containing $s$ and the first face of $P$ that does not contain $s$ it is not
possible for either of these vertices to be the vertex $s$. Thus, $s$ will
only be represented by the vertex in the first column of $H$. By stopping the
construction of $H$ as soon as we add a vertex representing $t$, we ensure
that $H$ only contains one vertex corresponding to the vertex $t$ in $G$.
For $2\leq i\leq M-2$, consider the vertex $a_{i-1}$ in $G$ represented by a
vertex in the $i^{th}$ column of $H$. If $F_{i}=F_{i}^{a}$ then by definition
of $F_{i}^{a}$, $F^{\prime}_{i}$ does not contain $a_{i-1}$. Since
$(a_{i},b_{i})$ is an edge of $F^{\prime}_{i}$, $a_{i}\neq a_{i-1}$ and
$b_{i}\neq a_{i-1}$. Because the face $F^{\prime}_{i}$ is closer to $F_{t}$
than $F_{i}^{a}$, $a_{i-1}$ is not a vertex on any of the faces on the path
from $F^{\prime}_{i}$ to $F_{t}$. Thus, subsequent columns of $H$ will not
contain vertices representing the vertex $a_{i-1}$ in $G$.
If $F_{i}=F_{i}^{b}$ then by Lemma 11, $a_{i-1}$ is not in $F_{i}$ and since
$(a_{i},b_{i})$ is an edge of $F_{i}$, $a_{i}\neq a_{i-1}$ and $b_{i}\neq
a_{i-1}$. Because $F_{i}$ is a face on $P$ closer to $F_{t}$ than $F_{i-1}$ (a
face that contains $a_{i-1}$) it follows from Observation 2 that none of the
faces on $P$ from $F_{i-1}$ to $F_{t}$ will have the vertex $a_{i-1}$ on their
face and, thus, $a_{i-1}$ will not be represented by vertices in subsequent
columns of $H$.
By switching the roles of $a_{i-1}$ with $b_{i-1}$ in the above reasoning we
can see that this also holds for $b_{i-1}$.
###### Lemma 13
The number of vertices and edges of $H$ is $O(L)$.
Proof. By Observation 3 and Lemma 12, the number of columns of $H$ is at most
$L$. Since each column has at most two vertices, each of which having at most
four outgoing edges, the total number of vertices and edges of $H$ is $O(L)$.
Observe that the total preprocessing time is $O(n)$. For two query vertices
$s$ and $t$, the DAG, $H$, can be constructed in $O(L)$ time. Finally, the
dynamic programming algorithm on $H$ takes $O(L)$ time. Thus, we have proved
Theorem 2 for maximal outerplanar graphs that satisfy the generalized triangle
inequality.
## 5 Proof of Lemma 4
Let $G$ be a maximal outerplanar beer graph with $n$ vertices that satisfies
the generalized triangle inequality. We will first show how to compute
$\mathsf{dist}_{B}(u,u)$ for each vertex $u$ of $G$, and
$\mathsf{dist}_{B}(u,v)$ for each edge $(u,v)$ of $G$. Consider again the dual
$D(G)$ of $G$. We choose an arbitrary face of $G$ and make it the root of
$D(G)$.
Let $(u,v)$ be any edge of $G$. This edge divides $G$ into two outerplanar
subgraphs, both of which contain $(u,v)$ as an edge. Let $G_{uv}^{R}$ be the
subgraph that contains the face represented by the root of $D(G)$, and let
$G_{uv}^{\neg R}$ denote the other subgraph. Note that if $(u,v)$ is an
external edge, then $G_{uv}^{R}=G$ and $G_{uv}^{\neg R}$ consists of the
single edge $(u,v)$.
By the generalized triangle inequality, the shortest beer path between $u$ and
$v$ is completely in $G_{uv}^{R}$ or completely in $G_{uv}^{\neg R}$. The same
is true for the shortest beer path from $u$ to itself. This implies:
###### Observation 4
For each edge $(u,v)$ of $G$,
1. 1.
$\mathsf{dist}_{B}(u,v)=\min\left(\mathsf{dist}_{B}(u,v,G_{uv}^{R}),\mathsf{dist}_{B}(u,v,G_{uv}^{\neg
R})\right)$,
2. 2.
$\mathsf{dist}_{B}(u,u)=\min\left(\mathsf{dist}_{B}(u,u,G_{uv}^{R}),\mathsf{dist}_{B}(u,u,G_{uv}^{\neg
R})\right)$,
3. 3.
$\mathsf{dist}_{B}(v,v)=\min\left(\mathsf{dist}_{B}(v,l,G_{uv}^{R}),\mathsf{dist}_{B}(v,v,G_{uv}^{\neg
R})\right)$,
Thus, it suffices to first compute $\mathsf{dist}_{B}(u,v,G_{uv}^{\neg R})$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{\neg R})$, and
$\mathsf{dist}_{B}(v,v,G_{uv}^{\neg R})$ for all edges $(u,v)$, and then
compute $\mathsf{dist}_{B}(u,v,G_{uv}^{R})$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{R})$, and $\mathsf{dist}_{B}(v,v,G_{uv}^{R})$,
again for all edges $(u,v)$.
### 5.1 Recurrences for $\mathsf{dist}_{B}(u,v,G_{uv}^{\neg R})$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{\neg R})$, and
$\mathsf{dist}_{B}(v,v,G_{uv}^{\neg R})$
Let $(u,v)$ be an edge of $G$. Item 1. below presents the base cases, whereas
item 2. gives the recurrences.
Figure 9: Illustrating the post-order traversal for all cases in item 2.
1. 1.
Assume that $(u,v)$ is an external edge of $G$.
1. (a)
If both $u$ and $v$ are beer stores, then $\mathsf{dist}_{B}(u,v,G_{uv}^{\neg
R})=\omega(u,v)$, $\mathsf{dist}_{B}(u,u,G_{uv}^{\neg R})=0$, and
$\mathsf{dist}_{B}(v,v,G_{uv}^{\neg R})=0$.
2. (b)
If exactly one of $u$ and $v$, say $u$, is a beer store, then
$\mathsf{dist}_{B}(u,v,G_{uv}^{\neg R})=\omega(u,v)$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{\neg R})=0$, and
$\mathsf{dist}_{B}(v,v,G_{uv}^{\neg R})=2\cdot\omega(u,v)$.
3. (c)
If neither $u$ nor $v$ is a beer store, then
$\mathsf{dist}_{B}(u,v,G_{uv}^{\neg R})=\infty$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{\neg R})=\infty$, and
$\mathsf{dist}_{B}(v,v,G_{uv}^{\neg R})=\infty$.
2. 2.
Assume that $(u,v)$ is an internal edge of $G$. Let $w$ be the third vertex of
the face of $G_{uv}^{\neg R}$ that contains $(u,v)$ as an edge. All possible
cases are illustrated in Figure 9.
1. (a)
The value of $\mathsf{dist}_{B}(u,v,G_{uv}^{\neg R})$ is the minimum of
1. i.
$\mathsf{dist}_{B}(u,w,G_{uw}^{\neg R})+\omega(w,v)$,
2. ii.
$\omega(u,w)+\mathsf{dist}_{B}(w,v,G_{vw}^{\neg R})$,
3. iii.
$\mathsf{dist}_{B}(u,u,G_{uw}^{\neg R})+\omega(u,v)$,
4. iv.
$\omega(u,v)+\mathsf{dist}_{B}(v,v,G_{vw}^{\neg R})$.
2. (b)
The value of $\mathsf{dist}_{B}(u,u,G_{uv}^{\neg R})$ is the minimum of
1. i.
$\mathsf{dist}_{B}(u,u,G_{uw}^{\neg R})$,
2. ii.
$2\cdot\omega(u,w)+\mathsf{dist}_{B}(w,w,G_{vw}^{\neg R})$,
3. iii.
$2\cdot\omega(u,v)+\mathsf{dist}_{B}(v,v,G_{vw}^{\neg R})$.
The value of $\mathsf{dist}_{B}(v,v,G_{uv}^{\neg R})$ is obtained by swapping
$u$ and $v$ in i., ii., and iii.
These recurrences express $\mathsf{dist}_{B}(u,v,G_{uv}^{\neg R})$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{\neg R})$, and
$\mathsf{dist}_{B}(v,v,G_{uv}^{\neg R})$ in terms of values that are “further
down” in the tree $D(G)$. Therefore, by performing a post-order traversal of
$D(G)$, we obtain all these values, for all edges $(u,v)$ of $G$, in $O(n)$
total time.
### 5.2 Recurrences for $\mathsf{dist}_{B}(u,v,G_{uv}^{R})$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{R})$, and $\mathsf{dist}_{B}(v,v,G_{uv}^{R})$
Let $(u,v)$ be an edge of $G$. Item 1. below presents the base cases, whereas
item 2. gives the recurrences.
1. 1.
Assume that $(u,v)$ is an edge of the face representing the root of $D(G)$.
Let $w$ be the third vertex of this face.
1. (a)
The value of $\mathsf{dist}_{B}(u,v,G_{uv}^{R})$ is the minimum of
1. i.
$\mathsf{dist}_{B}(u,u,G_{uw}^{\neg R})+\omega(u,v)$,
2. ii.
$\mathsf{dist}_{B}(u,w,G_{uw}^{\neg R})+\omega(w,v)$,
3. iii.
$\omega(u,v)+\mathsf{dist}_{B}(v,v,G_{vw}^{\neg R})$,
4. iv.
$\omega(u,w)+\mathsf{dist}_{B}(w,v,G_{vw}^{\neg R})$.
2. (b)
The value of $\mathsf{dist}_{B}(u,u,G_{uv}^{R})$ is the minimum of222We do not
have to consider $W:=\omega(u,v)+\mathsf{dist}_{B}(v,w,G_{vw}^{\neg
R})+\omega(w,u)$, because the sum of the values in ii. and iii. is at most
$2W$. Therefore, the smaller of the values in ii. and iii. is at most $W$.
1. i.
$\mathsf{dist}_{B}(u,u,G_{uw}^{\neg R})$,
2. ii.
$2\cdot\omega(u,w)+\mathsf{dist}_{B}(w,w,G_{vw}^{\neg R})$,
3. iii.
$2\cdot\omega(u,v)+\mathsf{dist}_{B}(v,v,G_{vw}^{\neg R})$.
The value of $\mathsf{dist}_{B}(v,v,G_{uv}^{\neg R})$ is obtained by swapping
$u$ and $v$ in i., ii., and iii.
2. 2.
Assume that $(u,v)$ is not an edge of the face represented by the root of
$D(G)$. Let $w$ be the third vertex of the face of $G_{uv}^{R}$ that contains
$(u,v)$ as an edge. We may assume without loss of generality that $(v,w)$ is
an edge of the face represented by the parent of the face representing
$(u,v,w)$. All possible cases are illustrated in Figure 10.
Figure 10: Illustrating the pre-order traversal for all cases in item 2.
1. (a)
The value of $\mathsf{dist}_{B}(u,v,G_{uv}^{R})$ is the minimum of
1. i.
$\mathsf{dist}_{B}(u,u,G_{uw}^{\neg R})+\omega(u,v)$,
2. ii.
$\mathsf{dist}_{B}(u,w,G_{uw}^{\neg R})+\omega(w,v)$,
3. iii.
$\omega(u,v)+\mathsf{dist}_{B}(v,v,G_{vw}^{R})$,
4. iv.
$\omega(u,w)+\mathsf{dist}_{B}(w,v,G_{vw}^{R})$.
2. (b)
The value of $\mathsf{dist}_{B}(u,u,G_{uv}^{R})$ is the minimum of
1. i.
$\mathsf{dist}_{B}(u,u,G_{uw}^{\neg R})$,
2. ii.
$2\cdot\omega(u,v)+\mathsf{dist}_{B}(v,v,G_{vw}^{R})$,
3. iii.
$2\cdot\omega(u,w)+\mathsf{dist}_{B}(w,w,G_{vw}^{R})$.
3. (c)
The value of $\mathsf{dist}_{B}(v,v,G_{uv}^{R})$ is the minimum of
1. i.
$\mathsf{dist}_{B}(v,v,G_{vw}^{R})$,
2. ii.
$2\cdot\omega(v,u)+\mathsf{dist}_{B}(u,u,G_{uw}^{R})$,
3. iii.
$2\cdot\omega(v,w)+\mathsf{dist}_{B}(w,w,G_{uw}^{R})$.
These recurrences express $\mathsf{dist}_{B}(u,v,G_{uv}^{R})$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{R})$, and $\mathsf{dist}_{B}(v,v,G_{uv}^{R})$
in terms of values that are “higher up” in the tree $D(G)$ and values that
involve graphs with a superscript “$\neg R$”. These latter values have been
computed already. Thus, by performing a pre-order traversal of $D(G)$, we
obtain all values $\mathsf{dist}_{B}(u,v,G_{uv}^{R})$,
$\mathsf{dist}_{B}(u,u,G_{uv}^{R})$, and $\mathsf{dist}_{B}(v,v,G_{uv}^{R})$,
for all edges $(u,v)$ of $G$, in $O(n)$ total time. This completes the proof
of the first claim in Lemma 4.
Consider $\mathsf{dist}(u,v)$ where $u=v$ or $(u,v)$ is an edge of $G$. If $u$
or $v$ is a beer store, then store $nil$ with $\mathsf{dist}(u,v)$.
The values $\mathsf{dist}(u,v,G_{uv}^{R})$ and $\mathsf{dist}(u,v,G_{uv}^{\neg
R})$ are computed as the minimum of a set of path weights from $u$ to $v$
through a vertex $x$ such that $x$ is adjacent to both $u$ and $v$ or $x$ is
equal to one of these vertices and adjacent to the other. Either the subpath
from $u$ to $x$ is a beer path or the subpath from $x$ to $v$ is a beer path.
Whenever we take the minimum of a set of path weights in the above
computation, we store with that distance the vertex $x$ and a bit to indicate
which subpath is the beer path. When $u=v$ we can arbitrarily choose which
subpath is the beer path. After taking the minimum of
$\mathsf{dist}(u,v,G_{uv}^{R})$ and $\mathsf{dist}(u,v,G_{uv}^{\neg R})$, we
are left with a vertex, $x$, on the shortest beer path from $u$ to $v$ and the
bit indicating which subpath is a beer path.
We recursively compute $\mathsf{SP}_{B}(u,v)$ where either $(u,v)$ is an edge
of $G$ or $u=v$ as follows.
1. 1.
If $nil$ is stored with $\mathsf{dist}_{B}(u,v)$ and $u=v$,
$\mathsf{SP}_{B}(u,v)=(u)$.
2. 2.
If $nil$ is stored with $\mathsf{dist}_{B}(u,v)$ and $(u,v)$ is an edge then
$\mathsf{SP}_{B}(u,v)=(u,v)$.
3. 3.
If a vertex $x$ is stored with $(u,v)$ and the subpath from $u$ to $x$ is a
beer path then recursively compute $\mathsf{dist}_{B}(u,x)$.
$\mathsf{SP}_{B}(u,v)=\mathsf{SP}_{B}(u,x)\cup(v)$.
4. 4.
Otherwise $x$ is stored with $(u,v)$ and the subpath from $x$ to $v$ is a beer
path. Recursively compute $\mathsf{dist}_{B}(x,v)$.
$\mathsf{SP}_{B}(u,v)=(u)\cup\mathsf{SP}_{B}(x,v)$.
Note that a constant amount of work is done at each level of the recurrence
excluding the time spent in recursive calls. In each recursive call, except
potentially the last call, we get one new vertex on the shortest beer path.
Thus, constructing the whole path requires a total of $O(L)$ time.
## 6 Extension to Arbitrary Outerplanar Graphs
### 6.1 Maximal Outerplanar
Let $G$ be an outerplanar beer graph with $n$ vertices. Assume that the outer
face of $G$ is not a Hamiltonian cycle. We traverse $G$ along the outer face
in a clockwise manner, and mark each vertex when we encounter it for the first
time. Each time we visit a marked vertex $v$, we take note of $v$’s current
counterclockwise neighbor, $\mathsf{CCW}(v)$. Then we continue from $v$ to the
next clockwise vertex on the outer face and add an edge from this vertex to
$\mathsf{CCW}(v)$. We continue this process until we have returned to the
vertex we started from and all vertices have been marked.
At this moment, the outer face is a Hamiltonian cycle. For every interior face
that is not a triangle, we pick a vertex $u$ on that face and add edges
connecting $u$ with all vertices of the face that are not already adjacent to
$u$.
The resulting graph is a maximal outerplanar graph. Each edge that has been
added is given a weight of infinity. Observe that each shortest (beer) path in
the resulting graph corresponds to a shortest (beer) path in the original
graph, and vice versa.
### 6.2 Generalized Triangle Inequality
Let $G$ be a maximal outerplanar graph with an edge weight function $\omega$.
In order to convert $G$ to a graph that satisfies the generalized triangle
inequality, we need to compute $\mathsf{dist}(u,v)$ for every edge $(u,v)$ in
$G$, and we need to be able to construct $\mathsf{SP}(u,v)$ for each edge
$(u,v)$.
Let $D(G)$ be the dual of $G$ rooted at an arbitrary interior face of $G$. For
each edge $(u,v)$, we initialize $\delta(u,v)=\omega(u,v)$. (At the end,
$\delta(u,v)$ will be equal to $\mathsf{dist}(u,v)$.) For each edge $(u,v)$,
we also maintain a parent vertex, $p(u,v)$, initialized to $nil$.
We first conduct a post-order traversal of $D(G)$, processing each associated
face in $G$. Then we conduct a pre-order traversal of $D(G)$, again processing
each associated face in $G$. Let $F$ be a face of $G$. We process $F$ as
follows. Let $(u,v)$ be the edge of $F$ that is shared with the predecessor
face $F^{\prime}$ in the traversal, and let $w$ be the third vertex of
$F^{\prime}$. If $\delta(u,v)>\delta(u,w)+\delta(w,v)$, we set
$\delta(u,v)=\delta(u,w)+\delta(w,v)$ and $p(u,v)=w$.
After these traversals, $\delta(u,v)=\mathsf{dist}(u,v)$ for every edge
$(u,v)$ in $G$. If $p(u,v)=nil$, then $\mathsf{SP}(u,v)=(u,v)$; otherwise,
$\mathsf{SP}(u,v)$ is the concatenation of $\mathsf{SP}(u,p(u,v))$ and
$\mathsf{SP}(p(u,v),v)$, both of which can be computed recursively.
## 7 Single Source Shortest Beer Path
In this section we will describe how to compute the single source shortest
beer path from a source vertex $s$ on a maximal outerplanar graph $G$ that
satisfies the generalized triangle inequality. In order to do this we first
precompute (i) $\mathsf{dist}_{B}(u,v)$ for every edge $(u,v)$ and
$\mathsf{dist}_{B}(u,u)$ for every vertex $u$ and (ii) $\mathsf{dist}(s,v)$
for every vertex $v$. By Lemma 4, we can compute (i) in $O(n)$ time and in
[7], Maheshwari and Zeh present a single source shortest path algorithm for
undirected outerplanar graphs which gives us (ii) in $O(n)$ time.
Let $D(G)$ be the dual of $G$ and let $F_{s}$ be an arbitrary interior face of
$G$ containing $s$. Root $D(G)$ at the node $F_{s}$ and then conduct a pre-
order traversal of $D(G)$. Let $F$ be the current node of $D(G)$ being
processed during this traversal.
1. 1.
If $F=F_{s}$, let $u$ and $v$ be the vertices of $F_{s}$ that are not $s$.
Since $(s,u)$ and $(s,v)$ are both edges of $G$, $\mathsf{dist}_{B}(s,s)$,
$\mathsf{dist}_{B}(s,u)$ and $\mathsf{dist}_{B}(s,v)$ were precomputed in (i).
2. 2.
If $F\neq F_{s}$, let $a$ and $b$ be the vertices of $F$ shared with the face
$F^{\prime}$, where $F^{\prime}$ is the parent of $F$ in $D(G)$. This implies
that by this step we have already computed $\mathsf{dist}_{B}(s,a)$ and
$\mathsf{dist}_{B}(s,b)$. Let $c$ be the third vertex of $F$. The value of
$\mathsf{dist}_{B}(s,c)$ is the minimum of:
1. (a)
$\mathsf{dist}(s,a)+\mathsf{dist}_{B}(a,c)$,
2. (b)
$\mathsf{dist}_{B}(s,a)+\omega(a,c)$,
3. (c)
$\mathsf{dist}(s,b)+\mathsf{dist}_{B}(b,c)$,
4. (d)
$\mathsf{dist}_{B}(s,b)+\omega(b,c)$.
Since $(a,c)$ and $(b,c)$ are edges of $G$ we precomputed
$\mathsf{dist}_{B}(a,c)$ and $\mathsf{dist}_{B}(b,c)$ in (i). Lastly, we
computed $\mathsf{dist}(s,a)$ and $\mathsf{dist}(s,b)$ in (ii) so each of the
values listed above can be computed in constant time.
The correctness of this algorithm follows from Observation 3 and the
generalized triangle inequality. Since we do a constant amount of work at each
face in the traversal of $D(G)$ and the number of interior faces of a maximal
outerplanar graph is $n-2$, this algorithm takes $O(n)$ time.
Let $L$ the the number of vertices on the shortest beer path from $s$ to $v$.
If $\mathsf{dist}_{B}(s,v)$ was found in step 1, then by Lemma 4,
$\mathsf{SP}_{B}(s,v)$ can be constructed in $O(L)$ time. If this is not the
case, then $v=c$ in some iteration of step 2. At this step we store a vertex
$p(v)$ such that $p(v)=a$ if (a) or (b) was the minimum of step 2 and $p(v)=b$
otherwise. We also store a bit to indicate if the subpath from $s$ to $p(v)$
is the shortest path (as in cases (a) and (c)) or the shortest beer path (as
in cases (b) and (d)). If the subpath from $s$ to $p(v)$ is the shortest path,
we use the method described in [7] to find $\mathsf{SP}(s,p(v))$ and use Lemma
4 to find $\mathsf{SP}_{B}(p(v),v)$ and then concatenate $\mathsf{SP}(s,p(v))$
and $\mathsf{SP}_{B}(p(v),v)$ to get $\mathsf{SP}_{B}(s,v)$. Both
$\mathsf{SP}(s,p(v))$ and $\mathsf{SP}_{B}(p(v),v)$ are found in time
proportional to the number of vertices on their paths, so this takes $O(L)$
time. If the subpath from $s$ to $p(v)$ is the shortest beer path, then we
recursively find $\mathsf{SP}_{B}(s,p(v))$ and concatenate it with the edge
$(p(v),v)$ (which is $\mathsf{SP}(p(v),v)$ by the generalized triangle
inequality). Each iteration of the recursive step takes time proportional to
the number of new vertices of the path found in that step. Thus, we find
$\mathsf{SP}_{B}(s,v)$ in a total of $O(L)$ time.
## References
* [1] N. Alon and B. Schieber. Optimal preprocessing for answering on-line product queries. Technical Report 71/87, Tel-Aviv University, 1987.
* [2] M. A. Bender and M. Farach-Colton. The LCA problem revisited. In Proceedings of the 4th Latin American Symposium on Theoretical Informatics, volume 1776 of Lecture Notes in Computer Science, pages 88–94, Berlin, 2000. Springer-Verlag.
* [3] T. M. Chan, M. He, J. I. Munro, and G. Zhou. Succinct indices for path minimum, with applications. Algorithmica, 78(2):453–491, 2017.
* [4] B. Chazelle. Computing on a free tree via complexity-preserving mappings. Algorithmica, 2:337–361, 1987.
* [5] H. Djidjev, G. E. Pantziou, and C. D. Zaroliagis. Computing shortest paths and distances in planar graphs. In Automata, Languages and Programming, 18th International Colloquium, ICALP91, volume 510 of Lecture Notes in Computer Science, pages 327–338. Springer, 1991.
* [6] D. Harel and R. E. Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. Comput., 13(2):338–355, 1984.
* [7] A. Maheshwari and N. Zeh. I/o-optimal algorithms for outerplanar graphs. Journal of Graph Algorithms and Applications, 8(1):47–87, 2004\.
* [8] S. Pettie. An inverse-Ackermann type lower bound for online minimum spanning tree verification. Combinatorica, 26(2):207–230, 2006.
* [9] M. Thorup. Parallel shortcutting of rooted trees. Journal of Algorithms, 32:139–159, 1997.
|
# Finite Bubble Statistics Constrain Late Cosmological Phase Transitions
Gilly Elor Weinberg Institute, Department of Physics, University of Texas at
Austin, Austin, TX 78712, USA Ryusuke Jinno Research Center for the Early
Universe (RESCEU), Graduate School of Science, The University of Tokyo, Tokyo
113-0033, Japan Soubhik Kumar Center for Cosmology and Particle Physics,
Department of Physics, New York University, New York, NY 10003, USA Berkeley
Center for Theoretical Physics, Department of Physics, University of
California, Berkeley, CA 94720, USA Theoretical Physics Group, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA Robert McGehee William
I. Fine Theoretical Physics Institute, School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA Leinweber Center for
Theoretical Physics, Department of Physics,
University of Michigan, Ann Arbor, MI 48109, USA Yuhsin Tsai Department of
Physics, University of Notre Dame, IN 46556, USA
###### Abstract
We consider first order cosmological phase transitions (PT) happening at late
times, below Standard Model (SM) temperatures $T_{\rm PT}\lesssim$ GeV. The
inherently stochastic nature of bubble nucleation and the finite number of
bubbles associated with a late-time PT lead to superhorizon fluctuations in
the PT completion time. We compute how such fluctuations eventually source
curvature fluctuations with universal properties, independent of the
microphysics of the PT dynamics. Using Cosmic Microwave Background (CMB) and
Large Scale Structure (LSS) measurements, we constrain the energy released in
a dark-sector PT. For 0.1 eV $\lesssim T_{\rm PT}\lesssim$ keV this constraint
is stronger than both the current bound from additional neutrino species
$\Delta N_{\rm eff}$, and in some cases, even CMB-S4 projections. Future
measurements of CMB spectral distortions and pulsar timing arrays will also
provide competitive sensitivity for keV $\lesssim T_{\rm PT}\lesssim$ GeV.
††preprint: FTPI-MINN-23-20††preprint: UTWI-39-2023
PTs have been studied extensively for decades in models of baryogenesis Cohen
_et al._ (1990); Fromme _et al._ (2006); Hall _et al._ (2020); Elor _et
al._ (2022), (asymmetric) dark matter Cohen _et al._ (2008); Zurek (2014);
Baker and Kopp (2017); Hall _et al._ (2023); Asadi _et al._ (2021a, b); Hall
_et al._ (2022); Elor _et al._ (2023); Asadi _et al._ (2022), extended Higgs
sectors Profumo _et al._ (2007); Chacko _et al._ (2006); Schwaller (2015);
Ivanov (2017), and spontaneously broken conformal symmetry Creminelli _et
al._ (2002); Randall and Servant (2007); Nardini _et al._ (2007); Konstandin
_et al._ (2010); Konstandin and Servant (2011); Baratella _et al._ (2019);
Agashe _et al._ (2020, 2021); Ares _et al._ (2020); Levi _et al._ (2023);
Mishra and Randall (2023), among others. PTs may also generate gravitational
waves (GW) Kosowsky _et al._ (1992a, b); Kosowsky and Turner (1993);
Kamionkowski _et al._ (1994); Caprini _et al._ (2016, 2020); Caldwell _et
al._ (2022); Auclair _et al._ (2023) that can be observed in the near future.
_Late-time_ PTs have garnered attention due to their possible connections to
puzzling observations from PTAs Agazie _et al._ (2023); Antoniadis _et al._
(2023a) and the ‘$H_{0}$ tension’: a discrepancy between the direct
measurements of the Hubble constant $H_{0}$ Riess _et al._ (2022) and its
value inferred from the CMB Aghanim _et al._ (2020) (for a review see
Schöneberg _et al._ (2022)). These PTs occur below SM temperatures of a GeV
(redshift $z\sim 10^{13}$) and before matter-radiation equality ($z\approx
3400$). For instance, a PT around $z\approx 10^{4}$ is motivated by the
proposed New Early Dark Energy (NEDE) solution to the $H_{0}$ problem
Wetterich (2004); Doran and Robbers (2006); Poulin _et al._ (2019);
Niedermann and Sloth (2021, 2020). Since the Hubble tension favors such
“early-time” solutions Kamionkowski and Riess (2022), other ideas also use
such PTs Niedermann and Sloth (2022); Freese and Winkler (2021). A PT at
$z\sim 10^{10}$ has also been proposed as a source of the observed stochastic
GW background measured by PTAs Agazie _et al._ (2023); Afzal _et al._
(2023); Antoniadis _et al._ (2023b); Reardon _et al._ (2023); Xu _et al._
(2023). Even later PTs may ameliorate the cosmological constant problem Bloch
_et al._ (2020).
Due to constraints from big bang nucleosynthesis (BBN) and the CMB, late-time
PTs that occur entirely in a dark sector with no significant reheating to SM
particles are favored Bai and Korwar (2022). Thus, we focus on PTs that only
release GWs and other forms of dark radiation (DR). The gravitational
backreaction on the SM sector is the only way to identify and constrain such
dark-sector PTs. For larger couplings between the dark sector and SM,
constraints stronger than our model-independent ones may exist. A well-known
constraint on post-BBN dark PTs is the bound on the number of additional
neutrinos, $\Delta N_{\rm eff}<0.29$ at $95\%$ CL Aghanim _et al._ (2020);
Cielo _et al._ (2023), derived from Baryon Acoustic Oscillations (BAO) and
CMB measurements. This places an upper bound on the fraction of DR energy
density compared to the total radiation energy density $f_{\rm
DR}\equiv\rho_{\rm DR}/\rho_{\text{tot}}\lesssim 0.04$.
A PT proceeds via nucleation of bubbles of true vacuum inside the metastable
phase. To estimate the typical number of bubbles, consider a comoving volume
corresponding to multipole $\ell\sim 10^{3}$. If $T_{\text{PT}}\sim\text{
TeV}$, there are an enormous number of bubbles inside that comoving volume:
$N_{b}\sim\left[a({\rm TeV})H({\rm
TeV})/(10^{3}a_{0}H_{0})\right]^{3}\sim\left[\tau_{0}/(10^{3}\tau({\rm
TeV}))\right]^{3}\sim 10^{34}$. Here $\tau({\rm TeV})$ and $\tau_{0}$ are the
conformal times at $T={\rm TeV}$ and today, respectively, with the
corresponding scale factors denoted by $a({\rm TeV})$ and $a_{0}$. However, if
$T_{\text{PT}}\sim\text{ keV}$, for example, the number of bubbles is much
less, $N_{b}\sim 10^{6}$.
In this Letter, we demonstrate that the finite number of bubbles involved in a
PT gives rise to super-horizon perturbations in PT completion time $\propto
1/\sqrt{N_{b}}$ which eventually contribute to curvature perturbations. We
present the first calculation of these perturbations using gauge-invariant
observables. The curvature power spectrum on large length scales follows a
universal power-law scaling in the PT parameters, independent of its
microscopic details.111While previous studies have addressed the effect of a
PT on curvature perturbations at a parametric level Freese and Winkler (2023,
2022); Liu _et al._ (2023), including the _big bubble problem_ in
inflationary models Copeland _et al._ (1994), they have not precisely
determined the magnitude of the power spectrum. Even for a dark PT with no
direct coupling to the SM, CMB and LSS measurements constrain the resulting
curvature perturbations. This in turn constrains $f_{\rm DR}$ even more than
the current and projected $\Delta N_{\rm eff}$ limits when
$T_{\text{PT}}\lesssim 1\leavevmode\nobreak\ \text{keV}$. Additionally, the
power-law scale dependence of this new contribution to curvature perturbations
can create distinct signatures in the CMB and matter power spectrum, enabling
us to identify the origin of the perturbation as a late-time PT.
#### Super-horizon fluctuations in percolation time from number of bubbles.
—A PT proceeds through bubble nucleation and expansion; a point in
space transitions to the true vacuum when a bubble engulfs it. However, this
process is inherently stochastic, and therefore, not all points in space
transition to the true vacuum at the same time. To quantify this, the bubble
nucleation rate per unit time and volume is Hogan (1983); Enqvist _et al._
(1992); Hindmarsh _et al._ (2015)
$\begin{split}\Gamma=\Gamma_{0}e^{-S(t)}\approx\Gamma_{0}e^{-S(t_{f})}e^{\beta(t-t_{f})},\end{split}$
(1)
with $S$ the bounce action for nucleating a critical bubble and
$\beta\approx-{\rm d}S(t_{f})/{\rm d}t$. Here $t_{f}$ is a reference time to
measure the progression of the PT. If the nucleation rate increases with time,
the fraction of the Universe in the metastable phase can be determined as
Enqvist _et al._ (1992),
$\begin{split}h(t)=\exp\left(-e^{\beta(t-t_{f})}\right).\end{split}$ (2)
This implies that for $\beta\gg H_{\rm PT}$, the Hubble scale at $t_{f}$, the
PT completes very rapidly around the time $t_{f}$, within a small fraction of
Hubble time. However, as mentioned above, the PT does not complete exactly at
the same time everywhere. Given the factor of $\beta$ in Eqs. (1) and (2), we
expect the variance of the PT completion time to be inversely proportional to
$\beta$, i.e., a slower PT with a small value of $\beta$ will exhibit a larger
variance and vice versa.
To characterize the variation in the PT completion time, we denote the time at
which a point $\vec{x}$ transitions to true vacuum by $t_{c}(\vec{x})$, and
compute the two-point function $H_{\rm PT}^{2}\langle\delta
t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle$. $\delta
t_{c}(\vec{x})=t_{c}(\vec{x})-\bar{t}_{c}$ with $\bar{t}_{c}$ the average time
of conversion (see the Supplementary Material for a detailed definition). We
include factors of $H_{\rm PT}$ so that $H_{\rm PT}\delta t_{c}$ is
dimensionless. Practically, it is easier to write $H_{\rm PT}^{2}\langle\delta
t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle=(H_{\rm
PT}/\beta)^{2}\times\beta^{2}\langle\delta t_{c}(\vec{x})\delta
t_{c}(\vec{y})\rangle$ and calculate the last factor, thus separating the
effect of cosmic expansion from the PT dynamics. It is more convenient to work
with the dimensionless Fourier transformed two-point function denoted by:
$\displaystyle\mathcal{P}_{\delta
t}(k)=\frac{k^{3}}{2\pi^{2}}\left(\frac{H_{\rm PT}}{\beta}\right)^{2}\int{\rm
d}^{3}r\leavevmode\nobreak\ e^{i\vec{k}\cdot\vec{r}}\beta^{2}\langle\delta
t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle,$ (3)
where $\vec{r}=\vec{x}-\vec{y}$. Physically, this characterizes the
correlation of $t_{c}$ between any two points separated by a distance $\sim
1/k$. $\mathcal{P}_{\delta t}(k)$ has two distinct contributions: single-
bubble and double-bubble, corresponding to $t_{c}(\vec{x})$ and
$t_{c}(\vec{y})$ being set by the same bubble or two bubbles (see the
Supplementary Material).
$\mathcal{P}_{\delta t}(k)$ changes qualitatively for modes smaller or larger
than the typical bubble size $v_{w}/\beta$, where $v_{w}$ is the bubble wall
velocity. For modes smaller than or comparable to the bubble size, we
analytically compute $\mathcal{P}_{\delta t}(k)$ in the Supplementary Material
assuming constant wall velocity. However, due to intricate fluid dynamics and
magnetohydrodynamics effects, a translation between $\mathcal{P}_{\delta
t}(k)$ and sourced curvature perturbations is involved and model-dependent.
Therefore, the dependence of the curvature power spectrum on $k$ for
$k_{p}\equiv k/a_{\rm PT}\gtrsim\beta/v_{w}$ is less universal and varies as
the properties of the PT change. (Here $k_{p}$ is a physical wavenumber and
$a_{\rm PT}$ is the scale factor at $t_{f}$.) On the other hand, scales
$k_{p}\ll\beta/v_{w}$ have many bubbles contributing to the correlation
function within a spatial volume of linear size $1/k_{p}$. Thus,
$\mathcal{P}_{\delta t}$ is more universal and less sensitive to the details
of the PT thanks to the central limit theorem.
We can understand the behavior of $\mathcal{P}_{\delta t}$ for
$k_{p}\ll\beta/v_{w}$ as follows. The average separation between bubbles is
Enqvist _et al._ (1992); Hindmarsh _et al._ (2015),
$d_{b}\approx(8\pi)^{1/3}v_{w}/\beta$. In a given volume $V$, there are $N\sim
V/d_{b}^{3}$ independent regions where bubble nucleation can take place in an
uncorrelated fashion. As a result, the standard deviation in PT completion
time, when averaged over this entire volume, scales as $1/\sqrt{N}$. Thus,
given $N\sim 1/(k_{p}d_{b})^{3}$ for a scale $k_{p}$, we expect ${\cal
P}_{\delta t}\propto(k_{p}d_{b})^{3}$. Also, the combination $\beta\times t$
is what appears in the nucleation rates in Eqs. (1) and (2), so ${\cal
P}_{\delta t}\propto 1/\beta^{2}$. Thus, for $k_{p}\ll\beta/v_{w}$ and $H_{\rm
PT}\ll\beta$, the dimensionless power spectrum scales as
$\begin{split}{\cal P}_{\delta t}&\sim(8\pi)\left(\frac{H_{\rm
PT}}{\beta}\right)^{2}\left(\frac{v_{w}k_{p}}{\beta}\right)^{3}\\\
&=c(8\pi)v_{w}^{3}(k\tau_{\rm PT})^{3}\left(\frac{H_{\rm
PT}}{\beta}\right)^{5}.\end{split}$ (4)
We have noted that $\tau_{\rm PT}=1/(a_{\rm PT}H_{\rm PT})$ for PTs during
radiation domination and introduced a constant prefactor $c$. From the
qualitative arguments above we expect $c\sim 1$; a detailed calculation in the
Supplementary Material gives $c\approx 2.8$. Another way to understand the
$k^{3}$ scaling of ${\cal P}_{\delta t}$ for small $k$ is through Eq. (3). We
show in the Supplementary Material that the correlation $\langle\delta
t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle\sim e^{-\beta ra_{\rm PT}/2}$ for
$\beta ra_{\rm PT}\gg 1$. As a result for $k_{p}\ll\beta/v_{w}$, the
exponential phase in Eq. (3) does not contribute, and the $k$-dependence of
${\cal P}_{\delta t}$ comes solely from the $k^{3}$ prefactor.
The result for ${\cal P}_{\delta t}$ is in Fig. 1, where $\xi\equiv
k_{p}d_{b}=(8\pi)^{1/3}v_{w}(k\tau_{\rm PT})(H_{\rm PT}/\beta)$. For $\xi\ll
1$, ${\cal P}_{\delta t}\sim\xi^{3}(H_{\rm PT}/\beta)^{2}$, as expected from
Eq. (4). However, close to $\xi\sim 1$, we see a deviation from that scaling.
Below, we only use the result for ${\cal P}_{\delta t}$ for $\xi\leq 1$ since
the regime $\xi\gg 1$ is sensitive to turbulence and magnetohydrodynamics
effects. However, these sub-horizon inhomogeneities also give rise to density
perturbations. In dark sectors in which sound waves dominantly source the GWs,
the resulting constraints may even be stronger Ramberg _et al._ (2023).
We have described how the PT completion time fluctuates on superhorizon and
‘superbubble’ scales. While such fluctuations are of ‘isocurvature’ type
initially (since they do not induce a change in energy density), eventually
they source curvature perturbations.
Figure 1: Dimensionless power spectrum of phase transition time fluctuation,
re-scaled by $(\beta/H_{\rm PT})^{2}$ and plotted against comoving wavenumber
ratio $\xi$ representing the perturbation mode relative to typical bubble
separation. The PT spectrum (red) derived in the Supplementary Material is
independent of $(\beta,\tau_{\rm PT})$, unlike adiabatic perturbations
(purple, orange).
#### Curvature perturbations from fluctuations in percolation time.
—Consider two acausal patches, $A$ and $B$, where the PT takes place.
Since our analysis relies on $\xi\sim 0.1-1$ and $\beta/H_{\rm PT}\lesssim
10^{3}$, we are in a regime where ${\cal P}_{\delta t}\gg A_{s}$, where
$A_{s}=2.1\times 10^{-9}$ Akrami _et al._ (2020) is the magnitude of the
inflationary scalar power spectrum (we express ${\cal P}_{\delta t}$ in terms
of gauge-invariant observables below). Thus, we can ignore effects due to
$A_{s}$ and assume the PT takes place in a Universe that is a priori
homogeneous in different patches. We will see how ${\cal P}_{\delta t}$ leads
to inhomogeneities in the dark sector and how they then feed back into the SM
sector, weighted by factors of $f_{\rm DR}$.
Take the two patches $A$ and $B$ to each have size $v_{w}/\beta$ and equal
energy density. $\rho_{F}$ is their (equal) false vacuum energy density and
$t_{c}^{A}$ and $t_{c}^{B}$ their respective PT completion times.
$t_{c}^{A}\neq t_{c}^{B}$ in general and we define the difference
$t_{c}^{B}-t_{c}^{A}\equiv\delta t_{c}\ll t_{c}^{A,B}\sim 1/H_{\rm PT}$, with
$\delta t_{c}>0$. When $A$ and $B$ undergo the PT, $\rho_{F}$ is converted
into DR with an energy density $\rho_{\rm DR}$. Right at $t_{c}^{A}$, the
energy densities in $A$ and $B$ are identical and the curvature perturbation
is still zero. However, there is a non-zero isocurvature perturbation in DR at
this time. This subsequently induces curvature perturbations as time evolves
since DR and vacuum energy redshift differently. In other words, the equation
of state of the Universe is not barotropic, i.e., the total pressure is not a
definite function of the total energy, $p\neq p(\rho)$. As a result, the
curvature perturbation is not constant (see e.g. Garcia-Bellido and Wands
(1996); Wands _et al._ (2000)) and evolves with time after the PT occurs.
We assume that (i) the PT takes place when the dark-sector energy density is
dominated by the false vacuum and (ii) $\rho_{F}$ is entirely converted to
$\rho_{\rm DR}$ after the PT. We can then write the DR energy density in the
two patches at a later time $t_{\rm fin}$ as
$\begin{split}\rho_{\rm DR}^{A,B}(t_{\rm
fin})=\rho_{F}\left(\frac{t_{c}^{A,B}}{t_{\rm fin}}\right)^{2}\,.\end{split}$
(5)
This shows that the energy densities of DR in the two patches are different
and a nonzero DR density perturbation has been sourced by the DR isocurvature
perturbation.222The different values of $\rho_{\rm DR}$ in $A$ and $B$ changes
Hubble in the two patches, altering the energy-density redshift, but this
correction is $\mathcal{O}(\delta\rho_{\rm DR}/\rho_{\rm SM})$ and negligible
for our leading-order analysis. We can compute this density perturbation using
Eq. (5), $\delta\rho_{\rm DR}/\rho_{\rm DR}=2\delta t_{c}/t_{c}$. Since we are
working to leading order in perturbations, $\rho_{\text{DR}}\equiv\rho_{\rm
DR}^{A,B}(t_{\rm fin})$ and likewise for $t_{c}$ in the denominators of the
previous expression.
To compute the associated curvature perturbation, we can use the spatially
flat gauge (for a review, see Malik and Wands (2009)), which amounts to
comparing the energy densities in patches $A$ and $B$ at a common time $t_{\rm
fin}$ when the scale factors are identical. Then the curvature perturbation
(on uniform-density hypersurfaces) is
$\begin{split}\zeta=-\frac{H_{\rm PT}\delta\rho_{\rm DR}}{(\dot{\rho}_{\rm
DR}+\dot{\rho}_{\rm SM})}&=\frac{1}{4}f_{\rm DR}\frac{\delta\rho_{\rm
DR}}{\rho_{\rm DR}}=\frac{1}{2}f_{\rm DR}\frac{\delta
t_{c}}{t_{c}}\,.\end{split}$ (6)
So far, we have been assuming the dark-sector PT takes place when the dark
sector is vacuum-energy dominated, i.e., the latent heat released in the PT is
much larger than the temperature of the dark radiation bath. If not, we can
include the standard parameter $\alpha_{\rm PT}=\rho_{F}/\rho_{\rm
DR*}\lesssim 1$ where $\rho_{\rm DR*}$ is the energy density in the dark
radiation bath at the percolation time. Including this factor, we arrive at
the curvature power spectrum,
$\mathcal{P}_{\zeta}(k)=\frac{1}{4}\left(\frac{\alpha_{\rm PT}}{1+\alpha_{\rm
PT}}\right)^{2}f_{\rm DR}^{2}\mathcal{P}_{\delta t}(k)+\mathcal{P}_{\rm
ad}(k)\,.$ (7)
In the last term, we have reintroduced the uncorrelated adiabatic perturbation
$\mathcal{P}_{\rm ad}$. We take the pivot scale $k_{*}=0.05$ Mpc-1,
$A_{s}=2.1\times 10^{-9}$, and tilt $n_{s}=0.966$ Aghanim _et al._ (2020)
when calculating $\mathcal{P}_{\rm ad}$. The constraints on
$\mathcal{P}_{\zeta}(k)$ can then be used to bound $f_{\rm DR}$ for different
dark PT parameters. For an alternate derivation that relates ${\cal P}_{\delta
t}$ to ${\cal P}_{\zeta}$ without relying on the separate Universe approach
followed here, together with a derivation using the $\delta N$-formalism
Langlois _et al._ (2008), see the Supplementary Material.
Since we ignored the presence of inflationary, adiabatic perturbations while
analyzing $\mathcal{P}_{\delta t}$, Eq. (7) is valid only for
$\mathcal{P}_{\delta t}\gg{\cal P}_{\rm ad}$. In practice, given the current
precision $\Delta{\cal P}_{\zeta}/{\cal P}_{\zeta}\sim 5\%$ on CMB scales, the
above restriction puts an upper bound $f_{\rm DR}\lesssim 0.4$, above which
the effects of $A_{s}$ would be relevant for determining ${\cal P}_{\delta
t}$. Once the PT ends, all the dominant energy densities are in radiation, and
superhorizon $\zeta$-modes remain constant until they enter the horizon. We
note that PT also generates DR isocurvature, with a size roughly given by
${\cal P}_{\delta t}$, implying that isocurvature vanishes in the limit of
${\cal P}_{\delta t}\rightarrow 0$. The Planck constraint on DR isocurvature
Ghosh _et al._ (2022) is similar to the constraints on curvature
perturbation. Therefore, we will not consider the effect of isocurvature
perturbations separately, but rather study their effects via the constraints
on ${\cal P}_{\zeta}$.
Figure 2: $2\sigma$ exclusion bounds on DR energy density fraction from
current observations, as derived in this work. We show bounds using the CMB
(Planck 2018 Akrami _et al._ (2020)) and Lyman-$\alpha$ Bird _et al._ (2011)
(pumpkin orange), as well as the FIRAS constraint Fixsen and Mather (2002)
(candy corn yellow). The light grey region represents the existing bound
$\Delta N_{\rm eff}\geq 0.29$ Aghanim _et al._ (2020), and the dotted grey
lines, the projected bounds from the Simons Observatory (SO) $\Delta N_{\rm
eff}\geq 0.1$ Ade _et al._ (2019) and CMB-S4 $\Delta N_{\rm eff}\geq 0.03$
Abazajian _et al._ (2016). Future projections from SuperPIXIE Chluba _et
al._ (2012a) (red), assuming sensitivity of
$\Delta\rho_{\gamma}/\rho_{\gamma}\sim 10^{-8}$, and PTA Lee _et al._ (2021)
(maroon) are also depicted. We display the NEDE model’s preferred region
Niedermann and Sloth (2020) (darker grey) and the PT generating the potential
stochastic GW background Afzal _et al._ (2023). To assess existing NEDE model
bounds, focus on $\beta/H_{\rm PT}$ values within the indicated bounds and
disregard the grey region representing the $\Delta N_{\rm eff}$ bound.
#### Time evolution of curvature perturbations.
—Perturbation modes with $k\tau_{\rm PT}\ll 1$ are outside the horizon
when the PT takes place and we can characterize their subsequent cosmological
evolution by just specifying ${\cal P}_{\zeta}(k)$. However, modes with
$\xi\approx 1$ correspond to $k\tau_{\rm PT}\approx(8\pi)^{-1/3}(\beta/H_{\rm
PT})v_{w}^{-1}\gtrsim 1$ for $\beta/H_{\rm PT}\gtrsim 10$, implying such modes
are already inside the horizon when the PT takes place. To derive constraints
based on ${\cal P}_{\zeta}(k)$ for a sub-horizon $k$-mode, we need to take
into account that there is no sub-horizon evolution for a time
$\Delta\tau\sim\tau_{\rm PT}-k^{-1}$, between mode reentry and the PT.
CMB temperature perturbations undergo diffusion damping while inside the
horizon. A delay in subhorizon evolution by an amount $\Delta\tau$ implies PT-
induced perturbations undergo less damping for a given $k$ compared to
$\Lambda$CDM expectations. Starting with the same value of ${\cal
P}_{\zeta}(k)$, the CMB anisotropies are larger in the PT scenario compared to
$\Lambda$CDM. In this work, we take a conservative approach by not including
this enhancement and leave a more precise computation for future work.
Perturbations in dark matter experience logarithmic growth in the radiation-
dominated era upon horizon reentry. In $\Lambda$CDM cosmology, the power in a
$k$-mode at the time of matter-radiation equality ($\tau_{\rm eq}\approx 110$
Mpc) is $\Delta_{\rm DM}(1/k,\tau_{\rm eq})^{2}{\cal P}_{\zeta}(k)$, where
$\Delta_{\rm DM}$ denotes the matter transfer function: $\Delta_{\rm
DM}(\tau_{i},\tau)\approx 6.4\ln(0.44\tau/\tau_{i})$ Hu and Sugiyama (1996);
Dodelson (2003). For the PT scenario, the analogous expression is $\Delta_{\rm
DM}(\tau_{\rm PT},\tau_{\rm eq})^{2}{\cal P}_{\zeta}(k)$. We use a rescaled
and weaker constraint ${\cal P}_{\zeta}(k)\times(\Delta_{\rm DM}(\tau_{\rm
PT},\tau_{\rm eq})/\Delta_{\rm DM}(1/k,\tau_{\rm eq}))^{2}$ for
$k>\tau^{-1}_{\rm PT}$ to take into account dark matter clustering bounds such
as from Lyman-$\alpha$ and future PTA constraint on DM clustering.
#### Cosmological Constraints.
—In Fig. 2, we present $2\sigma$ exclusion bounds on $f_{\text{DR}}$
using current constraints on $\mathcal{P}_{\zeta}$ and projected future
sensitivities. We translate the comoving time $\tau_{\rm PT}$ in Eq. (4) into
the SM temperature and redshift at the time of the PT. CMB Akrami _et al._
(2020) and Lyman-$\alpha$ Bird _et al._ (2011) measurements set upper bounds
on $\mathcal{P}_{\zeta}$ for $k$-modes up to $k\lesssim 3$ Mpc-1. Our analysis
excludes the pumpkin orange regions for various $\beta/H_{\rm PT}$ since in
those regions, the PT contribution to ${\cal P}_{\zeta}$ is too large. Other
constraints from ultracompact minihalos impacting PTAs may be relevant for
$T_{\text{PT}}\gtrsim 1\text{ MeV}$, but have unknown uncertainties related to
the time of DM collapse Clark _et al._ (2016a, b).
The $T_{\rm PT}$ dependence of the constraints in Fig. 2 can be understood as
follows. During a radiation-dominated epoch, $k_{\rm peak}\propto T_{\rm PT}$,
where $k_{\rm peak}$ is the comoving wavenumber of the peak in ${\cal
P}_{\delta t}(k)$. The constraint on $f_{\rm DR}$ for a given $T_{\rm PT}$
then depends on whether $k_{\rm peak}$ or the IR tail of ${\cal P}_{\delta
t}(k)$ lies within the range probed by a given observable. Suppose that for a
range of $T_{\rm PT}$, the corresponding range of $k_{\rm peak}$ is directly
constrained by an observable. Then if the constraint on ${\cal P}_{\zeta}(k)$
over that range of $k_{\rm peak}$ is flat, the associated constraint on
$f_{\rm DR}$ is also flat with respect to $T_{\rm PT}$, resulting in the
plateaus in Fig. 2. This is because ${\cal P}_{\delta t}(k_{\rm peak})$ does
not change as $T_{\rm PT}$ varies. This is what happens for the CMB bound for
$T_{\rm PT}\lesssim 1$ eV for $\beta/H_{\rm PT}=10$. For larger $T_{\rm PT}$,
$k_{\rm peak}$ lies outside the region probed by ${\cal P}_{\zeta}(k)$
constraints; constraints are only sensitive to the tail of the ${\cal
P}_{\delta t}$ distribution and $\propto f_{\rm DR}^{2}k^{3}$ (from (4) and
(7)). In those regions, as $T_{\rm PT}$ is increased, the bound on $f_{\rm
DR}$ goes as $1/T_{\rm PT}^{3/2}$ (since $T_{\rm PT}\propto k$). A similar
transition from a plateau behavior is also seen at $\sim$100 eV for
Ly-$\alpha$ and at $\sim$$100$ MeV for PTA.
Notably, for $\beta/H_{\rm PT}\lesssim 400$, the bounds we derive from DR
inhomogeneities are stronger than current $\Delta N_{\rm eff}$ constraints
that track the homogeneous abundance of DR. For PTs that occur before BBN,
there is a stricter bound of $\Delta N_{\rm eff}\lesssim 0.23$ when applying
more observational constraints; for those PTs after, the constraint is
slightly weaker at $\Delta N_{\rm eff}\lesssim 0.31$ Yeh _et al._ (2022).
Since our new CMB$+$Ly-$\alpha$ bounds are in the latter range, and we want to
show analogous $\Delta N_{\rm eff}$ constraints for CMB-S4, we plot Fig. 2
using the well-known $\Delta N_{\rm eff}<0.29$ Aghanim _et al._ (2020). For
$\beta/H_{\rm PT}\sim 10$ and $T_{\rm PT}\lesssim$ keV, our analysis using
CMB+Ly-$\alpha$ constrains such PTs as much as or better than the future
Simons Observatory (SO) Ade _et al._ (2019) and CMB-S4 projections on $\Delta
N_{\rm eff}$ Abazajian _et al._ (2016).
The NEDE model in Niedermann and Sloth (2021, 2020) favors $\alpha_{\rm
PT}f_{\rm DR}$ in the upper (lower) dark gray region ($\pm 1\sigma$) from the
Planck18+BAO+LSS fit with (without) SH0ES data. While large values of
$\beta/H_{\rm PT}$ generally require extra model-building, the model in
Niedermann and Sloth (2020) assumes $\beta/H_{\rm PT}\gtrsim 100$ and permits
$\beta/H_{\rm PT}$ as large as $\sim 10^{3}$ by including a field to trigger
the PT. Still, our $\mathcal{P}_{\zeta}$ bound effectively disfavors the
preferred NEDE region in $\left(T_{\text{PT}},f_{\text{DR}}\right)$ for all
values of $\beta/H_{\rm PT}\lesssim$ 320 (230) with (without) SH0ES data.
For a large $\mathcal{P}_{\zeta}(k)$ with $k\lesssim 5400$ Mpc-1 the PT can
impact the dissipation of acoustic modes in photon-baryon perturbations,
altering the photon’s blackbody spectrum and inducing $\mu$\- and
$y$-distortions Chluba _et al._ (2012b); Hooper _et al._ (2023):
$\displaystyle X\\!\\!\simeq$ $\displaystyle\\!\\!A\displaystyle{\int_{k_{\rm
min}}^{\infty}}\frac{{\rm
d}k}{k}\mathcal{P}_{\zeta}(k)\left[Be^{-\frac{k}{5400/{\rm
Mpc}}}-Ce^{-(\frac{k}{31.6/{\rm Mpc}})^{2}}\right]$ (8)
where $k_{\rm min}=1$ Mpc-1, $(A,B,C)_{X}=(2.2,1,1)_{\mu}$ and
$(0.4,0,-1)_{y}$. Comparing this to the FIRAS bound of $|\mu|<9.0\times
10^{-5}$ and $|y|<1.5\times 10^{-5}$ Mather _et al._ (1994); Fixsen _et al._
(1996), we derive the exclusion bound labelled as ‘FIRAS’. When lowering
$T_{\rm PT}$, the $y$-distortion bound takes over the $\mu$-bound around
$T_{\rm PT}=10^{3}$ $(10^{2})$ eV for the $\beta/H_{\rm PT}=10$ $(100)$ case.
In contrast to Ref. Liu _et al._ (2023), our findings indicate that the FIRAS
constraint is less stringent than the $\Delta N_{\rm eff}$ constraint, even
for PT with small $\beta=10H_{\rm PT}$.333Ref. Liu _et al._ (2023) assumes
the spatial energy density perturbation directly equals $\alpha_{\rm PT}f_{\rm
DR}(\beta/H_{\rm PT})^{-5/2}$, omitting the numerical prefactor derived in the
Supplementary Material. Their procedure in deriving $\mathcal{P}_{\zeta}(k)$
is also sensitive to the choice of a window function, while ours is not.
Instead, our analysis derives $\mathcal{P}_{\zeta}(k)$ by starting with the
primordial fluctuations and following the energy-momentum conservation
equation for linear perturbations Malik and Wands (2009). This could explain
why our bounds differ.
Current $\mathcal{P}_{\zeta}$ measurements are less sensitive to PTs than the
$\Delta N_{\rm eff}$ constraint for $T_{\rm PT}\gtrsim\text{ keV}$, but
several proposed searches can constrain $\mathcal{P}_{\zeta}$ more powerfully
and constrain weaker dark PTs. Super-PIXIE aims to measure the CMB with a
sensitivity of $\Delta\rho_{\gamma}/\rho_{\gamma}\sim 10^{-8}$ Chluba _et
al._ (2019) and the associated constraint is shown in red. PTAs can also probe
$\mathcal{P}_{\zeta}$ by observing the phase shift in periodic pulsar signals
mainly caused by the Doppler effect induced by an enhanced dark matter
structure that accelerates Earth or a pulsar. The PTA sensitivity curves
(maroon) use $\mathcal{P}_{\zeta}$ sensitivity derived in Lee _et al._ (2021)
that assumes 20 years of observations of 200 pulsars. This future sensitivity
to PTs with $T_{\text{PT}}\gtrsim\text{ MeV}$ may test exotic DM models which
rely on them Elor _et al._ (2023). Also shown is the $2\sigma$-preferred PT
region for the GW background hinted at by NANOGrav Agazie _et al._ (2023);
Afzal _et al._ (2023) (darker grey, see, e.g., Franciolini _et al._ (2023)
for alternative GW spectrum assumptions). At face value, this region conflicts
with the $\Delta N_{\rm eff}$ constraint, but this prominent GW signal could
largely originate from supermassive black hole mergers. With enhanced PTA
measurements, we might still detect the PT signal within a comparable $T_{\rm
PT}$ range. Then PTA measurements of $\mathcal{P}_{\zeta}$ could complement
the GW detection.
#### Discussion.
—We have demonstrated that finite bubble statistics can lead to
superhorizon fluctuations in the PT completion time, regardless of the PT
details. These fluctuations source curvature perturbations that affect the
CMB, LSS, and other observables. Utilizing these, we find our constraints are
in tension with some of the best fit regions of the NEDE models proposed to
ameliorate the Hubble tension. At superhorizon scales, the (dimensionless)
power spectrum of these fluctuations has a $k^{3}$-model-independent scaling
since it is just determined by Poisson statistics. This contribution makes the
total curvature perturbation scale non-invariant. Thus, the associated CMB
phenomenology shares some similarities with the scale non-invariant effects
due to ‘primordial features’ Chluba _et al._ (2015) produced during inflation
and models with ‘blue-tilted’ curvature perturbation Kasuya and Kawasaki
(2009); Kawasaki _et al._ (2013); Chung and Tadepalli (2022); Ebadi _et al._
(2023).
In our analysis, we have kept the $\Lambda$CDM parameters fixed. However,
given the model-independent shape, one can do a joint analysis where both
dark-sector and $\Lambda$CDM parameters are varied. We have also not
considered constraints from modes that are smaller than typical bubbles as
those are more model dependent. However, in the context of specific models one
can obtain stronger constraints from such modes. We leave these for future
work.
_Acknowledgments._ We thank Matthew Buckley, Zackaria Chacko, Peizhi Du, Anson
Hook, Gordan Krnjaic, Toby Opferkuch, Davide Racco, Albert Stebbins, Gustavo
Marques-Tavares, and Neal Weiner for helpful comments on the draft and
discussions. The research of GE is supported by the National Science
Foundation (NSF) Grant Number PHY-2210562, by a grant from University of Texas
at Austin, and by a grant from the Simons Foundation. The work of RJ is
supported by JSPS KAKENHI Grant Number 23K19048. SK is supported partially by
the NSF grant PHY-2210498 and the Simons Foundation. RM was supported in part
by DOE grant DE-SC0007859. YT is supported by the NSF grant PHY-2112540. RJ,
SK, RM, and YT thank the Mainz Institute for Theoretical Physics (MITP) of the
Cluster of Excellence PRISMA+ (project ID 39083149) for their hospitality
while a portion of this work was completed. GE, SK, and YT thank Aspen Center
for Physics (supported by NSF grant PHY-2210452) for their hospitality while
this work was in progress.
## References
* Cohen _et al._ (1990) A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Phys. Lett. B 245, 561 (1990).
* Fromme _et al._ (2006) L. Fromme, S. J. Huber, and M. Seniuch, JHEP 11, 038 (2006), arXiv:hep-ph/0605242 .
* Hall _et al._ (2020) E. Hall, T. Konstandin, R. McGehee, H. Murayama, and G. Servant, JHEP 04, 042 (2020), arXiv:1910.08068 [hep-ph] .
* Elor _et al._ (2022) G. Elor _et al._ , in _Snowmass 2021_ (2022) arXiv:2203.05010 [hep-ph] .
* Cohen _et al._ (2008) T. Cohen, D. E. Morrissey, and A. Pierce, Phys. Rev. D 78, 111701 (2008), arXiv:0808.3994 [hep-ph] .
* Zurek (2014) K. M. Zurek, Phys. Rept. 537, 91 (2014), arXiv:1308.0338 [hep-ph] .
* Baker and Kopp (2017) M. J. Baker and J. Kopp, Phys. Rev. Lett. 119, 061801 (2017), arXiv:1608.07578 [hep-ph] .
* Hall _et al._ (2023) E. Hall, T. Konstandin, R. McGehee, and H. Murayama, Phys. Rev. D 107, 055011 (2023), arXiv:1911.12342 [hep-ph] .
* Asadi _et al._ (2021a) P. Asadi, E. D. Kramer, E. Kuflik, G. W. Ridgway, T. R. Slatyer, and J. Smirnov, Phys. Rev. Lett. 127, 211101 (2021a), arXiv:2103.09822 [hep-ph] .
* Asadi _et al._ (2021b) P. Asadi, E. D. Kramer, E. Kuflik, G. W. Ridgway, T. R. Slatyer, and J. Smirnov, Phys. Rev. D 104, 095013 (2021b), arXiv:2103.09827 [hep-ph] .
* Hall _et al._ (2022) E. Hall, R. McGehee, H. Murayama, and B. Suter, Phys. Rev. D 106, 075008 (2022), arXiv:2107.03398 [hep-ph] .
* Elor _et al._ (2023) G. Elor, R. McGehee, and A. Pierce, Phys. Rev. Lett. 130, 031803 (2023), arXiv:2112.03920 [hep-ph] .
* Asadi _et al._ (2022) P. Asadi _et al._ , (2022), arXiv:2203.06680 [hep-ph] .
* Profumo _et al._ (2007) S. Profumo, M. J. Ramsey-Musolf, and G. Shaughnessy, JHEP 08, 010 (2007), arXiv:0705.2425 [hep-ph] .
* Chacko _et al._ (2006) Z. Chacko, H.-S. Goh, and R. Harnik, Phys. Rev. Lett. 96, 231802 (2006), arXiv:hep-ph/0506256 .
* Schwaller (2015) P. Schwaller, Phys. Rev. Lett. 115, 181101 (2015), arXiv:1504.07263 [hep-ph] .
* Ivanov (2017) I. P. Ivanov, Prog. Part. Nucl. Phys. 95, 160 (2017), arXiv:1702.03776 [hep-ph] .
* Creminelli _et al._ (2002) P. Creminelli, A. Nicolis, and R. Rattazzi, JHEP 03, 051 (2002), arXiv:hep-th/0107141 .
* Randall and Servant (2007) L. Randall and G. Servant, JHEP 05, 054 (2007), arXiv:hep-ph/0607158 .
* Nardini _et al._ (2007) G. Nardini, M. Quiros, and A. Wulzer, JHEP 09, 077 (2007), arXiv:0706.3388 [hep-ph] .
* Konstandin _et al._ (2010) T. Konstandin, G. Nardini, and M. Quiros, Phys. Rev. D 82, 083513 (2010), arXiv:1007.1468 [hep-ph] .
* Konstandin and Servant (2011) T. Konstandin and G. Servant, JCAP 12, 009 (2011), arXiv:1104.4791 [hep-ph] .
* Baratella _et al._ (2019) P. Baratella, A. Pomarol, and F. Rompineve, JHEP 03, 100 (2019), arXiv:1812.06996 [hep-ph] .
* Agashe _et al._ (2020) K. Agashe, P. Du, M. Ekhterachian, S. Kumar, and R. Sundrum, JHEP 05, 086 (2020), arXiv:1910.06238 [hep-ph] .
* Agashe _et al._ (2021) K. Agashe, P. Du, M. Ekhterachian, S. Kumar, and R. Sundrum, JHEP 02, 051 (2021), arXiv:2010.04083 [hep-th] .
* Ares _et al._ (2020) F. R. Ares, M. Hindmarsh, C. Hoyos, and N. Jokela, JHEP 21, 100 (2020), arXiv:2011.12878 [hep-th] .
* Levi _et al._ (2023) N. Levi, T. Opferkuch, and D. Redigolo, JHEP 02, 125 (2023), arXiv:2212.08085 [hep-ph] .
* Mishra and Randall (2023) R. K. Mishra and L. Randall, (2023), arXiv:2309.10090 [hep-ph] .
* Kosowsky _et al._ (1992a) A. Kosowsky, M. S. Turner, and R. Watkins, Phys. Rev. Lett. 69, 2026 (1992a).
* Kosowsky _et al._ (1992b) A. Kosowsky, M. S. Turner, and R. Watkins, Phys. Rev. D 45, 4514 (1992b).
* Kosowsky and Turner (1993) A. Kosowsky and M. S. Turner, Phys. Rev. D 47, 4372 (1993), arXiv:astro-ph/9211004 .
* Kamionkowski _et al._ (1994) M. Kamionkowski, A. Kosowsky, and M. S. Turner, Phys. Rev. D 49, 2837 (1994), arXiv:astro-ph/9310044 .
* Caprini _et al._ (2016) C. Caprini _et al._ , JCAP 04, 001 (2016), arXiv:1512.06239 [astro-ph.CO] .
* Caprini _et al._ (2020) C. Caprini _et al._ , JCAP 03, 024 (2020), arXiv:1910.13125 [astro-ph.CO] .
* Caldwell _et al._ (2022) R. Caldwell _et al._ , Gen. Rel. Grav. 54, 156 (2022), arXiv:2203.07972 [gr-qc] .
* Auclair _et al._ (2023) P. Auclair _et al._ (LISA Cosmology Working Group), Living Rev. Rel. 26, 5 (2023), arXiv:2204.05434 [astro-ph.CO] .
* Agazie _et al._ (2023) G. Agazie _et al._ (NANOGrav), Astrophys. J. Lett. 951, L8 (2023), arXiv:2306.16213 [astro-ph.HE] .
* Antoniadis _et al._ (2023a) J. Antoniadis _et al._ , (2023a), arXiv:2306.16214 [astro-ph.HE] .
* Riess _et al._ (2022) A. G. Riess _et al._ , Astrophys. J. Lett. 934, L7 (2022), arXiv:2112.04510 [astro-ph.CO] .
* Aghanim _et al._ (2020) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
* Schöneberg _et al._ (2022) N. Schöneberg, G. Franco Abellán, A. Pérez Sánchez, S. J. Witte, V. Poulin, and J. Lesgourgues, Phys. Rept. 984, 1 (2022), arXiv:2107.10291 [astro-ph.CO] .
* Wetterich (2004) C. Wetterich, Phys. Lett. B 594, 17 (2004), arXiv:astro-ph/0403289 .
* Doran and Robbers (2006) M. Doran and G. Robbers, JCAP 06, 026 (2006), arXiv:astro-ph/0601544 .
* Poulin _et al._ (2019) V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Phys. Rev. Lett. 122, 221301 (2019), arXiv:1811.04083 [astro-ph.CO] .
* Niedermann and Sloth (2021) F. Niedermann and M. S. Sloth, Phys. Rev. D 103, L041303 (2021), arXiv:1910.10739 [astro-ph.CO] .
* Niedermann and Sloth (2020) F. Niedermann and M. S. Sloth, Phys. Rev. D 102, 063527 (2020), arXiv:2006.06686 [astro-ph.CO] .
* Kamionkowski and Riess (2022) M. Kamionkowski and A. G. Riess, (2022), arXiv:2211.04492 [astro-ph.CO] .
* Niedermann and Sloth (2022) F. Niedermann and M. S. Sloth, Phys. Rev. D 105, 063509 (2022), arXiv:2112.00770 [hep-ph] .
* Freese and Winkler (2021) K. Freese and M. W. Winkler, Phys. Rev. D 104, 083533 (2021), arXiv:2102.13655 [astro-ph.CO] .
* Afzal _et al._ (2023) A. Afzal _et al._ (NANOGrav), Astrophys. J. Lett. 951, L11 (2023), arXiv:2306.16219 [astro-ph.HE] .
* Antoniadis _et al._ (2023b) J. Antoniadis _et al._ (EPTA), (2023b), arXiv:2306.16214 [astro-ph.HE] .
* Reardon _et al._ (2023) D. J. Reardon _et al._ , Astrophys. J. Lett. 951, L6 (2023), arXiv:2306.16215 [astro-ph.HE] .
* Xu _et al._ (2023) H. Xu _et al._ , Res. Astron. Astrophys. 23, 075024 (2023), arXiv:2306.16216 [astro-ph.HE] .
* Bloch _et al._ (2020) I. M. Bloch, C. Csáki, M. Geller, and T. Volansky, JHEP 12, 191 (2020), arXiv:1912.08840 [hep-ph] .
* Bai and Korwar (2022) Y. Bai and M. Korwar, Phys. Rev. D 105, 095015 (2022), arXiv:2109.14765 [hep-ph] .
* Cielo _et al._ (2023) M. Cielo, M. Escudero, G. Mangano, and O. Pisanti, (2023), arXiv:2306.05460 [hep-ph] .
* Freese and Winkler (2023) K. Freese and M. W. Winkler, Phys. Rev. D 107, 083522 (2023), arXiv:2302.11579 [astro-ph.CO] .
* Freese and Winkler (2022) K. Freese and M. W. Winkler, Phys. Rev. D 106, 103523 (2022), arXiv:2208.03330 [astro-ph.CO] .
* Liu _et al._ (2023) J. Liu, L. Bian, R.-G. Cai, Z.-K. Guo, and S.-J. Wang, Phys. Rev. Lett. 130, 051001 (2023), arXiv:2208.14086 [astro-ph.CO] .
* Copeland _et al._ (1994) E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart, and D. Wands, Phys. Rev. D 49, 6410 (1994), arXiv:astro-ph/9401011 .
* Hogan (1983) C. J. Hogan, Physics Letters B 133, 172 (1983).
* Enqvist _et al._ (1992) K. Enqvist, J. Ignatius, K. Kajantie, and K. Rummukainen, Phys. Rev. D 45, 3415 (1992).
* Hindmarsh _et al._ (2015) M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Phys. Rev. D 92, 123009 (2015), arXiv:1504.03291 [astro-ph.CO] .
* Ramberg _et al._ (2023) N. Ramberg, W. Ratzinger, and P. Schwaller, JCAP 02, 039 (2023), arXiv:2209.14313 [hep-ph] .
* Akrami _et al._ (2020) Y. Akrami _et al._ (Planck), Astron. Astrophys. 641, A10 (2020), arXiv:1807.06211 [astro-ph.CO] .
* Garcia-Bellido and Wands (1996) J. Garcia-Bellido and D. Wands, Phys. Rev. D 53, 5437 (1996), arXiv:astro-ph/9511029 .
* Wands _et al._ (2000) D. Wands, K. A. Malik, D. H. Lyth, and A. R. Liddle, Phys. Rev. D 62, 043527 (2000), arXiv:astro-ph/0003278 .
* Malik and Wands (2009) K. A. Malik and D. Wands, Phys. Rept. 475, 1 (2009), arXiv:0809.4944 [astro-ph] .
* Langlois _et al._ (2008) D. Langlois, F. Vernizzi, and D. Wands, JCAP 12, 004 (2008), arXiv:0809.4646 [astro-ph] .
* Ghosh _et al._ (2022) S. Ghosh, S. Kumar, and Y. Tsai, JCAP 05, 014 (2022), arXiv:2107.09076 [astro-ph.CO] .
* Bird _et al._ (2011) S. Bird, H. V. Peiris, M. Viel, and L. Verde, Mon. Not. Roy. Astron. Soc. 413, 1717 (2011), arXiv:1010.1519 [astro-ph.CO] .
* Fixsen and Mather (2002) D. J. Fixsen and J. C. Mather, Astrophys. J. 581, 817 (2002).
* Ade _et al._ (2019) P. Ade _et al._ (Simons Observatory), JCAP 02, 056 (2019), arXiv:1808.07445 [astro-ph.CO] .
* Abazajian _et al._ (2016) K. N. Abazajian _et al._ (CMB-S4), (2016), arXiv:1610.02743 [astro-ph.CO] .
* Chluba _et al._ (2012a) J. Chluba, R. Khatri, and R. A. Sunyaev, Mon. Not. Roy. Astron. Soc. 425, 1129 (2012a), arXiv:1202.0057 [astro-ph.CO] .
* Lee _et al._ (2021) V. S. H. Lee, A. Mitridate, T. Trickle, and K. M. Zurek, JHEP 06, 028 (2021), arXiv:2012.09857 [astro-ph.CO] .
* Hu and Sugiyama (1996) W. Hu and N. Sugiyama, Astrophys. J. 471, 542 (1996), arXiv:astro-ph/9510117 .
* Dodelson (2003) S. Dodelson, _Modern Cosmology_ (Academic Press, Elsevier Science, 2003).
* Clark _et al._ (2016a) H. A. Clark, G. F. Lewis, and P. Scott, Mon. Not. Roy. Astron. Soc. 456, 1394 (2016a), [Erratum: Mon.Not.Roy.Astron.Soc. 464, 2468 (2017)], arXiv:1509.02938 [astro-ph.CO] .
* Clark _et al._ (2016b) H. A. Clark, G. F. Lewis, and P. Scott, Mon. Not. Roy. Astron. Soc. 456, 1402 (2016b), [Erratum: Mon.Not.Roy.Astron.Soc. 464, 955–956 (2017)], arXiv:1509.02941 [astro-ph.CO] .
* Yeh _et al._ (2022) T.-H. Yeh, J. Shelton, K. A. Olive, and B. D. Fields, JCAP 10, 046 (2022), arXiv:2207.13133 [astro-ph.CO] .
* Chluba _et al._ (2012b) J. Chluba, A. L. Erickcek, and I. Ben-Dayan, Astrophys. J. 758, 76 (2012b), arXiv:1203.2681 [astro-ph.CO] .
* Hooper _et al._ (2023) D. Hooper, A. Ireland, G. Krnjaic, and A. Stebbins, (2023), arXiv:2308.00756 [astro-ph.CO] .
* Mather _et al._ (1994) J. C. Mather, E. S. Cheng, D. A. Cottingham, J. Eplee, R. E., D. J. Fixsen, T. Hewagama, R. B. Isaacman, K. A. Jensen, S. S. Meyer, P. D. Noerdlinger, S. M. Read, L. P. Rosen, R. A. Shafer, E. L. Wright, C. L. Bennett, N. W. Boggess, M. G. Hauser, T. Kelsall, J. Moseley, S. H., R. F. Silverberg, G. F. Smoot, R. Weiss, and D. T. Wilkinson, Astrophys. J. 420, 439 (1994).
* Fixsen _et al._ (1996) D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A. Shafer, and E. L. Wright, Astrophys. J. 473, 576 (1996), arXiv:astro-ph/9605054 .
* Chluba _et al._ (2019) J. Chluba _et al._ , Bull. Am. Astron. Soc. 51, 184 (2019), arXiv:1903.04218 [astro-ph.CO] .
* Franciolini _et al._ (2023) G. Franciolini, D. Racco, and F. Rompineve, (2023), arXiv:2306.17136 [astro-ph.CO] .
* Chluba _et al._ (2015) J. Chluba, J. Hamann, and S. P. Patil, Int. J. Mod. Phys. D 24, 1530023 (2015), arXiv:1505.01834 [astro-ph.CO] .
* Kasuya and Kawasaki (2009) S. Kasuya and M. Kawasaki, Phys. Rev. D 80, 023516 (2009), arXiv:0904.3800 [astro-ph.CO] .
* Kawasaki _et al._ (2013) M. Kawasaki, N. Kitajima, and T. T. Yanagida, Phys. Rev. D 87, 063519 (2013), arXiv:1207.2550 [hep-ph] .
* Chung and Tadepalli (2022) D. J. H. Chung and S. C. Tadepalli, Phys. Rev. D 105, 123511 (2022), arXiv:2110.02272 [astro-ph.CO] .
* Ebadi _et al._ (2023) R. Ebadi, S. Kumar, A. McCune, H. Tai, and L.-T. Wang, (2023), arXiv:2307.01248 [astro-ph.CO] .
* Jinno and Takimoto (2019) R. Jinno and M. Takimoto, JCAP 01, 060 (2019), arXiv:1707.03111 [hep-ph] .
* Jinno and Takimoto (2017) R. Jinno and M. Takimoto, Phys. Rev. D 95, 024009 (2017), arXiv:1605.01403 [astro-ph.CO] .
Supplementary Material for Finite Bubble Statistics Constrain Late
Cosmological Phase Transitions
Gilly Elor, Ryusuke Jinno, Soubhik Kumar, Robert McGehee, and Yuhsin Tsai
## I Power spectrum of the transition time
In this Supplementary Material we derive analytic expressions for the power
spectrum of the transition time $t_{c}(\vec{x})$. The calculation is similar
to Refs. Jinno and Takimoto (2019, 2017). We subtract the average transition
time $\bar{t}_{c}$ and define $\delta t_{c}(\vec{x})\equiv
t_{c}(\vec{x})-\bar{t}_{c}$, and calculate the power spectrum of the
dimensionless quantity $\beta\delta t_{c}(\vec{x})$. It is handy to introduce
the dimensionless power spectrum as usual
$\displaystyle{\cal P}_{\delta t}$ $\displaystyle\equiv{\cal P}_{\beta\delta
t_{c}}(k)=\frac{k^{3}}{2\pi^{2}}P_{\beta\delta t_{c}}(k),$ (S1) $\displaystyle
P_{\beta\delta t_{c}}(k)$ $\displaystyle=\int d^{3}r\leavevmode\nobreak\
e^{i\vec{k}\cdot\vec{r}}\beta^{2}\langle\delta t_{c}(\vec{x})\delta
t_{c}(\vec{y})\rangle=\int_{0}^{\infty}4\pi r^{2}dr\leavevmode\nobreak\
\frac{\sin(kr)}{kr}\beta^{2}\langle\delta t_{c}\delta t_{c}\rangle(r).$ (S2)
Though $\langle\delta t_{c}\delta t_{c}\rangle(r)$ depends only on
$r\equiv|\vec{x}-\vec{y}|$, we sometimes use the notation $\langle\delta
t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle$.
Since we consider the two-point function of the transition time, we have two
distinct contributions in $\langle\delta t_{c}(\vec{x})\delta
t_{c}(\vec{y})\rangle$. They are illustrated in Fig. S1. We fix the evaluation
points $\vec{x}$ and $\vec{y}$, and consider different realizations of bubble
nucleation history. Since the transition time for each spatial point is
determined by the bubble whose wall passes that point for the first time, the
possibilities are that the transition times for the two points are determined
by one single bubble, or that they are determined by two different bubbles.
These two cases correspond to the left and right panels of Fig. S1.
Let us consider the probability for $t_{c}(\vec{x})$ and $t_{c}(\vec{y})$ to
be in the infinitesimal intervals $t_{x}<t_{c}(\vec{x})<t_{x}+dt_{x}$ and
$t_{y}<t_{c}(\vec{y})<t_{y}+dt_{y}$, respectively. For this to happen, note
that $\vec{x}$ and $\vec{y}$ must be in the false vacuum at $t_{x}$ and
$t_{y}$. Thus we need (1) and “(2-s) or (2-d)” in the following:
* (1)
$\vec{x}$ and $\vec{y}$ remain in the false vacuum at $t_{x}$ and $t_{y}$,
respectively.
* (2)
* (2-s)
One bubble nucleates at the intersection of the two past cones (red diamond in
the right panel of Fig. S2).
* (2-d)
Two bubbles nucleate on the surface of the past cones, one being in the
$\vec{x}$ side and the other in the $\vec{y}$ side (blue bands in the right
panel of Fig. S2).
Hereafter we assume luminal walls $v_{w}=1$ for simplicity, and thus the cones
are light-cones. Once we obtain the probability for these to happen, the
correlator $\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle$ reduces
to $\sum_{t_{x}}\sum_{t_{y}}$ (probability) $\times$ (value of $\delta
t_{x}\delta t_{y}$). Since the two contributions are distinct, we may
decompose the correlator into two terms
$\displaystyle\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle$
$\displaystyle=\langle\delta t_{c}(\vec{x})\delta
t_{c}(\vec{y})\rangle^{(s)}+\langle\delta t_{c}(\vec{x})\delta
t_{c}(\vec{y})\rangle^{(d)},$ (S3)
with each being called the single- and double-bubble contribution. Note that
the condition (1) is common to (2-s) and (2-d). In the following we use the
nucleation rate
$\displaystyle\Gamma(t_{n})$ $\displaystyle=\Gamma_{*}e^{\beta(t_{n}-t_{*})},$
(S4)
with $\beta$ being constant and $t_{*}$ being the typical transition time.
Hereafter we adopt $\beta=1$ unit. We sometimes use a shorthand notation
$x\equiv(t_{x},\vec{x})$ and $y\equiv(t_{y},\vec{y})$.
Now let us move on to the detailed calculation. We first calculate the average
transition time $\bar{t}_{c}$. For a spatial point $\vec{x}$ to experience the
transition between $[t_{x},t_{x},dt_{x}]$, it must be in the false vacuum at
$t_{x}$. Such a probability, which we call “survival probability”, is given by
$\displaystyle P_{\rm surv}(x)$ $\displaystyle=e^{-\int_{z\in({\rm
inside\leavevmode\nobreak\ the\leavevmode\nobreak\ past\leavevmode\nobreak\
cone\leavevmode\nobreak\ of\leavevmode\nobreak\ }x)}d^{4}z\leavevmode\nobreak\
\Gamma(t_{z})}$
$\displaystyle=\exp\left[-\int_{-\infty}^{t_{x}}dt_{n}\leavevmode\nobreak\
4\pi(t_{x}-t_{n})^{2}\Gamma_{*}e^{-(t_{n}-t_{*})}\right]$
$\displaystyle=\exp\left[-8\pi\Gamma_{*}e^{t_{x}-t_{*}}\right].$ (S5)
Hence, the average transition time is obtained as
$\displaystyle\bar{t}_{c}$
$\displaystyle=\int_{-\infty}^{\infty}dt_{x}\leavevmode\nobreak\
t_{x}\int_{-\infty}^{t_{x}}dt_{n}\leavevmode\nobreak\
4\pi(t_{x}-t_{n})^{2}\Gamma(t_{n})P_{\rm surv}(x)$
$\displaystyle=t_{*}-\ln(8\pi\Gamma_{*})-\gamma,$ (S6)
with $\gamma$ being the Euler-Mascheroni constant.
We next calculate the probability for (1) to happen. Following a similar
procedure as above, we get
$\displaystyle P_{\rm surv}(x,y)$
$\displaystyle=e^{-I(x,y)}=e^{-\int_{z\in({\rm inside\leavevmode\nobreak\
the\leavevmode\nobreak\ past\leavevmode\nobreak\ cones\leavevmode\nobreak\
of\leavevmode\nobreak\ }\vec{x}{\rm\leavevmode\nobreak\
and\leavevmode\nobreak\ }\vec{y})}d^{4}z\leavevmode\nobreak\
\Gamma(t_{z})}=e^{-\Gamma_{*}{\cal I}(x,y)e^{t_{\langle x,y\rangle}-t_{*}}},$
(S7) $\displaystyle{\cal I}(x,y)$
$\displaystyle=8\pi\left[e^{t_{x,y}/2}+e^{-t_{x,y}/2}+\frac{t_{x,y}^{2}-(r^{2}+4r)}{4r}e^{-r/2}\right].$
(S8)
Here we defined $t_{\langle x,y\rangle}\equiv(t_{x}+t_{y})/2$, $t_{x,y}\equiv
t_{x}-t_{y}$, $t_{\langle x,y\rangle,n}\equiv t_{\langle x,y\rangle}-t_{n}$,
$t_{\langle x,y\rangle,xn}\equiv t_{\langle x,y\rangle}-t_{xn}$, and
$t_{\langle x,y\rangle,yn}\equiv t_{\langle x,y\rangle}-t_{yn}$. In the
following we calculate the probability for “(1) and (2-s)” and that for “(1)
and (2-d)”, and estimate the correlator $\beta^{2}\langle\delta
t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle$. The final expression is the sum
of Eqs. (S11) and (S16).
Figure S1: Two-dimensional illustration of the single-bubble (left) and
double-bubble (right) contributions. The black dots are the evaluation points
$\vec{x}$ and $\vec{y}$, while the stars denote the nucleation points of the
bubbles. The color corresponds to the transition time for each spatial point.
For the left case, the transition time at both points is determined by one
single bubble nucleating around the center, while for the right case it is
determined by two different bubbles nucleating at different locations.
Figure S2: Past-cone geometry in 3D spacetime. Embedded are 2D spatial slices
of past cones. Left: Past cone of $\vec{x}$. This geometry is used to
calculate the average transition time $\bar{t}_{c}$. The survival probability
$P(x)$ is defined as the probability for no bubble to nucleate inside the
cone. A bubble nucleating on the surface of the cone passes $\vec{x}$ at the
evaluation time $t_{x}$. Right: Past cones of $\vec{x}$ and $\vec{y}$. This
geometry is used to calculate the two-point correlation function
$\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle$. For the single-
bubble case one single bubble nucleates at the intersection of the cone, while
for the double-bubble two different bubbles nucleate on the surface of the
each past cone.
Single-bubble contribution. The single-bubble contribution corresponds to the
cases in which one single bubble nucleates at the red diamond in the right
panel of Fig. S2. Noting that the red diamond forms a circle along the $\phi$
direction in three-dimensional space, we obtain
$\displaystyle\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle^{(s)}$
$\displaystyle=\int_{|t_{x}-t_{y}|<r}dt_{x}dt_{y}\leavevmode\nobreak\
(t_{x}-\bar{t}_{c})(t_{y}-\bar{t}_{c})\int_{-\infty}^{(t_{x}+t_{y}-r)/2}dt_{n}\int_{0}^{2\pi}d\phi\leavevmode\nobreak\
\frac{(t_{x}-t_{n})(t_{y}-t_{n})}{r}P_{\rm surv}(x,y)\Gamma(t_{n})$
$\displaystyle=\int_{-\infty}^{\infty}dt_{\langle
x,y\rangle}\int_{-r}^{r}dt_{x,y}\int_{r/2}^{\infty}dt_{\langle
x,y\rangle,n}\leavevmode\nobreak\ \frac{2\pi}{r}e^{-\Gamma_{*}{\cal
I}(x,y)e^{t_{\langle x,y\rangle}-t_{*}}}\Gamma_{*}e^{t_{\langle
x,y\rangle}-t_{*}}e^{-t_{\langle x,y\rangle,n}}$
$\displaystyle\qquad\times\left(t_{\langle
x,y\rangle}+\frac{t_{x,y}}{2}-\bar{t}_{c}\right)\left(t_{\langle
x,y\rangle}-\frac{t_{x,y}}{2}-\bar{t}_{c}\right)\left(t_{\langle
x,y\rangle,n}+\frac{t_{x,y}}{2}\right)\left(t_{\langle
x,y\rangle,n}-\frac{t_{x,y}}{2}\right).$ (S9)
We first integrate out $t_{\langle x,y\rangle}$ and get
$\displaystyle\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle^{(s)}$
$\displaystyle=\int_{-r}^{r}dt_{x,y}\int_{r/2}^{\infty}dt_{\langle
x,y\rangle,n}\leavevmode\nobreak\ \frac{2\pi e^{-t_{\langle
x,y\rangle,n}}}{r\,{\cal I}(x,y)}\left(t_{\langle
x,y\rangle,n}^{2}-\frac{t_{x,y}^{2}}{4}\right)\left[\left(\ln\left(\frac{{\cal
I}(x,y)}{8\pi}\right)\right)^{2}-\frac{t_{x,y}^{2}}{4}+\frac{\pi^{2}}{6}\right],$
(S10)
and then integrate out $t_{\langle x,y\rangle,n}$ and get
$\displaystyle\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle^{(s)}$
$\displaystyle=\int_{-r}^{r}dt_{x,y}\leavevmode\nobreak\ \frac{2\pi
e^{-r/2}}{r\,{\cal
I}(x,y)}\left(\frac{r^{2}}{4}+r+2-\frac{t_{x,y}^{2}}{4}\right)\left[\left(\ln\left(\frac{{\cal
I}(x,y)}{8\pi}\right)\right)^{2}-\frac{t_{x,y}^{2}}{4}+\frac{\pi^{2}}{6}\right].$
(S11)
Double-bubble contribution. The single-bubble contribution corresponds to the
cases in which two different bubbles nucleate on the surface of the past cones
in Fig. S2.
$\displaystyle\langle\Delta t_{c}(\vec{x})\Delta t_{c}(\vec{y})\rangle^{(d)}$
$\displaystyle=\int_{|t_{x}-t_{y}|<r}dt_{x}dt_{y}\leavevmode\nobreak\
(t_{x}-\bar{t}_{c})(t_{y}-\bar{t}_{c})\int_{-\infty}^{t_{x}}dt_{xn}\int_{-\infty}^{t_{y}}dt_{yn}\int
dc_{x}\int
dc_{y}\int_{0}^{2\pi}d\phi_{x}\int_{0}^{2\pi}d\phi_{y}\leavevmode\nobreak\ $
$\displaystyle\qquad(t_{x}-t_{xn})^{2}(t_{y}-t_{yn})^{2}P_{\rm
surv}(x,y)\Gamma(t_{xn})\Gamma(t_{yn})$
$\displaystyle=\int_{-\infty}^{\infty}dt_{\langle
x,y\rangle}\int_{-r}^{r}dt_{x,y}\int_{0}^{\infty}dt_{x,xn}\int_{0}^{\infty}dt_{y,yn}\leavevmode\nobreak\
16\pi^{2}e^{-\Gamma_{*}{\cal I}(x,y)e^{t_{\langle
x,y\rangle}-t_{*}}}\Gamma_{*}^{2}e^{2t_{\langle
x,y\rangle}-2t_{*}}e^{-(t_{x,xn}+t_{y,yn})}$ $\displaystyle\qquad\times
t_{x,xn}^{2}t_{y,yn}^{2}\left(t_{\langle
x,y\rangle}+\frac{t_{x,y}}{2}-\bar{t}_{c}\right)\left(t_{\langle
x,y\rangle}-\frac{t_{x,y}}{2}-\bar{t}_{c}\right)f_{x}f_{y}.$ (S12)
Note that the integration ranges for $c_{x}$ and $c_{y}$ can be nontrivial
since the blue bands in the right panel of Fig. S2 do not necessarily form
complete circles (i.e., complete spheres in three dimensions). The fractions
$f_{x}$ and $f_{y}$ in the last line take account of this
$\displaystyle f_{x}$
$\displaystyle=1\quad\left(t_{x,xn}<\frac{t_{x,y}+r}{2}\right),\quad\frac{(t_{x,xn}+r)^{2}-(t_{x,xn}-t_{x,y})^{2}}{4rt_{x,xn}}\quad\left(t_{x,xn}>\frac{t_{x,y}+r}{2}\right),$
(S13) $\displaystyle f_{y}$
$\displaystyle=1\quad\left(t_{y,yn}<\frac{-t_{x,y}+r}{2}\right),\quad\frac{(t_{y,yn}+r)^{2}-(t_{y,yn}+t_{x,y})^{2}}{4rt_{y,yn}}\quad\left(t_{y,yn}>\frac{-t_{x,y}+r}{2}\right).$
(S14)
We first integrate out $t_{\langle x,y\rangle}$ and get
$\displaystyle\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle^{(d)}$
$\displaystyle=\int_{-r}^{r}dt_{x,y}\int_{0}^{\infty}dt_{x,xn}\int_{0}^{\infty}dt_{y,yn}\leavevmode\nobreak\
\frac{16\pi^{2}e^{-(t_{x,xn}+t_{y,yn})}}{{\cal I}^{2}(x,y)}$
$\displaystyle\qquad\times
t_{x,xn}^{2}t_{y,yn}^{2}\left[\left(\ln\left(\frac{{\cal
I}(x,y)}{8\pi}\right)-1\right)^{2}-\frac{t_{x,y}^{2}}{4}+\frac{\pi^{2}}{6}-1\right]f(x)f(y),$
(S15)
and then integrate out $t_{x,xn}$ and $t_{y,yn}$ and get
$\displaystyle\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle^{(d)}$
$\displaystyle=\int_{-r}^{r}dt_{x,y}\leavevmode\nobreak\
\frac{16\pi^{2}}{{\cal I}^{2}(x,y)}$
$\displaystyle\qquad\times\left[4-\frac{e^{-t_{x,y}/2-r/2}}{2r}(r+t_{x,y}+4)(r-t_{x,y})-\frac{e^{t_{x,y}/2-r/2}}{2r}(r-t_{x,y}+4)(r+t_{x,y})\right.$
$\displaystyle\qquad\left.+\frac{e^{-r}}{16r^{2}}((r+4)^{2}-t_{x,y}^{2})(r^{2}-t_{x,y}^{2})\right]\left[\left(\ln\left(\frac{{\cal
I}(x,y)}{8\pi}\right)-1\right)^{2}-\frac{t_{x,y}^{2}}{4}+\frac{\pi^{2}}{6}-1\right].$
(S16)
Final expressions. The final expression for the correlator
$\beta^{2}\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle$ is Eqs.
(S11) and (S16) substituted into Eq. (S3). These are one-dimensional integrals
and easy to evaluate. Fig. S3 shows the two contributions and their sum. As
expected, both decay exponentially at large $r$. Now it is straightforward to
evaluate $P_{\beta\delta t_{c}}$. Fig. S4 shows $P_{\beta\delta t_{c}}$
calculated from Eq. (S2). For small or large $k$, one may approximate the
spectrum with ${\cal P}_{\beta\delta t_{c}}\simeq 70(k/\beta)^{3}$ or ${\cal
P}_{\beta\delta t_{c}}\simeq 0.7(k/\beta)^{-3}$, respectively.
Figure S3: Single-bubble (blue) and double-bubble (red) contributions to the
correlator $\beta^{2}\langle\delta t_{c}(\vec{x})\delta t_{c}(\vec{y})\rangle$
and their sum (yellow). Figure S4: Power spectrum ${\cal P}_{\delta
t}\equiv{\cal P}_{\beta\delta t_{c}}$. The dashed lines are ${\cal
P}_{\beta\delta t_{c}}=100(k/\beta)^{3}$ and ${\cal P}_{\beta\delta
t_{c}}=(k/\beta)^{-3}$ for comparison.
## II Alternative Derivation of Curvature Perturbation
Here we provide a derivation of (7) using the superhorizon evolution equation
for the curvature perturbation $\zeta$. On large scales, where spatial
gradients can be ignored, $\zeta$ follows the equation Garcia-Bellido and
Wands (1996); Wands _et al._ (2000)
$\begin{split}\dot{\zeta}=-\frac{H}{\rho+p}\delta p_{\rm nad},\end{split}$
(S17)
where
$\displaystyle\delta p_{\rm nad}=\dot{p}\left(\frac{\delta
p}{\dot{p}}-\frac{\delta\rho}{\dot{\rho}}\right),$ (S18)
and $H$ the Hubble rate, $\rho$ the total energy density and $p$ the total
pressure. Now consider a dark sector PT that takes place at time $t_{c}$ that
instantaneously converts the false vacuum energy into DR. This assumption of
instantaneousness is justified since percolation takes place within a time
window much narrower than a Hubble time, for $\beta\gg H_{\rm PT}$, as we
consider.
We can then write the total energy density as,
$\displaystyle\rho=\rho_{F}\Theta(t_{c}-t)+\rho_{F}(a_{c}/a)^{4}\Theta(t-t_{c})+\rho_{\rm
SM},$ (S19)
and
$\displaystyle
p=-\rho_{F}\Theta(t_{c}-t)+(\rho_{F}/3)(a_{c}/a)^{4}\Theta(t-t_{c})+\rho_{\rm
SM}/3.$ (S20)
Here $\Theta(x)$ is the Heaviside theta function and $a_{c}$ is the scale
factor at $t_{c}$. From these we can compute the time derivatives:
$\displaystyle\dot{\rho}$
$\displaystyle=-\rho_{F}\delta(t_{c}-t)+(-4H)\rho_{F}(a_{c}/a)^{4}\Theta(t-t_{c})+\rho_{F}(a_{c}/a)^{4}\delta(t-t_{c})+\dot{\rho}_{\rm
SM},$ (S21)
$\displaystyle=(-4H)\rho_{F}(a_{c}/a)^{4}\Theta(t-t_{c})+\dot{\rho}_{\rm SM},$
(S22) $\displaystyle\equiv B+\dot{\rho}_{\rm SM},$ (S23)
and
$\displaystyle\dot{p}$
$\displaystyle=\rho_{F}\delta(t_{c}-t)+(-4H)(\rho_{F}/3)(a_{c}/a)^{4}\Theta(t-t_{c})+(\rho_{F}/3)(a_{c}/a)^{4}\delta(t-t_{c})+\dot{\rho}_{\rm
SM}/3,$ (S24)
$\displaystyle=(4/3)\rho_{F}\delta(t_{c}-t)+(-4H)(\rho_{F}/3)(a_{c}/a)^{4}\Theta(t-t_{c})+\dot{\rho}_{\rm
SM}/3,$ (S25) $\displaystyle=(4/3)\rho_{F}\delta(t_{c}-t)+B/3+\dot{\rho}_{\rm
SM}/3.$ (S26)
We note $B\ll\dot{\rho}_{\rm SM}$ since $f_{\rm DR}\ll 1$ for our analysis.
Using this we can approximate,
$\displaystyle\frac{\dot{p}}{\dot{\rho}}=\frac{(4/3)\rho_{F}\delta(t_{c}-t)+B/3+\dot{\rho}_{\rm
SM}/3}{B+\dot{\rho}_{\rm
SM}}\approx\frac{1}{3}+\frac{4\rho_{F}\delta(t-t_{c})}{3\dot{\rho}_{\rm SM}}.$
(S27)
So far we have computed the time derivatives of the homogeneous quantities
$\rho$ and $p$ with time. To evaluate $\delta p_{\rm nad}$, we also compute
the variations $\delta\rho$ and $\delta p$, which are variations in the total
energy density and pressure, respectively, as we compare different acausal
regions. As explained in the main text, we neglect inflationary fluctuations
and therefore $\delta\rho_{\rm SM}=0$. However, since the PT does not complete
everywhere at the same time, both $\delta t_{c}\neq 0$ and $\delta a_{c}\neq
0$. It is these variations that eventually source a non-zero curvature
perturbation $\zeta$. The variation in the total energy density is given by,
$\displaystyle\delta\rho$ $\displaystyle=\rho_{F}\delta(t_{c}-t)\delta
t_{c}+(4\delta
a_{c}/a_{c})\rho_{F}(a_{c}/a)^{4}\Theta(t-t_{c})+\rho_{F}(a_{c}/a)^{4}\delta(t-t_{c})(-\delta
t_{c})$ (S28) $\displaystyle=(4\delta
a_{c}/a_{c})\rho_{F}(a_{c}/a)^{4}\Theta(t-t_{c}),$ (S29)
and in the total pressure by,
$\displaystyle\delta p$ $\displaystyle=-\rho_{F}\delta(t_{c}-t)\delta
t_{c}+4(\delta
a_{c}/a)(\rho_{F}/3)(a_{c}/a)^{4}\Theta(t-t_{c})+(\rho_{F}/3)(a_{c}/a)^{4}\delta(t-t_{c})(-\delta
t_{c}),$ (S30) $\displaystyle=-(4/3)\rho_{F}\delta(t_{c}-t)\delta
t_{c}+4(\delta a_{c}/a)(\rho_{F}/3)(a_{c}/a)^{4}\Theta(t-t_{c}).$ (S31)
We now evaluate, to leading order in $f_{\rm DR}$,
$\displaystyle\delta
p-\frac{\dot{p}}{\dot{\rho}}\delta\rho\approx-\frac{4}{3}\rho_{F}\delta(t_{c}-t)\delta
t_{c}.$ (S32)
Hence we can finally evaluate the change in $\zeta$ using (S17),
$\displaystyle\dot{\zeta}$ $\displaystyle\approx-\frac{3H}{4\rho_{\rm
SM}}\left(-\frac{4}{3}\rho_{F}\delta(t_{c}-t)\delta t_{c}\right)$ (S33)
$\displaystyle\approx H\frac{\rho_{F}}{\rho_{\rm SM}}\delta(t_{c}-t)\delta
t_{c}$ (S34) $\displaystyle\approx\frac{\rho_{F}}{\rho_{\rm
SM}}\delta(t_{c}-t)\frac{\delta t_{c}}{2t_{c}}.$ (S35)
Since the curvature perturbation before the PT is zero (since we assume
inflationary fluctuations to be much smaller than the fluctuations induced by
the dark PT), we can integrate the above to arrive at
$\displaystyle\zeta=\frac{\rho_{F}}{\rho_{\rm SM}}\frac{\delta
t_{c}}{2t_{c}}=f_{\rm DR}\frac{\delta t_{c}}{2t_{c}}\leavevmode\nobreak\
\text{(following the PT)}.$ (S36)
This matches with the expression in (6), obtained using the separate Universe
approach.
For subhorizon modes, additional gradient terms would appear in Eq. (S17)
Malik and Wands (2009). However, those terms are homogeneous in $\zeta$ and
the Newtonian potential, and therefore do not source $\zeta$ via $t_{c}$
fluctuations. Hence, we do not consider the gradient terms when evaluating how
$\delta t_{c}$ sources $\zeta$. Once $\zeta$ is generated, however, we
estimate the subhorizon evolution as mentioned in the main text.
### II.1 Derivation using the $\delta N$ formalism
Figure S5: An illustration of the energy density evolution of dark PT.
$\rho_{r,{\rm SM}}$ is the energy density of SM particles dominated by
radiation. False vacuum energy of the two patches $A$ and $B$ reheats into DR
at $t_{c}^{A,B}$. The total (DR and SM) energy density of the two patches are
equal at $t_{f}^{A,B}$.
We can also derive the form of $\zeta$ by comparing the energy densities in
the two patches $A$ and $B$, as discussed in the main text, and using the
$\delta N$ formalism. The evolution of the energy densities is shown
schematically in Fig. S5. The points $\alpha$ and $\beta$ denote the time of
PT for patch $A$ and $B$, respectively. At the point $\delta$, patch $A$ has a
smaller energy density compared to patch $B$, as DR redshifts from point
$\alpha$ to $\delta$. This energy difference can be thought to arise from a
hypothetical curvaton field. In particular, we can translate the energy
difference between $\beta$ and $\delta$, to an energy density difference at
$t_{c}^{A}$, between $\alpha$ and $\gamma$. Extrapolating to an even earlier
time, we can imagine this difference to arise from primordial fluctuations of
a curvaton field $\chi$, on a spatially flat hypersurface. Given Eq. (5), we
find $\delta\rho_{\chi}=(2\delta t_{c}/t_{c})\bar{\rho}_{\chi}$, where
$\bar{\rho}_{\chi}$ is the homogeneous energy density in $\chi$. The advantage
of this approach is that we can now directly utilize the standard expressions
derived in the curvaton scenario, see, e.g., Langlois _et al._ (2008).
We denote the energy densities of the SM radiation and $\chi$ by $\rho_{r,{\rm
SM}}$ and $\rho_{\chi}$, respetively, and write them in terms of curvature
perturbation on uniform density hypersurfaces for each species Langlois _et
al._ (2008):
$\rho_{r,{\rm SM}}=\bar{\rho}_{r,{\rm SM}}e^{4(\zeta_{\rm inf}-\delta
N)}\,,\quad\rho_{\chi}=\bar{\rho}_{\chi}e^{3(\zeta_{\chi}-\delta N)}\,.$ (S37)
Here $\zeta_{\rm inf}$ is the curvature perturbation of the SM radiation
(sourced by inflaton), and $\zeta_{\chi}$ is the curvature perturbation of
$\chi$ right before its decay (sourced by PT time fluctuation). On the uniform
density hypersurface, the total radiation energy density, $\rho_{r,{\rm
SM}}+\rho_{\rm DR}\equiv\rho_{r}=\bar{\rho}_{r}\exp[4(\zeta_{r}-\delta N)]$,
right after the curvaton decay equals $\bar{\rho}_{r}$, giving
$\zeta_{r}=\delta N$. Given that $\rho_{\rm DR}=\rho_{\chi}$ right after the
decay, we have
$f_{\rm DR}e^{3(\zeta_{\chi}-\zeta_{r})}+(1-f_{\rm DR})e^{4(\zeta_{\rm
inf}-\zeta_{r})}=1\,,\quad f_{\rm DR}=\frac{\bar{\rho}_{\rm
DR}}{\bar{\rho}_{r,{\rm SM}}+\bar{\rho}_{\rm DR}}\,.$ (S38)
The leading order expansion of the above equation in $\zeta_{\chi}$ and
$\zeta_{\rm inf}$ gives
$\zeta_{r}=\frac{3f_{\rm DR}}{4-f_{\rm DR}}\zeta_{\chi}+\frac{4(1-f_{\rm
DR})}{4-f_{\rm DR}}\zeta_{\rm inf}\,.$ (S39)
If $f_{\rm DR}\to 0$ ($f_{\rm DR}=1$), $\zeta_{r}=\zeta_{\rm inf}$
($\zeta_{r}=\zeta_{\chi}$) as expected. In the post-inflation era, where the
curvaton energy density is very subdominant, the uniform energy density
hypersurface is characterized by $\delta N=\zeta_{\rm inf}$, and the local
density of $\chi$ is
$\bar{\rho}_{\chi}e^{3(\zeta_{\chi}-\delta
N)}=\bar{\rho}_{\chi}e^{3(\zeta_{\chi}-\zeta_{\rm
inf})}=\bar{\rho}_{\chi}+\delta\rho_{\chi}\,.$ (S40)
The leading order expansion gives
$3\zeta_{\chi}=\frac{\delta\rho_{\chi}}{\bar{\rho}_{\chi}}+3\zeta_{\rm
inf}=2\frac{\delta t_{c}}{\bar{t}_{c}}+3\zeta_{\rm inf}\,,$ (S41)
and the total curvature perturbation
$\zeta_{r}=\frac{2f_{\rm DR}}{4-f_{\rm DR}}\frac{\delta
t_{c}}{\bar{t}_{c}}+\zeta_{\rm inf}\,.$ (S42)
Since the PT perturbation is uncorrelated to the inflaton perturbation, we
have for $f_{\rm DR}\ll 1$,
$\mathcal{P}_{\zeta_{r}}\approx\frac{1}{4}f_{\rm DR}^{2}\mathcal{P}_{\delta
t}+\mathcal{P}_{\rm ad}\,,$ (S43)
where we have denoted the power spectrum of $\zeta_{\rm inf}$ by
$\mathcal{P}_{\rm ad}$. This reproduces Eq. (7) for $\alpha_{\rm PT}\gtrsim
1$.
|
# Short proofs on $k$-extendible graphs
Shenwei Huang College of Computer Science, Nankai University, Tianjin 300350,
China. Partially supported by the National Natural Science Foundation of China
(11801284) and Natural Science Foundation of Tianjin (20JCYBJC01190). Yongtang
Shi Center for Combinatorics and LPMC, Nankai University, Tianjin 300071,
China. Partially supported by the National Natural Science Foundation of China
(No. 11922112) and Natural Science Foundation of Tianjin (Nos. 20JCZDJC00840,
20JCJQJC00090).
(October 7, 2021)
###### Abstract
In this note, we give short inductive proofs of two known results on
$k$-extendible graphs based on a property proved in [Qinglin Yu, A note on
$n$-extendable graphs. Journal of Graph Theory, 16:349-353, 1992].
## 1 Introduction
A graph $G$ is $k$-extendible if it satisfies the following conditions:
* •
$|G|\geq 2k+2$;
* •
$G$ is connected;
* •
$G$ has a perfect matching;
* •
for every matching $M_{k}$ of $G$ of size $k$, there is a perfect matching of
$G$ containing $M_{k}$.
The notion of $k$-extendible graphs was first defined and studied by Plummer
[5]. In particular, 2-extendible bipartite graphs play an important role in
the study of Pólya’s permanent problem [6] whose solution was obtained by
Robertsen, Seymour and Tomas [7] and independently by McCuaig [4]. We refer to
the monograph of Lovász and Plummer [3] for a detailed account of 1-extendible
graphs.
Our Contribution. Based on a property of $k$-extendible graphs proved by Yu
[8], we give short inductive proofs of two known results on $k$-extendible
graphs. Our proofs are much simpler than the existing proofs due to the fact
that the property allows us to apply the inductive hypothesis on subgraphs of
the given $k$-extendible graph rather than on the given graph itself.
## 2 Preliminaries
Let $G=(V,E)$ be a graph. For $S\subseteq V$ and a subgraph $H$ of $G$, the
neighborhood of $S$ in $H$, denoted by $N_{H}(S)$, is the set of vertices in
$H$ that are adjacent to some vertex in $S$. The size of $N_{H}(S)$ is denoted
by $d_{H}(S)$. If $S=\\{v\\}$, we simply write $N_{H}(v)$ and $d_{H}(v)$
instead of $N_{H}(\\{v\\})$ and $d_{H}(\\{v\\})$, respectively. For a subset
$S\subseteq V$, we denote by $G[S]$ the subgraph induced by $S$. Given a
matching $M$ of $G$, we denote by $V(M)$ the set of vertices that are
endvertices of edges in $M$. The minimum degree and the matching number of $G$
are denoted by $\delta(G)$ and $\alpha^{\prime}(G)$, respectively. For other
standard terminology we refer to [1].
We start with two simple propositions of $k$-extendible graphs that were
obtained by Plummer [5]. Since the proofs are short, we include them here for
the sake of completeness.
###### Proposition 1 ([5]).
Every $k$-extendible graph is $(k-1)$-extendible.
###### Proof.
Let $G$ be a $k$-extendible graph. By contradiction, let
$M=\\{a_{1}b_{1},\ldots,a_{k-1}b_{k-1}\\}$ be a matching of size $k-1$ that is
not contained in a perfect matching of $G$. Since $G$ is $k$-extendible, $M$
is a maximal matching of $G$. This implies that
$S=V(G)\setminus\\{a_{1},b_{1},\ldots,a_{k-1},b_{k-1}\\}$
is independent. Since $|G|\geq 2k+2$, it follows that $|S|\geq 4$. Since $M$
is not a maximum matching of $G$, it follows from Berge’s Theorem (see [1])
that there exists an $M$-augmenting path, that is, an $u$-$v$ path $P$ such
that $u,v\notin V(M)$ and edges of $P$ are alternating between $E(G)\setminus
M$ and $M$, starting with an edge not in $M$. Then $M^{\prime}=(E(P)\setminus
M)\cup(M\setminus E(P))$ is a matching of size $k$ with
$V(M^{\prime})=V(M)\cup\\{u,v\\}$. Since $G-V(M^{\prime})=S\setminus\\{u,v\\}$
is an independent set of size at least 2, $M^{\prime}$ cannot be extended to a
perfect matching of $G$. This is a contradiction. ∎
###### Proposition 2 ([5]).
Every 1-extendible graph is 2-connected.
###### Proof.
Suppose by contradiction that $G$ is a 1-extendible graph but not 2-connected.
Then there exists a cut vertex $v$ such that $G-v$ has components
$C_{1},\ldots,C_{t}$ for some $t\geq 2$. Since $G$ is connected, $v$ has a
neighbor $u_{i}\in C_{i}$ for each $1\leq i\leq t$. Since $G$ is 1-extendible,
there is a perfect matching containing $vu_{1}$ and this implies that
$|C_{1}|$ is odd. On the other hand, there is a perfect matching containing
$vu_{2}$ and this implies that $|C_{1}|$ is even. This is a contradiction. ∎
The following property of $k$-extendible graphs was proved by Yu [8] whose
proof used Theorem 1 below. Here we give a new proof that avoids the use of
Theorem 1.
###### Proposition 3 ([8]).
Let $G$ be a $k$-extendible graph with $k\geq 2$. Then for every edge $uv\in
E(G)$, $G-\\{u,v\\}$ is $(k-1)$-extendible.
###### Proof.
Let $e=uv\in E(G)$ and $G^{\prime}=G-\\{u,v\\}$. By Propostion 1, $G$ is
1-extendible and so 2-connected by Propostion 2. Since $G$ is $k$-extendible,
every matching of size $k-1$ of $G^{\prime}$ can be extended to a perfect
matching of $G^{\prime}$. So it remains to show that $G^{\prime}$ is
connected. Suppose by contradiction that $G^{\prime}$ has components
$C_{1},C_{2},\ldots,C_{t}$ for $t\geq 2$. Since $G$ is 2-connected, each of
$u$ and $v$ has a neighbor in each component $C_{i}$. Let $s\in C_{1}$ be a
neighbor of $u$ and $t\in C_{2}$ be a neighbor of $v$. Since $G$ is
$k$-extendible with $k\geq 2$, there is a perfect matching of $G$ containing
$\\{us,vt\\}$ by Propostion 1. This implies that $|C_{1}|$ is odd. Then there
is no perfect matching of $G$ containing $uv$. This contradicts that $G$ is
1-extendible. Therefore, $G-\\{u,v\\}$ is $(k-1)$-extendible. ∎
## 3 New Proofs
In this section, we present our new proofs of two known results on
$k$-extendible graphs. The first result was proved by Plummer [5] on the
connectivity of $k$-extendible graphs. The overall strategy of Plummer [5] was
to apply the inductive hypothesis on the input graph (due to Propostion 1) and
then used a variation of Menger’s Theorem. Our proof below, on the other hand,
is simpler due to the fact that we were able to apply the inductive hypothesis
on subgraphs of the input graph due to Propostion 3.
###### Theorem 1 ([5]).
Every $k$-extendible graph is $(k+1)$-connected.
###### Our Proof.
Let $G$ be a $k$-extendible graph. We prove by induction on $k$. The base case
is Propostion 2. Now suppose that $k\geq 2$ and the statement is true for
$(k-1)$-extendible graphs. By Propostion 3, $G-\\{u,v\\}$ is
$(k-1)$-extendible for every edge $uv\in E(G)$ and so is $k$-connected by the
inductive hypothesis. By Propostion 1 and Propostion 2, it follows that
$\delta(G)\geq 2$. For any vertex $v\in V(G)$, let $u$ be a neighbor of $v$.
Since $d(u)\geq 2$, $u$ has a neighbor $w$ other than $v$. Since
$H=G-\\{u,w\\}$ is $k$-connected, $d_{H}(v)\geq k$ and thus $d_{G}(v)\geq
k+1$. This shows that $\delta(G)\geq k+1$.
Now let $S\subseteq V(G)$ be an arbitrary set with $|S|=k$. Let $s\in S$ and
$t$ be a neighbor of $s$. We show that $G-S$ is connected.
Case 1. $t\in S$. Then
$G-S=(G-\\{s,t\\})-(S\setminus\\{s,t\\})$
is connected, since $G-\\{s,t\\}$ is $k$-connected.
Case 2. $t\notin S$. Let $G^{\prime}=G-(S\cup\\{t\\})$. Note that
$G^{\prime}=(G-\\{s,t\\})-(S\setminus\\{s\\}).$
Since $G-\\{s,t\\}$ is $k$-connected, $G^{\prime}$ is connected. Since
$\delta(G)\geq k+1$, $t$ has a neighbor in $G^{\prime}$. Therefore,
$G-S=G[V(G^{\prime})\cup\\{t\\}]$
is connected. ∎
The second result is on $k$-extendible bipartite graphs. The celebrated Hall’s
Theorem gives a necessary and sufficient condition for a balanced bipartite
graph to have a perfect matching. It turns out that $k$-extendible bipartite
graphs have a similar characterization.
###### Theorem 2 ([2]).
Let $G=(X,Y)$ be a connected bipartite graph with a perfect matching and
$|G|\geq 2k+2$. Then $G$ is $k$-extendible if and only if $|N(A)|\geq|A|+k$
for every subset $A\subseteq X$ with $1\leq|A|\leq|X|-k$.
Theorem 2 was first stated and proved by Brualdi and Perfect [2] in the
language of matrices (Theorem 2.1 in [2]). Here we give two graph-theoretical
proofs. The first one relies on Propostion 3 while the second one is based on
the König-Ore Formula.
###### Theorem 3 (The König-Ore Formula).
Let $G=(X,Y)$ be a bipartite graph. Then
$\alpha^{\prime}(G)=|X|-\max_{S\subseteq X}(|S|-|N(S)|).$
###### Our First Proof of Theorem 2.
We first prove the sufficiency. Take a matching
$M=\\{x_{1}y_{1},\ldots,x_{k}y_{k}\\}$
of size $k$. Let $X^{\prime}=X\setminus\\{x_{1},\ldots,x_{k}\\}$ and
$Y^{\prime}=Y\setminus\\{y_{1},\ldots,y_{k}\\}$. Denote by $H$ the subgraph of
$G$ induced by $X^{\prime}\cup Y^{\prime}$. Note that every nonempty subset
$A$ of $X^{\prime}$ has $1\leq|A|\leq|X|-k$. It follows from the assumption
that $|N_{G}(A)|\geq|A|+k$. This implies that $|N_{H}(A)|\geq|A|$. By the
König-Ore Formula, $H$ has a perfect matching $M_{H}$. It follows that
$M_{H}\cup M$ is a perfect matching of $G$ containing $M$. This shows that $G$
is $k$-extendible.
We now prove the necessity by induction on $k$.
Base Case: $k=1$. By contradiction, let $A$ be a subset of $X$ with
$1\leq|A|\leq|X|-1$ such that $|N(A)|<|A|+1$. Since $G$ has a perfect
matching, $|N(A)|\geq|A|$. It then follows that $|N(A)|=|A|$. Since $G$ is
connected, there is an edge $e=xy$ between $N(A)$ and $X\setminus A$. So there
is no perfect matching of $G$ containing $e$, simply because there are not
enough vertices in $G-\\{x,y\\}$ to match vertices in $A$.
Inductive Step: We assume that $k\geq 2$ and the statement is true for $k-1$.
let $A$ be an arbitrary subset of $X$ with $1\leq|A|\leq|X|-k$. If $N(A)=Y$,
then
$|N(A)|=|Y|=|X|\geq|A|+k.$
So we may assume that $N(A)\neq Y$. Since $G$ is connected, there is an edge
$xy\in E(G)$ such that $y\in N(A)$ and $x\in X\setminus A$. Let
$G^{\prime}=G-\\{x,y\\}=(X\setminus\\{x\\},Y\setminus\\{y\\})$. By Propostion
3, $G^{\prime}$ is $(k-1)$-extendible. On the other hand, $A\subseteq
X\setminus\\{x\\}$ has
$1\leq|A|\leq|X|-k=(|X|-1)-(k-1).$
By the inductive hypothesis, $|N_{G^{\prime}}(A)|\geq|A|+(k-1)$. Since
$N_{G}(A)=N_{G^{\prime}}(A)\cup\\{y\\}$, it follows that
$|N_{G}(A)|\geq|A|+k$. ∎
###### Our Second Proof of Theorem 2.
The difference lies in the inductive step of the necessity. We assume that
$k\geq 2$ and the statement is true for $k-1$. By contradiction, let $A$ be a
subset of $X$ with $1\leq|A|\leq|X|-k$ such that $|N(A)|<|A|+k$. By Propostion
1, $G$ is $(k-1)$-extendible. By the inductive hypothesis,
$|N(A)|\geq|A|+(k-1)$. It follows that
$|N(A)|=|A|+(k-1).$ (1)
Let $X^{\prime}=X\setminus A$ and write $B=N(A)$. Note that $|B|\geq k$ for
otherwise Equation 1 would be contradicted. Denote by $H$ the subgraph induced
by $B\cup X^{\prime}$. If $H$ has a matching of size $k$, then it cannot be
extended to a perfect matching of $G$ (because there are not enough vertices
to match vertices in $A$). So the matching number of $H$ is at most $k-1$. By
the König-Ore Formula,
$\alpha^{\prime}(H)=|B|-\max_{S\subseteq B}(|S|-|N_{H}(S)|)\leq k-1.$
So there exists a subset $S\subseteq B$ such that
$|S|-|N_{H}(S)|\geq|B|-(k-1)$. Since $|B|\geq k$, $|S|\geq 1$. Moreover,
$|N_{H}(S)|\leq|S|+(k-1)-|B|$. Therefore,
$|N_{G}(S)|\leq|A|+|N_{H}(S)|\leq|A|+|S|+(k-1)-|B|=|S|,$
where the last equality follows from Equation 1. Since $|A|\leq|X|-k$, it
follows that $|B|=|N(A)|=|A|+(k-1)\leq|X|-1$. Hence, $S\subseteq B$ violates
the condition for $G$ to be 1-extendible. ∎
## 4 Concluding Remarks
The fact that our proof of Propostion 3 does not use Theorem 1 makes our new
proof of Theorem 1 self-contained. To the best of our knowledge, our first
proof of Theorem 2 is new and self-contained. The second proof, in essence, is
the graph counterpart of the proof given in [2] stated in matrix language.
That proof used the Frobenius-König Theorem which is the matrix counterpart of
the the König-Ore Formula. However, we feel that it may be convenient for
graph theorists to have a graph-theoretical proof. So we include our second
proof here as well.
## References
* [1] J. A. Bondy and U. S. R. Murty. Graph Theory. Springer, 2008.
* [2] R. A. Brualdi and H. Perfect. Extension of partial diagonals of matrices I. Monatshefte für Mathematik, 75:385–379, 1971.
* [3] L. Lovász and M. D. Plummer. Matching Theory. North–Holland, 1986.
* [4] W. McCuaig. Pólya’s permanent problem. The Electronic Journal of Combinatorics, 11:R79, 2004.
* [5] M. D. Plummer. On $n$-extendible graphs. Discrete Mathematics, 31:201–210, 1980.
* [6] G. Pólya. Aufgabe 424. Arch. Math. Phys., 20:271, 1913.
* [7] N. Robertsen, P. Seymour, and R. Thomas. Permanents, Pfaffian orientations, and even directed circuits. Annals of Mathematics, 150:929–975, 1999.
* [8] Q. Yu. A note on $n$-extendable graphs. Journal of Graph Theory, 16:349–353, 1992.
|
11institutetext: Department of Mathematics and Geosciences, University of
Trieste, Trieste, Italy
11email<EMAIL_ADDRESS>22institutetext: Department of
Economics, Business, Mathematics and Statistics, University of Trieste, Italy
22email<EMAIL_ADDRESS>33institutetext: LMU Munich, Munich,
Germany 44institutetext: Munich Center for Machine Learning (MCML), Munich,
Germany
44email<EMAIL_ADDRESS>55institutetext: Department of Engineering and
Architecture, University of Trieste, Trieste, Italy
55email<EMAIL_ADDRESS>
# Decision Predicate Graphs: Enhancing Interpretability in Tree Ensembles
Leonardo Arrighi 11 0009-0006-2494-0349 Luca Pennella 22 0009-0006-2721-1248
Gabriel Marques Tavares 33 4 4 0000-0002-2601-8108 Sylvio Barbon Junior 55
0000-0002-4988-0702
###### Abstract
Understanding the decisions of tree-based ensembles and their relationships is
pivotal for machine learning model interpretation. Recent attempts to mitigate
the human-in-the-loop interpretation challenge have explored the extraction of
the decision structure underlying the model taking advantage of graph
simplification and path emphasis. However, while these efforts enhance the
visualisation experience, they may either result in a visually complex
representation or compromise the interpretability of the original ensemble
model. In addressing this challenge, especially in complex scenarios, we
introduce the Decision Predicate Graph (DPG) as a model-agnostic tool to
provide a global interpretation of the model. DPG is a graph structure that
captures the tree-based ensemble model and learned dataset details, preserving
the relations among features, logical decisions, and predictions towards
emphasising insightful points. Leveraging well-known graph theory concepts,
such as the notions of centrality and community, DPG offers additional
quantitative insights into the model, complementing visualisation techniques,
expanding the problem space descriptions, and offering diverse possibilities
for extensions. Empirical experiments demonstrate the potential of DPG in
addressing traditional benchmarks and complex classification scenarios.
###### Keywords:
Ensemble Learning Explainable Artificial Intelligence Interpretability
Explainability Tree-based Ensemble Method Graph Random Forest
## 1 Introduction
Artificial intelligence, although still under strong development, is now a
consolidated and widely used tool. This is thanks to the continuous growth of
computing power, which allows the use of increasingly complex and
computationally expensive machine learning (ML) methods. The challenges
presented by modern-world problems are growing in complexity as well as the
proposed solutions.
Dealing with large quantities of data and frequently encountering unbalanced
datasets are still significant obstacles in addressing many real-world issues;
however, tree-based ensemble algorithms offer several advantages in overcoming
these challenges, including robustness to noise and outliers, scalability to
large datasets, automatic handling of missing values, and the ability to
capture complex relationships and interactions within the data [1, 2].
Without delving into the intricacies of various algorithms, exhaustively
described by Hastie et al. [3], the process of learning tree-based ensemble
models involves training multiple decision trees to combine their predictions,
optimising performance, and enhancing generalisation capabilities. While these
models indeed offer concrete solutions to a diverse array of problems,
developers and users are confronted with new challenges. The training outcome
yields an exceptionally complex model, commonly referred to as an opaque-box
model (also known as the black-box problem), whose internal workings are not
transparent or easily interpretable [4]. In response to this context, the
Decision Predicate Graph (DPG) is proposed in this paper.
Drawing inspiration from the expanding theme of eXplainable Artificial
Intelligence (XAI), we designed a graph structure to tackle transparency and
explainability challenges inherent in tree-based ensemble models. This
facilitates a better understanding of the intricate choices underlying these
ML models. DPG is created with inspiration from the concept of aggregating
random forests (RF) [5, 6], as introduced by Gossen and Steffen [7]. The
approach proposed in [7] suggests visualising the decisions within the RF by
combining the branches of the tree base learners into a single and compact
decision diagram. The concept behind DPG is to convert a generic tree-based
ensemble model for classification into a graph, a defined and studied
structure with known properties. In this graph, nodes represent predicates,
i.e., the feature-value associations present in each node of every tree, while
edges denote the frequency with which these predicates are satisfied during
the model training phase by the samples of the dataset. The DPG structure
enables comprehending the choices made by the model, enhancing transparency
and understandability. Moreover, it allows the exploitation of graph
properties to develop metrics and algorithms facilitating the analysis of the
ensemble model. This, in turn, aids in understanding the decisions it makes,
easing the task of visualising the graph which can be vast and complicated for
larger models with numerous tree base learners.
DPG serves as a model-agnostic tool offering a comprehensive interpretation of
tree-based ensemble models. It provides descriptive metrics that enhance the
understanding of the decisions inherent in the model, offering valuable
insights. This tool proves particularly useful for models that are _a priori_
considered satisfactory in terms of performance.
Our work contributes in the following ways:
* •
we introduce DPG, a novel interpretability structure that transforms an
opaque-box tree-based ensemble model into an enriched graph;
* •
we present the algorithm used to create DPG, accompanied by pseudo-code to
enhance understanding and facilitate replication, complete with its asymptotic
complexity;
* •
we provide the interpretation of three metrics from graph theory, enriching
the model comprehension and gaining insights;
* •
we demonstrate the use of the proposed method through two case studies: the
application of DPG to two RF models, respectively, on the Iris dataset [8] and
a challenging dataset.
It is important to highlight that these results are achieved in a generic
fashion, utilising a standard classifier on well-established datasets.
Significantly, we intentionally avoided incorporating scenario-specific
heuristics. Therefore, we posit that our aggregation approach has the
potential for widespread application across a diverse range of related
scenarios.
## 2 Literature Review
Complex yet effective models have become increasingly prevalent, especially in
fields where the outcomes bear significant importance and require a heightened
level of sensitivity. In these scenarios, there is a growing demand for models
to exhibit transparency, accountability, and a comprehensive understanding.
Consequently, the discussion on XAI has experienced significant growth,
leading to an extensive body of literature [9, 10, 11].
In XAI, as suggested by Dwivedi et al. [12], classifications are often based
on scope, distinguishing between global interpretability, which reveals
overall data trends and provides insights into the entire model, and local
interpretability, which elucidates the reasoning behind specific predictions
for individual instances. Another classification criterion involves how
interpretability is achieved: intrinsic interpretability relies on
straightforward model structures (e.g., concise decision trees or linear
models), while post hoc interpretability involves methods applied after model
training [11]. Interpretation methods are further categorised as model-
specific or model-agnostic. Model-specific interpretation tools are tailored
to specific model classes, limiting their applicability. In contrast, model-
agnostic tools are versatile, capable of being employed with any ML model, and
are applied post-training (e.g., SHAP [13], LIME [14], PDP [15], and Anchors
[16]).
In this context, XAI tools, especially those providing global interpretations,
become valuable instruments for understanding tree-based ensemble models.
These models are widely used in addressing diverse problem domains, as
highlighted in several surveys [17, 18, 19]. As a result, the number of
studies delving into model-specific techniques tailored for ensembles of trees
has also increased.
The first significant study is proposed by Mashayekhi and Gras [20]. The
authors introduced _RF+HC_ , an approach that employs a hill climbing
algorithm in RF to search for a decision set. This rule set reduces the number
of decisions dramatically, which significantly improves the comprehensibility
of the underlying model built by RF. Similarly, Hara and Hayashi [21] exploits
Bayesian model selection to extract the decision set. These approaches share
similarities with our method; however, our proposal extends beyond the
extraction of decisions from the RF. Visualising the decisions of tree-based
ensemble models and simultaneously complementing them with metrics developed
by graph theory makes DPG more adaptable and holistic. This approach allows
for obtaining insights into the model beyond just decision information.
Zhao et al. [22] proposed _iForest_ , a visual analysis system specifically
designed to interpret RF models and their predictions. They built a feature
view to illustrate the relationships between input features and outcome
predictions and proposed a design that summarises multiple decision paths
based on feature occurrences and ranges, allowing users to explore and
understand the partitioning logic of these paths. The iForest, like numerous
visualisation systems, faces significant challenges related to scalability and
interpretability when dealing with large ensembles. To overcome this
challenge, our approach does not prioritise visualisation but instead strives
for a comprehensive understanding of the ensemble’s logic.
Hatwell et al. [23] contributed with _Collection of High Importance Random
Path Snippets_ (CHIRPS), a method that incorporates the explanation of RF
classification for each data instance and extracts a decision path from each
tree in the forest, resulting in a set of decisions that elucidate the
classification process. However, this method is limited to rules extraction
and lacks insight into the model’s structure. Additionally, it does not
incorporate metrics to explain the logic of the tree-based ensemble model.
Another technique focused on visualising decisions underlying RF models is
introduced by Neto and Paulovich [24]. _Explainable Matrix_ (ExMatrix) employs
a matrix-like visual metaphor, where rows represent decisions, columns denote
features, and cells encapsulate decision predicates, thereby facilitating the
scrutiny of models and the audit of classification outcomes. The visualisation
capability of ExMatrix for global visualisation has limitations in terms of
scalability because the number of decisions increases significantly with the
number of trees in large ensembles. Moreover, ExMatrix layouts can rapidly
become challenging to explore, while the complexity of the model increases. As
mentioned earlier, our approach enables us to retrieve information without
relying solely on visualisation.
Dedja et al. [25] introduced an approach denoted as _Building Explanations
through a LocalLy AccuraTe Rule EXtractor_ (BELLATREX), which is designed to
explain the forest predictions for a given test instance by a set of logical
rules based on the features of the dataset. However, a potential limitation
lies in the computational complexity of the approach. While explaining a
single prediction is quick, applying the method to a complete dataset becomes
computationally expensive. Furthermore, BELLATREX focuses on instance-level
explanations rather than providing a global perspective. In addition,
BELLATREX uses clustering techniques to simplify the representation and
decision logic, whereas our approach uses graphs to avoid simplifications that
can lead to loss of information.
Various studies [26, 27, 28, 29] proposed several tree similarity metrics
through the process of clustering based on tree representations. However,
these approaches, while beneficial for interpretability, require simplifying
the model through techniques such as selection, pruning, and frequency
analysis, which can result in information loss.
A number of works [30, 31, 7, 32] established a connection between tree-based
ensemble models and graph theory. The foundational concept of these techniques
has been explored by [33, 34, 35, 36, 37, 38, 39]. These works established the
theory that introduces the transformation of decision trees into graphs,
aiming for more efficient and non-redundant tree structures. Nakahara et al.
[30] and Silva et al. [31] works focus on performance optimisation, with
Gossen and Steffen [7] and Murtovi et al. [32] being the sole contributors
that employed these techniques for interpretability purposes.
In particular, Gossen and Steffen [7] introduced the _Algebraic Decision
Diagram_ (ADD), aiming to transform tree-based ensemble models into bipartite
graphs. The ADD serves as an alternative construction to RF, providing an
additional predictive model that functions as a surrogate. ADD proves valuable
for specific aspects of interpretability, such as outcome explanation, logic
of _majority vote_ , and visualising the path. Acting as a surrogate model,
their primary contributions lie in the ability to provide a simple, optimised
model. In comparison, our objective is to extend their ideas by fully
leveraging graph theory. This goes beyond the transposition of tree-based
ensemble models into a graph; rather, it involves using graph theory and its
associated metrics to gain insights into the functionality of the models.
_ForestGUMP_ , an online tool developed by Murtovi et al. [32], is designed
for generating ADDs. This tool provides valuable information such as graph
visualisation, hypothetical sample path display, and logic of majority vote.
However, it has some limitations in terms of the number of usable trees (only
$20$) and in visualising problems that involve multiple features and
classification choices, making graph navigation complex. In our work, while we
enable visualisation, our focus is on utilising graph-related metrics, and we
do not incorporate the simplification of the analysed models.
## 3 Decision Predicate Graphs
DPGs capture the details of a tree-based ensemble model and learned dataset
specifics, emphasising predicate paths while preserving crucial decision
points. Essentially, DPG converts complex ensemble models into a graph
structure, where nodes represent predicates made by the model and edges denote
the occurrence of these predicates during model training.
In this segment, we present the formalisation of DPG, elucidate the algorithm
employed in its development through pseudo-code and its asymptotic complexity,
and provide a detailed exposition of various metrics and properties essential
for comprehending the tree-based ensemble model. Moreover, we meticulously
outline the advantages of metrics and articulate why they can serve as a
valuable complementary aid to graph visualisation, particularly in overcoming
its inherent limitations.
As previously mentioned in Section 1, DPG is tailored for tree-based ensemble
models designed specifically for classification tasks.
### 3.1 Definition
Let $\mathcal{M}_{n}$ be the tree-based ensemble model consisting of $n$ tree
base learners $T(x;\Theta_{b})$, where $x$ is a generic sample, and
$\Theta_{b}$ characterizes the $b$th learner in terms of split variables,
cutpoints at each node, and terminal-node values. More specifically,
$\Theta_{b}$ includes:
* •
all the splitting conditions associated with each $j$th internal node $n_{bj}$
based on a specific feature $f_{bj}$ and a threshold (for numerical features)
or a set of possible values (for categorical features) $t_{bj}$;
* •
the values assigned to leaf nodes $c_{b}$.
Let $\mathcal{D}$ be the training set on which $\mathcal{M}_{n}$ is trained.
Every base learner is trained on a dataset $\mathcal{D}_{b}$, where
$\mathcal{D}_{b}$ is a subset of $\mathcal{D}$. We define $\mathcal{O}$ as the
set of logical operations: $\mathcal{O}=\\{\leq,>,=,\neq\\}$.
A predicate set $\mathcal{P}(\mathcal{M}_{n})$, for an ensemble method
$\mathcal{M}_{n}$ is the set obtained by the union of the set of all the
triples $p=(f_{bj},o,t_{bj})$, where $o\in\mathcal{O}$, and $f_{bj}$,
$t_{bj}\in\Theta_{b}$, and the set of all leaf nodes $c_{b}$, for all $n$ tree
base learners of $\mathcal{M}_{n}$. The triples $p$ are called decisions,
while the elements of $\mathcal{P}(\mathcal{M}_{n})$ are called predicates.
A Decision Predicate Graph (DPG$(\mathcal{M}_{n})$) for a model
$\mathcal{M}_{n}$ is a directed weighted graph $(\mathcal{P},E)$ where:
* •
$\mathcal{P}$ is the set of nodes, which corresponds to the predicate set
$\mathcal{P}(\mathcal{M}_{n})$;
* •
$E$ is the set of edges, where each edge represents the frequency with which a
sample consecutively satisfies two predicates in a given base learner. This
frequency is computed considering all the samples of the training set
$\mathcal{D}$ for each base learner.
For conciseness, from this point onward, we will use the acronym DPG to refer
to the graph, indicating that it is constructed based on a model.
### 3.2 From Ensemble to a DPG
We introduce an algorithm, outlined in Algorithm 1, for constructing the DPG
based on a tree-based ensemble model, by traversing all tree base learners
with the training samples.
1
Input: Ensemble tree model $\mathcal{M}_{n}$, Training set $\mathcal{D}$
Output: DPG$(\mathcal{M}_{n})$
2
3Initialise empty set DPG$(\mathcal{M}_{n})$;
4
5foreach _$T$ (learner) in $\mathcal{M}_{n}$_ do
6
7 Initialise empty predicate set $\mathcal{P}$ and edge set ${E}$;
8
9 foreach _$x$ (sample) in $\mathcal{D}$_ do
10 Initialise empty predicate set $\mathcal{P}_{x}$ and edge set ${E_{x}}$;
// To obtain the predicates path for $x$ on the tree $T$
11 ($\mathcal{P}_{x}$, ${E_{x}}$) $\leftarrow$ TRAVERSING(_$T,x$_);
12 Add ($\mathcal{P}_{x}$, ${E_{x}}$) to $\mathcal{P}$ and ${E}$;
13
14
15
16DPG$(\mathcal{M}_{n})$ $\leftarrow$ AGGREGATING(_( $\mathcal{P}$, ${E}$)_);
17
18return _DPG $(\mathcal{M}_{n})$_;
Algorithm 1 Construct DPG from Ensemble Tree Model
The algorithm iterates over each base learner in the ensemble tree model
$\mathcal{M}_{n}$ and each training sample $x$ in the training set
$\mathcal{D}$.
To clarify, the TRAVERSING function follows the predicate path of a particular
input sample $x$ through the decision tree $T$, starting from the root node
and navigating to the appropriate leaf node based on the feature values of
$x$. Meanwhile, the AGGREGATING function processes the predicates and edges
obtained from TRAVERSING the decision trees for all samples into a single
graph representation, DPG$(\mathcal{M}_{n})$, by taking the union of
$\mathcal{P}$ and computing the frequency of elements of $E$.
The algorithm presents a systematic methodology for constructing DPG. The
overall asymptotic complexity can be formally expressed as follows:
$O(b\times s\times(k+k^{2}))=O(b\times s\times k^{2})$
where:
* •
$b$ is the number of learners in the ensemble,
* •
$s$ is the number of samples in the training set, $|\mathcal{D}|$, and
* •
$k$ represents the size of the ($\mathcal{P}_{x}$, ${E_{x}}$) processed by the
TRAVERSING and AGGREGATING functions.
This analysis takes into account the linear time complexity $O(k)$ for the
TRAVERSING function and the quadratic time complexity $O(k^{2})$ for the
AGGREGATING function. Our _Python_ 3.10 implementation is accessible
here111https://github.com/sbarbonjr/fhg/tree/main.
### 3.3 DPG interpretability
In this section, we enumerate and elucidate some of the advantages that DPG
can offer. The graph-based nature of DPG provides significant enhancements in
the direction of a complete mapping of the ensemble structure. Weighted
directed graphs, such as DPGs, are studied structures with well-established
properties that enable the identification or construction of useful metrics
and algorithms. It is crucial to emphasise that all the observations presented
in this section are valuable for comprehending and analysing the obtained
model.
#### 3.3.1 Visualisation.
DPG provides an immediate advantage by allowing the visualisation of the
entire tree-based ensemble models through a single comprehensive graph.
Similar to the idea proposed by Gossen and Steffen [7], consolidating all
individual basic learners within the model into a unified graph provides a
holistic representation of the decision-making process. This visualisation not
only elucidates the decisions made by the learners but also reveals the
intricate relationships between them. Consequently, it facilitates a
comprehensive understanding of the utilised features and, more importantly,
the associations between features and their values that enable the model to
accurately classify a sample into a specific class.
Another noteworthy aspect of DPG lies in the concept of representing edges as
frequencies. This feature enables a discerning analysis of the most
significant path through predicates, shedding light on decisions consistently
employed by numerous learners or across multiple samples. This insight not
only highlights the prevalence of certain decisions but also opens avenues for
targeted enhancement strategies, focusing on those influential aspects within
the model.
Moreover, by traversing all the possible paths between predicates in reverse,
starting from one of the classes, we can discern the essential characteristics
that a sample must possess to be classified into a specific class. This
capability facilitates the _a priori_ elimination of certain elements from the
dataset when considering the particular class.
Nevertheless, we acknowledge that while visualisation is a valuable tool, its
effectiveness diminishes with an increasing number of tree base learners. A
multitude of tree base learners implies an increase in decisions and,
consequently, an abundance of predicates. As a result, the size of the graph,
in terms of nodes, grows proportionally with the model’s scale. This expansion
can render the graph illegible or impractical to visualise due to its
intricate complexity. To address this challenge, we provide additional tools
that complement the visualisation, aiding in the extraction of model
properties and facilitating a more comprehensible understanding.
One approach is based on the desire and feasibility of determining the
specific characteristics a sample must exhibit to be assigned to a particular
class. Taking inspiration from the _outcome explanation problem_ introduced by
Gossen and Steffen [7], to enhance the immediacy and effectiveness of this
analysis, we provide an aggregation of predicates, referred to as _constraint_
, which represent intervals associated with the features of each class. The
constraints are defined as follows: for a given class identified in the DPG,
we list all nodes connected by a path originating from the node itself and
culminating in the class. For each feature within the node predicates, we
delineate the most extensive possible interval using the values associated
with the features. This interval is defined by two endpoints. The minor
endpoint is the smallest value within the set of values less than the feature,
while the major endpoint is the largest value within the set of values greater
than the feature. If either of these two sets is empty, the interval is deemed
infinite. Each class has its constraints for every feature contributing to the
classification of the samples.
#### 3.3.2 Centrality.
The centrality of a node is defined as a number or rank corresponding to the
node position within the network. By observing centrality, we can make
considerations that allow us to better understand the process hidden in the
ensemble method. The notion of centrality encompasses a wide range of metrics.
In this section, we explore those metrics that offer the most insightful
information.
According to Brandes [40], we define the _betweenness centrality_ (BC) of a
node as the fraction of all the shortest paths between every pair of nodes of
the graph passing through the considered node. Let DPG = ($\mathcal{P},E$) be
the graph and $s,t,v\in\mathcal{P}$ three vertexes of DPG, we can denote with
$\sigma(s,t)$ the number of shortest paths between $s$ and $t$ and with
$\sigma(s,t|v)$ the number of shortest paths between $s$ and $t$ passing
through $v$. Then, the BC of the node $v\in\mathcal{P}$ is defined as:
$BC(v)=\sum_{s,t\in\mathcal{P}}\frac{\sigma(s,t|v)}{\sigma(s,t)}.$
All details and observations about BC can be read in [40]. BC serves as a
relevant metric for gaining a deeper insight into the significance of
decisions within the ensemble model. We can observe that a node with a higher
BC value has a more significant influence on the flow of information within
the graph; nodes with high BC can be considered potential bottleneck nodes
because they play a crucial role in facilitating interactions between
different parts of the DPG. For this reason, we can assert that these nodes
are meaningful to understanding the tree-based ensemble models: in all tree
base learners, the decision contained in the node is essential to classify the
elements of the dataset. We highlight that the significance extends beyond the
characteristic itself; it encompasses the value associated with it.
According to Mones et al. [41], we define the _local reaching centrality_
(LRC) of a node $v$ of the DPG as the proportion of other nodes reachable from
node $v$ via outgoing edges. LRC can be generalised to weighted graphs by
measuring the average weight of a given directed path starting from node $v$
(more details are available on [41]). The LRC serves as a metric for assessing
the importance of DPG’s nodes. It gauges the extent to which decisions
contained in these nodes are employed by diverse tree base learners for
classifying samples in the training set. This, in turn, reflects the
importance of these decisions in the classification of new samples. The LRC
offers a comprehensive perspective on the concept of feature importance (FI)
by extending its definition to encompass the values associated with features
across various decisions. Additionally, the prominence of paths between
highlighted predicates indicates their frequent utilisation, providing
insights into how new samples can be classified with fewer decisions.
#### 3.3.3 Community.
While there is no single definition of a community, we can observe structures
similar to communities in the DPG. According to Radicchi et al. [42], we
define a _community_ as a subset of nodes of the DPG characterised by dense
interconnections between its elements and sparse connections with the other
nodes of the DPG that do not belong to the community. Based on the properties
of DPG, we employed _asynchronous label propagation_ algorithm, proposed by
Raghavan et al. [43], to detect communities within the graph. The core concept
of the algorithm involves each graph node determining its community membership
based on the majority of its neighbours. Without delving into details, which
can be appreciated in [43], the algorithm comprises a series of steps: each
node initially possesses a unique label. As these labels diffuse through the
graph, closely connected groups of nodes converge on a common label. These
consensus groups then expand outward until further expansion becomes
impractical. After this label propagation process, nodes sharing the same
labels are identified as belonging to the same community. This process is
iterated until each node in the network aligns its label with the community
that includes the maximum number of its neighbouring nodes. The algorithm is
defined asynchronous, as each node receives updates without waiting for
updates on the remaining nodes. Identifying communities in the DPG provides
insights into the ensemble model: visualising them allows us to discover
groups of nodes that similarly contribute to the classification of samples.
By employing the asynchronous label propagation algorithm, we observe that
each formed community is associated with a class. Within these communities,
the features utilised by the ensemble model to classify samples belonging to
the community’s class are emphasised. Once again, the association between
features and values plays a key role, highlighting the specific decisions made
by the learners. To quote Raghavan et al. [43]:
> Communities in social networks can provide insights about common
> characteristics or beliefs among people that make them different from other
> communities.
Similarly, we observe that communities within the DPG offer a valuable
understanding of the characteristics for samples to be assigned to a
particular community class. This intuition extends to identifying predominant
features and those that play a marginal role in the classification process.
Moreover, it is noteworthy that communities also provide insights into the
entire dataset and the complexity of the problem. A community comprising a
substantial number of nodes, each associated with different predicates often
involving distinct features, indicates that the model makes diverse decisions
to assign samples to the community class. This implies that the model
encounters challenges in classifying samples for this particular class, and
data from different classes are not easily distinguishable within the dataset.
Finally, communities, functioning as sub-graphs, can be used to visualise the
decisions made in the ensemble model, enabling the identification of a
specific class. This replaces the complex illustration of the DPG, especially
when we are focused on visualising a single class and dealing with many tree
base learners. We summarised the utility of discussed properties and metrics
in Table 1.
Table 1: Summary of Constraints, Betweenness Centrality (BC), Local Reaching Centrality (LRC), and Community, featuring provided definitions and their utility in offering insights into tree-based ensemble models. Property | Definition | Utility
---|---|---
Constraints | The intervals of values for each feature obtained from all predicates connected by a path that culminates in a given class. | Calculate the classification boundary values of each feature associated with each class.
BC | Quantifies the fraction of all the shortest paths between every pair of nodes of the graph passing through the considered node. | Identify potential bottleneck nodes that correspond to crucial decisions.
LRC | Quantifies the proportion of other nodes reachable from the local node through its outgoing edges. | Assess the importance of nodes similarly to feature importance, but enrich the information by encompassing the values associated with features across all decisions.
Community | A subset of nodes of the DPG which is characterised by dense interconnections between its elements and sparse connections with the other nodes of the DPG that do not belong to the community. | Understanding the characteristics of nodes to be assigned to a particular community class, identifying predominant predicates, and those that play a marginal role in the classification process.
## 4 Empirical Results and Discussion
In this section, we demonstrate the effectiveness of DPG to the well-known
Iris dataset [8] and a synthetic multiclass
dataset222https://github.com/sbarbonjr/fhg/tree/main/datasets. Each experiment
in this section was conducted using the RF classifier, with variations limited
to the number of tree base learners. The implementation is available
here333https://github.com/sbarbonjr/fhg. Finally, we discuss potential
enhancements to DPG and explore further development opportunities.
### 4.1 DPG: Iris insights
The first case study concerns the classification of the Iris dataset [8]. The
simplicity, manageability, versatility, and relevance of this dataset make it
an interesting and relevant resource for discussions and demonstrations of
interpretability in ML. The dataset comprises measurements of sepals and
petals for iris flowers encompassing three distinct species with a total of
four features and three classes. The RF was selected as the tree-based
ensemble model due to its well-established reputation and high-performance
capabilities. To conduct the classification, we partitioned the dataset into
training and test sets, following a $80$-$20$% proportion, respectively. A
seed value of $42$ was established for randomness control, and the number of
base tree learners was set to $5$. The RF performances, evaluated on the test
set, are summarised in the confusion matrix in Table 2. The model demonstrates
$100$% accuracy.
Table 2: Confusion matrix depicting the performance evaluation of the RF model with $5$ base tree learners on the test set. | Prediction
---|---
Ground truth | Class 0 | Class 1 | Class 2
Class 0 | 19 | 0 | 0
Class 1 | 0 | 13 | 0
Class 2 | 0 | 0 | 13
After training the model, we applied the algorithm outlined in Section 3.2 to
obtain the DPG, which can be visualised in Figure 1.
Figure 1: DPG of the RF composed of $5$ tree base learners trained on Iris
dataset.
Then, we can analyse the obtained graph using the metrics and algorithms
proposed in Section 3.3, verifying their utility and effectiveness in gaining
insights into the model. It is important to note that the DPG leads to the
calculation of both global metrics, referring to the overall graph, and
metrics at the level of individual nodes.
To illustrate the effectiveness and one of the advantages of employing DPG, we
highlight the constraints for the different classes in the Table 3. The class-
specific constraints delineate the necessary characteristics a sample must
exhibit to be assigned to that particular class by the tree-based ensemble
model. This insight contributes to a better understanding of how the model
utilises features for effective classification.
Table 3: Constraints for each class based on the DPG for an RF model within $5$ tree base learners. Class | Constraints
---|---
0 | petal width (cm) $\leq$ 1.65
petal length (cm) $\leq$ 2.50
1 | 5.25 < sepal length (cm) $\leq$ 6.05
0.80 < petal width (cm) $\leq$ 1.75
2.45 < petal length (cm) $\leq$ 5.35
2.75 < sepal width (cm) $\leq$ 2.90
2 | 5.75 < sepal length (cm) $\leq$ 6.05
0.80 < petal width (cm) $\leq$ 1.75
2.45 < petal length (cm) $\leq$ 5.35
2.75 < sepal width (cm) $\leq$ 3.10
The first metric under discussion is the BC of the nodes, as depicted in Table
4, where we identify potential bottleneck nodes. These nodes encapsulate
significant information, particularly representing decisions made by numerous
tree base learners. We quickly discern that the decision associated with petal
length (cm) and the value $4.85$ is pivotal, as it is frequently relied upon
by multiple basic learners and is essential for successful classification.
Table 4: Top eight predicates by evaluating their BC obtained from the DPG based on an RF model consisting of $5$ tree base learners. Predicate | BC
---|---
petal length (cm) > 4.85 | 0.053
petal length (cm) <= 4.85 | 0.036
petal width (cm) > 1.55 | 0.034
sepal length (cm) <= 6.05 | 0.032
petal length (cm) > 4.95 | 0.028
petal length (cm) > 4.65 | 0.022
petal width (cm) <= 1.75 | 0.022
petal width (cm) <= 1.55 | 0.021
Furthermore, the LRC metric provides additional information. Examining Table
5(a), we can emphasise the most crucial predicates influencing the decision-
making process of the tree-based ensemble model. This includes not only
identifying the most frequently used features but also recognising the
associated values that lead to significant and divisive splits in the dataset
across various basic learners. As observed in Table 5, a comparison between
the LRC of the nodes and the FI, calculated on the same model on which DPG is
based, suggests that the metric may rank the predicates similarly. FI is
calculated using the _Mean Decrease Impurity_ (MDI) algorithm introduced by
Scornet [44]. This comparison also provides additional information about the
values used in the decisions and the frequency of paths extending the concept
of FI.
Table 5: Comparison of the top eight predicates by evaluating their LRC
obtained from the DPG based on an RF model consisting of $5$ tree base
learners (Table 5(a)), alongside the FI of the same model (Table 5(b))
calculated exploiting MDI algorithm.
(a) LRC evaluation Predicate | LRC
---|---
petal width (cm) <= 1.65 | 1.531
petal length (cm) > 2.45 | 0.919
petal width (cm) > 0.80 | 0.874
petal length (cm) > 2.50 | 0.699
petal width (cm) > 1.65 | 0.618
petal length (cm) <= 5.20 | 0.565
petal width (cm) > 1.55 | 0.540
sepal length (cm) > 5.75 | 0.332
(b) FI evaluation Feature | FI
---|---
petal length (cm) | 0.550
petal width (cm) | 0.373
sepal length (cm) | 0.054
sepal width (cm) | 0.023
By employing the global metric community, we identified the presence of three
distinct communities. Table 6 illustrates the association between each
community and a distinct class obtained by applying the asynchronous label
propagation algorithm to the DPG. We can affirm that each node within a
community contains decisions that significantly contribute to the accurate
classification of samples belonging to a specific class. For instance, when
applying the predicates of Community 3 (comprising two features and two
predicates) to the test set and traversing from the root node, it achieved
$100$% accuracy for Class 0, the class delineated in the mentioned community.
Table 6: Communities obtained from the DPG based on an RF model composed of $5$ tree base learners. The table shows the number of predicates belonging to each community, the number of features in the community nodes, and the class involved in each community. Community | # Predicates | # Features | Class
---|---|---|---
Community 1 | 23 | 4 | 1
Community 2 | 18 | 4 | 2
Community 3 | 3 | 2 | 0
Finally, upon comparing the Table 6 and the Figure 2, we can state that the
communities facilitate the comprehension of how the model addresses the
classification problem. Examining the number of decisions and features
utilised in each community reveals that differentiating between Class 1
(Community 1) and Class 2 (Community 2) poses a greater challenge for the
model. This indicates the difficulty in effectively separating samples within
the dataset into their respective classes. This difficulty becomes apparent
when visualising the dataset across the features, as in Figure 2. Conversely,
the community encompassing Class 0 (Community 3) consists of fewer predicates,
signifying that it is more distinguishable from other classes, as confirmed in
the Figure 2.
Figure 2: Two-dimensional depiction of the Iris dataset, employing feature
pairs in each graph for visual representation.
### 4.2 Comparing to the Graph-based Solutions
As outlined in the Section 2, Gossen and Steffen [7], Murtovi et al. [32]
conducted studies on the interpretability of tree-based ensemble models
exploiting graph structures. We compared DPG with their proposed method by
examining their outcomes and potential insights. Using the same tree-based
ensemble model we studied in Section 4.1 as input, we generated the ADD
displayed in Figure 3. The first noticeable distinction from DPG lies in the
ADD structure, as it forms a bipartite graph. We can observe that the ADD is
generated from the trained model, albeit without fully leveraging the training
dataset. Consequently, the evaluation of connections between nodes and the
assessment of the significance of decisions made by different tree base
learners are not fully exploited. This implies that each branch carries equal
weight and impact in the diagram, and a classification error significantly
influences the structure by introducing an incorrect path.
Figure 3: ADD of an RF model with $5$ tree base learners induced for Iris
dataset.
Another crucial difference is that ADD does not provide graph metrics, leaving
the user the interpretation of the diagram and potentially missing out on
relevant information. Moreover, an additional limitation, as indicated in the
studies by [7, 32], involves the challenge that emerges when generating ADDs
and dealing with large ensembles. Visualisation becomes intricate, even with a
modest count of $20$ tree base learners. In contrast, DPG allows the
computation of both global and local metrics, even with a higher number of
tree base learners.
To further examine these aspects, we use two RF models, one with $20$ tree
learners and the other with $100$, to analyse a complex multiclass problem
with a dataset comprising $4$ classes, $1000$ samples, and $16$ features. This
introduces a four-class problem that was randomly generated. The dataset was
created using the make_classification444https://scikit-
learn.org/stable/modules/generated/sklearn.datasets.make_classification.html
function from scikit-learn. The following setup has been maintained for
training both models. We divided the dataset into training and test sets, with
an $80$-$20$% ratio. We fixed a seed value of $42$ for randomness control to
ensure reproducibility. The RF performances, assessed on the test set, are
summarised in the confusion matrices presented in Table 7. The model with
$100$ tree base learners shows better performance in every parameter, reaching
an overall accuracy of $66$%, outperforming the model with $20$ tree base
learners, which barely reaches $58$%.
Table 7: Confusion matrices of the RF models with $20$ tree base learners (RF 20) and with $100$ tree base learners (RF 100) tested on the synthetic dataset. | Prediction (RF 20) | Prediction (RF 100)
---|---|---
Ground truth | Class 0 | Class 1 | Class 2 | Class 3 | Class 0 | Class 1 | Class 2 | Class 3
Class 0 | 38 | 4 | 12 | 6 | 38 | 4 | 14 | 4
Class 1 | 5 | 31 | 3 | 5 | 4 | 32 | 2 | 6
Class 2 | 11 | 2 | 29 | 2 | 5 | 2 | 33 | 4
Class 3 | 10 | 13 | 10 | 19 | 9 | 9 | 5 | 29
We emphasise that even in this context, DPG is a useful tool. Both DPG and ADD
present intricate visualisations with $20$ tree base learners. However, DPG
overcomes this obstacle by providing metrics that can still offer valid
insights into the model. The first insight is displayed in Table 8, where we
provide constraints for the four classes of the dataset. Constraints, even in
this complex scenario, allow the visualisation of intervals where sample
features should be situated for precise classification into their respective
classes.
Table 8: Constraints for each class based on the DPG for an RF model within $100$ tree base learners. Class 0 | Class 1 | Class 2 | Class 3
---|---|---|---
$-5.87<\texttt{F1}\leq 5.74$ | $-5.79<\texttt{F1}\leq 5.72$ | $-5.76<\texttt{F1}\leq 5.72$ | $-5.76<\texttt{F1}\leq 5.72$
$-2.64<\texttt{F2}\leq 2.63$ | $-2.61<\texttt{F2}\leq 2.61$ | $-2.61<\texttt{F2}\leq 2.61$ | $-2.61<\texttt{F2}\leq 2.63$
$-5.24<\texttt{F3}\leq 3.75$ | $-5.38<\texttt{F3}\leq 3.75$ | $-5.24<\texttt{F3}\leq 3.75$ | $-5.24<\texttt{F3}\leq 3.75$
$-4.80<\texttt{F4}\leq 4.37$ | $-5.15<\texttt{F4}\leq 4.37$ | $-4.80<\texttt{F4}\leq 4.37$ | $-4.80<\texttt{F4}\leq 4.37$
$-2.61<\texttt{F5}\leq 2.44$ | $-2.61<\texttt{F5}\leq 2.71$ | $-2.61<\texttt{F5}\leq 2.44$ | $-2.61<\texttt{F5}\leq 2.44$
$-2.29<\texttt{F6}\leq 2.58$ | $-2.29<\texttt{F6}\leq 2.58$ | $-2.29<\texttt{F6}\leq 2.58$ | $-2.29<\texttt{F6}\leq 2.58$
$-2.82<\texttt{F7}\leq 2.47$ | $-2.82<\texttt{F7}\leq 2.47$ | $-2.82<\texttt{F7}\leq 2.47$ | $-2.82<\texttt{F7}\leq 2.86$
$-4.62<\texttt{F8}\leq 4.74$ | $-4.62<\texttt{F8}\leq 4.74$ | $-4.62<\texttt{F8}\leq 4.74$ | $-4.62<\texttt{F8}\leq 4.74$
$-2.40<\texttt{F9}\leq 2.59$ | $-2.40<\texttt{F9}\leq 2.59$ | $-2.40<\texttt{F9}\leq 2.71$ | $-2.40<\texttt{F9}\leq 2.59$
$-4.71<\texttt{F10}\leq 4.32$ | $-4.71<\texttt{F10}\leq 4.32$ | $-4.71<\texttt{F10}\leq 5.43$ | $-4.71<\texttt{F10}\leq 4.32$
$-2.87<\texttt{F11}\leq 2.77$ | $-2.87<\texttt{F11}\leq 2.77$ | $-2.87<\texttt{F11}\leq 2.86$ | $-2.87<\texttt{F11}\leq 2.77$
$-2.42<\texttt{F12}\leq 2.37$ | $-2.42<\texttt{F12}\leq 2.37$ | $-2.42<\texttt{F12}\leq 2.37$ | $-2.42<\texttt{F12}\leq 2.37$
$-4.28<\texttt{F13}\leq 5.01$ | $-4.28<\texttt{F13}\leq 5.01$ | $-4.28<\texttt{F13}\leq 5.01$ | $-4.28<\texttt{F13}\leq 5.01$
$-7.31<\texttt{F14}\leq 8.33$ | $-7.31<\texttt{F14}\leq 8.33$ | $-7.31<\texttt{F14}\leq 8.33$ | $-7.31<\texttt{F14}\leq 8.33$
$-4.46<\texttt{F15}\leq 4.19$ | $-4.46<\texttt{F15}\leq 4.19$ | $-4.46<\texttt{F15}\leq 4.19$ | $-4.97<\texttt{F15}\leq 4.5$
$-4.70<\texttt{F16}\leq 4.21$ | $-4.70<\texttt{F16}\leq 4.21$ | $-4.70<\texttt{F16}\leq 4.52$ | $-4.70<\texttt{F16}\leq 4.21$
The BC metric helps identifying potential bottleneck nodes. Upon observing
Table 9(a), we can see that there is not a large difference between the BC
values associated with the predicates. Therefore, we can assume that there are
no bottleneck nodes.
Examining the information provided in Table 9(b), the LRC underscores which
predicates significantly impact the decision-making process of the ensemble
model. The predicates F7 <= 1.62 and F1 <= $3.1$ are identified as
particularly crucial.
Table 9: Top eight predicates by evaluating their BC (Table 9(a)), and top
eight predicates by evaluating their LRC (Table 9(b)), both obtained from the
DPG based on an RF model consisting of $100$ tree base learners.
(a) BC evaluation Predicate | BC
---|---
F15 > 1.17 | 0.018
F15 <= 1.61 | 0.015
F12 > 0.20 | 0.015
F12 > 0.41 | 0.014
F4 <= 0.33 | 0.014
F8 > 0.36 | 0.014
F1 <= -1.10 | 0.013
F11 <= 1.16 | 0.013
(b) LRC evaluation Predicate | LRC
---|---
F7 <= 1.62 | 15.812
F1 <= 3.10 | 14.475
F14 > -1.78 | 13.313
F4 > -2.97 | 13.158
F5 > -1.92 | 13.065
F4 > -1.36 | 12.989
F1 <= 2.49 | 12.986
F13 > 1.98 | 12.920
Another insight can be obtained by employing the global metric community. In
this scenario, we identified the presence of four distinct communities,
displayed in Table 10. We note that each community contains a distinct class.
Furthermore, upon observing the table, we can conclude that each community
exhibits a high number of involved features and predicates, confirming the
complexity of the classification problem.
Table 10: Communities obtained from an RF model composed of $100$ tree base learners. The table shows the number of predicates belonging to each community, the number of features in the community nodes, and the class involved in each community. Community | # Predicates | # Features | Class
---|---|---|---
Community 1 | 7767 | 16 | 2
Community 2 | 2149 | 16 | 0
Community 3 | 2351 | 16 | 3
Community 3 | 2100 | 16 | 1
### 4.3 Potential Improvements
Several avenues await exploration in the future. The primary aim is to reduce
the computational cost of DPG, as many real-world problems involve large
datasets that do not scale well with the current implementation of DPG.
Expanding the application scope of DPG is another key goal, including its
utility in explaining models relevant to regression-type problems. Given DPG’s
applicability to any model and dataset, we aim to introduce new tests and use
cases to delve deeper into the method. This includes proposing applications to
novel datasets and exploring their compatibility with other tree-based
ensemble models. Furthermore, while this paper introduces certain metrics and
algorithms derived from graph theory, the field offers extensive possibilities
for future exploration. In the future, we plan to introduce new tools
associated with DPG to enhance the interpretation of tree-based ensemble
models.
## 5 Conclusion
In this paper, we introduced Decision Predicate Graphs (DPG) as a novel model-
agnostic tool for tree-based ensemble interpretability. DPG is obtained from a
trained model and data, ensuring the maintenance of its performance. The
concept behind DPG is to convert an opaque-box tree-based ensemble model into
an enriched graph. DPG enables qualitative measures and the identification of
its predicates, and facilitates comparisons between features and their
associated values, offering insights into the entire model. In particular, we
introduced Betweenness Centrality, Local Reaching Centrality, Community and
Constraints as useful metrics and properties towards improving and extending
the XAI interpretability approaches. While DPG is still considered an evolving
work, its potential is substantial, given the robust underlying theory and the
versatility of the tool. Furthermore, the effervescent research on graphs,
knowledge graphs, and complex networks might strengthen the possibilities
grounded in DPG. As the next step of our current research, we expect to apply
DPG to provide local interpretability and also contribute to improving the
interpretation of regression tasks.
## References
* [1] Arman Malekloo, Ekin Ozer, Mohammad AlHamaydeh, and Mark Girolami. Machine learning and structural health monitoring overview with emerging technology and high-dimensional data source highlights. 21(4):1906–1955. ISSN 1475-9217. doi: 10.1177/14759217211036880. Publisher: SAGE Publications.
* [2] Ibomoiye Domor Mienye and Yanxia Sun. A survey of ensemble learning: Concepts, algorithms, applications, and prospects. 10:99129–99149. ISSN 2169-3536. doi: 10.1109/ACCESS.2022.3207287. Conference Name: IEEE Access.
* [3] Trevor Hastie, Robert Tibshirani, and Jerome Friedman. Additive models, trees, and related methods. In Trevor Hastie, Robert Tibshirani, and Jerome Friedman, editors, _The Elements of Statistical Learning: Data Mining, Inference, and Prediction_ , Springer Series in Statistics, pages 295–336. Springer.
* Chimatapu et al. [2018] Ravikiran Chimatapu, Hani Hagras, Andrew Starkey, and Gilbert Owusu. Explainable ai and fuzzy logic systems. In _Theory and Practice of Natural Computing: 7th International Conference, TPNC 2018, Dublin, Ireland, December 12–14, 2018, Proceedings 7_ , pages 3–20. Springer, 2018.
* [5] Tin Kam Ho. Random decision forests. In _Proceedings of 3rd International Conference on Document Analysis and Recognition_ , volume 1, pages 278–282 vol.1. doi: 10.1109/ICDAR.1995.598994.
* [6] Leo Breiman. Random forests. 45(1):5–32. ISSN 1573-0565. doi: 10.1023/A:1010933404324.
* [7] Frederik Gossen and Bernhard Steffen. Algebraic aggregation of random forests: towards explainability and rapid evaluation. 25(3):267–285. ISSN 1433-2787. doi: 10.1007/s10009-021-00635-x.
* Fisher [1936] Ronald A Fisher. The use of multiple measurements in taxonomic problems. _Annals of eugenics_ , 7(2):179–188, 1936.
* Guidotti et al. [2018] Riccardo Guidotti, Anna Monreale, Salvatore Ruggieri, Franco Turini, Fosca Giannotti, and Dino Pedreschi. A survey of methods for explaining black box models. _ACM Comput. Surv._ , 51(5), aug 2018. ISSN 0360-0300. doi: 10.1145/3236009.
* [10] Ambreen Hanif, Xuyun Zhang, and Steven Wood. A survey on explainable artificial intelligence techniques and challenges. In _2021 IEEE 25th International Enterprise Distributed Object Computing Workshop (EDOCW)_ , pages 81–89. doi: 10.1109/EDOCW52865.2021.00036. ISSN: 2325-6605.
* [11] Amina Adadi and Mohammed Berrada. Peeking inside the black-box: A survey on explainable artificial intelligence (XAI). 6:52138–52160. ISSN 2169-3536. doi: 10.1109/ACCESS.2018.2870052. Conference Name: IEEE Access.
* [12] Rudresh Dwivedi, Devam Dave, Het Naik, Smiti Singhal, Rana Omer, Pankesh Patel, Bin Qian, Zhenyu Wen, Tejal Shah, Graham Morgan, and Rajiv Ranjan. Explainable AI (XAI): Core ideas, techniques, and solutions. 55(9):194:1–194:33. ISSN 0360-0300. doi: 10.1145/3561048.
* [13] Scott M. Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In _Proceedings of the 31st International Conference on Neural Information Processing Systems_ , NIPS’17, pages 4768–4777. Curran Associates Inc. ISBN 978-1-5108-6096-4.
* Ribeiro et al. [a] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. "why should i trust you?": Explaining the predictions of any classifier. In _Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , KDD ’16, pages 1135–1144. Association for Computing Machinery, a. ISBN 978-1-4503-4232-2. doi: 10.1145/2939672.2939778.
* [15] Jerome H. Friedman. Greedy function approximation: A gradient boosting machine. 29(5):1189–1232. ISSN 0090-5364. URL https://www.jstor.org/stable/2699986. Publisher: Institute of Mathematical Statistics.
* Ribeiro et al. [b] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Anchors: High-precision model-agnostic explanations. 32(1), b. ISSN 2374-3468. doi: 10.1609/aaai.v32i1.11491. Number: 1.
* [17] H. Chipman, Edward George, and Robert McCulloch. Making sense of a forest of trees. 29\.
* [18] Maissae Haddouchi and Abdelaziz Berrado. A survey of methods and tools used for interpreting random forest. In _2019 1st International Conference on Smart Systems and Data Science (ICSSD)_ , pages 1–6. doi: 10.1109/ICSSD47982.2019.9002770.
* [19] Massimo Aria, Corrado Cuccurullo, and Agostino Gnasso. A comparison among interpretative proposals for random forests. 6:100094. ISSN 2666-8270. doi: 10.1016/j.mlwa.2021.100094.
* [20] Morteza Mashayekhi and Robin Gras. Rule extraction from random forest: the RF+HC methods. In Denilson Barbosa and Evangelos Milios, editors, _Advances in Artificial Intelligence_ , Lecture Notes in Computer Science, pages 223–237. Springer International Publishing. ISBN 978-3-319-18356-5. doi: 10.1007/978-3-319-18356-5_20.
* [21] Satoshi Hara and Kohei Hayashi. Making tree ensembles interpretable: A bayesian model selection approach. In _Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics_ , pages 77–85. PMLR. ISSN: 2640-3498.
* [22] Xun Zhao, Yanhong Wu, Dik Lun Lee, and Weiwei Cui. iForest: Interpreting random forests via visual analytics. 25(1):407–416. ISSN 1941-0506. doi: 10.1109/TVCG.2018.2864475. Conference Name: IEEE Transactions on Visualization and Computer Graphics.
* [23] Julian Hatwell, Mohamed Medhat Gaber, and R. Muhammad Atif Azad. CHIRPS: Explaining random forest classification. 53(8):5747–5788. ISSN 1573-7462. doi: 10.1007/s10462-020-09833-6.
* [24] Mário Popolin Neto and Fernando V. Paulovich. Explainable matrix - visualization for global and local interpretability of random forest classification ensembles. 27(2):1427–1437. ISSN 1077-2626. doi: 10.1109/TVCG.2020.3030354. Publisher: IEEE Computer Society.
* [25] Klest Dedja, Felipe Kenji Nakano, Konstantinos Pliakos, and Celine Vens. BELLATREX: Building explanations through a LocaLly AccuraTe rule EXtractor. 11:41348 – 41367. ISSN 2169-3536. doi: 10.1109/ACCESS.2023.3268866.
* Van Assche and Blockeel [2007] Anneleen Van Assche and Hendrik Blockeel. Seeing the forest through the trees: Learning a comprehensible model from an ensemble. In _Machine Learning: ECML 2007: 18th European Conference on Machine Learning, Warsaw, Poland, September 17-21, 2007. Proceedings 18_ , pages 418–429. Springer, 2007.
* Zhou and Hooker [2016] Yichen Zhou and Giles Hooker. Interpreting models via single tree approximation, 2016.
* [28] Houtao Deng. Interpreting tree ensembles with inTrees. 7(4):277–287. ISSN 2364-4168. doi: 10.1007/s41060-018-0144-8.
* [29] Bogdan Gulowaty and Michał Woźniak. Extracting interpretable decision tree ensemble from random forest. In _2021 International Joint Conference on Neural Networks (IJCNN)_ , pages 1–8. doi: 10.1109/IJCNN52387.2021.9533601. ISSN: 2161-4407.
* [30] Hiroki Nakahara, Akira Jinguji, Simpei Sato, and Tsutomu Sasao. A random forest using a multi-valued decision diagram on an FPGA. In _2017 IEEE 47th International Symposium on Multiple-Valued Logic (ISMVL)_ , pages 266–271. doi: 10.1109/ISMVL.2017.40. ISSN: 2378-2226.
* [31] Olavo A. B. Silva, Alysson K. C. Silva, Ícaro G. S. Moreira, José A. M. Nacif, and Ricardo S. Ferreira. RDSF: Everything at same place all at once - a random decision single forest. In _Anais do Simpósio Brasileiro de Engenharia de Sistemas Computacionais (SBESC)_ , pages 31–36. SBC. ISSN: 2237-5430.
* [32] Alnis Murtovi, Alexander Bainczyk, Gerrit Nolte, Maximilian Schlüter, and Bernhard Steffen. Forest GUMP: a tool for verification and explanation. 25(3):287–299. ISSN 1433-2787. doi: 10.1007/s10009-023-00702-5.
* Oliver [1993] Jonathan J. Oliver. Decision graphs - an extension of decision trees. 1993\.
* [34] Dmitry Ignatov and Andrey Ignatov. Decision stream: Cultivating deep decision trees. In _2017 IEEE 29th International Conference on Tools with Artificial Intelligence (ICTAI)_ , pages 905–912. doi: 10.1109/ICTAI.2017.00140. ISSN: 2375-0197.
* [35] Scott Needham and David L. Dowe. Message length as an effective ockham’s razor in decision tree induction. In _International Workshop on Artificial Intelligence and Statistics_ , pages 216–223. PMLR. ISSN: 2640-3498.
* [36] Peter J. Tan and David L. Dowe. MML inference of decision graphs with multi-way joins and dynamic attributes. In Tamás (Tom) Domonkos Gedeon and Lance Chun Che Fung, editors, _AI 2003: Advances in Artificial Intelligence_ , Lecture Notes in Computer Science, pages 269–281. Springer. ISBN 978-3-540-24581-0. doi: 10.1007/978-3-540-24581-0_23.
* [37] Alexandre M. Florio, Pedro Martins, Maximilian Schiffer, Thiago Serra, and Thibaut Vidal. Optimal decision diagrams for classification. 37(6):7577–7585. ISSN 2374-3468. doi: 10.1609/aaai.v37i6.25920. Number: 6.
* [38] Bingzhao Zhu and Mahsa Shoaran. Tree in tree: from decision trees to decision graphs. In _Advances in Neural Information Processing Systems_ , volume 34, pages 13707–13718. Curran Associates, Inc.
* [39] Christopher Stewart Wallace. _Statistical and Inductive Inference by Minimum Message Length_. Information Science and Statistics. Springer-Verlag. ISBN 978-0-387-23795-4. doi: 10.1007/0-387-27656-4.
* [40] Ulrik Brandes. On variants of shortest-path betweenness centrality and their generic computation. 30(2):136–145. ISSN 0378-8733. doi: 10.1016/j.socnet.2007.11.001.
* [41] Enys Mones, Lilla Vicsek, and Tamás Vicsek. Hierarchy measure for complex networks. 7(3):e33799. ISSN 1932-6203. doi: 10.1371/journal.pone.0033799. Publisher: Public Library of Science.
* [42] Filippo Radicchi, Claudio Castellano, Federico Cecconi, Vittorio Loreto, and Domenico Parisi. Defining and identifying communities in networks. 101(9):2658–2663. doi: 10.1073/pnas.0400054101. Publisher: Proceedings of the National Academy of Sciences.
* [43] Usha Nandini Raghavan, Réka Albert, and Soundar Kumara. Near linear time algorithm to detect community structures in large-scale networks. 76(3):036106. doi: 10.1103/PhysRevE.76.036106. Publisher: American Physical Society.
* [44] Erwan Scornet. Trees, forests, and impurity-based variable importance.
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# A Precision Diagnostic Framework of Renal Cell Carcinoma on Whole-Slide
Images using Deep Learning
Jialun Wu1, Haichuan Zhang1,2,Zeyu Gao1, Xinrui Bao3, Tieliang Gong1, Chunbao
Wang4, and Chen Li1,∗ 1School of Computer Science and Technology, Xi’an
Jiaotong University, Xi’an, China
National Engineering Lab for Big Data Analytics, Xi’an Jiaotong University,
Xi’an, China 2School of Electrical Engineering and Computer Science,
Pennsylvania State University, University Park, PA, USA 3School of Automation
Science and Engineering, Xi’an Jiaotong University, Xi’an, China 4Department
of Pathology, the First Affiliated Hospital of Xi’an Jiaotong University,
Xi’an, China
Email<EMAIL_ADDRESS>
###### Abstract
Diagnostic pathology, which is the basis and gold standard of cancer
diagnosis, provides essential information on the prognosis of the disease and
vital evidence for clinical treatment. Tumor region detection, subtype and
grade classification are the fundamental diagnostic indicators for renal cell
carcinoma (RCC) in whole-slide images (WSIs). However, pathological diagnosis
is subjective, differences in observation and diagnosis between pathologists
is common in hospitals with inadequate diagnostic capacity. The main challenge
for developing deep learning based RCC diagnostic system is the lack of large-
scale datasets with precise annotations. In this work, we proposed a deep
learning-based framework for analyzing histopathological images of patients
with renal cell carcinoma, which has the potential to achieve pathologist-
level accuracy in diagnosis. A deep convolutional neural network (InceptionV3)
was trained on the high-quality annotated dataset of The Cancer Genome Atlas
(TCGA) whole-slide histopathological image for accurate tumor area detection,
classification of RCC subtypes, and ISUP grades classification of clear cell
carcinoma subtypes. These results suggest that our framework can help
pathologists in the detection of cancer region and classification of subtypes
and grades, which could be applied to any cancer type, providing auxiliary
diagnosis and promoting clinical consensus.
## I Introduction
Renal cell carcinoma (RCC) is one of the most common types of renal cancer,
accounting for 80% of early renal cancer. Renal cancer is the third most
common urological tumor after prostate cancer, and bladder cancer [1]. With
the development of medical imaging, the detection rate of early kidney cancer
is gradually increasing. The most common histopathological type of renal cell
carcinoma is clear cell carcinoma (ccRCC), papillary renal cell carcinoma
(pRCC), and chromophobe cell carcinoma (chRCC) [2]. In the pathological
diagnosis, subtypes and grades of renal cell carcinoma are the critical
diagnostic results, different subtypes of renal cell carcinoma can be treated
with different regimens (including chemotherapy and targeted therapy), the
higher the grading, the worse the prognosis, the greater the possibility of
recurrence and tumor metastasis, and the faster the course of the disease.
Pathology is the foundation and gold standard in the medical field. The
pathological diagnosis of tumor tissue by pathologists is the basis of
treatments and the cornerstone of clinical and drug research. Most carcinoma
identification requires microscopy-level image assessment for early tumor
discovery and for developing therapies based on diagnostic pathology [3].
Histopathological images have long been central to cancer diagnosis, staging,
and prognosis, and pathologists widely use them in clinical practice.
Diagnosing pathology slides is a complex task that requires years of
pathologist training, digital pathology slides are obtained at very high
resolution. However, with the rapid growth of cancer patients, the technical
requirements for pathologists also increase. At present, pathological
diagnosis is confronted with such problems as uneven distribution of medical
resources, huge workload of pathologists, uneven level and serious shortage of
doctors engaged in pathological work [4]. The diagnosis and treatment of some
diseases often lack precise pathological examination results and are only
based on speculation with incomplete evidence. It takes a significant amount
of time to train a pathologist and even longer for pathologist to become
experienced and be able to diagnose different types of tumor.
As a new tool in the field of pathology, artificial intelligence uses
intelligent pathology-assisted diagnosis technology to collect, manage and
analyze pathological information, which can help pathologists reduce a lot of
workload, effectively improve the efficiency and accuracy of pathological
diagnosis,provide better patient treatment and good support in clinical
teaching of pathology. The predictive models used by traditional medical
image-assisted diagnostic systems rely on the features extracted manually by
the pathologists, but the performance is often inadequate for clinical
practice. Deep learning (DL) is a powerful method for tumor region detection,
subtypes and grades classification of the whole-slide images in digital
pathology [5]. It has demonstrated high accuracy and universality in
diagnosing the most common human cancers, such as kidney cancer [6], lung
cancer [7], prostate cancer [8], colon cancer [9], breast cancer [10]. Deep
learning with convolutional neural networks (CNNs) has been shown to be a
powerful algorithm for prompting biomedical image analysis. There have been
works using convolutional neural networks [11] for tumor region detection and
cell segmentation [12, 13], with gridding the whole-slide images into small
patches for processing. A major obstacle to applying deep learning-based
algorithms in the analysis of pathological images is the lack of well-
annotated training datasets, which require professional pathologists to make a
large number of annotations at both patch level and pixel level.
To tackle these problems, we curated a set of pathology images from the TCGA
(the cancer genome atlas) project [14]. We scope our work to the renal cell
carcinoma (RCC) which is one of the most common malignant tumors in adult
kidney type, which has three common histologic subtypes in the dataset: clear
cell (TCGA-KIRC), papillary (TCGA-KIRP), and chromophobe (TCGA-KICH). In order
to validate the robustness of our framework, we apply the same technique to
the independent renal cell carcinoma dataset from the local hospital as well.
Each case in the dataset has its corresponding pathological report and
clinical information. The reports are made after pathologists observed the
histopathological sections stained with hematoxylin and eosin (H&E) [15] under
a microscope to make the diagnoses by inspecting the morphological
characteristics of histology slides. With the pathology report,
clinicians—with overall understanding of patients’ symptoms—can make informed
and precise decisions about treatments. We can verify the prediction results
of the model with the corresponding content of the pathology report to test
the accuracy of our framework [16].
In this work, we first constructed large, high-quality, fine-grained annotated
renal cancer datasets to address the lack of large, accurately annotated
datasets in the study of digital pathology research. Based on this dataset, a
deep learning model is used to detect cancer regions, classify tumor subtype
and grade on the whole-slide image, and finally generate a precise whole-case
report. We test our framework on the dataset from the hospital and invite
experienced pathologists to help additional validation. Our work could
fundamentally alleviate the problem of long training cycles for pathologists,
and further research will help demonstrate that the same approach for other
types of cancer (such as prostate cancer, lung cancer).
The organization of this paper is as follows: the second part discusses the
dataset of pathological images and describes our experimental process in
detail. In the third part, the results section introduces the performance of
our framework, and in the conclusion part, we discuss the experimental results
and the future work.
## II Materials and Methods
In order to detect cancer regions, classify cancer subtypes and cancer grades
on the whole-slide images, and finally generate a whole-case report, a group
of population with similar histology in different dataset project is needed
for each case.
### II-A Data sources and annotations
We selected renal cell carcinoma in our study and evaluated our frameworks on
two datasets. It is worth mentioning that the whole-slide images we used here
are the digital diagnostic slides of each patient, each patient was given a
corresponding pathological image. Our dataset comes from NCI Genomic Data
Commons [17], which provides an online research platform for uploading,
searching, viewing and downloading cancer-related data. All free whole-slide
images of kidney cancer are uploaded from this source. The first dataset
derived from the TCGA database we name as TCGA RCC. This dataset we used
contains three TCGA projects (i.e., KIRC, KIRP, KICH) and totally has 667
WSIs. All of the whole-slide images were scanned at 40x magnification, and the
data were cleaned by experienced pathologists before they were annotated to
remove the blurred, non-cancerous images.
As part of this work, we introduced a new RCC dataset which is obtained from
the Department of Pathology, the First Affiliated Hospital of Xi’an Jiaotong
University, China. We name as LH RCC. We totally collected 632 WSIs from 153
patients in hospital, which is about 50 patients for each subtype. These WSIs
are paraffin-embedded tissue sections and scanned at both 20x and 40x
magnification by the digital slide scanner of KFBIO. The labels provided by
the TCGA and Local hospital datasets were used as our gold standard. Those
labels were the result of a consensus as explained by the data curator: first,
the submitting institutions were asked to review each sample prior sending it
to confirm the diagnosis. Then, a slide from the sample was reviewed by an
expert pathologist. In the event of a disagreement, the slide would be
reviewed by one or more other expert pathologists.
Figure 1: Introduction of the two datasets. (a) The bar chart shows and counts
the detailed of different grades and different subtypes in the two datasets.
(b) The original digital pathological whole-slide image, a whole-slide image
annotated by the pathologist, distinguishes the cancer area from the normal
area, and provides fine-grained annotations of subtypes and grades. (c)
Examples of patches in different types include three different subtypes and
four different grades. Figure 2: The convolution network structure of the
tumor region detection subtask in the framework and a visualization example of
each step. (a) The input of the network is different types of images (cancer
patch and normal patch) for training, 512*512px in size, with stain
normalization. (b) We used the Inception V3 from Google with initialized our
network parameters to the best parameter set that was achieved on ImageNet
competition. The output expresses the probability of different classes (Tumor
and normal), If the probability of patch currently predicted as cancer area is
less than 0.2 or greater than 0.8, the results will be output. (c) If the
probability of patch prediction as cancer area is between 0.2 and 0.8, three
different methods (Rotation and flip, magnification, neighbor ensemble) are
used for second judgment, and the final output is obtained by voting on the
three results. (d) The selected WSI staining was not ideal, with obvious
cutting edges and artifacts. (e) After visualizing output, it could be seen
that there were obvious ”wrong judgment” areas except the cancer areas. We
used three methods to make the second prediction and vote to integrate the
results. (f) The final tumor area map is generated by overlaying the
probability map on the WSI slide to facilitate the pathologist’s
interpretation of the detected tumor area.
### II-B Annotation principles
OpenHI [18] is an open-source annotation and processing software which
specially designed for histopathology images. Now, it supports several types
of annotation and can record the time of annotation. We performed all
operations including tumor region selection and region annotation on the
OpenHI platform. For the tumor region detection and subtype recognition
subtasks of this framework, two experienced pathologists first selected the
tumor area for the whole-slide images, and we could take patches in the normal
and tumor areas respectively. After selecting the tumor regions, pathologists
annotated 123, 88, and 46 WSIs for ccRCC, pRCC, and chRCC respectively in the
TCGA dataset, and four well-trained annotators were invited to make the
complete region annotation on the remaining WSIs for the tumor region and
three subtypes. For the dataset in the local hospital, we randomly selected
ten patients of each subtype, 180 WSIs in total, then asked two pathologists
to do the same annotation on these WSIs. It is worth mentioning that all the
cases in these two datasets have corresponding pathology reports and clinical
records which contain the diagnostic subtype labels and the grading score. All
the annotation we have done is to annotate the cancerous and non-cancerous
regions.
For the grades classification subtask in this framework, from the WHO
guideline we can know that, the renal cell in tumor area of the whole-slide
images are divided into 4 grades based on the ISUP grading standard [19]. This
indicator is particularly important in the diagnosis of clear cell carcinoma.
We chose 200 slides image of renal cell carcinoma and chose 10 diagnostic
areas at 40X magnification on each slide in both TCGARCC dataset and part of
the LHRCC dataset at 40x magnification. The size of each selected region is
512*512 pixels. Six rigorously trained annotators which were divided into two
groups to annotate the chosen regions. After completing the data annotation,
we compute the inter-rater reliability between different annotators in a
group. According to the interpretability of kappa statics [20], the confidence
of different annotators is at moderate level. In order to achieve high quality
annotation, the class of each instance was decided based on the decision of
the majority which consist of three annotators in one group. The details of
these two datasets are shown in Figure 1.
### II-C Stain normalization
The normalization of haematoxylin and eosin stain was made at the training and
validation stages using a reference image and the Macenko algorithm [21].
### II-D The proposed framework
Figure 2 depicts our data analysis workflow for tumor region classification
subtask, Figure 3 depicts our subtask for tumor subtype and grade
classification. With the following sections describing the information for
each major step in our framework.
### II-E Tumor and Non-tumor region classification
For the tumor area detection task, we used part of the annotated data in the
two datasets as the training data, and use the rest of the data as the
validation data. In each annotated WSI, 5 patches were selected for tumor
region and non-tumor region respectively as two different classes. After
staining normalization, the patches were input into the convolutional network
model. In the selection of the model, we trained several different image
classification neural networks (VGGs, ResNet, MobileNet and InceptionV3).
According to the result we find that Inception V3 provided the best
performance and was used for all the final experiments.
We used 80% of those patches for training, 20% for validation and testing. We
based our model on Inception V3. This architecture makes use of inception
modules. The initial 5 convolution nodes are combined with 2 max pooling
operations and followed by 11 stacks of inception modules. The architecture
ends with a fully-connected layer and then a SoftMax output layer. The last
three layers of the model are discarded, and then the result of the bottleneck
layer is used as the feature extraction result of the model. After obtaining
the feature vector, it needs to be input into a fully-connected layer for
classification. We initialized our network parameters to the best parameter
set that was achieved on ImageNet competition. We use exponential decay
learning rate:
$global_{s}tep$ is a counter, counting from 0 to the number of iterations of
training; $learning_{r}ate$ is the initial learning rate;
$decayed_{l}earning_{r}ate$ decays as $global_{s}tep$ increases;
$decay_{s}teps$ is used to control the decay speed.
Inspired by the work of prostate cancer classification [8], the accuracy of
convolutional network can be improved by 0.7% during the training and
validating of stain normalized patches. The result of the convolutional neural
network calculation is the probabilistic quantitative information of different
categories of patches. According to the pathologist’s experience in diagnosis,
the probability of tumor diagnosis in the patch with malignant cancer is more
than 0.9. In the diagnosis results of our convolutional neural network, three
different augmentation strategies were added to our framework to improve the
accuracy of the model during validation and reduce the misdiagnosis that
should not occur, in order to avoid the wrong diagnosis of normal tissue as a
tumor region due to incorrect threshold selection. The condition to trigger
the data enhancement strategy is that the probability of the current patch
being diagnosed as tumor tissue by the convolutional neural network is between
0.2 and 0.8.
The first method of data enhancement is to create seven derivations (three
spins and four reversals) on the patch (Rotation and flip), test the derived
patch with the convolutive network, calculate the tumor probability of the
eight patches including the original patch, take the median and confirm the
final classification of the tumor probability with 0.5 as the threshold. The
second method uses the convolutional network model with the same principle of
high accuracy training to select a higher amplification factor for the current
patch training (Magnification), which predominately through additional yield
on tumor tissue recognition accuracy. The third method to improve diagnosis
accuracy is to find the auxiliary environment patch which surrounds the
current patch (Neighbor ensemble). For the current patch, we choose its
geometric center and get four environment patches at the same magnification
and size in the region near the geometric center, then using the convolution
network training four environmental patches to calculate the probability of
diagnosis of tumor. We average the four different results as the final
probability and take 0.5 for the threshold. The reason for adopting this
approach instead of selecting eight patches around the current patch is to
maximum retain the semantic information in the current image and improve the
computing speed as much as possible.
In our experiment, training with the above three methods of data enhancement
for each patch does not improve the final diagnostic accuracy, it is worth
mentioning that we do not enhance the data for each patch in the model but
only carry out secondary verification for the patches that meet the
requirements (probability between 0.2 and 0.8).
### II-F Tumor subtype and grading classification
For both subtype and grade classification subtasks, we confirmed the selected
tumor areas in the previous section to determine the subtypes and grades of
the cancer areas. We meshed the tumor areas and ensured that each area
validated experimentally included tumor areas and tumor boundaries. The
classification network we used is similar with the network for tumor region
classification. The difference is that, for subtype classification task, our
input and output correspondingly become the probability of images and patches
of three different subtypes belonging to the three categories respectively.
For the grade classification task, our input and output are the images
corresponding to ISUP grade 1 to grade 4 and the probability values obtained.
In our experiments for subtype classification, we have well-delineating tumor
architecture and trained models for classification using three classes (ccRCC,
pRCC and chRCC). For the grading classification we trained models using four
classes (G1, G2, G3, G4). According to the experience and the observation
during the annotation process of the pathologists, there are almost one types
of subtypes in one patch, but different grades of tumor cells will appear in
one patch. As a training dataset, we only used patches with pure and clear
patterns from tumor areas, each patch contained only one subtype and one
grade, so as to reduce the differences among observers. Three classes for pure
subtype (ccRCC, pRCC and chRCC) and four classes for pure grade (G1, G2, G3,
G4) were used for training. We used argmax strategy for the patches in the
final classification of subtypes, however we found that the best result for
ISUP grading do not stem from the final classification of patches using the
argmax strategy. Patches with different grade of tumor cells represents the
intermediate representation of tumor structure, not just the mixture of
different grades. Such statistics can well describe the tumor differentiation
of renal cell carcinoma and more accurately summarize the overall ISUP grade
of the cancer area from the grading probability of individual patches. During
validation, the grading experiments included the calculation of the percentage
of single grade. In the validation dataset, the International Society of
Uropathology (ISUP) grade groups 4 were pooled.
Figure 3: The convolution network structure of the subtype and grading subtask
in the framework. (a) Similar to the classification of cancer regions, we used
similar network structures for subtype classification and classification.
Subtype classification included three different subtypes, and classification
included four different ISUP grades in clear cell renal carcinoma (ccRCC). (b)
For subtype classification, the result is the probability that each patch
belongs to different subtypes. For grade classification, first judge whether
it is grade dour, and then calculate the probability that each patch belongs
to three grades. (c) The Whole Case Report contains the detailed information
of patients in the current Whole-slide image by our framework: (1) The tumor
detection heatmap; important WSI metrics (tissue Square, tumor Square and
tumor) are generated from tissue classification results; (2) The subtype
classification heatmap; the statistical probability of different subtypes in
the Whole-slide image; (3) Statistical information of ISUP classification.
### II-G Whole-case report generation
From the result of the classification convolutional network model, we can get
the tissue area of the whole-slide image, the area of tumor area, the
proportion of tumor area, the detailed information of subtypes, and the
detailed information of classification. Pathological report can clarify the
diagnosis of the disease and provide accurate basis for doctors to treat the
disease, evaluate the prognosis and explain the symptoms. Therefore, the
department of pathology should not only ensure the timeliness of the report,
but also ensure its accuracy. Precise calculation of the different metrics
which is naturally difficult for human pathologists could be used for
generation of the whole case report.
### II-H Data availability
All the relevant data used for training during the current study are available
through the Genomic Data Commons portal (https://gdc-portal.nci.nih.gov).
These datasets were generated by TCGA Research Network
(http://cancergenome.nih.gov/), and they have made them publicly available.
Other datasets analyzed during the current study are available from the
corresponding author on reasonable request.
## III Result
### III-A Classifying tumor versus non-tumor tissue
Among the perfect accuracy convolutional network in image classification, the
best convolutional neural network architecture which perform the best accuracy
was InceptionV3. The size of each patch representing different magnifications
were tested from range ×20 to ×40. The best result for classification were
achieved for the magnification of 500 × 500 px (patch size of 128 × 128 px
scaled from originally generated 600 × 600 px patches). The classification
accuracy of the native convolutional network was 93.8% for training and 92.2%
and 92.7% for validation dataset TCGARCC and LHRCC respectively, and could be
further improved through the additional deep learning-based strategies.
For each patch, the convolutional neural network calculates the quantitative
information of probability belonging to a certain class (tumor or non-
tumor).In most of the tumor patches, the probability of tumor determination is
0.90-1.0.However, if the patch contains normal tissue, if the tumor lesion is
small or well differentiated, or if there is mis operation and artifact in the
process of making slides by the pathologist, it will be less likely to judge
the patch as a tumor. In the ”Methods section”, we proposed three strategies
to enhance the output of convolutional neural network to further improve the
overall accuracy of the framework. The trigger factor for the secondary
diagnosis and analysis and calculation of each patch is that the probability
of belonging to the tumor category output by convoluted network is between 0.2
and 0.8. From the experiment we find that 3.9% patches in the verification
dataset TCGA and 3.6% of validation dataset Local hospital need to have the
secondary validation by the three different strategies.
Among our three strategies, the first strategy which creating seven
derivations (three spins and four reversals) on the patches (Rotation and
flip) shows impressive result and greatly improves the accuracy of the model
by 0.73%. The second strategy which selecting a higher amplification factor
for the current patch training (Magnification) also improves the accuracy of
the model by 0.68%. The third strategy which finding the auxiliary environment
patch which surrounds the current patch (Neighbor ensemble), by analyzing the
four patches around the model, greatly reduces the time and cost required for
calculation, and also achieves good results in tumor image classification. As
shown in Figure 2, a complete workflow was developed for WSI analysis using
trained convolutional network to realize model prediction and enhancement
strategy for part of patches and provide important tumor indicators for
pathological reports. We selected a pathological image with unsatisfied
staining, obvious artificial cut marks and irregular artifacts for the
framework process display, and the final tumor thermogram was superimposed on
the WSI slide to facilitate the pathologist to observe and interpret the
detected tumor area. We show that the output accuracy of the model can be
significantly improved through P8, ENS and ENV strategies. In tumor area
detection, important WSI indicators (the area of WSI, the area of the tumor
area, and the proportion of the tumor area) are generated from the
classification results and used to generate complete case reports that cannot
be accurately calculated by the pathologist during the diagnosis.
TABLE I: Performance of different segmentation models for patch-level on the data set Deep learning model | Cancer Region Detection
---|---
| ccRCC | pRCC | chRCC
ResNet-34 [22] | 0.87 | 0.82 | 0.84
DenseNet [23] | 0.89 | 0.83 | 0.89
U-Net [24] | 0.81 | 0.78 | 0.84
DeepLab v2 [25] | 0.91 | 0.84 | 0.87
Inception v3 [26] | 0.93 | 0.86 | 0.91
### III-B Tumor subtype classification
Similar to the classification of tumor tissue regions, we adopted the same
convolutional network structure in the classification of subtypes and got the
probability of three classes (ccRCC, pRCC and chRCC). Figure 3 shows the
convolutional network and data output. The patch size we selected was 1000 ×
1000 px. In the process of subtyping tumors, the selection of patch size needs
to consider the model separately, because the characteristics of subtypes need
to be considered in the field of vision. If the patch is too small, then the
characteristics of papillary renal cell carcinoma cannot be well presented. We
trained on two datasets and selected cancer images that were not used for
training for verification. Since each whole-slide image has the corresponding
basic information of the patient which provides labels on subtypes and grades,
the results our model predicted can be compared with ground truth. In Figure
3, we use the same pathological images as in Figure 2, and it can be seen that
only the cancerous region is separated by different colors, and different
colors represent the probability of a single patch being output through the
convolutional network. Precise calculations of different indicators (the
proportion and area of each subtype) are naturally difficult for human
pathologists and can be used to generate whole-case reports.
TABLE II: Performance of different classification models for patch-level on the data set Deep learning model | Classification
---|---
| 3-Class Subtyping | 4-Class Subtyping | 3-Class Grading | 4-Class Grading
ResNet-34 [22] | 0.78 | 0.82 | 0.84 | 0.82
DenseNet [23] | 0.86 | 0.89 | 0.87 | 0.84
U-Net [24] | 0.77 | 0.78 | 0.81 | 0.79
DeepLab v2 [25] | 0.86 | 0.89 | 0.86 | 0.87
Inception v3 [26] | 0.87 | 0.90 | 0.91 | 0.88
### III-C Tumor grading classification
Tumor classification and subtype classification are multi-classification
problems, so we chose the same convolutional network structure to obtain the
probability of four types belonging to four categories (G1, G2, G3 and G4). In
Figure 3, we showed the workflow of the two subtasks together. Different from
the subtype classification task, based on the understanding that the
pathologist indicates the cancer level is continuous. In the classification of
tumor grades, to maximize the retention of the phenotypes of the different
tumor cells, especially the hierarchical tumor cells, are described in the WHO
guideline, tumors showing extreme nuclear pleomorphism and/or containing tumor
giant cells and/or the presence of any proportion of tumor showing sarcomatoid
and/or Rhabdoid differentiation. In the preprocess of treatment, the tumor
area was cut into patches of 1000px*1000px, and then the probabilities were
analyzed and calculated respectively through the convolutional network. When
grading and classifying patches in each cancer region, we first adopted
dichotomy and divided each patch into grade IV cancer cell nuclei and non-
grade IV cancer cells. For non-grade IV cancer patches, the three-
classification model is adopted, and the results are the probability value of
each patch at grade I II III respectively. For the result of classification
task, each patch needs to keep its own probability value which belong to
different classification categories, and in the calculation of the cancer
grade of the whole-slide image, the whole tumor area should be aggregated for
calculation, instead of each patch being nominated to an independent
classification (each patch in the subtype classification has its own
independent corresponding subtype). By aggregating the entire tumor area for
calculation, information about cancer cells of different grades can be
retained to the greatest extent.
### III-D Whole-case report generation
A detailed pathological report can help clinicians to carry out more targeted
treatment measures, which is of great significance in both pathology and
clinical medicine. The size and proportion of tumor areas in the whole tissue
image, the size and proportion of different subtypes in the whole image and
the proportion of cancerous areas are accurately calculated by convolutional
network classification, and the cancer level area and proportion of tumor
areas are calculated by statistical method. It is difficult for human
pathologists to quantify these different indicators, which can be used to
generate whole-case reports.
## IV Discussion
In artificial intelligence for cancer diagnosis, deep learning is applied to
histological image detection of tumor tissues and standard subtype
classification, grading, and other pathological tasks in the whole-slide
images. The main direction of research is semantic-based image segmentation
using different methods for patch-level classification of histological
structure. However, the lack of publicly annotated datasets slows the
development of artificial intelligence in pathology. We have created a large
training dataset from nearly 1300 slides representing different patients in
different institutions and public datasets with extensive, high-quality
annotations. We agreed that building datasets with accurate annotations are
necessary to improve model accuracy using deep learning models, and we will
continue to expand our data sets in future work to include more cancer types
and cancer images.
This study is a new application of deep learning algorithms in renal cell
carcinoma to recognize and classify tumor tissues in histological digital
slides. Our study shows that a convolutional neural network can be used for
histopathological slides in the diagnosis of renal cell carcinoma, which
correctly distinguished normal tissue from tumor tissue, identified subtypes
and grades of renal cancer with high accuracy, and achieved a sensitivity and
specificity comparable to that of the pathologists.
In analyzing the prediction results, we noticed that several whole-slides
images were misclassified by both the neural network and pathologists. We
consulted pathologists about the occurrence of this condition, after several
diagnoses of experienced pathologists, some of the erroneous results were due
to errors in the TCGA diagnosis, as well as different results due to different
grading methods. Similarly, we selected some of the pathologist’s
misclassified images in the diagnosis result, then we used our framework to
predict the diagnosis, and showed that our diagnosis was identical to the
ground truth, which shows that our framework could help auxiliary diagnosis to
the pathologists and make the correct decisions.
In our work, we used three different strategies to improve the accuracy of
tumor detection. Tumor area detection is critical in our framework because the
subsequent subtypes and grading classification work is predicated on the tumor
area These strategies need to add calculation amount for the patch that meets
the conditions, but they improve classification accuracy, which is an
acceptable increase in calculation amount. Besides, the post-training of deep
learning-based models with new data can greatly improve the accuracy of
models, which is very important for the continuous development of models.
There are limitations to our work. First of all, we asked the pathologists to
perform multiple rounds of high-quality renal carcinoma pathologic imaging for
one year, but the quantity is far from enough. We need to incorporate new data
sets (more images of kidney cancer and other types of cancer) into our work to
further validate and develop the integrity of our work. Second, further
research should include work on the classification of the types in PRCC and
the classification of the fourth-grade phenotypes in CCRCC (rhabdomyosoid,
sarcomatoid, necrotic, and hemorrhagic). In papillary renal cell carcinoma,
different types (type I and type II) have different morphological
characteristics, which contain different prognostic significance. For example,
PRCC type II shows a higher grade of pathologic T staging and ISUP grade, and
this type is more prone to sinus/perirenal adipose infiltration and
sarcomatoid differentiation. There is a lot of morphological overlap between
type I and type II in PRCC, and type II patients have a worse prognosis. It is
of great significance for the prognostic evaluation of PRCC to correctly carry
out tissue-credit type.
Overall, our work shows that convolutional neural networks can help
pathologists classify whole-slide images of renal cell carcinoma and generate
whole-case reports in deep learning. Pathologists can use this information to
treat patients more specifically. We hope to apply this framework to identify
papillary renal cell carcinoma and other kind of cancer in future work. Also,
identifying some critical features of grade IV clear cell carcinoma
(rhabdomyosoid, sarcomatoid, necrotic, and hemorrhagic) and some critical
diagnostic features summarized by pathologists is also important to the
diagnosis and prognosis of the pathologic diagnostic procedure. In addition to
renal cell carcinoma, our model can be applied to other cancers in the future.
By expanding the robustness of our model to recognize a wider range of
histological features through various types of cancer, complete and accurate
pathologically assisted diagnosis can be made, effectively helping
pathologists in daily tasks and specific cases.
## V Conclusion
In this study, we developed a precise framework for analyzing whole-slide
images of renal cell carcinoma patients based on a deep learning model, which
has the potential to achieve pathologist-level accuracy and interpretable
auxiliary diagnosis. This framework is realized in the pipeline of digital
pathology which provides all the relevant tumor metrics and all the necessary
information for generating a pathology report and assists the pathologist in
their daily work. Further validation of diagnostic histopathology workflows in
the real world is necessary. The pathologist can understand these prediction
result from the deep learning method when performing the second examination
and visual examination. Our approach is data-agnostic. With the success of
deep learning, we believe that our approach has a strong universality for
learning the complex tissue structure of different types of cancer. Further
research will help to demonstrate its efficacy against other types of cancer,
such as lung cancer, breast cancer and prostate cancer. Exploring the use of
multiple types of information for diagnosis is clinically necessary.
## Acknowledgment
This work has been supported by National Natural Science Foundation of China
(61772409); the Innovative Research Group of the National Natural Science
Foundation of China (61721002); and the consulting research project of the
Chinese Academy of Engineering (The Online and Offline Mixed Educational
Service System for “The Belt and Road” Training in MOOC China); The results
shown here are in whole or part based upon data generated by the TCGA Research
Network: https://www.cancer.gov/tcga.
## References
* [1] J. Hsieh, M. Purdue, S. Signoretti, C. Swanton, L. Albiges, M. Schmidinger, D. Heng, J. Larkin, and V. Ficarra, “Renal cell carcinoma. nature reviews disease primers. 2017; 3: 17009,” Epub 2017/03/10. https://doi. org/10.1038/nrdp. 2017.9 PMID: 28276433, Tech. Rep., 2017.
* [2] B. Ljungberg, K. Bensalah, S. Canfield, S. Dabestani, F. Hofmann, M. Hora, M. A. Kuczyk, T. Lam, L. Marconi, A. S. Merseburger _et al._ , “Eau guidelines on renal cell carcinoma: 2014 update,” _European urology_ , vol. 67, no. 5, pp. 913–924, 2015.
* [3] Z. Zhang, P. Chen, M. McGough, F. Xing, C. Wang, M. Bui, Y. Xie, M. Sapkota, L. Cui, J. Dhillon _et al._ , “Pathologist-level interpretable whole-slide cancer diagnosis with deep learning,” _Nature Machine Intelligence_ , vol. 1, no. 5, pp. 236–245, 2019.
* [4] D. M. Metter, T. J. Colgan, S. T. Leung, C. F. Timmons, and J. Y. Park, “Trends in the us and canadian pathologist workforces from 2007 to 2017,” _JAMA network open_ , vol. 2, no. 5, pp. e194 337–e194 337, 2019.
* [5] K. Bera, K. A. Schalper, D. L. Rimm, V. Velcheti, and A. Madabhushi, “Artificial intelligence in digital pathology—new tools for diagnosis and precision oncology,” _Nature reviews Clinical oncology_ , vol. 16, no. 11, pp. 703–715, 2019.
* [6] S. Tabibu, P. Vinod, and C. Jawahar, “Pan-renal cell carcinoma classification and survival prediction from histopathology images using deep learning,” _Scientific reports_ , vol. 9, no. 1, pp. 1–9, 2019.
* [7] N. Coudray, P. S. Ocampo, T. Sakellaropoulos, N. Narula, M. Snuderl, D. Fenyö, A. L. Moreira, N. Razavian, and A. Tsirigos, “Classification and mutation prediction from non–small cell lung cancer histopathology images using deep learning,” _Nature medicine_ , vol. 24, no. 10, pp. 1559–1567, 2018.
* [8] Y. Tolkach, T. Dohmgörgen, M. Toma, and G. Kristiansen, “High-accuracy prostate cancer pathology using deep learning,” _Nature Machine Intelligence_ , vol. 2, no. 7, pp. 411–418, 2020.
* [9] J. N. Kather, J. Krisam, P. Charoentong, T. Luedde, E. Herpel, C.-A. Weis, T. Gaiser, A. Marx, N. A. Valous, D. Ferber _et al._ , “Predicting survival from colorectal cancer histology slides using deep learning: A retrospective multicenter study,” _PLoS medicine_ , vol. 16, no. 1, p. e1002730, 2019.
* [10] H. D. Couture, L. A. Williams, J. Geradts, S. J. Nyante, E. N. Butler, J. Marron, C. M. Perou, M. A. Troester, and M. Niethammer, “Image analysis with deep learning to predict breast cancer grade, er status, histologic subtype, and intrinsic subtype,” _NPJ breast cancer_ , vol. 4, no. 1, pp. 1–8, 2018.
* [11] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” _nature_ , vol. 521, no. 7553, pp. 436–444, 2015.
* [12] B. E. Bejnordi, M. Veta, P. J. Van Diest, B. Van Ginneken, N. Karssemeijer, G. Litjens, J. A. Van Der Laak, M. Hermsen, Q. F. Manson, M. Balkenhol _et al._ , “Diagnostic assessment of deep learning algorithms for detection of lymph node metastases in women with breast cancer,” _Jama_ , vol. 318, no. 22, pp. 2199–2210, 2017.
* [13] G. Litjens, C. I. Sánchez, N. Timofeeva, M. Hermsen, I. Nagtegaal, I. Kovacs, C. Hulsbergen-Van De Kaa, P. Bult, B. Van Ginneken, and J. Van Der Laak, “Deep learning as a tool for increased accuracy and efficiency of histopathological diagnosis,” _Scientific reports_ , vol. 6, no. 1, pp. 1–11, 2016.
* [14] K. Tomczak, P. Czerwińska, and M. Wiznerowicz, “The cancer genome atlas (tcga): an immeasurable source of knowledge,” _Contemporary oncology_ , vol. 19, no. 1A, p. A68, 2015.
* [15] A. H. Fischer, K. A. Jacobson, J. Rose, and R. Zeller, “Hematoxylin and eosin staining of tissue and cell sections,” _Cold spring harbor protocols_ , vol. 2008, no. 5, pp. pdb–prot4986, 2008.
* [16] J. Wu, K. Tang, H. Zhang, C. Wang, and C. Li, “Structured information extraction of pathology reports with attention-based graph convolutional network,” in _2020 IEEE International Conference on Bioinformatics and Biomedicine (BIBM)_. IEEE, 2020, pp. 2395–2402.
* [17] R. L. Grossman, A. P. Heath, V. Ferretti, H. E. Varmus, D. R. Lowy, W. A. Kibbe, and L. M. Staudt, “Toward a shared vision for cancer genomic data,” _New England Journal of Medicine_ , vol. 375, no. 12, pp. 1109–1112, 2016\.
* [18] P. Puttapirat, H. Zhang, J. Deng, Y. Dong, J. Shi, P. Lou, C. Wang, L. Yao, X. Zhang, and C. Li, “Openhi: open platform for histopathological image annotation,” _International Journal of Data Mining and Bioinformatics_ , vol. 22, no. 4, pp. 328–349, 2019.
* [19] B. Delahunt, J. C. Cheville, G. Martignoni, P. A. Humphrey, C. Magi-Galluzzi, J. McKenney, L. Egevad, F. Algaba, H. Moch, D. J. Grignon _et al._ , “The international society of urological pathology (isup) grading system for renal cell carcinoma and other prognostic parameters,” _The American journal of surgical pathology_ , vol. 37, no. 10, pp. 1490–1504, 2013.
* [20] M. L. McHugh, “Interrater reliability: the kappa statistic,” _Biochemia medica_ , vol. 22, no. 3, pp. 276–282, 2012.
* [21] M. Macenko, M. Niethammer, J. S. Marron, D. Borland, J. T. Woosley, X. Guan, C. Schmitt, and N. E. Thomas, “A method for normalizing histology slides for quantitative analysis,” in _2009 IEEE International Symposium on Biomedical Imaging: From Nano to Macro_. IEEE, 2009, pp. 1107–1110.
* [22] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.
* [23] G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, “Densely connected convolutional networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 4700–4708.
* [24] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in _International Conference on Medical image computing and computer-assisted intervention_. Springer, 2015, pp. 234–241.
* [25] L.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. L. Yuille, “Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs,” _IEEE transactions on pattern analysis and machine intelligence_ , vol. 40, no. 4, pp. 834–848, 2017.
* [26] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 2818–2826.
|
eigenvalue}-e^{i\omega_{0}}&:\;\;\;\phi_{a0}^{\prime}\otimes d_{+1}\text{ and
}\phi_{b+}\otimes d_{0},\\\ \text{Group 3,
eigenvalue}-e^{-i\omega_{0}}&:\;\;\;\phi_{a0}^{\prime}\otimes d_{-1}\text{ and
}\phi_{b+}\otimes d_{+1}.\end{split}$ (168)
Therefore, any linear combination of those two eigen-states belonging to the
same group will not change the matrix representation of $C_{3}$ in Eq. (167).
By choosing specific transformations, the 6-by-6 matrix representation of
$S_{2x}$ in Eq. (167) can be reduced into a direct sum of three 2-by-2
matrices. To show that, we define the following new basis
$\displaystyle\begin{cases}\Phi_{t_{2g}}^{0,1}&=i\frac{1}{\sqrt{3}}e^{-i\omega_{0}}\phi_{a0}^{\prime}\otimes
d_{0}+\sqrt{\frac{2}{3}}e^{i\omega_{0}}\phi_{b+}\otimes d_{-1},\\\
\Phi_{t_{2g}}^{0,2}&=-i\sqrt{\frac{2}{3}}e^{i\omega_{0}}\phi_{a0}^{\prime}\otimes
d_{0}+\frac{1}{\sqrt{3}}\phi_{b+}\otimes d_{-1},\\\ \end{cases}$ (169)
$\displaystyle\begin{cases}\Phi_{t_{2g}}^{+,1}&=-i\sqrt{\frac{2}{3}}e^{i\omega_{0}}\phi_{a0}^{\prime}\otimes
d_{+1}+\frac{1}{\sqrt{3}}\phi_{b+}\otimes d_{0},\\\
\Phi_{t_{2g}}^{+,2}&=\frac{1}{\sqrt{3}}\phi_{a0}^{\prime}\otimes
d_{+1}-i\sqrt{\frac{2}{3}}e^{-i\omega_{0}}\phi_{b+}\otimes d_{0},\\\
\end{cases}$ (170)
$\displaystyle\begin{cases}\Phi_{t_{2g}}^{-,1}&=i\frac{1}{\sqrt{3}}e^{i\omega_{0}}\phi_{a0}^{\prime}\otimes
d_{-1}+\sqrt{\frac{2}{3}}\phi_{b+}\otimes d_{+1},\\\
\Phi_{t_{2g}}^{-,2}&=\sqrt{\frac{2}{3}}\phi_{a0}^{\prime}\otimes
d_{-1}+i\frac{1}{\sqrt{3}}e^{-i\omega_{0}}\phi_{b+}\otimes d_{+1},\\\
\end{cases}$ (171)
where the indices $0,+,-$ labels the eigenvalues
$1,e^{i\omega_{0}},e^{-i\omega_{0}}$ of the matrix $C_{3}$, and the indices
$1,2$ label the number of wave functions. Within the new basis
$\displaystyle\\{\Phi_{t_{2g}}^{0,1},\;\Phi_{t_{2g}}^{+,1},\;\Phi_{t_{2g}}^{0,2},\;\Phi_{t_{2g}}^{-,1},\;\Phi_{t_{2g}}^{+,2},\;\Phi_{t_{2g}}^{-,2}\\},$
(172)
the 6-by-6 matrix representation of $S_{2x}$ in Eq. (167) becomes
$\displaystyle S_{2x}$ $\displaystyle=\begin{pmatrix}S_{2x,1}&0&0\\\
0&S_{2x,2}&0\\\ 0&0&S_{2x,3}\end{pmatrix},$ (173)
where
$\displaystyle
S_{2x,1}=\begin{pmatrix}-\frac{i}{\sqrt{3}}&-\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{6}}\\\
\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{6}}&\frac{i}{\sqrt{3}}\end{pmatrix},S_{2x,2}$
$\displaystyle=\begin{pmatrix}\frac{i}{\sqrt{3}}&-\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{6}}\\\
\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{6}}&-\frac{i}{\sqrt{3}}\end{pmatrix},S_{2x,3}=\begin{pmatrix}-\frac{i}{\sqrt{3}}&-\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{6}}\\\
\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{6}}&\frac{i}{\sqrt{3}}\end{pmatrix}.$ (174)
Therefore, the three 2D Irreps generated by $C_{3}$ and $S_{2x}$ are
$\displaystyle A-\text{type:
}\\{\Phi_{t_{2g}}^{0,1},\;\Phi_{t_{2g}}^{+,1}\\}\text{ and
}\\{\Phi_{t_{2g}}^{0,2},\;\Phi_{t_{2g}}^{-,1}\\},$ (175a) $\displaystyle
B-\text{type: }\\{\Phi_{t_{2g}}^{+,2},\;\Phi_{t_{2g}}^{-,2}\\}.$ (175b)
Moreover, the time-reversal symmetry leads to the 4D Irrep,
$\displaystyle A_{2}-\text{Irrep:
}\\{\Phi_{t_{2g}}^{0,1},\;\Phi_{t_{2g}}^{+,1}\\}\,\oplus\\{\Phi_{t_{2g}}^{0,1},\;\Phi_{t_{2g}}^{+,1}\\}^{\ast},$
(176a) $\displaystyle A_{3}-\text{Irrep:
}\\{\Phi_{t_{2g}}^{0,2},\;\Phi_{t_{2g}}^{-,1}\\}\,\oplus\\{\Phi_{t_{2g}}^{0,2},\;\Phi_{t_{2g}}^{-,1}\\}^{\ast},$
(176b) $\displaystyle B_{2}-\text{Irrep:
}\\{\Phi_{t_{2g}}^{+,2},\;\Phi_{t_{2g}}^{-,2}\\}\oplus\\{\Phi_{t_{2g}}^{+,2},\;\Phi_{t_{2g}}^{-,2}\\}^{\ast}.$
(176c)
We further check the matrix representations for $C_{3}$, $S_{2x}$, $S_{2y}$
and $\mathcal{T}$ for both the A2 Irrep and A3 Irrep as
$\displaystyle C_{3}$ $\displaystyle=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\exp\left(i\omega_{0}\right)&0&0\\\ 0&0&1&0\\\
0&0&0&\exp\left(-i\omega_{0}\right)\\\
\end{array}\right),\;\;\;\;\;\;\;\;S_{2x}=\left(\begin{array}[]{cccc}-\tfrac{i}{\sqrt{3}}&-\sqrt{\tfrac{2}{3}}&0&0\\\
\sqrt{\tfrac{2}{3}}&\tfrac{i}{\sqrt{3}}&0&0\\\
0&0&\tfrac{i}{\sqrt{3}}&-\sqrt{\tfrac{2}{3}}\\\
0&0&\sqrt{\tfrac{2}{3}}&-\tfrac{i}{\sqrt{3}}\\\ \end{array}\right),$ (185)
$\displaystyle S_{2y}$
$\displaystyle=\left(\begin{array}[]{cccc}-\frac{i}{\sqrt{3}}&\frac{1-i\sqrt{3}}{\sqrt{6}}&0&0\\\
\frac{-1-i\sqrt{3}}{\sqrt{6}}&\frac{i}{\sqrt{3}}&0&0\\\
0&0&\frac{i}{\sqrt{3}}&\frac{1+i\sqrt{3}}{\sqrt{6}}\\\
0&0&\frac{-1+i\sqrt{3}}{\sqrt{6}}&-\frac{i}{\sqrt{3}}\\\
\end{array}\right),\;\;\;\;\;\;\;\;\mathcal{T}=\left(\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\\\ \end{array}\right)\mathcal{K}.$ (194)
They are completely the same as the matrix representation of the A1 Irrep in
Eq. (156i). Please notice that both $A_{2}$-Irrep and $A_{3}$-Irrep are the
$R_{1}R_{3}$-Irrep, which is the same as the $A_{1}$-Irrep constructed in Sec.
B.2.2, and the $B_{2}$-Irrep is the $R_{2}R_{2}$-Irrep that is the same as the
$B_{1}$-Irrep in Sec. B.2.2.
## Appendix C Approach I for the hierarchy of the quasi-symmetry
In this section, we use the perturbation theory to identify the hierarchy of
quasi-symmetry. This is Approach I mentioned in the main text. Here we explain
the necessary details. The linear-$k$-order SOC-free Hamiltonian
$\mathcal{H}_{1}({\bf k})$,
$\displaystyle\mathcal{H}_{1}(\mathbf{k})$
$\displaystyle=C_{0}\sigma_{0}\tau_{0}+2A_{1}(\mathbf{k}\cdot\mathbf{L}),$
(195)
where the angular momentum operators are
$L_{x}=\tfrac{1}{2}\sigma_{y}\tau_{0},\;L_{y}=\tfrac{1}{2}\sigma_{x}\tau_{y},\;L_{z}=-\tfrac{1}{2}\sigma_{z}\tau_{y}$
defined in Eq. (28). Notice that $\mathcal{H}_{1}(\mathbf{k})$ is invariant
under the spin SU${}_{\text{s}}$(2) symmetry group. Moreover, we notice that
an additional hidden SU${}_{\text{o}}$(2) symmetry also exists for
$\mathcal{H}_{1}(\mathbf{k})$ in the “orbital” space that can be generated by
the following operators
$\displaystyle\mathcal{M}_{1,2,3}=\tfrac{1}{2}\\{s_{0}\sigma_{y}\tau_{z},s_{0}\sigma_{y}\tau_{x},s_{0}\sigma_{0}\tau_{y}\\},$
(196)
which is shown in the main text (see Eq. (2)), and $\mathcal{M}_{1,2,3}$ all
commute with $\mathcal{H}_{1}(\mathbf{k})$ and satisfy the commutation
relations
$\displaystyle[\mathcal{M}_{i},\mathcal{M}_{j}]$
$\displaystyle=i\epsilon_{i,j,k}\mathcal{M}_{k},$ (197a)
$\displaystyle\\{\mathcal{M}_{i},\mathcal{M}_{j}\\}$
$\displaystyle=\frac{1}{2}\delta_{i,j}.$ (197b)
Here $i,j=1,2,3$, $\epsilon_{i,j,k}$ is the 3D Levi-Civita symbol, and
$\delta_{i,j}$ is the Kronecker delta function. Thus, we refer to it as the
orbital SU${}_{\text{o}}$(2) quasi-symmetry group for $\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}$ because of
$\displaystyle[\mathcal{M}_{i},\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}]=0,$ (198)
where the on-site SOC Hamiltonian is given by Eq. 51,
$\displaystyle\mathcal{H}_{\text{soc}}=4\lambda_{0}(\mathbf{S}\cdot\mathbf{L}),$
(199)
where we use ${\bf S}=\tfrac{1}{2}(s_{x},s_{y},s_{z})$ for the spin-1/2
angular momentum operators. And $[S_{i},S_{j}]=i\epsilon_{ijk}S_{k}$. In
addition, we consider the $k^{2}$-order Hamiltonian that can be represented in
a compact form,
$\displaystyle\mathcal{H}_{2}(\mathbf{k})=\mathcal{H}_{2,\mathcal{M}_{1}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{2}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{3}}(\mathbf{k}),$
(200)
where each part of $\mathcal{H}_{2}(\mathbf{k})$ is given by
$\displaystyle\mathcal{H}_{2,\mathcal{M}_{i}}(\mathbf{k})={\bf g}_{i}\cdot{\bf
J}_{i},$ (201)
for $i=1,2,3$. Here we define the $k$-dependent vectors as
$\displaystyle\begin{split}{\bf g}_{1}({\bf
k})&=(C_{2}k_{x}k_{y},-C_{3}k_{x}k_{z},C_{1}k_{y}k_{z}),\\\ {\bf g}_{2}({\bf
k})&=(C_{3}k_{x}k_{y},C_{1}k_{x}k_{z},-C_{2}k_{y}k_{z}),\\\ {\bf g}_{3}({\bf
k})&=(C_{1}k_{x}k_{y},C_{2}k_{x}k_{z},-C_{3}k_{y}k_{z}).\end{split}$ (202)
And the corresponding vectors of operators
$\displaystyle\begin{split}{\bf
J}_{1}&=(\sigma_{x}\tau_{x},-\sigma_{z}\tau_{x},\sigma_{0}\tau_{z}),\\\ {\bf
J}_{2}&=(\sigma_{x}\tau_{z},\sigma_{z}\tau_{z},\sigma_{0}\tau_{x}),\\\ {\bf
J}_{3}&=(\sigma_{z}\tau_{0},\sigma_{x}\tau_{0},\sigma_{y}\tau_{y}).\end{split}$
(203)
In addition, we also realize that the $k^{2}$ terms of $\mathcal{H}_{2}({\bf
k})$ break this orbital SU${}_{\text{o}}$(2) quasi-symmetry generated by
$\\{\mathcal{M}_{1,2,3}\\}$ and lead to the splitting of all bands. However,
different parts of the entire $k^{2}$-terms can lead to the reduction from
SU${}_{\text{o}}$(2) to a orbital U(1). To show that, as we discussed in the
main text, we find that
$\displaystyle[{\bf J}_{i},\mathcal{M}_{i}]=0\text{ and }\\{{\bf
J}_{i},\mathcal{M}_{j}\\}=0\text{ for }i\neq j,$ (204)
which implies
$\displaystyle[\mathcal{H}_{2,\mathcal{M}_{i}}({\bf k}),\mathcal{M}_{i}]=0.$
(205)
This can be also found in the main text and Sec. A.2. Below, we focus on how
to identity the quasi-symmetry based on the effective perturbation theory
(Approach I). Note that alternative Approach II will be presented in Sec. D.
Furthermore, we notice there is a conservation of total angular momentum at
fixed ${\bf k}$ for $\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}=C_{0}+2A_{1}({\bf
k}\cdot\mathbf{L})+4\lambda_{0}({\bf S}\cdot{\bf L})$. As we have discussed in
Sec. A.4, the total angular momentum ${\bf S}+{\bf L}$ is conserved at ${\bf
k}=0$. But it does not commute with the Hamiltonian $\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}$ at the nonzero ${\bf k}$. However, we notice
that
$\displaystyle[({\bf S}+{\bf L})\cdot\vec{n}_{\bf k},\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}]=0,$ (206)
where $\vec{n}_{\bf k}=\frac{{\bf k}}{k}$ is the direction of the momentum
${\bf k}$. And $({\bf S}+{\bf L})\cdot\vec{n}_{\bf k}$ is physically similar
to the helicity operator at nonzero ${\bf k}$. Also, Eq. (206) can be proved
as follows,
* •
Spin and orbital are independent degree of freedoms, so we have $[{\bf S},{\bf
L}]=0$. Therefore, $[({\bf S}+{\bf L})\cdot\vec{n}_{\bf
k},\mathcal{H}_{1}({\bf k})]=[{\bf L}\cdot\vec{n}_{\bf k},C_{0}+2A_{1}({\bf
k}\cdot{\bf L})]=0$.
* •
We can define total angular momentum ${\bf L}_{\text{tot}}={\bf S}+{\bf L}$,
so that $\mathcal{H}_{\text{soc}}=2\lambda_{0}({\bf L}_{\text{tot}}^{2}-{\bf
S}^{2}-{\bf L}^{2})=2\lambda_{0}({\bf
L}_{\text{tot}}^{2}-\frac{3}{4}-\frac{3}{4})$. Then one can check
$\displaystyle[{\bf L}_{\text{tot},i},{\bf
L}_{\text{tot},j}]=[S_{i},S_{j}]+[L_{i},L_{j}]=i\epsilon_{ijk}S_{k}+i\epsilon_{ijk}L_{k}=i\epsilon_{ijk}{\bf
L}_{\text{tot},k}.$ (207)
Besides, we have $[{\bf L}_{\text{tot}},{\bf L}_{\text{tot}}^{2}]=0$ (i.e.,
the square of the angular momentum commutes with any of the components), which
leads to $[{\bf L}_{\text{tot}},\mathcal{H}_{\text{soc}}]=0$. Therefore, we
have
$\displaystyle[{\bf L}_{\text{tot}}\cdot\vec{n}_{{\bf k}},{\bf
L}_{\text{tot}}^{2}]=0.$ (208)
This proves that ${\bf L}_{\text{tot}}\cdot\vec{n}_{{\bf k}}$ is a symmetry
operator of $\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}$ at any nonzero
${\bf k}$.
### C.1 Approach I: the U(1) quasi-symmetry protected nodal planes
We solve the eigen-problem of the Hamiltonian $\mathcal{H}_{1}(\mathbf{k})$ in
Eq. (26) or Eq. (28). Due to the full rotation symmetry, we choose the
spherical coordinate with the momentum ${\bf
k}=(k_{x},k_{y},k_{z})=k(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$.
The eigen-energies of $\mathcal{H}_{1}(\mathbf{k})$ have two branches
$E_{\pm}=C_{0}\pm A_{1}k$ with each branch twofold degeneracy (fourfold if
spin degeneracy is involved). In this work, we assume $A_{1}>0$, and the two
degenerate eigen-wave functions $|\Psi_{A/B+}(\theta,\phi)\rangle$ of the
positive energy branch ($E_{+}$) and $|\Psi_{A/B-}(\theta,\phi)\rangle$ of the
negative energy branch ($E_{-}$) are given by
$\displaystyle\mathcal{H}_{1}(\mathbf{k})\begin{cases}|\Psi_{A/B+}(\theta,\phi)\rangle=E_{+}|\Psi_{A/B+}(\theta,\phi)\rangle,\\\
|\Psi_{A/B-}(\theta,\phi)\rangle=E_{-}|\Psi_{A/B-}(\theta,\phi)\rangle,\end{cases}$
(209)
where the index $\pm$ represent the eigenvalues of ${\bf L}\cdot\vec{n}_{{\bf
k}}$, and the eigen-states in the spherical coordinator are given by
$\displaystyle\begin{cases}|\Psi_{A+}(\theta,\phi)\rangle&=\tfrac{1}{\sqrt{2}}\left(\cos\theta\cos\phi-i\sin\phi,-\cos\theta\sin\phi-i\cos\phi,0,\sin\theta\right)^{T},\\\
|\Psi_{B+}(\theta,\phi)\rangle&=\tfrac{1}{\sqrt{2}}\left(-i\sin\theta\cos\phi,i\sin\theta\sin\phi,1,i\cos\theta\right)^{T}.\end{cases}$
(210a)
$\displaystyle\begin{cases}|\Psi_{A-}(\theta,\phi)\rangle&=\tfrac{1}{\sqrt{2}}\left(i\sin\theta\cos\phi,-i\sin\theta\sin\phi,1,-i\cos\theta\right)^{T},\\\
|\Psi_{B-}(\theta,\phi)\rangle&=\tfrac{1}{\sqrt{2}}\left(\cos\theta\cos\phi+i\sin\phi,-\cos\theta\sin\phi+i\cos\phi,0,\sin\theta\right)^{T}.\end{cases}$
(210b)
The solution is not unique, since there exists a twofold degeneracy between
$|\Psi_{A+}(\theta,\phi)\rangle$ and $|\Psi_{B+}(\theta,\phi)\rangle$ at
arbitrary ${\bf k}$, protected by the orbital SU(2) symmetry generated by
$\\{\mathcal{M}_{1},\mathcal{M}_{2},\mathcal{M}_{3}\\}$ in Eq. (196).
Moreover, the subscript $A(B)$ can represent the eigenvalues of the quasi-
symmetry operator defined as
$(\mathcal{M}_{1},\mathcal{M}_{2},\mathcal{M}_{3})\cdot\vec{n}$ with a
specific real normalized vector $\vec{n}=(n_{1},n_{2},n_{3})$. For the basis
in Eq. (210a) and Eq. (210b), the $\vec{n}$-vector reads
$\displaystyle\vec{n}=\left(-\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),-\cos(\theta)\right).$
(211)
And one can check
$\displaystyle({\bf\mathcal{M}}\cdot\vec{n})|\Psi_{A+}(\theta,\phi)\rangle$
$\displaystyle=\;\;\,|\Psi_{A+}(\theta,\phi)\rangle,$ (212a)
$\displaystyle({\bf\mathcal{M}}\cdot\vec{n})|\Psi_{A-}(\theta,\phi)\rangle$
$\displaystyle=\;\;\,|\Psi_{A-}(\theta,\phi)\rangle,$ (212b)
$\displaystyle({\bf\mathcal{M}}\cdot\vec{n})|\Psi_{B+}(\theta,\phi)\rangle$
$\displaystyle=-|\Psi_{B+}(\theta,\phi)\rangle,$ (212c)
$\displaystyle({\bf\mathcal{M}}\cdot\vec{n})|\Psi_{B-}(\theta,\phi)\rangle$
$\displaystyle=-|\Psi_{B-}(\theta,\phi)\rangle.$ (212d)
But $\vec{n}$ can be arbitrary due to this twofold degeneracy (fourfold if
spin degeneracy is accounted). This means the choice of eigen-states are not
unique. Besides, another set of wavefunctions chosen as eigen-states of
$\mathcal{M}_{3}$ will be shown in Sec. D.2 by fixing $\vec{n}=(0,0,1)$.
Furthermore, we can project the angular momentum operator $\mathbf{L}$ into
the eigenstate subspace,
$\displaystyle\langle\Psi_{A\pm}(\theta,\phi)|\mathbf{L}|\Psi_{A\pm}(\theta,\phi)\rangle$
$\displaystyle=\langle\Psi_{B\pm}(\theta,\phi)|\mathbf{L}|\Psi_{B\pm}(\theta,\phi)\rangle=\pm\frac{\mathbf{k}}{2k},$
(213)
$\displaystyle\langle\Psi_{A\pm}(\theta,\phi)|\mathbf{L}|\Psi_{B\pm}(\theta,\phi)\rangle$
$\displaystyle=\langle\Psi_{B\pm}(\theta,\phi)|\mathbf{L}|\Psi_{A\pm}(\theta,\phi)\rangle=0.$
(214)
Here, Eqs. (213) and (214) mean the emergent angular momentum operator
$\mathbf{L}$ is along the momentum direction after the projection. Besides,
with involving the spin degree of freedom, the corresponding four-fold
degenerate wave-functions are labelled as
$\displaystyle|\Psi_{+}\rangle=\\{|\Psi_{A+\uparrow}(\theta,\phi)\rangle,|\Psi_{B+\uparrow}(\theta,\phi)\rangle,|\Psi_{A+\downarrow}(\theta,\phi)\rangle,|\Psi_{B+\downarrow}(\theta,\phi)\rangle\\},$
(215)
where
$\displaystyle\begin{split}|\Psi_{A+\uparrow}(\theta,\phi)\rangle&=(1,0)^{T}\otimes|\Psi_{A+}(\theta,\phi)\rangle,\\\
|\Psi_{B+\uparrow}(\theta,\phi)\rangle&=(1,0)^{T}\otimes|\Psi_{B+}(\theta,\phi)\rangle,\\\
|\Psi_{A+\downarrow}(\theta,\phi)\rangle&=(0,1)^{T}\otimes|\Psi_{A+}(\theta,\phi)\rangle,\\\
|\Psi_{B+\downarrow}(\theta,\phi)\rangle&=(0,1)^{T}\otimes|\Psi_{B+}(\theta,\phi)\rangle,\end{split}$
(216)
where $(1,0)^{T}$ and $(0,1)^{T}$ label the spin-up and spin-down wave
functions, respectively. The above set of wave functions serves as the basis
for the projected 4-band perturbation model via the first-order perturbation
theory, dubbed as “the $P$-model”. We treat both the on-site SOC in Eq. (51)
and the $k^{2}$-terms in Eq. (29) as perturbations,
$\displaystyle\mathcal{H}_{perb}({\bf k})$
$\displaystyle=\mathcal{H}_{\text{soc}}+s_{0}\otimes\mathcal{H}_{2}(\mathbf{k}).$
(217)
The projected Hamiltonian
$\left\langle\Psi_{+}|\mathcal{H}_{perb}|\Psi_{+}\right\rangle$ is given by
$\displaystyle\mathcal{H}_{P}^{eff(1)}({\bf k})$
$\displaystyle=(E_{+}+B_{1}k^{2})s_{0}\omega_{0}+\mathcal{H}_{\text{soc},+}^{eff(1)}({\bf
k})+\mathcal{H}_{k^{2},+}^{eff(1)}({\bf k}),$ (218)
which is marked as the $P$-model around $R$-point. And
$\displaystyle\mathcal{H}_{\text{soc},+}^{eff(1)}({\bf k})$
$\displaystyle=\lambda_{0}\left(\lambda_{x}s_{x}+\lambda_{y}s_{y}+\lambda_{z}s_{z}\right)\otimes\omega_{0},$
(219a) $\displaystyle\mathcal{H}_{k^{2},+}^{eff(1)}({\bf k})$
$\displaystyle=\tilde{C}k^{2}s_{0}\otimes\left(d_{x}\omega_{x}+d_{y}\omega_{y}+d_{z}\omega_{z}\right).$
(219b)
where $\tilde{C}=C_{1}-C_{2}+C_{3}$, and $\omega_{x,y,z}$ are Pauli matrices
for the $\\{A+,B+\\}$ band subspace. The coefficients $\lambda_{x,y,z}$ are
defined as
$\displaystyle(\lambda_{x},\lambda_{y},\lambda_{z})$
$\displaystyle=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)=\tfrac{\mathbf{k}}{|\mathbf{k}|},$
(220)
On the other hand, the coefficients $d_{x,y,z}$ are given by
$\displaystyle\begin{split}d_{x}&=\tfrac{1}{4}\sin\theta\sin(2\theta)\sin(2\phi)(\cos\phi+\sin\phi),\\\
d_{y}&=\tfrac{1}{4}\sin\theta\sin(2\theta)\sin(2\phi)(\sin\theta+\cos\theta(\cos\phi-\sin\phi)),\\\
d_{z}&=\tfrac{1}{4}\sin\theta\sin(2\theta)\sin(2\phi)(\cos\theta-\sin\theta(\cos\phi-\sin\phi)).\end{split}$
(221)
Furthermore, we use the symmetry to understand the above first-order
perturbation Hamiltonian. The basis function in Eq. (215) can be labeled by
eigen-values of symmetries,
$\displaystyle|\Psi_{A+\uparrow}(\theta,\phi)\rangle$
$\displaystyle=|\uparrow\rangle\otimes|p=+\tfrac{1}{2},q=+1\rangle,$ (222a)
$\displaystyle|\Psi_{B+\uparrow}(\theta,\phi)\rangle$
$\displaystyle=|\uparrow\rangle\otimes|p=+\tfrac{1}{2},q=-1\rangle,$ (222b)
$\displaystyle|\Psi_{A+\downarrow}(\theta,\phi)\rangle$
$\displaystyle=|\downarrow\rangle\otimes|p=+\tfrac{1}{2},q=+1\rangle,$ (222c)
$\displaystyle|\Psi_{B+\downarrow}(\theta,\phi)\rangle$
$\displaystyle=|\downarrow\rangle\otimes|p=+\tfrac{1}{2},q=-1\rangle,$ (222d)
Here we take $p=\pm 1/2$ as the eigenvalues of ${\bf L}\cdot\vec{n}_{{\bf k}}$
and $q=\pm 1$ as the eigenvalues of $\vec{\mathcal{M}}\cdot\vec{n}$. All these
four states are degenerate with eigen-energy of $\mathcal{H}_{1}$ as
$C_{0}+A_{1}k$. The linear-$k$ Hamiltonian has both orbital SU(2) symmetry and
spin SU(2) symmetry. Specifically, the orbital SU(2) symmetry generated by
$\\{\mathcal{M}_{1,2,3}\\}$ indicates that the vector $\vec{n}$ is arbitrary.
And the spin SU(2) symmetry implies that $|\uparrow/\downarrow\rangle$ can be
any direction in the spin subspace. Then, to show the origin of the hidden
quasi-symmetry of the $P$-model, we individually do the projection for the on-
site SOC and $k^{2}$-order Hamiltonian,
* •
Only do the projection of the on-site SOC Hamiltonian. In this case, the
orbital SU(2) symmetry generated by $\\{\mathcal{M}_{1,2,3}\\}$ preserves,
because of $[\mathcal{M}_{i},\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}]=0$ with $i=1,2,3$. Thus, we obtain
$\displaystyle\text{SU}_{\text{s}}(2)\times\text{SU}_{\text{o}}(2)\stackrel{{\scriptstyle\left\langle\mathcal{H}_{\text{soc}}\right\rangle_{\mathcal{H}_{1}}}}{{\xhookrightarrow{\quad\quad\quad\quad\quad}}}\text{U}_{\text{s}}(1)\times\text{SU}_{\text{o}}(2),$
(223)
where the spin U(1) symmetry generator depends on ${\bf k}$. To understand
this, we recall the conservation of $({\bf S}+{\bf L})\cdot\vec{n}_{{\bf k}}$
for $\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}=C_{0}+2A_{1}{\bf
k}\cdot{\bf L}+4\lambda_{0}{\bf S}\cdot{\bf L}$. Then, for a fixed nonzero
${\bf k}$, one can take the eigen-states in Eq. (222) as eigen-states of the
helicity operator $({\bf S}+{\bf L})\cdot\vec{n}_{{\bf k}}$ by choosing
$\displaystyle|\Psi_{A+\uparrow}(\theta,\phi)\rangle$
$\displaystyle=|s=+\tfrac{1}{2}\rangle\otimes|p=+\tfrac{1}{2},q=+1\rangle,$
(224a) $\displaystyle|\Psi_{B+\uparrow}(\theta,\phi)\rangle$
$\displaystyle=|s=+\tfrac{1}{2}\rangle\otimes|p=+\tfrac{1}{2},q=-1\rangle,$
(224b) $\displaystyle|\Psi_{A+\downarrow}(\theta,\phi)\rangle$
$\displaystyle=|s=-\tfrac{1}{2}\rangle\otimes|p=+\tfrac{1}{2},q=+1\rangle,$
(224c) $\displaystyle|\Psi_{B+\downarrow}(\theta,\phi)\rangle$
$\displaystyle=|s=-\tfrac{1}{2}\rangle\otimes|p=+\tfrac{1}{2},q=-1\rangle.$
(224d)
Here we use $s=\pm 1/2$ as eigenvalues of ${\bf S}\cdot\vec{n}_{{\bf k}}$.
Here we use $\vec{n}_{{\bf k}}$, $\vec{n}_{{\bf k}}^{\prime}$, and
$\vec{n}_{{\bf k}}^{\prime\prime}$ to be the set of 3D orthogonal coordinates
at the fixed ${\bf k}$ and $\vec{n}_{{\bf k}}$ is the direction of ${\bf k}$
(i.e., $\vec{n}_{{\bf
k}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$). Namely,
$|\vec{n}_{{\bf k}}|=|\vec{n}_{{\bf k}}^{\prime}|=|\vec{n}_{{\bf
k}}^{\prime\prime}|=1$, and $\vec{n}_{{\bf k}}\cdot\vec{n}_{{\bf
k}}^{\prime}=\vec{n}_{{\bf k}}\cdot\vec{n}_{{\bf
k}}^{\prime\prime}=\vec{n}_{{\bf k}}^{\prime}\cdot\vec{n}_{{\bf
k}}^{\prime\prime}=0$. Then, the first-order projection of ${\bf S}\cdot{\bf
L}=[({\bf S}\cdot\vec{n}_{{\bf k}})({\bf L}\cdot\vec{n}_{{\bf k}})+({\bf
S}\cdot\vec{n}_{{\bf k}}^{\prime})({\bf L}\cdot\vec{n}_{{\bf
k}}^{\prime})+({\bf S}\cdot\vec{n}_{{\bf k}}^{\prime\prime})({\bf
L}\cdot\vec{n}_{{\bf k}}^{\prime\prime})]$ onto these four states in Eq. (224)
leads to
$\displaystyle\begin{split}[\left\langle\mathcal{H}_{\text{soc}}\right\rangle_{\mathcal{H}_{1}}]_{i,j}&=4\lambda_{0}\langle\Psi_{i}(\theta,\phi)|{\bf
S}\cdot{\bf L}|\Psi_{j}(\theta,\phi)\rangle,\\\
&=4\lambda_{0}\langle\Psi_{i}(\theta,\phi)|({\bf S}\cdot\vec{n}_{{\bf
k}})({\bf L}\cdot\vec{n}_{{\bf k}})+({\bf S}\cdot\vec{n}_{{\bf
k}}^{\prime})({\bf L}\cdot\vec{n}_{{\bf k}}^{\prime})+({\bf
S}\cdot\vec{n}_{{\bf k}}^{\prime\prime})({\bf L}\cdot\vec{n}_{{\bf
k}}^{\prime\prime})|\Psi_{j}(\theta,\phi)\rangle,\\\
&=4\lambda_{0}\langle\Psi_{i}(\theta,\phi)|({\bf S}\cdot\vec{n}_{{\bf
k}})({\bf L}\cdot\vec{n}_{{\bf k}})|\Psi_{j}(\theta,\phi)\rangle,\end{split}$
(225)
where $i,j=\\{A+\uparrow,B+\uparrow,A+\downarrow,B+\downarrow\\}$. Thus, the
first-order perturbation for on-site SOC Hamiltonian is
$\displaystyle\left\langle\mathcal{H}_{\text{soc}}\right\rangle_{\mathcal{H}_{1}}=\lambda_{0}\begin{pmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-1\end{pmatrix}.$ (226)
Please notice that
$\left\langle\mathcal{H}_{\text{soc}}\right\rangle_{\mathcal{H}_{1}}$ can be
also obtained after diagonalizing $\mathcal{H}_{\text{soc},+}^{eff(1)}({\bf
k})$ in Eq. (219). At nonzero ${\bf k}$, we find that the fourfold degenerate
states [$C_{0}+A_{1}k$ for $\mathcal{H}_{1}({\bf k})$] are split by the on-
site SOC Hamiltonian $\mathcal{H}_{\text{soc}}$ into two states and each state
has tow fold degeneracy stemming from the orbital SU(2) symmetry,
$E=C_{0}+A_{1}k\pm\lambda_{0}$. And the eigen-states are just Eq. (224). And
the spin-polarization along the direction of ${\bf k}$ represents the spin
U(1) symmetry.
* •
Only do the projection of the $k^{2}$-order Hamiltonian. In this case, the
spin SU(2) symmetry preserves. Since the $k^{2}$-order Hamiltonian is spin-
independent. Thus,
$\displaystyle\text{SU}_{\text{s}}(2)\times\text{SU}_{\text{o}}(2)\stackrel{{\scriptstyle\left\langle\mathcal{H}_{2}\right\rangle_{\mathcal{H}_{1}}}}{{\xhookrightarrow{\quad\quad\quad\quad\quad}}}\text{SU}_{\text{s}}(2)\times\text{U}_{\text{o}}(1).$
(227)
We do not have an elegant picture for the
$\left\langle\mathcal{H}_{2}\right\rangle_{\mathcal{H}_{1}}$. But we know it
does not dependent on spin, as shown in Eq. (219). The four-fold degeneracy is
split into two states, and each state has two fold degeneracy (i.e. spin
degeneracy). And, the two-by-two matrix in the $\\{A,B\\}$-subspace itself
severs as the orbital U(1) symmetry.
Therefore, up to the first order perturbation, we obtain a hierarchy of quasi-
symmetry for CoSi mentioned in the main text (see Eq. (7)),
$\displaystyle\text{SU}_{\text{s}}(2)\times\text{SU}_{\text{o}}(2)\stackrel{{\scriptstyle\left\langle\mathcal{H}_{\text{soc}}+\mathcal{H}_{2}\right\rangle_{\mathcal{H}_{1}}}}{{\xhookrightarrow{\quad\quad\quad\quad\quad\quad\quad}}}\text{U}_{\text{s}}(1)\times\text{U}_{\text{o}}(1).$
(228)
And $[\text{U}_{\text{s}}(1),\text{U}_{\text{o}}(1)]=0$, as shown below.
### C.2 The analytical properties of the effective perturbation 4-band
$P$-model
The $P$-model has the self-commuting structure at every ${\bf k}$. Explicitly,
$\displaystyle\left[\mathcal{H}_{\text{soc},+}^{eff(1)}({\bf
k}_{1}),\mathcal{H}_{k^{2},+}^{eff(1)}({\bf k}_{2})\right]=0,\quad\forall{\bf
k}_{1}\;\&\;\forall{\bf k}_{2}\text{ in the whole momentum space}$ (229)
as discussed in the main text, which is known as the projected stabilizer code
Hamiltonian and directly leads to the U(1) quasi-symmetry group. Below, we
provide some analytical study on the properties of the effective $P$-model.
The eigen-energies of the $P$-model in Eq. (218) are
$\displaystyle\begin{split}E_{\alpha\beta}(k,\theta,\phi)=C_{0}+B_{1}k^{2}+A_{1}k+\alpha\lambda_{0}+\beta\tfrac{\sqrt{3}}{4}\tilde{C}k^{2}|\sin
2\phi\sin 2\theta\sin\theta|,\end{split}$ (230)
where $\alpha=\pm$ and $\beta=\pm$. Here $\sin 2\phi\sin
2\theta\sin\theta=4k_{x}k_{y}k_{z}/k^{3}$, indicating that there are two-fold
degeneracy on high symmetry planes ($k_{x}=0$ or $k_{y}=0$ or $k_{z}=0$).
For the $\Gamma-R-M$ plane, $\phi=\pi/4$ so $\sin 2\phi=1$. The eigen-energies
along high symmetry lines are listed as follows:
* 1.)
Along the $R-M$ line.
$\theta=0$, so $E_{\alpha}=C_{0}+B_{1}k^{2}+A_{1}k+\alpha\lambda_{0}$. All the
bands are two-fold degenerate.
* 2.)
Along the $R-Z$ line.
$\theta=\pi/2$, so $E_{\alpha}=C_{0}+B_{1}k^{2}+A_{1}k+\alpha\lambda_{0}$. All
the bands are two-fold degenerate.
Figure 5: The four distinct Fermi surfaces of CoSi are numerically
calculated, as shown in (a) for the $\Gamma-R-M$ plane. These high-symmetry
points are marked in the first Brillouin zone [left panel in (a)]. And these
four Fermi surfaces are labeled by four different colors. The middle two have
a “crossing” behavior with a tiny gap. And the analytical results for the band
dispersions (see Eq. (230)) of the $P$-model along the $R-M$ line in (b) and
the $R-Z$ line in (c). The two dashes black lines represents the Fermi
surfaces with $\alpha=\pm 1$ and $\beta=0$. The $\beta$-term leads to the
quadratic band splitting around the $R-M$ line in (a) with
$\delta_{k}\sim\delta_{\theta}^{2}$, and the linear band splitting around the
$R-Z$ line in (b) with $\delta_{k}\sim\delta_{\theta}$. Here, the orange line
is for $\alpha=-1,\beta=-1$, the blue line is for $\alpha=-1,\beta=+1$, the
green line is for $\alpha=+1,\beta=-1$, and the yellow line is for
$\alpha=+1,\beta=+1$. Once the band splitting caused by the $\beta$-term is
large enough, the band crossing between the blue line ($\beta=+1$) and the
green line ($\beta=-1$) may happen at arbitrary momenta, which are exact and
protected by the quasi-symmetry.
We then analyze the perturbation along these two high symmetry lines (i.e.,
the $R-M$ line and the $R-Z$ line) to identify the band crossing types of the
Fermi surface (linear or quadratic). For a given $E_{f}$, the Fermi surface
shape of the momentum $k_{\pm}$ for the upper four bands is determined by the
quadratic equation
$\displaystyle B_{1}k^{2}+A_{1}k+C_{\alpha}=0,$ (231)
where $C_{\alpha}=\alpha\lambda_{0}+C_{0}-E_{f}$ with $\alpha=\pm$. This
equation leads to the solution
$\displaystyle\begin{split}k_{\alpha,\theta=0}=k_{\alpha,\theta=\pi/2}=\frac{1}{2B_{1}}\left(-A_{1}+\sqrt{A_{1}^{2}-4B_{1}C_{\alpha}}\right).\end{split}$
(232)
We firstly focus on the $R-M$ line by expanding Eq. (230) around $\theta=0$.
In this case, we have $\theta=0+\delta_{\theta}$ and
$k=k_{\alpha,\theta=0}+\delta_{k}$. Then, the Eq. (231) should be replaced by
$\displaystyle
0=C_{\alpha}+B_{1}k^{2}+A_{1}k+\beta\tfrac{\sqrt{3}}{4}\tilde{C}k^{2}|\sin
2\phi\sin 2\theta\sin\theta|,$ (233)
which leads to
$\displaystyle\begin{split}0=C_{\alpha}+B_{1}\left(k_{\alpha,\theta=0}+\delta_{k}\right)^{2}+A_{1}\left(k_{\alpha,\theta=0}+\delta_{k}\right)+\beta\tfrac{\sqrt{3}}{4}\tilde{C}\left(k_{\alpha,\theta=0}+\delta_{k}\right)^{2}|\sin
2\phi|(2\delta_{\theta}^{2}).\end{split}$ (234)
After neglecting the $\delta^{2}_{k}$ terms, we find
$\displaystyle\begin{split}\delta_{k,\beta}=-\beta\frac{\sqrt{3}}{2}\frac{\tilde{C}|\sin
2\phi|\delta_{\theta}^{2}(k_{\alpha,\theta=0})^{2}}{2B_{1}k_{\alpha,\theta=0}+A_{1}+\beta\sqrt{3}\tilde{C}|\sin
2\phi|\delta_{\theta}^{2}k_{\alpha,\theta=0}}.\end{split}$ (235)
To the $\delta_{\theta}^{2}$ order, we have
$\displaystyle\delta_{k,\beta}=-\beta\frac{\sqrt{3}}{2}\frac{\tilde{C}|\sin
2\phi|(k_{\alpha,\theta=0})^{2}}{2B_{1}k_{\alpha,\theta=0}+A_{1}}\times\delta_{\theta}^{2},$
(236)
which indicates that the exact crossing along the $R-M$ line is quadratic in
momentum $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$, as illustrated in Fig. 5 (a) around
the $R-M$ line.
For $\theta=\pi/2$ (i.e., the $R-Z$ line), we then expand Eq. (230) around
$\theta=\pi/2+\delta_{\theta}$. Because
$\displaystyle\begin{split}\sin
2\theta\sin\theta\to\sin(\pi+2\delta_{\theta})\sin(\pi/2+\delta_{\theta})\to-2\delta_{\theta}+\mathcal{O}(\delta_{\theta}^{3}),\end{split}$
(237)
It leads to the solution for $k$ at a fixed $E_{f}$,
$\displaystyle\delta_{k,\beta}=-\beta\frac{\sqrt{3}}{2}\frac{\tilde{C}|\sin
2\phi|(k_{\alpha,\theta=\pi/2})^{2}}{2B_{1}k_{\alpha,\theta=0}+A_{1}}\times|\delta_{\theta}|,$
(238)
which indicate the exact crossing along $k_{z}$ is linear, as illustrated in
Fig. 5 (b) around the $R-Z$ line.
Figure 6: Comparison of the Fermi surfaces with the $R$-model in (a) and the
$P$-model in (b) around the quasi nodal points in the $\Gamma-R-M$ plane. (a)
shows a tiny gap in the blue circle, and (b) shows the exact quasi-symmetry
protected degeneracy.
In addition, we discuss the emergent nodal lines on Fermi surfaces for the
$P$-model. The comparison of the FSs between the 8-band ${\bf k}\cdot{\bf p}$
effective Hamiltonian, the $R$-model (e.g. see Eq. (54) in Sec. A.3) and the
first-order-perturbation 4-band Hamiltonian, the $P$-model (e.g. see Eq. (218)
in Sec. C.1) are shown in Fig. 6. The $P$-model shows exact degeneracy at non-
high-symmetry points. The crossings from two bands can be obtained from the
constraint equation
$\displaystyle E_{\alpha=+1,\beta=-1}=E_{\alpha=-1,\beta=+1},$ (239)
of which the solution
$\displaystyle\lambda_{0}=\tfrac{\sqrt{3}}{4}\tilde{C}k^{2}|\sin 2\phi\sin
2\theta\sin\theta|=\sqrt{3}\tilde{C}\frac{|k_{x}k_{y}k_{z}|}{k}$ (240)
generally leads to nodal planes. The obtained nodal planes do not intersect
with high symmetry planes, e.g. $k_{x}=0$ or $k_{y}=0$ or $k_{z}=0$. At the
Fermi energy, we require an additional constraint equation
$\displaystyle E_{f}=E_{\alpha=+1,\beta=-1}=E_{\alpha=-1,\beta=+1}.$ (241)
By solving the above equation, we find the curve equation for the emergent
nodal lines at the Fermi energy
$\displaystyle\begin{split}\sqrt{3}\tilde{C}\frac{|k_{x}k_{y}k_{z}|}{k^{3}}=\tfrac{\sqrt{3}}{4}\tilde{C}|\sin
2\phi\sin
2\theta\sin\theta|=\frac{A_{1}^{2}\lambda_{0}}{2(E_{f}-C_{0})^{2}}\Big{[}1+\frac{2B_{1}(E_{f}-C_{0})}{A_{1}^{2}}+\sqrt{1+\frac{4B_{1}(E_{f}-C_{0})}{A_{1}^{2}}}\Big{]}.\end{split}$
(242)
For a fixed Fermi energy $E_{f}$, solving Eq. (242) generally gives rise to a
line solution in the $\theta-\phi$ plane. It represents a nodal line because
of Eq. (240): solving $\theta$ and $\phi$ at fixed $E_{f}$ from Eq. (242) will
fix $k$ simultaneously. However, the existence of such a line solution for Eq.
(242) depends on the value of $E_{f}$. For a critical $E_{f}$, there is no
line solution from the curve equation (242), instead, we can only get a point
solution. To find this minimal $E_{f}$, we set $\phi=\pi/4$, and notice that
the function $|\sin 2\theta\sin\theta|$ reaches its maximum when
$\theta\to\arcsin\sqrt{\tfrac{2}{3}}$, we find the minimal $E_{f}$ as
$\displaystyle
E_{f,c}=\frac{A_{1}\sqrt{C_{4}\lambda_{0}}+B_{1}\lambda_{0}+C_{0}C_{4}}{C_{4}}.$
(243)
Thus, $E_{f,c}$ is the energy for the single nodes with twofold degeneracy on
the corresponding nodal planes. Based on this analysis, we realize that the
quasi-nodal-line will emerge into single nodes when decreasing the Fermi
energy. For $E_{f}<E_{f,c}$, no solution of Eq. (242) can be obtained anymore.
Therefore, one can conclude that each nodal line emerges into a single point
after decreasing $E_{f}$ down to $E_{f,c}$. Moreover, when the nodal plane is
split due to high-order perturbations, this point becomes a Weyl point pinned
along the (111)-axis. Thus, we conclude that, nodal lines exist only when
$E_{f}\geq E_{f,c}$.
### C.3 The second-order perturbation: gap out the quasi-nodal planes
Next we consider the second-order perturbation corrections for the 4-band
$P$-model (e.g. see Eq. (218) in Sec. C.1), which can open a tiny gap for the
emergent nodal lines obtained from Eq. (242) at generic momenta. Recall that
the first order perturbation Hamiltonian, the $P$-model, is based on the basis
in Eq. (215)
$\displaystyle|\Psi_{+}\rangle\triangleq|\Psi_{upper}\rangle=\\{(1,0)^{T}\otimes|\Psi_{A+}(\theta,\phi)\rangle,(1,0)^{T}\otimes|\Psi_{B+}(\theta,\phi)\rangle,(0,1)^{T}\otimes|\Psi_{A+}(\theta,\phi)\rangle,(0,1)^{T}\otimes|\Psi_{B+}(\theta,\phi)\rangle\\},$
(244)
which are all eigen-states of $\mathcal{H}_{1}$ with the same eigen-energy
$E_{+}=C_{0}+A_{1}k$. And the spinless wavefunctions
$|\Psi_{A/B+}(\theta,\phi)\rangle$ are given by Eq. (210a),
$\displaystyle|\Psi_{A+}(\theta,\phi)\rangle$
$\displaystyle=\tfrac{1}{\sqrt{2}}\left(\cos\theta\cos\phi-i\sin\phi,-\cos\theta\sin\phi-i\cos\phi,0,\sin\theta\right)^{T},$
(245a) $\displaystyle|\Psi_{B+}(\theta,\phi)\rangle$
$\displaystyle=\tfrac{1}{\sqrt{2}}\left(-i\sin\theta\cos\phi,i\sin\theta\sin\phi,1,i\cos\theta\right)^{T},$
(245b)
where $A/B$ represent the eigen-values of $\vec{\mathcal{M}}\cdot\vec{n}$. The
second-order perturbation theory has been presented in the supplementary
materials in Ref. [1]. To make sure the completeness of this appendix, we
repeat the discussion of the second-order perturbation in this sub-section.
Here, we consider the inter-band correction via second-order perturbation. The
fourfold degenerate eigen-states of $\mathcal{H}_{1}$ with lower energy
($E_{-}=C_{0}-A_{1}k$) are given by
$\displaystyle|\Psi_{lower}\rangle=\left\\{(1,0)^{T}\otimes|\Psi_{B-}(\theta,\phi)\rangle,(1,0)^{T}\otimes|\Psi_{A-}(\theta,\phi)\rangle,(0,1)^{T}\otimes|\Psi_{B-}(\theta,\phi)\rangle,(0,1)^{T}\otimes|\Psi_{A-}(\theta,\phi)\rangle\right\\},$
(246)
which are all eigen-states of of $\mathcal{H}_{1}$ with the same eigen-energy
$E_{-}=C_{0}-A_{1}k$. And the spinless wavefunctions
$|\Psi_{A/B-}(\theta,\phi)\rangle$ are given by Eq. (210b),
$\displaystyle|\Psi_{A-}(\theta,\phi)\rangle$
$\displaystyle=\tfrac{1}{\sqrt{2}}\left(i\sin\theta\cos\phi,-i\sin\theta\sin\phi,1,-i\cos\theta\right),$
(247a) $\displaystyle|\Psi_{B-}(\theta,\phi)\rangle$
$\displaystyle=\tfrac{1}{\sqrt{2}}\left(\cos\theta\cos\phi+i\sin\phi,-\cos\theta\sin\phi+i\cos\phi,0,\sin\theta\right)^{T}.$
(247b)
Therefore, the second-order perturbed Hamiltonian is given by
$\displaystyle\Delta\mathcal{H}_{P}^{eff(2)}({\bf k})=\frac{1}{\Delta
E}\left(\langle\Psi_{upper}|(\mathcal{H}_{soc}+\mathcal{H}_{2}(\mathbf{k}))\hat{P}_{lower}(\mathcal{H}_{soc}+\mathcal{H}_{2}(\mathbf{k}))|\Psi_{upper}\rangle\right),$
(248)
where $\Delta E=E_{+}-E_{-}=2A_{1}k$ is the energy difference between the
upper-energy-band the lower-energy-band, and the projection operator
$\hat{P}_{lower}=|\Psi_{lower}\rangle\langle\Psi_{lower}|$ onto the lower four
bands in Eq. (246). The mixed terms of $\mathcal{H}_{soc}$ and
$\mathcal{H}_{2}(\mathbf{k})$ for the second order perturbation are given by
$\displaystyle\Delta\mathcal{H}_{P}^{eff(2)}({\bf k})=\frac{1}{\Delta
E}\left(\langle\Psi_{upper}|\mathcal{H}_{soc}|\Psi_{lower}\rangle\langle\Psi_{lower}|\mathcal{H}_{2}(\mathbf{k})|\Psi_{upper}\rangle\right)+\text{h.c.},$
(249)
with the matrix elements of $\Delta\mathcal{H}_{P}^{eff(2)}$
$\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{1,1}$
$\displaystyle=2\sin^{2}(\theta)\sin(\phi)\cos(\phi)\left(\sin^{3}(\theta)\sin(\phi)+\sin(\theta)\cos^{2}(\theta)\cos(\phi)-\cos^{3}(\theta)\right),$
(250) $\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{1,2}$
$\displaystyle=\sin^{2}(\theta)\cos(\phi)(\sin^{2}(\theta)\sin(2\phi)+2i\cos(\theta)\sin(\phi)(\cos(\theta)(\cos(\theta)\cos(\phi)$
$\displaystyle+\sin(\theta))+\sin(\phi)(\sin^{2}(\theta)+i\cos(\theta)))),$
(251) $\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{1,3}$
$\displaystyle=\tfrac{1}{4}e^{-i\phi}\sin(\theta)\cos(\theta)(\sin(2\phi)(\sin(3\theta)\sin(\phi)+4\cos^{3}(\theta))-8\sin(\theta)\cos^{2}(\theta)$
$\displaystyle\sin(\phi)\cos^{2}(\phi)+8i\cos(\theta)\sin^{2}(\phi)+(-\tfrac{3}{2}+6i)\sin(\theta)\cos(\phi)+(\tfrac{3}{2}+2i)\sin(\theta)$
$\displaystyle\cos(3\phi)),$ (252)
$\displaystyle~{}\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{1,4}$
$\displaystyle=e^{-i\phi}\sin(\theta)((\cos(\theta)-1)(\cos^{2}(\theta)\cos(\phi)+i\sin(\theta)\sin(\phi)\cos(\phi)(-\cos^{2}(\theta)$
$\displaystyle+\sin(\theta)(\cos(\theta)+i)\cos(\phi))+\sin(\theta)\cos(\theta)\sin^{2}(\phi)(-1-i\sin(\theta)\cos(\phi)))$
$\displaystyle-2i\cos^{2}(\tfrac{\theta}{2})(-i\cos^{2}(\theta)\cos(\phi)\cos(2\phi)+\sin(\phi)\cos(\phi)((\cos^{3}(\theta)+\cos(\theta))$
$\displaystyle\cos(\phi)-i\sin^{2}(\theta)\cos(\phi)+\sin(\theta)\cos^{2}(\theta))+\sin(\theta)\cos(\theta)\sin^{2}(\phi)(\sin(\theta)$
$\displaystyle\cos(\phi)+i)))$ (253)
and
$\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{2,2}$
$\displaystyle=-\tfrac{1}{4}\sin^{5}(\theta)\sin(\phi)\sin(2\phi)$
$\displaystyle~{}~{}~{}~{}\left(\cot^{2}(\theta)(-4\cot(\theta)\csc(\phi)+4\cot(\phi)-1)+\csc^{2}(\theta)+3\right),$
(254) $\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{2,3}$
$\displaystyle=\tfrac{1}{2}e^{-i\phi}\sin(\theta)\cos(\theta)(-4\sin(\phi)(\sin(\theta)\sin(\phi)+(2+2i)\sin^{2}(\frac{\theta}{2})\cos^{2}(\phi))+$
$\displaystyle\cos^{2}(\theta)\sin(2\phi)(-4i\sin^{2}(\tfrac{\theta}{2})\cos(\phi)+2i\sin(\theta)+\csc(\phi))+\cos(\theta)\csc(\phi)$
$\displaystyle(-(2+2i)\sin^{2}(\tfrac{\theta}{2})\sin^{2}(2\phi)+\sin^{2}(\tfrac{\theta}{2})\sin(4\phi)+4i\sin^{2}(\theta)\sin^{3}(\phi)\cos(\phi)$
$\displaystyle+\sin(2\phi))),$ (255)
$\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{2,4}$
$\displaystyle=-\tfrac{1}{8}e^{-i\phi}\sin(\theta)\cos(\theta)(8\cos^{3}(\theta)\sin(2\phi)-16\sin(\theta)\cos^{2}(\theta)\sin(\phi)\cos^{2}(\phi)$
$\displaystyle+16i\cos(\theta)\sin^{2}(\phi)+\cos(\phi)(4\sin(3\theta)\sin^{2}(\phi)+(-3+12i)\sin(\theta))$
$\displaystyle+(3+4i)\sin(\theta)\cos(3\phi)),$ (256)
and
$\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{3,3}$
$\displaystyle=-\tfrac{1}{4}\sin^{5}(\theta)\sin(\phi)\sin(2\phi)$
$\displaystyle\left(\cot^{2}(\theta)(-4\cot(\theta)\csc(\phi)+4\cot(\phi)-1)+\csc^{2}(\theta)+3\right),$
(257) $\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{3,4}$
$\displaystyle=-i\sin^{2}(\theta)\sin(\phi)\cos(\phi)(2\cos^{3}(\theta)\cos(\phi)+2\cos^{2}(\theta)(\sin(\theta)+i\sin(\phi))$
$\displaystyle+\sin(\theta)(\sin(2\theta)\sin(\phi)-2i\sin(\theta)\cos(\phi))),$
(258)
and
$\displaystyle\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{4,4}$
$\displaystyle=2\sin^{2}(\theta)\sin(\phi)\cos(\phi)(\sin^{3}(\theta)\sin(\phi)+\sin(\theta)\cos^{2}(\theta)\cos(\phi)-\cos^{3}(\theta)).$
(259)
The other parts are related by complex conjugation
$\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{i,j}=\left[\Delta\mathcal{H}_{P}^{eff(2)}\right]_{j,i}^{\ast}$.
### C.4 Additional terms due to the linear $k$ SOC
In the first-order perturbation for the $P$-model in Sec. C.1, we only
consider the contribution from on-site SOC Hamiltonian. Here, we further
discuss the additional first-order perturbation corrections stemming from the
linear-$k$ SOC in Eq. (53),
$\displaystyle\begin{split}\mathcal{H}_{k,\text{soc}}({\bf
k})&=\lambda_{1}(k_{x}s_{x}+k_{y}s_{y}+k_{z}s_{z})\otimes\sigma_{0}\tau_{0}+\lambda_{2}(k_{x}s_{y}\sigma_{x}\tau_{x}-k_{y}s_{z}\sigma_{0}\tau_{x}+k_{z}s_{x}\sigma_{x}\tau_{0})\\\
&+\lambda_{3}(k_{y}s_{x}\sigma_{x}\tau_{x}-k_{z}s_{y}\sigma_{0}\tau_{x}+k_{x}s_{z}\sigma_{x}\tau_{0})+\lambda_{4}(k_{x}s_{y}\sigma_{x}\tau_{z}-k_{y}s_{z}\sigma_{y}\tau_{y}-k_{z}s_{x}\sigma_{z}\tau_{x})\\\
&+\lambda_{5}(k_{y}s_{x}\sigma_{x}\tau_{z}-k_{z}s_{y}\sigma_{y}\tau_{y}-k_{x}s_{z}\sigma_{z}\tau_{x})+\lambda_{6}(k_{x}s_{y}\sigma_{z}\tau_{0}+k_{y}s_{z}\sigma_{0}\tau_{z}+k_{z}s_{x}\sigma_{z}\tau_{z})\\\
&+\lambda_{7}(k_{y}s_{x}\sigma_{z}\tau_{0}+k_{z}s_{y}\sigma_{0}\tau_{z}+k_{x}s_{z}\sigma_{z}\tau_{z}).\end{split}$
(260)
Then, we only need to compute $\langle\Psi_{+}|\mathcal{H}_{k,\text{soc}}({\bf
k})|\Psi_{+}\rangle$ based on the basis in Eq. (216),
$\displaystyle|\Psi_{+}\rangle=\\{(1,0)^{T}\otimes|\Psi_{A+}(\theta,\phi)\rangle,(1,0)^{T}\otimes|\Psi_{B+}(\theta,\phi)\rangle,(0,1)^{T}\otimes|\Psi_{A+}(\theta,\phi)\rangle,(0,1)^{T}\otimes|\Psi_{B+}(\theta,\phi)\rangle\\},$
(261)
which are all eigen-states of $\mathcal{H}_{1}({\bf k})$ with the same eigen-
energy $E_{+}=C_{0}+A_{1}k$. And the spinless wavefunctions
$|\Psi_{A/B+}(\theta,\phi)\rangle$ are given by Eq. (210a).
First, a term denoted by $\lambda_{1}$ only renormalizes $\lambda_{0}$ as
$\lambda_{0}\to\lambda_{0}+\lambda_{1}k$. Second, all the others terms denoted
by $\lambda_{2,3,4,5,6,7}$ generally gap the quasi-nodal plane obtained from
Eq. (240), i.e., $\lambda_{0}=\tfrac{\sqrt{3}}{4}\tilde{C}k^{2}|\sin 2\phi\sin
2\theta\sin\theta|$. For an illustration, we consider the $\lambda_{2}$ term
as an example, and the projected matrix is
$\displaystyle\mathcal{H}_{k,\lambda_{2},eff}(k,\theta,\phi)=\lambda_{2}k\left(\begin{array}[]{cccc}h_{11}&h_{12}&h_{13}&h_{14}\\\
h_{21}&h_{22}&h_{23}&h_{24}\\\ h_{31}&h_{32}&h_{33}&h_{34}\\\
h_{41}&h_{42}&h_{43}&h_{44}\\\ \end{array}\right),$ (266)
where the spherical coordinator is used, and
$\displaystyle\begin{split}h_{11}&=-\sin^{3}(\theta)\sin^{2}(\phi)\cos(\phi),\\\
h_{12}&=\frac{1}{2}\sin^{2}(\theta)\sin(2\phi)(-\cos(\phi)-i\cos(\theta)\sin(\phi)),\\\
h_{13}&=-\frac{1}{2}\sin(2\theta)(\cos(\theta)\sin(\phi)+i\sin(\theta)\cos^{2}(\phi)),\\\
h_{14}&=\sin(\theta)\cos(\theta)(\cos(\theta)\cos^{2}(\phi)+i\sin(\phi)(\sin(\theta)+\cos(\phi))),\\\
h_{22}&=\sin^{3}(\theta)\sin^{2}(\phi)\cos(\phi),\\\
h_{23}&=-\sin(\theta)\cos(\theta)(\cos(\theta)\cos^{2}(\phi)-i\sin(\phi)(\cos(\phi)-\sin(\theta))),\\\
h_{24}&=\frac{1}{2}\sin(2\theta)\left(\cos(\theta)\sin(\phi)+i\sin(\theta)\cos^{2}(\phi)\right),\\\
h_{33}&=\sin^{3}(\theta)\sin^{2}(\phi)\cos(\phi),\\\
h_{34}&=\frac{1}{2}\sin(2\phi)\sin^{2}(\theta)(\cos(\phi)+i\cos(\theta)\sin(\phi)),\\\
h_{44}&=-\sin^{3}(\theta)\sin^{2}(\phi)\cos(\phi).\end{split}$ (267)
The remaining parts can be obtained by complex conjugation. We find
$\displaystyle[\mathcal{M}_{eff},\mathcal{H}_{k,\lambda_{2},eff}({\bf k})]\neq
0\text{ for nonzero }{\bf k},$ (268)
where the U(1) quasi-symmetry $\mathcal{M}_{eff}$ is given by
$\mathcal{H}_{\text{soc},+}^{eff(1)}({\bf k})$ or
$\mathcal{H}_{k^{2},+}^{eff(1)}({\bf k})$. This suggests that the nodal planes
are generally gapped by involving the linear $k$ SOC terms. But this effect is
negligible.
## Appendix D Approach II for the hierarchy of the quasi-symmetry
In this section, we discuss the approach II for the hierarchy of the quasi-
symmetry and the perturbation theory based on the solution of
$\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}$ for the
$\mathcal{H}_{2}({\bf k})$. Please note that the algebra of the orbital SU(2)
quasi-symmetry defined in Eq. (196) in Sec. D.1,
$\displaystyle\mathcal{M}_{1,2,3}=\tfrac{1}{2}\\{s_{0}\sigma_{y}\tau_{z},s_{0}\sigma_{y}\tau_{x},s_{0}\sigma_{0}\tau_{y}\\},$
(269)
which all commute with $\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}$,
but do not commute between themselves. And
$[\mathcal{M}_{i},\mathcal{M}_{j}]=i\epsilon_{i,j,k}\mathcal{M}_{k}$.
Moreover, we can also defined the rotation for this orbital SU(2) quasi-
symmetry group
$\displaystyle\begin{split}\mathcal{U}(\theta,\phi)&=e^{i\theta\mathcal{M}_{3}}e^{i\phi\mathcal{M}_{2}}\\\
&=\cos\left(\tfrac{\theta}{2}\right)\cos\left(\tfrac{\phi}{2}\right)s_{0}\sigma_{0}\tau_{0}+i\sin\left(\tfrac{\theta}{2}\right)\cos\left(\tfrac{\phi}{2}\right)\mathcal{M}_{3}+i\cos\left(\tfrac{\theta}{2}\right)\sin\left(\tfrac{\phi}{2}\right)\mathcal{M}_{2}+i\sin\left(\tfrac{\theta}{2}\right)\sin\left(\tfrac{\phi}{2}\right)\mathcal{M}_{1}.\end{split}$
(270)
Clearly, $\mathcal{U}(\theta,\phi)$ commutes with
$s_{0}\otimes\mathcal{H}_{1}(\mathbf{k})+\mathcal{H}_{\text{soc}}$ for any
values of $\theta$ and $\phi$. The $\mathcal{U}(0,\frac{\pi}{2})$ can rotate
$\mathcal{M}_{1}$ to $\mathcal{M}_{3}$,
$\displaystyle\begin{split}&\mathcal{U}(0,\frac{\pi}{2})=e^{i\frac{\pi}{2}\mathcal{M}_{2}}=\frac{\sqrt{2}}{2}s_{0}\otimes\left(\sigma_{0}\tau_{0}+i\sigma_{y}\tau_{x}\right)\\\
\Rightarrow\quad&\mathcal{U}(0,\frac{\pi}{2})\mathcal{M}_{1}\mathcal{U}^{\dagger}(0,\frac{\pi}{2})=\mathcal{M}_{3}.\end{split}$
(271)
For better readability of this section, here, we first repeat the linear-$k$
Hamiltonian, on-site SOC Hamiltonian and $k^{2}$-order Hamiltonian,
$\displaystyle\mathcal{H}_{1}(\mathbf{k})$
$\displaystyle=C_{0}\sigma_{0}\tau_{0}+2A_{1}(\mathbf{k}\cdot\mathbf{L}),$
(272a) $\displaystyle\mathcal{H}_{\text{soc}}$
$\displaystyle=4\lambda_{0}(\mathbf{S}\cdot\mathbf{L}),$ (272b)
$\displaystyle\mathcal{H}_{2}(\mathbf{k})$
$\displaystyle=\mathcal{H}_{2,\mathcal{M}_{1}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{2}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{3}}(\mathbf{k}),$
(272c)
where the spin angular momentum ${\bf S}=\tfrac{1}{2}(s_{x},s_{y},s_{z})$, and
the orbital angular momentum operators are ${\bf
L}=(\tfrac{1}{2}\sigma_{y}\tau_{0},\tfrac{1}{2}\sigma_{x}\tau_{y},-\tfrac{1}{2}\sigma_{z}\tau_{y})$.
And each part of $\mathcal{H}_{2}(\mathbf{k})$ is given by
$\displaystyle\mathcal{H}_{2,\mathcal{M}_{i}}(\mathbf{k})={\bf g}_{i}({\bf
k})\cdot{\bf J}_{i},$ (273)
for $i=1,2,3$. Here we define the $k$-dependent vectors as
$\displaystyle\begin{split}{\bf g}_{1}({\bf
k})&=(C_{2}k_{x}k_{y},-C_{3}k_{x}k_{z},C_{1}k_{y}k_{z}),\\\ {\bf g}_{2}({\bf
k})&=(C_{3}k_{x}k_{y},C_{1}k_{x}k_{z},-C_{2}k_{y}k_{z}),\\\ {\bf g}_{3}({\bf
k})&=(C_{1}k_{x}k_{y},C_{2}k_{x}k_{z},-C_{3}k_{y}k_{z}).\end{split}$ (274)
and the corresponding vectors of operators
$\displaystyle\begin{split}{\bf
J}_{1}&=(\sigma_{x}\tau_{x},-\sigma_{z}\tau_{x},\sigma_{0}\tau_{z}),\\\ {\bf
J}_{2}&=(\sigma_{x}\tau_{z},\sigma_{z}\tau_{z},\sigma_{0}\tau_{x}),\\\ {\bf
J}_{3}&=(\sigma_{z}\tau_{0},\sigma_{x}\tau_{0},\sigma_{y}\tau_{y}).\end{split}$
(275)
In addition, we also realize that the $k^{2}$ terms of $\mathcal{H}_{2}({\bf
k})$ break this orbital SU${}_{\text{o}}$(2) quasi-symmetry generated by
$\\{\mathcal{M}_{1,2,3}\\}$ and lead to the splitting of all bands. However,
different parts of the entire $k^{2}$-terms can lead to the reduction from
SU${}_{\text{o}}$(2) to a orbital U(1). To show that, as we discussed in the
main text, we find that
$\displaystyle[{\bf J}_{i},\mathcal{M}_{i}]=0\text{ and }\\{{\bf
J}_{i},\mathcal{M}_{j}\\}=0\text{ for }i\neq j,$ (276)
which implies
$\displaystyle[\mathcal{H}_{2,\mathcal{M}_{i}}({\bf k}),\mathcal{M}_{i}]=0.$
(277)
### D.1 The algebra for the orbital SU(2) quasi-symmetry for the
$k^{2}$-order Hamiltonian
In this subsection, we discuss the algebra of the orbital SU(2) quasi-symmetry
in Eq. (196). To show the generality of the breaking of the orbital SU(2)
quasi-symmetry down to U(1) quasi-symmetry by the $k^{2}$-order Hamiltonian.
We can rewrite the $\mathcal{H}_{2}({\bf k})$ into the form
$\displaystyle\mathcal{H}_{2}(\mathbf{k})=k_{x}k_{y}\left(\vec{C}_{\mathcal{J}}\cdot\vec{\mathcal{J}}\right)+k_{x}k_{z}\left(\vec{C}_{\mathcal{P}}\cdot\vec{\mathcal{P}}\right)+k_{y}k_{z}\left(\vec{C}_{\mathcal{Q}}\cdot\vec{\mathcal{Q}}\right),$
(278)
where the three ${\bf k}$-independent parameter-vectors are
$\displaystyle\begin{split}\vec{C}_{\mathcal{J}}&=(C_{2},-C_{3},C_{1}),\\\
\vec{C}_{\mathcal{P}}&=(-C_{3},C_{1},C_{2}),\\\
\vec{C}_{\mathcal{Q}}&=(C_{1},-C_{2},-C_{3}),\\\ \end{split}$ (279)
and the corresponding operator-vectors are given by
$\displaystyle\begin{split}\vec{\mathcal{J}}&=(\sigma_{x}\tau_{x},-\sigma_{x}\tau_{z},\sigma_{z}\tau_{0}),\\\
\vec{\mathcal{P}}&=(-\sigma_{z}\tau_{x},\sigma_{z}\tau_{z},\sigma_{x}\tau_{0}),\\\
\vec{\mathcal{Q}}&=(\sigma_{0}\tau_{z},\sigma_{0}\tau_{x},\sigma_{y}\tau_{y}).\end{split}$
(280)
Moreover, we notice that
$\displaystyle\begin{cases}\;[\mathcal{J}_{a},\mathcal{M}_{b}]=i\epsilon_{abc}\mathcal{J}_{c},\\\
\;[\mathcal{P}_{a},\mathcal{M}_{b}]=i\epsilon_{abc}\mathcal{P}_{c},\\\
\;[\mathcal{Q}_{a},\mathcal{M}_{b}]=i\epsilon_{abc}\mathcal{Q}_{c},\end{cases}$
(281)
where $\epsilon_{abc}$ is the three-dimensional Levi-Civita symbol with
$a,b,c=1,2,3$. Therefore, for arbitrary real normalized vector
$\vec{n}=(n_{1},n_{2},n_{3})$, we have the following commutation relations
$\displaystyle[\vec{\mathcal{J}}\cdot\vec{n},\vec{\mathcal{M}}\cdot\vec{n}]=[\vec{\mathcal{P}}\cdot\vec{n},\vec{\mathcal{M}}\cdot\vec{n}]=[\vec{\mathcal{Q}}\cdot\vec{n},\vec{\mathcal{M}}\cdot\vec{n}]=0.$
(282)
These can be easily shown, for example,
$\displaystyle\begin{split}[\vec{\mathcal{J}}\cdot\vec{n},\vec{\mathcal{M}}\cdot\vec{n}]&=\sum_{a=1}^{3}\sum_{b=1}^{3}n_{a}n_{b}[\mathcal{J}_{a},\mathcal{M}_{b}]=\sum_{a=1}^{3}\sum_{b=1}^{3}n_{a}n_{b}\left(i\epsilon_{abc}\mathcal{J}_{c}\right)\\\
&=\sum_{a=1}^{3}\sum_{b=a+1}^{3}n_{a}n_{b}\left(i\epsilon_{abc}\mathcal{J}_{c}+i\epsilon_{bac}\mathcal{J}_{c}\right)=0.\end{split}$
(283)
Here we have used $\epsilon_{abc}+\epsilon_{bac}=0$. Therefore, we find a
general symmetry-breaking case with the orbital SU(2) quasi-symmetry down to
the U(1) quasi-symmetry. For any three normalized and orthogonal vector,
$\vec{n}$, $\vec{n}^{\prime}$ and $\vec{n}^{\prime\prime}$, satisfy
$|\vec{n}|=|\vec{n}^{\prime}|=|\vec{n}^{\prime\prime}|=1$ and
$\vec{n}\perp\vec{n}^{\prime}$, $\vec{n}\perp\vec{n}^{\prime\prime}$ and
$\vec{n}^{\prime}\perp\vec{n}^{\prime\prime}$. Then we can do the projection
for the $k^{2}$-order Hamiltonian,
$\displaystyle\mathcal{H}_{2}({\bf k})=\mathcal{H}_{2,\vec{n}}({\bf
k})+\mathcal{H}_{2,\vec{n}^{\prime}}({\bf
k})+\mathcal{H}_{2,\vec{n}^{\prime\prime}}({\bf k}),$ (284)
where we project the parameter-vectors
($\vec{C}_{\mathcal{J}},\vec{C}_{\mathcal{P}},\vec{C}_{\mathcal{Q}}$) into the
$\\{\vec{n},\vec{n}^{\prime},\vec{n}^{\prime\prime}\\}$ space,
$\displaystyle\vec{C}_{\mathcal{J}}=\vec{n}(\vec{C}_{\mathcal{J}}\cdot\vec{n})+\vec{n}^{\prime}(\vec{C}_{\mathcal{J}}\cdot\vec{n}^{\prime})+\vec{n}^{\prime\prime}(\vec{C}_{\mathcal{J}}\cdot\vec{n}^{\prime\prime}),$
(285a)
$\displaystyle\vec{C}_{\mathcal{P}}=\vec{n}(\vec{C}_{\mathcal{P}}\cdot\vec{n})+\vec{n}^{\prime}(\vec{C}_{\mathcal{P}}\cdot\vec{n}^{\prime})+\vec{n}^{\prime\prime}(\vec{C}_{\mathcal{P}}\cdot\vec{n}^{\prime\prime}),$
(285b)
$\displaystyle\vec{C}_{\mathcal{Q}}=\vec{n}(\vec{C}_{\mathcal{Q}}\cdot\vec{n})+\vec{n}^{\prime}(\vec{C}_{\mathcal{Q}}\cdot\vec{n}^{\prime})+\vec{n}^{\prime\prime}(\vec{C}_{\mathcal{Q}}\cdot\vec{n}^{\prime\prime}),$
(285c)
Therefore, the first term in Eq. (284) is given by
$\displaystyle\mathcal{H}_{2,\vec{n}}({\bf
k})=k_{x}k_{y}(\vec{C}_{\mathcal{J}}\cdot\vec{n})\times\left(\vec{n}\cdot\vec{\mathcal{J}}\right)+k_{x}k_{z}(\vec{C}_{\mathcal{P}}\cdot\vec{n})\times\left(\vec{n}\cdot\vec{\mathcal{P}}\right)+k_{y}k_{z}(\vec{C}_{\mathcal{Q}}\cdot\vec{n})\left(\vec{n}\cdot\vec{\mathcal{Q}}\right),$
(286)
which commutes with $\mathcal{M}\cdot\vec{n}$,
$\displaystyle[\mathcal{M}\cdot\vec{n},\mathcal{H}_{2,\vec{n}}({\bf k})]=0.$
(287)
Especially, in the main text, we have mentioned three cases,
* •
$\vec{n}=(1,0,0)$. The $\mathcal{H}_{2,\vec{n}}({\bf k})$ Hamiltonian is given
by $\mathcal{H}_{2,\mathcal{M}_{1}}({\bf k})$ in Eq. (200).
* •
$\vec{n}=(0,1,0)$. The $\mathcal{H}_{2,\vec{n}}({\bf k})$ Hamiltonian is given
by $\mathcal{H}_{2,\mathcal{M}_{2}}({\bf k})$ in Eq. (200).
* •
$\vec{n}=(0,0,1)$. The $\mathcal{H}_{2,\vec{n}}({\bf k})$ Hamiltonian is given
by $\mathcal{H}_{2,\mathcal{M}_{3}}({\bf k})$ in Eq. (200).
Moreover, the U(1) quasi-symmetry protected nodal-plane for
$\mathcal{H}_{2,\mathcal{M}_{1}}$ is discussed in the main text. More details
will be discussed in the following Sec. D.2, Sec. D.3, and Sec. D.4.
### D.2 Analytical solutions by using the U(1) quasi-symmetry
Furthermore, we show the important role of the orbital SU(2) quasi-symmetry
operators (i.e. $\mathcal{M}_{1,2,3}$) in analytically solving the eigen-state
problem. Here, we take $\mathcal{H}_{1}(\mathbf{k})$ in Eq. (26) or Eq. (28)
as an example,
$\displaystyle\mathcal{H}_{1}(\mathbf{k})=C_{0}+A_{1}(k_{x}\sigma_{y}\tau_{0}+k_{y}\sigma_{x}\tau_{y}-k_{z}\sigma_{z}\tau_{y}),$
(288)
which commutes with the orbital SU(2) quasi-symmetry. Here we choose the
eigen-states of $\mathcal{H}_{1}(\mathbf{k})$ to be the common eigen-state of
$\mathcal{M}_{3}=\sigma_{0}\tau_{y}$ in Eq. (196). It is equivalently to apply
a unitary transformation
$\displaystyle\mathcal{U}=\frac{1}{\sqrt{2}}s_{0}\otimes\sigma_{0}\otimes\begin{pmatrix}1&-i\\\
1&i\end{pmatrix},$ (289)
to the $R$-model, which only leads to a rotation in the
$(\tau_{x},\tau_{y},\tau_{z})$ subspace
$\displaystyle\mathcal{U}\tau_{0}\mathcal{U}^{\dagger}=\tau_{0},\mathcal{U}\tau_{x}\mathcal{U}^{\dagger}=\tau_{y},\mathcal{U}\tau_{y}\mathcal{U}^{\dagger}=\tau_{z},\mathcal{U}\tau_{z}\mathcal{U}^{\dagger}=\tau_{x}.$
(290)
Therefore, the quasi-symmetry ${\cal M}_{3}$ becomes ${\cal U}{\cal
M}_{3}{\cal U}^{\dagger}=\sigma_{0}\tau_{z}$. Note that the spin Pauli matrix
is dropped here. As a result, the linear-$k$ Hamiltonian becomes
$\displaystyle{\cal U}[\mathcal{H}_{1}(\mathbf{k})-C_{0}]{\cal U}^{\dagger}$
$\displaystyle=\left[\mathcal{H}_{1,\mathcal{M}_{3}=+1}(\mathbf{k})\right]_{2\times
2}\oplus\left[\mathcal{H}_{1,\mathcal{M}_{3}=-1}(\mathbf{k})\right]_{2\times
2}$
$\displaystyle=\begin{pmatrix}\mathcal{H}_{1,\mathcal{M}_{3}=+1}(\mathbf{k})&0\\\
0&\mathcal{H}_{1,\mathcal{M}_{3}=-1}(\mathbf{k})\end{pmatrix},$ (291a)
$\displaystyle\mathcal{H}_{1,\mathcal{M}_{3}=+1}(\mathbf{k})$
$\displaystyle=A_{1}(k_{x}\sigma_{y}+k_{y}\sigma_{x}-k_{z}\sigma_{z}),$ (291b)
$\displaystyle\mathcal{H}_{1,\mathcal{M}_{3}=-1}(\mathbf{k})$
$\displaystyle=A_{1}(k_{x}\sigma_{y}-k_{y}\sigma_{x}+k_{z}\sigma_{z}).$ (291c)
It is easy to analytically find the eigen-states of the 2-by-2 Hamiltonian
$\mathcal{H}_{1,\mathcal{M}_{3}=\pm 1}(\mathbf{k})$, in the spherical
coordinate with the momentum ${\bf
k}=(k_{x},k_{y},k_{z})=k(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$.
The eigen-states are given by
$\displaystyle\mathcal{H}_{1,\mathcal{M}_{3}=+1}(\mathbf{k})|A-\rangle=-A_{1}k|A-\rangle,\;\text{
with
}|A-\rangle=\frac{1}{\sqrt{2}\sqrt{1+\cos\theta}}(i(\cos\theta+1),e^{-i\phi}\sin\theta)^{T},$
(292a)
$\displaystyle\mathcal{H}_{1,\mathcal{M}_{3}=+1}(\mathbf{k})|A+\rangle=\;\;\;A_{1}k|A+\rangle,\;\text{
with
}|A+\rangle=\frac{1}{\sqrt{2}\sqrt{1-\cos\theta}}(i(\cos\theta-1),e^{-i\phi}\sin\theta)^{T},$
(292b)
and by substituting the replacement
$(k_{x},k_{y},k_{z})\to(k_{x},-k_{y},-k_{z})$ (i.e.,
$(\theta,\phi)\to(\theta+\pi,\pi-\phi)$), then
$\cos\theta\to-\cos\theta,\sin\theta\to-\sin\theta,e^{-i\phi}\to-e^{i\phi}$
into the above solution, it results in
$\displaystyle\mathcal{H}_{1,\mathcal{M}_{3}=-1}(\mathbf{k})|B-\rangle=-A_{1}k|B-\rangle,\;\text{
with
}|B-\rangle=\frac{1}{\sqrt{2}\sqrt{1-\cos\theta}}(i(-\cos\theta+1),e^{i\phi}\sin\theta)^{T},$
(293a)
$\displaystyle\mathcal{H}_{1,\mathcal{M}_{3}=-1}(\mathbf{k})|B+\rangle=\;\;\;A_{1}k|B+\rangle,\;\text{
with
}|B+\rangle=\frac{1}{\sqrt{2}\sqrt{1+\cos\theta}}(i(-\cos\theta-1),e^{i\phi}\sin\theta)^{T},$
(293b)
where the subscripts $A(B)$ represent the eigenvalues $+1(-1)$ of the quasi-
symmetry $\mathcal{M}_{3}$. Therefore, the eigen-state of
$\mathcal{H}_{1}(\mathbf{k})$ are given by
$\displaystyle\mathcal{H}_{1}(\mathbf{k})|\Psi_{A/B+}(\theta,\phi)\rangle=E_{+}|\Psi_{A/B+}(\theta,\phi)\rangle,$
(294a)
$\displaystyle\mathcal{H}_{1}(\mathbf{k})|\Psi_{A/B-}(\theta,\phi)\rangle=E_{-}|\Psi_{A/B-}(\theta,\phi)\rangle,$
(294b)
where $E_{\pm}=\pm A_{1}k$ and the corresponding eigen-states are given by
$\displaystyle|\Psi_{A-}(\theta,\phi)\rangle$
$\displaystyle=|A-\rangle\otimes(-\frac{i}{\sqrt{2}},\frac{1}{\sqrt{2}})^{T},$
(295a) $\displaystyle|\Psi_{A+}(\theta,\phi)\rangle$
$\displaystyle=|A+\rangle\otimes(-\frac{i}{\sqrt{2}},\frac{1}{\sqrt{2}})^{T},$
(295b) $\displaystyle|\Psi_{B-}(\theta,\phi)\rangle$
$\displaystyle=|B-\rangle\otimes(\frac{i}{\sqrt{2}},\frac{1}{\sqrt{2}})^{T},$
(295c) $\displaystyle|\Psi_{B+}(\theta,\phi)\rangle$
$\displaystyle=|B+\rangle\otimes(\frac{i}{\sqrt{2}},\frac{1}{\sqrt{2}})^{T}.$
(295d)
Due to the presence of the orbital SU(2) symmetry, one can also find the
common eigen-states of ${\cal H}_{1}({\bf k})$ and ${\cal M}\cdot\vec{n}$ for
arbitrary real vector $\vec{n}$, which is a rotation acting on the solution in
Eq. 295. Thus, the eigen-state solution is not unique due to this twofold
degeneracy, protected by the orbital SU(2) symmetry. Note that the degeneracy
will be doubled if spin degeneracy is taken into account.
Similarly, we can further analytically solve $\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}$ by using the U(1) quasi-symmetry generator
$\mathcal{M}_{3}$. The $R$-model in Eq. (54) becomes
$\displaystyle\mathcal{U}\mathcal{H}_{R}\mathcal{U}^{\dagger}=\mathcal{U}\left[s_{0}\otimes\mathcal{H}_{1}(\mathbf{k})+\mathcal{H}_{\text{soc}}+s_{0}\otimes\mathcal{H}_{2}(\mathbf{k})\right]\mathcal{U}^{\dagger}.$
(296)
The first two parts, $s_{0}\otimes\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}$, preserves the orbital SU(2) symmetry, while
some specific terms of the $k^{2}$-order Hamiltonian can break the SU(2)
quasi-symmetry down to U(1). This shows the hierarchy structure of the quasi-
symmetry, which will be discussed in details in Sec. D. Here we focus on
$s_{0}\otimes\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}$, which becomes
block-diagonal after this unitary transformation defined in Eq. (289),
$\displaystyle\begin{split}\mathcal{U}\left[s_{0}\otimes\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}\right]\mathcal{U}^{\dagger}=C_{0}+\begin{pmatrix}\mathcal{H}_{A}(\mathbf{k})&0\\\
0&\mathcal{H}_{B}(\mathbf{k})\end{pmatrix},\end{split}$ (297)
where the index $A(B)$ represent the eigenvalues $+1(-1)$ of quasi-symmetry
${\cal M}_{3}$, and $\mathcal{H}_{A}(\mathbf{k})$ and
$\mathcal{H}_{B}(\mathbf{k})$ are given by
$\displaystyle\mathcal{H}_{A}(\mathbf{k})$
$\displaystyle=A_{1}s_{0}\otimes\left(k_{x}\sigma_{y}+k_{y}\sigma_{x}-k_{z}\sigma_{z}\right)+\lambda_{0}\left(s_{x}\sigma_{y}+s_{y}\sigma_{x}-s_{z}\sigma_{z}\right),$
(298) $\displaystyle\mathcal{H}_{B}(\mathbf{k})$
$\displaystyle=A_{1}s_{0}\otimes\left(k_{x}\sigma_{y}-k_{y}\sigma_{x}+k_{z}\sigma_{z}\right)+\lambda_{0}\left(s_{x}\sigma_{y}-s_{y}\sigma_{x}+s_{z}\sigma_{z}\right).$
(299)
First, one can check that the Hamiltonian can be reduced back to that in Eq.
(291) by setting $\lambda_{0}=0$. In addition, $\mathcal{H}_{A}(\mathbf{k})$
and $\mathcal{H}_{B}(\mathbf{k})$ are related to each other by time-reversal
symmetry
$\displaystyle\mathcal{H}_{B}(\mathbf{k})=\mathcal{T}[\mathcal{H}_{A}(-\mathbf{k})]\mathcal{T}^{\dagger}.$
(300)
Here $\mathcal{T}=is_{y}\mathcal{K}$ with $\mathcal{K}$ the complex conjugate.
Therefore, we only need to solve the four eigen-states for
$\mathcal{H}_{A}(\mathbf{k})$. After straightforward calculation, the four
eigen-energies of $\mathcal{H}_{A}(\mathbf{k})$ are given by
$\displaystyle E({\bf k})=C_{0}+\left\\{\pm
A_{1}k+\lambda_{0},\quad\pm\sqrt{A_{1}^{2}k^{2}+4\lambda_{0}^{2}}-\lambda_{0}\right\\}.$
(301)
Drop the constant $C_{0}$ for short, the two positive upper bands are
$\displaystyle E_{A,+}(\mathbf{k})$ $\displaystyle=A_{1}k+\lambda_{0},$ (302a)
$\displaystyle E_{A,-}(\mathbf{k})$
$\displaystyle=\sqrt{A_{1}^{2}k^{2}+4\lambda_{0}^{2}}-\lambda_{0},$ (302b)
which both have increasing energy as $k$ increases, and $\pm$ are the band
index. Note we have assumed $A_{1}>0$. And we notice that
$E_{A,+}(\mathbf{k}=0)=E_{A,-}(\mathbf{k}=0)$ and
$E_{A,+}(\mathbf{k})>E_{A,-}(\mathbf{k})$ for any nonzero ${\bf k}$. It
indicates that the SOC-induced gap between them,
$E_{A,+}(\mathbf{k})-E_{A,-}(\mathbf{k})$, approaches to $2\lambda_{0}$ as
$k\to\infty$. By time-reversal symmetry, the four eigen-energies of
$\mathcal{H}_{B}(\mathbf{k})$ are the same. Therefore, each state has twofold
degeneracy at any nonzero ${\bf k}$. This is due to the presence of the
orbital SU(2) symmetry.
Moreover, we can also solve the eigen wavefunctions of
$\mathcal{U}\left[s_{0}\otimes\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}\right]\mathcal{U}^{\dagger}$. For instance, for
the two positive upper bands of $\mathcal{H}_{A}(\mathbf{k})$, the
corresponding wavefunctions are given by,
$\displaystyle\mathcal{H}_{A}(\mathbf{k})\begin{cases}|E_{A,+}(\mathbf{k})\rangle=E_{A,+}(\mathbf{k})|E_{A,+}(\mathbf{k})\rangle,\\\
|E_{A,-}(\mathbf{k})\rangle=E_{A,-}(\mathbf{k})|E_{A,-}(\mathbf{k})\rangle.\end{cases}$
(303)
And the corresponding eigen-wavefunctions are given by
$\displaystyle|E_{A,+}(\mathbf{k})\rangle=\frac{1}{\mathcal{N}_{A,+}(\mathbf{k})}\left(-i,\frac{k_{x}-ik_{y}}{k-k_{z}},\frac{-ik_{x}+k_{y}}{k+k_{z}},1\right)^{T},$
(304a)
$\displaystyle|E_{A,-}(\mathbf{k})\rangle=\frac{1}{\mathcal{N}_{A,-}(\mathbf{k})}\Big{(}i(-A_{1}k_{z}-\lambda_{0}+E_{A,-}(\mathbf{k})),A_{1}(-k_{x}+ik_{y}),A_{1}(-ik_{x}+k_{y}),A_{1}k_{z}-\lambda_{0}+E_{A,-}(\mathbf{k})\Big{)}^{T},$
(304b)
where the normalization factors are
$\displaystyle\mathcal{N}_{A,+}(\mathbf{k})$
$\displaystyle=2k/\sqrt{k_{x}^{2}+k_{y}^{2}},$ (305a)
$\displaystyle\mathcal{N}_{A,-}(\mathbf{k})$
$\displaystyle=2\sqrt{(E_{A,-}(\mathbf{k}))^{2}-\lambda_{0}^{2}}.$ (305b)
In addition, we discuss the band index $\pm$ that are actually eigen-values of
symmetry. To show that, we emphasize that one can also obtain the common
eigen-states for $\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}$ and
${\cal M}\cdot\vec{n}$. As we discussed in Eq. (206), there is a helicity-type
symmetry operator that commutes with $\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}$, which indicates the index $\pm$ in the eigen-
state solution $|E_{A,\pm}(\mathbf{k})\rangle$ are eigen-values of $({\bf
S}+{\bf L})\cdot\vec{n}_{{\bf k}}$. Explicitly, one can check that
$\displaystyle[({\bf S}+{\bf L})\cdot\vec{n}_{{\bf
k}}]|E_{A,+}(\mathbf{k})\rangle$ $\displaystyle=|E_{A,+}(\mathbf{k})\rangle,$
(306a) $\displaystyle[({\bf S}+{\bf L})\cdot\vec{n}_{{\bf
k}}]|E_{A,-}(\mathbf{k})\rangle$ $\displaystyle=0.$ (306b)
It represents the eigen-values of the z-component angular moment of the total
angular momentum at nonzero ${\bf k}$.
### D.3 Approach II: The U(1) quasi-symmetry protected nodal planes
As discussed in Sec. D.1, parts of the entire $k^{2}$-order Hamiltonian break
the SU(2) quasi-symmetry down to the U(1) quasi-symmetry. For example, we
consider
$\displaystyle\mathcal{H}_{2,\vec{n}}({\bf
k})=k_{x}k_{y}(\vec{C}_{\mathcal{J}}\cdot\vec{n})\times\left(\vec{n}\cdot\vec{\mathcal{J}}\right)+k_{x}k_{z}(\vec{C}_{\mathcal{P}}\cdot\vec{n})\times\left(\vec{n}\cdot\vec{\mathcal{P}}\right)+k_{y}k_{z}(\vec{C}_{\mathcal{Q}}\cdot\vec{n})\left(\vec{n}\cdot\vec{\mathcal{Q}}\right),$
(307)
which commutes with $\mathcal{M}\cdot\vec{n}$,
$\displaystyle[\mathcal{M}\cdot\vec{n},\mathcal{H}_{2,\vec{n}}({\bf k})]=0.$
(308)
For an illustration, we can choose $\vec{n}=(0,0,1)$ without loss of
generality, so that $\mathcal{M}_{3}$ is the symmetry generator for the
remaining U(1) quasi-symmetry. Therefore, we consider the Hamiltonian
$\mathcal{H}_{qsR}({\bf k})$ that commute with $\mathcal{M}_{3}$ as,
$\displaystyle\mathcal{H}_{qsR}({\bf k})$ $\displaystyle=\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}+\mathcal{H}_{2,\mathcal{M}_{3}}(\mathbf{k}),$
(309a) $\displaystyle\mathcal{H}_{1}({\bf k})$
$\displaystyle=C_{0}\sigma_{0}\tau_{0}+2A_{1}(\mathbf{k}\cdot\mathbf{L}),$
(309b) $\displaystyle\mathcal{H}_{\text{soc}}$
$\displaystyle=4\lambda_{0}(\mathbf{S}\cdot\mathbf{L}),$ (309c)
$\displaystyle\mathcal{H}_{2,\mathcal{M}_{3}}(\mathbf{k})$ $\displaystyle={\bf
g}_{3}\cdot{\bf
J}_{3}=(C_{1}k_{x}k_{y},C_{2}k_{x}k_{z},-C_{3}k_{y}k_{z})\cdot(\sigma_{z}\tau_{0},\sigma_{x}\tau_{0},\sigma_{y}\tau_{y}),$
(309d)
where
$L_{x}=\tfrac{1}{2}\sigma_{y}\tau_{0},\;L_{y}=\tfrac{1}{2}\sigma_{x}\tau_{y},\;L_{z}=-\tfrac{1}{2}\sigma_{z}\tau_{y}$.
Notice that $\mathcal{M}_{3}=\sigma_{0}\tau_{y}$ is the quasi-symmetry
operator. It is easy to check $[\mathcal{M}_{3},\mathcal{H}_{qsR}({\bf
k})]=0$. We next solve the common eigen-states of $\mathcal{M}_{3}$ and
$\mathcal{H}_{qsR}({\bf k})$. To do that, we only need to diagonalize the
$\tau_{y}$-term for $\mathcal{H}_{qsR}({\bf k})$. In other words, we apply a
unitary transformation
$\displaystyle\mathcal{U}_{\mathcal{M}_{3}}=\frac{1}{\sqrt{2}}s_{0}\otimes\sigma_{0}\otimes\begin{pmatrix}1&-i\\\
1&i\end{pmatrix},$ (310)
which leads to the rotating in the $(\tau_{x},\tau_{y},\tau_{z})$ subspace
$\displaystyle\mathcal{U}_{\mathcal{M}_{3}}\tau_{0}\mathcal{U}_{\mathcal{M}_{3}}^{\dagger}=\tau_{0},\mathcal{U}_{\mathcal{M}_{3}}\tau_{x}\mathcal{U}_{\mathcal{M}_{3}}^{\dagger}=\tau_{y},\mathcal{U}_{\mathcal{M}_{3}}\tau_{y}\mathcal{U}_{\mathcal{M}_{3}}^{\dagger}=\tau_{z},\mathcal{U}_{\mathcal{M}_{3}}\tau_{z}\mathcal{U}_{\mathcal{M}_{3}}^{\dagger}=\tau_{x}.$
(311)
Therefore, after this unitary transformation, the U(1) quasi-symmetry
generator $\mathcal{M}_{3}$ becomes
$\displaystyle\mathcal{U}_{\mathcal{M}_{3}}\mathcal{M}_{3}\mathcal{U}_{\mathcal{M}_{3}}^{\dagger}\triangleq\mathcal{M}_{qs}=\begin{pmatrix}\sigma_{0}&0\\\
0&-\sigma_{0}\end{pmatrix}.$ (312)
And we take this unitary transformation on the $\mathcal{H}_{qsR}(\mathbf{k})$
Hamiltonian and obtain
$\displaystyle\mathcal{U}_{\mathcal{M}_{3}}\mathcal{H}_{qsR}(\mathbf{k})\mathcal{U}_{\mathcal{M}_{3}}^{\dagger}\triangleq\mathcal{H}_{qsR}(\mathbf{k})=C_{0}+B_{1}k^{2}+\begin{pmatrix}\mathcal{H}_{+}^{\prime}(\mathbf{k})&0\\\
0&\mathcal{H}_{-}^{\prime}(\mathbf{k})\end{pmatrix},$ (313a)
where
$\displaystyle\mathcal{H}_{+}^{\prime}(\mathbf{k})$
$\displaystyle=s_{0}\otimes\left[(A_{1}k_{x}-C_{3}k_{y}k_{z})\sigma_{y}+(A_{1}k_{y}+C_{2}k_{x}k_{z})\sigma_{x}-(A_{1}k_{z}-C_{1}k_{x}k_{y})\sigma_{z}\right]+\lambda_{0}\left(s_{x}\sigma_{y}+s_{y}\sigma_{x}-s_{z}\sigma_{z}\right),$
(314) $\displaystyle\mathcal{H}_{-}^{\prime}(\mathbf{k})$
$\displaystyle=s_{0}\otimes\left[(A_{1}k_{x}+C_{3}k_{y}k_{z})\sigma_{y}-(A_{1}k_{y}-C_{2}k_{x}k_{z})\sigma_{x}+(A_{1}k_{z}+C_{1}k_{x}k_{y})\sigma_{z}\right]+\lambda_{0}\left(s_{x}\sigma_{y}-s_{y}\sigma_{x}+s_{z}\sigma_{z}\right),$
(315)
This means that we have chosen the eigen-states of
$\mathcal{H}_{qsR}(\mathbf{k})$ to be the common eigen-state of
$\mathcal{M}_{3}$ or $\mathcal{M}_{qs}$. Due to the presence of the U(1)
quasi-symmetry, we dubbed $\mathcal{H}_{qsR}$ as the quasi-symmetric R-model
(“qsR”). Therefore, the subscript $\pm$ for
$\mathcal{H}_{\pm}^{\prime}(\mathbf{k})$ represent the different eigenvalues
of $\mathcal{M}_{3}$ or $\mathcal{M}_{qs}$. Besides,
$\mathcal{H}_{+}^{\prime}(\mathbf{k})$ and
$\mathcal{H}_{-}^{\prime}(\mathbf{k})$ are related to each other by TR
$\mathcal{T}=is_{y}\mathcal{K}$ with $\mathcal{K}$ complex conjugate,
$\displaystyle\mathcal{H}_{-}^{\prime}(\mathbf{k})=\mathcal{T}\mathcal{H}_{+}^{\prime}(-\mathbf{k})\mathcal{T}^{\dagger}.$
(316)
Next, we compute the eigen-energy of the $qsR$-model. The energy of the upper
four bands are given by
$\displaystyle E_{+,1}(\mathbf{k})$
$\displaystyle=C_{0}+B_{1}k^{2}+\sqrt{A_{1}^{2}k^{2}+C_{1}^{2}k_{x}^{2}k_{y}^{2}+C_{2}^{2}k_{x}^{2}k_{z}^{2}+C_{3}^{2}k_{y}^{2}k_{z}^{2}-2A_{1}\tilde{C}k_{x}k_{y}k_{z}}+\lambda_{0},$
(317) $\displaystyle E_{+,2}(\mathbf{k})$
$\displaystyle=C_{0}+B_{1}k^{2}+\sqrt{A_{1}^{2}k^{2}+C_{1}^{2}k_{x}^{2}k_{y}^{2}+C_{2}^{2}k_{x}^{2}k_{z}^{2}+C_{3}^{2}k_{y}^{2}k_{z}^{2}-2A_{1}\tilde{C}k_{x}k_{y}k_{z}+4\lambda_{0}^{2}}-\lambda_{0},$
(318) $\displaystyle E_{-,1}(\mathbf{k})$
$\displaystyle=C_{0}+B_{1}k^{2}+\sqrt{A_{1}^{2}k^{2}+C_{1}^{2}k_{x}^{2}k_{y}^{2}+C_{2}^{2}k_{x}^{2}k_{z}^{2}+C_{3}^{2}k_{y}^{2}k_{z}^{2}+2A_{1}\tilde{C}k_{x}k_{y}k_{z}}+\lambda_{0},$
(319) $\displaystyle E_{-,2}(\mathbf{k})$
$\displaystyle=C_{0}+B_{1}k^{2}+\sqrt{A_{1}^{2}k^{2}+C_{1}^{2}k_{x}^{2}k_{y}^{2}+C_{2}^{2}k_{x}^{2}k_{z}^{2}+C_{3}^{2}k_{y}^{2}k_{z}^{2}+2A_{1}\tilde{C}k_{x}k_{y}k_{z}+4\lambda_{0}^{2}}-\lambda_{0},$
(320)
where $\tilde{C}=C_{1}-C_{2}+C_{3}$. The $qsR$-model breaks the $C_{3,(111)}$
rotation symmetry because of $C_{1}\neq C_{2}\neq C_{3}$. Moreover, the $\pm$
index for $E_{\pm,i}$ means the eigenvalues of $\mathcal{M}_{qs}$,
$\displaystyle\begin{split}&\mathcal{H}_{qsR}(\mathbf{k})|E_{\pm,i}(\mathbf{k})\rangle=E_{\pm,i}(\mathbf{k})|E_{\pm,i}(\mathbf{k})\rangle,\\\
&\mathcal{M}_{qs}|E_{\pm,i}(\mathbf{k})\rangle=\pm|E_{\pm,i}(\mathbf{k})\rangle,\end{split}$
(321)
where $i=1,2$ is the band index. $E_{+,i}(\mathbf{k})=E_{-,i}(-\mathbf{k})$ is
required by TR symmetry. The quasi-symmetry protected nodal planes are given
by
$\displaystyle\begin{cases}k_{x}k_{y}k_{z}>0,E_{+,1}(\mathbf{k})=E_{-,2}(\mathbf{k}),\\\
k_{x}k_{y}k_{z}<0,E_{+,2}(\mathbf{k})=E_{-,1}(\mathbf{k}).\end{cases}$ (322)
The crossings are between the bands with different eigenvalue of quasi-
symmetry $\mathcal{M}_{3}$ or $\mathcal{M}_{qs}$, and thus we find nodal
planes at generic momenta with the protection from the quasi-symmetry
$\mathcal{M}_{qs}$. By further imposing the Fermi energy constraint, there
will be nodal lines on the Fermi surface, shown in Fig. 7. In Fig. 7(a), the
four Fermi surfaces are plotted in the $\Gamma-R-M$ plane, where two inner FSs
intersect with each other and generate the quasi-symmetry protected exact
crossings (marked as purple circles). In Fig. 7 (b) and (c), the solution of
Eq. (322) are explicitly depicted, showing the exact nodal lines on the FSs.
Figure 7: The exact crossings of the 8-band $qsR$-model. In (a), the Fermi
surfaces with exact crossing are shown in the $\Gamma-R-M$ plane. In (b) and
(c), we show the exact nodal lines on the FSs. These nodal lines are protected
by the U(1) quasi-symmetry.
Next, let us discuss the terms that break the quasi-symmetry. By including the
remaining $k^{2}$-order Hamiltonian
$\mathcal{H}_{2,\mathcal{M}_{1}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{2}}(\mathbf{k})$,
the full Hamiltonian is given by
$\displaystyle\mathcal{H}_{R}({\bf k})=\mathcal{H}_{qsR}({\bf
k})+\mathcal{U}_{\mathcal{M}_{3}}[\mathcal{H}_{2,\mathcal{M}_{1}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{2}}(\mathbf{k})]\mathcal{U}_{\mathcal{M}_{3}}^{\dagger},$
(323)
where the U(1) quasi-symmetry breaking terms are given by
$\displaystyle\mathcal{H}_{2,qsb}(\mathbf{k})$
$\displaystyle=\mathcal{U}_{\mathcal{M}_{3}}[\mathcal{H}_{2,\mathcal{M}_{1}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{2}}(\mathbf{k})]\mathcal{U}_{\mathcal{M}_{3}}^{\dagger}=\begin{pmatrix}0&s_{0}\otimes
H_{(2)}\\\ s_{0}\otimes H_{(2)}^{\dagger}&0\end{pmatrix},$ (324a)
$\displaystyle H_{(2)}$
$\displaystyle=C_{1}(k_{y}k_{z}\sigma_{0}+k_{x}k_{z}\sigma_{z})+C_{2}(-ik_{x}k_{y}\sigma_{x}+ik_{y}k_{z}\sigma_{0})+C_{3}(k_{x}k_{y}\sigma_{x}+ik_{x}k_{z}\sigma_{z})$
$\displaystyle+(C_{1}+iC_{2})k_{y}k_{z}\sigma_{0}+(C_{1}+iC_{3})k_{x}k_{z}\sigma_{z}+(-iC_{2}+C_{3})k_{x}k_{y}\sigma_{x}.$
(324b)
Therefore, we have
$\displaystyle[\mathcal{M}_{qs},\mathcal{H}_{2,qsb}(\mathbf{k})]=2\begin{pmatrix}0&h_{(2)}(\mathbf{k})\\\
-h_{(2)}^{\dagger}(\mathbf{k})&0\end{pmatrix}\neq 0.$ (325)
where
$\displaystyle\begin{split}h_{(2)}=(C_{1}+iC_{2})k_{y}k_{z}\sigma_{0}+(C_{1}+iC_{3})k_{x}k_{z}\sigma_{z}+(-iC_{2}+C_{3})k_{x}k_{y}\sigma_{x}.\end{split}$
(326)
For this Hamiltonian, $[\mathcal{M}_{qs},\mathcal{H}_{2,qsb}(\mathbf{k})]=0$
only occurs for $\mathbf{k}=(0,0,0)$ (the $R$-point). Therefore, for any Fermi
surface that does not cross the $R$-point, the nodal planes will be gapped.
This leads to the quasi-symmetry hierarchy mentioned in the main text (see Eq.
(10)),
$\displaystyle\text{SU}_{\text{s}}(2)\times\text{SU}_{\text{o}}(2)\stackrel{{\scriptstyle\mathcal{H}_{\text{soc}}}}{{\xhookrightarrow{\quad\quad\quad\;}}}\text{SU}_{\text{o}}(2)\stackrel{{\scriptstyle\mathcal{H}_{2,\mathcal{M}_{i}}}}{{\xhookrightarrow{\quad\quad\quad\quad\;}}}\text{U}_{\text{o}}(1).$
(327)
More interestingly, as we discussed in Sec. D.1, we have shown that the choice
of $\mathcal{H}_{2,\mathcal{M}_{3}}$ is just one specific case, instead, for
any real vector $\vec{n}$ of $\mathcal{H}_{2,\vec{n}}$ that commute with
$\mathcal{M}\cdot\vec{n}$ (see Sec. D.1 for details) can generally lead to the
same quasi-symmetry hierarchy results. Because we have
$\displaystyle[\mathcal{M}\cdot\vec{n},s_{0}\otimes\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}+s_{0}\otimes\mathcal{H}_{2,\vec{n}}({\bf k})]=0.$
(328)
Thus, one can conclude that this analysis for the reduction from SU(2) quasi-
symmetry down to U(1) quasi-symmetry is general, which can help to protect
quasi-nodal-plane at generic momenta.
To show this nodal plane ude to the remaining U(1) quasi-symmetry in our
approach II more explicitly, we next consider the perturbation to
$\mathcal{H}_{2,\vec{n}}({\bf k})$. Here we emphasize that one can also obtain
the common eigen-states for $\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}$ and ${\cal M}\cdot\vec{n}$. Different from
Approach I in Sec. C, here we take the eigen-state solution of
$s_{0}\otimes\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}$, and do the
perturbation for the $k^{2}$-order Hamiltonian. Below we choose
$\vec{n}=(0,0,1)$ and the corresponding U(1) quasi-symmetry opeator is ${\cal
M}\cdot\vec{n}={\cal M}_{3}$. Recall that $\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}$ becomes block-diagonal after this unitary
transformation defined in Eq. (289),
$\displaystyle\begin{split}\mathcal{U}\left[s_{0}\otimes\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}\right]\mathcal{U}^{\dagger}=C_{0}+\begin{pmatrix}\mathcal{H}_{A}(\mathbf{k})&0\\\
0&\mathcal{H}_{B}(\mathbf{k})\end{pmatrix},\end{split}$ (329)
where $\mathcal{H}_{A/B}(\mathbf{k})$ are given by
$\displaystyle\mathcal{H}_{A}(\mathbf{k})$
$\displaystyle=A_{1}s_{0}\otimes\left(k_{x}\sigma_{y}+k_{y}\sigma_{x}-k_{z}\sigma_{z}\right)+\lambda_{0}\left(s_{x}\sigma_{y}+s_{y}\sigma_{x}-s_{z}\sigma_{z}\right),$
(330) $\displaystyle\mathcal{H}_{B}(\mathbf{k})$
$\displaystyle=A_{1}s_{0}\otimes\left(k_{x}\sigma_{y}-k_{y}\sigma_{x}+k_{z}\sigma_{z}\right)+\lambda_{0}\left(s_{x}\sigma_{y}-s_{y}\sigma_{x}+s_{z}\sigma_{z}\right).$
(331)
Again, note that the index $A(B)$ represent the eigenvalues $+1(-1)$ of the
quasi-symmetry ${\cal M}_{3}$. Therefore, after this unitary transformation,
the U(1) quasi-symmetry generator $\mathcal{M}_{3}$ becomes
$\displaystyle\mathcal{U}_{\mathcal{M}_{3}}\mathcal{M}_{3}\mathcal{U}_{\mathcal{M}_{3}}^{\dagger}\triangleq\mathcal{M}_{qs}=\begin{pmatrix}\sigma_{0}&0\\\
0&-\sigma_{0}\end{pmatrix}.$ (332)
The analytical solution for the two upper bands (i.e. Eq. (304a)) of
$\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}$ in Sec. D.2 are given by
$\displaystyle|E_{A,+}(\mathbf{k})\rangle=\frac{1}{\mathcal{N}_{A,+}(\mathbf{k})}\left(-i,\frac{k_{x}-ik_{y}}{k-k_{z}},\frac{-ik_{x}+k_{y}}{k+k_{z}},1\right)^{T},$
(333a)
$\displaystyle|E_{A,-}(\mathbf{k})\rangle=\frac{1}{\mathcal{N}_{A,-}(\mathbf{k})}\Big{(}i(-A_{1}k_{z}-\lambda_{0}+E_{A,-}(\mathbf{k})),A_{1}(-k_{x}+ik_{y}),A_{1}(-ik_{x}+k_{y}),A_{1}k_{z}-\lambda_{0}+E_{A,-}(\mathbf{k})\Big{)}^{T}.$
(333b)
where the index $\pm$ in the eigen-state solution
$|E_{A,\pm}(\mathbf{k})\rangle$ are eigen-values of $({\bf S}+{\bf
L})\cdot\vec{n}_{{\bf k}}$ for nonzero ${\bf k}$. And the normalization
factors are given by
$\mathcal{N}_{A,+}(\mathbf{k})=2k/\sqrt{k_{x}^{2}+k_{y}^{2}}$, and
$\mathcal{N}_{A,-}(\mathbf{k})=2\sqrt{(E_{A,-}(\mathbf{k}))^{2}-\lambda_{0}^{2}}$.
Moreover, note that they are the two positive upper bands with eigen-energies,
$\displaystyle E_{A,+}(\mathbf{k})$ $\displaystyle=A_{1}k+\lambda_{0},$ (334a)
$\displaystyle E_{A,-}(\mathbf{k})$
$\displaystyle=\sqrt{A_{1}^{2}k^{2}+4\lambda_{0}^{2}}-\lambda_{0},$ (334b)
which have increasing energy as $k$ increases. And, the eigen-states of the
$\mathcal{H}_{B}(\mathbf{k})$ block can be related to those of
$\mathcal{H}_{A}(\mathbf{k})$ by time-reversal symmetry,
$\displaystyle|E_{B,+}(\mathbf{k})\rangle$
$\displaystyle=\mathcal{T}|E_{A,+}(-\mathbf{k})\rangle,$ (335a)
$\displaystyle|E_{B,-}(\mathbf{k})\rangle$
$\displaystyle=\mathcal{T}|E_{A,-}(-\mathbf{k})\rangle,$ (335b)
and
$\displaystyle\mathcal{H}_{B}(\mathbf{k})|E_{B,\pm}(\mathbf{k})\rangle=E_{A,\pm}(\mathbf{k})|E_{B,\pm}(\mathbf{k})\rangle,$
(336)
where $E_{A,\pm}(-\mathbf{k})=E_{A,\pm}(\mathbf{k})$ has been used. Therefore,
we have the four bands as a basis,
$\displaystyle\\{|\Psi_{+}(\mathbf{k})\rangle\\}$
$\displaystyle=\Big{\\{}(1,0)^{T}\otimes|E_{A,+}(\mathbf{k})\rangle,(1,0)^{T}\otimes|E_{A,-}(\mathbf{k})\rangle,(0,1)^{T}\otimes|E_{B,+}(\mathbf{k})\rangle,(0,1)^{T}\otimes|E_{B,-}(\mathbf{k})\rangle\Big{\\}}.$
(337)
In this basis, the $\mathcal{H}_{1}({\bf k})+\mathcal{H}_{\text{soc}}$ is
diagonal,
$\displaystyle\begin{split}\mathcal{H}_{1}({\bf
k})+\mathcal{H}_{\text{soc}}=C_{0}+\text{Diag}[&E_{A,+}(\mathbf{k}),E_{A,-}(\mathbf{k}),E_{A,+}(\mathbf{k}),E_{A,-}(\mathbf{k})].\end{split}$
(338)
After straightforward calculation, we next project the $k^{2}$-term
$\mathcal{H}_{2,\mathcal{M}_{3}}({\bf k})$ onto the basis in Eq. (337), and we
arrive at the quasi-symmetry $P$-model ($qsP$-model) as,
$\displaystyle\mathcal{H}_{qsP}(\mathbf{k})=\begin{pmatrix}E_{A,+}^{\prime}(\mathbf{k})+f_{1}(\mathbf{k})&d_{1}(\mathbf{k})-id_{2}(\mathbf{k})&0&0\\\
d_{1}(\mathbf{k})+id_{2}(\mathbf{k})&E_{A,-}^{\prime}(\mathbf{k})+f_{2}(\mathbf{k})&0&0\\\
0&0&E_{A,+}^{\prime}(\mathbf{k})-f_{1}(\mathbf{k})&d_{1}(\mathbf{k})-id_{2}(\mathbf{k})\\\
0&0&d_{1}(\mathbf{k})+id_{2}(\mathbf{k})&E_{A,-}^{\prime}(\mathbf{k})-f_{2}(\mathbf{k})\end{pmatrix},$
(339)
where $E_{A,\pm}^{\prime}(\mathbf{k})=C_{0}+B_{1}k^{2}+E_{A,\pm}(\mathbf{k})$.
For the projected four-band $qsP$-model, we check the protection from quasi-
symmetry as
$\displaystyle[\mathcal{M}_{sq},\mathcal{H}_{qsP}({\bf k})]=0.$ (340)
And all the other components are given by
$\displaystyle f_{1}(\mathbf{k})$
$\displaystyle=-\tilde{C}\frac{k_{x}k_{y}k_{z}}{k},$ (341a) $\displaystyle
f_{2}(\mathbf{k})$
$\displaystyle=-A_{1}\tilde{C}\frac{k_{x}k_{y}k_{z}}{\sqrt{A_{1}^{2}k^{2}+4\lambda_{0}^{2}}},$
(341b) $\displaystyle d_{1}(\mathbf{k})$
$\displaystyle=\frac{2(E_{A,+}(\mathbf{k})-E_{A,-}(\mathbf{k}))k_{x}k_{y}}{\mathcal{N}_{A,+}\mathcal{N}_{A,-}}\left(C_{1}+(C_{2}-C_{3})\frac{k_{z}^{2}}{k_{x}^{2}+k_{y}^{2}}\right),$
(341c) $\displaystyle d_{2}(\mathbf{k})$
$\displaystyle=\frac{2(E_{A,+}(\mathbf{k})-E_{A,-}(\mathbf{k}))k_{z}k}{\mathcal{N}_{A,+}\mathcal{N}_{A,-}}\frac{C_{2}k_{x}^{2}+C_{3}k_{y}^{2}}{k_{x}^{2}+k_{y}^{2}}.$
(341d)
The eigen-energies of the $qsP$-model are given by
$\displaystyle E_{\alpha,\beta}(\mathbf{k})$
$\displaystyle=\frac{1}{2}\left[\Delta
E_{+}(\alpha\mathbf{k})+\beta\sqrt{(\Delta
E_{-}(\alpha\mathbf{k}))^{2}+E_{d_{12}}}\right],$ (342)
with $\alpha,\beta=\pm$. And $\alpha=\pm$ are eigenvalues of
$\mathcal{M}_{qs}$ and $\beta=\pm$ are for the band index. Here we have
defined
$\displaystyle E_{d_{12}}(\mathbf{k})$
$\displaystyle=4\left[(d_{1}(\mathbf{k}))^{2}+(d_{2}(\mathbf{k}))^{2}\right],$
(343a) $\displaystyle\Delta E_{\pm}(\mathbf{k})$
$\displaystyle=E_{A,+}^{\prime}(\mathbf{k})+f_{1}(\mathbf{k})\pm(E_{A,-}^{\prime}(\mathbf{k})+f_{2}(\mathbf{k})).$
(343b)
Similar to the discussion for the nodal plane of the $qsR$-model (see Eq.
(322)), the quasi-symmetry protected nodal planes of the $qsP$-model are give
by
$\displaystyle\begin{cases}k_{x}k_{y}k_{z}<0,E_{+,-}(\mathbf{k})=E_{-,+}(\mathbf{k}),\\\
k_{x}k_{y}k_{z}>0,E_{+,+}(\mathbf{k})=E_{-,-}(\mathbf{k}).\end{cases}$ (344)
The crossings happen for bands with different eigenvalue of quasi-symmetry
$\mathcal{M}_{qs}$. We numerically check these exact crossings on the $\Gamma-
R-M$ planes, which is consistent with the results in Fig. 7 (a).
### D.4 Perturbation theory for tiny gap of the nodal plane in Approach II
The above discussion on the nodal planes in the approach II requires a choice
of specific $k^{2}$-order terms, but in real materials, all the coefficients
before the $k^{2}$-order terms can generally be non-zero and at the same
order. Thus, our current approach does not directly explain the near nodal
plane seen in real materials. We notice that we treat the SOC terms accurately
in our approach II without any approximation, while the existence of near
nodal planes in real materials actually require the SOC strength to be much
smaller than the Fermi energy. Therefore, we below consider the limit of the
SOC strength $\lambda_{0}\ll A_{1}k_{F}$ ($k_{F}$ is the Fermi momentum) in
our Approach II, taking into account all non-zero $k^{2}$-order terms.
Following the same procedure of the perturbation projection as in the last
section, we project all the $k^{2}$-order terms
$\mathcal{H}_{2,\mathcal{M}_{1}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{2}}(\mathbf{k})+\mathcal{H}_{2,\mathcal{M}_{3}}(\mathbf{k})$
onto the basis in Eq. (337), and get the entire perturbation Hamiltonian,
$\displaystyle\mathcal{H}_{qsP+qsB}({\bf
k})=\begin{pmatrix}E_{A,+}^{\prime}(\mathbf{k})+f_{1}(\mathbf{k})&d_{1}(\mathbf{k})-id_{2}(\mathbf{k})&g_{1}(\mathbf{k})&g_{2}(\mathbf{k})\\\
d_{1}(\mathbf{k})+id_{2}(\mathbf{k})&E_{A,-}^{\prime}(\mathbf{k})+f_{2}(\mathbf{k})&g_{3}(\mathbf{k})&g_{4}(\mathbf{k})\\\
g_{1}^{\ast}(\mathbf{k})&g_{3}^{\ast}(\mathbf{k})&E_{A,+}^{\prime}(\mathbf{k})-f_{1}(\mathbf{k})&d_{1}(\mathbf{k})-id_{2}(\mathbf{k})\\\
g_{2}^{\ast}(\mathbf{k})&g_{4}^{\ast}(\mathbf{k})&d_{1}(\mathbf{k})+id_{2}(\mathbf{k})&E_{A,-}^{\prime}(\mathbf{k})-f_{2}(\mathbf{k})\end{pmatrix}.$
(345)
Here the terms $d_{1},d_{2},f_{1},f_{2}$ have been given by Eq. (341). In
addition, the off-diagonal terms $g_{1,2,3,4}$ are generally to open a gap for
the quasi-nodal plane. Because these off-diagonal terms break the quasi-
symmetry $\mathcal{M}_{qs}$. And, they are given by
$\displaystyle g_{1}(\mathbf{k})$
$\displaystyle=(1-i)\tilde{C}\frac{k_{x}k_{y}k_{z}}{k},$ (346a) $\displaystyle
g_{4}(\mathbf{k})$
$\displaystyle=(i-1)A_{1}\tilde{C}\frac{k_{x}k_{y}k_{z}}{E_{A,-}(\mathbf{k})+\lambda_{0}},$
(346b) $\displaystyle g_{2}(\mathbf{k})$
$\displaystyle=\frac{2}{(k_{x}^{2}+k_{y}^{2})\mathcal{N}_{A,+}(\mathbf{k})\mathcal{N}_{A,-}(\mathbf{k})}\Big{\\{}(C_{2}+iC_{3})k_{x}k_{y}(k_{x}^{2}+k_{y}^{2})(E_{A,-}(\mathbf{k})-E_{A,+}(\mathbf{k}))$
$\displaystyle+(C_{1}+iC_{3})k_{x}k_{z}(A_{1}k_{x}k^{2}-ik_{y}k_{z}(E_{A,-}(\mathbf{k})-E_{A,+}(\mathbf{k}))-k_{x}k(E_{A,-}(\mathbf{k})-\lambda_{0}))$
$\displaystyle+(C_{2}-iC_{1})k_{y}k_{z}(A_{1}k_{y}k^{2}+ik_{x}k_{z}(E_{A,-}(\mathbf{k})-E_{A,+}(\mathbf{k}))-k_{y}k(E_{A,-}(\mathbf{k})-\lambda_{0})),$
(346c) $\displaystyle g_{3}(\mathbf{k})$
$\displaystyle=\frac{2}{(k_{x}^{2}+k_{y}^{2})\mathcal{N}_{A,+}(\mathbf{k})\mathcal{N}_{A,-}(\mathbf{k})}\Big{\\{}-(C_{2}+iC_{3})k_{x}k_{y}(k_{x}^{2}+k_{y}^{2})(E_{A,-}(\mathbf{k})-E_{A,+}(\mathbf{k}))$
$\displaystyle+(C_{1}+iC_{3})k_{x}k_{z}(A_{1}k_{x}k^{2}+ik_{y}k_{z}(E_{A,-}(\mathbf{k})-E_{A,+}(\mathbf{k}))-k_{x}k(E_{A,-}(\mathbf{k})-\lambda_{0}))$
$\displaystyle+(C_{2}-iC_{1})k_{y}k_{z}(A_{1}k_{y}k^{2}-ik_{x}k_{z}(E_{A,-}(\mathbf{k})-E_{A,+}(\mathbf{k}))-k_{y}k(E_{A,-}(\mathbf{k})-\lambda_{0}))\Big{\\}}.$
(346d)
Here $g_{3}(\mathbf{k})=-g_{2}(-\mathbf{k})$. The nodal planes are completely
gapped out. Now we use perturbation to explain why the gap is tiny by
realizing that the SOC in CoSi is weak enough for doing a perturbation
expansion for the coefficients $f_{1,2}({\bf k})$, $d_{1,2}({\bf k})$, and
$g_{1,2,3,4}({\bf k})$. By setting $\lambda_{0}/A_{1}k_{F}\to 0$, we obtain
$\displaystyle f_{1}({\bf k})$ $\displaystyle=f_{2}({\bf
k})=-\tilde{C}\frac{k_{x}k_{y}k_{z}}{k},$ (347a) $\displaystyle
d_{1}(\mathbf{k})$ $\displaystyle=d_{2}(\mathbf{k})=0,$ (347b) $\displaystyle
g_{1}(\mathbf{k})$
$\displaystyle=-g_{2}(\mathbf{k})=(1-i)\tilde{C}\frac{k_{x}k_{y}k_{z}}{k},$
(347c) $\displaystyle g_{2}(\mathbf{k})$ $\displaystyle=g_{3}(\mathbf{k})=0.$
(347d)
for the zeroth-order terms. Please notice that only the diagonal energies
$E_{A,\pm}^{\prime}$ that are eigen-energies of
$s_{0}\otimes\mathcal{H}_{1}+\mathcal{H}_{\text{SOC}}$ involve the SOC
$\lambda_{0}$. As a result, the perturbation Hamiltonian in Eq. (345) to the
zeroth-order in $\lambda_{0}$ becomes
$\displaystyle\mathcal{H}^{(0)}_{qsP+qsB}({\bf k})$
$\displaystyle=\begin{pmatrix}E_{A,+}^{\prime}(\mathbf{k})+f_{1}(\mathbf{k})&0&g_{1}(\mathbf{k})&0\\\
0&E_{A,-}^{\prime}(\mathbf{k})+f_{1}(\mathbf{k})&0&-g_{1}(\mathbf{k})\\\
g_{1}^{\ast}(\mathbf{k})&0&E_{A,+}^{\prime}(\mathbf{k})-f_{1}(\mathbf{k})&0\\\
0&-g_{1}^{\ast}(\mathbf{k})&0&E_{A,-}^{\prime}(\mathbf{k})-f_{1}(\mathbf{k})\end{pmatrix},$
(348a)
$\displaystyle=\begin{pmatrix}E_{A,+}^{\prime}(\mathbf{k})+f_{1}(\mathbf{k})&g_{1}(\mathbf{k})\\\
g_{1}^{\ast}(\mathbf{k})&E_{A,+}^{\prime}(\mathbf{k})-f_{1}(\mathbf{k})\end{pmatrix}\oplus\begin{pmatrix}E_{A,-}^{\prime}(\mathbf{k})+f_{1}(\mathbf{k})&-g_{1}(\mathbf{k})\\\
-g_{1}^{\ast}(\mathbf{k})&E_{A,-}^{\prime}(\mathbf{k})-f_{1}(\mathbf{k})\end{pmatrix}$
(348b)
where the diagonal terms are
$E_{A,+}^{\prime}(\mathbf{k})=C_{0}+B_{1}k^{2}+A_{1}k+\lambda_{0}$ and
$E_{A,-}^{\prime}(\mathbf{k})=C_{0}+B_{1}k^{2}+\sqrt{A_{1}^{2}k^{2}+4\lambda_{0}^{2}}-\lambda_{0}$.
Thus, the eigen-energies are given by
$\displaystyle E_{1,\pm}(\mathbf{k})$
$\displaystyle=C_{0}+B_{1}k^{2}+A_{1}k+\lambda_{0}\pm\sqrt{|f_{1}({\bf
k})|^{2}+|g_{1}({\bf k})|^{2}},$ (349a) $\displaystyle E_{2,\pm}(\mathbf{k})$
$\displaystyle=C_{0}+B_{1}k^{2}+\sqrt{A_{1}^{2}k^{2}+4\lambda_{0}^{2}}-\lambda_{0}\pm\sqrt{|f_{1}({\bf
k})|^{2}+|g_{1}({\bf k})|^{2}},$ (349b)
which leads to the equation for the nodal-plane solution
$\displaystyle E_{1,-}(\mathbf{k})=E_{2,+}(\mathbf{k})\quad\Rightarrow\quad
A_{1}k-\sqrt{A_{1}^{2}k^{2}+4\lambda_{0}^{2}}+2\lambda_{0}=2\sqrt{|f_{1}({\bf
k})|^{2}+|g_{1}({\bf k})|^{2}}.$ (350)
In the $k\to\infty$ limit, this equation becomes
$\displaystyle\lambda_{0}=\sqrt{|f_{1}({\bf k})|^{2}+|g_{1}({\bf
k})|^{2}}=\sqrt{3}\tilde{C}\left|\frac{k_{x}k_{y}k_{z}}{k}\right|.$ (351)
Note that $\tilde{C}>0$ in this work. And it is exactly the same Eq. (240)
obtained from the Approach I in Sec. C. At small ${\bf k}$, they differs from
each other. Moreover, the first-order correction from $\lambda_{0}$ will open
a tiny gap for the nodal planes, which is actually the second-order
perturbation theory in Approach I. Based on this analysis, we conclude that
the results of Approach II is equivalent from those of Approach I.
## Appendix E The projected two-band model
In this section, we further project the first-order-perturbation 4-band
Hamiltonian, the $P$-model (e.g. see Eq. (218) in Sec. C.1) onto an effective
model that only consists of the two bands forming the nodal plane. To do that,
we consider the two eigen-states of the $P$-model Hamiltonian in Eq. (218),
$\displaystyle\mathcal{H}_{P}^{eff(1)}({\bf k})$
$\displaystyle=(E_{+}+B_{1}k^{2})s_{0}\omega_{0}+\mathcal{H}_{\text{soc},+}^{eff(1)}({\bf
k})+\mathcal{H}_{k^{2},+}^{eff(1)}({\bf k}),$ (352)
which is marked as the $P$-model around $R$-point. And
$\displaystyle\begin{split}\mathcal{H}_{\text{soc},+}^{eff(1)}({\bf
k})&=\lambda_{0}\left(\lambda_{x}s_{x}+\lambda_{y}s_{y}+\lambda_{z}s_{z}\right)\omega_{0},\\\
\mathcal{H}_{k^{2},+}^{eff(1)}({\bf
k})&=\tilde{C}k^{2}s_{0}\left(d_{x}\omega_{x}+d_{y}\omega_{y}+d_{z}\omega_{z}\right).\end{split}$
(353)
where $\tilde{C}=C_{1}-C_{2}+C_{3}$, and $\omega_{x,y,z}$ are Pauli matrices
for the $\\{A+,B+\\}$ band subspace. The eigen-energies are give by Eq. (230)
$\displaystyle\begin{split}E_{\alpha\beta}(k,\theta,\phi)=C_{0}+B_{1}k^{2}+A_{1}k+\alpha\lambda_{0}+\beta\tfrac{\sqrt{3}}{4}\tilde{C}k^{2}|\sin
2\phi\sin 2\theta\sin\theta|,\end{split}$ (354)
where $\alpha=\pm$, $\beta=\pm$, and $\sin 2\phi\sin
2\theta\sin\theta=4k_{x}k_{y}k_{z}/k^{3}$. Therefore, these two states
$\\{|\alpha=+,\beta=-\rangle,|\alpha=-,\beta=+\rangle\\}$ that give the eigen-
energies $E_{\alpha=+,\beta=-}$ and $E_{\alpha=-,\beta=+}$. The basis eigen-
wave function for the two-band model can be derived by solving the four-band
$P$ model, $\mathcal{H}_{P}^{eff(1)}$. After straightforward calculation, we
find
$\displaystyle|E_{+,-}\rangle$
$\displaystyle=\frac{1}{N_{+,-}}(\lambda_{z}+E_{\lambda},\lambda_{x}+i\lambda_{y})^{T}\otimes(d_{z}-E_{d},d_{x}+id_{y})^{T},$
(355a) $\displaystyle|E_{-,+}\rangle$
$\displaystyle=\frac{1}{N_{-,+}}(\lambda_{z}-E_{\lambda},\lambda_{x}+i\lambda_{y})^{T}\otimes(d_{z}+E_{d},d_{x}+id_{y})^{T},$
(355b)
where we have
$\displaystyle N_{s,\omega}^{2}$
$\displaystyle=((\lambda_{z}+sE_{\lambda})^{2}+\lambda_{x}^{2}+\lambda_{y}^{2})((d_{z}+\omega
E_{d})^{2}+d_{x}^{2}+d_{y}^{2}),$ (356a) $\displaystyle E_{\lambda}$
$\displaystyle=\sqrt{\lambda_{x}^{2}+\lambda_{y}^{2}+\lambda_{z}^{2}},$ (356b)
$\displaystyle E_{d}$ $\displaystyle=\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}},$
(356c)
where we define $\lambda_{x}=\sin\theta\cos\phi$,
$\lambda_{y}=\sin\theta\sin\phi$, $\lambda_{z}=\cos\theta$,
$d_{x}=\cos\theta\sin^{2}\theta\sin\phi\cos\phi(\cos\phi+\sin\phi)$,
$d_{y}=\cos\theta\sin^{2}\theta\sin\phi\cos\phi(\sin\theta+\cos\theta(\cos\phi-\sin\phi))$,
and
$d_{z}=\sin^{2}\theta\sin\phi\cos\phi(\cos^{2}\theta+\sin\theta\cos\theta(-\cos\phi+\sin\phi))$.
Within the sub-space formed by these two eigen-states, the effective
Hamiltonian $\mathcal{H}_{P}^{eff(1)}$ can be reduced to a two-band model
$\displaystyle\mathcal{H}_{eff}=\epsilon_{0}+d_{z}(\mathbf{k})\sigma_{z},$
(357)
where $\epsilon_{0}=C_{0}+B_{1}k^{2}+A_{1}k$ and
$d_{z}(\mathbf{k})=\lambda_{0}-\tfrac{\sqrt{3}}{4}\tilde{C}k^{2}|\sin
2\phi\sin
2\theta\sin\theta|=\lambda_{0}-\sqrt{3}\tilde{C}\frac{|k_{x}k_{y}k_{z}|}{k}$.
This Hamiltonian commutes with $\sigma_{z}$, which in turn generates the U(1)
quasi-symmetry group for the two-band model. The physical viewpoint of this
quasi-symmetry is related to the spin texture of the Fermi surfaces in the
momentum space. Due to the SOC term with $\mathbf{k}\cdot\mathbf{s}$ in the
$P$-model, the spin texture is in a hedgehog form on each Fermi surface
($\langle\mathbf{s}\rangle\sim\alpha\mathbf{k}$ with $\alpha=\pm$). It
indicates that the crossings between these two Fermi surfaces have opposite
spin-polarization. Therefore, according to the low-energy effective
Hamiltonian, the quasi-symmetry that protects the approximate nodal planes can
be viewed as the spin U(1) symmetry along the momentum direction.
Now we discuss the breaking of the quasi-symmetry by second-order perturbation
corrections. To see that, we can further project $\mathcal{H}_{P}^{eff(2)}$
into the subspace and the effective Hamiltonian includes two more terms
$\displaystyle\mathcal{H}_{eff(2)}=\delta d_{x}(\mathbf{k})\sigma_{x}+\delta
d_{y}(\mathbf{k})\sigma_{y},$ (358)
which are due to the spin-flipping terms. As a result, the co-dimension of the
nodal plane becomes 3 instead of 1, explaining the gap opening. While the
detailed expressions for $\delta d_{x}$ and $\delta d_{y}$ are complex and not
orthogonality, we give an estimate of the typical magnitude of the gap
$\Delta_{eff(2)}=2\sqrt{(\delta d_{x})^{2}+(\delta d_{y})^{2}}\sim 0.8$ meV
for $k_{F}=0.13$ Å-1, being the same order with the DFT estimations [1].
Therefore, perturbation theory leads to a general two-band model
$\displaystyle\mathcal{H}_{2\times
2}=\delta_{d_{x}}\sigma_{x}+\delta_{d_{y}}\sigma_{y}+\delta_{d_{z}}\sigma_{z}.$
(359)
The gap of this model is given by
$\displaystyle\Delta_{k,\text{soc}}=2\sqrt{(\delta_{d_{x}})^{2}+(\delta_{d_{y}})^{2}+(\delta_{d_{z}})^{2}}.$
(360)
If we impose the constraint of the quasi-symmetry $\sigma_{z}$ that commutes
with $\mathcal{H}_{2\times 2}$, we can always obtain the constraint equation
for the nodal planes by solving $\delta_{d_{z}}=0$.
Next, let us consider the other terms that break the quasi-symmetry, by taking
Eq. (266) as an example. The projected two-band model is then numerically
calculated directly. The three components are given by
$\displaystyle\begin{split}\delta_{d_{x}}&=\text{Re}[\langle
E_{+,-}|\mathcal{H}_{k,\lambda_{2},eff}|E_{-,+}\rangle],\\\
\delta_{d_{y}}&=-\text{Im}[\langle
E_{+,-}|\mathcal{H}_{k,\lambda_{2},eff}|E_{-,+}\rangle],\\\
\delta_{d_{z}}&=\frac{1}{2}[\langle
E_{+,-}|\mathcal{H}_{k,\lambda_{2},eff}|E_{+,-}\rangle-\langle
E_{-,+}|\mathcal{H}_{k,\lambda_{2},eff}|E_{-,+}\rangle],\end{split}$ (361)
The numerical results are shown in Fig. 8. The results only depend on $\theta$
and $\phi$, then, on the $k_{r}=0.15$ Å-1 sphere with $\lambda_{2}=0.005$ eV,
the numerical calculated $\delta_{d_{x}}$ and $\delta_{d_{y}}$ are shown in
(a) and (b), respectively. In the whole space, $\delta_{d_{x}}\neq 0$ and
$\delta_{d_{y}}$ expect the Weyl point along the $\Gamma-R$ line, which is
shown in Fig. 8 (c) by log-plotting the inverse of the gap
$\log[\frac{1}{\sqrt{\delta_{d_{x}}^{2}+\delta_{d_{y}}^{2}}}]$ (i.e., the gap
vanishes only at the Weyl point). The entire quasi-nodal plane are gapped out
by including the general liner-$k$ SOC Hamiltonian.
Figure 8: The projected two-band model with $\delta d_{x}$ in (a), $\delta
d_{y}$ in (b) and the corresponding gap in (c). We find that the gap only
closes at the Weyl point. It indicates the quasi-symmetry protected nodal
plane is completely gapped out by the linear-k SOC Hamiltonian. Here
$k_{r}=0.15$ Å-1 and $\lambda_{2}=0.005$ eV. |
# Partial Dyck paths with Air Pockets
Helmut Prodinger Helmut Prodinger, Mathematics Department, Stellenbosch
University, 7602 Stellenbosch, South Africa, and NITheCS (National Institute
for Theoretical and Computational Sciences), South Africa<EMAIL_ADDRESS>
###### Abstract.
Dyck paths with air pockets are obtained from ordinary Dyck paths by
compressing maximal runs of down-steps into giant down-steps of arbitrary
size. Using the kernel method, we consider partial Dyck paths with air
pockets, both, from left to right and from right to left.
In a last section, the concept is combined with the concept of skew Dyck
paths.
## 1\. Introduction
In a paper that was posted on valentine’s day [1], Baril et al. introduced a
new family of Dyck-like paths, called _Dyck paths with air pockets_. Many of
the usual parameters that one could think of are investigated in this paper.
The paths have the usual up-steps $(1,1)$ and down-steps $(1,-k)$ for any
$k=1,2,\dots$, but no such down-steps may follow each other. Otherwise, they
cannot go into negative territory, and must end at the $x$-axis, as usual. One
could just think about ordinary Dyck paths, and each (maximal) run of down-
steps is condensed into one (giant) downstep.
The Figure 1 explains the actions readily.
Figure 1. Graphical description of Dyck paths with air pockets. Top layer
describes the situation after an up-step, bottom layer after a down-step.
We introduce generating functions $f_{k}(z)$ and $g_{k}(z)$ where the
coefficient of $z^{n}$ in one of these functions counts paths ending in the
respective state according to the number of steps. The function
$f_{0}(z)+g_{0}(z)$ counts the Dyck paths with air pockets, as the zero in the
index just means that they returned to the $x$-axis.
In this short paper, we will enumerate _partial_ Dyck paths with air pockets,
namely we allow the path to end at level $i$. In other words, we compute all
$f_{k}(z)$ and $g_{k}(z)$.
Our instrument of choice is the kernel method, as can be found in the popular
account [2].
## 2\. Generating functions
Just looking at Figure 1, we find the following recursion, where we write
$f_{k}$ for $f_{k}(z)$ for simplicity:
$\displaystyle f_{0}$ $\displaystyle=1,$ $\displaystyle f_{k}$
$\displaystyle=zf_{k-1}+zg_{k-1},\quad k\geq 1,$ $\displaystyle g_{k}$
$\displaystyle=zf_{k+1}+zf_{k+2}+zf_{k+3}+\cdots,$
and now we introduce bivariate generating functions
$F(u,z)=F(u)=\sum_{k\geq 0}u^{k}f_{k}(z),\quad G(u,z)=G(u)=\sum_{k\geq
0}u^{k}g_{k}(z).$
Summing the recursions,
$F(u)=1+zuF(u)+zuG(u)$
and
$G(u)=\sum_{k\geq
0}u^{k}z\sum_{j>k}f_{j}=z\sum_{j>0}f_{j}\sum_{k=0}^{j-1}u^{k}=z\sum_{j>0}f_{j}\frac{1-u^{k}}{1-u}=\frac{z}{1-u}(F(1)-F(u)).$
Eliminating one function, we are left to analyze
$F(u)=1+zuF(u)+\frac{z^{2}u}{1-u}(F(1)-F(u)).$
Solving, we find
$F(u)=\frac{1-u+{z}^{2}uF(1)}{-zu+z{u}^{2}+{z}^{2}u+1-u}=\frac{1-u+{z}^{2}uF(1)}{z(u-s_{1})(u-s_{2})},$
with
$\displaystyle s_{1}$
$\displaystyle=\frac{1+z-{z}^{2}+\sqrt{-{z}^{2}-2{z}^{3}-2z+{z}^{4}+1}}{2z},$
$\displaystyle s_{2}$
$\displaystyle=\frac{1+z-{z}^{2}-\sqrt{-{z}^{2}-2{z}^{3}-2z+{z}^{4}+1}}{2z}.$
Note that $s_{1}s_{2}=\frac{1}{z}$. We still need to compute $F(1)$. Before we
can plug in $u=1$ and compute it, we must cancel the bad factor of both,
numerator and denominator. In this case, this is the factor $u-s_{2}$, since
the reciprocal of it would not allow a Taylor expansion around $u=1$. The
result is
$F(u)=\frac{-1+z^{2}F(1)}{zs_{2}-z+z^{2}-1+zu},$
from which we now can compute $F(1)$ by plugging in $u=1$. We get
$F(1)=\frac{-1+z^{2}F(1)}{zs_{2}+z^{2}-1}=\frac{1}{1-zs_{2}}$
and therefore
$F(u)=\frac{1-s_{1}}{1-zs_{2}}\frac{1}{u-s_{1}}=-\frac{1}{s_{1}}\frac{1-s_{1}}{1-zs_{2}}\frac{1}{1-u/s_{1}}.$
Reading off the coefficient of $u^{k}$, we further get
$f_{k}=-\frac{1}{s_{1}^{k+1}}\frac{1-s_{1}}{1-zs_{2}}=-z^{k+1}s_{2}^{k+1}\frac{1-1/(zs_{2})}{1-zs_{2}}=-z^{k}s_{2}^{k}\frac{zs_{2}-1}{1-zs_{2}}=z^{k}s_{2}^{k}.$
Since $G(u)=\dfrac{F(u)-1-zuF(u)}{zu}$, we also find
$g_{k}=\frac{1}{z}f_{k+1}-f_{k}=z^{k}(s_{2}^{k+1}-s_{2}^{k}).$
We can also compute $\textsc{total}(z)=F(1,z)+G(1,z)$ which counts path that
end anywhere, and the result is
$\textsc{total}(z)={\frac{1-z-{z}^{2}-\sqrt{-{z}^{2}-2{z}^{3}-2z+{z}^{4}+1}}{2{z}^{3}}}=\frac{1}{z^{2}}g_{0}.$
In retrospective, this is not surprising, since if we consider paths that end
at state 0 in the bottom layer, and we go back the last 2 steps, we could have
been indeed in any state.
It is worthwhile to notice that
$f_{0}+g_{0}=1+z^{2}+z^{3}+2z^{4}+4z^{5}+8z^{6}+17z^{7}+37z^{8}+82z^{9}+185z^{10}+423z^{11}+\cdots$
and the coefficients $1,1,2,4,8,17,\dots$ are sequence A004148 in [4].
###### Theorem 1.
The generating functions describing partial Dyck paths with air pockets,
landing in state $k$ of the upper/lower layer, are given by
$f_{k}=z^{k}s_{2}^{k},\quad g_{k}=z^{k}(s_{2}^{k+1}-s_{2}^{k}).$
In particular, $f_{k}+g_{k}=z^{k}s_{2}^{k+1}$ is the generating function of
partial paths ending at level $k$.
## 3\. Right to left model
Reading Dyck paths with air pockets from right to left means to have arbitrary
long up-steps, but only one at the time. While the enumeration for those paths
that end at the $x$-axis is the same as before, this is not the case for
_partial_ paths.
Figure 2 explains the concept. The generating functions $a_{k}$ refer to the
top layer and $b_{k}$ to the bottom layer.
Figure 2. Graphical description of Dyck paths with air pockets. Top layer
describes the situation after a down-step, bottom layer after an up-step.
The recursions are111Iverson’s notation is used here.
$\displaystyle a_{k}$ $\displaystyle=[k=0]+zb_{k+1},$ $\displaystyle b_{k}$
$\displaystyle=zb_{k+1}+z\sum_{0\leq j<k}a_{j}.$
With bivariate generating functions analogously to before, we find by summing
$A(u)=1+\frac{z}{u}(B(u)-b_{0})$
and
$B(u)=\frac{z}{u}(B(u)-b_{0})+z\sum_{0\leq
j<k}a_{j}u^{k}=\frac{z}{u}(B(u)-b_{0})+\frac{zu}{1-u}A(u).$
One variable can be eliminated:
$B(u)=\frac{z}{u}(B(u)-b_{0})+\frac{zu}{1-u}+\frac{z^{2}}{1-u}(B(u)-b_{0}).$
Solving
$B(u)={\frac{z\left(B(0)-B(0)u-{u}^{2}+zB(0)u\right)}{z-zu+{z}^{2}u-u+{u}^{2}}}$
The denominator factors as $(u-s_{1}^{-1})(u-s_{2}^{-1})$. The bad factor is
this time $(u-s_{1}^{-1})$. Dividing it out,
$B(u)=\frac{z\left(-us_{1}-B(0)s_{1}+B(0)s_{1}z-1\right)}{us_{1}-zs_{1}+{z}^{2}s_{1}-s_{1}+1}$
and further
$B(0)=b_{0}=\frac{z}{s_{1}-1}=s_{2}-1.$
Thus, after some simplifications,
$B(u)=-z+\frac{zs_{1}}{(s_{1}-1)(1-\frac{u}{zs_{1}})},$
or
$B(u)=-z+\frac{1}{s_{2}(s_{1}-1)(1-s_{2}u)}=-z+\frac{s_{2}-1}{zs_{2}(1-s_{2}u)}$
and then
$b_{k}=\frac{s_{2}-1}{z}s_{2}^{k-1},\quad k\geq 1.$
The functions $a_{k}$ could be computed from here as well, but for the partial
paths only the functions $b_{k}$ are of relevance, if we don’t consider the
empty path.
###### Theorem 2.
The generating functions of partial Dyck paths with air pockets in the right
to left model are
$\displaystyle 1+b_{0}=s_{2}$
and
$b_{k}=\frac{s_{2}-1}{z}s_{2}^{k-1},\quad k\geq 1.$
To consider the total does not make sense, since in just 1 or 2 steps, every
state can be reached, so a sum over $b_{k}$ would not converge.
## 4\. Skew Dyck paths with air pockets
The walks according to Figure 3 are related to skew Dyck paths [3]; the red
down-steps are modeled to stand for south-west steps, and the way they are
arranged, there are no overlaps of such a path. See [3] and the references
cited there.
Figure 3. Three layers of states according to the type of steps leading to
them (up, down-black, down-red).
Now we combine this model with air pockets. Each maximal sequence of black
down-steps is condensed into one giant down-step, depicted in dashed grey in
Figure 4
Figure 4. Three layers of states according to the type of steps leading to
them (up, down-black, down-red). Black down-steps are condensed into giant
grey down-steps.
Introducing generating functions, according to the three layers, we find the
following recursions by inspection;
$\displaystyle a_{0}$ $\displaystyle=1,\quad a_{k+1}=za_{k}+zb_{k},\ k\geq 0,$
$\displaystyle b_{k}$ $\displaystyle=z\sum_{j>k}a_{j}+z\sum_{j>k}c_{j},$
$\displaystyle c_{k}$ $\displaystyle=zb_{k+1}+zc_{k+1}.$
Translating these into bivariate generating functions, we further have
$\displaystyle A(u)=1+zuA(u)+zuB(u),$ $\displaystyle
B(u)=\frac{z}{1-u}[A(1)-A(u)]+\frac{z}{1-u}[C(1)-C(u)],$ $\displaystyle
C(u)=zuB(u)+zuC(u).$
Solving,
$\displaystyle A(u)$
$\displaystyle={\frac{{z}^{3}{u}^{2}A(1)+{z}^{3}{u}^{2}C(1)-z{u}^{2}-{z}^{2}uC(1)-{z}^{2}u-{z}^{2}uA(1)+zu+u-1}{\left(-1+zu\right)\left(z{u}^{2}+2{z}^{2}u-zu-u+1\right)}},$
$\displaystyle B(u)$
$\displaystyle=-{\frac{\left(-A(1)-C(1)+zuA(1)+zuC(1)+1\right)z}{z{u}^{2}+2{z}^{2}u-zu-u+1}},$
$\displaystyle C(u)$
$\displaystyle={\frac{{z}^{2}u\left(-A(1)-C(1)+zuA(1)+zuC(1)+1\right)}{\left(-1+zu\right)\left(z{u}^{2}+2{z}^{2}u-zu-u+1\right)}}.$
We factor $z{u}^{2}+2{z}^{2}u-zu-u+1=(u-s_{1})(u-s_{2})$ with
$s_{2}={\frac{-2{z}^{2}+z+1-\sqrt{4{z}^{4}-4{z}^{3}-3{z}^{2}-2z+1}}{2z}},\quad
s_{1}=\frac{1}{zs_{2}}.$
Since $A(u)-C(u)=\frac{1}{1-zu}$, we have $A(1)-C(1)=\frac{1}{1-z}$, and we
only need to compute one of them. Dividing the (bad) factor $(u-s_{2})$ out,
plugging in $u=1$ and solving leads to
$A(1)={\frac{-s_{2}z+2-z}{2\left(1-s_{2}z\right)\left(1-z\right)}}=\frac{1}{2(1-z)}+\frac{1}{2(1-zs_{2})}$
and
$C(1)=-\frac{1}{2(1-z)}+\frac{1}{2(1-zs_{2})}.$
Using these values, we find
$A(u)+B(u)+C(u)=\frac{s_{2}(1-z^{2}-zs_{2})}{(1-zs_{2})(1-uzs_{2})}$
and furthermore
$[u^{k}](A(u)+B(u)+C(u))=\frac{z^{k}s_{2}^{k+1}(1-z^{2}-zs_{2})}{(1-zs_{2})}$
These functions describe all skew Dyck paths with air pockets, ending at level
$k$. For $k=0$, this yields
$1+z^{2}+z^{3}+3z^{4}+7z^{5}+17z^{6}+45z^{7}+119z^{8}+323z^{9}+893z^{10}+2497z^{11}+\cdots.$
## References
* [1] Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, Vincent Vajnovszki, Enumeration of Dyck paths with air pockets, arXiv:2202.06893.
* [2] H. Prodinger, The kernel method: A collection of examples, Sém. Lothar. Combin., B50f (2004), 19 pages.
* [3] H. Prodinger, Partial skew Dyck paths—a kernel method approach, preprint, 2021.
* [4] N. J. A. Sloane et al., The On-line Encyclopedia of Integer Sequences, 2022.
|
# A General Approach to Coding in Early Olfactory and Visual Neural
Populations
###### Abstract
Recent experimental and theoretical work on neural populations belonging to
two separate early sensory systems, olfaction and vision, has challenged the
notion that the two operate under different computational paradigms by
providing evidence for the respective neural population codes having three
central, common features: they are highly redundant; they are organized such
that information is carried in the identity, and not the relative timing, of
the active neurons; they are capable of error correction. We present the first
model that captures these three properties in a general manner, making it
possible to investigate whether similar structure is present in other
population codes. Our model also makes specific predictions about additional,
as yet unseen, structure in such codes. If these predictions are found in real
data, this would provide new evidence that such population codes are operating
under more general computational principles.
## A General Approach to Coding in Early Olfactory and Visual Neural
Populations
William T. Redman
University of California, Santa Barbara
<EMAIL_ADDRESS>
## Introduction
Because of their relative ease to record from and their importance in being
the input to upstream brain areas, the coding properties of neural populations
belonging to early sensory systems have been the focus of a large body of
experimental and theoretical literature. This work has explored the precision
of retinal spike trains [1], the correlation structure of individual retinal
ganglion cells and their population wide collective states [2], [3], the
possibility of criticality in retinal populations [4], [5], the preservation
of odor identity representation by glomeruli (structures in the olfactory bulb
that receive projections from olfactory sensory neurons - OSNs) across varying
odor concentrations [6], [7], [8], [9] and the combinatorial nature of the
glomeruli code [10], [11]. While the respective work has followed a similar
mission (to understand the nature of the population codes), the results have
caused a divergence in the belief of the coding principles at use. Yet, in
both systems, the exact structure of the exact nature of the population code
has been unclear.
Recent work has taken advantage of advances in recording and manipulation
technology, as well as statistical methods, to probe more intricately at this
question of the exact structure of the retinal ganglion and the glomeruli
population codes [12], [13], [14], [15]. Despite the differences in system and
approach, the two have converged on three similar principles for the
respective codes. First, the codes are highly redundant, as only a small
subset of the neurons are responsible for carrying the information about the
identity of any given stimulus (whether it be some feature of the visual field
or an odor). Second, it is the identity of these neurons in the specific
subset (and not their relative timing) that carries the relevant information.
Finally, a neural population code with those two features is endowed with the
capability to be robust to noise, in the sense that the code is capable of
error correction. These core similarities clearly challenge the assumption
that the two systems are operating under different coding paradigms.
In the retinal ganglion population code, these features come about by the fact
that the probability space of population responses is populated by geometric
objects identified as ridges. These ridges correspond to unique “codewords”
that a downstream system maps all responses in the ridge to [13]. The identity
of these ridges is determined by an active set of neurons (those neurons that
were active in the states that make up a given ridge) and a silent set (those
neurons that weren’t active). The mapping from a given neural response to the
appropriate, ridge-specific codeword was hypothesized to be achieved by an
additional layer of neurons, each one firing if a certain fraction of neurons
in a given active set fire and none of neurons in the corresponding silent set
fire (see [13] Fig. 13). Under this simple model, it’s easy to see that there
could be a number of population response states that get mapped to the same
codeword.
In the glomeruli population code, these features arise from the fact that the
code has been found operate under the “Primacy Hypothesis”, namely that it is
the first $n$ active glomeruli that are responsible for encoding odor identity
[14]. Therefore, all population responses that have the same first $n$ active
glomeruli (where the order of the first $n$ active glomeruli doesn’t matter
[14]), are recognized as being the same odor. Any of those given states can
therefore be seen as a codeword. The relevant time scale (which determines the
relevant $n$) was found to be $<100$ ms [14]. Whether $n$ is fixed, or whether
it can vary for various odors and odor mixtures, has yet to be determined.
Recent theoretical work has suggested that the aforementioned three properties
might be a more universal feature of neural population codes [16]. In
particular, brain regions such as MT and V1, which have similar firing rates
and pairwise correlations to the retinal ganglion cells, were hypothesized to
have these same properties in their neural codes. A more general framework in
which to talk about neural population codes with these three properties is
therefore desirable, especially if it can make predictions to specific
structures that these codes should have.
The rest of this paper will focus on such a framework. In the Method section,
we outline our model for mapping arbitrary neural population responses to
codewords. Our model utilizes the universal property of free groups (UPFG). We
then provide an example of our model by mapping a specific set of neural
responses to a given set codewords. We compare our methods mapping with the
mapping generated by using Generalized Minimum Distance (GMD) decoding [17], a
well known method in coding theory. This example is given to provide a more
intuitive feel for our model (especially for those unfamiliar with group
theory) and illustrate how it compares to other known (but not directly
implicated in neural population coding) error correction algorithms. Finally,
in the Discussion section, we discuss the implications of our model with
regards to real neural population data and possible future directions for this
work.
## Why use the UPFG?
There are two reasons why we turn to using such a formal, and mathematically
abstract, language as group theory for this problem. First, for those with a
background in group theory, we believe that our model is fairly intuitive.
Second, and more pertinently, theoretical neuroscience has seen great advances
when appropriate theories/descriptions from formal physics and mathematics
have been applied to neural problems. For instance, by borrowing ideas and
analysis techniques from statistical physics, attractor neural networks (ANNs)
were able to be developed and thoroughly explored [18], [19]. In this vein, we
feel that when considering redundant neural codes that “collapse” different
population response states onto the same output (or recognized) state, the
theory associated with homomorphisms (and group theory), is an appropriate
language to use. For those unfamiliar with group theory, see some basic
discussion in the Material and Methods section for a brief discussion of the
simple group theoretic definitions and notions used in our model.
## Model
We start by defining the set of generators, $G$, as $G=S+B$, where
$S=\\{1,2,...,n\\}$, $B$ is a subset of the power set of $S$ (i.e.
$B\subset\mathcal{P}(S)$), and $+$ is the set concatenation operator. We refer
to $B$ as the basis set. The relevance of $B$ will be discussed below.
A simple example of this is, for $n=5$, $G=\\{1,2,3,4,5,12,145\\}$, where the
last two elements are the elements of the basis set (here $12$ stands for
$\\{1,2\\}$).
We define the group of codewords, $(C,*_{C})$, as
$C=\\{c_{1},c_{2},...,c_{m}\\}$ and $*_{C}$ is some operation on the elements
of $C$ that meets the standard group criteria [20].
Finally, the restricted free group of G, $\tilde{F}(G)$, is defined to be the
abeleanized group of all elements in the free group of $G$, $F(G)$, that are
made up of, at most, each element of $G$ once. For instance, while $121\in
F(G)$, $121\notin\tilde{F}(G)$ because it has $1$ twice.
Using the universal property of free groups [20], we have the following
diagram,
${G}$${F(G)}$${C}$$i$$g$$\varphi$ (1)
where $i$ is the inclusion map (i.e. $i(x)=x$ for all $x\in G$), $g$ is a
group function determining $C$ from $G$, and $\varphi$ is a unique
homomorphism (given a specific $g$) from $F(G)$ to $C$. The UPFG tells us that
we can relate $g$ to $\varphi$ by
$\varphi(A)=g(a_{1})*_{C}...*_{C}g(a_{n})$ (2)
where $A\in F(G)$ and $A=a_{1}...a_{n}$, such that $a_{i}\in G$ for all $1\leq
i\leq n$.
A final point on our model must be made. We define, for all $s\in S$,
$g(s)=\text{id}_{C}$ (3)
where idC is the identity element of the group of code words (i.e. for all
$a\in C$, $a\hskip 2.84526pt*_{C}$ id${}_{C}=a$).
With this definition, it is more clear what the role of the basis set, $B$, is
(as it is unnecessary for “building” $F(G)$); it is the generator of
$(C,*_{C})$ under $g$.
We now consider an element (or “word”), $\omega$, of $\tilde{F}(G)$ (we
consider $\tilde{F}(G)$ because of it represents all possible neural responses
that were considered in the neural data from [12] \- [15], but the same holds
true, with small modifications, for any element in $F(G)$). We want to find
the codeword (i.e. the element in $(C,*_{C})$ that corresponds to $\omega$. In
particular, for any given $\omega$, there exists a subset of $B$,
$\\{b_{1},...,b_{k}\\}$, and a subset of $S$, $\\{s_{1},..,s_{m}\\}$
($s_{j}\in S$), s.t.
$\omega=b_{1}...b_{k}s_{1}...s_{m}$ (4)
(the exact ordering of the $b_{i}$’s and $s_{j}$’s doesn’t matter because
$\tilde{F}(G)$ is abelian). Of course, this decomposition of $\omega$ is by no
means unique. We therefore define the decomposition of $\omega$ as the pair
$(\\{b_{i}\\},\\{s_{j}\\})$, s.t. the number of elements of $\\{b_{i}\\}$ is
the (if possible) non-zero minimum of all the possible decompositions of
$\omega$. From the neural perspective, this is equivalent to demanding that
every word is represented as simply as possible by the activity states that
make up the basis set.
With this, we can now look at the mapping of $\omega$ to its relevant
codeword, which is given by $\varphi(\omega)$ (since
$\varphi:\tilde{F}(G)\rightarrow C$).
$\varphi(\omega)=\varphi(b_{1}...b_{k}s_{1}...s_{l})$ (5)
$=g(b_{1})*_{C}...*_{C}g(b_{k})*_{C}g(s_{1})*_{C}...*_{C}g(s_{l})$
$=g(b_{1})*_{C}...*_{C}g(b_{k})$
By the defined decomposition of $\omega$, this decoding is unique.
Note therefore that if two elements, $\omega_{1}$ and $\omega_{2}$, in
$\tilde{F}(G)$ have the same basis elements, $\\{b_{i}\\}$, in their
decomposition, then the decoding of the two elements is equivalent
$\varphi(\omega_{1})=g(b_{1})*_{C}...*_{C}g(b_{k})=\varphi(\omega_{2})$ (6)
### Example
To illustrate our model, we map an example neural response space onto to
example codewords. We also compare this mapping to an existing error
correction method, Generalized Minimal Distance (GMD) decoding [17], as a way
to show the possible strengths of our method in reference to existing methods.
We will consider $G=\\{1,2,3,4,12,24\\}$,
$C=\big{(}\\{0,12,24,14\\},+\big{)}$. For simplicity, we will convert each
element of $G$ and $C$ into a binary string. This corresponds to
$G=\\{1000,0100,0010,0001,1100,0101\\}$ and
$C=\big{(}\\{0000,1100,0101,1001\\},+$ $\text{mod(2)}\big{)}$, if we take each
number, $1,..,4$, to be a position in a four bit string that has a value of
$1$.
The result of applying our mapping method and applying GMD decoding (where the
decoding is determined by the codeword that has the minimal Hamming distance
from the word we are trying to decode) is given in Table 1. Note that we are
assuming that every word is equally likely to be received. From this, we see
that, first and foremost, the number of three way ties (as denoted by $?$) is
significantly less using the UPFG decoding as opposed to GMD decoding (one vs.
eight). Additionally, in every determined decoding, the two methods agree.
Table 1: Comparison of our model (UPFG mapping) and GMD decoding on an example response space Word | UPFG decoding | GMD decoding
---|---|---
1111 | ? | ?
1110 | 1100 | 1100
1101 | 1100 | ?
1011 | 1001 | 1001
0111 | 0101 | 0101
1100 | 1100 | 1100
1001 | 1001 | 1001
0011 | 0000 | ?
1010 | 0000 | ?
0101 | 0101 | 0101
0110 | 0000 | ?
1000 | 0000 | ?
0100 | 0000 | ?
0010 | 0000 | 0000
0001 | 0000 | ?
0000 | 0000 | 0000
Table 2:
## Discussion
Recent work investigating the structure of the retinal ganglion and glomeruli
population codes, [12] \- [15], has challenged a number of widely held
theoretical biophysical and neural beliefs with their three central (and
convergent) conclusions: the respective neural population codes are redundant,
in the sense that multiple neural response states are interpreted as encoding
the same information; it is the identity of the relevant subset of neurons
that fire, and not the relative timing of their firing, that encodes the
identity of the stimulus; such codes with the previous two properties are
endowed with the capability of error correction.
The fact that these neural population codes were found to be redundant
contradicts the belief that neural codes should be very efficient. Such
conclusions will force theorists to reconsider what, if anything, neural
population codes are optimized for. Additionally, the finding that the first
two properties allow for error correction marks a transition from the focus of
biophysical and neural theory on how systems can output robustly despite noise
(for a few of many examples, see [21], [22], [23], [24]) to the idea that the
output of neural systems can be noisy, but it is the subsequent mapping (or
“interpretation”) of that output that is robust to noise.
Finally, that these conclusions are reached by very different experimental and
theoretical methods in the retinal ganglion and glomeruli populations suggest
that the two sensory systems (vision and olfaction), despite being previously
believed to be operating under different coding principles, are, in fact,
using the same principles. This surprising (and powerful) fact, coupled with
theoretical work [16] arguing that these coding principles might be used in
more than just early sensory systems, and in higher brain areas like MT and
V1, highlights the need for a general framework in which to discuss codes with
such three properties.
To see that our model indeed is a general framework that meets all three of
these facets of neural population codes, note first that it is only the basis
elements that determine the decoding of any word. Therefore, words that vary
by elements that don’t affect the basis elements are seen as equivalent (i.e.
eq. 6). This can also be clearly seen in Table 1 (e.g. 1100 and 1101 have the
same mapping). Second, the fact that we restrict ourselves to looking
specifically at the restricted free group of $G$, $\tilde{F}(G)$, we are not
only restricting ourself to the more reasonable neural case where neurons are
consider to have fired at most once in a time bin, we are also restricting
ourselves to only the identity of the basis elements mattering in decoding and
not the ordering, as $\tilde{F}(G)$ is abelian, and hence, the words
$\omega_{1}\omega_{2}$ and $\omega_{2}\omega_{1}$ are equivalent. Finally, as
noted before, because only the basis elements in the decomposition of a given
word determine the decoding, our method is capable of error correction.
Our model, while capturing the three main principles of the aforementioned
work, also makes predictions about aspects of the early sensory systems’
coding. In particular it predicts the existence of a basis set. In the context
of the glomeruli population code, there exist two possibilities. Either one,
$n$ (the number of relevant active glomeruli for odor classification) is fixed
and the basis set is the set of codewords, which gives no new insight into the
code. If $n$ is not fixed, then there exists the possibility of all codewords
being able to be built from some smaller set of response states that are
themselves codewords. For example, if $n=2,3,$ and $4$ are all allowed (e.g.
$n$ is a function of odor complexity), then any $n=4$ codeword could be built
from two codewords that are $n=2$. This greatly reduces the amount of elements
needed for describing the codewords, and sheds light onto possible downstream
decoding mechanisms.
Similarly, for the retinal population code, the existence of a non-trivial
basis set would shed light on downstream decoding schemes. In particular, it
would provide possible adjustments to the model hypothesized in [13] (i.e.
Fig. 13). It is important to note that looking for the basis set in the
retinal population code may require a switch in perspective, where it is the
neurons in a given codewords silent set that might be the relevant feature.
Future work will focus on methods for searching for basis sets, as well as
specific implications of the existence of basis sets for downstream decoding
schemes and how such schemes might develop in a natural way. Finding such
basis sets in both the glomeruli and retinal population codes would extend
even further the growing understanding of how similar the two codes are, and
provide more evidence for the two operating under more universal computing
principles.
One clear failure of our model is that is relies on group theoretic notions
that most in the neuroscience community aren’t familiar with. While we believe
that continuing to think in terms of this language will be useful (and indeed,
we hope that our model convinces others in the mathematical community to
consider the possible role free groups, and the UPFG, might play in error
correction and neural coding), we hope to translate this model into a more
clear, and less mathematically technical, model that still captures the main
principles, but can more easily be communicated to others.
We hope that we have made clear the similarities of the two early sensory
systems’ population codes, the need for a general framework in which to
explore arbitrary codes that share the properties exhibited by those two
codes, and the possible utility of using group theory as a language to talk
about such codes.
### Materials and Methods
We provide here the basic group theory definitions and notions that are
required for understanding our model. We have tried to make this as easy to
understand as possible for the reader not familiar with group theory. Note
that therefore some of the extra complications or subtleties are swept under
the rug at the discretion of the author, especially if they are not believed
to be relevant for understanding our model. Everything written below can be
found in the following two references [20], [25].
### Groups
A group is defined as the pair $(G,*_{G})$, where $G$ is a set of elements and
$*_{G}$ is a binary operation that satisfies the following three properties:
1. 1.
There exists an identity element in $G$ under $*_{G}$. That is, there is an
element idG such that id${}_{G}*_{G}x=x$ for all $x\in G$.
2. 2.
The elements of $G$ are self contained under $*_{G}$. That is, for all $x,y\in
G$, $(x*_{G}y)\in G$.
3. 3.
There exists an inverse element for all elements of $G$. That is, for all
$x\in G$, there exists $x^{-1}$ such that $x*_{G}x^{-1}=$ idG.
An example of a group is the integers under addition. It is easily verified
that all three conditions are met, where the identity element is $0$ and the
inverse of $n$ is $-n$.
For simplicity, groups will now be referred to as just $G$, where $*_{G}$ is
implicitly assumed.
### Abelian
A group is said to be abelian if the elements of $G$ commute. That is, if for
all $x,y\in G$, $x*_{G}y=y*_{G}x$.
For the integers under addition, this is clearly the case. But most groups are
not abelian (e.g. the set of matrices with unit eigenvalues under matrix
multiplication are not abelian).
### Homomorphisms
A homomorphism is a map, $\varphi$, between two groups, $G$ and $H$, such that
$\varphi(x*_{G}y)=\varphi(x)*_{H}\varphi(y)$ for all $x,y\in G$. Note that the
binary operations are different on each side of the equation.
In simpler terms, a homomorphism is a map that collapses one group onto
another, while preserving some structure. For instance, the parity map (i.e.
the map that returns $0$ if the argument is even and $1$ if the argument is
odd), $\varphi$, from the integers to the integers modulus $2$ (i.e.
$(\\{0,1\\},+\text{mod}(2))$, where $1+1=0$ mod$(2)$) is a homomorphism.
$\varphi$ clearly collapses the integers (it reduces them to a set with only
two elements), but it preserves some structure (namely, parity).
### Free groups
A free group, $F(G)$, is an infinite group (in the sense that the set that
comprises the elements of the free group is infinite) that is comprised of
every possible combination of the elements of the set $G$, using the binary
operation $*$. For instance, if $G=\\{a,b,a^{-1},b^{-1}\\}$, then the set
comprising $F(G)$ is given by
$\\{a,b,a*a,a*b,b*a,a^{-1}*b,a*b^{-1},a^{-1}*b^{-1},a*a*a,...\\}$.
### Acknowledgements
We thank Sylvain Cappell for introducing us to free groups and for his clear
explanation of the UFPG, Sanchit Chaturvedi and Roy Rinberg for their
insightful discussions, and Nick Verga for inviting us to present our work
early on. Finally, we thank Michael Berry and Dima Rinberg for discussing
their work with us.
## References
* [1] Berry M, Warland D, Meister M (1997) The structure and precision of retinal spike trains. Proc. Natl. Acad. Sci.
* [2] Schneidman E, Berry II M, Segev R, Bialek W (2006) Weak pairwise correlations imply strongly correlated network states in a neural population. Nature.
* [3] Tkacik G, et al. (2014) Searching for collective behavior in a large network of sensory neurons. PLoS Computational Biology.
* [4] Tkacik G, et al. (2015) Thermodynamics and signatures of criticality in a network of neurons. Proc. Natl. Acad. Sci.
* [5] Mora T, Deny S, Marre O (2015) Dynamical criticality in the collective activity of a population of retinal neurons. Physics Review Letters.
* [6] Gross-Isseroff R, Smith B (1988) Concentration-dependent changes of percieved odor quality. Chem. Senses.
* [7] Bhagavan S, Smith B (1997) Olfactory conditioning in the honey bee, apis mellifera: effects of odor intensity. Physiol. Behav.
* [8] Uchida N, Mainen Z (2007) Odor concentration invariance by chemical ratio coding. Front. Syst. Neurosci.
* [9] Cleland T, et al. (2011) Sequential mechanisms underlying concentration invariance in biological olfaction. Front. Neuroeng.
* [10] Malnic B, Hirono J, Sato T, Buck LB (1999) Combinatorial receptor codes for odors. Cell.
* [11] Saito H, Chi Q, Zhuang H, Matusnami H, Mainland JD (2009) Odor coding by a mammalian receptor repertoire. Sci. Signal.
* [12] Prentice J, et al. (2016) Error-robust modes of the retinal population code. PLoS Computational Biology.
* [13] Loback A, Prentice J, Ioffe M, Berry II M (2017) Noise-robust modes of the retinal population code have the geometry of “ridges” and correspond to neuronal communities. Neural Computation 29(12):3119–3180.
* [14] Wilson C, Serrano G, Koulakov A, Rinberg D (2017) A primacy code for odor identity. Nature communications 8(1).
* [15] Giaffar H, Rinberg D, Koulakov A (2018) Primacy model and the evolution of the olfactory receptor repertoire. bioRxiv.
* [16] Ioffe M, Berry II M (2017) The structured ‘low temperature’ phase of the retinal population code. PLoS Computational Biology.
* [17] Forney G (1996) Generalized minimum distance decoding. IEEE Transactions on Information Theory.
* [18] Hopfield J (1982) Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci.
* [19] Amit D (1989) Modeling Brain Function: The World of Attractor Neural Networks. (Cambridge University Press).
* [20] Dummit D, Foote R (1991) Abstract Algebra. (Prentice Hall, Engelwood Cliffs, N.J.).
* [21] Wolpert L (1969) Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol.
* [22] Gregor T, Tank D, Wieschaus E, Bialek W (2007) Probing the limits to positional information. Cell.
* [23] Elowitz M, Levine A, Siggia E, Swain P (2002) Stochastic gene expression in a single cell. Science.
* [24] Losick R, Desplan C (2008) Stochasticity and cell fate. Science.
* [25] Herstein IN (1975) Topics in Algebra. (John Wiley & Sons), 2 edition.
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# Constructing the space of quasisymmetric stellarators
E. Rodríguez [ Max Planck Institute for Plasma Physics, Greifswald, Germany,
17491 W. Sengupta [ Department of Astrophysical Sciences, Princeton
University, Princeton, NJ, 08543 Princeton Plasma Physics Laboratory,
Princeton, NJ, 08540 A. Bhattacharjee [ Department of Astrophysical
Sciences, Princeton University, Princeton, NJ, 08543 Princeton Plasma Physics
Laboratory, Princeton, NJ, 08540
###### Abstract
A simplified view of the space of optimised stellarators has the potential to
guide and aid the design efforts of magnetic confinement configurations
suitable for future fusion reactors. We present one such view for the class of
quasisymmetric stellarators based on their approximate description near their
centre (magnetic axis). The result is a space that captures existing designs
and presents new ones, providing a common framework to study them. Such a
simplified construction offers a basic topological approach, guided by certain
theoretical and physical choices, which this paper presents in detail.
††preprint: AIP/123-QED
## I Introduction
Designing magnetic configurations suitable for holding a thermonuclear plasma
is central to fusion research. However, finding the appropriate shape of
magnetic fields with the desired properties is challenging. There are
primarily two reasons for this difficulty. First, one must select an
appropriate set of objectives, imposed both by physics and technology, which
may not be mutually compatible. Second, the potential parameter space for
three-dimensional fields is very large.
Most three-dimensional configurations are not attractive candidates, mainly
because of their poor confinement properties. In an inhomogeneous magnetic
field, charged particles generally drift away from field lines unless the
field is carefully designed. The class of stellarators where charged particles
are, on average, confined (collisionlessly), are called omnigeneous[1, 2, 3,
4, 5]. This paper focuses on a subgroup of omnigeneous stellarators known as
quasisymmetric stellarators [6, 7, 8]. In these configurations, the magnitude
of the magnetic field (but not the full vector field) has a direction of
symmetry. The conventional approach to finding this particular subset of
configurations remains an extensive search in the space of all possible
stellarators. The search attempts to ‘minimise’ the asymmetries in the
magnetic field magnitude and has successfully provided multiple designs.[9,
10, 11, 12, 13] However, the optimisation procedure remains, to a large
extent, a black box. This leaves important questions about quasisymmetry and
its implications unanswered. Moreover, and perhaps most importantly, the black
box can miss out on a significant number of undiscovered designs of
potentially high value.
This paper presents an attempt to shed some light on these questions by
considering an alternative view on the configuration space of quasisymmetric
stellarators. This alternative view consists of a model based on an
approximate description of stellarators close to their magnetic axis, where
the complexity of the stellarator is significantly reduced. In such a
framework, potential quasisymmetric stellarators are reduced to a combination
of a magnetic axis shape and the choice of two scalar parameters, in terms of
which many relevant properties can be expressed. Such a description provides
the space with a basic topological structure that enables deeper
understanding. Several choices are necessary for the axes of such a model to
represent quasisymmetric stellarators. This paper presents the formal and
physical basis underlying such choices, as well as the resulting highlights of
the approach.
Section II introduces the basics of the truncated near-axis expansion that
constitutes the basic quasisymmetric stellarator model used. Sections III-V
take the elements of such a model and explore their physical implications,
guiding the appropriate choice of parameters for the model. The focus here
will not be on the axis shapes, on which a thorough discussion may be found
elsewhere[14], but rather on choosing the remaining parameters that complete
the model. Section VI then presents an example of QS configuration space,
which we analyse to illustrate the potential of this approach. We conclude
with some final remarks and open questions.
## II Quasisymmetry and near-axis expansions
We begin by introducing the notion of quasisymmetry(QS). This hidden symmetry
is the minimal property of a magnetic field that provides the dynamics of
charged particles with an approximate conserved dynamical quantity (Tamm’s
theorem) to leading order in the gyroradius.111We simplify the picture by
assuming that the electrostatic potential shares the QS to leading gyro-order
and do not include it in our considerations. Such a conserved momentum
prevents particles from freely escaping the magnetic field, making the concept
naturally attractive for magnetic confinement.[16, 17, 8] The condition can be
formally expressed as $\nabla\psi\times\nabla
B\cdot\nabla(\mathbf{B}\cdot\nabla B)=0$, where $2\pi\psi$ is the toroidal
flux and $\mathbf{B}$ is the magnetic field. However, this form hides the
underlying nature of QS, which is to make the contours of $|\mathbf{B}|$
symmetric. Under the assumption of ideal magnetohydrostatic equilibrium,
$\mathbf{j}\times\mathbf{B}=\nabla p$, where $\mathbf{j}$ is the current
density, in Boozer coordinates[6], QS implies[18] that
$|\mathbf{B}|=B(\psi,\chi=\theta-N\phi)$ is a function that depends on a
linear combination of Boozer angles.222For a more general form of equilibrium,
a formally analogous approach exists in terms of so-called generalised Boozer
coordinates, details of which may be found in [24]. As a result, many of the
properties of quasisymmetric stellarators in this paper are independent of a
particular form of equilibrium. Here, $N\in\mathbb{Z}$ describes the pitch of
the symmetry, which leads to the distinction between quasi-axisymmetric (QA,
$N=0$) and quasi-helically symmetric (QH, $N\neq 0$) stellarators.
Our goal is to describe stellarators with the property of QS (in the second
form discussed above) close to the magnetic axis. The magnetic axis is the
centre of the stellarator, a closed magnetic field line around which magnetic
flux surfaces accrue. Because we are considering a description of the
stellarator near its axis, it is natural to use the axis as reference for our
coordinate system. We describe flux surfaces (i.e., constant $\psi$ surfaces,
which we assume to be nested) using the Frenet-Serret basis[20, 21, 22]
$\\{\hat{b},\hat{\kappa},\hat{\tau}\\}$ and Boozer coordinates
$\\{\psi,\theta,\phi\\}$, so that with respect to the axis $\mathbf{r}_{0}$,
we write
$\mathbf{x}=\mathbf{r}_{0}+X(\psi,\theta,\phi)\hat{\kappa}+Y(\psi,\theta,\phi)\hat{\tau}+Z(\psi,\theta,\phi)\hat{b},$
(1)
where $X,~{}Y$, and $Z$ are functions of all Boozer coordinates. This paper
uses the notation in [22], including the convention on the sign of the
torsion. Not every flux surface shape described by Equation (1) is consistent
with a given divergenceless magnetic field, which must be in equilibrium and
be quasisymmetric. Let us start by imposing the condition that the magnetic
field is solenoidal ($\nabla\cdot\mathbf{B}=0$) and lives on flux surfaces
($\mathbf{B}\cdot\nabla\psi=0$) formally[23] by writing both the covariant and
contravariant forms of $\mathbf{B}$ and using Boozer coordinates as
independent coordinates, [20, 22, 24],
$\displaystyle(B_{\alpha}(\psi)-$
$\displaystyle\bar{\iota}B_{\theta})\frac{\partial\mathbf{x}}{\partial\psi}\times\frac{\partial\mathbf{x}}{\partial\chi}+B_{\theta}\frac{\partial\mathbf{x}}{\partial\phi}\times\frac{\partial\mathbf{x}}{\partial\psi}+B_{\psi}\frac{\partial\mathbf{x}}{\partial\chi}\times\frac{\partial\mathbf{x}}{\partial\phi}=$
$\displaystyle=\frac{\partial\mathbf{x}}{\partial\phi}+\bar{\iota}(\psi)\frac{\partial\mathbf{x}}{\partial\chi},$
(2)
where $\mathbf{x}$ is defined in Eq. (1). Here $\bar{\iota}=\iota-N$ and
$\iota$ is the rotational transform, and $B_{i}$ are the covariant components
of the magnetic field.
In Boozer coordinates, a quasisymmetric magnetic field satisfies
$\frac{B_{\alpha}(\psi)^{2}}{B(\psi,\chi)^{2}}=\left|\frac{\partial\mathbf{x}}{\partial\phi}+\bar{\iota}\frac{\partial\mathbf{x}}{\partial\chi}\right|^{2}.$
(3)
As given, Eqs. (2) and (3) constitute a coupled set of partial differential
equations (PDEs) describing a quasisymmetric magnetic field, not only near the
magnetic axis but everywhere. Unless we consider them close to the axis, the
system of equations is overly complicated.
The asymptotic description of the fields near the magnetic axis is known as
the near-axis expansion[25, 26], in the form here presented pioneered by [20].
This procedure entails expanding all relevant fields in the problem as power
series in the distance from the magnetic axis. A pseudo-radial coordinate
$\epsilon=\sqrt{\psi}$ is defined, where $2\pi\psi$ is the toroidal flux,
which serves as the appropriate ordering parameter.333For simplicity, we have
not normalised $\psi$ respect to the magnetic field on the axis and its
curvature as it is often customary[20, 21]. Doing so simply introduces
constant rescaling factors in various quantities involved. Because of its
radial nature, the expansion in $\epsilon$ must be carefully coupled to the
poloidal-angle, $\theta$, behaviour. To avoid a coordinate singularity on the
magnetic axis, all physical quantities must have the following asymptotic
form,
$f=\sum_{n=0}^{\infty}\epsilon^{n}{\sum_{m=0|1}^{n}}\left[f_{nm}^{c}(\phi)\cos
m\chi+f_{nm}^{s}(\phi)\sin m\chi\right].$ (4)
If the function $f$ is a flux function, the expansion reduces to a Taylor
expansion in $\epsilon^{2}$. In this paper, we will use this subscript
notation repeatedly.
Expanding all functions in Eqs. (2)-(3) as the coupled Fourier (in $\chi$)
-Taylor (in $\epsilon$) series described by Eq. (4), the PDEs are reduced into
an (a priori) infinite ordered set of ordinary differential and algebraic
equations. By order $n$, we are referring here to all the elements in the
problem that have the same power $\epsilon^{n}$. At each order, various
parameters and functions are needed as inputs to the equations to uniquely
determine the solution, as summarised in Table 1. The main elements are the
magnetic field on the axis ($B_{0}$) and its leading variations nearby
($\eta,~{}B_{22}^{C}$ and $B_{22}^{S}$), the shape of the axis, the leading
contribution of the toroidal current ($B_{\theta 20}$), the stellarator-
symmetry breaking ($\sigma(0)$) and the pressure gradient ($B_{\alpha 2}$).
Each choice represents a different stellarator; in that sense, the framework
described here serves as a reduced stellarator model. We do not present the
detailed order-by-order set of equations that constitute the near-axis
description, as these may be found elsewhere, both in the form that concerns
us here[21, 28] and in the context of more general equilibria[29, 30].
Instead, we focus on the choice of parameters that make the near-axis model
represent optimised quasisymmetric configurations.
Order | Params.
---|---
$0$ | $B_{0}$, axis ($\kappa,~{}\tau,~{}l)$
$1$ | $B_{\theta 20}$, $\sigma(0)$, $\eta$
$2$ | $B_{22}^{C}$, $B_{22}^{S}$, $B_{\alpha 2}$
Table 1: Quasisymmetric configuration characterising parameters. The table
gathers the free parameters (and functions) defining the leading order form of
quasisymmetric configurations.
## III Zeroth order: magnetic axis
Let us start with the most basic element in the model: the shape of the
magnetic axis (see Tab. 1). At a fundamental level, the near-axis model
identifies every configuration with a three-dimensional closed curve (magnetic
axis). Every configuration that shares the same magnetic axis must then also
share certain properties. A detailed discussion on the role of the magnetic
axis in the context of quasisymmetric stellarators was presented in [14]. Here
we reproduce the essential elements concerned in constructing our
quasisymmetric stellarator model.
From the set of all smooth, three-dimensional closed curves, those with
inflection points, that is, points of vanishing curvature, must be excluded.
To see why this is the case, interpret the curvature of the axis, $\kappa$, as
a measure of the scale of the magnetic field magnitude gradient near the axis
(from equilibrium $\nabla_{\perp}(B^{2}/2)=B^{2}\vec{\kappa}$). Thus, to
support any finite magnetic field variation on a flux surface around a point
where $\kappa\approx 0$, an unphysical, nearly infinitely elongated flux
surface is necessary[31, 14]. Under such conditions, the set of excluded axis
shapes becomes physically interesting only outside the rigorous realm of QS,
as in the case of quasi-isodynamic stellarators[3, 32, 33]. In QS, which is
the focus of the present paper, the requirement of specialising to regular
curves makes the Frenet-Serret frame well-defined everywhere.
With such a frame defined, we can construct a self-linking number[14, 34, 35,
36, 37] ($S_{L}$), which is the number of times the curvature vector of the
axis encircles itself in a full toroidal excursion. This number is precisely
the integer $N\in\mathbb{Z}$ that appears in the quasisymmetric form of
$|\mathbf{B}|=B(\psi,\theta-N\phi)$. Thus, the shape of the axis (a local
feature) fully determines the class of the quasisymmetric stellarator (a
global feature)[14, 31].
Besides its global implications, the strength of this association is that
$S_{L}$ is a topological invariant under regular isotopies. The property
remains unchanged if the magnetic axis is continuously deformed provided that
the curvature does not vanish anywhere along the way. Thus, the space of all
closed curves is partitioned into regions identified by an integer value of
$S_{L}$. Each quasisymmetry class thus behaves as a quasisymmetric phase,
whose nature may only be changed by crossing the separating manifolds, phase-
transitions, made up of curves with inflection points. It is then natural to
see each phase as a distinct class.
Lacking a theory that directly relates axis shapes to quasisymmetric quality
of the stellarator, we must consider all (regular) axis shapes as part of our
space of potential QS configurations. Elements from higher orders in the near-
axis expansion will be needed to tell different shapes apart and delve into
additional properties of the configurations. It is important to note the
difference between this model space and the one more traditionally employed in
optimisation: the space of toroidal surface shapes. The consideration of the
axis reduces the dimensionality of the space and, crucially, provides such a
space with topological structure [14]. It is evident from the latter that an
optimisation that starts in a particular phase will remain within it.
There is no unique way to represent this space of closed curves, and we
consider a Fourier description of the curves in cylindrical coordinates
$(R(\phi),Z(\phi))$ for simplicity. Here, $\phi$ is the cylindrical angle, and
$R=\sum_{n}R_{n}\cos nN\phi$ and $Z=\sum_{n}Z_{n}\sin nN\phi$. We specialise
to stellarator-symmetric configurations. This parametrisation of the curves
guarantees they are closed and have $N-$fold symmetry. However, the torsion,
$\tau$, and curvature, $\kappa$, which are most directly involved in the near-
axis construction become byproducts that (as is the case of torsion) can be
pretty sensitive to the choice of Fourier harmonics. Parametrising the curves
providing $\kappa$ and $\tau$, which we may call the Frenet approach, would be
more natural (and in agreement with the fundamental theorem of curves[38]) but
suffer from the issue of closing the curve. Other possible alternatives, such
as the use of control points of splines, have generated renewed interest[39,
40]. The story of representations is mixed: the advantages of one appears to
be the Achilles heel of the other. However, the foundations of this paper are
independent of the particular form of a representation.
## IV First order: elliptic shaping
With the magnetic axis in place, let us proceed to the first order. Following
Table 1, we must choose appropriate values for three parameters: the toroidal
current $B_{\theta 20}$, the stellarator-symmetry breaking $\sigma(0)$, and
the $|\mathbf{B}|$ variation, $\eta$. Different choices will describe
different configurations, thereby lifting part of the degeneracy that results
from identifying stellarators with their magnetic axis. In this order, the
model introduces elements of flux surface shaping and the rotational
transform. We will see how the choice of parameters in this order affects the
behaviour of these features, and provide guidelines for appropriate choices.
### IV.0.1 Toroidal current, $B_{\theta 20}$
The coefficient $B_{\theta 20}$ controls the plasma current density on the
axis and is, in more familiar notation, the leading contribution to
$B_{\theta}=I(\psi)$. In constructing most QS configurations, the trivial
assumption $B_{\theta 20}=0$ is often made. We shall do so here as well. Of
course, such a choice is exact when modelling a vacuum field, but it is not
always the appropriate limit for configurations that support a finite plasma
pressure. Plasma currents, such as bootstrap currents, may be present in that
case, even without having to drive them externally. Their evaluation generally
requires a separate kinetic consideration [41, 42]. Bearing this caveat in
mind, we specialise to vacuum magnetic fields and thus take $B_{\theta 20}=0$.
The problem does not generally show singular behaviour in this limit. Thus,
this simplifying assumption is justified for illustrating the approach in this
paper.
### IV.0.2 Surface shaping and rotational transform: $\eta$ and $\sigma(0)$
We now consider the relevant parameters $\eta$ and $\sigma(0)$. By definition,
the parameter $\eta=-B_{11}^{C}/2B_{0}$ is a measure of the variation of the
magnetic field over flux surfaces. The other parameter, $\sigma(0)$, is
$\sigma=Y_{11}^{C}/Y_{11}^{S}$, a quantity related to the shaping of flux
surfaces at the point of stellarator symmetry $\phi=0$. Beyond these
definitions, both of these parameters may be connected to natural elements of
the geometry of flux surfaces (see [43] for more details). The shapes of the
flux surfaces at the first order are elliptical and can be characterised on
the plane orthogonal to the magnetic axis by an elongation $\mathcal{E}$, and
rotation angle, $\vartheta$,
$\displaystyle\mathcal{E}=\frac{F}{\eta^{2}/\kappa^{2}}\left[1+\sqrt{1-\frac{\eta^{4}/\kappa^{4}}{F^{2}}}\right],$
(5a) $\displaystyle\tan
2\vartheta=\frac{\sigma\eta^{2}/\kappa^{2}}{\eta^{4}/4\kappa^{4}-1-\sigma^{2}},$
(5b)
where $F=1+\sigma^{2}+\eta^{4}/4\kappa^{4}$. Here elongation is defined as the
ratio of the major to the minor radius, and $\vartheta$ is the angle between
the major radius and the positive $X$ direction (i.e., the normal vector).
From Eq. (5b), it is clear that the parameter $\sigma(0)$ serves as a measure
of the misalignment of the ellipse with the Frenet-Serret frame at the origin,
$\phi=0$. That is, it is a measure of the up-down asymmetry of this cross-
section and, generally, of stellarator asymmetry. If we specialise to the
stellarator-symmetric choice ($\sigma(0)=0$), a single parameter is then left
to choose, $\eta$.
From the form of $\mathcal{E}$, Eq. (5a), in the limit of $\sigma=0$, where
$\mathcal{E}\sim\eta^{2}/2\kappa^{2},~{}2\kappa^{2}/\eta^{2}$ (the latter for
$\eta<\kappa\sqrt{2}$), we may interpret $\eta$ as an approximate measure of
elongation. A large $\eta$ indicates a large elongation in the direction of
the curvature vector, while a small $\eta$ corresponds to large elongation
along the binormal. This correspondence is only approximate because even with
the stellarator-symmetric choice $\sigma(0)=0$, $\sigma$ is generally a non-
zero function of $\phi$, which makes Eq. (5a) highly non-trivial.
This naturally takes us to the equation that governs the form of $\sigma$, a
first-order non-linear differential equation,
$\frac{\mathrm{d}\sigma}{\mathrm{d}\phi}=-\bar{\iota}_{0}\left[1+\sigma^{2}-\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}\right]+\frac{B_{\alpha
0}}{2}(2\tau+B_{\theta 20})\left(\frac{\eta}{\kappa}\right)^{2},$ (6)
referred to as the $\sigma$ Riccati equation[28, 21, 29]. The choice of $\eta$
affects $\sigma$ non-trivially, and thus the shaping of the flux surfaces.
Following Mercier[25], we then expect the rotational transform of the
configuration on the axis, $\iota_{0}$ (or more generally in quasisymmetry,
$\bar{\iota}_{0}=\iota-N$, where $N$ is the self-linking number of the axis)
to depend on $\eta$ as well. Formally, $\bar{\iota}_{0}$ forms part of the
solution to the Riccati equation, Eq. (6), as there is a unique value for
which $\sigma(\phi)$ is periodic[21]. Thus, through the lens of this equation,
we may regard the rotational transform as a function of $\eta$ for a fixed
axis shape (see Figure 1).
Figure 1: Example of $\bar{\iota}_{0}$ as a function of $\eta$. Example of the
rotational transform as a function of $\eta$ for a fixed axis shape. This
example corresponds to a quasiaxisymmetric configuration, with an axis like
the precise QA in [44]. The broken lines indicate the asymptotic behaviour
described in the text.
No closed form expression can generally be obtained for
$\bar{\iota}_{0}(\eta)$. However, the behaviour of $\bar{\iota}_{0}(\eta)$ for
large and small values of $\eta$ can be determined. In the small $\eta$ limit,
the dominant balance $\sigma\sim\eta^{2}$ and $\bar{\iota}_{0}\sim\eta^{2}$ of
Eq. (6) yields, upon integration,
$\bar{\iota}_{0}\sim\bar{\iota}_{-}\eta^{2}=\frac{B_{\alpha
0}}{4\pi}\eta^{2}\int_{0}^{2\pi}\frac{2\tau+B_{\theta
20}}{\kappa^{2}}\mathrm{d}\phi.$ (7)
Thus, the rotational transform on the axis tends to $\iota_{0}\rightarrow N$.
In the case of a quasi-axisymmetric (QA) configuration, this leads to a
vanishing rotational transform. Physically, this is a result of making flux
surfaces very elongated in the curvature direction in a way that, from
Mercier’s perspective, the rotational transform on the axis is driven only by
the rotating-ellipse contribution. Of course, in the QA case, the ellipse has
no net rotation (recall the meaning of the self-linking number).444From the
Mercier perspective on rotational transform[5],
$\bar{\iota}_{0}=-\frac{1}{2\pi}\int_{0}^{L}\frac{\cosh\bar{\eta}-1}{\cosh\bar{\eta}}(d^{\prime}+\tau)\mathrm{d}l+\frac{1}{2\pi}\int_{0}^{L}\tau\mathrm{d}l,$
in the large ellipticity $\bar{\eta}\rightarrow\infty$ limit,
$\bar{\iota}_{0}=-d/2\pi$, where $d$ is the angle of rotation of the ellipse
with respect to the curvature vector. Because $\sigma\sim\eta^{2}\rightarrow
0$ in this limit, the cross-sections align with the Frenet-Serret frame. Thus,
the net rotation $d=0$. Thus, $\iota_{0}=0$. As $\eta$ increases, the torsion
and current contributions play the same role in driving rotational transform.
In a QA configuration, this growth in $\bar{\iota}_{0}$ is equivalent to
growth in rotational transform. However, in the case of a quasi-helically
symmetric (QH) configuration, this depends on the relative sign of Eq. (7) and
the QS helicity $N$, which tends, in practice (and zero current), to be
negative. In the limit of large $\eta$, the physical scenario is similar to
that in the small $\eta$ limit. The shaping becomes large, and
$\bar{\iota}_{0}$ also tends to zero. This follows formally from the dominant
balance $\bar{\iota}_{0}=\bar{\iota}_{0}^{*}/\eta^{2}$ and
$\sigma=\eta^{2}\sigma^{*}$ in Eq. (6). The asymptotic forms of
$\\{\sigma^{*},\bar{\iota}_{0}^{*}\\}$ are solutions to a ‘universal’
$\eta$-independent Riccati equation, unique to each axis shape
($\kappa,~{}\tau$).
By the mean value theorem, it follows that $\iota_{0}(\eta)$ must have at
least one turning point (except in the marginal case of $\int(2\tau+B_{\theta
20})/\kappa^{2}=0$). In Appendix A we prove that this extremum exists, its
value $\eta=\eta^{*}$ is unique and satisfies the condition,
$\int_{0}^{2\pi}\left[\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}-1\right]E\mathrm{d}\varphi=0,$
(8)
where
$E=\exp[{2\bar{\iota}_{0}\int_{0}^{\varphi}\sigma\mathrm{d}\varphi^{\prime}}]$.
From this expression, it follows that there is a value of $\eta$ in
$(0,\sqrt{2}\kappa_{\mathrm{max}}B_{0}^{1/4}]$ that extremises the rotational
transform (see Fig. 1).
To illustrate further the meaning of $\eta^{*}$, and to further connect it to
shaping, consider the familiar limit of axisymmetry. In that case, the Riccati
equation, Eq. (6), can be solved exactly (taking $B_{\alpha 0}=1=R_{0}$) to
yield
$\iota_{0}=\frac{1}{2}\frac{B_{\theta 20}\eta^{2}}{1+\eta^{4}/4}.$ (9)
The solution in Eq. (9) exhibits all the properties of $\bar{\iota}_{0}$ we
studied in the general case, including the uniqueness of $\eta^{*}=\sqrt{2}$.
This choice of $\eta$ corresponds to having a circular cross-scetion. This
suggests that in order to maximise the use of the toroidal current to generate
rotational transform, one must simultaneously minimise the amount of shaping.
The correspondence between the behaviour of the rotational transform and the
shaping of surfaces prevails beyond axisymmetry. This can be seen by
investigating the behaviour of elongation $\mathcal{E}$, Eq. (5a), with $\eta$
in an analogous form to the case of $\bar{\iota}_{0}$. Its asymptotic
behaviour was already touched upon before, as we saw that elongation diverged
both for small $\eta$ ($\mathcal{E}\approx 2\sqrt{B_{0}}(\kappa/\eta)^{2}$)
and for large $\eta$ ($\mathcal{E}\sim
2\eta^{2}\sqrt{B_{0}}\kappa^{2}[\tilde{\sigma}^{2}+1/4B_{0}\kappa^{4}]$).
Thus, there must be some minimally elongated configuration somewhere in
between.
Figure 2: Difference in behaviour between elongation measures. Plot showing
the behaviour of different elongation measures with $\eta$ for the magnetic
axis of the ‘precise QA’ configuration[44]. The plot shows the difference in
behaviour between different global scalar measures for elongation:
$\mathcal{E}(0)$ is the value of elongation at $\varphi=0$,
$\langle\mathcal{E}\rangle$ is the $\phi$-average of elongation,
$\mathrm{max}_{\varphi}[\mathcal{E}]$ is the maximum of elongation and
$\langle\mathcal{E}/\kappa^{2}\rangle$ is the weighted average of elongation.
The broken line represents the value of $\eta$ obtained from the global
solution for comparison.
However, this notion must be qualified, as $\mathcal{E}$ is not a single
scalar but a function of $\varphi$. This allows for different ways in which to
construct a single measure for the shaping of the configuration. Figure 2
shows some possibilities. It is evident that the behaviour changes from one
definition to another. Among these measures, there is one that is particularly
insightful. Consider the weighted average
$\bar{\mathcal{E}}=(1/2\pi)\int(\mathcal{E}/\kappa^{2})\mathrm{d}\phi$ which
penalises elongation in the straighter sections of the configuration. This
particular elongation measure has the same asymptotic behaviour (up to an
$\eta-$independent factor) as $\bar{\iota}_{0}$. This formal analogy makes the
choice of $\eta^{*}$ also a choice that roughly minimises elongation (see the
comparison in Fig. 8 or Table 2 for the implications in practice).
Besides all the physical implications of $\eta^{*}$, this choice also
represents a least sensitive choice, so much so that it corresponds to a
turning point. That is, a set of near-axis constructions in some range
$(\eta^{*}-\delta,\eta^{*}+\delta)$ have roughly the same properties, making
their properties more robust. Following [29], this resilience of the
rotational transform makes the magnetic shear (a quantity that would be, in
general, a third-order quantity) independent of third-order parameter choices
(see Appendix B). Thus, our model can predict the behaviour of magnetic shear
as well.
| ARIESCS | ESTELL | GAR | HSX | NCSX | QHS48 | Precise QA | Precise QH
---|---|---|---|---|---|---|---|---
$\eta_{\mathrm{VMEC}}$ | 0.11 | 0.79 | 0.51 | 1.75 | 0.62 | 0.21 | 0.96 | 2.12
$\eta_{\mathrm{VMEC}}/\eta^{*}$ | 1.00 | 0.91 | 0.95 | 0.87 | 0.92 | 0.88 | 0.95 | 0.91
$\eta_{\mathrm{VMEC}}/\eta_{\langle\nabla\mathbf{B}\rangle}$ | 1.03 | 0.12 | 0.57 | 0.99 | 0.73 | 1.07 | 0.64 | 1.04
$\eta_{\mathrm{VMEC}}/\eta_{\bar{\mathcal{E}}}$ | 1.07 | 0.95 | 1.01 | 0.91 | 0.97 | 0.91 | 1.03 | 0.95
$N$ | 0 | 0 | 0 | 4 | 0 | 4 | 0 | 4
Table 2: Comparison of $\eta$ choices in some quasisymmetric designs.
Comparison of the $\eta$ values for many quasisymmetric designs: ARIESCS[11],
ESTELL[46], GAR[47, 48], HSX[9], NCSX[10], QHS48[12], Precise QA and QH[44].
$\eta_{\mathrm{VMEC}}$ corresponds to the value of $\eta$ obtained from the
$|\mathbf{B}|$ of the global VMEC[49] equilibrium solution of the
quasisymmetric designs. $\eta^{*}$ is the parameter value that extremises
$\bar{\iota}$ for a fixed axis obtained from VMEC.
$\eta_{\langle\nabla\mathbf{B}\rangle}$ corresponds to the maxima of the
average of $L_{\nabla\mathbf{B}}$ (see Appendix C), and finally
$\eta_{\bar{\mathcal{E}}}$ from the minimum of the weighted average elongation
$\bar{\mathcal{E}}$. The integer $N$ denotes the symmetry class.
In summary, the choice of $\eta=\eta^{*}$ is a formally convenient and
representative choice for $\eta$. It is unique and always exists, maximises
rotational transform in QA configurations, and regularises the shaping of flux
surfaces, preventing them from having extreme shaping. With such a choice,
every axis in our structured space of configurations will have an associated
natural choice of $\eta$. We do not need to keep this parameter explicitly as
an added dimension in this space, reducing its complexity. A practical design
would benefit from an additional tweaking of this parameter in the
neighbourhood of $\eta^{*}$. This refinement could be seen as a subsequent
optimisation in which one can consider higher-order properties or more
sophisticated construction specifications.
To present some evidence that backs the suitability of the choice of
$\eta^{*}$, we present in Table 2 the value of $\eta$ extracted from global
equilibrium solutions designed through other means compared to different
choices of $\eta$. This shows that $\eta^{*}$ is a reasonable representative
choice across the board. In the Table, we also include a choice
$\eta_{\nabla\mathbf{B}}$, which we have not mentioned in the discussion. This
choice of $\eta$ is taken to be the value that minimises the magnitude of
$||\nabla\mathbf{B}||$, a measure of the gradients of the magnetic field. This
measure has been used by other researchers[50, 44, 51] to good effect in
optimising near-axis QS configurations. The rationale behind this measure is
that the characteristic length scale
$L_{\nabla}\stackrel{{\scriptstyle\cdot}}{{=}}1/||\nabla\mathbf{B}||$, may be
interpreted to approximately set the distance to the field-generating coils,
and thus to indicate a measure of the range of validity of the near-axis
model.[50] Guiding the choice of $\eta$ for a fixed axis could be considered
as a guiding principle in choosing $\eta$. This is successful in many cases
(see Tab. 2), but it lacks the robustness and generality of $\eta^{*}$, and
fails in the QA phase. See Appendix C for a more detailed discussion.
## V Second order: surface triangularity and QS breaking
So far, we have identified quasisymmetric configurations with a model
consisting of an axis and elliptical cross-sections. For every curve in our
space of configurations, the shape of the latter results from the choice of
the parameter $\eta^{*}$. However, we note that this model is exactly
quasisymmetric up to this point, and that there is no way of differentiating
which configurations will exhibit better QS globally. Consideration of the
second order in the near-axis expansion is needed for this.
When incorporating the second order, two important parameter choices must be
made: (i) the plasma pressure gradient (in the notation here, related directly
to $B_{\alpha 1}$) and (ii) the second order variation of the magnetic field
magnitude, $B_{2}$. For every choice made, the model gains a different flux-
surface shaping in the form of triangularity, Shafranov shift, and a different
degree of QS. In line with the simplifying vacuum and stellarator-symmetric
assumptions, we choose $B_{\alpha 1}=0$ and $B_{22}^{S}=0$ to reduce the
number of free parameter choices to one. Only one of the harmonics of $B_{2}$
remains. Following the approach of [21], this parameter is taken to be
$B_{22}^{C}$. This leaves the remaining component of $B$ at second order,
$B_{20}$, to be found self-consistently. This lack of freedom in $B_{20}$
follows from the necessity of satisfying force-balance and the appropriate
magnetic equations simultaneously, which generally requires $B_{20}$ to have a
toroidal angle $\varphi$ dependence.[28] Of course, such a variation violates
QS, and the variation of $B_{20}$, which we may define as $\Delta B_{20}$
bottom-to-peak, becomes a measure of QS quality at this order.
Formally, $B_{20}$ is the solution to a second-order, linear differential
equation. The $B_{20}$ equation may be found in the appendix of [30], and may
be written as,
$\mathcal{A}\left(\frac{B_{20}}{B_{0}}\right)^{\prime\prime}+\mathcal{B}\left(\frac{B_{20}}{B_{0}}\right)^{\prime}+\mathcal{C}\frac{B_{20}}{B_{0}}+\mathcal{D}=0,$
(10)
where
$\displaystyle\mathcal{A}=-\frac{B_{\alpha
0}\eta}{2\kappa^{2}\bar{\iota}_{0}l^{\prime}}\left[1+\frac{4B_{0}\kappa^{4}}{\eta^{4}}(1+\sigma^{2})\right],$
(11a) $\displaystyle\mathcal{B}=\frac{2B_{\alpha
0}\eta}{\bar{\iota}_{0}l^{\prime}}\frac{\kappa^{\prime}}{\kappa^{3}}-\frac{4l^{\prime}\sigma}{\bar{\iota}_{0}\eta}\tau,$
(11b) $\displaystyle\mathcal{C}=-\frac{l^{\prime}}{2B_{\alpha
0}\eta^{3}\kappa^{2}}\left[\bar{\iota}_{0}\left(4\kappa^{4}(1+\sigma^{2})-\frac{3\eta^{4}}{B_{0}}\right)+8B_{\alpha
0}\eta^{2}\kappa^{2}\tau\right],$ (11c)
and $\mathcal{D}$ is a complicated expression given in Appendix D.2, Eq.
(D25d) of [30] (see also Appendix D).
Relaxing QS through $B_{20}$ and not other components of $|\mathbf{B}|$ is
only a choice, not a requirement. It is, however, not a whimsical choice. For
one, it has the benefit of making the self-consistent $B_{20}$ regular, in the
sense of Fuchs criteria[52]. This follows from $\mathcal{A}\neq 0$, as Eq.
(11a) is proportional to a sum of squares. This local consideration on
$B_{20}$ is not to say that a solution to the equation that satisfies the
condition of periodicity exists. Proving that consistent periodic solutions to
Eq. (10) exist is a more challenging problem. To explore this question, let us
first consider the axisymmetric limit.
In the axisymmetric limit, everything is by definition $\varphi$-independent,
and thus Equation (10) reduces (using the axisymmetric simplification of
$[1+(1/4B_{0})\eta^{4}/\kappa^{4}]\bar{\iota}_{0}=B_{\alpha 0}B_{\theta
20}\eta^{2}/2\kappa^{2}$) to,
$-\bar{\iota}_{0}^{2}\left(\eta^{4}-12B_{0}\kappa^{4}\right)\frac{B_{20}}{B_{0}}-\frac{3}{\eta^{2}\kappa^{2}}(\eta^{4}-4B_{0}\kappa^{4})B_{22}^{C}=\dots.$
(12)
The dots denote second-order independent terms. Solving the equation for
$B_{20}$ (the approach described above[21]) clearly shows ill behaviour for
$\eta^{4}=12\kappa^{4}B_{0}$, seemingly an arbitrary choice of elliptical
shape. In that scenario, $B_{20}$ decouples from the equation, leaving only
$B_{22}^{C}$ to satisfy the equation. The roles of $B_{20}$ and $B_{22}^{C}$
can be reversed to solve Eq. (12) for $B_{22}^{C}$. However, in this case, the
construction fails again at another value of $\eta$, i.e., whenever circular
cross-sections are considered. To avoid excluding this case, solving for
$B_{20}$ appears to be the more convenient choice.
The breakdown of solutions for this ‘critical’ value of $\eta$ persists beyond
axisymmetry. Numerical evidence of this is presented in Fig. 3. We plot
$\Delta B_{20}$ for a fixed axis shape as a function of $\eta$ and
$B_{22}^{C}$, which shows a clear critical value for $\eta$.
Figure 3: Quasisymmetry breaking as a function of $\eta$ and $B_{22}^{C}$ for
an example. Variation in $B_{20}$ ($\Delta B_{20}$) as a function of the
choice of parameters $\\{\eta,B_{22}^{C}\\}$ computed using the code pyQSC.
The dashed lines correspond to the $\eta_{\mathrm{VMEC}}$ (from the global
equilibrium) and $\eta^{\star}$ values (see Tab. 2).
Formally, the appearance of such singularity may be explained through the
Fredholm alternative theorem for the existence of solutions. The singularity
occurs when there is no solution to Eq. (10), which occurs whenever a solution
exists to the adjoint problem. The adjoint problem does not generally have a
solution (and thus, a solution to Eq. (10) exists and is unique), but one may
rigorously prove that there is at least one critical value
$\eta_{\mathrm{crit}}$ for which this is the case. If this critical value is
unique (which numerical evidence suggests to be the case), it follows that
$\eta^{*}<\eta_{\mathrm{crit}}$. Thus, our choice of $\eta^{*}$ at lower order
guarantees the existence of a unique solution to Eq. (10). Details on this may
be found in Appendix E.
The existence of a solution and this critical value $\eta_{\mathrm{crit}}$
have told us little about the influence of $B_{22}^{C}$, and how to choose it.
However, the choice of $B_{22}^{C}$ follows naturally when we try to maximise
the quality of QS. That is, we choose $B_{22}^{C}$ so that it minimises the
deviation of $B_{20}(\varphi)$ from being a constant. This will make our model
represent the ‘most’ quasisymmetric configuration. Because the parameter
$B_{22}^{C}$ appears only in the inhomogeneous term of Eq. (10), the choice of
$B_{22}^{C}$ has no dramatic effect on $B_{20}$. Therefore, we expect the
search of $B_{22}^{C}$ that minimises the QS residual to be smooth. More
quantitatively, we write the dependence of $\mathcal{D}$, Eq. (10), on
$B_{22}^{C}$,
$\mathcal{D}=\frac{3\kappa^{2}}{B_{\alpha
0}B_{0}(l^{\prime})^{2}\eta^{3}}\left[\frac{\bar{\iota}_{0}}{2}\left(B_{\alpha
0}^{2}\frac{\eta^{4}}{\kappa^{4}}+4(\sigma^{2}-1)\right)-\right.\\\
\left.-4\sigma^{\prime}-8\bar{\iota}_{0}\sigma^{2}\right]B_{22}^{C}+\dots,$
which in the limit of large $|B_{22}^{C}|$, makes $B_{20}$ scale with
$B_{22}^{C}$, with a solution of the ‘universal’ form,
$[\mathcal{A}\partial_{\phi}^{2}+\mathcal{B}\partial_{\phi}+\mathcal{C}]\frac{B_{20,\mathrm{univ}}}{B_{0}}+\mathcal{D}_{C}=0,$
(13)
where $B_{20}=B_{22}^{C}B_{20,\mathrm{univ}}$. This is universal in the sense
that each first-order construction has a single solution
$B_{20,\mathrm{univ}}$. This solution provides a measure of the effect of
$B_{22}^{C}$ on symmetry- breaking (see Figure 4). The influence of
$B_{22}^{C}$ decreases as $B_{20,\mathrm{univ}}$ becomes closer to a constant
to vanish for axisymmetry. Away from the neighbourhood of axisymmetry (where
$B_{22}^{C}$ has little effect on QS), this means there must be at least one
local minimum at a finite value of $B_{22}^{C}$. This makes minimising $\Delta
B_{20}(B_{22}^{C})$ modifying $B_{22}^{C}$ a well-posed 1D search problem.
This way of choosing $B_{22}^{C}$ can be shown to be representative of
optimised stellarators in practice (see Figure 4).
Figure 4: Example of $B_{22}^{C}$ choice for precise QS designs. Plots showing
the variation of the QS residual as a function of $B_{22}^{C}$ for the NAE
models of the precise QS designs [44] (top) and their respective universal
solutions $B_{20,\mathrm{univ}}$ (bottom). The plots correspond to the QA
(left) and QH (right). The scatter points represent the values of $B_{22}^{C}$
obtained from the $|\mathbf{B}|$ spectrum of the global solutions.
Although choosing $B_{22}^{C}$ this way constitutes a well-posed problem, we
should not disregard other aspects of the stellarator that $B_{22}^{C}$ also
affects. This includes MHD stability near the axis[53, 43], the shaping of
flux surfaces[43], and as a result, the smallest effective aspect ratio of the
configuration[50]. Characterising the influence of second-order parameters on
geometry in a clear way is non-trivial. Detailed analysis on shaping was
presented as part of [54], which we refer the reader to for a detailed
discussion. However, one can show that in the large $|B_{22}^{C}|$ limit, flux
surfaces become increasingly shaped to limit their achievable aspect ratio.
For this to be lower than 10, we shall limit, quite crudely,
$|B_{22}^{C}|\lesssim 10$ (see Appendix F). This is only a rough estimate, but
it provides a useful domain to perform the search of $B_{22}^{C}$ to minimise
$\Delta B_{20}$. This will leave out any good QS configuration outside the
allowed range of $B_{22}^{C}$. To include some of those cases, one may proceed
a posteriori by relaxing the parameter choices.
## VI Space of quasisymmetric configurations
Following the arguments in the previous sections, we have a prescription to
complete a second-order near-axis model for every regular axis shape. Every
point in our space of curves represents a stellarator-symmetric, vacuum field
stellarator that tries to be as quasisymmetric as possible while preserving
some minimal requirements on the rotational transform and shaping. To
illustrate the power of the approach, we consider the space of configurations
spanned by magnetic axes described by two Fourier harmonics. We do not keep
$\\{Z_{n}\\}$ explicitly in this space, and instead, for each set
$\\{R_{n}\\}$ we look for the most quasisymmetric solution performing a
Nelder-Mead optimisation [55, Ch. 9.5] on $\\{R_{n}\\}$ under the constraint
that $Z_{n}\sim R_{n}$. This is consistent with the evidence gathered from
optimised QS designs. This way, each point in the $\\{R_{n}\\}$ space
corresponds to an optimal QS axis shape.
This reduced space is adequate to capture common QS designs, as was explicitly
shown in [14], and may be seen in Figure 5. The spaces for a number of
different field periods are presented with the colormap representing the
quality of QS, $\Delta B_{20}$. The spaces were generated using the C++
libraries qsc555See https://github.com/landreman/qsc. The script used to
obtain the main plot in Fig. 5 can be found in the Zenodo repository
associated to this paper. The same may be achieved, albeit slower, using
pyQSC, which was how it was originally done and is also included there. and
gsl for the near-axis calculations and optimisation, respectively. On average,
the evaluation of each point in this space takes less than a second running on
a single CPU in a laptop666An 11th Gen i7-11850H core was used for this
purpose. The main space in Fig. 5 (which is 300x300) took a total of about 14
hrs. Most time is devoted to the optimisation sub-problems at each point
(search for $\eta^{*}$, $B_{22}^{C}$, and $\\{Z_{n}\\}$). Of course, the
construction of such a space is trivially parallelisable.
Figure 5: Quality of quasisymmetry for the two-harmonic quasisymmetric
configuration space. The figure shows $\Delta B_{20}$ (in logarithmic scale)
for the space of configurations spanned by two harmonic magnetic axes for
several field periods $N=2,~{}3,~{}4,~{}5$ in the top figures and $N=4$ for
the lower one. As the text describes, the $\\{Z_{n}\\}$ harmonics have been
optimised at each point of space to minimise the quasisymmetry residual. The
coloured scatter points represent typical quasisymmetric designs in our
reduced space (see legend). To represent them all in the $N=4$ space, the
magnetic axis harmonics are rescaled as
$R_{4n}=R_{nN}(1+n^{2}N^{2})/(1+16n^{2})$ following the insight in [14].
Typical designs lie close to the bands of good quasisymmetry, which show the
power of the approach. The black lines represent phase transition curves for
$R_{n}=Z_{n}$. The dark purple point represents a new QH design, construction
presented in Figure 7. The gap at $R_{4}$ is numerical, as the numerical
evaluation of the space was performed in two separate runs.
The space exhibits two remarkable features. First, the QS phase structure
studied in the context of the magnetic axis makes itself clear (see phase
transitions as solid black lines). Such features, as well as others, remain
largely unchanged as the number of field periods changes. This allows us to
represent the QS designs in Table 2 together in Fig. 5. The second important
feature of this configuration space is the appearance of what we call
quasisymmetric branches. These branches consist of well-distinguished regions
of configuration space with excellent QS. (We leave a more precise definition
for the future, a definition that will be necessary for a more systematic
study of the branches.) Importantly, these branches agree with the location of
typical QS designs.[11, 46, 47, 48, 9, 10, 12, 44]. It proves the predictive
power of the approach and the role of our model as a unifying framework.
With the QS branches identified, we have all the tools from the near-axis
framework to investigate their properties. To illustrate what can be learned
from such an analysis, let us focus on the dominant branch in the QA phase
(the phase that includes the origin) that grows from the origin in the
direction of the lowest harmonic $R_{n}$. Many (if not all) standard QA
designs belong to this branch, irrespective of $N$.
Figure 6: Properties of the configurations in the main QA branch. Properties
of configurations in the main QA branch for $N=2,~{}3,~{}4$ (blue, orange,
yellow respectively) plotted against[14] $s_{N}=R_{N}(1+N^{2})$. The
properties shown are the rotational transform on the axis, the QS residual,
the limit on aspect ratio, the $B_{22}^{C}$ parameter, the average $B_{20}$
for the magnetic well criterion, and the magnetic shear. The latter is defined
so that the change in rotational transform is $\bar{\iota}_{1}$ times the
inverse aspect ratio squared. These are obtained by following the branches in
the 2-harmonic space of Fig. 5. The shaded area indicates a region where the
construction of the NAE model is not well-behaved due to proximity to
axisymmetry (see main text).
Without delving into the origin of the branches, which we leave for future
work, we now describe the properties and trends of this class of
configurations as evident from Fig. 6. These properties should be interpreted
as representative of the class, noting that many features may be changed by
additional tweaking of the axis shape and parameter choice. However, before
looking at these, note that in the region closest to the origin (i.e., near
axisymmetry), the model presents ill behaviour, as it fails to show $\Delta
B_{20}\rightarrow 0$. Such misbehaviour aligns with the uncertainty in the
choice of $B_{22}^{C}$ that arises close to axisymmetry (see $B_{22}^{C}$ in
Fig. 5). The reason for this type of behavior may be deeper than we choose to
pursue here, see [58].
Away from this region, some physically interesting trends are observed. The QS
quality degrades as the QA-QH phase transition is approached and the
rotational transform and the magnetic shear grow. The degradation may be seen
as a result of an increasingly shaped configuration driven by an increasingly
twisted axis, a behaviour predicted in [14]. This also leads to an increase in
the minimum effective aspect ratio $\sqrt{2}/\epsilon_{\mathrm{max}}$, as it
does with the number of field periods. An aspect ratio of roughly $5-7$
appears possible for $N=2$ but increases to $15-25$ for $N=4$. Thus, although
the rotational transform grows with $N$, which is of interest, the limitation
in the compactness (and the quality of QS) restricts the configurations of
interest to the lower $N$ values, as observed in optimisation efforts[10, 44].
Compared to the rest of the QA phase space, the shear, the QS residual, and
$1/\epsilon_{\mathrm{max}}$ are small along the QA branch. This appears to
match the observation that the magnetic shear of QS configurations tends to be
small in practice[44]. The magnetic shear also presents a sign opposite to
that of $\iota_{0}$, and thus the rotational transform profiles are tokamak-
like: the rotational transform decreases towards the edge of the
configuration. The magnetic well criterion[53, 43] (via the sign of $\langle
B_{20}\rangle$) shows that the majority of the QA branch gives rise to a
magnetic hill, and thus is MHD unstable. This aligns with the conclusions
reached in [44], where an additional effort was made to reach a solution with
a magnetic well. Those configurations along the QA branch further away from
the origin will be more easily stabilised by additional tweaking of the
configuration since their magnetic hill is shallow.
This discussion of the QA branch is only a partial account of the full power
of the present approach to understand optimised QA configurations. A similar
effort could be devoted to the branches in the QH phase, but we shall not do
that here. Instead, we content ourselves with a few general observations. We
call the branch the HSX branch as it lies close to the HSX design[9]. The
other branch appears not to include any existing QH design, and thus we shall
refer to it as the new QH branch. The QH space is significantly more sensitive
to the parameter choices than the QA phase, which in particular leads to the
location of the new QH branch in phase space changing under different choices
of $\eta$. If $\eta$ were to be treated as another optimisation parameter, the
branch would become a broader region bounded by large elongation
configurations. This latter branch is particularly interesting. Configurations
along it lack the common bean-shaped cross-section and exhibit a natural
magnetic well. Nevertheless, most remarkably, no existing QS-optimised design
belongs to this class. This illustrates the power of this approach in
exploring QS-optimised configurations.
We present for completeness an example of a stellarator belonging to this
branch, which we call ‘new QH’. We construct a global equilibrium solving the
equilibrium problem inside a fixed outer surface, constructed from a finite
aspect ratio evaluation of the near-axis model (see Fig. 5) following [21].
This form of linking the near-axis and global solutions is not the best, as it
uses the worst-described feature of the near-axis model (the ‘outer’ surface)
as an intermediary. However, it suffices as a first approximation. The
particular configuration in the space of Fig. 5 was chosen to allow for a
reasonable aspect ratio (an aspect ratio of $A\sim 13.5$ given by VMEC, which
has difficulty initialising the solver in more compact scenarios). To
construct the final form of the configuration, some additional refinement of
the axis shape was made within the near-axis framework by allowing for three
small additional harmonic components. The configuration in Fig. 7 shows good
QS behaviour as expected from the near-axis model. This is especially
important, considering that no optimisation has been performed in the space of
global equilibria.
This configuration serves as an example of the approach’s potential. The
complications encountered by the numerical solvers (including the large axis
excursion) could be a reason behind the design efforts missing out
stellarators belonging to the new QH branch. However, the exact reason is hard
to pinpoint. It could have also resulted from other constraints (such as
aspect ratio) or the initial guesses
Other authors have recently found stellarator designs considering optimisation
in the space of near-axis configurations[51]. The result is a multitude of
designs, many of which have previously unseen features. The approach there may
be regarded complementary to that presented here, as this should serve to
structure, guide (e.g., seeding the search to be more exhaustive), and
understand the other approach.
Figure 7: Example of new QH branch configuration. a) Different projections of
the 3D boundary of a configuration from the new QH branch. The colormap
represents contours of constant $|\mathbf{B}|$. The global equilibria were
solved using VMEC given the surface of an NAE construction with an axis given
by the following $\\{R_{n},Z_{n}\\}$ components
$R_{n}=\\{0.426,~{}0.044,~{}-6.36\times 10^{-11},~{}2.85\times
10^{-5},~{}3.89\times 10^{-8}\\}$ and $Z_{n}=\\{0.411,~{}0.043,~{}6.53\times
10^{-5},~{}1.36\times 10^{-5},~{}1.16\times 10^{-5}\\}$. The higher harmonics
have been chosen as small last tweaking choices to minimise further $\Delta
B_{20}$, which within the QS framework, we make $\Delta B_{20}\sim 7\times
10^{-3}$. b) Magnetic field magnitude on the last flux surface of the global
solution. c) Quasisymmetric residual and $\epsilon_{\mathrm{eff}}$ (a measure
of particle transport) [59, 60] as a function of radius in the global solution
showing the quasisymmetric nature of the configuration.
## VII Conclusion
In this paper, we have presented the construction of a model for
quasisymmetric stellarators based on the near-axis expansion, in which
configurations can be identified with their axis shapes only. Doing so enables
us to represent the space of configurations in a form that inherits the
topological structure of a space of closed regular curves. To concoct such a
model requires a careful choice of parameters that form part of the near-axis
expansion.
In the case of vacuum field, stellarator-symmetric stellarators (although the
extension to more general cases should be straightforward), there are two such
parameters: $\eta$ and $B_{22}^{C}$. The former is chosen to guarantee that
flux surfaces are not extremely elongated, maximising the rotational transform
in quasiaxisymmetric configurations. The choice as presented always exists and
is unique. The $B_{22}^{C}$ parameter is then chosen so that each axis shape
is represented by the ‘most quasisymmetric’ configuration. That way, we can
construct an example space of quasisymmetric stellarators.
The model constructed is a powerful tool that reproduces stellarator designs
optimised for quasisymmetry by other approaches. It does so naturally,
grouping them into families that we call quasisymmetric branches. We show
explicitly how the model may be used to study the properties of one such
branch, namely the main QA branch. We also showcase how new configurations
appear from this approach. The structure of the space and simplicity of the
model opens the door to a fundamental study of quasisymmetric stellarators and
their properties, as well as a practical exploration of designs complementary
to recent attempts[51]. However, there remains significant room for future
work, including a systematic way of identifying branches, initialising
optimisation from these, and a more robust way of connecting the near-axis
construction to global equilibrium solvers.
## Acknowledgements
The authors would like to acknowledge fruitful discussions with M. Landreman,
R. Jorge, R. Nies, S. Buller, and E. Paul. This research was primarily
supported by a grant from the Simons Foundation/SFARI (560651, AB) and DoE
Contract No DE-AC02-09CH11466. ER was also partially supported by the
Charlotte Elizabeth Procter Fellowship at Princeton University.
## Data availability
The data supporting this study’s findings are available at the following
Zenodo repository https://doi.org/10.5281/zenodo.7817884.
## Appendix A Definition and uniqueness of $\eta^{*}$
The choice of $\eta$ modifies the competition between the different
contributions to the rotational transform on the axis. We learned in the main
text that at large and small $\eta$, the rotational transform
$\bar{\iota}_{0}$ vanishes. Thus, by the mean value theorem under the
assumption of continuity, it must be that the function $\bar{\iota}_{0}(\eta)$
has at least a turning point. Let us see what we can learn about this point.
Consider linearising the Riccati equation, Eq. (6), with respect to $\eta$,
$\frac{\mathrm{d}}{\mathrm{d}\phi}\frac{\delta\sigma}{\delta\eta}=-2\bar{\iota}_{0}\sigma\frac{\delta\sigma}{\delta\eta}-\bar{\iota}_{0}\frac{\eta^{3}}{B_{0}\kappa^{4}}+\frac{\eta
B_{\alpha 0}}{\kappa^{2}}(2\tau+B_{\theta
20})-\frac{\delta\bar{\iota}_{0}}{\delta\eta}\left[1+\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}\right].$
(14)
Looking at the extremum
$\delta\bar{\iota}_{0}/\delta\eta\stackrel{{\scriptstyle!}}{{=}}0$ makes Eq.
(14) an ODE on $\delta\sigma/\delta\eta$, the change in the solution $\sigma$
upon infinitesimal change of the parameter $\eta$. Thus,
$\delta\sigma/\delta\eta$ must, as $\sigma$, be periodic for
$\sigma+\delta\sigma$ to be so. Because $\sigma(0)=0$ is kept fixed,
$\delta\sigma/\delta\eta=0$ at $\phi=0$. The linearised ODE can then be solved
by an integration factor,
$\frac{\delta\sigma}{\delta\eta}=e^{-2\bar{\iota}_{0}\int_{0}^{\phi}\sigma\mathrm{d}\phi^{\prime}}\int_{0}^{\phi}e^{2\bar{\iota}_{0}\int_{0}^{\phi^{\prime}}\sigma\mathrm{d}\phi^{\prime\prime}}\frac{2}{\eta}\left[\frac{B_{\alpha
0}}{2}(2\tau+B_{\theta
20})\left(\frac{\eta}{\kappa}\right)^{2}-\frac{\bar{\iota}_{0}}{2B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}\right]\mathrm{d}\phi^{\prime}.$
(15)
Imposing periodicity, using the Riccati equation, Eq. (6), and integrating by
parts (assuming $\bar{\iota}_{0}\neq 0$ and stellarator symmetry to drop the
boundary term), we obtain
$\int_{0}^{2\pi}\left[\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}-1\right]E\mathrm{d}\phi=0,$
(16)
where
$E=\exp[{2\bar{\iota}_{0}\int_{0}^{\phi}\sigma\mathrm{d}\phi^{\prime}}]$.
Definition of $\eta^{*}$: the first-order parameter $\eta$ extremises the
rotational transform on the axis of a stellarator symmetric construction iff,
$\int_{0}^{2\pi}\left[\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}-1\right]E\mathrm{d}\phi=0,$
(17)
where
$E=\exp[{2\bar{\iota}_{0}\int_{0}^{\phi}\sigma\mathrm{d}\phi^{\prime}}]$.
Although exact, the condition in Eq. (8) is implicit through $\sigma$. This
prevents a closed form of $\eta^{*}$, which has to be found numerically. To
this end, finding bounds on $\eta^{*}$ from Eq. (8) is helpful. Because
$\sigma^{2}$ and $E$ are both positive quantities, in order for Eq. (8) to
hold, the integrand in square brackets must cross zero somewhere. However,
when $\eta\geq(4B_{0})^{1/4}\kappa_{\mathrm{max}}$, the integrand is always
positive, and thus there cannot be a solution. This serves as an upper bound
on $\eta^{*}$. In the small $\eta$ limit (with $\sigma\sim\eta^{2}$), the $-1$
piece dominates, and the equation cannot be satisfied. We may then rigorously
give the interval $\eta^{*}\in(0,\sqrt{2}\kappa_{\mathrm{max}}B_{0}^{1/4}]$,
or with less rigour, change the lower bound to
$\eta^{*}>\sqrt{2}\kappa_{\mathrm{min}}B_{0}^{1/4}$, condition below which
$\sigma^{2}$ in Eq. (8) is the only term that may balance the other two. Given
the differences in curvature between the QA and QH phases (see [14]),
$\eta^{*}$ will tend to be larger in QH configurations.
Although the existence of $\eta^{*}$ is guaranteed by the form of the $\eta$
asymptotes and the mean value theorem, we do not know whether such an extremum
is unique. To prove so, we will investigate the second variation of the
Riccati equation, Eq. (6), and its sign at $\eta=\eta^{*}$. Taking the
variation of Eq. (14) (and looking at the extrema),
$\frac{\mathrm{d}}{\mathrm{d}\phi}\frac{\delta^{2}\sigma}{\delta\eta^{2}}=-2\bar{\iota}_{0}\left[\left(\frac{\delta\sigma}{\delta\eta}\right)^{2}+\sigma\frac{\delta^{2}\sigma}{\delta\eta^{2}}\right]-3\bar{\iota}_{0}\frac{\eta^{2}}{B_{0}\kappa^{4}}+\\\
+\frac{B_{\alpha 0}}{\kappa^{2}}(2\tau+B_{\theta
20})-\frac{\delta^{2}\bar{\iota}_{0}}{\delta\eta^{2}}\left[1+\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}\right].$
(18)
The periodicity requirement in $\delta^{2}\sigma/\delta\eta^{2}$ may be
written in the following solvability form: eliminating $\tau$ using the
Riccati $\sigma$-equation, Eq. (6), integrating by parts, using the $\eta^{*}$
condition, Eq. (8), and using the odd parity of $\sigma$ in stellarator
symmetry, we obtain
$\int_{0}^{2\pi}E\left\\{-2\bar{\iota}_{0}\left[\left(\frac{\delta\sigma}{\delta\eta}\right)^{2}+\frac{\eta^{2}}{B_{0}\kappa^{4}}\right]-P\frac{\delta^{2}\bar{\iota}_{0}}{\delta\eta^{2}}\right\\}\mathrm{d}\phi=0,$
(19)
where $P=1+\sigma^{2}+(\eta/\kappa)^{4}/4B_{0}$. The term with the square
bracket has a sign of $-\mathrm{sgn}(\bar{\iota}_{0})$. As $P,E>0$, for the
integral to vanish, it must be the case that the sign of the second variation
of $\bar{\iota}_{0}$ (which is not a function of $\phi$) satisfies,
$\mathrm{sgn}\left(\frac{\delta^{2}\bar{\iota}_{0}}{\delta\eta^{2}}\right)=-\mathrm{sgn}(\bar{\iota}_{0}).$
(20)
As the sign of $\bar{\iota}_{0}$ is set by the sign of the combination
$\int(B_{\alpha 0}/\kappa^{2})(2\tau+B_{\theta 20})\mathrm{d}\phi$ (and thus
cannot change with $\eta$), the extrema of $\bar{\iota}_{0}$ can only be
either maxima or minima (but only one of these), as the sign of
$\delta^{2}\bar{\iota}_{0}/\delta\eta^{2}$ is fixed. Thus, $\eta^{*}$ is
unique.
## Appendix B Magnetic shear decoupling for $\eta^{*}$
It was shown in [29] that, within the near-axis framework, one could evaluate
magnetic shear (and higher derivatives of rotational transform) as solvability
conditions of first-order periodic ODEs, so-called generalised $\sigma$
equations. Physically, this is reasonable since the rotational transform must
be self-consistently chosen given an average toroidal current profile. Here we
shall focus on magnetic shear, writing
$\bar{\iota}=\bar{\iota}_{0}+\psi\bar{\iota}_{1}+\dots$,
$\mathrm{d}\bar{\iota}/\mathrm{d}\psi=\bar{\iota}_{1}$ [5, Sec. 2.8]. Having
an understanding of this quantity is important, as it affects important
properties of the stellarator, such as ballooning stability [61, Ch. 6.14].
From [29], it is clear that magnetic shear is a third-order quantity in the
near-axis framework (or rather 2.5 order) in a vacuum given by
$\bar{\iota}_{1}=\frac{\int_{0}^{2\pi}\mathrm{d}\phi^{\prime}e^{2\bar{\iota}_{0}\int_{0}^{\phi^{\prime}}\sigma\mathrm{d}\phi^{\prime\prime}}\tilde{\Lambda}_{3}}{\int_{0}^{2\pi}\mathrm{d}\phi^{\prime}e^{2\bar{\iota}_{0}\int_{0}^{\phi^{\prime}}\sigma\mathrm{d}\phi^{\prime\prime}}\left[1+\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}\right]}.$
(21)
where,
$\displaystyle\tilde{\Lambda}_{3}=$
$\displaystyle\frac{1}{(Y^{S}_{1,1}){}^{2}}\left\\{2l^{\prime}\left[\tau\left(X^{C}_{3,1}Y^{S}_{1,1}+2X^{C}_{2,2}Y^{S}_{2,2}+X^{C}_{1,1}Y^{S}_{3,1}-\right.\right.\right.$
$\displaystyle\left.\left.\left.X^{S}_{3,1}Y^{C}_{1,1}-2X^{S}_{2,2}Y^{C}_{2,2}\right)+\kappa\left(2X^{C}_{2,2}Z^{S}_{2,2}+X^{C}_{1,1}Z^{S}_{3,1}-\right.\right.\right.$
$\displaystyle\left.\left.\left.-2X^{S}_{2,2}Z^{C}_{2,2}\right)\right]-2\iota_{0}\left(2X^{C}_{2,2}{}^{2}+X^{C}_{1,1}X^{C}_{3,1}+2X^{S}_{2,2}{}^{2}+\right.\right.$
$\displaystyle\left.\left.+2Y^{C}_{2,2}{}^{2}+2Y^{S}_{2,2}{}^{2}+Y^{S}_{1,1}Y^{S}_{3,1}+2Z^{C}_{2,2}{}^{2}+2Z^{S}_{2,2}{}^{2}\right)-\right.$
$\displaystyle\left.-X^{S}_{3,1}X^{C}_{1,1}{}^{\prime}-2X^{S}_{2,2}X^{C}_{2,2}{}^{\prime}+2X^{C}_{2,2}X^{S}_{2,2}{}^{\prime}+X^{C}_{1,1}X^{S}_{3,1}{}^{\prime}\right.$
$\displaystyle\left.-Y^{S}_{3,1}Y^{C}_{1,1}{}^{\prime}-2Y^{S}_{2,2}Y^{C}_{2,2}{}^{\prime}+2Y^{C}_{2,2}Y^{S}_{2,2}{}^{\prime}+Y^{C}_{1,1}Y^{S}_{3,1}{}^{\prime}\right.$
$\displaystyle\left.-2Z^{S}_{2,2}Z^{C}_{2,2}{}^{\prime}+2Z^{C}_{2,2}Z^{S}_{2,2}{}^{\prime}\right\\}.$
(22)
The weighted integral of this quantity $\tilde{\Lambda}_{3}$ drives the
magnetic shear, Eq. (21).
This quantity depends on third-order quantities, for which closed forms may be
found in the literature[22, 21, 28]. Although Eq. (22) is of the third order,
most elements in it may be written exclusively in terms of lower order
quantities, except for $X_{3}$, which explicitly introduces parameters
$B_{31}^{C}$ and $B_{31}^{S}$. Therefore, let us focus on the elements in the
shear that depend on $B_{31}^{C}$.
This component of the magnetic field may be seen as the variation of the
parameter $\eta$ with radius. An approximate variation of ellipticity of flux
surfaces with radius. It is then unsurprising that, given the central role of
$\eta$ in determining the rotational transform on the axis, $B_{31}^{C}$
directly affects the shear. Explicitly, the dependence of
$\tilde{\Lambda}_{3}$ on $B_{31}^{C}$ is,
$\tilde{\Lambda}_{3}=\left[\frac{\bar{\iota}_{0}}{B_{0}\eta}\left(\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}-1\right)-\frac{\sigma^{\prime}}{2B_{0}\eta}\right]B_{31}^{C}+\dots,$
(23)
from the expressions for $X_{31}$ (see Appendix F in [29]). Define the factor
$\mathcal{M}$ as the multiplicative factor modulating the contribution of
$B_{31}^{C}$ to the magnetic shear. Using Eq. (23) in Eq. (21), integrating by
parts and assuming stellarator symmetry, we obtain
$\mathcal{M}=\frac{\bar{\iota}_{0}}{B_{0}\eta}\left[1-\frac{2\int_{0}^{2\pi}e^{2\bar{\iota}_{0}\int\sigma\mathrm{d}\phi^{\prime}}\mathrm{d}\phi}{\int_{0}^{2\pi}\left[1+\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}\right]e^{2\bar{\iota}_{0}\int\sigma\mathrm{d}\phi^{\prime}}\mathrm{d}\phi}\right].$
(24)
The expression in the large square brackets is a number between $(-1,1)$ sets
a hard upper bound to the effect of the third-order modulation on the shear.
Furthermore, it restricts the rotational transform on the axis not to exceed
$(B_{31}^{C}/B_{0})/\eta$. The minimum of $\mathcal{M}$ with finite (non-zero)
$\eta$ is achieved when the expression in brackets vanishes. Rewriting it into
a single fraction, we get
$\mathcal{M}=\frac{\bar{\iota}_{0}}{B_{0}\eta}\frac{\int_{0}^{2\pi}\left[\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}-1\right]E\mathrm{d}\phi}{\int_{0}^{2\pi}\left[1+\sigma^{2}+\frac{1}{4B_{0}}\left(\frac{\eta}{\kappa}\right)^{4}\right]E\mathrm{d}\phi},$
(25)
where $E=\exp[2\bar{\iota}_{0}\int\sigma\mathrm{d}\phi^{\prime}]$. The
numerator is precisely the extremum condition for $\eta^{*}$ in Eq. (8). Thus,
the resilience to $\eta$ presents itself by making the magnetic shear
independent of third-order choices. Thus, $\iota_{1}$ becomes a second-order
quantity upon choosing $\eta^{*}$. This makes the truncated near-axis model,
as constructed in this paper, more complete. Note that $\mathcal{M}$ vanishing
does not equal vanishing of the magnetic shear. There remains a generally non-
zero ‘intrinsic’ contribution from lower-order pieces. However, if
configurations with a reasonably low aspect ratio are sought, then the second-
order shaping should remain small, and so will the magnetic shear.
## Appendix C Choosing $\eta$ to maximise $L_{\nabla}$
Let us consider briefly in this Appendix some of the intricacies of the
measure proposed in [50] as a guiding principle in the choice of the parameter
$\eta$. The measure is the magnitude of $||\nabla\mathbf{B}||$, using the
Frobenius norm $||M_{ij}||^{2}=\sum_{i,j}(M_{ij})^{2}$ of the
$\nabla\mathbf{B}$ tensor. This provides a characteristic length scale of the
field, approximately that corresponding to the maximum distance at which a
coil may be placed[50]. This measure has been used as a proxy of the radius of
applicability of the near-axis construction, which appears to work when
applied to optimisation within the near-axis framework in practice[51, 62].
From this perspective, it appears to be a natural choice of $\eta$ that
maximises the length scale
$L_{\nabla}\stackrel{{\scriptstyle\cdot}}{{=}}1/||\nabla\mathbf{B}||$. To
investigate this, we write the gradient of $\mathbf{B}$ at first order (in the
notation used in this paper),
$\nabla\mathbf{B}=\nabla\psi\partial_{\psi}\mathbf{B}+\nabla\chi\partial_{\chi}\mathbf{B}+\nabla\phi\partial_{\phi}\mathbf{B}.$
(26)
Using
$\mathbf{B}=\mathcal{J}^{-1}(\partial_{\phi}+\bar{\iota}\partial_{\chi})\mathbf{x}$
and the dual relations[20], we may then expand the expression in the near-axis
fashion,
$\nabla\mathbf{B}\approx\mathcal{J}_{0}^{-1}\partial_{\chi}\mathbf{x}_{1}\times\partial_{\phi}\mathbf{x}_{0}\partial_{\psi}\mathbf{B}_{1}+\mathcal{J}_{0}^{-1}\partial_{\phi}\mathbf{x}_{0}\times\partial_{\psi}\mathbf{x}_{1}\partial_{\chi}\mathbf{B}_{1}+\\\
\mathcal{J}_{0}^{-1}\partial_{\psi}\mathbf{x}_{1}\times\partial_{\chi}\mathbf{x}_{1}\partial_{\phi}\mathbf{B}_{0}.$
(27)
For $\mathbf{x}$ we take Eq. (1), to construct expressions like,
$\partial_{\chi}\mathbf{x}_{1}\times\partial_{\phi}\mathbf{x}_{0}=\frac{\mathrm{d}l}{\mathrm{d}\phi}(\partial_{\chi}Y_{1}\hat{\kappa}-\partial_{\chi}X_{1}\hat{\tau}),$
and
$\partial_{\chi}\mathbf{B}_{1}=\mathcal{J}^{-1}\left[-\kappa
l^{\prime}\partial_{\chi}X_{1}\hat{b}+(\partial_{\chi}Y_{1}^{\prime}-\bar{\iota}Y_{1}-\tau
l^{\prime}\partial_{\chi}X_{1})\hat{\tau}+\right.\\\
\left.(\partial_{\chi}X_{1}^{\prime}-\bar{\iota}X_{1}+\tau
l^{\prime}\partial_{\chi}Y_{1})\right].$
With this,
$\nabla\mathbf{B}=\frac{1}{B_{\alpha
0}B_{0}}\left[\frac{1}{2\sqrt{B_{0}}}\left(Y_{11}^{S}X_{11}^{C}{}^{\prime}+\bar{\iota}_{0}Y_{11}^{C}X_{11}^{C}\right)\hat{\kappa}\hat{\kappa}+\right.\\\
\left.+\left(l^{\prime}\tau-\frac{\bar{\iota}_{0}}{2l^{\prime}}(X_{11}^{C})^{2}\right)\hat{\tau}\hat{\kappa}+\right.\\\
\left.+\left(\frac{Y_{11}^{C}{}^{\prime}Y_{11}^{S}-Y_{11}^{S}{}^{\prime}Y_{11}^{C}+\bar{\iota}_{0}[(Y_{11}^{S})^{2}+(Y_{11}^{C})^{2}]}{2\sqrt{B_{0}}}-l^{\prime}\tau\right)\hat{\kappa}\hat{\tau}+\right.\\\
+\left.\frac{1}{2\sqrt{B_{0}}}(X_{11}^{C}Y_{11}^{S}{}^{\prime}-\bar{\iota}_{0}X_{11}^{C}Y_{11}^{C})\hat{\tau}\hat{\tau}\right]+\frac{\kappa}{\sqrt{B_{0}}}(\hat{b}\hat{\kappa}+\hat{\kappa}\hat{b}).$
(28)
which is consistent with [50, Eq. (3.12)], differences in notation provided.
Eq. (28) may be interpreted as a function of $\eta$ for a fixed axis shape,
much in the same way as we did for $\bar{\iota}_{0}$. As in that case, a basic
understanding of the behaviour of $||\nabla\mathbf{B}||$ can be gained from
the large and small $\eta$ asymptotics. For stellarator symmetry and
$\eta\rightarrow 0$, (taking for ease of notation $B_{0}\sim 1$),
$||\nabla\mathbf{B}||=\sqrt{2\kappa^{2}+\frac{2}{(l^{\prime})^{2}}\left(\frac{\kappa^{\prime}}{\kappa}\right)^{2}+\tau^{2}+(\tau+B_{\theta
20})^{2}},$ (29)
while for large $\eta$,
$||\nabla\mathbf{B}||=\sqrt{2\kappa^{2}+\frac{2}{(l^{\prime})^{2}}\left(\frac{\kappa^{\prime}}{\kappa}-\sigma\bar{\iota}_{0}\right)^{2}+\left(\tau-\frac{\bar{\iota}_{0}}{2l^{\prime}}\left(\frac{\eta}{\kappa}\right)^{2}\right)^{2}+\left(\tau-\frac{\bar{\iota}_{0}}{2l^{\prime}}\left(\frac{\eta}{\kappa}\right)^{2}+B_{\theta
20}\right)^{2}}.$ (30)
These are both generally different constant values, provided $\sigma\neq 0$,
which does not guarantee the existence of an extremum. An example of the
special case is that of axisymmetry, for which the asymptotes are
$||\nabla\mathbf{B}||\sim\sqrt{2+B_{\theta 20}^{2}}$, and the extremum of
$||\nabla\mathbf{B}||$ coincides with $\eta^{*}$, the circular cross-section.
The consequence of this uneven asymptotic behaviour is generally a small
prominence of the extrema, if there is one. Examples of this behaviour are
shown in Figure 8, where the $L_{\nabla}$ metric is compared to other first-
order measures such as the rotational transform and the weighted averaged
elongation for a selection of quasisymmetric designs. This uncertain behaviour
of $||\nabla\mathbf{B}||$, which shows excellent behaviour in the QH cases,
makes this gradient length not as robust a measure as the other choices
proposed (see Table 2). This excellent behaviour in the case of QH
configurations may result from the tendency of such configurations to develop
small-scale features. Like $\mathcal{E}$, $L_{\nabla}$ is also a function of
$\phi$, and differences arise from different considerations.
Figure 8: Comparison of $\eta$ measures. The main measures guiding the choice
of $\eta$ discussed in the text are compared for the same configurations as
Table 2. These include $|\bar{\iota}_{0}|$ whose extremum is $\eta^{*}$
(blue), the weighted elongation whose extremum is $\eta_{\bar{\mathcal{E}}}$
(orange), the mean $L_{\nabla}$ whose maximum is $\eta_{\nabla\mathbf{B}}$
(green), and for comparison the minimum of $L_{\nabla}$ (red). Green and red
correspond to the right-axis scale. The blue point represents $\eta$ from the
global optimised designs, namely $\eta_{\mathrm{VMEC}}$. In many designs, all
considerations provide similar $\eta$ values (see Table 2), but the
shortcoming of $L_{\nabla}$ is also patent (especially in QA examples such as
ESTELL [46]). These also show the similarity in behaviour of
$\langle\mathcal{E}/\kappa^{2}\rangle^{-1}$ and $|\bar{\iota}_{0}|$.
## Appendix D Second order ODE on $B_{20}$
The seed of overdetermination at the second order comes from the magnetic
field having to satisfy both the QS and equilibrium conditions. The common way
to avoid this overdetermination at the second order is to relax the condition
of quasisymmetry partially. Doing so requires solving a second-order, regular
ODE for $B_{20}$, the second-order, $\theta$ independent change in
$1/|\mathbf{B}|^{2}$. We write this equation explicitly here for a
stellarator-symmetric vacuum field:
$\mathcal{A}\frac{\mathrm{d}^{2}}{\mathrm{d}\phi^{2}}\left(\frac{B_{20}}{B_{0}}\right)+\mathcal{B}\frac{\mathrm{d}}{\mathrm{d}\phi}\left(\frac{B_{20}}{B_{0}}\right)+\mathcal{C}\frac{B_{20}}{B_{0}}+\mathcal{D}=0,$
(31)
where,
$\displaystyle\mathcal{A}=-\frac{B_{\alpha
0}\eta^{2}}{2\kappa^{2}\bar{\iota}_{0}l^{\prime}}\left[1+\frac{4B_{0}\kappa^{4}}{\eta^{4}}(1+\sigma^{2})\right],$
(32a) $\displaystyle\mathcal{B}=\frac{2B_{\alpha
0}\eta^{2}}{\bar{\iota}_{0}l^{\prime}}\frac{\kappa^{\prime}}{\kappa^{3}}-\frac{4l^{\prime}\sigma}{\bar{\iota}_{0}}\tau,$
(32b) $\displaystyle\mathcal{C}=-\frac{l^{\prime}}{2B_{\alpha
0}\eta^{2}\kappa^{2}}\left[\bar{\iota}_{0}\left(4\kappa^{4}(1+\sigma^{2})-\frac{3\eta^{4}}{B_{0}}\right)+8B_{\alpha
0}\eta^{2}\kappa^{2}\tau\right],$ (32c)
and $B_{\alpha 0}^{2}B_{0}=(\mathrm{d}l/\mathrm{d}\phi)^{2}$. The homogeneous
operator only depends on zeroth and first-order considerations. However, the
inhomogeneous term $D$ is given by,
$\mathcal{D}=\frac{\mathcal{Y}_{0}^{S}}{\mathcal{Y}_{0}^{C}}\beta_{R}^{C}+\mathcal{Y}_{1}^{S}\left(\frac{\beta_{R}^{C}}{\mathcal{Y}_{0}^{C}}\right)^{\prime}+\beta_{R}^{S},$
(33)
where,
$\displaystyle\mathcal{Y}_{0}^{C}=\frac{4\bar{\iota}_{0}\kappa}{\eta},$ (34a)
$\displaystyle\mathcal{Y}_{0}^{S}=\frac{4\kappa^{\prime}-8\bar{\iota}_{0}\kappa\sigma}{\eta},$
(34b) $\displaystyle\mathcal{Y}_{1}^{S}=-\frac{4\kappa}{\eta}$ (34c)
$\beta_{R}^{C}=\frac{1}{2l^{\prime}\kappa}\left[4\eta^{2}B_{\alpha
0}^{2}\left(Y_{22}^{C}Z_{22}^{S}+Y_{22}^{S}\left(\tilde{Z}_{20}-Z_{22}^{C}\right)\right)+2B_{\alpha
0}\left(\eta^{2}Y_{22}^{C}\tau l^{\prime}+\right.\right.\\\
\left.\left.+\eta^{2}{X_{22}^{C}}^{\prime}-\kappa
l^{\prime}\tilde{Z}_{20}\left(\eta^{2}-4\kappa\left(X_{22}^{C}-\sigma
X_{22}^{S}\right)\right)-4l^{\prime}\kappa^{2}\sigma
X_{22}^{C}Z_{22}^{S}-4l^{\prime}\kappa^{2}\tilde{X}_{20}Z_{22}^{C}+\right.\right.\\\
\left.\left.+4l^{\prime}\kappa^{2}\sigma\tilde{X}_{20}Z_{22}^{S}+2\eta^{2}\iota_{0}X_{22}^{S}+4l^{\prime}\kappa^{2}\sigma
X_{22}^{S}Z_{22}^{C}+\eta^{2}\kappa
l^{\prime}Z_{22}^{C}-\eta^{2}\tilde{X}_{20}{}^{\prime}\right)+\right.\\\
\left.+4\kappa^{2}l^{\prime}\left(\sigma\tilde{X}_{20}\tau
l^{\prime}-\sigma\tau l^{\prime}X_{22}^{C}-\tau
l^{\prime}X_{22}^{S}+\sigma\tilde{Y}_{22}^{C}{}^{\prime}-2\iota_{0}(Y_{22}^{C}-\sigma
Y_{22}^{S})+\tilde{Y}_{22}^{S}{}^{\prime}\right)\right]$ (34d)
$\beta_{R}^{S}=\frac{1}{2\kappa l^{\prime}}\left[2B_{\alpha
0}\left(\eta^{2}Y_{22}^{S}\tau
l^{\prime}-2\eta^{2}\iota_{0}X_{22}^{C}+4\kappa^{2}\sigma\tilde{Z}_{20}X_{22}^{C}l^{\prime}-\right.\right.\\\
\left.\left.-4\kappa^{2}X_{22}^{C}Z_{22}^{S}-4\kappa^{2}\sigma\tilde{X}_{20}Z_{22}^{C}l^{\prime}-4\kappa^{2}\tilde{X}_{20}Z_{22}^{S}l^{\prime}+\eta^{2}{X_{22}^{S}}^{\prime}+4\kappa^{2}X_{22}^{S}Z_{22}^{C}l^{\prime}+\right.\right.\\\
\left.\left.+4\kappa^{2}\tilde{Z}_{20}X_{22}^{S}l^{\prime}+\eta^{2}\kappa
Z_{22}^{S}l^{\prime}\right)+\kappa l^{\prime}\left(4\kappa\tilde{X}_{20}\tau
l^{\prime}+4\kappa l^{\prime}\tau X_{22}^{C}-4\kappa l^{\prime}\sigma\tau
X_{22}^{S}-4\kappa\tilde{Y}_{22}^{C}{}^{\prime}-8\iota_{0}\kappa(\sigma\tilde{Y}_{22}^{C}+Y_{22}^{S})+\right.\right.\\\
\left.\left.+4\kappa\sigma\tilde{Y}_{22}^{S}{}^{\prime}\right)-4B_{\alpha
0}^{2}\eta^{2}\tilde{Y}_{22}^{C}\tilde{Z}_{20}\right],$ (34e)
The various forms of $X$, $Y$, and $Z$ are needed to complete the expression
for $D$ (remembering that for the expressions we should make $B_{20}=0$ as we
have dealt with these explicitly already). can be found explicitly in [22] in
the simplifying vacuum limit. More explicitly, the forms of $X$ are found in
Eqs. (14)-(15) and Appendix C; the expressions for $Y$ in Eqs. (27)-(28); and
the expressions for $Z$ in Eq. (24) and following. For a systematic way to
obtain these expressions, see [29]. This leaves a contribution to $D$
proportional to $B_{22}^{C}$.
## Appendix E Solution existence for $B_{20}$
In this Appendix, we consider some existence and uniqueness properties of the
$B_{20}$ equation, Eq. (10). We do so by assessing the second-order
differential equation by applying the Fredholm Alternative theorem.
Fredholm Alternative Theorem Let $\mathbb{L}$ be a linear operator with
adjoint $\mathbb{L}^{\dagger}$. Then exactly one of the following is true:
* •
The inhomogeneous problem $\mathbb{L}y=f$ has a unique solution $y$.
* •
The homogeneous adjoint problem $\mathbb{L}^{\dagger}y=0$ has a non-trivial
solution.
In the event of the latter, the inhomogeneous equation has either no solution
or infinitely many. If the solvability condition $\langle y_{0},f\rangle=0$,
then there are an infinite number of solutions.
For application of the Fredholm Alternative theorem, write Eq. (10) as,
$\mathbb{L}\frac{B_{20}}{B_{0}}+\mathcal{D}=[\mathcal{A}\partial_{\phi}^{2}+\mathcal{B}\partial_{\phi}+\mathcal{C}]\frac{B_{20}}{B_{0}}+\mathcal{D}=0,$
(35)
and complete the problem with periodic boundary conditions on $B_{20}$.
It is thus clear that we first need to study the homogeneous operator
$\mathbb{L}$. To construct the adjoint of $\mathbb{L}$, it is convenient to
rewrite Eq. (35) in a self-adjoint form. This can be achieved by writing the
second-order ODE in the form of a Hill equation,
$\psi^{\prime\prime}+Q\psi=-\frac{D}{\mathcal{A}}e^{\int\frac{\mathcal{B}}{2\mathcal{A}}\mathrm{d}\phi},$
(36)
where,
$\displaystyle
Q=-\frac{1}{2}\left(\frac{\mathcal{B}}{\mathcal{A}}\right)^{\prime}-\frac{1}{4}\left(\frac{\mathcal{B}}{\mathcal{A}}\right)^{2}+\frac{C}{\mathcal{A}},$
(37a)
$\displaystyle\psi=B_{20}e^{-\int\left(\frac{\mathcal{B}}{2\mathcal{A}}\right)\mathrm{d}\phi}.$
(37b)
Using the coefficients $\mathcal{A}$ and $\mathcal{B}$, Eqs. (11a)-(11b),
$\frac{\mathcal{B}}{\mathcal{A}}=2\bar{\iota}_{0}\sigma+\left[\ln\left(-\frac{\bar{\beta}_{1}^{C}}{\mathcal{Y}_{1}^{S}}\right)\right]^{\prime},$
where $\mathcal{Y}_{1}^{S}=-4\kappa/\eta$ and $\bar{\beta}_{1}^{C}=B_{\alpha
0}\eta^{2}l^{2}\left[1+4\kappa^{4}(1+\sigma^{2})/\eta^{4}B_{\alpha
0}^{2}\right]/8\kappa^{3}\bar{\iota}_{0}$. The exponential factor that maps
$B_{20}\rightarrow\psi$ is then by construction periodic. Therefore, solving
the $B_{20}$ equation is equivalent to solving Eq. (36) with periodic boundary
conditions on $\psi$. This mapping preserves the even parity of $B_{20}$ in
stellarator symmetry, and thus we may focus on Eq. (36).
As the homogeneous operator of Eq. (36) is self-adjoint, the adjoint problem
for the Fredholm Alternative is $\psi^{\prime\prime}+Q\psi=0$. The problem is
nothing but a periodic, time-independent Schrödinger equation. With this
interpretation, the ‘quantum potential’ is $V=-Q/2$ (formally taking
$\hbar,m=1$). Due to periodicity, the potential can be thought to represent a
‘crystal’ (or periodic lattice) of period $\Delta\phi=2\pi/N$. The structure
of this potential is determined by the axis shape and choice of $\eta$.
Whenever this adjoint periodic Schrödinger equation supports a zero energy
state, then by the Fredholm Alternative, there will be no unique solution to
the original $B_{20}$ equation. This may mean that no solution or an infinite
number of them exists, a distinction in which the second-order choices will
intervene through $\mathcal{D}$.
For a generally shaped potential (see Fig. 9), there is no closed form for the
energy eigenstates of the adjoint. However, the discrete nature of energy
eigenvalues and the very special requirement for a vanishing energy state
suggests that, in general, such an eigenstate will not exist. Thus, a solution
to the direct $B_{20}$ problem will. Only for very particular choices will the
zero energy eigenstate exist, leading to the divergence observed in Fig. 3.
Such special value for a fixed axis shape corresponds to
$\eta_{\mathrm{crit}}$. We shall illustrate these abstract statements with an
example in which the existence or not of a solution is considered.
Figure 9: Example of effective quantum potential for the adjoint problem. This
shows examples of the potential $-Q/2=V$ for (a) a QA with $R=1+0.0144\cos
4\phi$, $Z=0.0144\sin 4\phi$, $N=4$ and $\eta=2.1$ and (b) a QH with
$R=1+0.1\cos 4\phi$, $Z=0.1\sin 4\phi$, $N=4$ and $\eta=2$. The broken line in
(a) shows a quadratic potential centred around the minimum. These are two
examples of what the effective potential wells of the adjoint problem look
like.
For example, the near-axis construction corresponding to Fig. 9, in which the
potential has a simple finite depth and width well in each lattice cell. In
general, more complex features will be present, as also shown in the figure.
Naturally, the first guess to the form of the eigenstates are states
corresponding to single finite depth well, a standard textbook problem [63,
Chap. 2.6]. Of course, the solution will be significantly more involved than
this. For once, in a lattice, wavefunctions are generally not isolated to each
cell, but there is a hopping energy and an overlap between wavefunctions in
nearby lattice sites that lead to the splitting of the energy states.
Secondly, the shapes of the potential wells change the eigenstructure of the
problem. Of course, not having an exact solution, we would like to estimate
the energy spectrum of the problem without depending excessively on the model
used. This is achieved through a variational approach in quantum mechanics.
From the orthogonality of the energy eigenstates and focusing on the ground-
state energy, this can be estimated by considering
$E_{0}\leq\frac{\int\left[(\psi^{\prime})^{2}-Q\psi^{2}\right]\mathrm{d}\phi}{\int\psi^{2}\mathrm{d}\phi},$
(38)
where $\psi$ can be any (here real) function. To get a tighter bound on the
energy, an informed guess of $\psi$ is needed. Leaving some parameters of
$\psi$ as unknowns (say concatenated single-well wavefunctions with free depth
and width parameters), a minimum to the expression may be sought, with the
resulting value being an approximation to the energy state.
Figure 10: Ground-state energy of adjoint problem as a function of $\eta$. The
plot shows the evaluation of the adjoint ground-state energies ($E_{0}$
normalised to $Q$ at the bottom of the potential troughs, $Q_{0}$) as a
function of the $\eta$ parameter for the QA example in Fig. 9. The blue curve
shows the estimate of the ground-state using the wavefunction guess from
$B_{20}$, and the orange shows the finite well variational approach. The
estimated ground-state can be considered the smallest of these two curves. The
crossing with the 0 for the $B_{20}$ estimate is precisely the point at which
the resonance occurs in Fig. 9.
For most values of $\eta$ (see the domain in which $E_{0}<0$ in Fig. 10), the
ground-state lies below the zero energy level, and the $B_{20}$ solution is
unique. This assumes that none of the higher energy states resonate, which is
true given the proximity of the energy $E_{0}$ to 0 (compared to the well
depth) and the parity requirement on the solution, which makes the next energy
level solution lie significantly higher. Although we do not prove it here, we
shall assume that only the ground-state is relevant here. The precise location
of the 0-energy crossing will vary with $\psi_{\mathrm{guess}}$, and thus only
a lower bound may be given for $\eta_{\mathrm{crit}}$. A more precise crossing
value can be obtained by constructing $\psi$ from the solution of $B_{20}$.
This is generally not a great guess for the adjoint problem, except in the
region close to where the existence of the solution starts to break down: the
critical $\eta$ value. If this critical value indicates the second alternative
of the Fredholm Alternative, then in the neighbourhood of this critical
resonance, the solution $B_{20}$ will be dominated by the ground-state
solution of the adjoint problem, and thus the zero crossing will be exact (see
Fig. 10). At this point, the choice of $B_{22}^{C}$ is important, as it
determines whether the inhomogeneous term in Eq. (36) is orthogonal to the
ground-state solution or not, controlling whether there exist none or an
infinite number of solutions, where the infinite family would correspond to
the addition of an arbitrary multiple of the ground-state $\psi_{0}$.
Solutions do exist in general and are unique, except for particular critical
values of $\eta$ for which the ground-state of the adjoint problem has
vanishing energy.
Figure 11: Diagram describing the idea behind the existence proof.
Illustration of the rationale behind the proof of existence of a singular
value $\eta_{\mathrm{crit}}$ to the $B_{20}$ equation.
The existence of such a value has not been proven yet. We do so now by showing
that the energy state of the adjoint homogeneous equation necessarily crosses
the zero. Assuming continuity with $\eta$, it will suffice to show that the
energy level is positive (negative) for small (large) $\eta$ (see Fig. 11).
Let us start with the small $\eta$ limit, for which $\mathcal{A}\sim
O(1/\eta^{5})$, $\mathcal{B}\sim O(1/\eta)$ and $\mathcal{C}\sim O(1/\eta)$,
from Eqs. (11a)-(LABEL:eqn:CB20). To leading order $O(\eta^{4})$ then
$Q\sim-(\mathcal{B}/\mathcal{A})^{\prime}/2+\mathcal{C}/\mathcal{A}$. In this
limit, it is sufficient to show that the estimate of the ground-state energy
is negative. Choose the wavefunction to be constant, $\psi=\psi_{0}$. Then,
$E_{0}\leq\frac{\int_{0}^{2\pi}\left[(\psi^{\prime})^{2}-Q\psi^{2}\right]\mathrm{d}\phi}{\int_{0}^{2\pi}\psi^{2}\mathrm{d}\phi}=-\frac{1}{2\pi}\int_{0}^{2\pi}Q\mathrm{d}\phi.$
Given the asymptotic form of $Q$,
$\int_{0}^{2\pi}Q\mathrm{d}\phi=\int_{0}^{2\pi}\frac{\mathcal{C}}{\mathcal{A}}\mathrm{d}\phi\sim\frac{3\bar{\iota}_{0}^{2}}{B_{\alpha
0}(l^{\prime})^{2}}2\pi\geq 0.$
Thus, for a small (but finite $\eta$), the energy state of the adjoint
homogeneous equation has ground-state energy $E_{0}<0$.
In the large $\eta$ limit, $\mathcal{A}\sim O(\eta^{3})$, $\mathcal{B}\sim
O(\eta^{3})$ and $\mathcal{C}\sim O(1/\eta)$. Thus, the potential
$Q\sim-(\mathcal{B}/\mathcal{A})^{\prime}/2-(\mathcal{B}/\mathcal{A})^{2}/4$
becomes independent of $\mathcal{C}$ allowing one to rewrite the adjoint
problem defining $y=\psi\exp[-\int(\mathcal{B}/2\mathcal{A})\mathrm{d}\phi]$,
$y^{\prime\prime}+\frac{\mathcal{B}}{\mathcal{A}}y^{\prime}\sim 0,$ (39)
which can be solved exactly by
$y\sim\mathbb{C}\int\exp[-\int(\mathcal{B}/\mathcal{A})\mathrm{d}\phi^{\prime}]\mathrm{d}\phi$.
The exponential must be positive; thus, its integral is non-periodic, making
$y$ non-periodic as well. Thus, the only solution is the trivial $y=0$. Due to
the adjoint problem only having a trivial solution, in the large $\eta$ limit,
the solution to the $B_{20}$ equation is once again unique. However, this does
not provide us with the ground-state energy of the adjoint equation. To learn
what that is, write $y^{\prime\prime}+(\mathcal{B}/\mathcal{A})y^{\prime}\sim-
E_{0}y$, where we are explicitly including the energy eigenvalue associated
with the ground-state. We multiply the equation by $y$ and integrate over
$\phi$, so that
$y\left[Ey^{\prime}\right]^{\prime}\sim-E_{0}Ey^{2}\rightarrow
E_{0}\sim\frac{\int(y^{\prime})^{2}E\mathrm{d}\phi}{\int
y^{2}E\mathrm{d}\phi}>0.$
Here $E=\exp[2\bar{\iota}_{0}\int\sigma\mathrm{d}\phi]$. Regardless of the
choice of $y$, the ground-state is necessarily positive. Thus, as the ground-
state energy of the adjoint problem in these two limits has opposite signs, by
the mean-value theorem, it must cross the zero at some value of
$\eta=\eta_{\mathrm{crit}}$. This proof does not guarantee the uniqueness of
this singularity (see Fig. 3), but it guarantees its presence. Given the
numerical evidence, we shall, for the remainder of this Appendix, assume it to
be unique.
All of this has introduced a new special $\eta$ value, $\eta_{\mathrm{crit}}$,
which appears not to bear any relation to the choice of $\eta^{*}$ introduced
in the main text. We would like to relate the two. For a unique
$\eta_{\mathrm{crit}}$, the sign of $E_{0}$ at a given $\eta$ determines its
position relative to $\eta_{\mathrm{crit}}$: $E_{0}<0$ for
$\eta<\eta_{\mathrm{crit}}$ and $E_{0}>0$ for $\eta>\eta_{\mathrm{crit}}$.
Thus, we would like to assess the sign of the ground-state energy associated
with $\eta^{*}$ to situate it to the right or left of $\eta_{\mathrm{crit}}$.
Let us again exploit the Ritz variational form of energy, Eq. (38), in this
case using a trial wavefunction
$\psi\sim\exp\left[\int(\mathcal{B}/2\mathcal{A})\mathrm{d}\phi\right]$. A
wavefunction of this form is suggested by the defining condition of
$\eta^{*}$, Eq. (8). For the ground-state energy threshold, the following are
required,
$\displaystyle\int\left(\frac{\mathcal{B}}{\mathcal{A}}\right)^{2}\psi^{2}\mathrm{d}\phi=4\int(\psi^{\prime})^{2}\mathrm{d}\phi,$
$\displaystyle\int\left(\frac{\mathcal{B}}{\mathcal{A}}\right)^{\prime}\psi^{2}\mathrm{d}\phi=-4\int(\psi^{\prime})^{2}\mathrm{d}\phi,$
$\displaystyle\int\frac{\mathcal{C}}{\mathcal{A}}\psi^{2}\mathrm{d}\phi=\frac{|\bar{\iota}_{0}|}{4B_{\alpha
0}\eta^{*}}\int
e^{2\bar{\iota}_{0}\int\sigma\mathrm{d}\phi^{\prime}}\mathrm{d}\phi.$
For the latter, the forms of $\mathcal{A}$ and $\mathcal{B}$ in Eqs.
(11a)-(11b) were used, as well as integration by parts using Eq. (8). Putting
them together,
$\int\left[(\psi^{\prime})^{2}-Q\psi^{2}\right]\mathrm{d}\phi=-\int\frac{\mathcal{C}}{\mathcal{A}}\psi^{2}\mathrm{d}\phi<0,$
(40)
which thus yields a ground-state energy $E_{0}(\eta^{*})<0$. Thus, it follows
that $\eta^{*}<\eta_{\mathrm{crit}}$. The choice of $\eta^{*}$ thus naturally
avoids the divergence of the $B_{20}$ equation, as will a search in
$0<\eta<\eta^{*}$. This observation may be taken as additional evidence in
favour of the choice of $\eta^{*}$ first order.
## Appendix F Bounding $B_{22}^{C}$ search
We argued in the main text that to avoid extreme second-order shaping, we
should put some bounds on the allowable $B_{22}^{C}$. Following [43], it is
straightforward to see that both the Shafranov shift as well as the standard
measure of triangularity increase linearly with $B_{22}^{C}$ in the large
$|B_{22}^{C}|$ limit. Furthermore, with this dependence, the construction
becomes unphysical if $|B_{22}^{C}|$ is too large.
To make things more quantitative, consider an up-down symmetric cross-section
in the stellarator as representative of the behaviour of shaping in the
configuration, and focus on the behaviour of its Shafranov shift. This
describes the relative displacement of the centres of cross-sections when
going from one flux surface to the next. If the relative displacement of the
centres of the cross-sections at different $\psi$ is too large, then
eventually flux surfaces will intersect each other (this phenomenon of flux
surface intersection gives rise to the measure $r_{c}$, the largest value of
$\epsilon$ in which the near-axis description is sensible, presented in [50]).
This is unacceptable and thus can be leveraged to estimate a bound on
$B_{22}^{C}$. We reproduce from [54] a simple estimate of when this situation
is reached.
Start for simplicity with a second-order construction in which, besides the
elliptic shape, the second-order shaping has $X_{2}=X_{20}+X_{22}^{C}\cos
2\chi$. Focus then on the centre-line of the cross-section, about which it is
up-down symmetric. Looking at $\chi=0$, $X=\epsilon
X_{11}^{C}+\epsilon^{2}(X_{20}+X_{22}^{C})$. Now the intersection occurs
whenever $\partial_{\epsilon}X=0$, which can be solved for the critical
Shafranov shift,
$X_{20}+X_{22}^{C}=\frac{X_{11}^{C}}{2\epsilon}.$ (41)
Whenever the Shafranov shift exceeds the value on the RHS, then the cross-
sections at the stellarator symmetric point will intersect.
Let us see how a large $|B_{22}^{C}|$ affects the Shafranov shift. To do that,
we need to learn about the behaviour of $X_{20}$ and $X_{22}^{C}$ at $\phi=0$.
These expressions may be found by careful consideration of the symmetry
properties of the various functions involved. As a result,
$X_{20}+X_{22}^{C}\sim-\frac{1}{2\kappa}\frac{B_{22}^{C}}{B_{0}}\left(1+\frac{B_{20}^{\mathrm{univ}}}{B_{0}}\right).$
(42)
Thus, a critical $B_{22}^{C}$,
$\frac{|B_{22}^{C}|}{B_{0}}\sim\frac{\kappa
X_{11}^{C}}{\epsilon}\left(1+\frac{B_{20}^{\mathrm{univ}}}{B_{0}}\right)^{-1}.$
(43)
Crudely, taking $\eta=\kappa X_{11}^{C}\sim 1$ and ignoring the expression in
the bracket, we get $|B_{22}^{C}|/B_{0}\sim 1/\epsilon$. That is, we may
roughly consider it proportional to the ‘aspect ratio,’ $1/\epsilon$, of the
configuration. With a reasonable aspect ratio of $1/\epsilon\sim 10$, we
obtain the $B_{22}^{C}$ limit in the text (which could be relaxed by allowing
for other larger values like, e.g., 20).
It should be clear from this approach that the estimate is but a crude one,
yet nevertheless useful. For instance, we are ignoring
$B_{20}^{\mathrm{univ}}$, which could relax this bound significantly depending
on the situation. In practice, the construction of this crude bound allows us
to perform bounded optimisation. In most of the relevant space (where the most
reasonably shaped, quasisymmetric configurations lie), it has not had much of
an effect on the result (the $B_{22}^{C}$ minimum is well within the interval,
as an example, see Fig. 4).
## References
* Bernardin, Moses, and Tataronis [1986] M. P. Bernardin, R. W. Moses, and J. A. Tataronis, “Isodynamical (omnigenous) equilibrium in symmetrically confined plasma configurations,” The Physics of Fluids 29, 2605–2611 (1986).
* Cary and Shasharina [1997] J. R. Cary and S. G. Shasharina, “Omnigenity and quasihelicity in helical plasma confinement systems,” Physics of Plasmas 4, 3323–3333 (1997).
* Hall and McNamara [1975] L. S. Hall and B. McNamara, “Three‐dimensional equilibrium of the anisotropic, finite‐pressure guiding‐center plasma: Theory of the magnetic plasma,” The Physics of Fluids 18, 552–565 (1975).
* Landreman and Catto [2012] M. Landreman and P. J. Catto, “Omnigenity as generalized quasisymmetry,” Physics of Plasmas 19, 056103 (2012).
* Helander [2014] P. Helander, “Theory of plasma confinement in non-axisymmetric magnetic fields,” Reports on Progress in Physics 77, 087001 (2014).
* Boozer [1983] A. H. Boozer, “Transport and isomorphic equilibria,” The Physics of Fluids 26, 496–499 (1983).
* Nührenberg and Zille [1988] J. Nührenberg and R. Zille, “Quasi-helically symmetric toroidal stellarators,” Physics Letters A 129, 113 – 117 (1988).
* Rodríguez, Helander, and Bhattacharjee [2020] E. Rodríguez, P. Helander, and A. Bhattacharjee, “Necessary and sufficient conditions for quasisymmetry,” Physics of Plasmas 27, 062501 (2020).
* Anderson _et al._ [1995] F. S. B. Anderson, A. F. Almagri, D. T. Anderson, P. G. Matthews, J. N. Talmadge, and J. L. Shohet, “The Helically Symmetric Experiment (HSX) goals, design and status,” Fusion Technology 27, 273–277 (1995).
* Zarnstorff _et al._ [2001] M. Zarnstorff, L. Berry, A. Brooks, E. Fredrickson, G. Fu, S. Hirshman, S. Hudson, L. Ku, E. Lazarus, D. Mikkelsen, _et al._ , “Physics of the compact advanced stellarator ncsx,” Plasma Physics and Controlled Fusion 43, A237 (2001).
* Najmabadi _et al._ [2008] F. Najmabadi, A. Raffray, S. Abdel-Khalik, L. Bromberg, L. Crosatti, L. El-Guebaly, P. Garabedian, A. Grossman, D. Henderson, A. Ibrahim, _et al._ , “The ARIES-CS compact stellarator fusion power plant,” Fusion Science and Technology 54, 655–672 (2008).
* Ku and Boozer [2010] L. Ku and A. Boozer, “New classes of quasi-helically symmetric stellarators,” Nuclear Fusion 51, 013004 (2010).
* Bader _et al._ [2019] A. Bader, M. Drevlak, D. T. Anderson, B. J. Faber, C. C. Hegna, K. M. Likin, J. C. Schmitt, and J. N. Talmadge, “Stellarator equilibria with reactor relevant energetic particle losses,” Journal of Plasma Physics 85, 905850508 (2019).
* Rodríguez, Sengupta, and Bhattacharjee [2022] E. Rodríguez, W. Sengupta, and A. Bhattacharjee, “Phases and phase-transitions in quasisymmetric configuration space,” Plasma Physics and Controlled Fusion 64, 105006 (2022).
* Note [1] We simplify the picture by assuming that the electrostatic potential shares the QS to leading gyro-order and do not include it in our considerations.
* Tessarotto _et al._ [1996] M. Tessarotto, J. L. Johnson, R. B. White, and L. Zheng, “Quasi‐helical magnetohydrodynamic equilibria in the presence of flow,” Physics of Plasmas 3, 2653–2663 (1996).
* Burby, Kallinikos, and MacKay [2021] J. W. Burby, N. Kallinikos, and R. S. MacKay, “Approximate symmetries of guiding-centre motion,” Journal of Physics A: Mathematical and Theoretical 54, 125202 (2021).
* Rodríguez, Paul, and Bhattacharjee [2022] E. Rodríguez, E. Paul, and A. Bhattacharjee, “Measures of quasisymmetry for stellarators,” Journal of Plasma Physics 88, 905880109 (2022).
* Note [2] For a more general form of equilibrium, a formally analogous approach exists in terms of so-called generalised Boozer coordinates, details of which may be found in [rodrigGBC]. As a result, many of the properties of quasisymmetric stellarators in this paper are independent of a particular form of equilibrium.
* Garren and Boozer [1991a] D. A. Garren and A. H. Boozer, “Magnetic field strength of toroidal plasma equilibria,” Physics of Fluids B: Plasma Physics 3, 2805–2821 (1991a).
* Landreman, Sengupta, and Plunk [2019] M. Landreman, W. Sengupta, and G. G. Plunk, “Direct construction of optimized stellarator shapes. part 2. numerical quasisymmetric solutions,” Journal of Plasma Physics 85, 905850103 (2019).
* Rodríguez and Bhattacharjee [2021] E. Rodríguez and A. Bhattacharjee, “Solving the problem of overdetermination of quasisymmetric equilibrium solutions by near-axis expansions. i. generalized force balance,” Physics of Plasmas 28, 012508 (2021).
* Kruskal and Kulsrud [1958] M. D. Kruskal and R. M. Kulsrud, “Equilibrium of a magnetically confined plasma in a toroid,” The Physics of Fluids 1, 265–274 (1958).
* Rodríguez, Sengupta, and Bhattacharjee [2021] E. Rodríguez, W. Sengupta, and A. Bhattacharjee, “Generalized boozer coordinates: A natural coordinate system for quasisymmetry,” Physics of Plasmas 28, 092510 (2021).
* Mercier [1964] C. Mercier, “Equilibrium and stability of a toroidal magnetohydrodynamic system in the neighbourhood of a magnetic axis,” Nuclear Fusion 4, 213 (1964).
* Solov’ev and Shafranov [1970] L. S. Solov’ev and V. D. Shafranov, _Reviews of Plasma Physics 5_ (Consultants Bureau, New York - London, 1970).
* Note [3] For simplicity, we have not normalised $\psi$ respect to the magnetic field on the axis and its curvature as it is often customary[20, 21]. Doing so simply introduces constant rescaling factors in various quantities involved.
* Garren and Boozer [1991b] D. A. Garren and A. H. Boozer, “Existence of quasihelically symmetric stellarators,” Physics of Fluids B: Plasma Physics 3, 2822–2834 (1991b).
* Rodríguez, Sengupta, and Bhattacharjee [2022] E. Rodríguez, W. Sengupta, and A. Bhattacharjee, “Weakly quasisymmetric near-axis solutions to all orders,” Physics of Plasmas 29, 012507 (2022).
* Rodriguez [2022] E. Rodriguez, _Quasisymmetry_ , Ph.D. thesis (2022).
* Landreman and Sengupta [2018] M. Landreman and W. Sengupta, “Direct construction of optimized stellarator shapes. part 1. theory in cylindrical coordinates,” Journal of Plasma Physics 84, 905840616 (2018).
* Plunk, Landreman, and Helander [2019] G. G. Plunk, M. Landreman, and P. Helander, “Direct construction of optimized stellarator shapes. part 3. omnigenity near the magnetic axis,” Journal of Plasma Physics 85, 905850602 (2019).
* Rodriguez and Plunk [2023] E. Rodriguez and G. G. Plunk, “Higher order theory of quasi-isodynamicity near the magnetic axis of stellarators,” arXiv preprint arXiv:2303.06038 (2023).
* Oberti and Ricca [2016] C. Oberti and R. L. Ricca, “On torus knots and unknots,” Journal of Knot Theory and Its Ramifications 25, 1650036 (2016).
* Aicardi [2000] F. Aicardi, “Self-linking of spatial curves without inflections and its applications,” Funct Anal Its Appl 34, 79–85 (2000).
* Fuller [1999] J. Fuller, Edgar J., _The geometric and topological structure of holonomic knots_ (1999) p. 112.
* Moffatt and Ricca [1992] H. K. Moffatt and R. L. Ricca, “Helicity and the cälugäreanu invariant,” Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439, 411–429 (1992).
* Fenchel [1951] W. Fenchel, “On the differential geometry of closed space curves,” Bulletin of the American Mathematical Society 57, 44–54 (1951).
* Paz-Soldan [2020] C. Paz-Soldan, “Non-planar coil winding angle optimization for compatibility with non-insulated high-temperature superconducting magnets,” Journal of Plasma Physics 86, 815860501 (2020).
* Lonigro and Zhu [2022] N. Lonigro and C. Zhu, “Stellarator coil design using cubic splines for improved access on the outboard side,” Nuclear Fusion 62, 066009 (2022).
* Helander and Sigmar [2005] P. Helander and D. Sigmar, _Collisional Transport in Magnetized Plasmas_ , Cambridge Monographs on Plasma Physics (Cambridge University Press, 2005).
* Ware _et al._ [2006] A. S. Ware, D. A. Spong, L. A. Berry, S. P. Hirshman, and J. F. Lyon, “Bootstrap current in quasi-symmetric stellarators,” Fusion Science and Technology 50, 236–244 (2006).
* Rodriguez [2023a] E. Rodriguez, “Mhd stability and the effects of shaping: a near-axis view for tokamaks and quasisymmetric stellarators,” arXiv preprint arXiv:2302.03359 (2023a).
* Landreman and Paul [2022] M. Landreman and E. Paul, “Magnetic fields with precise quasisymmetry for plasma confinement,” Physical Review Letters 128, 035001 (2022).
* Note [4] From the Mercier perspective on rotational transform[5], $\bar{\iota}_{0}=-\frac{1}{2\pi}\intop\ilimits@_{0}^{L}\frac{\mathop{cosh}\nolimits\bar{\eta}-1}{\mathop{cosh}\nolimits\bar{\eta}}(d^{\prime}+\tau)\mathrm{d}l+\frac{1}{2\pi}\intop\ilimits@_{0}^{L}\tau\mathrm{d}l,$ in the large ellipticity $\bar{\eta}\rightarrow\infty$ limit, $\bar{\iota}_{0}=-d/2\pi$, where $d$ is the angle of rotation of the ellipse with respect to the curvature vector. Because $\sigma\sim\eta^{2}\rightarrow 0$ in this limit, the cross-sections align with the Frenet-Serret frame. Thus, the net rotation $d=0$.
* Drevlak _et al._ [2013] M. Drevlak, F. Brochard, P. Helander, J. Kisslinger, M. Mikhailov, C. Nührenberg, J. Nührenberg, and Y. Turkin, “ESTELL: A quasi-toroidally symmetric stellarator,” Contributions to Plasma Physics 53, 459–468 (2013).
* Garabedian [2008] P. R. Garabedian, “Three-dimensional analysis of tokamaks and stellarators,” Proceedings of the National Academy of Sciences 105, 13716–13719 (2008).
* Garabedian and McFadden [2009] P. R. Garabedian and G. B. McFadden, “Design of the DEMO fusion reactor following ITER,” Journal of research of the National Institute of Standards and Technology 114, 229 (2009).
* Hirshman and Whitson [1983] S. P. Hirshman and J. C. Whitson, “Steepest‐descent moment method for three‐dimensional magnetohydrodynamic equilibria,” The Physics of Fluids 26, 3553–3568 (1983).
* Landreman [2021] M. Landreman, “Figures of merit for stellarators near the magnetic axis,” Journal of Plasma Physics 87, 905870112 (2021).
* Landreman [2022] M. Landreman, “Mapping the space of quasisymmetric stellarators using optimized near-axis expansion,” Journal of Plasma Physics 88, 905880616 (2022).
* Bender and Orszag [1999] C. M. Bender and S. A. Orszag, _Advanced mathematical methods for scientists and engineers: I: Asymptotic methods and perturbation theory_ (Springer, 1999).
* Landreman and Jorge [2020] M. Landreman and R. Jorge, “Magnetic well and mercier stability of stellarators near the magnetic axis,” Journal of Plasma Physics 86, 905860510 (2020).
* Rodriguez [2023b] E. Rodriguez, “Mhd stability and the effects of shaping: a near-axis view for tokamaks and quasisymmetric stellarators,” arXiv preprint arXiv:2302.03359 (2023b).
* Wright, Nocedal _et al._ [1999] S. Wright, J. Nocedal, _et al._ , “Numerical optimization,” Springer Science 35, 7 (1999).
* Note [5] See https://github.com/landreman/qsc. The script used to obtain the main plot in Fig. 5 can be found in the Zenodo repository associated to this paper. The same may be achieved, albeit slower, using pyQSC, which was how it was originally done and is also included there.
* Note [6] An 11th Gen i7-11850H core was used for this purpose. The main space in Fig. 5 (which is 300x300) took a total of about 14 hrs. Most time is devoted to the optimisation sub-problems at each point (search for $\eta^{*}$, $B_{22}^{C}$, and $\\{Z_{n}\\}$). Of course, the construction of such a space is trivially parallelisable.
* Plunk and Helander [2018] G. Plunk and P. Helander, “Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum,” Journal of Plasma Physics 84, 905840205 (2018).
* Nemov _et al._ [1999] V. V. Nemov, S. V. Kasilov, W. Kernbichler, and M. F. Heyn, “Evaluation of $1/\nu$ neoclassical transport in stellarators,” Physics of Plasmas 6, 4622–4632 (1999).
* Nemov, Kasilov, and Kernbichler [2014] V. V. Nemov, S. V. Kasilov, and W. Kernbichler, “Collisionless high energy particle losses in optimized stellarators calculated in real-space coordinates,” Physics of Plasmas 21, 062501 (2014).
* Wesson [2011] J. Wesson, _Tokamaks; 4th ed._ , International series of monographs on physics (Oxford Univ. Press, Oxford, 2011).
* Jorge _et al._ [2022] R. Jorge, G. Plunk, M. Drevlak, M. Landreman, J.-F. Lobsien, K. Camacho Mata, and P. Helander, “A single-field-period quasi-isodynamic stellarator,” Journal of Plasma Physics 88, 175880504 (2022).
* Griffiths [2004] D. J. Griffiths, _Introduction to Quantum Mechanics (2nd Edition)_ , 2nd ed. (Pearson Prentice Hall, 2004).
|
# Extreme Variability Quasars in Their Various States. II: Spectral Variation
Revealed with Multi-epoch Spectra
Wenke Ren CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy, University of Science and Technology of China, Hefei,
Anhui 230026, China School of Astronomy and Space Science, University of
Science and Technology of China, Hefei 230026, China Junxian Wang CAS Key
Laboratory for Research in Galaxies and Cosmology, Department of Astronomy,
University of Science and Technology of China, Hefei, Anhui 230026, China
School of Astronomy and Space Science, University of Science and Technology of
China, Hefei 230026, China Zhenyi Cai CAS Key Laboratory for Research in
Galaxies and Cosmology, Department of Astronomy, University of Science and
Technology of China, Hefei, Anhui 230026, China School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China
Xufan Hu CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy, University of Science and Technology of China, Hefei,
Anhui 230026, China School of Astronomy and Space Science, University of
Science and Technology of China, Hefei 230026, China
(Received September 10, 2023; Revised November 26, 2023; Accepted December 14,
2023)
###### Abstract
We previously built a sample of 14,012 extremely variable quasars (EVQs) based
on SDSS and Pan-STARRS1 photometric observations. In this work we present the
spectral fitting to their SDSS spectra, and study the spectral variation in
1,259 EVQs with multi-epoch SDSS spectra (after prudently excluding spectra
with potentially unreliable spectroscopic photometry). We find clear “bluer-
when-brighter” trend in EVQs, consistent with previous findings of normal
quasars and AGNs. We detect significant intrinsic Baldwin effect (iBeff, i.e.,
smaller line EW at higher continuum flux in individual AGNs) in the broad Mg
II and C IV lines of EVQs. Meanwhile, no systematical iBeff is found for the
broad H$\beta$ line, which could be attributed to strong host contamination at
longer wavelengths. Remarkably, by comparing the iBeff slope of EVQs with
archived changing-look quasars (CLQs), we show that the CLQs identified in
literature are mostly likely a biased (due to its definition) sub-population
of EVQs, rather than a distinct population of quasars. We also found no
significant broad line breathing of either H$\beta$, Mg II or C IV, suggesting
the broad line breathing in quasars may disappear at longer timescales ($\sim$
3000 days).
Quasars (1319), Time domain astronomy (2109), Active galactic nuclei (16),
Black hole physics (159)
††journal: ApJ††facilities: Sloan, PS1, PO:1.2m††software: PyQSOFit (Guo et
al., 2018), astropy (Astropy Collaboration et al., 2013; Price-Whelan et al.,
2018; Astropy Collaboration et al., 2022), pandas (pandas development team,
2020; Wes McKinney, 2010), scipy (Virtanen et al., 2020), numpy (Harris et
al., 2020),
## 1 Introduction
One of the hallmarks of quasars is their universal variability across all
wavelengths. Quasar light curves typically exhibit variations with an
amplitude of $\sim$0.2 mag in the rest-frame UV/optical band, occurring on
timescales of months to years (e.g., Vanden Berk et al., 2004; Sesar et al.,
2007; MacLeod et al., 2010). These variations are often described by a damped
random walk (DRW) model first proposed by Kelly et al. (2009). Reverberation
mapping (RM) studies have revealed strong correlations between the variation
of the continuum and the broad emission lines (BELs) (e.g., Grier et al.,
2017, 2019; Homayouni et al., 2020), suggesting that the optical-UV
variability of quasars is driven by the innermost region of the accretion disk
(e.g., Ross et al., 2018). However, due to the stochastic and complex nature
of quasar variability (e.g. Cai et al., 2018; Sun et al., 2020), the physical
processes underlying these variations are still a topic of debate.
In the cutting-edge field of quasar variability research, the phenomenon of
changing-look (CL) is particularly intriguing. It has long been noted that a
few AGNs exhibit drastic changes in their broad emission lines, with their
spectral types transitioning between Type 1 and Type 2 (e.g., Cohen et al.,
1986; Goodrich, 1995). Initially, due to limited data sizes and time spans,
the discovered CLQs are very rare and usually transient. The observed changes
in broad lines were often attributed to dust extinction (Risaliti et al.,
2007), tidal disruption events, or supernovae (Aretxaga et al., 1999).
However, with the accumulation of data, an increasing number of CLQs
($\sim$100) have been identified (e.g., Ruan et al., 2016; MacLeod et al.,
2019; Green et al., 2022). Most CL phenomenon are accompanied by extreme
continuum variations, and their new spectral states can keep for months to
years. These observational results indicate that the CL phenomenon should be
driven by intrinsic but violent, accretion dominated activity. What are the
underlying mechanisms for such violent process? Are they due to possible
accretion transition in analogy to X-ray binaries (e.g. Ruan et al., 2019;
Yang et al., 2023), or just more violent version of normal variability? Much
larger samples of sources with violent variability are desired to distinguish
such possibilities.
Since the identification of CLQs relies on repeated spectral observations
showing emerging/disappearing of broad emission lines, which are relatively
rare, some related studies initially focus on quasars with extremely strong
continuum variability (MacLeod et al., 2016; Rumbaugh et al., 2018; Sheng et
al., 2020). Compared with CLQs, much larger samples of such quasars could be
easily built solely based on broadband photometric variability, and such
samples could provide essential clues to the physical nature of violent
variability and its relation with changing-look and normal (less extreme)
variability.
Through systematic research on extremely variable quasars (EVQs), Rumbaugh et
al. (2018) found that the equivalent width (EW) of the broad emission lines of
EVQs are larger than that of normal quasars. Note Kang et al. (2021) further
discovered a general correlation between the variability and emission line EW
in large sample of quasars. It is clear that quasar variability has an
underlying influence on the broad-line regions (BLRs) and thus is worth
investing more efforts. Furthermore, studies with small samples of AGNs have
illustrated that emission lines change along with continuum variations, i.e.,
the EW of broad lines decreases as the continuum increases (intrinsic Baldwin
effect, iBeff, e.g. Goad et al., 2004; Rakić et al., 2017); the full width at
half maximum (FWHM) of different lines also exhibits varying correlations with
continuum luminosity (e.g. Wang et al., 2020). However, restricted by the
signal-to-noise ratio (SNR) of available survey spectra, such changes are
often not eminent enough to distinguish from noise when extending to quasars,
as most of which exhibit low intrinsic variability. In this sense, studying
the emission line variability in large samples of EVQs could also shed new
light on the BLR physics.
In our previous work Ren et al. (2022) (hereafter, Paper I, ), we constructed
a sample of EVQs consisting of 14,012 sources using photometric data from the
Sloan Digital Sky Survey (SDSS) and Pan-STARRS (PS1). We obtained the
composite spectra of the EVQs in various states (according to the relative
brightness of each EVQ during SDSS spectroscopic observation compared with the
mean brightness from multi epoch photometric observations) and of control
samples with matched redshift, luminosity and black hole mass. We verified
that EVQs tend to have stronger emission lines in H$\beta$, [O III], Mg II,
and C IV. The EW changes between different states shows a clear iBeff in Mg II
and C IV, but not in H$\beta$. Furthermore, we found that EVQs have stronger
broad line wings compared to normal quasars, suggesting that EVQs could launch
more gas from the inner disk into the very broad line region (VBLR).
In this work, we focus on the EVQs with multi-epoch SDSS spectra to explore
the spectral variations in individual EVQs. Based on the EVQ sample we
constructed in Paper I, we select a sub-sample with repeated spectral
observations in SDSS. The structure of this paper is as follows. In §2, we
describe the sample selection and methods of spectral measurement. The results
are presented in §3. We then discuss our results in §4 and provide a summary
in §5. Throughout this paper, we adopt a flat $\rm{\Lambda}$CDM cosmology with
$\Omega_{\Lambda}=0.7$, $\Omega_{m}=0.3$, and
$H_{0}=70~{}\rm{km~{}s^{-1}~{}Mpc^{-1}}$.
## 2 The Data and Sample
### 2.1 The EVQ Sample
We start from the EVQ catalog constructed by Paper I, which is selected from
the SDSS data release 14 (DR14) quasar catalog (DR14Q, Pâris et al., 2018).
Here we provide a brief overview of the selection criteria and spectra fitting
procedure. Readers are advised to refer to Paper I for detailed description.
Based on all photometric observations ($g$ and $r$) from SDSS and PS1 database
for each quasar in DR14Q, we amend the minor differences in filter
transmission between two surveys, reject epochs with potentially unreliable
photometric measurements, and build clean light curves for every source.
Sources with robust detection of extreme variability in both $g$ and $r$ light
curves (with $\Delta g_{max}>1$ and $\Delta r_{max}>0.8$) are selected as
EVQs. In total, we select 14,012 EVQs with 20,069 archived SDSS spectra. Each
spectrum is processed using PyQSOFit (Guo et al., 2018) following the
procedure described in Paper I. The catalog of the EVQs and the spectral
fitting results are now presented in Table 1 of Appendix A.
Among the 14,012 EVQs, 1354 sources have repeated spectral observations in
SDSS archive with a continuum SNR $>3$. In Appendix §B, we develop a method to
identify spectra with potentially unreliable spectroscopic photometry (such as
due to fiber-drop) by comparing the observed spectroscopic photometry with the
expected value derived through modeling the photometric light curves with a
damped random walk (DRW) process. After excluding such possibly dubious
spectra, 1259 multi-spectra EVQs are remained for further analyses.
### 2.2 Spectral measurements
The spectra are fitted following Shen et al. (2011) using PyQSOFit (Guo et
al., 2018). For each spectrum, we fit with a power-law
($f_{\lambda}\propto\lambda^{\alpha}$) and a broadened Fe II template for
continuum within tens of separated emission line-free windows and give the
measurement of continuum and Fe II properties. Then, we cut the spectrum
around three luminous broad emission lines: H$\beta$, Mg II, and C IV if they
are within the spectral coverage and then fit the lines locally. The continuum
is re-fitted using the windows on both sides of the emission lines. After
detracting the continuum, we use a few groups of Gaussian to fit the
residuals: 4 Gaussians (2 cores + 2 wings) for [O III] doublet, 1 for narrow
H$\beta$ and three for broad H$\beta$ in H$\beta$ region; 1 Gaussian for
narrow and 3 for broad Mg II region; and 3 Gaussians for only broad C IV
component in C IV region. The detailed procedure and parameter settings can be
found in Paper I.
For each emission line component, we calculate the fitted peak wavelength
(PEAK), integrated flux (FLUX), equivalent width (EW), line dispersion (second
moment, SIGMA), and full width at half maximum (FWHM). Since the broad
component of H$\beta$, Mg II, and C IV (consisting of triple Gaussians) may
have asymmetric shapes indicating complex structures of BLRs, we further
measure the following properties. The bisectional center (BISECT, Sun et al.,
2018), where the wavelength splits the line into two equal areas in flux, has
been used in Paper I to qualitatively reflect the wing structures in a broad
line. To further parameterize the shape and asymmetry of the broad component,
we adopt the: i) full width at half maximum (FWHM), ii) full width at quarter
maximum (FWQM), and iii) full width at 10% maximum (FW10M) to denote the BLRs
from different distance to the SMBH. We also measure the corresponding shifts
of the centers of the above three widths, Z50, Z25, and Z10 to show the
intrinsic gravitational redshift or inflow/outflow of BLRs (cf. Fig. 6 in
Rakić 2022).
We employ a Monte Carlo approach to assess the statistical uncertainties of
the parameters we measured. This is done through generating 50 mock spectra by
adding Gaussian noise to each original spectrum using the reported flux
density errors, fitting the mock spectra with the same routines, and deriving
the scatter of each parameter.
We adopt the bolometric corrections (BCs) and the calibration parameters in
Shen et al. (2011) to estimate the bolometric luminosity $L_{bol}$ and the BH
mass $M_{BH}$ of a single-epoch spectrum. We take fiducial $L_{bol}$ and
$M_{BH}$ calculated from $L_{5100}$ and H$\beta$ for sources at $0.08\leq
z<0.7$, from $L_{3000}$ and Mg II at $0.7\leq z<1.9$, and from $L_{3000}$ and
Mg II at $1.9\leq z<4.0$ respectively. Our following results will be based on
such three sub-samples. The Eddington ratio is also given in our catalog. We
adopt the mean value from multiple-epoch measurements for each target and plot
the general physical property distribution of multi-spectra EVQs in Fig. 1.
Figure 1: The redshift, $L_{bol}$, $M_{BH}$, and $\eta_{Edd}$ distribution of
multi-spectra EVQs. The mean values from multiple spectra are adopted for each
EVQ. We split the sample into three groups according to redshift. The source
number of each group is noted in the parenthesis in the legend. As a
comparison, we also plot the re-normalized distribution of SDSS DR14Q (Rakić,
2022), divided by the same redshift criterion and drawn with corresponding
colored shades.
## 3 Results
Figure 2: The correlation between the variability of spectral properties and
that of continuum luminosity. The variation of continuum power-law slope
($\alpha$), broad line EW and FWHM are shown in rows sequentially. The
variability for each parameter is derived through subtracting the mean value
of multi-epoch spectral measurements for the same EVQ from a single-epoch
measurement. The Spearman correlation coefficients and the p-values (derived
through bootstrapping the sample, hereafter the same, see the 2nd parapha of
§3 for details) are shown in the upper right of each plot. For correlations
with p-value$<$0.01, we use a black solid line to show the orthogonal distance
regression result and present the derived regression slope $k$ with 1$\sigma$
uncertainties derived through bootstrapping (hereafter the same).
In this section, we explore how the spectral properties of EVQs vary along
with the continuum flux, in three distinct samples split by redshift to
guarantee the emission line measurements. In total, we have 329 sources with
920 spectra for the H$\beta$ sample, 676 sources with 2206 spectra for the Mg
II sample, and 254 sources with 799 spectra for the C IV sample. About $2/3$
of the objects have only two spectra, and there are 15 sources with more than
20 observations which are all from the SDSS-RM program (Shen et al., 2014).
To present sources with different luminosities together and explore their
average variability, our results are primarily expressed in terms of
variation. To fully leverage each observation, we derive the deviation of each
single-epoch measurement from the mean value for each EVQ. In this case, one
single-epoch spectrum will contribute one data point to our following plots.
We note that the various data points from the same EVQ are thus not completely
independent to each other (since calculating the mean value consumes one
degree of freedom for each EVQ) and the traditional p-value of the Spearman
correlation evaluated from Student’s t-test is no longer valid. Therefore, we
estimate the p-value by bootstrapping the quasar sample instead. Note since we
run bootstrap 100,000 times, thus we are unable to give exact p-value smaller
than 1.e-5.
### 3.1 Color variation
The so-called “bluer-when-brighter” (BWB) pattern has been widely seen in
quasars and AGNs (e.g. Trevese & Vagnetti, 2002; Sun et al., 2014; Guo & Gu,
2016). In Paper I, the composite spectra of EVQs indirectly support such a
pattern. We quantitatively test this conclusion with the multi-spectra EVQs by
exploring the spectral slope variability in individual sources.
The first row in Fig. 2 shows the changes of spectral slope versus the changes
of log-luminosity. The anti-correlation between $\Delta\alpha$ and
$\Delta\log{L}$ is found in all three samples, denoting a consistent BWB
pattern up to $z\sim 4$. We use the orthogonal distance regression (Boggs &
Rogers, 1989) method provided by scipy.odr package to fit the data with the
data errors considered (hereafter the same, unless otherwise stated). The
regression slope $k$ for the three samples are $-1.6\pm 0.3$, $-0.90\pm 0.12$,
and $-1.4\pm 0.3$ ((with the 1$\sigma$ statistical uncertainties derived from
bootstrapping the samples, hereafter the same), respectively, roughly
consistent with the results from PG QSOs (e.g. Pu et al., 2006).
The slightly steeper slope from the H$\beta$ sample could be due to the
contamination of the host galaxy as we do not decompose the host galaxy
component while fitting the continuum. As a result, for those low-z sources
for which the host contamination could be more prominent, the
$\Delta\log{L_{5100}}$ will be underestimated, and the $\alpha$ at dim states
will be overestimated. Besides, the relative flux of the host galaxy component
may also vary depending on the observation conditions (Zhang, 2013; Hu et al.,
2015). Nevertheless, given their high luminosity as quasars (see Fig. 1) and
the strong correlation at higher redshift, the host alone should not be fully
responsible for the observed spectral changes. Indeed, it has been well
demonstrated that the host contamination can not dominate the observed color
variation in SDSS quasars, as host contamination can not reproduce the
observed timescale-dependence of the color variation (Sun et al., 2014; Cai et
al., 2016; Zhu et al., 2016).
### 3.2 Variation of line EW and FWHM
In the second row of Fig. 2, we explore the iBeff of the three most luminous
broad lines. Strong anti-correlations between the variation of line EW and
that of the continuum luminosity are found in Mg II and C IV while only
marginal anti-correlation in H$\beta$. The averaged iBeff slope for Mg II and
C IV are comparable but slightly steeper than the results revealed with the
stacked spectra in Paper I and those from literature (Pogge & Peterson, 1992;
Kong et al., 2006; Homan et al., 2020; Zajaček et al., 2020). The steeper
iBeff of Mg II and C IV could likely be attributed to the large variability of
our sample, as Homan et al. (2020) also found a steeper slope in their more
variable sample.
We note that the missing anti-correlation between $EW_{H\beta}$ and $L_{5100}$
does not means the absence of iBeff in H$\beta$, as the host component has not
been properly considered in our spectral fitting. The blend of the host galaxy
in the spectra causes significant underestimation of $EW_{H\beta}$,
particularly in dim states, thereby weakening the underlying anti-
correlations. Subtracting the host component from such low SNR individual
spectra, which is beyond the scope of this work, would be highly unreliable
and would introduce great uncertainties to spectral measurements (e.g. Wu &
Shen, 2022).
Figure 3: The broad H$\beta$ iBeff is visible in a sub-sample of more luminous
EVQs, suggesting the absence of broad H$\beta$ iBeff shown in the lower left
panel of Fig. 2 is due to host contamination. The labels are similar to those
in Fig. 2.
To alleviate the host contamination, we build a luminous sub-sample from our
EVQs which contains about 1/3 of the total sample with
$\log\overline{L}_{5100}>44.5$. The broad H$\beta$ iBeff of this sub-sample is
shown in Fig. 3. A robust negative correlation is observed in this sub-sample,
indicating that the disappearance of the iBeff of H$\beta$ line in the full
sample is likely due to the host contamination. We would further discuss the
effects of host contamination in §4.1, where the connection between the EVQs
and CLQs are investigated.
We also explore the so-called ”breathing” of the broad lines, e.g., how the
line width changes with the luminosity (see the third row of Fig. 2). Only
weak and marginal anti-breathing (positive correlation) is found in the C IV
sample, and no clear sign of breathing is observed in the Mg II sample. These
results are generally consistent to those reported in Wang et al. (2020).
However, we find that the H$\beta$ sample also shows no breathing according to
Spearman test, contradicting previous findings (e.g., Denney et al., 2008;
Park et al., 2011; Wang et al., 2020). This paradox will be discussed in §4.2.
### 3.3 Line profile and asymmetry
Figure 4: The variation of broad line profile and asymmetry (see text for
description) versus that of continuum luminosity. The labels are similar to
those in Fig 2
In Paper I, we found that the EVQs have slightly stronger broad wings compared
with control samples. Here we investigate whether the broad line profiles vary
with continuum flux in EVQs with multi-epoch spectra. We use the line width
ratio (FWQM/FWHM and FW10M/FWHM) to denote the line profile and the shift of
line center (Z50, Z25, and Z10) to quantify the line asymmetry. The changes of
above parameters with the continuum luminosity are shown in Fig. 4.
According to the Spearman correlation coefficient and its p-value, we do not
find any strong dependency between line profiles and continuum flux. This
suggests the BLR structure does not systematically change with continuum
variation. Such a result also indicates that the EVQs are not an unusual
population with exceptionally large variability.
## 4 Discussion
### 4.1 EVQs and CLQs
In §3.2 we found the H$\beta$ EW of EVQs does not vary with continuum flux
(but with considerable scatter), i.e., showing no clear iBeff on average. In
such cases, there will naturally be some EVQs that have lower H$\beta$ EW in
their noisy dim-state spectra which may yield the so-called “vanishment” of
broad Balmer lines (i.e., CLQs). In this section, through this entry point, we
will delve deeper into whether CLQ is a distinct population or simply a biased
sub-population of EVQ.
#### 4.1.1 Archived CLQs
The study of CLQs has been heating up in recent years, decades after the
discovery of Changing-look AGNs (e.g., Tholine & Osterbrock, 1976; Cohen et
al., 1986). So far, nearly a hundred of CLQs have been claimed, but a unique
classification is still lacking. The term “Changing-look” is an iconic
description (highly qualitative but not quantitative) firstly used to denote
the appearance or disappearance of broad emission lines. Since the CL
phenomenon is relatively rare at the early stage, it was feasible to use AGN
types or merely by visual checks to identify CL-AGNs. Such subjective
identification method is inherited to the research of CLQs.
Since the CL phenomenon is usually accompanied by dramatic continuum changes,
one developed approach to efficiently search for CLQs is to find candidates
from quasars with strong continuum variation, and then pick CLQs through
visual check (e.g. MacLeod et al., 2016). Obviously, the process of visual
check can be easily influenced by individual subjective judgements.
With the accumulation of multi-epoch spectra, researchers have explored
another method for hunting CLQs. For the spectra of CLQs in dim states usually
have no broad lines, they were often classified as galaxies by the initial
pipelines. Therefore, one can search for CLQs by directly examining sources
with repeated spectra but with different classifications (e.g. Ruan et al.,
2016; Yang et al., 2018). Such semi-quantitatively search for CLQs is highly
dependent to the classification pipeline, and in many cases still dependent on
visual inspection (e.g., 19 confirmed CLQs out of 10,204 candidates by visual
check in Yang et al., 2018).
In order to further standardize the selection of CLQs, a few recent works
start to focus on the change of emission line flux itself. MacLeod et al.
(2019) proposed that a quasar could be deemed as a CLQ if the SNR (signal-to-
noise ratio) of the broad line flux change between two epochs is greater than
3. Meanwhile Yang et al. (2018) and Sheng et al. (2020) directly use the SNR
of broad line to decide whether a broad component is apparent or not. The most
recent work by Green et al. (2022) combined the quantitative criteria and
visual inspection. These efforts indeed make the search for CLQs more
objective and consistent. However, the criterion above highly rely on the SNR
of the spectra, and they are unstable when applying to spectra observed in
different conditions and could be sensitive to the quasar luminosity. For
example, as MacLeod et al. (2019) have noted that their methods may select
quasars with minor variation in H$\beta$ line flux when their spectra have
very high SNR. Furthermore, applying quantitative criteria on the broad line
flux is not straightforward, as the broad line flux is expected to vary with
continuum in all quasars and AGNs (the cornerstone of reverberation mapping
studies).
To explore whether there are intrinsic differences between currently archived
CLQs and EVQs, we collected all archived H$\beta$ CLQs and refitted their
spectra using the same method we applied to our EVQs. Note in literature there
are only 3 MgII CLQs and 3 C IV CLQs claimed (Guo et al., 2019, 2020; Ross et
al., 2020, see also §4.1.3), thus here we focus on Hb select CLQs only. We
gathered a total of 79 securely-identified CLQs from LaMassa et al. (2015);
Ruan et al. (2016); MacLeod et al. (2016); Yang et al. (2018); MacLeod et al.
(2019); Sheng et al. (2020); Green et al. (2022); López-Navas et al. (2022).
To ensure the consistency of analyses, we only retained CLQs with repeated
SDSS spectra in both dim and bright states. As a result, 36 sources are
preserved, all with $z<0.65$. Among those CLQs, 14 are included in our EVQ
sample. The other 22 CLQs do not satisfy our strict selection criteria of EVQs
(i.e., $\Delta g_{max}$ $>$ 1 mag and $\Delta r_{max}$ $>$ 0.8 mag, in SDSS &
PS1 photometric light curves well cleaned against potential photometric
defects, see Ren et al. 2022).
Indeed, our independent spectral fitting reveal broad H$\beta$ lines in the
dim states of 33 CLQs, indicating in most archived CLQs the broad H$\beta$
indeed does not completely disappear in the dim state. In the remaining three
cases, two of them have their H$\beta$ lines in the dim states completely
submerged by the host galaxy, leaving an obvious absorption feature. The dim-
state spectrum of the third CLQ is flawed without enough wavelength coverage
around H$\beta$. These 3 dim-state spectra are therefore excluded from further
analysis below.
#### 4.1.2 Comparing the iBeff of EVQs with CLQs
Figure 5: The orthogonal distance regression correlation between the changes
of continuum luminosity and the changes of line EW. The variation of continuum
luminosity and line EW are calculated by subtracting the mean value of each
object from the measurement of each epoch, which is identical to that in Fig.
2. Upper: the iBeff of H$\beta$ of our EVQs and CLQs collected from
literature. Lower: the Mg II iBeff for the sources plotted in the upper panel
but with Mg II line measurements. The fitting results and the Spearman
correlation coefficient $r$ (with p-values) of each sample is listed. For
correlations with p-value$<$0.01, we use a black solid line to show the
orthogonal distance regression fitting result. Figure 6: The distribution of
H$\beta$ EW and iBeff slope versus the mean continuum luminosity. The iBeff
slope is given by the $\Delta EW_{Hb}$/$\Delta L_{5100}$ between the brightest
and dimmest states. The histogram distributions of the parameters are shown
aside the scatter plot. The median of each distribution is shown with
vertical/horizon lines in accordance style. From the EVQ sample, the luminous
sub-sample with $\log\overline{L}_{5100}>44.5$ are plotted with gray lines in
the histograms. Figure 7: Upper: the correlation between the total luminosity
at 5100$\AA$ and the host galaxy fraction from the RM spectra of Shen et al.
(2014). The dashed line represents a simple linear regression (adopting
$L_{5100}$ as the independent variable), which is then utilized to estimate
the host fraction at given input $L_{5100}$ for our EVQs/CLQs plotted in the
lower panel. Lower: the iBeff of H$\beta$ of the EVQs and CLQs with the host
contamination corrected. The orthogonal distance regression fitting results
and the Spearman correlation coefficient $r$ (with p-values) of each sample
are listed.
In Fig. 5, we plot the variation of line EW along with the continuum of our
H$\beta$ EVQ sample, in comparison with the 33 archived H$\beta$-selected
CLQs. As already shown in Fig. 2, there is no clear correlation between
$\Delta EW_{Hb}$ and $\Delta L_{5100}$ in the EVQ sample, i.e., no iBeff.
Meanwhile a prominent positive correlation is observed in the CLQ sample
(i.e., a reverse iBeff). This is actually expected as the definition of
changing-look automatically requires much weaker H$\beta$ broad line in dim-
state CLQs. However, when it comes to the Mg II line for the same
H$\beta$-selected CLQs (i.e., lower panel of Fig. 5), CLQs also exhibit iBeff
(suggesting CLQs are physically similar to normal quasars), but with a slope
marginally flatter than EVQs.
We further examine the mean broad H$\beta$ EW versus the mean 5100Å
luminosity, and H$\beta$ iBeff slope versus the mean 5100Å luminosity, for the
EVQs and H$\beta$-selected CLQs (Fig. 6). The iBeff slope is calculated using
the brightest and dimmest epochs of each source. Similar to the results in
Fig. 5, the majority of CLQs have a reverse iBeff slope, with a median of
$0.60\pm 0.18$ 111We note that the only study in literature discussing the
change in EW of CLQs is presented in Green et al. (2022), which reports a
notably higher average iBeff slope of 2.05 compared to ours. However, their
result could be severely influenced by an individual target
J091234.00+262828.32, which has an iBeff slope of $\sim$107 due to its minor
change in $L_{5100}$, primarily affected by the host galaxy. The average slope
of the rest CLQs of Green et al. (2022) is comparable with our result. , while
the median for EVQs is $-0.11\pm 0.05$. The overall luminosity and mean
H$\beta$ EW of CLQs are lower than those of EVQs. However we do not see clear
correlation between mean broad H$\beta$ EW and 5100Å luminosity, or between
iBeff slope and 5100Å luminosity, for our uniformly selected EVQ sample. The
smaller mean H$\beta$ EW and 5100Å luminosity of CLQs (compared with EVQs) are
likely caused by observational biases, that EVQs with weaker broad H$\beta$
line and lower 5100Å luminosity are more likely be classified as CLQs (due to
limited spectral SNR).
Could the different iBeff slope of CLQs (compared with EVQs, as shown in Fig.
5 and 6) be due to potentially stronger host contamination in CLQs? Given the
overall low SNR of the SDSS spectra, directly decomposing the host component
in individual sources through spectral fitting is yet unreliable.
Nevertheless, by referencing to the results from Shen et al. (2014), we can
estimate the host fraction as a simple function of the luminosity at 5100Å and
give a rough correction for host contamination. In the upper panel of Fig. 7,
we present the correlation between the total luminosity (AGN plus host) at
5100Å, $L_{5100,tot}$, and the host fraction at 5100Å, $f_{host,5100}$,
decomposed by Shen et al. (2014) utilizing their reverberation mapping data.
Through simple linear regression we yield the expected average $f_{host,5100}$
as a function of $L_{5100,tot}$. For each EVQ/CLQ, we conservatively estimate
its host contribution using the $L_{5100,tot}$ from its dimmest epoch and
subtract the host contribution from the spectra of all epochs. The corrected
result is shown in the lower panel of Fig. 7, in which we could see clear
H$\beta$ iBeff in our EVQs, with a slope similar to that from the luminous
sub-sample (Fig. 3) and closer to the results in literature (Goad et al.,
2004; Rakić et al., 2017). This further suggests the absence of H$\beta$ iBeff
in EVQs shown in Fig. 2 is due to host contamination. However, after the
correction, the average iBeff slope of CLQs remains statistically different
from that of EVQs, which could be an expected consequence of the definition of
“changing-look”.
#### 4.1.3 Are CLQs a distinct population or just a biased sub-population of
EVQs?
As we have shown above, the initial definition of “changing-look” requires the
disappearance of broad H$\beta$ line in the spectrum of dim state, but note
the definition is qualitative but not quantitative. In reality, the broad
H$\beta$ lines of CLQs selected in literature do weaken in the dim state, but
not “completely disappear” in most of them.
Meanwhile, the broad H$\beta$ line EW of our EVQs does not systematically vary
with continuum, i.e., on average with no iBeff, but with large scatter in the
iBeff slope in individual sources. Thus individual EVQs with reverse iBeff
slope (smaller broad H$\beta$ line EW in dimmer state) could more likely be
identified as “changing-look” due to the limited SNR of the spectra of dim
state. Meanwhile the CLQs selected in literature do exhibit reverse H$\beta$
iBeff as expected.
In this sense, CLQs are not necessarily a distinct population, but a sub-
population (biased by its definition) of EVQs. In this scheme, the fundamental
driven mechanism for both CLQs and EVQs is the continuum variation.
Previous researches implied that the lower Eddington ratio should be
responsible for the dramatic change of the BELs in CLQs (e.g. MacLeod et al.,
2016). Similarly, as shown in Fig. 6, we show CLQs have systematically lower
luminosity and lower H$\beta$ EW, compared with our EVQs. However, all these
facts could be attributed to selection biases: 1) the stronger host
contamination in sources with lower luminosity/Eddington ratio could hinder
the detection of broad H$\beta$ in the noisy dim state spectra; 2)
intrinsically weaker H$\beta$ lines are more difficult to be detected in the
noisy dim state spectra.
We note in literature while most CLQs were selected based on H$\beta$ line, Mg
II or C IV CLQs were rarely found (Guo et al., 2020; Ross et al., 2020). This
could also be naturally be attributed to the different iBeff slopes of
H$\beta$, Mg II and C IV we shown in Fig. 2. Since both Mg II and C IV of EVQs
exhibit strong iBeff, sources in the dim state tend to have larger EW of Mg II
and C IV, thus they are less likely be classified as changing-look. Meanwhile,
as EVQs exhibit no apparent iBeff on average in H$\beta$ line (like due to
host contamination), because of the large scatter in the iBeff slope from
source to source, some EVQs could exhibit reverse iBeff in H$\beta$ and are
more likely to identified as changing-look in their dim states.
The large scatter in the observed iBeff slope may partially be attributed to
the lag between broad line and continuum variation. Meanwhile, the fact that
the observed optical continuum variation does not necessarily coordinate with
that of the UV ionizing continuum (Xin et al., 2020; Gaskell et al., 2021)
could also play a significant role. Note such non-coordinated variations in
different bands (see also Sou et al. 2022) are natural subsequence of the
inhomogeneous disc fluctuation model (Cai et al., 2018, 2020).
To conclude, though we can not completely rule out dramatic vanishment of BLR
in some CLQs, most CLQs selected in literature appear consistent with a biased
sub-population of normal quasars with extreme continuum variability, i.e., the
essence of the “changing-look” phenomena is the extreme continuum variability.
### 4.2 Line Breathing
As shown in Fig. 2, the breathing of H$\beta$ is missing in our EVQ sample,
contradicting the results of previous works on AGNs and quasars (e.g., Denney
et al., 2008; Park et al., 2011; Wang et al., 2020). In addition, the
breathing relation in the Hb line within CLQ sample is also insignificant,
with a Spearman test $r=-0.29$ and $p=1.0\times 10^{-2}$.
It is not uncommon for AGNs to be caught undergoing changes in their breathing
relations. For example, Lu et al. (2022) reveal that the H$\beta$ of NGC 5548
well breathes in the most recent 5 years, however, previous decades of data
deviate significantly. Another interesting example is Mrk 50, in which a
breathing mode transition (from anti-breathing to normal breathing) was
captured during an $\sim$ 80-day RM observation (cf. Fig. 16 and Fig. 21 in
Barth et al. 2015). A more interesting fact is that the breathing period of
NGC 5548 coincides with its luminous stage, while Mrk 50 restored its
breathing when it was at its dimmest state, thus it seems the breathing mode
transition may not be simply one-way related to the flux level. Furthermore,
although the majority of sources in Wang et al. (2020) exhibit H$\beta$
breathing, a considerable proportion of their sources exhibit anti-breathing.
This suggests that breathing mode transitions might be frequent and the
structure of the BLRs could be more complex than expected in the simple
breathing model.
Meanwhile, it is notable that more than half of the sources in our EVQ sample
have a maximum spectral gap $>$3000 days which is much longer than the
duration of common RM projects. Therefore the absence of significant
breathing/anti-breathing on average in H$\beta$ (and in Mg II, and marginal in
C IV) in our EVQs could suggest possibly different evolution of the BLR (i.e.,
no breathing on average) on longer timescales.
## 5 Conclusions
In this work, we measured all 20,069 available SDSS spectra for the 14,012
EVQs built in Paper I with PyQSOFit. The catalog is presented in §A. From this
sample, we select 1259 EVQs with multi-epoch SDSS spectra after eliminating
spectra with suspicious calibrations, and study the spectral variations in
individual EVQs.
We find clear ”bluer-when-brighter” relation in multi-epoch spectra of EVQs,
consistent with previous works on normal quasars and AGNs. The iBeff of the
broad Mg II and C IV in EVQs are significantly seen. However, no robust iBeff
of H$\beta$ is detected, which could be attributed to stronger host
contamination at longer rest-frame wavelengths. Meanwhile, no systematical
variation of the broad line shape and asymmetry with continuum flux is found.
We do not detect statistically significant broad line breathing of H$\beta$,
Mg II or C IV either, suggesting the BLR evolution on longer timescales
($\sim$ 3000 days in the observed frame) could be different from those on
shorter timescales.
More interestingly, through comparing the iBeff of our EVQs with CLQs selected
in literature, we show that CLQs are more likely a biased (because of its
definition) sub-population of normal quasars with extreme continuum variation,
instead of a distinct population.
The work is supported by National Natural Science Foundation of China (grants
No. 11890693, 12033006, & 12192221), and the Cyrus Chung Ying Tang
Foundations.
## Appendix A The EVQ Catalog
We publish the EVQs catalog along with all measured spectral quantities in
this paper. The format of the catalog is described in Table 1 and the full
catalog is available at https://doi.org/10.5281/zenodo.8328175.
Table 1: FITS catalog format
Number | Column Name | Format | Unit | Description
---|---|---|---|---
0 | SDSS_NAME | string | | Unique identifier from the SDSS DR14 quasar catalog
1 | RA | float32 | deg | Right ascension (J2000)
2 | DEC | float32 | deg | Declination (J2000)
3 | Z | float32 | | Redshift
4 | MEANMAG_G | float32 | mag | Weighted mean magnitude of g-band light curve
5 | NG | int32 | | Number of observations in g-band light curve
6 | PLATE | int32 | | SDSS plate number
7 | MJD | int32 | | MJD when spectrum was observed
8 | FIBER | int32 | | SDSS fiber ID
9 | SPECPRIMARY | bool | | If the spectrum is the primary observation of object
10 | SPECMAG_G | float32 | mag | Spectrum projected onto g filters
11 | SPECMAG_G_ERR | float32 | mag | Error in SPECMAG_G
12 | SPECMAG_R | float32 | mag | Spectrum projected onto r filters
13 | SPECMAG_R_ERR | float32 | mag | Error in SPECMAG_R
14 | SN_CONTI | float32 | | Mean signal-to-noise ratio per pixel of continuum estimated at wavelength around 1350, 3000, and 5100 depending on the spectral coverage
15 | PL_SLOPE | float32 | | Slope of AGN power law
16 | PL_SLOPE_ERR | float32 | | Error in PL_SLOPE
17 | LOGL5100 | float32 | erg s-1 | Logarithmic continuum luminosity at rest-frame 5100Å
18 | LOGL5100_ERR | float32 | erg s-1 | Error in LOGL5100
19 | LOGL3000 | float32 | erg s-1 | Logarithmic continuum luminosity at rest-frame 3000Å
20 | LOGL3000_ERR | float32 | erg s-1 | Error in LOGL3000
21 | LOGL1350 | float32 | erg s-1 | Logarithmic continuum luminosity at rest-frame 1350Å
22 | LOGL1350_ERR | float32 | erg s-1 | Error in LOGL1350
23 | FE_2240_2650_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Rest-frame flux of the UV Fe II complex within the 2240-2650Å
24 | FE_2240_2650_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in FE_2240_2650_FLUX
25 | FE_2240_2650_EW | float32 | Å | Rest-frame EW of the UV Fe II complex within the 2240-2650Å
26 | FE_2240_2650_EW_ERR | float32 | Å | Error in FE_2240_2650_EW
27 | FE_4435_4685_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Rest-frame flux of the optical Fe II complex within the 4435-4685Å
28 | FE_4435_4685_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in FE_4435_4685_FLUX
29 | FE_4435_4685_EW | float32 | Å | Rest-frame EW of the optical Fe II complex within the 4435_4685Å
30 | FE_4435_4685_EW_ERR | float32 | Å | Error in FE_4435_4685_EW
31 | HB_BR_PEAK | float32 | Å | Peak wavelength of Hb broad component
32 | HB_BR_PEAK_ERR | float32 | Å | Error in HB_BR_PEAK
33 | HB_BR_BISECT | float32 | Å | Wavelength bisect the area of Hb broad component
34 | HB_BR_BISECT_ERR | float32 | Å | Error in HB_BR_BISECT
35 | HB_BR_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of Hb broad component
36 | HB_BR_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in HB_BR_FLUX
37 | HB_BR_EW | float32 | Å | Rest-frame EW of Hb broad component
38 | HB_BR_EW_ERR | float32 | Å | Error in HB_BR_EW
39 | HB_BR_SIGMA | float32 | km s-1 | Line dispersion of Hb broad component
40 | HB_BR_SIGMA_ERR | float32 | km s-1 | Error in HB_BR_SIGMA
41 | HB_BR_FWHM | float32 | km s-1 | FWHM of Hb broad component
42 | HB_BR_FWHM_ERR | float32 | km s-1 | Error in HB_BR_FWHM
43 | HB_BR_FWQM | float32 | km s-1 | FWQM of Hb broad component
44 | HB_BR_FWQM_ERR | float32 | km s-1 | Error in HB_BR_FWQM
45 | HB_BR_FW10M | float32 | km s-1 | FW10M of Hb broad component
46 | HB_BR_FW10M_ERR | float32 | km s-1 | Error in HB_BR_FW10M
47 | HB_BR_Z50 | float32 | km s-1 | The half maximum center shift of Hb broad component
48 | HB_BR_Z50_ERR | float32 | km s-1 | Error in HB_BR_Z50
49 | HB_BR_Z25 | float32 | km s-1 | The quarter maximum center shift of Hb broad component
50 | HB_BR_Z25_ERR | float32 | km s-1 | Error in HB_BR_Z25
51 | HB_BR_Z10 | float32 | km s-1 | The 10 percent maximum center shift of Hb broad component
52 | HB_BR_Z10_ERR | float32 | km s-1 | Error in HB_BR_Z10
53 | HB_NA_PEAK | float32 | Å | Peak wavelength of Hb narrow component
54 | HB_NA_PEAK_ERR | float32 | Å | Error in HB_NA_PEAK
55 | HB_NA_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of Hb narrow component
56 | HB_NA_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in HB_NA_FLUX
57 | HB_NA_EW | float32 | Å | Rest-frame EW of Hb narrow component
58 | HB_NA_EW_ERR | float32 | Å | Error in HB_NA_EW
59 | HB_NA_SIGMA | float32 | km s-1 | Line dispersion of Hb narrow component
60 | HB_NA_SIGMA_ERR | float32 | km s-1 | Error in HB_NA_SIGMA
61 | HB_NA_FWHM | float32 | km s-1 | FWHM of Hb narrow component
62 | HB_NA_FWHM_ERR | float32 | km s-1 | Error in HB_NA_FWHM
63 | OIII4959C_PEAK | float32 | Å | Peak wavelength of OIII4959 core component
64 | OIII4959C_PEAK_ERR | float32 | Å | Error in OIII4959C_PEAK
65 | OIII4959C_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of OIII4959 core component
66 | OIII4959C_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in OIII4959C_FLUX
67 | OIII4959C_EW | float32 | Å | Rest-frame EW of OIII4959 core component
68 | OIII4959C_EW_ERR | float32 | Å | Error in OIII4959C_EW
69 | OIII4959C_SIGMA | float32 | km s-1 | Line dispersion of OIII4959 core component
70 | OIII4959C_SIGMA_ERR | float32 | km s-1 | Error in OIII4959C_SIGMA
71 | OIII4959C_FWHM | float32 | km s-1 | FWHM of OIII4959 core component
72 | OIII4959C_FWHM_ERR | float32 | km s-1 | Error in OIII4959C_FWHM
73 | OIII4959W_PEAK | float32 | Å | Peak wavelength of OIII4959 wing component
74 | OIII4959W_PEAK_ERR | float32 | Å | Error in OIII4959W_PEAK
75 | OIII4959W_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of OIII4959 wing component
76 | OIII4959W_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in OIII4959W_FLUX
77 | OIII4959W_EW | float32 | Å | Rest-frame EW of OIII4959 wing component
78 | OIII4959W_EW_ERR | float32 | Å | Error in OIII4959W_EW
79 | OIII4959W_SIGMA | float32 | km s-1 | Line dispersion of OIII4959 wing component
80 | OIII4959W_SIGMA_ERR | float32 | km s-1 | Error in OIII4959W_SIGMA
81 | OIII4959W_FWHM | float32 | km s-1 | FWHM of OIII4959 wing component
82 | OIII4959W_FWHM_ERR | float32 | km s-1 | Error in OIII4959W_FWHM
83 | OIII5007C_PEAK | float32 | Å | Peak wavelength of OIII5007 core component
84 | OIII5007C_PEAK_ERR | float32 | Å | Error in OIII5007C_PEAK
85 | OIII5007C_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of OIII5007 core component
86 | OIII5007C_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in OIII5007C_FLUX
87 | OIII5007C_EW | float32 | Å | Rest-frame EW of OIII5007 core component
88 | OIII5007C_EW_ERR | float32 | Å | Error in OIII5007C_EW
89 | OIII5007C_SIGMA | float32 | km s-1 | Line dispersion of OIII5007 core component
90 | OIII5007C_SIGMA_ERR | float32 | km s-1 | Error in OIII5007C_SIGMA
91 | OIII5007C_FWHM | float32 | km s-1 | FWHM of OIII5007 core component
92 | OIII5007C_FWHM_ERR | float32 | km s-1 | Error in OIII5007C_FWHM
93 | OIII5007W_PEAK | float32 | Å | Peak wavelength of OIII5007 wing component
94 | OIII5007W_PEAK_ERR | float32 | Å | Error in OIII5007W_PEAK
95 | OIII5007W_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of OIII5007 wing component
96 | OIII5007W_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in OIII5007W_FLUX
97 | OIII5007W_EW | float32 | Å | Rest-frame EW of OIII5007 wing component
98 | OIII5007W_EW_ERR | float32 | Å | Error in OIII5007W_EW
99 | OIII5007W_SIGMA | float32 | km s-1 | Line dispersion of OIII5007 wing component
100 | OIII5007W_SIGMA_ERR | float32 | km s-1 | Error in OIII5007W_SIGMA
101 | OIII5007W_FWHM | float32 | km s-1 | FWHM of OIII5007 wing component
102 | OIII5007W_FWHM_ERR | float32 | km s-1 | Error in OIII5007W_FWHM
103 | MGII_BR_PEAK | float32 | Å | Peak wavelength of MgII broad component
104 | MGII_BR_PEAK_ERR | float32 | Å | Error in MGII_BR_PEAK
105 | MGII_BR_BISECT | float32 | Å | Wavelength bisect the area of MgII broad component
106 | MGII_BR_BISECT_ERR | float32 | Å | Error in MGII_BR_BISECT
107 | MGII_BR_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of MgII broad component
108 | MGII_BR_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in MGII_BR_FLUX
109 | MGII_BR_EW | float32 | Å | Rest-frame EW of MgII broad component
110 | MGII_BR_EW_ERR | float32 | Å | Error in MGII_BR_EW
111 | MGII_BR_SIGMA | float32 | km s-1 | Line dispersion of MgII broad component
112 | MGII_BR_SIGMA_ERR | float32 | km s-1 | Error in MGII_BR_SIGMA
113 | MGII_BR_FWHM | float32 | km s-1 | FWHM of MgII broad component
114 | MGII_BR_FWHM_ERR | float32 | km s-1 | Error in MGII_BR_FWHM
115 | MGII_BR_FWQM | float32 | km s-1 | FWQM of MgII broad component
116 | MGII_BR_FWQM_ERR | float32 | km s-1 | Error in MGII_BR_FWQM
117 | MGII_BR_FW10M | float32 | km s-1 | FW10M of MgII broad component
118 | MGII_BR_FW10M_ERR | float32 | km s-1 | Error in MGII_BR_FW10M
119 | MGII_BR_Z50 | float32 | km s-1 | The half maximum center shift of MgII broad component
120 | MGII_BR_Z50_ERR | float32 | km s-1 | Error in MGII_BR_Z50
121 | MGII_BR_Z25 | float32 | km s-1 | The quarter maximum center shift of MgII broad component
122 | MGII_BR_Z25_ERR | float32 | km s-1 | Error in MGII_BR_Z25
123 | MGII_BR_Z10 | float32 | km s-1 | The 10 percent maximum center shift of MgII broad component
124 | MGII_BR_Z10_ERR | float32 | km s-1 | Error in MGII_BR_Z10
125 | MGII_NA_PEAK | float32 | Å | Peak wavelength of MgII narrow component
126 | MGII_NA_PEAK_ERR | float32 | Å | Error in MGII_NA_PEAK
127 | MGII_NA_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of MgII narrow component
128 | MGII_NA_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in MGII_NA_FLUX
129 | MGII_NA_EW | float32 | Å | Rest-frame EW of MgII narrow component
130 | MGII_NA_EW_ERR | float32 | Å | Error in MGII_NA_EW
131 | MGII_NA_SIGMA | float32 | km s-1 | Line dispersion of MgII narrow component
132 | MGII_NA_SIGMA_ERR | float32 | km s-1 | Error in MGII_NA_SIGMA
133 | MGII_NA_FWHM | float32 | km s-1 | FWHM of MgII narrow component
134 | MGII_NA_FWHM_ERR | float32 | km s-1 | Error in MGII_NA_FWHM
135 | CIV_BR_PEAK | float32 | Å | Peak wavelength of CIV broad component
136 | CIV_BR_PEAK_ERR | float32 | Å | Error in CIV_BR_PEAK
137 | CIV_BR_BISECT | float32 | Å | Wavelength bisect the area of CIV broad component
138 | CIV_BR_BISECT_ERR | float32 | Å | Error in CIV_BR_BISECT
139 | CIV_BR_FLUX | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Flux of CIV broad component
140 | CIV_BR_FLUX_ERR | float32 | $10^{-17}\text{erg cm${}^{-2}$ s${}^{-1}$}$ | Error in CIV_BR_FLUX
141 | CIV_BR_EW | float32 | Å | Rest-frame EW of CIV broad component
142 | CIV_BR_EW_ERR | float32 | Å | Error in CIV_BR_EW
143 | CIV_BR_SIGMA | float32 | km s-1 | Line dispersion of CIV broad component
144 | CIV_BR_SIGMA_ERR | float32 | km s-1 | Error in CIV_BR_SIGMA
145 | CIV_BR_FWHM | float32 | km s-1 | FWHM of CIV broad component
146 | CIV_BR_FWHM_ERR | float32 | km s-1 | Error in CIV_BR_FWHM
147 | CIV_BR_FWQM | float32 | km s-1 | FWQM of CIV broad component
148 | CIV_BR_FWQM_ERR | float32 | km s-1 | Error in CIV_BR_FWQM
149 | CIV_BR_FW10M | float32 | km s-1 | FW10M of CIV broad component
150 | CIV_BR_FW10M_ERR | float32 | km s-1 | Error in CIV_BR_FW10M
151 | CIV_BR_Z50 | float32 | km s-1 | The half maximum center shift of CIV broad component
152 | CIV_BR_Z50_ERR | float32 | km s-1 | Error in CIV_BR_Z50
153 | CIV_BR_Z25 | float32 | km s-1 | The quarter maximum center shift of CIV broad component
154 | CIV_BR_Z25_ERR | float32 | km s-1 | Error in CIV_BR_Z25
155 | CIV_BR_Z10 | float32 | km s-1 | The 10 percent maximum center shift of CIV broad component
156 | CIV_BR_Z10_ERR | float32 | km s-1 | Error in CIV_BR_Z10
157 | LOGLBOL_5100 | float32 | erg s-1 | Logarithmic bolometric luminosity estimated based on LOGL5100
158 | LOGLBOL_5100_ERR | float32 | erg s-1 | Error in LOGLBOL_5100
159 | LOGLBOL_3000 | float32 | erg s-1 | Logarithmic bolometric luminosity estimated based on LOGL3000
160 | LOGLBOL_3000_ERR | float32 | erg s-1 | Error in LOGLBOL_3000
161 | LOGLBOL_1350 | float32 | erg s-1 | Logarithmic bolometric luminosity estimated based on LOGL1350
162 | LOGLBOL_1350_ERR | float32 | erg s-1 | Error in LOGLBOL_1350
163 | LOGMBH_HB | float32 | $M_{\odot}$ | Logarithmic black hole mass estimated based on broad Hb line
164 | LOGMBH_HB_ERR | float32 | $M_{\odot}$ | Error in LOGMBH_HB
165 | LOGMBH_MGII | float32 | $M_{\odot}$ | Logarithmic black hole mass estimated based on broad MgII line
166 | LOGMBH_MGII_ERR | float32 | $M_{\odot}$ | Error in LOGMBH_MGII
167 | LOGMBH_CIV | float32 | $M_{\odot}$ | Logarithmic black hole mass estimated based on broad CIV line
168 | LOGMBH_CIV_ERR | float32 | $M_{\odot}$ | Error in LOGMBH_CIV
169 | LOGLBOL | float32 | erg s-1 | The adopted fiducial bolometric luminosity
170 | LOGLBOL_ERR | float32 | erg s-1 | Error in LOGLBOL
171 | LOGMBH | float32 | $M_{\odot}$ | The adopted fiducial black hole mass
172 | LOGMBH_ERR | float32 | $M_{\odot}$ | Error in LOGMBH
173 | LOGREDD | float32 | | Logarithmic Eddington ratio based on fiducial BH mass and bolometric luminosity
Note. — We provide all 20,069 available spectral measurements of 14,012 EVQs
selected in Ren et al. 2022. Repeated observed spectra of a same EVQ will have
the same SDSS_NAME with different spectral info (PLATE,MJD, and FIBER). We
include the SPECPRIMARY flag to indicate if the spectrum is the best
observation of this object. The unmeasurable parameters are set to -999. The
errors are obtained from 100 iterations of Monte Carlo simulation. The
complete table is available on https://doi.org/10.5281/zenodo.8328175
## Appendix B Evaluate spectral calibration
The fiber-drop (Dawson et al., 2013) is an unavoidable problem in SDSS
spectrophotometry leading to varying degrees of underestimation of spectral
flux and has been widely reported in literature (e.g. Shen et al., 2014; Sun
et al., 2015; Guo et al., 2020). In certain cases the spectrophotometry could
also be overestimated due to contamination to the line of sight by spurious or
unrelated signals. Such effects may produce artificial spectral variability
which is hard to distinguish from the intrinsic spectral variation. This would
be particularly more relevant if one aims to search for rare events (such as
CLQs) out of a huge number of spectra. In this work our EVQs are pre-selected
based on multi-epoch photometric observations. Below we develop a technique to
evaluate the reliability of the SDSS spectrophotometry of the EVQs through
comparing the spectroscopic photometry with the photometric light curve
modeled with damped random walk (DRW). The spectroscopic photometry which
severely deviates from the DRW yielded ranges would be excluded as potentially
unreliable.
### B.1 Extend the light curves
The light curves we built for EVQ selection, consisted of data from SDSS and
PS1, only cover a span of 1998 to 2014. About half of the SDSS spectra were
conducted by SDSS IV since 2014 (Blanton et al., 2017), significantly beyond
the time span of our photometric light curves. We then introduce the PTF/iPTF
and ZTF observations to extend the photometric light curves to cover the
epochs when SDSS IV spectra were obtained.
The Palomar Transient Factory (PTF) is a fully-automated, wide-field survey
conducted by the Palomar 48-inch Samuel Oschin Schmidt telescope with
12K$\times$8K CCD array during 2009 to 2012 (Rau et al., 2009; Law et al.,
2009). The intermediate Palomar Transient Factory (iPTF) is the successor PTF
ran from 2013 to 2017 with a relatively higher cadence. The Zwicky Transient
Facility (ZTF) is a next-generation optical time-domain survey build upon the
PTF/iPTF. It started from 2018, scanned 3750 $\rm deg^{2}/hour$, and is still
in operation (Masci et al., 2018). The PTF/iPTF and ZTF surveys are shallower
than the SDSS and the PS1 with relatively larger photometry error.
Nevertheless, among the 3042 EVQs with repeated SDSS spectra, only 11 of them
have no detection in either PTF, iPTF or ZTF.
### B.2 Evaluating the reliability of spectroscopic photometry
We use the celerite (Foreman-Mackey et al., 2017) to model our $g$\- and $r$\-
band light curves and estimate their DRW parameters (timescale $\tau$ and
variability amplitude $\sigma$) with maximum likelihood estimation (MLE)
method in scipy. The mean magnitude $\bar{m}$ of DRW model is assumed to be
the mean of the light curve. The detailed explanation of fitting a light curve
with DRW process is presented in Zu et al. (2011, 2013, 2016).
Next, we utilize the DRW parameters to determine the probability distribution
function of the magnitude at a given epoch. According to the DRW process, the
next epoch signal ($s_{n+1}$ at $t=t_{n+1}$) is only relative to the present
one $s_{n}$, which can be expressed as:
$p(s_{n+1}=x_{n+1}|s_{n}=x_{n})=N(\mu_{n+1},\eta_{n+1}^{2}),$ (B1)
where
$\mu_{n+1}=e^{-\frac{\Delta t}{\tau}}x_{n}+\bar{m}(1-e^{-\frac{\Delta
t}{\tau}}),$ (B2) $\eta_{n+1}^{2}=\sigma^{2}(1-e^{-\frac{2\Delta
t}{\tau}})+err_{n}^{2}+err_{n+1}^{2}.$ (B3)
The $s_{n}$ and $s_{n+1}$ are the magnitude of the couple of adjacent light
curve data at $t_{n}$ and $t_{n+1}$ respectively, $err$ the observational
error, and $\Delta t=t_{n+1}-t_{n}$.
In any gap of the light curve, with the magnitude known at both ends, we can
calculate the probability distribution function (PDF) of the magnitude at any
epoch ($t_{n}<t_{0}<t_{n+1}$) within this duration. According to Bayes
formula, the PDF of the signal at $t_{0}$ given its adjacent data can be
determined as:
$\begin{split}&p(s_{0}=x_{0}|s_{n}=x_{n},~{}s_{n+1}=x_{n+1})\\\
=&\frac{p(s_{0}=x_{0},~{}s_{n+1}=x_{n+1}|s_{n}=x_{n})}{p(s_{n+1}=x_{n+1}|s_{n}=x_{n})}\\\
=&\frac{p(s_{0}=x_{0}|s_{n}=x_{n})p(s_{n+1}=x_{n+1}|s_{0}=x_{0},~{}s_{n}=x_{n})}{p(s_{n+1}=x_{n+1}|s_{n}=x_{n})}\\\
=&\frac{p(s_{0}=x_{0}|s_{n}=x_{n})p(s_{n+1}=x_{n+1}|s_{0}=x_{0})}{p(s_{n+1}=x_{n+1}|s_{n}=x_{n})}.\end{split}$
(B4)
To evaluate the reliability of a spectroscopic photometry, we first set
$t_{0}$ to the date of the spectrum and find its adjacent photometric data in
the corresponding light curve to calculate the PDF of the magnitude at this
epoch. By integrating this PDF, we can give the confidence interval of the
true magnitude of this spectrum. The sketch of the 99.7% confidence interval
is shown in Fig. 8.
Figure 8: An example of $g$-band light curve showing the 99.7% confidence
intervals of the magnitude in the light curve gaps. The observed photometric
magnitudes from different surveys are plotted with distinct colors and marks.
The spectroscopic magnitudes of three SDSS spectra are marked with stars and
vertical dashed lines. As can be seen, the spectroscopic magnitude of the
first spectrum is out of the 99.7% confidence intervals, which is likely due
to the fiber-drop effect.
In this work, we consider the spectroscopic photometry reliable if both its
$g$\- and $r$\- band spectroscopic magnitude fall within the 99.7% confidence
intervals. Under this criterion, 301 spectra are excluded as potentially
unreliable, leaving 1259 sources with repeated observations.
### B.3 The validity of the approach
Precisely evaluating whether this method can properly rule out unreliable
spectroscopic photometry could be challenging. However, we can provide an
indirect approach to demonstrate its validity. We further select a sample of
low-variability quasars (LVQs) with $|\Delta g|_{\text{max}}<0.2$ mag, where
$|\Delta g|_{\text{max}}$ represents the maximum difference in magnitude
between any two epochs in the light curves constructed by Paper I. Since we do
not expect a sudden change in flux for these low-variability quasars, spectra
with significantly deviating flux from their light curve means in LVQs are
likely caused by calibration issues.
In Fig. 9, we show the normalized distribution of
$g_{\text{spec}}-g_{\text{mean}}$ (the difference between the $g$-band
magnitude from the spectrum and the light curve mean, see §3.1 in Paper I for
details) for our sample and the LVQs. The distribution of
$g_{\text{spec}}-g_{\text{mean}}$ of LVQs (for which we only expect small
deviation from zero) does span a broad range, indicating the calibration
issues are not negligible at large $|g_{\text{spec}}-g_{\text{mean}}|$. For
our EVQs, our screening process does exclude considerable fraction of epochs
at large $|g_{\text{spec}}-g_{\text{mean}}|$ but minor fraction at small
$|g_{\text{spec}}-g_{\text{mean}}|$ (the ratio of blue to yellow line in Fig.
9 could represent the excluded fraction as a function of
$g_{\text{spec}}-g_{\text{mean}}$). At large
$|g_{\text{spec}}-g_{\text{mean}}|$ (e.g. $>$ 0.8), we have excluded
relatively more epochs than the prediction from LVQs (blue line lying above
the green line), suggesting our approach is somehow conservative that some
good spectra could have been excluded (which is however unavoidable).
Figure 9: Normalized distribution of $g_{\text{spec}}-g_{\text{mean}}$ for
multi-spectra EVQs (yellow) and LVQs (green). The blue solid line plots the
excluded portion of the yellow line. A logarithmic y-axis is used to better
visualize the outliers, i.e., with large $|g_{\text{spec}}-g_{\text{mean}}|$.
## References
* Aretxaga et al. (1999) Aretxaga, I., Joguet, B., Kunth, D., Melnick, J., & Terlevich, R. J. 1999, The Astrophysical Journal, 519, L123, doi: 10.1086/312114
* Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
* Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
* Barth et al. (2015) Barth, A. J., Bennert, V. N., Canalizo, G., et al. 2015, ApJS, 217, 26, doi: 10.1088/0067-0049/217/2/26
* Blanton et al. (2017) Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, AJ, 154, 28, doi: 10.3847/1538-3881/AA7567
* Boggs & Rogers (1989) Boggs, P. T. P., & Rogers, J. J. E. 1989, Nistir 89-4197, 1989, doi: 10.6028/nist.ir.89-4197
* Cai et al. (2016) Cai, Z.-Y., Wang, J.-X., Gu, W.-M., et al. 2016, ApJ, 826, 7, doi: 10.3847/0004-637X/826/1/7
* Cai et al. (2020) Cai, Z.-Y., Wang, J.-X., & Sun, M. 2020, ApJ, 892, 63, doi: 10.3847/1538-4357/ab7991
* Cai et al. (2018) Cai, Z.-Y., Wang, J.-X., Zhu, F.-F., et al. 2018, ApJ, 855, 117, doi: 10.3847/1538-4357/aab091
* Cohen et al. (1986) Cohen, R. D., Puetter, R. C., Rudy, R. J., Ake, T. B., & Foltz, C. B. 1986, ApJ, 311, 135, doi: 10.1086/164758
* Dawson et al. (2013) Dawson, K. S., Schlegel, D. J., Ahn, C. P., et al. 2013, AJ, 145, 10, doi: 10.1088/0004-6256/145/1/10
* Denney et al. (2008) Denney, K. D., Peterson, B. M., Dietrich, M., Vestergaard, M., & Bentz, M. C. 2008, ApJ, 692, 246, doi: 10.1088/0004-637X/692/1/246
* Foreman-Mackey et al. (2017) Foreman-Mackey, D., Agol, E., Ambikasaran, S., & Angus, R. 2017, The Astronomical Journal, 154, 220, doi: 10.3847/1538-3881/AA9332
* Gaskell et al. (2021) Gaskell, C. M., Bartel, K., Deffner, J. N., & Xia, I. 2021, MNRAS, 508, 6077, doi: 10.1093/mnras/stab2443
* Goad et al. (2004) Goad, M. R., Korista, K. T., & Knigge, C. 2004, MNRAS, 352, 277, doi: 10.1111/j.1365-2966.2004.07919.x
* Goodrich (1995) Goodrich, R. W. 1995, The Astrophysical Journal, 440, 141, doi: 10.1086/175256
* Green et al. (2022) Green, P. J., Pulgarin-Duque, L., Anderson, S. F., et al. 2022, ApJ, 933, 180, doi: 10.3847/1538-4357/ac743f
* Grier et al. (2017) Grier, C. J., Trump, J. R., Shen, Y., et al. 2017, ApJ, 851, 21, doi: 10.3847/1538-4357/aa98dc
* Grier et al. (2019) Grier, C. J., Shen, Y., Horne, K., et al. 2019, ApJ, 887, 38, doi: 10.3847/1538-4357/ab4ea5
* Guo & Gu (2016) Guo, H., & Gu, M. 2016, ApJ, 822, 26, doi: 10.3847/0004-637x/822/1/26
* Guo et al. (2018) Guo, H., Shen, Y., & Wang, S. 2018, PyQSOFit: Python code to fit the spectrum of quasars, Astrophysics Source Code Library. http://ascl.net/1809.008
* Guo et al. (2019) Guo, H., Sun, M., Liu, X., et al. 2019, ApJ, 883, L44, doi: 10.3847/2041-8213/ab4138
* Guo et al. (2020) Guo, H., Shen, Y., He, Z., et al. 2020, ApJ, 888, 58, doi: 10.3847/1538-4357/ab5db0
* Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
* Homan et al. (2020) Homan, D., MacLeod, C. L., Lawrence, A., Ross, N. P., & Bruce, A. 2020, MNRAS, 496, 309, doi: 10.1093/mnras/staa1467
* Homayouni et al. (2020) Homayouni, Y., Trump, J. R., Grier, C. J., et al. 2020, ApJ, 901, 55, doi: 10.3847/1538-4357/ababa9
* Hu et al. (2015) Hu, C., Du, P., Lu, K.-X., et al. 2015, ApJ, 804, 138, doi: 10.1088/0004-637X/804/2/138
* Kang et al. (2021) Kang, W.-Y., Wang, J.-X., Cai, Z.-y., & Ren, W.-K. 2021, The Astrophysical Journal, 911, 148, doi: 10.3847/1538-4357/abeb69
* Kelly et al. (2009) Kelly, B. C., Bechtold, J., & Siemiginowska, A. 2009, ApJ, 698, 895, doi: 10.1088/0004-637X/698/1/895
* Kong et al. (2006) Kong, M. Z., Wu, X. B., Wang, R., Liu, F. K., & Han, J. L. 2006, A&A, 456, 473, doi: 10.1051/0004-6361:20064871
* LaMassa et al. (2015) LaMassa, S. M., Cales, S., Moran, E. C., et al. 2015, ApJ, 800, 144, doi: 10.1088/0004-637X/800/2/144
* Law et al. (2009) Law, N. M., Kulkarni, S. R., Dekany, R. G., et al. 2009, Publications of the Astronomical Society of the Pacific, 121, 1395, doi: 10.1086/648598/XML
* López-Navas et al. (2022) López-Navas, E., Martínez-Aldama, M. L., Bernal, S., et al. 2022, Mon. Not. R. Astron. Soc. Lett., 513, L57, doi: 10.1093/mnrasl/slac033
* Lu et al. (2022) Lu, K.-X., Bai, J.-M., Wang, J.-M., et al. 2022, ApJS, 263, 10, doi: 10.3847/1538-4365/ac94d3
* MacLeod et al. (2010) MacLeod, C. L., Ivezić, Ž., Kochanek, C. S., et al. 2010, The Astrophysical Journal, 721, 1014, doi: 10.1088/0004-637X/721/2/1014
* MacLeod et al. (2016) MacLeod, C. L., Ross, N. P., Lawrence, A., et al. 2016, MNRAS, 457, 389, doi: 10.1093/mnras/stv2997
* MacLeod et al. (2019) MacLeod, C. L., Green, P. J., Anderson, S. F., et al. 2019, ApJ, 874, 8, doi: 10.3847/1538-4357/ab05e2
* Masci et al. (2018) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2018, Publications of the Astronomical Society of the Pacific, 131, 018003, doi: 10.1088/1538-3873/AAE8AC
* pandas development team (2020) pandas development team, T. 2020, pandas-dev/pandas: Pandas, latest, Zenodo, doi: 10.5281/zenodo.3509134
* Pâris et al. (2018) Pâris, I., Petitjean, P., Aubourg, É., et al. 2018, A&A, 613, A51, doi: 10.1051/0004-6361/201732445
* Park et al. (2011) Park, D., Woo, J.-H., Treu, T., et al. 2011, ApJ, 747, 30, doi: 10.1088/0004-637X/747/1/30
* Pogge & Peterson (1992) Pogge, R. W., & Peterson, B. M. 1992, AJ, 103, 1084, doi: 10.1086/116127
* Price-Whelan et al. (2018) Price-Whelan, A. M., Sipőcz, B. M., Günther, H. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
* Pu et al. (2006) Pu, X., Bian, W., & Huang, K. 2006, MNRAS, 372, 246, doi: 10.1111/j.1365-2966.2006.10850.x
* Rakić (2022) Rakić, N. 2022, MNRAS, 516, 1624, doi: 10.1093/mnras/stac2259
* Rakić et al. (2017) Rakić, N., La Mura, G., Ilić, D., et al. 2017, A&A, 603, A49, doi: 10.1051/0004-6361/201630085
* Rau et al. (2009) Rau, A., Kulkarni, S. R., Law, N. M., et al. 2009, Publications of the Astronomical Society of the Pacific, 121, 1334, doi: 10.1086/605911/XML
* Ren et al. (2022) Ren, W., Wang, J., Cai, Z., & Guo, H. 2022, ApJ, 925, 50, doi: 10.3847/1538-4357/ac3828
* Risaliti et al. (2007) Risaliti, G., Elvis, M., Fabbiano, G., et al. 2007, The Astrophysical Journal, 659, L111, doi: 10.1086/517884
* Ross et al. (2020) Ross, N. P., Graham, M. J., Calderone, G., et al. 2020, MNRAS, 498, 2339, doi: 10.1093/mnras/staa2415
* Ross et al. (2018) Ross, N. P., Ford, K. E., Graham, M., et al. 2018, Monthly Notices of the Royal Astronomical Society, 480, 4468, doi: 10.1093/mnras/sty2002
* Ruan et al. (2019) Ruan, J. J., Anderson, S. F., Eracleous, M., et al. 2019, doi: 10.48550/arXiv.1909.04676
* Ruan et al. (2016) Ruan, J. J., Anderson, S. F., Cales, S. L., et al. 2016, ApJ, 826, 188, doi: 10.3847/0004-637X/826/2/188
* Rumbaugh et al. (2018) Rumbaugh, N., Shen, Y., Morganson, E., et al. 2018, ApJ, 854, 160, doi: 10.3847/1538-4357/aaa9b6
* Sesar et al. (2007) Sesar, B., Ivezić, Ž., Lupton, R. H., et al. 2007, The Astronomical Journal, 134, 2236, doi: 10.1086/521819
* Shen et al. (2011) Shen, Y., Richards, G. T., Strauss, M. A., et al. 2011, ApJS, 194, 45, doi: 10.1088/0067-0049/194/2/45
* Shen et al. (2014) Shen, Y., Brandt, W. N., Dawson, K. S., et al. 2014, ApJS, 216, 4, doi: 10.1088/0067-0049/216/1/4
* Sheng et al. (2020) Sheng, Z., Wang, T., Jiang, N., et al. 2020, ApJ, 889, 46, doi: 10.3847/1538-4357/ab5af9
* Sou et al. (2022) Sou, H., Wang, J.-X., Xie, Z.-L., Kang, W.-Y., & Cai, Z.-Y. 2022, MNRAS, 512, 5511, doi: 10.1093/mnras/stac738
* Sun et al. (2018) Sun, M., Xue, Y., Richards, G. T., et al. 2018, ApJ, 854, 128, doi: 10.3847/1538-4357/aaa890
* Sun et al. (2015) Sun, M., Trump, J. R., Shen, Y., et al. 2015, ApJ, 811, 42, doi: 10.1088/0004-637X/811/1/42
* Sun et al. (2020) Sun, M., Xue, Y., Brandt, W. N., et al. 2020, The Astrophysical Journal, 891, 178, doi: 10.3847/1538-4357/ab789e
* Sun et al. (2014) Sun, Y.-H., Wang, J.-X., Chen, X.-Y., & Zheng, Z.-Y. 2014, ApJ, 792, 54, doi: 10.1088/0004-637X/792/1/54
* Tholine & Osterbrock (1976) Tholine, J. E., & Osterbrock, D. E. 1976, ApJ, 210, L117, doi: 10.1086/182317
* Trevese & Vagnetti (2002) Trevese, D., & Vagnetti, F. 2002, ApJ, 564, 624, doi: 10.1086/324541
* Vanden Berk et al. (2004) Vanden Berk, D. E., Wilhite, B. C., Kron, R. G., et al. 2004, The Astrophysical Journal, 601, 692, doi: 10.1086/380563
* Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
* Wang et al. (2020) Wang, S., Shen, Y., Jiang, L., et al. 2020, ApJ, 903, 51, doi: 10.3847/1538-4357/abb36d
* Wes McKinney (2010) Wes McKinney. 2010, in Proceedings of the 9th Python in Science Conference, ed. Stéfan van der Walt & Jarrod Millman, 56 – 61, doi: 10.25080/Majora-92bf1922-00a
* Wu & Shen (2022) Wu, Q., & Shen, Y. 2022, Astrophys. J. Suppl., 263, 42, doi: 10.3847/1538-4365/ac9ead
* Xin et al. (2020) Xin, C., Charisi, M., Haiman, Z., & Schiminovich, D. 2020, MNRAS, 495, 1403, doi: 10.1093/mnras/staa1258
* Yang et al. (2018) Yang, Q., Wu, X.-B., Fan, X., et al. 2018, ApJ, 862, 109, doi: 10.3847/1538-4357/aaca3a
* Yang et al. (2023) Yang, Q., Green, P. J., MacLeod, C. L., et al. 2023, The Astrophysical Journal, 953, 61, doi: 10.3847/1538-4357/acdedd
* Zajaček et al. (2020) Zajaček, M., Czerny, B., Martinez–Aldama, M. L., et al. 2020, ApJ, 896, 146, doi: 10.3847/1538-4357/ab94ae
* Zhang (2013) Zhang, X. G. 2013, MNRAS, 435, 2141, doi: 10.1093/mnras/stt1432
* Zhu et al. (2016) Zhu, F.-F., Wang, J.-X., Cai, Z.-Y., & Sun, Y.-H. 2016, ApJ, 832, 75, doi: 10.3847/0004-637X/832/1/75
* Zu et al. (2016) Zu, Y., Kochanek, C. S., Kozłowski, S., & Peterson, B. M. 2016, ApJ, 819, 122, doi: 10.3847/0004-637X/819/2/122
* Zu et al. (2013) Zu, Y., Kochanek, C. S., Kozłowski, S., & Udalski, A. 2013, ApJ, 765, 106, doi: 10.1088/0004-637X/765/2/106
* Zu et al. (2011) Zu, Y., Kochanek, C. S., & Peterson, B. M. 2011, ApJ, 735, 80, doi: 10.1088/0004-637X/735/2/80
|
# On the Level Crossing Rate
of Fluid Antenna Systems
Priyadarshi Mukherjee, Constantinos Psomas, and Ioannis Krikidis Department
of Electrical and Computer Engineering, University of Cyprus
Email: {mukherjee.priyadarshi, psomas<EMAIL_ADDRESS>
###### Abstract
Multiple-input multiple-output (MIMO) technology has significantly impacted
wireless communication, by providing extraordinary performance gains. However,
a minimum inter-antenna space constraint in MIMO systems does not allow its
integration in devices with limited space. In this context, the concept of
fluid antenna systems (FASs) appears to be a potent solution, where there is
no such restriction. In this paper, we investigate the average level crossing
rate (LCR) of such FASs. Specifically, we derive closed-form analytical
expressions of the LCR of such systems and extensive Monte-Carlo simulations
validate the proposed analytical framework. Moreover, we also demonstrate that
under certain conditions, the LCR obtained coincides with that of a
conventional selection combining-based receiver. Finally, the numerical
results also provide insights regarding the selection of appropriate
parameters that enhance the system performance.
###### Index Terms:
Fluid antenna systems, spatial correlation, level crossing rate, selection
combining, diversity.
## I Introduction
Multiple-input multiple-output (MIMO) can be considered as one of the most
popular wireless technologies in recent years. The concepts of diversity and
multiplexing gain form the basis of MIMO systems, which led to its
extraordinary performance for wireless communication links. However, there
must be a minimum distance of $\frac{\lambda}{2}$ between the antennas in MIMO
systems, where $\lambda$ is the transmission wavelength [1]. This limits the
integration of MIMO systems inside mobile devices such as tablets and mobile
phones, where the physical space is very limited.
To overcome the aforementioned limitation of MIMO, the novel concept of fluid
antenna system (FAS) was proposed in [2]. FAS is essentially a single antenna
system with $N$ fixed locations (referred as “port”) distributed over a given
space. The idea of FAS is originally motivated by the increasing trend of
using ionized solutions or liquid metals for antennas [3, 4, 5]. The most
interesting aspect of FAS is that an antenna element is no longer kept fixed
at a particular location, but it can switch to a relatively more favorable
location inside the boundaries, if required. Apparently, the objective of FAS
resembles that of traditional transmit antenna selection (TAS) systems [6],
where multiple antennas are deployed at different locations and the antenna
with the strongest signal is selected. However, unlike TAS systems, the single
antenna element in FAS can change position among the predetermined ports. In
this way, an FAS exploits the phenomenon of spatial diversity and the received
signal from the port with the strongest channel condition is selected.
Furthermore, there is no limitation of maintaining a minimum inter-port
distance and as a result, the space making up the FAS may be small with large
number of ports. The work in [2] evaluates the theoretical performance of such
systems in terms of outage probability. It is shown that even with a small
space and a practically feasible number of ports, FAS can significantly
outperform conventional maximum ratio combining-based systems. The work in [7]
investigates the second order statistics of a FAS-based receiver. In
particular, the work evaluates its performance in terms of the level crossing
rate (LCR), average fade duration (AFD), and also ergodic capacity.
As stated above, FAS does not have any constraint on the inter-port distance.
This, unlike in conventional MIMO systems, makes the aspect of spatial
correlation a crucial factor in characterizing its performance limits.
However, the work in [7] does not make this consideration. Motivated by this,
in this work, we investigate the FAS second order statistics by taking the
spatial correlation into account. Specifically, we derive closed-form
analytical expressions of the LCR as a function of both the number of ports
and the associated spatial correlation. We demonstrate that in certain
scenarios, the FAS LCR coincides with that of a conventional selection
combining (SC)-based system with independent and identically distributed
(i.i.d.) channels. To the best of our knowledge, this is the first work that
presents a complete analytical framework to characterize the LCR for a FAS, by
taking into account its practical constraints and limitations.
## II System model
Figure 1: The considered topology consisting of a single antenna transmitter
and a FAS-based mobile device.
We consider a simple point-to-point topology, where we have a single antenna
transmitter and a $N$-port FAS-based mobile receiver, as depicted in Fig. 1.
As shown in the figure, a typical FAS-based receiver is essentially a single
antenna system with a single radio-frequency (RF) chain. The $N$ ports are
evenly distributed in a linear space of $W\lambda$, where $\lambda$ is the
transmission wavelength and the antenna can switch locations instantly among
the ports [2]. Fig. 1 illustrates that the first port is the reference and the
liquid antenna is always switched to the port with best channel conditions.
The channels at these ports are characterized as [2]
$\displaystyle\begin{cases}\\!h_{1}=\sigma x_{0}+j\sigma y_{0}\\\
\\!h_{k}=\sigma\left({\sqrt{1-\mu_{k}^{2}}x_{k}+\mu_{k}x_{0}}\right)\\\
\quad\>\>+j\sigma\left({\sqrt{1-\mu_{k}^{2}}y_{k}+\mu_{k}y_{0}}\right)\>\>\text{for
}k=2,{\cdots},N,\end{cases}$ (1)
where $x_{0},\cdots,x_{N},y_{0},\cdots,y_{N}$ are independent Gaussian random
variables with zero mean and variance $\frac{1}{2}$, and $\mu_{k}$ $\forall$
$k$ are the parameters that control the spatial correlation between the
channels. Accordingly, we have $\mathbb{E}[|h_{k}|^{2}]=\sigma^{2}$ $\forall$
$k$ and [2]
$\mu_{k}=J_{0}\left(\frac{2\pi(k-1)}{N-1}W\right),\qquad\text{for
}k=2,{\dots},N,$ (2)
where $J_{0}(\cdot)$ is the zero-order Bessel function of the first kind. Note
that due to the oscillatory nature of $J_{0}(\cdot)$, for a given $N$,
$\mu_{k}$ $\forall$ $k$ is not a monotonically decreasing function of $W$. It
may happen that, multiple values of $W$ result in identical $\mu_{k}$. Hence,
to obtain a reasonable range of $W,$ we consider $J_{0}(\cdot)$ till the point
where it reaches the first zero. Accordingly, irrespective of $N$, we obtain
$W\in[0,0.38]$. The FAS always selects the port with the strongest channel
condition, i.e.
$|h_{\mathrm{FAS}}|=\max\\{|h_{1}|,|h_{2}|,{\dots},|h_{N}|\\}.$ (3)
As the ports in an FAS are very close to each other, the aspect of spatial
correlation plays a crucial role in determining which of the $N$ ports is
selected. Accordingly, the joint probability distribution function (PDF) of
$|h_{1}|,|h_{2}|,\cdots,|h_{N}|$ is [2]
$\displaystyle p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{N})$
$\displaystyle\quad=\prod_{\begin{subarray}{c}k=1\\\ (\mu_{1}\triangleq
0)\end{subarray}}^{N}\\!\\!\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{1}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}I_{0}\left({\frac{2~{}\mu_{k}x_{1}x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)\\!,$
(4)
for $x_{1},{\dots},x_{N}\geq 0$, where $I_{0}(\cdot)$ is the zero-order
modified Bessel function of the first kind. It is important to note that the
mutual coupling does not affect an FAS, as only one antenna element is
activated at each time. Hence, (II) is not a conventional $N$-variate random
variable, but a product of $N$ bi-variate random variables.
## III Level Crossing Rate for FAS
In this section, we characterize the LCR of an FAS, which is an important
parameter in characterizing the dynamics of any random process. It facilitates
to evaluate the impact of the time-varying channel on the FAS performance. The
LCR enables to estimate the statistics of error occurrence in signal
detection. The LCR of a random process $r$ at threshold $r_{\rm th}$
essentially gives the number of times per unit duration that $r$ crosses
$r_{\rm th}$ in the negative (or positive) direction [8]. Mathematically it
defined as
$L(r_{\rm th})=\int_{0}^{\infty}\dot{r}p_{\dot{R}R}(\dot{r},r_{\rm
th})d\dot{r},$ (5)
where $\dot{r}$ is the time derivative of $r$ and $p_{\dot{R}R}(\dot{r},r)$ is
the joint PDF of $r(t)$ and $\dot{r}(t)$ in an arbitrary instant $t$. For an
isotropic scattering scenario, the time derivative of the signal envelope is
Gaussian distributed with zero mean, irrespective of the fading distribution
[9]. As we aim to analyze the LCR of a FAS, we propose the following theorem
in this direction.
###### Theorem 1.
The LCR for a $N$-port FAS is given by (III).
$\displaystyle L(x_{\rm th})$ $\displaystyle=\frac{\sqrt{2\pi}x_{\rm
th}f_{D}}{\sigma}\Biggl{\\{}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\prod_{k=2}^{N}\left[1-Q_{1}\left(\sqrt{\frac{2\mu_{k}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th},\sqrt{\frac{2}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th}\right)\right]+\sum_{i=2}^{N}\frac{1}{(1-\mu_{i}^{2})}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}$
$\displaystyle\quad\times\int_{0}^{x_{\rm
th}}\frac{2x_{1}}{\sigma^{2}}e^{-\frac{x_{1}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}I_{0}\left({\frac{2~{}\mu_{i}x_{\rm
th}x_{1}}{\sigma^{2}(1-\mu_{i}^{2})}}\right)\prod_{\begin{subarray}{c}k=2\\\
k\neq
i\end{subarray}}^{N}\left[1-Q_{1}\left(\sqrt{\frac{2\mu_{k}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}x_{1},\sqrt{\frac{2}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th}\right)\right]dx_{1}\Biggr{\\}}.$ (6)
###### Proof.
See Appendix A. ∎
We observe from (III) that the LCR is a function of the spatial correlation,
the number of ports, the decision threshold, and the maximum Doppler frequency
of the channel. This LCR of an FAS is different from that of a conventional
SC-based receiver, primarily because of the unique PDF of the channels at the
$N$ ports (as it can be seen from (II)) and also the aspect of the associated
spatial correlation. For the sake of completeness, we consider the following
two extreme cases: $\mu_{k}=0,1$ $\forall$ $k=1,\cdots,N$. The LCR $L(x_{\rm
th})$ corresponding to $\mu_{k}=0$ $\forall$ $k$, i.e. for a no spatial
correlation scenario, is given below.
###### Corollary 1.
For a scenario without spatial correlation, i.e. $\mu_{k}=0$ $\forall$ $k$,
$L(x_{\rm th})$ is
$L(x_{\rm th})=N\sqrt{2\pi}f_{D}\frac{x_{\rm th}}{\sigma}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\left(1-e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\right)^{N-1}.$ (7)
The above corollary follows directly from Theorem 1, by replacing $\mu_{k}=0$
$\forall$ $k=1,\cdots,N$ and using
$Q_{1}(0,b)=\int_{b}^{\infty}xe^{-\frac{x^{2}}{2}}dx=e^{-\frac{b^{2}}{2}}\quad\text{for}\quad
b\geq 0.$ (8)
Without a spatial correlation, for a given set of system parameters, $L(x_{\rm
th})$ obtained in (7) coincides with the LCR of a conventional SC-based
receiver with i.i.d. channels [10, Eq. 18]. The case with $\mu_{k}\\!=\\!1$
$\forall$ $k$ essentially corresponds to the scenario where the $N$ ports are
identical, i.e., there is no need of any switching of the liquid antenna among
the ports.
###### Corollary 2.
For a scenario with $\mu_{k}\\!=\\!1$ $\forall$ $k$, we have
$L(x_{\rm th})=\frac{\sqrt{2\pi}}{\sigma}f_{D}x_{\rm th}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}.$ (9)
###### Proof.
See Appendix B. ∎
It is interesting to note from the above corollary, that in case of identical
channels, the FAS LCR is independent of $N$. Nevertheless, the analytical
expression of $L(x_{\rm th})$ derived in Theorem 1, being too involved, does
not provide any insightful analysis. Hence, we consider a simple case of
$N=2$.
###### Corollary 3.
For a two-port FAS, the LCR is given by
$\displaystyle L(x_{\rm th})$ $\displaystyle=\frac{2\sqrt{2\pi}f_{D}x_{\rm
th}}{\sigma^{3}(1-\mu^{2})}e^{\left(\\!\\!-\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu^{2})}\right)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times\sum_{k=0}^{\infty}\frac{\left(\mu
x_{\rm
th}\right)^{2k}}{(k!)^{2}\left(\sigma^{2}(1-\mu^{2})\right)^{k-1}}\gamma\left(k+1,\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu^{2})}\right).$ (10)
###### Proof.
See Appendix C. ∎
The above corollary demonstrates the effect of parameters such as $x_{\rm
th},\mu,$ and $\sigma$ on the LCR. For example, we observe from (3) that
$L(x_{\rm th})$ is the product of an unimodal function and an increasing
function with respect to $x_{\rm th}$. This implies that $L(x_{\rm th})$ is
also unimodal, i.e. $L(x_{\rm th})$ initially increases with $x_{\rm th}$, but
it starts decreasing after a certain point.
## IV Numerical Results
We now validate our theoretical analysis with extensive Monte-Carlo
simulations. Without any loss of generality, we consider unit power channels,
i.e. $\mathbb{E}[|h_{k}|^{2}]=\sigma^{2}=1$ $\forall$ $k=1,\cdots,N$, where
$N$ is the number of ports in the FAS and a carrier frequency of $900$ MHz.
Figure 2: Verification of proposed analysis via Monte-Carlo simulations.
Fig. 2 demonstrates the variation of the normalized LCR (NLCR) $L(x_{\rm
th})/f_{D}$ versus the decision threshold $x_{\rm th}$ for two scenarios with
$W=0.1$ and $0.3$, respectively. In this figure, we have considered a $N$-port
FAS, with $N=2,3,$ and $4$, respectively. Note that the values considered are
solely for illustration. We observe that the theoretical results (lines) match
very closely with the simulation results (markers); this verifies our proposed
analytical framework. The figure supports our claim that LCR depends on both
$x_{\rm th}$ and $\mu$. It can be further noted that the LCR increases with
increase of $x_{\rm th}$ until it reaches its maximum and then it decreases.
Moreover, this particular value of $x_{\rm th}$ depends on the value of $W$,
which corroborates the claims made in [11] regarding the effect of optimum
threshold selection. Furthermore, we observe that, the choice of $W$ also has
an impact on the NLCR. Finally, the closed-form expression for NLCR derived in
[7] significantly deviates from the simulation results.
Figure 3: Impact of spatial correlation on NLCR; $x_{\rm th}=28$ dBm.
Fig. 3 illustrates the effect of $N$ on the NLCR performance. For an identical
$N$, a marginal increase of $W$ leads to a significant improvement in
performance; for example, observe the performance gap at $N=16$ between
$W=0.1,0.2,$ and $0.3,$ respectively. It is worth to note, that the best
performance is observed when the channels are independent at the ports, i.e.
$\mu_{k}=0$ $\forall$ $k=1,\cdots,N$. Finally, the figure demonstrates that
irrespectively of the choice of $W$, an FAS asymptotically attains the optimal
performance in terms of NLCR, as $N\rightarrow\infty$; greater the value of
$W$, faster is the convergence. However, from (2), we know that $W$ cannot be
increased arbitrarily due to the oscillatory nature of $J_{0}(\cdot)$; in this
way, the spatial correlation affects the FAS performance. Furthermore, we also
observe that the change of $W$ or $N$ does not affect the NLCR, as in [7].
Fig. 4 depicts the variation of NLCR with $N$ for multiple values of $x_{\rm
th}$. We observe that for both the cases, i.e. (III) and [7], a lower
threshold results in a lower NLCR. Furthermore, we observe that, as also seen
in Fig. 3, [7] is invariant to $N$ (from the dashed lines). On the contrary,
NLCR as derived in (III) is significantly affected with increasing $N$; the
NLCR decreases with $N$. This demonstrates the key benefit of an FAS, where it
is advantageous to have higher $N$ without any inter-port distance constraint.
## V Conclusion
In this paper, motivated by the practical constraints of a FAS, we proposed a
novel and general analytical framework for the exact evaluation of an
important second order statistical parameter of FAS, namely the LCR. In
particular, by considering the effect of the time-varying nature of fading
channels, we investigated the aspect of spatial correlation in characterizing
this performance metric. Closed form expressions for the LCR were analytically
derived and it was demonstrated that in certain scenarios, they coincide with
the LCR of an SC-based receiver with i.i.d. channels. Finally, we validate our
proposed framework by extensive Monte-Carlo simulations. Based on the
framework presented in this paper, an immediate extension of this work is to
investigate the second order statistics of a multiple FAS-based topology with
a heterogeneous geometry.
## Appendix
### V-A Proof of Theorem 1
Figure 4: Impact of spatial correlation on NLCR; $W=0.2$.
In this context, the $N$-variate joint PDF $p_{|\dot{h}|,|h|}(\dot{x},x)$ is
given by [12, 8.42]
$\displaystyle p_{|\dot{h}|,|h|}(\dot{x},x)$
$\displaystyle=\sum_{i=1}^{N}p_{|\dot{h_{i}}|}(\dot{x})$
$\displaystyle\\!\\!\times\underbrace{\int_{0}^{x}\\!\\!\cdots\\!\\!\int_{0}^{x}}_{(N-1)-{\rm
fold}}\\!\\!\\!\\!p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x,\cdots,x_{N})$
$\displaystyle\times\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq i\end{subarray}},$ (11)
where $|\dot{h}_{i}|$ is the time derivative of the signal envelope at the
$i$-th port. Hence, from (5), we obtain the LCR as
$\displaystyle L(x_{\rm th})$
$\displaystyle=\int_{0}^{\infty}\dot{x}p_{|\dot{h}|,|h|}(\dot{x},x)d\dot{x}$
$\displaystyle=\int_{0}^{\infty}\\!\\!\\!\\!\dot{x}\sum_{i=1}^{N}p_{|\dot{h_{i}}|}(\dot{x})\\!\\!$
$\displaystyle\times\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\\!\\!\cdots\\!\\!\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}\\!\\!\\!\\!\\!\\!p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm
th},\cdots,x_{N})$ $\displaystyle\times\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq
i\end{subarray}}d\dot{x}.$ (12)
Furthermore, in case of an identically distributed Rayleigh fading scenario,
$p_{|\dot{h_{i}}|}(\dot{x})$ $\forall$ $i=1,\cdots,N,$ follows a zero mean
Gaussian PDF [9] with variance
$\sigma_{\dot{X}}^{2}=\pi^{2}\sigma^{2}f_{D}^{2}$, where $f_{D}$ is the
maximum Doppler frequency. As a result, we obtain
$\int_{0}^{\infty}\\!\\!\dot{x}p_{|\dot{h_{i}}|}(\dot{x})d\dot{x}=\frac{\sigma_{\dot{X}}}{\sqrt{2\pi}}=\sqrt{\frac{\pi}{2}}\sigma
f_{D},\>\>\forall i=1,\cdots,N.$ (13)
By combining (V-A) and (13), we have
$\displaystyle L(x_{\rm th})=\sqrt{\frac{\pi}{2}}\sigma f_{D}$
$\displaystyle\times\sum_{i=1}^{N}\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\cdots\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm th},\cdots,x_{N})$
$\displaystyle\times\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq i\end{subarray}}.$ (14)
The summation term in (V-A) is alternatively written as
$\displaystyle\sum_{i=1}^{N}\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\cdots\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm th},\cdots,x_{N})$
$\displaystyle\quad\times\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq i\end{subarray}}$
$\displaystyle=\underbrace{\int_{0}^{x_{\rm th}}\\!\\!\cdots\int_{0}^{x_{\rm
th}}}_{(N-1)-{\rm fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1}=x_{\rm
th},\cdots,x_{N})\underbrace{dx_{2}\cdots dx_{N}}_{(N-1)-{\rm fold}}$
$\displaystyle\quad+\sum_{i=2}^{N}\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\cdots\\!\\!\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}\\!\\!p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm
th},\cdots,x_{N})$ $\displaystyle\quad\times\underbrace{dx_{1}\cdots
dx_{k}\cdots dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq
i\end{subarray}}.$ (15)
The first term of (V-A) is evaluated as
$\displaystyle\underbrace{\int_{0}^{x_{\rm th}}\cdots\int_{0}^{x_{\rm
th}}}_{(N-1)-{\rm fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1}=x_{\rm
th},\cdots,x_{N})\underbrace{dx_{2}\cdots dx_{N}}_{(N-1)-{\rm fold}}$
$\displaystyle\overset{(a)}{=}\frac{2x_{\rm th}}{\sigma^{2}}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\int_{0}^{x_{\rm th}}\cdots\int_{0}^{x_{\rm
th}}\prod_{k=2}^{N}\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{\rm
th}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}$ $\displaystyle\quad\times
I_{0}\left({\frac{2~{}\mu_{k}x_{\rm
th}x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)dx_{2}\cdots dx_{N}$
$\displaystyle=\frac{2x_{\rm th}}{\sigma^{2}}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\prod_{k=2}^{N}\int_{0}^{x_{\rm
th}}\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{\rm
th}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}$ $\displaystyle\quad\times
I_{0}\left({\frac{2~{}\mu_{k}x_{\rm
th}x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)dx_{k}$
$\displaystyle\overset{(b)}{=}\frac{2x_{\rm th}}{\sigma^{2}}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\prod_{k=2}^{N}\left[1-Q_{1}\left(\sqrt{\frac{2\mu_{k}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th},\right.\right.$
$\displaystyle\quad\left.\left.\sqrt{\frac{2}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th}\right)\right],$ (16)
where $(a)$ follows from (II), $Q_{1}(\cdot,\cdot)$ is the first-order Marcum
$Q$-function, and $(b)$ follows from [13, Eq. 10]. Hereafter, the second term
of (V-A) is expanded as (V-A), where $(c)$ is based on [13, Eq. 10] and the
fact that the joint PDF stated in (II) is not a regular multivariate Rayleigh
PDF. This distribution is a product of $N$ pairs of bi-variate Rayleigh PDFs,
with the first port of the FAS being the reference point for all the remaining
$N-1$ ports. By combining (V-A), (V-A), (V-A), and (V-A), we obtain (III),
which completes the proof.
$\displaystyle\sum_{i=2}^{N}\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\cdots\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm
th},\cdots,x_{N})\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq i\end{subarray}}$
$\displaystyle=\sum_{i=2}^{N}\int_{0}^{x_{\rm
th}}\\!\\!\cdots\\!\\!\int_{0}^{x_{\rm th}}\\!\\!\frac{2x_{\rm
th}}{\sigma^{2}(1-\mu_{i}^{2})}e^{-\frac{x_{\rm
th}^{2}+\mu_{i}^{2}x_{1}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}I_{0}\left({\frac{2~{}\mu_{i}x_{\rm
th}x_{1}}{\sigma^{2}(1-\mu_{i}^{2})}}\right)\\!\\!\prod_{\begin{subarray}{c}k=1\\\
k\neq
i\end{subarray}}^{N}\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{1}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}I_{0}\left({\frac{2~{}\mu_{k}x_{k}x_{1}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)dx_{1}\cdots
dx_{k}\cdots dx_{N}$ $\displaystyle\overset{(c)}{=}\sum_{i=2}^{N}\frac{2x_{\rm
th}}{\sigma^{2}(1-\mu_{i}^{2})}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}\\!\\!\int_{0}^{x_{\rm
th}}\frac{2x_{1}}{\sigma^{2}}e^{-\frac{x_{1}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}I_{0}\left(\\!{\frac{2~{}\mu_{i}x_{\rm
th}x_{1}}{\sigma^{2}(1-\mu_{i}^{2})}}\\!\right)\\!\\!\prod_{\begin{subarray}{c}k=2\\\
k\neq
i\end{subarray}}^{N}\left[\\!1-Q_{1}\left(\sqrt{\frac{2\mu_{k}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}x_{1},\sqrt{\frac{2}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th}\right)\\!\right]dx_{1}.$ (17)
### V-B Proof of Corollary 2
The case of $\mu_{k}=1$ $\forall$ $k$ implies identical channels at all the
ports. As a result, we obtain
$\displaystyle L(x_{\rm th})$
$\displaystyle=\int_{0}^{\infty}\\!\\!\dot{x}p_{|\dot{h}|,|h|}(\dot{x},x_{\rm
th})d\dot{x}\overset{(a)}{=}p_{|h|}(x_{\rm
th})\\!\\!\int_{0}^{\infty}\dot{x}p_{|\dot{h}|}(\dot{x})d\dot{x}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{2x_{\rm
th}}{\sigma^{2}}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\\!\\!\int_{0}^{\infty}\\!\\!\dot{x}p_{|\dot{h_{i}}|}(\dot{x})d\dot{x}\overset{(b)}{=}\frac{\sqrt{2\pi}}{\sigma}f_{D}x_{\rm
th}e^{-\frac{x_{\rm th}^{2}}{\sigma^{2}}},$ (18)
where $(a)$ follows from
$p_{|\dot{h}|,|h|}(\dot{x},x)=p_{|\dot{h}|}(\dot{x})p_{|h|}(x)$ [9, Eq. 2.97]
and $(b)$ follows from (13). Hence, the proof.
### V-C Proof of Corollary 3
Since we are considering a two-port scenario, we take into account the joint
PDF of the channel at these two ports. Hence, by replacing $N=2$ in (II), the
joint PDF becomes
$\displaystyle p_{|h_{1}|,|h_{2}|}(x_{1},x_{2})$
$\displaystyle=\frac{4x_{1}x_{2}}{\sigma^{4}(1-\mu_{2}^{2})}e^{-\frac{x_{1}^{2}+x_{2}^{2}}{\sigma^{2}(1-\mu_{2}^{2})}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\times
I_{0}\left({\frac{2\mu_{2}x_{1}x_{2}}{\sigma^{2}(1-\mu_{2}^{2})}}\right)\\!,\quad\text{for}\quad
x_{1},x_{2}\geq 0.$ (19)
Thus, by replacing $N=2$ in (III) and after some trivial algebraic
manipulations, we obtain
$\displaystyle L(x_{\rm th})$ $\displaystyle=\sqrt{\frac{\pi}{2}}\sigma
f_{D}\left(\int_{0}^{x_{\rm th}}p_{|h_{1}|,|h_{2}|}(x_{\rm
th},x_{2})dx_{2}\right.$ $\displaystyle\quad\left.+\int_{0}^{x_{\rm
th}}p_{|h_{1}|,|h_{2}|}(x_{1},x_{\rm th})dx_{1}\right)$
$\displaystyle\overset{(a)}{=}\sqrt{2\pi}\sigma f_{D}\\!\\!\int_{0}^{x_{\rm
th}}p_{|h_{1}|,|h_{2}|}(x_{\rm th},x_{2})dx_{2}$
$\displaystyle=\frac{4\sqrt{2\pi}\sigma f_{D}x_{\rm
th}}{\sigma^{4}(1-\mu^{2})}\exp\left(\\!\\!-\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu^{2})}\right)\int_{0}^{x_{\rm th}}\\!\\!x_{2}$
$\displaystyle\quad\times\\!\exp\left(\\!\\!-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\\!\\!\right)\\!\\!I_{0}\left(\\!{\frac{2\mu
x_{\rm th}x_{2}}{\sigma^{2}(1-\mu^{2})}}\\!\right)\\!dx_{2},$ (20)
where $(a)$ follows from (V-C) and the integral $\int_{0}^{x_{\rm
th}}x_{2}\\!\exp\left(\\!\\!-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\\!\\!\right)I_{0}\left(\\!{\frac{2\mu
x_{\rm th}x_{2}}{\sigma^{2}(1-\mu^{2})}}\\!\right)\\!dx_{2}$ is obtained as
$\displaystyle\int_{0}^{x_{\rm
th}}x_{2}\exp\left(-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\right)I_{0}\left({\frac{2\mu
x_{\rm th}x_{2}}{\sigma^{2}(1-\mu^{2})}}\right)dx_{2}$
$\displaystyle\overset{(b)}{=}\\!\\!\int_{0}^{x_{\rm
th}}\\!\\!x_{2}\exp\left(\\!\\!-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\right)\\!\\!\sum_{k=0}^{\infty}\frac{1}{(k!)^{2}}\left(\frac{\mu
x_{\rm th}x_{2}}{\sigma^{2}(1-\mu^{2})}\right)^{\\!\\!2k}\\!\\!dx_{2}$
$\displaystyle\overset{(c)}{=}\sum_{k=0}^{\infty}\frac{1}{(k!)^{2}}\left(\frac{\mu
x_{\rm th}}{\sigma^{2}(1-\mu^{2})}\right)^{2k}$
$\displaystyle\quad\times\int_{0}^{x_{\rm
th}}\exp\left(-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\right)x_{2}^{2k+1}dx_{2}$
$\displaystyle=\frac{1}{2}\\!\sum_{k=0}^{\infty}\\!\frac{\left(\mu x_{\rm
th}\right)^{2k}}{(k!)^{2}\left(\sigma^{2}(1-\mu^{2})\right)^{k-1}}\gamma\left(\\!k+1,\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu^{2})}\\!\right).$ (21)
Here $(b)$ follows from [14, 8.445] and by assuming $0<\mu<1$, $(c)$ follows
from changing the order of summation and integration, and
$\gamma(\cdot,\cdot)$ denotes the lower incomplete Gamma function. As a
result, by substituting (V-C) in (V-C), we finally obtain (3).
## Acknowledgment
This work was co-funded by the European Regional Development Fund and the
Republic of Cyprus through the Research and Innovation Foundation, under the
project INFRASTRUCTURES/1216/0017 (IRIDA). It has also received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant agreement No. 819819).
## References
* [1] A. Goldsmith, _Wireless communications_. Cambridge University Press, 2005.
* [2] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna systems,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 3, pp. 1950–1962, Mar. 2021.
* [3] A. Dey, R. Guldiken, and G. Mumcu, “Microfluidically reconfigured wideband frequency-tunable liquid-metal monopole antenna,” _IEEE Trans. Antennas Propag._ , vol. 64, no. 6, pp. 2572–2576, June 2016.
* [4] C. Borda-Fortuny, K.-F. Tong, A. Al-Armaghany, and K.-K. Wong, “A low-cost fluid switch for frequency-reconfigurable vivaldi antenna,” _IEEE Antennas Wireless Propag. Lett._ , vol. 16, pp. 3151–3154, 2017.
* [5] A. Singh, I. Goode, and C. E. Saavedra, “A multistate frequency reconfigurable monopole antenna using fluidic channels,” _IEEE Antennas Wireless Propag. Lett._ , vol. 18, no. 5, pp. 856–860, 2019.
* [6] I. Bahceci, T. Duman, and Y. Altunbasak, “Antenna selection for multiple-antenna transmission systems: performance analysis and code construction,” _IEEE Trans. Inf. Theory_ , vol. 49, no. 10, pp. 2669–2681, Oct. 2003.
* [7] K. K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Performance limits of fluid antenna systems,” _IEEE Commun. Lett._ , vol. 24, no. 11, pp. 2469–2472, Nov. 2020.
* [8] S. O. Rice, “Mathematical analysis of random noise,” _Bell Syst. Tech. J._ , vol. 24, pp. 46–156, 1945.
* [9] G. L. Stüber, _Principles of Mobile Communications_. Boston: Kluwer Academic Publishers, 1996.
* [10] C.-D. Iskander and P. Takis Mathiopoulos, “Analytical level crossing rates and average fade durations for diversity techniques in Nakagami fading channels,” _IEEE Trans. Commun._ , vol. 50, no. 8, pp. 1301–1309, Aug. 2002\.
* [11] A. S. Panajotović, M. . Stefanović, D. L. Draca, and N. M. Sekulović, “Average level crossing rate of dual selection diversity in correlated Rician fading with Rayleigh cochannel interference,” _IEEE Commun. Lett._ , vol. 14, no. 7, pp. 605–607, July 2010.
* [12] L. Yang and M.-S. Alouini, _Average outage duration of wireless communication systems_. Kluwer Academic Publishers, 2004, ch. 8, pp. 209–240.
* [13] G. Corazza and G. Ferrari, “New bounds for the Marcum Q-function,” _IEEE Trans. Inf. Theory_ , vol. 48, no. 11, pp. 3003–3008, Nov. 2002.
* [14] I. S. Gradshteyn and I. M. Ryzhik., _Table of Integrals, Series, and Products_. Elsevier, 2007.
|
# Reconfigurable Intelligent Surface-induced Randomness for mmWave Key
Generation
Shubo Yang, Han Han, Yihong Liu, , Weisi Guo, , Zhibo Pang, and Lei Zhang S.
Yang is with the Glasgow College, University of Glasgow (e-mail:
2429400y@student.gla.ac.uk). H. Han is with Electrical and Computer
Engineering Department, University of Toronto, ON M5S 3G4, Canada (e-mail:
johnny.han@mail.utoronto.ca). Y. Liu and L. Zhang are with James Watt School
of Engineering, University of Glasgow, Glasgow G12 8QQ, U.K. (e-mail:
<EMAIL_ADDRESS>lei.zhang@glasgow.ac.uk). W. Guo is with School of
Aerospace, Transport and Manufacturing, Cranfield University, MK43 0AL
Cranfield, U.K. (e-mail: weisi.guo@cranfield.ac.uk). Z. Pang is with
Department of Automation Technology, ABB Corporate Research Sweden, Vasteras,
Sweden, and Department of Intelligent Systems, Royal Institute of Technology
(KTH), Stockholm, Sweden. (e-mail<EMAIL_ADDRESS>zhibo@kth.se).
Funding information: Zhibo Pang’s work is partly funded by the Swedish
Foundation for Strategic Research (SSF) through the project APR20-0023.
###### Abstract
Secret key generation in physical layer security exploits the unpredictable
random nature of wireless channels. The millimeter-wave (mmWave) channels have
limited multipath and channel randomness in static environments. In this
paper, for mmWave secret key generation of physical layer security, we use a
reconfigurable intelligent surface (RIS) to induce randomness directly in
wireless environments, without adding complexity to transceivers. We consider
RIS to have continuous individual phase shifts (CIPS) and derive the RIS-
assisted reflection channel distribution with its parameters. Then, we propose
continuous group phase shifts (CGPS) to increase the randomness specifically
at legal parties. Since the continuous phase shifts are expensive to
implement, we analyze discrete individual phase shifts (DIPS) and derive the
corresponding channel distribution, which is dependent on the quantization
bit. We then derive the secret key rate (SKR) to evaluate the randomness
performance. With the simulation results verifying the analytical results,
this work explains the mathematical principles and lays a foundation for
future mmWave evaluation and optimization of artificial channel randomness.
###### Index Terms:
Physical layer security, secret key generation, reconfigurable intelligent
surface, intelligent reflecting surface.
## I Introduction
Wireless networks are becoming ubiquitous nowadays and in the future Internet
of Things (IoT) systems. However, their broadcast nature makes them vulnerable
to malicious attacks. Classic encryption schemes, such as advanced encryption
standard and public key cryptography, are dependent on cryptography
computation techniques [1]. The applications of classic schemes to IoT devices
and wireless sensor networks (WSNs) bring challenges, since the devices and
sensor nodes have small sizes and limited computational capability. Thus,
extensive research is carried out in secret key generation of physical layer
security, where the legitimate users extract keys from their correlated
observations of the reciprocal channel in a lightweight manner [2]. The
correlation of channels makes it possible to generate keys without key
exchange, and the dynamic uniqueness of the channel prevents eavesdroppers
from mimicking. While in dynamic environments the movements of users or
objects are sufficient to produce randomness, the randomness is limited in
static environments, such as in open terrain with no moving objects. Besides,
the millimeter wave (mmWave) communication is envisioned as a significant
technology for the fifth generation (5G) networks and beyonds [3]. Thus, the
security in mmWave static environments needs to be researched.
The newly developed reconfigurable intelligent surface (RIS), also known as
Intelligent Reflecting Surface (IRS), has the potential to produce artificial
randomness in mmWave static environments. RIS is a two-dimensional surface
consisting of a large number of passive low-cost reflecting elements [4]. Each
scattering element of RIS is independently capable of altering the amplitude
and/or phase of the incident signals. Additionally, RIS becomes more important
in high-frequency band communications, e.g., mmWave and THz communications
that have severe coverage issues. The existing RIS applications mainly target
indoor static scenarios [5]. Therefore, RIS can be easily incorporated in
mmWave static environments. By adding RIS, artificial randomness can be
produced. The randomness does not rely on dynamic environments and is produced
directly in the channel, without needing increased transceiver costs.
Research has been done on RIS-assisted key generation. Random shifting RIS is
applied to increase the secure transmission rate, and the time allocation for
key generation and transmission is designed [6]. RIS with discrete phase
shifts is adopted to generate secret keys, and the secret key rate (SKR) is
derived [7]. The practical implementation of using RIS in the OFDM system is
conducted [8]. However, most existing literature focuses on sub-6 GHz systems
and models on Gaussian channels. The mmWave channels have a poor scattering
nature and exhibit limited multipath, so they may not conform to Gaussian
channels. Moreover, the channel distribution resulted from RIS phase shifts
and element number is still unknown. Besides, most previous works assume each
RIS element phase shift is independent and identically distributed (i.i.d),
without changing phase shift distributions to improve randomness.
Therefore, in this paper, the RIS-assisted mmWave key generation is proposed.
We model RIS-induced randomness in mmWave key generation, and we focus on the
fundamental analytical derivations of channel distribution with random RIS
weights. For the random weights, we consider applying continuous individual
phase shifts (CIPS) on each element and continuous group phase shifts (CGPS)
on elements in groups to produce higher randomness. In addition, we consider
channels for both continuous111The continuous RIS weights are difficult to
implement. This is because more levels of weight phase shift result in more
costs, which is not scalable to a large number of elements [9]. The continuous
weights’ performance is the upper limit for discrete weights when weights’
quantization bits approach infinity. and discrete phase shifts and compare
their performance. To summarize, the main contributions of this paper are as
follows.
* •
The RIS-induced channel distribution and its parameters are derived, given RIS
is a uniform rectangular array (URA) and the elements have CIPS. As the
result, an artificial Rayleigh/Rician fading is induced directly in the
environments.
* •
To increase the amount of induced randomness, CGPS is proposed and the channel
is derived. The channel variance for legal parties increases, and there is a
tradeoff between group number and group size. The discrete individual phase
shifts (DIPS) is also discussed based on quantization bits.
* •
The SKR is derived for CIPS, CGPS, and DIPS, to evaluate the performance of
artificial randomness.
_Notations:_ Bold-faced letters are used to denote matrix or column vectors,
while lightfaced letters are used to denote scalar quantities. Superscripts
$(\cdot)^{T}$, $(\cdot)^{*}$, and $(\cdot)^{H}$ represent the transpose,
conjugate and conjugate transpose operations, respectively. $\odot$ denotes
the point-wise multiplication. We use the notations shown in Table I in this
paper.
TABLE I: Notations
Symbol | Definition
---|---
$\lambda$ | Wavelength
$k$ | Wave number, where $k=\frac{2\pi}{\lambda}$
$d$ | Element spacing in RIS
$M$ | Element number of the RIS
$m$ | An integer in range $[1,M]$
$\phi_{m}$ | The phase shift of the $m$-th RIS element weight
$\psi$ | The incident or reflected azimuth angle
$\theta$ | The incident or reflected elevation angle
$B$ | RIS discrete weight quantization bit
$\bm{R}$ | The covariance between the real and imaginary parts of a complex Gaussian distribution
$R_{s}$ | Secret key rate
$U(a,b)$ | Indicate a uniform distribution on interval $(a,b)$
$N(\mu,\sigma^{2})$ | Indicate a Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$
$h_{(\cdot)}$ | Indicate the channel
$(\cdot)_{x},(\cdot)_{y}$ | Indicate physical quantities on $x$-axis and $y$-axis, respectively
$(\cdot)_{real},(\cdot)_{imag}$ | Indicate quantities of the real and imaginary parts, respectively
$(\cdot)_{cos},(\cdot)_{sin}$ | Indicate quantities of cosine and sine functions, respectively
$(\cdot)_{i},(\cdot)_{o}$ | Indicate incident and reflected angles, respectively
$(\cdot)_{CI}$ | Indicate quantities when RIS weights have continuous individual phase shifts
$(\cdot)_{CG}$ | Indicate quantities when RIS weights have continuous group phase shifts
$(\cdot)_{DI}$ | Indicate quantities when RIS weights have discrete individual phase shifts
## II System Model
Consider an RIS-assisted wireless communication between legitimate parties,
Alice and Bob. There is also an eavesdropper Eve, who passively listens to the
communication. Assume all the parties are operated under a mmWave static
environment. We consider the narrowband line-of-sight (LOS) transmission in
the far field between Alice/Bob and RIS.
Figure 1: RIS-assisted secret key generation model.
This assumption can be justified since the multipath effect is constrained and
the high path loss makes non-line-of-sight (NLOS) paths’ power small. The RIS
provides cascaded LOS paths. Additionally, the path loss is a constant related
to distance and can be easily predicted by eavesdropper [7]. Since the
transmission channel between Alice and Bob is reciprocal, we only consider the
situation where Alice is transmitting signals to Bob to avoid redundancy.
As shown in Fig. 1. Alice and Bob aim at establishing a random secret
cryptographic key based on their reciprocal communication channel and reducing
the key leakage to Eve. They measure the common channel through their
exchanged signals and generate keys using parameters of the received signals,
e.g. channel state information (CSI) and received signal strength (RSS). When
Alice transmits signal $s$, the received signal at Bob can be expressed as
$y=(h_{ab}+\bm{h_{rb}^{H}}\bm{W}\bm{h_{ar}})s+z\ ,$ (1)
where $h_{ab}\in\mathbb{C}^{1\times 1}$, $\bm{h_{ar}}\in\mathbb{C}^{M\times
1}$, $\bm{h_{rb}}\in\mathbb{C}^{M\times 1}$ represent the direct channel
between Alice and Bob, the channel between Alice and RIS, and the channel
between RIS and Bob, $z\sim\mathcal{CN}(0,\sigma_{z}^{2})$ is the additive
white Gaussian noise (AWGN), and $\bm{W}$ is the diagonal weight matrix with
each entity on the diagonal being the RIS weight of each surface element. The
weight of the $m$-th surface element is represented as $e^{j\phi_{m}}$.
To generate shared keys in static environments and reduce the key leakage to
Eve, it is crucial to increase the channel randomness through RIS. As in (1),
though $\bm{h_{ar}}$ and $\bm{h_{rb}}$ are static in mmWave static
environments, when RIS weights $\bm{W}$ vary,
$\bm{h_{rb}^{H}}\bm{W}\bm{h_{ar}}$ becomes dynamic. Thus, RIS becomes the
major source of channel randomness through varying the reflection channel
$\widetilde{H}=\bm{h_{rb}^{H}}\bm{W}\bm{h_{ar}}$. In this case, we focus on
$\widetilde{H}$, which can be expressed using steering vectors and weight
vector [10] as
$\widetilde{H}=\bm{w}^{H}\cdot[\bm{a}(\bm{\Omega}_{i})\odot\bm{a}(\bm{\Omega}_{o})]\
,$ (2)
where $\bm{w}$ is the weight vector with each entity being the diagonal entity
of $\bm{W}$, i.e., weight of each surface element. Besides, the steering
vector of the channels between Alice and RIS and between RIS and Bob are
$\bm{a}(\bm{\Omega}_{i})=[a(\bm{\Omega}_{i,1}),\cdots,a(\bm{\Omega}_{i,m}),\cdots,a(\bm{\Omega}_{i,M})]^{T}$
(3)
$\bm{a}(\bm{\Omega}_{o})=[a(\bm{\Omega}_{o,1}),\cdots,a(\bm{\Omega}_{o,m}),\cdots,a(\bm{\Omega}_{o,M})]^{T}\
,$ (4)
where $\bm{\Omega}_{i,m}$ and $\bm{\Omega}_{o,m}$ are the terms characterizing
the incident spatial information from Alice at RIS $m$-th element and the
reflected spatial information from $m$-th element to Bob, respectively. Since
the environment is static, $\bm{\Omega}_{i}$ and $\bm{\Omega}_{o}$ are fixed,
and $\widetilde{H}$ becomes a function of solely $\bm{w}$. Note that averaging
$\widetilde{H}$ derived from all possible pairs of $\bm{\Omega}_{i}$ and
$\bm{\Omega}_{o}$ provides an estimation of the overall channel distribution.
## III RIS-Assisted Key Generation
In this section, we first derive the probability density function (p.d.f) of
reflection channel $\widetilde{H}$ with a given random RIS having CIPS and
CGPS. Individual shifting means that the RIS phase shifts
$\\{\bm{\phi}=[\phi_{1},\cdots,\phi_{m},\cdots,\phi_{M}],m\in[1,M]\\}$ are
i.i.d. Group shifting means the elements phase shifts in a group are first
implemented maximum ratio transmission (MRT) and then added a random phase
shift. Next, we consider DIPS. The channel for 1-bit is found different from
other quantization bits. Since group phase shifts need MRT to compensate for
the phase difference caused by element distance, the discrete group phase
shifts (DGPS) need multiple levels to realize the compensation. Implementing
multiple-level discrete phase shifts is expensive compared to low-level phase
shifts. Thus, we only discuss CGPS, and DGPS will be our future work.
Note that a uniform linear array (ULA) can be considered a special case of URA
while URA is a typical planar array structure. Thus, we consider RIS in URA
configuration, which can be further extended into other complex RIS shapes.
### III-A Continuous Individual Phase Shifts
Consider a URA RIS with $M_{x}$ and $M_{y}$ elements equally spaced on x-axis
and y-axis, respectively, where the total element number is $M=M_{x}\cdot
M_{y}$. The incident and reflected angles are described by
$\bm{\Omega}_{i}=(\psi_{i},\theta_{i})$ and
$\bm{\Omega}_{o}=(\psi_{o},\theta_{o})$. Assume the i.i.d RIS weight phase
shifts $\bm{\phi}$ are continuous uniformly distribution $\phi_{m}\sim
U(0,2\pi)$. Then, the multiplication of steering vectors in (2) with elements
spacing along x-axis $d_{x}$ and y-axis $d_{y}$ can be expressed as
$\displaystyle\bm{a}(\bm{\Omega}_{i})\odot\bm{a}(\bm{\Omega}_{o})=$
$\displaystyle[1,\cdots,e^{j(\xi_{x}\cdot(m_{x}-1)+\xi_{y}\cdot(m_{y}-1))},$
(5)
$\displaystyle\cdots,e^{j(\xi_{x}\cdot(M_{x}-1)+\xi_{y}\cdot(M_{y}-1))}]^{T}\
,$
where $m_{x}\in[1,M_{x}]$ and $m_{y}\in[1,M_{y}]$ are integers [11].
Additionally, $k=\frac{2\pi}{\lambda}$ is the wave number and
$\xi_{x}=kd_{x}(\cos\psi_{i}\sin\theta_{i}+\cos\psi_{o}\sin\theta_{o})\ ,$ (6)
$\xi_{y}=kd_{y}(\sin\psi_{i}\sin\theta_{i}+\sin\psi_{o}\sin\theta_{o})\ .$ (7)
Consequently, the reflection channel following (2) can be constructed as
$\displaystyle\widetilde{H}\\!\\!=$ $\displaystyle
1+\cdots+\\!e^{j[\phi_{m}+(\xi_{x}\cdot(m_{x}-1)+\xi_{y}\cdot(m_{y}-1))]}$ (8)
$\displaystyle+\cdots+e^{j[\phi_{M}+(\xi_{x}\cdot(M_{x}-1)+\xi_{y}\cdot(M_{y}-1))]}$
$\displaystyle=$
$\displaystyle\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}e^{j(\phi_{m}+\alpha)}$
$\displaystyle=$
$\displaystyle\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\cos(\phi_{m}+\alpha)+j\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\sin(\phi_{m}+\alpha)\
,$
where
$m=(m_{y}-1)\cdot M_{x}+m_{x}\ ,\
\alpha=(m_{x}-1)\cdot\xi_{x}+(m_{y}-1)\cdot\xi_{y}\ .$ (9)
According to the central limit theorem (CLT), $\widetilde{H}$ converges to a
complex Gaussian distribution when $M$ is large enough. This assumption could
be justified since a practical RIS usually has an extremely large number of
elements, e.g. more than tens of elements [5]. Therefore, the real and
imaginary parts both converge to Gaussian distributions that are fully
determined by means and variances. The mean and variance of the real part of
$\widetilde{H}_{CI}$ can be expressed as
$\mu_{real,CI}\\!=\\!\sum_{m_{y}=1}^{M_{y}}\\!\\!\sum_{m_{x}=1}^{M_{x}}\\!\\!\mu_{cos,CI}\
,\
\sigma^{2}_{real,CI}\\!=\\!\\!\sum_{m_{y}=1}^{M_{y}}\\!\\!\sum_{m_{x}=1}^{M_{x}}\\!\\!\sigma^{2}_{cos,CI}\
,$ (10)
where
$\mu_{cos,CI}=\int_{0}^{2\pi}\cos(\phi_{m}+\alpha)f(\phi_{m})d\phi_{m}=0\ .$
(11)
$\sigma^{2}_{cos,CI}\\!\\!=\\!\\!\int_{0}^{2\pi}\\!\\!\\!\\!\cos^{2}(\phi_{m}+\alpha)f(\phi_{m})d\phi_{m}-\mu_{cos,CI}^{2}\\!=\\!\frac{1}{2}.$
(12)
The imaginary part can be derived similar to the real part. By substituting
(11) (12) into (10), the channel real and imaginary parts are
$\Re{(\tilde{H}_{CI})}\sim N(0,\frac{M}{2})\ ,\ \Im{(\tilde{H}_{CI})}\sim
N(0,\frac{M}{2})\ .$ (13)
Since the real and imaginary parts of $\widetilde{H}_{CI}$ constitute a joint
Gaussian distribution and the covariance of them equals 0, the real and
imaginary parts are independent. The distribution of $\widetilde{H}_{CI}$ is
also independent of $\bm{\Omega}_{i}$ and $\bm{\Omega}_{o}$. With any pair of
incident and reflected angles, the mean of $\widetilde{H}_{CI}$ is
$\mu_{CI}=0$ while the variance of $\widetilde{H}_{CI}$ can be calculated as
$\sigma^{2}_{CI}=\mathbb{E}[(\widetilde{H}_{CI}-\mu_{CI})(\widetilde{H}_{CI}-\mu_{CI})^{*}]=M\
.$ (14)
Due to the channel parameters CSI and RSS are normally used to generate keys,
the magnitude and phase distributions are also important. According to the
joint Gaussian distribution, the result magnitude p.d.f and phase p.d.f of
$\widetilde{H}_{CI}$ are Rayleigh and uniform distributions. When the direct
path and reflection path are both considered, the direct path adds a non-zero
mean, and the induced Rayleigh fading will turn into Rician fading. Hence, the
CIPS RIS produces randomness by inducing an artificial Rayleigh/Rician fading
in the environments.
### III-B Continuous Group Phase Shifts
Different from the i.i.d elements in CIPS, group shifting means first using
MRT to achieve maximum transmission rate at Alice and Bob, then shifting the
phase randomly. This induces more randomness at Alice and Bob.
The total elements $M$ is divided into groups of $q$ elements. There are
$N=\lfloor\frac{M}{q}\rfloor$ groups in total. The $n$-th group has the added
random phase $\phi_{n,rand}\sim U(0,2\pi)$. When the $m$-th element is in
$n$-th group, the phase shift for $m$-th element is
$\phi_{m,CG}=\phi_{m,MRT}+\phi_{n,rand}\ ,$ (15)
where
$\phi_{m,MRT}=\frac{[\bm{a}(\bm{\Omega}_{i,m})\cdot\bm{a}(\bm{\Omega}_{o,m})]^{H}}{||\bm{a}(\bm{\Omega}_{i,m})\cdot\bm{a}(\bm{\Omega}_{o,m})||}\
.$ (16)
The $\phi_{m,MRT}$ compensates the steering vector phase difference of
elements in the same group $n$ to achieve larger signal strength. The random
phase shift $\phi_{n,rand}$ produces the randomness. Then, according to the
channel in (8), the channel for CGPS can be expressed as
$\widetilde{H}_{CG}=\sum_{n=1}^{N}q\cos(\phi_{n,rand})+\
j\sum_{n=1}^{N}q\sin(\phi_{n,rand})\ .$ (17)
When RIS has a large number of elements, the number of groups $N$ can be large
when given a proper value of $q$. Then, $\widetilde{H}_{CG}$ converges to a
complex Gaussian distribution. Similarly to Section III-A, the mean
$\mu_{real,CG}=0$ and the variance of the real part can be expressed as
$\displaystyle\sigma^{2}_{real,CG}=\sum_{n=1}^{N}\sigma^{2}[q\cos(\phi_{n,rand})]=\frac{Nq^{2}}{2}\
.$ (18)
Thus, the channel distribution for CGPS RIS is
$\Re{(\tilde{H}_{CG})}\sim N(0,\frac{Nq^{2}}{2})\ ,\Im{(\tilde{H}_{CG})}\sim
N(0,\frac{Nq^{2}}{2})\ .$ (19)
The channel variance can be then calculated as
$\sigma^{2}_{CG}=Nq^{2}\ .$ (20)
There is a tradeoff between $N=\lfloor\frac{M}{q}\rfloor$ and $q$. When $q$ is
large the variance increases quadratically, but $N$ may become too small to
use CLT. The Gaussian distribution is the distribution that maximizes the
entropy at a given variance. The CGPS variance $\sigma^{2}_{CG}=Nq^{2}$ can
achieve larger randomness than the CIPS at Alice and Bob.
### III-C Discrete Individual Phase Shifts
Since continuous phase shifts are expensive to implement, discrete phase
shifts RIS need be researched. Assume RIS has discrete weight phase shifts
$\bm{\phi}$ with quantization bit $B$. $\bm{\phi}$ is i.i.d uniformly on
discrete values
$\\{0,\frac{2\pi}{2^{B}},\cdots,\frac{2\pi(2^{B}-1)}{2^{B}}\\}$. When $B$
approaches to infinity, the channel distribution approaches to the
distribution assisted by RIS with continuous phase shifts. Similar to the
CIPS, the DIPS reflection channel can be expanded as in (8) and (10). The mean
and variance of the real part $\mu_{cos,DI}$ and $\sigma^{2}_{cos,DI}$ are
expressed as
$\mu_{cos,DI}=\mathbb{E}\\{\cos(\phi_{m}+\alpha)\\}=0\ ,$ (21)
$\displaystyle\\!\\!\sigma^{2}_{cos,DI}$
$\displaystyle=\\!\mathbb{E}\\{\cos^{2}(\phi_{m}+\alpha)\\}-\mu_{cos,DI}^{2}$
(22)
$\displaystyle=\\!\frac{1}{2}+\frac{1}{2}\mathbb{E}\\{\cos(2\phi_{m}+2\alpha)\\}$
$\displaystyle=\\!\\!\frac{1}{2}\\!\\!+\\!\\!\frac{1}{2}\cos(2\alpha)\mathbb{E}\\{\\!\cos(2\phi_{m})\\!\\}\\!-\\!\frac{1}{2}\sin(2\alpha)\mathbb{E}\\{\\!\sin(2\phi_{m})\\!\\}\\!\
\\!,$
where the means of $\cos(2\phi_{m})$ and $\sin(2\phi_{m})$ determine
$\sigma^{2}_{cos,DI}$. According to quantization bits $B$, there are two cases
of $\sigma^{2}_{cos,DI}$.
#### III-C1 $B\geq 2$
$\mathbb{E}\\{\cos(2\phi_{m})\\}=0$ and $\mathbb{E}\\{\sin(2\phi_{m})\\}=0$,
which leads to $\sigma^{2}_{cos,DI}=\frac{1}{2}$. Therefore, same as CIPS, the
channel is a complex Gaussian random variable with real and imaginary parts
being same as in (13), that are, $\Re{(\tilde{H}_{DI})}\sim N(0,\frac{M}{2})$
and $\Im{(\tilde{H}_{DI})}\sim N(0,\frac{M}{2})$. The variance is the same as
in (14) $\sigma^{2}_{DI}=M$.
#### III-C2 $B=1$
$\mu_{real,DI}=\mu_{imag,DI}=0$. Different from $B\geq 2$ case, since
$\phi_{m}$ only takes values $\\{0,\pi\\}$,
$\mathbb{E}\\{\cos(2\phi_{m})\\}=1$ and $\mathbb{E}\\{\sin(2\phi_{m})\\}=0$.
Thus, according to (22), the real and imaginary parts of $\widetilde{H}_{DI}$
can be expressed as
$\displaystyle\Re{(\tilde{H}_{DI})}\sim
N(0,\frac{M}{2}+\frac{1}{2}\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\cos(2\alpha))\
,$ (23) $\displaystyle\Im{(\tilde{H}_{DI})}\sim
N(0,\frac{M}{2}-\frac{1}{2}\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\cos(2\alpha))\
.$
The real and imaginary parts of $\widetilde{H}_{DI}$ are correlated, with
covariance $\mathbf{R}_{DI}$ expressed as
$\displaystyle\mathbf{R}_{DI}$
$\displaystyle=\mathbb{E}\\{\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\cos(\phi_{m}+\alpha)\sin(\phi_{m}+\alpha)\\}$
(24)
$\displaystyle=\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\frac{1}{2}\sin(2\alpha)\
.$
The real and imaginary parts are independent when $\mathbf{R}_{DI}=0$. Assume
$m_{x}=m_{y}$ and $d_{x}=d_{y}$, which is a typical configuration of URA RIS.
Then, $\mathbf{R}_{DI}=0$ when the following condition is satisfied
$\gamma=\frac{b\pi}{2kd_{x}}\ ,b\in\mathbb{Z}\ ,$ (25)
$\gamma=\cos\psi_{i}\sin\theta_{i}+\cos\psi_{o}\sin\theta_{o}+\sin\psi_{i}\sin\theta_{i}+\sin\psi_{o}\sin\theta_{o}\
.$ (26)
Additionally, the reflection channel variance is $\sigma^{2}_{DI}\leq M$. The
less-equal sign is due to the covariance between the real and imaginary parts.
When $\mathbf{R}_{DI}=0$, 1-bit RIS provides the same variance. The magnitude
and phase distributions for 1-bit DIPS can be turned into deriving the
envelope and phase of correlated Gaussian quadratures according to [12].
## IV Secret Key Rate
SKR is the upper bound of the information bits per channel sample generated
between Alice and Bob. Since when Eve is several wavelengths away from Alice
and Bob, its channel is considered uncorrelated with the channel between Alice
and Bob, which is tested in practical experiments [13]. The SKR can be
expressed as
$R_{s}=I(y_{A};y_{B}|y_{E})=I(y_{A};y_{B})\ ,$ (27)
where $y_{A}$, $y_{B}$, and $y_{E}$ are the channels estimated by Alice, Bob,
and Eve, respectively.
### IV-A CIPS, CGPS, and $B\geq 2$ DIPS
The mutual information between $y_{A}$ and $y_{B}$ is determined by their
variances. The direct channel adds a mean to the reflection channel, without
influencing the variance. Therefore, based on the channel expressions in (1)
and Section III-A, the SKR $R_{s}$ for CIPS and $B\geq 2$ DIPS can be derived
as
$R_{s,CI}=\log_{2}(1+\frac{M/2}{2\sigma_{z}^{2}+\frac{2\sigma_{z}^{4}}{M}})\
,\vspace{-3mm}$ (28)
where $\sigma_{z}^{2}$ is the noise power.
The SKR for CGPS can be calculated as (28) by substituting $\sigma^{2}_{CI}=M$
by $\sigma^{2}_{CG}=Nq^{2}$.
(a) Variance when RIS has different element number.
(b) Magnitude distribution.
(c) Phase distribution.
Figure 2: The channel variance of CIPS, $q=2,4$ CGPS, and $B=3$ DIPS. The
magnitude distribution and phase distribution of the CIPS and $B=3$ DIPS.
### IV-B $B=1$ DIPS
The estimated SKR under $B=1$ DIPS can be expressed as
$\displaystyle R_{s,DI}$
$\displaystyle\leq\\{I(\Re{(y_{A})};\Re{(y_{B})})+I(\Im{(y_{A})};\Im{(y_{B})})\\}$
(29)
$\displaystyle=\log_{2}\\{(1\\!+\\!\frac{\sigma^{2}_{real,DI}}{2\sigma_{z}^{2}+\frac{\sigma_{z}^{4}}{\sigma^{2}_{real,DI}}})(1\\!+\\!\frac{\sigma^{2}_{imag,DI}}{2\sigma_{z}^{2}+\frac{\sigma_{z}^{4}}{\sigma^{2}_{imag,DI}}})\\},$
where $\sigma^{2}_{real,D,I}$ and $\sigma^{2}_{imag,D,I}$ are the variances of
the real and imaginary parts in (23). The less-than sign is because in (24),
the 1-bit DIPS RIS channel real and imaginary parts are correlated. The sign
takes eqivalence when the real and imaginary parts are independent taking the
condition in (25).
(a) Variance when RIS has different element number.
(b) Real part distribution.
(c) Imaginary part distribution.
Figure 3: The variance, real part distribution and imaginary part distribution
of the channel assisted by $B=1$ DIPS RIS.
## V Simulation Results
In this section, the analytical channel distributions, channel parameters, and
SKR for CIPS, CGPS, and DIPS are verified by simulations.
### V-A Verification for CIPS and CGPS Channel
Element spacing is set to $d=\frac{\lambda}{2}$. Incident and reflection
angles are set as $\bm{\Omega_{i}}=(30^{\circ},30^{\circ})$, and
$\bm{\Omega_{o}}=(150^{\circ},60^{\circ})$. For CGPS, set group size $q=2,4$.
#### V-A1 Channel Variance
The simulation for the reflection channel variance with respect to $M$ is
plotted in Fig. 2(a). The simulation results match analytical results in (14)
(20). The variance means the RIS has a great potential to induce channel
variation since it is usually implemented with a huge number of elements. The
variance for the CGPS is larger than the CIPS, and $q=4$ CGPS variance is
larger than $q=2$ CGPS because the variance increases quadratically with $q$.
#### V-A2 Magnitude and Phase Distributions
Set $M=64$, with $M_{x}=M_{y}=8$. The magnitude and phase distributions for
CIPS are presented in Fig. 2(b) and Fig. 2(c), respectively. The magnitude and
phase distribution simulation results match the analytical results. The
magnitude and phase are a Rayleigh distribution and a uniform distribution
being independent of $\bm{\Omega}_{i}$ and $\bm{\Omega}_{o}$, respectively.
The independence of $\bm{\Omega}_{i}$ and $\bm{\Omega}_{o}$ means that
whatever angles Alice and Bob locate in, the channel randomness remains the
same. Note that the sharp drops around 0 and $2\pi$ in Fig. 2(c) result from
several adjacent probability density values outside the bound $[0,2\pi)$,
being equal to 0, are averaged to plot a smoother p.d.f. Additionally, the
channel distribution Eve receives stays unchanged, and no more information
leakage dependent on angles. Note that if Eve locates at the same angle as
Bob222The situation Eve locates at the same angle as Bob do not happen in most
cases., the information is fully leaked. This is because, in the far-field
environment, Eve and Bob are differentiated by their locating angles. The
uniform phase distribution means that the values in $[0,2\pi)$ are taken with
equal probability. Thus, when CSI is utilized to generate keys, it greatly
prevents Eve from guessing the particular phase of the channel between Alice
and Bob.
### V-B Verification for DIPS
#### V-B1 RIS Weight $B\geq 2$
Set $B=3$. Other parameters remain the same as the continuous case. The
simulations for the variance, magnitude, and phase of the reflection channel
are plotted in Fig. 2(a), Fig. 2(b), and Fig. 2(c). The simulation results are
the same as the ideal CIPS. This means that the ideal channel distribution and
uniform phase distribution can be achieved in a more practical situation.
#### V-B2 RIS Weight $B=1$
Set $B=1$, and two pairs of input and output angles, with case 1:
$\bm{\Omega_{i}}=(30^{\circ},30^{\circ})$,
$\bm{\Omega_{o}}=(150^{\circ},60^{\circ})$, and case 2:
$\bm{\Omega_{i}}=(110^{\circ},50^{\circ})$,
$\bm{\Omega_{o}}=(310^{\circ},20^{\circ})$. Other parameters are the same as
continuous case. The simulation for the reflection channel variance is plotted
in Fig. 3(a), where the variances increase linearly with respect to $M$.
The real and imaginary part of the reflection channel for two cases when
$M_{x}=M_{y}=8$ are plotted in Figure. 3(b) and Figure. 3(c), respectively.
The simulation results match the analytical result in (23) well. the real and
imaginary part distributions are dependent on values of $\bm{\Omega}_{i}$ and
$\bm{\Omega}_{o}$.
### V-C Secret Key Rate
The RIS element number is set to $M=64$. The _Information Theoretical
Estimators (ITE) Toolbox_ is used to simulate the mutual information [14]. The
SKR for CIPS, $q=2$ CGPS, $B=2$ DIPS, and $B=1$ DIPS against signal-to-noise
ratio (SNR) are plotted in Fig. 4. The simulation results match the analytical
results in (28) and (29). The CGPS SKR is larger than the individual shifting.
Figure 4: Secret key rate when RIS is CIPS, $q=2$ CGPS, $B=2$ DIPS, and $B=1$
DIPS.
## VI Conclusions
In this paper, we utilize RIS for mmWave physical layer security secret key
generation, and completes the underlying mathematical principles by deriving
the channel distribution. Though mmWave static channels lack randomness and
multipath, the RIS can induce an artificial Rayleigh/Rician fading randomness,
without adding transceiver costs. Specifically, based on the RIS-assisted
secret key generation model, we consider RIS weights have CIPS, CGPS, and DIPS
to produce artificial channel randomness. We derive the channel distribution
and its parameters. The CGPS is able to produce more randomness at legal
parties Alice and Bob, compared to individual shifting. The DIPS channel
distribution is dependent on the quantization bit. The SKR for the above RIS
settings is derived to evaluate the performance. Our theoretical conclusions
are verified by simulations. While this work focuses more on the channel
between Alice and Bob, the detailed analysis of leakage to eavesdroppers and
DGPS will be our future work.
## References
* [1] J. Zhang, S. Rajendran, Z. Sun, R. Woods, and L. Hanzo, “Physical Layer Security for the Internet of Things: Authentication and Key Generation,” _IEEE Wireless Communications_ , vol. 26, no. 5, pp. 92–98, 2019.
* [2] J. Zhang, R. Woods, T. Q. Duong, A. Marshall, Y. Ding, Y. Huang, and Q. Xu, “Experimental Study on Key Generation for Physical Layer Security in Wireless Communications,” _IEEE Access_ , vol. 4, pp. 4464–4477, 2016.
* [3] A. M. Al-samman, M. H. Azmi, and T. A. Rahman, “A Survey of Millimeter Wave (mm-Wave) Communications for 5G: Channel Measurement Below and Above 6 GHz,” in _Recent Trends in Data Science and Soft Computing_ , F. Saeed, N. Gazem, F. Mohammed, and A. Busalim, Eds. Cham: Springer International Publishing, 2019, pp. 451–463.
* [4] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent Reflecting Surface Aided Wireless Communications: A Tutorial,” _IEEE Transactions on Communications_ , pp. 1–1, 2021.
* [5] S. Gong, X. Lu, D. T. Hoang, D. Niyato, L. Shu, D. I. Kim, and Y.-C. Liang, “Toward Smart Wireless Communications via Intelligent Reflecting Surfaces: A Contemporary Survey,” _IEEE Communications Surveys Tutorials_ , vol. 22, no. 4, pp. 2283–2314, 2020.
* [6] Z. Ji, P. Lep Yeoh, G. Chen, C. Pan, Y. Zhang, Z. He, H. Yin, and Y. Li, “Random Shifting Intelligent Reflecting Surface for OTP Encrypted Data Transmission,” _arXiv e-prints_ , p. arXiv:2010.14268, Oct. 2020.
* [7] X. Hu, L. Jin, K. Huang, X. Sun, Y. Zhou, and J. Qu, “Intelligent Reflecting Surface-Assisted Secret Key Generation With Discrete Phase Shifts in Static Environment,” _IEEE Wireless Communications Letters_ , vol. 10, no. 9, pp. 1867–1870, 2021.
* [8] P. Staat, H. Elders-Boll, M. Heinrichs, R. Kronberger, C. Zenger, and C. Paar, “Intelligent Reflecting Surface-Assisted Wireless Key Generation for Low-Entropy Environments,” in _2021 IEEE 32nd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC)_ , 2021, pp. 745–751.
* [9] Q. Wu and R. Zhang, “Beamforming Optimization for Wireless Network Aided by Intelligent Reflecting Surface With Discrete Phase Shifts,” _IEEE Transactions on Communications_ , vol. 68, no. 3, pp. 1838–1851, 2020.
* [10] Liu, Yihong and Zhang, Lei and Yang, Bowen and Guo, Weisi and Imran, Muhammad Ali, “Programmable Wireless Channel for Multi-user MIMO Transmission using Meta-surface,” in _2019 IEEE Global Communications Conference_ , 2019, pp. 1–6.
* [11] H. Han, Y. Liu, and L. Zhang, “On Half-Power Beamwidth of Intelligent Reflecting Surface,” _IEEE Communications Letters_ , pp. 1–1, 2020.
* [12] V. A. Aalo, G. P. Efthymoglou, and C. Chayawan, “On the Envelope and Phase Distributions for Correlated Gaussian Quadratures,” _IEEE Communications Letters_ , vol. 11, no. 12, pp. 985–987, 2007.
* [13] S. Sun, H. Yan, J. MacCartney, George R., and T. S. Rappaport, “Millimeter Wave Small-Scale Spatial Statistics in an Urban Microcell Scenario,” _arXiv e-prints_ , p. arXiv:1703.08239, Mar. 2017.
* [14] Z. Szabó, “Information Theoretical Estimators Toolbox,” _Journal of Machine Learning Research_ , vol. 15, pp. 283–287, 2014.
|
# Global Optimisation in Hilbert Spaces using the Survival of the Fittest
Algorithm.
Andrew Yu. Morozov1,2,3
Oleg Kuzenkov3
Simran K. Sandhu1∗
###### Abstract
Global optimisation problems in high-dimensional and infinite dimensional
spaces arise in various real-world applications such as engineering, economics
and finance, geophysics, biology, machine learning, optimal control, etc.
Among stochastic approaches to global optimisation, biology-inspired methods
are currently very popular in the literature. Bio-inspired approaches imitate
natural ecological and evolutionary processes and are reported to be efficient
in a large number of practical study cases. On the other hand, many of bio-
inspired methods can possess some vital drawbacks. For example, due to their
semi-empirical nature, convergence to the globally optimal solution cannot
always be guaranteed. Another major obstacle is that the existing methods
often struggle with the high dimensionality of space (approximating the
underlying functional space), showing a slow convergence. It is often
difficult to adjust the dimensionality of the space of parameters in the
corresponding computer code for a practical realisation of the optimisation
method. Here, we present a bio-inspired global stochastic optimisation method,
applicable in Hilbert function spaces. The proposed method is an evolutionary
algorithm inspired by Darwin’s’ famous idea of the survival of the fittest and
is, therefore, referred to as the ‘Survival of the Fittest Algorithm’ (SoFA).
Mathematically, the convergence of SoFA is a consequence of a fundamental
property of localisation of probabilistic measure in a Hilbert space, and we
rigorously prove the convergence of the introduced algorithm for a generic
class of functionals. The approach is simple in terms of practical coding. As
an insightful, real-world problem, we apply our method to find the globally
optimal trajectory for the daily vertical migration of zooplankton in the
ocean and lakes, the phenomenon considered to be the largest synchronised
movement of biomass on Earth. We maximise fitness in a function space derived
from a von-Foerster stage-structured population model with biologically
realistic parameters. We show that for problems of fitness maximisation in
high-dimensional spaces, SoFA provides better performance as compared to some
other stochastic global optimisation algorithms. We highlight the links
between the new optimisation algorithm and natural selection process in
ecosystems occurring within a population via gradual exclusion of competitive
con-specific strains.
1\. Department of Mathematics, University of Leicester, LE1 7RH, UK;
2\. Institute of Ecology and Evolution, Russian Academy of Sciences, Moscow,
Russia
3\. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
$\ast$ Corresponding author; e-mail<EMAIL_ADDRESS>
## 1 Introduction
Global optimisation problems in high-dimensional finite as well as infinite-
dimensional Hilbert spaces, arise in a large number of applications within
different research areas such as mechanics, optics, geophysics, economics,
finance, machine learning, biology, etc. Stochastic approaches are currently
widely implemented in global optimisation. In particular, the tremendous
progress in molecular biology, genetics, microbiology, and foraging ecology
made within the recent 30-40 years has led to the emergence of a series of new
stochastic optimisation techniques known as bio-inspired methods. Such
algorithms of optimisation imitate ecological processes (e.g. functioning of
ant or bees colonies, populations of bacteria, swarms of krill, etc.) as well
as those in genetics and biochemistry (e.g. replication and mutation of DNA)
[1, 2, 3, 4]. The objective functional in bio-inspired optimisation is
biological fitness, which, is understood in a broad sense. Among bio-inspired
optimisation methods, evolutionary algorithms - such as genetic algorithms and
differential evolution - are considered to be the key ones. These methods try
to mimic long-term biological evolution, including the processes of mutation,
recombination, selection and reproduction [1]. Genetic algorithms encode
information about an individual via a binary representation, whereas
differential evolution considers a non-binary representation [5, 6]. The other
types of bio-inspired algorithms include swarm intelligence, which is based
off interacting searching agents sharing information [7, 3], and bacterial
foraging, which imitate the social foraging behaviour of Escherichia coli or
other microorganisms ([2]) or even humans ([8]).
Despite the existence of dozens of promising optimisation techniques imitating
biological processes, there are still crucial challenges of their usage [3].
The central question is about the eventual convergence of a particular method
towards the optimal solution for an increasing number of iterations. Indeed,
despite an algorithm having a good record of previously successful
implementations, it cannot be guaranteed, without rigorous analytical proof,
that the considered method would locate the optimal solution for some new
class of problems. The other shortcoming of the existing bio-inspired
techniques is that most of them are not well-designed to find the optimal
solution in function spaces characterised by an infinitely large number of
dimensions (in practice, of course, there is still an upper limit for the
maximal dimensionality). For example, swarm intelligence algorithms are best
fit to choose the optimal path out of the existing combinations. On the other
hand, for problems with complex constraints on the parameters or/and
functions, optimisation methods can get stuck in non-feasible domains (e.g.
producing non-viable mutants in the course of evolution) which substantially
reduces the speed of optimisation. Adjustment of evolutionary algorithms to
deal with the unfeasible realisation of parameters can be achieved, in
principle [1]; however, this would increase the complexity of the methods
preventing their extensive applicability.
In this paper, we introduce a novel method of bio-inspired stochastic global
optimisation in high-dimensional or even infinite-dimension Hilbert spaces,
named Survival of the Fittest Algorithm (SoFA), which can successfully cope
with the pre-mentioned difficulties. SoFA is close to the genetic algorithms
of search, however unlike previously proposed methods, it takes advantage of
the the fundamental principles of population ecology, in particular, the
competitive exclusion principle [9]. As the algorithm name suggests, that is
the quantification of Darwin’s’ famous idea of the survival of the fittest
[10]. Mathematically, the method uses the recently developed approach of
modelling biological evolution based on measure dynamics and measure
localisation around the point(s) with the maximal fitness [11, 12, 13, 14]. A
crucial advantage of SoFA is its convergence which holds for an arbitrary
positive objective functional (fitness function) in a Hilbert space, and here
we present the corresponding rigorous proof. The convergence rate of SoFA is
easily adjustable by tuning the parameters describing mutation rates. Finally,
the computational algorithm of the method is extremely straightforward in its
usage and does not require specific knowledge of the subject, further
improving its practical applicability.
To demonstrate the great potential of the presented method in solving
optimisation problems in complex biological systems (this method was in fact
inspired by such systems), we predict the globally optimal trajectory for the
diel vertical migration (DVM) of zooplankton in the ocean. Note that although
DVM of zooplankton is recognised to be the largest synchronised movement of
biomass on the planet [15, 16], this phenomenon is still not well-understood
and mathematical modelling would play a role to bridge the current gaps in the
area. In our model, each developmental stage of zooplankton is mathematically
characterised as a continuous function describing the vertical position of an
organism in the water column, so the problem requires the use of Hilbert
spaces. The objective functional, defined by population fitness, is derived
from the well-known von-Foerster stage-structured population model. We use
biological relevant parameters describing DVM of a dominant zooplankton
herbivorous species in the north-eastern Black Sea. We compare our method with
some other stochastic global optimisation techniques including ESCH [17, 18],
CRS [19, 20, 21], and MLSL [22, 23] which are well-known in the optimisation
literature. We show that SoFA exhibits a higher performance (in terms of
reduction of the average error with the number of iterations), in the case
where the parametric space is highly dimensional. In particular, the proposed
method deals well with biological constraints required all functionals (e.g.
mortality, reproduction, maturation, etc.) in the model to be positive which
is not the case of the considered ESCH, CRS and MLSL methods.
The paper is structured as follows. In Section 2, we introduce the basic
algorithm of the method (Section 2.1) and explain how to apply the
optimisation procedure to reveal the optimal trajectory of diel vertical
migration (DVM) of zooplankton (Section 2.2). In the Results section, we
rigorously prove the convergence of the global optimisation method (Section
3.1) and construct the optimal trajectories of DVM of zooplankton based on the
stage-structured model (Section 3.2). We also compare the efficiency of the
proposed method with the existing global optimisation methods (Section 3.3).
The Discussion section addresses the connection between the new optimisation
method and biological evolution processes. The Summary section concludes the
study.
## 2 Methodology
### 2.1 Description of the basic SoFA framework
Consider a standard Hilbert space $l_{2}$ of functions $z$ where infinite
sequences are square-summable, i.e.
$l_{2}=\left\\{z:z=(x_{1},x_{2},\ldots,x_{k},\ldots),\left(\sum_{n=1}^{\infty}x_{n}^{2}\right)^{1/2}<\infty\right\\}.$
(1)
We take an element $c=(c_{1},c_{2},\ldots c_{k},\ldots)$ with positive
components and denote
$R=\big{(}\sum_{n=1}^{\infty}c_{n}^{2}\big{)}^{1/2}<\infty$. Based on the
above element we introduce the following infinite dimensional cube
(parallelepiped) $\Pi=\\{z:|x_{n}-a_{n}|\leq c_{n}/2,n=1,2,\ldots\\}$. Suppose
there is some continuous positive function (functional) $J(z)$ which is
defined in $\Pi$. Assuming that $J(z)$ has an unique point of maximum (denoted
by $z^{*}$), it can be located by implementing a novel optimisation method.
The method is outlined in the following algorithm (SoFA). The algorithm will
use the following auxiliary function determined by
$f(r)=-\frac{r^{2}}{2R^{2}}$.
The formal description of the algorithm is the following.
1. 1.
Randomly choose a point $z_{1}=(x_{11},0,0,...)$ assuming a uniform
distribution in the one-dimensional projection $\Pi_{1}=[-c_{1}/2,c_{1}/2]$ of
the cube $\Pi$. The functional $J$ is evaluated at this point, i.e.
$J(z_{1})$.
2. 2.
The point $z_{1}$ is taken as the current reference point (denoted by
$\bar{z}$) to find the $z_{2}$. The second iteration point
$z_{2}=(x_{12},x_{22},0,0,...)$ is randomly chosen in the two-dimensional
projection $\Pi_{2}=[-c_{1}/2,c_{1}/2]\times[-c_{2}/2,c_{2}/2]$ of the cube
$\Pi$. The probability density distribution of $z_{2}$ is given by
$\frac{2^{f(\|z-\bar{z}\|)}}{\int_{\Pi_{2}}\,2^{f(\|z-\bar{z}\|)}\,d\Pi_{2}}.$
The functional is evaluated at the point $J(z_{2})$.
3. 3.
The new reference point $\bar{z}$ is randomly chosen out of $z_{i}$ ($i=1,2$)
with the probability defined by $J^{2}(z_{i})/(J^{2}(z_{1})+J^{2}(z_{2}))$.
Using the updated reference point $\bar{z}$, the third iteration point
$z_{2}=(x_{13},x_{23},x_{33},0,0,...)$ is obtained, which is random vector
generated in the three-dimensional projection
$\Pi_{3}=[-c_{1}/2,c_{1}/2]\times[-c_{2}/2,c_{2}/2]\times[-c_{3}/2,c_{3}/2]$
of the cube $\Pi$. The probability density distribution of $z_{3}$ is given by
$\frac{3^{f(\|z-\bar{z}\|)}}{\int_{\Pi_{3}}\,3^{f(\|z-\bar{z}\|)}\,d\Pi_{3}}.$
The functional is evaluated at the point $J(z_{3})$.
4. 4.
Assume $k$ steps of the method have already been completed and therefore one
has $k$ points $z_{i}$, $i=1,\ldots,k$, with known corresponding values of
$J(z_{1}),\ldots,J(z_{k})$. Then randomly select some reference point
$\bar{z}$ out of all previously found points $z_{i}$. The probability of
selecting each of these available points is given by
$\frac{J^{k}(z_{i})}{J^{k}(z_{1})+\ldots+J^{k}(z_{k})}$, $i=1,\ldots,k$.
5. 5.
Using the above reference point $\bar{z}$, select a new point $z_{k+1}$ which
is a random vector generated in the (k+1)-dimensional projection
$\Pi_{k+1}=[-c_{1}/2,c_{1}/2]\times[-c_{2}/2,c_{2}/2]\dotsm\times[-c_{k+1}/2,c_{k+1}/2]$
of the cube $\Pi$. The probability density distribution of $z_{k+1}$ is given
by
$\frac{(k+1)^{f(\|z-\bar{z}\|)}}{\int_{\Pi_{k+1}}\,(k+1)^{f(\|z-\bar{z}\|)}\,d\Pi_{k+1}}.$
(2)
The functional is evaluated at the point $J(z_{k+1})$, then the above steps
are repeated.
6. 6.
The method will terminate once the approximation satisfies some initially
prescribed requirement. In the simplest case, this can be a maximum number of
iterations, although this simple criterion cannot necessarily guarantee an
appropriate approximation of the maximal point. Another criterion can be that
the distribution of points around the optimal should approach to the delta
function (e.g. the standard deviation of the distribution should be smaller
than a certain threshold).
###### Remark.
Note that, technically, one can use a simplified version of the optimisation
algorithm defined above (see Appendix A for details). In particular, we
approximate the multi-dimensional function (2) in the probability density
distribution for the point $z_{k+1}$ by a product of simpler one-dimensional
functions each of which given by
$\frac{A^{j}_{k+1}}{\epsilon_{k+1}+(x_{j,k+1}-\bar{x}_{j})^{2}},$
where $A^{j}_{k+1}$ is a normalising constant, $\epsilon_{k+1}$ is the
parameter characterising the standard deviation of the distribution,
$\bar{x}_{j}$ is the $j^{th}$ coordinate of the current reference point
$\bar{z}$. By choosing a particular parameterisation of $\epsilon_{k+1}$, one
can regulate the rate of convergence of the optimisation algorithm. Finally,
in practice, one can add extra dimensions only after a certain number of
iterations (i.e. not at every iteration as in the basic version of SoFA); the
dimensions can be added in blocks (i.e. adding several dimensions at some
iteration step). After achieving a high dimensionality, one can keep the same
number of dimensions until the end of the optimisation procedure, this is
because the negative effects of computational noise in a higher dimensional
system can undermine the positive effects of adding extra coordinates when
approximating the function space with a finite number of dimensions.
###### Remark.
The main idea of SoFA is combining the processes of selection and mutation at
each step. Unlike genetic algorithms and methods of differential evolution,
SoFA does not mimic the phenomenon of crossover.
In SoFA, evolutionary selection within the population is mimicked when
choosing the reference point $\bar{z}$ in each iteration step. We can explain
the underlying biological rationale in a non-rigorous way by considering the
following simple discrete population model. We suggest that we have $k$
competing subpopulations within the population. The growth of the
subpopulation $i$ ($i=1,..,k$) with density $y_{i}$ from the start of year $s$
to the end of the same year denoted by $s^{+}$ is described by
$y_{i}(s^{+})=y_{i}(s)(1+a_{i}),$
where $a_{i}>0$ is the per capita population growth rate for the given
subpopulation. For simplicity, it is assumed to be constant and the generation
time equals one year. We consider that by the end of each year the population
is harvested proportionally to the current densities of each subpopulation in
a way the total population density is kept at some carrying capacity (we
assume it to be unity without the loss of generality). In this case, in the
beginning of year $s+1$ the population density $y_{i}$ will be given by
$y_{i}(s+1)=\frac{y_{i}(s^{+})}{\sum_{j}y_{j}(s^{+})}=\frac{y_{i}(s)(1+a_{i})}{\sum_{j}y_{j}(s)(1+a_{j})}.$
We start with the initial densities of the subpopulations such that their sum
is exactly the carrying capacity of the system. In this case, one can easily
derive the density $y_{i}(k^{+})$ at the end of year $k$
$y_{i}(k^{+})=\frac{(1+a_{i})^{k}}{\sum_{j}(1+a_{j})^{k}}.$
By denoting $J_{i}=1+a_{i}$, we can re-write the above ratio as
$y_{i}(k^{+})=\frac{J_{i}^{k}}{\sum_{j}J_{j}^{k}}.$
One can see that with long times (large $k$), the density of the subpopulation
$y_{i}$ having the maximal fitness $\max J_{i}$ will tend to unity, whereas
the proportion of the others will tend to zero. This fact explains the idea of
the method where at each iteration, the selected reference point is in the
proposed ratio form.
The mutation in the new optimisation method is modelled by an integral term
(2) with a Gaussian kernel. The centre of this kernel corresponds to the
parent strain with the life-history trait $\bar{z}$, which produces mutant
offspring. Note the choice of the parent strain is based on its reproductive
success determined by $\frac{J_{i}^{k}}{\sum_{j}J_{j}^{k}}$.
### 2.2 Modelling optimal DVM of zooplankton
We apply the above optimisation method to model the particularly thought-
provoking ecological case study, which is the regular daily vertical migration
of zooplankton. Diel vertical migration (DVM) of marine and freshwater
zooplankton in the water column, is regarded to be the largest synchronised
movement of biomass on Earth [15, 16]. It is also vital to develop our
understanding of patterns of DVM as it heavily impacts the biochemical cycles
in the ocean, playing a fundamental role in the carbon exchange between the
deep and surface waters, the ocean’s biological pump and thus, the climate
[24, 25, 26, 27]. The typical pattern of DVM consists of the zooplankton
organisms ascending to the phytoplankton rich surface waters for feeding at
night, then descending to deeper depths and remaining there during the day
[15, 28]. There are currently several explanations of what is the ultimate
cause of this mass migration. The most widely accepted hypothesis is that the
zooplankton performs DVM to avoid visual predation (mostly by planktivorous
fish) by spending daylight hours in the deeper darker waters and migrating up
at night when these visual predators cannot see them [28, 29, 30, 31].
The phenomenon of DVM has been studied extensively, both empirically [28, 29,
30] and theoretically through the use of several mathematical models [32, 33,
34]; however, there are still some important gaps in our knowledge of
zooplankton DVM. The major challenge in modelling zooplankton DVM is to
properly define the criterion of optimality since the use of different
criteria may result in distinct predictions [35, 36, 37, 38]. Recently, there
has been the proposal of a new rigorous approach to identify the evolutionary
fitness in systems with inheritance [38, 14]. The approach considers the long-
term dynamics of competing subpopulations which are described by different
inherited units. Evolutionary fitness is defined based on the comparative
ranking order of the subpopulations characterised by different behavioural
strategies or life-history traits. In other words, a subpopulation will
outcompete all other subpopulations possessing a lower ranking, with this
ranking order being defined by the fitness function. The proposed idea of
modelling biological evolution in [38, 14] is similar to that of SoFA
framework which makes it natural to apply of the new optimisation method to
maximise evolutionary fitness, in particular in DVM of zooplankton.
Using the above theoretical approach, the expression for evolutionary fitness
was determined for some age-structured population models [14, 39]. The need
for age-structured models is justified by the fact that different zooplankton
developmental stages exhibit distinct migration patterns of DVM [35, 39].
Using a von-Foerster-type equation [40, 41] with continuous age but discrete
stages Morozov et al. derived the following expression for evolutionary
fitness [39]
$J(v)=\frac{\max_{i}{\mathbb{R}(\lambda_{i}(v))}}{R(v)},$ (3)
where $v$ is a continuous vector function of time describing the daily
vertical trajectory for the considered developmental stages, $\lambda_{i}$ is
the eigenvalue of the appropriate characteristic equation (see [14, 39] for
details) and $\mathbb{R}$ denotes the real part of this eigenvalue. $R(v)$ is
the functional describing intraspecific competition within the population. For
simplicity we assume $R(v)\equiv 1$. Note that Morozov and co-authors [39]
explored the the optimal trajectories in of DVM only based on a piecewise
linear approximation, moreover, the authors used local rather than global
optimisation methods. As such, it would be important to reveal the smooth
underlying DVM trajectories which are ‘the exact’ solution of the global
optimisation problem (3).
As in [39], we consider three different developmental stages of herbivorous
zooplankton: young stages, juveniles and adults. In this case, the
characteristic equation for the eigenvalue $\lambda$ is given by [39]:
$\lambda=b\exp{\left(-a_{Y}\tau_{1}-a_{J}(\tau_{J}-\tau_{Y})\right)}\left[\exp{\left(-\tau_{J}\lambda\right)}-\exp{\left(-\tau_{A}\lambda-
a_{2}(\tau_{A}-\tau_{J})\right)}\right]-a_{A},$ (4)
where $i=Y,J,A$ correspond to young, juvenile and adult stages, respectively.
Here, $a_{i}$ is the mortality rates which accounts for the natural losses a
due to visual predators (varying throughout the day due to dependence on light
intensity), additional morality at the boundaries of unfavourable zones and
natural non-predatory based mortality. $\tau_{i}$ are the maturation times of
each stage; $\tau_{A}$ denotes the maximal reproduction age of adults. $b(v)$
is the reproduction coefficient which quantifies the number of eggs produced
by a female in any given day. The terms $b(v)$ and $\tau_{i}$ incorporate the
energy gained from feeding on phytoplankton, loses due to basal metabolism and
the metabolic cost spent on active movements in the water when feeding and
moving upwards while ascending.
Maximisation of $J$ defined by (4) will locate the optimal strategies for all
three developmental stages $v=(v_{Y},v_{J},v_{A})$. The parameterisation of
the integrals for computing $a_{i}$, $\tau_{i}$ and $b$ are assumed to be the
same as in [39], for brevity, we do not include here the corresponding
expressions. The model parameters used in the integrals are defined to be the
default values from Table 1 of the cited work. The main difference between
this study and that of Morozov et al. [39] is that here the zooplankton is
considered to be feeding when at a shallow enough depth and with their
vertical speed being less than than the minimum threshold given by
$c_{0}=10m/h$. Whereas Morozov et al. assumed that zooplankton grazers only
feed when staying at shallowest depths during DVM and therefore did not allow
for feeding during any vertical movement.
To reconstruct the trajectory in the Hilbert function space, we consider the
following Fourier expansion
$v_{s}(t)=v_{s,1}+\sum^{N}_{m=1}\left(v_{s,2m}\sin(2\pi
tm)+v_{s,2m+1}\cos(2\pi tm)\right)$ (5)
for $s=Y,J,A$. The maximal order $n=2N+1$ ($N=0,1,2,...$) in the above Fourier
expansion can be set as large as possible. We substitute $v_{s}(t)$ in the
integrals for $a_{i}$, $\tau_{i}$ and $b$ in the equation for fitness. We
combine the trajectories of the all considered developmental stages in a
single vector $v$ of dimension $3n$ setting $v_{Y}=v_{1},....,v_{n}$,
$v_{J}=v_{n+1},...,v_{2n}$ and $v_{A}=v_{2n+1},...,v_{3n}$ to perform
optimisation in the space of dimension $D=3n$.
To evaluate the effectiveness of our method, along with the other long-
standing optimisation methods, we introduce the following error definition.
###### Definition 2.1.
Function Error Let the true maximal Fourier coefficients to be defined by the
vector $v^{*}$ of dimension $D=3n$, then we define the function error of the
approximated trajectory with Fourier coefficients given by $v$ as
$Err=J(v^{*})-J(v)$ (6)
We can also define the probability of convergence of an optimisation algorithm
in the following way.
###### Definition 2.2.
Probability of Convergence. Let the true maximal point be $v^{*}$ and the
approximation of the optimal solution be $v$. Then the probability of
convergence of the fitness to the ’true’ optimal value $J^{*}$ with error
$\delta>0$ is defined as follows
$P_{\delta}=P(|J(v^{*})-J(v)|<\delta)$ (7)
By fixing the value of $\delta$, we evaluate the performance of optimisation
by plotting the ratio of successful approximations (i.e. those below the
tolerance $\delta$) for an increasing number of iterations. Note that, usually
the exact value of $v^{*}$ is unknown, so in this study, we use an
approximation to the optimal solution by running simulations for a large
number of iterations.
Along with SoFA, we also implement three other well-known stochastic
optimisation algorithms: ESCH (Evolutionary Strategy with Cauchy distribution)
[17, 18], CRS (Controlled Random Search with local mutation) [19, 20, 21], and
MLSL (Multi-Level Single-Linkage) [22, 23]. The comparison of the efficiency
of optimisation techniques is made based on the above-introduced error and the
probability of convergence. Note that the maximal eigenvalue in the implicit
equation for fitness (4) is, numerically, found using the Levenberg-Marquardt
algorithm. Finally, it is also important to mention that within some given
domains of $v$, it is possible to have biologically irrelevant results with,
for example, negative mortality or maturation rates. Here, we denote these
points $v$ as unfeasible points. We will plot the percentage of such points to
effectively compare the efficiency of different global optimisation methods
used to reveal the optimal DVM of zooplankton.
Finally, to compare the optimisation in an infinite-dimensional function space
we also construct the optimal trajectory based on piecewise linear
approximation consider in the study of Morozov et al. [39]. The corresponding
equations are given by
$\bar{v_{s}}(t)=\left\\{\begin{array}[]{ll}H_{i0}&0\leq t<t_{i0}\\\
c_{i0}(t-t_{i0})+H_{i0}&t_{i0}\leq t<t_{i1}\\\ H_{i1}&t_{i1}\leq t<t_{i2}\\\
c_{i1}(t-t_{i2})+H_{i1}&t_{i2}\leq t<t_{i3}\\\ H_{i0}&t_{i3}\leq t\leq
1\end{array}\right.$ (8)
for $s=Y,J,A$. Here, the shallowest and deepest depths are given by $H_{i0}$
and $H_{i1}$, respectively; the speeds of ascending and descending are
$c_{i0}$ and $c_{i1}$, respectively; the times of the end of each movement
phase is described by $t_{ij}$. In this case, the number of unknown parameters
is 3 for each stage which makes the overall number of parameters to be $D=9$.
This is because in piecewise linear setting the optimal trajectories are
symmetrical with respect to $t=0.5$ (see [39] for detail)
We choose the domains for each of the Fourier coefficients $v_{s,i}$ in (5) to
be the smallest possible domain such that both the ‘true’ optimal point
$v^{*}_{s,i}$ and the initial starting points (determined by the Fourier
expansions of the piecewise linear ‘true’ optimal trajectory).
## 3 Results
### 3.1 Proof of convergence of SoFA
Here we rigorously demonstrate the convergence of the Survival of the Fittest
Algorithm (SoFA) in the cube $\Pi$ in Hilbert space introduced in Section 2.1.
The proof consists of the two following theorems.
###### Theorem 1.
The sequence $(z_{1},...,z_{k},...)$ is everywhere dense with the probability
of unity, i.e. for any $z\in\Pi$ the probability to have a point $z_{i}$ in
its neighbourhood $O_{\epsilon}(z)$ tends to unity for large $m$.
###### Proof.
As $f(r)=-\frac{r^{2}}{2R^{2}}$, it is easy to see that $f(0)=0$ and for any
two elements $h$ and $g$ within the cube $\Pi$ we have $\left\|h-g\right\|<R$,
thus $f(\left\|h-g\right\|)>f(R)=-1/2$ (we define the constant $R$ in Section
2.1).
We take a positive number $\epsilon<1/2$ and arbitrary element
$z=(x_{1},\ldots,x_{n},\ldots)\in\Pi$, thus we can introduce the $\epsilon$
neighbourhood $O_{\epsilon}(z)$ of $z$. Since the series
$\sum_{n=1}^{\infty}c_{n}^{2}$ is convergent ($c_{n}$ are introduced in
Section 2.1), there should exist a number $N$ such that
$\sum\limits_{n=N+1}^{\infty}c_{n}^{2}\leq\frac{\epsilon^{2}}{2}$. Consider a
point $y=(y_{1},\ldots,y_{n},\ldots)\in\Pi$. In the case its components
satisfy the condition $|y_{n}-x_{n}|\leq\frac{\epsilon}{2^{n}},n=1,\ldots,N,$
then the point $y$ should belong to $O_{\epsilon}(z)$. Indeed, one can see
that
$\sum\limits_{n=1}^{\infty}(x_{n}-y_{n})^{2}=\sum\limits_{n=1}^{N}(x_{n}-y_{n})^{2}+\sum\limits_{n=N+1}^{\infty}(x_{n}-y_{n})^{2}\leq\sum\limits_{n=1}^{N}(\frac{\epsilon}{2^{n}})^{2}+\sum\limits_{n=N+1}^{\infty}(c_{n})^{2}\leq\epsilon^{2}.$
Therefore, the set $\omega=\\{y=(y_{1},\ldots
y_{n},\ldots):\left|y_{n}-x_{n}\right|\leq\frac{\epsilon}{2^{n}},n=1,\ldots,N\\}$
is actually a subset of $O_{\epsilon}(z)$.
Now assume that $k\geq N$ iterations of the method have been completed. The
probability $P_{k+1}(\omega)$ for the point $z_{k+1}$ to lend in $\omega$ is
given by
$P_{k+1}(\omega)=\frac{\int\limits_{\omega}(k+1)^{f(\left\|z-\overline{z}\right\|)}d\Pi_{k+1}}{\int\limits_{\Pi_{k+1}}(k+1)^{f(\left\|z-\overline{z}\right\|)}d\Pi_{k+1}}.$
We estimate the lower bound of the above probability;
$P_{k+1}(\omega)=\prod\limits_{n=1}^{N}\frac{\int\limits_{x_{n}-\frac{\epsilon}{2^{n}}}^{x_{n}+\frac{\epsilon}{2^{n}}}(k+1)^{f(\left|y_{n}-\bar{x}_{n}\right|)}dy_{n}}{\int\limits_{-c_{n}/2}^{c_{n}/2}(k+1)^{f(\left|y_{n}-\bar{x}_{n}\right|)}dy_{n}}\geq(k+1)^{-\frac{1}{2}}\prod\limits_{n=1}^{N}\frac{2\epsilon}{c_{n}2^{n}}=(k+1)^{-\frac{1}{2}}\mu^{*},$
where $\mu^{*}=\prod\limits_{n=1}^{N}\frac{2\epsilon}{c_{n}2^{n}}$ is a
constant which does not depend on the number of iteration; $\bar{x}_{n}$ are
the coordinates of the reference point $\bar{z}$. The probability to not land
in the set $\omega$ within $M$ consecutive iterations starting from $N+1$ to
$N+M$ can be estimated as
$P_{M}\leq(1-\mu^{*}(M+N)^{-\frac{1}{2}})^{M}.$
Since we have
$\lim\limits_{M\to\infty}(1-\mu^{*}(M+N)^{-\frac{1}{2}})^{M}=0,$
then the probability of non-choosing a point from $\omega$ for $M$ iterations
tends to zero with $M\to\infty$.
Therefore, the sequence of points $z_{k}$ is everywhere dense in $\Pi$ with
the probability of unity. ∎
###### Theorem 2.
Let a continuous positive function $J(z)$ defined in $\Pi$ has an unique point
of maximum given by $z^{*}=(x_{1}^{*},x_{2}^{*},\ldots x_{k}^{*},\ldots)$.
Then for any $\epsilon>0$ the probability of choosing a point $z_{k}$ from the
neighbourhood $O_{\epsilon}(z^{*})$ tends to unity when $k$ becomes infinitely
large. In other words, the optimisation algorithm converges.
###### Proof.
As it was shown in the proof of the previous theorem, the set
$\omega(z^{*})=\\{y=(y_{1},\ldots
y_{n},\ldots):\left|y_{n}-x^{*}_{n}\right|\leq\frac{\epsilon}{2^{n}},n=1,\ldots,N\\}$
is actually a subset of the neighbourhood $O_{\varepsilon}(z^{*})$. We
estimate the probability of landing of the point $z_{k}$ in $\omega(z^{*})$.
We introduce the following definitions:
$J_{0}=\sup\limits_{z\in\Pi\backslash\omega(z^{*})}J(z)$; $I_{m}$ is the set
of indexes of the points $z_{i};i=1,\ldots,k$ which get into $\omega(z^{*})$;
$\overline{I_{k}}$ is the set of indexes of the points $z_{i},i=1,\ldots,k$
which do not get into $\omega(z^{*})$.
From Theorem 1, $z_{k}$ is an everywhere dense sequence (with the probability
of unity) in $\Pi$. Since we assume that $J$ is a continuous function, there
will always be a point $z_{p}\in\omega(z^{*})$ such that we have
$J(z_{p})>J_{0}$.
For $k>p$, the probability to choose the point $z_{j}$ as a reference point
$\overline{z}$ with the index $j$ from the set $\overline{I_{k}}$, can be
estimated as follows;
$\frac{\sum\limits_{j\in\overline{I_{k}}}J^{k}(z_{j})}{\sum\limits_{j=1}^{k}J^{k}(z_{j})}\leq\frac{\sum\limits_{j\in\overline{I_{k}}}J_{0}^{k}J^{-k}(z_{p})}{\sum\limits_{j=1}^{k}J^{k}(z_{j})J^{-k}(z_{p})}<\left(\frac{J_{0}}{J(z_{p})}\right)^{k}\xrightarrow{k\to\infty}0$
Since the above probability tends to zero, then the probability of choosing
the point $z_{j}$ as a reference point $\overline{z}$ with the index $j$ from
the set $I_{k}$ tends to unity.
For $k>N$ we estimate the probability to choose the point $z_{k+1}$ under the
condition that the reference point we obtain $z_{j}$ with the index $j$ from
the set $I_{k}$, i.e. $\overline{z}\in\omega(z^{*})$.
$P_{k+1}(\omega(z^{*}))=\prod\limits_{n=1}^{N}\frac{\int\limits_{x_{n}-\frac{\epsilon}{2^{n}}}^{x_{n}+\frac{\epsilon}{2^{n}}}(k+1)^{f(\left|y_{n}-\overline{x_{n}}\right|)}dy_{n}}{\int\limits_{-c_{n}/2}^{c_{n}/2}(k+1)^{f(\left|y_{n}-\overline{x_{n}}\right|)}dy_{n}}\geq$
$\geq\Phi\Big{(}\frac{x^{*}_{n}+\frac{\epsilon}{2^{n}}-\overline{x_{n}}}{R}\sqrt{\ln(k+1)}\Big{)}-\Phi\Big{(}\frac{x^{*}_{n}-\frac{\epsilon}{2^{n}}-\overline{x_{n}}}{R}\sqrt{\ln(k+1)}\Big{)},$
where
$\Phi(t)=\frac{2}{\sqrt{\pi}}\int\limits_{0}^{t}e^{-\tau^{2}}d\tau.$
Since $x^{*}_{n}+\frac{\epsilon}{2^{n}}-\overline{x_{n}}>0$, we have
$\frac{x^{*}_{n}+\frac{\epsilon}{2^{n}}-\overline{x_{n}}}{R}\sqrt{\ln(k+1)}\xrightarrow{k\to\infty}+\infty,\quad\Phi(\frac{x^{*}_{n}+\frac{\epsilon}{2^{n}}-\overline{x_{n}}}{R}\sqrt{\ln(k+1)})\xrightarrow{k\to\infty}1.$
Since $x^{*}_{n}-\frac{\epsilon}{2^{n}}-\overline{x_{n}}<0$, we have
$\frac{x^{*}_{n}-\frac{\epsilon}{2^{n}}-\overline{x_{n}}}{R}\sqrt{\ln(k+1)}\xrightarrow{k\to\infty}-\infty,\quad\Phi(\frac{x^{*}_{n}-\frac{\epsilon}{2^{n}}-\overline{x_{n}}}{R}\sqrt{\ln(k+1)})\xrightarrow{k\to\infty}0.$
Therefore,
$P_{k+1}(\omega(z^{*}))\geq\Phi\Big{(}\frac{x^{*}_{n}+\frac{\epsilon}{2^{n}}-\overline{x_{n}}}{R}\sqrt{\ln(k+1)}\Big{)}-\Phi\Big{(}\frac{x^{*}_{n}-\frac{\epsilon}{2^{n}}-\overline{x_{n}}}{R}\sqrt{\ln(k+1)}\Big{)}\xrightarrow{k\to\infty}1.$
The above limit finalises the proof of convergence of the proposed
optimisation method. ∎
### 3.2 Simulating optimal trajectories of DVM of zooplankton
We apply SoFA to reveal in optimal trajectories of DVM of zooplankton using
the functional $J$ given by (4). The infinite-dimensional space will be
approximated by the $n=2N+1$ first terms in Fourier series (5). The number of
the considered Fourier terms will determine the accuracy of the approximation
of the optimal DVM patterns. One key advantage of SoFA is that one can adjust
the speed of convergence by appropriately selecting the rate of decrease in
the function $\epsilon(k)$ (we implement the simplified version of SoFA, see
the first Remark in Section 2.1). Here we consider the parameterisation given
by $\epsilon(k)=k^{-\alpha(k)}$, where $\alpha(k)$ is an increasing function
of $k$. For simplicity, we consider the linear dependence $\alpha(k)=a+bk$
(with $a,b>0$); in Section 4 we briefly discuss the role of parameterisation
of $\epsilon(k)$ on the efficiency of the method.
Figure 1: Optimal patterns of DVM of zooplankton obtained using SoFA for
varying order of the Fourier series (5) shown for each of the three
developmental stages. (A) Trajectories are each defined by 5 Fourier terms
with the globally maximum strategy being that with the overall shallower depth
($v_{1}$), unlike the corresponding piecewise linear optimal trajectories
($\bar{v}$). (B-D) Trajectories are each defined by 15, 27 and 55 Fourier
terms, respectively, with the globally maximum strategy being at deeper depths
($v_{2}$), similar to the corresponding piecewise optimal trajectories
($\bar{v}$). Time is re-scaled such that t=0 is equivalent to midnight and
t=0.5 is midday. The strategy for the first local maximum
($v_{1}=(v_{Y1},v_{J1},v_{A1})$) is indicated by the blue, orange and yellow
curves (representing youths, juveniles and adults respectively), the strategy
for the second local maximum ($v_{2}=(v_{Y2},v_{J2},v_{A2})$) is indicated by
the purple, green and cyan curves. The dashed curves represent the optimal DVM
for the case when trajectories are symmetric piecewise linear functions.
The optimal trajectories of DVM, constructed using SoFA, are presented in
Fig.1 shown for a progressively increasing number of Fourier terms in (5). The
figure also shows the approximation of the trajectories using a piecewise
linear function (8), depicted by the dashed lines. Finally, we show other
local maximum possible in the system to demonstrate the need for usage of
global optimisation. Technically, to find a local maximum we implemented the
local methods (realised in the MATLAB function fminsearch based on the Nelder-
Mead simplex algorithm) starting from different initial conditions.
From Fig.1 one can conclude that only the older developmental stages of
zooplankton (denoted by $J$ and $A$) exhibit pronounced vertical migrations,
whereas, the youngest stage ($Y$) remains in the surface waters all day. This
pattern is observed for both the piecewise linear and smooth approximation of
fitness. Fig.1 shows that the two local maxima of the fitness function $J$
give two very different trajectories; the first
($v_{1}=(v_{Y1},v_{J1},v_{A1})$) with a shallower depth of around 70m, unlike
the piecewise linear optimal DVM patterns ($\bar{v}$). The second
($v_{2}=(v_{Y2},v_{J2},v_{A2})$) goes to much deeper depths of approximately
110m, as was observed with the piecewise linear trajectories ($\bar{v}$). In
the case with only $n=5$ Fourier terms used for each of the three age groups,
the global fitness is achieved by the shallower patterns of DVM, dissimilar to
the optimal piecewise linear DVM patterns. By increasing the order to $n=15$
terms, the globally maximal fitness is attained, by the trajectories similar
to the optimal piecewise linear DVM patterns, i.e. those that travel to a
deeper depth. Therefore, these two results demonstrate that the addition of
more Fourier terms (meaning an increase in order) results in a drastic change
in optimal trajectories, causing a switch in global maximum between the two
local maxima. Following this, we also investigated very high orders of Fourier
expansions such as $n=27$ or even $n=55$ terms to describe each trajectory.
Fig.1(C, D) shows that for both these higher Fourier orders there exists two
local maxima, furthermore, the global maximum is achieved by the trajectories
that migrate to deeper depths.
We investigated the influence of the total number of Fourier terms, per
trajectory, $n$ used (i.e. the system dimensionality $D=3n$) on the absolute
value of evolutionary fitness $J$, with the results shown in Fig.2. We found
that $J$ initially increases with $n$ up to around 27 terms (with $D=81$) and
then remains almost constant. Note that, the small oscillations in $J$ for a
large number of Fourier terms, can be partially explained by effects of noise
on the system. In the same figure, we also indicate (dashed yellow line) the
fitness corresponding to piecewise linear approximation (8). The figure shows
that fitness is much lower with a piecewise linear trajectory (yellow dashed
line) when compared to that of the global maximum based on smooth curves
(solid red curve). An important conclusion can be drawn from Fig.2, that using
too many terms in the approximation of fitness seems to be computationally
inefficient due to effects of noise as well as a long computational time
needed to operate a high-dimensional system. For this reason, we consider
Fourier approximations containing either 15 or 27 terms for the comparison
with the other optimisation methods.
Figure 2: Dependence of fitness $J$ of different strategies of DVM of
zooplankton on the number of Fourier terms used in (5). Fitness of the optimal
piecewise linear trajectories $J(\bar{v})$, is indicated by the dashed yellow
curve. The two local maxima, $J(v_{1})$ and $J(v_{2})$, shown by the blue and
red curves, respectively, with the global maximum $J(v^{*})$ given by the
maximum of these two curves.
It is important to compare the efficiency of SoFA with some existing bio-
inspired global algorithms such as ESCH, CRS and MLSL. For each of these
methods, we run 200 realisations, with each of these realisations run for
$2\times 10^{5}$ iterations. Fig.3 presents the comparison of the efficiency
of the fore-mentioned global optimisation algorithms for the Fourier series
approximating the DVM trajectory. As a comparison metric, we use the error
function $Err$ (shown in upper left panel of Fig.3) and the probability of
convergence $P_{\delta}$ for different values of the accuracy $\delta$ (shown
in the bottom panels of Fig.3). To understand possible slowness of a method,
we also plot the percentage iterations which produce non-viable biological
quantities (e.g. negative values of reproduction, maturation time, etc.). The
percentage of unfeasible trajectories, displayed in the upper right panel of
Fig.3, the greater this percentage, the larger the number of ‘wasted’
iterations that give an approximation of $v$ that are biologically irrelevant.
For the method introduced in this paper, we found the percentage of unfeasible
trajectories is zero. From Fig.3, one can see that SoFA shows the best
performance both in terms of the functional error and the probability of
convergence. The panels (A) and (B) only differ in the overall number of the
Fourier terms used. The figure shows that the novel SoFA optimisation is
effective at converging towards the maximum as demonstrated through the steady
decrease in the function error. Implementing SoFA results in a rapid
convergence towards the global maximum, reaching an error less than $2\times
10^{-4}$ in $100\%$ of the 200 repetitions of the algorithm when $n=15$ (with
about $80\%$ when $n=27$). Thus, for the considered problem of optimal DVM,
SoFA with $\epsilon(k)=k^{-\alpha(k)}$ provides a far superior approximation
than the ESCH, CRS and MLSL methods (orange, yellow and purple curves
respectively) in terms of the speed and accuracy of its approximation.
Figure 3: Implementation of multiple global optimisation algorithms to
maximise fitness $J(v)$, with each trajectory $v_{s}$ being of the form (5).
(A) represent the results for an optimisation problem with a dimension of
$D=45$ (i.e. $n=15$ Fourier terms per trajectory), and (B) are that but with
dimension $D=81$ (i.e. $n=27$ Fourier terms per trajectory). For each block
(A) and (B) the upper left panel shows the function error of the approximation
for each iteration defined by (6). The upper right panel shows the probability
of each iteration giving an unfeasible approximation to the global maximum.
The bottom panels show the probabilities of convergence to $\delta=10^{-3}$
and $\delta=5$x$10^{-4}$, as defined by (7). The algorithms implemented are:
SoFA with $\epsilon(k)=k^{-(a+bk)}$ with $a=0.7$ and $b=2.5$x$10^{-6}$ (blue
curve) along with ESCH, CRS and MLSL (orange, yellow and purple curves
respectively). The presented results are the average of 200 repetitions of
each algorithm.
Interestingly, SoFA always provides a feasible iteration, where both the ESCH
and CRS methods take some time to begin to produce some feasible points and
consequently a valid approximation to the maximum. We tested the influence of
increasing complexity of the system on the performance of the new optimisation
method further by increasing the number of the Fourier expansions from $n=15$
to $n=27$ (see Fig.3 (B)). The drastic increase in the dimensionality of the
optimisation problem, $D=45$ to $D=81$, results in an overall reduction of
accuracy for the fixed number of iterations (i.e. one needs more iterations to
achieve the same accuracy), however, SoFA shows better performance as compared
to the other optimisation methods used in this study. We found that this trend
is observed for a further increase of dimensionality of the optimisation space
(we do not show the corresponding figures for brevity).
## 4 Discussion
Currently, methods of global optimisation are used intensively in various
areas of research in applied mathematics, physics, economics, finance and
biology. Despite the abundance of the promising techniques, novel optimisation
frameworks are still being developed, in particular concerning bio-inspired
algorithms. The urgent need for new methods is necessary as existing methods
possess some crucial drawbacks and new practical applications require more
specific problem-oriented algorithms, for example, to optimise complex
biological systems in infinite-dimensional spaces. The need for developing
optimisation methods efficiently working in function spaces is justified by
the fact more traditional semi-analytical methods deriving Euler-Lagrange
equations, Bellman equations or Pontryagin’s principle of maximum with their
further numerical solution are not efficient in the case where the objective
functional is non-linear, or given by a transcendental equation as in (4).
In this study, we introduce a bio-inspired global optimisation method named
SoFA, which uses the famous idea of the survival of the fittest in biological
evolution uncovered by Charles Darwin [10]. Note that, the underlying concept
of SoFA was inspired by some earlier works [42]. An important advantage of the
proposed method is that one can rigorously guarantee its convergence for an
extensive class of objective functions. Note that, for some bio-inspired
techniques the existence of convergence has been an issue (in terms of formal
proof and some simply not guaranteeing convergence with a probability of
unity). Another significant advantage of SoFA is that this framework can
efficiently work in higher dimensional spaces and allows a gradual increase in
dimensionality to deal with infinite-dimension spaces. Our tests based on a
complex biological system (4) showed high efficiency of SoFA compared to
several other bio-inspired optimisation methods (see Fig.3). Finally, the
algorithm itself is simple in terms of practical coding. Moreover, the given
algorithm allows us to use parallel programming, for example one can implement
a modification of SoFA by introducing new dimensions by entire blocks and
fulfil optimisation for entire blocks.
As it follows from its nature, the algorithm can be applied to uncover the
optimal fitness in complex biological systems with multiple maxima, as the
considered example of plankton DVM. However, the method can be naturally
implemented to other non-biological systems, for example, in problems of
optimal control of heat transfer [42] or deformations in a metal rod [43].
Mathematically, the convergence of our method is guaranteed by the property of
the localisation of the probability measure in a Hilbert space, which, was
reported in previous studies of modelling biological evolution [11, 12, 13,
14]. In this case, the measure of available strategies (life-history traits)
within the population should be the indicator of the presence of a set of
strategies in this population. This measure (in the simplest case it is the
amount organisms using a particular strategy) will evolve with time, i.e. with
an increase of the number of iterations. As a result, after a long time, only
strategies with the fitness close to the optimal one would survive in the
system. Eventually, the distribution (e.g. the density function) of strategies
would tend to a delta function with a centre corresponding to the maximum of
fitness. We call this phenomenon the localisation of the measure.
The phenomenon of the localisation of measure in SoFA can be seen better in
Fig. 4, which shows the evolution of the distribution of the probability
density function for the point $z_{k+1}$ after $k$ first iterations shown in
the figure label. For simplicity, we show the evolution of the constant term
$v_{Y,1}$ in the Fourier expansion for the stage Y. The resulting
distributions in Fig. 4 are obtained using the total probability law theorem:
the conditional probability to find $z_{k+1}$ for a particular choice of the
reference point $\bar{z}$ is averaged across all possible reference points
with weights given by $\frac{J^{k}(z_{i})}{J^{k}(z_{1})+\ldots+J^{k}(z_{k})}$,
$i=1,\ldots,k$. One can see from the figure that the distribution of the
population of strategies is drifting on a logarithmic time scale towards a
certain final value of parameter value and the shape of the distribution is
approaching a delta function. As a result, the measure (the integral over the
density) is eventually being localised at the point corresponding to the
maximum of fitness. Note that the closeness of the distribution to the delta
function can be a working criterion of terminating interactions in the
considered algorithm. Interestingly, plotting the distributions of iteration
points as those in Fig. 4 can be useful for assessing the standard deviation
of life-history traits within a population as a function of the number of
population generations. This will of practical biological (i.e. not only
mathematical) interest in the case a biologist needs to estimate the scatting
of the considered life-history trait of evolving organisms from the expected
eventual optimal value.
Figure 4: Probability density distribution of the parameter $v_{Y,1}$ (the
average daily depth in the DVM trajectory of early developmental stage Y).
Here the new optimisation method is implemented with $\epsilon(k)=k^{-(a+bk)}$
with $a=0.7$ and $b=5$x$10^{-6}$ to the maximisation of the fitness $J(v)$,
with each trajectory $v_{s}$ being of the form (5) each with $n=15$ terms.
Interestingly, our implementation of the SoFA framework to reveal the DVM of
herbivorous zooplankton provides some important ecological insights. Note that
the model parameters describing the growth, morality, maturation, energy
losses, etc. are biologically meaningful and correspond to those in [39]. For
these realistic parameters, we reveal the existence of another (suboptimal)
scenario of DVM where fitness has its local maximum. This scenario is
characterised by zooplankton staying in shallower depths during the day time
(see Fig.1). Biologically, the shallow depths scenario signifies that
zooplankton grazers reach the minimal depth, where the visual predation by
fish becomes negligible. Whereas, empirical observations confirm that, in
reality, the other scenario where zooplankton stay during the day time at much
deeper depths is realised [39]. The deep waters DVM scenario corresponds to
the global maximum of population fitness $J$ shown in Fig.1. The relevant
biological conclusion is that the optimal migration depth in the considered
ecosystem (the north-eastern Black Sea), is determined by metabolic costs
(which are negligible in deep waters) rather than the predation threat which
becomes negligible already at depths of 40-50m. Another interesting conclusion
from the implementation of SoFA is that the smooth nonlinear optimal
trajectories of DVM modelled in the entire function space are close to those
given by linear functions at the ascending, descending and deep waters phase
of DVM. On the contrary, near the water surface (during the night time), a
constant adjustment of depth by a zooplankter results in a higher value of
fitness, compared to the scenario where the organism remains at a constant
depth.
Note that as a stochastic optimisation method, SoFA is close to the
evolutionary algorithms, such as differential evolution [6]. However, an
important difference of SoFA with the existing evolutionary algorithms is that
it uses the information about the trial points obtained from the all previous
iterations (i.e. using the entire evolution history) when choosing the
reference point at the current iteration step. As such, the selection of
strategies in SoFA occurs on a slower time scale. Also, unlike genetic
algorithms and differential evolution, which are semi-empirical, SoFA assures
the convergence for an arbitrary positive continuous functional $J$.
The proposed method is an extension of a Monte-Carlo classical approach,
which, uses a non-uniform and adjustable density of distribution of trial
points.
Following from our numerical experiments, the convergence rate of SoFA largely
depends on how quickly the value of the standard deviation in the distribution
of $z_{k}$ decreases with the number of iterations. A fast drop in the
mentioned standard deviation results in an improvement of the convergence
rate; however, this also increases the risk of being stuck for a long time
near a suboptimal local maximum. In the case the objective function does not
contain abrupt spikes, the risk of being trapped by a non-global maximum is
low. For a sufficiently smooth function $J$, one can substantially accelerate
the convergence rate via increasing the rate of decay of the standard
deviation of the distribution of $z_{k}$. Therefore, the method has a
potential of improving the convergence rate via an a priori knowledge about
the behaviour of the objective function. The required a priori knowledge of
$J$ can be obtained as a part of the implementation of the algorithm, for
example by estimating the upper bound of the Lipschitz constant for different
parts of the parameter space. Using the evaluation of $J$ at each trial point,
one can estimate the local Lipschitz constant. Using the obtained estimates of
local Lipschitz constants, one can group the trial points $z_{k}$ by splitting
the whole parameter space into subdomains with different patterns of behaviour
of the functions: ‘flat valleys’ and ‘sharp peaks’. For different subdomains,
one can consider distinct rates of increase of the standard deviations, which
will allow for the enhancing of the convergence properties without risk to get
tapped by a non-global maximum.
The intrinsic connection between SoFA and biological evolution can be
uncovered when considering the growth of the dimensionality of the search
space, which biologically signifies a gradual increase in the complexity of
competing species, macroevolution. The method generates mutations not only by
modifying the values of the existing model parameters but also producing new
types of mutants with characteristics absent in ancestors via adding new
dimensions. This is related to the increase of the number of inherited life-
history traits and the overall length of the underlying genetic code. Since
the increase in dimensionality in the method is not bounded, the evolutionary
processes modelled by SoFA admit an unlimited enhancement of quality species
and an unrestricted increase in the complexity of their organisation.
Finally, the presented optimisation method would contribute to the resolving
of the curse of dimensionality problem in optimisation. For a hyperball in $n$
dimensions, its volume is concentrated near its boundary. Signifying that in
the case where we generate trial points based on a uniform distribution inside
some $n$-dimensional hyperball, most of the points will lend within a very
thin boundary layer of this hyperball, whereas, only a small proportion of
them will target the central part [44]. However, from the classical theory of
calculus of variations, it is well-known that the solution to the optimisation
problem is usually an internal point of the functional space. Therefore, a
numerical solution of the optimisation problem in high dimensional spaces
approximating the underlying Hilbert space requires approaching to some
internal point of the considered multi-dimensional hyperball. All stochastic
methods based on the uniform distribution of trial points become largely
inefficient in this situation, whereas SoFA would be a better candidate to
cope with the challenge. Indeed, with a further increase of dimensionality of
the search space, the distribution of generated points in SoFA experiences a
permanent evolution of the shape. This guarantees the localisation of the
trial points in internal parts (i.e. located far from the boundary) of the
multi-dimensional hyperballs (see Fig. 4 as an illustrative example).
## 5 Summary
In this paper, we present a bio-inspired method of global optimisation which
quantifies Darwin’s’ famous idea of the survival of the fittest (the Survival
of the Fittest Algorithm, SoFA). The method has multiple advantages as
compared to other bio-inspired stochastic optimisation algorithms. In
particular, its convergence is guaranteed for any positive continuous
objective function(al) and one can apply the method can cope with increasing
dimensionality of space: it can find the optimal solution in an infinite-
dimensional functional space. Based on an insightful motivating example,
maximisation of the fitness functional in a stage-dependent population model,
we demonstrate the better performance of SoFA, as compared with some other
stochastic algorithms of global optimisation in the case when the
dimensionality of the parameter space is high.
## Appendix A. Simplified algorithm of the optimisation method.
Here, we provided a simplified version of the SoFA framework presented in
Section 2.1.
Suppose there is some continuous positive function $J(z)$ which is defined on
the rectangular domain
$P=\\{z=[z^{1},\ldots,z^{n}]:a^{i}<z^{i}<b^{i},i=1:n\\}$. Assuming that $J(z)$
has a unique point of maximum (denoted by $z^{*}$), such a point can be
located by implementing a simplification of the novel optimisation method
described in section 1. This simplified method is outlined in the following
algorithm.
1. 1.
Assume $k$ steps of the method have already been completed and therefore one
has $k$ points in $P=\\{z_{1},..,z_{k}\\}$, with known corresponding values of
$J(z_{1}),\ldots,J(z_{k})$.
2. 2.
Randomly select some reference point $\bar{z}$ out of all the available points
in $P$. The probability of selecting each of these available points is given
by $\frac{J^{k}(z_{i})}{J^{k}(z_{1})+\ldots+J^{k}(z_{k})}$
3. 3.
Using this reference point, select a new point $z_{k+1}$ by setting each
coordinate of this point to be a random variable in the interval
$[a^{j},b^{j}]$ with the probability density function given by
$\frac{A^{j}_{k+1}}{\epsilon_{k+1}+((z_{k+1}^{j})-\bar{z}^{j})^{2}}$ where the
constants $A^{j}_{K+1}$ are chosen to normalise the probability over the
interval $[a^{j},b^{j}]$. Here $\bar{z}^{j}$ is $j^{th}$ coordinate the
reference point $\bar{z}$. Note that we use the same notation for the
reference point $\bar{z}$ for the sake of simplicity: at each step the
reference point might be different.
4. 4.
The function is evaluated at $z_{k+1}$ giving the value of $J(z_{k+1})$, then
the steps can be repeated.
5. 5.
The method will terminate once the approximation satisfies some initially
prescribed requirement (for example, the distribution of $z_{k+1}$ is
sufficiently close to a delta function, see Fig.4).
## References
* [1] K. Deb, Multi-objective optimization using evolutionary algorithms, Vol. 16, John Wiley & Sons, 2001.
* [2] K. M. Passino, Bacterial foraging optimization, in: Innovations and Developments of Swarm Intelligence Applications, IGI Global, 2012, pp. 219–234.
* [3] M. Mavrovouniotis, C. Li, S. Yang, A survey of swarm intelligence for dynamic optimization: Algorithms and applications, Swarm and Evolutionary Computation 33 (2017) 1–17.
* [4] D. Rai, K. Tyagi, Bio-inspired optimization techniques: a critical comparative study, ACM SIGSOFT Software Engineering Notes 38 (4) (2013) 1–7.
* [5] T. Back, Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms, Oxford university press, 1996.
* [6] R. Storn, K. Price, Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces, Journal of global optimization 11 (4) (1997) 341–359.
* [7] F. Di Patti, D. Fanelli, F. Piazza, Optimal search strategies on complex multi-linked networks, Scientific reports 5 (1) (2015) 1–6.
* [8] D. Volchenkov, J. Helbach, M. Tscherepanow, S. Kühnel, Exploration–exploitation trade-off features a saltatory search behaviour, Journal of The Royal Society Interface 10 (85) (2013) 20130352.
* [9] E. P. Odum, G. W. Barrett, Fundamentals of ecology, Vol. 3, Saunders Philadelphia, 1971.
* [10] C. Darwin, W. F. Bynum, The origin of species by means of natural selection: or, the preservation of favored races in the struggle for life, AL Burt New York, 2009.
* [11] A. N. Gorban, Selection theorem for systems with inheritance, Mathematical Modelling of Natural Phenomena 2 (4) (2007) 1–45.
* [12] O. Kuzenkov, E. Ryabova, Limit possibilities of solution of a hereditary control system, Differential Equations 51 (4) (2015) 523–532.
* [13] O. Kuzenkov, A. Novozhenin, Optimal control of measure dynamics, Communications in Nonlinear Science and Numerical Simulation 21 (1) (2015) 159 – 171.
* [14] O. Kuzenkov, A. Morozov, Towards the construction of a mathematically rigorous framework for the modelling of evolutionary fitness, Bulletin of Mathematical Biology 81 (11) (2019) 4675–4700.
* [15] G. C. Hays, A review of the adaptive significance and ecosystem consequences of zooplankton diel vertical migrations, in: Migrations and Dispersal of Marine Organisms, Springer, 2003, pp. 163–170.
* [16] M. J. Kaiser, M. J. Attrill, S. Jennings, D. N. Thomas, D. K. Barnes, et al., Marine ecology: processes, systems, and impacts, Oxford University Press, 2011\.
* [17] C. H. da Silva Santos, M. S. Goncalves, H. E. Hernandez-Figueroa, Designing novel photonic devices by bio-inspired computing, IEEE Photonics Technology Letters 22 (15) (2010) 1177–1179.
* [18] C. H. da Silva Santos, Parallel and bio-inspired computing applied to analyze microwave and photonic metamaterial strucutures., Ph.D. thesis, University of Campinas, Brazil (2010).
* [19] P. Kaelo, M. Ali, Some variants of the controlled random search algorithm for global optimization, Journal of optimization theory and applications 130 (2) (2006) 253–264.
* [20] W. Price, Global optimization by controlled random search, Journal of Optimization Theory and Applications 40 (3) (1983) 333–348.
* [21] W. L. Price, A controlled random search procedure for global optimisation, The Computer Journal 20 (4) (1977) 367–370.
* [22] A. R. Kan, G. T. Timmer, Stochastic global optimization methods part i: Clustering methods, Mathematical programming 39 (1) (1987) 27–56.
* [23] A. R. Kan, G. T. Timmer, Stochastic global optimization methods part ii: Multi level methods, Mathematical Programming 39 (1) (1987) 57–78.
* [24] H. W. Ducklow, D. K. Steinberg, K. O. Buesseler, Upper ocean carbon export and the biological pump, OCEANOGRAPHY-WASHINGTON DC-OCEANOGRAPHY SOCIETY- 14 (4) (2001) 50–58.
* [25] K. O. Buesseler, C. H. Lamborg, P. W. Boyd, P. J. Lam, T. W. Trull, R. R. Bidigare, J. K. Bishop, K. L. Casciotti, F. Dehairs, M. Elskens, et al., Revisiting carbon flux through the ocean’s twilight zone, Science 316 (5824) (2007) 567–570.
* [26] D. Bianchi, E. D. Galbraith, D. A. Carozza, K. Mislan, C. A. Stock, Intensification of open-ocean oxygen depletion by vertically migrating animals, Nature Geoscience 6 (7) (2013) 545–548.
* [27] A. N. Hansen, A. W. Visser, Carbon export by vertically migrating zooplankton: an optimal behavior model, Limnology and Oceanography 61 (2) (2016) 701–710.
* [28] M. D. Ohman, J. A. Runge, Sustained fecundity when phytoplankton resources are in short supply: omnivory by calanus finmarchicus in the gulf of st. lawrence, Limnology and Oceanography 39 (1) (1994) 21–36.
* [29] M. Fortier, L. Fortier, H. Hattori, H. Saito, L. Legendre, Visual predators and the diel vertical migration of copepods under arctic sea ice during the midnight sun, Journal of Plankton Research 23 (11) (2001) 1263–1278.
* [30] S. Pearre, Jr, Eat and run? the hunger/satiation hypothesis in vertical migration: history, evidence and consequences, Biological Reviews 78 (1) (2003) 1–79.
* [31] W. Lampert, The adaptive significance of diel vertical migration of zooplankton, Functional ecology 3 (1) (1989) 21–27.
* [32] C. W. Clark, M. Mangel, et al., Dynamic state variable models in ecology: methods and applications, Oxford University Press on Demand, 2000.
* [33] J. Ringelberg, Diel vertical migration of zooplankton in lakes and oceans: causal explanations and adaptive significances, Springer Science & Business Media, 2009.
* [34] A. Morozov, E. Arashkevich, A. Nikishina, K. Solovyev, Nutrient-rich plankton communities stabilized via predator—prey interactions: revisiting the role of vertical heterogeneity, Mathematical Medicine and Biology: a Journal of the IMA 28 (2) (2011) 185–215.
* [35] Ø. Fiksen, J. Giske, Vertical distribution and population dynamics of copepods by dynamic optimization, ICES Journal of Marine Science 52 (3-4) (1995) 483–503. doi:10.1016/1054-3139(95)80062-X.
* [36] B.-P. Han, M. Straškraba, Control mechanisms of diel vertical migration: theoretical assumptions, Journal of Theoretical Biology 210 (3) (2001) 305–318.
* [37] S.-H. Liu, S. Sun, B.-P. Han, Diel vertical migration of zooplankton following optimal food intake under predation, Journal of Plankton Research 25 (9) (2003) 1069–1077.
* [38] A. Y. Morozov, O. A. Kuzenkov, Towards developing a general framework for modelling vertical migration in zooplankton, Journal of Theoretical Biology 405 (2016) 17–28.
* [39] A. Morozov, O. A. Kuzenkov, E. G. Arashkevich, Modelling optimal behavioural strategies in structured populations using a novel theoretical framework, Scientific reports 9 (1) (2019) 1–15.
* [40] J. M. Cushing, An introduction to structured population dynamics, SIAM, 1998.
* [41] L. W. Botsford, B. D. Smith, J. F. Quinn, Bimodality in size distributions: the red sea urchin strongylocentrotus franciscanus as an example, Ecological Applications 4 (1) (1994) 42–50.
* [42] O. A. Kuzenkov, V. A. Grishagin, Global optimization in hilbert space, AIP Conference Proceedings 1738 (1) (2016) 400007. doi:10.1063/1.4952195.
* [43] A. Irkhina, O. Kuzenkov, Identification of the distribution of deformations in a rod as a problem of optimal control, Journal of Computer and Systems Sciences International 44 (5) (2005) 689–694.
* [44] A. N. Gorban, I. Y. Tyukin, Blessing of dimensionality: mathematical foundations of the statistical physics of data, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376 (2118) (2018) 20170237.
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# Quintessence in the Weyl-Gauss-Bonnet Model
José Jaime Terente Díaz Konstantinos Dimopoulos Mindaugas Karčiauskas
Antonio Racioppi
###### Abstract
Quintessence models have been widely examined in the context of scalar-Gauss-
Bonnet gravity, a subclass of Horndeski’s theory, and were proposed as viable
candidates for Dark Energy. However, the relatively recent observational
constraints on the speed of gravitational waves $c_{\textrm{GW}}$ have
resulted in many of those models being ruled out because they predict
$c_{\textrm{GW}}\neq c$ generally. While these were formulated in the metric
formalism of gravity, it was found later that some Horndeski models could be
rescued in the Palatini formalism, where the connection is independent of the
metric and the underlying geometry no longer corresponds to the pseudo-
Riemannian one. Motivated by this and the relation between scalar-Gauss-Bonnet
gravity and Horndeski’s theory, we put forward a new quintessence model with
the scalar-Gauss-Bonnet action but in Weyl geometry. We find the fixed points
of the dynamical system under some assumptions and determine their stability
via linear analysis. Although the past evolution of the Universe as we know it
is correctly reproduced, the constraints on $c_{\textrm{GW}}$ are shown to be
grossly violated for the coupling function under consideration. The case of
$c_{\textrm{GW}}=c$ is regarded also, but no evolution consistent with other
cosmological observations is obtained.
## 1 Introduction
Late Dark Energy (DE) models have become quite popular following the
observational discovery of the current accelerated expansion of the Universe
from Type Ia supernovae [1, 2]. _Planck_ Collaboration in particular finds
that the equation of state (EoS) parameter of this DE is consistent with a
cosmological constant (CC) [3], arguably the simplest candidate for DE. This
CC is interpreted as vacuum energy in particle physics [4, 5, 6], but no
convincing scenario has been found where the actual observed energy scale of
DE can be naturally accounted for by vacuum energy ascribed to a particle
physics model [7, 8].
_Planck_ measurements also leave the door open to a dynamical scalar field
(named ‘quintessence’ [9, 10]), with changing DE EoS parameter. This latter
possibility allows for a more sensible mechanism to explain DE [11, 12, 13]
and paves the way for a unified description (called ‘quintessential inflation’
[14, 15, 16, 17]) of the late accelerated expansion era and early inflation.
The inflationary phase in the early evolution of the Universe serves to
explain the primordial origin of the Large Scale Structures in the Universe
while resolving the horizon and flatness problems of the Hot Big Bang model
[18, 19, 20].
Many of those quintessence DE models rely on modifications of gravity at large
and fall within the scalar-tensor theories chiefly [21, 25, 26, 24, 22, 23].
Consequently, they have been (and still are) under close examination after the
detection of gravitational waves (GWs) made by the Advanced LIGO and Virgo
detectors [27, 28, 29] (see Refs. [30, 31, 32, 33, 34, 35]). The multi-
messenger observation of a binary neutron star system’s merger [36] placed
stringent bounds on the relative difference between the speed of propagation
of GWs, $c_{\textrm{GW}}$, and the speed of light in vacuum, $c$ [37].
Defining the parameter
$\alpha_{T}\equiv c_{\textrm{GW}}^{2}-1~{},$ (1.1)
in units where $c=1$, those bounds yield
$|\alpha_{T}|<10^{-15},$ (1.2)
given the delay between arrival times of the GWs and $\gamma$-rays emitted by
the system’s merger.
Horndeski’s theory [38] is an example of theory of gravity that has been
strongly constrained by Eq. (1.2).111Regarding this bound, its consistency
across the whole spectrum of GW frequencies has been discussed within the
context of Horndeski’s theory in Ref. [39]. This theory is the most general
one in four dimensions leading to second-order field equations of the metric
tensor $g_{\mu\nu}(x)$ and a scalar field $\phi(x)$. It can be shown that, in
order for GWs to propagate at the speed of light ($\alpha_{T}=0$), only the
standard, field-dependent conformal coupling to gravity $G(\phi)R$, where $R$
is the Ricci scalar, is allowed. This reduces significantly the number of
subclasses of Horndeski’s theory conducive to modelling quintessence [40, 31].
Among those subclasses affected by the constraint (1.2), the scalar-Gauss-
Bonnet (SGB) model, the scalar-tensor generalisation of Einstein-Gauss-Bonnet
(EGB) gravity [41], is a well-known case [40] in cosmology. This model
consists of a field-dependent coupling $\xi(\phi)$ to the Gauss-Bonnet (GB)
term
$\mathcal{G}\equiv
R^{2}-4R^{\mu\nu}R_{\mu\nu}+R^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}~{},$
(1.3)
where $R_{\mu\nu}$ and $\tensor{R}{{}^{\alpha}_{\beta\mu\nu}}$ are the Ricci
and Riemann tensors, respectively, in addition to the Einstein-Hilbert action
of General Relativity (GR). The upper bound on $|\alpha_{T}|$ is satisfied if
$\xi$ is constant. This is not surprising as the GB term is a topological term
in four dimensions and does not contribute to the field equations of the
metric [41]. Hence, one is left with Einstein’s theory of GR, which predicts
that GWs propagate at the speed of light. Another possibility is to impose the
following condition:
$\ddot{\xi}=H\dot{\xi}~{},$ (1.4)
which restricts the functional form of the coupling $\xi(t)$, such that
$\dot{\xi}(t)\propto a(t)$. $H(t)\equiv\dot{a}(t)/a(t)$ is the Hubble
parameter and $a(t)$ the scale factor. Overdots denote time derivatives and a
homogeneous, isotropic and spatially-flat spacetime, described by the flat
Friedmann-Lemaître-Robertson-Walker (FLRW) metric
$g_{\mu\nu}=\textrm{diag}\left[-1,a^{2}(t),a^{2}(t),a^{2}(t)\right]$, was
assumed. The consequences of the scaling of $\xi$ after Eq. (1.4) have been
studied in the literature in the context of cosmic inflation [42, 44, 43],
despite the fact that the neutron star system’s merger is a low-redshift
event, significantly lower than the estimated redshift value at the onset of
DE domination [45] and at a time much later than inflation. For this latter
reason, we considered Gauss-Bonnet Dark Energy (GBDE) models with
$\alpha_{T}=0$ in Ref. [46]. Unfortunately, we found that one cannot make
$c_{\textrm{GW}}=1$ if the density parameters of matter and DE are such that
$\Omega_{\textrm{M}}\sim\Omega_{\textrm{DE}}$.
Leaving those two possibilities aside, there are works in which the effects of
the constraint on $|\alpha_{T}|$ are examined and illustrated [48, 47], the
first reference including a case of kinetic coupling to curvature besides the
coupling to the GB term. In Ref. [46], those effects were explored in the GBDE
models mentioned above. We came to the realisation that the bound does not
seem very constraining for those models. Contrary to this intuition however,
it was found that the constraint is easily violated at present
time.222Modifications to the SGB model including a coupling between the matter
sector and the quintessence field or a non-minimal coupling of the scalar
field to gravity seem to satisfy the constraint (1.2) for very specific
initial conditions set in radiation domination and $\xi(\phi)\propto\phi^{2}$.
See results reported in Refs. [49, 50].
Additionally, there has been an intensive research on Horndeski models in the
framework of _the Palatini formalism_ [51, 52, 53, 54, 55].333Recent
quintessential inflation models have been proposed in this formalism in $f(R)$
and $f(\phi,R)$ theories [56, 57]. In the case of $f(\phi)R$ gravity, an
example of quintessence scalar field couplings with metric and torsion can be
found in Ref. [58], where the growth of linear matter perturbations is
studied. A striking result of that investigation has been the realisation that
Horndeski models in this formalism, with a conformal coupling depending on the
kinetic term
$X\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ (that is,
$G(\phi,X)R$), do allow GWs propagating at the speed of light [53]. This is in
contrast to what the metric one prescribes. Similar attempts were made to
address this question for more complicated Horndeski models [54, 55], but the
connection field equations turn out to be _differential_ rather than
algebraic, so the connection $\Gamma^{\alpha}_{\mu\nu}$ becomes a propagating
degree of freedom of the theory. This entails relevant new aspects in the
physical content of the theory [51]. Even the fact of whether the SGB model
remains a subclass of Horndeski theory in the Palatini formalism is an
unexplored issue in the literature to the best of our knowledge.
Motivated by that relation between the SGB model and Horndeski’s theory in the
metric formalism, and the results of the referenced works involving the speed
of GWs, we propose a quintessence model with the SGB action assuming Weyl
geometry, and determine the behaviour of the $\alpha_{T}$ parameter. A
previous analysis of EGB gravity with a Weyl connection in higher dimensions
was carried out in Ref. [59], but the present work considers a field-dependent
coupling function $\xi(\phi)$ instead. On top of this, we find that the very
same homogeneous and tensor perturbation equations derived here apply to
another connection which, as opposed to the Weyl connection, has non-vanishing
torsion and is metric compatible. The role the projective transformations play
to achieve this is discussed in Sec. C of the Appendix.
Once the equations are derived, we perform a dynamical systems analysis
assuming an exponential potential, in line with Ref. [46]. This potential is
very common in quintessence scenarios [60, 61, 57, 62, 11, 12]. For an
exponential coupling function $\xi(\phi)$, the fixed points are calculated,
those being relevant from the cosmological standpoint. We analyse their
stability and compute the $\alpha_{T}$ parameter along trajectories that
reproduce the observationally constrained values of the density parameter of
matter and the effective EoS parameter of the DE fluid at the present time.
While in the SGB model, Eq. (1.4) must be satisfied in order for $\alpha_{T}$
to vanish, here a different equation is obtained and its stability around the
de Sitter and scaling regimes is examined.
Natural units for which $M_{\textrm{Pl}}\equiv(8\pi G)^{-1/2}$,
$M_{\textrm{Pl}}=2.43\times 10^{18}\,$GeV being the reduced Planck mass, have
been assumed in this work. $G$ is Newton’s gravitational constant.
## 2 Theoretical Framework
### 2.1 Quadratic Gravity in the Palatini Formalism
In the metric formalism of gravity, the connection is set to be
$\mathring{\Gamma}^{\alpha}_{\mu\nu}\equiv\frac{1}{2}g^{\alpha\beta}\left(\partial_{\mu}g_{\beta\nu}+\partial_{\nu}g_{\beta\mu}-\partial_{\beta}g_{\mu\nu}\right),$
(2.1)
which is the well-known ‘Levi-Civita’ (LC) connection [63].444Rings
(overcircles) are used to denote geometric quantities defined with respect to
the LC connection and its first derivatives. We shall adopt this convention
hereon, unless otherwise stated. It is torsion free and metric compatible.
Consequently
$\displaystyle\mathring{\Gamma}^{\alpha}_{\mu\nu}=\mathring{\Gamma}^{\alpha}_{\nu\mu}~{},$
(2.2) $\displaystyle\mathring{\nabla}_{\alpha}g_{\mu\nu}=0~{},$ (2.3)
respectively, where $\mathring{\nabla}_{\mu}$ is the covariant derivative
induced by the LC connection.
In the Palatini formalism of gravity however, the connection
$\Gamma^{\alpha}_{\mu\nu}(x)$ does not depend on the metric tensor
$g_{\mu\nu}(x)$. The dynamics of $\Gamma^{\alpha}_{\mu\nu}$ is governed by
appropriate field equations. While $\Gamma^{\alpha}_{\mu\nu}$ does not
necessarily fulfill Eqs. (2.2) and (2.3), one can relate it to the LC
connection by
$\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}\equiv\Gamma^{\alpha}_{\mu\nu}-\mathring{\Gamma}^{\alpha}_{\mu\nu}~{}.$
(2.4)
Since $\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}$ is the difference of two
connections, it transforms as a tensor under coordinate transformations. This
tensor is called the ‘distortion tensor’ [64]. By defining the torsion and
non-metricity tensors as
$\displaystyle\tensor{T}{{}_{\mu\nu}^{\alpha}}\equiv\Gamma^{\alpha}_{\left[\mu\nu\right]}=\Gamma^{\alpha}_{\mu\nu}-\Gamma^{\alpha}_{\nu\mu}~{},$
(2.5) $\displaystyle Q_{\alpha\mu\nu}\equiv\nabla_{\alpha}g_{\mu\nu}~{},$
(2.6)
respectively, we find the following equations:
$\displaystyle\tensor{T}{{}_{\mu\nu}^{\alpha}}=\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}-\tensor{\kappa}{{}_{\nu\mu}^{\alpha}},$
(2.7) $\displaystyle
Q_{\alpha\mu\nu}=-\kappa_{\alpha\mu\nu}-\kappa_{\alpha\nu\mu}~{},$ (2.8)
where indices are raised and lowered with respect to the metric tensor
$g_{\mu\nu}$. We see that $\tensor{T}{{}_{\mu\nu}^{\alpha}}$ is antisymmetric
in its first two indices, while $Q_{\alpha\mu\nu}$ is symmetric in the last
two.
In this work, we consider $\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}$ of the form
$\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}=\delta_{\mu}^{\alpha}A_{\nu}+\delta_{\nu}^{\alpha}A_{\mu}-g_{\mu\nu}A^{\alpha}.$
(2.9)
$A_{\mu}$ in the above expression is a vector field that we call ‘Weyl vector’
and $\Gamma^{\alpha}_{\mu\nu}$ with that distortion tensor is known as the
‘Weyl connection’ [59]. Then, the corresponding torsion and non-metricity
tensors, using Eqs. (2.7) and (2.8), are respectively,
$\tensor{T}{{}^{\alpha}_{\mu\nu}}=0$ and
$Q_{\alpha\mu\nu}=-2g_{\mu\nu}A_{\alpha}~{}.$ (2.10)
Weyl connection then has non-vanishing non-metricity tensor. A more general
connection is regarded in App. A.
Given the GB term in the metric formalism $\mathring{\mathcal{G}}$
$\mathring{\mathcal{G}}\equiv\mathring{R}^{2}-4\mathring{R}^{\mu\nu}\mathring{R}_{\mu\nu}+\mathring{R}^{\alpha\beta\mu\nu}\mathring{R}_{\alpha\beta\mu\nu}~{},$
(2.11)
one might regard a similar collection of quadratic curvature scalars as the GB
term in the Palatini formalism. However, given the lack of symmetries in the
latter formalism (see App. A), one must take into account that the most
general combination of quadratic terms is more complicated. In fact, such a
combination can be written as [65]
$\displaystyle\mathcal{G}=\alpha
R^{2}+R^{\mu\nu}\left(\beta_{1}R_{\mu\nu}+\beta_{2}R_{\nu\mu}\right)+\tilde{R}^{\mu\nu}\left(\beta_{3}R_{\mu\nu}+\beta_{4}R_{\nu\mu}+\beta_{5}\tilde{R}_{\mu\nu}+\beta_{6}\tilde{R}_{\nu\mu}\right)+\bar{R}^{\mu\nu}\left(\beta_{7}R_{\mu\nu}+\right.$
$\displaystyle\left.+\beta_{8}\tilde{R}_{\mu\nu}+\beta_{9}\bar{R}_{\mu\nu}\right)+R^{\alpha\beta\mu\nu}\left(\gamma_{1}R_{\alpha\beta\mu\nu}+\gamma_{2}R_{\beta\alpha\mu\nu}+\gamma_{3}R_{\mu\nu\alpha\beta}+\gamma_{4}R_{\alpha\mu\nu\beta}+\gamma_{5}R_{\mu\alpha\nu\beta}+\right.$
$\displaystyle\left.+\gamma_{6}R_{\nu\beta\mu\alpha}+\gamma_{7}R_{\beta\nu\mu\alpha}\right),$
(2.12)
where $\tilde{R}_{\mu\nu}$ and $\bar{R}_{\mu\nu}$ are the co-Ricci tensor and
the homothetic curvature tensor, defined in Eqs. (A.5) and (A.6)
respectively.555We refer the reader to Ref. [59], where ‘co-Ricci’ and
‘homothetic’ are used to name the respective tensors as well. In the above,
the constant coefficients $\alpha$, $\left\\{\beta_{i}\right\\}_{i=1,...,6}$
and $\left\\{\gamma_{i}\right\\}_{i=1,...,7}$ are arbitrary. In order to
recover Eq. (2.11) when $\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}=0$ (i.e.
$\Gamma^{\alpha}_{\mu\nu}=\mathring{\Gamma}^{\alpha}_{\mu\nu}$), we set
$(\alpha,\beta,\gamma)=(1,-4,1)$, where
$\displaystyle\beta\equiv\sum^{6}_{i=1}\beta_{i}~{},$ (2.13)
$\displaystyle\gamma\equiv\gamma_{1}-\gamma_{2}+\gamma_{3}~{}.$ (2.14)
The rest of $\gamma$ coefficients are chosen such that
$\gamma_{4}=\gamma_{5}=\gamma_{6}=\gamma_{7}$ [65]. Since
$\mathring{\bar{R}}=0$, there is no need to fix $\beta_{7}$, $\beta_{8}$ and
$\beta_{9}$.
With that parametrisation, $\mathcal{G}$ becomes the Lagrangian of quadratic
terms that we use in the action of the SGB model in the Palatini formalism,
which reads
$S=\int\textrm{d}^{4}x\sqrt{-g}\left[\frac{M^{2}_{\textrm{Pl}}}{2}R-\frac{1}{2}\xi(\phi)\mathcal{G}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right].$
(2.15)
Also, $R=\mathring{R}+\kappa$ (see Eq. (A.9) in App. A). $V(\phi)$ is the
scalar field potential and $\xi(\phi)$ the GB coupling function.
### 2.2 Weyl-Gauss-Bonnet Action
Using the tensors defined in Eqs. (A.2)-(A.7), their symmetry properties that
were shown in App. A, and the scalar quantity $\kappa$ defined in Eq. (A.9),
we can write $\mathcal{G}$ as
$\displaystyle\mathcal{G}=\mathring{\mathcal{G}}+2\mathring{R}\kappa-8\mathring{R}^{\mu\nu}\tilde{\kappa}_{\mu\nu}+2\mathring{R}^{\alpha\beta\mu\nu}\bar{\kappa}_{\alpha\beta\mu\nu}+\kappa^{2}-4\tilde{\kappa}^{\mu\nu}\tilde{\kappa}_{\mu\nu}+\bar{\kappa}^{\alpha\beta\mu\nu}\bar{\kappa}_{\alpha\beta\mu\nu}-\frac{1}{4}\Upsilon\bar{\kappa}^{\mu\nu}\bar{\kappa}_{\mu\nu}~{},$
(2.16)
where $\mathring{\mathcal{G}}$ is defined in Eq. (2.11) and $\Upsilon$ is
given by
$\displaystyle\Upsilon\equiv
1-\beta_{1}+\beta_{2}-2\beta_{7}-4\beta_{9}-2\gamma_{1}+\gamma_{4}~{}.$ (2.17)
As one can see, $\Upsilon$ depends on the arbitrary coefficients, whereas the
rest of the terms in Eq. (2.16) are oblivious to them.
In terms of the Weyl vector and its derivatives, $\mathcal{G}$ reads
$\displaystyle\mathcal{G}=\mathring{\mathcal{G}}+8\left[\left(\mathring{G}^{\mu\nu}-\mathring{\nabla}^{\nu}A^{\mu}\right)\mathring{\nabla}_{\mu}A_{\nu}-\left(\mathring{R}^{\mu\nu}-2\mathring{\nabla}^{\mu}A^{\nu}\right)A_{\mu}A_{\nu}+\left(\mathring{\nabla}_{\sigma}A^{\sigma}+A_{\sigma}A^{\sigma}\right)\mathring{\nabla}_{\rho}A^{\rho}-\right.$
$\displaystyle\left.-\Upsilon\mathring{\nabla}^{\mu}A^{\nu}\left(\mathring{\nabla}_{\mu}A_{\nu}-\mathring{\nabla}_{\nu}A_{\mu}\right)\right].$
(2.18)
Plugging this $\mathcal{G}$ into Eq. (2.15), we obtain the action of what we
call the ‘Weyl-Gauss-Bonnet’ (WGB) model,
$\displaystyle
S_{\textrm{WGB}}=\frac{1}{2}\int\textrm{d}^{4}x\sqrt{-g}\left[M^{2}_{\textrm{Pl}}\mathring{R}-\xi(\phi)\mathring{\mathcal{G}}\right]-4\int\textrm{d}^{4}x\sqrt{-g}\left\\{\frac{3}{4}M^{2}_{\textrm{Pl}}A_{\sigma}A^{\sigma}-\left(\mathring{G}^{\mu\nu}-\mathring{\nabla}^{\mu}A^{\nu}\right)A_{\mu}\partial_{\nu}\xi-\right.$
$\displaystyle\left.-\left(\mathring{\nabla}_{\sigma}A^{\sigma}+A_{\sigma}A^{\sigma}\right)\partial_{\rho}\xi
A^{\rho}-\xi(\phi)\Upsilon\mathring{\nabla}^{\mu}A^{\nu}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\right)\right\\}-\int\textrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+V(\phi)\right]+$
$\displaystyle+\int\textrm{d}^{4}x\sqrt{-g}\mathcal{L}_{\textrm{M}}(g_{\mu\nu},\Psi)~{},$
(2.19)
where we have integrated by parts and used some well-known properties of the
curvature tensors in the metric formalism [63]. We drop the rings from now on
as it is clear that every geometric quantity is defined with respect to the LC
connection and its derivatives once the variables depending on the distortion
tensor have been rewritten in terms of the Weyl vector and its covariant
derivative. Notice also that we have included the matter action
$S_{\textrm{M}}$,
$S_{\textrm{M}}[g_{\mu\nu},\Psi]=\int\textrm{d}^{4}x\sqrt{-g}\mathcal{L}_{\textrm{M}}(g_{\mu\nu},\Psi)~{},$
(2.20)
given our interest in the late Universe. ‘$\Psi$’ is used to denote
collectively the matter fields and $\mathcal{L}_{\textrm{M}}$ is the
corresponding Lagrangian. As we see, the matter fields are minimally coupled
to gravity and the action $S_{\textrm{M}}$ is _not_ a functional of the
connection, only of the metric tensor. This choice is made for simplicity,
although one might explore the possibility of matter fields coupled to the
connection. In that case, besides the energy-momentum tensor, one obtains the
so-called ‘hypermomentum tensor’ from the variation of the action with respect
to the connection [66].
### 2.3 Homogeneous Field Equations
We now present the homogeneous equations of the WGB model. The field equation
of the vector field $A_{\mu}$, derived from Eq. (2.19), is (remember that
rings are dropped for simplicity)
$\displaystyle\frac{3}{2}\left(M^{2}_{\textrm{Pl}}-\frac{4}{3}\partial_{\nu}\xi
A^{\nu}\right)A_{\mu}=\left(\tensor{G}{{}_{\mu}^{\nu}}-2\nabla_{\mu}A^{\nu}\right)\partial_{\nu}\xi+\left(2\nabla_{\sigma}A^{\sigma}+A_{\sigma}A^{\sigma}\right)\partial_{\mu}\xi+\frac{1}{2}\Upsilon\nabla_{\nu}\left(\xi\tensor{\bar{\kappa}}{{}_{\mu}^{\nu}}\right).$
(2.21)
A vector field of the form $A_{\mu}(t)\equiv(-\Phi(t),\mathbf{0})$ is
considered. That is, we take the spatial components of the vector field to
vanish at the background level. This is required to keep the expansion of the
Universe isotropic (see App. B). Such a choice implies
$\bar{\kappa}_{\mu\nu}=0$ in the background. Consequently, $\Upsilon$ does not
play any role in the dynamics of $\Phi$. In particular, the homogeneous part
of Eq. (2.21) reads
$\Phi=2\frac{\dot{\xi}}{M^{2}_{\textrm{Pl}}}\left(H-\Phi\right)^{2}.$ (2.22)
For $\dot{\xi}\neq 0$, one can write this equation as
$\Phi^{2}-2\Phi\left(H+\frac{M^{2}_{\textrm{Pl}}}{4\dot{\xi}}\right)+H^{2}=0~{}.$
(2.23)
It can be noticed from Eq. (2.22) that $\Phi=0$ for $\dot{\xi}=0$ (constant
$\xi$), and the LC connection is recovered at the background level. Another
possibility that leads to $\Phi=0$ is $H=\Phi$ (see Eq. (2.22)), but this
implies $H=0$, which is not a valid solution when we regard a time dependent
scale factor.
Eq. (2.23) is an algebraic equation that can be readily solved for
$\dot{\xi}\neq 0$
$\Phi=H\left[1+\frac{1}{4\mu}\left(1\pm\sqrt{1+8\mu}\right)\right],$ (2.24)
where $\mu$ is defined as
$\mu\equiv\frac{\dot{\xi}H}{M_{\textrm{Pl}}^{2}}~{}.$ (2.25)
Notice that $\mu$ cannot be smaller than $-1/8$, otherwise $\Phi$ becomes
complex. Since we shall consider expanding universes ($H>0$), this bound has
to be taken into account if $\dot{\xi}$ turns negative.
The _Weyl parameter_ $W\equiv H-\Phi$ is introduced, such that
$\frac{W}{H}=-\frac{1}{4\mu}\left(1-\sqrt{1+8\mu}\right),$ (2.26)
where the positive root in Eq. (2.24) has been discarded. The reason for this
is that if one takes $\dot{\xi}=0$ in Eq. (2.22), then $\Phi=0$. But only the
negative root in Eq. (2.24) yields $\Phi\simeq 0$ (i.e. $W\simeq H$) when
$|\mu|\ll 1$. On the other hand, $|\mu|\gg 1$ in Eq. (2.26) leads to
$\frac{W}{H}\simeq\frac{1}{\sqrt{2\mu}}~{},$ (2.27)
and hence $W\ll H$. For $W$ to be real, $\mu>0$ when $|\mu|\gg 1$ as was
pointed out above in relation to $\Phi$.
Later we will make use of the inverse relation to Eq. (2.26). Using Eq.
(2.22), we can write the $H(W,\dot{\xi})$ function as
$H=W+\Phi=W\left(1+2\frac{\dot{\xi}W}{M^{2}_{\textrm{Pl}}}\right).$ (2.28)
The homogeneous field equations of the metric and the scalar field are given
by (see App. B)
$\displaystyle
H^{2}=\frac{1}{3M^{2}_{\textrm{Pl}}}\left[\frac{1}{2}\dot{\phi}^{2}+V(\phi)+\rho_{\textrm{M}}+12\dot{\xi}W^{3}\right]+\Phi^{2},$
(2.29)
$\displaystyle\dot{H}=-\frac{1}{2M^{2}_{\textrm{Pl}}}\left[\dot{\phi}^{2}+\rho_{\textrm{M}}+P_{\textrm{M}}+4\dot{\xi}W^{3}-4\left(\dot{\xi}W^{2}\right)^{\bullet}-12\dot{\xi}\Phi
W^{2}\right]-3\Phi^{2},$ (2.30)
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=-12\xi_{,\phi}W^{2}\left(\dot{W}+HW\right).$
(2.31)
$\rho_{\textrm{M}}$ and $P_{\textrm{M}}$ are the homogeneous energy density
and isotropic pressure of the matter fields, respectively. The energy density
satisfies the following continuity equation:
$\dot{\rho}_{\textrm{M}}=-3H\rho_{\textrm{M}}\left(1+w_{\textrm{M}}\right),$
(2.32)
where $w_{\textrm{M}}\equiv P_{\textrm{M}}/\rho_{\textrm{M}}$ is the EoS
parameter of the background fluid under consideration, which we will assume to
lie in the range $0\leq w_{\textrm{M}}<1$. This range includes the values
corresponding to pressureless matter ($w_{\textrm{M}}=w_{m}=0$) and radiation
($w_{\textrm{M}}=w_{r}=1/3$).
If $\Phi=0$, Eqs. (2.29)-(2.31) are the same as those of the SGB model [44]
$\displaystyle
H^{2}=\frac{1}{3M^{2}_{\textrm{Pl}}}\left[\frac{1}{2}\dot{\phi}^{2}+V(\phi)+\rho_{\textrm{M}}+12\dot{\xi}H^{3}\right],$
(2.33)
$\displaystyle\dot{H}=-\frac{1}{2M^{2}_{\textrm{Pl}}}\left[\dot{\phi}^{2}+\rho_{\textrm{M}}+P_{\textrm{M}}+4\dot{\xi}H^{3}-4\left(\dot{\xi}H^{2}\right)^{\bullet}\right],$
(2.34)
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=-12\xi_{,\phi}H^{2}\left(\dot{H}+H^{2}\right).$
(2.35)
This is reasonable as we have not made use of the background equation of
$\Phi$, Eq. (2.22), yet. Now, using Eq. (2.22), Eqs. (2.29)-(2.31) take on the
form
$\displaystyle
W^{2}=\frac{1}{3M^{2}_{\textrm{Pl}}}\left[\frac{1}{2}\dot{\phi}^{2}+V(\phi)+\rho_{\textrm{M}}\right],$
(2.36) $\displaystyle\dot{W}+\Phi
W=-\frac{1}{2M^{2}_{\textrm{Pl}}}\left(\dot{\phi}^{2}+\rho_{\textrm{M}}+P_{\textrm{M}}\right),$
(2.37)
$\displaystyle\ddot{\phi}+3W\dot{\phi}+V_{,\phi}=-12\xi_{,\phi}W^{2}\left(W^{2}-\rho_{\textrm{M}}\frac{1+w_{\textrm{M}}}{2M^{2}_{\textrm{Pl}}}\right).$
(2.38)
For $\Phi=0$, these equations reduce to
$\displaystyle
H^{2}=\frac{1}{3M^{2}_{\textrm{Pl}}}\left[\frac{1}{2}\dot{\phi}^{2}+V(\phi)+\rho_{\textrm{M}}\right],$
(2.39)
$\displaystyle\dot{H}=-\frac{1}{2M^{2}_{\textrm{Pl}}}\left(\dot{\phi}^{2}+\rho_{\textrm{M}}+P_{\textrm{M}}\right),$
(2.40) $\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0~{}.$ (2.41)
We recover the equations of GR above because $\Phi=0$ implies $\dot{\xi}=0$
(unless $H=0$) by virtue of Eq. (2.22).
### 2.4 Equations for Tensor Perturbations
We derived the homogeneous equations and solved the only non-trivial equation
(2.22) that governs the evolution of $A_{\mu}$. Now, we perturb about the flat
FLRW metric and consider tensor perturbations
$\textrm{d}s^{2}=-\textrm{d}t^{2}+a^{2}(t)\left[\delta_{ij}+h_{ij}(t,\mathbf{x})\right]\textrm{d}x^{i}\textrm{d}x^{j},$
(2.42)
where $h_{ij}(t,\mathbf{x})$ are the traceless-transverse tensor components,
such that $\tensor{h}{{}^{i}_{i}}=0=\partial_{i}\tensor{h}{{}^{i}_{j}}$.
Perturbing the action (2.19) up to quadratic terms, we derive the equation of
tensor perturbations666We assume there is no contribution from the matter
action in the form of an anisotropic stress tensor.
$\left(M^{2}_{\textrm{Pl}}-4\dot{\xi}W\right)\ddot{h_{ij}}+\left[3H\left(M^{2}_{\textrm{Pl}}-4\dot{\xi}W\right)-4\left(\ddot{\xi}W+\dot{\xi}\dot{W}\right)\right]\dot{h_{ij}}-\left[M^{2}_{\textrm{Pl}}-4\left(\ddot{\xi}+\dot{\xi}\Phi\right)\right]a^{-2}\partial_{k}\partial^{k}h_{ij}=0~{},$
(2.43)
where $\partial_{k}\partial^{k}\equiv\delta^{kl}\partial_{k}\partial_{l}$.
_Assuming that_ $4\dot{\xi}W\neq M^{2}_{\textrm{Pl}}$, the equation may be
written as
$\ddot{h_{ij}}+\left(3+\alpha_{M}\right)H\dot{h_{ij}}-\left(1+\alpha_{T}\right)a^{-2}\partial_{k}\partial^{k}h_{ij}=0~{},$
(2.44)
where $\alpha_{M}$ and $\alpha_{T}$ are given by
$\displaystyle\alpha_{M}\equiv-\frac{4}{H}\frac{\ddot{\xi}W+\dot{\xi}\dot{W}}{M^{2}_{\textrm{Pl}}-4\dot{\xi}W}~{},$
(2.45)
$\displaystyle\alpha_{T}\equiv-4\frac{\ddot{\xi}-\left(H-2\Phi\right)\dot{\xi}}{M^{2}_{\textrm{Pl}}-4\dot{\xi}W}~{}.$
(2.46)
The speed of GWs predicted by the WGB model is then given by
$c_{\textrm{GW}}^{2}=1+\alpha_{T}$, with $\alpha_{T}$ of Eq. (2.46). Using Eq.
(2.22), we may write the $\alpha_{T}$ parameter entirely in terms of $W$,
$\dot{\xi}$ and $\ddot{\xi}$
$\alpha_{T}=-\frac{4}{M^{2}_{\textrm{Pl}}}\frac{M^{2}_{\textrm{Pl}}\left(\ddot{\xi}-\dot{\xi}W\right)+2\dot{\xi}^{2}W^{2}}{M^{2}_{\textrm{Pl}}-4\dot{\xi}W}~{}.$
(2.47)
## 3 Dynamical Systems Analysis
### 3.1 Dynamical System
Given the homogeneous field equations (2.36)-(2.38), we analyse the generic
behaviour of the dynamical system. We find the corresponding fixed points and
determine their stability keeping in mind the known history of the Universe.
To that end, the following dimensionless variables are defined:
$\displaystyle x\equiv\frac{\dot{\phi}}{\sqrt{6}WM_{\textrm{Pl}}}~{},~{}\
~{}y\equiv\frac{1}{WM_{\textrm{Pl}}}\sqrt{\frac{V}{3}}~{},~{}\
~{}z\equiv\frac{1}{WM_{\textrm{Pl}}}\sqrt{\frac{\rho_{\mathrm{M}}}{3}}~{},$
(3.1)
such that Eq. (2.36), which is the analogue of the Friedmann equation in GR,
can be written as
$1=x^{2}+y^{2}+z^{2}.$ (3.2)
It is noticed that, in contrast to the definitions of $x$, $y$ and $z$ in Ref.
[46], these are given in terms of the Weyl parameter, $W$, not the Hubble one.
$W$ and $H$ are related by Eq. (2.28). Using the dimensionless variables, this
relation can be written as
$H=W\left(1+ux\right),$ (3.3)
where another dimensionless variable,
$u\equiv 2\sqrt{6}\frac{\xi_{,\phi}W^{2}}{M_{\textrm{Pl}}}~{},$ (3.4)
was defined.
A second difference between those definitions and the ones discussed in Ref.
[46] is that $u$ is not constrained by Eq. (3.2). The reason for this
independence is related to the fact that the modifications of gravity are
included in the definitions of $x$, $y$ and $z$ in Eq. (3.1) via the Weyl
parameter. In Ref. [46], on the other hand, the effects of modified gravity
are accounted for solely by the $u$ variable (see Eq. (17) in that paper).
With the definitions of the dimensionless variables above, we can write the
density parameter of the background fluid $\Omega_{\textrm{M}}$ as
$\Omega_{\textrm{M}}\equiv\frac{\rho_{\textrm{M}}}{3H^{2}M^{2}_{\textrm{Pl}}}=\frac{1-x^{2}-y^{2}}{(1+ux)^{2}}~{},$
(3.5)
where Eqs. (3.2) and (3.3) were used.
Using those very same equations, we define the density parameter of the scalar
field $\phi$ too and write it in terms of the dimensionless variables (3.1)
and (3.4)
$\Omega_{\phi}\equiv\frac{\frac{1}{2}\dot{\phi}^{2}+V}{3H^{2}M_{\textrm{Pl}}^{2}}=\frac{x^{2}+y^{2}}{(1+ux)^{2}}~{}.$
(3.6)
Finally, the ‘effective’ density parameter associated with $\xi$ is
$\Omega_{\xi}\equiv\frac{(2+ux)ux}{(1+ux)^{2}}~{},$ (3.7)
such that $1=\Omega_{\phi}+\Omega_{\textrm{M}}+\Omega_{\xi}$.
Assuming an exponential potential
$V(\phi)=V_{0}e^{-\lambda\phi/M_{\textrm{Pl}}},$ (3.8)
such that $V_{,\phi}=-\frac{\lambda}{M_{\textrm{Pl}}}V$ and $\lambda>0$ is a
non-zero constant, the dynamical equations become self-similar given the
definitions in Eqs. (3.1) and (3.4). This class of potentials gives rise to
scaling solutions and they are ubiquitous in quintessence models (see e.g.
Ref. [25], where the exponential potential is the large field limit of the one
proposed therein). Then, taking the derivatives of $x$, $y$, $z$ and $u$, and
using Eq. (2.31), the scalar field equation, the continuity equation (2.32),
and Eq. (3.3), we arrive at
$\displaystyle
x^{\prime}=-\frac{1}{1+ux}\left[(3x+u)(1+ux)-\sqrt{\frac{3}{2}}\lambda
y^{2}+(x+u)\frac{\dot{W}}{W^{2}}\right],$ (3.9) $\displaystyle
y^{\prime}=-\frac{y}{1+ux}\left(\frac{\dot{W}}{W^{2}}+\sqrt{\frac{3}{2}}\lambda
x\right),$ (3.10) $\displaystyle
z^{\prime}=-z\left[\frac{\dot{W}}{W^{2}}\frac{1}{1+ux}+\frac{3}{2}\left(1+w_{\mathrm{M}}\right)\right],$
(3.11) $\displaystyle
u^{\prime}=\frac{2}{1+ux}\left(u\frac{\dot{W}}{W^{2}}+6\xi_{,\phi\phi}W^{2}x\right).$
(3.12)
$\dot{W}/W^{2}$ can be expressed in terms of the dimensionless variables using
Eq. (2.37). Plugging Eq. (2.22) into Eq. (2.37) we find
$-\frac{\dot{W}}{W^{2}}=x\left(3x+u\right)+\frac{3}{2}z^{2}\left(1+w_{\textrm{M}}\right).$
(3.13)
Primes denote derivatives with respect to the number of elapsing e-folds,
which we choose to be $\textrm{d}N\equiv\textrm{d}\ln a=H\textrm{d}t$.
$\dot{W}/W^{2}$ can be related to the Hubble flow parameter $\epsilon_{H}$,
which is defined as [67]
$\epsilon_{H}\equiv-\frac{\dot{H}}{H^{2}}~{}.$ (3.14)
$\epsilon_{H}$ parametrises the acceleration of spatial slices. Using Eq.
(2.28) we find
$\epsilon_{H}=-\frac{1}{1+ux}\left[\frac{\dot{W}}{W^{2}}+\frac{(ux)^{\bullet}}{H}\right].$
(3.15)
Notice that no assumption regarding $\xi(\phi)$ has been made so far. All the
previous equations are valid for any coupling function, including the
exponential one
$\xi(\phi)=\xi_{0}e^{\kappa\phi/M_{\textrm{Pl}}},$ (3.16)
such that $\xi_{,\phi\phi}=\frac{\kappa}{M_{\textrm{Pl}}}\xi_{,\phi}$, where
$\kappa>0$ is a non-zero constant.777The strength of the exponential $\kappa$
in Eq. (3.16) should not be confused with the scalar $\kappa$ defined in Eq.
(A.9), discussed in Sec. 2.2 and App. A. Assuming such a coupling function,
Eq. (3.12) is a self-similar equation too and the explicit dependence on $W$
drops out of the equation888For a linear coupling function
$\xi(\phi)\propto\phi$, the explicit dependence on the Weyl parameter $W$ also
disappears from Eq. (3.12). However, as we will argue later on (see footnote
12), the linear coupling case can be discarded on the basis of our knowledge
of the past history of the Universe.
$u^{\prime}=\frac{2u}{1+ux}\left(\frac{\dot{W}}{W^{2}}+\sqrt{\frac{3}{2}}\kappa
x\right).$ (3.17)
### 3.2 Fixed Points
Fixed points of the dynamical system of equations (3.9), (3.10) and (3.17) are
denoted by $x_{c}$, $y_{c}$ and $u_{c}$, respectively, and correspond to
regions of the phase space where $x^{\prime}=y^{\prime}=u^{\prime}=0$. Among
those, we can list the following ones:
* •
M: $(x_{c},y_{c},u_{c})=(0,0,0)$ and
$\left.\epsilon_{H}\right|_{c}=\frac{3}{2}\left(1+w_{\textrm{M}}\right)$;
* •
K$\pm$: $(x_{c},y_{c},u_{c})=(\pm 1,0,0)$ and
$\left.\epsilon_{H}\right|_{c}=3$;
* •
I:
$(x_{c},y_{c},u_{c})=\left(\frac{\lambda}{\sqrt{6}},\sqrt{1-\frac{\lambda^{2}}{6}},0\right)$
and $\left.\epsilon_{H}\right|_{c}=\frac{\lambda^{2}}{2}$;
* •
ScI:
$(x_{c},y_{c},u_{c})=\left(\sqrt{\frac{3}{2}\frac{\left(1+w_{\textrm{M}}\right)^{2}}{\lambda^{2}}},\sqrt{\frac{3}{2}\frac{1-w^{2}_{\textrm{M}}}{\lambda^{2}}},0\right)$
and $\left.\epsilon_{H}\right|_{c}=\frac{3}{2}\left(1+w_{\textrm{M}}\right)$;
which are fixed points in GR too because $W_{c}=H_{c}$ at those fixed points
(see Eq. (3.3)). $z_{c}$ is determined from the constraint equation (3.2), so
we do not really need to impose $z^{\prime}=0$; it follows from the constraint
equation.999Notice from Eq. (3.11) that, in order for $z^{\prime}=0$, either
$z_{c}=0$ or
$\left.\epsilon_{H}\right|_{c}=\frac{3}{2}\left(1+w_{\textrm{M}}\right)$ (see
Eq. (3.15)). Both are satisfied (although not simultaneously) by the fixed
points above and by the ones we discuss later. Besides this, we remark that,
unlike $z$, $u^{\prime}=0$ is imposed as $x^{\prime}=y^{\prime}=0$ are
satisfied for $u=u(N)$ and $x_{c}=\pm 1$ and $y_{c}=0$, for example. As was
argued already, this is due to the fact that the modifications of gravity are
included in all the definitions of the dimensionless variables, Eqs. (3.1) and
(3.4), whilst in our previous work, Ref. [46], these were only represented by
$u$, which is defined in Eq. (17) of that paper.
M represents the regime of the background fluid domination. K$\pm$ stands for
the period of kination (dominance of the kinetic energy of the scalar field
$\dot{\phi}^{2}/2$) and I corresponds to a regime of power law inflation (when
$\lambda<\sqrt{2}$, so that $\left.\epsilon_{H}\right|_{\textrm{{I}}}<1$)
where matter fields do not contribute ($z_{\textrm{{I}}}=0$; see Eq. (3.2)).
Notice that this latter fixed point exists if $\lambda<\sqrt{6}$, so
$y_{\textrm{{I}}}$ is real (remember that $\lambda>0$). Lastly, we have the
scaling regime (ScI). In this case, the EoS parameter of the scalar field,
$w_{\phi}\equiv\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)}~{},$
(3.18)
becomes equal to that of the background fluid, $w_{\textrm{M}}$. This fixed
point yields
$x^{2}_{\textrm{{ScI}}}+y^{2}_{\textrm{{ScI}}}=\frac{3}{\lambda^{2}}\left(1+w_{\textrm{M}}\right),$
(3.19)
and, by virtue of Eq. (3.2), we obtain the following existence condition:
$\lambda>\sqrt{3(1+w_{\textrm{M}})}\geq\sqrt{3}~{}.$ (3.20)
Therefore, we observe that I and ScI coexist whenever
$\sqrt{6}>\lambda>\sqrt{3}$. If I corresponds to power law inflation, then
both fixed points cannot coexist because $\lambda<\sqrt{2}$ is required, as
was pointed out already.
In addition to the fixed points listed above, we find five other points:
* •
dS: $(x_{c},y_{c},u_{c})=\left(0,1,\sqrt{\frac{3}{2}}\lambda\right)$ and
$\left.\epsilon_{H}\right|_{c}=0$;
* •
ScII$\pm$:
$(x_{c},y_{c},u_{c})=\left(\frac{\kappa}{\sqrt{6}}\frac{1+3w_{\textrm{M}}}{3(1-w^{2}_{\textrm{M}})}\Delta_{\pm}(\kappa,w_{\textrm{M}}),0,\frac{\kappa}{\sqrt{6}}\frac{1}{1+w_{\textrm{M}}}\Delta_{\pm}(\kappa,w_{\textrm{M}})\right)$,
where
$\Delta_{\pm}(\kappa,w_{\textrm{M}})\equiv
1\pm\sqrt{1-\frac{18}{\kappa^{2}}\frac{(1+w_{\textrm{M}})(1-w^{2}_{\textrm{M}})}{1+3w_{\textrm{M}}}}~{},$
(3.21)
and $\left.\epsilon_{H}\right|_{c}=\frac{3}{2}\left(1+w_{\textrm{M}}\right)$;
* •
K̂$\pm$: $(x_{c},y_{c},u_{c})=\left(\pm 1,0,\sqrt{\frac{3}{2}}\kappa\mp
3\right)$ and
$\left.\epsilon_{H}\right|_{c}=\frac{\mp\frac{3\kappa}{2\sqrt{6}}}{1\mp\frac{3\kappa}{2\sqrt{6}}}$.
The first fixed point corresponds to the de Sitter expansion, where the total
energy density consists only of the CC. This fixed point exists in GBDE models
even for $\lambda\neq 0$. It is also found in the SGB model (see Refs. [46,
68]) because $x_{\textrm{{dS}}}=0$ and $W_{\textrm{{dS}}}=H_{\textrm{{dS}}}$
(see Eq. (3.3)).
Besides the de Sitter fixed point, we have a second scaling regime which
consists of two fixed points (ScII$\pm$), depending on the root sign in Eq.
(3.21). These fixed points exist if
$\kappa^{2}\geq
18\frac{(1+w_{\textrm{M}})(1-w^{2}_{\textrm{M}})}{1+3w_{\textrm{M}}}~{},$
(3.22)
so that $x_{\textrm{{ScII}$\pm$}}$ and $u_{\textrm{{ScII}$\pm$}}$ are real.
Then $1\leq\Delta_{+}<2$ and $0<\Delta_{-}\leq 1$. Also, Eq. (3.2) demands
$x^{2}_{\textrm{{ScII}$\pm$}}=\frac{\kappa^{2}}{54}\frac{(1+3w_{\textrm{M}})^{2}}{(1-w^{2}_{\textrm{M}})^{2}}\Delta_{\pm}^{2}<1~{}.$
(3.23)
Combining it with Eq. (3.22) we obtain
$\Delta_{\pm}^{2}<\frac{3(1-w_{\textrm{M}})}{1+3w_{\textrm{M}}}~{}.$ (3.24)
Given that $1\leq\Delta_{+}^{2}<4$ and $0<\Delta_{-}^{2}\leq 1$, we have
$0\leq w_{\textrm{M}}<\frac{1}{3}$ for $\Delta_{+}$ and $0\leq
w_{\textrm{M}}<1$ for $\Delta_{-}$ (we remind the reader that the whole range
of values of $w_{\textrm{M}}$ is assumed to be $0\leq w_{\textrm{M}}<1$; see
below Eq. (2.32)). Therefore, the scaling fixed point ScII$+$ in particular
does not exist if the background fluid under consideration corresponds to
radiation for example, where $w_{\textrm{M}}=w_{r}=1/3$. The scaling regime is
still represented by ScII$-$ in that case though.
Finally, the fourth and fifth fixed points listed above have vanishing
potential energy and the energy density of the background fluid is zero. We
denote them by ‘K̂$\pm$’ and call the corresponding regime ‘pseudo-kination’
since
$3W^{2}_{\textrm{{\^{K}}}\pm}M^{2}_{\textrm{Pl}}=\dot{\phi}_{\textrm{{\^{K}}}\pm}^{2}/2$,
although $W_{\textrm{{\^{K}}}\pm}\neq H_{\textrm{{\^{K}}}\pm}$. Irrespective
of the sign, if $\kappa\gg 1$, then
$\left.\epsilon_{H}\right|_{\textrm{{\^{K}}}\pm}\rightarrow 1$. This value of
the Hubble flow parameter, for large $\kappa$, leads into a universe that does
not accelerate.
For $x_{c}=+1$ (i.e. K̂$+$) and $\kappa<2\sqrt{\frac{2}{3}}$,
$\left.\epsilon_{H}\right|_{\textrm{{\^{K}}}+}<0$ (remember that $\kappa$ is
defined to be positive; see below Eq. (3.16)). This implies that the Universe
accelerates faster than during de Sitter, where it accelerates
exponentially.101010In order to see why it accelerates faster than
exponential, we solve $\epsilon_{H}=-\alpha$, where $\alpha>0$ is constant
($\epsilon_{H}$ is constant at the fixed points K̂$\pm$, indeed)
$H(t)=\frac{H_{0}}{1-\alpha H_{0}t}~{},$ (3.25) where $H_{0}=H(t=0)$. Solving
for $a(t)$ we obtain $a(t)=\frac{a_{0}}{(1-\alpha H_{0}t)^{1/\alpha}}~{},$
(3.26) such that $a_{0}=a(t=0)$. Assuming an expanding universe, $H_{0}>0$ and
$t<t_{s}\equiv 1/(\alpha H_{0})$. Then, $a(t)$ diverges (approaches
$a(t_{s})$) faster than if $a(t)\propto e^{H_{0}t}$.
Lastly, choosing $\lambda=\kappa$ allows for all $x_{c}$, $y_{c}$ and $u_{c}$
being non-zero at once as can be seen from Eqs. (3.10) and (3.17). The same
occurs in the SGB model (see Ref. [46]). The corresponding fixed ‘curves’ are
* •
ScIII:
$(x,y,u)=\left(\frac{\frac{3}{2}(1+w_{\textrm{M}})}{\sqrt{\frac{3}{2}}\lambda-\frac{3}{2}u_{c}(1+w_{\textrm{M}})},\sqrt{\frac{\sqrt{\frac{3}{2}}\lambda
u_{c}+\frac{3}{2}(1+w_{\textrm{M}})\left(\frac{3}{2}(1-w_{\textrm{M}})-\sqrt{\frac{3}{2}}\lambda
u_{c}+\frac{1}{2}(1+3w_{\textrm{M}})u_{c}^{2}\right)}{\left(\sqrt{\frac{3}{2}}\lambda-\frac{3}{2}u_{c}(1+w_{\textrm{M}})\right)^{2}}},u_{c}\right)$
and $\epsilon_{H}|_{c}=\frac{3}{2}(1+w_{\textrm{M}})$;
* •
$\widehat{\textrm{{IV}}}$:
$(x,y,u)=\left(x_{c},\sqrt{1-x_{c}^{2}},-3x_{c}+\sqrt{\frac{3}{2}}\lambda\right)$
and $\epsilon_{H}|_{c}=\frac{\sqrt{\frac{3}{2}}\lambda
x_{c}}{1-3x_{c}^{2}+\sqrt{\frac{3}{2}}\lambda x_{c}}$.
We then have a third scaling solution, ScIII, and a regime of zero energy
density of the background fluid, corresponding to the fixed ‘curve’
$\widehat{\textrm{{IV}}}$. This curve is denoted that way following the
convention in Ref. [46] (see Table I in that paper), although neither of those
two fixed ‘curves’ are found in the SGB model strictly because $W_{c}\neq
H_{c}$, so carets are used on the latter to differentiate it from the fixed
curve in the SGB model (similarly to what was done with the pseudo-kination
fixed points we just discussed).
Those two regimes (especially the scaling one) could be of interest, although
in this work we shall consider $\kappa\neq\lambda$. We leave the stability
analysis of these two fixed curves and possible solutions for future works.
### 3.3 Stability
To perform the stability analysis of the fixed points shown above, we perturb
Eqs. (3.9), (3.10) and (3.17) linearly such that
$\displaystyle\left(1+u_{c}x_{c}\right)\delta
x^{\prime}=-\left[3-9x_{c}^{2}-2u_{c}x_{c}-\frac{3}{2}(1+w_{\textrm{M}})\left(1-3x^{2}_{c}-y_{c}^{2}-2u_{c}x_{c}\right)\right]\delta
x+$
$\displaystyle+y_{c}\left[\sqrt{6}\lambda-3(1+w_{\textrm{M}})(x_{c}+u_{c})\right]\delta
y-\left[1-x_{c}^{2}-\frac{3}{2}\left(1+w_{\textrm{M}}\right)\left(1-x_{c}^{2}-y_{c}^{2}\right)\right]\delta
u~{},$ (3.27) $\displaystyle(1+u_{c}x_{c})\delta
y^{\prime}=-y_{c}\left[\sqrt{\frac{3}{2}}\lambda-
u_{c}-3x_{c}(1-w_{\textrm{M}})\right]\delta
x-\left[\left(\sqrt{\frac{3}{2}}\lambda-3x_{c}-u_{c}\right)x_{c}-\right.$
$\displaystyle\left.-\frac{3}{2}(1+w_{\textrm{M}})\left(1-x_{c}^{2}-3y_{c}^{2}\right)\right]\delta
y+y_{c}x_{c}\delta u~{},$ (3.28) $\displaystyle\left(1+u_{c}x_{c}\right)\delta
u^{\prime}=2u_{c}\left[\sqrt{\frac{3}{2}}\kappa-
u_{c}-3x_{c}\left(1-w_{\textrm{M}}\right)\right]\delta
x+6u_{c}y_{c}\left(1+w_{\textrm{M}}\right)\delta y+$
$\displaystyle+2\left[\left(\sqrt{\frac{3}{2}}\kappa-3x_{c}-2u_{c}\right)x_{c}-\frac{3}{2}\left(1+w_{\textrm{M}}\right)\left(1-x_{c}^{2}-y_{c}^{2}\right)\right]\delta
u~{},$ (3.29)
where the constraint equation (3.2) was used to replace $z^{2}$ in Eq. (3.13).
We begin our stability analysis with the fixed points M and K$\pm$, given the
relevance of the latter in particular in early stages in models of
quintessential inflation. In the case of M, which corresponds to the stage of
the background fluid domination, as was commented already, the eigenvalues,
which we denote by ‘$m$’, are the following ones:
$\displaystyle m_{1}=-\frac{3}{2}\left(1-w_{\textrm{M}}\right),$ (3.30)
$\displaystyle m_{2}=\frac{3}{2}\left(1+w_{\textrm{M}}\right),$ (3.31)
$\displaystyle m_{3}=-3\left(1+w_{\textrm{M}}\right).$ (3.32)
One of the eigenvalues, $m_{2}$, which corresponds to the eigenvector
$\mathbf{v}=(0,1,0)$,111111We order eigenvector components as
$\mathbf{v}=(v_{x},v_{y},v_{u})$. is non-negative, whereas $m_{1},m_{3}<0$. In
view of this, M is a saddle point and is unstable in the $y$ direction.
Regarding K$\pm$, the eigenvalues are given by
$\displaystyle m_{1}=3\left(1-w_{\textrm{M}}\right),$ (3.33) $\displaystyle
m_{2}=3\mp\sqrt{\frac{3}{2}}\lambda~{},$ (3.34) $\displaystyle
m_{3}=-6\pm\sqrt{6}\kappa~{},$ (3.35)
where the upper sign corresponds to the fixed point K$+$. $m_{1}$,
corresponding to the eigenvector $\mathbf{v}=(1,0,0)$, is always positive,
irrespective of $\lambda$ and $\kappa$. Thus, the kination period is unstable
in the $x$ direction. The stability of these fixed points, M and K$\pm$, that
is shown here, resembles the one in the SGB model [68, 46].
We now analyse the stability of the fixed points ScI and dS, which stand out
amongst the rest of the fixed points given our interest in quintessence
cosmology. For ScI, the eigenvalues can be easily calculated to be
$\displaystyle
m_{1}=3(1+w_{\textrm{M}})\left(\frac{\kappa}{\lambda}-1\right),$ (3.36)
$\displaystyle
m_{2,3}=-\frac{3}{4}(1-w_{\textrm{M}})\left\\{1\mp\sqrt{1-8\frac{1+w_{\textrm{M}}}{1-w_{\textrm{M}}}\left[1-\frac{3}{\lambda^{2}}(1+w_{\textrm{M}})\right]}\right\\}.$
(3.37)
Since $\lambda>\sqrt{3(1+w_{\textrm{M}})}$ (see Eq. (3.19)), $m_{2,3}$ are
complex eigenvalues whose real part is always negative. On the other hand, if
$\kappa>\lambda$, $m_{1}$ will be positive, and the fixed point ScI will be a
saddle point. This same feature is shared with the SGB model [46, 68], and it
is an important one because the scaling regime, in which the expansion of the
Universe does not accelerate in contrast to observations [69], is brought to
an end.121212For a linear coupling function $\xi(\phi)\propto\phi$, we have
that $\kappa=0$ because $\xi_{,\phi\phi}=0$. This implies that $m_{1}<0$ in
Eq. (3.36), irrespective of $\lambda$. Then, the scaling fixed point is
stable. Moreover, we see that dS is unstable (see Eq. (3.39), where
$m_{2}>0$). The stability of the scaling fixed point ScI contradicts
observations indicating that the Universe undergoes a phase of current
accelerating expansion, which cannot correspond to the scaling regime with
$\epsilon_{H}=\frac{3}{2}(1+w_{\textrm{M}})$ as commented already. Therefore,
the linear coupling can be discarded. Also, $\kappa=0$ implies that either
$u_{c}=0$ or $\left.\epsilon_{H}\right|_{c}=0$ (see Eqs. (3.17) and (3.15)).
This implies that no other scaling regime with $u_{c}\neq 0$ exists. The same
conclusions can be drawn in the SGB model (see Ref. [46]).
The Universe can enter the de Sitter phase depending on the stability of this
fixed point. The eigenvalues are
$\displaystyle m_{1}=-3\left(1+w_{\textrm{M}}\right),$ (3.38) $\displaystyle
m_{2,3}=-\frac{3}{2}\left[1\mp\sqrt{1-\frac{4}{3}\lambda^{2}\left(\frac{\kappa}{\lambda}-1\right)}\right].$
(3.39)
As long as $\kappa>\lambda$, all the eigenvalues will have negative real
parts, and dS will be a stable fixed point (an attractor). The condition
$\kappa>\lambda$ was sufficient in order for ScI to be unstable (see Eq.
(3.36)), and therefore, the Universe may transition from ScI to dS. This
reproduces the evolution of the Universe as we know it: from radiation/matter
domination to an eventual exponentially expanding universe. The fact that dS
may be a stable fixed point is a well-known result in the SGB model as well
[68, 46].
For completeness, we include the stability analysis of I before getting to the
other two fixed points with $u_{c}\neq 0$. In this case, the eigenvalues read
$\displaystyle m_{1}=\lambda^{2}\left(\frac{\kappa}{\lambda}-1\right),$ (3.40)
$\displaystyle m_{2}=\frac{1}{2}\left(\lambda^{2}-6\right),$ (3.41)
$\displaystyle m_{3}=\lambda^{2}-3\left(1+w_{\textrm{M}}\right).$ (3.42)
Since $\lambda^{2}<6$ in order for I to exist (so that $y_{\textrm{{I}}}$ is
real, as was commented earlier), we have that $m_{2}<0$. Given the range of
values of $w_{\textrm{M}}$, $m_{3}$ may be non-negative ($\lambda^{2}$ would
have to be smaller than $3(1+w_{\textrm{M}})$ otherwise). Considering power
law inflation though, such that $\lambda^{2}<2$, we see that $m_{3}$ is
definitely negative. Regardless of $w_{\textrm{M}}$ and $\lambda$, $m_{1}$
will be positive if $\kappa>\lambda$. Then, I will be unstable if
$\kappa>\lambda$.
The fixed point K̂$+$ could lead to a phase of faster than exponential
acceleration, $\left.\epsilon_{H}\right|_{\textrm{{\^{K}}}+}<0$, if
$\kappa<2\sqrt{\frac{2}{3}}$ (remember that we are assuming that $\kappa$ does
not vanish; see Eq. (3.16)). We study the stability of the two fixed points
K̂$\pm$. The eigenvalues are
$\displaystyle
m_{1}=-\frac{2}{2\mp\sqrt{\frac{3}{2}}\kappa}\left[\pm\sqrt{\frac{3}{2}}\kappa+\frac{3}{2}\left(1+w_{\textrm{M}}\right)\left(2\mp\sqrt{\frac{3}{2}}\kappa\right)\right],$
(3.43) $\displaystyle
m_{2}=\frac{\mp\sqrt{\frac{3}{2}}\lambda}{2\mp\sqrt{\frac{3}{2}}\kappa}\left(\frac{\kappa}{\lambda}-1\right),$
(3.44) $\displaystyle
m_{3}=-\frac{2}{2\mp\sqrt{\frac{3}{2}}\kappa}\left(3\mp\sqrt{\frac{3}{2}}\kappa\right),$
(3.45)
the upper sign corresponding to K̂$+$. In the case of K̂$-$, the fixed point
will be unstable whenever $\kappa>\lambda$ (because $m_{2}>0$), regardless of
the magnitude of $\kappa$ (given that $m_{1}$ and $m_{3}$ are always negative
for K̂$-$). For K̂$+$ and $\kappa<2\sqrt{\frac{2}{3}}$, $m_{1}$ and $m_{3}$
are both negative too. If we assume that $\kappa>\lambda$, then $m_{2}$,
corresponding to the eigenvector $\mathbf{v}=(0,1,0)$, will be negative and
the phase of super-acceleration will be _stable_.131313‘Super-acceleration’ in
relation to $\epsilon_{H}<0$ is used, for example, in Refs. [70, 71]. If not,
it will be unstable in the $y$ direction. Irrespective of the sign of $x_{c}$,
if $\kappa\gg 1$, but $\lambda>\kappa$, the asymptotic phase of zero
acceleration, corresponding to
$\left.\epsilon_{H}\right|_{\textrm{{\^{K}}}\pm}=1$, will be stable as well.
Notice that, for that super-accelerated phase (see Eq. (3.3)),
$\left.\frac{H}{W}\right|_{\textrm{{\^{K}}}+}=1+u_{\textrm{{\^{K}}}+}x_{\textrm{{\^{K}}}+}=1+\left(\sqrt{\frac{3}{2}}\kappa-3\right)$
(3.46)
is negative (remember that $\kappa<2\sqrt{\frac{2}{3}}$). For the rest of the
fixed points treated above, the ratio $H/W$ is $1$ (positive in the case of
ScII$\pm$ as one can check from the values of $x$ and $u$ at those fixed
points; see before Eq. (3.21)) because $u_{c}=0$. The same is true for K̂$-$
given that
$\left.\frac{H}{W}\right|_{\textrm{{\^{K}}}-}=1+u_{\textrm{{\^{K}}}-}x_{\textrm{{\^{K}}}-}=1-\left(\sqrt{\frac{3}{2}}\kappa+3\right)$
(3.47)
is always negative for $\kappa>0$.
Finally, for the second scaling solution, ScII$\pm$, the eigenvalues are
$\displaystyle
m_{1}=\frac{3}{2}\left(1+w_{\textrm{M}}\right)\left(1-\frac{\lambda}{\kappa}\right),$
(3.48) $\displaystyle
m_{2,3}=-\frac{3}{4}\left(1-w_{\textrm{M}}\right)\left\\{1\mp\sqrt{1+8\frac{1+w_{\textrm{M}}}{1-w_{\textrm{M}}}\left[1-\frac{1-3w_{\textrm{M}}}{3(1-w_{\textrm{M}})}\Delta_{\pm}-\frac{12}{\kappa^{2}}\frac{\left(1+w_{\textrm{M}}\right)^{2}}{1+3w_{\textrm{M}}}\right]}\right\\}.$
(3.49)
We see that $m_{1}$, which corresponds to the eigenvector
$\mathbf{v}=(0,1,0)$, is positive (and hence neither of the scaling fixed
points ScII$\pm$ is stable) whenever $\kappa>\lambda$ (same condition as that
of the former scaling solution; see Eq. (3.36)). The instability is in the $y$
direction. On the other hand, $m_{2,3}$ do not depend on the magnitude of
$\lambda$.
From the existence condition of the fixed points (see Eq. (3.22)) we find
$\frac{1-3w_{\textrm{M}}}{3(1-w_{\textrm{M}})}\Delta_{\pm}+\frac{12}{\kappa^{2}}\frac{(1+w_{\textrm{M}})^{2}}{1+3w_{\textrm{M}}}\leq\frac{(1-3w_{\textrm{M}})\Delta_{\pm}+2}{3(1-w_{\textrm{M}})}~{}.$
(3.50)
Also, since $1\leq\Delta_{+}<2$ and $0\leq w_{\textrm{M}}<\frac{1}{3}$ for the
fixed point ScII$+$ (see Eq. (3.24)), then
$\frac{1-3w_{\textrm{M}}}{3(1-w_{\textrm{M}})}\Delta_{+}+\frac{12}{\kappa^{2}}\frac{(1+w_{\textrm{M}})^{2}}{1+3w_{\textrm{M}}}<\frac{2}{3}\frac{2-3w_{\textrm{M}}}{1-w_{\textrm{M}}}\leq\frac{4}{3}~{}.$
(3.51)
In the case of ScII$-$, we assume that $0\leq w_{\textrm{M}}<\frac{1}{3}$ as
well, although $w_{\textrm{M}}$ was constrained to be smaller than $1$. Since
$0<\Delta_{-}\leq 1$, we have
$\frac{1-3w_{\textrm{M}}}{3(1-w_{\textrm{M}})}\Delta_{-}+\frac{12}{\kappa^{2}}\frac{(1+w_{\textrm{M}})^{2}}{1+3w_{\textrm{M}}}\leq
1~{}.$ (3.52)
Thus only in the case of ScII$-$ will $m_{2,3}$ always be real (again,
assuming $0\leq w_{\textrm{M}}<\frac{1}{3}$). For complex $m_{2,3}$, the real
part of both eigenvalues is negative and ScII$+$ is stable if
$\lambda>\kappa$. For the rest of the possibilities, that depends on the value
of $w_{\textrm{M}}$ under consideration.
| | Fixed point: $(x_{c},y_{c},u_{c})$ | Stability
---|---|---|---
M | $(0,0,0)$ | Unstable
K$\pm$ | $(\pm 1,0,0)$ | Unstable
I | $\left(\frac{\lambda}{\sqrt{6}},\sqrt{1-\frac{\lambda^{2}}{6}},0\right)$ | Unstable whenever $\kappa>\lambda$
ScI | $\left(\sqrt{\frac{3}{2}\frac{(1+w_{\textrm{M}})^{2}}{\lambda^{2}}},\sqrt{\frac{3}{2}\frac{1-w^{2}_{\textrm{M}}}{\lambda^{2}}},0\right)$ | Unstable whenever $\kappa>\lambda$
dS | $\left(0,1,\sqrt{\frac{3}{2}}\lambda\right)$ | Stable unless $\kappa<\lambda$
ScII$\pm$ | $\left(\frac{\kappa}{\sqrt{6}}\frac{1+3w_{\textrm{M}}}{3(1-w^{2}_{\textrm{M}})}\Delta_{\pm},0,\frac{\kappa}{\sqrt{6}}\frac{1}{1+w_{\textrm{M}}}\Delta_{\pm}\right)$ | Unstable whenever $\kappa>\lambda$
K̂$\pm$ | K̂$+$ | $\left(+1,0,\sqrt{\frac{3}{2}}\kappa-3\right)$ | Unstable whenever $\kappa<\lambda$ and $\kappa<2\sqrt{\frac{2}{3}}$
K̂$-$ | $\left(-1,0,\sqrt{\frac{3}{2}}\kappa+3\right)$ | Unstable whenever $\kappa>\lambda$
Table 1: Summary of the fixed points considered in this work and their
stability according to $\kappa$ and $\lambda$ for an exponential coupling
function (see Eq. (3.16) and remember that $\kappa\neq\lambda$ is assumed). We
determine the stability of each fixed point assuming an EoS parameter of the
background fluid in the range $0\leq w_{\textrm{M}}<1$.
$\Delta_{\pm}(\kappa,w_{\textrm{M}})$ is defined in Eq. (3.21).
A summary of all the fixed points we studied in this section, and their
stability, is provided in Table 1.
### 3.4 Numerical Simulations
In view of the fixed points shown in Sec. 3.2 and their stability, we are
interested in the trajectories that begin close to kination, pass near the
scaling regime ScI and approach the de Sitter fixed point, depicting the
evolution of the Universe as suggested by the observations. To that end, Eqs.
(3.9), (3.10) and (3.17) are solved numerically, taking the constraint
equation (3.2) into account. We select those trajectories in the phase space
that satisfy $\Omega_{\textrm{M}}=0.3147\pm 0.0074$ and
$w_{\textrm{f}}=-0.957\pm 0.080$ [3] at $N=0$, which corresponds to the
present time. To define $w_{\textrm{f}}$, we use the relation [46]
$\epsilon_{H}=\frac{3}{2}\frac{\rho_{\phi}(1+w_{\textrm{f}})+\rho_{\textrm{M}}(1+w_{\textrm{M}})}{\rho_{\phi}+\rho_{\textrm{M}}}~{}.$
(3.53)
$w_{\textrm{f}}$ can be interpreted as the EoS parameter of the DE fluid
causing the accelerated expansion of the Universe [69]. Then
$w_{\textrm{f}}=-1+\frac{1}{x^{2}+y^{2}}\left[\frac{2}{3}\epsilon_{H}-\left(1-x^{2}-y^{2}\right)\left(1+w_{\textrm{M}}\right)\right],$
(3.54)
where $\epsilon_{H}$ is given in Eq. (3.15). The background fluid EoS
parameter is chosen to be that of pressureless matter,
$w_{\textrm{M}}=w_{m}=0$ (therefore $\Omega_{\textrm{M}}=\Omega_{m}$).
As initial values, we set $u_{0}=10^{-23}$ and $y_{0}=10^{-3}$. Such a small
initial value of $u$ is imposed to ensure a negligible initial contribution
from the GB coupling, given that the trajectories in phase space begin near
the fixed points K$\pm$ and M, in accordance with the evolution prescribed by
the quintessence models. The initial value of $x$ is chosen differently within
the range $(-1,1)$. All the trajectories obtained from the simulations end up
at the de Sitter fixed point $y_{\textrm{{dS}}}=1$ (these trajectories
correspond to the different values assigned to $x_{0}$). Since $\delta
u\propto\exp\left[3\left(\frac{\kappa}{\lambda}-1\right)N\right]$ during the
scaling regime (see Eq. (3.36), where $w_{\textrm{M}}=w_{m}=0$),
$\kappa/\lambda$ can be chosen to be of order $\mathcal{O}(1)$, so it takes
longer for the Universe to abandon the scaling regime.
We see in Figs. 1a and 1b the results of some simulations for
$\kappa/\lambda=1.4$ (so the ratio $\kappa/\lambda$ is of order
$\mathcal{O}(1)$, as argued above) depicting the evolution of interest as
explained at the beginning of this section. The $3$D phase portrait is shown
in Fig. 1e. In Figs. 1a and 1e, the M fixed point has been represented by a
square. The kination fixed points K$\pm$ correspond to the left and right
leaning triangles, depending on the sign. The de Sitter fixed point is given
by the circle instead, and the scaling regime corresponds to a star onto which
the trajectories converge. It can be noticed that, before the de Sitter phase
is reached, the density parameter of the scalar field (see Eq. (3.6)) becomes
larger than $1$ given that $\Omega_{\xi}$ (see Eq. (3.7)) turns negative. We
obtained this in many other simulations, with different values of $\lambda$
and $\kappa$. Likewise, the growth of $w_{\textrm{f}}$ reaches the value of
the EoS parameter of pressureless matter and surpasses it for a brief moment
of time. There is an equally short period where it takes on phantom values
before settling down at $w_{\textrm{{dS}}}=-1$, which is the expected value
during de Sitter. Note that slightly phantom DE is preferred by the
observational data [3].
We observe another feature shared with other simulations: the density
parameter $\Omega_{\xi}$ is always the dominant contribution to DE at the
present time. This occurs unless the scalar field energy density takes over
during the scaling regime, which is discarded given that the field is supposed
to be light and may alter the successful predictions of Big Bang
Nucleosynthesis (BBN).
The Universe may enter the second scaling regime ScII$\pm$ instead, where
$y_{\textrm{{ScII}$\pm$}}=0$, starting from kination. If $\Omega_{\textrm{M}}$
is larger than the sum of the rest of the density parameters,
$\Omega_{\phi}+\Omega_{\xi}$, during the scaling regime, then
$\left.\Omega_{\textrm{M}}\right|_{\textrm{{ScII}$\pm$}}>0.5$.141414One might
argue that this is not necessarily true if $\Omega_{\xi}$ is negative.
However, this is discarded in this situation given that $\Omega_{\xi}<0$
implies $-2<ux<0$ (see Eq. (3.7)). At ScII$\pm$, we obtain
$u_{\textrm{{ScII}$\pm$}}x_{\textrm{{ScII}$\pm$}}=\frac{1+3w_{\textrm{M}}}{3(1-w_{\textrm{M}})}u_{\textrm{{ScII}$\pm$}}^{2}~{},$
(3.55) and because $0\leq w_{\textrm{M}}<1$,
$\left.\Omega_{\xi}\right|_{\textrm{{ScII}$\pm$}}$ is never negative. For
pressureless matter ($w_{\textrm{M}}=w_{m}=0$),
$\left.\Omega_{m}\right|_{\textrm{{ScII}$\pm$}}$ reads
$\left.\Omega_{m}\right|_{\textrm{{ScII}$\pm$}}=\frac{54\left(54-\kappa^{2}\Delta_{\pm}^{2}\right)}{\left(54+3\kappa^{2}\Delta_{\pm}^{2}\right)^{2}}~{}.$
(3.56)
Then $\left.\Omega_{m}\right|_{\textrm{{ScII}$\pm$}}>0.5$ implies
$\kappa\Delta_{\pm}<\sqrt{6}$. The fixed point ScII$+$ can be discarded
because $\kappa>3\sqrt{2}$ in order for the fixed point to exist, and
$1\leq\Delta_{+}$ (see Eq. (3.22) and discussion below), which means that
$\kappa\leq\kappa\Delta_{+}<\sqrt{6}$. Regarding the fixed point ScII$-$, we
have (see Eq. (3.21))
$\kappa\Delta_{-}=\kappa-\sqrt{\kappa^{2}-18}<\sqrt{6}~{},$ (3.57)
and hence $\kappa>2\sqrt{6}$ for $\Omega_{m}$ to be larger than
$\Omega_{\phi}+\Omega_{\xi}$ during the scaling regime, in agreement with the
BBN predictions.
(a) Phase portrait for $\lambda=4.47\sqrt{\frac{2}{3}}$ and
$\kappa/\lambda=1.4$. (b) $\Omega$’s and $w_{\textrm{f}}$ drawn from the red
trajectory.
(c) Phase portrait for $\lambda=0.72\sqrt{3}$ and $\kappa/\lambda=4.41$. (d)
$\Omega$’s and $w_{\textrm{f}}$ drawn from the red trajectory.
(e) $3$D phase portrait for $\lambda=4.47\sqrt{\frac{2}{3}}$ and
$\kappa/\lambda=1.4$. (f) $3$D phase portrait for $\lambda=0.72\sqrt{3}$ and
$\kappa/\lambda=4.41$.
Figure 1: Phase portraits and density parameters and $w_{\textrm{f}}$ for some
values of the parameters $\lambda$ and $\kappa$. In Fig. 1a, all the
trajectories begin either close to K$\pm$ or near M. They all converge at ScI
and transition to dS. In Fig. 1c, the red and cyan trajectories in particular
pass near the scaling regime ScII$-$ (triangle pointing downwards) before
reaching dS. The rest converge at I instead (the diamond in between K$+$ and M
in the plot). The stripe indicates the forbidden region explicitly due to the
constraint equation (3.2) and, the triangle pointing upwards, the scaling
fixed point ScII$+$. Same phase portraits in $3$D are shown in Figs. 1e and
1f.
If the fixed points ScII$\pm$ are unstable only in the $y$ direction (see
discussion below Eq. (3.49)), then $m_{2,3}$ should be complex, given that the
real part in that case is always negative (see Eq. (3.49)). Unfortunately,
this cannot happen in the case of ScII$-$ (see Eq. (3.52) and discussion
below), and only at this fixed point can $\Omega_{m}$ dominate over
$\Omega_{\phi}+\Omega_{\xi}$.
An example of trajectories going through the second scaling solution
(corresponding to the fixed point ScII$-$, which is being represented by a
triangle pointing downwards) are the red and cyan ones in Fig. 1c, where
$\kappa/\lambda=4.41$ and $u_{0}=10^{-5}$. The purple line no longer passes
near the scaling regime, but approaches the I fixed point, represented by the
diamond in the plot. The corresponding $3$D phase portrait is shown in Fig.
1f. In Fig. 1c we show the density parameters and $w_{\textrm{f}}$ associated
with the red trajectory, which are compatible with the constraints on
$\Omega_{m}$ and $w_{\textrm{f}}$. Unfortunately, the scaling regime does not
last long enough for this particular case, although we see that $\Omega_{m}$
takes on the value that we expect in view of Eq. (3.56) and the corresponding
value of $\kappa$. Also, in line with what we explained earlier, the fixed
point ScII$+$ falls within the forbidden region because $\kappa>3\sqrt{2}$.
Such a fixed point is represented by a triangle pointing upwards. The
forbidden region corresponds to the pale red coloured stripe, which excludes
values of $x>1$ that violate the constraint equation (3.2).
### 3.5 The Speed of Gravitational Waves
In this last section we examine the evolution of the $\alpha_{T}$ parameter.
We first write it in terms of the dimensionless variables defined in Eqs.
(3.1) and (3.4), and of the Hubble flow parameter $\epsilon_{H}$ (see Eqs.
(2.47) and (3.15))
$\alpha_{T}=-\frac{2}{1-2ux}\left\\{(1+2ux)(ux)^{\prime}-ux\left[1-ux-\epsilon_{H}\left(1+ux\right)\right]\right\\}.$
(3.58)
It can be readily seen that
$\left.\alpha_{T}\right|_{\textrm{{M}}}=\left.\alpha_{T}\right|_{\textrm{{K}}\pm}=\left.\alpha_{T}\right|_{\textrm{{I}}}=\left.\alpha_{T}\right|_{\textrm{{ScI}}}=\left.\alpha_{T}\right|_{\textrm{{dS}}}=0~{}.$
(3.59)
For the rest of the fixed points, the $\alpha_{T}$ parameter is given by
$\displaystyle\left.\alpha_{T}\right|_{\textrm{{ScII}$\pm$}}=-\frac{18(1+w_{\textrm{M}})(1-w^{2}_{\textrm{M}})-2\kappa^{2}(1+3w_{\textrm{M}})\Delta_{\pm}}{27(1+w_{\textrm{M}})(1-w_{\textrm{M}}^{2})-2\kappa^{2}(1+3w_{\textrm{M}})\Delta_{\pm}}\left[2-\frac{\kappa^{2}}{18}\frac{(1+3w_{\textrm{M}})(5+3w_{\textrm{M}})}{(1+w_{\textrm{M}})(1-w^{2}_{\textrm{M}})}\Delta_{\pm}\right],$
(3.60)
$\displaystyle\left.\alpha_{T}\right|_{\textrm{{\^{K}}}\pm}=-\frac{(6\mp\sqrt{6}\kappa)(4\mp\sqrt{6}\kappa)}{7\mp\sqrt{6}\kappa}~{}.$
(3.61)
$\left.\alpha_{T}\right|_{\textrm{{\^{K}}}+}$ in particular is zero if
$\kappa=3\sqrt{\frac{2}{3}}$, which does not satisfy the bound
$\kappa<2\sqrt{\frac{2}{3}}$ and falls out of the super-accelerated phase
region. The other possibility is $\kappa=2\sqrt{\frac{2}{3}}$, but in that
case $u_{c}x_{c}=-1$ and $H_{c}=0$ (see Eq. (3.3)). Since $\kappa>0$,
$\left.\alpha_{T}\right|_{\textrm{{\^{K}}}-}$ never vanishes exactly.
Regarding $\left.\alpha_{T}\right|_{\textrm{{ScII}$\pm$}}$, in the case of the
fixed point ScII$+$, the $\alpha_{T}$ parameter cannot be zero due to the
existence conditions of the fixed point derived below Eq. (3.21) and the range
of values of $w_{\textrm{M}}$ under consideration. One might have
$\left.\alpha_{T}\right|_{\textrm{{ScII}$-$}}=0$ though. However, given
$w_{\textrm{M}}=w_{m}=0$, $\left.\alpha_{T}\right|_{\textrm{{ScII}$-$}}=0$ if
$\kappa^{2}\Delta_{-}(\kappa,0)=36/5$ or $\kappa^{2}\Delta_{-}(\kappa,0)=9$
(see Eq. (3.60)). We do not obtain solutions for $\kappa$ in those cases.
Thus, $\alpha_{T}$ never vanishes at ScII$-$ when $w_{\textrm{M}}=0$.
Although the value of $|\alpha_{T}|$ at the fixed points is useful to have an
indication of its evolution, the current state of the Universe cannot be
described by any of those fixed points. Moreover, $|\alpha_{T}|$ does not have
to be exactly zero, instead it must satisfy the bound in Eq. (1.2). Then, we
show the evolution of $\alpha_{T}$ connecting the different fixed points and
make sure it does not surpass the upper bound from the observational
constraint.
Figure 2: $|\alpha_{T}|$ predicted by the WGB model for some values of the
parameters $\lambda$ and $\kappa$ and different initial conditions of $x$. The
stripe in the smallest graph on the left indicates the excluded region
starting at redshift $z=0.00980\pm 0.00079$ (see Ref. [45]). We see that the
bound $|\alpha_{T}|<10^{-15}$ is grossly violated for the cases under
consideration. These cases have been selected given that they did satisfy the
constraints on $\Omega_{\textrm{M}}$ and $w_{\textrm{f}}$.
That bound is applicable in the relatively recent history of the Universe. To
be precise, $|\alpha_{T}|$ is constrained from redshift151515If one consults
Ref. [72] the redshift value, $z=0.0099$, is the geocentric redshift of the
host galaxy. The cosmological redshift by contrast, which is the one shown
here and what we need in this case, is given in Ref. [45]. $z=0.00980\pm
0.00079$ until today ($z=0$).161616Do not confuse ‘$z$’ here denoting the
redshift with the dimensionless variable defined in Eq. (3.1) and related with
the energy density of the background fluid. Selecting models which reproduce
trajectories that go through the scaling solution ScI, before entering de
Sitter, with different values of the parameters $\kappa$ and $\lambda$, and
different initial condition $x_{0}$, we plot the evolution of the
$|\alpha_{T}|$ parameter over the number of e-folds $N$ (see Fig. 2). In that
plot, we zoom in on the excluded region, marked by a vertical, pale red
coloured stripe, and we observe that the bound is violated by many orders of
magnitude, similarly to the SGB model [46]. The cases shown in the plot were
chosen given that they did satisfy the constraints on $\Omega_{m}$ and
$w_{\textrm{f}}$ at the present time.
If we examine Eq. (3.58), we can understand why the bound is so strongly
violated. We notice that for $ux$ of order
$\mathcal{O}(0.1)$-$\mathcal{O}(1)$, then $\Omega_{\xi}$ is of order
$\mathcal{O}(0.1)$ (see Eq. (3.7)). We observed numerically that
$\Omega_{\xi}$ is always the density parameter that dominates over the rest by
the time $\Omega_{m}$ and $w_{\textrm{f}}$ satisfy the constraints. In
addition to this, we remark that it grows at that time. Then $ux\rightarrow
1/2$ (i.e. $\Omega_{\xi}\rightarrow 0.6$) while the numerator in Eq. (3.58)
remains of order $\mathcal{O}(0.1)$-$\mathcal{O}(1)$. This causes
$|\alpha_{T}|$ to increase near the present time, although it does not diverge
because $(ux)^{\prime}$ increases eventually as well, even though
$\Omega_{\xi}$ remains constant. This can be seen by simply taking the
derivative of Eq. (3.7)
$\Omega_{\xi}^{\prime}=\frac{2(1-\Omega_{\xi})}{1+ux}(ux)^{\prime}.$ (3.62)
When $\Omega_{\xi}\rightarrow 1$, $\Omega_{\xi}^{\prime}\rightarrow 0$.
The fact that $|\alpha_{T}|$ does not remain below the observational bound
$10^{-15}$ while we approach the present time indicates that the exponential
coupling function in the WGB model must be discarded. Similar results are
obtained in the SGB model [46].
### 3.6 The Case of $\alpha_{T}=0$
In this section, instead of specifying the functional form of $\xi(\phi)$ and
consider examples other than the exponential case, we impose the condition
$\alpha_{T}=0$. In the SGB model, such a condition translates into Eq. (1.4).
In the WGB model, however, $\alpha_{T}=0$ leads to the following equation (see
Eq. (2.47)):
$\ddot{\xi}=\dot{\xi}W\left(1-2\frac{\dot{\xi}W}{M^{2}_{\textrm{Pl}}}\right).$
(3.63)
Using Eq. (2.26), Eq. (3.63) becomes
$\ddot{\xi}=-\frac{\dot{\xi}H}{2\mu}\left(1+2\mu-\sqrt{1+8\mu}\right).$ (3.64)
Now, the equation of $\dot{H}$ in the SGB model when $\ddot{\xi}=H\dot{\xi}$
reads (see Eq. (2.34))
$-2\dot{H}\left(M^{2}_{\textrm{Pl}}-4\dot{\xi}H\right)=\dot{\phi}^{2}+\rho_{\textrm{M}}\left(1+w_{\textrm{M}}\right).$
(3.65)
This implies that in de Sitter, $\dot{\phi}=0$ (so $\dot{H}=0$). However, this
value of $\dot{\phi}$ is problematic when we integrate Eq. (1.4). Indeed, the
integral on the LHS of the equation
$\int_{0}^{\dot{\xi}_{1}}\textrm{d}\ln(|\dot{\xi}|)=\int_{a_{\textrm{{dS}}}}^{a_{1}}\textrm{d}\ln
a~{},$ (3.66)
does not converge. In the WGB model however, since we have $\Phi$ in addition
to $H$, there is a chance that $\dot{H}=0$ but still $\dot{\phi}\neq 0$. To
verify this, we must write an equation analogous to Eq. (3.65). Bearing Eq.
(3.64) in mind, we write Eq. (2.30) as
$-\frac{2M^{2}_{\textrm{Pl}}\dot{H}}{\sqrt{1+8\mu}}=\dot{\phi}^{2}+\rho_{\textrm{M}}(1+w_{\textrm{M}})-4\ddot{\xi}H^{2}\frac{(1-\sqrt{1+8\mu})^{2}}{16\mu^{2}\sqrt{1+8\mu}}-4\dot{\xi}H^{3}\frac{(1-\sqrt{1+8\mu})^{3}}{64\mu^{3}}~{},$
(3.67)
where Eq. (2.26) was used. Plugging Eq. (3.64) into Eq. (3.67), we obtain
$-\frac{2M^{2}_{\textrm{Pl}}\dot{H}}{\sqrt{1+8\mu}}=\dot{\phi}^{2}+\rho_{\textrm{M}}\left(1+w_{\textrm{M}}\right)+3H^{2}M^{2}_{\textrm{Pl}}\frac{\left(1-\sqrt{1+8\mu}\right)^{2}}{16\mu^{2}\sqrt{1+8\mu}}\left(1+4\mu-\sqrt{1+8\mu}\right).$
(3.68)
We see that the last term vanishes when $|\mu|\ll 1$, recovering Eq (3.65) for
$|\dot{\xi}H|\ll M_{\textrm{Pl}}^{2}$ ($\mu$ was defined in Eq. (2.25)).
Unfortunately, each term on the RHS is non-negative meaning that $\dot{H}=0$
implies $\dot{\phi}=0$ and hence $\dot{\xi}=0$, same as in the SGB model.
Knowing this, we proceed to analyse the stability of the de Sitter and scaling
fixed points, dS and ScI respectively, when the new condition $\alpha_{T}=0$
is imposed, in line with was done in Ref. [46]. Firstly, we write Eq. (3.63)
as
$(ux)^{\prime}=\frac{ux}{1+ux}\left[1-\left(3x+2u\right)x-\frac{3}{2}\left(1+w_{\textrm{M}}\right)\left(1-x^{2}-y^{2}\right)\right],$
(3.69)
where we used Eq. (3.13), the constraint equation (3.2) and Eqs. (3.4) and
(3.12), given that we do not assume a coupling function form in particular.
This is equivalent to $\alpha_{T}=0$ in Eq. (3.58). We however make it clear
that we use Eq. (3.12) to write it that way.
We perturb Eq. (3.69) around the de Sitter fixed point (which is still a fixed
point of the system despite Eq. (3.69)), that can only correspond to $x=0$ and
$z=0$ as we just argued
$\delta x^{\prime}=\delta x~{}.$ (3.70)
This equation implies that $\delta x\propto e^{N}$. The rest of the equations
read171717Notice that, although we have not assumed a specific coupling
function $\xi(\phi)$ in this section,
$u_{\textrm{{dS}}}=\sqrt{\frac{3}{2}}\lambda$, as in the case of the
exponential function. This can be seen from Eq. (3.9) with
$x_{\textrm{{dS}}}=0$ and $y_{\textrm{{dS}}}=1$.
$\displaystyle\delta x^{\prime}=-3\delta
x-\sqrt{\frac{3}{2}}\lambda\left(1+3w_{\textrm{M}}\right)\delta y-\delta
u~{},$ (3.71) $\displaystyle\delta y^{\prime}=-3(1+w_{\textrm{M}})\delta
y~{}.$ (3.72)
The second equation gives $\delta y\propto e^{-3(1+w_{\textrm{M}})N}$. Using
Eq. (3.70), Eq. (3.71) gives $\delta u$. Since $\delta y$ dies out but $\delta
x$ grows, $\delta u$ will increase too. Then we see that the de Sitter fixed
point dS is unstable, as in the SGB model [46].
Regarding the scaling regime ScI, since $u_{\textrm{{ScI}}}=0$, this is still
a fixed point despite Eq. (3.69). Linearising Eq. (3.69), we obtain
$\delta u^{\prime}=-\frac{1}{2}\left(1+3w_{\textrm{M}}\right)\delta u~{}.$
(3.73)
Hence $\delta u\propto e^{-\frac{1}{2}(1+3w_{\textrm{M}})N}$, and given the
eigenvalues in Eq. (3.37) (which remain the same despite $\alpha_{T}=0$), we
have that ScI becomes stable, a result similar to the one obtained in the SGB
model in Ref. [46] and in quintessence models in GR. When it comes to the
second scaling regime, where $y_{\textrm{{ScII}$\pm$}}=0$, we can see that
they are not even fixed points of the new system of equations that includes
Eq. (3.69).
Consequently, the WGB model with $\alpha_{T}=0$ cannot reproduce a long
scaling regime with small $u$ followed by a period of accelerated expansion,
and hence can be discarded.
## 4 Summary and Conclusions
Gauss-Bonnet (GB) gravity with $\xi(\phi)\mathcal{G}$ (the scalar-Gauss-Bonnet
(SGB) model), where $\mathcal{G}$ is the GB term, has become a popular dark
energy (DE) model and it is known to predict the propagation speed of
gravitational waves (GWs) $c_{\textrm{GW}}\neq 1$ (in units where $c=1$). For
simple coupling functions $\xi(\phi)$ which reproduce the past evolution of
the Universe as we know it, the SGB model does not satisfy the observational
constraints on $|\alpha_{T}|\equiv|c_{\textrm{GW}}^{2}-1|$ from GRB170817A
[37] when applied to the late Universe in particular [46]. The same occurs
with Horndeski’s theory [38]. The model is actually a subclass of that theory
and one can recover the SGB action for a non-trivial choice of the functions
of the theory [40].
In spite of those well-known results, it was shown recently that Horndeski’s
theory where $G(\phi,X)R$ with
$X\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ does
predict GWs propagating at the speed of light in vacuum when formulated in the
Palatini formalism [53]. All the previous claims of $c_{\textrm{GW}}\neq 1$
were made assuming the metric case instead, where the connection is the Levi-
Civita (LC) one, which fully depends on the metric and its first derivative.
In the Palatini formalism however, the connection is independent of the metric
and obeys some field equations, meaning that the geometry differs from the
pseudo-Riemannian one. Given this, in the present work we considered the SGB
action but in Weyl geometry where the connection (the Weyl connection) has
zero torsion but it is not metric compatible. We named this model Weyl-Gauss-
Bonnet. A similar action was analysed in Ref. [59] but ours considers the
field-dependent coupling to the GB term and restricts to four dimensions.
Thus, the goal in our work was to investigate whether the SGB gravity, as a DE
model, when formulated in a different geometry, fulfilled the constraint (1.2)
and yet displayed a period of matter domination followed by an accelerated
expanding phase of the Universe in accordance with other constraints on the
density and equation of state (EoS) parameters.
We assumed a spatially-flat FLRW metric and a background Weyl vector with
temporal component $\Phi(t)$ and no spatial components breaking spatial
isotropy. The resulting action once the LC and non-Riemannian parts were
written separately was given in Eq. (2.19). The $\alpha_{T}$ parameter was
calculated in Sec. 2.4. In Sec. 3, we performed a dynamical systems analysis
assuming an exponential potential, which is common in quintessence scenarios.
The dimensionless variables in Eqs. (3.1) and (3.4) were formulated in terms
of the newly defined Weyl parameter $W\equiv H-\Phi$, thereby including the
modifications to Einstein’s gravity in contrast to the metric formalism where
these are accounted for by $u$ alone. Fixed points of the dynamical system
were found in Sec. 3.2 under the assumption that $\xi(\phi)$ is an exponential
function (see Eq. (3.16)). This choice makes all the equations self-similar so
the explicit dependence on $W$ disappears. Another possibility was the linear
GB coupling (because $\xi_{,\phi\phi}=0$; see Eq. (3.12)), but it was argued
that this did not lead to the expected evolution of the Universe, as was
noticed in the metric formalism as well [46]. A stability analysis of the
fixed points was carried out in Sec. 3.3.
Among the fixed points calculated in this work, the scaling ScI (where $u=0$)
and de Sitter dS ones had been found in the SGB model of the metric formalism
too [46], and were shown to satisfy the same stability conditions in both the
WGB and SGB models. While the scaling fixed point is unstable if
$\kappa>\lambda$, dS becomes an attractor and the Universe is predicted to
abandon a period of matter domination and enter a regime of accelerated
expansion, where $\kappa$ and $\lambda$ are the strength of the exponentials
involved in the GB coupling and the scalar potential, as shown in Eqs. (3.16)
and (3.8) respectively. If $\kappa<\lambda$ however, the roles are switched
over and de Sitter becomes unstable (a saddle point) whereas ScI is stable. On
top of these and other known fixed points, we found a second scaling regime
consisting of two fixed points ScII$\pm$ (where $y=0$) and a regime of
‘pseudo-kination’ where the Weyl parameter, $W_{c}$, satisfied
$3W_{c}^{2}M^{2}_{\textrm{Pl}}=\dot{\phi}^{2}_{c}/2$ but $W_{c}\neq H_{c}$.
This regime is associated with two fixed points K̂$\pm$ as well. In the case
of ScII$+$, it was shown not to exist if the EoS parameter of the background
fluid fell within the range $\frac{1}{3}\leq w_{\textrm{M}}<1$, thereby
excluding background radiation but including pressureless matter
$w_{\textrm{M}}=0$. Regarding the pseudo-kination regime, we found that K̂$+$
led to a universe that accelerated faster than during de Sitter, where it
accelerates exponentially, if $\kappa<2\sqrt{\frac{2}{3}}$. Also, regardless
of the sign of $x_{\textrm{{\^{K}}$\pm$}}$, if $\kappa\gg 1$, then the
Universe was predicted not to accelerate. Moreover, both situations, the
super-accelerated and uniform expansion phases, were shown to be stable if
$\kappa>\lambda$ and $\kappa<\lambda$, respectively. The scaling regime
ScII$\pm$ was determined to be unstable whenever $\kappa>\lambda$, same as in
the case of ScI. The instability in the former case was found to be in the $y$
direction of the phase space. We found also that ScII$+$ is always stable only
if $\lambda>\kappa$ because two of the three eigenvalues, which do not depend
on $\lambda$, are complex with negative real parts. The summary of the fixed
points and their stability was provided in Table 1.
In Sec. 3.4 we performed some numerical simulations assuming pressureless
background matter $w_{\textrm{M}}=w_{m}=0$ and checked trajectories crossing
both scaling regimes. When it comes to those approaching ScI, we plotted the
2D phase portrait in Fig. 1a and the 3D one in Fig. 1e for some values of
$\lambda$ and $\kappa$. These were chosen among other simulations given that
they satisfied the constraints on $\Omega_{m}$ and $w_{\textrm{f}}$ [3] at
present time ($N=0$), where $w_{\rm f}$ is the EoS parameter of the DE fluid
responsible for the current accelerated expansion of the Universe. The
evolution of the density parameters and $w_{\textrm{f}}$ for one of the
trajectories in the phase portrait was displayed in Fig. 1b. The system’s
behaviour is reproduced as expected: starting from a period of kination, the
Universe enters a long period of background fluid domination which gives way
to the present epoch of DE domination. An eventual period of de Sitter is
predicted after a transient epoch where the effective density parameter
associated with the GB coupling $\xi(\phi)$, $\Omega_{\xi}$, dominates and
$w_{\textrm{f}}$ takes on phantom values (i.e. $w_{\textrm{f}}<-1$). We
pointed out that a common feature of all the simulations was that, during DE
domination, $\Omega_{\xi}$ was always dominant over the density parameter of
the scalar field $\Omega_{\phi}$.
A similar but much shorter period of matter domination was reproduced in the
case of the second scaling solution ScII$-$ (see Fig. 1d). The corresponding
2D phase diagram was shown in Fig. 1c and the 3D phase space in Fig. 1f. In
order for $\Omega_{m}$ to dominate over $\Omega_{\phi}+\Omega_{\xi}$ during
the scaling regime (so BBN predictions are not affected), it was demonstrated
that $\kappa>2\sqrt{6}$. As to ScII$+$, the bound translated into
$\kappa<\sqrt{6}$ and was at odds with the existence condition of that fixed
point. Indeed, we saw that ScII$+$ was located in the excluded region of the
phase portrait 1c (the coloured stripe).
The $\alpha_{T}$ parameter was written in terms of the dimensionless
variables, defined in Eqs. (3.1) and (3.4), their derivatives, and the Hubble
flow parameter $\epsilon_{H}$ in Eq. (3.58). We noticed that it is always non-
vanishing for K̂$-$, and for K̂$+$ only if one considers the super-accelerated
regime. Also, $\alpha_{T}$ never vanishes at ScII$-$ if $w_{\textrm{M}}=0$.
However, it was argued that the value of $|\alpha_{T}|$ at the fixed points
was not enough because the current Universe had to be somewhere in between the
scaling and de Sitter fixed points in the phase space. We plotted in Fig. 2
the evolution of $|\alpha_{T}|$ for different values of the constants
$\lambda$ and $\kappa$ and initial conditions $x_{0}$. Unfortunately, the
constraint in Eq. (1.2) was found to be grossly violated at present time, when
$\Omega_{m}$ and $w_{\textrm{f}}$ satisfy the aforementioned constraints. It
is to that moment that Eq. (1.2) applies approximately given that the binary
neutron star system’s merger occurred in the relatively recent past history of
the Universe. This was indicated in an inset plot, where the constrained
region corresponds to the coloured narrow stripe.
Despite the negative results, similar to the ones we obtained in [46] for the
SGB model in the metric formalism, the analysis performed in this work did not
consider coupling functions other than the exponential and linear cases (so
the equations are self-similar), which might lead to different conclusions. We
then regarded the case of $\alpha_{T}=0$, without making any assumption on the
coupling function, and analysed the stability of the scaling and de Sitter
fixed points in particular given their importance in late time cosmology. We
found that the latter was only possible if $\dot{\phi}=0$, same as in the SGB
model, despite the presence of the new variable $\Phi(t)$ in the equations.
The subsequent analysis of the stability indicated that, whereas the de Sitter
regime became unstable, the scaling one was stable, contradicting the known
evolution of the Universe. Consequently, when $\alpha_{T}=0$ for a non-
constant coupling function $\xi(\phi)$, we encountered the same stability of
ScI and dS as that in Ref. [46].
Assuming that the Weyl vector has vanishing spatial components, one can apply
the very same results of this work to a connection compatible with the metric
but with non-zero torsion. We showed this in App. C. In App. A we introduced
such a connection and the geometric structures used in this manuscript
briefly. The background equations used in Sec. 2 and those including the
homogeneous spatial components of the Weyl vector were detailed in App. B.
As mentioned above, we have regarded only exponential and linear coupling
functions in this work, given that the equations were self-similar and the
dynamical systems analysis could be easily done. On top of this, the case of
$\kappa=\lambda$, which was analytically considered in Ref. [73] in the metric
case, was not tackled here (because, again, it was not amenable to the
presented methods of analysis). It would be interesting to apply an analysis
similar to that of the referenced paper, although two fixed ‘curves’ were
calculated in Sec. 3.2 already, one of them corresponding to a third scaling
solution of cosmological interest. Also, despite that the homogeneous spatial
components of the Weyl vector were fully neglected under the assumption of
spatial isotropy, one might regard the presence of such components so that
they are relevant in the dynamical analysis but subdominant enough not to
generate a large scale anisotropy. During inflation, this and even the fully
dominant presence of the vector field were shown to be consistent with
observations in Ref. [74]. We leave these questions for future publications.
With this work, we hope to have brought attention to the fact that SGB gravity
could be rescued in the Palatini formalism with the aim of addressing the late
time evolution of the Universe, where the stringent observational bounds on
the speed of propagation of GWs are an unavoidable test for scalar-tensor
theories at large.
## Acknowledgments
We are grateful to T. Koivisto for useful comments while this work was being
completed. K.D. is supported (in part) by the STFC consolidated grant:
ST/X000621/1. M.K. is supported by the María Zambrano grant, provided by the
Ministry of Universities from the Next Generation funds of the European Union.
This work is also partially supported by the MICINN (Spain) projects
PID2019-107394GB-I00/AEI/10.13039/501100011033 and PID2022-139841NB-I00
(AEI/FEDER, UE). A.R. is supported by the Estonian Research Council grant
PRG1055.
For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) license to any Author Accepted Manuscript version arising.
## Appendix A Palatini Formalism and Curvature Tensors
In this section of the Appendix, we turn our attention to various geometric
structures in the Palatini formalism. The Riemann tensor is defined only with
respect to the connection and its first derivatives as [63]
$\tensor{R}{{}^{\alpha}_{\beta\mu\nu}}\equiv\partial_{\mu}\Gamma^{\alpha}_{\nu\beta}-\partial_{\nu}\Gamma^{\alpha}_{\mu\beta}+\Gamma^{\alpha}_{\mu\lambda}\Gamma^{\lambda}_{\nu\beta}-\Gamma^{\alpha}_{\nu\lambda}\Gamma^{\lambda}_{\mu\beta}~{}.$
(A.1)
In the metric formalism, the above expression can be written in terms of the
second derivatives of the metric by virtue of Eq. (2.1). In the Palatini
formalism, on the other hand, the Riemann tensor does not depend on the
derivatives of the metric tensor. However, in that case, we can write Eq.
(A.1) as the sum of the LC part, and the distortion tensor (defined in Eq.
(2.4)) and its derivatives
$\tensor{R}{{}^{\alpha}_{\beta\mu\nu}}=\tensor{\mathring{R}}{{}^{\alpha}_{\beta\mu\nu}}+\tensor{\kappa}{{}^{\alpha}_{\beta\mu\nu}}~{},$
(A.2)
where we introduced a tensor $\tensor{\kappa}{{}^{\alpha}_{\beta\mu\nu}}$,
given by
$\tensor{\kappa}{{}^{\alpha}_{\beta\mu\nu}}\equiv\mathring{\nabla}_{\mu}\tensor{\kappa}{{}_{\nu\beta}^{\alpha}}-\mathring{\nabla}_{\nu}\tensor{\kappa}{{}_{\mu\beta}^{\alpha}}+\tensor{\kappa}{{}_{\mu\lambda}^{\alpha}}\tensor{\kappa}{{}_{\nu\beta}^{\lambda}}-\tensor{\kappa}{{}_{\nu\lambda}^{\alpha}}\tensor{\kappa}{{}_{\mu\beta}^{\lambda}}.$
(A.3)
An important second-rank tensor that is related to the Riemann one is the
Ricci tensor, which is defined as
$R_{\mu\nu}\equiv\tensor{R}{{}^{\sigma}_{\mu\sigma\nu}}=\mathring{R}_{\mu\nu}+\kappa_{\mu\nu}~{},$
(A.4)
where $\kappa_{\mu\nu}\equiv\tensor{\kappa}{{}^{\sigma}_{\mu\sigma\nu}}$.
While $\mathring{R}_{\mu\nu}$ is symmetric, $R_{\mu\nu}$ does not possess any
symmetry _a priori_.
Another related tensor, which in the metric formalism coincides with the Ricci
tensor, is the co-Ricci tensor
$\tilde{R}_{\mu\nu}\equiv
g^{\alpha\beta}R_{\mu\alpha\nu\beta}=\mathring{R}_{\mu\nu}+\tilde{\kappa}_{\mu\nu}~{},$
(A.5)
where $\tilde{\kappa}_{\mu\nu}\equiv
g^{\alpha\beta}\kappa_{\mu\alpha\nu\beta}$.
The last second-rank tensor that can be obtained from the Riemann tensor is
the ‘homothetic curvature tensor’, $\bar{R}_{\mu\nu}$. In the Palatini
formalism, the Riemann tensor is not generically antisymmetric in its first
two indices. The only symmetry it has is the antisymmetry in the last two
indices, which can be inferred from Eq. (A.1). Therefore, $\bar{R}_{\mu\nu}$
is defined as
$\bar{R}_{\mu\nu}\equiv\tensor{R}{{}^{\sigma}_{\sigma\mu\nu}}=\bar{\kappa}_{\mu\nu}~{},$
(A.6)
where
$\bar{\kappa}_{\mu\nu}\equiv\tensor{\kappa}{{}^{\sigma}_{\sigma\mu\nu}}=\mathring{\nabla}_{\mu}\tensor{\kappa}{{}_{\nu\sigma}^{\sigma}}-\mathring{\nabla}_{\nu}\tensor{\kappa}{{}_{\mu\sigma}^{\sigma}}.$
(A.7)
This tensor is fully antisymmetric. Consequently, the Ricci scalar
$R\equiv
g^{\mu\nu}R_{\mu\nu}=\tensor{\tilde{R}}{{}^{\sigma}_{\sigma}}=\mathring{R}+\kappa~{},$
(A.8)
is the only non-vanishing scalar one can extract from the Riemann tensor. In
the above expression, the scalar $\kappa$ is defined as
$\kappa\equiv
g^{\mu\nu}\kappa_{\mu\nu}=\tensor{\tilde{\kappa}}{{}^{\sigma}_{\sigma}}~{}.$
(A.9)
The Weyl connection considered in Eq. (2.9) is a concrete example of a more
general one with distortion tensor
$\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}=c_{1}\delta_{\mu}^{\alpha}A_{\nu}+c_{2}\delta_{\nu}^{\alpha}A_{\mu}-c_{3}g_{\mu\nu}A^{\alpha},$
(A.10)
where $c_{1}$, $c_{2}$ and $c_{3}$ are constants. The torsion and non-
metricity tensors read (see Eqs. (2.7) and (2.8), respectively)
$\displaystyle\tensor{T}{{}_{\mu\nu}^{\alpha}}=\left(c_{1}-c_{2}\right)\left(\delta_{\mu}^{\alpha}A_{\nu}-\delta_{\nu}^{\alpha}A_{\mu}\right),$
(A.11) $\displaystyle
Q_{\alpha\mu\nu}=-2c_{2}g_{\mu\nu}A_{\alpha}-\left(c_{1}-c_{3}\right)\left(g_{\mu\alpha}A_{\nu}+g_{\nu\alpha}A_{\mu}\right).$
(A.12)
We see that for $(c_{1},c_{2},c_{3})=(1,1,1)$, the torsion tensor vanishes
although the non-metricity tensor is not zero, leading to Eq. (2.10). Besides,
the choice $(c_{1},c_{2},c_{3})=(1,0,1)$ leads to a connection compatible with
the metric (i.e. with vanishing non-metricity tensor), but with non-zero
torsion. A connection with non-zero torsion and non-metricity tensors
corresponds to $(c_{1},c_{2},c_{3})=(0,1,0)$. This latter connection is well-
known for being the most general solution to the connection field equations
derived from the Einstein-Hilbert action in the Palatini formalism [75, 76].
In the case of the Weyl connection, the contributions from the distortion
tensor to the Ricci, co-Ricci and homothetic curvature tensors read
$\displaystyle\kappa_{\mu\nu}=\mathring{\nabla}_{\mu}A_{\nu}-3\mathring{\nabla}_{\nu}A_{\mu}+2A_{\mu}A_{\nu}-g_{\mu\nu}\left(\mathring{\nabla}_{\sigma}A^{\sigma}+2A_{\sigma}A^{\sigma}\right),$
(A.13)
$\displaystyle\tilde{\kappa}_{\mu\nu}=-\mathring{\nabla}_{\mu}A_{\nu}-\mathring{\nabla}_{\nu}A_{\mu}+2A_{\mu}A_{\nu}-g_{\mu\nu}\left(\mathring{\nabla}_{\sigma}A^{\sigma}+2A_{\sigma}A^{\sigma}\right),$
(A.14)
$\displaystyle\bar{\kappa}_{\mu\nu}=4\left(\mathring{\nabla}_{\mu}A_{\nu}-\mathring{\nabla}_{\nu}A_{\mu}\right),$
(A.15)
respectively. One can easily show that the relation among the three tensors
above can be written as
$\kappa_{\mu\nu}=\tilde{\kappa}_{\mu\nu}+\frac{1}{2}\bar{\kappa}_{\mu\nu}~{}.$
(A.16)
Notice that $\tilde{\kappa}_{\mu\nu}$ (and thus $\tilde{R}_{\mu\nu}$) is
symmetric in this particular case. Moreover, $\tilde{\kappa}_{\mu\nu}$ is the
symmetric part of $\kappa_{\mu\nu}$; i.e.
$\kappa_{(\mu\nu)}=\tilde{\kappa}_{\mu\nu}$. The antisymmetric part, on the
other hand, is given by $\kappa_{[\mu\nu]}=\frac{1}{2}\bar{\kappa}_{\mu\nu}$.
The scalar $\kappa$, defined in Eq. (A.9), becomes
$\kappa=-6\left(\mathring{\nabla}_{\sigma}A^{\sigma}+A_{\sigma}A^{\sigma}\right).$
(A.17)
Regarding $\tensor{\kappa}{{}^{\alpha}_{\beta\mu\nu}}$, we have the following
symmetry relations:
$\displaystyle\kappa_{\alpha\beta\mu\nu}=\bar{\kappa}_{\alpha\beta\mu\nu}+\frac{1}{4}g_{\alpha\beta}\bar{\kappa}_{\mu\nu}~{},$
(A.18)
$\displaystyle\kappa_{\alpha\beta\mu\nu}-\kappa_{\mu\nu\alpha\beta}=\frac{1}{4}\left(g_{\alpha\beta}\bar{\kappa}_{\mu\nu}-g_{\mu\nu}\bar{\kappa}_{\alpha\beta}-g_{\beta\nu}\bar{\kappa}_{\mu\alpha}+g_{\mu\alpha}\bar{\kappa}_{\beta\nu}-g_{\mu\beta}\bar{\kappa}_{\alpha\nu}+g_{\alpha\nu}\bar{\kappa}_{\mu\beta}\right),$
(A.19)
where $\bar{\kappa}_{\alpha\beta\mu\nu}\equiv\kappa_{[\alpha\beta]\mu\nu}$.
Among the various options discussed below Eq. (A.12), the second connection
with torsion is addressed in App. C.
## Appendix B Homogeneous Spatial Vector
For completeness, we derive the homogeneous equations when the Weyl vector has
non-zero homogeneous spatial components. This induces background anisotropy,
as expected. To that end, we vary the action with respect to the inverse
metric first.
The variation of the scalar-Gauss-Bonnet part of the action in Eq. (2.19) is
$\displaystyle\delta
S_{\textrm{SGB}}=\frac{M^{2}_{\textrm{Pl}}}{2}\int\textrm{d}^{4}x\sqrt{-g}\left\\{G_{\mu\nu}-\frac{4}{M^{2}_{\textrm{Pl}}}\left(\tensor{R}{{}_{\mu}^{\alpha}{}_{\nu}^{\beta}}+2\tensor{S}{{}^{\alpha}_{\mu}}\delta_{\nu}^{\beta}-g_{\mu\nu}G^{\alpha\beta}-R_{\mu\nu}g^{\alpha\beta}\right)\nabla_{\alpha}\partial_{\beta}\xi-\right.$
$\displaystyle\left.-\frac{1}{M^{2}_{\textrm{Pl}}}\left[\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+2V\right)\right]\right\\}\delta
g^{\mu\nu},$ (B.1)
where $S_{\mu\nu}$ is the traceless tensor defined as
$S_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R~{}.$ (B.2)
We remind the reader we dropped the rings (overcircles) here because, as was
argued below Eq. (2.19), every geometric quantity is defined with respect to
the LC connection once the LC and non-Riemannian parts (the latter depending
on the Weyl vector and its derivatives) have been written separately. Then
$\displaystyle\delta
S_{\textrm{SGB}}=\frac{M^{2}_{\textrm{Pl}}}{2}\int\textrm{d}^{4}x\sqrt{-g}\left\\{\left[3H^{2}-12\frac{\dot{\xi}}{M^{2}_{\textrm{Pl}}}H^{3}-\frac{1}{M^{2}_{\textrm{Pl}}}\rho_{\phi}\right]\delta
g^{00}-\left[2\dot{H}+3H^{2}-4\frac{\ddot{\xi}}{M^{2}_{\textrm{Pl}}}H^{2}-\right.\right.$
$\displaystyle\left.\left.-8\frac{\dot{\xi}}{M^{2}_{\textrm{Pl}}}H^{3}-8\frac{\dot{\xi}}{M^{2}_{\textrm{Pl}}}H\dot{H}+\frac{1}{M^{2}_{\textrm{Pl}}}P_{\phi}\right]g_{ij}\delta
g^{ij}\right\\},$ (B.3)
where the spatially-flat FLRW metric was assumed, and $\rho_{\phi}$ and
$P_{\phi}$ are defined as the energy density and pressure of the canonical
scalar field $\phi$
$\displaystyle\rho_{\phi}\equiv\frac{1}{2}\dot{\phi}^{2}+V(\phi)~{},$ (B.4)
$\displaystyle P_{\phi}\equiv\frac{1}{2}\dot{\phi}^{2}-V(\phi)~{},$ (B.5)
respectively.
Regarding the action of the vector field $A_{\mu}$, which we shall call
‘$S_{A}$’, such that $S_{\textrm{WGB}}=S_{\textrm{SGB}}+S_{A}+S_{\textrm{M}}$
(see Eq. (2.20) where $S_{\textrm{M}}$ was introduced) and
$\displaystyle
S_{A}=-4\int\textrm{d}^{4}x\sqrt{-g}\left[\frac{3}{4}M^{2}_{\textrm{Pl}}A_{\sigma}A^{\sigma}-\left(G^{\mu\nu}-\nabla^{\mu}A^{\nu}\right)A_{\mu}\partial_{\nu}\xi-\left(\nabla_{\sigma}A^{\sigma}+A_{\sigma}A^{\sigma}\right)\partial_{\rho}\xi
A^{\rho}-\right.$ $\displaystyle\left.-\frac{1}{2}\xi\Upsilon
F^{\mu\nu}F_{\mu\nu}\right],$ (B.6)
where we defined
$\displaystyle
F_{\mu\nu}\equiv\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}~{},$
(B.7)
i.e. $F_{\mu\nu}=\frac{1}{4}\bar{\kappa}_{\mu\nu}$, we have
$\displaystyle\delta
S_{A}=-4\int\textrm{d}^{4}x\sqrt{-g}\left\\{\left[\frac{3}{8}M^{2}_{\textrm{Pl}}\Phi^{2}-\frac{1}{2}\dot{\xi}Wa^{-2}A_{i}A^{i}-\frac{9}{2}\dot{\xi}H\Phi
W-\frac{3}{2}\dot{\xi}\Phi^{3}-\frac{1}{2}\xi\Upsilon
a^{-2}\dot{A_{i}}\dot{A^{i}}+\right.\right.$
$\displaystyle\left.\left.+\frac{3}{8}M^{2}_{\textrm{Pl}}a^{-2}A_{i}A^{i}\right]\delta
g^{00}-\frac{3}{2}M^{2}_{\textrm{Pl}}\left[\Phi-2\frac{\dot{\xi}}{M^{2}_{\textrm{Pl}}}\left(W^{2}-\frac{1}{3}a^{-2}A_{j}A^{j}\right)\right]A_{i}\delta
g^{0i}+\left[\frac{1}{2}\left(\frac{3}{2}M^{2}_{\textrm{Pl}}+\right.\right.\right.$
$\displaystyle\left.\left.\left.+\ddot{\xi}+3H\dot{\xi}-2\dot{\xi}\Phi\right)A_{i}A_{j}+\dot{\xi}\dot{A_{i}}A_{j}+\xi\Upsilon\dot{A_{i}}\dot{A_{j}}+\left(\ddot{\xi}H\Phi-\frac{1}{2}\ddot{\xi}\Phi^{2}+\frac{3}{8}M^{2}_{\textrm{Pl}}\Phi^{2}-\right.\right.\right.$
$\displaystyle\left.\left.\left.-\frac{3}{8}M^{2}_{\textrm{Pl}}a^{-2}A_{k}A^{k}+\dot{\xi}H\dot{\Phi}+\dot{\xi}\dot{H}\Phi-\dot{\xi}\Phi\dot{\Phi}-\frac{1}{2}\dot{\xi}\Phi^{3}+\frac{3}{2}\dot{\xi}H^{2}\Phi-\frac{1}{2}\dot{\xi}Wa^{-2}A_{k}A^{k}-\right.\right.\right.$
$\displaystyle\left.\left.\left.-\frac{1}{2}\xi\Upsilon
a^{-2}\dot{A_{k}}\dot{A^{k}}\right)g_{ij}\right]\delta g^{ij}\right\\},$ (B.8)
after some tedious calculations. $A_{i}(t)$ are the homogeneous spatial
components of the Weyl vector; i.e.
$A_{\mu}(t)\equiv\left(-\Phi(t),A_{i}(t)\right).$ (B.9)
Spatial indices are raised and lowered with respect to $\delta_{ij}$.
The off-diagonal part in Eq. (B.8) vanishes because of the $\Phi$ equation
which, in the presence of the homogeneous spatial component, is
$\Phi=2\frac{\dot{\xi}}{M^{2}_{\textrm{Pl}}}\left(W^{2}-\frac{1}{3}a^{-2}A_{j}A^{j}\right).$
(B.10)
The equation of $A_{i}$, which can be obtained from Eq. (2.21), reads
$\Upsilon\left[\xi\ddot{A_{i}}+\left(\dot{\xi}+\xi
H\right)\dot{A_{i}}\right]=\left(\frac{3}{4}M^{2}_{\mathrm{Pl}}+\dot{\xi}W\right)A_{i}~{}.$
(B.11)
The $M^{2}_{\textrm{Pl}}G_{ij}=T^{\textrm{eff}}_{ij}$ metric field equations,
$T^{\textrm{eff}}_{\mu\nu}$ being the effective energy-momentum tensor, can be
read off from Eqs. (B.3) and (B.8) (including the matter fields)
$\displaystyle-M^{2}_{\textrm{Pl}}\left(2\dot{H}+3H^{2}\right)g_{ij}=-\left(4\ddot{\xi}H^{2}+8\dot{\xi}H^{3}+8\dot{\xi}H\dot{H}-P_{\phi}-P_{\textrm{M}}\right)g_{ij}+4\left(\frac{3}{2}M^{2}_{\textrm{Pl}}+\ddot{\xi}+3H\dot{\xi}-\right.$
$\displaystyle\left.-2\Phi\dot{\xi}\right)A_{i}A_{j}+8\dot{\xi}\dot{A}_{\left(i\right.}A_{\left.j\right)}+8\xi\Upsilon\dot{A_{i}}\dot{A_{j}}+8\left(\ddot{\xi}H\Phi-\frac{1}{2}\ddot{\xi}\Phi^{2}+\frac{3}{8}M^{2}_{\textrm{Pl}}\Phi^{2}-\frac{3}{8}M^{2}_{\textrm{Pl}}a^{-2}A_{k}A^{k}+\dot{\xi}H\dot{\Phi}+\right.$
$\displaystyle\left.+\dot{\xi}\dot{H}\Phi-\dot{\xi}\Phi\dot{\Phi}-\frac{1}{2}\dot{\xi}\Phi^{3}+\frac{3}{2}\dot{\xi}H^{2}\Phi-\frac{1}{2}\dot{\xi}Wa^{-2}A_{k}A^{k}-\frac{1}{2}\xi\Upsilon
a^{-2}\dot{A_{k}}\dot{A^{k}}\right)g_{ij}~{}.$ (B.12)
Assuming that the homogeneous spatial vector lies along the $z$ direction
without the loss of generality, $\mathbf{A}(t)=(0,0,A_{z}(t))$, we see that
$T^{\textrm{eff}}_{xx}=T^{\textrm{eff}}_{yy}\neq T^{\textrm{eff}}_{zz}$, which
means spatial anisotropy (and the off-diagonal components vanish).
Finally, $M^{2}_{\textrm{Pl}}G_{00}=T^{\textrm{eff}}_{00}$ reads
$3H^{2}M^{2}_{\textrm{Pl}}=\rho_{\phi}+\rho_{\textrm{M}}+3\left(\Phi^{2}+a^{-2}A_{i}A^{i}\right)M^{2}_{\textrm{Pl}}-4\xi\Upsilon
a^{-2}\dot{A_{i}}\dot{A^{i}}+12\dot{\xi}W\left(W^{2}-\frac{1}{3}a^{-2}A_{i}A^{i}\right).$
(B.13)
## Appendix C Projective Transformations and Torsion
In this section of the Appendix, the homogeneous equations (2.36)-(2.38) and
the tensor perturbation equation (2.43) are shown to be valid for a connection
that is metric compatible but has non-zero torsion. For this purpose, we
review the so-called ‘projective transformations’, which are given by (see
Ref. [77])
$\displaystyle\Gamma^{\alpha}_{\mu\nu}\rightarrow\Gamma^{\alpha}_{\mu\nu}+B_{\mu}\delta_{\nu}^{\alpha}~{},$
(C.1)
where $B_{\mu}$ is an arbitrary vector. This amounts to
$\displaystyle\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}\rightarrow\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}+B_{\mu}\delta_{\nu}^{\alpha}~{}.$
(C.2)
The $\kappa$ tensors defined in App. A transform as
$\displaystyle\tensor{\kappa}{{}^{\alpha}_{\beta\mu\nu}}\rightarrow\tensor{\kappa}{{}^{\alpha}_{\beta\mu\nu}}+\delta_{\beta}^{\alpha}\mathring{F}_{\mu\nu}~{},$
(C.3)
$\displaystyle\kappa_{\mu\nu}\rightarrow\kappa_{\mu\nu}+\mathring{F}_{\mu\nu}~{},$
(C.4)
$\displaystyle\tilde{\kappa}_{\mu\nu}\rightarrow\tilde{\kappa}_{\mu\nu}-\mathring{F}_{\mu\nu}~{},$
(C.5)
$\displaystyle\bar{\kappa}_{\mu\nu}\rightarrow\bar{\kappa}_{\mu\nu}+4\mathring{F}_{\mu\nu}~{},$
(C.6)
where
$\mathring{F}_{\mu\nu}\equiv\mathring{\nabla}_{\mu}B_{\nu}-\mathring{\nabla}_{\nu}B_{\mu}$.
The scalar $\kappa$ is invariant under these transformations because the
tensor $\mathring{F}_{\mu\nu}$ is antisymmetric. Notice also that the tensor
$\bar{\kappa}_{\alpha\beta\mu\nu}$, which was defined as the
antisymmetrisation of the first two indices of $\kappa_{\alpha\beta\mu\nu}$
below Eq. (A.19), is invariant as well.
Now, let us apply this transformation to the Weyl connection. The
$\mathcal{G}$ in Eq. (2.16) becomes
$\displaystyle\mathcal{G}\rightarrow\mathcal{G}-4\mathring{F}^{\mu\nu}\mathring{F}_{\mu\nu}-4\Upsilon\left(\frac{1}{2}\bar{\kappa}^{\mu\nu}+\mathring{F}^{\mu\nu}\right)\mathring{F}_{\mu\nu}~{}.$
(C.7)
Under this transformation, the distortion tensor of the Weyl connection is
given by
$\displaystyle\tensor{\kappa}{{}_{\mu\nu}^{\alpha}}\rightarrow\delta_{\mu}^{\alpha}A_{\nu}+\delta_{\nu}^{\alpha}\left(A_{\mu}+B_{\mu}\right)-g_{\mu\nu}A^{\alpha}.$
(C.8)
If $B_{\mu}=-A_{\mu}$, we obtain the connection compatible with the metric but
with non-zero torsion which was the second of the three cases extracted from
Eq. (A.10) with $(c_{1},c_{2},c_{3})=(1,0,1)$. Furthermore, if the projective
transformation is applied to the LC connection, the resulting connection
corresponds to the third example, which had non-vanishing torsion and non-
metricity tensors, $(c_{1},c_{2},c_{3})=(0,1,0)$. For $B_{\mu}=-A_{\mu}$, we
have
$\displaystyle\mathring{F}_{\mu\nu}=-\frac{1}{4}\bar{\kappa}_{\mu\nu}~{},$
(C.9)
hence the $\mathcal{G}$ with torsion and vanishing non-metricity is
$\displaystyle\mathcal{G}=\mathcal{G}^{\textrm{Weyl}}-\frac{1-\Upsilon}{4}\bar{\kappa}^{\mu\nu}\bar{\kappa}_{\mu\nu}~{}.$
(C.10)
We see that $\mathcal{G}$ and $\mathcal{G}^{\textrm{Weyl}}$ only differ by the
term $\bar{\kappa}^{\mu\nu}\bar{\kappa}_{\mu\nu}$. Actually, we can see that
the new $\mathcal{G}$ does not depend on $\Upsilon$
$\displaystyle\mathcal{G}=\mathring{\mathcal{G}}+2\mathring{R}\kappa-8\mathring{R}^{\mu\nu}\tilde{\kappa}_{\mu\nu}+2\mathring{R}^{\alpha\beta\mu\nu}\bar{\kappa}_{\alpha\beta\mu\nu}+\kappa^{2}-4\tilde{\kappa}^{\mu\nu}\tilde{\kappa}_{\mu\nu}+\bar{\kappa}^{\alpha\beta\mu\nu}\bar{\kappa}_{\alpha\beta\mu\nu}-\frac{1}{4}\bar{\kappa}^{\mu\nu}\bar{\kappa}_{\mu\nu}~{}.$
(C.11)
So the calculations are the same as in the case of the Weyl connection if we
simply set $\Upsilon=1$ (see Eq. (2.16)).
If $A_{i}(t)=0$, the $\Upsilon$ term does not contribute to the homogeneous
and tensor perturbations equations as it multiplies $\bar{\kappa}_{\mu\nu}$
(which vanishes in that case; see Eq. (2.21)). Consequently, for a background
Weyl vector that has solely a non-zero temporal component (so it does not
break spatial isotropy), the results derived in this work are the same as
those obtained assuming the $(c_{1},c_{2},c_{3})=(1,0,1)$ connection with non-
zero torsion and metric compatible.
## References
* [1] A. G. Riess et al. [Supernova Search Team], _Observational evidence from supernovae for an accelerating universe and a cosmological constant_ , _Astron. J._ 116 1009-1038 (1998) [astro-ph/9805201].
* [2] S. Perlmutter et al. [Supernova Cosmology Project], _Measurements of $\Omega$ and $\Lambda$ from $42$ high redshift supernovae_, _Astrophys. J._ 517 565-586 (1999) [astro-ph/9812133].
* [3] N. Aghanim et al. [Planck Collaboration], _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 A6 (2020) [Erratum: Astron. Astrophys. 652, C4 (2021)] [1807.06209].
* [4] S. M. Carroll, _The Cosmological constant_ , _Living Rev. Rel._ 4, 1 (2001) [astro-ph/0004075].
* [5] J. Martin, _Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask)_ , _Comptes Rendus Physique_ 13, 566-665 (2012) [1205.3365].
* [6] A. Joyce, B. Jain, J. Khoury and M. Trodden, _Beyond the Cosmological Standard Model_ , _Phys. Rept._ 568, 1-98 (2015) [1407.0059].
* [7] S. Weinberg, _The Cosmological Constant Problem_ , _Rev. Mod. Phys._ 61, 1-23 (1989).
* [8] S. Weinberg, _The Cosmological constant problems_ , _Contribution to 4th International Symposium on Sources and Detection of Dark Matter in the Universe_ (DM 2000) (2000) [astro-ph/0005265].
* [9] R. R. Caldwell, R. Dave and P. J. Steinhardt, _Cosmological imprint of an energy component with general equation of state_ , _Phys. Rev. Lett._ 80, 1582-1585 (1998) [astro-ph/9708069].
* [10] I. Zlatev, Li-Min Wang and P. J. Steinhardt, _Quintessence, cosmic coincidence, and the cosmological constant_ , _Phys. Rev. Lett._ 82, 896-899 (1999) [astro-ph/9807002].
* [11] E. J. Copeland, M. Sami and S. Tsujikawa, _Dynamics of dark energy_ , _Int. J. Mod. Phys. D_ 15, 1753-1936 (2006) [hep-th/0603057].
* [12] S. Tsujikawa, _Dark energy: investigation and modeling_ , _ASSL_ Vol. 370, 331-402, Springer, Dordrecht (2011) [1004.1493].
* [13] S. Tsujikawa, _Quintessence: A Review_ , _Class. Quant. Grav._ 30, 214003 (2013) [1304.1961].
* [14] P. J. E. Peebles and A. Vilenkin, _Quintessential inflation_ , _Phys. Rev. D_ 59, 063505 (1999) [astro-ph/9810509].
* [15] K. Dimopoulos and J. W. F. Valle, _Modeling Quintessential Inflation_ , _Astropart. Phys._ 18 287-306 (2002) [astro-ph/0111417].
* [16] J. de Haro, _On the viability of quintessential inflation models from observational data_ , _Gen. Rel. Grav._ 49, 1, 6 (2017) [1602.07138].
* [17] K. Dimopoulos, _Jointly modelling Cosmic Inflation and Dark Energy_ , _J. Phys. Conf. Ser._ 2105, 1, 012001 (2021) [2106.14966].
* [18] A. A. Starobinsky, _A New Type of Isotropic Cosmological Models Without Singularity_ , _Phys. Lett. B_ 91, 99-102 (1980).
* [19] A. H. Guth, _The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems_ , _Phys. Rev. D_ 23 347-356 (1981).
* [20] A. D. Linde, _A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems_ , _Phys. Lett. B_ 108 389-393 (1982).
* [21] S. Capozziello, _Curvature quintessence_ , _Int. J. Mod. Phys. D_ 11, 483-492 (2002) [gr-qc/0201033].
* [22] S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, _Is cosmic speed - up due to new gravitational physics?_ , _Phys. Rev. D_ 70, 043528 (2004) [astro-ph/0306438].
* [23] T. Koivisto and D. F. Mota, _Cosmology and Astrophysical Constraints of Gauss-Bonnet Dark Energy_ , _Phys. Lett. B_ 644, 104-108 (2007) [astro-ph/0606078].
* [24] S. Nojiri, S. D. Odintsov and V. K. Oikonomou, _Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution_ , _Phys. Rept._ 692, 1-104 (2017) [1705.11098].
* [25] C. van de Bruck, K. Dimopoulos, C. Longden and C. Owen, _Gauss-Bonnet-coupled Quintessential Inflation_ , 1707.06839.
* [26] J. de Haro and L. A. Saló, _A Review of Quintessential Inflation_ , _Galaxies_ 9, 4, 73 (2021)[2108.11144].
* [27] B. P. Abbott et al. [LIGO Scientific, Virgo et al. Collaboration], _Observation of Gravitational Waves from a Binary Black Hole Merger_ , _Phys. Rev. Lett._ 116, 6, 061102 (2016) [1602.03837].
* [28] B. P. Abbott et al. [LIGO Scientific, Virgo et al. Collaboration], _GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence_ , _Phys. Rev. Lett._ 116, 24, 241103 (2016) [1606.04855].
* [29] B. P. Abbott et al. [LIGO Scientific, Virgo et al. Collaboration], _GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral_ , _Phys. Rev. Lett._ 119, 16, 161101 (2017) [1710.05832].
* [30] J. Sakstein and B. Jain, _Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories_ , _Phys. Rev. Lett._ 119, 25, 251303 (2017) [1710.05893].
* [31] J. M. Ezquiaga and M. Zumalacárregui, _Dark Energy After GW170817: Dead Ends and the Road Ahead_ , _Phys. Rev. Lett._ 119, 25, 251304 (2017) [1710.05901].
* [32] T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller and I. Sawicki, _Strong Constraints on Cosmological Gravity from GW170817 and GRB 170817A_ , _Phys. Rev. Lett._ 119, 25, 251301 (2017) [1710.06394].
* [33] S. Arai and A. Nishizawa, _Generalized framework for testing gravity with gravitational-wave propagation. II. Constraints on Horndeski theory_ , _Phys. Rev. D_ 97, 10, 104038 (2018) [1711.03776].
* [34] P. Creminelli and F. Vernizzi, _Dark Energy after GW170817 and GRB170817A_ , _Phys. Rev. Lett._ 119, 25, 251302 (2017) [1710.05877].
* [35] D. Langlois, R. Saito, D. Yamauchi and K. Noui, _Scalar-tensor theories and modified gravity in the wake of GW170817_ , _Phys. Rev. D_ 97, 6, 061501 (2018) [1711.07403].
* [36] B. P. Abbott et al. [LIGO Scientific, Virgo et al. Collaboration], _Multi-messenger Observations of a Binary Neutron Star Merger_ , _Astrophys. J. Lett._ 848, 2, L12 (2017) [1710.05833].
* [37] B. P. Abbott et al. [LIGO Scientific, Virgo et al. Collaboration], _Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A_ , _Astrophys. J. Lett._ 848, 2, L13 (2017) [1710.05834].
* [38] G. W. Horndeski, _Second-order scalar-tensor field equations in a four-dimensional space_ , _Int. J. Theor. Phys._ 10, 363-384 (1974).
* [39] C. de Rham and S. Melville, _Gravitational Rainbows: LIGO and Dark Energy at its Cutoff_ , _Phys. Rev. Lett._ 121, 22, 221101 (2018) [1806.09417].
* [40] T. Kobayashi, M. Yamaguchi and J. Yokoyama, _Generalized G-inflation: Inflation with the most general second-order field equations_ , _Prog. Theor. Phys._ 126, 511-529 (2011) [1105.5723].
* [41] P. G. S. Fernandes, P. Carrilho, T. Clifton and D. J. Mulryne, _The 4D Einstein-Gauss-Bonnet theory of gravity: a review_ , _Class. Quant. Grav._ 39, 6, 063001 (2022) [2202.13908].
* [42] S. D. Odintsov and V. K. Oikonomou, _Inflationary Phenomenology of Einstein Gauss-Bonnet Gravity Compatible with GW170817_ , _Phys. Lett. B_ 797, 134874 (2019) [1908.07555].
* [43] S. D. Odintsov, V. K. Oikonomou and F. P. Fronimos, _Rectifying Einstein-Gauss-Bonnet Inflation in View of GW170817_ , _Nucl. Phys. B_ 958, 115135 (2020) [2003.13724].
* [44] S. D. Odintsov and V. K. Oikonomou, _Swampland implications of GW170817-compatible Einstein-Gauss-Bonnet gravity_ , _Phys. Lett. B_ 805, 135437 (2020) [2004.00479].
* [45] J. Hjorth et al., _The Distance to NGC 4993: The Host Galaxy of the Gravitational-wave Event GW170817_ , _Astrophys. J. Lett._ 848, 2, L31 (2017) [1710.05856].
* [46] J. J. Terente Díaz, K. Dimopoulos, M. Karčiauskas and A. Racioppi, _Gauss-Bonnet Dark Energy and the speed of gravitational waves_ , _JCAP_ 10, 031 (2023) [2307.06163].
* [47] L. N. Granda and D. F. Jimenez, _The speed of gravitational waves and power-law solutions in a scalar-tensor model_ , _Astropart. Phys._ 103, 115-121 (2018) [1802.03781].
* [48] Y. Gong, E. Papantonopoulos and Z. Yi, _Constraints on scalar-tensor theory of gravity by the recent observational results on gravitational waves_ , _Eur. Phys. J. C_ 78, 9, 738 (2018) [1711.04102].
* [49] H. Mohseni Sadjadi, _Scalar-Gauss-Bonnet model, the coincidence problem and the gravitational wave speed_ , 2309.07816.
* [50] H. Mohseni Sadjadi, _Non-minimally coupled quintessence in the Gauss-Bonnet model, symmetry breaking, and cosmic acceleration_ , 2310.03697.
* [51] T. Helpin and M. S. Volkov, _Varying the Horndeski Lagrangian within the Palatini approach_ , _JCAP_ 01, 044 (2020) [1906.07607].
* [52] T. Helpin and M. S. Volkov, _A metric-affine version of the Horndeski theory_ , _Int. J. Mod. Phys. A_ 35, 02n03, 2040010 (2020) [1911.12768].
* [53] M. Kubota, Kin-Ya Oda, K. Shimada and M. Yamaguchi, _Cosmological Perturbations in Palatini Formalism_ , _JCAP_ 03, 006 (2021) [2010.07867].
* [54] Yu-Qi. Dong and Yu-Xiao Liu, _Polarization modes of gravitational waves in Palatini-Horndeski theory_ , _Phys. Rev. D_ 105, 6, 6 (2022) [2111.07352].
* [55] Yu-Qi Dong, Yu-Qiang Liu and Yu-Xiao Liu, _Constraining Palatini-Horndeski theory with gravitational waves after GW170817_ , _Eur. Phys. J. C_ 83, 8, 702 (2023) [2211.12056].
* [56] K. Dimopoulos and S. Sánchez López, _Quintessential inflation in Palatini $f(R)$ gravity_, _Phys. Rev. D_ 103, 4, 043533 (2021) [2012.06831].
* [57] K. Dimopoulos, A. Karam, S. Sánchez López and E. Tomberg, _Palatini $R^{2}$ quintessential inflation_, _JCAP._ 10, 076 (2022) [2206.14117].
* [58] M. Kumar Sharma and S. Sur, _Growth of matter perturbations in an interacting dark energy scenario emerging from metric-scalar-torsion couplings_ , 2102.01525.
* [59] J. B. Jimenez and T. S. Koivisto, _Extended Gauss-Bonnet gravities in Weyl geometry_ , _Class. Quant. Grav._ 31, 135002 (2014) [1402.1846].
* [60] E. J. Copeland, A. R. Liddle and D. Wands, _Exponential potentials and cosmological scaling solutions_ , _Phys. Rev. D_ 57, 4686-4690 (1998) [gr-qc/9711068].
* [61] P. G. Ferreira and M. Joyce, _Cosmology with a primordial scaling field_ , _Phys. Rev. D_ 58, 023503 (1998) [astro-ph/9711102].
* [62] T. Barreiro, E. J. Copeland and N. J. Nunes, _Quintessence arising from exponential potentials_ , _Phys. Rev. D_ 61, 127301 (2000) [astro-ph/9910214].
* [63] S. M. Carroll, _Lecture Notes On General Relativity_ , gr-qc/9712019.
* [64] S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. L. Said, J. Mifsud and E. Di Valentino, _Teleparallel gravity: from theory to cosmology_ , _Rept. Prog. Phys._ 86, 2, 026901 (2023) [2106.13793].
* [65] M. Borunda, B. Janssen and M. Bastero-Gil, _Palatini versus metric formulation in higher curvature gravity_ , _JCAP_ 11, 008 (2008) [0804.4440].
* [66] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, _Modified Gravity and Cosmology_ , _Phys. Rept._ 513, 1-189 (2012) [1106.2476].
* [67] D. Baumann, _Inflation_ , _Physics of the Large and the Small_ 523-686 (2011) [0907.5424].
* [68] T. Koivisto and D. F. Mota, _Gauss-Bonnet Quintessence: Background Evolution, Large Scale Structure and Cosmological Constraints_ , _Phys. Rev. D_ 75, 023518 (2006) [hep-th/0609155].
* [69] D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hirata, A. G. Riess and E. Rozo, _Observational Probes of Cosmic Acceleration_ , _Phys. Rept._ 530, 87-255 (2013) [1201.2434].
* [70] L. Jarv, P. Kuusk and M. Saal, _Remarks on (super-)accelerating cosmological models in general scalar-tensor theory_ , _Proc. Est. Acad. Sci. Phys. Math._ 59, 306-312 (2010) [0903.4357].
* [71] H. Mohseni Sadjadi, _On cosmic acceleration in four dimensional Einstein-Gauss-Bonnet gravity_ , _Phys. Dark Univ._ 30, 100728 (2020) [2005.10024].
* [72] B. P. Abbott et al. [LIGO Scientific, Virgo et al. Collaboration], _Properties of the binary neutron star merger GW170817_ , _Phys. Rev. X_ 9, 1, 011001 (2019) [1805.11579].
* [73] S. Nojiri, S. D. Odintsov and M. Sasaki, _Gauss-Bonnet dark energy_ , _Phys. Rev. D_ 71, 123509 (2005) [hep-th/0504052].
* [74] K. Dimopoulos, _Can a vector field be responsible for the curvature perturbation in the Universe?_ , _Phys. Rev. D_ 74, 083502 (2006) [hep-ph/0607229].
* [75] N. Dadhich and J. M. Pons, _On the equivalence of the Einstein-Hilbert and the Einstein-Palatini formulations of general relativity for an arbitrary connection_ , _Gen. Rel. Grav._ 44, 2337-2352 (2012) [1010.0869].
* [76] A. N. Bernal, B. Janssen, A. Jimenez-Cano, J. A. Orejuela, M. Sanchez and P. Sanchez-Moreno, _On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism_ , _Phys. Lett. B_ 768, 280-287 (2017) [1606.08756].
* [77] J. A. Schouten, _Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications_ , _Springer-Verlag_ , Berlin and Heidelberg (1954).
|
# A Full Resolution of the 450 $\micron$ Extragalactic Background Light
Qi-Ning Hsu Graduate Institute of Astronomy, National Tsing Hua University,
No. 101, Section 2, Kuang-Fu Road, Hsinchu 30013, Taiwan Academia Sinica
Institute of Astronomy and Astrophysics (ASIAA), No. 1, Section 4, Roosevelt
Rd., Taipei 10617, Taiwan L. L. Cowie Institute for Astronomy, University of
Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA Chian-Chou Chen Academia
Sinica Institute of Astronomy and Astrophysics (ASIAA), No. 1, Section 4,
Roosevelt Rd., Taipei 10617, Taiwan A. J. Barger Department of Astronomy,
University of Wisconsin-Madison, 475 N. Charter Street, Madison, WI 53706, USA
Department of Physics and Astronomy, University of Hawaii, 2505 Correa Road,
Honolulu, HI 96822, USA Institute for Astronomy, University of Hawaii, 2680
Woodlawn Drive, Honolulu, HI 96822, USA
###### Abstract
The extragalactic background light (EBL) is the cumulative radiation outside
the Milky Way. The determination of its corresponding primary emitting sources
as well as its total energy level across the entire electromagnetic spectrum
has profound implications for both cosmology and galaxy formation. However,
the detailed origin of the EBL at far-infrared wavelengths, particularly those
close to the peak of the cosmic infrared background, remains unclear. Here we
report the results of our ongoing SCUBA-2 450 $\micron$ survey of 10 massive
galaxy cluster fields. By exploiting the strong gravitational lensing offered
by these clusters, we obtain significant counts down to an unprecedented depth
of $\sim$0.1 mJy at this wavelength, about ten times deeper than that reached
by any other previous survey. The cumulative energy density based on the
counts is 138.1${}^{+23.9}_{-19.3}$ Jy degree-2, or 0.45${}^{+0.08}_{-0.06}$
MJy sr-1. Comparing our measurements to those made by the COBE and Planck
satellites, we find that at this flux density level, the 450 $\micron$ EBL is
entirely resolved by our SCUBA-2 observations. Thus, we find for the first
time that discrete sources produce fully to the 450 $\micron$ EBL, and that
about half of it comes from sources with sub-mJy flux densities. Our deep
number counts provide strong constraints on galaxy formation models.
galaxies: clusters — galaxies: number counts — EBL — submillimeter galaxies
††facilities: JCMT(SCUBA-2)††software: astropy (Astropy Collaboration et al.
2013, 2018), LENSTOOL (Kneib et al., 2011)
## 1 Introduction
The cosmic energy budget, encompassing the total amount of energy radiated
throughout the Universe, is a fundamental aspect of modern astrophysics.
Central to this budget is the extragalactic background light (EBL), which
represents the integrated emission across the electromagnetic spectrum from
astrophysical sources outside the Milky Way (see, e.g., the review by Cooray
2016). Observational measurements of the EBL at different wavelengths allow
one to understand its energy distribution, and, in principle, can provide
insights on the dominant contributors to the EBL (see, e.g., the discussion in
Hill et al. 2018).
One common way to put constraints on the EBL is to perform imaging surveys
using ground-based facilities, which allow one to construct source number
counts and calculate the integrated energy densities (e.g., the early results
in the far-infrared (FIR)/submillimeter by Smail et al. 1997; Barger et al.
1999; Cowie et al. 2002). By comparing with satellite measurements having
lower spatial resolution, which, in principle, take into account any diffuse
emission, one can estimate the energy contributions to the EBL from galaxies
and understand whether galaxies are the dominant contributors. Surveys have
found that the optical and near-infrared EBL primarily originate from the
directly observed star formation in galaxies, while the FIR/submillimeter EBL
mainly comes from the thermal emission from interstellar dust reradiated
starlight (Puget et al., 1996; Fixsen et al., 1998; Dole et al., 2006). These
dusty galaxies, sometimes called submillimeter galaxies (SMGs), are typically
characterized by intense dust emission indicating high rates of star formation
(see, e.g., review articles by Blain et al. 2002; Casey et al. 2014).
The advent of space-based telescopes like the Herschel Space Observatory and
ground-based facilities like the Atacama Large Millimeter/submillimeter Array
(ALMA) has revolutionized our ability to perform either wide-field or deep FIR
surveys, enabling unprecedented studies of galaxy number counts (Oliver et
al., 2010; Berta et al., 2011; Gómez-Guijarro et al., 2022; Cowie et al.,
2023; Chen et al., 2023). These observations, coupled with theoretical models,
provide a comprehensive view of the FIR Universe and its connection to other
astrophysical phenomena (e.g., Lacey et al. 2016; Lagos et al. 2019; Hayward
et al. 2021). However, due to the confusion limits of Herschel and the
inefficient survey capability of ALMA with its small field-of-view, the number
counts at wavelengths close to the peak of the FIR EBL ($\sim 200-300$ µm;
e.g., Odegard et al. 2019) remain limited to the brightest end (e.g., Oliver
et al. 2010).
SCUBA-2, a state-of-the-art submillimeter camera, offers unmatched sensitivity
and high angular resolution at 450 $\micron$. Its optimized design for deep
submillimeter observations makes it an excellent tool for studying dust-rich
galaxies (Holland et al., 2013). By harnessing the sensitivity of SCUBA-2 at
submillimeter wavelengths, we can detect and characterize faint sources at 450
$\micron$, enabling comprehensive studies of galaxy number counts close to the
peak of the FIR EBL and a deeper exploration of the submillimeter Universe
(Geach et al. 2013; Casey et al. 2013; Zavala et al. 2017; Wang et al. 2017;
Lim et al. 2020; Barger et al. 2022; Gao et al. 2024).
Additionally, the technique of gravitational lensing provides a unique means
for studying galaxy number counts at the fainter end (Smail et al., 1997;
Cowie et al., 2002; Knudsen et al., 2008; Johansson et al., 2011; Chen et al.,
2013; Hsu et al., 2016; Cowie et al., 2022). Gravitational lensing occurs when
the gravitational field of a massive object, such as a rich cluster of
galaxies, bends and magnifies the light from distant galaxies. By exploiting
the lensing effect, we can effectively boost the observed flux of background
galaxies, allowing us to probe deeper into the Universe and uncover fainter
sources that would otherwise remain undetected in blank-field surveys.
In this study, we utilize the deep data obtained with SCUBA-2 on 10 massive
galaxy cluster fields to derive robust number counts at 450 $\micron$. Our
analysis properly accounts for selection biases, completeness, and
uncertainties. By constructing the deepest 450 $\mu$m number counts ever, we
aim to unravel the relative contributions of SMGs to the 450 $\mu$m EBL. In
Section 2, we describe our SCUBA-2 data and data reduction. In Section 3, we
describe our methodology, including source extraction, Monte Carlo
simulations, and number counts calculations. In Section 4, we present our
number counts and the integrated energy density. We summarize our results in
Section 5.
## 2 Data and Data Reduction
We retrieved the SCUBA-2 450 $\mu$m data from the CADC archive. We used the
data taken under weather band 1 and band 2 conditions ($\tau_{225~{}\rm
GHz}~{}<$ 0.08) between October 2011 and July 2022 (Cowie et al., 2022). The
scan pattern used for these 10 cluster fields was CV DAISY, which has a
roughly circular field size of $\simeq 6^{\prime}$ in radius. We summarize the
data in Table 1.
We reduced the data following Chen et al. (2013). We used the Dynamic
Iterative Map Maker (DIMM) method in the SMURF package contained in the
STARLINK software (Chapin et al., 2013). This method models individual
components that make up the time-series recorded by the bolometer to produce
science maps. We adopted the “blank field” configuration file, which is
suitable for detecting faint point sources in extragalactic surveys. For each
cluster field, we produced scan maps with a pixel scale of $1^{\prime\prime}$
and then applied the recommended Flux Conversion Factors (FCFs) from Mairs et
al. (2021) to convert the pixel unit from picowatts to Jansky per beam. After
calibration, we used the MOSAIC_JCMT_IMAGES recipe from the Pipeline for
Combing and Analyzing Reduced Data (PICARD) to co-add and mosaic calibrated
scans for each field.
To improve source detection, we applied a matched filter using the
SCUBA2_MATCHED_FILTER recipe in PICARD. This recipe first convolves the map
with a Gaussian to estimate the low spatial frequency noise and then subtracts
it from the original map. We adopted the default $20^{\prime\prime}$ FWHM
value for the Gaussian profile. We verified the flux recovery capability of
SCUBA2_MATCHED_FILTER following Lim et al. (2020), and we adopted a mean
upward correction of $5.3\%$ for the flux loss to our 450 $\mu$m data.
Table 1: SUMMARY of SCUBA-2 450 $\micron$ DATA Field | RA | Dec | Exposure | Central RMS
---|---|---|---|---
| | | (hours) | (mJy)
A370 | 39.9604 | -1.5856 | 29.1 | 2.19
A1689 | 197.8729 | -1.3411 | 27.7 | 2.16
A2390 | 328.3979 | 17.6867 | 39.6 | 2.28
A2744 | 3.5788 | -30.3894 | 24.5 | 3.08
MACS J0416.1-2403 | 64.0349 | -24.0724 | 22.1 | 2.42
MACS J0717.5+3745 | 109.4020 | 37.7564 | 51.5 | 1.51
MACS J1149.5+2223 | 117.3962 | 22.4030 | 35.0 | 1.46
MACS J1423.8+2404 | 215.9486 | 24.0778 | 43.8 | 1.76
MACS J2129.40741 | 322.3592 | -7.6906 | 12.6 | 4.07
RX J1347.5-1145 | 206.8775 | -11.7528 | 20.0 | 2.27
Figure 1: Left: An example SCUBA-2 450 $\micron$ flux density map of the
cluster MACS J1149.5+2223 with a 6′ radius circular footprint. The red circles
on the 450 $\mu$m map show the extracted $>3\sigma$ sources. The orange boxes
mark the sources below 1 mJy after delensing. The white contours represent the
CATS lens model for $z=1.5$ at magnification values of 1.5, 2.0, and 2.5
(moving inwards). Right: Histogram of the SNR values based on the map of MACS
J1149.5+2223. The orange region shows the SNR distribution in the jackknife
map. The blue represents detections in the data map.
## 3 Analysis
### 3.1 Source Extraction
Before source extraction, for each field we generated SCUBA-2 PSF models by
stacking the 10-20 highest signal-to-noise ratio (SNR) source images without
neighboring sources. This method inherently assumes that the typical source
size is much smaller than the beam size of SCUBA-2 in the submillimeter ($\sim
7\farcs 5$ at 450 $\micron$), which is supported by recent ALMA observations
that found typical sizes of subarcseconds (e.g., Simpson et al. 2015; Hodge et
al. 2016; Fujimoto et al. 2017; Gullberg et al. 2018; Tadaki et al. 2020). We
then fitted a double Gaussian profile to model the PSFs, and we used the best-
fit model PSFs for source extraction.
We performed the source extractions as in Hsu et al. (2016). We searched for
the maximum SNR pixel in the central circular region ($6^{\prime}$ radius).
(The pointing centers of the maps are normally close to the cluster
centroids.) We recorded the location and flux density of the pixel, then
subtracted a rescaled PSF centered at this pixel and searched for the next
maximum SNR pixel. Following Hsu et al. (2016), we used a 3$\sigma$ threshold,
which allows us to obtain a better SNR in the number counts at the faint end.
We repeated the extraction process until we reached this threshold. In Figure
1, we show one of our cluster fields as an example for the source extraction.
### 3.2 Delensed Raw Number Counts
We calculated delensed differential number counts at 450 $\micron$ by using
sources with SNR $>3.0$. To compute the demagnified flux densities, we used
the public software LENSTOOL (Kneib et al., 2011) to generate magnification
maps for our cluster fields. Since we do not have redshifts for these sources,
we corrected the flux densities of the sources by adopting estimated median
redshifts of 1.5 based on previous 450 $\micron$ studies (e.g., Chen et al.
2013; Casey et al. 2013). We took our lens models from the LENSTOOL developers
and Hubble Frontier Fields Archive (Ammons et al. 2014; Bradač et al. 2009;
Caminha et al. 2017; Diego et al. 2015; Grillo et al. 2015; Hoag et al. 2016;
Ishigaki et al. 2015; Jullo et al. 2007; Jullo & Kneib 2009; Jauzac et al.
2012, 2014; Johnson et al. 2014; Keeton 2010; Kawamata et al. 2018;
Liesenborgs et al. 2006; Lotz et al. 2017; Merten et al. 2011; McCully et al.
2014; Mohammed et al. 2014; Oguri 2010; Richard et al. 2014; Zitrin et al.
2013).
We determined the magnification factors $\mu_{\rm i}$ of the point sources
from the magnification maps. The demagnified flux density of each source can
be obtained from
$S_{\rm demag,\emph{i}}=\frac{S_{\rm obs,\emph{i}}}{\mu_{i}}\,,$ (1)
where $S_{\rm demag,\emph{i}}$ and $S_{\rm obs,\emph{i}}$ are demagnified and
observed flux densities, respectively. We calculated the effective area
A${}_{\rm eff,\emph{i}}$ of each source on the source plane. We summed over
the pixels whose SNRs were greater than 3$\sigma$ and then converted pixels to
square degrees. We then calculated the delensed raw counts at the $j$–th flux
bin as
$\frac{dN_{\rm raw,j}}{dS_{j}}=\frac{1}{\Delta S_{j}}\sum_{i}^{n}X_{i}$ (2)
and
$X_{i}=\frac{1}{A_{\rm eff,\emph{i}}}\,.$ (3)
Here, Xi represents the number density contribution of each source within that
flux bin. We based the error calculation on Poisson statistics.
### 3.3 Simulations
Corrections for flux boosting, false detections, and incompleteness are needed
in order to obtain the intrinsic number counts. To do this, we ran Monte Carlo
simulations to find the underlying models for our fields. We used the
Schechter function form as our number counts model:
$\rm\frac{dN}{dS}=\Bigl{(}\frac{N_{0}}{S_{0}}\Bigr{)}~{}\Bigl{(}\frac{S}{S_{0}}\Bigr{)}^{\alpha}~{}exp\Bigl{(}-\frac{S}{S_{0}}\Bigr{)}\,.$
(4)
We generated artificial sources whose flux densities were assigned according
to the underlying models for each cluster field. We then randomly distributed
these sources in the source plane. Next, we projected the simulated sources
onto the image plane using LENSTOOL. The outputs of LENSTOOL contain the new
flux densities and positions of the simulated sources in the image plane. We
convolved the simulated sources with the PSF and added them into the jackknife
maps to produce mock observation maps. We produced jackknife maps following
Hsu et al. (2016) by coadding even and odd scans separately. We then
subtracted these two coadded maps and rescaled the value of each pixel by a
factor of $\sqrt{t_{\rm even}}\times\sqrt{t_{\rm odd}}$/(teven \+ todd), where
teven and todd represent the integration times of each pixel from the even and
odd coadded maps, respectively. We also applied matched-filtering to the
jackknife maps, as we did for the real data images.
In order to estimate the true number counts, we adopted an iterative procedure
in our simulations. We generated 15–20 mock maps in each iteration step. We
then performed source extraction and computed the averaged recovered counts of
the mock maps. Next, we corrected the raw counts by using the ratio between
the averaged recovered counts and the raw counts. Finally, we did a $\chi^{2}$
fit to the corrected counts by using the Schechter function. This fit will be
the new counts model for the next iteration. We iterated until the recovered
counts converged with the raw counts to within the 1$\sigma$ uncertainties for
all the flux bins.
Once we obtained the intrinsic number counts models, we used these to produce
500 mock images and then performed the source extractions to create the source
catalogs. Following Gao et al. (2024), we cross-matched the input and output
catalogs within a 1/2 beam FWHM as our search area to find the brightest
counterparts. We considered Soutput/S${}_{\rm input}\leq 3$ a match in this
analysis, which is similar to what has been adopted previously (e.g., Geach et
al. 2017). Following Gao et al. (2024), we estimated the boosting factors,
false detection rates, and completeness of the point sources using a
two–dimensional binning method in our cross-matched catalogs.
### 3.4 Delensed Corrected Number Counts
To derive delensed corrected number counts, we first deboosted the fluxes of
the sources. We then delensed their fluxes and estimated their effective areas
in the source plane. Finally, we derived the delensed corrected number counts
using
$\frac{dN_{\rm corr,j}}{dS_{j}}=\frac{1}{\Delta S_{j}}\sum_{i}^{n}X_{\rm
corr,\emph{i}}$ (5)
and
$X_{\rm corr,\emph{i}}=\frac{1-p_{\rm false,\emph{i}}}{C_{i}A_{\rm
eff,\emph{i}}}\,,$ (6)
where p${}_{\rm false,\emph{i}}$ is the false detection rate, and Ci is the
completeness. We confirmed that the delensed corrected number counts agree
with the intrinsic counts models. We estimated the statistical uncertainties
from the source positions and took these uncertainties into account in the
analyses.
### 3.5 Systematic Uncertainties
Systematic uncertainties related to the modeling of gravitational lensing need
to be taken into account for a proper assessment of the number counts error
budget. In the following, we give our results with estimates of the systematic
uncertainties caused by the redshift distribution of the background lensed
SMGs, the lens models, and the clustering of the source distributions.
In our methodology, we calculated corrected number counts by assuming a median
redshift of 1.5. However, it is expected that the SMGs have a redshift
distribution, which could affect the magnification estimates. This leads to
additional uncertainties in the number counts. To address this, we randomly
assigned redshifts to our sources using the redshift distribution from the
STUDIES survey, which is the deepest 450 $\micron$ blank-field survey (Wang et
al. 2017; Dudzevičiūtė et al. 2021; Gao et al. 2024). We calculated the
corrected number counts again and compared the standard deviations of the
number counts in the randomized redshift sample with the Poisson noise. We
found that the uncertainties caused by the assumptions concerning the redshift
are subdominant, and they are, on average, about 25% of the Poisson
uncertainties. Nevertheless, we include this error budget in the total error
budget.
Table 2: Combined Differential Number Counts at 450 $\micron$
S${}_{450~{}\micron}$ | log10(dN/dS) | $<\mu>^{1}$ | N${}_{\rm total}^{2}$
---|---|---|---
(mJy) | (mJy-1 deg-2) | |
0.13 | 6.2${}_{-1.4}^{+0.4}$ | 34.4 | 3
0.25 | 5.6${}_{-99.9}^{+0.6}$ | 22.4 | 5
0.45 | 5.1${}_{-0.7}^{+0.3}$ | 14.9 | 12
0.81 | 4.7${}_{-99.9}^{+0.5}$ | 7.4 | 3
1.48 | 4.2${}_{-0.5}^{+0.3}$ | 4.1 | 26
2.70 | 3.5${}_{-0.2}^{+0.2}$ | 2.1 | 108
4.91 | 3.0${}_{-0.1}^{+0.1}$ | 1.6 | 232
8.94 | 2.4${}_{-0.1}^{+0.1}$ | 1.2 | 358
16.28 | 1.5${}_{-0.2}^{+0.2}$ | 1.1 | 228
29.65 | 0.6${}_{-0.8}^{+0.4}$ | 1.2 | 23
11footnotetext: Averaged gravitational magnification
22footnotetext: Total number of sources in each flux bin
To estimate the systematic uncertainties caused by the lens modeling, we
utilized the various lens models provided for the five Hubble Frontier Fields
(A370, A2744, MACS J0416.1-2403, MACS J0717.5+3745, and MACS J1149.5+2223). We
ran the same Monte Carlo simulations as those done on the real data but using
different lens models. We then estimated the systematic uncertainties by
calculating the standard deviations of the corrected counts obtained using the
different lens models. We found the systematic uncertainties to be
subdominant, again about 25% of the Poisson uncertainties. We include this
error budget in the total error budget.
We note that in our Monte Carlo simulations, we randomly distributed the
positions of the artificial sources. This might bias our counts, since we do
not take clustering effects into account when we calculate the boosting
factors, false detection rates, and completeness from the mock maps. To test
whether neglecting clustering could significantly alter our counts results, we
used the empirical catalogs produced by the SIDES simulation (Béthermin et
al., 2017), which inherently include clustering effects, since the simulation
builds upon dark matter lightcones.
Before assessing the clustering effects, we first validated our methodology
for estimating intrinsic counts by performing the following test. We clipped
the original SIDES 2 deg2 simulated map into a set of 25 smaller cutout maps
with a size similar to our SCUBA-2 footprint, and we treated them as different
cluster fields in the source planes. We then lensed these simulated maps to
the image planes using LENSTOOL and used our methodology to find the corrected
counts. We compared the corrected counts to the true counts provided by SIDES.
In this test, we adopted the three-dimensional source positions from the SIDES
catalog instead of having the sources randomly distributed on the sky with an
assigned redshift. We show our results in Figure 2, where the averaged
corrected number counts over the 25 cutout maps are consistent with the true
number counts.
After validating our methodology, we moved on with a similar test. This time
we randomly distributed the source positions to calculate the corrected number
counts using the smaller SIDES cutout maps. We show our results in Figure 2.
While not taking clustering into account can lead to larger uncertainties,
evidently with a larger scatter in the averaged counts, on average, there is
no significant difference between the SIDES counts and the corrected counts.
Our results are therefore consistent with previous studies—either from other
SCUBA-2 450 $\micron$ surveys (Wang et al., 2017), or from Herschel studies
with similar beam sizes but at slightly shorter wavelengths (Béthermin et al.,
2015)—where no significant impact of clustering on the number counts was
found.
Figure 2: Ratios between the corrected number counts and the model counts,
where the model counts are based on the SIDES simulation source catalog, and
the corrected counts were obtained by applying our methodology to the mock
images that were made based on the SIDES source catalog. The top panel shows
the results for the case where the source positions are adopted from SIDES,
which includes clustering. The bottom panel shows the results for the case
where the sky positions of the sources are randomized.
## 4 Results and Discussion
We present our corrected differential number counts for the 10 lensing cluster
fields in the left panel of Figure 3. The solid black curve is the best-fit
Schechter function for our corrected counts, which can be parameterized by
Equation 4 with the following parameters: N0 = 4437.5$\pm$1399.5, S0 =
10.4$\pm$1.7, and $\alpha$=-1.9$\pm$0.1. The uncertainties on each data point
include Poisson noise and the uncertainties from the redshift distribution and
the lens model. Thanks to the powerful effects of strong gravitational
lensing, we have pushed the detection limit down to $\sim$0.1 mJy at 450
$\micron$, a factor of $>$10 improvement over the deepest blank-field counts
(Wang et al. 2017; Gao et al. 2024).
We also calculate the weighted average counts by combining all 10 fields. We
show the results in the right panel of Figure 3, and we provide the
corresponding values in Table 2. Our results are consistent with the previous
measurements shown in Figure 3. However, the various physical or empirical
models (dotted and dashed curves in the right panel) tend to overpredict
source densities at the bright end at 450 $\micron$ ($\gtrsim$1 mJy) by about
10-30%. On the other hand, at the faint end ($\lesssim$1 mJy), our counts are
slightly higher, although not significantly. The physical reasons for the
disagreement between the measurements and the models at the bright end need to
be investigated. Gao et al. (2024) proposed that it could be due to the
mismatch in halo masses between those inferred from clustering measurements
(Lim et al., 2020) and those in the models. Studies of the physical properties
of the 450 $\micron$ sources, such as stellar mass, could potentially shed
more light on this issue.
Figure 3: Left: Corrected differential number counts for all 10 cluster
fields at 450 $\mu$m. The solid black curve shows the best-fit Schechter
function, while the shaded gray region is the 68% confidence interval. The
uncertainties include Poisson noise, as well as those caused by the underlying
redshift distributions and lens models. Right: Weighted average counts of all
10 cluster fields in filled black circles, along with previous 450 $\micron$
surveys in open symbols (Chen et al. 2013; Hsu et al. 2016; Zavala et al.
2017; Wang et al. 2017; Gao et al. 2024). Model predictions from SIDES
(Béthermin et al., 2017), GALFORM (Cowley et al., 2019), and SHARK (Lagos et
al., 2020) are shown as dotted and dashed curves.
To estimate the contributions of the detected sources to the 450 $\micron$
EBL, we integrated the best-fit Schechter model, which we show as a function
of flux in Figure 4. Above 0.1 mJy where significant counts are obtained, we
find the total energy density at 450 $\mu$m to be 138.1${}^{+23.9}_{-19.3}$ Jy
deg-2, or 0.45${}^{+0.08}_{-0.06}$ MJy sr-1. This corresponds to
103${}^{+18}_{-14}\%$ of the total 450 $\mu$m EBL reported by Odegard et al.
(2019).
Past works have suggested that a broken power law could be a viable
alternative to the Schechter function for the underlying counts models. It can
be described as
$\frac{dN}{dS}=\begin{cases}N_{0}(\frac{S}{S_{0}})^{-\alpha},&if\ S\leq
S_{0}\\\ N_{0}(\frac{S}{S_{0}})^{-\beta},&if\ S>S_{0}\end{cases}$ (7)
We ran further counts analyses based on this model form and found consistent
results compared to those obtained based on Schechter functions. The best-fit
parameters of the broken power law are N0=151.0$\pm$14.6, S0=9.6$\pm$0.4,
$\alpha$=2.0$\pm$0.1, $\beta$=5.6$\pm$1.1. We plot the cumulative energy
density of the best-fit broken power law in Figure 4. The cumulative energy
density based on the broken power law is 126.9${}^{+41.6}_{-41.6}$ Jy deg2.
Our work demonstrates for the first time that discrete sources, are the
dominant contributors to the 450 $\micron$ EBL. Interestingly, about half of
the contribution comes from sources that are fainter than $\sim$1 mJy, below
the typical confusion limit of SCUBA-2 450 $\micron$ images (Gao et al.,
2024).
Noticeably, as shown in Figure 4, the integrated energy density does not
converge when integrating down to 0.14 mJy, suggesting that deeper data are
needed in order to put a tighter constraint on the faint end counts and thus
obtain a converged constraint on the EBL contributions from discrete sources.
Figure 4: Cumulative EBL as a function of flux density at 450 $\mu$m. The
black solid curve was calculated based on our best-fit Schechter function with
the 1$\sigma$ uncertainties in gray shading. The blue solid curve was
calculated based on a broken power law function. The black dotted curve was
calculated by combining our results with the deepest blank-field counts from
Gao et al. (2024). The horizontal red dashed line is the EBL measured by the
COBE$+$Planck satellites (Odegard et al., 2019). The red shaded region shows
the range from COBE estimates (e.g., Puget et al. 1996; Fixsen et al. 1998;
Gispert et al. 2000).
## 5 Summary
In summary, our research represents a significant step forward in the study of
the 450 $\micron$ EBL. Through the innovative combination of gravitational
lensing and the unparalled capability of SCUBA-2, we have achieved a complete
resolution of the 450 $\micron$ EBL and established the dominance of discrete
SMGs as its main contributors. These findings provide a broader understanding
of the FIR/submillimeter regime, the cosmic energy distribution, and the
interplay between galaxies and the diffuse background radiation. Our
measurements could also be helpful for the design of the next generation
submillimeter facilities, such as LST (Kohno et al., 2020) and AtLAST
(Klaassen et al., 2020), which aim to obtain wide blank-field images to a
depth that is similar to what has been reached by this work.
We thank the reviewer for a useful report that has improved the manuscript.
Q.-N.H. and C.-C.C acknowledge support from the National Science and
Technology Council of Taiwan (NSTC 109-2112-M-001-016-MY3 and
111-2112M-001-045-MY3), as well as Academia Sinica through the Career
Development Award (AS-CDA-112-M02). L.L.C acknowledges support from NASA grant
80NSSC22K0483. A.J.B. acknowledges support from a Kellett Mid-Career Award and
a WARF Named Professorship from the University of Wisconsin-Madison Office of
the Vice Chancellor for Research and Graduate Education with funding from the
Wisconsin Alumni Research Foundation. The James Clerk Maxwell Telescope is
operated by the East Asian Observatory on behalf of The National Astronomical
Observatory of Japan; Academia Sinica Institute of Astronomy and Astrophysics;
the Korea Astronomy and Space Science Institute; the National Astronomical
Research Institute of Thailand; Center for Astronomical Mega-Science (as well
as the National Key R&D Program of China with No. 2017YFA0402700). Additional
funding support is provided by the Science and Technology Facilities Council
of the United Kingdom and participating universities and organizations in the
United Kingdom and Canada. Additional funds for the construction of SCUBA-2
were provided by the Canada Foundation for Innovation. The authors wish to
recognize and acknowledge the very significant cultural role and reverence
that the summit of Maunakea has always had within the indigenous Hawaiian
community. We are most fortunate to have the opportunity to conduct
observations from this mountain. This work utilizes gravitational lensing
models produced by PIs Bradač, Natarajan & Kneib (CATS), Merten & Zitrin,
Sharon, Williams, Keeton, Bernstein and Diego, and the GLAFIC group. This lens
modeling was partially funded by the HST Frontier Fields program conducted by
STScI. STScI is operated by the Association of Universities for Research in
Astronomy, Inc. under NASA contract NAS 5-26555. The lens models were obtained
from the Mikulski Archive for Space Telescopes (MAST).
## Appendix A Appendix A
In Figure 5 we show the imaging data for the remaining nine fields. The
symbols follow those used in Figure 1.
Figure 5: Flux density maps for the remaining nine fields. The format follows
that adopted in the left panel of Figure 1.
## References
* Ammons et al. (2014) Ammons, S. M., Wong, K. C., Zabludoff, A. I., & Keeton, C. R. 2014, ApJ, 781, 2, doi: 10.1088/0004-637X/781/1/2
* Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
* Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
* Barger et al. (2022) Barger, A. J., Cowie, L. L., Blair, A. H., & Jones, L. H. 2022, ApJ, 934, 56, doi: 10.3847/1538-4357/ac67e7
* Barger et al. (1999) Barger, A. J., Cowie, L. L., & Richards, E. A. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 191, Photometric Redshifts and the Detection of High Redshift Galaxies, ed. R. Weymann, L. Storrie-Lombardi, M. Sawicki, & R. Brunner, 279, doi: 10.48550/arXiv.astro-ph/9907022
* Berta et al. (2011) Berta, S., Magnelli, B., Nordon, R., et al. 2011, A&A, 532, A49, doi: 10.1051/0004-6361/201116844
* Béthermin et al. (2015) Béthermin, M., Daddi, E., Magdis, G., et al. 2015, A&A, 573, A113, doi: 10.1051/0004-6361/201425031
* Béthermin et al. (2017) Béthermin, M., Wu, H.-Y., Lagache, G., et al. 2017, A&A, 607, A89, doi: 10.1051/0004-6361/201730866
* Blain et al. (2002) Blain, A. W., Smail, I., Ivison, R. J., Kneib, J. P., & Frayer, D. T. 2002, Phys. Rep., 369, 111, doi: 10.1016/S0370-1573(02)00134-5
* Bradač et al. (2009) Bradač, M., Treu, T., Applegate, D., et al. 2009, ApJ, 706, 1201, doi: 10.1088/0004-637X/706/2/1201
* Caminha et al. (2017) Caminha, G. B., Grillo, C., Rosati, P., et al. 2017, A&A, 600, A90, doi: 10.1051/0004-6361/201629297
* Casey et al. (2014) Casey, C. M., Narayanan, D., & Cooray, A. 2014, Phys. Rep., 541, 45, doi: 10.1016/j.physrep.2014.02.009
* Casey et al. (2013) Casey, C. M., Chen, C.-C., Cowie, L. L., et al. 2013, MNRAS, 436, 1919, doi: 10.1093/mnras/stt1673
* Chapin et al. (2013) Chapin, E. L., Berry, D. S., Gibb, A. G., et al. 2013, MNRAS, 430, 2545, doi: 10.1093/mnras/stt052
* Chen et al. (2013) Chen, C.-C., Cowie, L. L., Barger, A. J., et al. 2013, ApJ, 776, 131, doi: 10.1088/0004-637X/776/2/131
* Chen et al. (2023) Chen, J., Ivison, R. J., Zwaan, M. A., et al. 2023, MNRAS, 518, 1378, doi: 10.1093/mnras/stac2989
* Cooray (2016) Cooray, A. 2016, Royal Society Open Science, 3, 150555, doi: 10.1098/rsos.150555
* Cowie et al. (2023) Cowie, L. L., Barger, A. J., & Bauer, F. E. 2023, ApJ, 952, 28, doi: 10.3847/1538-4357/acd763
* Cowie et al. (2022) Cowie, L. L., Barger, A. J., Bauer, F. E., et al. 2022, ApJ, 939, 5, doi: 10.3847/1538-4357/ac91d2
* Cowie et al. (2002) Cowie, L. L., Barger, A. J., & Kneib, J. P. 2002, AJ, 123, 2197, doi: 10.1086/339978
* Cowley et al. (2019) Cowley, W. I., Lacey, C. G., Baugh, C. M., et al. 2019, MNRAS, 487, 3082, doi: 10.1093/mnras/stz1398
* Diego et al. (2015) Diego, J. M., Broadhurst, T., Benitez, N., et al. 2015, MNRAS, 446, 683, doi: 10.1093/mnras/stu2064
* Dole et al. (2006) Dole, H., Lagache, G., Puget, J. L., et al. 2006, A&A, 451, 417, doi: 10.1051/0004-6361:20054446
* Dudzevičiūtė et al. (2021) Dudzevičiūtė, U., Smail, I., Swinbank, A. M., et al. 2021, MNRAS, 500, 942, doi: 10.1093/mnras/staa3285
* Fixsen et al. (1998) Fixsen, D. J., Dwek, E., Mather, J. C., Bennett, C. L., & Shafer, R. A. 1998, ApJ, 508, 123, doi: 10.1086/306383
* Fujimoto et al. (2017) Fujimoto, S., Ouchi, M., Shibuya, T., & Nagai, H. 2017, ApJ, 850, 83, doi: 10.3847/1538-4357/aa93e6
* Gao et al. (2024) Gao, Z.-K., Lim, C.-F., Wang, W.-H., et al. 2024, ApJ, submitted
* Geach et al. (2013) Geach, J. E., Chapin, E. L., Coppin, K. E. K., et al. 2013, MNRAS, 432, 53, doi: 10.1093/mnras/stt352
* Geach et al. (2017) Geach, J. E., Dunlop, J. S., Halpern, M., et al. 2017, MNRAS, 465, 1789, doi: 10.1093/mnras/stw2721
* Gispert et al. (2000) Gispert, R., Lagache, G., & Puget, J. L. 2000, A&A, 360, 1, doi: 10.48550/arXiv.astro-ph/0005554
* Gómez-Guijarro et al. (2022) Gómez-Guijarro, C., Elbaz, D., Xiao, M., et al. 2022, A&A, 658, A43, doi: 10.1051/0004-6361/202141615
* Grillo et al. (2015) Grillo, C., Suyu, S. H., Rosati, P., et al. 2015, ApJ, 800, 38, doi: 10.1088/0004-637X/800/1/38
* Gullberg et al. (2018) Gullberg, B., Swinbank, A. M., Smail, I., et al. 2018, ApJ, 859, 12, doi: 10.3847/1538-4357/aabe8c
* Hayward et al. (2021) Hayward, C. C., Sparre, M., Chapman, S. C., et al. 2021, MNRAS, 502, 2922, doi: 10.1093/mnras/stab246
* Hill et al. (2018) Hill, R., Masui, K. W., & Scott, D. 2018, Applied Spectroscopy, 72, 663, doi: 10.1177/0003702818767133
* Hoag et al. (2016) Hoag, A., Huang, K. H., Treu, T., et al. 2016, ApJ, 831, 182, doi: 10.3847/0004-637X/831/2/182
* Hodge et al. (2016) Hodge, J. A., Swinbank, A. M., Simpson, J. M., et al. 2016, ApJ, 833, 103, doi: 10.3847/1538-4357/833/1/103
* Holland et al. (2013) Holland, W. S., Bintley, D., Chapin, E. L., et al. 2013, MNRAS, 430, 2513, doi: 10.1093/mnras/sts612
* Hsu et al. (2016) Hsu, L.-Y., Cowie, L. L., Chen, C.-C., Barger, A. J., & Wang, W.-H. 2016, ApJ, 829, 25, doi: 10.3847/0004-637X/829/1/25
* Ishigaki et al. (2015) Ishigaki, M., Kawamata, R., Ouchi, M., et al. 2015, ApJ, 799, 12, doi: 10.1088/0004-637X/799/1/12
* Jauzac et al. (2012) Jauzac, M., Jullo, E., Kneib, J.-P., et al. 2012, MNRAS, 426, 3369, doi: 10.1111/j.1365-2966.2012.21966.x
* Jauzac et al. (2014) Jauzac, M., Clément, B., Limousin, M., et al. 2014, MNRAS, 443, 1549, doi: 10.1093/mnras/stu1355
* Johansson et al. (2011) Johansson, D., Sigurdarson, H., & Horellou, C. 2011, A&A, 527, A117, doi: 10.1051/0004-6361/201016138
* Johnson et al. (2014) Johnson, T. L., Sharon, K., Bayliss, M. B., et al. 2014, ApJ, 797, 48, doi: 10.1088/0004-637X/797/1/48
* Jullo & Kneib (2009) Jullo, E., & Kneib, J. P. 2009, MNRAS, 395, 1319, doi: 10.1111/j.1365-2966.2009.14654.x
* Jullo et al. (2007) Jullo, E., Kneib, J. P., Limousin, M., et al. 2007, New Journal of Physics, 9, 447, doi: 10.1088/1367-2630/9/12/447
* Kawamata et al. (2018) Kawamata, R., Ishigaki, M., Shimasaku, K., et al. 2018, ApJ, 855, 4, doi: 10.3847/1538-4357/aaa6cf
* Keeton (2010) Keeton, C. R. 2010, General Relativity and Gravitation, 42, 2151, doi: 10.1007/s10714-010-1041-1
* Klaassen et al. (2020) Klaassen, P. D., Mroczkowski, T. K., Cicone, C., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11445, Ground-based and Airborne Telescopes VIII, ed. H. K. Marshall, J. Spyromilio, & T. Usuda, 114452F, doi: 10.1117/12.2561315
* Kneib et al. (2011) Kneib, J.-P., Bonnet, H., Golse, G., et al. 2011, LENSTOOL: A Gravitational Lensing Software for Modeling Mass Distribution of Galaxies and Clusters (strong and weak regime), Astrophysics Source Code Library, record ascl:1102.004. http://ascl.net/1102.004
* Knudsen et al. (2008) Knudsen, K. K., van der Werf, P. P., & Kneib, J. P. 2008, MNRAS, 384, 1611, doi: 10.1111/j.1365-2966.2007.12820.x
* Kohno et al. (2020) Kohno, K., Kawabe, R., Tamura, Y., et al. 2020, in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy X, ed. J. Zmuidzinas & J.-R. Gao (SPIE), doi: 10.1117/12.2561238
* Lacey et al. (2016) Lacey, C. G., Baugh, C. M., Frenk, C. S., et al. 2016, MNRAS, 462, 3854, doi: 10.1093/mnras/stw1888
* Lagos et al. (2020) Lagos, C. d. P., da Cunha, E., Robotham, A. S. G., et al. 2020, MNRAS, 499, 1948, doi: 10.1093/mnras/staa2861
* Lagos et al. (2019) Lagos, C. d. P., Robotham, A. S. G., Trayford, J. W., et al. 2019, MNRAS, 489, 4196, doi: 10.1093/mnras/stz2427
* Liesenborgs et al. (2006) Liesenborgs, J., De Rijcke, S., & Dejonghe, H. 2006, MNRAS, 367, 1209, doi: 10.1111/j.1365-2966.2006.10040.x
* Lim et al. (2020) Lim, C.-F., Wang, W.-H., Smail, I., et al. 2020, ApJ, 889, 80, doi: 10.3847/1538-4357/ab607f
* Lotz et al. (2017) Lotz, J. M., Koekemoer, A., Coe, D., et al. 2017, ApJ, 837, 97, doi: 10.3847/1538-4357/837/1/97
* Mairs et al. (2021) Mairs, S., Dempsey, J. T., Bell, G. S., et al. 2021, AJ, 162, 191, doi: 10.3847/1538-3881/ac18bf
* McCully et al. (2014) McCully, C., Keeton, C. R., Wong, K. C., & Zabludoff, A. I. 2014, MNRAS, 443, 3631, doi: 10.1093/mnras/stu1316
* Merten et al. (2011) Merten, J., Coe, D., Dupke, R., et al. 2011, MNRAS, 417, 333, doi: 10.1111/j.1365-2966.2011.19266.x
* Mohammed et al. (2014) Mohammed, I., Liesenborgs, J., Saha, P., & Williams, L. L. R. 2014, MNRAS, 439, 2651, doi: 10.1093/mnras/stu124
* Odegard et al. (2019) Odegard, N., Weiland, J. L., Fixsen, D. J., et al. 2019, ApJ, 877, 40, doi: 10.3847/1538-4357/ab14e8
* Oguri (2010) Oguri, M. 2010, PASJ, 62, 1017, doi: 10.1093/pasj/62.4.1017
* Oliver et al. (2010) Oliver, S. J., Wang, L., Smith, A. J., et al. 2010, A&A, 518, L21, doi: 10.1051/0004-6361/201014697
* Puget et al. (1996) Puget, J. L., Abergel, A., Bernard, J. P., et al. 1996, A&A, 308, L5
* Richard et al. (2014) Richard, J., Jauzac, M., Limousin, M., et al. 2014, MNRAS, 444, 268, doi: 10.1093/mnras/stu1395
* Simpson et al. (2015) Simpson, J. M., Smail, I., Swinbank, A. M., et al. 2015, ApJ, 807, 128, doi: 10.1088/0004-637X/807/2/128
* Smail et al. (1997) Smail, I., Ivison, R. J., & Blain, A. W. 1997, ApJ, 490, L5, doi: 10.1086/311017
* Tadaki et al. (2020) Tadaki, K.-i., Belli, S., Burkert, A., et al. 2020, ApJ, 901, 74, doi: 10.3847/1538-4357/abaf4a
* Wang et al. (2017) Wang, W.-H., Lin, W.-C., Lim, C.-F., et al. 2017, ApJ, 850, 37, doi: 10.3847/1538-4357/aa911b
* Zavala et al. (2017) Zavala, J. A., Aretxaga, I., Geach, J. E., et al. 2017, MNRAS, 464, 3369, doi: 10.1093/mnras/stw2630
* Zitrin et al. (2013) Zitrin, A., Meneghetti, M., Umetsu, K., et al. 2013, ApJ, 762, L30, doi: 10.1088/2041-8205/762/2/L30
|
$\displaystyle\inf_{x\in K}\mathbb{P}_{x}(n<\tau_{\partial})\geq\inf_{x\in
K}\mathbb{P}_{x}(n+k_{0}<\tau_{\partial})\geq
c_{1}c\,\mathbb{P}_{\nu}(n<\tau_{\partial})$
$\displaystyle\geq\frac{c_{1}c}{c_{2}}\sup_{y\in
E}\mathbb{P}_{y}(n<\tau_{\partial}).$
This implies (E3) and that $\inf_{x\in
K}\mathbb{P}_{x}(k_{0}<\tau_{\partial})>0$. Hence, (E1) follows from (A1) with
the probability measure $\frac{\nu(\cdot\cap K)}{\nu(K)}$. Moreover, for any
$n$ large enough to have $C\alpha^{n}\leq 1/2$ where the constants $C$ and
$\alpha$ are those of (3.6), we have $\mathbb{P}_{x}(X_{n}\in K\mid
t<\tau_{\partial})\geq\nu_{QSD}(K)-C\alpha^{n}\geq 1/2-\varepsilon>0$ and
hence (E4) is satisfied. The last computation also entails (3.7) with
$n^{\prime}_{4}=n$.
It remains to construct a function $\varphi_{1}$ satisfying (E2). For all
$x\in E$,
$\displaystyle\mathbb{P}_{x}(X_{n}\in E\setminus K\mid n<\tau_{\partial})$
$\displaystyle\leq\nu_{QSD}(E\setminus
K)+C\alpha^{n}\leq\varepsilon+C\alpha^{n}.$
Using (11.14), we deduce that
$\displaystyle\mathbb{P}_{x}(X_{n}\in E\setminus K)$ $\displaystyle\leq
C^{\prime}(\varepsilon+C\alpha^{n})\theta_{0}^{n},$
so that there exists $n_{0}$ large enough such that
$\displaystyle\mathbb{P}_{x}(n_{0}<T_{K}\wedge\tau_{\partial})$
$\displaystyle\leq\frac{1}{3}\theta_{0}^{n_{0}}=\left(\frac{\theta_{0}}{3^{1/n_{0}}}\right)^{n_{0}}.$
From this follows that, for all $k\in\mathbb{N}$ and all $x\in E$,
$\displaystyle\mathbb{P}_{x}(kn_{0}<T_{K}\wedge\tau_{\partial})$
$\displaystyle\leq\left(\frac{\theta_{0}}{3^{1/n_{0}}}\right)^{kn_{0}}.$
In particular, for $\theta_{1}:=\theta_{0}/2^{1/n_{0}}$,
$\displaystyle\varphi_{1}(x):=\mathbb{E}_{x}\left(\theta_{1}^{-T_{K}\wedge\lceil\tau_{\partial}\rceil}\right),\quad\forall
x\in E,$
is a bounded function on $E$ and Lemma 3.3 implies that, for all $x\in E$,
$\displaystyle
P_{1}\varphi_{1}(x)\leq\theta_{1}\varphi_{1}(x)+\|\varphi_{1}\|_{\infty}\mathbbm{1}_{K}(x).$
Since $\theta_{1}<\theta_{0}$, (E2) is proved.
## 12 Proof of the results of Section 4.1
In order to prove Theorem 4.1, we check Condition (F). The goal of Subsection
12.1 is to give the construction of the process $X$ and to check (F0) with
$L=K_{k}$ for some $k\geq 1$. In Subsection 12.2, we explain how (F3) can be
deduced from general Harnack inequalities. Finally, Subsection 12.3 completes
the proof of Theorem 4.1. The proof of Corollary 4.2 is then given in
Subsection 12.4.
### 12.1 Construction of the diffusion process $X$ and Markov property
The goal of this section is to construct a weak solution $X$ to the SDE (4.1)
with absorption out of $D$, and prove that it is Markov and satisfies a strong
Markov property at appropriate stopping times, enough to entail Condition (F0)
for $L=K_{k}$ for some $k\geq 1$. We introduce the natural path space for the
process $X$ as
$\mathcal{D}:=\Biggl{\\{}w:\mathbb{R}_{+}\rightarrow D\cup\\{\partial\\}:\
\forall k\geq 1,\ w\text{ is continuous on }[0,\tau_{k}(w)]\\\ \left.\text{
and }w(t)=\partial,\ \forall t\geq\sup_{k\geq 1}\tau_{k}(w)\right\\},$
where $\tau_{k}(w):=\inf\\{t\geq 0:w_{t}\in D\setminus K_{k}\\}$. Note that
$\mathcal{D}$ contains functions which are not càdlàg since they may not have
a left limit at $\tau_{\partial}-$ and, indeed, it is easy to construct
examples where $X$ is not càdlàg $\mathbb{P}$-a.s.222For example, one may
consider $D$ the open disc of radius 1 centered at 0 in $\mathbb{R}^{2}$,
$\sigma=\text{Id}$ and $b(x)=(-x_{2}\beta(|x|),x_{1}\beta(|x|))$ where
$x=(x_{1},x_{2})\in D$. Decomposing the process in polar coordinates
$(R_{t},\theta_{t}):=(|X_{t}|,\arctan(X^{(1)}_{t}/X^{(2)}_{t}))$, the radius
$R_{t}$ is a 2-dimensional Bessel process, and $X_{t}$ is sent to $\partial$
when $R_{t}$ hits 1 (in a.s. finite time). The angle $\theta_{t}$ is solution
to $d\theta_{t}=R_{t}^{-1}dW_{t}-\beta(R_{t})dt$ before $\tau_{\partial}$, for
some Brownian motion $W$. Hence, if $\beta(r)$ converges sufficiently fast to
$+\infty$ when $r\rightarrow 1$, $\theta_{t}$ a.s. converges to $-\infty$ when
$t\rightarrow\tau_{\partial}-$, so $X$ does not admit a left limit at time
$\tau_{\partial}$. Note also that this definition means that we are looking
for a process $X$ such that
$\displaystyle\tau_{\partial}:=\sup_{k\geq 1}\tau_{D\setminus K_{k}},$
which is the natural definition of $\tau_{\partial}$ when the left limit of
$X$ at time $\tau_{\partial}$ does not exist.
We endow the path space $\mathcal{D}$ with its natural filtration
$\mathcal{F}_{t}=\sigma(w_{s},s\leq t)=\bigvee_{n\geq 1,0\leq
t_{1}<t_{2}<\ldots<t_{n}\leq t}\sigma(w_{t_{1}},w_{t_{2}},\ldots,w_{t_{n}})$
and we follow the usual method which consists in constructing for all $x\in D$
a probability measure $\mathbb{P}_{x}$ on $\mathcal{D}$ and a stochastic
process $(B_{t},t\geq 0)$ on
$\mathcal{D}\times\mathcal{C}(\mathbb{R}_{+},\mathbb{R}^{r})$, such that $B$
is a standard $r$-dimensional Brownian motion under
$\mathbb{P}_{x}\otimes\mathbb{W}^{r}$, where $\mathbb{W}^{r}$ is the
$r$-dimensional Wiener measure and such that $w_{0}=x\ $
$\mathbb{P}_{x}\otimes\mathbb{W}^{r}$-almost surely and the canonical process
$(w_{t},t\geq 0)$ solves the SDE (4.1) for this Brownian motion $B$ on the
time interval $[0,\sup_{k}\tau_{k}(w))$ 333Since $\sigma(x)$ is non-degenerate
for all $x\in D$, the space $\mathcal{C}(\mathbb{R}_{+},\mathbb{R}^{r})$
equipped with the Wiener measure $\mathbb{W}^{r}$ is only used to construct
the Brownian path $B_{t}$ after time $\sup_{k}\tau_{k}(w)$ and could be
omitted for our purpose since we only need to construct the process $B$ up to
time $\sup_{k}\tau_{k}(w)$..
For this construction, we use the fact that $b$ and $\sigma$ can be extended
out of $K_{k}$ to $\mathbb{R}^{d}$ as globally Hölder and bounded functions
$b_{k}$ and $\sigma_{k}$ and such that $\sigma_{k}$ is uniformly elliptic on
$\mathbb{R}^{d}$. Hence (see e.g. [53, Rk. 5.4.30]) the martingale problem is
well-posed for the SDE
$\displaystyle dX^{k}_{t}=b_{k}(X^{k}_{t})dt+\sigma_{k}(X^{k}_{t})dB_{t}.$
Let us denote by $\mathbb{P}^{k}_{x}$ the solution to this martingale problem
for the initial condition $x\in\mathbb{R}^{d}$. This is a probability measure
on $\mathcal{C}:=\mathcal{C}(\mathbb{R}_{+},\mathbb{R}^{d})$, equipped with
its canonical filtration $(\mathcal{G}_{t})_{t\geq 0}$.
For all $k\geq 1$, we define $\tau^{\prime}_{k}(w)=\inf\\{t\geq
0,w_{t}\not\in\text{int}(K_{k})\\}$, where $\text{int}(K_{k})$ is the interior
of $K_{k}$. Since the paths $w\in\mathcal{D}$ or $\mathcal{C}$ are continuous
at time $\tau^{\prime}_{k}$ and $\mathbb{R}^{d}\setminus\text{int}(K_{k})$ is
closed, it is standard to prove that $\tau^{\prime}_{k}$ is a stopping time
for the canonical filtration $(\mathcal{F}_{t})_{t\geq 0}$ on $\mathcal{D}$
and for the canonical filtration $(\mathcal{G}_{t})_{t\geq 0}$ on
$\mathcal{C}$. We define as usual the stopped $\sigma$-fields
$\mathcal{F}_{\tau^{\prime}_{k}}$ and $\mathcal{G}_{\tau^{\prime}_{k}}$, and
we define for all $x\in\text{int}(K_{k})$ the restriction of $\mathbb{P}_{x}$
to $\mathcal{F}_{\tau^{\prime}_{k}}$ as the restriction of
$\mathbb{P}^{k}_{x}$ to $\mathcal{G}_{\tau^{\prime}_{k}}$, where we can
identify the events of the two filtrations since they both concern continuous
parts of the paths. This construction is consistent for $k$ and $k+1$ (meaning
that if $x\in K_{k}$, they give the same probability to events of
$\mathcal{F}_{\tau_{k}}$) by uniqueness of the solutions $\mathbb{P}^{k}_{x}$
and $\mathbb{P}^{k+1}_{x}$ to the above martingale problems. Hence there
exists a unique extension $\mathbb{P}_{x}$ of the above measures to
$\bigvee_{k\geq 1}\mathcal{F}_{\tau^{\prime}_{k}}$. Note that, because of the
specific structure of the path space $\mathcal{D}$, we have
$\bigvee_{k\geq 1}\mathcal{F}_{\tau^{\prime}_{k}}=\mathcal{F}_{\infty}.$
(12.1)
To check this, it suffices to observe that, for all $t\geq 0$ and all
measurable $A\subset D\cup\\{\partial\\}$,
$\displaystyle\\{w_{t}\in A\\}$ $\displaystyle=\\{t<\tau_{\partial},\ w_{t}\in
A\cap D\\}\cup\\{\tau_{\partial}\leq t,\ \partial\in A\\}$
$\displaystyle=\left(\bigcup_{k\geq 1}\\{t<\tau^{\prime}_{k},\ w_{t}\in A\cap
D\\}\right)\cup\left(\bigcap_{k\geq 1}\\{\tau^{\prime}_{k}\leq t,\partial\in
A\\}\right),$ (12.2)
hence $\\{w_{t}\in A\\}\in\bigvee_{k\geq 1}\mathcal{F}_{\tau^{\prime}_{k}}$,
and, proceeding similarly, the same property holds for events of the form
$\\{w_{t_{1}}\in A_{1},\ldots,w_{t_{n}}\in A_{n}\\}$.
We recall (see [53, Section 5.4]) that
$(\mathbb{P}^{k}_{x})_{x\in\mathbb{R}^{d}}$ forms a strong Markov family on
the canonical space $\mathcal{C}$. Our goal is now to prove that the family of
probability measures $(\mathbb{P}_{x})_{x\in D\cup\\{\partial\\}}$, where
$\mathbb{P}_{\partial}$ is defined as the Dirac measure on the constant path
equal to $\partial$, forms a Markov kernel of probability measures, for which
the strong Markov property applies at well-chosen stopping times.
We first need to prove that $(\mathbb{P}_{x})_{x\in D}$ defines a kernel of
probability measures, i.e. that $x\mapsto\mathbb{P}_{x}(\Gamma)$ is measurable
for all events $\Gamma$ of $\mathcal{F}_{\infty}$. We prove it for an event of
the form $\\{w_{t}\in A\\}$, the extension to events of the form
$\\{w_{t_{1}}\in A_{1},\ldots,w_{t_{n}}\in A_{n}\\}$, and hence to all events
of $\mathcal{F}_{\infty}$, being easy. This follows from (12.2):
$\displaystyle\mathbb{P}_{x}(w_{t}\in A)$
$\displaystyle=\lim_{k\rightarrow+\infty}\mathbb{P}_{x}(t<\tau^{\prime}_{k},\
w_{t}\in A\cap D)+\mathbbm{1}_{\partial\in
A}\lim_{k\rightarrow+\infty}\mathbb{P}_{x}(\tau^{\prime}_{k}\leq t)$
$\displaystyle=\lim_{k\rightarrow+\infty}\mathbb{P}^{k+1}_{x}(t<\tau^{\prime}_{k},\
w_{t}\in A\cap D)+\mathbbm{1}_{\partial\in
A}\lim_{k\rightarrow+\infty}\mathbb{P}^{k+1}_{x}(\tau^{\prime}_{k}\leq t).$
Since all the probabilities in the right-hand side are measurable functions of
$x$, so is $x\mapsto\mathbb{P}_{x}(w_{t}\in A)$.
Now, let us prove that $(X_{t},t\geq 0)$ is Markov. It is well-known that this
is implied by the following property: for all $n\geq 1$ and $0\leq
t_{1}\leq\ldots\leq t_{n+1}$ and $A_{1},\ldots,A_{n+1}$ measurable subsets of
$D\cup\\{\partial\\}$,
$\displaystyle\mathbb{P}_{x}(w_{t_{1}}\in A_{1},\ldots,w_{t_{n+1}}\in
A_{n+1})=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A_{1},\ldots,w_{t_{n}}\in A_{n}}\mathbb{P}_{w_{t_{n}}}(w_{t_{n+1}-t_{n}}\in
A_{n+1})\right].$
We prove this property only for $n=1$. It is easy to extend the proof to all
values of $n\geq 1$. We have
$\mathbb{P}_{x}(w_{t_{1}}\in A_{1},w_{t_{2}}\in
A_{2})=\mathbb{P}_{x}(w_{t_{1}}\in A_{1},w_{t_{2}}\in
A_{2},\tau_{\partial}>t_{2})\\\ +\mathbb{P}_{x}(w_{t_{1}}\in
A_{1},t_{1}<\tau_{\partial}\leq t_{2})\mathbbm{1}_{\partial\in
A_{2}}+\mathbb{P}_{x}(\tau_{\partial}\leq t_{1})\mathbbm{1}_{\partial\in
A_{1}\cap A_{2}}.$
Now, using that $(\mathbb{P}^{k}_{x})_{x\in\mathbb{R}^{d}}$ is a Markov family
for all $k\geq 1$,
$\mathbb{P}_{x}(w_{t_{1}}\in A_{1},w_{t_{2}}\in
A_{2},\tau_{\partial}>t_{2})\\\ \begin{aligned}
&=\lim_{k\rightarrow\infty}\mathbb{P}_{x}(w_{t_{1}}\in A_{1},w_{t_{2}}\in
A_{2},\tau_{k}>t_{2})\\\
&=\lim_{k\rightarrow\infty}\mathbb{P}^{k}_{x}(w_{t_{1}}\in A_{1},w_{t_{2}}\in
A_{2},\tau_{k}>t_{2})\\\
&=\lim_{k\rightarrow\infty}\mathbb{E}^{k}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A_{1},t_{1}<\tau_{k}}\mathbb{P}^{k}_{w_{t_{1}}}(w_{t_{2}-t_{1}}\in
A_{2},\tau_{k}>t_{2}-t_{1})\right]\\\
&=\lim_{k\rightarrow\infty}\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A_{1},t_{1}<\tau_{k}}\mathbb{P}_{w_{t_{1}}}(w_{t_{2}-t_{1}}\in
A_{2},\tau_{k}>t_{2}-t_{1})\right]\\\
&=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A_{1},t_{1}<\tau_{\partial}}\mathbb{P}_{w_{t_{1}}}(w_{t_{2}-t_{1}}\in
A_{2},\tau_{\partial}>t_{2}-t_{1})\right]\end{aligned}$
and similarly
$\displaystyle\mathbb{P}_{x}(w_{t_{1}}\in A_{1},t_{1}<\tau_{\partial}\leq
t_{2})\mathbbm{1}_{\partial\in A_{2}}$
$\displaystyle=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A_{1},t_{1}<\tau_{\partial}}\mathbb{P}_{w_{t_{1}}}(\tau_{\partial}\leq
t_{2}-t_{1})\right]\mathbbm{1}_{\partial\in A_{2}}$
$\displaystyle=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A_{1},t_{1}<\tau_{\partial}}\mathbb{P}_{w_{t_{1}}}(\tau_{\partial}\leq
t_{2}-t_{1},w_{t_{2}-t_{1}}\in A_{2})\right].$
Since
$\displaystyle\mathbb{P}_{x}(\tau_{\partial}\leq
t_{1})\mathbbm{1}_{\partial\in A_{1}\cap A_{2}}$
$\displaystyle=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A_{1},\tau_{\partial}\leq t_{1}}\mathbb{P}_{w_{t_{1}}}(w_{t_{2}-t_{1}}\in
A_{2})\right],$
we have proved that $\mathbb{P}_{x}(w_{t_{1}}\in A_{1},w_{t_{2}}\in
A_{2})=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A_{1}}\mathbb{P}_{w_{t_{1}}}(w_{t_{2}-t_{1}}\in A_{2})\right]$. This ends the
proof of the Markov property.
To conclude this subsection, let us prove that the strong Markov property
holds for all stopping times $\tau_{F}$ where $F\subset D$ is closed in $D$.
Note that $\tau_{F}$ is indeed a stopping time for the filtration
$\mathcal{F}_{t}$ since
$\tau_{F}=\sup_{k}\tau_{F}\wedge\tau^{\prime}_{k}=\sup_{k}\tau_{(F\cup
D^{c})\cup\text{int}(K_{k})^{c}}$, where the complement is understood in
$\mathbb{R}^{d}$, $(F\cup D^{c})\cup\text{int}(K_{k})^{c}$ is a closed subset
of $\mathbb{R}^{d}$ and all $w\in\mathcal{D}$ is continuous at time
$\tau_{(F\cup D^{c})\cup\text{int}(K_{k})^{c}}$. Let $x\in D$,
$t_{1},t_{2},s\geq 0$ and $A,B\subset D$ be measurable sets. We proceed as
above: first, observe that
$\\{w_{t_{1}}\in A,\ t_{1}<\tau_{F}\leq t_{2},\ w_{\tau_{F}+s}\in B\\}\\\
\begin{aligned} &=\bigcup_{\ell\geq 1}\\{w_{t_{1}}\in A,\ t_{1}<\tau_{F}\leq
t_{2},\ w_{\tau_{F}+s}\in B,\ w_{r}\in K_{\ell}\ \forall
r\in[0,\tau_{F}+s]\\}\\\ &=\bigcup_{\ell\geq 1}\\{w_{t_{1}}\in A,\
t_{1}<\tau_{F}\wedge\tau^{\prime}_{\ell}\leq t_{2},\
w_{\tau_{F}\wedge\tau^{\prime}_{\ell}+s}\in B,\
\tau^{\prime}_{\ell}>\tau_{F}+s\\}.\end{aligned}$
Since $\tau_{F}\wedge\tau^{\prime}_{\ell}$ is a $\mathcal{G}_{t}$-stopping
time on $\mathcal{C}(\mathbb{R}_{+},\mathbb{R}^{d})$ and using the strong
Markov property under $\mathbb{P}^{\ell}$, we deduce that
$\mathbb{P}_{x}(w_{t_{1}}\in A,\ t_{1}<\tau_{F}\leq t_{2},\ w_{\tau_{F}+s}\in
B)\\\ \begin{aligned}
&=\lim_{\ell\rightarrow+\infty}\mathbb{P}^{\ell}_{x}(w_{t_{1}}\in A,\
t_{1}<\tau_{F}\wedge\tau^{\prime}_{\ell}\leq t_{2},\
w_{\tau_{F}\wedge\tau^{\prime}_{\ell}+s}\in B,\
\tau^{\prime}_{\ell}>\tau_{F}+s)\\\
&=\lim_{\ell\rightarrow+\infty}\mathbb{E}^{\ell}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A,\ t_{1}<\tau_{F}\wedge\tau^{\prime}_{\ell}\leq
t_{2}}\mathbb{P}^{\ell}_{w_{\tau_{F}\wedge\tau^{\prime}_{\ell}}}(w_{s}\in B,\
s<\tau^{\prime}_{\ell})\right]\\\
&=\lim_{\ell\rightarrow+\infty}\mathbb{E}^{\ell}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in
A,\ t_{1}<\tau_{F}\leq\tau^{\prime}_{\ell}\wedge
t_{2}}\mathbb{P}^{\ell}_{w_{\tau_{F}}}(w_{s}\in B,\
s<\tau^{\prime}_{\ell})\right]\\\
&=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in A,\
t_{1}<\tau_{F}\leq\tau_{\partial}\wedge
t_{2}}\mathbb{P}_{w_{\tau_{F}}}(w_{s}\in B,\
s<\tau_{\partial})\right].\end{aligned}$
Similarly,
$\displaystyle\mathbb{P}_{x}(w_{t_{1}}\in A,\ t_{1}<\tau_{F}\leq t_{2},\ $
$\displaystyle w_{\tau_{F}+s}=\partial)$
$\displaystyle=\lim_{\ell\rightarrow+\infty}\mathbb{P}^{\ell}_{x}(w_{t_{1}}\in
A,\ t_{1}<\tau_{F}\leq t_{2}\wedge\tau^{\prime}_{\ell},\
\tau^{\prime}_{\ell}\leq\tau_{F}+s)$
$\displaystyle=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in A,\
t_{1}<\tau_{F}\leq
t_{2}\wedge\tau_{\partial}}\mathbb{P}_{w_{\tau_{F}}}(w_{s}=\partial)\right]$
and thus
$\mathbb{P}_{x}(w_{t_{1}}\in A,\ t_{1}<\tau_{F}\leq t_{2},\ w_{\tau_{F}+s}\in
B)=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in A,\ t_{1}<\tau_{F}\leq
t_{2}\wedge\tau_{\partial}}\mathbb{P}_{w_{\tau_{F}}}(w_{s}\in B)\right]$
for all $A,B\subset D\cup\\{\partial\\}$ measurable. The previous computation
extends without difficulty to prove
$\mathbb{P}_{x}\left(w_{t_{1}}\in A_{1},\ldots,w_{t_{n}}\in A_{n},\
t_{n}<\tau_{F}\leq t_{n+1},\ w_{\tau_{F}+s_{1}}\in
B_{1},\ldots,w_{\tau_{F}+s_{m}}\in B_{m}\right)\\\
=\mathbb{E}_{x}\left[\mathbbm{1}_{w_{t_{1}}\in A_{1},\ldots,w_{t_{n}}\in
A_{n},\ t_{n}<\tau_{F}\leq t_{n+1}}\mathbb{P}_{w_{\tau_{F}}}(w_{s_{1}}\in
B_{1},\ldots,w_{s_{m}}\in B_{m})\right]$ (12.3)
for all $n,m\geq 1$, $0\leq t_{1}\leq\ldots\leq t_{n+1}$, $0\leq
s_{1}\leq\ldots\leq s_{m}$ and $A_{1},\ldots,A_{n},B_{1},\ldots,B_{m}\subset
D\cup\\{\partial\\}$ measurable. This implies the strong Markov property at
time $\tau_{F}$, in the sense that, for all $k\geq 1$, all $x\in E$ and all
$\Gamma\in\mathcal{F}_{\infty}$,
$\displaystyle\mathbb{P}_{x}\left(w^{\tau_{F}}\in\Gamma\mid\mathcal{H}_{\tau_{F}}\right)=\mathbb{P}_{w_{\tau_{F}}}(\Gamma),\quad\mathbb{P}_{x}\text{-almost
surely},$
where $w^{\tau_{F}}=(w_{\tau_{F}+s},s\geq 0)$ and
$\mathcal{H}_{\tau_{F}}=\sigma\Big{(}\left\\{w_{t_{1}}\in
A_{1},\ldots,w_{t_{n}}\in A_{n},t_{n}<\tau_{F}\leq t_{n+1}\right\\},\\\ 0\leq
t_{1}\leq\ldots\leq t_{n+1},\ A_{1},\ldots,A_{n}\in D\text{
measurable}\Big{)}.$
This form of strong Markov property at time $\tau_{F}$ is enough for our
purpose, since it entails (F0) for $L=K_{k}$ for all $k\geq 1$. It will be
also needed in the next section.
### 12.2 Harnack inequalities
Our goal here is to check Conditions (F1) and (F3) for the diffusion process
constructed above. We will make use of general Harnack inequalities of Krylov
and Safonov [59].
###### Proposition 12.1.
There exist a probability measure $\nu$ on $D$ and a constant $t_{\nu}>0$ such
that, for all $k\geq 1$, there exists a constant $b_{k}>0$ such that
$\displaystyle\mathbb{P}_{x}(X_{t_{\nu}}\in\cdot)\geq b_{k}\nu(\cdot),\
\forall x\in K_{k}.$ (12.4)
Moreover, for all $k\geq 1$ such that $K_{k}$ is non-empty,
$\displaystyle\inf_{t\geq 0}\frac{\inf_{x\in
K_{k}}\mathbb{P}_{x}(t<\tau_{\partial})}{\sup_{x\in
K_{k}}\mathbb{P}_{x}(t<\tau_{\partial})}>0.$ (12.5)
###### Proof.
Consider a bounded measurable function $f:D\rightarrow\mathbb{R}$ with
$\|f\|_{\infty}\leq 1$ and define the application
$u:(t,x)\in\mathbb{R}_{+}\times
E\mapsto\mathbb{E}_{x}[\mathbbm{1}_{t<\tau_{\partial}}f(X_{t})]$. It is proved
in [18] using [59] that, for all $k\geq 1$, there exist two constants
$N_{k}>0$ and $\delta_{k}>0$, which do not depend on $f$ (provided
$\|f\|_{\infty}\leq 1$), such that
$\displaystyle u(\delta_{k}+\delta_{k}^{2},x)\leq
N_{k}u(\delta_{k}+2\delta_{k}^{2},y),\text{ for all $x,y\in K_{k}$ such that
}|x-y|\leq\delta_{k}.$ (12.6)
Note that the proof given in [18] makes use of the following strong Markov
property: for all open ball $B$ such that $B\subset K_{k}$ for some $k\geq 1$,
all $x\in B$, $t\geq 0$ and all measurable
$f:D\cup\\{\partial\\}\rightarrow\mathbb{R}_{+}$,
$\mathbb{E}_{x}\left[f(X_{t})\mathbbm{1}_{\tau_{D\setminus B}\leq
t}\right]=\mathbb{E}_{x}\left[\mathbbm{1}_{\tau_{D\setminus B}\leq
t}\mathchoice{{\mathbb{E}_{X_{\tau_{D\setminus
B}}}\left[f(X_{t-u})\right]\,\smash{\vrule
height=6.00002pt,depth=3.13307pt}}_{\,u=\tau_{D\setminus
B}}}{{\mathbb{E}_{X_{\tau_{D\setminus
B}}}\left[f(X_{t-u})\right]\,\smash{\vrule
height=6.00002pt,depth=3.13307pt}}_{\,u=\tau_{D\setminus
B}}}{{\mathbb{E}_{X_{\tau_{D\setminus
B}}}\left[f(X_{t-u})\right]\,\smash{\vrule
height=4.20001pt,depth=2.6684pt}}_{\,u=\tau_{D\setminus
B}}}{{\mathbb{E}_{X_{\tau_{D\setminus
B}}}\left[f(X_{t-u})\right]\,\smash{\vrule
height=3.0pt,depth=2.6684pt}}_{\,u=\tau_{D\setminus B}}}\right].$
This property follows from (12.3).
Step 1 : Proof of (12.4)
Fix $x_{1}\in D$ and $k_{1}\geq 1$ such that $x_{1}\in\text{int}(K_{k_{1}})$.
Let $\nu$ denote the conditional law
$\mathbb{P}_{x_{1}}(X_{\delta_{k_{1}}+\delta_{k_{1}}^{2}}\in\cdot\mid\delta_{k_{1}}+\delta_{k_{1}}^{2}<\tau_{\partial})$.
Then, for all measurable $A\subset D\cup\\{\partial\\}$, Harnack’s inequality
(12.6) with $f=\mathbbm{1}_{A}$ entails that, for all $x\in D$ such that
$|x-x_{1}|<\delta_{k_{1}}\wedge d(x_{1},D\setminus K_{k_{1}})$,
$\displaystyle\mathbb{P}_{x}(X_{\delta_{k_{1}}+2\delta_{k_{1}}^{2}}\in
A)\geq\frac{\mathbb{P}_{x_{1}}(\delta_{k_{1}}+\delta_{k_{1}}^{2}<\tau_{\partial})}{N_{k_{1}}}\,\nu(A).$
Since the diffusion is locally elliptic and $D$ is connected, for all $k\geq
1$, there exists a constant $d_{k}>0$ such that
$\displaystyle\inf_{x\in K_{k}}\mathbb{P}_{x}(X_{1}\in
B(x_{1},\delta_{k_{1}}\wedge d(x_{1},D\setminus K_{k_{1}}))\geq d_{k}.$
This and Markov’s property entail that, for all $x\in K_{k}$,
$\displaystyle\mathbb{P}_{x}(X_{1+\delta_{k_{1}}+2\delta_{k_{1}}^{2}}\in\cdot)\geq
d_{k}\frac{\mathbb{P}_{x_{1}}(\delta_{k_{1}}+\delta_{k_{1}}^{2}<\tau_{\partial})}{N_{k_{1}}}\,\nu.$
This implies the first part of Proposition 12.1.
Step 2 : Proof of (12.5)
Fix $k\geq 1$ such that $K_{k}$ is non-empty and consider $\ell>k$ such that
$K_{k}$ is included in one connected component of $\text{int}(K_{\ell})$. For
all $t\geq\delta_{\ell}+2\delta_{\ell}^{2}$, the inequality (12.6) applied to
$f(x)=\mathbb{P}_{x}(t-\delta_{\ell}-2\delta_{\ell}^{2}<\tau_{\partial})$ and
the Markov property entail that
$\displaystyle\mathbb{P}_{x}(t-\delta_{\ell}^{2}<\tau_{\partial})\leq
N_{\ell}\mathbb{P}_{y}(t<\tau_{\partial}),\text{ for all $x,y\in K_{\ell}$
such that }|x-y|\leq\delta_{\ell}.$
Since $s\mapsto\mathbb{P}_{x}(s<\tau_{\partial})$ is non-increasing, we deduce
that
$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})\leq
N_{\ell}\mathbb{P}_{y}(t<\tau_{\partial}),\text{ for all $x,y\in K_{\ell}$
such that }|x-y|\leq\delta_{\ell}.$
Since $K_{k}$ has a finite diameter and is included in a connected component
of $K_{\ell}$, we deduce that there exists $N^{\prime}_{k}$ equal to some
power of $N_{\ell}$ such that, for all
$t\geq\delta_{\ell}+2\delta_{\ell}^{2}$,
$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})\leq
N^{\prime}_{k}\mathbb{P}_{y}(t<\tau_{\partial}),\text{ for all $x,y\in
K_{k}$.}$
Now, for $t\leq\delta_{\ell}+2\delta_{\ell}^{2}$, we simply use the fact that
$x\mapsto\mathbb{P}_{x}(\delta_{\ell}+2\delta_{\ell}^{2}<\tau_{\partial})$ is
uniformly bounded from below on $K_{k}$ by a constant
$1/N^{\prime\prime}_{k}>0$. In particular,
$\displaystyle\mathbb{P}_{x}(t<\tau_{\partial})\leq 1\leq
N^{\prime\prime}_{k}\mathbb{P}_{y}(\delta_{\ell}+2\delta_{\ell}^{2}<\tau_{\partial})\leq
N^{\prime\prime}_{k}\mathbb{P}_{y}(t<\tau_{\partial}),\text{ for all $x,y\in
K_{k}$.}$
This concludes the proof of Proposition 12.1. ∎
### 12.3 Proof of Theorem 4.1
Our aim is to prove that Condition (F) holds true with $L=K_{k}$ for some
$k\geq 1$. We have already proved (F0), (F1) and (F3) with $L=K_{k}$ for any
$k\geq 1$. Hence we only have to check (F2). Fix
$\rho_{1}\in(\lambda_{0},\lambda_{1})$, $\rho_{2}\in(\lambda_{0},\rho_{1})$
and $p\in(1,\lambda_{1}/\rho_{1})$ and define
$\psi_{1}(x)=\varphi(x)^{1/p},\ \forall x\in D.$
Fix $\rho^{\prime}_{1}\in(\rho_{1},\lambda_{1}/p)$ and
$t_{2}\geq\frac{2s_{1}(C+\lambda_{1})}{\lambda_{1}-p\rho^{\prime}_{1}}\,\vee\,\frac{\log
2}{\rho^{\prime}_{1}-\rho_{1}},$
where the constant $C$ comes from (4.5). Set $L=K_{k_{0}}$ with $k_{0}$ large
enough so that $\nu(K_{k_{0}})>0$ and, using (4.6),
$\mathbb{P}_{x}(s_{1}<\tau_{K_{k_{0}}}\wedge\tau_{\partial})\leq
e^{-(\rho^{\prime}_{1}+C/p)t_{2}}$
for all $x\in D_{0}$.
From the definition of $\lambda_{0}$ and applying the same argument as in Step
2 of the proof of Proposition 12.1 with
$f(x)=\mathbb{P}_{x}(X_{t-\delta_{\ell}-2\delta_{\ell}^{2}}\in L)$ with $\ell$
large enough to have $K_{k_{0}}$ included in one connected component of
$K_{\ell}$, we deduce that
$\displaystyle\liminf_{t\rightarrow+\infty}e^{\rho_{2}t}\inf_{x\in
L}\mathbb{P}_{x}(X_{t}\in L)=+\infty,$
and hence the last line of (F2) is proved with $\gamma_{2}=e^{-\rho_{2}}$.
Let us now check that the first line of Assumption (F2) holds true for all
$x\in D_{0}$ and then for all $x\in D\setminus D_{0}$. For all $x\in D_{0}$,
we have $\psi_{1}(x)\leq\sup_{x\in D_{0}}\\\ varphi^{1/p}(x)<+\infty$, and
hence, for all $t\in[s_{1},t_{2}]$, using Hölder’s inequality and the
definition of $k_{0}$,
$\displaystyle\mathbb{E}_{x}\left(\psi_{1}(X_{t})\mathbbm{1}_{t<\tau_{L}\wedge\tau_{\partial}}\right)$
$\displaystyle\leq\mathbb{E}_{x}\left(\mathbbm{1}_{t<\tau_{\partial}}\varphi(X_{t})\right)^{1/p}\mathbb{P}_{x}(t<\tau_{L}\wedge\tau_{\partial})^{\frac{p-1}{p}}$
$\displaystyle\leq\varphi(x)^{1/p}e^{Ct_{2}/p}\mathbb{P}_{x}(s_{1}<\tau_{L}\wedge\tau_{\partial})^{\frac{p-1}{p}}$
(12.7) $\displaystyle\leq e^{-\rho^{\prime}_{1}t_{2}}\leq
e^{-\rho_{1}t_{2}}\psi_{1}(x).$
To prove (12.7), we used the fact that $\mathcal{L}\varphi\leq C\leq C\varphi$
and Itô’s formula to obtain $P_{t}\varphi\leq e^{Ct}\varphi$. Since this
argument is used repeatedly in the sequel, we give it in details for sake of
completeness. It follows from Itô’s formula that, for all $k\geq 1$,
$\mathbb{P}_{x}$-almost surely,
$\displaystyle
e^{-C\left(t\wedge\tau_{K_{k}^{c}}\right)}\varphi\left(X_{t\wedge\tau_{K_{k}^{c}}}\right)=\varphi(x)$
$\displaystyle+\int_{0}^{t}\mathbbm{1}_{s\leq\tau_{K_{k}^{c}}}e^{-Cs}\left(\mathcal{L}\varphi(X_{s})-C\varphi(X_{s})\right)ds$
$\displaystyle+\int_{0}^{t}\mathbbm{1}_{s\leq\tau_{K_{k}^{c}}}e^{-Cs}\nabla\varphi(X_{s})^{*}\sigma(X_{s})dB_{s}.$
Since $\nabla\varphi(x)$ and $\sigma(x)$ are uniformly bounded on $K_{k}$, the
last term has zero expectation, and thus
$\displaystyle\mathbb{E}_{x}\left[e^{-C\left(t\wedge\tau_{K_{k}^{c}}\right)}\varphi\left(X_{t\wedge\tau_{K_{k}^{c}}}\right)\right]$
$\displaystyle\leq\varphi(x).$
Letting $k\rightarrow+\infty$, we deduce form Fatou’s lemma that
$\displaystyle\mathbb{E}_{x}\left[e^{-Ct}\mathbbm{1}_{t<\tau_{\partial}}\varphi(X_{t})\right]$
$\displaystyle\leq\varphi(x)$ (12.8)
as claimed.
This proves the second line of (F2) for all $x\in D_{0}$ and
$\gamma_{1}=e^{-\rho_{1}}$.
Now, for all $x\in D\setminus D_{0}$, since $D_{0}$ is closed in $D$, it
follows from the strong Markov property (12.3) at time $\tau_{D_{0}}$ that
$\mathbb{E}_{x}\left(\psi_{1}(X_{t_{2}})\mathbbm{1}_{t_{2}<\tau_{L}\wedge\tau_{\partial}}\right)=\mathbb{E}_{x}\left(\mathbbm{1}_{t_{2}-s_{1}<\tau_{L}\wedge\tau_{\partial}\wedge\tau_{D_{0}}}\mathbb{E}_{X_{t_{2}-s_{1}}}\left(\psi_{1}(X_{s_{1}})\mathbbm{1}_{s_{1}<\tau_{L}\wedge\tau_{\partial}}\right)\right)\\\
+\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{D_{0}}\leq
t_{2}-s_{1}}\mathchoice{{\mathbb{E}_{X_{\tau_{D_{0}}}}\left(\psi_{1}(X_{t_{2}-u})\mathbbm{1}_{t_{2}-u<\tau_{\partial}\wedge\tau_{L}}\right)\,\smash{\vrule
height=6.00002pt,depth=3.5163pt}}_{\,u=\tau_{D_{0}}}}{{\mathbb{E}_{X_{\tau_{D_{0}}}}\left(\psi_{1}(X_{t_{2}-u})\mathbbm{1}_{t_{2}-u<\tau_{\partial}\wedge\tau_{L}}\right)\,\smash{\vrule
height=6.00002pt,depth=3.5163pt}}_{\,u=\tau_{D_{0}}}}{{\mathbb{E}_{X_{\tau_{D_{0}}}}\left(\psi_{1}(X_{t_{2}-u})\mathbbm{1}_{t_{2}-u<\tau_{\partial}\wedge\tau_{L}}\right)\,\smash{\vrule
height=4.20001pt,depth=3.05162pt}}_{\,u=\tau_{D_{0}}}}{{\mathbb{E}_{X_{\tau_{D_{0}}}}\left(\psi_{1}(X_{t_{2}-u})\mathbbm{1}_{t_{2}-u<\tau_{\partial}\wedge\tau_{L}}\right)\,\smash{\vrule
height=3.0pt,depth=3.05162pt}}_{\,u=\tau_{D_{0}}}}\right).$ (12.9)
Using Hölder’s inequality and (12.8), we deduce that, for all $y\in D$,
$\displaystyle\mathbb{E}_{y}\left(\psi_{1}(X_{s_{1}})\mathbbm{1}_{s_{1}<\tau_{L}\wedge\tau_{\partial}}\right)\leq\mathbb{E}_{y}\left(\varphi(X_{s_{1}})\mathbbm{1}_{s_{1}<\tau_{\partial}}\right)^{1/p}\leq
e^{\frac{s_{1}C}{p}}\,\varphi(y)^{1/p}=e^{\frac{s_{1}C}{p}}\,\psi_{1}(y).$
Hence, the first term in the right-hand side of (12.9) satisfies
$\displaystyle\mathbb{E}_{x}\left(\mathbbm{1}_{t_{2}-s_{1}<\tau_{L}\wedge\tau_{\partial}\wedge\tau_{D_{0}}}\mathbb{E}_{X_{t_{2}-s_{1}}}\left(\psi_{1}(X_{s_{1}})\mathbbm{1}_{s_{1}<\tau_{L}\wedge\tau_{\partial}}\right)\right)\leq
e^{\frac{s_{1}C}{p}}\mathbb{E}_{x}\left(\mathbbm{1}_{t_{2}-s_{1}<\tau_{L}\wedge\tau_{\partial}\wedge\tau_{D_{0}}}\psi_{1}(X_{t_{2}-s_{1}})\right).$
As a consequence, using again Hölder’s inequality and applying as above Itô’s
formula using that $\mathcal{L}\varphi(x)\leq-\lambda_{1}\varphi(x)$ for all
$x\notin D_{0}$, one has
$\displaystyle\mathbb{E}_{x}\left(\mathbbm{1}_{t_{2}-s_{1}<\tau_{L}\wedge\tau_{\partial}\wedge\tau_{D_{0}}}\mathbb{E}_{X_{t_{2}-s_{1}}}\left(\psi_{1}(X_{s_{1}})\mathbbm{1}_{s_{1}<\tau_{L}\wedge\tau_{\partial}}\right)\right)$
$\displaystyle\leq
e^{-\lambda_{1}\frac{t_{2}-s_{1}}{p}}e^{\frac{s_{1}C}{p}}\,\varphi(x)^{1/p}$
$\displaystyle\leq
e^{-t_{2}\frac{\rho^{\prime}_{1}+\lambda_{1}/p}{2}}\psi_{1}(x),$
where we used in the last inequality that
$t_{2}\geq\frac{2s_{1}(C+\lambda_{1})}{\lambda_{1}-p\rho^{\prime}_{1}}$.
Moreover, using (12.7), we obtain that the second term in the right-hand side
of (12.9) satisfies
$\mathbb{E}_{x}\left(\mathbbm{1}_{\tau_{D_{0}}\leq
t_{2}-s_{1}}\mathchoice{{\mathbb{E}_{X_{\tau_{D_{0}}}}\left(\psi_{1}(X_{t_{2}-u})\mathbbm{1}_{t_{2}-u<\tau_{\partial}\wedge\tau_{L}}\right)\,\smash{\vrule
height=6.00002pt,depth=3.5163pt}}_{\,u=\tau_{D_{0}}}}{{\mathbb{E}_{X_{\tau_{D_{0}}}}\left(\psi_{1}(X_{t_{2}-u})\mathbbm{1}_{t_{2}-u<\tau_{\partial}\wedge\tau_{L}}\right)\,\smash{\vrule
height=6.00002pt,depth=3.5163pt}}_{\,u=\tau_{D_{0}}}}{{\mathbb{E}_{X_{\tau_{D_{0}}}}\left(\psi_{1}(X_{t_{2}-u})\mathbbm{1}_{t_{2}-u<\tau_{\partial}\wedge\tau_{L}}\right)\,\smash{\vrule
height=4.20001pt,depth=3.05162pt}}_{\,u=\tau_{D_{0}}}}{{\mathbb{E}_{X_{\tau_{D_{0}}}}\left(\psi_{1}(X_{t_{2}-u})\mathbbm{1}_{t_{2}-u<\tau_{\partial}\wedge\tau_{L}}\right)\,\smash{\vrule
height=3.0pt,depth=3.05162pt}}_{\,u=\tau_{D_{0}}}}\right)\\\ \leq
e^{-\rho^{\prime}_{1}t_{2}}\mathbb{P}_{x}(\tau_{D_{0}}\leq t_{2}-s_{1})\leq
e^{-\rho^{\prime}_{1}t_{2}}\psi_{1}(x).$
We finally deduce from (12.9) and from the definition of $L=K_{k_{0}}$ that
$\displaystyle\mathbb{E}_{x}\left(\psi_{1}(X_{t_{2}})\mathbbm{1}_{t_{2}<\tau_{L}\wedge\tau_{\partial}}\right)\leq
2e^{-\rho^{\prime}_{1}t_{2}}\psi_{1}(x)\leq e^{-\rho_{1}t_{2}}\psi_{1}(x),$
where we used that $t_{2}\geq\log 2/(\rho^{\prime}_{1}-\rho_{1})$. This
concludes the proof that the second line of (F2) holds true.
Since $\varphi$ is locally bounded, $\sup_{L}\varphi<\infty$, and hence, using
again (12.8), we deduce that, for all $t\geq 0$,
$\displaystyle\sup_{x\in L}\
\mathbb{E}_{x}(\psi_{1}(X_{t})\mathbbm{1}_{t<\tau_{\partial}})\leq\sup_{x\in
L}\ \mathbb{E}_{x}(\varphi(X_{t})\mathbbm{1}_{t<\tau_{\partial}})\leq
e^{Ct}\,\sup_{x\in L}\varphi(x)<\infty,$
which implies the third line of Assumption (F2).
In addition, because of the local uniform ellipticity of the diffusion $X$,
for all $n_{0}\geq 1$, $\psi_{2}:=\sum_{k=0}^{n_{0}}P_{k}\mathbbm{1}_{L}$ is
uniformly bounded away from zero on all compact subsets of $D$. This and
Theorem 3.5 concludes the proof of Theorem 4.1.
### 12.4 Proof of Corollary 4.2
Using Theorem 3.5, there exists $\lambda^{\prime}_{0}$ such that, for all
$x\in D$,
$\eta(x)=\lim_{t\rightarrow+\infty}e^{\lambda^{\prime}_{0}t}\mathbb{P}_{x}(t<\tau_{\partial}).$
We choose in the definition of $\lambda_{0}$ a ball $B$ such that
$\nu_{QSD}(B)>0$ (recall that $\lambda_{0}$ is independent of the choice of
$B$). Given $x\in D$ such that $\eta(x)>0$,
$\lim_{t\rightarrow+\infty}e^{\lambda^{\prime}_{0}t}\mathbb{P}_{x}(X_{t}\in
B)=\eta(x)\nu_{QSD}(B)\in(0,+\infty).$
Hence, $\lambda_{0}=\lambda^{\prime}_{0}$ and the infimum in the definition of
$\lambda_{0}$ is a minimum. The rest of the properties stated in Corollary 4.2
are direct consequences of Theorem 3.5.
Let us now prove that $\eta$ is $\mathcal{C}^{2}$. First, it follows from [79,
Theorem 7.2.4] that $e^{\lambda_{0}t}\mathbb{P}_{x}(t<\tau_{\partial})$ is
continuous for all $t\geq 0$ (see [18] for a detailed proof). Hence the
uniform convergence in Theorem 2.5 implies that $\eta$ is continuous on $D$.
Now, let $B$ be any non-empty open ball such that $\overline{B}\subset D$. We
consider the following initial-boundary value problem (in the terminology of
[39]) associated to the differential operator $\mathcal{L}$ defined in (4.3)
$\begin{cases}\partial_{t}u(t,x)-\mathcal{L}u(t,x)-\lambda_{0}u(t,x)=0&\text{for
all }(t,x)\in(0,T]\times B,\\\ u(0,x)=\eta(x)&\text{for all }x\in B,\\\
u(t,x)=\eta(x)&\text{for all }(t,x)\in(0,T]\times\partial B.\end{cases}$
Since the coefficients of $\mathcal{L}$ are Hölder and uniformly elliptic in
$\overline{B}$ and since $\eta$ is continuous, we can apply Corollary 1 of
Chapter 3 of [39] to obtain the existence and uniqueness of a solution $u$ to
the above problem, continuous on $[0,T]\times\overline{B}$ and
$\mathcal{C}^{1,2}((0,T]\times B)$. Now, we can apply Itô’s formula to
$e^{\lambda_{0}s}u(T-s,X_{s})$: for all $s\leq\tau_{B^{c}}\wedge T$ and all
$x\in B$, $\mathbb{P}_{x}$-almost surely,
$\displaystyle e^{\lambda_{0}s}u(T-s,X_{s})$
$\displaystyle=u(T,x)+\int_{0}^{s}e^{\lambda_{0}r}\left(-\frac{\partial
u}{\partial t}+\mathcal{L}u+\lambda_{0}u\right)(T-r,X_{r})\,dr$
$\displaystyle+\int_{0}^{s}e^{\lambda_{0}r}\nabla
u(T-r,X_{r})\sigma(X_{r})\,dB_{r}.$
Since $u$ is bounded and continuous on $[0,T]\times\overline{B}$ and $\nabla
u(t,x)$ is locally bounded in $(0,T]\times B$, it follows from standard
localization arguments that
$\displaystyle u(T,x)$
$\displaystyle=\mathbb{E}_{x}\left[e^{\lambda_{0}(T\wedge\tau_{B^{c}})}u(T-(T\wedge\tau_{B^{c}}),X_{T\wedge\tau_{B^{c}}})\right]$
$\displaystyle=\mathbb{E}_{x}\left[e^{\lambda_{0}(T\wedge\tau_{B^{c}})}\eta(X_{T\wedge\tau_{B^{c}}})\right].$
Now, the Markov property and the fact that $P_{t}\eta=e^{-\lambda_{0}t}\eta$
entail that $e^{\lambda_{0}t}\eta(X_{t})$ is a martingale on
$(\mathcal{D},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P}_{x})$, hence
$\eta(x)=\mathbb{E}_{x}\left[e^{\lambda_{0}(T\wedge\tau_{B^{c}})}\eta(X_{T\wedge\tau_{B^{c}}})\right]=u(T,x).$
Therefore, $\eta\in\mathcal{C}^{2}(D)$ and
$\mathcal{L}\eta(x)=-\lambda_{0}\eta(x)$ for all $x\in D$.
## References
* [1] E. Arjas, E. Nummelin, and R. L. Tweedie. Semi-Markov processes on a general state space: $\alpha$-theory and quasistationarity. J. Austral. Math. Soc. Ser. A, 30(2):187–200, 1980/81.
* [2] K. B. Athreya and P. E. Ney. Branching processes. Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196.
* [3] R. Azaï s, J.-B. Bardet, A. Génadot, N. Krell, and P.-A. Zitt. Piecewise deterministic Markov process—recent results. In Journées MAS 2012, volume 44 of ESAIM Proc., pages 276–290. EDP Sci., Les Ulis, 2014.
* [4] M. Baudel and N. Berglund. Spectral theory for random poincaré maps. SIAM J. Appl. Math., 49(6):4319–4375, 2017.
* [5] N. Berglund and D. Landon. Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model. Nonlinearity, 25(8):2303–2335, 2012.
* [6] G. Birkhoff. Extensions of Jentzsch’s theorem. Trans. Amer. Math. Soc., 85:219–227, 1957.
* [7] F. M. Buchmann. Simulation of stopped diffusions. J. Comput. Phys., 202(2):446–462, 2005.
* [8] D. L. Burkholder. Martingale transforms. Ann. Math. Statist., 37:1494–1504, 1966.
* [9] P. Cattiaux, P. Collet, A. Lambert, S. Martínez, S. Méléard, and J. San Martín. Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab., 37(5):1926–1969, 2009.
* [10] P. Cattiaux and S. Méléard. Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non-extinction. J. Math. Biol., 60(6):797–829, 2010.
* [11] J. A. Cavender. Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Probab., 10(3):570–586, 1978.
* [12] N. Champagnat, K. Coulibaly-Pasquier, and D. Villemonais. Exponential convergence to quasi-stationary distribution for multi-dimensional diffusion processes. ArXiv e-prints, Mar. 2016.
* [13] N. Champagnat, P. Diaconis, and L. Miclo. On Dirichlet eigenvectors for neutral two-dimensional Markov chains. Electron. J. Probab., 17:no. 63, 41, 2012.
* [14] N. Champagnat and S. Rœlly. Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions. Electron. J. Probab., 13:no. 25, 777–810, 2008.
* [15] N. Champagnat and D. Villemonais. Exponential convergence to quasi-stationary distribution and Q-process. Probab. Theory Related Fields, 164(1):243–283, 2016.
* [16] N. Champagnat and D. Villemonais. Population processes with unbounded extinction rate conditioned to non-extinction. ArXiv e-prints, Nov. 2016.
* [17] N. Champagnat and D. Villemonais. Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing. ALEA Lat. Am. J. Probab. Math. Stat., 14(1):177–199, 2017.
* [18] N. Champagnat and D. Villemonais. Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes. ArXiv e-prints, Apr. 2017.
* [19] N. Champagnat and D. Villemonais. Uniform convergence of conditional distributions for absorbed one-dimensional diffusions. J. Appl. Probab., June 2017. To appear.
* [20] N. Champagnat and D. Villemonais. Uniform convergence of penalized time-inhomogeneous Markov processes. ESAIM Probab. Stat., Mar. 2018. To appear.
* [21] S. D. Chatterji. An $L^{p}$-convergence theorem. Ann. Math. Statist., 40:1068–1070, 1969.
* [22] J.-R. Chazottes, P. Collet, and S. Méléard. Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes. Probab. Theory Related Fields, 164(1-2):285–332, 2016.
* [23] J.-R. Chazottes, P. Collet, and S. Méléard. On time scales and quasi-stationary distributions for multitype birth-and-death processes. ArXiv e-prints, Feb. 2017.
* [24] P. Collet, S. Martínez, and J. San Martín. Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Probab., 23(3):1300–1314, 1995.
* [25] P. Collet, S. Martínez, and J. San Martín. Quasi-stationary distributions. Probability and its Applications (New York). Springer, Heidelberg, 2013\. Markov chains, diffusions and dynamical systems.
* [26] C. Coron. Stochastic modeling and eco-evolution of a diploid population. PhD thesis, Palaiseau, Ecole polytechnique, 2013.
* [27] C. Coron, S. Méléard, E. Porcher, and A. Robert. Quantifying the mutational meltdown in diploid populations. The American Naturalist, 181(5):623–636, 2013.
* [28] J. N. Darroch and E. Seneta. On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probab., 2:88–100, 1965.
* [29] J. N. Darroch and E. Seneta. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probab., 4:192–196, 1967.
* [30] E. B. Davies and B. Simon. Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal., 59(2):335–395, 1984.
* [31] P. Del Moral. Feynman-Kac formulae. Probability and its Applications (New York). Springer-Verlag, New York, 2004. Genealogical and interacting particle systems with applications.
* [32] P. Del Moral. Mean field simulation for Monte Carlo integration, volume 126 of Monographs on Statistics and Applied Probability. CRC Press, Boca Raton, FL, 2013.
* [33] P. Del Moral and D. Villemonais. Exponential mixing properties for time inhomogeneous diffusion processes with killing. Bernoulli, 24(2):1010–1032, 2018.
* [34] M. Faure and S. J. Schreiber. Quasi-stationary distributions for randomly perturbed dynamical systems. Ann. Appl. Probab., 24(2):553–598, 2014.
* [35] P. A. Ferrari, H. Kesten, and S. Martínez. $R$-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata. Ann. Appl. Probab., 6(2):577–616, 1996.
* [36] P. A. Ferrari, H. Kesten, S. Martínez, and P. Picco. Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab., 23(2):501–521, 1995.
* [37] P. A. Ferrari, S. Martínez, and P. Picco. Some properties of quasi-stationary distributions in the birth and death chains: a dynamical approach. In Instabilities and nonequilibrium structures, III (Valparaíso, 1989), volume 64 of Math. Appl., pages 177–187. Kluwer Acad. Publ., Dordrecht, 1991.
* [38] P. A. Ferrari, S. Martínez, and P. Picco. Existence of nontrivial quasi-stationary distributions in the birth-death chain. Adv. Appl. Probab., 24(4):795–813, 1992.
* [39] A. Friedman. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
* [40] E. Gobet. Weak approximation of killed diffusion using Euler schemes. Stochastic Process. Appl., 87(2):167–197, 2000.
* [41] E. Gobet. Euler schemes and half-space approximation for the simulation of diffusion in a domain. ESAIM Probab. Statist., 5:261–297, 2001.
* [42] G. L. Gong, M. P. Qian, and Z. X. Zhao. Killed diffusions and their conditioning. Probab. Theory Related Fields, 80(1):151–167, 1988.
* [43] P. Good. The limiting behavior of transient birth and death processes conditioned on survival. J. Austral. Math. Soc., 8:716–722, 1968.
* [44] F. Gosselin. Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology. Ann. Appl. Probab., 11(1):261–284, 2001.
* [45] M. Hairer. Convergence of markov processes (lecture notes). www.hairer.org/notes/Convergence.pdf, 2010.
* [46] T. E. Harris. The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119. Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963.
* [47] C. R. Heathcote, E. Seneta, and D. Vere-Jones. A refinement of two theorems in the theory of branching processes. Teor. Verojatnost. i Primenen., 12:341–346, 1967.
* [48] A. Hening and M. Kolb. Quasistationary distributions for one-dimensional diffusions with singular boundary points. ArXiv e-prints, Sept. 2014.
* [49] G. Hinrichs, M. Kolb, and V. Wachtel. Persistence of one-dimensional AR(1)-sequences. ArXiv e-prints, Jan. 2018.
* [50] K. Itô and H. P. McKean, Jr. Diffusion processes and their sample paths, volume 125 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 1974. Second printing, corrected.
* [51] A. Joffe and F. Spitzer. On multitype branching processes with $\rho\leq 1$. J. Math. Anal. Appl., 19:409–430, 1967.
* [52] O. Kallenberg. Foundations of modern probability. Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2002.
* [53] I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.
* [54] S. Karlin and J. McGregor. The classification of birth and death processes. Trans. Amer. Math. Soc., 86:366–400, 1957.
* [55] M. Kijima and E. Seneta. Some results for quasi-stationary distributions of birth-death processes. J. Appl. Probab., 28(3):503–511, 1991.
* [56] J. F. C. Kingman. The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3), 13:337–358, 1963.
* [57] R. Knobloch and L. Partzsch. Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Anal., 33:107–136, 2010.
* [58] M. Kolb and D. Steinsaltz. Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Probab., 40(1):162–212, 2012.
* [59] N. V. Krylov and M. V. Safonov. A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat., 44(1):161–175, 239, 1980.
* [60] A. Lambert. Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab., 12:no. 14, 420–446, 2007.
* [61] J. Littin C. Uniqueness of quasistationary distributions and discrete spectra when $\infty$ is an entrance boundary and 0 is singular. J. Appl. Probab., 49(3):719–730, 2012.
* [62] M. Lladser and J. San Martín. Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process. J. Appl. Probab., 37(2):511–520, 2000.
* [63] P. Maillard. The $\lambda$-invariant measures of subcritical Bienaymé-Galton-Watson processes. Bernoulli, 24(1):297–315, 2018.
* [64] P. Mandl. Spectral theory of semi-groups connected with diffusion processes and its application. Czechoslovak Math. J., 11 (86):558–569, 1961.
* [65] R. Mannella. Absorbing boundaries and optimal stopping in a stochastic differential equation. Phys. Lett. A, 254(5):257–262, 1999.
* [66] S. Martínez and J. San Martín. Classification of killed one-dimensional diffusions. Ann. Probab., 32(1A):530–552, 2004.
* [67] S. Martínez, J. San Martín, and D. Villemonais. Existence and uniqueness of a quasistationary distribution for Markov processes with fast return from infinity. J. Appl. Probab., 51(3):756–768, 2014.
* [68] S. Méléard and D. Villemonais. Quasi-stationary distributions and population processes. Probab. Surv., 9:340–410, 2012.
* [69] S. Meyn and R. L. Tweedie. Markov chains and stochastic stability. Cambridge University Press, Cambridge, second edition, 2009. With a prologue by Peter W. Glynn.
* [70] S. P. Meyn and R. L. Tweedie. Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab., 25(3):518–548, 1993.
* [71] Y. Miura. Ultracontractivity for Markov semigroups and quasi-stationary distributions. Stoch. Anal. Appl., 32(4):591–601, 2014.
* [72] Y. Ogura. Asymptotic behavior of multitype Galton-Watson processes. J. Math. Kyoto Univ., 15(2):251–302, 1975.
* [73] R. G. Pinsky. On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes. Ann. Probab., 13(2):363–378, 1985.
* [74] R. G. Pinsky. Explicit and almost explicit spectral calculations for diffusion operators. J. Funct. Anal., 256(10):3279–3312, 2009.
* [75] P. E. Protter. Stochastic integration and differential equations, volume 21 of Applications of Mathematics (New York). Springer-Verlag, Berlin, second edition, 2004. Stochastic Modelling and Applied Probability.
* [76] B. Roynette, P. Vallois, and M. Yor. Some penalisations of the Wiener measure. Jpn. J. Math., 1(1):263–290, 2006.
* [77] E. Seneta and D. Vere-Jones. On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab., 3:403–434, 1966.
* [78] D. Steinsaltz and S. N. Evans. Markov mortality models: Implications of quasistationarity and varying initial conditions. Theo. Pop. Bio., 65(65):319–337, 2004.
* [79] D. W. Stroock and S. R. S. Varadhan. Multidimensional diffusion processes. Classics in Mathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1997 edition.
* [80] E. A. van Doorn. Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Probab., 23(4):683–700, 1991.
* [81] E. A. van Doorn. Conditions for the existence of quasi-stationary distributions for birth-death processes with killing. Stochastic Process. Appl., 122(6):2400–2410, 2012.
* [82] E. A. van Doorn and P. K. Pollett. Quasi-stationary distributions for discrete-state models. European J. Oper. Res., 230(1):1–14, 2013.
* [83] A. Velleret. Non-uniform exponential convergence to qsd, with a possibly stabilizing extinction. In preparation, 2018.
* [84] D. Vere-Jones. Ergodic properties of nonnegative matrices. I. Pacific J. Math., 22:361–386, 1967.
* [85] D. Villemonais. Minimal quasi-stationary distribution approximation for a birth and death process. Electron. J. Probab., 20:no. 30, 18, 2015.
* [86] J. Wang. First eigenvalue of one-dimensional diffusion processes. Electron. Commun. Probab., 14:232–244, 2009.
* [87] J. Wang. Sharp bounds for the first eigenvalue of symmetric Markov processes and their applications. Acta Math. Sin. (Engl. Ser.), 28(10):1995–2010, 2012.
* [88] A. M. Yaglom. Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.), 56:795–798, 1947. |
# Probing Pretrained Language Models for Lexical Semantics
Ivan Vulić♠ Edoardo M. Ponti♠ Robert Litschko♢ Goran Glavaš♢ Anna Korhonen♠
♠Language Technology Lab, University of Cambridge, UK
♢Data and Web Science Group, University of Mannheim, Germany
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract
The success of large pretrained language models (LMs) such as BERT and RoBERTa
has sparked interest in probing their representations, in order to unveil what
types of knowledge they implicitly capture. While prior research focused on
morphosyntactic, semantic, and world knowledge, it remains unclear to which
extent LMs also derive lexical type-level knowledge from words in context. In
this work, we present a systematic empirical analysis across six typologically
diverse languages and five different lexical tasks, addressing the following
questions: 1) How do different lexical knowledge extraction strategies
(monolingual versus multilingual source LM, out-of-context versus in-context
encoding, inclusion of special tokens, and layer-wise averaging) impact
performance? How consistent are the observed effects across tasks and
languages? 2) Is lexical knowledge stored in few parameters, or is it
scattered throughout the network? 3) How do these representations fare against
traditional static word vectors in lexical tasks? 4) Does the lexical
information emerging from independently trained monolingual LMs display latent
similarities? Our main results indicate patterns and best practices that hold
universally, but also point to prominent variations across languages and
tasks. Moreover, we validate the claim that lower Transformer layers carry
more type-level lexical knowledge, but also show that this knowledge is
distributed across multiple layers.
## 1 Introduction and Motivation
Language models (LMs) based on deep Transformer networks Vaswani et al.
(2017), pretrained on unprecedentedly large amounts of text, offer unmatched
performance in virtually every NLP task Qiu et al. (2020). Models such as BERT
Devlin et al. (2019), RoBERTa Liu et al. (2019c), and T5 Raffel et al. (2019)
replaced task-specific neural architectures that relied on static word
embeddings (WEs; Mikolov et al., 2013b; Pennington et al., 2014; Bojanowski et
al., 2017), where each word is assigned a single (type-level) vector.
While there is a clear consensus on the effectiveness of pretrained LMs, a
body of recent research has aspired to understand why they work Rogers et al.
(2020). State-of-the-art models are “probed” to shed light on whether they
capture task-agnostic linguistic knowledge and structures Liu et al. (2019a);
Belinkov and Glass (2019); Tenney et al. (2019); e.g., they have been
extensively probed for syntactic knowledge (Hewitt and Manning, 2019; Jawahar
et al., 2019; Kulmizev et al., 2020; Chi et al., 2020, inter alia) and
morphology Edmiston (2020); Hofmann et al. (2020).
In this work, we put focus on uncovering and understanding how and where
lexical semantic knowledge is coded in state-of-the-art LMs. While preliminary
findings from Ethayarajh (2019) and Vulić et al. (2020) suggest that there is
a wealth of lexical knowledge available within the parameters of BERT and
other LMs, a systematic empirical study across different languages is
currently lacking.
We present such a study, spanning six typologically diverse languages for
which comparable pretrained BERT models and evaluation data are readily
available. We dissect the pipeline for extracting lexical representations, and
divide it into crucial components, including: the underlying source LM, the
selection of subword tokens, external corpora, and which Transformer layers to
average over. Different choices give rise to different extraction
configurations (see Table 1) which, as we empirically verify, lead to large
variations in task performance.
We run experiments and analyses on five diverse lexical tasks using standard
evaluation benchmarks: lexical semantic similarity (LSIM), word analogy
resolution (WA), bilingual lexicon induction (BLI), cross-lingual information
retrieval (CLIR), and lexical relation prediction (RELP). The main idea is to
aggregate lexical information into static type-level “BERT-based” word
embeddings and plug them into “the classical NLP pipeline” Tenney et al.
(2019), similar to traditional static word vectors. The chosen tasks can be
seen as “lexico-semantic probes” providing an opportunity to simultaneously 1)
evaluate the richness of lexical information extracted from different
parameters of the underlying pretrained LM on intrinsic (e.g., LSIM, WA) and
extrinsic lexical tasks (e.g., RELP); 2) compare different type-level
representation extraction strategies; and 3) benchmark “BERT-based” static
vectors against traditional static word embeddings such as fastText Bojanowski
et al. (2017).
Our study aims at providing answers to the following key questions: Q1) Do
lexical extraction strategies generalise across different languages and tasks,
or do they rather require language- and task-specific adjustments?; Q2) Is
lexical information concentrated in a small number of parameters and layers,
or scattered throughout the encoder?; Q3) Are “BERT-based” static word
embeddings competitive with traditional word embeddings such as fastText?; Q4)
Do monolingual LMs independently trained in multiple languages learn
structurally similar representations for words denoting similar concepts
(i.e., translation pairs)?
We observe that different languages and tasks indeed require distinct
configurations to reach peak performance, which calls for a careful tuning of
configuration components according to the specific task–language combination
at hand (Q1). However, several universal patterns emerge across languages and
tasks. For instance, lexical information is predominantly concentrated in
lower Transformer layers, hence excluding higher layers from the extraction
achieves superior scores (Q1 and Q2). Further, representations extracted from
single layers do not match in accuracy those extracted by averaging over
several layers (Q2). While static word representations obtained from
monolingual LMs are competitive or even outperform static fastText embeddings
in tasks such as LSIM, WA, and RELP, lexical representations from massively
multilingual models such as multilingual BERT (mBERT) are substantially worse
(Q1 and Q3). We also demonstrate that translation pairs indeed obtain similar
representations (Q4), but the similarity depends on the extraction
configuration, as well as on the typological distance between the two
languages.
## 2 Lexical Representations from Pretrained Language Models
Classical static word embeddings (Bengio et al., 2003; Mikolov et al., 2013b;
Pennington et al., 2014) are grounded in distributional semantics, as they
infer the meaning of each word type from its co-occurrence patterns. However,
LM-pretrained Transformer encoders have introduced at least two levels of
misalignment with the classical approach Peters et al. (2018); Devlin et al.
(2019). First, representations are assigned to word tokens and are affected by
the current context and position within a sentence (Mickus et al., 2020).
Second, tokens may correspond to subword strings rather than complete word
forms. This begs the question: do pretrained encoders still retain a notion of
lexical concepts, abstracted from their instances in texts?
Analyses of lexical semantic information in large pretrained LMs have been
limited so far, focusing only on the English language and on the task of word
sense disambiguation. Reif et al. (2019) showed that senses are encoded with
finer-grained precision in higher layers, to the extent that their
representation of the same token tends not to be self-similar across different
contexts (Ethayarajh, 2019; Mickus et al., 2020). As a consequence, we
hypothesise that abstract, type-level information could be codified in lower
layers instead. However, given the absence of a direct equivalent to a static
word type embedding, we still need to establish how to extract such type-level
information.
In prior work, contextualised representations (and attention weights) have
been interpreted in the light of linguistic knowledge mostly through probes.
These consist in learned classifier predicting annotations like POS tags
(Pimentel et al., 2020) and word senses (Peters et al., 2018; Reif et al.,
2019; Chang and Chen, 2019), or linear transformations to a space where
distances mirror dependency tree structures (Hewitt and Manning, 2019).111The
interplay between the complexity of a probe and its accuracy, as well as its
effect on the overall procedure, remain controversial (Pimentel et al., 2020;
Voita and Titov, 2020).
Figure 1: Illustration of the components denoting adopted extraction strategies, including source LM (top right), presence of context (bottom right), special tokens (top left), and layer-wise averaging (bottom left). Component | Label | Short Description
---|---|---
Source LM | mono | Language-specific (i.e., monolingually pretrained) BERT
multi | Multilingual BERT, pretrained on 104 languages (with shared subword vocabulary)
Context | iso | Each vocabulary word $w$ is encoded in isolation, without any external context
aoc-M | Average-over-context: average over word’s encodings from $M$ different contexts/sentences
Subword Tokens | nospec | Special tokens [CLS] and [SEP] are excluded from subword embedding averaging
all | Both special tokens [CLS] and [SEP] are included into subword embedding averaging
withcls | [CLS] is included into subword embedding averaging; [SEP] is excluded
Layerwise Avg | avg(L$\leq$n) | Average representations over all Transformer layers up to the $n$-th layer $L_{n}$ (included)
L$=$n | Only the representation from the layer $L_{n}$ is used
Table 1: Configuration components of word-level embedding extraction,
resulting in 24 possible configurations.
In this work, we explore several unsupervised word-level representation
extraction strategies and configurations for lexico-semantic tasks (i.e.,
probes), stemming from different combinations of the components detailed in
Table 1 and illustrated in Figure 1. In particular, we assess the impact of:
1) encoding tokens with monolingual LM-pretrained Transformers vs. with their
massively multilingual counterparts; 2) providing context around the target
word in input; 3) including special tokens like [CLS] and [SEP]; 4) averaging
across several layers as opposed to a single layer.222For clarity of
presentation, later in §4 we show results only for a representative selection
of configurations that are consistently better than the others
## 3 Experimental Setup
Pretrained LMs and Languages. Our selection of test languages is guided by the
following constraints: a) availability of comparable pretrained (language-
specific) monolingual LMs; b) availability of evaluation data; and c)
typological diversity of the sample, along the lines of recent initiatives in
multilingual NLP (Gerz et al., 2018; Hu et al., 2020; Ponti et al., 2020,
inter alia). We work with English (en), German (de), Russian (ru), Finnish
(fi), Chinese (zh), and Turkish (tr). We use monolingual uncased BERT Base
models for all languages, retrieved from the HuggingFace repository Wolf et
al. (2019).333https://huggingface.co/models; the links to the actual BERT
models are in the appendix. All BERT models comprise 12 768-dimensional
Transformer layers $\\{L_{1}\,\text{(bottom
layer)},\dots,L_{12}\,\text{(top)}\\}$ plus the input embedding layer
($L_{0}$), and 12 attention heads. We also experiment with multilingual BERT
(mBERT) Devlin et al. (2019) as the underlying LM, aiming to measure the
performance difference between language-specific and massively multilingual
LMs in our lexical probing tasks.
Word Vocabularies and External Corpora. We extract type-level representations
in each language for the top 100K most frequent words represented in the
respective fastText (FT) vectors, which were trained on lowercased monolingual
Wikipedias by Bojanowski et al. (2017). The equivalent vocabulary coverage
allows a direct comparison to fastText vectors, which we use as a baseline
static WE method in all evaluation tasks. To retain the same vocabulary across
all configurations, in aoc variants we back off to the related iso variant for
words that have zero occurrences in external corpora.
For all aoc vector variants, we leverage 1M sentences of maximum sequence
length 512, which we randomly sample from external corpora: Europarl Koehn
(2005) for en, de, fi, available via OPUS Tiedemann (2009); the United Nations
Parallel Corpus for ru and zh Ziemski et al. (2016), and monolingual tr WMT17
data Bojar et al. (2017).
Evaluation Tasks. We carry out the evaluation on five standard and diverse
lexical semantic tasks:
Task 1: Lexical semantic similarity (LSIM) is the most widespread intrinsic
task for evaluation of traditional word embeddings Hill et al. (2015). The
evaluation metric is the Spearman’s rank correlation between the average of
human-elicited semantic similarity scores for word pairs and the cosine
similarity between the respective type-level word vectors. We rely on the
recent comprehensive multilingual LSIM benchmark Multi-SimLex Vulić et al.
(2020), which covers 1,888 pairs in 13 languages. We focus on en, fi, zh, ru,
the languages represented in Multi-SimLex.
Task 2: Word Analogy (WA) is another common intrinsic task. We evaluate our
models on the Bigger Analogy Test Set (BATS) Drozd et al. (2016) with 99,200
analogy questions. We resort to the standard vector offset analogy resolution
method, searching for the vocabulary word $w_{d}\in V$ such that its vector
$d$ is obtained by $argmax_{d}(cos(d,c−a+b))$, where $a$, $b$, and $c$ are
word vectors of words $w_{a}$, $w_{b}$, and $w_{c}$ from the analogy
$w_{a}:w_{b}=w_{c}:x$. The search space comprises vectors of all words from
the vocabulary $V$, excluding $a$, $b$, and $c$. This task is limited to en,
and we report Precision@1 scores.
Task 3: Bilingual Lexicon Induction (BLI) is a standard task to evaluate the
“semantic quality” of static cross-lingual word embeddings (CLWEs) Gouws et
al. (2015); Ruder et al. (2019). We learn “BERT-based” CLWEs using a standard
mapping-based approach Mikolov et al. (2013a); Smith et al. (2017) with VecMap
Artetxe et al. (2018). BLI evaluation allows us to investigate the
“alignability” of monolingual type-level representations extracted for
different languages. We adopt the standard BLI evaluation setup from Glavaš et
al. (2019): 5K training word pairs are used to learn the mapping, and another
2K pairs as test data. We report standard Mean Reciprocal Rank (MRR) scores
for 10 language pairs spanning en, de, ru, fi, tr.
Task 4: Cross-Lingual Information Retrieval (CLIR). We follow the setup of
Litschko et al. (2018, 2019) and evaluate mapping-based CLWEs (the same ones
as on BLI) in a document-level retrieval task on the CLEF 2003
benchmark.444All test collections comprise 60 queries. The average document
collection size per language is 131K (ranging from 17K documents for ru to
295K for de). We use a simple CLIR model which showed competitive performance
in the comparative studies of Litschko et al. (2019) and Glavaš et al. (2019).
It embeds queries and documents as IDF-weighted sums of their corresponding
WEs from the CLWE space, and uses cosine similarity as the ranking function.
We report Mean Average Precision (MAP) scores for 6 language pairs covering
en, de, ru, fi.
Task 5: Lexical Relation Prediction (RELP). We probe if we can recover
standard lexical relations (i.e., synonymy, antonymy, hypernymy, meronymy,
plus no relation) from input type-level vectors. We rely on a state-of-the-art
neural model for RELP operating on type-level embeddings Glavaš and Vulić
(2018): the Specialization Tensor Model (STM) predicts lexical relations for
pairs of input word vectors based on multi-view projections of those
vectors.555Note that RELP is structurally different from the other four tasks:
instead of direct computations with word embeddings, called metric learning or
similarity-based evaluation Ruder et al. (2019), it uses them as features in a
neural architecture. We use the WordNet-based Fellbaum (1998) evaluation data
of Glavaš and Vulić (2018): they contain 10K annotated word pairs balanced by
class. Micro-averaged $F_{1}$ scores, averaged across 5 runs for each input
vector space (default STM setting), are reported for en and de.
(a) English
(b) Finnish
(c) Mandarin Chinese
(d) Russian
Figure 2: Spearman’s $\rho$ correlation scores for the lexical semantic
similarity task (LSIM) in four languages. For the representation extraction
configurations in the legend, see Table 1. Thick solid horizontal lines denote
performance of standard monolingual fastText vectors trained on Wikipedia
dumps of the respective languages.
(e) Summary BLI results
(f) Summary CLIR results
Figure 3: Summary results for the two cross-lingual evaluation tasks: (a) BLI
(MRR scores) and (b) CLIR (MAP scores). We report average scores over all
language pairs; individual results for each language pair are available in the
appendix. Thick solid horizontal lines denote performance of standard fastText
vectors in exactly the same cross-lingual mapping setup.
(a) RELP: English
(b) RELP: German
(c) WA: English
Figure 4: Micro-averaged $F_{1}$ scores in the RELP task for (a) en and (b)
de. The scores with 768-dim vectors randomly initalized via Xavier init Glorot
and Bengio (2010) are 0.473 (en) and 0.512 (de); (c) en WA results.
## 4 Results and Discussion
A summary of the results is shown in Figure 4 for LSIM, in Figure 2(e) for
BLI, in Figure 2(f) for CLIR, in Figure 3(a) and Figure 3(b) for RELP, and in
Figure 3(c) for WA. These results offer multiple axes of comparison, and the
ensuing discussion focuses on the central questions Q1-Q3 posed in §1.666Full
results are available in the appendix.
Monolingual versus Multilingual LMs. Results across all tasks validate the
intuition that language-specific monolingual LMs contain much more lexical
information for a particular target language than massively multilingual
models such as mBERT or XLM-R Artetxe et al. (2020). We see large drops
between mono.* and multi.* configurations even for very high-resource
languages (en and de), and they are even more prominent for fi and tr.
Encompassing 100+ training languages with limited model capacity, multilingual
models suffer from the “curse of multilinguality” Conneau et al. (2020): they
must trade off monolingual lexical information coverage (and consequently
monolingual performance) for a wider language coverage.777For a particular
target language, monolingual performance can be partially recovered by
additional in-language monolingual training via masked language modeling
Eisenschlos et al. (2019); Pfeiffer et al. (2020). In a side experiment, we
have also verified that the same holds for lexical information coverage.
How Important is Context? Another observation that holds across all
configurations concerns the usefulness of providing contexts drawn from
external corpora, and corroborates findings from prior work Liu et al.
(2019b): iso configurations cannot match configurations that average subword
embeddings from multiple contexts (aoc-10 and aoc-100). However, it is worth
noting that 1) performance gains with aoc-100 over aoc-10, although
consistent, are quite marginal across all tasks: this suggests that several
word occurrences in vivo are already sufficient to accurately capture its
type-level representation. 2) In some tasks, iso configurations are only
marginally outscored by their aoc counterparts: e.g., for
mono.*.nospec.avg(L$\leq$8) on en–fi BLI or de–tr BLI, the respective scores
are 0.486 and 0.315 with iso, and 0.503 and 0.334 with aoc-10. Similar
observations hold for fi and zh LSIM, and also in the RELP task.
In RELP, it is notable that ‘BERT-based’ embeddings can recover more lexical
relation knowledge than standard FT vectors. These findings reveal that
pretrained LMs indeed implicitly capture plenty of lexical type-level
knowledge (which needs to be ‘recovered’ from the models); this also suggests
why pretrained LMs have been successful in tasks where this knowledge is
directly useful, such as NER and POS tagging Tenney et al. (2019); Tsai et al.
(2019). Finally, we also note that gains with aoc over iso are much more
pronounced for the under-performing multi.* configurations: this indicates
that mono models store more lexical information even in absence of context.
How Important are Special Tokens? The results reveal that the inclusion of
special tokens [CLS] and [SEP] into type-level embedding extraction
deteriorates the final lexical information contained in the embeddings. This
finding holds for different languages, underlying LMs, and averaging across
various layers. The nospec configurations consistently outperform their all
and withcls counterparts, both in iso and aoc-{10, 100} settings.888For this
reason, we report the results of aoc configurations only in the nospec
setting.
Our finding at the lexical level aligns well with prior observations on using
BERT directly as a sentence encoder Qiao et al. (2019); Singh et al. (2019);
Casanueva et al. (2020): while [CLS] is useful for sentence-pair
classification tasks, using [CLS] as a sentence representation produces
inferior representations than averaging over sentence’s subwords. In this
work, we show that [CLS] and [SEP] should also be fully excluded from subword
averaging for type-level word representations.
How Important is Layer-wise Averaging? Averaging across layers bottom-to-top
(i.e., from $L_{0}$ to $L_{12}$) is beneficial across the board, but we notice
that scores typically saturate or even decrease in some tasks and languages
when we include higher layers into averaging: see the scores with
*.avg(L$\leq$10) and *.avg(L$\leq$12) configurations, e.g., for fi LSIM; en/de
RELP, and summary BLI and CLIR scores. This hints to the fact that two
strategies typically used in prior work, either to take the vectors only from
the embedding layer $L_{0}$ Wu et al. (2020); Wang et al. (2019) or to average
across all layers Liu et al. (2019b), extract sub-optimal word representations
for a wide range of setups and languages.
Figure 5: CKA similarity scores of type-level word representations extracted
from each layer (using different extraction configurations, see Table 1) for a
set of 7K translation pairs in en–de, en–fi, and en–tr from the BLI
dictionaries of Glavaš et al. (2019). Additional heatmaps (where random words
from two languages are paired) are available in the appendix.
(a) en–ru: Word translation pairs
(b) en–ru: Random word pairs
Figure 6: CKA similarity scores of type-level word representations extracted
from each layer for a set of (a) 7K en–ru translation pairs from the BLI
dictionaries of Glavaš et al. (2019); (b) 7K random en–ru pairs.
Figure 7: Self-similarity heatmaps: linear CKA similarity of representations
for the same word extracted from different Transformer layers, averaged across
7K words for English and Finnish. mono.aoc-100.nospec.
$L_{0}$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$
$L_{9}$ $L_{10}$ $L_{11}$ $L_{12}$ LSIM en .503 .513 .505 .510 .505 .484 .459
.435 .402 .361 .362 .372 .390 fi .445 .466 .445 .436 .430 .434 .421 .404 .374
.346 .333 .324 .286 WA en .220 .272 .293 .285 .293 .261 .240 .217 .199 .171
.189 .221 .229 BLI en–de .310 .354 .379 .400 .394 .393 .373 .358 .311 .272
.273 .264 .287 en–fi .309 .339 .360 .367 .369 .345 .329 .303 .279 .252 .231
.194 .192 de–fi .211 .245 .268 .283 .289 .303 .291 .292 .288 .282 .262 .219
.236 CLIR en–de .059 .060 .059 .060 .043 .036 .036 .036 .027 .024 .027 .035
.038 en–fi .038 .040 .022 .018 .011 .008 .006 .006 .005 .002 .003 .002 .007
de–fi .054 .057 .028 .015 .016 .022 .017 .021 .020 .023 .015 .008 .030
Table 2: Task performance of word representations extracted from different
Transformer layers for a selection of tasks, languages, and language pairs.
Configuration: mono.aoc-100.nospec. Highest scores per row are in bold.
The sweet spot for $n$ in *.avg(L$\leq$n) configurations seems largely task-
and language-dependent, as peak scores are obtained with different $n$-s.
Whereas averaging across all layers generally hurts performance, the results
strongly suggest that averaging across layer subsets (rather than selecting a
single layer) is widely useful, especially across bottom-most layers: e.g.,
$L\leq 6$ with mono.iso.nospec yields an average score of 0.561 in LSIM, 0.076
in CLIR, and 0.432 in BLI; the respective scores when averaging over the 6 top
layers are: 0.218, 0.008, and 0.230. This evidence implies that, although
scattered across multiple layers, type-level lexical information seems to be
concentrated in lower Transformer layers. We investigate these conjectures
further in §4.1.
Comparison to Static Word Embeddings. The results also offer a comparison to
static FT vectors across languages. The best-performing extraction
configurations (e.g., mono.aoc-100.nospec) outperform FT in monolingual
evaluations on LSIM (for en, fi, zh), WA, and they also display much stronger
performance in the RELP task for both evaluation languages. While the
comparison is not strictly apples-to-apples, as FT and LMs were trained on
different (Wikipedia) corpora, these findings leave open a provocative
question for future work: Given that static type-level word representations
can be recovered from large pretrained LMs, does this make standard static WEs
obsolete, or are there applications where they are still useful?
The trend is opposite in the two cross-lingual tasks: BLI and CLIR. While
there are language pairs for which ‘BERT-based’ WEs outperform FT (i.e., en–fi
in BLI, en–ru and fi–ru in CLIR) or are very competitive to FT’s performance
(e.g., en–tr, tr–BLI, de–ru CLIR), FT provides higher scores overall in both
tasks. The discrepancy between results in monolingual versus cross-lingual
tasks warrants further investigation in future work. For instance, is using
linear maps, as in standard mapping approaches to CLWE induction, sub-optimal
for ‘BERT-based’ word vectors?
Differences across Languages and Tasks. Finally, while we observe a
conspicuous amount of universal patterns with configuration components (e.g.,
mono $>$ multi; aoc $>$ iso; nospec $>$ all, withcls), best-performing
configurations do show some variation across different languages and tasks.
For instance, while en LSIM performance declines modestly but steadily when
averaging over higher-level layers (avg(L$\leq n$), where $n>4$), performance
on en WA consistently increases for the same configurations. The BLI and CLIR
scores in Figures 2(e) and 2(f) also show slightly different patterns across
layers. Overall, this suggests that 1) extracted lexical information must be
guided by task requirements, and 2) config components must be carefully tuned
to maximise performance for a particular task–language combination.
### 4.1 Lexical Information in Individual Layers
Evaluation Setup. To better understand which layers contribute the most to the
final performance in our lexical tasks, we also probe type-level
representations emerging from each individual layer of pretrained LMs. For
brevity, we focus on the best performing configurations from previous
experiments: {mono, mbert}.{iso, aoc-100}.nospec.
In addition, tackling Q4 from §1, we analyse the similarity of representations
extracted from monolingual and multilingual BERT models using the centered
kernel alignment (CKA) as proposed by Kornblith et al. (2019). The linear CKA
computes similarity that is invariant to isotropic scaling and orthogonal
transformation. It is defined as
$\displaystyle\text{CKA}(X,Y)=\frac{\left\lVert
Y^{\top}X\right\rVert^{2}_{\text{F}}}{\left(\left\lVert
X^{\top}X\right\rVert_{\text{F}}\left\lVert
Y^{\top}Y\right\rVert_{\text{F}}\right)}.$ (1)
$X,Y\in\mathbb{R}^{s\times d}$ are input matrices spanning $s$
$\ell_{2}$-normalized and mean-centered examples of dimensionality $d=768$. We
use CKA in two different experiments: 1) measuring self-similarity where we
compute CKA similarity of representations extracted from different layers for
the same word; and 2) measuring bilingual layer correspondence where we
compute CKA similarity of representations extracted from the same layer for
two words constituting a translation pair. To this end, we again use BLI
dictionaries of Glavaš et al. (2019) (see §3) covering 7K pairs (training +
test pairs).
Discussion. Per-layer CKA similarities are provided in Figure 7 (self-
similarity) and Figure 5 (bilingual), and we show results of representations
extracted from individual layers for selected evaluation setups and languages
in Table 2. We also plot bilingual layer correspondence of true word
translations versus randomly paired words for en–ru in Figure 6. Figure 7
reveals very similar patterns for both en and fi, and we also observe that
self-similarity scores decrease for more distant layers (cf., similarity of
$L_{1}$ and $L_{2}$ versus $L_{1}$ and $L_{12}$). However, despite structural
similarities identified by linear CKA, the scores from Table 2 demonstrate
that structurally similar layers might encode different amounts of lexical
information: e.g., compare performance drops between $L_{5}$ and $L_{8}$ in
all evaluation tasks.
The results in Table 2 further suggest that more type-level lexical
information is available in lower layers, as all peak scores in the table are
achieved with representations extracted from layers $L_{1}-L_{5}$. Much lower
scores in type-level semantic tasks for higher layers also empirically
validate a recent hypothesis of Ethayarajh (2019) “that contextualised word
representations are more context-specific in higher layers.” We also note that
none of the results with L=$n$ configurations from Table 1 can match best
performing avg(L$\leq$n) configurations with layer-wise averaging. This
confirms our hypothesis that type-level lexical knowledge, although
predominantly captured by lower layers, is disseminated across multiple
layers, and layer-wise averaging is crucial to uncover that knowledge.
Further, Figure 5 and Figure 6 reveal that even LMs trained on monolingual
data learn similar representations in corresponding layers for word
translations (see the mono.aoc columns). Intuitively, this similarity is much
more pronounced with aoc configurations with mBERT. The comparison of scores
in Figure 6 also reveals much higher correspondence scores for true
translation pairs than for randomly paired words (i.e., the correspondence
scores for random pairings are, as expected, random). Moreover, multi CKA
similarity scores turn out to be higher for more similar language pairs (cf.
en–de versus en–tr multi.aoc columns). This suggests that, similar to static
WEs, type-level ‘BERT-based’ WEs of different languages also display
topological similarity, often termed approximate isomorphism Søgaard et al.
(2018), but its degree depends on language proximity. This also clarifies why
representations extracted from two independently trained monolingual LMs can
be linearly aligned, as validated by BLI and CLIR evaluation (Table 2 and
Figure 4).999Previous work has empirically validated that sentence
representations for semantically similar inputs from different languages are
less similar in higher Transformer layers Singh et al. (2019); Wu and Dredze
(2019). In Figure 5, we demonstrate that this is also the case for type-level
lexical information; however, unlike sentence representations where highest
similarity is reported in lowest layers, Figure 5 suggests that highest CKA
similarities are achieved in intermediate layers $L_{5}$-$L_{8}$.
We also calculated the Spearman’s correlation between CKA similarity scores
for configurations mono.aoc-100.nospec.avg(L$\leq$n), for all $n=0,\ldots,12$,
and their corresponding BLI scores on en–fi, en–de, and de–fi. The
correlations are very high: $\rho=1.0,0.83,0.99$, respectively. This further
confirms the approximate isomorphism hypothesis: it seems that higher
structural similarities of representations extracted from monolingual
pretrained LMs facilitate their cross-lingual alignment.
## 5 Further Discussion and Conclusion
What about Larger LMs and Corpora? Aspects of LM pretraining, such as the
number of model parameters or the size of pretraining data, also impact
lexical knowledge stored in the LM’s parameters. Our preliminary experiments
have verified that en BERT-Large yields slight gains over the en BERT-Base
architecture used in our work (e.g., peak en LSIM scores rise from 0.518 to
0.531). In a similar vein, we have run additional experiments with two
available Italian (it) BERT-Base models with identical parameter setups, where
one was trained on 13GB of it text, and the other on 81GB. In en (BERT-
Base)–it BLI and CLIR evaluations we measure improvements from 0.548 to 0.572
(BLI), and from 0.148 to 0.160 (CLIR) with the 81GB it model. In-depth
analyses of these factors are out of the scope of this work, but they warrant
further investigations.
Opening Future Research Avenues. Our study has empirically validated that
(monolingually) pretrained LMs store a wealth of type-level lexical knowledge,
but effectively uncovering and extracting such knowledge from the LMs’
parameters depends on several crucial components (see §2). In particular, some
universal choices of configuration can be recommended: i) choosing monolingual
LMs; ii) encoding words with multiple contexts; iii) excluding special tokens;
iv) averaging over lower layers. Moreover, we found that type-level WEs
extracted from pretrained LMs can surpass static WEs like fastText (Bojanowski
et al., 2017).
This study has only scratched the surface of this research avenue. In future
work, we plan to investigate how domains of external corpora affect aoc
configurations, and how to sample representative contexts from the corpora. We
will also extend the study to more languages, more lexical semantic probes,
and other larger underlying LMs. The difference in performance across layers
also calls for more sophisticated lexical representation extraction methods
(e.g., through layer weighting or attention) similar to meta-embedding
approaches Yin and Schütze (2016); Bollegala and Bao (2018); Kiela et al.
(2018). Given the current large gaps between monolingual and multilingual LMs,
we will also focus on lightweight methods to enrich lexical content in
multilingual LMs Wang et al. (2020); Pfeiffer et al. (2020).
## Acknowledgments
This work is supported by the ERC Consolidator Grant LEXICAL: Lexical
Acquisition Across Languages (no 648909) awarded to Anna Korhonen. The work of
Goran Glavaš and Robert Litschko is supported by the Baden-Württemberg
Stiftung (AGREE grant of the Eliteprogramm).
## References
* Artetxe et al. (2018) Mikel Artetxe, Gorka Labaka, and Eneko Agirre. 2018. A robust self-learning method for fully unsupervised cross-lingual mappings of word embeddings. In _Proceedings of ACL_ , pages 789–798.
* Artetxe et al. (2020) Mikel Artetxe, Sebastian Ruder, and Dani Yogatama. 2020. On the cross-lingual transferability of monolingual representations. In _Proceedings of ACL_ , pages 4623–4637.
* Belinkov and Glass (2019) Yonatan Belinkov and James R. Glass. 2019. Analysis methods in neural language processing: A survey. _Transactions of the Association of Computational Linguistics_ , 7:49–72.
* Bengio et al. (2003) Yoshua Bengio, Réjean Ducharme, Pascal Vincent, and Christian Jauvin. 2003. A neural probabilistic language model. _Journal of Machine Learning Research_ , 3:1137–1155.
* Bojanowski et al. (2017) Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov. 2017. Enriching word vectors with subword information. _Transactions of the ACL_ , 5:135–146.
* Bojar et al. (2017) Ondřej Bojar, Rajen Chatterjee, Christian Federmann, Yvette Graham, Barry Haddow, Shujian Huang, Matthias Huck, Philipp Koehn, Qun Liu, Varvara Logacheva, Christof Monz, Matteo Negri, Matt Post, Raphael Rubino, Lucia Specia, and Marco Turchi. 2017. Findings of the 2017 Conference on Machine Translation (WMT17). In _Proceedings of WMT_ , pages 169–214.
* Bollegala and Bao (2018) Danushka Bollegala and Cong Bao. 2018. Learning word meta-embeddings by autoencoding. In _Proceedings of COLING_ , pages 1650–1661.
* Casanueva et al. (2020) Iñigo Casanueva, Tadas Temčinas, Daniela Gerz, Matthew Henderson, and Ivan Vulić. 2020. Efficient intent detection with dual sentence encoders. In _Proceedings of the 2nd Workshop on Natural Language Processing for Conversational AI_ , pages 38–45.
* Chang and Chen (2019) Ting-Yun Chang and Yun-Nung Chen. 2019. What does this word mean? Explaining contextualized embeddings with natural language definition. In _Proceedings of EMNLP-IJCNLP_ , pages 6064–6070.
* Chi et al. (2020) Ethan A. Chi, John Hewitt, and Christopher D. Manning. 2020. Finding universal grammatical relations in multilingual BERT. In _Proceedings of ACL_ , pages 5564–5577.
* Conneau et al. (2020) Alexis Conneau, Kartikay Khandelwal, Naman Goyal, Vishrav Chaudhary, Guillaume Wenzek, Francisco Guzmán, Edouard Grave, Myle Ott, Luke Zettlemoyer, and Veselin Stoyanov. 2020. Unsupervised cross-lingual representation learning at scale. In _Proceedings of ACL_ , pages 8440–8451.
* Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. BERT: Pre-training of deep bidirectional transformers for language understanding. In _Proceedings of NAACL-HLT_ , pages 4171–4186.
* Drozd et al. (2016) Aleksandr Drozd, Anna Gladkova, and Satoshi Matsuoka. 2016. Word embeddings, analogies, and machine learning: Beyond king - man + woman = queen. In _Proceedings of COLING_ , pages 3519–3530.
* Edmiston (2020) Daniel Edmiston. 2020. A systematic analysis of morphological content in BERT models for multiple languages. _CoRR_ , abs/2004.03032.
* Eisenschlos et al. (2019) Julian Eisenschlos, Sebastian Ruder, Piotr Czapla, Marcin Kardas, Sylvain Gugger, and Jeremy Howard. 2019. MultiFiT: Efficient multi-lingual language model fine-tuning. In _Proceedings of EMNLP-IJCNLP_ , pages 5701–5706.
* Ethayarajh (2019) Kawin Ethayarajh. 2019. How contextual are contextualized word representations? Comparing the geometry of BERT, ELMo, and GPT-2 embeddings. In _Proceedings of EMNLP-IJCNLP_ , pages 55–65.
* Fellbaum (1998) Christiane Fellbaum. 1998. _WordNet_. MIT Press.
* Gerz et al. (2018) Daniela Gerz, Ivan Vulić, Edoardo Maria Ponti, Roi Reichart, and Anna Korhonen. 2018. On the relation between linguistic typology and (limitations of) multilingual language modeling. In _Proceedings of EMNLP_ , pages 316–327.
* Glavaš and Vulić (2018) Goran Glavaš and Ivan Vulić. 2018. Discriminating between lexico-semantic relations with the specialization tensor model. In _Proceedings of NAACL-HLT_ , pages 181–187.
* Glavaš et al. (2019) Goran Glavaš, Robert Litschko, Sebastian Ruder, and Ivan Vulić. 2019. How to (properly) evaluate cross-lingual word embeddings: On strong baselines, comparative analyses, and some misconceptions. In _Proceedings of ACL_ , pages 710–721.
* Glorot and Bengio (2010) Xavier Glorot and Yoshua Bengio. 2010. Understanding the difficulty of training deep feedforward neural networks. In _Proceedings of AISTATS_ , pages 249–256.
* Gouws et al. (2015) Stephan Gouws, Yoshua Bengio, and Greg Corrado. 2015. BilBOWA: Fast bilingual distributed representations without word alignments. In _Proceedings of ICML_ , pages 748–756.
* Hewitt and Manning (2019) John Hewitt and Christopher D. Manning. 2019. A structural probe for finding syntax in word representations. In _Proceedings of NAACL-HLT_ , pages 4129–4138.
* Hill et al. (2015) Felix Hill, Roi Reichart, and Anna Korhonen. 2015. SimLex-999: Evaluating semantic models with (genuine) similarity estimation. _Computational Linguistics_ , 41(4):665–695.
* Hofmann et al. (2020) Valentin Hofmann, Janet B. Pierrehumbert, and Hinrich Schütze. 2020. Generating derivational morphology with BERT. _CoRR_ , abs/2005.00672.
* Hu et al. (2020) Junjie Hu, Sebastian Ruder, Aditya Siddhant, Graham Neubig, Orhan Firat, and Melvin Johnson. 2020. XTREME: A massively multilingual multi-task benchmark for evaluating cross-lingual generalization. In _Proceedings of ICML_.
* Jawahar et al. (2019) Ganesh Jawahar, Benoît Sagot, and Djamé Seddah. 2019. What does BERT learn about the structure of language? In _Proceedings of ACL_ , pages 3651–3657.
* Kiela et al. (2018) Douwe Kiela, Changhan Wang, and Kyunghyun Cho. 2018. Dynamic meta-embeddings for improved sentence representations. In _Proceedings of EMNLP_ , pages 1466–1477.
* Koehn (2005) Philipp Koehn. 2005. Europarl: A parallel corpus for statistical machine translation. In _Proceedings of the 10th Machine Translation Summit (MT SUMMIT)_ , pages 79–86.
* Kornblith et al. (2019) Simon Kornblith, Mohammad Norouzi, Honglak Lee, and Geoffrey E. Hinton. 2019. Similarity of neural network representations revisited. In _Proceedings of ICML_ , pages 3519–3529.
* Kulmizev et al. (2020) Artur Kulmizev, Vinit Ravishankar, Mostafa Abdou, and Joakim Nivre. 2020. Do neural language models show preferences for syntactic formalisms? In _Proceedings of ACL_ , pages 4077–4091.
* Litschko et al. (2018) Robert Litschko, Goran Glavaš, Simone Paolo Ponzetto, and Ivan Vulić. 2018\. Unsupervised cross-lingual information retrieval using monolingual data only. In _Proceedings of SIGIR_ , pages 1253–1256.
* Litschko et al. (2019) Robert Litschko, Goran Glavaš, Ivan Vulić, and Laura Dietz. 2019. Evaluating resource-lean cross-lingual embedding models in unsupervised retrieval. In _Proceedings of SIGIR_ , pages 1109–1112.
* Liu et al. (2019a) Nelson F. Liu, Matt Gardner, Yonatan Belinkov, Matthew E. Peters, and Noah A. Smith. 2019a. Linguistic knowledge and transferability of contextual representations. In _Proceedings of NAACL-HLT_ , pages 1073–1094.
* Liu et al. (2019b) Qianchu Liu, Diana McCarthy, Ivan Vulić, and Anna Korhonen. 2019b. Investigating cross-lingual alignment methods for contextualized embeddings with token-level evaluation. In _Proceedings of CoNLL_ , pages 33–43.
* Liu et al. (2019c) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019c. RoBERTa: A robustly optimized BERT pretraining approach. _CoRR_ , abs/1907.11692.
* Mickus et al. (2020) Timothee Mickus, Denis Paperno, Mathieu Constant, and Kees van Deemter. 2020. What do you mean, BERT? Assessing BERT as a distributional semantics model. _Proceedings of the Society for Computation in Linguistics_ , 3(34).
* Mikolov et al. (2013a) Tomas Mikolov, Quoc V. Le, and Ilya Sutskever. 2013a. Exploiting similarities among languages for machine translation. _arXiv preprint, CoRR_ , abs/1309.4168.
* Mikolov et al. (2013b) Tomas Mikolov, Ilya Sutskever, Kai Chen, Gregory S. Corrado, and Jeffrey Dean. 2013b. Distributed representations of words and phrases and their compositionality. In _Proceedings of NeurIPS_ , pages 3111–3119.
* Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014. Glove: Global vectors for word representation. In _Proceedings of EMNLP_ , pages 1532–1543.
* Peters et al. (2018) Matthew Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke Zettlemoyer. 2018. Deep contextualized word representations. In _Proceedings of NAACL-HLT_ , pages 2227–2237.
* Pfeiffer et al. (2020) Jonas Pfeiffer, Ivan Vulić, Iryna Gurevych, and Sebastian Ruder. 2020. MAD-X: An adapter-based framework for multi-task cross-lingual transfer. In _Proceedings of EMNLP_.
* Pimentel et al. (2020) Tiago Pimentel, Josef Valvoda, Rowan Hall Maudslay, Ran Zmigrod, Adina Williams, and Ryan Cotterell. 2020. Information-theoretic probing for linguistic structure. In _Proceedings of ACL_ , pages 4609–4622.
* Ponti et al. (2020) Edoardo Maria Ponti, Goran Glavaš, Olga Majewska, Qianchu Liu, Ivan Vulić, and Anna Korhonen. 2020. XCOPA: A multilingual dataset for causal commonsense reasoning. In _Proceedings of EMNLP_.
* Qiao et al. (2019) Yifan Qiao, Chenyan Xiong, Zheng-Hao Liu, and Zhiyuan Liu. 2019. Understanding the behaviors of BERT in ranking. _CoRR_ , abs/1904.07531.
* Qiu et al. (2020) Xipeng Qiu, Tianxiang Sun, Yige Xu, Yunfan Shao, Ning Dai, and Xuanjing Huang. 2020\. Pre-trained models for natural language processing: A survey. _CoRR_ , abs/2003.08271.
* Raffel et al. (2019) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2019. Exploring the limits of transfer learning with a unified text-to-text transformer. _CoRR_ , abs/1910.10683.
* Reif et al. (2019) Emily Reif, Ann Yuan, Martin Wattenberg, Fernanda B. Viegas, Andy Coenen, Adam Pearce, and Been Kim. 2019. Visualizing and measuring the geometry of BERT. In _Proceedings of NeurIPS_ , pages 8594–8603.
* Rogers et al. (2020) Anna Rogers, Olga Kovaleva, and Anna Rumshisky. 2020. A primer in BERTology: what we know about how BERT works. _Transactions of the ACL_.
* Ruder et al. (2019) Sebastian Ruder, Ivan Vulić, and Anders Søgaard. 2019. A survey of cross-lingual embedding models. _Journal of Artificial Intelligence Research_ , 65:569–631.
* Singh et al. (2019) Jasdeep Singh, Bryan McCann, Richard Socher, and Caiming Xiong. 2019. BERT is not an interlingua and the bias of tokenization. In _Proceedings of the 2nd Workshop on Deep Learning Approaches for Low-Resource NLP (DeepLo 2019)_ , pages 47–55.
* Smith et al. (2017) Samuel L. Smith, David H.P. Turban, Steven Hamblin, and Nils Y. Hammerla. 2017. Offline bilingual word vectors, orthogonal transformations and the inverted softmax. In _Proceedings of ICLR (Conference Track)_.
* Søgaard et al. (2018) Anders Søgaard, Sebastian Ruder, and Ivan Vulić. 2018. On the limitations of unsupervised bilingual dictionary induction. In _Proceedings of ACL_ , pages 778–788.
* Tenney et al. (2019) Ian Tenney, Dipanjan Das, and Ellie Pavlick. 2019. BERT rediscovers the classical NLP pipeline. In _Proceedings of ACL_ , pages 4593–4601.
* Tiedemann (2009) Jörg Tiedemann. 2009. News from OPUS - A collection of multilingual parallel corpora with tools and interfaces. In _Proceedings of RANLP_ , pages 237–248.
* Tsai et al. (2019) Henry Tsai, Jason Riesa, Melvin Johnson, Naveen Arivazhagan, Xin Li, and Amelia Archer. 2019. Small and practical BERT models for sequence labeling. In _Proceedings EMNLP-IJCNLP_ , pages 3632–3636.
* Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In _Proceedings of NeurIPS_ , pages 6000–6010.
* Voita and Titov (2020) Elena Voita and Ivan Titov. 2020. Information-theoretic probing with minimum description length. In _Proceedings of EMNLP_.
* Vulić et al. (2020) Ivan Vulić, Simon Baker, Edoardo Maria Ponti, Ulla Petti, Ira Leviant, Kelly Wing, Olga Majewska, Eden Bar, Matt Malone, Thierry Poibeau, Roi Reichart, and Anna Korhonen. 2020. Multi-Simlex: A large-scale evaluation of multilingual and cross-lingual lexical semantic similarity. _Computational Linguistics_.
* Wang et al. (2020) Ruize Wang, Duyu Tang, Nan Duan, Zhongyu Wei, Xuanjing Huang, Jianshu Ji, Guihong Cao, Daxin Jiang, and Ming Zhou. 2020. K-Adapter: Infusing knowledge into pre-trained models with adapters. _CoRR_ , abs/2002.01808.
* Wang et al. (2019) Yuxuan Wang, Wanxiang Che, Jiang Guo, Yijia Liu, and Ting Liu. 2019. Cross-lingual BERT transformation for zero-shot dependency parsing. In _Proceedings of EMNLP-IJCNLP_ , pages 5721–5727.
* Wolf et al. (2019) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, R’emi Louf, Morgan Funtowicz, and Jamie Brew. 2019. HuggingFace’s Transformers: State-of-the-art natural language processing. _ArXiv_ , abs/1910.03771.
* Wu et al. (2020) Shijie Wu, Alexis Conneau, Haoran Li, Luke Zettlemoyer, and Veselin Stoyanov. 2020\. Emerging cross-lingual structure in pretrained language models. In _Proceedings of ACL_ , pages 6022–6034.
* Wu and Dredze (2019) Shijie Wu and Mark Dredze. 2019. Beto, bentz, becas: The surprising cross-lingual effectiveness of BERT. In _Proceedings of EMNLP_ , pages 833–844.
* Yin and Schütze (2016) Wenpeng Yin and Hinrich Schütze. 2016. Learning word meta-embeddings. In _Proceedings of ACL_ , pages 1351–1360.
* Ziemski et al. (2016) Michal Ziemski, Marcin Junczys-Dowmunt, and Bruno Pouliquen. 2016. The United Nations Parallel Corpus v1.0. In _Proceedings of LREC_.
## Appendix A Appendix
URLs to the models and external corpora used in our study are provided in
Table 3 and Table 4, respectively. URLs to the evaluation data and task
architectures for each evaluation task are provided in Table 5. We also report
additional and more detailed sets of results across different tasks, word
embedding extraction configurations/variants, and language pairs:
* •
In Table 6 and Table 7, we provide full BLI results per language pair. All
scores are Mean Reciprocal Rank (MRR) scores (in the standard scoring
interval, 0.0–1.0).
* •
In Table 8, we provide full CLIR results per language pair. All scores are
Mean Average Precision (MAP) scores (in the standard scoring interval,
0.0–1.0).
* •
In Table 9, we provide full relation prediction (RELP) results for en and de.
All scores are micro-averaged $F_{1}$ scores over 5 runs of the relation
predictor Glavaš and Vulić (2018). We also report standard deviation for each
configuration.
Finally, in Figures 8-10, we also provide heatmaps denoting bilingual layer
correspondence, computed via linear CKA similarity Kornblith et al. (2019),
for several en–$L_{t}$ language pairs (see §4.1), which are not provided in
the main paper
Language URL en https://huggingface.co/bert-base-uncased de
https://huggingface.co/bert-base-german-dbmdz-uncased ru
https://huggingface.co/DeepPavlov/rubert-base-cased fi
https://huggingface.co/TurkuNLP/bert-base-finnish-uncased-v1 zh
https://huggingface.co/bert-base-chinese tr https://huggingface.co/dbmdz/bert-
base-turkish-uncased Multilingual https://huggingface.co/bert-base-
multilingual-uncased it https://huggingface.co/dbmdz/bert-base-italian-uncased
https://huggingface.co/dbmdz/bert-base-italian-xxl-uncased
Table 3: URLs of the models used in our study. The first part of the table
refers to the models used in the main experiments throughout the paper, while
the second part refers to the models used in side experiments.
Language URL en http://opus.nlpl.eu/download.php?f=Europarl/v8/moses/de-
en.txt.zip de http://opus.nlpl.eu/download.php?f=Europarl/v8/moses/de-
en.txt.zip ru http://opus.nlpl.eu/download.php?f=UNPC/v1.0/moses/en-ru.txt.zip
fi http://opus.nlpl.eu/download.php?f=Europarl/v8/moses/en-fi.txt.zip zh
http://opus.nlpl.eu/download.php?f=UNPC/v1.0/moses/en-zh.txt.zip tr
http://data.statmt.org/wmt18/translation-task/news.2017.tr.shuffled.deduped.gz
it http://opus.nlpl.eu/download.php?f=Europarl/v8/moses/en-it.txt.zip
Table 4: Links to the external corpora used in the study. We randomly sample
1M sentences of maximum sequence length 512 from the corresponding corpora.
Task Evaluation Data and/or Model Link LSIM Multi-SimLex Data:
multisimlex.com/ WA BATS Data: vecto.space/projects/BATS/ BLI Data:
Dictionaries from Glavaš et al. (2019) Data: github.com/codogogo/xling-
eval/tree/master/bli_datasets Model: VecMap Model: github.com/artetxem/vecmap
CLIR Data: CLEF 2003 Data: catalog.elra.info/en-
us/repository/browse/ELRA-E0008/ Model: Agg-IDF from Litschko et al. (2019)
Model: github.com/rlitschk/UnsupCLIR RELP Data: WordNet-based RELP data Data:
github.com/codogogo/stm/tree/master/data/wn-ls Model: Specialization Tensor
Model Model: github.com/codogogo/stm
Table 5: Links to evaluation data and models.
Configuration en–de en–tr en–fi en–ru de–tr de–fi de–ru fasttext.wiki 0.610
0.433 0.488 0.522 0.358 0.435 0.469 mono.iso.nospec avg(L$\leq$2) 0.390 0.332
0.392 0.409 0.237 0.269 0.291 avg(L$\leq$4) 0.430 0.367 0.438 0.447 0.269
0.311 0.338 avg(L$\leq$6) 0.461 0.386 0.476 0.472 0.299 0.359 0.387
avg(L$\leq$8) 0.472 0.390 0.486 0.487 0.315 0.387 0.407 avg(L$\leq$10) 0.461
0.386 0.483 0.488 0.321 0.395 0.416 avg(L$\leq$12) 0.446 0.379 0.471 0.473
0.323 0.395 0.412 mono.aoc-10.nospec avg(L$\leq$2) 0.399 0.342 0.386 0.403
0.242 0.269 0.292 avg(L$\leq$4) 0.457 0.379 0.448 0.433 0.283 0.322 0.343
avg(L$\leq$6) 0.503 0.399 0.480 0.458 0.315 0.369 0.380 avg(L$\leq$8) 0.527
0.414 0.499 0.461 0.332 0.394 0.391 avg(L$\leq$10) 0.534 0.415 0.498 0.459
0.337 0.401 0.394 avg(L$\leq$12) 0.534 0.416 0.492 0.453 0.337 0.401 0.376
mono.aoc-100.nospec avg(L$\leq$2) 0.401 0.343 0.391 0.398 0.239 0.269 0.293
avg(L$\leq$4) 0.459 0.381 0.449 0.437 0.288 0.325 0.343 avg(L$\leq$6) 0.504
0.403 0.484 0.459 0.318 0.373 0.382 avg(L$\leq$8) 0.532 0.418 0.503 0.462
0.334 0.394 0.389 avg(L$\leq$10) 0.540 0.422 0.504 0.459 0.338 0.402 0.393
avg(L$\leq$12) 0.542 0.426 0.500 0.454 0.343 0.401 0.378 mono.iso.all
avg(L$\leq$2) 0.352 0.289 0.351 0.374 0.230 0.265 0.283 avg(L$\leq$4) 0.375
0.317 0.391 0.393 0.264 0.302 0.331 avg(L$\leq$6) 0.386 0.330 0.406 0.407
0.289 0.350 0.376 avg(L$\leq$8) 0.372 0.327 0.409 0.413 0.291 0.370 0.392
avg(L$\leq$10) 0.352 0.320 0.396 0.402 0.290 0.370 0.383 avg(L$\leq$12) 0.313
0.310 0.373 0.394 0.283 0.358 0.371 mono.iso.withcls avg(L$\leq$2) 0.367 0.306
0.368 0.386 0.236 0.272 0.285 avg(L$\leq$4) 0.394 0.339 0.408 0.410 0.267
0.307 0.331 avg(L$\leq$6) 0.406 0.344 0.428 0.425 0.294 0.353 0.381
avg(L$\leq$8) 0.393 0.344 0.430 0.431 0.306 0.369 0.400 avg(L$\leq$10) 0.371
0.336 0.421 0.421 0.303 0.382 0.395 avg(L$\leq$12) 0.331 0.329 0.403 0.409
0.302 0.375 0.387 multi.iso.nospec avg(L$\leq$2) 0.293 0.176 0.176 0.147 0.216
0.203 0.160 avg(L$\leq$4) 0.304 0.184 0.190 0.164 0.219 0.214 0.178
avg(L$\leq$6) 0.315 0.189 0.203 0.198 0.223 0.225 0.198 avg(L$\leq$8) 0.325
0.193 0.209 0.228 0.224 0.235 0.217 avg(L$\leq$10) 0.330 0.194 0.210 0.243
0.220 0.234 0.226 avg(L$\leq$12) 0.333 0.193 0.206 0.248 0.219 0.231 0.227
multi.aoc-10.nospec avg(L$\leq$2) 0.309 0.171 0.172 0.146 0.208 0.200 0.156
avg(L$\leq$4) 0.350 0.186 0.189 0.186 0.224 0.214 0.191 avg(L$\leq$6) 0.389
0.219 0.215 0.240 0.241 0.243 0.225 avg(L$\leq$8) 0.432 0.246 0.251 0.287
0.255 0.263 0.254 avg(L$\leq$10) 0.448 0.258 0.264 0.306 0.260 0.282 0.272
avg(L$\leq$12) 0.456 0.267 0.272 0.316 0.260 0.292 0.284 multi.iso.all
avg(L$\leq$2) 0.292 0.173 0.175 0.143 0.209 0.203 0.154 avg(L$\leq$4) 0.301
0.176 0.188 0.155 0.211 0.213 0.171 avg(L$\leq$6) 0.307 0.181 0.198 0.186
0.216 0.221 0.193 avg(L$\leq$8) 0.315 0.184 0.202 0.207 0.213 0.228 0.208
avg(L$\leq$10) 0.318 0.182 0.197 0.216 0.208 0.226 0.215 avg(L$\leq$12) 0.319
0.181 0.189 0.220 0.209 0.220 0.213 mono.iso.nospec (reverse) avg(L$\geq$12)
0.104 – 0.054 – – 0.077 – avg(L$\geq$10) 0.119 – 0.061 – – 0.063 –
avg(L$\geq$8) 0.144 – 0.108 – – 0.095 – avg(L$\geq$6) 0.230 – 0.223 – – 0.238
– avg(L$\geq$4) 0.308 – 0.318 – – 0.335 – avg(L$\geq$2) 0.365 – 0.385 – –
0.372 – avg(L$\geq$0) 0.446 – 0.471 – – 0.395 –
Table 6: Results in the BLI task across different language pairs and word
vector extraction configurations. MRR scores reported. For clarity of
presentation, a subset of results is presented in this table, while the rest
(and the averages) are presented in Table 7. avg(L$\leq$n) means that we
average representations over all Transformer layers up to the $n$th layer
(included), where $L=0$ refers to the embedding layer, $L=1$ to the bottom
layer, and $L=12$ to the final (top) layer. Different configurations are
described in §2 and Table 1. Additional diagnostic experiments with top-to-
bottom layerwise averaging configs (reverse) are run for a subset of
languages: {en, de, fi }.
Configuration tr–fi tr–ru fi–ru average fasttext.wiki 0.358 0.364 0.439 0.448
mono.iso.nospec avg(L$\leq$2) 0.237 0.217 0.290 0.306 avg(L$\leq$4) 0.279
0.261 0.337 0.348 avg(L$\leq$6) 0.311 0.288 0.372 0.381 avg(L$\leq$8) 0.334
0.315 0.387 0.398 avg(L$\leq$10) 0.347 0.317 0.392 0.401 avg(L$\leq$12) 0.352
0.319 0.387 0.396 mono.aoc-10.nospec avg(L$\leq$2) 0.247 0.221 0.284 0.308
avg(L$\leq$4) 0.288 0.263 0.331 0.355 avg(L$\leq$6) 0.319 0.294 0.366 0.388
avg(L$\leq$8) 0.334 0.311 0.375 0.404 avg(L$\leq$10) 0.340 0.311 0.379 0.407
avg(L$\leq$12) 0.344 0.310 0.360 0.402 mono.aoc-100.nospec avg(L$\leq$2) 0.244
0.220 0.285 0.308 avg(L$\leq$4) 0.288 0.261 0.333 0.356 avg(L$\leq$6) 0.322
0.291 0.367 0.390 avg(L$\leq$8) 0.338 0.309 0.376 0.406 avg(L$\leq$10) 0.348
0.314 0.377 0.410 avg(L$\leq$12) 0.349 0.311 0.361 0.407 mono.iso.all
avg(L$\leq$2) 0.226 0.212 0.284 0.287 avg(L$\leq$4) 0.270 0.254 0.328 0.322
avg(L$\leq$6) 0.302 0.274 0.358 0.348 avg(L$\leq$8) 0.318 0.296 0.371 0.356
avg(L$\leq$10) 0.328 0.303 0.373 0.352 avg(L$\leq$12) 0.328 0.306 0.368 0.340
mono.iso.withcls avg(L$\leq$2) 0.232 0.217 0.285 0.295 avg(L$\leq$4) 0.274
0.257 0.331 0.332 avg(L$\leq$6) 0.307 0.279 0.362 0.358 avg(L$\leq$8) 0.327
0.303 0.377 0.368 avg(L$\leq$10) 0.334 0.314 0.383 0.366 avg(L$\leq$12) 0.340
0.317 0.373 0.357 multi.iso.nospec avg(L$\leq$2) 0.170 0.131 0.127 0.180
avg(L$\leq$4) 0.180 0.135 0.138 0.191 avg(L$\leq$6) 0.188 0.147 0.151 0.204
avg(L$\leq$8) 0.189 0.152 0.164 0.214 avg(L$\leq$10) 0.188 0.153 0.165 0.216
avg(L$\leq$12) 0.188 0.158 0.163 0.217 multi.aoc-10.nospec avg(L$\leq$2) 0.165
0.127 0.130 0.178 avg(L$\leq$4) 0.176 0.146 0.139 0.200 avg(L$\leq$6) 0.192
0.174 0.162 0.230 avg(L$\leq$8) 0.210 0.192 0.185 0.258 avg(L$\leq$10) 0.219
0.198 0.200 0.271 avg(L$\leq$12) 0.223 0.198 0.206 0.277 multi.iso.all
avg(L$\leq$2) 0.163 0.126 0.123 0.176 avg(L$\leq$4) 0.175 0.128 0.133 0.185
avg(L$\leq$6) 0.179 0.139 0.142 0.196 avg(L$\leq$8) 0.182 0.144 0.152 0.203
avg(L$\leq$10) 0.178 0.141 0.153 0.203 avg(L$\leq$12) 0.175 0.143 0.150 0.202
Table 7: Results in the bilingual lexicon induction (BLI) task across
different language pairs and word vector extraction configurations: Part II.
MAP scores reported. For clarity of presentation, a subset of results is
presented in this table, while the rest (also used to calculate the averages)
is provided in Table 6 in the previous page. avg(L$\leq$n) means that we
average representations over all Transformer layers up to the $n$th layer
(included), where $L=0$ refers to the embedding layer, $L=1$ to the bottom
layer, and $L=12$ to the final (top) layer. Different configurations are
described in §2 and Table 1.
Configuration en–de en–fi en–ru de–fi de–ru fi–ru average fasttext.wiki 0.193
0.136 0.118 0.221 0.112 0.105 0.148 mono.iso.nospec avg(L$\leq$2) 0.059 0.075
0.106 0.126 0.086 0.123 0.096 avg(L$\leq$4) 0.061 0.069 0.098 0.111 0.075
0.106 0.087 avg(L$\leq$6) 0.052 0.061 0.079 0.112 0.068 0.102 0.079
avg(L$\leq$8) 0.042 0.048 0.075 0.112 0.063 0.105 0.074 avg(L$\leq$10) 0.036
0.043 0.067 0.107 0.065 0.080 0.066 avg(L$\leq$12) 0.032 0.034 0.059 0.097
0.077 0.083 0.064 mono.aoc-10.nospec avg(L$\leq$2) 0.069 0.078 0.094 0.109
0.078 0.108 0.089 avg(L$\leq$4) 0.076 0.105 0.119 0.112 0.098 0.117 0.104
avg(L$\leq$6) 0.086 0.090 0.129 0.122 0.098 0.125 0.108 avg(L$\leq$8) 0.092
0.073 0.137 0.105 0.100 0.114 0.103 avg(L$\leq$10) 0.095 0.073 0.147 0.102
0.102 0.135 0.109 avg(L$\leq$12) 0.104 0.073 0.139 0.100 0.105 0.131 0.109
mono.aoc-100.nospec avg(L$\leq$2) 0.073 0.081 0.097 0.111 0.078 0.106 0.091
avg(L$\leq$4) 0.078 0.107 0.115 0.107 0.100 0.115 0.104 avg(L$\leq$6) 0.087
0.087 0.127 0.132 0.103 0.123 0.110 avg(L$\leq$8) 0.091 0.076 0.137 0.118
0.101 0.106 0.105 avg(L$\leq$10) 0.099 0.074 0.161 0.103 0.104 0.104 0.107
avg(L$\leq$12) 0.106 0.076 0.146 0.105 0.106 0.100 0.106 mono.iso.all
avg(L$\leq$2) 0.044 0.045 0.076 0.095 0.067 0.098 0.071 avg(L$\leq$4) 0.039
0.042 0.079 0.094 0.066 0.100 0.070 avg(L$\leq$6) 0.024 0.034 0.069 0.089
0.066 0.094 0.063 avg(L$\leq$8) 0.018 0.020 0.039 0.068 0.059 0.092 0.049
avg(L$\leq$10) 0.016 0.016 0.030 0.048 0.058 0.067 0.039 avg(L$\leq$12) 0.014
0.013 0.033 0.034 0.064 0.061 0.036 mono.iso.withcls avg(L$\leq$2) 0.050 0.057
0.086 0.106 0.071 0.108 0.080 avg(L$\leq$4) 0.046 0.055 0.084 0.104 0.071
0.102 0.077 avg(L$\leq$6) 0.032 0.042 0.076 0.103 0.066 0.097 0.069
avg(L$\leq$8) 0.025 0.028 0.046 0.086 0.059 0.101 0.057 avg(L$\leq$10) 0.021
0.030 0.037 0.072 0.057 0.079 0.049 avg(L$\leq$12) 0.020 0.016 0.032 0.052
0.045 0.072 0.040 multi.iso.nospec avg(L$\leq$2) 0.110 0.009 0.045 0.057 0.020
0.013 0.042 avg(L$\leq$4) 0.100 0.007 0.075 0.044 0.025 0.011 0.044
avg(L$\leq$6) 0.098 0.007 0.046 0.043 0.029 0.030 0.042 avg(L$\leq$8) 0.088
0.008 0.052 0.043 0.032 0.031 0.042 avg(L$\leq$10) 0.084 0.008 0.051 0.042
0.034 0.026 0.041 avg(L$\leq$12) 0.082 0.006 0.048 0.039 0.037 0.024 0.039
multi.aoc-10.nospec avg(L$\leq$2) 0.127 0.013 0.049 0.027 0.019 0.009 0.041
avg(L$\leq$4) 0.123 0.018 0.055 0.032 0.029 0.008 0.044 avg(L$\leq$6) 0.120
0.018 0.055 0.051 0.042 0.009 0.049 avg(L$\leq$8) 0.123 0.018 0.057 0.053
0.049 0.016 0.053 avg(L$\leq$10) 0.127 0.019 0.062 0.050 0.051 0.018 0.054
avg(L$\leq$12) 0.128 0.021 0.065 0.049 0.052 0.019 0.056 multi.iso.all
avg(L$\leq$2) 0.072 0.005 0.032 0.014 0.016 0.004 0.024 avg(L$\leq$4) 0.075
0.004 0.027 0.014 0.022 0.005 0.024 avg(L$\leq$6) 0.065 0.004 0.026 0.015
0.027 0.007 0.024 avg(L$\leq$8) 0.054 0.004 0.035 0.015 0.032 0.008 0.025
avg(L$\leq$10) 0.054 0.005 0.032 0.017 0.035 0.007 0.025 avg(L$\leq$12) 0.058
0.004 0.034 0.018 0.032 0.006 0.025 mono.iso.nospec (reverse) avg(L$\geq$12)
0.005 0.012 – 0.001 – – – avg(L$\geq$10) 0.002 0.002 – 0.001 – – –
avg(L$\geq$8) 0.004 0.002 – 0.002 – – – avg(L$\geq$6) 0.014 0.006 – 0.004 – –
– avg(L$\geq$4) 0.020 0.012 – 0.016 – – – avg(L$\geq$2) 0.024 0.019 – 0.043 –
– – avg(L$\geq$0) 0.032 0.034 – 0.097 – – –
Table 8: Results in the CLIR task across different language pairs and word
vector extraction configurations. MAP scores reported; avg(L$\leq$n) means
that we average representations over all Transformer layers up to the $n$th
layer (included), where $L=0$ refers to the embedding layer, $L=1$ to the
bottom layer, and $L=12$ to the final (top) layer. Different configurations
are described in §2 and Table 1. Additional diagnostic experiments with top-
to-bottom layerwise averaging configs (reverse) are run for a subset of
languages: {en, de, fi }.
Configuration en de fasttext.wiki $0.660_{\pm 0.008}$ $0.601_{\pm 0.007}$
random.xavier $0.473_{\pm 0.003}$ $0.512_{\pm 0.008}$ mono.iso.nospec
avg(L$\leq$2) $0.688_{\pm 0.007}$ $0.649_{\pm 0.002}$ avg(L$\leq$4)
$0.698_{\pm 0.002}$ $0.664_{\pm 0.004}$ avg(L$\leq$6) $0.699_{\pm 0.007}$
$0.677_{\pm 0.006}$ avg(L$\leq$8) $0.706_{\pm 0.003}$ $0.674_{\pm 0.016}$
avg(L$\leq$10) $0.718_{\pm 0.002}$ $0.679_{\pm 0.008}$ avg(L$\leq$12)
$0.714_{\pm 0.012}$ $0.673_{\pm 0.003}$ mono.aoc-10.nospec avg(L$\leq$2)
$0.690_{\pm 0.007}$ $0.657_{\pm 0.005}$ avg(L$\leq$4) $0.705_{\pm 0.006}$
$0.671_{\pm 0.009}$ avg(L$\leq$6) $0.714_{\pm 0.008}$ $0.675_{\pm 0.014}$
avg(L$\leq$8) $0.722_{\pm 0.004}$ $0.681_{\pm 0.010}$ avg(L$\leq$10)
$0.719_{\pm 0.007}$ $0.682_{\pm 0.007}$ avg(L$\leq$12) $0.720_{\pm 0.005}$
$0.680_{\pm 0.007}$ mono.aoc-100.nospec avg(L$\leq$2) $0.692_{\pm 0.007}$
$0.655_{\pm 0.007}$ avg(L$\leq$4) $0.709_{\pm 0.007}$ $0.670_{\pm 0.005}$
avg(L$\leq$6) $0.718_{\pm 0.009}$ $0.672_{\pm 0.008}$ avg(L$\leq$8)
$0.717_{\pm 0.003}$ $0.680_{\pm 0.006}$ avg(L$\leq$10) $0.721_{\pm 0.009}$
$0.678_{\pm 0.004}$ avg(L$\leq$12) $0.715_{\pm 0.003}$ $0.678_{\pm 0.006}$
mono.iso.all avg(L$\leq$2) $0.688_{\pm 0.008}$ $0.654_{\pm 0.012}$
avg(L$\leq$4) $0.698_{\pm 0.011}$ $0.662_{\pm 0.008}$ avg(L$\leq$6)
$0.711_{\pm 0.005}$ $0.664_{\pm 0.005}$ avg(L$\leq$8) $0.709_{\pm 0.008}$
$0.663_{\pm 0.015}$ avg(L$\leq$10) $0.712_{\pm 0.006}$ $0.669_{\pm 0.003}$
avg(L$\leq$12) $0.704_{\pm 0.005}$ $0.666_{\pm 0.013}$ mono.iso.withcls
avg(L$\leq$2) $0.693_{\pm 0.004}$ $0.649_{\pm 0.016}$ avg(L$\leq$4)
$0.699_{\pm 0.004}$ $0.664_{\pm 0.006}$ avg(L$\leq$6) $0.709_{\pm 0.002}$
$0.671_{\pm 0.006}$ avg(L$\leq$8) $0.710_{\pm 0.003}$ $0.679_{\pm 0.006}$
avg(L$\leq$10) $0.713_{\pm 0.006}$ $0.670_{\pm 0.007}$ avg(L$\leq$12)
$0.705_{\pm 0.005}$ $0.676_{\pm 0.006}$ multi.iso.nospec avg(L$\leq$2)
$0.671_{\pm 0.009}$ $0.628_{\pm 0.013}$ avg(L$\leq$4) $0.669_{\pm 0.006}$
$0.640_{\pm 0.004}$ avg(L$\leq$6) $0.684_{\pm 0.010}$ $0.637_{\pm 0.009}$
avg(L$\leq$8) $0.680_{\pm 0.005}$ $0.647_{\pm 0.006}$ avg(L$\leq$10)
$0.676_{\pm 0.006}$ $0.629_{\pm 0.008}$ avg(L$\leq$12) $0.681_{\pm 0.005}$
$0.637_{\pm 0.004}$ multi.aoc-10.nospec avg(L$\leq$2) $0.674_{\pm 0.005}$
$0.635_{\pm 0.011}$ avg(L$\leq$4) $0.681_{\pm 0.006}$ $0.630_{\pm 0.007}$
avg(L$\leq$6) $0.692_{\pm 0.008}$ $0.649_{\pm 0.010}$ avg(L$\leq$8)
$0.695_{\pm 0.004}$ $0.652_{\pm 0.011}$ avg(L$\leq$10) $0.704_{\pm 0.005}$
$0.657_{\pm 0.012}$ avg(L$\leq$12) $0.702_{\pm 0.005}$ $0.661_{\pm 0.008}$
multi.iso.all avg(L$\leq$2) $0.674_{\pm 0.004}$ $0.626_{\pm 0.014}$
avg(L$\leq$4) $0.682_{\pm 0.009}$ $0.640_{\pm 0.009}$ avg(L$\leq$6)
$0.680_{\pm 0.002}$ $0.632_{\pm 0.007}$ avg(L$\leq$8) $0.683_{\pm 0.003}$
$0.638_{\pm 0.010}$ avg(L$\leq$10) $0.678_{\pm 0.007}$ $0.638_{\pm 0.015}$
avg(L$\leq$12) $0.676_{\pm 0.013}$ $0.636_{\pm 0.005}$ mono.iso.nospec
(reverse) avg(L$\geq$12) $0.683_{\pm 0.007}$ $0.628_{\pm 0.009}$
avg(L$\geq$10) $0.692_{\pm 0.014}$ $0.628_{\pm 0.008}$ avg(L$\geq$8)
$0.688_{\pm 0.016}$ $0.648_{\pm 0.007}$ avg(L$\geq$6) $0.704_{\pm 0.015}$
$0.658_{\pm 0.006}$ avg(L$\geq$4) $0.704_{\pm 0.008}$ $0.668_{\pm 0.007}$
avg(L$\geq$2) $0.707_{\pm 0.008}$ $0.667_{\pm 0.004}$ avg(L$\geq$0)
$0.714_{\pm 0.012}$ $0.673_{\pm 0.003}$
Table 9: Results in the relation prediction task (RELP) across different word
vector extraction configurations. Micro-averaged $F_{1}$ scores reported ,
obtained as averages over 5 experimental runs for each configuration; standard
deviation is also reported. avg(L$\leq$n) means that we average
representations over all Transformer layers up to the $n$th layer (included),
where $L=0$ refers to the embedding layer, $L=1$ to the bottom layer, and
$L=12$ to the final (top) layer. Different configurations are described in §2
and Table 1. random.xavier are 768-dim vectors for the same vocabularies,
randomly initialised via Xavier initialisation Glorot and Bengio (2010).
(a) en–de: Word translation pairs
(b) en–de: Random word pairs
Figure 8: CKA similarity scores of type-level word representations extracted
from each layer (using different extraction configurations, see Table 1) for a
set of (a) 7K en–de translation pairs from the BLI dictionaries of Glavaš et
al. (2019); (b) 7K random en–de pairs.
(a) en–fi: Word translation pairs
(b) en–fi: Random word pairs
Figure 9: CKA similarity scores of type-level word representations extracted
from each layer (using different extraction configurations, see Table 1) for a
set of (a) 7K en–fi translation pairs from the BLI dictionaries of Glavaš et
al. (2019); (b) 7K random en–fi pairs.
(a) en–tr: Word translation pairs
(b) en–tr: Random word pairs
Figure 10: CKA similarity scores of type-level word representations extracted
from each layer (using different extraction configurations, see Table 1) for a
set of (a) 7K en–tr translation pairs from the BLI dictionaries of Glavaš et
al. (2019); (b) 7K random en–tr pairs.
|
ngerman-x-latestngermanngerman-x-latest german-x-latestgermangerman-x-latest
# Approaches for Biobjective Integer Linear Robust Optimization
Fabian Chlumsky-Harttmann111supported by DFG under grant SCHO 1140/11-1
Faculty of Mathematics, University of Kaiserslautern-Landau Marie Schmidt
Faculty of Mathematics and Computer Science, University of Würzburg Anita
Schöbel Faculty of Mathematics, University of Kaiserslautern-Landau
Fraunhofer Institute for Industrial Mathematics ITWM
#### Abstract
Real-world optimization problems often do not just involve multiple objectives
but also uncertain parameters. In this case, the goal is to find Pareto-
optimal solutions that are robust, i.e., reasonably good under all possible
realizations of the uncertain data. Such solutions have been studied in many
papers within the last ten years and are called _robust efficient_. However,
solution methods for finding robust efficient solutions are scarce. In this
paper, we develop three algorithms for determining robust efficient solutions
to biobjective mixed-integer linear robust optimization problems.
To this end, we draw from methods for both multiobjective optimization and
robust optimization: dichotomic search for biobjective mixed-integer
optimization problems and an optimization-pessimization approach from (single-
objective) robust optimization, which iteratively adds scenarios and thereby
increases the uncertainty set. We propose two algorithms that combine
dichotomic search with the optimization-pessimization method as well as a
dichotomic search method for biobjective linear robust optimization that
exploits duality. On the way we derive some other results: We extend
dichotomic search from biobjective linear problems to biobjective linear
minmax problems and generalize the optimization-pessimization method from
single-objective to multi-objective robust optimization problems.
We implemented and tested the three algorithms on linear and integer linear
instances and discuss their respective strengths and weaknesses.
## 1 Introduction
Real-world optimization problems are often complicated by two issues: First,
in many cases decision makers have not only one but multiple objectives.
Second, the optimization problems may involve uncertainty – be it through
prediction errors about parameters like demand, that will only be known in the
future, or measurement errors. These two issues are treated in the fields of
multiobjective optimization and robust optimization.
In order to do deal with problems that are both uncertain and multiobjective,
multiobjective robust optimization has been studied for more than ten years
leading to various models and theoretical results. However, research into
methods of actually solving such problems is still in its initial stages. In
this paper we propose three algorithms for computing robust efficient
solutions for uncertain biobjective mixed-integer linear optimization
problems.
In order to find a good solution for an uncertain multiobjective problem, a
notion of what constitutes a _robust efficient_ solution has to be formulated
first. This is not trivial since there is no straightforward way to generalize
the concept of Pareto optimality used in multiobjective optimization to
uncertain multiobjective problems or to generalize the notion of robustness to
multiobjective problems. Over the years, several concepts for robust
multiobjective efficiency have been proposed, (see [IS16, WD16] for surveys).
The oldest among them is the notion of _flimsily efficient_ (sometimes:
_possibly efficient_) and _highly efficient_ (sometimes: _necessarily
efficient_) solutions (see, e.g., [Bit80, IS96, Kuh+16, ES20]) describing
solutions that are efficient for at least one or for all considered scenarios,
respectively. Other notable concepts include _regret-robust efficiency_ (see
[RY13, Xid+17, GW22]), _multi-scenario efficiency_ (see [BS19]), _lightly
robust efficiency_ (see [Kuh+16, IS16]), _local efficiency w.r.t. the robust
counterpart_ (see [Chu20]), and three different generalizations of minmax
robustness to multiobjective problems called _set-based_ (see [EIS14]), _hull-
based_ (see [BF17]) and _point-based minmax efficiency_. The latter concept
has been introduced by Kuroiwa and Lee (see [KL12]) and is used in this paper.
For point-based minmax robust efficiency, many theoretical results exist:
Goberna, Jeyakumar, Li and Vicente-Pérez consider specific forms of data
uncertainty (box data uncertainty, norm data uncertainty, ellipsoidal
uncertainty) and provide deterministic reformulations (see [Gob+15]). Box
uncertainty with a limited sum of deviations has been considered in [HNS13].
In [Ant+20] necessary and sufficient conditions for robust
$\varepsilon$-efficient solutions for uncertain nonsmooth multiobjective
optimization problems are established, but no algorithmic method is provided.
In [WCL20a, WCL20] separation results and some characterizations of optimality
are developed, and the robustness gap for point-based minmax robust efficiency
has been introduced in [Krü+23]. The _price of robustness_ has been defined in
[SZK21]. Point-based minmax robust efficiency has been generalized to
_efficiency w.r.t. to a general cone_ (see [WLC15, Ide+14]) and it has been
applied to decision robustness in [EKS17].
As general algorithmic idea, many authors suggest scalarization approaches
transferring a robust multiobjective problem to a single-objective robust
problem, e.g, [EIS14, Ide+14, Gob+15], but the approaches proposed in those
papers are still on an abstract level and only capable of finding some robust
efficient solutions while in this paper we give concrete algorithms for
determining a representative set of all supported robust efficient solutions.
Other algorithmic approaches consider special cases, e.g., cardinality-
constrained uncertainty for combinatorial problems (see [Rai+18a]), uncertain
multiobjective shortest-path problems [Rai+18] or cardinality-constrained box
uncertainty in the context of portfolio selection problems [HNS14].
The remainder of the paper is organized as follows. In Section 2 we derive a
biobjective integer linear minmax optimization problem as robust counterpart
and collect other necessary preliminaries. Section 3.1 considers the problem
first and foremost as a biobjective problem. The well-known dichotomic search
algorithm for biobjective problems is briefly summarized before we show how it
can be extended to robust problems. The opposite approach is taken in Section
4.1, where the problem is considered from a robust optimization perspective.
An optimization-pessimization approach for (single-objective) robust
optimization is reviewed and then extended to multiobjective problems.
In Section 5 we combine dichotomic search and optimization-pessimization and
receive two different methods for finding robust efficient solutions. For the
special case of a bilinear continuous objective function, we additionally
develop a dual approach together with dichotomic search in Section 5.3.
Numerical results are given in Section 6 and, finally, some conclusions are
drawn and suggestions for further research are formulated in Section 7.
## 2 Problem definition and preliminaries
In this section we briefly review multiobjective robust optimization. We start
with restating some definitions from robust optimization and multiobjective
optimization which we then combine to the emerging topic of multiobjective
robust optimization. We define what a robust efficient solution to an
uncertain multi-objective problem is and from this we derive the biobjective
mixed-integer linear robust optimization problem (BRO) — the problem to be
solved in this paper. We finally recall some concepts of multiobjective
optimization which are needed later.
#### Single-objective robust optimization.
Robust optimization deals with uncertain optimization problems, i.e., problems
with some uncertain parameters $\xi\in\mathbb{R}^{m}$ which depend on
measurements, future developments, delays or other uncertainties. Every $\xi$
is called a _scenario_. As usual in robust optimization, we assume that the
set $\mathcal{U}\subseteq\mathbb{R}^{m}$ of all possible scenarios is known.
We call $\mathcal{U}$ _uncertainty set_. A single-objective uncertain problem
is described by a family of parameterized optimization problems
$\left\\{\min_{x\in\mathcal{X}}h(x,\xi)\right\\}_{\xi\in\mathcal{U}}$ (1)
with $\mathcal{X}\subseteq\mathbb{R}^{n}$ and
$h\colon\mathcal{X}\times\mathcal{U}\to\mathbb{R}$.
There is usually no solution that is optimal for all scenarios. Instead one
aims to find _robust solutions_ which are reasonably good for all (or most)
scenarios. Out of many robustness concepts that have been defined (see, e.g.,
[GS16] for an overview on different robustness concepts), minmax robustness is
one of the most commonly used. For a detailed account of the subject, we refer
to [BTEN09]. A solution to problem (1) is called _(minmax) robust optimal_ if
it is an optimal solution to its _robust counterpart_
$\displaystyle\textup{P}^{\textup{single}}(\mathcal{U})$
$\displaystyle\min_{x\in\mathcal{X}}\sup_{\xi\in\mathcal{U}}h(x,\xi).$ (2)
#### Multiobjective (deterministic) problems.
Now let us turn to multiobjective problems
$\min_{x\in\mathcal{X}}\underbrace{\begin{pmatrix}g_{1}(x)\\\ g_{2}(x)\\\
\vdots\\\ g_{p}(x)\end{pmatrix}}_{\eqqcolon g(x)}$ (3)
with $\mathcal{X}\subseteq\mathbb{R}^{n}$ and
$g_{i}\colon\mathcal{X}\to\mathbb{R}$, $i=1,2,\dots,p$.
Instead of minimizing a scalar objective function, we have to compare vectors
in order to find an optimal solution. As common in multi-objective
optimization, we use the following vector relations: For two vectors
$y,\bar{y}\in\mathbb{R}^{p}$, the ordering relations $<$ and $\leq$ are meant
to be component-wise. Furthermore, if $y\leq\bar{y}$ and $y\not=\bar{y}$, we
write $y\preceq\bar{y}$ and say that $y$ _dominates_ $\bar{y}$. Accordingly,
we define $\mathbb{R}_{\succeq}^{p}\coloneqq\\{r\in\mathbb{R}^{p}\colon
r\succeq 0\\}$ and $\mathbb{R}^{p}_{\geq},\mathbb{R}^{p}_{>}$. Biobjective
optimization is the special case of multiobjective optimization with $p=2$.
The most important concept for multiobjective optimization is _efficiency_
(also called _Pareto optimality_). Given a multiobjective problem (3) a
solution $x\in\mathcal{X}$ is called _efficient_ and its image
$g(x)\in\mathcal{Y}\coloneqq g(\mathcal{X})\subset\mathbb{R}^{p}$ is called
_nondominated_ if no solution $x^{\prime}\in\mathcal{X}\setminus\\{x\\}$
exists, such that $g(x^{\prime})$ dominates $g(x)$. By
$\mathcal{Y}_{\textup{N}}$ we denote the set of nondominated points. These
points form the _Pareto frontier_.
#### Multiobjective robust optimization.
Real-world optimization problems often have multiple objective functions _and_
uncertain parameters. We consider multiobjective uncertain optimization
problems which depend on a scenario $\xi\in\mathcal{U}\subseteq\mathbb{R}^{m}$
$\displaystyle\textup{P}(\xi)$
$\displaystyle\min_{x\in\mathcal{X}}\underbrace{\begin{pmatrix}f_{1}(x,\xi)\\\
f_{2}(x,\xi)\\\ \vdots\\\ f_{p}(x,\xi)\end{pmatrix}}_{\eqqcolon f(x,\xi)}$ (4)
with $\mathcal{X}\subseteq\mathbb{R}^{n}$,
$f_{i}\colon\mathcal{X}\times\mathcal{U}\to\mathbb{R}$, $i=1,2,\dots,p$.
Analogously to single-objective optimization, one calls the parameterized
family
$\\{P(\xi)\colon\xi\in\mathcal{U}\\}$ (5)
an _uncertain multiobjective optimization problem_. We are interested in
finding efficient solutions to the uncertain multiobjective optimization
problem, which are robust.
Several ways to generalize minmax robustness to multiobjective uncertain
problems have been proposed (see [IS16, WD16] for surveys). In this article we
aim to find _point-based minmax robust efficient_ solutions as defined in
[KL12, Kuh+16]. A solution is called _point-based minmax robust efficient_
(from now on: _robust efficient_) if it is an efficient solution to
$\displaystyle P(\mathcal{U})$
$\displaystyle\min_{x\in\mathcal{X}}\begin{pmatrix}\sup_{\xi\in\mathcal{U}}f_{1}(x,\xi)\\\
\sup_{\xi\in\mathcal{U}}f_{2}(x,\xi)\\\ \vdots\\\
\sup_{\xi\in\mathcal{U}}f_{p}(x,\xi)\\\ \end{pmatrix}.$ (6)
$P(\mathcal{U})$ is called the _robust counterpart_ of the uncertain
multiobjective optimization problem (5) or just _multiobjective robust_
problem. Since $\mathcal{U}$ is varied within some of the proposed algorithms,
we refer to the specific set $\mathcal{U}$ in the notation $P(\mathcal{U})$.
To improve readability, we define
$f_{i}^{\mathcal{U}}(x)\coloneqq\sup_{\xi\in\mathcal{U}}f_{i}(x,\xi)$,
$i=1,2,\dots,p$ and set
$f^{\mathcal{U}}(x)\eqqcolon(f_{1}^{\mathcal{U}}(x),f_{2}^{\mathcal{U}}(x),\dots,f_{p}^{\mathcal{U}}(x))^{t}$
as the vector containing the $p$ objective functions. Problem (6) can hence be
interpreted as a (deterministic) multiobjective problem of type (3) with
$g\coloneqq f^{\mathcal{U}}$ as objective function. This point of view is used
in Section 3 and in Algorithms 5.2 and 5.3.
Finally, note that
$f^{\mathcal{U}^{\prime}}(x)\leq f^{\mathcal{U}}(x)\textup{ for
}\mathcal{U}^{\prime}\subseteq\mathcal{U}\textup{.}$ (7)
#### The problem to be solved: (BRO) – biobjective mixed-integer linear robust
optimization.
We consider uncertain biobjective optimization problems with $p=2$. Their
robust counterpart is given in (6), i.e., we receive the following biobjective
mixed-integer linear program with minmax objective function,
$\displaystyle\textup{BRO}(\mathcal{U})$
$\displaystyle\min_{x\in\mathcal{X}}\begin{pmatrix}\sup_{\xi\in\mathcal{U}}f_{1}(x,\xi)\\\
\sup_{\xi\in\mathcal{U}}f_{2}(x,\xi)\end{pmatrix}\textup{.}$ (BRO)
Our goal is to determine the Pareto frontier and the associated efficient
solutions of $\textup{BRO}(\mathcal{U})$.
For $\textup{BRO}(\mathcal{U})$ we always assume the following:
* •
(BRO-1) a feasible set
$\mathcal{X}=P\cap(\mathbb{Z}^{k}\times\mathbb{R}^{n-k})$ where
$P\subseteq\mathbb{R}^{n}$ is a polytope and $0\leq k\leq n$,
* •
(BRO-2) a polytope or finite set $\mathcal{U}\subseteq\mathbb{R}^{m}$, and
* •
(BRO-3) functions $f_{1},f_{2}\colon\mathcal{X}\times\mathcal{U}\to\mathbb{R}$
which are linear in $x$ for every fixed $\xi\in\mathcal{U}$ and quasi-convex
and continuous in $\xi$ for every fixed $x\in\mathcal{X}$.
Under the latter two assumptions, (BRO-2) and (BRO-3), the supremum in the
definition of (BRO) is always attained and we can write maximum instead, i.e.,
$f^{\mathcal{U}}_{i}(x)=\max_{\xi\in\mathcal{U}}f_{i}(x,\xi)$ for
$x\in\mathcal{X}$, $i=1,2$. (BRO-3) guarantees that
$f_{i}\colon\mathcal{X}\times\mathcal{U}\to\mathbb{R}$ is jointly continuous
in $(x,\xi)$ (see, e.g., [KD69]). Finally, the feasible set $\mathcal{X}$
determines the type of the problem at hand: For $k=0$ the problem is a (pure)
_linear_ minmax problem, for $k=n$ the problem is an _integer linear_ minmax
problem and for $1\leq k<n$ we have a _mixed-integer linear_ minmax problem.
#### Concepts from multi-objective optimization.
We recall some concepts from multiobjective optimization which we need in this
paper. Consider the deterministic multiobjective problem (3). We first define
two special types of efficient solutions, namely _supported efficient_ and
_extreme supported efficient_ solutions. There exist slightly different
characterizations of these solutions. We use the definitions of Özpeynirci and
Köksalan (see [ÖK10]) and call a point $y\in\mathcal{Y}$ _extreme supported
nondominated_ , if there is no convex combination of nondominated points
$y^{(1)},y^{(2)},\dots,y^{(n)}\in\mathcal{Y}\setminus\\{y\\}$ such that
$\sum_{i=1}^{n}\lambda_{i}y^{(i)}\leq y$. We call a point _supported
nondominated_ , if there is no convex combination of nondominated points
$y^{(1)},y^{(2)},\dots,y^{(n)}\in\mathcal{Y}\setminus\\{y\\}$ such that
$\sum_{i=1}^{n}\lambda_{i}y^{(i)}<y$. A solution $x\in\mathcal{X}$ is called
_(extreme) supported efficient_ , if $y=g(x)$ is (extreme) supported
nondominated. A supported efficient solution can be found by solving the
scalarized weighted-sum problem
$\displaystyle P(\lambda)$
$\displaystyle\min_{x\in\mathcal{X}}\lambda_{1}g_{1}(x)+\lambda_{2}g_{2}(x)+\dots+\lambda_{p}g_{p}(x)$
for some weight vector $\lambda\in\mathbb{R}^{p}_{\succeq 0}$. We use
$\mathcal{Y}_{\textup{ESN}}$ to denote the set of extreme supported
nondominated points. Any subset of $\mathcal{X}$ whose image under $g$ is the
set of nondominated points $\mathcal{Y}_{\textup{N}}$ is called a
_representative set_ ; a set whose image under $g$ equals the set of extreme
supported nondominated points $\mathcal{Y}_{\textup{ESN}}$ is called
_representative set for the extreme supported efficient solutions_.
In the following we state two properties that are essential to prove some of
our results. The first is the existence of the _ideal point_
$y^{\text{I}}\coloneqq\begin{pmatrix}\min_{x\in\mathcal{X}}g_{1}(x)\\\
\min_{x\in\mathcal{X}}g_{2}(x)\\\ \vdots\\\
\min_{x\in\mathcal{X}}g_{p}(x)\end{pmatrix}$
for (3). We say that the _ideal point property_ is satisfied if an ideal point
exists, i.e.,
$\min_{x\in\mathcal{X}}g_{i}(x)\textup{ exists for }i=1,2,\dots,p.$ (ideal)
The second property we need is the _domination property_ (see [Hen86]).
For all $y\in\mathcal{Y}\setminus\mathcal{Y}_{\textup{N}}$, there exists a
point $y^{\prime}\in\mathcal{Y}_{\textup{N}}$ with $y^{\prime}\preceq y$.
(dom)
The following result is well known.
###### Lemma 1.
Let a multiobjective problem (3) be given. If $\mathcal{X}$ is finite, or if
$\mathcal{X}$ is compact and $g$ is continuous, then both, (ideal) and (dom)
hold.
###### Proof.
For (ideal) this is due to Weierstrass’ Extreme Value Theorem, for (dom) we
refer to [Hen86]. ∎
#### Domination and ideal point property for multiobjective robust
optimization problems.
We conclude this section by discussing under which assumptions (ideal) and
(dom) are satisfied for _robust_ multiobjective problems (see (6)), i.e., for
the case that the objective functions of (3) are given as $g=f^{\mathcal{U}}$.
For a discussion of (dom) in the context of multiobjective robust
optimization, see also [SZK21].
###### Theorem 2.
Let either
1. (i)
$\mathcal{X}$ and $\mathcal{U}$ be compact and $f$ jointly continuous in
$\mathcal{X}$ and $\mathcal{U}$,
2. (ii)
$\mathcal{X}$ be finite, $\mathcal{U}$ compact and $f$ continuous in
$\mathcal{U}$ for every fixed $x\in\mathcal{X}$,
3. (iii)
$\mathcal{U}$ be finite, $\mathcal{X}$ compact and $f$ continuous in
$\mathcal{X}$ for every fixed $\xi\in\mathcal{U}$, or
4. (iv)
$\mathcal{X}$ and $\mathcal{U}$ both be finite.
Then both, (dom) and (ideal) are satisfied for a multiobjective robust
optimization problem (3).
###### Proof.
We set $g_{i}(x)\coloneqq\sup_{\xi\in\mathcal{U}}f_{i}(x,\xi)$,
$i=1,2,\dots,p$, and distinguish two cases:
* (a)
$\mathcal{X}$ is finite: Due to Lemma 1, (dom) and (ideal) hold if
$g_{i}(x)=\sup_{\xi\in\mathcal{U}}f_{i}(x,\xi)$ exists for all
$x\in\mathcal{X}$. This is the case since either $\mathcal{U}$ is finite or
$\mathcal{U}$ is compact and $f_{i}(x,\cdot)$ continuous for every fixed
$x\in\mathcal{X}$.
* (b)
$\mathcal{X}$ is compact: In this case, Lemma 1 requires that $g_{i}(x)$ is
continuous. This holds since
* –
either $\mathcal{U}$ is finite, hence $g(x)$ is continuous as the maximum of a
finite set of continuous functions $f(\cdot,\xi)$, $\xi\in\mathcal{U}$,
* –
or $\mathcal{U}$ is compact and $f$ is jointly continuous in $(x,\xi)$ and
hence again, $g_{i}(x)$ is continuous.
∎
We conclude that (dom) and (ideal) hold for (BRO).
###### Corollary 3.
(BRO) satisfies both, (dom) and (ideal).
###### Proof.
By the assumptions of (BRO), $\mathcal{X}$ and $\mathcal{U}$ are both compact
(see (BRO-1) and (BRO-2)) and
$f_{i}:\mathcal{X}\times\mathcal{U}\to\mathbb{R}$ is jointly continuous in
$(x,\xi)$ for $i=1,2$ (BRO-3). Theorem 2 hence gives the result. ∎
## 3 Dichotomic search for biobjective minmax optimization
In this section we view our problem (BRO) as a deterministic biobjective
(mixed-integer) linear minmax problem. First, in Section 3.1, we repeat
dichotomic search from literature. In Section 3.2 we generalize this method
from biobjective mixed-integer linear optimization to biobjective mixed-
integer linear _minmax_ optimization, i.e., to problems of type (BRO).
### 3.1 Dichotomic search for biobjective mixed-integer linear optimization
We consider a special case of (3), namely biobjective linear mixed-integer
optimization problems,
$\min_{x\in\mathcal{X}}\begin{pmatrix}g_{1}(x)\\\
g_{2}(x)\end{pmatrix}\text{.}$ (8)
The feasible set $\mathcal{X}\subseteq\mathbb{R}^{n}$ is a polyhedron and as
in (BRO) it is intersected with $\mathbb{Z}^{k}\times\mathbb{R}^{n-k}$. The
objective functions $g_{1},g_{2}\colon\mathcal{X}\to\mathbb{R}$ are linear
functions.
A well-known approach to solve such problems is _dichotomic search_ ,
formulated in Algorithm 3.1. The method has first been published by Aneja and
Nair in 1979 (see [AN79]) and Cohon (see [Coh78]) for more specific problem
classes and is now part of multi-objective folklore and sometimes also known
as _Aneja and Nair’s bicriteria method_ (e.g., [UT94]) or _CAN method_ (e.g.,
[ÖK10]). Most frequently, it is used to solve biobjective _linear_ problems.
However, it can also be applied to biobjective _mixed-integer linear_ problems
where it determines all extreme supported efficient nondominated points
$Y^{\ast}$ and a representative set of extreme supported nondominated
solutions $X^{\ast}$. Dichotomic search takes advantage of the fact that in
$\mathbb{R}^{2}$ sorting nondominated solutions with respect to their first
coordinates is the same as reverse sorting by the second coordinate, i.e., for
two nondominated solutions $y^{l},y^{r}\in\mathcal{Y}\subset\mathbb{R}^{2}$,
$y^{l}_{1}<y^{r}_{1}$ implies $y^{r}_{2}>y^{l}_{2}$. The idea is to start with
the lexicographically optimal solutions and then in each step find a supported
non-dominated point “between” two given supported non-dominated points. The
method proceeds iteratively until all extreme supported nondominated points
are identified. Algorithmically, first, the lexicographic optimal solutions
$x^{L},x^{R}$ for (8) are computed. After that, in each iteration, a tuple
$(y^{l},y^{r})$ of two points known to be supported nondominated is taken and
$\lambda=(y^{l}_{2}-y^{r}_{2},y^{r}_{1}-y^{l}_{1})$, corresponding to the
slope $\frac{y^{l}_{2}-y^{r}_{2}}{y^{r}_{1}-y^{l}_{1}}$ of the line segment
from $y^{l}$ to $y^{r}$, is chosen. Solving the corresponding weighted-sum
(scalarized) problem
$\min_{x\in\mathcal{X}}\lambda^{t}g(x)$
either finds a new supported nondominated point between $y^{l}$ and $y^{r}$ or
certifies that there is no such point. The algorithm terminates when all
extreme supported nondominated points – each with a corresponding extreme
supported efficient solution – have been discovered. It might find also
supported nondominated points which are not extreme supported nondominated,
but these can be easily identified and removed.
0: Biobjective mixed-integer linear optimization problem (8).
0: Feasible set $\mathcal{X}$ is a polyhedron intersected with
$\mathbb{R}^{n-k}\times\mathbb{Z}^{k}$ for some $k\in\\{0,\ldots,n\\}$.
1: Initialize $\mathcal{L}\coloneqq\emptyset$. {$\mathcal{L}$ will contain
list of tuple images $(y^{l},y^{r})$ satisfying
$y^{l}_{1}<y^{r}_{1},y^{l}_{2}>y^{r}_{2}$}
2: Compute $\varepsilon_{1}\coloneqq\min_{x\in\mathcal{X}}g_{1}(x)$.
3: Determine
$x^{L}\in\operatorname*{arg\,min}_{x\in\mathcal{X}}\\{g_{2}(x)\colon
g_{1}(x)\leq\varepsilon_{1}\\}$.
4: Set $y^{L}\coloneqq g(x^{L})$.
5: Compute $\varepsilon_{2}\coloneqq\min_{x\in\mathcal{X}}g_{2}(x)$.
6: Determine
$x^{R}\in\operatorname*{arg\,min}_{x\in\mathcal{X}}\\{g_{1}(x)\colon
g_{2}(x)\leq\varepsilon_{2}\\}$.
7: Set $y^{R}\coloneqq g(x^{R})$.
8: if $y^{L}=y^{R}$ then
9: STOP. Only one nondominated image found.
10: return $Y^{\ast}=\\{y^{L}\\},X^{\ast}=\\{x^{L}\\}$.
11: else
12:
$Y^{\ast}=\\{y^{L},y^{R}\\},X^{\ast}=\\{x^{L},x^{R}\\},\mathcal{L}=\\{(y^{L},y^{R})\\}$.
13: end if
14: while $L\not=\emptyset$ do
15: Remove element $(y^{l},y^{r})$ from $\mathcal{L}$.
16: Compute $\lambda\coloneqq(y^{l}_{2}-y^{r}_{2},y^{r}_{1}-y^{l}_{1})$.
17: Determine
$x^{\ast}\in\operatorname*{arg\,min}_{x\in\mathcal{X}}\lambda^{T}g(x)$.
18: Set $y^{\ast}\coloneqq g(x^{\ast})$.
19: if $\lambda^{T}y^{\ast}<\lambda^{T}y^{l}$. then
20: Add $y^{\ast}$ to $Y^{\ast}$, add $x^{\ast}$ to $X^{\ast}$.
21: Add $(y^{l},y^{\ast}),(y^{\ast},y^{r})$ to $\mathcal{L}$.
22: end if
23: end while
24: return $Y^{\ast}$: contains all extreme supported nondominated points.
25: return $X^{\ast}$: contains a representative set of extreme supported
efficient solutions.
Determine lexicographic solutions
Solve weighted-sum problem for weights $\lambda$
Algorithm 3.1 Dichotomic search
Finiteness and correctness of Algorithm 3.1 follow from the considerations
above which are derived from the literature (e.g., [PKL19, ÖK10]) and are
stated in the following lemma. The lemma is valid if (8) satisfies (ideal).
This is a slight generalization to [ÖK10] who assumed that (8) is bounded by
the origin, i.e., $g_{i}(x)$, $i=1,2$, are non-negative for all
$x\in\mathcal{X}$.
###### Lemma 4 (e.g., [ÖK10]).
Let a biobjective problem as in (8) be given, i.e.,
* •
with linear objectives $g_{1}$, $g_{2}$ and
* •
a feasible set $\mathcal{X}$ that is a polyhedron intersected with
$\mathbb{R}^{n-k}\times\mathbb{Z}^{k}$.
* •
Furthermore, let (ideal) hold for (8).
Then Algorithm 3.1 returns a set $Y^{\ast}$ containing all extreme supported
nondominated points and a set $X^{\ast}$ containing a representative set of
extreme supported efficient solutions after $2|Y^{\ast}|-3$ iterations (lines
15–22) if $|Y^{\ast}|>2$ and zero iterations if $|Y^{\ast}|=1$.
It is known that in the case of biobjective linear optimization problems, the
set of all extreme supported nondominated points and a representative set of
extreme supported efficient solutions can be used to construct all
nondominated points and a representative set of efficient solutions,
respectively. We will show a related result for (BRO) in Lemma 16 in Section
5.
### 3.2 Dichotomic search for biobjective mixed-integer linear _minmax_
optimization
Our goal is to apply dichotomic search to (BRO), i.e., to a biobjective mixed-
integer linear robust optimization problem which is given as the minmax
problem introduced in Section 2
$\displaystyle\textup{BRO}(\mathcal{U})$
$\displaystyle\min_{x\in\mathcal{X}}\begin{pmatrix}\sup_{\xi\in\mathcal{U}}f_{1}(x,\xi)\\\
\sup_{\xi\in\mathcal{U}}f_{2}(x,\xi)\end{pmatrix}$ (BRO revisited)
Recall that the functions $f_{1}$ and $f_{2}$ are linear in $x$ for every
fixed $\xi\in\mathcal{U}$ and
$\mathcal{X}=P\cap(\mathbb{Z}^{k}\times\mathbb{R}^{n-k})$ for a polyhedron $P$
and $0\leq k\leq n$, i.e., without the supremum (BRO) would satisfy the
requirements of Lemma 4. However, since the functions
$f_{i}^{\mathcal{U}}\colon\mathcal{X}\to\mathbb{R}$,
$x\mapsto\sup_{\xi\in\mathcal{U}}f_{i}(x,\xi)$, $i=1,2$, are not linear, we
aim to transform (BRO) to a biobjective mixed-integer linear optimization
problem, i.e., to a problem of type (8) for which we can apply dichotomic
search.
We proceed in two steps. The first step is to transform (BRO) to its
bottleneck version, i.e., to
$\displaystyle\textup{BRO}_{\textup{BN}}(\mathcal{U})$
$\displaystyle\min\begin{pmatrix}y_{1}\\\ y_{2}\end{pmatrix}$
$\displaystyle\text{s.t. }y_{1}$ $\displaystyle\geq f_{1}(x,\xi)$
$\displaystyle\;\forall\xi\in\mathcal{U}$ $\displaystyle y_{2}$
$\displaystyle\geq f_{2}(x,\xi)$ $\displaystyle\;\forall\xi\in\mathcal{U}$
$\displaystyle x$ $\displaystyle\in\mathcal{X}$ $\displaystyle y$
$\displaystyle\in\mathbb{R}^{2}$
This is justified by the following lemma which regards the relationship of
$\textup{BRO}(\mathcal{U})$ and $\textup{BRO}_{\textup{BN}}(\mathcal{U})$.
###### Lemma 5.
Let a problem of type (BRO) be given. In particular, let $\mathcal{U}$ be
compact and
$f_{i}(x,\cdot)\colon\operatorname*{conv}(\mathcal{U})\to\mathbb{R}$,
$x\in\mathcal{X}$, $i=1,2$, be continuous. Then
1. (i)
$\\{(x,y):x\in\mathcal{X},y\geq f^{\mathcal{U}}(x)\\}\not=\emptyset$ is the
set of feasible solutions for $\textup{BRO}_{\textup{BN}}(\mathcal{U})$.
2. (ii)
$X\subseteq\mathcal{X}$ is the set of efficient solutions to
$\textup{BRO}(\mathcal{U})$ if and only if $\\{(x,y):x\in
X,y=f^{\mathcal{U}}(x)\\}$ is the set of efficient solutions to
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$. In particular, the set of
nondominated points for $\textup{BRO}(\mathcal{U})$ and
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$ coincide.
3. (iii)
The set of extreme supported nondominated points for
$\textup{BRO}(\mathcal{U})$ and $\textup{BRO}_{\textup{BN}}(\mathcal{U})$
coincide.
4. (iv)
$X\subseteq\mathcal{X}$ is a representative set of extreme supported efficient
solutions to $\textup{BRO}(\mathcal{U})$ if and only if $\\{(x,y)\colon x\in
X,y=f^{\mathcal{U}}(x)\\}$ is a representative set of extreme supported
efficient solutions to $\textup{BRO}_{\textup{BN}}(\mathcal{U})$.
###### Proof.
1. (i)
Directly by definition of $\textup{BRO}_{\textup{BN}}(\mathcal{U})$. The
feasible set of $\textup{BRO}_{\textup{BN}}(\mathcal{U})$ is not empty due to
compactness of $\mathcal{U}$.
2. (ii)
Let $(x,y)$ be efficient for $\textup{BRO}_{\textup{BN}}(\mathcal{U})$. We
show that this yields $y=f^{\mathcal{U}}(x)$: Clearly, $y\geq
f^{\mathcal{U}}(x)$ otherwise $(x,y)$ is not feasible for
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$, (see (i)). Now assume that
$y_{i}>\max_{\xi\in\mathcal{U}}f_{i}(x,\xi)$ for $i\in\\{1,2\\}$. Then $(x,y)$
is dominated by the feasible solution $(x,f^{\mathcal{U}}(x))$ and hence not
efficient. The set of efficient solutions to
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$ hence is contained in
$\\{(x,f^{\mathcal{U}}(x)):x\in\mathcal{X}\\}$.
Note that $f^{\mathcal{U}}(x)$ is the objective function value of $x$ in
$\textup{BRO}(\mathcal{U})$ and also of $(x,f^{\mathcal{U}}(x))$ in
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$. This yields that $x$ is efficient
to $\textup{BRO}(\mathcal{U})$ if and only if $(x,f^{\mathcal{U}}(x))$ is
efficient to $\textup{BRO}_{\textup{BN}}(\mathcal{U})$. Hence, $X$ is the set
of efficient solutions to $\textup{BRO}(\mathcal{U})$ if and only if
$\\{(x,f^{\mathcal{U}}(x)):x\in X\\}$ is the set of efficient solutions to
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$ and the sets of nondominated points
of both problems coincide.
3. (iii)
The definition of extreme supported nondominated solutions only uses the set
of nondominated points in objective space. Due to (ii) the set of nondominated
points for $\textup{BRO}(\mathcal{U})$ and
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$ coincide, hence also their extreme
supported nondominated points.
4. (iv)
Let $X\subseteq\mathcal{X}$ be a representative set of extreme supported
efficient solutions to $\textup{BRO}(\mathcal{U})$. Then $f^{\mathcal{U}}(X)$
is the set of extreme supported nondominated points for
$\textup{BRO}(\mathcal{U})$. According to (iii), $f^{\mathcal{U}}(X)$ is also
the set of extreme supported nondominated points to
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$. Since $f^{\mathcal{U}}(X)$ is the
image of $\\{(x,f^{\mathcal{U}}(x)):x\in X\\}$ for
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$, the latter set is a representative
set of extreme supported efficient solutions to
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$.
Let there be a representative set of extreme supported efficient solutions to
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$. By (ii), it takes the form
$\\{(x,y)\colon x\in X,y=f^{\mathcal{U}}(x)\\}$ for some
$X\subseteq\mathcal{X}$.
Its image $f^{\mathcal{U}}(X)$ then is the set of extreme supported
nondominated solutions to $\textup{BRO}_{\textup{BN}}(\mathcal{U})$, and
according to (iii), also to $\textup{BRO}(\mathcal{U})$. Consequently, $X$ is
a representative set of extreme supported efficient solutions to
$\textup{BRO}(\mathcal{U})$.
∎
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$ has linear objective functions.
However, to ensure its feasible set meets the requirements of Lemma 4, we
additionally need that the feasible set of
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$ is a polyhedron intersected with
$\mathbb{Z}^{k}\times\mathbb{R}^{n-k}$ for $0\leq k\leq n$. Then Algorithm 3.1
can be applied to $\textup{BRO}_{\textup{BN}}(\mathcal{U})$ and determines all
its extreme supported nondominated points and a representative set of extreme
supported efficient solutions. In the following lemma we show more, namely
that we do not need the bottleneck version but can apply Algorithm 3.1
directly to $\textup{BRO}(\mathcal{U})$ to receive the extreme supported
nondominated points and a representative set of extreme supported efficient
solutions of $\textup{BRO}(\mathcal{U})$, if the set $\mathcal{U}$ of
scenarios is finite.
###### Lemma 6.
Let a problem of type (BRO) be given and let (BRO-1) and (BRO-3) hold. We
assume that $\mathcal{U}$ is non-empty and finite.
Then Algorithm 3.1 applied to (BRO) returns a set $Y^{\ast}$ containing all
extreme supported nondominated points and a set $X^{\ast}$ containing a
representative set of extreme supported efficient solutions after
$2|Y^{\ast}|-3$ iterations (lines 14-23) if $|Y^{\ast}|>2$ and zero iterations
if $|Y^{\ast}|=1$.
###### Proof.
The proof is in two parts: First, we show that dichotomic search _applied to
the bottleneck version_ $\textup{BRO}_{\textup{BN}}(\mathcal{U})$ of
$\textup{BRO}(\mathcal{U})$ returns a representative set of extreme supported
efficient solutions and the set of all extreme supported nondominated points
for the (non-bottleneck) problem $\textup{BRO}(\mathcal{U})$. Second, we show
that applying dichotomic search directly to $\textup{BRO}(\mathcal{U})$ yields
the exact same solutions as applying it to the bottleneck version
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$.
For the first part we use that the bottleneck version of the problem, i.e.,
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$, meets the requirements of Lemma 4:
We use the assumptions made for (BRO) and see that
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$ is a biobjective problem with two
linear objectives $y_{1}$ and $y_{2}$. For the feasible set note that the
original feasible set $\mathcal{X}$ of $\textup{BRO}(\mathcal{U})$ is given as
$\mathcal{X}=P\cap(\mathbb{R}^{n-k}\times\mathbb{Z}^{k})$. Since we add two
variables and two linear constraints for each scenario from the finite set
$\mathcal{U}$ (see part (i) of Lemma 5) also the resulting feasible set for
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$ can be written as
$P^{\prime}\cap(\mathbb{R}^{n^{\prime}-k}\times\mathbb{Z}^{k})$ with a new
polyhedron $P^{\prime}$ and dimension $n^{\prime}=n+2$. Furthermore, (ideal)
holds due to Corollary 3.
Thus, due to Lemma 4, dichotomic search (Algorithm 3.1) can be applied and a
set $Y^{\ast}_{BN}$ containing all extreme supported nondominated points and a
representative set of extreme supported efficient solutions $X^{\ast}_{BN}$
for $\textup{BRO}_{\textup{BN}}(\mathcal{U})$ are determined after
$2|Y^{\ast}|-3$ iterations (lines 14-23) if $|Y^{\ast}|>2$ and zero iterations
if $|Y^{\ast}|=1$.
Lemma 5 (iv) shows that $X^{*}_{BN}=\\{(x,f^{\mathcal{U}}(x))\colon x\in X\\}$
for some set $X\subseteq\mathcal{X}$ which is a representative set of extreme
supported efficient solutions of $\textup{BRO}(\mathcal{U})$.
For the second part, note that the difference between using
$\textup{BRO}(\mathcal{U})$ or $\textup{BRO}_{BN}(\mathcal{U})$ concerns lines
2, 3, 5, 6, and each iteration of line 17 of Algorithm 3.1. However, there is
no difference between applying these steps to $\textup{BRO}(\mathcal{U})$ and
$\textup{BRO}_{\textup{BN}}(\mathcal{U})$: the feasible set of the latter
problem is of higher dimension than the feasible set of the former but their
outcomes in the objective space $\mathbb{R}^{2}$ coincide (see Lemma 5) and
only those are needed for subsequent computations. ∎
The lemma above justifies the application of dichotomic search to our problem
of interest (BRO) if $\mathcal{U}$ is finite. However, in (BRO) $\mathcal{U}$
may be a polytope. On the other hand, in (BRO-3) we made the additional – and
thus far unnecessary – assumption that
$f_{i}(x,\cdot)\colon\mathcal{U}\to\mathbb{R}$, $i=1,2$ are quasi-convex.
Utilizing this additional requirement, we now show that Lemma 6 is still valid
if $\mathcal{U}$ is a polytope instead of a finite set.
###### Lemma 7.
Let a problem of type (BRO) be given and let (BRO-1) and (BRO-3) hold. We
assume that $\mathcal{U}$ is a polytope.
Then Algorithm 3.1 applied to (BRO) returns a set $Y^{\ast}$ containing all
extreme supported nondominated points and a set $X^{\ast}$ containing a
representative set of extreme supported efficient solutions after
$2|Y^{\ast}|-3$ iterations (lines 14-23) if $|Y^{\ast}|>2$ and zero iterations
if $|Y^{\ast}|=1$.
###### Proof.
If $\mathcal{U}$ is a polytope it has a finite number of (not necessarily
known) extreme points $\xi_{1},\ldots,\xi_{l}$. Since the functions
$f_{1}(x,\cdot),f_{2}(x,\cdot)\colon\operatorname*{conv}(\mathcal{U})\to\mathbb{R}$,
$x\in\mathcal{X}$ are quasi-convex, according to [EIS14, Theorem 5.9],
$\textup{BRO}(\mathcal{U})$ and $\textup{BRO}(\\{\xi_{1},\ldots,\xi_{l}\\})$
are equivalent since their objective functions $f^{\mathcal{U}}$ and
$f^{\\{\xi_{1},\ldots,\xi_{l}\\}}$ are the same.
Lemma 6 justifies that we can apply Algorithm 3.1 to
$\textup{BRO}(\\{\xi_{1},\ldots,\xi_{l}\\})$ and get all extreme supported
nondominated points and a representative set of extreme supported efficient
solutions of $\textup{BRO}(\\{\xi_{1},\ldots,\xi_{l}\\})$ and hence also of
$\textup{BRO}(\mathcal{U})$ in $2|Y^{\ast}|-3$ iterations if $|Y^{\ast}|>2$
and zero iterations if $|Y^{\ast}|=1$. This, however, requires that
$\xi_{1},\ldots,\xi_{l}$ are known. Since finding the vertices of a given
polytope, known as _vertex enumeration_ , is a hard problem (see [Kha+09]), we
apply Algorithm 3.1 directly to $\textup{BRO}(\mathcal{U})$ without using the
extreme points of $\mathcal{U}$. Luckily, this can be done by using the
equivalence of $\textup{BRO}(\\{\xi_{1},\ldots,\xi_{l}\\})$ and
$\textup{BRO}(\mathcal{U})$ once more:
Namely, we replace $\textup{BRO}(\\{\xi_{1},\ldots,\xi_{l}\\})$ by
$\textup{BRO}(\mathcal{U})$ whenever it occurs in Algorithm 3.1, i.e, in Steps
2,3,6,7 and in Step 17 and note that it does not change any result.
Summarizing, we can also apply Algorithm 3.1 directly to
$\textup{BRO}(\mathcal{U})$. ∎
## 4 Optimization-pessimization for biobjective optimization
In the previous section, we conceived the problem (BRO) primarily as a
biobjective problem – with the more complicated objective function
$f^{\mathcal{U}}$ – and suggested biobjective optimization methods. In this
section, we take the perspective of a robust optimizer and apply a method
known from robust optimization. More precisely, we use a cutting plane
approach, called _optimization-pessimization_ , which is designed to find
minmax robust solutions of uncertain (but single-objective) optimization
problems. The approach is reviewed in Section 4.1 and extended to multi-
objective optimization problems in Section 4.2.
### 4.1 Optimization-pessimization for single-objective robust optimization
This section deals with uncertain (single-objective) optimization problems,
$\left\\{\min_{x\in\mathcal{X}}h(x,\xi)\colon\xi\in\mathcal{U}\right\\}\text{.}$
(1 revisited)
More specifically, we want to determine minmax robust solutions for such
problems and, to that end, solve the robust counterpart,
$\displaystyle\textup{P}^{\textup{single}}(\mathcal{U})$
$\displaystyle\min_{x\in\mathcal{X}}\sup_{\xi\in\mathcal{U}}h(x,\xi)\text{.}$
(2 revisited)
We assume that for every fixed $x\in\mathcal{X}$ the function
$h(x,\cdot)\colon\operatorname*{conv}(\mathcal{U})\to\mathbb{R}$ is continuous
and quasi-convex and that $\mathcal{U}$ is compact. Hence,
$\sup_{\xi\in\mathcal{U}}h(x,\xi)$ is attained for all $x\in\mathcal{X}$ and
from now on we can write $\max_{\xi\in\mathcal{U}}h(x,\xi)$ instead. Let us
denote
$z(\mathcal{U})\coloneqq\min_{x\in\mathcal{X}}\max_{\xi\in\mathcal{U}}h(x,\xi)$
as optimal objective function value of (2) for a given uncertainty set
$\mathcal{U}$.
There exist many approaches for solving problem (2), which are grouped in
[GYd15] into two classes: The first class of algorithms is based on
reformulations to avoid the maximum over an (often infinite) set. We follow
this approach in Section 5.3. The algorithms of the second class proceed
iteratively. They start with a small set of scenarios and add scenarios step
by step. These approaches are known under various names such as _cutting set
method_ ([MB09]), _cutting plane method_ ([BDL16]), _scenario relaxation
procedure_ ([Ass+08], [ABV09]), _outer approximation method_ ([Ree94] [BNA13]
[GS16]), _(modified) Benders decomposition approach_ ([Mon06], [SAG11]), or
_implementor-adversarial framework_ ([Bie07]).
We refer to it as _optimization-pessimization_. The idea is to utilize that
robust optimization problems are easier to solve for (very) small uncertainty
sets: The routine starts with a reduced set of scenarios
$\mathcal{U}^{\prime}$ for which a robust solution is determined. For this
solution, the routine determines a worst-case scenario out of the full
uncertainty set $\mathcal{U}$ which is added to $\mathcal{U}^{\prime}$. For
the new scenario set, a new robust solution is found. This procedure is
repeated until the quality of the solution found is good enough, see Figure 1
for an illustration.
Optimization: Determine robust solution $x^{\ast}\in\mathcal{X}$ of
$\textup{P}^{\textup{single}}(\mathcal{U}^{\prime})$ finding
$x^{\ast}\in\operatorname*{arg\,min}_{x\in\mathcal{X}}\max_{\xi\in\mathcal{U}^{\prime}}h(x,\xi)$
Pessimization: Determine worst-case scenario $\xi^{\ast}\in\mathcal{U}$ for
given $x^{\ast}$ finding
$\xi^{\ast}\in\operatorname*{arg\,max}_{\xi\in\mathcal{U}}h(x^{\ast},\xi)$ Add
scenario: $\mathcal{U}^{\prime}\coloneqq\mathcal{U}^{\prime}\cup\\{\xi\\}$
Figure 1: Optimization-pessimization for robust single-objective optimization
problems
Formally, the optimization and pessimization problems are defined as follows:
For any $\mathcal{U}^{\prime}\subseteq\mathcal{U}$ the _optimization problem_
is defined as
$\displaystyle\textup{P}^{\textup{single}}(\mathcal{U}^{\prime})$
$\displaystyle
z(\mathcal{U}^{\prime})\coloneqq\min_{x\in\mathcal{X}}\max_{\xi\in\mathcal{U}^{\prime}}h(x,\xi).$
It is a relaxation of $\textup{P}^{\textup{single}}(\mathcal{U})$ and, thus,
yields a lower bound for $\textup{P}^{\textup{single}}(\mathcal{U})$, i.e.,
$z(\mathcal{U}^{\prime})\leq z(\mathcal{U})\text{.}$ (9)
For a given $x\in\mathcal{X}$, the _pessimization problem_
$\displaystyle\text{Pess}(x)$ $\displaystyle
h^{\mathcal{U}}(x)\coloneqq\max_{\xi\in\mathcal{U}}h(x,\xi)$
evaluates $x$ over the complete set of scenarios $\mathcal{U}$ and, thus,
provides an upper bound for $z(\mathcal{U})$, i.e.,
$h^{\mathcal{U}}(x)\geq z(\mathcal{U})\textup{.}$ (10)
Algorithm 4.1 describes how this method can be put to use algorithmically if
$\mathcal{U}$ is a polytope or finite.
0: Robust optimization problem $P(\mathcal{U})$ as in (2).
0: Finite initial set $\mathcal{U}^{(0)}\subseteq\mathcal{U}$.
0: Either $\mathcal{U}$ finite or $\mathcal{U}$ a polytope and $h(x,\cdot)$
continuous and quasi-convex.
Set $k\coloneqq 0$.
repeat
Set $\mathcal{U}^{(k+1)}\coloneqq\mathcal{U}^{(k)}$.
Determine
$x^{k}\in\operatorname*{arg\,min}_{x\in\mathcal{X}}\\{\max_{\xi\in\mathcal{U}^{k}}h(x,\xi)\\}$.
Set $z(\mathcal{U}^{(k)})\coloneqq\max_{\xi\in\mathcal{U}^{(k)}}h(x^{k},\xi)$.
For given $x^{k}$ determine solution
$\xi^{k}\in\operatorname*{arg\,max}_{\mathcal{U}}h(x^{k},\xi^{k})$. Set
$h^{\mathcal{U}}(x^{k})\coloneqq h(x^{k},\xi^{k})$.
Add $\xi^{k}$ to $\mathcal{U}^{(k+1)}$.
Set $k\coloneqq k+1$.
until $h^{\mathcal{U}}(x^{k-1})=z(\mathcal{U}^{(k-1)})$.
return robust solution $x^{\ast}$.
return set of worst-case scenarios
$\mathcal{U}^{\textup{FINAL}}\coloneqq\mathcal{U}^{k}$.
Optimization
Pessimization
Algorithm 4.1 Optimization-pessimization for single-objective robust
optimization
The routine produces a sequence of sets
$\mathcal{U}^{(0)}\subseteq\mathcal{U}^{(1)}\subseteq\mathcal{U}^{(2)}\subseteq\dots\subseteq\mathcal{U}.$
(11)
According to (9) we receive a sequence of lower bounds
$z(\mathcal{U}^{(0)})\leq z(\mathcal{U}^{(1)})\leq
z(\mathcal{U}^{(2)})\leq\dots\leq z(\mathcal{U})$ (12)
and, a feasible solution $x^{k}$ in each iteration from which we can derive an
upper bound according to (10), i.e.,
$z(\mathcal{U}^{(k)})\leq z(\mathcal{U})\leq h^{\mathcal{U}}(x^{k})\textup{.}$
(13)
We stop when lower and upper bound coincide. Then an optimal solution to (2)
and thus a (minmax) robust optimal solution to (1) has been found. For more
detailed discussions of the method we refer to [BDL16, ABV09, PS20]. The
finiteness of Algorithm 4.1 for uncertainty sets $\mathcal{U}$ that are
polytopes is shown in the following lemma in part (ii).
###### Lemma 8.
Assume that $\textup{P}^{\textup{single}}(\mathcal{U})$ has an optimal
solution and $\textup{P}^{\textup{single}}(\mathcal{U}^{\prime})$ has an
optimal solution for all finite $\mathcal{U}^{\prime}\subseteq\mathcal{U}$.
1. (i)
Let $\mathcal{U}$ be finite. Then Algorithm 4.1 returns a solution to
$\textup{P}^{\textup{single}}(\mathcal{U})$ in at most $|\mathcal{U}|$
iterations.
2. (ii)
Let $\mathcal{U}$ be a polytope or finite and let
$\operatorname*{ext}(\mathcal{U})$ be its set of extreme points. Let
$h(x,\cdot)\colon\operatorname*{conv}(\mathcal{U})\to\mathbb{R}$,
$x\in\mathcal{X}$, be continuous and quasi-convex. Then Algorithm 4.1 returns
a solution to $\textup{P}^{\textup{single}}(\mathcal{U})$ in at most
$|ext(\mathcal{U})|$ iterations if we choose an algorithm for the
pessimization problem $\textup{Pess}(x)$ which always finds an extreme point
of $\mathcal{U}$.
###### Proof.
Algorithm 4.1 stops if the lower and upper bound for $z(\mathcal{U})$ coincide
(see line 8 of Algorithm 4.1), i.e., if
$h^{\mathcal{U}}(x^{k})=z(\mathcal{U}^{(k)})$. We hence have that $x^{k}$ is
an optimal solution. Note that
$\max_{\xi\in\mathcal{U}}h(x^{k},\xi)=h^{\mathcal{U}}(x^{k})=z(\mathcal{U}^{(k)})=\max_{\xi\in\mathcal{U}^{(k)}}h(x^{k},\xi)\textup{,}$
(14)
if at least one worst-case scenario of $\mathcal{U}$ for $x^{k}$ is already
contained in $\mathcal{U}^{(k)}$. For a finite uncertainty set, in every
iteration either a new worst-case scenario is added or (14) holds and the
procedure stops. The latter happens after at most $|\mathcal{U}|$ iterations
which shows (i).
For (ii), consider the pessimization problem $\textup{Pess}(x^{k})$: here we
maximize a continuous function over a compact set $\mathcal{U}$, i.e., a
maximum always exists. Since $h(x,\cdot)$ is quasi-convex, a maximum is always
attained at an extreme point of $\mathcal{U}$. If we choose an algorithm that
returns an extreme point for such optimization problems, we add a new extreme
point in each iteration. Since the number of extreme points of $\mathcal{U}$
is finite the procedure stops when (14) holds. As in part (i) this happens
after at most $|\operatorname*{ext}(\mathcal{U})|$ iterations. ∎
We remark that Algorithm 4.1 also converges for bounded non-polyhedral sets
$\mathcal{U}$ under uniform Lipschitz-continuity in $x$ for all fixed values
of $\xi$ (see [MB09]).
### 4.2 Optimization-pessimization for multi-objective robust optimization
#### Optimization and pessimization problem in the multiobjective case.
In order to apply optimization-pessimization to (BRO), we need to generalize
it to biobjective problems. In this section we go a step further and consider
minmax problems with $p$ objective functions, i.e.,
$\displaystyle P(\mathcal{U})$
$\displaystyle\min_{x\in\mathcal{X}}\begin{pmatrix}\sup_{\xi\in\mathcal{U}}f_{1}(x,\xi)\\\
\sup_{\xi\in\mathcal{U}}f_{2}(x,\xi)\\\ \vdots\\\
\sup_{\xi\in\mathcal{U}}f_{p}(x,\xi)\\\ \end{pmatrix}\text{.}$ (6 revisited)
for which we aspire to determine a representative set of extreme supported
efficient solutions. With this purpose in mind, we develop a generalized
version of optimization-pessimization of Section 4.1.
The optimization problem $P(\mathcal{U}^{\prime})$ for
$\mathcal{U}^{\prime}\subseteq\mathcal{U}$ is the multiobjective optimization
problem
$\displaystyle P(\mathcal{U}^{\prime})$ $\displaystyle
z(\mathcal{U}^{\prime})\coloneqq\min_{x\in\mathcal{X}}\begin{pmatrix}\sup_{\xi\in\mathcal{U}^{\prime}}f_{1}(x,\xi)\\\
\sup_{\xi\in\mathcal{U}^{\prime}}f_{2}(x,\xi)\\\ \vdots\\\
\sup_{\xi\in\mathcal{U}^{\prime}}f_{p}(x,\xi)\end{pmatrix}\text{.}$ (15)
The pessimization problem
$\displaystyle\textup{Pess}(x)$ $\displaystyle
f^{\mathcal{U}}(x)\coloneqq\begin{pmatrix}\sup_{\xi\in\mathcal{U}}f_{1}(x,\xi)\\\
\sup_{\xi\in\mathcal{U}}f_{2}(x,\xi)\\\ \vdots\\\
\sup_{\xi\in\mathcal{U}}f_{p}(x,\xi)\end{pmatrix}$ (16)
for given $x\in\mathcal{X}$ consists of $p$ indepedent pessimization problems.
#### Lower and upper bounds provided by the optimization and the pessimization
problem.
We first discuss the optimization and pessimization problems in relation to
(6) which we are interested to solve.
For single-objective problems (6), the solutions to
$\textup{P}^{\textup{single}}(\mathcal{U}^{\prime})$ and $\textup{Pess}(x)$
provide lower and upper bounds to (6). In the multi-objective setting we do
not evaluate single solutions, but we need to evaluate (Pareto) sets. Sets can
be compared by set order relations, one of the most common ones is the _upper
setless order_ : For two sets $Y_{1},Y_{2}\subset\mathbb{R}^{p}$ it is defined
as follows:
$Y_{1}\preceq^{upp}Y_{2}\mbox{ if for all }y\in Y_{2}\mbox{ there exists
}\tilde{y}\in Y_{1}\mbox{ with }\tilde{y}\leq y.$
In this sense, we can say that $Y_{1}$ is an (upper setless) lower bound on
$Y_{2}$. We now use the upper setless order to generalize (13) showing that
for multi-objective optimization we also get lower and upper bounds on (6)
when solving (15) and (16) for a subset $\mathcal{U}^{\prime}$ of
$\mathcal{U}$. More precisely, let $X^{\ast}(\mathcal{U})$ be the set of
efficient solutions to (6). Then $\\{f^{\mathcal{U}}(x)\colon x\in
X^{\ast}(\mathcal{U})\\}$ describes the Pareto frontier of (6). It can be
bounded based on the solutions of the relaxation $P(\mathcal{U}^{\prime})$ as
follows.
###### Lemma 9.
Let $\mathcal{U}^{\prime}\subseteq\mathcal{U}$ and denote
$X^{\ast}(\mathcal{U}^{\prime})$ and $X^{\ast}(\mathcal{U})$ the set of
efficient solutions of $P(\mathcal{U}^{\prime})$, and $P(\mathcal{U})$,
respectively. Assume that $P(\mathcal{U}^{\prime})$, and $P(\mathcal{U})$ both
satisfy the domination property (dom). Then the following holds for the upper
setless order $\preceq^{upp}$:
$\\{f^{\mathcal{U}^{\prime}}(x)\colon x\in
X^{\ast}(\mathcal{U}^{\prime})\\}\preceq^{upp}\\{f^{\mathcal{U}}(x)\colon x\in
X^{\ast}(\mathcal{U})\\}\preceq^{upp}\\{f^{\mathcal{U}}(x)\colon x\in
X^{\ast}(\mathcal{U}^{\prime})\\}$ (17)
###### Proof.
We first show the left hand side of (17). To this end, take $x\in
X^{\ast}(\mathcal{U})$. We want to show that there exists $\tilde{x}\in
X^{\ast}(\mathcal{U}^{\prime})$ such that
$f^{\mathcal{U}^{\prime}}(\tilde{x})\leq f^{\mathcal{U}}(x).$ (18)
From $\mathcal{U}^{\prime}\subseteq\mathcal{U}$ we get that
$f^{\mathcal{U}^{\prime}}(x)\leq f^{\mathcal{U}}(x)$, see (7). Hence, if $x\in
X^{\ast}(\mathcal{U}^{\prime})$ we set $\tilde{x}\coloneqq x$ and are done.
Otherwise, $x\not\in X^{\ast}(\mathcal{U}^{\prime})$, i.e., $x$ is not an
efficient solution to $P(\mathcal{U}^{\prime})$. Then, due to the domination
property, there exists $\tilde{x}\in X^{\ast}(\mathcal{U}^{\prime})$ with
$f^{\mathcal{U}^{\prime}}(\tilde{x})\leq f^{\mathcal{U}^{\prime}}(x)\leq
f^{\mathcal{U}}(x)$ and (18) holds.
For the right hand side, we take $x\in X^{\ast}(\mathcal{U}^{\prime})$. The
goal is to find $\tilde{x}\in X^{\ast}(\mathcal{U})$ such that
$f^{\mathcal{U}}(\tilde{x})\leq f^{\mathcal{U}}(x)\textup{.}$
Similar as above, if $x\in X^{\ast}(\mathcal{U})$ we set $\tilde{x}\coloneqq
x$ and are done. Otherwise, $x$ is not efficient for $P(\mathcal{U})$ and due
to the domination property we find $\tilde{x}\in X^{\ast}(\mathcal{U})$ with
$f^{\mathcal{U}}(\tilde{x})\leq f^{\mathcal{U}}(x)$ which finishes the proof.
∎
The statement in (17) is the multiobjective analog of (13).
#### Reduction of the scenario set.
In this paragraph, we examine the conditions under which a reduced uncertainty
set $\mathcal{U}^{\prime}\subset\mathcal{U}$ already contains all relevant
scenarios, such that the efficient solutions $P(\mathcal{U}^{\prime})$ are the
same as those of $P(\mathcal{U})$. In the single-objective setting this is the
case if for an efficient solution to $P(\mathcal{U}^{\prime})$ a worst-case
scenario is already included in $\mathcal{U}^{\prime}$ (see (14)). We call
this the worst-case property (wc). The following theorem formalizes the above
considerations and shows when the efficient solutions of $P(\mathcal{U})$ and
$P(\mathcal{U}^{\prime})$ coincide.
###### Theorem 10.
Let $\mathcal{U}^{\prime}\subset\mathcal{U}$. Consider $x\in\mathcal{X}$. If
we have
$\displaystyle\sup_{\xi\in\mathcal{U}^{\prime}}f_{i}(x,\xi)=\sup_{\xi\in\mathcal{U}}f_{i}(x,\xi)\text{
for all }i=1,2,\dots,p\text{,}$ (wc)
then the following holds:
$x\text{ is efficient for }P(\mathcal{U}^{\prime})\Rightarrow x\text{ is
efficient for }P(\mathcal{U})\text{.}$
Additionally, if the domination property (dom) holds for
$P(\mathcal{U}^{\prime})$ and _all_ solutions $x\in\mathcal{X}$ that are
efficient for $P(\mathcal{U}^{\prime})$ satisfy (wc), then the following
holds:
$\displaystyle x\text{ is efficient for
}P(\mathcal{U}^{\prime})\Leftrightarrow x\text{ is efficient for
}P(\mathcal{U})\text{.}$
###### Proof.
$\Rightarrow$: Let $x$ be efficient for $P(\mathcal{U}^{\prime})$ and satisfy
(wc), i.e., $f^{\mathcal{U}^{\prime}}(x)=f^{\mathcal{U}}(x)$. Assume to the
contrary that $x$ is not efficient for $P(\mathcal{U})$, i.e., there exists
$x^{\prime}\in\mathcal{X}$, such that
$f^{\mathcal{U}}(x^{\prime})\preceq f^{\mathcal{U}}(x)\text{.}$ (19)
$\mathcal{U}^{\prime}\subseteq\mathcal{U}$, hence
$f^{\mathcal{U}^{\prime}}(x^{\prime})\leq f^{\mathcal{U}}(x^{\prime})$, see
(7). This leads to
$f^{\mathcal{U}^{\prime}}(x^{\prime})\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\leq}}f^{\mathcal{U}}(x^{\prime})\stackrel{{\scriptstyle\eqref{eq:pb-
proof01}}}{{\preceq}}f^{\mathcal{U}}(x)\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}f^{\mathcal{U}^{\prime}}(x),$
which contradicts efficiency of $x$ for $P(\mathcal{U}^{\prime})$.
$\Leftarrow$: Let (wc) hold for all solutions which are efficient for
$P(\mathcal{U}^{\prime})$ and let $x$ be efficient for $P(\mathcal{U})$.
Assume to the contrary that $x\in\mathcal{X}$ is not efficient for
$P(\mathcal{U}^{\prime})$. Then, since the domination property holds, there is
a solution $x^{\prime}\in\mathcal{X}$ that is efficient for
P($\mathcal{U}^{\prime}$) such that
$f^{\mathcal{U}^{\prime}}(x^{\prime})\preceq
f^{\mathcal{U}^{\prime}}(x)\text{.}$ (20)
Note that since $x^{\prime}$ is efficient for $P(\mathcal{U}^{\prime})$, it
satisfies (wc). Together with $\mathcal{U}^{\prime}\subseteq\mathcal{U}$ we
receive
$f^{\mathcal{U}}(x^{\prime})\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}f^{\mathcal{U}^{\prime}}(x^{\prime})\stackrel{{\scriptstyle\eqref{eq:pb-
proof02}}}{{\preceq}}f^{\mathcal{U}^{\prime}}(x)\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\leq}}f^{\mathcal{U}}(x)\text{.}$
This contradicts the assumption of $x$ being efficient for $P(\mathcal{U})$. ∎
Checking _all_ efficient solutions of a multiobjective problem is
computationally hard (or even impossible). Thus, in the next result we
strengthen the above theorem in a fashion that (wc) must only be satisfied for
all solutions from a representative set.
###### Theorem 11.
Let the domination property (dom) be satisfied for $P(\mathcal{U})$ and
$P(\mathcal{U}^{\prime})$. If there is a representative set
$\mathcal{R}^{\prime}$ of efficient solutions for $P(\mathcal{U}^{\prime})$
whose elements satisfy (wc), then we have:
1. (i)
$x\in\mathcal{R}^{\prime}$ $\Rightarrow$ $x$ is efficient for
$P(\mathcal{U})$,
2. (ii)
$x$ is efficient for $P(\mathcal{U})$ $\Rightarrow$ $x$ is efficient for
$P(\mathcal{U}^{\prime})$, and
3. (iii)
$\mathcal{R}^{\prime}$ is a representative set of efficient solutions to
$P(\mathcal{U})$.
###### Proof.
1. (i)
Let $x\in\mathcal{R}^{\prime}$. In particular, $x$ is efficient for
$P(\mathcal{U}^{\prime})$ and by assumption it satisfies (wc). We can hence
apply Theorem 10 and conclude that $x$ is efficient for $P(\mathcal{U})$.
2. (ii)
Let $x$ be efficient for $P(\mathcal{U})$ and assume $x$ is not efficient for
$P(\mathcal{U}^{\prime})$. Due to the domination property, there is is a
solution $x^{\prime}$ that satisfies
$f^{\mathcal{U}^{\prime}}(x^{\prime})\preceq f^{\mathcal{U}^{\prime}}(x)$.
Moreover, since $\mathcal{R}^{\prime}$ is a representative set for
$P(\mathcal{U}^{\prime})$ we can choose $x^{\prime}\in\mathcal{R}^{\prime}$.
Hence, (wc) holds for $x^{\prime}$ and we receive
$f^{\mathcal{U}}(x^{\prime})\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}f^{\mathcal{U}^{\prime}}(x^{\prime})\stackrel{{\scriptstyle}}{{\preceq}}f^{\mathcal{U}^{\prime}}(x)\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\leq}}f^{\mathcal{U}}(x)\text{.}$
This contradicts efficiency of $x$ for $P(\mathcal{U})$.
3. (iii)
Let $\mathcal{R}\subset\mathcal{X}$ be a representative set of efficient
solutions for $P(\mathcal{U})$. We show that
$f^{\mathcal{U}}(\mathcal{R}^{\prime})=f^{\mathcal{U}}(\mathcal{R})$.
$\subset$: Let $y^{\prime}\in f^{\mathcal{U}}(\mathcal{R}^{\prime})$. Then
$y^{\prime}=f^{\mathcal{U}}(x^{\prime})$ for some
$x^{\prime}\in\mathcal{R}^{\prime}$. According to (i), $x^{\prime}$ is
efficient for $P(\mathcal{U})$, hence $y^{\prime}\in
f^{\mathcal{U}}(\mathcal{R})$.
$\supset$: Let $y\in f^{\mathcal{U}}(\mathcal{R})$. Then
$y=f^{\mathcal{U}}(x)$ for some $x$ that is efficient for $P(\mathcal{U})$.
According to (ii), $x$ is also efficient for $P(\mathcal{U}^{\prime})$. Hence,
$x^{\prime}\in\mathcal{R}^{\prime}$ exists such that
$f^{\mathcal{U}^{\prime}}(x)=f^{\mathcal{U}^{\prime}}(x^{\prime})$. This leads
to
$y=f^{\mathcal{U}}(x)\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\geq}}f^{\mathcal{U}^{\prime}}(x)=f^{\mathcal{U}^{\prime}}(x^{\prime})\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}f^{\mathcal{U}}(x^{\prime})\textup{.}$
Since by assumption $y$ is nondominated for $P(\mathcal{U})$, equality must
hold true. Thus, $y=f^{\mathcal{U}}(x^{\prime})$ for
$x^{\prime}\in\mathcal{R}^{\prime}$ and, consequently, $y\in
f^{\mathcal{U}}(\mathcal{R}^{\prime})$.
∎
Theorem 11 shows that it is not necessary to check the worst-case property
(wc) for _all_ efficient solutions, rather it is sufficient to check it only
for a representative set. However, a representative set may still be infinite,
even for linear problems.
In Theorem 14 we show that the statement of Theorem 11 remains valid if we
replace the set of all efficient solutions not only by a representative set,
but even by a representative set of only their extreme supported solutions.
Recall that in Section 3.1 with dichotomic search we provided an algorithm for
computing such a representative set of extreme supported solutions. In
preparation for Theorem 14 we need the following lemma and corollary that
investigate the relation of extreme supported nondominated points with the set
of all images $\mathcal{Y}=f^{\mathcal{U}}(\mathcal{X})$.
###### Lemma 12.
Let a multi-objective optimization problem (3) with
$\mathcal{Y}\subsetneq\mathbb{R}^{p}$ compact be given and let
$\mathcal{Y}_{\textup{ESN}}\not=\emptyset$ be its set of extreme supported
nondominated points. We assume that $\mathcal{Y}_{\textup{ESN}}$ is finite.
Then
$\mathcal{Y}\subseteq\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}^{p}_{\geq}$
holds.
###### Proof.
Assume there is a $\bar{y}\in\mathcal{Y}$ that does not lie in
$\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}^{p}$. We then can
show that there is also $\hat{y}$ outside of
$\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}^{p}$ which is
extreme supported nondominated, a contradiction.
So, assume to the contrary that
$\bar{y}\in\mathcal{Y}\setminus(\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}^{p}_{\geq})$
exists. Then the sets $\\{\bar{y}\\}$ and
$\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}^{p}_{\geq 1}$ are
disjoint, nonempty, closed and convex sets. Hence, a separating hyperplane
exists (see [BV04]), i.e., $\nu\in\mathbb{R}^{p}\setminus\\{0\\}$ and
$s\in\mathbb{R}$ exist such that
$\nu^{t}\bar{y}<s<\nu^{t}y,\;\forall
y\in\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}^{p}_{\geq}\textup{.}$
(21)
The elements of
$\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}^{p}_{\geq}$ can
get arbitrarily big in each component, hence $\nu_{i}\geq 0$ for all
$i=1,2,\dots,p$. Let now $z_{\nu}\coloneqq\min\\{\nu^{t}y\colon
y\in\mathcal{Y}\\}$ and
$\mathcal{Y}_{\nu}\coloneqq\operatorname*{arg\,min}\\{\nu^{t}y\colon
y\in\mathcal{Y}\\}$. Since $\bar{y}\in\mathcal{Y}$, we get
$\nu^{t}y^{\ast}\leq\nu^{t}\bar{y}<s,\;\forall y^{\ast}\in\mathcal{Y}_{\nu}.$
(22)
Together with (21) this shows that the elements in $\mathcal{Y}_{\nu}$ can be
separated from
$\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}_{\geq}^{p}$ and,
hence, cannot be extreme supported nondominated themselves. Specifically, the
lexicographic minimum,
$\hat{y}\coloneqq\operatorname*{lex\,min}_{y\in\mathcal{Y}_{\nu}}$, i.e.,
$\hat{y}_{j}=\min\\{y_{j}\colon
y\in\mathcal{Y}_{\nu},y_{1}=\hat{y}_{1},\dots,y_{j-1}=\hat{y}_{j-1}\\}$,
$j=1,2,\dots,p$, is not extreme supported nondominated (the existence of this
point follows from compactness of $\mathcal{Y}$).
Hence, a nontrivial convex combination of nondominated points
$y^{(1)},\dots,y^{(n)}\in\mathcal{Y}$ exists such that
$\hat{y}\geq\sum_{i=1}^{n}\lambda_{i}y^{(i)}\textup{.}$ (23)
Now we assume that $y^{(i)}\not\in\mathcal{Y}_{\nu}$ for at least one
$i=1,2,\dots,n$. Wlog., assume $y^{(1)}\not\in\mathcal{Y}_{\nu}$. Then
$\nu^{t}\sum_{i=1}^{n}\lambda_{i}y^{(i)}=\lambda_{1}\underbrace{\nu^{t}y^{(1)}}_{>z_{\nu}}+\sum_{i=2}^{n}\underbrace{\nu^{t}\lambda_{i}y^{(i)}}_{\geq
z_{\nu}}>\sum_{i=1}^{n}\lambda_{i}z_{\nu}=z_{\nu}=\nu^{t}\hat{y}\textup{.}$
(24)
Since $\nu_{i}\geq 0$, $i=1,2,\dots,n$, (24) contradicts (23). Thus, our
assumption that $y^{(i)}\not\in\mathcal{Y}_{\nu}$ for at least one
$i=1,2,\dots,n$ is contradicted and we have that $y^{(i)}\in\mathcal{Y}_{\nu}$
for all $i=1,2,\dots,n$.
Consequently, a nontrivial convex combination only consisting of nondominated
points $y^{(1)},\dots,y^{(n)}\in\mathcal{Y}_{\nu}\subseteq\mathcal{Y}$ exists
such that (23) holds. This, however, is not possible since, by definition,
$\hat{y}$ is the lexicographic minimum of $\mathcal{Y}_{\nu}$ and thus all
other elements of $\mathcal{Y}_{\nu}$ lie in the lexicographic cone
$\hat{y}+\\{y\in\mathbb{R}^{p}\colon y_{1}=y_{2}=\dots
y_{i}=0,y_{i+1}>0\textup{ for some }i=0,1,\dots,p\\}$. ∎
The following corollary will be used in the proof of the subsequent theorem.
###### Corollary 13.
Under the assumptions of Lemma 12 for any
$y\in\mathcal{Y}\setminus\mathcal{Y}_{\textup{ESN}}$, a nontrivial convex
combination
$\sum_{i=1}^{n}\lambda_{i}y^{(i)}\leq y$
with $y^{(1)},\dots,y^{(n)}\in\mathcal{Y}_{\textup{ESN}}$ exists.
We can now utilize the above corollary and show that the statement of Theorem
11 remains valid even if only representative sets of extreme supported
efficient solutions are considered.
###### Theorem 14.
Let the domination property (dom) be satisfied for $P(\mathcal{U})$ and
$P(\mathcal{U}^{\prime})$. If $\mathcal{Y}$ is compact and there is a finite
representative set $\mathcal{R}^{\prime}_{\text{ESE}}$ of extreme supported
efficient solutions for $P(\mathcal{U}^{\prime})$ whose elements satisfy (wc),
then
1. (i)
$x\in\mathcal{R}^{\prime}_{\text{ESE}}$ $\Rightarrow$ $x$ is extreme supported
efficient for $P(\mathcal{U})$,
2. (ii)
$x$ is extreme supported efficient for $P(\mathcal{U})$ $\Rightarrow$ $x$ is
extreme supported efficient for $P(\mathcal{U}^{\prime})$, and
3. (iii)
$\mathcal{R}^{\prime}_{\text{ESE}}$ is a representative set of extreme
supported efficient solutions to $P(\mathcal{U})$.
###### Proof.
1. (i)
Let $x\in\mathcal{R}^{\prime}_{\text{ESE}}$. Assume to the contrary that $x$
is not extreme supported efficient for $P(\mathcal{U})$, i.e., there exists a
nontrivial convex combination of solutions efficient for $P(\mathcal{U})$
$x^{\prime}_{1},\dots,x^{\prime}_{n}\in\mathcal{X}$, and
$\lambda\in\mathbb{R}_{\geq 0}$, $\sum_{i=1}^{n}\lambda_{i}=1$ such that
$\sum_{i=1}^{n}\lambda_{i}f^{\mathcal{U}}(x^{\prime}_{i})\leq
f^{\mathcal{U}}(x)\text{,}$ (25)
and $f^{\mathcal{U}}(x^{\prime}_{i})\not=f^{\mathcal{U}}(x)$ for all
$i=1,2,\dots,n$.
$\mathcal{U}^{\prime}\subseteq\mathcal{U}$, hence
$f^{\mathcal{U}^{\prime}}(x^{\prime}_{i})\leq
f^{\mathcal{U}}(x^{\prime}_{i})$, $i=1,2\dots,n$, see (7). This leads to
$\displaystyle\sum_{i=1}^{n}\lambda_{i}f^{\mathcal{U}^{\prime}}(x^{\prime}_{i})\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\leq}}\sum_{i=1}^{n}\lambda_{i}f^{\mathcal{U}}(x^{\prime}_{i})\stackrel{{\scriptstyle\eqref{eq:pb-
proof11}}}{{\leq}}f^{\mathcal{U}}(x)\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}f^{\mathcal{U}^{\prime}}(x).$
(26)
Hence, extreme supported efficiency of $x$ for $P(\mathcal{U}^{\prime})$ is
contradicted or
$f^{\mathcal{U}^{\prime}}(x^{\prime}_{i})=f^{\mathcal{U}^{\prime}}(x)\textup{
for at least one }i=1,2,\dots,n$ (27)
must hold. Assume that (27) holds. Then
$f^{\mathcal{U}}(x)\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}f^{\mathcal{U}^{\prime}}(x)\stackrel{{\scriptstyle\eqref{eq:pb-
proof13}}}{{=}}f^{\mathcal{U}^{\prime}}(x_{i}^{\prime})\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\leq}}f^{\mathcal{U}}(x_{i}^{\prime})$
follows. Since $x_{i}^{\prime}$ is efficient for $P(\mathcal{U})$, equality
holds. Hence, $f^{\mathcal{U}}(x^{\prime}_{i})\not=f^{\mathcal{U}}(x)$ for all
$i=1,2,\dots,n$ is contradicted.
2. (ii)
Let $x$ be extreme supported efficient for $P(\mathcal{U})$. Assume to the
contrary that $x\in\mathcal{X}$ is not extreme supported efficient for
$P(\mathcal{U}^{\prime})$. Then Corollary 13 can be applied to the problem
$P(\mathcal{U}^{\prime})$ with
$\mathcal{Y}=f^{\mathcal{U}^{\prime}}(\mathcal{X})$ and there exists a
nontrivial convex combination
$x^{\prime}_{1},\dots,x^{\prime}_{n}\in\mathcal{R}^{\prime}_{\textup{ESE}}$,
and $\lambda\in\mathbb{R}_{\geq 0}$, $\sum_{i=1}^{n}\lambda_{i}=1$ such that
$\sum_{i=1}^{n}\lambda_{i}f^{\mathcal{U}^{\prime}}(x^{\prime}_{i})\leq
f^{\mathcal{U}^{\prime}}(x)$ (28)
and $f^{\mathcal{U}^{\prime}}(x^{\prime}_{i})\not=f^{\mathcal{U}^{\prime}}(x)$
for all $i=1,2,\dots,n$.
Note that since $x^{\prime}_{i}\in\mathcal{R}^{\prime}_{\textup{ESE}}$,
$i=1,2,\dots,n$, they satisfy (wc). Together with
$\mathcal{U}^{\prime}\subseteq\mathcal{U}$ we receive
$\sum_{i=1}^{n}\lambda_{i}f^{\mathcal{U}}(x^{\prime}_{i})\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}\sum_{i=1}^{n}\lambda_{i}f^{\mathcal{U}^{\prime}}(x^{\prime}_{i})\stackrel{{\scriptstyle\eqref{eq:pb-
proof14}}}{{\leq}}f^{\mathcal{U}^{\prime}}(x)\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\leq}}f^{\mathcal{U}}(x)\text{.}$
(29)
This contradicts the assumption of $x$ being extreme supported efficient for
$P(\mathcal{U})$ or
$f^{\mathcal{U}}(x^{\prime}_{i})=f^{\mathcal{U}}(x)\textup{ for at least one
}i=1,2,\dots,n$ (30)
must hold. Assume (30) holds.
Then
$f^{\mathcal{U}^{\prime}}(x)\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\leq}}f^{\mathcal{U}}(x)\stackrel{{\scriptstyle\eqref{eq:pb-
proof16}}}{{=}}f^{\mathcal{U}}(x_{i}^{\prime})\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}f^{\mathcal{U}^{\prime}}(x_{i}^{\prime})$
follows. Since $x_{i}^{\prime}$ is efficient for $P(\mathcal{U}^{\prime})$,
equality holds. Hence,
$f^{\mathcal{U}^{\prime}}(x^{\prime}_{i})\not=f^{\mathcal{U}^{\prime}}(x)$ for
all $i=1,2,\dots,n$ is contradicted.
3. (iii)
Let $\mathcal{R}_{\text{ESE}}\subset\mathcal{X}$ be a representative set of
extreme supported efficient solutions for $P(\mathcal{U})$. Analogously to the
proof of Theorem 11 (iii) we show that
$f^{\mathcal{U}}(\mathcal{R}_{\text{ESE}}^{\prime})=f^{\mathcal{U}}(\mathcal{R}_{\text{ESE}})$.
$\subset$: Let $y^{\prime}\in
f^{\mathcal{U}}(\mathcal{R}_{\text{ESE}}^{\prime})$. Then
$y^{\prime}=f^{\mathcal{U}}(x^{\prime})$ for some
$x^{\prime}\in\mathcal{R}_{\text{ESE}}^{\prime}$. According to (i),
$x^{\prime}$ is extreme supported efficient for $P(\mathcal{U})$, hence
$y^{\prime}\in f^{\mathcal{U}}(\mathcal{R}_{\text{ESE}})$.
$\supset$: Let $y\in f^{\mathcal{U}}(\mathcal{R}_{\text{ESE}})$. Then
$y=f^{\mathcal{U}}(x)$ for some $x$ that is extreme supported efficient for
$P(\mathcal{U})$. According to (ii), $x$ is also extreme supported efficient
for $P(\mathcal{U}^{\prime})$. Hence,
$x^{\prime}\in\mathcal{R}^{\prime}_{\text{ESE}}$ exists such that
$f^{\mathcal{U}^{\prime}}(x)=f^{\mathcal{U}^{\prime}}(x^{\prime})$. This leads
to
$y=f^{\mathcal{U}}(x)\stackrel{{\scriptstyle\eqref{eq:U'relaxation}}}{{\geq}}f^{\mathcal{U}^{\prime}}(x)=f^{\mathcal{U}^{\prime}}(x^{\prime})\stackrel{{\scriptstyle\eqref{eq:WCincluded}}}{{=}}f^{\mathcal{U}}(x^{\prime})\textup{.}$
Since by assumption $y$ is extreme supported nondominated for
$P(\mathcal{U})$, equality must hold true. Thus,
$y=f^{\mathcal{U}}(x^{\prime})$ for
$x^{\prime}\in\mathcal{R}_{\text{ESE}}^{\prime}$ and, consequently, $y\in
f^{\mathcal{U}}(\mathcal{R}_{\text{ESE}}^{\prime})$.
∎
We can now formulate the multiobjective generalization of optimization-
pessimization.
#### Adaption of optimization-pessimization.
In order to deal with the multiobjective setting algorithmically, we modify
optimization-pessimization for multiobjective problems as it is described in
the following (see also Figure 2):
Optimization: Determine representative set of extreme supported efficient
solutions $X^{\ast}$ of $P(\mathcal{U}^{\prime})$ Pessimization: For all $x\in
X^{\ast}$, $i=1,2,\dots,p$: Determine worst-case scenario
$\xi^{\ast}\in\operatorname*{arg\,max}_{\xi\in\mathcal{U}}f_{i}(x^{\ast},\xi)$
Add scenarios to $\mathcal{U}^{\prime}$ Figure 2: Optimization-pessimization
for robust multiobjective optimization problems
When solving the _optimization problem_ $P(\mathcal{U}^{\prime})$ we do not
only determine one optimal solution, but a representative set $X^{\prime\ast}$
of extreme supported efficient solutions. In the subsequent _pessimization
step_ we consider _all_ solutions $x\in X^{\prime\ast}$. For each of them we
determine not just one worst-case scenario, but a worst-case scenario for each
of the $p$ objective functions independently. All of these
$p\cdot|X^{\prime\ast}|$ worst-case scenarios are then added to the
uncertainty set.
Algorithm 4.2 describes the exact procedure and the following lemma shows its
correctness.
0: Multi-objective robust optimization problem $P(\mathcal{U})$ as in (6).
0: Finite initial set $\mathcal{U}^{(0)}\subseteq\mathcal{U}$.
0: Either $\mathcal{U}$ finite or $\mathcal{U}$ a polytope and
$f_{i}(x,\cdot)$, $i=1,2,\dots,p$ continuous and quasi-convex.
0: (dom), (ideal) hold for $P(\mathcal{U})$ and for $P(\mathcal{U}^{\prime})$
for any finite subset $\mathcal{U}^{\prime}\subseteq\mathcal{U}$.
Set $k\coloneqq 0$.
repeat
Set $\mathcal{U}^{(k+1)}\coloneqq\mathcal{U}^{(k)}$.
Determine representative set for extreme supported efficient solutions
$X^{(k)\ast}$ and representative set for extreme supported nondominated points
$Y^{(k)\ast}$ of $P(\mathcal{U}^{(k)})$.
for all $x^{\ast}\in X^{(k)\ast}$ do
for all $i=1,2,\dots,p$ do
Determine
$\xi^{\ast}\in\operatorname*{arg\,max}_{\mathcal{U}}f_{i}(x^{\ast},\xi)$.
Add $\xi^{\ast}$ to $\mathcal{U}^{(k+1)}$.
end for
end for
$k\coloneqq k+1$
until $f^{\mathcal{U}}(x^{\ast})=f^{\mathcal{U}^{(k-1)}}(x^{\ast})$ for all
$x^{\ast}\in X^{(k-1)\ast}$.
return $X^{(k-1)\ast}$: representative set of extreme supported efficient
solutions of $P(\mathcal{U})$.
return $Y^{(k-1)\ast}$: set of extreme supported nondominated points of
$P(\mathcal{U})$.
return $\mathcal{U}^{\textup{FINAL}}\coloneqq\mathcal{U}^{k}$: set of worst-
case scenarios.
Algorithm 4.2 Optimization-pessimization for multi-objective robust
optimization
Optimization
Pessimization
###### Lemma 15.
Let (dom), (ideal) hold for $P(\mathcal{U})$ and for $P(\mathcal{U}^{\prime})$
for any finite subset $\mathcal{U}^{\prime}\subseteq\mathcal{U}$.
1. (i)
Let $\mathcal{U}$ be finite. Then Algorithm 4.2 returns a representative set
of extreme supported efficient solutions to $P(\mathcal{U})$ in at most
$|\mathcal{U}|$ iterations.
2. (ii)
Let $\mathcal{U}$ be a polytope or finite and
$f_{i}(x,\cdot)\colon\operatorname*{conv}(\mathcal{U})\to\mathbb{R}$,
$i=1,2,\dots,p$, be continuous and quasi-convex. Then Algorithm 4.2 returns a
representative set of extreme supported efficient solutions to (6) in at most
$k$ iterations where $k$ is the number of extreme points of $\mathcal{U}$, if
we choose an algorithm for the pessimization problem which always finds an
extreme point of $\mathcal{U}$.
###### Proof.
Algorithm 4.2 determines a representative set of extreme supported efficient
solutions to $\mathcal{U}^{(k-1)}$ in step $k$. It stops if
$f^{\mathcal{U}}(x^{\ast})=f^{\mathcal{U}^{(k-1)}}(x^{\ast})$ (31)
for all $x^{\ast}\in X^{(k-1)\ast}$.
Hence, $\mathcal{R}_{\textup{ESN}}=X^{(k-1)\ast}$ is a representative set of
extreme supported efficient solutions to $P(\mathcal{U}^{\prime})$ for
$\mathcal{U}^{(k-1)}$ whose elements satisfy (wc). Furthermore,
$\mathcal{Y}=f^{\mathcal{U}}(x)$ is compact, since it is the image of a
compact set under the function $\max_{\xi\in\mathcal{U}^{\prime}}f(x,\xi)$
that is continuous since $\mathcal{U}^{\prime}$ is finite. We can thus apply
Theorem 14 for $\mathcal{U}^{\prime}=\mathcal{U}^{(k-1)}\subseteq\mathcal{U}$
and, after termination, $X^{(k-1)\ast}$ is a representative set of extreme
supported efficient solutions to $P(\mathcal{U})$.
We now show the bounds on the number of iterations.
1. ad (i)
In every iteration, either at least one new worst-case scenario is added or
(31) holds and the procedure stops. Since $\mathcal{U}$ is finite, the latter
happens after at most $|\mathcal{U}|$ iterations.
2. ad (ii)
Consider the pessimization problem $\textup{Pess}(x^{k})$: here we maximize a
continuous function over a compact set $\mathcal{U}$, i.e., a maximum always
exists. Since $f(x,\cdot)$ is quasi-convex, the maximum is always attained at
an extreme point of $\mathcal{U}$. If we choose an algorithm that returns an
extreme point for such optimization problems, we add a new extreme point in
each iteration until (18) holds as in part (i).
∎
Algorithm 4.2 provides a method to solve problem (BRO) under the stated
assumptions. However, this is still challenging since in each iteration a
representative set for all extreme supported efficient solutions to
$P(\mathcal{U}^{\prime})$ for some $\mathcal{U}^{\prime}\subset\mathcal{U}$
needs to be found. In Section 5.2 we employ dichotomic search for this
purpose.
## 5 Algorithms for robust biobjective optimization
In Sections 3.2 and 4.2 algorithms known from (deterministic) biobjective and
(single-objective) robust optimization, respectively, have been generalized.
However, in each iteration of the proposed dichotomic search method (Algorithm
3.1, Lemma 7) a robust problem has to be solved and, similarly, in each
iteration of the proposed optimization-pessimization method (Algorithm 4.2,
Lemma 15) a multiobjective problem has to be solved. So far, we treated these
steps as if they were performed by an oracle.
In this section we put these steps into concrete terms and, in doing so,
present algorithms designed to solve uncertain biobjective problems, more
specifically the problem (BRO) as defined in Section 2. Throughout this
section we always assume that the assumptions of (BRO), i.e., (BRO-1),
(BRO-2), and (BRO-3) (see page BRO), hold.
Specifically, three different approaches to find minmax robust solutions for
$P(\mathcal{U})$ are presented:
* •
A robust optimizer’s approach (ROA): We view the problem (BRO) primarily as a
_robust_ optimization problem – just with the added difficulty that it has two
objective functions – and, consequently, apply a method from robust
optimization, namely the generalized optimization-pessimization method
(Algorithm 4.2), to the problem $\textup{BRO}(\mathcal{U})$. The subproblem to
be solved in each iteration is a _biobjective_ problem
$\textup{BRO}(\mathcal{U}^{\prime})$ with a small uncertainty set
$\mathcal{U}^{\prime}\subseteq\mathcal{U}$ which we tackle by the generalized
version of dichotomic search (Algorithm 3.1). This algorithm is presented in
Section 5.1.
* •
A multiobjective optimizer’s approach (MOA): We view the problem (BRO)
primarily as a _biobjective_ optimization problem – with the added difficulty
that we aim to find a _robust_ solution and the objective functions, thus,
contain a maximum – and, consequently, apply a method from biobjective
optimization, namely the generalized version of dichotomic search (Algorithm
3.1) to the problem $\textup{BRO}(\mathcal{U})$. The subproblem to be solved
in each iteration is a single-objective but _uncertain_ problem
$P(\mathcal{U},\lambda)$ which we tackle by the optimization-pessimization
method (Algorithm 4.1). This algorithm is presented in Section 5.2.
* •
A multiobjective optimizer’s approach for bilinear problems using dualization
(DA): As in the aforementioned approach, we take the multiobjective
optimizer’s perspective and apply the generalized version of dichotomic search
(Algorithm 3.1) to the problem $\textup{BRO}(\mathcal{U})$. The subproblem
$P(\mathcal{U},\lambda)$ is directly solved through a reformulation in each
iteration. This algorithm is presented in Section 5.3.
Algorithms 5.1, 5.2, and 5.3 each determine all extreme supported nondominated
points and a corresponding representative set of extreme supported efficient
solution for $\textup{BRO}(\mathcal{U})$. The following lemma shows that these
sets can be used to determine _all_ nondominated points and a representative
set for all efficient solutions of $\textup{BRO}(\mathcal{U})$.
###### Lemma 16.
Let $\textup{BRO}(\mathcal{U})$ be given and let $\mathcal{X}$ be a polytope.
Further, let $\mathcal{Y}_{\textup{ESN}}$,
$|\mathcal{Y}_{\textup{ESN}}|<\infty$, be its set of nondominated extreme
supported points and $\mathcal{X}_{\textup{ESE}}$ a representative set of
extreme supported efficient solutions. Let
$\mathcal{X}_{\textup{ESE}}=\left\\{x^{(1)},x^{(2)},\dots,x^{(n)}\right\\}$,
$\mathcal{Y}_{\textup{ESN}}=\left\\{y^{(1)},y^{(2)},\dots,y^{(n)}\right\\}$,
$y^{(1)}_{1}<y^{(2)}_{1}<\dots<y^{(n)}_{1}$ and $f(x^{(i)})=y^{(i)}$ for
$i=1,2,\dots,n$ . Then
$\displaystyle\mathcal{X}^{\ast}$
$\displaystyle\coloneqq\bigcup_{i=1,2,\dots,n-1}\left\\{\lambda
x^{(i)}+(1-\lambda)x^{(i+1)}\colon\lambda\in(0,1)\right\\}$ is a
representative set (of efficient solutions) and
$\displaystyle\mathcal{Y}^{\ast}$
$\displaystyle\coloneqq\bigcup_{i=1,2,\dots,n-1}\left\\{\lambda
y^{(i)}+(1-\lambda)y^{(i+1)}\colon\lambda\in(0,1)\right\\}$
is the set of nondominated points of $\textup{BRO}(\mathcal{U})$.
###### Proof.
Let $\bar{x}\in\mathcal{X}^{\ast}$. Then $\bar{x}=\lambda
x^{(i)}+(1-\lambda)x^{(i+1)}$ for some $i=1,2,\dots,n-1$, $\lambda\in(0,1)$,
and
$\displaystyle\bar{y}\coloneqq f^{\mathcal{U}}(\bar{x})$
$\displaystyle=\max_{\xi\in\mathcal{U}}f(\lambda
x^{(i)}+(1-\lambda)x^{(i+1)},\xi)$
$\displaystyle=\max_{\xi\in\mathcal{U}}\left\\{\lambda
f(x^{(i)},\xi)+(1-\lambda)f(x^{(i+1)},\xi)\right\\}$
$\displaystyle\leq\max_{\xi\in\mathcal{U}}\lambda
f(x^{(i)},\xi)+\max_{\xi\in\mathcal{U}}(1-\lambda)f(x^{(i+1)},\xi)$
$\displaystyle=\lambda
f^{\mathcal{U}}(x^{(i)})+(1-\lambda)f^{\mathcal{U}}(x^{(i+1)})$
$\displaystyle=\lambda y^{(i)}+(1-\lambda)y^{(i+1)}.$
However, since by Lemma 12 we have
$\mathcal{Y}\subseteq\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}_{\geq}^{2}$
and since $\\{\lambda y^{(i)}+(1-\lambda)y^{(i+1)}\\}$ is a facet of
$\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})$, there is no
$y\in\mathcal{Y}$ with $y\preceq\lambda y^{(i)}+(1-\lambda)y^{(i+1)}$. Thus,
we have $\bar{y}=\lambda y^{(i)}+(1-\lambda)y^{(i+1)}$ and $\bar{y}$ is
nondominated. This shows that the solutions in $\mathcal{X}^{\ast}$ are
efficient and the points in $\mathcal{Y}^{\ast}$ are nondominated.
It remains to be shown that all nondominated points are included in
$\mathcal{Y}^{\ast}\cup\mathcal{Y}_{\textup{ESN}}$. This, however, follows
directly from the fact that, by Lemma 12
$\mathcal{Y}\subseteq\operatorname*{conv}(\mathcal{Y}_{\textup{ESN}})+\mathbb{R}_{g}eq^{2}$.
∎
### 5.1 A robust optimizer’s approach
The robust optimizer’s approach is based on the idea of applying the
generalization of optimization-pessimization (Algorithm 4.2). In the $k$-th
iteration a representative set of extreme supported efficient solutions to
$P(\mathcal{U}^{(k)})$ has to be determined. For this purpose in Algorithm 5.1
we employ dichotomic search for robust biobjective linear mixed-integer
optimization problems as shown possible in Section 3.2.
0: Biobjective mixed-integer linear robust optimization problem (BRO).
0: Finite initial set $\mathcal{U}^{(0)}\subseteq\mathcal{U}$.
0: Feasible set $\mathcal{X}$ is a polyhedron intersected with
$\mathbb{R}^{n-k}\times\mathbb{Z}^{k}$ for some $k\in\\{0,\ldots,n\\}$.
0: $\mathcal{U}$ finite or $\mathcal{U}$ a polytope and $f_{i}(x,\cdot)$,
$i=1,2,\dots,p$ continuous and quasi-convex.
0: (dom), (ideal) hold for $P(\mathcal{U})$ and for $P(\mathcal{U}^{\prime})$
for any finite subset $\mathcal{U}^{\prime}\subseteq\mathcal{U}$.
Set $k\coloneqq 0$.
repeat
Set $\mathcal{U}^{(k+1)}\coloneqq\mathcal{U}^{(k)}$.
Call dichotomic search (Algorithm 3.1) for $\textup{BRO}(\mathcal{U}^{k})$ to
determine representative set for extreme supported efficient solutions
$X^{(k)\ast}$ and representative set for extreme supported nondominated points
$Y^{(k)\ast}$.
for all $x^{\ast}\in X^{\ast}$ do
for all $i=1,2$ do
Determine one
$\xi^{\ast}\in\operatorname*{arg\,max}_{\mathcal{U}}f_{i}(x^{\ast},\xi)$.
Add $\xi^{\ast}$ to $\mathcal{U}^{(k+1)}$.
end for
end for
$k\coloneqq k+1$
until $f^{\mathcal{U}}(x^{\ast})=f^{\mathcal{U}^{(k-1)}}(x^{\ast})$ for all
$x^{\ast}\in X^{(k-1)\ast}$.
return $X^{(k-1)\ast}$: representative set of extreme supported efficient
solutions of $P(\mathcal{U})$.
return $Y^{(k-1)\ast}$: set of extreme supported nondominated points of
$P(\mathcal{U})$.
return $\mathcal{U}^{\textup{FINAL}}\coloneqq\mathcal{U}^{k}$: set of worst-
case scenarios.
Algorithm 5.1 Robust optimizer’s approach (ROA)
Optimization
Pessimization
Note that Algorithm 5.1 is just Algorithm 4.2 with the optimization step
performed by dichtomic search (Algorithm 3.1). Consequently, the requirements
correspond to those of Algorithm 4.2 and Algorithm 3.1 as formulated in Lemma
15 and Lemma 7, respectively. This is stated in the following lemma.
###### Lemma 17.
Let $\textup{BRO}(\mathcal{U})$ be given.
1. (i)
Let $\mathcal{U}$ be finite. Then Algorithm 5.1 returns a representative set
of extreme supported efficient solutions to (BRO) in at most $|\mathcal{U}|$
iterations.
2. (ii)
Let $\mathcal{U}$ be a polytope or finite and
$f_{i}(x,\cdot)\colon\operatorname*{conv}(\mathcal{U})\to\mathbb{R}$,
$i=1,2,\dots,p$, be continuous and quasi-convex. Then Algorithm 5.1 returns a
representative set of extreme supported efficient solutions to (BRO), in at
most $k$ iterations (where $k$ is the number of extreme points of
$\mathcal{U}$) if we choose an algorithm for the pessimization problem which
always finds an extreme point of $\mathcal{U}$.
###### Proof.
By Corollary 3, $\textup{BRO}(\mathcal{U})$ satisfies (dom) and (ideal).
Algorithm 5.1 is the same as Algorithm 4.2, but for $p=2$ and with dichotomic
search (Algorithm 3.1) specified in the optimization step. Lemma 7 justifies
that dichotomic search works correctly for BRO. Consequently, we may use
dichotomic search in line 4 of Algorithm 4.2. Under (dom) and (ideal) for
$\textup{BRO}(\mathcal{U})$ and $\textup{BRO}(\mathcal{U}^{\prime})$ for all
finite sets $\mathcal{U}^{\prime}\subseteq\mathcal{U}$ Lemma 15 gives us
correctness of Algorithm 4.2 and hence also of Algorithm 5.1. ∎
Note that if Algorithm 5.1 is stopped before the stopping criterion in line 12
is met, the set $\\{f^{\mathcal{U}}(k-1)(x):x\in\mathcal{X}^{(k-1)\ast}\\}$
and $\\{f^{\mathcal{U}}(x):x\in X^{(k-1)\ast}\\}$ provide lower and upper
bounds with respect to the upper setless order, as we have shown in Lemma 9.
Using convex combinations of subsequent points in these sets like we did in
Lemma 16 for $Y^{\ast}$, we obtain bounds on the region in which the Pareto
frontier $Y^{\ast}$ will lie. In this sense, Algorithm 5.1 can be used as an
approximation algorithm for (BRO).
### 5.2 A multiobjective optimizer’s approach
The multiobjective optimizer’s approach is based on the idea of applying
dichotomic search (Algorithm 3.1) as introduced in Section 3.1 directly to
$P(\mathcal{U})$. In each iteration of dichotomic search, we have to solve the
scalarized weighted-sum problem
$\displaystyle P(\mathcal{U},\lambda)$
$\displaystyle\min_{x\in\mathcal{X}}\lambda_{1}f_{1}^{\mathcal{U}}(x)+\lambda_{2}f_{2}^{\mathcal{U}}(x)+\dots+\lambda_{p}f_{p}^{\mathcal{U}}(x)$
(32)
for $p=2$ and given weights $\lambda\in\mathbb{R}_{\succeq 0}$. In order to do
this, we utilize optimization-pessimization for single-objective robust
optimization as reviewed in Section 4.1: We solve a sequence of problems
$P(\mathcal{U}^{0},\lambda),P(\mathcal{U}^{1},\lambda),\dots,P(\mathcal{U}^{k},\lambda)$
until it is guaranteed that $P(\mathcal{U}^{k},\lambda)$ and
$P(\mathcal{U},\lambda)$ share a representative set of extreme supported
minmax robust efficient solutions. As in Section 4 we exploit the fact, that
for finite sets $\mathcal{U}^{\prime}$ a problem
$P(\mathcal{U}^{\prime},\lambda)$ is easier to solve than
$P(\mathcal{U},\lambda)$ as it can be written as a problem with finitely many
constraints. For solving the scalarization we assumed an oracle in Algorithm
3.1. Now we want to be more specific. We first reformulate problem (32) such
that we can apply optimization-pessimization (see Section 4.1) for its
solution. This is done in the next lemma.
###### Lemma 18.
Let $\lambda\in\mathbb{R}^{2}_{\succeq 0}$ be fixed. Then
$P(\mathcal{U},\lambda)$ can be transformed to
$\displaystyle\bar{P}(\mathcal{U},\lambda)$
$\displaystyle\min_{x\in\mathcal{X}}\sup_{\bar{\xi}\in\bar{\mathcal{U}}}\bar{f}_{\lambda}(x,\bar{\xi})\textup{,}$
(33)
i.e., a problem of type $\textup{P}^{\textup{single}}$ as introduced in (2),
for $\bar{\mathcal{U}}\coloneqq\bigtimes_{i=1,2,\dots,p}\mathcal{U}$,
$\bar{\xi}\coloneqq(\xi_{1},\xi_{2},\dots,\xi_{p})$ and
$\bar{f}_{\lambda}(x,\bar{\xi})\coloneqq\sum_{i=1}^{p}\lambda_{i}f_{i}(x,\xi_{i})$.
###### Proof.
We reformulate (32) as follows:
$\displaystyle\min_{x\in\mathcal{X}}\left\\{\lambda_{1}f_{1}^{\mathcal{U}}(x)+\lambda_{2}f_{2}^{\mathcal{U}}(x)+\dots+\lambda_{p}f_{p}^{\mathcal{U}}(x)\right\\}$
$\displaystyle=\min_{x\in\mathcal{X}}\left\\{\sum_{i=1}^{p}\lambda_{i}\sup_{\xi\in\mathcal{U}}f_{i}(x,\xi)\right\\}$
$\displaystyle=\min_{x\in\mathcal{X}}\sup_{(\xi_{1},\xi_{2},\dots,\xi_{p})\in\mathcal{U}^{p}}\left\\{\sum_{i=1}^{p}\lambda_{i}f_{i}(x,\xi_{i})\right\\}$
$\displaystyle=\min_{x\in\mathcal{X}}\sup_{\bar{\xi}\in\bar{\mathcal{U}}}\bar{f}_{\lambda}(x,\bar{\xi})\text{.}$
∎
Lemma 18 shows that $P(\mathcal{U},\lambda)$ can be solved by solving a
single-objective robust optimization problem
$\bar{P}(\bar{\mathcal{U}},\lambda)$, i.e., of type $P^{\textup{single}}$, as
has been introduced in (2).
Algorithm 5.2 describes a _basic version_ of the multiobjective optimizer’s
approach. Its correctness is shown in the following lemma.
0: Biobjective mixed-integer linear robust optimization problem (BRO).
0: Finite initial set $\mathcal{U}^{(0)}\subseteq\mathcal{U}$.
0: Feasible set $\mathcal{X}$ is a polyhedron intersected with
$\mathbb{R}^{n-k}\times\mathbb{Z}^{k}$ for some $k\in\\{0,\ldots,n\\}$.
0: $\mathcal{U}$ finite or $\mathcal{U}$ a polytope and $f_{i}(x,\cdot)$,
$i=1,2,\dots,p$ continuous and quasi-convex.
0: (dom), (ideal) hold for $P(\mathcal{U})$ and for $P(\mathcal{U}^{\prime})$
for any finite subset $\mathcal{U}^{\prime}\subseteq\mathcal{U}$.
Initialize $\mathcal{L}\coloneqq\emptyset$ {$\mathcal{L}$ will contain list of
tuple images $(y^{l},y^{r})$ satisfying
$y^{l}_{1}<y^{r}_{1},y^{l}_{2}>y^{r}_{2}$}
Call optimization-pessimization (Algorithm 4.1) on
$\min_{x\in\mathcal{X}}f^{\mathcal{U}}_{1}(x)$ with initial set
$\mathcal{U}^{(0)}$ to determine $\varepsilon_{1}$,
$\mathcal{U}^{\text{FINAL}}$, and $\xi^{\textup{WC}}$.
Call optimization-pessimization (Algorithm 4.1) on
$\min_{x\in\mathcal{X}}\\{f^{\mathcal{U}}_{2}(x)\colon\max_{\xi\in\mathcal{U}}f_{1}(x,\xi)\leq\varepsilon_{1}\\}$
with initial set $\mathcal{U}^{\text{FINAL}}$ to determine optimal solution
$x^{L}$.
Set $y^{L}\coloneqq f^{\mathcal{U}}(x^{L})$.
Call optimization-pessimization (Algorithm 4.1) on
$\min_{x\in\mathcal{X}}f^{\mathcal{U}}_{2}(x)$ with initial set
$\mathcal{U}^{(0)}$ to determine $\varepsilon_{2}$,
$\mathcal{U}^{\text{FINAL}}$, and $\xi^{\textup{WC}}$.
Call optimization-pessimization (Algorithm 4.1) on
$\min_{x\in\mathcal{X}}\\{f^{\mathcal{U}}_{1}(x)\colon\max_{\xi\in\mathcal{U}}f_{2}(x,\xi)\leq\varepsilon_{2}\\}$
with initial set $\mathcal{U}^{\text{FINAL}}$ to determine optimal solution
$x^{R}$..
Set $y^{R}\coloneqq f^{\mathcal{U}}(x^{R})$.
if $y^{L}=y^{R}$ then
STOP. Only one nondominated image found.
return $Y^{\ast}=\\{y^{L}\\},X^{\ast}=\\{x^{L}\\}$.
else
$Y^{\ast}=\\{y^{L},y^{R}\\},X^{\ast}=\\{x^{L},x^{R}\\},\mathcal{L}=\\{(y^{L},y^{R})\\}$.
end if
while $L\not=\emptyset$ do
Remove element $(y^{l},y^{r})$ from $\mathcal{L}$.
Compute $\lambda\coloneqq(y^{l}_{2}-y^{r}_{2},y^{r}_{1}-y^{l}_{1})$.
Call optimization-pessimization (Algorithm 4.1) on
$\min_{x\in\mathcal{X}}\bar{f_{\lambda}}(x)$ with initial set
$\mathcal{U}^{(0)}$ to determine optimal solution $x^{\ast}$.
Set $y^{\ast}\coloneqq\bar{f_{\lambda}}(x^{\ast})$.
if $\lambda^{T}y^{\ast}\not=\lambda^{T}y^{l}$ then
Add $y^{\ast}$ to $Y^{\ast}$, add $x^{\ast}$ to $X^{\ast}$.
Add $(y^{l},y^{\ast}),(y^{\ast},y^{r})$ to $\mathcal{L}$
end if
end while
return $X^{\ast}$: representative set of extreme supported efficient solutions
of $P(\mathcal{U})$.
return $Y^{\ast}$: set of extreme supported nondominated points of
$P(\mathcal{U})$.
Algorithm 5.2 Multiobjective optimizer’s approach (MOA)
Determine lexicographic solutions
Solve weighted-sum problem $\bar{P}(\mathcal{U},\lambda)$
###### Lemma 19.
Let $\textup{BRO}(\mathcal{U})$ be given. Then Algorithm 5.2 returns a
representative set of extreme supported efficient solutions to (BRO) after a
finite number of iterations.
###### Proof.
Algorithm 5.2 is dichotomic search (Algorithm 3.1), where we specified the
algorithm for steps 2-3, 5-6, 17-18, namely by solving
$\textup{BRO}(\mathcal{U},\lambda)$ by optimization-pessimization (Algorithm
4.1) in each iteration. Since $\textup{BRO}(\mathcal{U})$ meets the
requirements of Lemma 6 (in case $\mathcal{U}$ is finite) or Lemma 7 (in case
$\mathcal{U}$ is a polytope), Algorithm 3.1 returns a representative set of
extreme supported efficient solutions and a set of extreme supported
nondominated solutions after finitely many iterations.
It remains to show that lines 2-3, 5-6 and 17-18 in Algorithm 5.2 are correct
specifications of the same lines of Algorithm 3.1.
For lines 2 and 5 this is straightforward as the problems
$\min_{x\in\mathcal{X}}f_{i}^{\mathcal{U}}(x),$ (34)
$i=1,2$, are single-objective robust optimization problems. Since
$\mathcal{U}$ is a polytope or finite and
$f_{i}(x,\cdot)\colon\mathcal{U}\to\mathbb{R}$, $i=1,2$, are continuous and
quasi-convex, Lemma 8 can be applied and optimization-pessimization (Algorithm
4.1) solves (34).
The problems in lines 3 and 6 are also of type (34) only with one additional
constraint, i.e., with feasible set is
$\mathcal{X}^{\prime}_{j}\coloneqq\\{x\in\mathcal{X}:\max_{\xi\in\mathcal{U}}f_{j}(x,\xi)\leq\varepsilon_{j}\\},j=2,1.$
In lines 17-18 of Algorithm 3.1 the problem $P(\mathcal{U},\lambda)$ is to be
solved for some $\lambda\in\mathbb{R}_{\succeq 0}$. By Lemma 18 this can be
done by solving $\bar{P}(\mathcal{U},\lambda)$ instead which is done in lines
17-18 of Algorithm 5.2. Since continuity and quasi-convexity of $\bar{f}$ are
inherited from continuity and quasi-convexity of $f_{1}$ and $f_{2}$, Lemma 15
can be applied and optimization-pessimization returns a robust solution to
$P(\mathcal{U},\lambda)$. ∎
#### Warm start modifications.
In the basic version of Algorithm 5.2 the cutting plane method is initialized
with $\mathcal{U}^{\text{(0)}}$ in lines 2,5 and 17. A possible modification
of Algorithm 5.2 is to start the cutting plane method with a larger set
$\mathcal{U}^{\prime}$ that includes some additional scenarios that have been
generated in previous iterations but that is still guaranteed to be finite.
This way, previously generated cutting planes are not forgotten. Specifically,
we propose two modifications:
* •
Variant 1 (MOA-ws1): We initialize optimization-pessimization with all
previously generated scenarios. To this end, we modify lines 5 and 17 such
that the cutting plane method is initialized with
$\mathcal{U}^{\text{FINAL}}$. This way, $\mathcal{U}^{\text{FINAL}}$ grows
monotonically.
* •
Variant 2 (MOA-ws2): We initialize the cutting plane method with those
scenarios that turned out to be worst-case scenarios for a previously found
solution optimal $x$. After lines 2-3, 5-6, and 17-18 the worst-case scenarios
$\xi^{\text{WC}}$ for $x^{L}$, $x^{R}$, and $x^{\ast}$, respectively, are
added to $\mathcal{U}^{\text{(0)}}$ and the set grows monotonically, but is
much smaller than the set in Variant 1.
As Lemmas 18 and 19 above only assume finiteness of the initial uncertainty
set their validity is not affected by these modifications.
### 5.3 A multiobjective optimizer’s approach for bilinear problems
In this section, we confine ourselves to a special class of problems:
biobjective mixed-integer linear robust optimization problems (BRO) which
satisfy not only (BRO-1), (BRO-2), and (BRO-3) as before, but also the
following additional properties:
* •
the uncertainty set $\mathcal{U}$ is as a polytope
$\mathcal{U}=\\{\xi\in\mathbb{R}^{m}\colon C\xi\leq d\\}$ for a matrix
$C\in\mathbb{R}^{m^{\prime}\times m}$ and a vector
$d\in\mathbb{R}^{m^{\prime}}$, and
* •
the functions $f_{1},f_{2}\colon\mathcal{X}\times\mathcal{U}\to\mathbb{R}$ are
not only linear in $x$ for every fixed $\xi\in\mathcal{U}$ as required in
(BRO-3), but also linear in $\xi$ for each $x$, i.e., they are _bilinear_
functions.
The following lemma shows that under these assumptions a biobjective mixed-
integer linear _minmax_ optimization problem can be reformulated as a
biobjective mixed-integer linear _minimization_ problem.
###### Lemma 20.
We consider the uncertain problem
$\displaystyle P(\mathcal{U})$
$\displaystyle\min_{x\in\mathcal{X}}f^{\mathcal{U}}(x)\textup{.}$ (6
revisited)
Let the uncertainty set be a non-empty polytope
$\mathcal{U}=\\{\xi\in\mathbb{R}^{m}\colon C\xi\leq d\\}$, with
$C\in\mathbb{R}^{m^{\prime}\times m}$, $d\in\mathbb{R}^{m^{\prime}}$, and let
the functions $f_{i}(x,\xi)$, $i=1,2,\dots,p$, be linear in $\xi$ for each
$x$, i.e.,
$f_{i}(x,\xi)\coloneqq\left[\hat{c_{i}}(x)\right]^{t}\xi$
for functions $\hat{c_{i}}\colon\mathcal{X}\to\mathbb{R}^{m}$,
$i=1,2,\dots,p$.
Let $\lambda\in\mathbb{R}_{\succeq}^{p}$. Then a solution
$x^{\ast}\in\mathcal{X}$ is optimal for the scalarized problem
$\displaystyle P(\mathcal{U},\lambda)$
$\displaystyle\min_{x\in\mathcal{X}}\lambda^{t}f^{\mathcal{U}}(x)$ (32
revisited)
if and only if there exist
$\pi^{(1)\ast},\dots,\pi^{(p)\ast}\in\mathbb{R}^{m^{\prime}}$ such that
$(x^{\ast},\pi^{(1)\ast},\dots,\pi^{(p)\ast})$ is optimal for
$\displaystyle D(\mathcal{U},\lambda)$
$\displaystyle\min_{x\in\mathcal{X},\pi^{(1)},\dots,\pi^{(p)}\in\mathbb{R}^{m^{\prime}}}\left\\{d^{t}\sum_{i=1}^{p}\lambda_{i}\pi^{(i)}\colon
C^{t}\pi^{(i)}=\hat{c_{i}}(x),\pi^{(i)}\geq 0,i=1,2,\dots,p\right\\}$
More precisely, let $x$ be fixed and let $(\pi^{(1)\ast},\dots,\pi^{(p)\ast})$
be an optimal solution to
$\min_{\pi^{(1)},\dots,\pi^{(p)}\in\mathbb{R}^{m^{\prime}}}\left\\{d^{t}\sum_{i=1}^{p}\lambda_{i}\pi^{(i)}\colon
C^{t}\pi^{(i)}=\hat{c_{i}}(x),\pi^{(i)}\geq 0,i=1,2,\dots,p\right\\}$
with optimal objective function value $z$. Then
$z=\lambda^{t}f^{\mathcal{U}}(x)$ and for all $i=1,2,\dots,p$ with
$\lambda_{i}>0$
$\displaystyle
d^{t}\pi^{(i)\ast}=\max_{\xi\in\mathcal{U}}\left[\hat{c_{i}}(x)\right]^{t}\xi\textup{.}$
(35)
###### Proof.
First note that $D(\mathcal{U},\lambda)$ is equivalent to
$\hskip
28.45274pt\min_{x\in\mathcal{X}}\min_{\pi^{(1)},\dots,\pi^{(p)}\in\mathbb{R}^{m^{\prime}}}\left\\{d^{t}\sum_{i=1}^{p}\lambda_{i}\pi^{(i)}\colon
C^{t}\pi^{(i)}=\hat{c_{i}}(x),\pi^{(i)}\geq 0,i=1,2,\dots,p\right\\}$
which can be interpreted as optimization problem
$\displaystyle\bar{D}(\mathcal{U},\lambda)$
$\displaystyle\min_{x\in\mathcal{X}}g_{\lambda}(x)$
with
$g_{\text{$\lambda$}}(x)\coloneqq\min_{\pi^{(1)},\dots,\pi^{(p)}\in\mathbb{R}^{m^{\prime}}}\left\\{d^{t}\sum_{i=1}^{p}\lambda_{i}\pi^{(i)}\colon
C^{t}\pi^{(i)}=\hat{c_{i}}(x),\pi^{(i)}\geq
0,i=1,2,\dots,p\right\\}\textup{.}$
We now need to show that the objective function and the feasible set of
$P(\mathcal{U},\lambda)$ and $\bar{D}(\mathcal{U},\lambda)$ coincide.
Specifically, we show
$\lambda^{t}f^{\mathcal{U}}(x)=g_{\lambda}(x)$
for all $x\in\mathcal{X}$.
We first note that $\mathcal{U}$ is a compact set, hence for any fixed
$x\in\mathcal{X}$ and any $i=1,2,\dots,p$ the linear program
$\max\left\\{\left[\hat{c_{i}}(x)\right]^{t}\xi\colon\xi\in\mathcal{U}\right\\}$
has an optimal solution. Using that $\mathcal{U}=\\{\xi\in\mathbb{R}^{m}\colon
C\xi\leq d\\}$ we hence get from linear programming duality for
$i=1,2,\dots,p$ and fixed $x\in\mathcal{X}$ that
$\displaystyle\max\left\\{\left[\hat{c_{i}}(x)\right]^{t}\xi\colon C\xi\leq
d,\xi\in\mathbb{R}^{m}\right\\}=\min\left\\{d^{t}\pi^{(i)}\colon
C^{t}\pi^{(i)}=\hat{c_{i}}(x),\pi^{(i)}\geq
0,\pi^{(i)}\in\mathbb{R}^{m^{\prime}}\right\\},$ (36)
i.e., for any fixed $x\in\mathcal{X}$ and $i=1,\ldots,p$, an optimal solution
$\pi^{(i)\ast}$ to the right hand side satisfies
$d^{t}\pi^{(i)\ast}=\max_{\xi\in\mathcal{U}}\left[\hat{c_{i}}(x)\right]^{t}\xi$
which shows (35). We can now derive
$\displaystyle\lambda^{t}f^{\mathcal{U}}(x)$
$\displaystyle=\sum_{i=1}^{p}\lambda_{i}\underbrace{\max_{\xi\in\mathbb{R}^{m}}\left\\{\left[\hat{c_{i}}(x)\right]^{t}\xi\colon
C\xi\leq d\right\\}}_{=f_{i}^{\mathcal{U}}(x)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:DualityMinIsMax+}}}{{=}}\sum_{i=1}^{p}\lambda_{i}\min_{\pi^{(i)}\in\mathbb{R}^{m^{\prime}}}\left\\{d^{t}\pi^{(i)}\colon
C^{t}\pi=\hat{c_{i}}(x),\pi^{(i)}\geq 0\right\\}$
$\displaystyle=\min_{\pi^{(1)},\dots,\pi^{(p)}\in\mathbb{R}^{m^{\prime}}}\left\\{d^{t}\sum_{i=1}^{p}\lambda_{i}\pi^{(i)}\colon
C^{t}\pi^{(i)}=\hat{c_{i}}(x),\pi^{(i)}\geq 0\right\\}\text{,}$
where the last step puts the single optimization problems together into a
bigger (still separable) problem. Thus, for any fixed $x$ the objective values
of $P(\mathcal{U},\lambda)$ and $\bar{D}(\mathcal{U},\lambda)$ coincide and
hence $x$ is optimal to $P(\mathcal{U},\lambda)$ if and only if it is optimal
to $\bar{D}(\mathcal{U},\lambda)$. ∎
As in Section 5.2, we apply dichotomic search to $P(\mathcal{U})$ and solve
$P(\mathcal{U},\lambda)$ for different weights
$\lambda\in\mathbb{R}^{p}_{\succeq}$. However, unlike in Section 5.2 we do not
solve $P(\mathcal{U},\lambda)$ with an iterative approach, but adopt the other
approach described by [GYd15]: reformulation of $P(\mathcal{U},\lambda)$. More
specifically, we weaponize Lemma 20 and choose to solve
$\displaystyle D(\mathcal{U},\lambda)$ $\displaystyle
z^{\ast}(\mathcal{U},\lambda)\coloneqq\min_{x\in\mathcal{X},\pi^{(1)},\dots,\pi^{(p)}\in\mathbb{R}^{m^{\prime}}}\left\\{d^{t}\sum_{i=1}^{p}\lambda_{i}\pi^{(i)}\colon
C^{t}\pi^{(i)}=\hat{c_{i}}(x),\pi^{(i)}\geq 0,i=1,2,\dots,p\right\\}$
instead of $P(\mathcal{U},\lambda)$.
This leads to Algorithm 5.3. The following lemma shows correctness.
0: Biobjective mixed-integer linear robust optimization problem (BRO).
0: Finite initial set $\mathcal{U}^{(0)}\subseteq\mathcal{U}$.
0: Feasible set $\mathcal{X}$ is a polyhedron intersected with
$\mathbb{R}^{n-k}\times\mathbb{Z}^{k}$ for some $k\in\\{0,\ldots,n\\}$.
0: $\mathcal{U}$ a polytope and $f_{i}(x,\cdot)$, $i=1,2,\dots,p$ continuous
and quasi-convex.
0: (dom), (ideal) hold for $P(\mathcal{U})$ and for $P(\mathcal{U}^{\prime})$
for any finite subset $\mathcal{U}^{\prime}\subseteq\mathcal{U}$.
0: $f(x,\cdot)\colon\mathcal{U}\to\mathbb{R}^{p}$ linear
Initialize $\mathcal{L}\coloneqq\emptyset$ {$\mathcal{L}$ will contain list of
tuple images $(y^{l},y^{r})$ satisfying
$y^{l}_{1}<y^{r}_{1},y^{l}_{2}>y^{r}_{2}$}
Determine optimal objective value $\varepsilon_{1}$ of $D(\mathcal{U},(1,0))$
Determine $x^{L}\in\operatorname*{arg\,min}_{\mathcal{X}}\\{g_{(0,1)}(x)\colon
g_{(1,0)}(x)\leq\varepsilon_{1}\\}$
Set $y^{L}\coloneqq(\varepsilon_{1},g_{(0,1)}(x^{L}))^{t}$
Determine optimal objective value $\varepsilon_{2}$ of $D(\mathcal{U},(0,1))$
Determine $x^{R}\in\operatorname*{arg\,min}_{\mathcal{X}}\\{g_{(1,0)}(x)\colon
g_{(0,1)}(x)\leq\varepsilon_{2}\\}$
Set $y^{R}\coloneqq(g_{(1,0)}(x^{L}),\varepsilon_{2})^{t}$
if $y^{L}=y^{R}$ then
STOP. Only one nondominated image found
return $Y^{\ast}=\\{y^{L}\\},X^{\ast}=\\{x^{L}\\}$
else
$Y^{\ast}=\\{y^{L},y^{R}\\},X^{\ast}=\\{x^{L},x^{R}\\},\mathcal{L}=\\{(y^{L},y^{R})\\}$
end if
while $L\not=\emptyset$ do
Remove element $(y^{l},y^{r})$ from $\mathcal{L}$
Compute $\lambda\coloneqq(y^{l}_{2}-y^{r}_{2},y^{r}_{1}-y^{l}_{1})$.
Find one optimal solution $(x^{\ast},\pi^{(1)},\dots,\pi^{(k)})$ for
$D(\mathcal{U},\lambda)$.
Set $y_{i}^{\ast}=d^{t}\pi^{(i)\ast}$ for $i=1,2$.
if $\lambda^{T}y^{\ast}\not=\lambda^{T}y^{l}$ then
Add $y^{\ast}$ to $Y^{\ast}$, add $x^{\ast}$ to $X^{\ast}$.
Add $(y^{l},y^{\ast}),(y^{\ast},y^{r})$ to $\mathcal{L}$
end if
end while
return $X^{\ast}$: representative set of extreme supported efficient solutions
of $P(\mathcal{U})$.
return $Y^{\ast}$: set of extreme supported nondominated points of
$P(\mathcal{U})$. Determine lexicographic solutions Solve
$D(\mathcal{U},\lambda)$
Algorithm 5.3 Multiobjective optimizer’s approach with dualization (DA)
###### Lemma 21.
Let $\textup{BRO}(\mathcal{U})$ with an nonempty polytope explicitly stated as
$\mathcal{U}=\\{\xi\in\mathbb{R}^{m}\colon C\xi\leq d\\}$ for a matrix
$C\in\mathbb{R}^{m^{\prime}\times m}$ and a vector
$d\in\mathbb{R}^{m^{\prime}}$ as uncertainty set and bilinear functions
$f_{1},f_{2}\colon\mathcal{X}\times\mathcal{U}\to\mathbb{R}$ be given. Then
Algorithm 5.3 solves (BRO).
###### Proof.
The assumptions of Lemma 7 are satisfied since (BRO-1) and (BRO-3) hold and
$\mathcal{U}$ is a polytope. Hence dichotomic search can be applied to
$\min_{x\in\mathcal{X}}f^{\mathcal{U}}(x)$. It remains to be shown that
$P(\mathcal{U},\lambda)$ is solved correctly throughout the algorithm. Lemma
20 shows that robust solutions of $P(\mathcal{U},\lambda)$ can be determined
by solving $D(\mathcal{U},\lambda)$ (lines 2, 5 17) and the corresponding
point on the Pareto front can be computed by $y_{i}^{\ast}=d^{t}\pi^{(i)\ast}$
(see line 16). ∎
## 6 Numerical results
We implemented Algorithms 5.1, 5.2 and 5.3 and conducted computational
experiments.
#### Structure of the problems.
We restricted ourselves to a certain class of biobjective optimization
problems: The objective functions
$f_{i}\colon\mathcal{X}\times\mathcal{U}\to\mathbb{R}^{2}$, $i=1,2$, were
assumed to be bilinear, and the feasible set and uncertainty set were
polytopes or discrete sets. More specifically, we considered problems
$\displaystyle P(\mathcal{U})$
$\displaystyle\left\\{\min_{x\in\mathcal{X}}\begin{pmatrix}\max_{\xi\in\mathcal{U}}\xi^{t}M_{1}x\\\
\max_{\xi\in\mathcal{U}}\xi^{t}M_{2}x\end{pmatrix}\right\\}_{\xi\in\mathcal{U}}$
with
$\displaystyle\mathcal{X}$ $\displaystyle=\\{x\in\mathbb{R}^{n}\colon
L^{x}\leq x_{i}\leq U^{x},Ax\leq b\\}\textup{ or }$ $\displaystyle\mathcal{X}$
$\displaystyle=\\{x\in\mathbb{Z}^{n}\colon L^{x}\leq x_{i}\leq U^{x},Ax\leq
b\\},$ $\displaystyle\mathcal{U}$ $\displaystyle=\\{\xi\in\mathbb{R}^{m}\colon
L^{\xi}\leq\xi_{i}\leq U^{\xi},C\xi\leq d\\}\textup{ or }$
$\displaystyle\mathcal{U}$ $\displaystyle=\\{\xi\in\mathbb{Z}^{m}\colon
L^{\xi}\leq\xi_{i}\leq U^{\xi},C\xi\leq d\\}.$
The lower and upper bounds $L^{x},U^{x},L^{\xi},U^{\xi}$ are added to ensure
that $\mathcal{X}$ and $\mathcal{U}$ are subsets of the boxes
$[L^{x},U^{x}]^{n}$ and $[L^{\xi},U^{\xi}]^{m}$, respectively, and, thus, are
bounded as it is required. We chose $L^{x}=1,U^{x}=200,L^{\xi}=-100$ and
$U^{\xi}=100$. By doing so we avoid problems where $0_{n}\in\mathcal{X}$ and
$0_{m}\in\operatorname*{int}(\mathcal{U})$, since this would imply that $x=0$
is a trivial minimizer of $f_{i}^{\mathcal{U}}(x)=\max_{\xi\in\mathcal{U}}\xi
M_{i}x$, $i=1,2$.
#### Generating instances
We created 100 instances of $\textup{BRO}(\mathcal{U})$ with
$A\in\mathbb{Z}^{30\times 5}$ and $C\in\mathbb{Z}^{30\times 5}$. To obtain
instances with smaller number of constraints, as used in our experiments, we
removed constraints from these initial instances. This makes it easier to draw
conclusions when comparing algorithm performance for different values of
$n^{\prime}$ and $m^{\prime}$. The entries of the matrices
$A\in\mathbb{Z}^{n^{\prime}\times n}$ and $C\in\mathbb{Z}^{m^{\prime}\times
m}$ as well as the entries of $M_{1},M_{2}\in\mathbb{Z}^{m\times n}$
determining the objective function are randomly and independently generated
uniformly distributed integers in $\\{-100,-99,\dots,99,100\\}$.
Equally, $\tilde{b}_{i}$, $i=1,2,\dots,n^{\prime}$ and $\tilde{d}_{j}$,
$j=1,2,\dots,m^{\prime}$ are randomly generated uniformly distributed integers
in $\\{50,51,\dots,99,100\\}$. We then set
$\bar{x}\coloneqq(100,100,\dots,100)^{t}\in\mathbb{Z}^{n}$ and
$\bar{\xi}\coloneqq(0,0,\dots,0)^{t}\in\mathbb{Z}^{m}$. Let $A_{i}$,
$i=1,2,\dots,n^{\prime}$ and $C_{j}$, $j=1,2,\dots,m^{\prime}$ denote the the
columns of $A$ and $C$. By setting the right hand-side coefficients
$b_{i}\coloneqq A_{i}^{t}x_{0}+\tilde{b}_{i}\left\|A_{i}\right\|_{2}$ for
$i=1,2,\dots,n^{\prime}$ and $d_{j}\coloneqq
C_{j}^{t}\xi_{0}+\tilde{d}_{j}\left\|C_{j}\right\|_{2}$ for
$j=1,2,\dots,m^{\prime}$, we guarantee that the spheres
$\\{x\in\mathbb{R}^{n}\colon\left\|x-\bar{x}\right\|_{2}\leq 50\\}$ and
$\\{\xi\in\mathbb{R}^{m}\colon\left\|\xi-\bar{\xi}\right\|_{2}\leq 50\\}$ are
included in $\mathcal{X}$ and $\mathcal{U}$, respectively. See [CV14] for more
on this.
#### Implementation
We used C++ to implement our algorithms. Whenever a linear or integer
optimization problem has to be solved, Gurobi 2.3 is called (with default
settings). We use Gurobi’s capacity to provide solution that are known to be
basic solutions. The implementations were tested on a computer with 16 GB RAM,
AMD Ryzen 5 PRO 2500U, 2.00 GHz.
### 6.1 Evaluation of the algorithms
In this section we evaluate the performance of the algorithms for instances of
different types (polytopal and discrete sets $\mathcal{X}$ and $\mathcal{U}$)
and different sizes by varying the number of considered constraints
$n^{\prime}$ and $m^{\prime}$, respectively.
#### Discrete feasible set and discrete uncertainty set
First, let us consider problems with a discrete feasible set and a discrete
uncertainty set. For such instances, the robust optimizer’s approach (ROA,
Algorithm 5.1) and the multiobjective optimizer’s approach (MOA, Algorithm
5.2) in its baseline version and with its two warm-start modifications are
available. The dualization approach (DA, Algorithm 5.3) cannot solve such
instances as it requires a polytope as uncertainty set.
Figure 3 shows the average running time of our algorithms. Each data point is
the average over 100 instances with $n^{\prime}$ constraints on the feasible
set. The number of variables for the feasible set $n$, the number of variables
for the uncertainty set $m$ and the number of constraints for the uncertainty
set $m^{\prime}$ are all fixed and set at 5.
Figure 3: Average running time of our four algorithms for 100 instances as a
function of $n^{\prime}$ with $n=m=m^{\prime}=5$ fixed, $\mathcal{X}$ and
$\mathcal{U}$ discrete (the lines for MOA-ws1 and MOA-ws2 overlap and are hard
to see)
Independently of $n^{\prime}$, the robust optimizer’s approach – where the
uncertainty set $\mathcal{U}^{(k)}$ increases monotonously – is faster than
the baseline version of the multiobjective optimizer’s approach. However, the
warm start modifications to the latter method turn out to be significant
improvements over the baseline version: with those the multiobjective
optimizer’s approach performs faster. We see a clear increase in running time
when going from 5 to 10 constraints for all tested methods, but above that
point an increasing number number of constraints does not seem to make the
problem much harder to solve.
Figure 4 shows how the number of constraints in the definition of the
uncertainty set $\mathcal{U}$ influences the running time.
Figure 4: Average running time of our four algorithms for 100 instances as a
function of $m^{\prime}$ with $n=m=n^{\prime}=5$ fixed, $\mathcal{X}$ and
$\mathcal{U}$ discrete
We observe the same pattern: The modified warm-start versions of MOA are by
far the fastest algorithms; ROA is still faster than the baseline version of
MOA. Clearly, the problem gets harder the more constraints are necessary to
describe $\mathcal{U}$. This leads us to conclude that the difficulty of the
problem is rooted much more in the complexity of $\mathcal{U}$ than in the one
of $\mathcal{X}$.
#### Discrete feasible set and polytopal uncertainty set
Now let us turn to problems with a polytope as uncertainty set. On those
instances all of the algorithms we introduced can be used. This includes the
dualization approach (DA), which is the only algorithm that does not use
optimization-pessimization but instead solves the scalarized problem
$P(\mathcal{U},\lambda)$ for each weight $\lambda$ directly (via the means of
dualization of the inner problem).
Figures 5 and 6 show the average running time of our algorithms on the same
instances as in Figures 3 and 4 – just with the integrality constraint for
$\mathcal{U}$ dropped.
Figure 5: Average running time of our five algorithms for 100 instances as a
function of $n^{\prime}$ with $n=m=m^{\prime}=5$ fixed, $\mathcal{X}$
discrete, $\mathcal{U}$ polytope Figure 6: Average running time of our five
algorithms for 100 instances as a function of $m^{\prime}$ with
$n=m=n^{\prime}=5$ fixed, $\mathcal{X}$ discrete, $\mathcal{U}$ polytope
Our experiments show that for such instances DA is effective, but not
noticeably better than the modified versions of MOA. The ranking of the other
algorithms is essentially the same as before: The modified warm-start versions
of MOA outperform ROA which is still faster than MOA’s baseline version.
Dropping the integrality constraint reduced the overall running time of all
algorithms by about factor two. This is while the number of extreme supported
nondominated points stayed roughly the same.
The apparent ranking of the proposed algorithms raises the question of whether
this applies only on average over a larger number of instances, or if it also
applies to each individual instance. For this we turn to Figure 7. In this
figure we display the objective values for the 5 different algorithms on the
first 10 of the tested 100 instances. Including all tested instances here does
not change the discussed findings, but decreases visibility, which is why we
included only the results of ten instances.
Figure 7: Running time of our five algorithms for 10 instances with
$n=n^{\prime}=m=5,m^{\prime}=30$, $\mathcal{X}$ discrete, $\mathcal{U}$
polytope
Each of the ten columns in Figure 7 represents one instance (with
$n=m=n^{\prime}=5,m^{\prime}=30$) on which we tested the algorithms. We can
see that for all instances either DA or the warm-start modifications of MOA
perform best and either ROA or the baseline version of MOA perform worst. The
ranking of the algorithms is not the same for all instances.
To get a deeper understanding of this we turn to Figure 8.
Figure 8: Running time vs. number of worst-case scenarios added for 10
instances with $n=n^{\prime}=m=5,m^{\prime}=30$, $\mathcal{X}$ discrete,
$\mathcal{U}$ polytope
For the ROA and all three versions of MOA it shows the running time plotted
against the number of times we add a worst-case scenario during the execution
of the algorithms. The strong correlation indicates that the number of
pessimization steps decisively determines the overall time required. The two
algorithms where the uncertainty set $\mathcal{U}^{(k)}$ grows monotonously,
namely MOA-ws1 and ROA, have similarly high costs per added scenario. This can
be explained by the fact that the resulting robust optimization problems are
harder to solve due to the number of scenarios in $\mathcal{U}^{(k)}$. Vice
versa, MOA and MOA-ws2 both “forget” scenarios. Consequently, they need to
(re)add more scenarios, but the optimization problems are simpler. For them
the ratio between runtime and added scenario is lower. This also explains why
the warm-start modifications pay off: Apparently, the additional cost of
starting with a larger scenario set $\mathcal{U}^{(k)}$ is more than offset by
less frequent need to execute of the pessimization step.
#### Evaluation for polytopal feasible sets
Additionally, we tested the algorithms on instances with feasible sets
$\mathcal{X}$ that are polytopes. In this case DA is faster. Apart from that,
the observations do not deviate significantly from the ones discussed in the
previous paragraphs except that if $\mathcal{U}$ is a polytope too, DA is
faster than MOA-ws1 and MOA-ws2 as can be seen in Figure 9.
Figure 9: Average running time of our five algorithms for 100 instances as a
function of $m^{\prime}$ with $n=m=n^{\prime}=5$ fixed, $\mathcal{X}$ and
$\mathcal{U}$ polytopes
#### The algorithms as approximation algorithms
Lastly, we want to investigate how soon the algorithms provide a reasonable
approximation of the Pareto front. For this we turn to Algorithm 5.1, which in
the $k$-th iteration determines (via dichotomic search) all extreme supported
nondominated points of $BRO(\mathcal{U}^{k})$ and then determines the worst-
case outcomes of those points under $\mathcal{U}$. Figure 10 shows for an
instance with $n=m=n^{\prime}=5,m^{\prime}=30$ and $\mathcal{X}$,
$\mathcal{U}$ both continuous, the lower and upper bound determined in the
second and fourth iteration and the robust solutions determined in the final
7th iteration. We can see that our method provides a good approximation to the
Pareto front early on.
Figure 10: Lower bound and upper bound determined in the second, fourth
iteration and in the final (7th) iteration in an instance with
$n=m=10,n^{\prime}=m^{\prime}=20$ and $\mathcal{X}$, $\mathcal{U}$ continuous
## 7 Conclusions and further research
In this paper, we have shown how biobjective mixed-integer linear optimization
problems, where both objective functions are the maximum of a set of linear
objective functions, can be solved. While we framed this as a method for
_robust_ biobjective optimization – specifically to determine _point-based
minmax robust efficient_ solutions for biobjective mixed-integer linear robust
optimization problems –, our methods are not limited to such problems. They
can be applied to any biobjective optimization problem of the described
structure.
Our solution method combines a well-known approach from biobjective
optimization, namely dichotomic search, with approaches used in robust
optimization, namely optimization-pessimization and reformulation. In our
numerical experiments, it has be shown that all our approaches are sensible
for some problems. We illustrate which approach is most suitable for which
situation: The robust optimizer’s approach provides a good approximation of
the set of extreme supported efficient solutions already early on; the warm-
start modifications improve the multiobjective optimizer’s approach such that
it is fastest on instances where $\mathcal{U}$ is discrete. If $\mathcal{X}$
and $\mathcal{U}$ are polytopes, the dualization approach is the fastest.
Many avenues for further research exist that use the framework that we
developed: First, other and more advanced solution methods for multiobjective
optimization can be used. More specifically, dichotomic search can be replaced
by any other enumeration method for extreme nondominated points (such as the
one proposed in [BM15]). That way, a method similar to the one proposed in
this paper can be used for problems with more than two objectives. Similarly,
solution methods for specific problems such as the multiobjective knapsack or
the multiobjective TSP (see [Vis+98, Ehr05]) can be combined with
optimization-pessimization to find robust solutions of these problems.
Second, extension to other robustness concepts for multiobjective
optimization, such as set-based minmax robust efficiency, would be desirable.
We plan to adapt the presented algorithms to the concept of regret robust
efficiency (see [GW22]).
## Acknowledgments
Fabian Chlumsky-Harttmann was supported by the DFG Research Grant “Robust
Multi-Objective Optimization: Analysis and Approaches”.
## References
* [ABV09] Hassene Aissi, Cristina Bazgan and Daniel Vanderpooten “Min–max and min–max regret versions of combinatorial optimization problems: A survey” In _European Journal of Operational Research_ 197.2, 2009, pp. 427–438
* [AN79] Y. P. Aneja and K. P. K. Nair “Bicriteria Transportation Problem” In _Management Science_ 25.1, 1979, pp. 73–78
* [Ant+20] Tadeusz Antczak, Yogendra Pandey, Vinay Singh and Shashi Kant Mishra “On approximate efficiency for nonsmooth robust vector optimization problems” In _Acta Mathematica Scientia_ 40.3 Springer, 2020, pp. 887–902
* [Ass+08] Tiravat Assavapokee, Matthew J Realff, Jane C Ammons and I-Hsuan Hong “Scenario relaxation algorithm for finite scenario-based min–max regret and min–max relative regret robust optimization” In _Computers & operations research_ 35.6 Elsevier, 2008, pp. 2093–2102
* [BDL16] Dimitris Bertsimas, Iain Dunning and Miles Lubin “Reformulation versus cutting-planes for robust optimization” In _Computational Management Science_ 13.2 Springer, 2016, pp. 195–217
* [BF17] R. Bokrantz and A. Fredriksson “Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization” In _European Journal of Operational Research_ 262.2, 2017, pp. 682–692
* [BM15] Fritz Bökler and Petra Mutzel “Output-Sensitive Algorithms for Enumerating the Extreme Nondominated Points of Multiobjective Combinatorial Optimization Problems” In _Algorithms - ESA 2015_ Berlin, Heidelberg: Springer Berlin Heidelberg, 2015, pp. 288–299
* [BNA13] Mathias Bürger, Giuseppe Notarstefano and Frank Allgöwer “A polyhedral approximation framework for convex and robust distributed optimization” In _IEEE Transactions on Automatic Control_ 59.2 IEEE, 2013, pp. 384–395
* [BS19] Marco Botte and Anita Schöbel “Dominance for multi-objective robust optimization concepts” In _European Journal of Operational Research_ 273.2, 2019, pp. 430–440
* [BTEN09] A. Ben-Tal, L. El Ghaoui and A.S. Nemirovski “Robust Optimization”, Princeton Series in Applied Mathematics Princeton University Press, 2009
* [BV04] Stephen P Boyd and Lieven Vandenberghe “Convex optimization” Cambridge university press, 2004
* [Bie07] Daniel Bienstock “Histogram models for robust portfolio optimization” In _Journal of computational finance_ 11.1 RISK PUBLICATIONS, 2007, pp. 1
* [Bit80] Gabriel R. Bitran “Linear Multiple Objective Problems with Interval Coefficients” In _Management Science_ 26, 1980, pp. 694–706
* [CV14] Karthekeyan Chandrasekaran and Santosh S Vempala “Integer feasibility of random polytopes: random integer programs” In _Proceedings of the 5th conference on Innovations in theoretical computer science_ , 2014, pp. 449–458
* [Chu20] Thai Doan Chuong “Robust Optimality and Duality in Multiobjective Optimization Problems under Data Uncertainty” In _SIAM Journal on Optimization_ 30.2, 2020, pp. 1501–1526
* [Coh78] J.L. Cohon “Multiobjective Programming and Planning” Academic Press, 1978
* [EIS14] M. Ehrgott, J. Ide and A. Schöbel “Minmax Robustness for Multi-objective Optimization Problems” In _European Journal of Operational Research_ 239, 2014, pp. 17–31
* [EKS17] G. Eichfelder, C. Krüger and A. Schöbel “Decision uncertainty in multiobjective optimization” online first In _Journal of Global Optimization_ , 2017, pp. 1–26
* [ES20] A. Engau and D. Sigler “Pareto Solutions in Multicriteria Optimization under Uncertainty” In _European Journal of Operational Research_ 281.2 Elsevier, 2020, pp. 357–368
* [Ehr05] M. Ehrgott “Multicriteria Optimization” Springer Nature, 2005
* [GS16] M. Goerigk and A. Schöbel “Algorithm Engineering in Robust Optimization” In _Algorithm Engineering: Selected Results and Surveys_ 9220, LNCS State of the Art, 2016, pp. 245–279
* [GW22] Patrick Groetzner and Ralf Werner “Multiobjective optimization under uncertainty: A multiobjective robust (relative) regret approach” In _European Journal of Operational Research_ 296.1, 2022, pp. 101–115
* [GYd15] Bram L. Gorissen, Yanıkoğlu and Dick den Hertog “A practical guide to robust optimization” In _Omega_ 53, 2015, pp. 124–137
* [Gob+15] M.A. Goberna, V. Jeyakumar, G. Li and J. Vicente-Pérez “Robust solutions to multi-objective linear programs with uncertain data” In _European Journal of Operational Research_ 242, 2015, pp. 730–743
* [HNS13] F. Hassanzadeh, H. Nemati and M. Sun “Robust Optimization for Multiobjective Programming Problems with Imprecise Information” In _Procedia Computer Science_ 17, 2013, pp. 357 –364
* [HNS14] Farhad Hassanzadeh, Hamid Nemati and Minghe Sun “Robust optimization for interactive multiobjective programming with imprecise information applied to R&D project portfolio selection” In _European Journal of Operational Research_ 238.1 Elsevier, 2014, pp. 41–53
* [Hen86] Mordechai I Henig “The domination property in multicriteria optimization” In _Journal of Mathematical Analysis and Applications_ 114.1, 1986, pp. 7–16
* [IS16] J. Ide and A. Schöbel “Robustness for uncertain multi-objective optimization: A survey and analysis of different concepts” In _OR Spectrum_ 38.1, 2016, pp. 235–271
* [IS96] Masahiro Inuiguchi and Masatoshi Sakawa “Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test” Fuzzy Multiple Criteria Decision Making In _Fuzzy Sets and Systems_ 78.2, 1996, pp. 231 –241
* [Ide+14] J. Ide, E. Köbis, D. Kuroiwa, A. Schöbel and C. Tammer “The relationship between multi-objective robustness concepts and set valued optimization” In _Fixed Point Theory and Applications_ 2014.83, 2014
* [KD69] RL Kruse and JJ Deely “Joint continuity of monotonic functions” In _The American Mathematical Monthly_ 76.1 Taylor & Francis, 1969, pp. 74–76
* [KL12] D. Kuroiwa and G. M. Lee “On Robust Multiobjective Optimization” In _Vietnam Journal of Mathematics_ 40.2&3, 2012, pp. 305–317
* [Kha+09] Leonid Khachiyan, Endre Boros, Konrad Borys, Vladimir Gurvich and Khaled Elbassioni “Generating all vertices of a polyhedron is hard” In _Twentieth Anniversary Volume: Discrete & Computational Geometry_ Springer, 2009, pp. 1–17
* [Krü+23] Corinna Krüger, Anita Schöbel, Lena Fritzen and Margaret M Wiecek “The point-based robustness gap for uncertain multiobjective optimization” In _Optimization_ Taylor & Francis, 2023, pp. 1–35
* [Kuh+16] K. Kuhn, A. Raith, M. Schmidt and A. Schöbel “Bicriteria robust optimization” In _European Journal of Operational Research_ 252, 2016, pp. 418–431
* [MB09] Almir Mutapcic and Stephen Boyd “Cutting-set methods for robust convex optimization with pessimizing oracles” In _Optimization Methods and Software_ 24.3 Taylor & Francis, 2009, pp. 381–406
* [Mon06] Roberto Montemanni “A Benders decomposition approach for the robust spanning tree problem with interval data” In _European Journal of Operational Research_ 174.3 Elsevier, 2006, pp. 1479–1490
* [ÖK10] Özgür Özpeynirci and Murat Köksalan “An exact algorithm for finding extreme supported nondominated points of multiobjective mixed integer programs” In _Management Science_ 56.12 INFORMS, 2010, pp. 2302–2315
* [PKL19] Anthony Przybylski, Kathrin Klamroth and Renaud Lacour “A simple and efficient dichotomic search algorithm for multi-objective mixed integer linear programs”, 2019 arXiv:1911.08937 [math.OC]
* [PS20] Julius Pätzold and Anita Schöbel “Approximate cutting plane approaches for exact solutions to robust optimization problems” In _European Journal of Operational Research_ 284.1, 2020, pp. 20–30
* [RY13] S. Rivaz and M.A. Yaghoobi “Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients” In _Central European Journal of Operations Research_ 21.3 Springer, 2013, pp. 625–649
* [Rai+18] Andrea Raith, Marie Schmidt, Anita Schöbel and Lisa Thom “Extensions of labeling algorithms for multi-objective uncertain shortest path problems” In _Networks_ 72.1 Wiley Online Library, 2018, pp. 84–127
* [Rai+18a] Andrea Raith, Marie Schmidt, Anita Schöbel and Lisa Thom “Multi-objective minmax robust combinatorial optimization with cardinality-constrained uncertainty” In _European Journal of Operational Research_ 267.2 Elsevier, 2018, pp. 628–642
* [Ree94] Rembert Reemtsen “Some outer approximation methods for semi-infinite optimization problems” In _Journal of Computational and Applied Mathematics_ 53.1 Elsevier, 1994, pp. 87–108
* [SAG11] Sauleh Siddiqui, Shapour Azarm and Steven Gabriel “A modified Benders decomposition method for efficient robust optimization under interval uncertainty” In _Structural and Multidisciplinary Optimization_ 44 Springer, 2011, pp. 259–275
* [SZK21] Anita Schöbel and Yue Zhou-Kangas “The price of multiobjective robustness: Analyzing solution sets to uncertain multiobjective problems” In _European Journal of Operational Research_ 291.2, 2021, pp. 782–793 |
# Are stripped envelope supernovae really deficient in 56Ni?
Ryoma Ouchi Keiichi Maeda Department of Astronomy, Kyoto University,
Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan Joseph P. Anderson
European Southern Observatory, Alonso de Córdova 3107, Casilla 19, Santiago,
Chile Ryo Sawada Department of Earth Science and Astronomy, Graduate School
of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo
153-8902, Japan
###### Abstract
Recent works have indicated that the 56Ni masses estimated for Stripped
Envelope SNe (SESNe) are systematically higher than those estimated for SNe
II. Although this may suggest a distinct progenitor structure between these
types of SNe, the possibility remains that this may be caused by observational
bias. One important possible bias is that SESNe with low 56Ni mass are dim,
and therefore they are more likely to escape detection. By investigating the
distributions of the 56Ni mass and distance for the samples collected from the
literature, we find that the current literature SESN sample indeed suffers
from a significant observational bias, i.e., objects with low 56Ni mass - if
they exist - will be missed, especially at larger distances. Note, however,
that those distant objects in our sample are mostly SNe Ic-BL. We also
conducted mock observations assuming that the 56Ni mass distribution for SESNe
is intrinsically the same with that for SNe II. We find that the 56Ni mass
distribution of the detected SESNe samples moves toward higher mass than the
assumed intrinsic distribution, because of the difficulty in detecting the
low-56Ni mass SESNe. These results could explain the general trend of the
higher 56Ni mass distribution (than SNe II) of SESNe found thus far in the
literature. However, further finding clear examples of low-56Ni mass SESNe
($\leq 0.01M_{\odot}$) is required to add weight to this hypothesis. Also, the
objects with high 56Ni mass ($\gtrsim 0.2M_{\odot}$) are not explained by our
model, which may require an additional explanation.
stars: massive — supernovae: general
††journal: ApJ
## 1 Introduction
Core collapse supernovae (SNe) are the explosions of massive stars, marking
the termination of their lives. A small fraction of the gravitational energy
of the collapsing iron core is converted into the kinetic and thermal energy
of the ejected matter (Woosley et al., 2002). Core collapse SNe are classified
into several categories, based on their spectra and light curves. SNe with
hydrogen lines in their spectra are classified as Type II SNe (SNe II), while
those lacking hydrogen lines are called Type I SNe (SNe I). Among SNe I, those
having He lines are called Type Ib SNe (SNe Ib) and those lacking He lines are
called SNe Ic. Type IIb supernovae (SNe IIb) are characterized by hydrogen
lines in their early phase spectra which gradually disappear, and by the He
lines which become increasingly strong at later phases (Filippenko, 1997). SNe
IIb, Ib and Ic are considered to originate from massive stars that have lost a
significant fraction of the envelope during their evolution, and thus they are
collectively called Stripped-Envelope SNe (SESNe) (Smartt et al., 2009).
It has been established that SNe IIP are the explosions of red supergiants
based on light curve models (Falk & Arnett, 1977; Elmhamdi et al., 2003;
Bersten et al., 2011) and also from the direct detection of the progenitors on
pre-SN images (Smartt, 2009, 2015). On the contrary, the progenitors of SESNe
are more uncertain. For SNe IIb/Ib/Ic, two possible progenitor channels have
been proposed. One is a massive WR star (with the main-sequence mass
$M_{\mathrm{ms}}\gtrsim 25M_{\odot}$) that has blown off the H-rich envelope
by its own stellar wind (Begelman & Sarazin, 1986; Georgy, 2012; Gräfener &
Vink, 2016). The other is a relatively low mass star which loses its envelope
by mass transfer to a binary companion (Podsiadlowski et al., 1992; Stancliffe
& Eldridge, 2009). Recent observational evidence favors the latter scenario.
The light curve modeling and direct progenitor detection indicate that the
progenitors are relatively low mass stars ($M_{\mathrm{ms}}\lesssim
18M_{\odot}$), being consistent with the binary scenario (Maund et al., 2011;
Bersten et al., 2014; Van Dyk et al., 2014; Folatelli et al., 2015). Also, for
some SESNe, companion star candidates have been detected, which indicates a
binary origin (Maund et al., 2004; Folatelli et al., 2014).
One of the most important power sources of SNe is newly synthesized 56Ni.
${}^{56}\mathrm{Ni}$ decays into ${}^{56}\mathrm{Co}$, and then into
${}^{56}\mathrm{Fe}$. This nuclear decay chain powers the tail phase of SNe II
and most of the light curve of SESNe. The 56Ni masses of SNe have been
estimated using several methods (Anderson, 2019). For SNe II, the tail
luminosity has mostly been used to estimate the 56Ni mass, assuming the
complete trapping of $\gamma$-rays produced from the nuclear decay. For SESNe,
on the contrary, the tail luminosity cannot be easily used due to the
incomplete trapping of the $\gamma$-ray photons, and the ‘Arnett-rule’ has
often been used instead (Arnett, 1982; Wheeler et al., 2015). This rule
dictates that the peak luminosity of SESNe should be equal to the
instantaneous energy deposition rate by the nuclear decay. For both types of
SNe, the mass of synthesized 56Ni has also been estimated from light curve
modeling (e.g. Utrobin & Chugai, 2011; Bersten et al., 2014).
---
Figure 1: Left: The time to peak ($t_{p}$) as a function of the 56Ni mass for
the sample of Meza-SESNe. The filled circles denote the samples excluding Type
Ic-BL, while open circles denote SNe Ic-BL. A solid line is the result of the
linear regression (i.e. equation 2), while dashed lines show the standard
error. Here, the standard error is estimated as
$\Sigma_{i=1}^{i=N}(t_{p,\mathrm{data},i}-t_{p,\mathrm{fit},i})^{2}/(N_{\mathrm{smaples}}-2)$.
Right: The peak luminosity ($L_{p}$) as a function of the 56Ni mass for the
sample of Meza-SESNe. Solid line is the prediction from the equations 1 and 2,
while dashed lines show the standard error.
Interestingly, mounting evidence has been accumulating to show that the masses
of synthesized 56Ni of the observed SESNe are systematically higher than those
of SNe II. This result was first formally outlined by Kushnir (2015). After
that, Anderson (2019) collected the 56Ni masses for 258 SNe from the published
literature and compared the 56Ni mass distributions for various types of SNe.
He found that the 56Ni masses estimated for SNe II are systematically lower
than SESNe; the median of the 56Ni masses is $0.032M_{\odot}$ for SNe II,
$0.102M_{\odot}$ for SNe IIb, $0.163M_{\odot}$ for SNe Ib, $0.155M_{\odot}$
for SNe Ic, and $0.369M_{\odot}$ for SNe Ic-broad line (SNe Ic-BL).
This result has important implications. The production of 56Ni is sensitive to
the explosion mechanism (Maeda & Tominaga, 2009; Suwa & Tominaga, 2015; Sawada
& Maeda, 2019) and the progenitor mass (Suwa et al., 2019). Indeed, this may
be qualitatively consistent with some indications that the progenitors of
SESNe may be more massive than SNe II either as an entire class or for the
particular SN Ic class (e.g. Anderson et al., 2012; Valenti et al., 2012; Fang
et al., 2019). The possible difference in the nature of the progenitors
between SNe II and SESNe may introduce some tension to the popular suggestion
for a binary origin for SESN progenitors, since the core structure should be
similar between SESNe and SNe II; the binarity mainly affects the outer
envelope but not the core structure (Yoon et al., 2010, 2017; Ouchi & Maeda,
2017). However, this picture may also be an oversimplification, since there
are several factors that can affect the nature of the progenitor even in the
binary scenario. For example, massive stars are claimed to be preferentially
formed in close binary systems (Moe & Di Stefano, 2017). The angular momentum
transfer may also have effects on the core structure and boost the synthesized
56Ni mass (Schneider et al., 2021). In any case, understanding the origin of
different 56Ni masses between SESNe and SNe II should help to clarify the
progenitors of SESNe.
Before concluding that the systematically different 56Ni mass between SESNe
and SNe II may be caused by a different structure in the progenitor cores,
systematic errors in calculating the 56Ni masses should be addressed
(Anderson, 2019). Indeed, the ‘Arnett-rule’, which has been widely used for
SESNe have been claimed to overestimate the 56Ni mass (Dessart et al., 2015,
2016; Khatami & Kasen, 2019). However, several works have concluded that even
by taking into account the different methods to derive the 56Ni mass and
various observational errors, a difference in 56Ni masses between SNeII and
SESNe remains (Afsariardchi et al., 2020; Meza & Anderson, 2020; Sharon &
Kushnir, 2020). Meza & Anderson (2020) further noted the possibility that
SESNe with a small amount of 56Ni might have been missed by the existing
surveys. Since the luminosity of SESNe is mostly powered by the radioactive
decay of 56Ni, the SESNe with the lowest 56Ni masses are the faintest (Lyman
et al., 2016). Thus, they can possibly escape from detection depending on the
survey depth. On the contrary, SNe II with a small amount of 56Ni can still
power themselves by diffusion of the thermal energy coming from the explosion
energy. Thus, SNe II can more easily be detected than SESNe, even if the 56Ni
mass is small. Indeed, several Ni-poor SESNe ($M_{\mathrm{Ni}}\lesssim
0.02M_{\odot}$) have been detected (Kasliwal et al., 2010; Shivvers et al.,
2016; Nakaoka et al., 2019). However, it should also be noted that none of
these examples are just a low-luminosity version of canonical SESNe as they
all show unusual properties.
The aim of this paper is to investigate how much observational bias may lie in
the 56Ni mass distribution of the samples collected from the published
literature. In section 2, we define the samples that are used throughout the
paper. Section 3 describes some equations that are used in this paper. In
section 4, we investigate whether there is an observational bias in the 56Ni
mass distribution of our data samples by examining the relation between
distance, luminosity and 56Ni mass. In section 5 and section 6, we conduct
mock observations of SESNe and theoretically investigate the effect of
observational bias on the ‘observed’ 56Ni mass distribution. We discuss the
results in section 7 and finally conclude the paper in section 8.
## 2 Data sample
In this section, we describe the observational samples used in this paper.
Throughout the paper, we use the samples of 56Ni estimates collected from the
published literature both for SESNe and SNe II. Anderson (2019) recently
compiled such samples, including 143 SESNe and 115 SNe II. Specifically, he
used the SAO/NASA ADS astronomy query
form111https://ui.adsabs.harvard.edu/classic-form, searching for articles with
“supernova” and “type II” that were published until August 2018, then
“supernova” and “type IIb” and so forth in manuscript abstracts. Then, he
identified those publications with published 56Ni mass estimates. In addition
to this sample, we add newly published objects between August 2018 and
November 2020. The newly added reference list can be found at the end of this
manuscript. Note that we do not include 56Ni estimates that are derived from
combined models, such as ‘magnetar + 56Ni model’ or ‘circumstellar interaction
+ 56Ni model’ (e.g. Gangopadhyay et al., 2020). We name these final samples
‘LS-SESNe (Large Sample SESNe)’ and ‘LS-SNeII (Large Sample SNe II)’,
respectively. The sample sizes are 187 and 115222The size of LS-SNeII is the
same as the sample of Anderson (2019). This occurred because this time we
excluded objects with only upper or lower limits for the 56Ni mass. The number
of thus removed events was by chance equal to that of the newly added events.
for LS-SESNe and LS-SNeII, respectively.
Several different methods have been used to derive the 56Ni masses in the
literature. For SNe II, the tail luminosity is commonly used to measure the
56Ni mass. For SESNe, on the contrary, 56Ni mass is often derived by feeding a
peak luminosity into the ‘Arnett-relation’. It is true that tail luminosity
has also been used for SESNe to constrain their 56Ni masses. However, since
the assumption of complete $\gamma$-ray trapping is usually not valid for
SESNe, the tail luminosity underestimates 56Ni mass (unless additional
modelling is employed; see e.g. Sharon & Kushnir (2020).)
In addition to LS-SESNe, we also use a different sample of SESNe, which we
call ‘Meza-SESNe’; this is the same sample as that used by Meza & Anderson
(2020). Those authors defined a SESN sample with well-sampled photometry at
optical and near-IR wavelengths. This led to a sample of 37 events. To obtain
peak luminosities, they applied a local polynomial regression with a Gaussian
kernel, using the public modules from PyQt-fit in Python4. Note that the
integration was done in the wavelength range from the $B$ band to $H$ band
without extrapolation outside. Therefore, their resulting light curves should
be considered to be pseudo-bolometric, and are a lower limit to the true
bolometric luminosity at all times. However, the wavelength coverage is
reasonably large, and therefore the error here is probably less significant
than that coming from the different methods to derive the 56Ni mass. In the
paper, they tested three different methods to derive the 56Ni mass. In the
first method, they used ‘Arnett-rule’. In the second method, they used a tail
luminosity. The third method employed that recently proposed by Khatami &
Kasen (2019), that overcomes several limitations of Arnett-like models. Meza &
Anderson (2020) showed that using the different methods does not change the
overall trends in the derived 56Ni masses and their conclusions. In the
present work, we mostly use the Arnett rule in our analysis (§3), but we show
that our results are not affected by this choice in Appendix B.
In section 4.2, in order to compare the luminosity function between SESNe and
SNe II, we use the sample of 57 SNeII taken from Hamuy (2003); Müller et al.
(2017); Pejcha & Prieto (2015). These events have published values of the mid-
plateau phase luminosity. These papers are included in the reference list of
Anderson (2019), and thus, this is a sub-sample of LS-SNeII. We call this
small sample ‘SS-SNeII (Small Sample SNe II)’.
Finally, in section 7, we will compare our results to a sample of candidate
ultra-stripped envelope SNe (USSNe). The data for the USSN candidates are also
collected from the published literature published before November 2020 by
searching “ultra-stripped” and “supernova” in the ADS abstract form. The
references for them are listed at the end of the manuscript 333Note that these
reference lists are not covering all the USSN candidates claimed so far, since
we only take into account those objects for which 56Ni masses have been
estimated.
For all these objects, we adopt the distance to the host galaxy from the
redshift independent measurement in NED444https://ned.ipac.caltech.edu. In
case there is no redshift independent measurement of the distance to the host,
we adopt the Hubble distance on NED, which includes the correction of Virgo,
GA, and Shapley. For the cosmological parameters, $H_{0}$ = 67.8 km/sec/Mpc,
$\Omega_{\mathrm{matter}}$ = 0.308, $\Omega_{\mathrm{vacuum}}$ = 0.692 have
been used. If the host was anonymous or the distance was not found in NED, we
take the distance from the individual published literature.
Figure 2: The 56Ni masses of our samples as a function of the distance. The
red points refer to LS-SESNe, while yellow points refer to LS-SNeII. Note that
red filled circles exclude SNe Ic-BL among LS-SESNe, while red open circles
are for SNe Ic-BL among LS-SESNe. For reference, the limiting distance for a
given 56Ni mass estimated in section 3 is also shown for the case of limiting
magnitude of $V_{\mathrm{lim}}=17$ (blue) and 19 mag (green).
## 3 The relations used in this paper
---
Figure 3: Left: The 56Ni mass distribution of LS-SESNe for the volume-limited
samples of different sizes. The red solid line represents the cumulative 56Ni
mass distribution of the samples among LS-SESNe whose distances are less than
5000 Mpc, while yellow, green and blue solid lines represent the cumulative
distributions of the objects whose distances are less than 100 Mpc, 60 Mpc and
20 Mpc, respectively. Dotted lines represent the same distributions, except
the we exclude SNe Ic-BL from the sample. Right: The same figure as the left
panel but for LS-SNeII. For both panels, the non-linear least square fit to
the LS-SNe II cumulative distribution is also shown with a gray line (See
section 5.1 for more detail).
### 3.1 The relations for the peak luminosity and the timescale of SESNe
Two important quantities that characterize the light curves of SESNe are the
the peak luminosity ($L_{p}$) and the time it takes from the explosion to the
peak ($t_{p}$). In the following analyses of this paper, we require the
relations that connect these values to the 56Ni mass.
For a given 56Ni mass, the peak luminosity is estimated from the formula shown
in Stritzinger & Leibundgut (2005), which is based on the ‘Arnett-rule’. This
rule assumes that the peak luminosity ($L_{p}$) of a SN powered by the decay
of 56Ni is equal to the instantaneous energy deposition rate by radioactive
decay at that time:
$\displaystyle L_{p}$ $\displaystyle=$ $\displaystyle
10^{43}\times(M_{\mathrm{Ni}}/M_{\odot})$ (1) $\displaystyle\times$
$\displaystyle(6.45\times e^{-t_{p}/8.8}+1.45\times e^{-t_{p}/111.3})\
[\mathrm{ergs^{-1}}].$
The timescale, $t_{p}$, is not necessarily determined by a 56Ni mass. However,
in this paper, we take a phenomenological approach using the observational
data, and express $t_{p}$ as a function of the 56Ni mass. For that purpose, we
derive a fitting formula for $t_{p}$ as a function of 56Ni mass using the
well-observed sample of Meza-SESNe taken from Meza & Anderson (2020). Since
this sample is composed of nearby objects (the median distance of their SESNe
sample, excluding Ic-BL, is 46.7 Mpc), the objects are considered to be less
affected by a possible observational bias than the other SESN samples (and
note that the observational bias we will discuss later would not much affect
this relation). Furthermore, the objects in their sample are chosen under the
condition that they contain the data around the peak. Thus, $L_{p}$ and
$t_{p}$ in their sample are considered to be relatively accurate.
As shown in the left panel of Fig. 1, $t_{p}$ and log $M_{\mathrm{Ni}}$
broadly follow a linear correlation. Thus, we conducted a linear regression to
the data, using the least square method. We did not use SNe Ic-BL samples for
the fit, since they may indeed involve a different explosion mechanism from
canonical SNe II and SESNe, and also they are taken at relatively distant
locations. The derived formula becomes:
$\displaystyle t_{p}$ $\displaystyle=$ $\displaystyle(9.41\pm
2.98)\times\mathrm{log}_{10}(M_{\mathrm{Ni}}/M_{\odot})$ (2)
$\displaystyle+(29.74\pm 3.42)\,[\mathrm{day}].$
Using this equation, together with equation 1, we can estimate $t_{p}$ for a
given 56Ni mass. In the right panel of Fig. 1, the peak luminosity calculated
for a given 56Ni mass using equation 1 and 2 are compared to the data points
of Meza-SESNe. It is seen that the data points for the peak luminosity are
well reproduced by our fitting formula.
### 3.2 Observable distance for a given luminosity
Now that we know the peak luminosity of a SESN for a given 56Ni mass, it is
also important to know out to what distance we can detect it assuming a fixed
limiting magnitude. For this purpose, we use the relation in Hamuy (2003):
$\displaystyle\mathrm{log}D_{\mathrm{lim}}[\mathrm{cm}]=\frac{1}{5}\times(2.5\
\mathrm{log}L[\mathrm{ergs}^{-1}]$
$\displaystyle+V_{\mathrm{lim}}-A_{t}+BC+8.14).$ (3)
Here, $V_{\mathrm{lim}}$ is the limiting magnitude in the $V$ band,
$D_{\mathrm{lim}}$ is the limiting distance, $A_{t}$ is the total extinction
and $BC$ is the bolometric correction. For simplicity, we assume zero both for
$A_{t}$ and $BC$. Using this relation, we can calculate the observable
distance for a given luminosity, assuming a fixed limiting magnitude.
## 4 Investigating observational bias in the data sample
In this section, we investigate whether there is an observational bias in the
56Ni mass distribution of our data samples by examining the relation between
distance, luminosity and 56Ni mass.
### 4.1 56Ni mass and distance
In order to clarify how the observational biases may affect the 56Ni mass
distribution in our samples, we look at the 56Ni mass of our samples as a
function of the distance. Figure 2 shows the 56Ni masses of our samples
plotted as a function of the distance. It can be seen that there is a strong
trend that the 56Ni mass decreases as the distance decreases for SESNe. This
suggests that the objects with low 56Ni masses (i.e. dim objects) and large
distance, if they exist, may be missed. It is, however, important to emphasize
that we are still lacking the SESNe with low 56Ni mass (log
$M_{\mathrm{Ni}}(M_{\odot})\lesssim-1.7$: i.e., $M_{\mathrm{Ni}}\lesssim
0.02M_{\odot}$) even at small distance (log distance (Mpc) $\lesssim 1$). For
SNe II, even though the 56Ni mass slightly decreases as the distance
decreases, the effect is much less significant than SESNe.
Figure 4: The average 56Ni mass in the volume-limited sample plotted as a
function of the threshold distance from 20 Mpc to 5000 Mpc. The red solid line
refers to LS-SESNe sample, while the red dotted line refers to LS-SESNe
excluding SNe Ic-BL. The yellow solid line refers to LS-SNeII.
These trends can be confirmed by looking at Figs. 3 and 4. Figure 3 shows how
the 56Ni mass distribution changes when we take different sizes of volume-
limited samples. It is expected that the 56Ni mass distribution approaches to
the intrinsic distribution as we take the volume-limited sample at a closer
location. It is seen that the 56Ni mass distribution of SESNe significantly
shifts to the lower mass when we take the smaller volume-limited
sample555Note, however, that the distributions for the lowest 20% of the 56Ni
masses are nearly the same for these different sizes of volume-limited
samples. This may indicate that the lack of canonical SESNe with relatively
low 56Ni mass ($\lesssim 0.02M_{\odot}$) is real.. On the contrary, the 56Ni
mass distribution of SNe II does not change notably for the different sizes of
volume-limited sample. From this, we can infer that the LS-SESNe may not trace
the intrinsic 56Ni mass distribution, while LS-SNeII nearly do. Figure 4 shows
the average 56Ni mass in the volume-limited samples plotted as a function of
the threshold distance. This figure, again, shows that LS-SESNe suffer from a
significant observational bias and the discrepancy between SESNe and SNe II
becomes smaller as we take the smaller volume-limited sample, and finally
becomes within a factor of three.
Note, however, that if we remove SNe Ic-BL from LS-SESNe, then, the trend that
56Ni mass decreases with distance is much weakened. This may indicate that SNe
Ic-BL, whose distances are larger than the other types of SNe, are heavily
affected by an observational bias, while other types of SESNe (e.g. SNe IIb,
Ib and Ic) are less affected by it.
Figure 5: The comparison of the distance distributions of LS-SESNe (red) and
LS-SNeII (yellow). The blue solid histogram represents LS-SESNe excluding SNe
Ic-BL, while the dotted one is for SNe Ic-BL among LS-SESNe.
Figure 5 compares the distance distribution between LS-SESNe and LS-SNeII. We
can see that the distance distribution is closer for SNe II than SESNe. Meza &
Anderson (2020) showed that their 35 SESNe sample, excluding two Ic-GRB
objects, have the mean distance (46.7 Mpc) similar to that of their SNe II
sample (42.7 Mpc). However, our significantly larger sample of LS-SESNe has
the larger mean distance of 226.6 Mpc, while LS-SNeII has the mean distance of
41.5 Mpc. Even if we remove Ic-BL and Ic-GRB from the SESNe sample, the mean
distance is 99.8 Mpc, which is more than twice the value of LS-SNeII. This
indicates that the SESNe samples are collected at more distant locations than
SNe II, where the objects suffer from more significant observational bias,
supporting the results derived above 666Although the samples in Meza &
Anderson (2020) were taken at small distances ($\approx 40-50$ Mpc), the SESNe
with low 56Ni mass ($\lesssim 0.02M_{\odot}$) were still lacking. We will
further discuss this issue in section 7.2.
### 4.2 Luminosity distribution
Figure 6: Left: The distributions of the peak luminosity of SESNe (blue
points) and mid-plateau luminosity of SNe II (yellow points) plotted as a
function of distance. For reference, the limiting luminosity as a function of
distance is plotted for the case of limiting magnitude of
$V_{\mathrm{lim}}=17$ (blue line) and 19 mag (green line), respectively.
Right: The comparison of the luminosity function of SESNe (blue) and SNe II
(yellow) sample. For SESNe we show the peak luminosity, while for SNeII we
show the mid-plateau luminosity.
In this section, we investigate the luminosity distribution of our samples. We
emphasise that the analysis in this section is not affected by the assumption
about the relation between the 56Ni mass and the peak luminosity of SESNe. As
noted in section 2, here, we only use the sample of 57 SNeII taken from Hamuy
(2003); Müller et al. (2017); Pejcha & Prieto (2015), which we call SS-SNeII.
Note that Hamuy (2003) only published the V-band magnitude, so, we convert it
to the bolometric luminosity assuming the bolometric correction to be zero,
following Goldberg et al. (2019). For SESNe, we use the sample of 37 from
Meza-SESNe.
The left panel of Fig. 6 shows the luminosity distribution as a function of
distance for these samples. For SESNe we show the peak luminosity, while we
show the mid-plateau luminosity for SNe II. There is a positive correlation
between the luminosity and distance both for SESNe and SNe II. Also, the
minimum luminosity for a fixed distance is similar between SNe II and SESNe.
This indicates that the ‘luminosity’ distributions seen in both of our samples
(SESNe and SNe II) may be suffering from the same observational selection
effect. The right panel of Fig. 6 compares the luminosity functions of SESNe
and SNe II. We notice that there is a luminosity cut off for both types at
around log $L$[erg s-1] $\sim 41.7$. The SNe II plateau phase and the SESNe
peak phase are powered by the different physical mechanisms, with the former
powered by the explosion energy and the latter powered by the radioactive
decay of 56Ni. It is true the plateau luminosity and the 56Ni mass of SNe II
are known to be positively correlated (Pejcha & Prieto, 2015), but it is
unlikely that the lower luminosity cut off are the same between the two groups
of SNe just in terms of physics. Thus, we speculate that this simultaneous cut
off of the luminosity functions for both types of SNe is caused by an
observational selection effect. This selection effect will introduce a bias in
the 56Ni mass distribution for SESNe, as the 56Ni mass is closely connected to
their peak luminosities.
In summary, the results derived in section 4 all point to the following
interpretation: The 56Ni masses of SESNe samples collected from the published
literature suffer from notable observational bias, i.e. the distant objects
with relatively low 56Ni mass are missed, meaning that the samples are biased
towards the luminous objects. On the contrary, the 56Ni masses of SNe II
samples suffer much less from such bias. We, again, emphasise that the
analyses in this section are not affected by the assumption about the relation
between the 56Ni mass and the peak luminosity of SESNe777We note, however,
that Meza & Anderson (2020) have shown that the statistical difference of 56Ni
mass between SESNe and SNe II remains even if they take relatively close
samples ($\approx 40-50$ Mpc), which is also confirmed by our analyses (Figs.
3 & 4). Thus, the observational bias alone may not be sufficient to explain
all of the statistical difference in the 56Ni mass (section 7)..
## 5 Method of Mock observations
In the following sections, we conduct mock observations of SESNe and
investigate the effect of observational bias on the 56Ni mass distribution of
detected SESNe. In the previous sections, we have found that the 56Ni mass
distribution in our SNe II sample is not notably suffering from the
observational bias. Therefore, below, we start additional analysis based on
the following two working hypotheses: (1) The 56Ni mass distribution in our
SNe II sample (LS-SNeII) represents the intrinsic 56Ni mass distribution of
SNe II, and (2) SESNe have the same intrinsic 56Ni mass distribution as that
of SNe II. The second assumption is based on the hypothesis that assuming a
binary origin for SESNe, progenitors of SESNe and SNe II are expected to share
the similar range in the initial mass (see section 1)888Note, however, that
there are indications that the progenitors of SESNe may be more massive than
SNe II either as an entire class or for the particular SN Ic class (e.g.
Anderson et al., 2012; Valenti et al., 2012; Fang et al., 2019).. Based on
these hypotheses, we conduct mock observations of SESNe. In the rest of this
section, we describe the procedure of the mock observation in more detail.
### 5.1 56Ni mass distribution
As noted above, here we assume that the intrinsic 56Ni mass distribution of
SESNe is the same as the 56Ni mass distribution of LS-SNe II. To simplify the
numerical analyses, we fit the cumulative histogram of 56Ni mass (denoted here
as $f(x)$) by the function of $f(x)=$tanh$(a_{0}\times x)$, using non-linear
least squares. We obtained $a_{0}=14.60$ as the best fit parameter. The
comparison of our fitted curve with our sample is shown in Fig. 7. Below, we
use the function of $f(x)$= tanh$(14.60\times x)$ to represent the assumed
intrinsic 56Ni mass distribution of SESNe.
Figure 7: The cumulative 56Ni mass distribution of LS-SNeII and the non-linear
least squares fit to it, assuming the function of $f(x)=$tanh$(a_{0}\times
x)$.
### 5.2 Simulating the observations
We simulate one SESN by selecting the 56Ni mass and distance from the given
probability distributions. We select a value of 56Ni mass from the
distribution derived in section 5.1. Then, for each 56Ni mass thus derived,
the distance is randomly chosen following the probability function of
$p\propto$ (distance)3, i.e., the volume size. The range of distance is set
from zero up to the limiting distance corresponding to the peak luminosity of
SESNe with a 56Ni mass of 1.0$M_{\odot}$, unless otherwise noted.
For each object with a given 56Ni mass and distance, we decide whether to add
it to the ‘detected’ sample or not based on the following procedure. First,
from the given 56Ni mass, we randomly pick up a value of $t_{p}$ based on the
distribution, taking into account the dispersion, derived from the fit in
section 3. Combining this value of $t_{p}$ with the chosen value of 56Ni mass,
we can estimate a peak luminosity of SESNe (see, equation 1). Next, we
estimate the limiting distance using equation 3, for the peak luminosity
derived above. If the selected distance is within the observable distance
corresponding to its peak luminosity, we consider it to be detected and add it
to the detected sample. Otherwise, we consider that the object escapes
detection and do not add it to the detected sample. Once the number of
detections reaches a specified number, we stop one iteration of the mock
observation. Below, the number of detections is set to be 100, unless
otherwise noted. The number of 100 is chosen to be consistent with the order
of magnitude of our LS-SESNe sample size. We iterate the procedure described
above $10^{3}$ times, in order to clarify the possible range of the
distributions by taking into account the statistical fluctuation due to the
limited sample size.
## 6 Results of mock observation
### 6.1 Luminosity function
Figure 8: The luminosity functions derived from the mock observation are
compared to the luminosity function of Meza-SESNe. The green histogram is the
luminosity function of the detected sample in the mock observation, assuming
the limiting magnitude of 19 mag. Among the green region, the solid line and
shaded region represent the mean distribution and standard deviation,
respectively, computed for the $10^{3}$ iterations. For reference, the yellow-
solid line represents the input intrinsic luminosity function of SESNe in the
mock observations (assuming that this is the same as that of SNeII).
Here, we show the results of the mock observation described in section 5. We
assume a fixed limiting magnitude of 19 mag in this section. Figure 8 compares
the luminosity functions derived from the mock observations to the luminosity
function of Meza-SESNe. Here, in order to make the direct comparison to Meza-
SESNe, we stop one iteration of mock observation when the number of detected
objects reaches 37 (i.e., the Meza-SESNe sample size), not 100. Then, we
repeat this $10^{3}$ times to clarify the possible range of the distributions.
Interestingly, we can see that the luminosity function in the ‘detected’
samples is shifted to high luminosity compared to the model intrinsic
luminosity function. As seen from Fig. 1, the objects with higher 56Ni mass,
in general, have higher peak luminosity. Thus, they have the larger observable
volume and dominates the detected sample. It is also worthwhile to note that
the luminosity function of our detected samples in the mock observation
roughly explains the observed luminosity function of Meza-SESNe. Especially,
the lower cut-off at around log $L\sim 41.7$ is naturally obtained.
### 6.2 56Ni mass distribution
Figure 9: Left: The 56Ni mass distribution in the detected sample of our mock observation (green) is compared to the different samples, excluding SNe Ic-BL. The green-solid line represents the mean distribution of the mock observation with the $10^{3}$ iterations (and 100 detected objects in one interaction), assuming the limiting magnitude of 19 mag. The shaded region represents the standard deviation obtained with $10^{3}$ iterations. Blue- and cyan-dotted lines denote the ‘Arnett mass’ and ‘Tail mass’ in Meza-SESNe, while a red-dotted line denotes the LS-SESNe. Right: the 56Ni mass distribution, excluding SNe Ic-BL (dotted) and including SNe Ic-BL (solid) are compared for the different samples. SN distributions ($N$: Number of samples) | $D$ | $p$
---|---|---
LS-SNe II (115), LS-SESNe (187) | 0.690 | 3.4 $\times 10^{-34}$
LS-SESNe (187), Meza-SESNe (37) | 0.453 | 2.6 $\times 10^{-6}$
LS-SNe II (115), Mock | 0.472 | 1.5 $\times 10^{-9}$
LS-SENe (187), Mock | 0.433 | 5.6 $\times 10^{-13}$
LS-SENe (w/o Ic-BL; 131), Mock | 0.317 | 5.9 $\times 10^{-5}$
Meza (Arnett, 37), Mock | 0.182 | 0.60
Meza (Arnett, w/o Ic-BL; 33), Mock | 0.220 | 0.46
Meza (Tail; 20), Mock | 0.462 | 6.7 $\times 10^{-2}$
Meza (Tail, w/o Ic-BL; 18), Mock | 0.523 | 3.9 $\times 10^{-2}$
Table 1: The KS statistical differences between the different 56Ni mass
distributions. In the first column, the two distributions being compared are
listed together with the number of samples in brackets. In the second column,
the D parameter is given, while in the last column the p value is presented.
‘Mock’ means the result of the mock observation assuming a limiting magnitude
of $V_{\mathrm{lim}}=$ 19 mag. Here, the number of the simulated objects in
one iteration is set to be the same as the size of the sample with which the
mock observation is compared. The $D$ and $p$ for ‘Mock’ are the mean values
computed with the $10^{3}$ iterations.
Figure 9 compares the cumulative 56Ni mass distribution in the detected
samples to different data samples. Note that, below, we stop one iteration of
mock observation when the number of detect objects reaches 100, unless
otherwise noted. The 56Ni mass distribution in the detected sample of our mock
observation is skewed to higher mass compared to the assumed intrinsic
distribution. This is due to the observational bias, being consistent with the
shift of the luminosity function discussed in the previous section.
To make quantitative comparisons of the different 56Ni mass distributions, we
conducted K-S statistical test as shown in Table 1. We used the library,
scipy.stats.kstest, for conducting a K-S statistical test. When we compared
the result of mock observation to a data sample, we set the number of
detections for one iteration to be the same as the number of samples being
compared to. Then, we took the mean of the $D$ parameter and the $p$ value for
$10^{3}$ iterations. The $p$ value between the result of our mock observation
and Meza-SESNe (Arnet) is quite high, being 0.60. This indicates that the 56Ni
mass distribution in Meza-SESNe may be explained by taking into account for an
observational bias on the intrinsic distribution similar to that of LS-SNe II.
Also, this result is consistent with the reasonable match of the 56Ni mass
distribution and luminosity function of our mock observation to Meza-SESNe
(Figs. 8 and 10).
However, the assumption that the 56Ni mass distributions from the mock
observations and LS-SESNe originate from the same distribution is rejected,
with a quite low $p$ value of $5.6\times 10^{-13}$. Even if we exclude SNe Ic-
BL from the sample, the $p$ value is still low, being $5.9\times 10^{-5}$. One
of the possible reasons for this is that our assumption for the intrinsic 56Ni
distribution (section 5.1) may have been too simplistic. Indeed, from the
intrinsic 56Ni mass distribution we assumed, a high value of 56Ni mass
($\gtrsim 0.2M_{\odot}$) is rarely produced, while such high values are found
in LS-SESNe (Fig. 9). Another possible reason is that, while our mock
observation is based on the ‘Arnett-rule’ (section 3.1), which is the same
method used in Meza-SESNe (Arnett), the LS-SESNe consists of the 56Ni masses
estimated by various kinds of methods. Thus, direct comparison of our mock
observation to LS-SESNe may not be appropriate.
In short, the 56Ni mass distribution taking into account an observational bias
is consistent with the well-observed sample of Meza-SESNe. However, further
explanation is needed for the discrepancy between our mock observation and a
larger sample of LS-SESNe. Indeed, the difference between the two samples,
Meza-SESNe and LS-SESNe, is intriguing. It suggests that the 56Ni mass
distribution of SESNe is dependent on how the sample is constructed; the
different samples may thus be contaminated by different degrees of possible
observational biases.
### 6.3 Effect of different limiting magnitudes
Figure 10: Left: The cumulative 56Ni mass distribution of the detected samples
in the mock observation for the different limiting magnitudes. Blue, green,
orange lines refer to the limiting magnitude of 17 mag, 19 mag, and 21 mag,
respectively. Each line is the mean of the distributions derived with $10^{3}$
iterations. For reference, the fit to the 56Ni mass distribution of LS-SNe II
is shown with a gray line. Right: The same figure for the result of the mock
observation, assuming the outer boundary as $80$ Mpc (i.e. volume limited
samples).
In the previous section, we have assumed a limiting magnitude of
$V_{\mathrm{lim}}=$ 19 mag. Next, we will see how the different values of the
limiting magnitudes affect our results. In the left panel of Fig. 10, we show
the 56Ni mass distribution of the detected samples in the mock observation for
the different limiting magnitudes. We can see that the 56Ni mass distribution
is quite insensitive to the different values of the limiting magnitudes. This
can be explained as follows. The 56Ni mass distribution in the observed sample
can be derived by multiplying the assumed intrinsic distribution of 56Ni mass
by the observable volume, i.e., $D_{\mathrm{lim}}(M_{\mathrm{Ni}})^{3}$. Here,
$D_{\mathrm{lim}}(M_{\mathrm{Ni}})$ is the limiting distance calculated using
equation 3. It is seen that the term of $V_{\mathrm{lim}}$ only changes the
scale of the 56Ni mass distribution, but does not affect the normalized
distribution. Note that this apparently counterintuitive result is obtained,
because we consider a magnitude-limited sample here. For comparison, we show
the result of mock observations, assuming the outer boundary of all the events
is 80Mpc, in the right panel of Fig. 10. This constructs the volume-limited
samples. In this case, the cumulative distribution of 56Ni mass approaches to
the intrinsic one as the limiting magnitude is set larger, which is consistent
with our intuition.
Figure 11: The distance distribution of the detected samples in the mock
observation for the different limiting magnitudes. Blue, green, orange lines
refer to the limiting magnitude of 17 mag, 19 mag, and 21 mag, respectively.
Each line is the mean of the distributions derived with $10^{3}$ iterations.
In Fig. 11, we show the distance distribution of the detected samples in the
mock observation for the different limiting magnitudes. We can see that the
distance distribution is sensitive to the different value of the limiting
magnitude. The higher the limiting magnitude is, the observable volume becomes
larger. Thus, the more distant objects dominates the observed sample.
---
Figure 12: Left: The 56Ni mass and distance of the detected samples in one
iteration (i.e., 100 detections) for the limiting magnitudes of 19 mag. Gray
points are all the events that were randomly picked up until the number of
detections reached 100. For reference, the limiting distance for a given 56Ni
mass estimated as in section 3 are also shown assuming the limiting magnitudes
of 19 (green) and 25 (red) mag. The latter represents the limiting magnitude
for the single-visit depth in LSST (Ivezić et al., 2019). Right: The 56Ni mass
and distance of the detected samples in one iteration for the different
limiting magnitudes. Blue, green, orange points refer to the case of limiting
magnitude of 17 mag, 19 mag, and 21 mag, respectively. We add the ultra-
stripped envelope SNe (USSNe) candidates with magenta-star symbols (see
section 7 for discussion of these events). The references for USSNe are listed
at the end of the manuscript.
The limiting distance for a given 56Ni mass estimated as in section 3 are also
shown assuming the different limiting magnitudes.
Figure 13: Left: The mean cumulative 56Ni mass distributions for the different
distance cuts are compared. For making this figure, we assumed a limiting
magnitude of $V_{\mathrm{lim}}=19$ mag, and set the number of detected samples
to be $10^{3}$. This panel is similar to Fig. 3. Right: The average 56Ni mass
for the detected samples inside a given distance are plotted as a function of
distance cuts. This panel is similar to Fig. 4 for our mock observation.
In the right panel of Fig. 12, the 56Ni masses and distances of the detected
samples for the different limiting magnitudes are over-plotted onto Fig.2. As
discussed above, the objects with low 56Ni mass are lacking compared to the
assumed intrinsic distribution, which is consistent with the data samples.
However, our predictions from the mock observations fail to explain the high
56Ni masses of $\gtrsim 0.2M_{\odot}$ (mostly SNe Ic-BL) that exist in the
data samples collected from the published literature, as already noted in
section 6.2. We will discuss this issue in section 7.3.
In order to compare our results to Fig. 4, i.e., the average 56Ni mass for the
volume-limited samples of different sizes, we conduct an additional analysis
as follows. We set the number of detections to be $10^{3}$ (not 100) and
repeat the mock observation conducted above. For this larger sample, we
investigate how the cumulative 56Ni mass distribution changes for the
different values of distance cuts. In Fig. 13, we show the results of such an
analysis. The right panel of Fig. 13 shows that the average 56Ni mass
decreases as the distance cut is decreased, just like Fig. 4. This behaviour
can be understood as follows. As seen in Figs. 8 and 10, the average peak
luminosity of the detected samples in a magnitude-limiting sample is $\sim$ a
few $10^{42}$ erg s-1 (i.e., $M_{\mathrm{Ni}}\sim 0.1M_{\odot}$), irrespective
of the values of limiting magnitude. However, those dominant objects are
detected at different distances depending on the limiting magnitudes.
Actually, Fig. 11 shows that such dominant objects are found at $\sim 30-40$
Mpc for $V_{\mathrm{lim}}=17$ mag, while they are found at $\sim 100$ Mpc for
$V_{\mathrm{lim}}=19$ mag. Thus, if we consider a sufficiently large volume-
limited sample, then, the average 56Ni mass becomes $\sim 0.1M_{\odot}$.
However, if we consider a distance cut smaller than a value (e.g., $\sim 100$
Mpc for $V_{\mathrm{lim}}=19$ mag), then, part of the dominant objects in a
magnitude-limited sample would be missed and the average 56Ni mass starts to
decrease.
Here, it is crucial to mention that the right panel of Fig. 13 does not
perfectly match to Fig. 4. It is true that the decrease of average 56Ni mass
for SESNe at $\sim 100$ Mpc is seen in Fig. 4, just like Fig. 13. However,
Fig. 13 implies that the average 56Ni mass should decrease down to $\sim
0.05M_{\odot}$ at sufficiently small distance ($\lesssim 100$ Mpc), which is
not the case in Fig. 4. These results imply either that an observational bias
alone may not be sufficient to explain the different 56Ni mass between SESNe
and SNe II or that the objects with low 56Ni mass may escape detection due to
other reasons (see section 7.2). Still, our result here suggests that an
observational bias is likely to be present at least for high 56Ni masses, and
one needs to consider this when discussing the 56Ni mass distribution of
SESNe.
### 6.4 Effects of different observational cadences
So far, we implicitly assumed an infinitely small observational cadence in the
mock observation. This means that an object is always detected as long as its
peak luminosity exceeds the observational limiting magnitudes. However,
existing surveys have a wide range of observational cadence from hours to a
few tens of days, depending on their scientific aims. Therefore, some objects
may be missed due to infrequent observations, even if the peak luminosity
exceeds the observational limiting magnitudes. Therefore, here we attempt to
take this into account. Following this, we investigate how the different
observational cadences affect our results. We fix the limiting magnitude as
19.0 mag in this section for simplicity.
To proceed with this investigation, we take a simplified approach. We assume
that the peak luminosity is maintained for the duration of $t_{p}$ calculated
using equation 2. In the mock observation, we add a procedure as follows, in
order to decide whether an object is detected or not: if the duration of the
event is less than the observational cadence, we add it to the observed sample
with the probability of $p=t_{p}/t_{\mathrm{cadence}}$. If the duration of the
event is longer than the observational cadence, we consider the object is
detected and add it to the observed sample.
Figure 14 compares the 56Ni mass distribution of the detected samples for the
different observational cadences. It is seen that the cumulative distribution
shifts to higher mass, as the observational cadence is set larger. This
happens because an object with relatively low 56Ni mass tends to escape the
detection for a large observational cadence, due to its short timescale.
However, the difference seen in the 56Ni mass distributions is almost
negligible. Thus, we conclude that our results are robust to the different
assumptions about the observational cadences.
Note, however, that our assumption that a peak luminosity maintains for
$t_{p}$ is quite simplistic. Thus, although our discussion clarifies the
qualitative effect of observational cadence, we do not consider that it has a
quantitative predictive power. Also, this investigation is based on the linear
fitting equations 2. This fitting is done using the samples with 56Ni masses
above $0.03M_{\odot}$, and the validity of the linear extrapolation to the
lower 56Ni masses is not trivial (see section 7).
## 7 Discussion
In this section, we firstly discuss some caveats about our analyses conducted
above. Next, we discuss the lack of low 56Ni mass objects ($\lesssim
0.02M_{\odot}$) in the literature. Then, we discuss the high 56Ni mass
($\gtrsim 0.2M_{\odot}$) objects, which are not explained by our model.
### 7.1 Caveats
In section 4, we found that the 56Ni masses of SESNe samples collected from
the published literature suffer from notable observational bias, while those
of SNe II samples suffer from much less bias. This may be because: (1) SNe II
samples are collected at closer distances compared to SESNe samples (Fig. 5),
meaning that the former suffer less bias; or (2) the luminosity of SESNe have
higher dependence on the 56Ni mass than SNe II. Indeed, the peak luminosity of
SESNe is theoretically expected to follow $L_{\mathrm{peak}}\propto
M_{\mathrm{Ni}}$ (Arnett, 1982), while the mid-plateau luminosity of SNe II is
phenomenologically known to follow $L_{\mathrm{plateau}}\propto
M_{\mathrm{Ni}}^{0.65}$ (Pejcha & Prieto, 2015). Moreover, at the early phase,
SNe II generally have higher luminosity than the mid-plateau phase. Thus, the
detectability of SNe II is affected much less by the 56Ni mass than SESNe.
We have assumed that SESNe share the same 56Ni mass distribution as SNe II.
However, as noted in section 1, there are several indications that at least a
fraction of SESNe progenitors may be more massive than those of SNe II (e.g.
Anderson et al., 2012; Maund, 2018; Fang et al., 2019). This allows a
possibility that SESNe indeed have higher 56Ni mass than SNe II in general. If
this is the case, then the 56Ni mass distribution from the mock observation
would shift to even higher mass with the observational bias effect found in
this paper. It may then match with the LS-SESN samples better.
The results derived in section 6 are based on the fitting relation for $t_{p}$
as a function of 56Ni mass (equation 2). These 56Ni masses have been derived
using ‘Arnett-rule’, which has been claimed to overestimate the value by a
factor of few (Dessart et al., 2016; Khatami & Kasen, 2019). In appendix B, we
have shown that even if we use the 56Ni mass derived from tail luminosity for
deriving the $t_{p}-M_{\mathrm{Ni}}$ relation, our main results do not change
notably. Thus, our results in section 6 is not affected by the methods for
deriving the 56Ni mass.
For the mock observation, we assumed zero both for $A_{t}$ and $BC$. In
reality, non-zero values of $A_{t}$ and $BC$ would shift the observable
distance for a given limiting magnitude. While the effect of the $BC$ is
probably not large given that their colors are similar around the peak999The
colors of SNe II in the plateau (e.g. de Jaeger et al., 2019) are similar to
those of SESNe around the peak (e.g. Drout et al., 2011), and BC is nearly the
same for a given color between SNe II and SESNe (Lyman et al., 2016)., the
extinction may be systematically different between SNe II and SESNe; the
latter are typically associated with a more active star-forming region
(Anderson et al., 2012). If we would assume a larger value for $A_{t}$ for
SESNe, the limiting distance for SESNe will be decreased and the effect of the
observational bias investigated in this paper will become even more
substantial.
Throughout the paper, we have contrasted the SNe II to SESNe in general.
Anderson (2019) has suggested that there may be difference in the 56Ni mass
distribution even among the different types of SESNe. Especially, SNe IIb seem
to have smaller 56Ni masses than SNe Ib/Ic. One possible observational bias
that might explain this behavior is that SNe IIb can be detected more easily
than SNe Ib/Ic due to their cooling emission. However, quantitatively
investigating this possibility is beyond the scope of this paper.
### 7.2 low 56Ni mass objects
We have conducted mock observations and shown that if we assume that the
intrinsic 56Ni mass distribution of SESNe is the same as that of LS-SNe II,
the 56Ni mass distribution of SESNe in the detected samples becomes more
massive compared to the assumed intrinsic distribution; the resulting
distribution is found to be very close to the distribution of Meza-SESNe. This
indicates that even if a significant number of SESNe with low 56Ni masses
(i.e. similar to those found in the SNe II samples) would exist, we would find
a difficulty in detecting them and thus they would be significantly
underrepresented in the current literature samples.
However, some problems still remain to be solved. It is true that our mock
observations predict that the detection of SESNe is dominated by relatively
luminous objects. This would predict that there should be at least a few SESNe
with a low 56Ni mass, $M_{\mathrm{Ni}}\lesssim 0.02M_{\odot}$, especially at
small distances, considering that many SNe II with such low 56Ni masses have
been detected and that the observed fraction of SESNe to that of SNe II is
0.52 (Li et al., 2011). However, in our samples, very few SESNe have been
found with such a low 56Ni mass. Of course, it may indicate that SESNe with
such low 56Ni masses actually would not exist and the statistical difference
of 56Ni mass between SESNe and SNe II is real. However, it is also possible
that the SESNe with low 56Ni masses would not appear as canonical SESNe but
instead appear as peculiar objects, and therefore they may not be labeled as
SESNe (and thus missing in the present ‘SESN’ samples).
First, such low-56Ni mass SESNe may be related to the so called rapidly
evolving transients. As shown in Fig. 15, there is a hint that the 56Ni mass
and the timescale of SESNe are positively correlated. Thus, SESNe with lower
56Ni mass are expected to have shorter timescales. Also shown in Fig. 15 are
USSNe candidates. Taking these objects into account, the timescale of SESNe
may decrease more rapidly than our prediction (Fig. 15). Thus, our linear fit
(section 3.1) may not be valid at small 56Ni masses, and it is possible that
SESNe with a low 56Ni mass ($\lesssim 0.02M_{\odot}$) are observed as rapidly
evolving transients with timescales shorter than 10 days101010Note, that some
of the rapidly evolving transient are known to be difficult to explain by only
considering the radioactive decay model (e.g. Drout et al., 2014). However,
the properties of the rapidly evolving the transients are diverse (Pursiainen
et al., 2018) and there are many that are compatible with the radioactive
decay scenario. Indeed, a recent compilation of the rapid transients found by
the ZTF shows that this population is indeed largely contaminated by rapidly
evolving SESNe (Ho et al., 2021). Actually, SN 2017czd in our sample is an SN
IIb with very small 56Ni mass of $0.003M_{\odot}$. This object was classified
as a rapidly evolving transient (Nakaoka et al., 2019). Drout et al. (2014)
have estimated that the rate of rapidly evolving transients is 4-7 % of the
core collapse SNe rate. Since the fraction of SESNe in the core collapse SNe
is 36.6 % (Smith et al., 2011), the rapidly evolving transients occupy 11-19 %
of SESNe. This number is comparable to the fraction of SESNe with
$M_{\mathrm{Ni}}\lesssim 0.01M_{\odot}$ assuming the same 56Ni mass
distribution as LS-SNeII. Since the events with short timescales ($\lesssim
10$ days) can be easily missed, this hypothesis may be consistent with the
lack of SESNe with low 56Ni masses ($\lesssim 0.02M_{\odot}$)111111Note, that
most of the rapidly evolving transient discovered so far have 56Ni mass of
$\gtrsim 0.03M_{\odot}$ (Drout et al., 2014; Pursiainen et al., 2018; Tampo et
al., 2020). However, considering that the number of samples detected so far is
limited ($\approx 100$) (Pursiainen et al., 2018), it is natural that they are
dominated by the relatively luminous objects as we have shown in section 6.
The SESNe with low 56Ni mass may also originate from the so-called ultra-
stripped envelope SNe (USSNe). Actually, the ejecta mass and 56Ni mass of
SESNe are known to be positively correlated (Lyman et al., 2016). Thus, the
ejecta mass of the SESNe with low 56Ni mass are expected to be small. In Fig.
12, the USSNe candidates are also shown. They have 56Ni masses lower than most
of our SESNe sample. Theoretical calculations also indicate that USSNe should
synthesize quite low 56Ni of $\sim 0.01M_{\odot}$ (Suwa et al., 2015; Moriya
et al., 2017). Specifically, SN2019dge, an USSNe candidate, has an estimated
56Ni mass of 0.017$M_{\odot}$ (Yao et al., 2020), which is quite low. The rate
of such events is estimated as 2-12% of core collapse supernova: i.e.,
5.6-33.3% of SESNe (Smith et al., 2011). This number is consistent with the
fraction of SESNe with $M_{\mathrm{Ni}}\lesssim 0.02M_{\odot}$ under the
distribution we assumed. Furthermore, the timescale of USSNe candidates are
known to be short ($\lesssim 10$ days), which is much less than our prediction
(Fig. 15). Such short-timescale objects may be systematically non-detected in
the existing surveys as noted in the previous paragraph. Note, however, that
the 56Ni masses of USSNe candidates discovered so far are in general not too
low: i.e., many SNe II have been detected with 56Ni mass lower than these
objects. Therefore, these objects alone may not be sufficient to explain the
deficit of SESNe with low 56Ni mass.
The SESNe with low 56Ni masses may also have the possible link to SNe Ibn,
which are not included in our samples. SNe Ibn are characterized by He
emission lines that are considered to originate from the interaction with the
He-rich circumstellar material. These objects are shown to eject less 56Ni
than the bulk of other SESNe (Moriya & Maeda, 2016). Further, Ho et al. (2021)
recently showed that SNe Ibn with short time scale do contaminate the ZTF
rapid transient sample substantially, together with the rapid (non-
interacting) SESNe.
When deep surveys like LSST are deployed in the future, we can test our
hypotheses. In the left panel of Fig. 12, we show the detection limit for a
limiting magnitude of 25 mag, representing the single-visit depth in LSST
(Ivezić et al., 2019). We can see that basically all the SESNe with low 56Ni
masses ($\lesssim 0.02M_{\odot}$) are detected if they occur closer than
$\approx 100$ Mpc121212Although there are many objects with
$M_{\mathrm{Ni}}\lesssim 10^{-3}M_{\odot}$ in the left panel of Fig.12, they
are considered to be an artifact caused by an analytical fitting to the
distribution, considering that there are no such objects in LS-SNe II (section
7). Thus, we will be able to construct a complete sample of SESNe in the local
universe. With such a sample, we can test whether the lack of SESNe with
$\lesssim 0.02M_{\odot}$ is real or not.
Figure 14: The 56Ni mass distribution of the detected samples for the
different observational cadences. Green, blue, yellow, and red points refer to
the cadence of 0, 10, 20, 30 days, respectively. Each line is the mean of the
distributions derived from $10^{3}$ iterations. For reference, the fit to the
56Ni mass distribution of LS-SNe II is shown with a gray line.
### 7.3 High 56Ni mass objects
As previously noted (Fig. 9 and 12), our predictions from the mock
observations fail to explain the objects with high 56Ni masses ($\gtrsim
0.2M_{\odot}$). One of the possible reasons for this is that such objects may
indeed represent a different population from other SESNe which does not have a
counterpart in SNe II. Indeed, most of those objects are SNe Ic-BL, for which
the natures of the progenitor and the explosion have been proposed to be
different from canonical SESNe. Thus, our analyses may not be applicable to
these objects 131313Among those objects with high 56Ni masses ($\gtrsim
0.2M_{\odot}$), there are several objects that are not SNe Ic-BL (Fig. 12).
The possible reason why they are not classified as SNe Ic-BL is that they are
highly off-axis, considering that SNe Ic-BL often have a jet-like structure
(Valenti et al., 2008). . Another possible reason is that our assumption about
the intrinsic 56Ni mass distribution (section 5.1) may be too simplistic.
Indeed, the intrinsic 56Ni mass distribution we assumed rarely produce such a
high value of 56Ni mass ($\gtrsim 0.2M_{\odot}$). Yet another possibility is
that the different amounts of the hydrogen-rich envelope may indeed affect the
56Ni production, even though the core structure would be similar between SNe
II and SESNe; SNe II have the thick hydrogen envelope outside the He core, and
the shock is decelerated while it is propagating through the envelope. Thus,
it is expected that SNe II suffer from a fall back of the inner material,
including 56Ni, more substantially than SESNe. In this case, the 56Ni mass
distribution of SNe II we have used may provide a lower limit for SESNe
(Sawada et al. in prep).
Figure 15: The estimated time to peak from the explosion ($t_{p}$)
of SESNe using equation 2 plotted as a function of 56Ni mass. For reference,
the observational data taken from Lyman et al. (2016) and Prentice et al.
(2019) are also shown with blue points. Among them, open circles are the SNe
Ic-BL, while the filled circles are the other types of SESNe. Also, the data
of USSNe candidates are also shown with magenta-star symbols, which are taken
from the references listed at the end of the manuscript.
## 8 Conclusions
The nuclear decay of 56Ni is one of the most important power sources of
supernovae (SNe). Recent works have indicated that the 56Ni masses estimated
for SESNe are systematically higher than those estimated for SNe II. Although
this may indicate a distinct progenitor structure or explosion mechanism
between these types of SNe, the possibility remains that this may be caused by
observational biases.
By investigating the distributions of 56Ni mass and distance for the data
samples collected from the literature, we have found that SESNe samples suffer
from significant observational bias; objects with low 56Ni masses may be
systematically missed, especially at larger distances. Thus, this work has
elucidated that the observational bias must be taken into account in
discussing the different 56Ni masses between SNe II and SESNe.
We also conducted mock observations assuming that the intrinsic 56Ni mass
distribution of SESNe is the same as the 56Ni mass distribution of SNe II
collected from the literature. We have found that the 56Ni distribution for
the detected samples of SESNe becomes more massive compared to the assumed
intrinsic distribution due to the observational bias. This result may, at
least partially, explain the lack of low 56Ni mass objects in the SESNe data
sample collected from the literature. Although this result relies on the
assumption noted above, this supports that at least a part of the
systematically different 56Ni masses between these types of SNe are due to the
observational bias.
We emphasize, however, that the SESNe with low 56Ni mass ($\lesssim
0.02M_{\odot}$) are still lacking even at small distances ($\lesssim$ 30 Mpc).
This may indicate that the observational bias alone may not be sufficient to
explain all of the statistical difference between SESNe and SNe II. Another
possibility is that the SESNe with low 56Ni mass appear as either rapidly
evolving transients or ultra-stripped SNe, which are difficult to detect due
to their short timescales.
## 9 acknowledgement
R.O. acknowledges support provided by Japan Society for the Promotion of
Science (JSPS) through KAKENHI grant (19J14158). K.M. acknowledges support
provided by Japan Society for the Promotion of Science (JSPS) through KAKENHI
grant (18H05223, 20H00174, and 20H04737). This work is partly supported by the
JSPS Open Partnership Bilateral Joint Research Project between Japan and
Chile.
## Appendix A Newly added reference list for 56Ni masses
Below, the newly added references to the reference list in Anderson (2019) are
listed.
SNe II: Utrobin & Chugai (2011), Bose et al. (2018), Lisakov (2018), Singh et
al. (2018, 2019a, 2019b), Afsariardchi et al. (2019), Andrews et al. (2019),
Buta & Keel (2019), Dastidar et al. (2019), Hillier & Dessart (2019), Martinez
& Bersten (2019); Martinez et al. (2020), Meza et al. (2019), Ricks &
Dwarkadas (2019), Szalai et al. (2019), Van Dyk et al. (2019), Bostroem et al.
(2019, 2020), Gutiérrez et al. (2020a, b), Jäger et al. (2020), Sharon &
Kushnir (2020), Müller-Bravo et al. (2020), Rodríguez et al. (2020), Zhang et
al. (2020).
SESNe: Margutti et al. (2017), Ashall et al. (2019), Fremling et al. (2019),
Ho et al. (2019, 2020a, 2020b), Nakaoka et al. (2019), Taddia et al. (2018,
2019a, 2019b), Prentice et al. (2018, 2019, 2020b), Singh et al. (2019), Wang
et al. (2019), Xiang et al. (2019), Moriya et al. (2020), Sharon & Kushnir
(2020), Stritzinger et al. (2020).
Ultra-stripped envelope SESNe candidates: Kasliwal et al. (2012) De et al.
(2018), De et al. (2018), Yao et al. (2020), Prentice et al. (2020a).
## Appendix B The case when using tail luminosity for deriving 56Ni masses
The results derived in section 6 are based on the fitting relation for $t_{p}$
as a function of 56Ni mass (equation 2). The 56Ni mass used for the fit has
been derived using the ‘Arnett-rule’ (below, the ‘Arnett mass’), which has
been claimed to have an uncertainty of a factor of a few (Dessart et al.,
2015, 2016; Khatami & Kasen, 2019). An alternative method to derive a 56Ni
mass is to use a tail luminosity (below, the ‘Tail mass’). This gives a lower
limit to the 56Ni mass of SESNe. In this Appendix, we investigate how these
different methods of deriving 56Ni mass affect the results in section 6. For
that purpose, we repeat the mock observations in section 5 and 6 using the
‘Tail mass’ when fitting $t_{p}$. In Meza & Anderson (2020), both ‘Tail mass’
and ‘Arnett mass’ have been measured for the same sample. Thus, we adopt the
‘Tail mass’ from this literature.
---
Figure 16: The same figure as Fig. 1, except that we use the ‘Tail mass’ for
the fit and plots.
Fig. 16 is the same figure as Fig. 1, except that we use the ‘Tail mass’ for
the fit and plots. That is, we first use the least square method and linearly
fit to $t_{p}$ as a function of log ‘Tail mass’, as shown in the left panel of
Fig. 16. Then, from this relation thus derived, we use equation 1 and
calculate the peak luminosity as a function of 56Ni mass, propagating the
errors. Since the ‘Tail mass’ gives the lower limit to the actual value, the
peak luminosity calculated from it should be lower than the observed value.
This is indeed the case as shown in the right panel of Fig. 16.
---
Figure 17: The same as Fig. 8 but using the ‘tail mass’. Right: The same Fig.
9 but using the ‘tail mass’.
In Fig. 17, we compare the luminosity function and 56Ni mass distribution for
the cases of using the ‘Tail mass’ and the ‘Arnett mass’. The 56Ni masses for
the ‘Tail mass’ case are slightly higher than the case of the ‘Arnett mass’.
However, it is seen that the difference in the luminosity function and the
56Ni mass distribution between these two cases are nearly indistinguishable.
From these analyses, we conclude that our results in section 6 are robust to
the different methods of deriving 56Ni mass. It is true that several works
model the incomplete $\gamma$-ray trapping when using the tail luminosity of
SESNe (Afsariardchi et al., 2020; Sharon & Kushnir, 2020). Since the 56Ni
masses derived from such methods typically lie between ‘Tail mass’ and ‘Arnett
mass’ (Sharon & Kushnir, 2020), the effect of using such methods on the mock
observation is covered by our discussion on the two cases above (i.e., ‘Arnett
mass’ and ‘Tail mass’).
## References
* Afsariardchi et al. (2019) Afsariardchi, N., Moon, D.-S., Drout, M. R., et al. 2019, ApJ, 881, 22
* Afsariardchi et al. (2020) Afsariardchi, N., Drout, M. R., Khatami, D., et al. 2020, arXiv:2009.06683
* Anderson et al. (2012) Anderson, J. P., Habergham, S. M., James, P. A., et al. 2012, MNRAS, 424, 1372. doi:10.1111/j.1365-2966.2012.21324.x
* Anderson (2019) Anderson, J. P. 2019, A&A, 628, A7
* Andrews et al. (2019) Andrews, J. E., Sand, D. J., Valenti, S., et al. 2019, ApJ, 885, 43
* Arnett (1982) Arnett, W. D. 1982, ApJ, 253, 785
* Ashall et al. (2019) Ashall, C., Mazzali, P. A., Pian, E., et al. 2019, MNRAS, 487, 5824
* Barnes et al. (2018) Barnes, J., Duffell, P. C., Liu, Y., et al. 2018, ApJ, 860, 38
* Bianco et al. (2014) Bianco, F. B., Modjaz, M., Hicken, M., et al. 2014, ApJS, 213, 19
* Begelman & Sarazin (1986) Begelman, M. C. & Sarazin, C. L. 1986, ApJ, 302, L59. doi:10.1086/184637
* Benvenuto et al. (2013) Benvenuto, O. G., Bersten, M. C., & Nomoto, K. 2013, ApJ, 762, 74. doi:10.1088/0004-637X/762/2/74
* Bersten et al. (2011) Bersten, M. C., Benvenuto, O., & Hamuy, M. 2011, ApJ, 729, 61
* Bersten et al. (2014) Bersten, M. C., Benvenuto, O. G., Folatelli, G., et al. 2014, AJ, 148, 68
* Bose et al. (2018) Bose, S., Dong, S., Kochanek, C. S., et al. 2018, ApJ, 862, 107
* Bostroem et al. (2019) Bostroem, K. A., Valenti, S., Horesh, A., et al. 2019, MNRAS, 485, 5120. doi:10.1093/mnras/stz570
* Bostroem et al. (2020) Bostroem, K. A., Valenti, S., Sand, D. J., et al. 2020, ApJ, 895, 31
* Buta & Keel (2019) Buta, R. J. & Keel, W. C. 2019, MNRAS, 487, 832
* Dastidar et al. (2019) Dastidar, R., Misra, K., Valenti, S., et al. 2019, MNRAS, 490, 1605
* de Jaeger et al. (2019) de Jaeger, T., Zheng, W., Stahl, B. E., et al. 2019, MNRAS, 490, 2799. doi:10.1093/mnras/stz2714
* De et al. (2018) De, K., Kasliwal, M. M., Ofek, E. O., et al. 2018, Science, 362, 201
* De et al. (2018) De, K., Kasliwal, M. M., Cantwell, T., et al. 2018, ApJ, 866, 72. doi:10.3847/1538-4357/aadf8e
* Dessart et al. (2015) Dessart, L., Hillier, D. J., Woosley, S., et al. 2015, MNRAS, 453, 2189. doi:10.1093/mnras/stv1747
* Dessart et al. (2016) Dessart, L., Hillier, D. J., Woosley, S., et al. 2016, MNRAS, 458, 1618. doi:10.1093/mnras/stw418
* Drout et al. (2011) Drout, M. R., Soderberg, A. M., Gal-Yam, A., et al. 2011, ApJ, 741, 97. doi:10.1088/0004-637X/741/2/97
* Drout et al. (2014) Drout, M. R., Chornock, R., Soderberg, A. M., et al. 2014, ApJ, 794, 23. doi:10.1088/0004-637X/794/1/23
* Elmhamdi et al. (2003) Elmhamdi, A., Danziger, I. J., Chugai, N., et al. 2003, MNRAS, 338, 939
* Falk & Arnett (1977) Falk, S. W., & Arnett, W. D. 1977, ApJS, 33, 515
* Fang et al. (2019) Fang, Q., Maeda, K., Kuncarayakti, H., et al. 2019, Nature Astronomy, 3, 434. doi:10.1038/s41550-019-0710-6
* Filippenko (1997) Filippenko, A. V. 1997, ARA&A, 35, 309
* Folatelli et al. (2014) Folatelli, G., Bersten, M. C., Benvenuto, O. G., et al. 2014, ApJ, 793, L22
* Folatelli et al. (2015) Folatelli, G., Bersten, M. C., Kuncarayakti, H., et al. 2015, ApJ, 811, 147. doi:10.1088/0004-637X/811/2/147
* Fremling et al. (2019) Fremling, C., Ko, H., Dugas, A., et al. 2019, ApJ, 878, L5
* Gangopadhyay et al. (2020) Gangopadhyay, A., Misra, K., Sahu, D. K., et al. 2020, MNRAS, 497, 3770. doi:10.1093/mnras/staa1821
* Georgy (2012) Georgy, C. 2012, A&A, 538, L8
* Goldberg et al. (2019) Goldberg, J. A., Bildsten, L., & Paxton, B. 2019, ApJ, 879, 3
* Gräfener & Vink (2016) Gräfener, G. & Vink, J. S. 2016, MNRAS, 455, 112
* Gutiérrez et al. (2020a) Gutiérrez, C. P., Sullivan, M., Martinez, L., et al. 2020, MNRAS, 496, 95
* Gutiérrez et al. (2020b) Gutiérrez, C. P., Pastorello, A., Jerkstrand, A., et al. 2020, MNRAS, 499, 974. doi:10.1093/mnras/staa2763
* Hamuy (2003) Hamuy, M. 2003, ApJ, 582, 905
* Heger et al. (2003) Heger, A., Fryer, C. L., Woosley, S. E., et al. 2003, ApJ, 591, 288
* Hillier & Dessart (2019) Hillier, D. J. & Dessart, L. 2019, A&A, 631, A8
* Ho et al. (2019) Ho, A. Y. Q., Goldstein, D. A., Schulze, S., et al. 2019, ApJ, 887, 169
* Ho et al. (2020a) Ho, A. Y. Q., Corsi, A., Cenko, S. B., et al. 2020, ApJ, 893, 132
* Ho et al. (2020b) Ho, A. Y. Q., Kulkarni, S. R., Perley, D. A., et al. 2020, ApJ, 902, 86. doi:10.3847/1538-4357/aba630
* Ho et al. (2021) Ho, A. Y. Q., Perley, D. A., Gal-Yam, A., et al. 2021, arXiv:2105.08811
* Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111. doi:10.3847/1538-4357/ab042c
* Jäger et al. (2020) Jäger, Z., Vinkó, J., Bíró, B. I., et al. 2020, MNRAS, 496, 3725
* Kasliwal et al. (2010) Kasliwal, M. M., Kulkarni, S. R., Gal-Yam, A., et al. 2010, ApJ, 723, L98
* Kasliwal et al. (2012) Kasliwal, M. M., Kulkarni, S. R., Gal-Yam, A., et al. 2012, ApJ, 755, 161. doi:10.1088/0004-637X/755/2/161
* Khatami & Kasen (2019) Khatami, D. K. & Kasen, D. N. 2019, ApJ, 878, 56. doi:10.3847/1538-4357/ab1f09
* Kilpatrick et al. (2018) Kilpatrick, C. D., Takaro, T., Foley, R. J., et al. 2018, MNRAS, 480, 2072
* Kushnir (2015) Kushnir, D. 2015, arXiv:1506.02655
* Lazzati et al. (2012) Lazzati, D., Morsony, B. J., Blackwell, C. H., et al. 2012, ApJ, 750, 68
* Li et al. (2011) Li, W., Leaman, J., Chornock, R., et al. 2011, MNRAS, 412, 1441. doi:10.1111/j.1365-2966.2011.18160.x
* Lisakov (2018) Lisakov, S. 2018, Ph.D. Thesis
* Lyman et al. (2016) Lyman, J. D., Bersier, D., James, P. A., et al. 2016, MNRAS, 457, 328
* Maeda & Tominaga (2009) Maeda, K., & Tominaga, N. 2009, MNRAS, 394, 1317
* Margutti et al. (2017) Margutti, R., Kamble, A., Milisavljevic, D., et al. 2017, ApJ, 835, 140
* Martinez & Bersten (2019) Martinez, L. & Bersten, M. C. 2019, A&A, 629, A124
* Martinez et al. (2020) Martinez, L., Bersten, M. C., Anderson, J. P., et al. 2020, A&A, 642, A143. doi:10.1051/0004-6361/202038393
* Maund et al. (2004) Maund, J. R., Smartt, S. J., Kudritzki, R. P., et al. 2004, Nature, 427, 129
* Maund et al. (2011) Maund, J. R., Fraser, M., Ergon, M., et al. 2011, ApJ, 739, L37
* Maund (2018) Maund, J. R. 2018, MNRAS, 476, 2629. doi:10.1093/mnras/sty093
* Meza et al. (2019) Meza, N., Prieto, J. L., Clocchiatti, A., et al. 2019, A&A, 629, A57
* Meza & Anderson (2020) Meza, N. & Anderson, J. P. 2020, A&A, 641, A177. doi:10.1051/0004-6361/201937113
* Milisavljevic et al. (2017) Milisavljevic, D., Patnaude, D. J., Raymond, J. C., et al. 2017, ApJ, 846, 50. doi:10.3847/1538-4357/aa7d9f
* Moe & Di Stefano (2017) Moe, M. & Di Stefano, R. 2017, ApJS, 230, 15. doi:10.3847/1538-4365/aa6fb6
* Moriya & Maeda (2016) Moriya, T. J. & Maeda, K. 2016, ApJ, 824, 100. doi:10.3847/0004-637X/824/2/100
* Moriya et al. (2017) Moriya, T. J., Mazzali, P. A., Tominaga, N., et al. 2017, MNRAS, 466, 2085. doi:10.1093/mnras/stw3225
* Moriya et al. (2020) Moriya, T. J., Suzuki, A., Takiwaki, T., et al. 2020, MNRAS, 497, 1619
* Müller et al. (2017) Müller, T., Prieto, J. L., Pejcha, O., et al. 2017, ApJ, 841, 127
* Müller-Bravo et al. (2020) Müller-Bravo, T. E., Gutiérrez, C. P., Sullivan, M., et al. 2020, MNRAS, 497, 361
* Nakaoka et al. (2019) Nakaoka, T., Moriya, T. J., Tanaka, M., et al. 2019, ApJ, 875, 76
* Ouchi & Maeda (2017) Ouchi, R. & Maeda, K. 2017, ApJ, 840, 90
* Pejcha & Prieto (2015) Pejcha, O., & Prieto, J. L. 2015, ApJ, 806, 225
* Prentice et al. (2016) Prentice, S. J., Mazzali, P. A., Pian, E., et al. 2016, MNRAS, 458, 2973
* Prentice et al. (2018) Prentice, S. J., Ashall, C., Mazzali, P. A., et al. 2018, MNRAS, 478, 4162
* Prentice et al. (2019) Prentice, S. J., Ashall, C., James, P. A., et al. 2019, MNRAS, 485, 1559
* Prentice et al. (2020a) Prentice, S. J., Maguire, K., Flörs, A., et al. 2020, A&A, 635, A186. doi:10.1051/0004-6361/201936515
* Prentice et al. (2020b) Prentice, S. J., Maguire, K., Boian, I., et al. 2020, MNRAS, 499, 1450. doi:10.1093/mnras/staa2947
* Podsiadlowski et al. (1992) Podsiadlowski, P., Joss, P. C., & Hsu, J. J. L. 1992, ApJ, 391, 246
* Pursiainen et al. (2018) Pursiainen, M., Childress, M., Smith, M., et al. 2018, MNRAS, 481, 894. doi:10.1093/mnras/sty2309
* Reynolds et al. (2020) Reynolds, T. M., Fraser, M., Mattila, S., et al. 2020, MNRAS, 493, 1761
* Ricks & Dwarkadas (2019) Ricks, W. & Dwarkadas, V. V. 2019, ApJ, 880, 59
* Rodríguez et al. (2020) Rodríguez, Ó., Pignata, G., Anderson, J. P., et al. 2020, MNRAS, 494, 5882
* Sawada & Maeda (2019) Sawada, R., & Maeda, K. 2019, ApJ, 886, 47
* Schneider et al. (2021) Schneider, F. R. N., Podsiadlowski, P., & Müller, B. 2021, A&A, 645, A5. doi:10.1051/0004-6361/202039219
* Shivvers et al. (2016) Shivvers, I., Zheng, W. K., Mauerhan, J., et al. 2016, MNRAS, 461, 3057
* Singh et al. (2018) Singh, A., Srivastav, S., Kumar, B., et al. 2018, MNRAS, 480, 2475
* Singh et al. (2019a) Singh, A., Sahu, D. K., Anupama, G. C., et al. 2019, ApJ, 882, L15
* Singh et al. (2019b) Singh, A., Kumar, B., Moriya, T. J., et al. 2019, ApJ, 882, 68
* Singh et al. (2019) Singh, M., Misra, K., Sahu, D. K., et al. 2019, MNRAS, 485, 5438
* Sharon & Kushnir (2020) Sharon, A. & Kushnir, D. 2020, MNRAS, 496, 4517
* Smartt (2009) Smartt, S. J. 2009, ARA&A, 47, 63
* Smartt et al. (2009) Smartt, S. J., Eldridge, J. J., Crockett, R. M., et al. 2009, MNRAS, 395, 1409
* Smartt (2015) Smartt, S. J. 2015, PASA, 32, e016
* Smith et al. (2011) Smith, N., Li, W., Filippenko, A. V., et al. 2011, MNRAS, 412, 1522
* Stancliffe & Eldridge (2009) Stancliffe, R. J. & Eldridge, J. J. 2009, MNRAS, 396, 1699
* Stritzinger & Leibundgut (2005) Stritzinger, M., & Leibundgut, B. 2005, A&A, 431, 423
* Stritzinger et al. (2020) Stritzinger, M. D., Taddia, F., Holmbo, S., et al. 2020, A&A, 634, A21
* Suwa et al. (2015) Suwa, Y., Yoshida, T., Shibata, M., et al. 2015, MNRAS, 454, 3073. doi:10.1093/mnras/stv2195
* Suwa & Tominaga (2015) Suwa, Y. & Tominaga, N. 2015, MNRAS, 451, 282
* Suwa et al. (2019) Suwa, Y., Tominaga, N., & Maeda, K. 2019, MNRAS, 483, 3607
* Szalai et al. (2019) Szalai, T., Vinkó, J., Könyves-Tóth, R., et al. 2019, ApJ, 876, 19
* Taddia et al. (2015) Taddia, F., Sollerman, J., Leloudas, G., et al. 2015, A&A, 574, A60
* Taddia et al. (2018) Taddia, F., Sollerman, J., Fremling, C., et al. 2018, A&A, 609, A106
* Taddia et al. (2019a) Taddia, F., Sollerman, J., Fremling, C., et al. 2019, A&A, 621, A64
* Taddia et al. (2019b) Taddia, F., Sollerman, J., Fremling, C., et al. 2019, A&A, 621, A71
* Tampo et al. (2020) Tampo, Y., Tanaka, M., Maeda, K., et al. 2020, ApJ, 894, 27. doi:10.3847/1538-4357/ab7ccc
* Utrobin & Chugai (2011) Utrobin, V. P. & Chugai, N. N. 2011, A&A, 532, A100. doi:10.1051/0004-6361/201117137
* Valenti et al. (2008) Valenti, S., Benetti, S., Cappellaro, E., et al. 2008, MNRAS, 383, 1485. doi:10.1111/j.1365-2966.2007.12647.x
* Valenti et al. (2012) Valenti, S., Taubenberger, S., Pastorello, A., et al. 2012, ApJ, 749, L28. doi:10.1088/2041-8205/749/2/L28
* Vallely et al. (2018) Vallely, P. J., Prieto, J. L., Stanek, K. Z., et al. 2018, MNRAS, 475, 2344. doi:10.1093/mnras/stx3303
* Van Dyk et al. (2014) Van Dyk, S. D., Zheng, W., Fox, O. D., et al. 2014, AJ, 147, 37
* Van Dyk et al. (2019) Van Dyk, S. D., Zheng, W., Maund, J. R., et al. 2019, ApJ, 875, 136
* Wang et al. (2019) Wang, S.-Q., Cano, Z., Li, L., et al. 2019, ApJ, 877, 20
* Wang & Li (2020) Wang, S.-Q. & Li, L. 2020, ApJ, 900, 83. doi:10.3847/1538-4357/aba6e9
* Wheeler et al. (2015) Wheeler, J. C., Johnson, V., & Clocchiatti, A. 2015, MNRAS, 450, 1295
* Woosley et al. (1994) Woosley, S. E., Eastman, R. G., Weaver, T. A., et al. 1994, ApJ, 429, 300. doi:10.1086/174319
* Woosley et al. (2002) Woosley, S. E., Heger, A., & Weaver, T. A. 2002, Reviews of Modern Physics, 74, 1015
* Xiang et al. (2019) Xiang, D., Wang, X., Mo, J., et al. 2019, ApJ, 871, 176
* Yao et al. (2020) Yao, Y., De, K., Kasliwal, M. M., et al. 2020, ApJ, 900, 46
* Yoon et al. (2010) Yoon, S.-C., Woosley, S. E., & Langer, N. 2010, ApJ, 725, 940
* Yoon et al. (2017) Yoon, S.-C., Dessart, L., & Clocchiatti, A. 2017, ApJ, 840, 10
* Zhang et al. (2020) Zhang, J., Wang, X., József, V., et al. 2020, MNRAS, 498, 84. doi:10.1093/mnras/staa2273
|
# Generalization of Weinstein’s Morphism
Andrés Pedroza Facultad de Ciencias
Universidad de Colima
Bernal Díaz del Castillo No. 340
Colima, Col., Mexico 28045<EMAIL_ADDRESS>
###### Abstract.
We present a generalization of Weinstein’s morphism defined on
$\pi_{2k-1}(\textup{Ham}(M,\omega))$. We use this morphism to show that for
$n\geq 2$ the Lie group $SU(2)$ induces an element in
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ of infinite
order.
###### Key words and phrases:
Weinstein’s morphism.
###### 1991 Mathematics Subject Classification:
Primary: 57S05, 53D35 Secondary: 57R17, 53D05
The author was supported by CONACYT-CB-2017-2018-A1-S-8830 grant
## 1\. Introduction
Let $(M,\omega)$ be a closed symplectic manifold of dimension $2n$. For $1\leq
k\leq n$ define the group $\mathcal{P}_{2k}(M,\omega)$ as the image of the
pairing $\langle\omega^{k},\cdot\rangle\colon
H_{2k}(M;\mathbb{Z})\to\mathbb{R}.$ Notice that for $k=1$,
$\mathcal{P}_{2}(M,\omega)$ is the well-known period group of the symplectic
manifold $(M,\omega)$. In [4], A. Weinstein defined a morphism
$\displaystyle\mathcal{A}\colon\pi_{1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2}(M,\omega)$
based on the action functional of a Hamiltonian function
$H:M\times[0,1]\to\mathbb{R}$ associated with the loop of Hamiltonian
diffeomorphisms. Inspired by Weinstein’s morphism, in this article we define a
group morphism for each $k\in\\{2,\ldots,n\\}$
$\displaystyle\mathcal{A}\colon\pi_{2k-1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega).$
In broad terms, the value of the morphism $\mathcal{A}$ on
$[\psi_{\underline{t}}]\in\pi_{2k-1}(\textup{Ham}(M,\omega))$, is the average
over $(M,\omega)$ of the $\omega^{k}$-area of $2k$-balls whose boundary is the
topological $(2k-1)$-sphere $\\{\psi_{\underline{t}}(p)|{\underline{t}}\in
S^{2k-1}\\}$ as $p$ varies in $(M,\omega)$.
As an application, for $n\geq 2$ we compute $\mathcal{A}$ for a particular
element in $\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$. Such
element is induced by the Lie group $SU(2)$, which topologically is a
3-sphere. For every $A\in SU(2)$ we define a Hamiltonian diffeomorphism
$\psi_{A}$ on $(\mathbb{C}P^{n},\omega_{\textup{FS}})$ and using the fact that
there is a symplectic embedding of the open unit $2n$-ball in the standard
symplectic Euclidean space into $(\mathbb{C}P^{n},\omega_{\textup{FS}})$ with
dense image, we compute the value of $\mathcal{A}([\\{\psi_{A}\\}]_{A\in
SU(2)})$ in Propositions 3.4 and 3.5. The description of such symplectic
embedding appears in the Appendix of [3]. Hence, the computation is performed
on the standard symplectic Euclidean space. Such computations give rise to the
main result of this article.
###### Theorem 1.1.
Let $n\geq 2$ and consider the $3$-sphere of Hamiltonian diffeomorphisms
$\\{\psi_{A}\\}_{A\in SU(2)}$ on $(\mathbb{C}P^{n},\omega_{\textup{FS}}).$
Then $[\\{\psi_{A}\\}_{A\in
SU(2)}]\in\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ is an
element of infinite order.
In [1], M. Gromov showed that
$\textup{Ham}(\mathbb{C}P^{2},\omega_{\textup{FS}})$ has the homotopy type of
$PSU(3)$, thus
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{2},\omega_{\textup{FS}}))\simeq\mathbb{Z}$.
Therefore, the element of Theorem 1.1 is a generator of
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{2},\omega_{\textup{FS}}))$.
The fact that $\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$
contains a subgroup isomorphic to $\mathbb{Z}$, for $n\geq 4$, in not new. In
[2, Theorem 1.1], J. Kȩdra and D. McDuff showed that if $(M,\omega)$ admits a
Hamiltonian circle action that is contractible in $\textup{Ham}(M,\omega)$
then $\pi_{3}(\textup{Ham}(M,\omega))$ contains an element of infinite order.
This result applies to $(\mathbb{C}P^{n},\omega_{\textup{FS}})$. The
difference with our approach is that we present a generalization of
Weinstein’s morphism that enables us to present an explicit generator of the
element of infinite order.
## 2\. Definition of the morphism $\mathcal{A}$
Fix $1\leq k\leq n$ and assume that the symplectic manifold $(M,\omega)$ is
such that $\pi_{2k-1}(M)$ is trivial. Let $D^{2k}$ stand for the
$2k$-dimensional closed ball. Then for any $\gamma\in\Omega_{2k-1}(M)$ there
exists a smooth map $u_{\gamma}\colon D^{2k}\to M$ that maps the boundary of
$D^{2k}$ onto $\gamma$. Define the map
$A\colon\Omega_{2k-1}(M)\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$ as
(1) $\displaystyle A(\gamma):=\int_{D^{2k}}u_{\gamma}^{*}(\omega^{k})$
where $u_{\gamma}\colon D^{2k}\to M$ is as above. Since $A$ takes values in
$\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$, it follows that $A(\gamma)$ is
independent of the choice of $u_{\gamma}$.
For $\psi_{\underline{t}}\in\Omega_{2k-1}(\textup{Ham}(M,\omega))$ and $p\in
M$ let $\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p):=\\{\psi_{\underline{t}}(p)|{\underline{t}}\in S^{2k-1}\\}$ the
trace of $p$ with respect to $\psi_{\underline{t}}$. Therefore,
$\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in S^{2k-1}},p)\subset M$
is a topological $(2k-1)$-sphere and hence it induces the element
$\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p)\in\Omega_{2k-1}(M)$. Define
$\mathcal{A}:\Omega_{2k-1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$
as the average of $A(\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p))$ over $(M,\omega)$,
(2)
$\displaystyle\mathcal{A}(\psi_{\underline{t}}):=\frac{1}{\textup{Vol}(M,\omega^{n}/n!)}\int_{M}A(\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p)))\frac{\omega^{n}}{n!}.$
Next, we show that homotopic elements of
$\Omega_{2k-1}(\textup{Ham}(M,\omega))$ have the same value under
$\mathcal{A}$.
###### Lemma 2.2.
If
$\psi^{(0)}_{\underline{t}},\psi^{(1)}_{\underline{t}}\in\Omega_{2k-1}(\textup{Ham}(M,\omega))$
are such that $[\psi^{(0)}_{\underline{t}}]=[\psi^{(1)}_{\underline{t}}]$ in
$\pi_{2k-1}(\textup{Ham}(M,\omega))$, then
$\mathcal{A}(\psi^{(0)}_{\underline{t}})=\mathcal{A}(\psi^{(1)}_{\underline{t}})$.
###### Proof.
Let ${\bf\Psi}\colon[0,1]\times S^{2k-1}\to\textup{Ham}(M,\omega)$ be a
homotopy between $\psi^{(0)}_{\underline{t}}$ and
$\psi^{(1)}_{\underline{t}}$. For $p\in M$ denote by $u_{0},u_{1}\colon
D^{2k}\to M$ be the caping maps of the $(2k-1)$-spheres
$\psi^{(0)}_{\underline{t}}(p)$ and $\psi^{(1)}_{\underline{t}}(p)$
respectively.
Evaluating at $p\in M$ the homotopy ${\bf\Psi}$ induces a homotopy in $M$,
${\bf\Psi}_{p}\colon[0,1]\times S^{2k-1}\to M$, between the $(2k-1)$-spheres
$\psi^{(0)}_{\underline{t}}(p)$ and $\psi^{(1)}_{\underline{t}}(p)$.
Therefore, the image of $u_{0},\overline{u_{1}}$ and ${\bf\Psi}_{p}$
represents a $2k$-sphere in $M$ and hence
$\displaystyle\int_{D^{2k}}u_{0}^{*}(\omega^{k})+\int_{[0,1]\times
S^{2k-1}}{\bf\Psi}_{p}^{*}(\omega^{k})+\int_{D^{2k}}\overline{u_{1}}^{*}(\omega^{k})=0$
in $\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$. Here $\overline{u_{1}}$ stands for
the same map as $u_{1}$ where the domain $D^{2k}$ has the opposite
orientation. Note that the integral of ${\bf\Psi}_{p}^{*}(\omega^{k})$ is
zero, since is the trace of a $(2k-1)$-sphere under Hamiltonian
diffeomorphisms. Therefore,
${A}(\gamma(\\{\psi^{(0)}_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p))={A}(\gamma(\\{\psi^{(1)}_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p))$ for any $p\in M$ and as a consequence
$\mathcal{A}(\psi^{(0)}_{\underline{t}})=\mathcal{A}(\psi^{(1)}_{\underline{t}}).$
∎
Hence there is an induced map
$\displaystyle\pi_{2k-1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$
that we still denote by $\mathcal{A}$. Furthermore, from the definition of $A$
in Equation (1) we have that $-A(\gamma_{1})=A(-\gamma_{1})$ and
$A(\gamma_{1}+\gamma_{2})=A(\gamma_{1})+A(\gamma_{2})$ for
$\gamma_{1},\gamma_{2}\in\Omega_{2k-1}(M)$. It then follows that $\mathcal{A}$
is a group morphism. Thus, we have a generalization of Weinstein’s morphism as
follows.
Let $(M,\omega)$ be a closed symplectic $(2n)$-manifold, fix
$k\in\\{2,\ldots,n\\}$ and assume that the group $\pi_{2k-1}(M)$ is trivial,
then define Weinstein’s morphism,
$\mathcal{A}\colon\pi_{2k-1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega),$
by
$\displaystyle\mathcal{A}(\psi_{\underline{t}}):=\frac{1}{\textup{Vol}(M,\omega^{n}/n!)}\int_{M}A(\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p)))\frac{\omega^{n}}{n!}.$
for $\psi_{\underline{t}}\in\pi_{2k-1}(\textup{Ham}(M,\omega))$.
An alternative definition to
$\displaystyle\mathcal{A}\colon\pi_{1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2}(M,\omega)$
is presented in [4]. Namely,
$\displaystyle\mathcal{A}(\psi^{H})=\int_{D^{2}}u_{x_{0}}^{*}(\omega)+\int_{0}^{1}H_{t}(x_{0})dt$
where $H_{t}$ is a normalized Hamiltonian generating the loop $\psi^{H}$,
$x_{0}\in M$ is a base point and $u_{x_{0}}:D^{2}\to M$ is a smooth function
that maps the boundary to $\\{\psi_{t}^{H}(x_{0})\\}$. This alternative
definition for $k=1$ does not admit a direct generalization for $k>1$ as the
one presented here.
## 3\. $\pi_{3}$ of the group of Hamiltonian diffeomorphisms of the complex
projective plane
Consider $(\mathbb{C}P^{n},\omega_{\textup{FS}})$ where the symplectic form is
normalized so that the symplectic area of the line is $\pi$. Thus, there is a
symplectic embedding
$j\colon(B^{2n}(1),\omega_{0})\to(\mathbb{C}P^{n},\omega_{\textup{FS}})$ of
the open ball $(B^{2n}(1),\omega_{0})$ into the complex projective space is
given by
$\displaystyle(z_{1},\ldots,z_{n})\to\left[z_{1}:\cdots:z_{n}:\sqrt{1-\sum_{j=1}^{n}|z_{j}|^{2}}\right].$
For a proof of this claim see for instance the Appendix in [3]. Furthermore,
the complement of the embedding corresponds to the hypersurface
$\mathbb{C}P^{n-1}$. Using Weinstein’s morphism $\mathcal{A}$ defined in the
previous section, we will exhibit a nontrivial element in
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ for $n\geq 2.$
Recall that through the Hopf fibration, $\pi_{3}(\mathbb{C}P^{n})$ is trivial
for $n\geq 2$. Also, since
$\langle\omega^{2}_{\textup{FS}},\mathbb{C}P^{2}\rangle=\pi^{2}/2$ and
$[\mathbb{C}P^{2}]$ is the only generator of
$H_{4}(\mathbb{C}P^{n};\mathbb{Z})$ then
$\mathcal{P}_{4}(\mathbb{C}P^{n},\omega_{\textup{FS}})=\langle\pi^{2}/2\rangle$
for $n\geq 2$.
Next, the $3$-sphere of Hamiltonian diffeomorphisms on
$(\mathbb{C}P^{n},\omega_{\textup{FS}})$ that we will consider corresponds to
the Lie group $SU(2)$. For, $A\in SU(2)$ define the map $\psi_{A}$ on
$(\mathbb{C}P^{n},\omega_{\textup{FS}})$ as
(3)
$\displaystyle\psi_{A}([z_{1}:\cdots:z_{n+1}])=[A(z_{1},z_{2}):z_{3}:\cdots:z_{n+1}]$
Here, $A(z_{1},z_{2})$ corresponds to the first two homogeneous coordinates of
$\psi_{A}([z_{1}:\cdots:z_{n+1}])\in\mathbb{C}P^{n}$. Notice that since $n\geq
2$, in the definition of $\psi_{A}$ in Equation (3) we consider the group
$SU(2)$ instead of the group $PSU(2)$. Also, note $\psi_{A}$ is well-defined
and a Hamiltonian diffeomorphism. Therefore $\\{\psi_{A}\\}_{A\in SU(2)}$
induces an element in
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ for each $n\geq
2.$
Next, take into account the action of $SU(2)$ on the first two entries of
$\mathbb{C}^{n}$;
$\displaystyle
A.(z_{1},\ldots,z_{n})\mapsto(A(z_{1},z_{2}),z_{3},\ldots,z_{n}).$
Thus, the unit ball $B^{2n}(1)$ is an invariant set. Furthermore, with this
action the symplectic embedding
$j\colon(B^{2n}(1),\omega_{0})\to(\mathbb{C}P^{n},\omega_{\textup{FS}})$ is
$SU(2)$-equivariant. Observe that if ${\bf z}=(z_{1},\ldots,z_{n})\in
B^{2n}(1)$ is such that $(z_{1},z_{2})\neq(0,0)$, then $\\{A.{\bf z}:A\in
SU(2)\\}\subset B^{2n}(1)$ is a $3$-sphere. Also, the volume of the 4-ball in
the $(z_{1},z_{2})$-plane that is bounded by $\\{A.{\bf z}:A\in
SU(2)\\}\subset B^{2n}(1)$ is
(4) $\displaystyle\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}.$
In the case when $(z_{1},z_{2})=(0,0)$, ${\bf z}$ is fixed by the action of
$SU(2)$. Since $j$ is a $SU(2)$-equivariant embedding, the same statements
hold for the points $j({\bf z})\in\mathbb{C}P^{n}$.
As above, for $[{\bf z}]\in\mathbb{C}P^{n}$ denote by
$\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf z}])$ the trace of $[{\bf z}]$ under
$\\{\psi_{A}\\}_{A\in SU(2)}$.
###### Lemma 3.3.
If $n\geq 2$ and $[{\bf z}]=[z_{1}:\cdots:z_{n+1}]\in\mathbb{C}P^{n}$ is such
that $z_{n+1}\neq 0$, then
$A(\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf
z}]))=\frac{\pi^{2}}{2}\left(\frac{\left|\frac{z_{1}}{z_{n+1}}\right|^{2}+\left|\frac{z_{2}}{z_{n+1}}\right|^{2}}{1+\sum_{j=1}^{n}\left|\frac{z_{j}}{z_{n+1}}\right|^{2}}\right)^{2}.$
###### Proof.
We perform the computation in $(B^{2n}(1),\omega_{0})$, then using the
symplectic embedding $j$ we obtain the assertion of the lemma.
For ${\bf z}\in B^{2n}(1)$ let $\gamma(SU(2),{\bf z})$ be the orbit of ${\bf
z}$ under the action of $SU(2)$ on $\mathbb{C}^{n}$. If
$(z_{1},z_{2})\neq(0,0)$, then $\gamma(SU(2),{\bf z})$ is a 3-sphere inside
$B^{2n}(1)$. Define $u_{\gamma(SU(2),{\bf z})}:D^{4}\to B^{2n}(1)$ as the
rescaling of the unit closed 4-ball to the closed 4-ball of radius
$\sqrt{|z_{1}|^{2}+|z_{2}|^{2}}$ followed by the inclusion into of the
$(z_{1},z_{2})$-plane into $B^{2n}(1)$. Thus $u_{\gamma(SU(2),{\bf
z})}:D^{4}\to B^{2n}(1)$ is such that the boundary is mapped to the 3-sphere
${\gamma(SU(2),{\bf z})}$. Therefore
$A({\gamma(SU(2),{\bf z})})=\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}.$
Notice that in the case when $(z_{1},z_{2})=(0,0)$, ${\gamma(SU(2),{\bf z})}$
consists of a single point and in this case the constant map
$u_{\gamma(SU(2),{\bf z})}:D^{4}\to B^{2n}(1)$ gives $A({\gamma(SU(2),{\bf
z})})=0$.
Since the embedding
$j\colon(B^{2n}(1),\omega_{0})\to(\mathbb{C}P^{n},\omega_{\textup{FS}})$ is
$SU(2)$-equivariant it follows that $\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf
z}])=j({\gamma(SU(2),{\bf z})})$. Hence the boundary of the image of the map
$j\circ u_{\gamma(SU(2),{\bf z})}:D^{4}\to\mathbb{C}P^{n}$ is precisely
$\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf z}]).$ Finally, since $j$ is a
symplectic embedding the claim of the lemma follows. ∎
We have computed $A(\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf z}]))$ for every
$[{\bf z}]\in\mathbb{C}P^{n}$ such that $z_{n+1}\neq 0.$ Thus we are in a
position to compute $\mathcal{A}([\psi_{A}])$. Once again we used the
symplectic embedding $j$ to perform the average on $(B^{2n}(1),\omega_{0})$
instead that on $(\mathbb{C}P^{n},\omega_{\textup{FS}})$. Henceforth,
$\displaystyle\mathcal{A}([\psi_{A}])$ $\displaystyle=$
$\displaystyle\frac{1}{\textup{Vol}({\mathbb{C}P^{n}},\omega_{\textup{FS}}^{n}/n!)}\int_{\mathbb{C}P^{n}}A(\psi_{A}([{\bf
z}]))\frac{\omega_{\textup{FS}}^{n}}{n!}$ $\displaystyle=$
$\displaystyle\frac{1}{\textup{Vol}(B^{2n}(1),\omega_{0}^{n}/n!)}\int_{B^{2n}(1)}A(\gamma_{{\bf
z}})\frac{\omega_{0}^{n}}{n!}$ $\displaystyle=$
$\displaystyle\frac{1}{\textup{Vol}(B^{2n}(1),\omega_{0}^{n}/n!)}\int_{B^{2n}(1)}\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}\frac{\omega_{0}^{n}}{n!}.$
By the nature of the expression $(|z_{1}|^{2}+|z_{2}|^{2})^{2}$, we must
consider two cases separately when integrating over $B^{2n}(1)$. Namely, the
cases when $n=2$ and $n\geq 3$.
###### Proposition 3.4.
Let $n\geq 3$ and $\\{\psi_{A}\\}$ be the $3$-sphere of Hamiltonian
diffeomorphisms on $(\mathbb{C}P^{n},\omega_{\textup{FS}})$ defined in (3).
Then,
$\displaystyle\mathcal{A}([\psi_{A}])=\frac{96}{n^{2}(n^{3}-4n^{2}-4n+16)}\pi^{n-1}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$
###### Proof.
The image of the symplectic embedding
$j\colon(B^{2n}(1),\omega_{0})\to(\mathbb{C}P^{n},\omega_{\textup{FS}})$ is
dense. Thus we compute $\mathcal{A}$ on $(B^{2n}(1),\omega_{0})$ instead of on
$(\mathbb{C}P^{n},\omega_{\textup{FS}})$. Recall that the volume of
$(B^{2n}(1),\omega_{0})$ is $\pi^{n}/n!$. Therefore,
$\displaystyle\mathcal{A}([\psi_{A}])$ $\displaystyle=$
$\displaystyle\frac{n!}{\pi^{n}}\int_{B^{2n}(1)}\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}\
\ \frac{\omega_{0}^{n}}{n!}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{n-2}}\cdot\frac{96\pi^{2}}{n^{2}(n^{3}-4n^{2}-4n+16)}\cdot\pi^{2n-6}\cdot
2\pi$ $\displaystyle=$
$\displaystyle\frac{96}{n^{2}(n^{3}-4n^{2}-4n+16)}\pi^{n-1}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$
∎
For the case $n=2$ we proceed similarly. Thus,
$\displaystyle\mathcal{A}([\psi_{A}])$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{2}}\int_{B^{4}(1)}\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}\
\ \frac{\omega_{0}^{2}}{2}$ $\displaystyle=$
$\displaystyle\int_{B^{4}(1)}(|z_{1}|^{2}+|z_{2}|^{2})^{2}\ \
\frac{\omega_{0}^{2}}{2}$ $\displaystyle=$
$\displaystyle\frac{97\pi^{3}+8\pi^{2}}{256}=\frac{97}{256}\pi^{3}+\frac{1}{32}\pi^{2}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$
###### Proposition 3.5.
Let $\\{\psi_{A}\\}$ be the $3$-sphere of Hamiltonian diffeomorphisms on
$(\mathbb{C}P^{2},\omega_{\textup{FS}})$ defined in (3). Then,
$\displaystyle\mathcal{A}([\psi_{A}])=\frac{97}{256}\pi^{3}+\frac{1}{32}\pi^{2}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$
The proof of the main result is now straightforward.
###### Proof of Theorem 1.1.
Recall that
$\mathcal{P}_{4}(\mathbb{C}P^{n},\omega_{\textup{FS}})=\langle\pi^{2}/2\rangle$
for all $n\geq 2$. Thus, only the case $n=3$ deserves attention since in this
case the expression $\mathcal{A}([\psi_{A}])$ is a rational multiple of
$\pi^{2}$. For $n=3$, Proposition 3.4 takes the form
$\displaystyle\mathcal{A}([\psi_{A}])=-\frac{32}{15}\pi^{2}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$
Henceforth,
$[\psi_{A}]\in\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ has
infinite order for all $n\geq 2$. ∎
## References
* [1] Gromov, M. Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 2 (1985), 307–347.
* [2] Kȩdra, J., and McDuff, D. Homotopy properties of Hamiltonian group actions. Geom. Topol. 9 (2005), 121–162.
* [3] McDuff, D., and Polterovich, L. Symplectic packings and algebraic geometry. Invent. Math. 115, 3 (1994), 405–434. With an appendix by Yael Karshon.
* [4] Weinstein, A. Cohomology of symplectomorphism groups and critical values of Hamiltonians. Math. Z. 201, 1 (1989), 75–82.
|
# Scalable variable selection for two-view learning tasks with projection
operators
Sandor Szedmak
Department of Computer Science
Aalto University
Espoo, Finland
<EMAIL_ADDRESS>
&Riikka Huusari
Department of Computer Science
Aalto University
Espoo, Finland
<EMAIL_ADDRESS>
&Tat Hong Duong Le
Department of Computer Science
Aalto University
Espoo, Finland
<EMAIL_ADDRESS>
&Juho Rousu
Department of Computer Science
Aalto University
Espoo, Finland
<EMAIL_ADDRESS>
###### Abstract
In this paper we propose a novel variable selection method for two-view
settings, or for vector-valued supervised learning problems. Our framework is
able to handle extremely large scale selection tasks, where number of data
samples could be even millions. In a nutshell, our method performs variable
selection by iteratively selecting variables that are highly correlated with
the output variables, but which are not correlated with the previously chosen
variables. To measure the correlation, our method uses the concept of
projection operators and their algebra. With the projection operators the
relationship, correlation, between sets of input and output variables can also
be expressed by kernel functions, thus nonlinear correlation models can be
exploited as well. We experimentally validate our approach, showing on both
synthetic and real data its scalability and the relevance of the selected
features.
Keywords: Supervised variable selection, vector-valued learning, projection-
valued measure, reproducing kernel Hilbert space
## 1 Introduction
Vector-valued, or more generally structured output learning tasks arising from
various domains have attracted much research attention in recent years
[Micchelli and Pontil, 2005, Deshwal et al., 2019, Brogat-Motte et al., 2022].
For both supervised but also unsupervised learning approaches, multi-view data
has been of interest [Hotelling, 1936, Xu et al., 2013, Minh et al., 2016a].
Despite many successful approaches for various multi-view and vector-valued
learning settings, including interpretability to these models has received
less attention. While there are various feature selection and dimensionality
reduction methods either for scalar-valued learning tasks, or unsupervised
methods for data represented in a single view [Zebari et al., 2020, Li et al.,
2017, Anette and Nokto, 2018, Bommert et al., 2020], there is scarcity of
methods suitable for when data is represented in two views, or arises from a
vector-valued learning task. From the point of view of interpretability,
especially feature selection methods are advantageous over dimensionality
reduction since the relevant features are directly obtained as a result and
not given only in (linear) combinations.
Recently, some feature selection methods have been proposed for structured
output learning tasks. [Brouard et al., 2022] proposed kernel-based non-linear
feature selection model relying on sparsity regularization. Another supervised
feature selection approach based on kernel methods is introduced in [Song et
al., 2012], this one relying instead on forward- and backward-selection
ideology. In addition, [Jordan et al., 2021] discusses feature selection in
conjunction with kernel-based models, obtaining sparsity implicitly via loss
function without explicit regularization term. An alternative, spline based,
approach to the non-linear feature selection is proposed by [Boyd et al.,
2018]. These methods, relying on the kernel evaluations between data samples
for both inputs and outputs, tend not to scale very well to large sample
sizes.
In this paper, we introduce a novel variable selection approach for vector-
valued, or two-view learning tasks, including CCA. Our method is based on
efficient iterative computation of projections of input variables to the
vector space intersection between the space spanned by the output variables
and the one of the previously selected input variables. In this space, the
input variables are then selected by a correlation-based criterion. Going one
step further, we also exploit a kernel-based representation of the variables,
allowing us to capture complex, nonlinear relationships. Here, we consider the
kernelised representation of the variables instead of data samples – in
essence, we model the co-variance on the features in the Hilbert space induced
by the kernel. Notably, both input and output features are captured with the
same kernelisation. This is in stark contrast to other proposed kernel-based
feature selection approaches in literature, where separate kernels are used
for data samples in input and output spaces [Brouard et al., 2022, Song et
al., 2012, Jordan et al., 2021]. We can more readily draw comparisons to
canonical correlation analysis (CCA) and its kernelized version, where the
correlations are computed between two sets of variables instead of pairs of
individual ones [Bie et al., 2005].
For many approaches, scalability in feature selection can be a challenge for
when the data dimensionality is extremely large. Some supervised linear
feature selection models adapted to this setting are proposed in [Fan et al.,
2009, Aghazadeh et al., 2018, Valcárcel et al., 2022]. We note, that all these
methods are for the supervised setting, but with scalar-valued output
variables. While scalability w.r.t the feature dimensionality is often
considered due to motivations arising from fat data, the scalability to large
sample sizes is less focused on. Traditionally, kernelized algorithms, while
powerful, are very poorly scalable due to the dependence to the kernel matrix,
especially if its inverse is required. Contrary to the usual, by leveraging
the recursive formulation of our algorithm and a trick with singular value
decomposition on the variable representation, our approach is extremely
scalable to large sample sizes - which we also demonstrate experimentally in
Section 5.
To summarize, our main contributions in this paper are as follows:
* •
we propose projective selection (ProjSe) algorithm, a novel approach for
variable selection for vector-valued or two-view learning problems that is
based on projection operators. In ProjSe the result of the feature selection
only depends on the subspace spanned by the outputs, not on the specific
values (invariance).
* •
our proposed iterative method offers high scalability even for the kernelised
formulation capturing non-linearities in the data, due to a trick with
singular value decomposition applied to the feature representation.
* •
we experimentally validate the proposed approach, showing both relevance of
the selected features and the efficiency of the algorithm.
Table 1: Some of the frequently used notation in this paper. $\mathcal{H}$ | is a Hilbert space - unless otherwise noted, it has finite dimension $d$, in which case $\mathcal{H}=\mathbb{R}^{d}$.
---|---
$\oplus$ | denotes the direct sum of subspaces
$\mathbf{I}$ | is the identity operator acting on $\mathcal{H}$.
$\mathcal{L}$ | is an arbitrary subspace of $\mathcal{H}$.
$\mathcal{L}_{\mathbf{X}}$ | is a subspace of $\mathcal{H}$, spanned by the columns of matrix $\mathbf{X}$.
$\mathcal{L}^{\perp}$ | is a subspace of $\mathcal{H}$, the orthogonal complement of $\mathcal{L}$.
$\mathbf{P}_{\mathcal{L}}$ | is an orthogonal projection operator into subspace $\mathcal{L}$, $\mathbf{P}_{\mathcal{L}}:\mathcal{H}\rightarrow\mathcal{L}$
$\mathcal{L}_{\mathbf{P}}$ | is the subspace corresponding to the projection operator $\mathbf{P}$.
$\mathbf{P}_{\mathcal{L}^{\perp}}$ | is an orthogonal projection operator into the orthogonal complement of subspace $\mathcal{L}$.
$\mathbf{P}_{\mathbf{X}}$ | is an orthogonal projection operator into the subspace of $\mathcal{H}$, spanned by the columns of matrix $\mathbf{X}$. It is the same as $\mathbf{P}_{\mathcal{L}_{X}}$.
$\mathbf{P}_{\mathbf{X}^{\perp}}$ | is an orthogonal projection operator into the subspace of $\mathcal{H}$ orthogonal to the subspace spanned by the columns of matrix $\mathbf{X}$. It is the same as $\mathbf{P}_{\mathcal{L}_{X}^{\perp}}$.
$\mathbf{A}^{+}$ | denotes the Moore-Penrose inverse of matrix $\mathbf{A}$.
$[n]$ | is a short hand notation for the set $\\{1,\dots,n\\}.$
$\mathbf{A}\circ\mathbf{B}$ | denotes pointwise(Hadamard) product of matrices $\mathbf{A}$ and $\mathbf{B}$.
$\mathbf{A}^{\circ n}$ | is the pointwise power of matrix $\mathbf{A}$.
$\mathbf{A}[:,\mathcal{I}]$ | selects the subset of columns of matrix $\mathbf{A}$ with indices in set $\mathcal{I}$.
The paper is organised as follows. In the next section we give overview of the
method, before moving to more rigorous treatment in Section 3. There we give a
brief introduction to projection operators and their matrix representation,
and discuss the key detail of our approach, expressing the projector into
intersection. We then move on to describing our large-scale kernelized
adaptation of the algorithm in Section 4. We validate our approach
experimentally in Section 5 before concluding.
## 2 Method overview
Figure 1: Illustration of the main steps of the algorithm
Our algorithm is designed to perform variable selection when there are
multiple dependent variables of interest.We denote the matrix containing the
data from which the variables are selected as
$\mathbf{X}\in\mathbb{R}^{m\times n_{x}}$, and the reference data as
$\mathbf{Y}\in\mathbb{R}^{m\times n_{y}}$ – the sample size is $m$, and the
number of features/variables are $n_{x}$ and $n_{y}$ (see other frequently
used notation in Table 1). Here $\mathbf{X}$ and $\mathbf{Y}$ could also
correspond to vector-valued inputs and outputs of some supervised learning
task. Our method is based on defining correlation via projection operators: we
define the correlation between a variable vector $\mathbf{x}\in\mathbb{R}^{m}$
(a column vector from $\mathbf{X}$ containing the values of a single input
variable for all data points) and a set of variables in columns of matrix
$\mathbf{Y}$, as
$\text{corr}(\mathbf{x},\mathbf{Y})=\left\|\mathbf{P}_{\mathcal{L}_{Y}}\frac{\mathbf{x}}{||\mathbf{x}||}\right\|=\left\langle\mathbf{P}_{\mathcal{L}_{Y}}\frac{\mathbf{x}}{||\mathbf{x}||},\mathbf{P}_{\mathcal{L}_{Y}}\frac{\mathbf{x}}{||\mathbf{x}||}\right\rangle^{\frac{1}{2}}=\left\langle\frac{\mathbf{x}}{||\mathbf{x}||},\mathbf{P}_{\mathcal{L}_{Y}}\frac{\mathbf{x}}{||\mathbf{x}||}\right\rangle^{\frac{1}{2}}$
(1)
where $\mathbf{P}_{\mathcal{L}_{Y}}$ (or $\mathbf{P}_{Y}$ in shorthand) is the
orthogonal projection operator into a subspace $\mathcal{L}_{Y}$ spanned by
the columns of $\mathbf{Y}$. This definition is motivated by the concept of
Projection-Valued Measure which plays a significant role in quantum mechanics
theory (see for example [Nielsen and Chuang, 2000]). Our approach selects
variables from input data $\mathbf{X}$ iteratively, such that correlation
between the selected variable and the outputs is high, while correlation to
the previously selected variables is low.
###### Remark 1.
For sake of simplicity, we assume that for all $\mathbf{x}\in\mathbb{R}^{m}$,
$\|\mathbf{x}\|=1$.
Our variable selection algorithm, ProjSe, is illustrated in Figure 1. The
first set of variables is is chosen simply to maximize the projection onto the
subspace spanned by columns of $\mathbf{Y}$, $\mathcal{L}_{Y}$. This is
illustrated with $\mathbf{x}_{1}$, which is projected with $\mathbf{P}_{Y}$ as
$\mathbf{P}_{Y}\mathbf{x}_{1}$. The second set of features chosen,
$\mathbf{x}_{2}$ in the figure, is projected into the intersection of
$\mathcal{L}_{Y}$, and the orthogonal complement of the chosen feature
$\mathbf{x}_{1}$, $\mathcal{L}_{\mathbf{x_{1}}^{\perp}}$. At this step, the
correlation is measured with the projection operator
$\mathbf{P}_{\mathcal{L}_{Y}\cap\mathcal{L}_{\mathbf{x}_{1}^{\perp}}}$.
Interestingly, it turns out that this projected feature,
$\mathbf{P}_{\mathbf{Y}\cap\mathbf{x}_{1}^{\perp}}\mathbf{x}_{2}$, lies also
in the intersection of $\mathcal{L}_{Y}$ and
$\mathcal{L}_{(\mathbf{P}_{Y}\mathbf{x}_{1})^{\perp}}$. This observation paves
the way for building our efficient, recursive algorithm for the feature
selection with projection operators.
The pseudo-code of the basic form of our proposed variable selection by
projection, ProjSe, algorithm is displayed in Figure 2. The approach is fully
deterministic without randomness, and thus practical to apply. Similarly to
CCA, our variable selection algorithm in a sense joins the variable spaces of
the inputs and outputs – both of them are considered in the same space. At the
same time, in order to our selection approach to work,
$\mathcal{L}_{\mathbf{X}}$ should not be fully orthogonal
$\mathcal{L}_{\mathbf{Y}}$. Additionally, due to the properties of the
projection operators, our approach promotes invariance: the selected
explanatory variables (input features) depend only on the subspace spanned by
the response variables (output features), and are independent on any
transformation on the response variables that would span the same subspace.
These transformations can be singular or even nonlinear, as long as they are
automorphisms of the output space.
In this basic form the algorithm is is scalable to medium-scale data, as it is
limited memory required to store the projection matrix.In the following
sections we present techniques that allow scaling to very large datasets, e.g.
$m>1000000$ and $m\gg n_{x},n_{y}$. A recursive representation of the
projection operators (see Section 3.2), and especially the singular vector
based form, (eq. (9)), significantly reduces the demand for resources, both
for memory and for computation time.
Generic algorithm of the variable selection by projection 1. Input: (a) A set
of output variables $\\{\mathbf{y}_{1},\dots,\mathbf{y}_{n_{y}}\\}$ of
$\mathbb{R}^{m}$, in $\mathbf{Y}\in\mathbb{R}^{m\times n_{y}}$. (b) A set of
input variables $\\{\mathbf{x}_{1},\dots,\mathbf{x}_{n_{x}}\\}$ of
$\mathbb{R}^{m}$, collected to $\mathbf{X}\in\mathbb{R}^{m\times n_{x}}$. (c)
$D\leq n_{y}$: number of variables to be chosen from $\mathbf{X}$. 2. Output:
Set $\mathcal{I}_{D}$ of indices of selected variables from $\mathbf{X}$ in
the selection order. The algorithm: 1. Let $t=0$ and
$\mathcal{I}_{t}=\emptyset$.
$\tilde{\mathbf{X}}_{t}=\mathbf{X}[:,\mathcal{I}_{t}]$; since
$\mathcal{I}_{t}=\emptyset$, $\tilde{\mathbf{X}}_{t}^{\perp}=\mathbb{R}^{m}$.
2. For $t\in\\{1,\dots,D\\}$ Do (a) Let
$\mathbf{P}_{\mathcal{L}_{Y}\cap\mathcal{L}_{\tilde{X}_{t}^{\perp}}}$ be the
projection into the intersection of $\mathcal{L}_{Y}$ and
$\mathcal{L}_{\tilde{X}_{t}^{\perp}}$. (b) Choose an index by
$k_{*}=\operatorname*{arg\,max}_{k\in\\{1,\dots,n_{x}\\}\setminus\mathcal{I}_{t}}\left\|\mathbf{P}_{\mathcal{L}_{Y}\cap\mathcal{L}_{\tilde{X}_{t}^{\perp}}}\tfrac{\mathbf{x}_{k}}{\|\mathbf{x}_{k}\|}\right\|^{2}$
(c) $\mathcal{I}_{t+1}=\mathcal{I}_{t}\cup\\{k_{*}\\}$, and
$\tilde{\mathbf{X}}_{t+1}=\mathbf{X}[:,\mathcal{I}_{t+1}]$.
Figure 2: The generic algorithm of supervised variable selection by projection
## 3 Projection operators
This section first introduces relevant background on projection operators and
their algebra. Then, two key points for our algorithm are discussed: the
matrix representation of the projectors, and how the projection into the
intersection can be expressed.
### 3.1 Projection operators, projectors
We now briefly introduce the mathematical framework describing the projection
operators of a Hilbert space. The proofs of the statements mentioned, as well
as further details, are presented for example by [Kreyszig, 1989].
Let $\mathbf{T}$ be a linear operator
$\mathbf{T}:\mathcal{H}\rightarrow\mathcal{H}$. Its adjoint
$\mathbf{T}^{*}:\mathcal{H}\rightarrow\mathcal{H}$ is defined by
$\braket{\mathbf{y},\mathbf{T}^{*}\mathbf{x}}=\braket{\mathbf{T}\mathbf{y},\mathbf{x}}$
for all $\mathbf{x},\mathbf{y}\in\mathcal{H}$. A linear operator $\mathbf{T}$
is self-adjoint, or Hermitian if $\mathbf{T}=\mathbf{T}^{*}$, unitary if
$\mathbf{T}^{*}=\mathbf{T}^{-1}$ and normal if
$\mathbf{T}\mathbf{T}^{*}=\mathbf{T}^{*}\mathbf{T}$. On the set of self-
adjoint operators of $\mathcal{H}$ one can define a partial order $\preceq$ by
$\mathbf{T}_{1}\preceq\mathbf{T}_{2}\Leftrightarrow\braket{\mathbf{T}_{1}\mathbf{x},\mathbf{x}}\leq\braket{\mathbf{T}_{2}\mathbf{x},\mathbf{x}},$
(2)
for all $\mathbf{x}\in\mathcal{H}$. An operator $\mathbf{T}$ is positive if
$\mathbf{0}\preceq\mathbf{T}\Leftrightarrow
0\leq\braket{\mathbf{T},\mathbf{x}},$ (3)
for all $\mathbf{x}\in\mathcal{H}$. As a consequence we have
$\mathbf{T}_{1}\preceq\mathbf{T}_{2}\Leftrightarrow\mathbf{0}\preceq\mathbf{T}_{2}-\mathbf{T}_{1}.$
(4)
Let $\mathcal{L}$ be a subspace of $\mathcal{H}$, the orthogonal complement of
$\mathcal{L}$ is given by
$\mathcal{L}^{\perp}=\\{\mathbf{x}|\mathbf{x}\perp\mathbf{z},\forall\mathbf{z}\in\mathcal{L},\mathbf{x}\in\mathcal{H}\\}$.
###### Theorem 2.
For any subspace $\mathcal{L}\subseteq\mathcal{H}$,
$\mathcal{H}=\mathcal{L}\oplus\mathcal{L}^{\perp}$.
A linear operator $\mathbf{P}$ is a projection operator if
$\mathbf{P}:\mathcal{H}\rightarrow\mathcal{L}$ for a subspace $\mathcal{L}$ of
$\mathcal{H}$. To highlight the connection between the subspace and
projection, they can be also denoted as $\mathcal{L}_{P}$ and
$\mathbf{P}_{L}$.
An operator $\mathbf{P}$ is idempotent if
$\mathbf{P}\mathbf{P}\mathbf{x}=\mathbf{P}\mathbf{x},\ \text{or}\
\mathbf{P}\mathbf{P}=\mathbf{P}$ holds for any $\mathbf{x}\in\mathcal{H}$. The
projection operators can be characterized by the following statements.
###### Theorem 3.
A linear operator $\mathbf{P}:\mathcal{H}\rightarrow\mathcal{H}$ is a
projection if it is self adjoint, $\mathbf{P}=\mathbf{P}^{*}$, and idempotent
$\mathbf{P}\mathbf{P}=\mathbf{P}$.
###### Proposition 4.
The map connecting the set of closed subspaces111In a finite dimensional
Hilbert space all subspaces are closed. of $\mathcal{H}$ and the set of the
corresponding orthogonal projections is bijective.
As a consequence of the idempotent and self-adjoint properties we have that
the range $\mathcal{R}(\mathbf{P})$ and the null space
$\mathcal{N}(\mathbf{P})$ of $\mathbf{P}$ are orthogonal, namely for any
$x,y\in\mathcal{H}$
$\braket{\mathbf{P}x,y-\mathbf{P}y}=\braket{\mathbf{P}^{2}x,y-\mathbf{P}y}=\braket{\mathbf{P}x,(\mathbf{P}-\mathbf{P}^{2})y}=0.$
(5)
The following theorems describe some algebraic properties of projection
operators we are going to exploit.
###### Theorem 5.
(Product of projections) Let $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ be
projections on $\mathcal{H}$. $\mathbf{P}=\mathbf{P}_{1}\mathbf{P}_{2}$ is
projection if and only if
$\mathbf{P}_{1}\mathbf{P}_{2}=\mathbf{P}_{2}\mathbf{P}_{1}$. Then
$\mathbf{P}:\mathcal{H}\rightarrow\mathcal{L}_{P_{1}}\cap\mathcal{L}_{P_{2}}$.
###### Theorem 6.
(Sum of projections) Let $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ be projections
on $\mathcal{H}$. $\mathbf{P}=\mathbf{P}_{1}+\mathbf{P}_{2}$ is projection if
and only if $\mathcal{L}_{P_{1}}\perp\mathcal{L}_{P_{2}}$. Then
$\mathbf{P}:\mathcal{H}\rightarrow\mathcal{L}_{P_{1}}\oplus\mathcal{L}_{P_{2}}$.
###### Theorem 7.
(Partial order) Let $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ be projections on
$\mathcal{H}$, and $\mathcal{N}(\mathbf{P}_{1})$ and
$\mathcal{N}(\mathbf{P}_{2})$ the corresponding null spaces. Then the
following statements are equivalent.
$\begin{array}[]{ll}\mathbf{P}_{1}\mathbf{P}_{2}&=\mathbf{P}_{2}\mathbf{P}_{1}=\mathbf{P}_{1},\\\
\mathcal{L}_{P_{1}}&\subseteq\mathcal{L}_{P_{2}},\\\
\mathcal{N}(\mathbf{P}_{1})&\supseteq\mathcal{N}(\mathbf{P}_{2}),\\\
||\mathbf{P}_{1}\mathbf{x}||&\leq||\mathbf{P}_{2}\mathbf{x}||,\\\
\mathbf{P}_{1}&\preceq\mathbf{P}_{2}.\\\ \end{array}$ (6)
###### Theorem 8.
(Difference of projections) Let $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ be
projections on $\mathcal{H}$. $\mathbf{P}=\mathbf{P}_{2}-\mathbf{P}_{1}$ is
projection if and only $\mathcal{L}_{P_{1}}\subseteq\mathcal{L}_{P_{2}}$. Then
$\mathbf{P}:\mathcal{H}\rightarrow\mathcal{L}_{P}$, where
$\mathcal{L}_{P_{2}}=\mathcal{L}_{P_{1}}\oplus\mathcal{L}_{P}$, namely
$\mathcal{L}_{P}$ is the orthogonal complement of $\mathcal{L}_{P_{1}}$ in
$\mathcal{L}_{P_{2}}$.
From the theorems above we can derive a simple corollary: if $\mathcal{L}$ is
a subspace, then the projection into its complement is equal to
$\mathbf{P}_{\mathcal{L}^{\perp}}=\mathbf{I}-\mathbf{P}_{\mathcal{L}}$.
###### Theorem 9.
(Monotone increasing sequence) Let $(\mathbf{P}_{n})$ be monotone increasing
sequence of projections defined on $\mathcal{H}$. Then:
1. 1.
$(\mathbf{P}_{n})$ is strongly operator convergent, and the limit
$\mathbf{P}$, $\mathbf{P}_{n}\rightarrow\mathbf{P}$, is a projection.
2. 2.
$\mathbf{P}:\mathcal{H}\rightarrow\cup_{n=1}^{\infty}\mathcal{L}_{\mathbf{P}_{n}}$.
3. 3.
$\mathcal{N}(\mathbf{P})=\cap_{n=1}^{\infty}\mathcal{N}(\mathbf{P}_{n})$.
If $\mathbf{S}$ is a self-adjoint operator and $\mathbf{P}$ is a projection
into the range of $\mathbf{S}$ then
$\mathbf{S}\mathbf{P}=\mathbf{P}\mathbf{S}$, see [Conway, 1997] for further
details.
Let $\mathbf{I}$ be projection into the entire space, and $\mathbf{0}$ its
complement. If $\mathbf{0}\leq\mathbf{S}\leq\mathbf{I}$, and
$\mathbf{T}\geq\mathbf{0}$ operators. If $\mathbf{P}$ is a projection into
$\mathbf{S}+\mathbf{T}$ then $\mathbf{P}$ commutes both $\mathbf{S}$ and
$\mathbf{T}$. See in [Hayashi and Nagaoka, 2002].
### 3.2 Matrix representation of projectors
If a basis of the Hilbert space $\mathcal{H}$ is fixed, then every linear
operator acting on $\mathcal{H}$ can be represented by a matrix. Let the
subspace $\mathcal{L}$ of $\mathcal{H}$ be spanned by the vectors
$\mathbf{a}_{1},\dots,\mathbf{a}_{k}$ of $\mathcal{H}$. Let us construct a
matrix $\mathbf{A}$ whose columns are equal to the vectors
$\mathbf{a}_{1},\dots,\mathbf{a}_{k}$. Here the linear independence of those
vectors is not assumed. The corresponding subspace is denoted by
$\mathcal{L}_{A}$
The matrix representing the orthogonal projection operator into to subspace
$\mathcal{L}_{A}$ can be expressed by a well-known minimization problem [Golub
and Loan, 2013],
$\arg\min_{\mathbf{w}}\|\mathbf{x}-\mathbf{A}\mathbf{w}\|^{2}=\mathbf{w}^{*}=(\mathbf{A}^{T}\mathbf{A})^{+}\mathbf{A}^{T}\mathbf{x},$
(7)
where + denotes the Moore-Penrose pseudo-inverse. Based on eq. (5) the vector
$\mathbf{A}^{T}\mathbf{w}^{*}$ is the orthogonal projection of $\mathbf{x}$
into $\mathcal{L}$. The orthogonal projection of $\mathbf{x}$ is equal to
$\mathbf{P}_{A}\mathbf{x}=\mathbf{A}(\mathbf{A}^{T}\mathbf{A})^{+}\mathbf{A}^{T}\mathbf{x}$.
Since this is true for any $\mathbf{x}\in\mathcal{H}$, the matrix
representation of the orthogonal projection operator $\mathbf{P}_{A}$ is given
by
$\mathbf{P}_{A}=\mathbf{A}(\mathbf{A}^{T}\mathbf{A})^{+}\mathbf{A}^{T}.$ (8)
This formula can be simplified by exploiting the properties of the Moore-
Penrose pseudo-inverse, see for example [Ben-Israel and Greville, 2003], via
the singular value decomposition
$\mathbf{U}_{A}\mathbf{S}_{A}\mathbf{V}_{A}^{T}$ of the matrix $\mathbf{A}$.
Here we assume that the matrix $\mathbf{A}\in\mathbb{R}^{m\times n_{A}}$,
$m>n_{A}$, and $\mathbf{V}_{A}$ is a square matrix, but $\mathbf{U}_{A}$
contains only those left singular vectors where the corresponding singular
values are not equal to zero. We have
$\framebox{$\mathbf{P}_{A}$}=\mathbf{A}(\mathbf{A}^{T}\mathbf{A})^{+}\mathbf{A}^{T}=\mathbf{A}\mathbf{A}^{+}=\mathbf{U}_{A}\mathbf{S}_{A}\mathbf{V}_{A}^{T}\mathbf{V}_{A}\mathbf{S}_{A}^{+}\mathbf{U}_{A}^{T}=\framebox{$\mathbf{U}_{A}\mathbf{U}_{A}^{T}$}.$
(9)
This representation of the projection operator plays a central role in our
variable selection algorithm. The following proposition ensures that the
projection operator does not depend on its representation.
###### Proposition 10.
Assume that two different matrices $\mathbf{A}$ and $\mathbf{B}$ span the same
subspace $\mathcal{L}$ of dimension $k$. Then the two representations
$\mathbf{P}_{A}=\mathbf{U}_{A}\mathbf{U}_{A}^{T}$ and
$\mathbf{P}_{B}=\mathbf{U}_{B}\mathbf{U}_{B}^{T}$ yield the same projection
operator.
###### Proof.
Since the columns of $\mathbf{U}_{B}$ as linear combinations of $\mathbf{B}$
are in the $\mathcal{L}$, thus $\mathbf{P}_{A}\mathbf{U}_{B}=\mathbf{U}_{B}$.
Multiplying both sides with $\mathbf{U}_{B}^{T}$ we obtain that
$\mathbf{P}_{A}\mathbf{U}_{B}\mathbf{U}_{B}^{T}=\mathbf{U}_{B}\mathbf{U}_{B}^{T}$
which is $\mathbf{P}_{A}\mathbf{P}_{B}=\mathbf{P}_{B}$. Because the right hand
side, $\mathbf{P}_{B}$, is a projection, the left hand side
$\mathbf{P}_{A}\mathbf{P}_{B}$ is also one. Following the same line we have
$\mathbf{P}_{B}\mathbf{P}_{A}=\mathbf{P}_{A}$ as well. From Theorem 5 we know
that if the product of projections is a projection, then the product of
projections is commutative,
$\mathbf{P}_{B}\mathbf{P}_{A}=\mathbf{P}_{A}\mathbf{P}_{B}$. Finally we can
conclude that
$\mathbf{P}_{A}=\mathbf{P}_{B}\mathbf{P}_{A}=\mathbf{P}_{A}\mathbf{P}_{B}=\mathbf{P}_{B}.$
∎
We also exploited that if $\mathcal{H}$ is finite dimensional and the
corresponding field is $\mathbb{R}$ then the adjoint of $\mathbf{P}^{*}$ is
represented by the transpose $\mathbf{P}^{T}$ of the matrix $\mathbf{P}$.
#### 3.2.1 Projection onto the intersection of subspaces - General view
Our algorithm hinges on the orthogonal projector of the intersection of a set
of subspaces
$\\{\mathcal{L}_{1},\mathcal{L}_{2},\dots,\mathcal{L}_{n_{L}}\\}$. To
introduce this concept, here we mainly follow the line presented by [Ben-
Israel, 2015]. We can start with some classical result, first we can recall
[von Neumann, 1950], who derived a solution in case of two subspaces as a
limit:
$\mathbf{P}_{\mathcal{L}_{1},\cap\mathcal{L}_{2}}=\lim_{n\rightarrow\infty}(\mathbf{P}_{\mathcal{L}_{1}}\mathbf{P}_{\mathcal{L}_{2}})^{n}.$
(10)
That result has been extended to arbitrary finite sets of subspaces by
[Halperin, 1962],:
$\mathbf{P}_{\mathcal{L}_{1},\cap\dots\cap\mathcal{L}_{n_{L}}}=\lim_{n\rightarrow\infty}(\mathbf{P}_{\mathcal{L}_{1}}\dots\mathbf{P}_{L_{n}})^{n}.$
(11)
[Anderson. and Duffin, 1969], gave an explicit formula for the case of two
subspaces by
$\mathbf{P}_{\mathcal{L}_{1},\cap\mathcal{L}_{2}}=2\mathbf{P}_{\mathcal{L}_{1}}(\mathbf{P}_{\mathcal{L}_{1}}+\mathbf{P}_{\mathcal{L}_{2}})^{\dagger}\mathbf{P}_{\mathcal{L}_{2}}.$
(12)
[Ben-Israel, 2015], provides an alternative to compute
$\mathbf{P}_{\mathcal{L}_{1},\cap\dots\cap\mathcal{L}_{n_{L}}}$ Here we rely
on the Lemma 4.1 and Corollary 4.2 of his work:
###### Proposition 11.
For $i=1,\dots,n_{L}$, let $\mathcal{L}_{i}$ be subspaces of $\mathcal{H}$,
$\mathbf{P}_{i}$ be the corresponding projectors,
$\mathbf{P}^{\perp}_{i}=\mathbf{I}-\mathbf{P}_{i}$, and $\lambda_{i}>0$.
Define $\mathbf{Q}:=\sum_{i=1}^{n_{L}}\lambda_{i}\mathbf{P}_{i}^{\perp}.$ then
we have
$\mathbf{P}_{\mathcal{L}_{1},\cap\dots\cap\mathcal{L}_{n_{L}}}=\mathbf{I}-\mathbf{Q}^{\dagger}\mathbf{Q}.$
With the particular choice $\sum_{i=1}^{n_{L}}\lambda_{i}=1$, $\mathbf{Q}$
might be written as
$\mathbf{Q}:=\mathbf{I}-\sum_{i=1}^{n_{L}}\lambda_{i}\mathbf{P}_{i},$
eliminating all the complements of the projectors.
By exploting that for any projector $\mathbf{P}$
$\mathbf{P}^{\perp}=\mathbf{I}-\mathbf{P}$, the $\mathbf{Q}_{t}$ corrsponding
to $\mathbf{P}_{\mathcal{L}_{V}\cap\mathcal{L}_{\tilde{X}_{t}^{\perp}}}$ can
be written as
$\mathbf{Q}_{t}=\lambda_{V}(\mathbf{I}-\mathbf{P}_{\mathcal{L}_{V}})+\sum_{\mathbf{x}\in\tilde{X}_{t}}\lambda_{x}\mathbf{P}_{L_{\\{\mathbf{x}\\}}}.$
(13)
The critical point is the computation of the Moore-Penrose inverse of
$\mathbf{Q}$.
### 3.3 Expressing the projector into intersection
To implement the proposed variable selection algorithm (Figure 2) the
projection into the intersection of an arbitrary subspace
$\mathcal{L}_{\mathbf{P}}$ and the complement of an arbitrary vector
$\mathbf{x}$, $\mathbf{P}_{\mathcal{L}\cap\mathbf{x}^{\perp}}$, has to be
computed. The projector $\mathbf{P}_{\mathcal{L}^{\perp}}$ to the complement
of a subspace $\mathcal{L}$ can be expressed as
$\mathbf{I}-\mathbf{P}_{\mathcal{L}}$, hence the projector into
$\mathbf{P}_{\mathbf{x}^{\perp}}$ is given by
$\mathbf{I}-\dfrac{\mathbf{x}\mathbf{x}^{T}}{||\mathbf{x}||^{2}}$. Since
$\mathcal{L}$ is arbitrary we use $\mathbf{P}$ instead of
$\mathbf{P}_{\mathcal{L}}$ for sake of simplicity.
While we have these two projectors, their product, according to Theorem 5, is
not a projection as it does not commute:
$\mathbf{P}\left(\mathbf{I}-\dfrac{\mathbf{x}\mathbf{x}^{T}}{||\mathbf{x}||^{2}}\right)=\mathbf{P}-\dfrac{\mathbf{P}\mathbf{x}\mathbf{x}^{T}}{||\mathbf{x}||^{2}}\neq\left(\mathbf{I}-\dfrac{\mathbf{x}\mathbf{x}^{T}}{||\mathbf{x}||^{2}}\right)\mathbf{P}=\mathbf{P}-\dfrac{\mathbf{x}\mathbf{x}^{T}\mathbf{P}}{||\mathbf{x}||^{2}},$
(14)
because in the general case
$\mathbf{P}\mathbf{x}\mathbf{x}^{T}\neq\mathbf{x}\mathbf{x}^{T}\mathbf{P}$. To
overcome this problem we can recognize that the intersection
$\mathcal{L}_{\mathbf{P}}\cap\mathcal{L}_{\mathbf{x}^{\perp}}$ can be
expressed after a simple transformation.
###### Lemma 12.
Let $\mathbf{P}$ be a projector and $\mathbf{x}$ be any vector, then the
intersections $\mathcal{L}_{\mathbf{P}}\cap\mathcal{L}_{\mathbf{x}^{\perp}}$
and $\mathcal{L}_{\mathbf{P}}\cap\mathcal{L}_{(\mathbf{P}\mathbf{x})^{\perp}}$
are the same subspaces of $\mathcal{L}_{\mathbf{P}}$.
###### Proof.
Any vector $\mathbf{u}$ is in $\mathcal{L}_{\mathbf{P}}$ if
$\mathbf{P}\mathbf{u}=\mathbf{u}$, $\mathbf{u}$ is in
$\mathcal{L}_{\mathbf{x}^{\perp}}$ if $\braket{\mathbf{x},\mathbf{u}}=0$, and
$\mathbf{u}$ is in $\mathcal{L}_{(\mathbf{P}\mathbf{x})^{\perp}}$ if
$\braket{\mathbf{P}\mathbf{x},\mathbf{u}}=0$. Since
$\mathbf{P}\mathbf{u}=\mathbf{u}$, therefore
$\braket{\mathbf{x},\mathbf{u}}=\braket{\mathbf{P}\mathbf{x},\mathbf{u}}=0$. ∎
By projecting $\mathbf{x}$ into $\mathcal{L}$ first, and then computing the
corresponding intersection, we can compute the projector into
$\mathcal{L}_{\mathbf{P}}\cap\mathcal{L}_{\mathbf{x}^{\perp}}$ in a simple
way.
###### Proposition 13.
Let $\|\mathbf{x}\|=1$ and
$\alpha=\|\mathbf{P}\mathbf{x}\|^{2}=\mathbf{x}^{T}\mathbf{P}\mathbf{P}\mathbf{x}=\mathbf{x}^{T}\mathbf{P}\mathbf{x}$.
If $\alpha>0$ then
$\framebox{$\mathbf{P}_{\mathcal{L}_{\mathbf{P}}\cap\mathcal{L}_{\mathbf{x}^{\perp}}}$}=\mathbf{P}_{\mathcal{L}_{\mathbf{P}}\cap\mathcal{L}_{(\mathbf{P}{\mathbf{x})^{\perp}}}}=\mathbf{P}\left(\mathbf{I}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}\right)=\framebox{$\mathbf{P}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}$}.$
(15)
When $\alpha=0$, which means $\mathbf{x}$ orthogonal to
$\mathcal{L}_{\mathbf{P}}$, then we have
$\mathbf{P}_{\mathcal{L}_{\mathbf{P}}\cap\mathcal{L}_{\mathbf{x}^{\perp}}}=\mathbf{P}.$
We can check that
$\mathbf{P}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}$ is
a real projector. It is idempotent, since
$\begin{array}[]{@{}l@{}}\left(\mathbf{P}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}\right)^{2}=\mathbf{P}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}+\tfrac{\alpha}{\alpha^{2}}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}=\
\mathbf{P}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}.\end{array}$
This agrees with Theorem 5 which states that the product of projections is a
projection, idempotent and adjoint, and it is map into the intersection of the
corresponding subspaces. Furthermore the orthogonality of $\mathbf{x}$ and the
projection of any $\mathbf{u}\in\mathcal{H}$ by
$\mathbf{P}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}$ can
also be verified, as
$\begin{array}[]{ll}\left\langle\mathbf{x},\left(\mathbf{P}-\tfrac{1}{\alpha}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}\right)\mathbf{u}\right\rangle=\mathbf{x}^{T}\mathbf{P}\mathbf{u}-\tfrac{1}{\alpha}\mathbf{x}^{T}\mathbf{P}\mathbf{x}\mathbf{x}^{T}\mathbf{P}\mathbf{u}=\mathbf{x}^{T}\mathbf{P}\mathbf{u}-\tfrac{1}{\alpha}\alpha\mathbf{x}^{T}\mathbf{P}\mathbf{u}=0.\end{array}$
###### Definition 14.
([Tjur, 1984]) Two subspaces $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are
orthogonal if for any vectors, $\mathbf{x_{1}}\in\mathcal{L}_{1}$ and
$\mathbf{x_{2}}\in\mathcal{L}_{2}$ $\braket{\mathbf{x}_{1},\mathbf{x}_{2}}=0$
holds. Two subspaces $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are geometrically
orthogonal if
$\mathcal{L}_{1}=(\mathcal{L}_{1}\cap\mathcal{L}_{2})\oplus\mathcal{C}_{1}$
and
$\mathcal{L}_{2}=(\mathcal{L}_{1}\cap\mathcal{L}_{2})\oplus\mathcal{C}_{2}$
and $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are orthogonal.
###### Lemma 15.
([Tjur, 1984]) Two subspaces $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are
geometrically orthogonal if and only if
$\mathbf{P}_{\mathcal{L}_{1}}\mathbf{P}_{\mathcal{L}_{2}}=\mathbf{P}_{\mathcal{L}_{2}}\mathbf{P}_{\mathcal{L}_{1}}$
and
$\mathcal{L}_{\mathbf{P}_{\mathcal{L}_{1}}\mathbf{P}_{\mathcal{L}_{2}}}=\mathcal{L}_{1}\cap\mathcal{L}_{2}$.
###### Proposition 16.
The subspaces $\mathcal{L}_{\mathbf{P}}$ and
$\mathcal{L}_{(\mathbf{P}\mathbf{x})^{\perp}}$ are geometrically orthogonal.
###### Proof.
It is a simple Corollary of Proposition 13. ∎
## 4 Selecting variables in RKHS
In order to take into account non-linear correlations in the data, we propose
a kernelized adaptation of the problem. Kernel methods are a group of varied
machine learning models, taking advantage of a symmetric and positive semi-
definite kernel function comparing data samples (sets of features)
$k:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$. The usage of a kernel
function allows including non-linearity to the models implicitly via a feature
map $\varphi:\mathcal{X}\rightarrow\mathcal{F}_{k}$: a kernel evaluated with
two samples corresponds to an inner product in this so-called feature space
(more specifically reproducing kernel Hilbert space, RKHS):
$k(x,z)=\langle\varphi(x),\varphi(z)\rangle_{\mathcal{F}_{k}}$. For more
thorough introduction to traditional kernel methods, we refer the reader e.g.
to [Hofmann et al., 2008].
We here propose to kernelize the variable representation. We consider
$\phi:\mathbb{R}^{m}\rightarrow\mathcal{H}$, where $\mathbb{R}^{m}$ is the
vector space containing all columns of $\mathbf{Y}\in\mathbb{R}^{m\times
n_{y}}$ and $\mathbf{X}\in\mathbb{R}^{m\times n_{x}}$, and $\mathcal{H}$ is a
RKHS. In essence, this corresponds to defining a kernel on the variable
vectors, $\kappa:\mathbb{R}^{m}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$ – in
fact, we assume that the $\phi$ is only given implicitly via $\kappa$. In
mathematical sense, this matrix can equally well be considered to be a kernel
matrix, since distinction between the rows and columns is by convention only.
Usually however the matrix built from inner products between the variables is
referred to as covariance operator. The covariance operators are also extended
to RKHS with various applications in machine learning tasks [Muandet et al.,
2017, Minh et al., 2016b]. Contrary to our approach, there the feature map and
kernel are defined on data space $\mathcal{X}$ instead of variable space
$\mathbb{R}^{m}$. We need to mention here also the Gaussian Process
Regression, [Rasmussen and Williams, 2005] where kernels are also used to
cover the covariance matrix, thus connecting the variables via inner product.
We highlight that as the kernel is defined on variables, we can easily
evaluate $\kappa(\mathbf{x}_{i},\mathbf{y}_{j})$. We use the following
shorthands for feature and kernel evaluations on the available training data:
$\phi(\mathbf{Y})=[\phi(\mathbf{y}_{1}),\dots,\phi(\mathbf{y}_{n_{y}})]$ with
$\phi(\mathbf{y}_{i}),i\in[n_{y}]$ a column vector, and
$\kappa(\mathbf{Y},\mathbf{x})=[\kappa(\mathbf{y}_{1},\mathbf{x}),\dots,\kappa(\mathbf{y}_{n_{y}},\mathbf{x})]^{\top}$
a column vector of kernel evaluations (similarly for $\phi(\mathbf{X})$). Note
that $\kappa(\mathbf{Y},\mathbf{Y})=\phi(\mathbf{Y})^{\top}\phi(\mathbf{Y})$
with this notation. We further denote
$\mathbf{K_{y}}=\kappa(\mathbf{Y},\mathbf{Y})$. We assume that
$\|\phi(\mathbf{x})\|=1$.
### 4.1 Expressing the Projection operator in RKHS
Based on Section 3.2, Equation (9), the projection $\mathbf{P}_{Y}$ is
represented with the left singular vectors of $\mathbf{P}_{Y}$,
$\mathbf{U}_{Y}$. This representation is also needed for the kernelized
algorithm. However calculating directly the singular value decomposition on
$\phi(\mathbf{Y})$,
$\phi(\mathbf{Y})=\mathbf{U}_{Y}\mathbf{S}_{Y}\mathbf{V}_{Y}^{T}$, might not
be feasible if the dimensionality of the feature space is large. Assuming that
$\mathcal{H}$ is finite dimensional222For clarity, we restrict the discussion
to finite dimensions and $\mathcal{H}=\mathbb{R}^{d}$ with $d<\infty$. We note
that the approach is equally valid also with infinite dimensions. with
dimension $d$, we have $\phi(\mathbf{Y}),\mathbf{U}_{Y}\in\mathbb{R}^{d\times
n_{y}}$, and $\mathbf{S}_{Y},\mathbf{V}_{Y}\in\mathbb{R}^{n_{y}\times n_{y}}$.
Therefore we can write
$\mathbf{S}_{Y}=\left[\begin{array}[]{c}\mathbf{D}_{Y}\\\
\boldsymbol{\varnothing}\end{array}\right],\mathbf{D}_{Y}\in\mathbb{R}^{n_{y}\times
n_{y}},\boldsymbol{\varnothing}\in[0]^{m-n_{y},n_{y}},$ (16)
and $\mathbf{D}_{Y}$ diagonal with nonnegative elements of singular values,
thus $\phi(\mathbf{Y})=\mathbf{U}_{Y}\mathbf{D}_{Y}\mathbf{V}_{Y}^{T}$. Again,
this decomposition can not be computed directly, however we can go on the
following line of computation.
To express the $\mathbf{U}_{Y}$ we can apply a similar approach to what is
exploited in the computation of the kernel principal component analysis
[Schölkopf et al., 1998]. Recall that the kernel matrix on columns of
$\phi(\mathbf{Y})$ is $\mathbf{K}_{Y}=\phi(\mathbf{Y})^{T}\phi(\mathbf{Y})$.
From the singular value decomposition we can derive that
$\mathbf{K}_{Y}=\mathbf{V}_{Y}\mathbf{S}_{Y}^{2}\mathbf{V}^{T}$. This kernel
has a reasonably small size, $n_{y}\times n_{y}$, thus its eigenvalue
decomposition can be computed, which yields $\mathbf{V}_{Y}$ and the squares
of the singular values of the diagonal elements of $\mathbf{S}_{Y}$. By
combining these expressions we have
$\phi(\mathbf{Y})\mathbf{V}_{Y}\mathbf{S}_{Y}=\mathbf{U}_{Y}\mathbf{S}_{Y}^{2}\
\Rightarrow\
\framebox{$\mathbf{U}_{Y}=\phi(\mathbf{Y})\mathbf{V}_{Y}\mathbf{S}_{Y}^{+}$}$
(17)
with the help of the Moore-Penrose generalized inverse. Our algorithm hinges
on evaluating products between projectors and the variable vectors. We can now
write the products of the $\mathbf{U}_{Y}^{T}$ with an arbitrary vector
represented in $\mathcal{H}$ as
$\begin{array}[]{ll}\framebox{$\mathbf{U}_{Y}^{T}\phi(\mathbf{x})$}=\mathbf{S}_{Y}^{-1}\mathbf{V}_{Y}^{T}\phi(\mathbf{Y})^{T}\phi(\mathbf{x})=\framebox{
$\mathbf{S}_{Y}^{-1}\mathbf{V}_{Y}^{T}\kappa(\mathbf{Y},\mathbf{x})$}.\end{array}$
(18)
Thus the product can be expressed with the help of the kernel on the variables
with complexity $O(n_{y}^{2})$ if the $\mathbf{K}_{Y}$, $\mathbf{V}_{Y}$ and
$\mathbf{S}_{Y}^{-1}$ are precomputed.
### 4.2 The recursive selection procedure
To calculate the projection operator efficiently in each iteration we can
exploit the structure of
$\mathbf{P}_{\mathcal{L}_{Y}\cap\mathcal{L}_{\tilde{X}_{t}^{\perp}}}\phi(\mathbf{x})$
introduced in Proposition 13. To this end, we define an intermediate operator,
projection into the complement subspace of vector
$\mathbf{q}\in\mathbb{R}^{n}$ as:
$\mathbf{Q}(\mathbf{q})=\mathbf{I}-\dfrac{\mathbf{q}\mathbf{q}^{T}}{||\mathbf{q}||^{2}}.$
(19)
Since $\mathbf{Q}(\mathbf{q})$ is a projection, we have
$\mathbf{Q}(\mathbf{q})=\mathbf{Q}(\mathbf{q})\mathbf{Q}(\mathbf{q})$ and
$\mathbf{Q}(\mathbf{q})=\mathbf{Q}(\mathbf{q})^{T}$. It can also be seen that
multiplying $\mathbf{Q}$ with a matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$,
$\mathbf{Q}(\mathbf{q})\mathbf{A}=\left(\mathbf{I}-\dfrac{\mathbf{q}\mathbf{q}^{T}}{||\mathbf{q}||^{2}}\right)\mathbf{A}=\mathbf{A}-\dfrac{\mathbf{q}(\mathbf{q}^{T}\mathbf{A})}{||\mathbf{q}||^{2}},$
(20)
has the complexity of only $O(n^{2})$ since only matrix-vector and outer
product are needed. We are also going to use the following recurrent matrix
products for a fixed $t$
$\tilde{\mathbf{U}}_{t}=\mathbf{U}_{Y}\prod_{s=1}^{t-1}\mathbf{Q}(\mathbf{q}_{s})=\tilde{\mathbf{U}}_{t-1}\mathbf{Q}(\mathbf{q}_{t-1}).$
(21)
Now we can write up the sequence of projections corresponding to the Algorithm
(2):
$\begin{array}[]{@{}l@{\:}l@{}}\text{Let}\hfil\>&\mathbf{U}_{0}=\mathbf{U}_{Y},\
\mathcal{I}_{0}=\emptyset,\\\
\framebox{$\mathbf{P}_{0}$}\hfil\>&=\mathbf{P}_{\phi(Y)}=\framebox{$\mathbf{U}_{0}\mathbf{U}_{0}^{T}$},\\\
\mathbf{q}_{1}\hfil\>&=\mathbf{U}_{0}^{T}\phi(\mathbf{x}_{k_{1*}}),\
k_{1*}=\arg\max_{k\in[n_{x}]\setminus\mathcal{I}_{0}}||\mathbf{P}_{0}\phi(\mathbf{x}_{k})||^{2},\
\mathcal{I}_{1}=\mathcal{I}_{0}\cup{k_{1^{*}}},\\\
\framebox{$\mathbf{P}_{1}$}\hfil\>&=\mathbf{U}_{0}\mathbf{U}_{0}^{T}-\dfrac{\mathbf{U}_{0}\mathbf{q}_{1}\mathbf{q}_{1}^{T}\mathbf{U}_{0}^{T}}{||\mathbf{q}_{1}||^{2}}=\mathbf{U}_{0}\mathbf{Q}(\mathbf{q}_{1})\mathbf{U}_{0}^{T}=\mathbf{U}_{0}\mathbf{Q}(\mathbf{q}_{1})\mathbf{Q}(\mathbf{q}_{1})\mathbf{U}_{0}^{T}=\framebox{$\mathbf{U}_{1}\mathbf{U}_{1}^{T}$},\\\
\mathbf{q}_{2}\hfil\>&=\mathbf{U}_{1}^{T}\phi(\mathbf{x}_{k_{2*}}),\
k_{2*}=\arg\max_{k\in[n_{x}]\setminus\mathcal{I}_{1}}||\mathbf{P}_{1}\phi(\mathbf{x}_{k})||^{2},\
\mathcal{I}_{2}=\mathcal{I}_{1}\cup{k_{2^{*}}},\\\ \vdots\hfil\>\\\
\framebox{$\mathbf{P}_{t}$}\hfil\>&=\mathbf{U}_{t-1}\mathbf{U}_{t-1}^{T}-\dfrac{\mathbf{U}_{t-1}\mathbf{q}_{t}\mathbf{q}_{t}^{T}\mathbf{U}_{t-1}^{T}}{||\mathbf{q}_{t}||^{2}}=\mathbf{U}_{t-1}\mathbf{Q}(\mathbf{q}_{t})\mathbf{U}_{t-1}^{T}\\\
\hfil\>&=\mathbf{U}_{t-1}\mathbf{Q}(\mathbf{q}_{t})\mathbf{Q}(\mathbf{q}_{t})\mathbf{U}_{t-1}^{T}=\framebox{$\mathbf{U}_{t}\mathbf{U}_{t}^{T}$},\\\
\mathbf{q}_{t+1}\hfil\>&=\mathbf{U}_{t}^{T}\phi(\mathbf{x}_{k_{(t+1)*}}),\
k_{(t+1)*}=\arg\max_{k\in[n_{x}]\setminus\mathcal{I}_{t}}||\mathbf{P}_{t}\phi(\mathbf{x}_{k})||^{2},\
\mathcal{I}_{t+1}=\mathcal{I}_{t}\cup{k_{(t+1)^{*}}},\\\
\vdots\hfil\>\end{array}$
###### Proposition 17.
The sequence of projections above correctly computes the projection operators
of Algorithm in Figure 2.
###### Proof.
We apply induction on $t$ to prove the statement. In case of $t=1$ we have by
Proposition 13, that
$\begin{array}[]{@{}l@{\;}l@{}}\mathbf{P}_{1}&=\mathbf{P}_{0}-\dfrac{\mathbf{P}_{0}\phi(\mathbf{x}_{k_{1*}})\phi(\mathbf{x}_{k_{1*}})^{T}\mathbf{P}_{0}}{||\mathbf{P}_{0}\phi(\mathbf{x}_{k_{1*}})||}=\mathbf{U}_{0}\mathbf{U}_{0}^{T}-\dfrac{\mathbf{U}_{0}\mathbf{U}_{0}^{T}\phi(\mathbf{x}_{k_{1*}})\phi(\mathbf{x}_{k_{1*}})^{T}\mathbf{U}_{0}\mathbf{U}_{0}^{T}}{||\mathbf{U}_{0}\mathbf{U}_{0}^{T}\phi(\mathbf{x}_{k_{1*})})||^{2}}\\\
&=\mathbf{U}_{0}\left(\mathbf{I}-\dfrac{\mathbf{q}_{1}\mathbf{q}_{1}^{T}}{||\mathbf{q}_{1}||^{2}}\right)\mathbf{U}_{0}^{T}=\mathbf{U}_{0}\mathbf{Q}(\mathbf{q}_{1})\mathbf{U}_{0}^{T}=\mathbf{U}_{0}\mathbf{Q}(\mathbf{q}_{1})\mathbf{Q}(\mathbf{q}_{1})\mathbf{U}_{0}^{T}=\framebox{$\mathbf{U}_{1}\mathbf{U}_{1}^{T}$}.\\\
\end{array}$ (22)
In transforming
$||\mathbf{U}_{0}\mathbf{U}_{0}^{T}\phi(\mathbf{x})_{t_{1*}}||^{2}$ into
$||\mathbf{q}_{1}||^{2}$ we exploited that $\mathbf{U}_{0}\mathbf{U}_{0}^{T}$
is a projection, hence it is idempotent. Let $t>1$ be arbitrary. Suppose that
$\begin{array}[]{l}\mathbf{P}_{t}=\mathbf{U}_{t-1}\mathbf{U}_{t-1}^{T}-\dfrac{\mathbf{U}_{t-1}\mathbf{q}_{t}\mathbf{q}_{t}^{T}\mathbf{U}_{t-1}^{T}}{||\mathbf{q}_{t}||^{2}}=\framebox{$\mathbf{U}_{t}\mathbf{U}_{t}^{T}$},\\\
\end{array}$ (23)
holds true. Now, computing the projector $t+1$ we obtain
$\begin{array}[]{@{}l@{\;}l@{}}\mathbf{P}_{t+1}&=\mathbf{P}_{t}-\dfrac{\mathbf{P}_{t}\phi(\mathbf{x}_{k_{(t+1)*}})\phi(\mathbf{x}_{k_{(t+1)*}})^{T}\mathbf{P}_{t}}{||\mathbf{P}_{t}\phi(\mathbf{x}_{k_{(t+1)*}})||^{2}}\\\
&=\mathbf{U}_{t}\mathbf{U}_{t}^{T}-\dfrac{\mathbf{U}_{t}\mathbf{U}_{t}^{T}\phi(\mathbf{x}_{k_{(t+1)*}})\phi(\mathbf{x}_{k_{(t+1)*}})^{T}\mathbf{U}_{t}\mathbf{U}_{t}^{T}}{||\mathbf{U}_{t}\mathbf{U}_{t}^{T}\phi(\mathbf{x}_{k_{(t+1)*}})||^{2}}=\mathbf{U}_{t}\left(\mathbf{I}-\dfrac{\mathbf{q}_{t+1}\mathbf{q}_{t+1}}{||\mathbf{q}_{t+1}||^{2}}\right)\mathbf{U}_{t}^{T}\\\
&=\mathbf{U}_{t}\mathbf{Q}(\mathbf{q}_{t+1})\mathbf{U}_{t}^{T}=\mathbf{U}_{t}\mathbf{Q}(\mathbf{q}_{t+1})\mathbf{Q}(\mathbf{q}_{t+1})\mathbf{U}_{t}^{T}=\framebox{$\mathbf{U}_{t+1}\mathbf{U}_{t+1}^{T}$}.\end{array}$
In the norm we again applied that $\mathbf{U}_{t}\mathbf{U}_{t}^{T}$ is
idempotent. ∎
1. Input: Output data matrix $\mathbf{Y}\in\mathbb{R}^{m\times n_{y}}$, input data matrix $\mathbf{X}\in\mathbb{R}^{m\times n_{x}}$, $D\leq n_{y}$ the number of variables to select, $\kappa$ kernel on variables 2. Output: Set $\mathcal{I}_{D}$ of indices of selected variables from $\mathbf{X}$ in the selection order. The algorithm: 1. Compute the cross-kernel matrix $\mathbf{K}_{YX}=[\kappa(y_{i},x_{j})],\ i\in[n_{y}],j\in[n_{x}]$. 2. Compute $\mathbf{K}_{YY}=[\kappa(y_{i},y_{j})],\ i,j\in[n_{x}]$; compute $\mathbf{K}_{YY}=\mathbf{V}_{Y}\mathbf{S}_{Y}^{2}\mathbf{V}_{Y}^{T}$ 3. Express $\mathbf{U}_{Y}^{T}\phi(X)^{T}$ as $\mathbf{R}^{(0)}=\mathbf{S}_{Y}^{+}\mathbf{V}_{Y}^{T}\mathbf{K}_{YX}$. 4. Let $t=0$, $\mathcal{I}_{t}=\emptyset$. 5. For $t\in\\{0,\dots,D-1\\}$ Do (a) If $t>0$ Then $\mathbf{R}^{(t)}=\mathbf{R}^{(t-1)}-\dfrac{\mathbf{q}_{t-1}\mathbf{q}_{t-1}^{T}\mathbf{R}^{(t-1)}}{||\mathbf{q}_{t-1}||^{2}}$ projection update (b) $k_{t}^{*}=\operatorname*{arg\,max}\limits_{j\in[n_{x}]\setminus\mathcal{I}_{t}}\left(\sum_{i=1}^{n_{y}}\left(\mathbf{R}^{(t)}\right)_{i,j}^{\circ 2}\right)$ (c) $\mathbf{q}_{t}=\mathbf{R}^{(t)}[:,k^{*}_{t}]$; $\mathcal{I}_{t}=\mathcal{I}_{t-1}\cup\\{k_{t}^{*}\\}$
Figure 3: Efficient implementation of the kernelized realization of supervised
variable selection by projection algorithm, ProjSe. Note the notation, e.g.
$\mathbf{R}^{(t)}=\mathbf{U}_{t}\phi(X)$.
We can express the main computation step, Step 2.b in Algorithm 2, by
exploiting the kernelized recursive iteration. From the sequential procedure
we can see that a key step of the computation is the calculation of the
vectors $\mathbf{q}_{i}$ via Equation (18),
$\mathbf{U}_{Y}^{T}\phi(\mathbf{x})=\mathbf{S}_{Y}^{-1}\mathbf{V}_{Y}^{T}\kappa(\mathbf{Y},\mathbf{x})$
for an arbitrary $\phi(\mathbf{x})\in\mathcal{H}$. In iteration $t$, we have
$\begin{array}[]{@{}l@{}l@{}}\mathbf{q}_{t+1}=\mathbf{U}_{t}^{T}\phi(\mathbf{x})=\left(\mathbf{U}_{Y}\prod_{s=1}^{t-1}\mathbf{Q}(\mathbf{q}_{s})\right)^{T}\phi(\mathbf{x})=\mathbf{Q}(\mathbf{q}_{t})\cdot\dots\cdot\mathbf{Q}(\mathbf{q}_{1})\underbrace{\mathbf{U}_{Y}^{T}\phi(\mathbf{x})}_{\mathbf{S}_{Y}^{-1}\mathbf{V}_{Y}^{T}\kappa(\mathbf{Y},\mathbf{x})}.\end{array}$
(24)
Taking advantage of the recursive definition of
$\mathbf{U}_{t}^{T}\phi(\mathbf{x})$ we also have that
$\begin{array}[]{ll}\mathbf{U}_{t+1}^{T}\phi(\mathbf{x})&=\mathbf{Q}(\mathbf{q}_{t+1})\mathbf{U}_{t}^{T}\phi(\mathbf{x})=\left(\mathbf{I}-\dfrac{\mathbf{q}_{t+1}\mathbf{q}_{t+1}^{T}}{||\mathbf{q}_{t+1}||^{2}}\right)\mathbf{U}_{t}^{T}\phi(\mathbf{x}),\end{array}$
(25)
where $\mathbf{q}_{t+1}=\mathbf{U}_{t}^{T}\phi(\mathbf{x}_{k_{(t+1)*}})$, thus
all terms relate to those computed in the previous iteration. The computation
of the norm $||\mathbf{q}_{t+1}||^{2}$ can also exploit the recursive nature
of the algorithm. Finally, all the feature representations $\phi(\mathbf{x})$
and $\phi(\mathbf{Y})$ are implicit, and are only expressed via kernel
evaluations since they only appear in inner products.
Based on these statements and Proposition 17 we can present a concrete
implementation of our algorithm in Figure 3. In the first step the kernels are
computed, where $\mathbf{K}_{YX}$ requires $O(mn_{y}n_{x})$, and
$\mathbf{K}_{Y}$ $O(mn_{y}^{2})$ operations in case of for example linear and
Gaussian kernels. For the eigenvalue decomposition of $\mathbf{K}_{Y}$ we need
$O(n_{y}^{3})$ operations, where $D\leq\min(n_{y},n_{x})$. In the algorithm,
the critical step is Step 4.a. Its complexity in step $t$ is
$O(n_{y}(n_{x}-t))$, thus, in general for selecting $D$ variables we need
$O(n_{y}n_{x}D)$ operations. Assuming that $m\gg n_{y},n_{x}$ , the dominating
part is the computation of the kernels, thus the entire complexity is equal to
$O(mn_{y}\max(n_{x},n_{y}))$.
## 5 Experiments
In this section we experimentally validate our approach.333The code for the
algorithm is available https://github.com/aalto-ics-kepaco/ProjSe. 444The
experiments are run on a machine with this parameters: 12th Gen Intel CoreTM
i5-12600K * 10. We first show demonstrations on our algorithm’s scalability on
synthetic data, before moving on to experimenting with real data and analysing
the stability of the feature selection.
### 5.1 Scalability demonstration with synthetic data
This test is implemented via a scheme presented by (27) in Figure 4 and by
(26). The components of the input matrix, $\mathbf{X}$ and the components of a
transformation matrix $\mathbf{W}$ are independently sampled from normal
distribution. Then output matrix is constructed, and finally random noise is
added to the output.
$\begin{array}[]{l@{\ }c@{\ }c@{\ }c@{\ }c}\text{Input}&&\text{Linear
transformation}&&\text{Noise}\\\
\hline\cr&&\mathbf{W}\sim[\mathcal{N}(0,\sigma)]^{n_{x}\times
n_{y}}&&\mathbf{E}\sim[\mathcal{N}(0,\sigma)]^{m\times n_{y}}\\\
&&\Downarrow&&\Downarrow\\\ \mathbf{X}\sim[\mathcal{N}(0,\sigma)]^{m\times
n_{x}}&\Longrightarrow&\mathbf{Y}=\mathbf{X}\mathbf{W}&\Longrightarrow&\tilde{\mathbf{Y}}=\mathbf{Y}+\mathbf{E}.\end{array}$
(26)
We apply ProjSe to this data with various sample sizes. Figure 4 presents the
dependence of the selection time on the sample size, where the maximum sample
size is $10$ million and the number of variables is $10$ – the variable
selection is performed in less than four seconds.
| $\begin{array}[]{ll}\mathbf{Y}=\mathbf{X}\mathbf{W}+\mathbf{E}\\\
\hline\cr\mathbf{X}\sim[\mathcal{N}(0,\sigma)]^{m\times n_{x}},&m=1,\dots,10\
\text{million},\\\ \mathbf{W}\sim[\mathcal{N}(0,\sigma)]^{n_{x}\times
n_{y}},&n_{x}=100,\\\ \mathbf{E}\sim[\mathcal{N}(0,\sigma)]^{m\times
n_{y}},&n_{y}=100.\\\ \end{array}$ (27)
---|---
Figure 4: The dependence of the variable selection time on the sample size is
shown in seconds on the left, and the random data generation scheme applied is
on right
### 5.2 Feature selection from biological data
In this set of experiments, we compare our approach to [Brouard et al., 2022]
– a kernel-based feature selection method, where kernels are considered
traditionally on data samples instead of features. We experiment with the two
gene expression datasets, ”Carcinoma”, ”Glioma”, considered there for
unsupervised feature selection. While this kind of setting with one-view fat
data is not the one our method was developed for, as the scalability we
propose is first and foremost for large sample sizes, these experiments still
serve for illustrative comparison of feature selection performance.
Figure 5: Clustering results (NMI) on Carcinoma (right) and Glioma (left)
datasets with ProjSe as functions of number of variables chosen, averaged over
20 runs of k-means. ”Full” refers to results when full set of features is
used. Table 2: ProjSe clustering results (NMI and time) with selected
features (10 or 300) compared to results reported in [Brouard et al., 2022].
Running time of ProjSe for choosing 300 features; running time and variation
of k-means is negligible.
| Carcinoma ($m$=174, $n_{x}$=9182, $C$=11) | Glioma ($m$=50, $n_{x}$=4434, $C$=4)
---|---|---
| NMI(10) | NMI(300) | t (s) | NMI(10) | NMI(300) | t (s)
lapl | 0.36 (0.02) | 0.64 (0.04) | 0.25 (0.04) | 0.50 (0.03) | 0.47 (0.06) | 0.02 (0.00)
NDFS | 0.22 (0.28) | 0.78 (0.03) | 6,162 (305) | 0.20 (0.04) | 0.36 (0.07) | 368 (21)
UKFS | 0.57 (0.03) | 0.75 (0.05) | 326 (52) | 0.26 (0.05) | 0.42 (0.05) | 23.74 (4.03)
ProjSe lin | 0.31 (0.01) | 0.58 (0.03) | 1,146111Running time of ProjSe for choosing 300 features; running time and variation of k-means is negligible. | 0.37 (0.01) | 0.52 (0.03) | 210111Running time of ProjSe for choosing 300 features; running time and variation of k-means is negligible.
ProjSe poly | 0.34 (0.01) | 0.79 (0.03) | 1,239111Running time of ProjSe for choosing 300 features; running time and variation of k-means is negligible. | 0.13 (0.02) | 0.34 (0.07) | 299111Running time of ProjSe for choosing 300 features; running time and variation of k-means is negligible.
ProjSe RBF | 0.33 (0.01) | 0.82 (0.03) | 1,263111Running time of ProjSe for choosing 300 features; running time and variation of k-means is negligible. | 0.38 (0.05) | 0.23 (0.06) | 284111Running time of ProjSe for choosing 300 features; running time and variation of k-means is negligible.
As the data is only available in one view in unsupervised setting, we apply
our algorithm by using the data in both views: as the reference/label view and
as the view the feature selection is performed on. Intuitively, this would
filter out the noise and redundant features in the view. In our method we
consider linear kernel, $k(\mathbf{x},\mathbf{z})=\mathbf{x}^{T}\mathbf{z}$,
polynomial kernel of degree 3,
$k(\mathbf{x},\mathbf{z})=(\mathbf{x}^{T}\mathbf{z})^{3}$, and RBF kernel,
$k(\mathbf{x},\mathbf{z})=\exp(\|\mathbf{x}-\mathbf{z}\|^{2}/(2\sigma^{2}))$
with the kernel parameter $\sigma$ set as mean of pairwise distances. We
assess the performance of the feature selection by measuring the normalised
mutual information (NMI) of k-means clustering results. Here the clusterer has
been given the amount of classes in the data as the number of clusters.
The results are displayed in Table 2, with comparison to selected methods from
[Brouard et al., 2022]: UKFS proposed there, as well as a scoring-based method
”lapl” [He et al., 2005] that performed well on Glioma dataset, and NDFS [Li
et al., 2012], a clustering-based approach that performed well with Carcinoma
dataset. Our method is very competitive with these, sometimes achieving better
performance. As in our method the kernel is calculated on features, the
running time is slower than for UKFS where the kernel on samples is
considered. However notably we are still competitive when compared to the
NDFS.
Additionally, Figure 2 displays more detailed clustering results with respect
to the number of variables chosen by ProjSe. These results also highlight the
differences that can be obtained by applying different kernels on the
features: with Carcinoma dataset the non-linear kernels, RBF and polynomial
kernel of degree 3, are clearly superior, while with Glioma linearity works
the best.
### 5.3 Feature selection from vector-valued output setting
Table 3: The time series classification datasets. Dtaset name | # tr. samples | # test samples | # features | # classes
---|---|---|---|---
Crop | 7200 | 16800 | 46 | 24
NonInvasiveFetalECGThorax1 | 1800 | 1965 | 750 | 42
ShapesAll | 600 | 600 | 512 | 60
Figure 6: Results with time series datasets. The top row reports the kernel
alignment of the input kernel with chosen variables (either RBF or linear) to
the ideal output kernel. The bottom row reports accuracy on test set, again
with both linear and RBF kernels for SVM; comparison is shown to randomly
selected features and to full feature set. The colors differentiate which
kernel is used on features in ProjSe, while the line style indicates if
traditional linear or RBF kernel is used on samples.
We next consider a setting more aligned with our method: supervised feature
selection with vector-valued output as the reference view. Here we consider
three datasets from the UEA & UCR Time Series Classification
Repository555http://www.timeseriesclassification.com/dataset.php, Crop,
NonInvasiveFetalECGThorax1 (”Thorax”), and ShapesAll, as detailed in Table 3.
These datasets are associated with multi-class classification tasks, and we
use the one-hot encoding of the labels as the vector-valued output to perform
the feature selection with ProjSe. As before, we consider linear, polynomial
and RBF kernels. We assess the success of the feature selection task by
performing classification with SVM with the selected features - here we
consider both linear and RBF kernels on the data samples.
The results are displayed in Figure 6, where both kernel alignment
($KA(\mathbf{K},\mathbf{K}^{\prime})=\langle\mathbf{K}_{c},\mathbf{K}^{\prime}_{c}\rangle_{F}/(\|\mathbf{K}_{c}\|_{F}\|\mathbf{K}^{\prime}_{c}\|_{F})$
where $c$ denotes centering) to the linear kernel on the one-hot-encoded
outputs, and accuracy of SVM classification are shown. The different kernels
used in feature selection give slightly different features; however the
performance on the subsequent classification task is mostly dependent on which
kernel is used on the samples. Especially for Thorax and ShapesAll datasets
with higher dimensionality, it can be seen that all the ProjSe results with
linear SVM outperform using the full set of features.
### 5.4 Experiments with two-view data
Table 4: The detailed computation times in seconds for the variable selection method, where 10, 20, 50 and 100 variables are extracted. | | MediaMill | Cifar 100
---|---|---
| Linear | RBF | Linear | RBF
Computing $K_{yx}$ | 0.015 | 0.019 | 0.221 | 0.470
Computing $K_{yy}$ | 0.007 | 0.015 | 0.006 | 0.025
Eigen decomp. of $K_{yy}$ | 0.003 | 0.001 | 0.001 | 0.001
Centralization of $\mathbf{K}_{xx}$ 111Optional | 0.005 | 0.014 | 1.462 | 2.065
10 variables | 0.035 | 0.045 | 1.848 | 2.640
20 variables | 0.024 | 0.048 | 1.793 | 2.657
50 variables | 0.037 | 0.050 | 1.804 | 2.671
100 variables | 0.049 | 0.061 | 1.833 | 2.706
In our last set of experiments, we consider the following datasets:
* •
MNIST handwritten digits [LeCun et al., 1998, LeCun, 1998]: This dataset
contains 60000 training and 10000 test examples of handwritten digits in
greyscale. The number of pixels in each image is $28\times 28=784$, resulting
in total to 784 variables. To construct the two sets of variables, the image
columns are split into half similarly as in [Andrew et al., 2013]. Thus both
views comprise of 392 variables.
* •
MediaMill dataset [Snoek et al., 2006]: This dataset contains 43907 examples
which are extracted from keyframes of video shots. There are two views in this
data: text annotations (101 variables) and visual features (120 variables).
* •
Cifar100 dataset [Krizhevsky, 2009]: This dataset, chosen to demonstrate the
scalability of ProjSe, contains 50000 training and 10000 test examples of
color images. The number of pixels in each image is $32\times 32=1024$, where
to each pixel 3 colors are assigned. The examples belong to 100 classes, where
each class contains 500 training and 100 test examples. The classes are
represted by indicator vectors.
We perform variable selection independently on both views in the datasets.
After the variable selection is performed on both sides, we compute canonical
correlations between all subset pairs of the extracted variables, starting
from the first ones and incrementally growing them to the entire sets. To
demonstrate the performance of the proposed variable selection algorithm
ProjSe, it is compared to the following methods: large-scale sparse kernel
canonical correlation analysis (GradKCCA), [Uurtio et al., 2019], deep
canonical correlation analysis (DCCA), [Andrew et al., 2013], randomized non-
linear CCA (RCCA), [Lopez-Paz et al., 2014], kernel non-linear orthogonal
iterations (KNOI), [Wang and Livescu, 2015], and CCA through Hilbert-Schmidt
independence criterion (SCCA-HSIC), [Uurtio et al., 2018].
These CCA variants explicitly or implicitly rely on singular value
decomposition, and their performance highly depends on the distribution of the
singular values of the data matrices. Since the data matrices have small
number of dominating singular values, we expect from a variable selection
method that it can capture a relatively small set of variables to reproduce
similar accuracy, measured in canonical correlation. We need to bear in mind
that the CCA-based methods add up all information represented by the full
collection of variables, however the projector based selection only relies a
relatively small subset of those variables.
(a) MNIST dataset
(b) MediaMill dataset
Figure 7: Variable selection results w.r.t the number of selected variables. Table 5: CCA comparison on the MNIST and MediaMill datasets. | | MNIST | MediaMill
---|---|---
| $\rho_{\text{TEST}}$ | TIME (s) | $\rho_{\text{TEST}}$ | TIME (s)
Generic CCA | 0.923 | 2.40 | 0.675 | 0.429
GradKCCA | 0.952 $\pm$ 0.001 | 56 $\pm$6 | 0.657 $\pm$ 0.007 | 8 $\pm$4
DCCA | 0.943 $\pm$ 0.003 | 4578 $\pm$203 | 0.633 $\pm$ 0.003 | 1280 $\pm$112
RCCA | 0.949 $\pm$ 0.010 | 78 $\pm$13 | 0.626 $\pm$ 0.005 | 23 $\pm$9
KNOI | 0.950 $\pm$ 0.005 | 878 $\pm$62 | 0.645 $\pm$ 0.003 | 218 $\pm$73
SCCA-HSIC | 0.934 $\pm$ 0.006 | 5611 $\pm$193 | 0.625 $\pm$ 0.002 | 1804$\pm$143
ProjSe 10 var. | 0.847 | | 0.542 |
ProjSe 20 var. | 0.890 | | 0.586 |
ProjSe 50 var. | 0.918 | | 0.631 |
ProjSe 100 var. | 0.935 | 0.45111The algorithm was run once with 100 variables, the smaller amounts were subsampled | 0.672 | 0.041111The algorithm was run once with 100 variables, the smaller amounts were subsampled
Figure 7 shows the performance of ProjSe on MNIST and MediaMill datasets with
both linear and Gaussian kernels, as functions of the number of selected
variables. The results are measured by canonical correlation between the
subsets of variables selected from the two views. Comparing the results to the
other CCA methods in Table 5 (taken from [Uurtio et al., 2019]), we observe
ProjSe obtaining comparable performance after 20 or 50 selected variables,
while being orders of magnitude faster than the other methods. Since ProjSe is
fully deterministic, there is no variance is reported for it. The heaviest
computation for ProjSe is in the beginning when eigenvalue decomposition of
$\mathbf{K}_{YY}$ is calculated (see Table 4). Thus the running time varies
only minimally when different number of variables is selected. This is also
demonstrated in 4 where the running times for MNIST and Cifar100 datasets are
detailed.
|
---|---
Figure 8: Means of the two stability scores on random sub-samples computed on
the data sets the MNIST (left) and the MediaMill (right), with linear kernel
As a variable selection method, we are interested in evaluating the stability
of the selection process. In order to measure this, we here consider the
stability index [Nogueira et al., 2018] and the average Pearson correlation of
relevance [Hamer and Dupont, 2021]. In both measures, higher values indicate
higher stability; the maximum value in both is 1.
First the number of extracted variables is chosen from $(1,2,\dots,5)$ in
MNIST, and from $(10,20,\dots,50)$ in MediaMill. For each number of selected
variables the subsamples are taken with the following percentage of the entire
training sets: $(10\%,20\%,\dots,50\%)$. Then random subsets are extracted
$10$ times of the size given above. The scores are computed for each number of
selected variables, and for each subsample size and finally averaged on all
random samples. They are shown in Figure 8, where the averages for all pairs
of subsets of variables and for all subsample sizes are presented.
## 6 Conclusion
In this paper we introduced a novel variable selection method for two-view
settings. Our method is deterministic, selecting variables based on
correlation defined with projection operators. The kernelised formulation of
our approach paves way for efficient and highly scalable implementation,
allowing the application of our method to datasets with millions of data
samples. We empirically demonstrated this efficiency and the suitability of
our approach for feature selection task, with both synthetic and real data.
## Declarations
The authors wish to acknowledge the financial support by Academy of Finland
through the grants 334790 (MAGITICS), 339421 (MASF) and 345802 (AIB), as well
as the Global Programme by Finnish Ministry of Education and Culture
## References
* [Aghazadeh et al., 2018] Aghazadeh, A., Spring, R., LeJeune, D., Dasarathy, G., Shrivastava, A., et al. (2018). Mission: Ultra large-scale feature selection using count-sketches. In ICML, pages 80–88. PMLR.
* [Anderson. and Duffin, 1969] Anderson., J. W. N. and Duffin, R. J. (1969). Series and parallel addition of matrices. SIAM J. Appl. Math., 26:576–594.
* [Andrew et al., 2013] Andrew, G., Arora, R., Bilmes, J., and Livescu, K. (2013). Deep canonical correlation analysis. In Dasgupta, S. and McAllester, D., editors, Proceedings of the 30th ICML, volume 28(3) of Proceedings of Machine Learning Research, pages 1247–1255, Atlanta, Georgia, USA. PMLR.
* [Anette and Nokto, 2018] Anette, K. and Nokto, D. (2018). A benchmark of prevalent feature selection algorithms on a diverse set of classification problems. Master’s thesis.
* [Ben-Israel, 2015] Ben-Israel, A. (2015). Projectors on intersections of subspaces. Contemporary Mathematics, 636.
* [Ben-Israel and Greville, 2003] Ben-Israel, A. and Greville, T. N. (2003). Generalized inverses: Theory and applications. Springer New York, NY, second edition.
* [Bie et al., 2005] Bie, T. D., Cristianini, N., and Rosipal, R. (2005). Eigenproblems in pattern recognition. In Handbook of Geometric Computing : Applications in Pattern Recognition, Computer Vision, Neuralcomputing, and Robotics, pages 129–170. Springer-Verlag.
* [Bommert et al., 2020] Bommert, A., Sun, X., Bischl, B., Rahnenführer, J., and Lang, M. (2020). Benchmark for filter methods for feature selection in high-dimensional classification data. Computational Statistics & Data Analysis, 143(C).
* [Boyd et al., 2018] Boyd, N., Hastie, T., Boyd, S., Recht, B., and Jordan, M. I. (2018). Saturating splines and feature selection. JMLR, 18(197):1–32.
* [Brogat-Motte et al., 2022] Brogat-Motte, L., Rudi, A., Brouard, C., Rousu, J., and d’Alché-Buc, F. (2022). Vector-valued least-squares regression under output regularity assumptions. JMLR, 23(344):1–50.
* [Brouard et al., 2022] Brouard, C., Mariette, J., Flamary, R., and Vialaneix, N. (2022). Feature selection for kernel methods in systems biology. NAR genomics and bioinformatics, 4(1):lqac014.
* [Conway, 1997] Conway, J. B. (1997). Course in Functional Analysis. Springer, 2nd edition.
* [Deshwal et al., 2019] Deshwal, A., Doppa, J. R., and Roth, D. (2019). Learning and inference for structured prediction: A unifying perspective. In IJCAI-19.
* [Fan et al., 2009] Fan, J., Samworth, R., and Wu, Y. (2009). Ultrahigh dimensional feature selection: Beyond the linear model. JMLR, 10:2013–2038.
* [Golub and Loan, 2013] Golub, G. H. and Loan, C. F. V. (2013). Matrix Computations. The Johns Hopkins University Press, Baltimore, MD, 4th edition.
* [Halperin, 1962] Halperin, I. (1962). The product of projection operators. Acta Sei. Math. (Szeged), 23:96–99.
* [Hamer and Dupont, 2021] Hamer, V. and Dupont, P. (2021). An importance weighted feature selection stability measure. JMLR, 22(116):1–57.
* [Hayashi and Nagaoka, 2002] Hayashi, M. and Nagaoka, H. (2002). A general formula for the classical capacity of a general quantum channel. In Proceedings IEEE International Symposium on Information Theory,. IEEE.
* [He et al., 2005] He, X., Cai, D., and Niyogi, P. (2005). Laplacian score for feature selection. Advances in neural information processing systems, 18.
* [Hofmann et al., 2008] Hofmann, T., Schölkopf, B., and Smola, A. J. (2008). Kernel methods in machine learning. The annals of statistics, pages 1171–1220.
* [Hotelling, 1936] Hotelling, H. (1936). Relations between two sets of variates. Biometrika, 28(3/4):321.
* [Jordan et al., 2021] Jordan, M. I., Liu, K., and Ruan, F. (2021). On the self-penalization phenomenon in feature selection. arXiv preprint arXiv:2110.05852.
* [Kreyszig, 1989] Kreyszig, E. (1989). Introductory Functional Analysis with Applications. Wiley.
* [Krizhevsky, 2009] Krizhevsky, A. (2009). Learning multiple layers of features from tiny images. Technical report, University of Totonto.
* [LeCun, 1998] LeCun, Y. (1998). The mnist database of handwritten digits. exdb.
* [LeCun et al., 1998] LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. (1998). Gradient-based learning applied to document recognition. In Proceedings of the IEEE, volume 86(11), pages 2278–2324.
* [Li et al., 2017] Li, J., Cheng, K., Wang, S., Morstatter, F., Trevino, R. P., Tang, J., and Liu, H. (2017). Feature selection. ACM Computing Surveys, 50(6):1–45.
* [Li et al., 2012] Li, Z., Yang, Y., Liu, J., Zhou, X., and Lu, H. (2012). Unsupervised feature selection using nonnegative spectral analysis. In Proceedings of the AAAI, volume 26, pages 1026–1032.
* [Lopez-Paz et al., 2014] Lopez-Paz, D., Sra, S., Smola, A., Ghahramani, Z., and Schoelkopf, B. (2014). Randomized nonlinear component analysis. In Xing, E. P. and Jebara, T., editors, Proceedings of the 31st ICML, volume 32(2) of Proceedings of Machine Learning Research, pages 1359–1367, Bejing, China. PMLR.
* [Micchelli and Pontil, 2005] Micchelli, C. A. and Pontil, M. (2005). On learning vector-valued functions. Neural computation, 17(1):177–204.
* [Minh et al., 2016a] Minh, H. Q., Bazzani, L., and Murino, V. (2016a). A unifying framework in vector-valued reproducing kernel hilbert spaces for manifold regularization and co-regularized multi-view learning. volume 17, pages 769–840. JMLR. org.
* [Minh et al., 2016b] Minh, H. Q., Biagio, M. S., Bazzani, L., and Murino, V. (2016b). Approximate log-hilbert-schmidt distances between covariance operators for image classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5195–5203.
* [Muandet et al., 2017] Muandet, K., Fukumizu, K., Sriperumbudur, B., Schölkopf, B., et al. (2017). Kernel mean embedding of distributions: A review and beyond. Foundations and Trends® in Machine Learning, 10(1-2):1–141.
* [Nielsen and Chuang, 2000] Nielsen, M. and Chuang, I. (2000). Quantum Computation and Quantum Information. Cambridge University Press.
* [Nogueira et al., 2018] Nogueira, S., Sechidis, K., and Brown, G. (2018). On the stability of feature selection algorithms. JMLR, 18(174):1–54.
* [Rasmussen and Williams, 2005] Rasmussen, C. E. and Williams, C. K. I. (2005). Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press.
* [Schölkopf et al., 1998] Schölkopf, B., Smola, A., and Müller, K.-R. (1998). Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation, 10(5):1299–1319.
* [Snoek et al., 2006] Snoek, C. G. M., Worring, M., van Gemert, J. C., Geusebroek, J.-M., and Smeulders, A. W. M. (2006). The challenge problem for automated detection of 101 semantic concepts in multimedia. In Proceedings of the 14th ACM International Conference on Multimedia, MM ’06, page 421–430.
* [Song et al., 2012] Song, L., Smola, A., Gretton, A., Bedo, J., and Borgwardt, K. (2012). Feature selection via dependence maximization. JMLR, 13(1):1393–1434.
* [Tjur, 1984] Tjur, T. (1984). Analysis of variance models in orthogonal designs. International Statistical Review, Revue Internationale de Statistique, 52:33–65.
* [Uurtio et al., 2018] Uurtio, V., Bhadra, S., and Rousu, J. (2018). Sparse non-linear cca through hilbert-schmidt independence criterion. In 2018 IEEE International Conference on Data Mining (ICDM), pages 1278–1283, Los Alamitos, CA, USA. IEEE Computer Society.
* [Uurtio et al., 2019] Uurtio, V., Bhadra, S., and Rousu, J. (2019). Large-scale sparse kernel canonical correlation analysis. In Chaudhuri, K. and Salakhutdinov, R., editors, Proceedings of the 36th ICML, volume 97 of Proceedings of Machine Learning Research, pages 6383–6391. PMLR.
* [Valcárcel et al., 2022] Valcárcel, L. V., San José-Enériz, E., Cendoya, X., Rubio, A., Agirre, X., Prósper, F., and Planes, F. J. (2022). Boso: A novel feature selection algorithm for linear regression with high-dimensional data. PLOS Computational Biology, 18(5):1–29.
* [von Neumann, 1950] von Neumann, J. (1950). Functional Operators. Vol II: The Geometry of Orthogonal Subspaces, volume 39. Annals of Math. Studies, Princeton University Press, Princeton.
* [Wang and Livescu, 2015] Wang, W. and Livescu, K. (2015). Large-scale approximate kernel canonical correlation analysis. CoRR, abs/1511.04773.
* [Xu et al., 2013] Xu, C., Tao, D., and Xu, C. (2013). A survey on multi-view learning. arXiv preprint arXiv:1304.5634.
* [Zebari et al., 2020] Zebari, R., Abdulazeez, A., Zeebaree, D., Zebari, D., and Saeed, J. (2020). A comprehensive review of dimensionality reduction techniques for feature selection and feature extraction. J. Appl. Sci. Technol. Trends, 1(2):56–70.
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# Accurate and precise measurement of the wavelength of
4d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$ transition of Kr II
Y. Dancheva The author to whom correspondence may be addressed:
<EMAIL_ADDRESS>P. Coniglio D. Pagano A. Garde F.
Scortecci Aerospazio Tecnologie Srl, via dei Tessili 99, 53040 Rapolano Terme
(SI), Italy
###### Abstract
Electric propulsion requires exhaustive ground test campaigns to obtain an
accurate characterization of the propulsion devices, or thrusters, used by the
spacecraft. Among the many plasma parameters, accurately measured during the
tests, that of the ion velocity is key, and can be measured using non-
intrusive tools such as Laser-Induced Fluorescence (LIF) diagnostics. The ion
velocity is inferred by Doppler shift measurements that presupposes a precise
and accurate knowledge of the wavelength of excitation of the ions at rest.
Today electric propulsion is moving towards the use of Krypton as a
propellant, due to the dramatic increase in cost of the more advantageous
Xenon gas propellant, commonly used until now. The utilization of Krypton
implies that LIF diagnostic tool used be adapted accordingly. In this work an
accurate measurement of the wavelength of
4d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$ transition of Kr II (one
of the most advantageous transitions for LIF diagnostics) is performed in the
plasma of a hollow cathode acting as a neutralizer. These measurements will
significantly improve accuracy in the determination of the ion velocity.
## I Introduction
Traditionally, Xenon has been the favorite propellant gas for electric
propulsion applications as it provides an optimal compromise between
performance and ease of handling. Although Xenon has numerous technical
advantages, its relative scarcity and resulting increase in price are now
posing considerable budgetary constrains for its use, especially for deep
space exploration Tirila_2023 ; Nakles_2011 . Therefore, a number of
alternative propellants are being examined intensively for use in
electrostatic spacecraft propulsion thrusters Duchemin_2009 . A more economic
alternative to Xe needs to be identified, while maintaining thruster
performance levels, and without overlooking performance optimization aspects.
Among various types of electric thrusters, the Hall Effect Thruster (HET) is
one of the most widely used owing to its simple structure, high reliability,
and long life Mazouffre_2016 ; Levchenko_2018 .
Like Xenon, Krypton is a noble gas and can be integrated easily into existing
spacecraft propellant management systems with minimal modifications. As this
gas has similar ionization potential it should not dramatically affect
thruster efficiency, and the lower atomic mass could possibly produce a 25%
increase in specific impulse due to the increased propellant exit velocity of
lighter ions Linnell_2007 ; Hargus_2012 . The higher specific impulse also
provides advantages in space missions such as maintaining spacecraft orbit
Kurzyna_2018 ; Lim2019 . A lower anode efficiency of approximately 5-15$\%$ is
expected with respect to Xenon propellant at the same operating conditions
Hargus_2012 ; Kurzyna_2018 ; Jorns_2022 . Krypton is also about ten times more
common in the atmosphere than Xenon, and hence less expensive. Currently,
Krypton is the most widely-used alternative propellant in HET both for space
applications (for example the Starlink constellation), as well as ground
testing Tirila_2023 . Table 1 summarizes the properties of Xenon and Krypton
specifically relevant to electrostatic spacecraft propulsion.
Table 1: Kr and Xe properties comparison Property | Xe | Kr
---|---|---
Atomic mass | 131.293 amu | 83.798 amu
First ionization energy | 12.1 eV | 14.0 eV
Atmospheric | |
concentration | 87 ppb | 1000 ppb
Stable isotopes | 9 | 6
Odd isotopes | 2 | 1
Critical pressure | 57.65 atm | 54.3 atm
Critical temperature | 290 K | 209 K
Boiling point (1 atm) | 161 K | 120 K
The characterization and qualification of ion thrusters requires extensive,
long-duration test campaigns in space simulators. A valuable tool for
characterizing thruster performance and comparing this with already-developed
models is Light-Induced Fluorescence (LIF) spectroscopy, which measures the
velocity of ions employing the Doppler effect. Many excellent works on LIF
characterization of ion thrusters operating with Xe propellant are available
in the literature, but only few of them consider operation with Krypton.
Advantageous transitions for LIF diagnostics are characterized by large
intensity and narrow intrinsic width (transitions involving long-living
levels). The Xe transition at 834 nm
(5d${}^{2}[4]_{7/2}\rightarrow$6p${}^{2}[3]^{\circ}_{5/2}$) is used for LIF
diagnostics by the majority of the electric propulsion research community. As
to Kr, such favoured transitions are at 820 nm (the
4d${}^{4}F_{7/2}\rightarrow$5p${}^{2}D^{\circ}_{5/2}$ transition) Lejeune_2012
and at 729 nm (the 4d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$
transition) Hargus_2011 ; Buorgeois_2011 . An accurate knowledge of the
features of the transition used for LIF spectroscopy is required for better
determination of the ion velocity.
The air wavelength of the Kr II
4d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$ transition has been
measured to be 728.982 nm with an accuracy of 0.7 pm Dzierżęga_2001 ;
Saloman_2007 . Such accuracy would introduce an error of about 300 m/s in the
determination of the Kr ion velocity. In this work, the vacuum/air wavelength
of the Kr II transition 5d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$
is measured with an improved accuracy (34 fm). For this purpose, the LIF
spectroscopy is applied in the plasma of a hollow cathode discharge, normally
used as a neutralizer in electric propulsion.
## II Set-up
The LIF is a particularly powerful optical-diagnostic tool providing valuable,
species selective information with excellent spatial resolution. A highly
coherent, single-frequency, laser source that can be tuned over a broad
spectral interval permits to establish a versatile laser-plasma interaction
tool. Given its remote detection nature, LIF diagnostic is minimally invasive.
The plasma is interrogated locally with an electromagnetic tool: all the
optical devices used to illuminate the plasma and to detect its fluorescence
are placed far from the interaction point, can have dimensions in the mm
scale, and the interaction volume can have sizes as small as few millimeters.
The ion velocity distribution function (IVDF) can be measured along different
directions with excellent spatial and time resolution. The measurement
procedure is based on detecting the intensity of the fluorescence emitted by
the plasma in the interaction point in response to the illumination by means
of narrow-band radiation characterized by an accurately and precisely known
and scanned frequency.
### II.1 The test facility
The measurements shown in this work are conducted in a non-magnetic stainless
steel vacuum chamber. Vacuum is obtained by a single stage cryogenic panel.
The base pressure of the vacuum chamber is as low as $10^{-7}$mbar and
increases up to about $10^{-5}$mbar during the cathode operation. The hollow
cathode is operated in diode regime (see Fig.1) at currents of about 3 A with
an anode positioned 20 mm away from it. The measurements are performed
positioning the LIF equipment outside the vacuum chamber illuminating the
detection region and detecting the emitted fluorescence through an optical
access (view port).
Figure 1: Schematics of the laser excitation and fluorescence collection
arrangement. FAS - fluorescence analyzing system, FCO - fluorescence
collection objective, VP - view port of the vacuum chamber, SMF, MMF - single-
mode and multi-mode fibers and AOM - acousto-optical modulator.
### II.2 The LIF set-up
The light source used to excite the
4d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$ Kr II transition is a
tunable diode laser in a primary-and-secondary configuration (the primary
laser is an extended-cavity diode laser and the secondary laser is a tapered
amplifier). The laser linewidth is better than 1 MHz and the power available
is in excess.
Figure 2: Schematic of the LIF set-up: Proportional-Integral-Derivative (PID)
controller; acousto-optical modulator (AOM); photo-multiplier tube (PMT); and
data acquisition card (DAQ).
A schematic of the LIF set-up is given in Fig.2 (a more detailed description
can be found in Refs.Dancheva_2022 ; Dancheva_2023 ). The laser wavelength is
locked and scanned with the help of a high accuracy ($\pm~10~$MHz) wavelength
meter and a proportional-integral-derivative (PID) controller. The wavelength
meter is calibrated periodically using a diode laser that is frequency
stabilized to the absorption profile of the Caesium D2 line and typically
drifts below 2 MHz/day.
The detection region is near to the cathode exit plane at the cathode tip
center, where the laser beam and the fluorescence collection objective (FCO)
view regions are crossing (see Fig.1). The laser beam is aligned at a small
angle with respect to the normal to cathode exit plane ($\alpha=6^{\circ}$ in
Fig.1) to excite the plasma at the tip center. The spatial resolution is
provided by the laser intensity amplitude modulation at 3 kHz by a fibre-
coupled, acousto-optical modulator (AOM). The AOM output is coupled to a
single-mode optical fiber (see Fig.2) that provides a laser beam with waist
diameter of 0.3 mm using a fiber collimator positioned at 0.6 m away from the
cathode.
The view spot of the FCO is about 2.5 mm in diameter at the region of
detection. The fluorescence signal at air wavelength of 473.9 nm
(corresponding to the transition
5p${}^{4}P^{\circ}_{5/2}\rightarrow$5s${}^{4}P_{5/2}$) is selected using a
grating monochromator. Subsequently, the signal at 473.9 nm is detected using
a photo-multiplier tube (PMT). The LIF signal demodulation is performed by a
lock-in amplifier, whose input is the PMT current, converted to voltage by a
trans-impedance amplifier. The poor signal-to-noise ratio of the LIF spectra
is counteracted by using a long lock-in integration time, implying a slow scan
of the laser wavelength and averaging over many measurements. Indeed, the
lock-in amplifier integration time is 300 ms, which with a third order low-
pass filter at the output determines a bandwidth of about 0.3 Hz .
The laser vacuum wavelength is locked and scanned by the wavelength meter
using a PID control (the schematics of the set-up are shown in Fig.2). A scan
of 20 pm (about 11 GHz) is performed backwards and forwards using the
wavelength meter in an overall time of 100 s. The bidirectional scanning
permits to account for possible shifts of the spectral line due to integration
time.
The wavelength meter used in this work is temperature and pressure stabilized.
A variation less than 50 MHz is expected for temperature variations of about
10∘C, that are well compensated for rates lower than
$\nicefrac{{2$^{\circ}$C}}{{hour}}$, which is higher than the typical
temperature excursion in the laboratory.
## III The Kr II 4d4D${}_{7/2}\rightarrow$5p4P${}^{\circ}_{5/2}$ line.
In this analysis the Kr isotopes with concentration higher than 0.3$\%$ are
considered and listed in Table 2. A theoretical line-shape is calculated and
used to fit the experimental data.
Table 2: Isotope shift and hyperfine coefficients | Relative shift |
---|---|---
Isotope | with respect to | Abundance $\alpha$
| 84Kr (MHz) | ($\%$)
78 | 1185.7 | 0.35
80 | 768.0 | 2.27
82 | 372.6 | 11.56
83 | 175.0 | 11.55
84 | 0 | 56.9
86 | -365.2 | 17.37
Electronic | A coefficient | B coefficient
state | (MHz) | (MHz)
4d4D7/2 | -43.513 | -294.921
5p4P5/2 | -167.2 | +91
Only one isotope (83Kr) has a non zero nuclear spin (I=9/2) and thus its
hyperfine structure should be included in the line-shape analysis. The
hyperfine constants of both the states of the 83Kr II transition have been
measured previously Scholl_1986 ; Schuessler_1992 and are given in Table 2.
Table 3: Kr hyperfine structure components | | Relative | Relative |
---|---|---|---|---
| | shift | shift $\delta_{i}$ | Intensity
F | F′ | (with respect | (with respect | $\beta$($\%$)
| | to J$\rightarrow$J′) (MHz) | to 84Kr) (MHz) |
8 | 7 | -1099.2 | -924.2 | 0.21
7 | 7 | -1559.6 | -1384.6 | 0.034
7 | 6 | -431.7 | -256.7 | 0.15
6 | 7 | -1901.1 | -1726.1 | 0.0031
6 | 6 | -773.2 | -598.2 | 0.055
6 | 5 | 223.2 | 398.2 | 0.10
5 | 6 | -1020.2 | -845.2 | 0.0083
5 | 5 | -23.8 | 151.2 | 0.064
5 | 4 | 827.3 | 1002.3 | 0.065
4 | 5 | -197.5 | -22.5 | 0.015
4 | 4 | 653.7 | 828.7 | 0.063
3 | 4 | 535.8 | 710.8 | 0.022
3 | 3 | 1230.4 | 1405.4 | 0.052
3 | 2 | 1759.3 | 1934.3 | 0.013
2 | 3 | 1154.3 | 1329.3 | 0.03
2 | 2 | 1683.2 | 1858.2 | 0.033
1 | 2 | 1638.3 | 1813.3 | 0.038
As can be seen from Fig.3 the lower 83Kr II state splits in 8 hyperfine
sublevels and the upper one in 6 with total angular momenta as denoted on the
energy schematic. The electric-dipole allowed 18 transitions are shown in the
plot. A detailed description about the hyperfine structure and the relevant
selection rules can be found elsewhere Arimondo_1977 ; Hargus_2010 ;
Hargus_2001 . The spectrum of the
4d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$ transition at Doppler
linewidth of 35 MHz (to highlight the line structure) and 1900 MHz (to show
the line-shape at ion temperatures near to the measured one) are reported in
Fig.3.
Figure 3: The transition line-shape at two different temperature/Doppler
widths. The hyperfine structure of 83Kr is shown in the insert.
## IV Results
Fig.4 shows the spectrum of the
4d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$ recorded when scanning
the laser frequency in both directions. Each trace is obtained as an average
over 36 laser wavelength scans and the cathode is kept at constant operational
parameters during all the measurements.
Figure 4: Kr II line at about 729 nm when scanning the laser wavelength in
both directions and measuring the vacuum a) and the air wavelength c). For the
sake of convenience, the measurements are also presented as a function of the
frequency - vacuum case, and as a function of $\nu n(p,T,\lambda)$ \- air
case, where n is the index of refraction of the dry air at the wavelength of
interest. Each measurement is an average over 36 laser scans. The residual is
shown for each case in plots b) and d).
The spectrum is registered measuring both the vacuum and the air wavelength
(at standard dry air: T=15∘C and p=760 mmHg). The experimental data are fitted
using an orthogonal distance regression algorithm Pallavi_2022 considering
the wavelength meter accuracy and the LIF signal noise outside the spectral
line as errors.
The fitting function is given by:
$\begin{split}G(\nu)=A\sum_{i}B_{i}\exp^{-4\ln{2}\big{(}\frac{\nu-(\nu_{0}\pm\delta_{i})}{\Delta\nu_{D}}\big{)}^{2}}+D,\\\
~\Delta\nu_{D}=\frac{\nu_{0}}{c}\sqrt{8\ln{2}\frac{k_{B}T}{m}},\end{split}$
(1)
where the $\delta_{i}$ is the relative shift for each spectral component
(given in Table 3), $B_{i}$ are the relative intensities
($B_{i}$=$\alpha_{i}\times\beta_{i}$), $k_{B}$ is the Boltzmann constant, $T$
is the ion temperature, $m$ is the ion mass, $A$ is a scaling factor, $c$ is
the speed of light, and $D$ accounts for possible data offset is used to fit
the experimental data (see Fig.4). Among the four fitting parameters ($A$,
$\nu_{0}$, $T$, and $D$) the line central frequency $\nu_{0}$ and the ion
temperature $T$ are of interest. The residuals, given in Fig.4, are indicative
for the fit goodness.
Figure 4 shows the measured $\nu_{0}$ when scanning the laser wavelength in
both directions. A shift, occurring due to the finite lock-in settling time of
about 100 MHz is estimated from the fitted $\nu_{0}$ values. As this
instrumental shift occurs symmetrically in both cases, the exact value of the
$\nu_{0}$ has to be estimated as the average of the two results.
The $\nu_{0}$ value is temperature dependent through the hyperfine and isotope
structure contribution. A negative systematic error in the determination of
$\nu_{0}$ is introduced at different Doppler widths (ion temperatures). The
calculated shift is given in Fig.5.
Figure 5: Kr line center shift as a function of the ion temperature (Doppler
width).
Considering a coverage factor of 2, for a confidence level of approximately
95$\%$, the measured central wavelength $\lambda_{0}$ and frequency $\nu_{0}$
are given in Table 4. Taking into account a Doppler width of 1900 MHz the
corrected $\lambda_{0}$ values are also provided.
Table 4: Measured central wavelength and frequency. HFS - hyperfine structure. | $\lambda_{0}$ | Uncertainty
---|---|---
Condition: | (nm) | (fm)
vacuum | 729.1794(87) | 34
air | 728.9786(02) | 34
| $\nu_{0}$ | Uncertainty
Condition: | (THz) | (MHz)
vacuum/air | 411.1367(14) | 20
| $\lambda_{0}$ | Uncertainty
Condition: | (nm) | (fm)
vacuum (corrected for | |
HFS and isotope shift) | 729.1795(04) | 34
air (corrected for | |
HFS and isotope shifts) | 728.9786(19) | 34
The fitted ion temperature results in the range from 3300 K to 3500 K and can
be related to the cathode wall temperature which is about 1.5 times lower.
## V Conclusions
The plasma of a hollow cathode has been used for accurate measurement of the
wavelength of the Kr II 4d${}^{4}D_{7/2}\rightarrow$5p${}^{4}P^{\circ}_{5/2}$
transition. By performing long-lasting measurements an improved signal to
noise is obtained as to make the residual noise contribute negligibly to the
error of the measurements. The theoretical spectrum of the transition is
calculated and used to fit the experimental data. Both the frequency and the
vacuum/air wavelength centers are given. Such measurements can be used, for
example, to perform velocity distribution measurements that would find
application in the field of the electric propulsion.
The systematic error due to the presence of hyperfine and isotope structure in
determination of the line centre is calculated and taken into account as well
as its variation with the ion temperature. In the field of electric
propulsion, the achieved accuracy in the characterization of the Kr II line at
729 nm will determine an uncertainty of the Kr ion velocity measurements of
only 14 m/s.
## VI Acknowledgements
This work has received funding from the European Union’s Horizon 2020 research
and innovation programme under grant agreement No 101004140. The authors would
like to thank Patricia Nugent for revising the English of the manuscript.
## VII Data availability statement
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
## References
* (1) V.-G. Tirila, A. Demairé, and C. N. Ryan, “Review of alternative propellants in Hall thrusters,” Acta Astronautica, vol. 212, pp. 284–306, 2023.
* (2) M. R. Nakles, W. A. Hargus Jr., J. J. Delgadoz, and R. L. Coreyx, “A performance comparison of xenon and krypton propellant on an SPT-100 Hall thruster,” International Electric Propulsion Conference, Weisbaden, Germany, pp. IEPC–2011–003, 2011.
* (3) O. Duchemin, D. Valentian, and N. Cornu, “Cryostorage of propellants for electric propulsion,” 45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 2009.
* (4) S.Mazouffre, “Electric propulsion for satellites and spacecraft: established technologies and novel approaches,” Plasma Sources Sci. Technol., vol. 25, p. 033002, 2016.
* (5) I. Levchenko, K. Bazaka, Y. Ding, Y. Raitses, S. Mazouffre, T. Henning, P. J. Klar, S. Shinohara, J. Schein, L. Garrigues, M. Kim, D. Lev, F. Taccogna, R. W. Boswell, C. Charles, H. Koizumi, Y. Shen, C. Scharlemann, M. Keidar, and S. Xu, “Space micropropulsion systems for Cubesats and small satellites: From proximate targets to furthermost frontiers,” Applied Physics Reviews, vol. 5, p. 011104, 02 2018.
* (6) J. A. Linnell, “An evaluation of krypton propellant in hall thrusters,” Ph. D. Thesis, University of Michigan, 2007.
* (7) W. A. Hargus Jr., G. M. Azarnia, and M. R. Nakles, “Kr II laser-induced fluorescence for measuring plasma acceleration,” Review of Scientific Instruments, vol. 83, p. 103111, 10 2012.
* (8) J. Kurzyna, M. Jakubczak, A. Szelecka, and K. Dannenmayer, “Performance tests of IPPLM’s krypton Hall thruster,” Laser and Particle Beams, vol. 36, pp. 105–114, 2018.
* (9) J. Lim, I. Levchenko, S. Huang, L. Xu, R. Sim, J. Yee, G. Potrivitu, Y. Sun, K. Bazaka, X. Wen, et al., “Plasma parameters and discharge characteristics of lab-based krypton-propelled miniaturized hall thruster,” Plasma Sources Science and Technology, vol. 28, no. 6, p. 064003, 2019.
* (10) L.Su, R.Marks, and B.Jorns, “Investigation into the efficiency gap between Krypton and Xenon operation on a magnetically shielded Hall thruster,” International Electric Propulsion Conference 2022, June 19-23, 2022, Boston, MA, 2022.
* (11) A. Lejeune, G. Bourgeois, and S. Mazouffre, “Kr II and Xe II axial velocity distribution functions in a cross-field ion source,” Physics of Plasmas, vol. 19, p. 073501, 07 2012.
* (12) W. Hargus JR., G. Azarnia, and M. Nakles, “Demonstration of laser-induced fluorescence on a krypton Hall effect thruster,” IEPC - Electric Rocket Society, pp. Paper No. 2011–018, 2011.
* (13) G. Bourgeois, A. Lejune, and S. Mazouffre, “Ion velocity evolution with channel width, magnetic topology and propellant in a 200 W Hall thruster,” IEPC - Electric Rocket Society, pp. Paper No. 2011–123, 2010\.
* (14) K. Dzierżęga, U. Griesmann, G. Nave, and Łukasz Bratasz, “Absolute transition rates for transitions from 5p levels in kr ii,” Physica Scripta, vol. 63, p. 209, mar 2001.
* (15) E. B. Saloman, “Energy Levels and Observed Spectral Lines of Krypton, Kr I through Kr XXXVI,” Journal of Physical and Chemical Reference Data, vol. 36, pp. 215–386, 02 2007.
* (16) Y. Dancheva, D. Pagano, S. Scaranzin, R. Mercatelli, M. Presi, F. Scortecci, and G. Castellini, “Non-intrusive tools for electric propulsion diagnostics,” CEAS Space Journal, 2022.
* (17) Y. Dancheva, P. Coniglio, M. Da Valle, and F. Scortecci, “Ion dynamic characterization using phase-resolved laser-induced fluorescence spectroscopy in a Hall effect thruster,” Review of Scientific Instruments, vol. 94, p. 073503, 07 2023.
* (18) T. J. Scholl, T. D. Gaily, R. A. Holt, and S. D. Rosner, “Fast-ion-beam laser and laser-rf double-resonance measurements of hyperfine structure in 83KrII,” Phys. Rev. A, vol. 33, pp. 2396–2400, Apr 1986.
* (19) H. A. Schuessler, A. Alousi, M. Idrees, Y. F. Li, F. Buchinger, R. M. Evans, and C. F. Fischer, “Isotope shifts and hyperfine-structure-splitting constants of the 4d-5p transition of Kr II at $\lambda$=729 nm,” Phys. Rev. A, vol. 45, pp. 6459–6467, May 1992.
* (20) E. Arimondo, M. Inguscio, and P. Violino, “Experimental determinations of the hyperfine structure in the alkali atoms,” Rev. Mod. Phys., vol. 49, pp. 31–75, Jan 1977.
* (21) W. Hargus, “A preliminary study of krypton laser-induced fluorescence,” 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, 25-28 July 2010, Nashvill TN.
* (22) W.A.Hargus and M.A.Cappelli, “Laser-induced fluorescence measurements of velocity within a Hall discharge,” Appl. Phys. B, vol. 72, pp. 961–9, 2001.
* (23) P. Pallavi, S. Joshi, D. Singh, M. Kaur, and H. Lee, “Comprehensive review of orthogonal regression and its applications in different domains,” Archives of Computational Methods in Engineering, vol. 29, p. 4027–4047, 2022\.
|
# Effect of Boundary Conditions on Second-Order Singularly-Perturbed Phase
Transition Models on $\mathbb{R}$
Thomas Lam
###### Abstract.
The second-order singularly-perturbed problem concerns the integral functional
$\int_{\Omega}\varepsilon_{n}^{-1}W(u)+\varepsilon_{n}^{3}\|\nabla^{2}u\|^{2}\,dx$
for a bounded open set $\Omega\subseteq\mathbb{R}^{N}$, a sequence
$\varepsilon_{n}\to 0^{+}$ of positive reals, and a function
$W:\mathbb{R}\to[0,\infty)$ with exactly two distinct zeroes. This functional
is of interest since it models the behavior of phase transitions, and its
Gamma limit as $n\to\infty$ was studied by Fonseca and Mantegazza. In this
paper, we study an instance of the problem for $N=1$. We find a different form
for the Gamma limit, and study the Gamma limit under the addition of boundary
data.
## 1\. Introduction
For a bounded open set $\Omega\subseteq\mathbb{R}^{N}$, we may imagine
$\Omega$ as a container for a liquid whose density is given by
$u:\Omega\to\mathbb{R}$. The potential energy of the liquid can be measured by
the integral functional $u\mapsto\int_{\Omega}W(u)\,dx$ where
$W:\mathbb{R}\to[0,\infty)$ is the energy per unit volume.
Suppose that $W$ is a two-welled potential, so that $W$ has exactly two
distinct zeroes $z_{1},z_{2}$ with $z_{1}<z_{2}$, which are the phases of the
liquid. Then the liquid will tend to take on the two densities $z_{1}$ and
$z_{2}$, and in particular its density will simply take the form $u=z_{1}\cdot
1_{E}+z_{2}\cdot 1_{\Omega\setminus E}$ in order to minimize the potential
$\int_{\Omega}W(u)\,dx$, which is unexciting.
However, in practice, such rapid changes in density induces high interfacial
energy between phases. To account for this, the Van der Waals-Cahn-Hilliard
theory of phase transitions [7] models the potential energy via an integral
functional of the form
$J_{\varepsilon}(u):=\int_{\Omega}W(u)+\varepsilon^{2}\|\nabla
u\|^{2}\,dx,\qquad u\in W^{1,2}(\Omega).$ (1)
The problem of interest is to minimize $J_{\varepsilon}(u)$ subject to a mass
constraint $\int_{\Omega}u\,dx=m$. A minimizing function $u_{\varepsilon}$
gives a stable density distribution that the liquid would likely conform to.
What sort of stable density distribution is approached as $\varepsilon\to
0^{+}$? Specifically, suppose that we have a sequence $\varepsilon_{n}\to
0^{+}$ such that the sequence of minimizers $u_{\varepsilon_{n}}$ converges to
a function $u$ in some reasonable sense. What properties must be satisfied by
$u$?
Using Gamma convergence (described in Section 2.4), Modica [11] and Sternberg
[13] independently proved that such a $u$ satisfies $u\in\\{z_{1},z_{2}\\}$
almost everywhere, and minimizes the perimeter of $u^{-1}(z_{1})$. In
particular, they prove that for each sequence of positive reals
$\varepsilon_{n}$ with $\varepsilon_{n}\to 0^{+}$, we have that
$\Gamma\text{-}\lim_{n\to\infty}(J_{\varepsilon_{n}}/\varepsilon_{n})(u)=\begin{cases}2\displaystyle\int_{z_{1}}^{z_{2}}\sqrt{W(x)}\,dx\cdot\operatorname{Per}_{\Omega}(u^{-1}(z_{1})),&u\in
BV(\Omega;\\{z_{1},z_{2}\\})\\\ +\infty,&\text{otherwise}\end{cases}$
under $L^{1}(\Omega)$ convergence, where $\operatorname{Per}_{\Omega}(E)$
denotes the perimeter of a set $E\subseteq\Omega$, defined generally as
$\operatorname{Per}_{\Omega}(E):=\sup\left\\{\int_{E}\operatorname{div}\varphi:\varphi\in
C_{0}^{\infty}(\mathbb{R}^{N}),|\varphi|\leq 1\right\\}.$
Owen, Rubinstein and Sternberg [12] studied the family of functionals
$J_{\varepsilon}$ under boundary conditions instead of a mass constraint. They
prove that if $\Omega$ has $C^{2}$ boundary, $h_{\varepsilon}\in
L^{p}(\partial\Omega)\cap L^{\infty}(\partial\Omega)$ is the trace of a
function in $W^{1,2}(\Omega)$ for each $\varepsilon>0$, $h_{\varepsilon}\to
h\in L^{1}(\partial\Omega)\cap L^{\infty}(\partial\Omega)$ in
$L^{1}(\partial\Omega)$ as $\varepsilon\to 0$,
$\int_{\partial\Omega}\left|\frac{\partial
h_{\varepsilon}}{\partial\sigma}\right|$ is bounded in $\varepsilon$ and
$\left\|\frac{\partial
h_{\varepsilon}}{\partial\sigma}\right\|_{L^{\infty}(\partial\Omega)}\leq
C\varepsilon^{-1/4}$ for a constant $C>0$ where $\sigma$ is a surface
parameter on $\partial\Omega$, and
$K_{\varepsilon}:L^{1}(\Omega)\to\overline{\mathbb{R}}$ is defined as
$K_{\varepsilon}:=\begin{cases}\int_{\Omega}\varepsilon^{-1}W(u)+\varepsilon\|\nabla
u\|^{2}\,dx,&u\in W^{1,2}(\Omega)\text{ and
}\operatorname{Tr}u=h_{\varepsilon}\\\ +\infty,&\text{otherwise}\end{cases},$
(2)
where $\overline{\mathbb{R}}=[-\infty,\infty]$, then
$\displaystyle\Gamma\text{-}\lim_{n\to\infty}$ $\displaystyle
K_{\varepsilon_{n}}(u)$
$\displaystyle=\begin{cases}\int_{\Omega}|\nabla\chi(u)|+\int_{\partial\Omega}|\chi(h(x))-\chi(\operatorname{Tr}(u)(x))|\,d\mathcal{H}^{N-1}(x),&u\in
BV(\Omega;\\{z_{1},z_{2}\\})\\\ +\infty,&\text{otherwise}\end{cases},$
where
$\chi(t):=2\int_{z_{1}}^{t}\sqrt{W(z)}\,dz$
and $\varepsilon_{n}\to 0^{+}$.
Fonseca and Mantegazza [6] consider a second-order derivative. To be precise,
they define
$H_{\varepsilon}(u):=\int_{\Omega}\varepsilon^{-1}W(u)+\varepsilon^{3}\|\nabla^{2}u\|^{2}\,dx$
(3)
and proved that for every sequence of positive reals $\varepsilon_{n}\to
0^{+}$, we have
$\Gamma\text{-}\lim_{n\to\infty}H_{\varepsilon_{n}}(u)=\begin{cases}c\cdot\operatorname*{Per}_{\Omega}(u^{-1}(z_{1})),&u\in
BV(\Omega,\\{z_{1},z_{2}\\})\\\ +\infty,\text{otherwise}\end{cases}$
under $L^{1}(\Omega)$ convergence, where
$c:=\min\left\\{\int_{\mathbb{R}}W(u)+|u^{\prime\prime}|^{2}\,dt:u\in
W^{2,2}_{\text{loc}}(\mathbb{R}),\lim_{t\to-\infty}u(t)=z_{1},\lim_{t\to\infty}u(t)=z_{2}\right\\}.$
(4)
For a treatment of more general functionals, see [2], [3], and [4].
Fonseca and Mantegazza’s proofs appeal to rather sophisticated constructs such
as the Young measure. In this paper, we consider a 1-dimensional instance of
the problem solved by Fonseca and Mantegazza, which will allow for alternative
and more elementary methodologies.
Our goal will ultimately be to combine the efforts of Owen, Rubinstein, and
Sternberg with those of Fonseca and Mantegazza by considering the addition of
boundary conditions as in 2 to the second-order problem as in 3, which will be
possible due to our alternative methodologies.
To wit, define
$\Phi(u):=\left(\int_{0}^{1}(u(y)^{2}-1)^{2}\,dy\right)^{3/4}\left(\int_{0}^{1}|u^{\prime\prime}(y)|^{2}\,dy\right)^{1/4}$
(5)
for all $u\in W^{2,p}(0,1)$. Define the families
$\mathscr{J}:=\\{u\in
W^{2,p}(0,1):u(0_{+})=-1,u(1_{-})=1,u^{\prime}(0_{+})=u^{\prime}(1_{-})=0\\}$
(6)
and
$\mathscr{J}^{\prime}(t):=\\{u\in
W^{2,p}(0,1):u(0_{+})=-1,u(1_{-})=t,u^{\prime}(0_{+})=0\\}$ (7)
for each $t\in\mathbb{R}$. Let
$\alpha:=\frac{2}{3^{3/4}}\inf_{u\in\mathscr{J}}\Phi(u)$ (8)
and
$\beta(t):=\frac{4}{3^{3/4}}\inf_{u\in\mathscr{J}^{\prime}(t)}\Phi(u)$ (9)
for all $t\in\mathbb{R}$. Then our main result is the following theorem.
###### Theorem 1.1.
Let $\Omega=(a,b)$ and let $1\leq p\leq 4$. Let $a_{0},b_{0}\in\mathbb{R}$,
and for each $\varepsilon>0$, let
$a_{\varepsilon},b_{\varepsilon}\in\mathbb{R}$ be such that
$a_{\varepsilon}\to a_{0}$ and $b_{\varepsilon}\to b_{0}$ as $\varepsilon\to
0^{+}$. For each $\varepsilon>0$ define a functional
$G_{\varepsilon}(u):L^{p}(\Omega)\to\overline{\mathbb{R}}$ via
$G_{\varepsilon}(u):=\begin{cases}\int_{\Omega}\varepsilon^{-1}(u^{2}-1)^{2}+\varepsilon^{3}|u^{\prime\prime}|^{2}\,dx,&u\in
W^{2,p}(\Omega),u(a_{+})=a_{\varepsilon},u(b_{-})=b_{\varepsilon}\\\
+\infty,&\text{otherwise}\end{cases}.$
Let $\varepsilon_{n}\to 0^{+}$ be a sequence of positive reals. Then, under
strong $L^{p}(\Omega)$ convergence, we have that
$\displaystyle\Gamma\text{-}\lim_{n\to\infty}$ $\displaystyle
G_{\varepsilon_{n}}(u)$
$\displaystyle=\begin{cases}\alpha\operatorname*{essVar}_{\Omega}u+\beta(-a_{0}\operatorname{sgn}u(a_{+}))+\beta(-b_{0}\operatorname{sgn}u(b_{-})),&u\in\text{BPV}(\Omega;\\{-1,1\\})\\\
+\infty,&\text{otherwise}\end{cases}.$
Here, $\operatorname*{essVar}$ denotes essential variation, which is described
in Section 2.2. The addition of boundary conditions is of particular interest
here because they can ensure at least one phase transition by preventing the
existence of trivial minimizers.
The utility of this Gamma convergence result is justified by the following
compactness result.
###### Theorem 1.2 (Compactness for Second Order Problem).
Let $\Omega=(a,b)$ and let $1\leq p\leq 4$. For each $\varepsilon>0$, define
the functional $F_{\varepsilon}:L^{p}(\Omega)\to\overline{\mathbb{R}}$ via
$F_{\varepsilon}(u):=\begin{cases}\int_{\Omega}\varepsilon^{-1}(u^{2}-1)^{2}+\varepsilon^{3}|u^{\prime\prime}|^{2}\,dx,&u\in
W^{2,p}(\Omega)\\\ +\infty,&\text{otherwise}\end{cases}.$ (10)
Let $\varepsilon_{n}$ be a sequence of positive reals with $\varepsilon_{n}\to
0^{+}$. If we have a sequence $u_{n}\in L^{p}(\Omega)$ with
$\sup_{n\in\mathbb{N}}|F_{\varepsilon_{n}}(u_{n})|<\infty$, then there exists
a subsequence $u_{n_{k}}$ for which $u_{n_{k}}\to u$ in $L^{p}(\Omega)$ for
some $u\in BPV(\Omega;\\{-1,1\\})$.
Proofs of this result are given in [6], and [3]. We will provide yet another
proof.
The structure of this paper is as follows. In Section 2, we review pointwise
variation and results in Sobolev spaces. We then define Gamma convergence and
motivate its study.
In Section 3, we prove Theorem 1.2, and then give a more elementary proof for
Fonseca and Mantegazza’s results in the one-dimensional case. Specifically,
let $\Omega=(a,b)$ and define the integral functional
$F_{\varepsilon}:L^{p}(\Omega)\to\overline{\mathbb{R}}$ as in (10). Let
$\varepsilon_{n}$ be a sequence of positive reals with $\varepsilon_{n}\to
0^{+}$, and let $1\leq p\leq 4$. We first prove that if $u_{n}\in
L^{2}(\Omega)$ is a sequence for which
$\sup_{n\in\mathbb{N}}F_{\varepsilon_{n}}(u_{n})<\infty$, then we can find a
subsequence $u_{n_{k}}$ such that $u_{n_{k}}\to u$ in $L^{p}(\Omega)$ for some
$u\in BPV(\Omega;\pm 1)$. Then, we prove that for $1\leq p<\infty$, we have
$\Gamma\text{-}\lim_{n\to+\infty}F_{\varepsilon_{n}}(u)=\begin{cases}\alpha\operatorname*{essVar}_{\Omega}u,&u\in
BPV(\Omega;\pm 1)\\\ +\infty,&\text{otherwise}\end{cases}$
under $L^{p}(\Omega)$ convergence, where $\alpha$ is defined as in
$\eqref{eq:alpha}$.
Lastly, in Section 4, we build off of the work done in Section 3 to prove our
main result.
The current work, and a slicing methodology, will be used to extend to the
$N$-dimensional case. [8]
## 2\. Preliminaries
### 2.1. Pointwise Variation and Essential Variation
For a function $u:I\to\mathbb{R}$, where $I$ is an interval, we may define the
pointwise variation of $u$ as
$\operatorname*{Var}_{I}u:=\sup\left\\{\sum_{i=1}^{n}|u(x_{i})-u(x_{i-1})|,x_{i}\in
I,x_{0}<x_{1}<\ldots<x_{n}\right\\}.$
If $\operatorname*{Var}_{I}u<\infty$ then we write $u\in BPV(I)$. A useful
property is that if $u$ is absolutely continuous, then
$\operatorname*{Var}_{I}u=\int_{I}|u^{\prime}|\,dx$
A family of functions having uniformly bounded pointwise variation is a
powerful property.
###### Theorem 2.1 (Helly’s Selection Theorem).
Let $I$ be an interval and $\mathcal{F}\subseteq BPV(I)$ be an infinite family
of functions $u:I\to\mathbb{R}$ such that
$\sup_{u\in\mathcal{F}}\operatorname*{Var}_{I}u\leq C$ for a constant $C>0$.
Assume moreover that there exists $x_{0}\in I$ such that the set
$\\{u(x_{0}):u\in\mathcal{F}\\}$ is bounded. Then there exists a sequence
$\\{u_{n}\\}_{n}\in\mathcal{F}$ that converges pointwise to some $u\in
BPV(I)$.
This is given as Theorem 2.44 in [10], where a proof can be found.
In the case that such a pointwise convergence is obtained, we can moreover
obtain a bound on $\operatorname*{Var}_{I}u$.
###### Theorem 2.2.
If $u_{n},u\in BPV(I)$ and $u_{n}\to u$ pointwise, then
$\liminf_{n\to\infty}\operatorname*{Var}_{I}u_{n}\geq\operatorname*{Var}_{I}u.$
See Proposition 2.47 in [10] for a proof.
Now suppose, say, $u\in L^{p}(I)$. Then there is no sense in speaking of
$\operatorname*{Var}_{I}u$ because the values of $u$ are not well-defined
pointwise. The workaround is to define the essential pointwise variation of
$u$.
###### Definition 2.3 (Essential Pointwise Variation).
Let $I$ be an interval and $u\in L^{1}_{\text{loc}}(I)$. Then the essential
pointwise variation of $u$ over $I$ is given by
$\operatorname*{essVar}_{I}u:=\inf\left\\{\operatorname*{Var}_{I}\tilde{u}:\tilde{u}\text{
is a representative of }u\right\\}.$
Particularly, if $u\in W^{1,1}(I)$ then $u$ has an absolutely continuous
representative $\tilde{u}$ and we can show that
$\operatorname*{essVar}_{I}u=\int_{I}|\tilde{u}^{\prime}|\,dx$. A nice
property is that the infimum in the definition for
$\operatorname*{essVar}_{I}$ is obtained, i.e. there is always a
representative $\tilde{u}$ for $u$ such that
$\operatorname*{essVar}_{I}u=\operatorname*{Var}_{I}\tilde{u}$. This can be
proven using Helly’s Selection Theorem.
For $u\in L^{p}(I)$, we write $u\in BPV(I)$ if $u$ has a representative
$\tilde{u}$ with $\tilde{u}\in BPV(I)$. By the property from above, we see
that $u\in BPV(I)$ if and only if $\operatorname*{essVar}_{I}u<\infty$.
### 2.2. Sobolev Spaces
We begin by defining weak differentiation.
###### Definition 2.4 (Weak Derivative).
Let $\Omega\subseteq\mathbb{R}$ be an open set and $1\leq p\leq\infty$. For
$u\in L^{p}(\Omega)$, we say that $u$ admits a weak derivative of order
$k\in\mathbb{N}$ if there exists $v\in L^{p}(\Omega)$ satisfying
$\int_{\Omega}u\varphi^{(k)}\,dx=(-1)^{k}\int_{\Omega}v\varphi\,dx$
for all $\varphi\in C^{\infty}_{c}(\Omega)$.
It is not too difficult to verify that the weak derivative is unique up to
almost-everywhere equivalence. If $u$ has a differentiable representative, we
call its derivative (in the traditional sense) the strong derivative, so as to
distinguish the two notions of derivative.
The weak derivative is also notated in the same way as the strong derivative.
For example, if $\Omega=(-1,1)$ and we take $u(x)=|x|$, then $u$ admits a weak
derivative given by $u^{\prime}(x)=\operatorname{sgn}(x)$.
Sobolev spaces consist of $L^{p}$ functions that admit weak derivatives.
###### Definition 2.5 (Sobolev Space).
Let $\Omega\subseteq\mathbb{R}$ be open, $1\leq p\leq\infty$, and
$k\in\mathbb{N}$. Then the Sobolev space $W^{k,p}(\Omega)$ is the normed space
of all $u\in L^{p}(\Omega)$ that admit weak derivatives up to order $k$, such
that $u^{(l)}\in L^{p}(\Omega)$ for all $1\leq l\leq k$. We may endow
$W^{k,p}(\Omega)$ with the following norm:
$\|u\|_{W^{k,p}(\Omega)}:=\|u\|_{L^{p}(\Omega)}+\sum_{l=1}^{k}\|u^{(l)}\|_{L^{p}(\Omega)}.$
This definition may be unwieldy for showing that a function belongs to a
Sobolev space, so we often work with the following equivalent condition.
###### Theorem 2.6 (ACL Condition for $k=1$ on $\mathbb{R}$).
Let $\Omega\subseteq\mathbb{R}$ be an open set and $1\leq p<\infty$. Then
$u\in W^{1,p}(\Omega)$ if and only if $u$ is absolutely continuous and
$u,u^{\prime}\in L^{p}(\Omega)$. Moreover, if $u\in W^{1,p}(\Omega)$ then the
strong and weak derivatives of $u$ agree.
Absolute continuity of $L^{p}$ functions is discussed in the sense that there
exists an absolutely continuous representative, and similarly for
differentiability. A proof of this condition may be found in [10].
An analogue of this condition exists for $k=2$.
###### Theorem 2.7 (ACL Condition for $k=2$ on $\mathbb{R}$).
Let $\Omega\subseteq\mathbb{R}$ be an open set and $1\leq p<\infty$. Then
$u\in W^{2,p}(\Omega)$ if and only if $u\in C^{1}(\Omega)$, $u^{\prime}$ is
absolutely continuous, and $u,u^{\prime},u^{\prime\prime}\in L^{p}(\Omega)$.
Moreover, if $u\in W^{2,p}(\Omega)$ then the first and second-order strong
derivatives of $u$ agree with their weak analogues.
A proof of this may be found in [9].
Various algebraic properties satisfied by strong derivatives have analogues
for weak derivatives. For instance, we have the following chain rule.
###### Theorem 2.8 (Chain Rule).
Let $\Omega\subseteq\mathbb{R}$ be an open, bounded interval. Suppose
$f:\mathbb{R}\to\mathbb{R}$ is Lipschitz and $u\in W^{1,p}(\Omega)$. Then
$f(u)\in W^{1,p}(\Omega)$ and
$\frac{d}{dx}f(u)=f^{\prime}(\tilde{u})u^{\prime}$, where $\tilde{u}$ is the
absolutely continuous representative of $u$, and we take
$f^{\prime}(\tilde{u}(x))u^{\prime}(x)$ to be 0 whenever $u^{\prime}(x)=0$.
This is a consequence of Exercise 11.51 in [10].
It is useful to obtain a bound for $\int_{a}^{b}|u^{\prime}|^{2}\,dx$ in terms
of $\int_{a}^{b}|u|^{2}\,dx$ and $\int_{a}^{b}|u^{\prime\prime}|^{2}\,dx$ for
$u\in W^{2,p}(a,b)$. The following two results are special cases of Lemma 7.38
and Theorem 7.37 in [10].
###### Lemma 2.9.
Let $I=(a,b)$. Suppose $u\in W^{2,1}(I)$ such that $u^{\prime}$ has at least
one zero in $[a,b]$. Then there exists a universal constant $c>0$ such that
$\int_{a}^{b}|u^{\prime}|^{2}\,dx\leq
c\left(\int_{a}^{b}u^{2}\,dx\right)^{1/2}\left(\int_{a}^{b}|u^{\prime\prime}|^{2}\,dx\right)^{1/2}.$
###### Theorem 2.10.
For an open interval $I$ and $u\in W^{2,1}(I)$, there exists a universal
constant $c>0$ such that
$\left(\int_{I}|u^{\prime}|^{2}\,dx\right)^{1/2}\leq
cl^{-1}\left(\int_{I}u^{2}\,dx\right)^{1/2}+cl\left(\int_{I}|u^{\prime\prime}|^{2}\,dx\right)^{1/2}$
for every $l$ with $0<l<\mathcal{L}^{1}(I)$.
We note that by applying the QM-AM inequality to the above theorem, we may
obtain the inequality
$\int_{I}|u^{\prime}|^{2}\,dx\leq
c^{\prime}l^{-2}\int_{I}u^{2}\,dx+c^{\prime}l^{2}\int_{I}|u^{\prime\prime}|^{2}\,dx$
under the same conditions but for a different universal constant
$c^{\prime}>0$.
### 2.3. Weak Convergence
###### Definition 2.11 (Weak Convergence in $L^{p}$).
Let $1\leq p<\infty$. Let $E\subseteq\mathbb{R}$ be measurable, and let
$u_{n}\in L^{p}(E)$ for all $n\in\mathbb{N}$. For $u\in L^{p}(E)$, we say that
$u_{n}$ converges weakly to $u$ in $L^{p}(E)$ if
$\lim_{n\to\infty}\int_{E}u_{n}v\,dx=\int_{E}uv\,dx$
for all $v\in L^{p^{\prime}}(E)$, where $p^{\prime}\in[1,\infty]$ is such that
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$, and we write $u_{n}\rightharpoonup u$ in
$L^{p}(E)$.
###### Definition 2.12 (Weak Convergence in $W^{k,p}$).
For $\Omega\subseteq\mathbb{R}$ open, $k\in\mathbb{N}$, $1\leq p<\infty$, and
$u_{n},u\in W^{k,p}(\Omega)$, we say that $u_{n}$ converges weakly in
$W^{k,p}(\Omega)$ if $u_{n}^{(l)}\rightharpoonup u^{(l)}$ in $L^{p}(\Omega)$
for every $0\leq l\leq k$, and we write $u_{n}\rightharpoonup u$ in
$W^{k,p}(\Omega)$.
It can be shown that weak limits are unique up to almost-everywhere
equivalence. Moreover, Hölder’s inequality implies that strong convergence in
$L^{p}(E)$ implies weak convergence in $L^{p}(E)$, and consequently strong
convergence in $W^{k,p}(\Omega)$ implies weak convergence in
$W^{k,p}(\Omega)$.
Proofs of the theorems that follow may be found in [5]. An important property
is that we may strengthen some convergences in the definition of weak
convergence in $W^{k,p}(\Omega)$.
###### Theorem 2.13.
Let $(a,b)$ be an interval, $k\in\mathbb{N}$, $1\leq p<\infty$, and suppose
$u_{n},u\in W^{k,p}(a,b)$ are such that $u_{n}\rightharpoonup u$ in
$W^{k,p}(a,b)$. Then $u_{n}^{(l)}\to u^{(l)}$ strongly in $L^{p}(a,b)$ for all
$0\leq l\leq k-1$.
Weak convergence enables us to consider a notion of weak compactness.
###### Theorem 2.14 (Weak Compactness in $W^{k,p}$).
Let $\Omega\subseteq\mathbb{R}$ be open, $k\in\mathbb{N}$, $1\leq p<\infty$,
and suppose that $u_{n}\in W^{k,p}(\Omega)$ is such that $\\{u_{n}\\}_{n}$ is
uniformly bounded on $W^{k,p}(\Omega)$. That is,
$\sup_{n\in\mathbb{N}}\|u_{n}\|_{W^{k,p}(\Omega)}<\infty$. Then there exist a
subsequence $\\{u_{n_{l}l}\\}_{l}$ and $u\in W^{k,p}(\Omega)$ such that
$u_{n_{l}}\rightharpoonup u$ in $W^{k,p}(\Omega)$.
Lastly, weak convergence is useful for obtaining a weak form of semi-
continuity for certain functions. A particular result we shall use is the
following.
###### Theorem 2.15.
Let $1\leq p<\infty$ and let $\Omega\subseteq\mathbb{R}^{N}$ be open and
bounded. Suppose that $f:\mathbb{R}\to\mathbb{R}$ is convex. Then the integral
functional $F:L^{p}(\Omega)\to\overline{\mathbb{R}}$ defined by
$F(u):=\int_{\Omega}f(u)\,dx$
is sequentially lower semi-continuous with respect to weak
$L^{p}_{\text{loc}}(\Omega)$ convergence. That is, we have
$F(u)\leq\liminf_{n\to\infty}F(u_{n})$
whenever $u_{n},u\in L^{p}(\Omega)$ such that $u_{n}\rightharpoonup u$ in
$L^{p}_{\text{loc}}(\Omega)$.
For a proof, see Theorem 5.14 in [5].
### 2.4. Gamma Limits
The Gamma limit is a notion of function convergence that is of interest
because it preserves minima.
###### Definition 2.16.
For a metric space $(X,d)$ and a sequence $f_{n}:X\to\overline{\mathbb{R}}$,
we say that $f_{n}$ Gamma converges to a function
$f:X\to\overline{\mathbb{R}}$ if for all $x_{0}\in X$:
1. (1)
$\liminf_{n\to\infty}f_{n}(x_{n})\geq f(x_{0})$ for all sequences $x_{n}$ with
$x_{n}\stackrel{{\scriptstyle d}}{{\to}}x_{0}$, and
2. (2)
$\limsup_{n\to\infty}f_{n}(x_{n})\leq f(x_{0})$ for some sequence $x_{n}$ with
$x_{n}\stackrel{{\scriptstyle d}}{{\to}}x_{0}$,
and we write $f=\Gamma\text{-}\lim_{n\to\infty}f_{n}$.
The two inequality conditions are referred to as the liminf inequality and
limsup inequality respectively.
The following properties are proven in Chapter 1 of [1].
* •
The Gamma limit, if it exists, is unique pointwise. Moreover, it is given
precisely by
$\left(\Gamma\text{-}\lim_{n\to\infty}f_{n}\right)(x)=\inf\left\\{\liminf_{n\to\infty}f_{n}(x_{n}):x_{n}\stackrel{{\scriptstyle
d}}{{\to}}x\right\\}.$
* •
It is not necessarily true that the Gamma limit coincides with the pointwise
limit, if both exist.
* •
If $f=\Gamma\text{-}\lim f_{n}$ exists and $x_{n}$ is a minimizer for $f_{n}$,
then every accumulation point of the sequence $\\{x_{n}\\}_{n}$ is a minimizer
for $f$.
As an example, let us define $\Omega:=(0,1)$ and consider the functional
$F_{\varepsilon}:W^{1,2}(\Omega)\to\mathbb{R}$ defined by
$F_{\varepsilon}(u):=\begin{cases}\int_{0}^{1}(u^{2}-1)^{2}+\varepsilon^{2}|u^{\prime}|^{2}\,dx,&\int_{\Omega}u\,dx=0\\\
+\infty,&\text{otherwise}\end{cases}$
for $\varepsilon>0$. Although it is quite non-trivial, it can be shown that
$F_{\varepsilon}$ has a minimizer.
The $\varepsilon^{2}|u^{\prime}|^{2}$ term is a complication. Can it be
removed without much change to the minimizers? As before, let
$\varepsilon_{n}\to 0^{+}$ be a sequence. We can show that
$\Gamma\text{-}\lim_{n\to\infty}F_{\varepsilon_{n}}(u)=\int_{0}^{1}(u^{2}-1)^{2}\,dx$
under $W^{1,2}(\Omega)$ convergence. Now let us attempt to repeat the previous
logic. If we take $u_{n}$ to be a minimizer of $F_{\varepsilon_{n}}$, then any
subsequence of $u_{n}$ converging in $W^{1,2}(\Omega)$ must converge to a
minimizer of $\int_{0}^{1}(u^{2}-1)^{2}\,dx$. However, we can show that
$\inf\left\\{\int_{0}^{1}(u^{2}-1)^{2}\,dx:\int_{\Omega}u\,dx=0\text{ and
}u\in W^{2,1}(\Omega)\right\\}=0,$
hence a minimizer $u_{\text{min}}$ would have to take the form
$u_{\text{min}}=1\cdot 1_{E}+(-1)\cdot 1_{\Omega\setminus E}$ for some $E$
with $\mathcal{L}^{1}(E)=\frac{1}{2}$, which cannot be consistent with the
requirement that $u\in W^{2,1}(\Omega)$. Hence, there is no such minimizer!
Our failure to deduce anything meaningful from this Gamma convergence result
is a consequence of missing a compactness result that guarantees the existence
of converging subsequences of $u_{n}$. Fortunately, without computing the
exact minimizers, we can still recover a reasonable Gamma limit result by
weakening the convergence. Specifically, we may sacrifice the strong
$W^{1,2}(\Omega)$ convergence for a weaker convergence called weak
$L^{4}(\Omega)$ convergence (and written as “$u_{n}\rightharpoonup u$ in
$L^{4}(\Omega)$"). The advantage of this weakening is that if we take $u_{n}$
to be a minimizer of $F_{\varepsilon_{n}}$, then we can use weak compactness
to prove that there is a subsequence $u_{n_{k}}$ and some $u\in L^{4}(\Omega)$
for which $u_{n_{k}}\rightharpoonup u$ in $L^{4}(\Omega)$, which is precisely
the sort of compactness result we seek. Under weak $L^{4}(\Omega)$
convergence, the Gamma limit changes to
$\Gamma\text{-}\lim_{n\to\infty}F_{\varepsilon_{n}}(u)=\int_{0}^{1}f^{**}(u)\,dx,$
where $f^{**}(z):=\begin{cases}(z^{2}-1)^{2},&|z|>1\\\ 0,&|z|\leq
1\end{cases}$ denotes the convex envelope of $f(z):=(z^{2}-1)^{2}$. Due to the
existence of a compactness result, we are guaranteed the existence of a
subsequence $u_{n_{k}}$ converging weakly in $L^{2}(\Omega)$ to a minimizer of
$\int_{\Omega}f^{**}(u)\,dx$, and indeed there are many minimizers of this
functional.
This demonstrates the importance of choosing the correct metric of convergence
for the Gamma limit. If the convergence is too strong, there may be no
compactness result. On the other hand, the weaker the convergence, the weaker
the result we end up proving.
## 3\. The Second-Order Singularly-Perturbed Problem
Our first goal is to prove Theorem 1.2. As a steppingstone, we first prove a
similar compactness result in the context of a first order singularly-
perturbed problem, which is of interest in itself.
###### Theorem 3.1 (Compactness for First Order Problem).
Let $\Omega=(a,b)$ be a non-empty open interval, and let $1\leq p<\infty$. Let
$\varepsilon_{n}$ be a sequence of positive reals with $\varepsilon_{n}\to
0^{+}$.
1. (i)
If we have a sequence $u_{n}\in W^{1,p}(\Omega)$ with
$C:=\sup_{n\in\mathbb{N}}\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx<\infty,$
then there exists a subsequence $u_{n_{k}}$ for which $u_{n_{k}}\to u$ in
$L^{p}(\Omega)$ for some $u\in BPV(\Omega;\\{-1,1\\})$.
2. (ii)
Moreover, such a $u$ must satisfy
$\operatorname*{essVar}_{\Omega}u\leq\frac{3}{4}C.$
###### Proof.
Since $\int_{\Omega}(u_{n}^{2}-1)^{2}\,dx\leq C\varepsilon_{n}\to 0$, we see
that $u_{n}^{2}-1\to 0$ in $L^{2}(\Omega)$. So, by extraction of a
subsequence, we may assume that $u_{n}^{2}-1\to 0$ almost everywhere. In
particular, we get that $|u_{n}|\to 1$ almost everywhere.
Applying the AM-GM inequality inside the integral gives
$C\geq\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\geq\int_{\Omega}2|u_{n}^{2}-1|\cdot|u_{n}^{\prime}|\,dx=\int_{\Omega}|U_{n}^{\prime}|\,dx$
for all $n$, where $U_{n}:=2u_{n}-\frac{2}{3}u_{n}^{3}$. Taking
$\tilde{U_{n}}$ to be the absolutely continuous representative of $U_{n}$, we
deduce that $\operatorname*{Var}_{\Omega}\tilde{U}_{n}\leq C$ for all $n$.
Moreover, since $|u_{n}|\to 1$ almost everywhere, we have that $|U_{n}|\to
2-\frac{2}{3}=\frac{4}{3}$ almost everywhere by continuity of $z\mapsto
2z-\frac{2}{3}z^{3}$, and so in particular there is some $x\in\Omega$ for
which $\tilde{U}_{n}(x)$ converges. From this, we may now apply Helly’s
Selection Theorem to find a subsequence $\tilde{U}_{n_{k}}$ that converges
pointwise to some $\tilde{U}\in BPV(\Omega)$, and
$C\geq\liminf_{n\to\infty}\operatorname*{Var}_{\Omega}\tilde{U}_{n}\geq\operatorname*{Var}\tilde{U}$
by Theorem 2.2.
It follows that $U_{n_{k}}$ converges to some $U$ almost everywhere that has
representative $\tilde{U}$, with $U=\pm\frac{4}{3}$ almost everywhere, and
moreover $\operatorname*{essVar}_{\Omega}U\leq\operatorname*{Var}\tilde{U}\leq
C<\infty$. Hence, by taking $u=\frac{3}{4}U$, we have that
$\operatorname*{essVar}_{\Omega}u\leq\frac{3}{4}C<\infty,$ (11)
and particularly $u$ has a representative $\tilde{u}\in BPV(\Omega;\pm 1)$.
To see that $u_{n_{k}}\to u$ almost everywhere, recall that $|u_{n_{k}}|\to 1$
almost everywhere. Taking representatives $\tilde{u}_{n_{k}}$, we have that
almost every $x_{0}\in\Omega$ satisfies $|\tilde{u}_{n_{k}}(x_{0})|\to 1$,
$2\tilde{u}_{n_{k}}-\frac{2}{3}\tilde{u}_{n_{k}}^{3}=\tilde{U}_{n_{k}}$ for
all $k$, $\tilde{U}_{n_{k}}(x_{0})\to\tilde{U}(x_{0})$, and
$\tilde{U}(x_{0})=\frac{4}{3}\tilde{u}(x_{0})$.
For every such $x_{0}$ we have that
$2\tilde{u}_{n_{k}}(x_{0})-\frac{2}{3}u_{n_{k}}(x_{0})^{3}\to\frac{4}{3}\tilde{u}(x_{0})$.
If $\tilde{u}(x_{0})=1$, then
$2\tilde{u}_{n_{k}}(x_{0})-\frac{2}{3}\tilde{u}_{n_{k}}(x_{0})^{3}-\frac{4}{3}\to
0$, and by factoring we obtain
$(\tilde{u}_{n_{k}}(x_{0})-1)^{2}(\tilde{u}_{n_{k}}(x_{0})+2)\to 0$. But
$|\tilde{u}_{n_{k}}(x_{0})|\to 1$, so
$\liminf_{k\to\infty}|\tilde{u}_{n_{k}}(x_{0})+2|\geq 1$, and hence it must
follow that $(\tilde{u}_{n_{k}}(x_{0})-1)^{2}\to 0$, i.e.
$\tilde{u}_{n_{k}}(x_{0})\to 1=\tilde{u}(x_{0})$. By a symmetrical argument we
see that if $\tilde{u}(x_{0})=-1$ then
$\tilde{u}_{n_{k}}(x_{0})\to-1=\tilde{u}(x_{0})$. This shows that
$u_{n_{k}}\to u$ almost everywhere.
Lastly, to obtain $L^{p}(\Omega)$ convergence, we may use the fact that
$\operatorname*{essVar}_{\Omega}U_{n_{k}}\leq C$ and that $U_{n_{k}}$
converges almost everywhere to deduce that $U_{n_{k}}$ is uniformly bounded by
some constant $C^{\prime}$. Then the inverse image of $[-C,C]$ under $z\mapsto
2z-\frac{2}{3}z^{3}$ is compact by continuity, thus
$|U_{n_{k}}|=|2u_{n_{k}}-\frac{2}{3}u_{n_{k}}^{3}|\leq C^{\prime}$ implies
that $|u_{n_{k}}|\leq C^{\prime\prime}$ for some constant $C^{\prime\prime}$.
In particular, we see that $u_{n_{k}}$ is uniformly bounded, so we may apply
dominated convergence to obtain the desired $L^{p}(\Omega)$ convergence. This
proves item (i), and the bound claimed in item (ii) was acquired in (11). ∎
Next, we will prove an essential interpolation inequality.
###### Theorem 3.2.
Let $\Omega=(a,b)$ be a non-empty open interval, let
$\Omega_{0}=(a_{0},b_{0})\Subset(a,b)$, and let $1\leq p\leq 4$. Let
$\varepsilon_{n}$ be a sequence of positive reals with $\varepsilon_{n}\to
0^{+}$. If we have a sequence $u_{n}\in W^{2,p}(\Omega)$ with
$\sup_{n\in\mathbb{N}}\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx<\infty,$
then for a universal constant $C^{\prime}>0$ we have that
$\int_{\Omega_{0}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq
C^{\prime}\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx$
for all large enough $n$.
We begin by proving two lemmas.
###### Lemma 3.3.
Let $\Omega$ be an open interval, $1\leq p\leq 4$, and $u_{n}\in
L^{p}(\Omega)$ such that
$\lim_{n\to\infty}\int_{a}^{b}(u_{n}^{2}-1)^{2}\,dx=0$. Then
$\\{|u_{n}|^{p}\\}_{n}$ is uniformly integrable.
###### Proof.
First, find some $c>0$ so small that $cz^{p}\leq(z^{2}-1)^{2}$ for all $z$
large enough; say, $z\geq M_{0}$. This is possible by virtue of order of
growth, since $1\leq p\leq 4$. Now
$\int_{\Omega}|u_{n}|^{p}\,dx=\int_{\\{|u_{n}|<M_{0}\\}}|u_{n}|^{p}\,dx+\int_{\\{|u_{n}|\geq
M_{0}\\}}|u_{n}|^{p}\,dx\leq M_{0}^{p}+\int_{\\{|u_{n}|\geq
M_{0}\\}}\frac{1}{c}(u_{n}^{2}-1)^{2}\,dx<\infty,$
so $|u_{n}|^{p}$ is integrable for all $n$.
Fix $\eta>0$. Find $N_{\eta}$ such that
$\int_{\Omega}(u_{n}^{2}-1)^{2}\,dx\leq c\eta$ for all $n>N_{\eta}$. Lastly,
by integrability of $|u_{n}|^{p}$ for $n=1,2,\cdots,N_{\eta}$, we choose
$M_{\eta}\geq M_{0}$ so large that $\int_{|u_{n}|\geq
M_{\eta}}|u_{n}|^{p}\,dx<\eta$ for all such $n$. This $M_{\eta}$ witnesses the
uniform integrability since for all $n\geq N_{\eta}$ we have
$\int_{|u_{n}|\geq M_{\eta}}|u_{n}|^{p}\,dx\leq\frac{1}{c}\int_{|u_{n}|\geq
M_{\eta}}(u_{n}^{2}-1)^{2}\,dx\leq\eta.$
∎
###### Lemma 3.4.
Let $\Omega$ be an open interval. Then for every $u\in W^{2,p}(\Omega)$, the
set $Z=\\{x\in\Omega:u(x)=0,u^{\prime}(x)\neq 0\\}$ is countable.
###### Proof.
Let $x_{0}\in Z$. It suffices to show that $x_{0}$ is an isolated point of
$Z$. We may assume without loss of generality that $u^{\prime}(x_{0})>0$.
Since $u\in W^{2,p}(\Omega)$ we know that $u^{\prime}$ is continuous (See
Theorem 2.7). Thus, there exists $\delta>0$ such that $u^{\prime}(x)>0$ for
all $x\in(x_{0}-\delta,x_{0}+\delta)$. We conclude that there is no
$x\in(x_{0}-\delta,x_{0}+\delta)$ for which $u(x)=0$ (so that in particular,
$x\not\in Z$), otherwise we obtain a contradiction from Rolle’s Theorem. ∎
Although it is not worth stating as a lemma, we will be using the inequality
$(|z|-1)^{2}\leq(z^{2}-1)^{2}$ quite liberally, which follows from factoring
$(z^{2}-1)^{2}$ and taking cases on the sign of $z$.
We may now prove Theorem 1.2.
###### Proof.
Let
$C=\sup_{n\in\mathbb{N}}\int_{a}^{b}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx.$
(12)
Since $(a_{0},b_{0})\Subset(a,b)$ and $\varepsilon_{n}\to 0$, it is sufficient
to obtain an interpolation inequality of the form
$\int_{\Omega_{\varepsilon_{n}}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq
C^{\prime}\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx,$
(13)
where $C^{\prime}$ is a universal constant and
$\Omega_{\varepsilon}:=(a+\varepsilon,b-\varepsilon)$ for all $\varepsilon>0$.
Note that if $\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}\,dx=0$, then
we must have either $u_{n}\equiv 1$ or $u_{n}\equiv-1$. Hence
$\int_{\Omega_{\varepsilon_{n}}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx=0$, so
the inequality (13) holds and there is nothing more to show. Thus, we may
assume the contrary, so that
$\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx>0.$
(14)
It is clear from (12) that
$\int_{\Omega}|u_{n}^{\prime\prime}|^{2}\,dx\leq\frac{C}{\varepsilon_{n}^{3}}$
(15)
and
$\int_{\Omega}(u_{n}^{2}-1)^{2}\,dx\leq\varepsilon_{n}C$
for all $n$.
We divide our analysis of $u_{n}$ into three sets: Where $u_{n}=0$, where
$u_{n}$ is near 0, and where $u_{n}$ is far from 0.
Firstly, the argument in Lemma 3.3 implies that $u_{n}\in L^{2}(\Omega)$, and
(15) implies that $u_{n}^{\prime\prime}\in L^{2}(\Omega)$. It follows by
Theorem 2.10 that $u_{n}^{\prime}\in L^{2}(\Omega)$. Hence, by this fact and
the assumption (14), there must exist $\alpha_{n}\in(0,\tfrac{1}{2})$ so small
that
$\int_{0<|u_{n}|\leq\alpha_{n}}|u_{n}^{\prime}|^{2}\,dx\leq\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx.$
(16)
Our three sets in question are then $u_{n}^{-1}(\\{0\\})$,
$u_{n}^{-1}\big{(}[-\alpha_{n},0)\cup(0,\alpha_{n}]\big{)}$, and
$u_{n}^{-1}\big{(}(-\infty,-\alpha_{n})\cup(\alpha_{n},+\infty)\big{)}$.
For the set $u_{n}^{-1}(\\{0\\})$, we claim that
$\int_{u_{n}=0}|u_{n}^{\prime}|\,dx=0.$ (17)
To prove this, it suffices to show that
$\int_{\\{y:u_{n}(y)=0,u_{n}^{\prime}(y)\neq 0\\}}|u_{n}^{\prime}|\,dx=0$
since the integral over the set where $u_{n}^{\prime}=0$ is clearly 0. But by
Lemma 3.4, the domain over which we integrate here is countable, so the
integral is indeed 0.
The set $u_{n}^{-1}\big{(}[-\alpha_{n},0)\cup(0,\alpha_{n}]\big{)}$ is already
handled by (16).
It remains to handle
$u_{n}^{-1}((-\infty,-\alpha_{n})\cup(\alpha_{n},+\infty))$. This set is open
by continuity, so it is the at-most countable union of disjoint open
intervals. Let those intervals that do not have either $a$ or $b$ as an
endpoint be enumerated as $\\{(a_{i},b_{i})\\}_{i\in I}$ for a countable,
possibly empty index set $I$. Let the union of these intervals be $E_{n}$, so
$E_{n}$ is where $|u_{n}|>\alpha_{n}$ except possibly near the endpoints of
$(0,1).$
Then for all $i\in I$ we have that $u_{n}(a_{i})=u_{n}(b_{i})$ by a continuity
argument. Thus by Rolle’s Theorem there is $c_{i}\in(a_{i},b_{i})$ for which
$u_{n}^{\prime}(c_{i})=0$. Moreover, $u_{n}\neq 0$ over $(a_{i},b_{i})$, thus
$v_{n}:=|u_{n}|-1$ is differentiable over $(a_{i},b_{i})$. If we assume
without loss of generality that $u_{n}>0$ over $(a_{i},b_{i})$, then in fact
$v_{n}=u_{n}-1$, so it is clear that $v_{n}\in W^{2,p}(a_{i},b_{i})$, with
$v_{n}^{\prime}=u_{n}^{\prime}$ and
$v_{n}^{\prime\prime}=u_{n}^{\prime\prime}$. By Lemma 2.9, it follows that
$\displaystyle\int_{a_{i}}^{b_{i}}|u_{n}^{\prime}|^{2}\,dx$ $\displaystyle\leq
c_{1}\left(\int_{a_{i}}^{b_{i}}(|u_{n}|-1)^{2}\,dx\right)^{1/2}\left(\int_{a_{i}}^{b_{i}}|u_{n}^{\prime\prime}|^{2}\,dx\right)^{1/2}$
$\displaystyle\leq
c_{1}\left(\int_{a_{i}}^{b_{i}}(u_{n}^{2}-1)^{2}\,dx\right)^{1/2}\left(\int_{a_{i}}^{b_{i}}|u_{n}^{\prime\prime}|^{2}\,dx\right)^{1/2}$
for a constant $c_{1}>0$. Multiplying by $\varepsilon_{n}$ gives
$\int_{a_{i}}^{b_{i}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq
c_{1}\left(\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}\,dx\right)^{1/2}\left(\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx\right)^{1/2},$
then applying the AM-GM inequality gives
$\int_{a_{i}}^{b_{i}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq\frac{c_{1}}{2}\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx.$
Summing this inequality over all $i\in I$, we conclude that
$\int_{E_{n}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq\frac{c_{1}}{2}\int_{E_{n}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx.$
(18)
We are almost done. Let
$S_{n}=\sup\\{x\in\Omega:|u_{n}(y)|\geq\alpha_{n}\,\forall a<y\leq x\\}$ and
$T_{n}=\inf\\{x\in\Omega:|u_{n}(y)|\geq\alpha_{n}\,\forall x\leq y<b\\}$. If
$\\{x\in\Omega:|u_{n}(y)|\geq\alpha_{n}\,\forall a<y\leq x\\}$ is empty then
we take $S_{n}=a$, and similarly for $T_{n}$. The intervals $(a,S_{n})$ and
$(T_{n},b)$, if distinct and non-empty, are the two intervals that we omitted
from $\\{x\in\Omega:u_{n}(x)>\alpha_{n}\\}$ to obtain $E_{n}$, so that
$E_{n}\cup(a,S_{n})\cup(T_{n},b)=\\{x\in\Omega:u_{n}(x)>\alpha_{n}\\}$. It
remains to handle these two intervals.
If $S_{n}-a\leq\varepsilon_{n}$, then clearly
$\int_{(a,S_{n})\cap\Omega_{\varepsilon_{n}}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx=0$.
If otherwise $S_{n}-a>\varepsilon_{n}$ then we may apply Theorem 2.10 to
$|u_{n}|-1\in W^{2,p}(a,S_{n})$ to get
$\displaystyle\int_{a}^{S_{n}}|u_{n}^{\prime}|^{2}\,dx$ $\displaystyle\leq
c_{2}\varepsilon_{n}^{-2}\int_{a}^{S_{n}}(|u_{n}|-1)^{2}\,dx+c_{1}\varepsilon_{n}^{2}\int_{a}^{S_{n}}|u_{n}^{\prime\prime}|^{2}\,dx$
$\displaystyle\leq
c_{2}\varepsilon_{n}^{-2}\int_{a}^{S_{n}}(u_{n}^{2}-1)^{2}\,dx+c_{1}\varepsilon_{n}^{2}\int_{a}^{S_{n}}|u_{n}^{\prime\prime}|^{2}\,dx,$
for a constant $c_{2}>0$. Multiplying by $\varepsilon_{n}$ gives
$\int_{(a,S_{n})\cap\Omega_{\varepsilon_{n}}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq\int_{a}^{S_{n}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq
c_{2}\int_{a}^{S_{n}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx.$
(19)
Thus the inequality (19) holds in either the cases
$S_{n}-a\leq\varepsilon_{n}$ and $S_{n}-a>\varepsilon_{n}$. Applying the same
arguments above to the interval $(T_{n},b)$, we see that in either of the
cases $b-T_{n}\leq\varepsilon_{n}$ and $b-T_{n}>\varepsilon_{n}$ we may obtain
the inequality
$\int_{(T_{n},b)\cap\Omega_{\varepsilon_{n}}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq
c_{2}\int_{T_{n}}^{b}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx.$
(20)
By summing (18), (19), and (20), we get the bound
$\displaystyle\int_{\\{|u_{n}|>\alpha_{n}\\}\cap\Omega_{\varepsilon_{n}}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx$
$\displaystyle\leq\int_{E_{n}\cup((a,S_{n})\cap\Omega_{\varepsilon_{n}})\cup((T_{n},b)\cap\Omega_{\varepsilon_{n}})}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx$
$\displaystyle\leq\max(c_{1}/2,c_{2})\int_{E_{n}\cup(a,S_{n})\cup(T_{n},b)}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx$
$\displaystyle=\max(c_{1}/2,c_{2})\int_{|u_{n}|>\alpha_{n}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx.$
(21)
At last, once we add up (16), (17), and (21), we can arrive at the inequality
$\int_{\Omega_{\varepsilon_{n}}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq
c_{3}\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx,$
where $c_{3}=1+\max(c_{1}/2,c_{2})$. This is of the form (13), so we are done.
∎
We now turn to the proof of Theorem 1.2.
###### Proof.
Let $C=\sup_{n\in\mathbb{N}}F_{\varepsilon_{n}}(u_{n})$. We first claim that
for each $\delta>0$ there exists a subsequence $u_{n_{k}}$ of $u_{n}$ such
that $u_{n_{k}}\to v$ in $L^{p}(a+\delta,b-\delta)$ for some function $v\in
BPV((a+\delta,b-\delta);\pm 1)$, with
$\operatorname*{essVar}_{(a+\delta,b-\delta)}v\leq C_{3}$ for a constant
$C_{3}$ not depending on $\delta$.
To see this, we apply Theorem 3.2 to the interval $(a+\delta,b-\delta)$ to
obtain the bound
$\int_{a+\delta}^{b-\delta}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq\int_{\Omega}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx\leq
C$
for all large enough $n$. It follows that
$\int_{a+\delta}^{b-\delta}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq
2C$
for all such $n$, and so by item (i) of Theorem 3.1 applied to the interval
$(a+\delta,b-\delta)$, we may find a subsequence $u_{n_{k}}$ with
$u_{n_{k}}\to v$ in $L^{p}(a+\delta,b-\delta)$ for some $v\in
BPV((a+\delta,b-\delta);\pm 1)$, and moreover by item (ii) of Theorem 3.1 we
must have that
$\operatorname*{essVar}_{(a+\delta,b-\delta)}v\leq\frac{3}{2}C$. Hence the
claim is proven with $C_{3}=\frac{3}{2}C$.
We finish with a diagonalization argument. Let
$\Omega_{m}=\left(a+\delta_{m},b-\delta_{m}\right)$ for a sequence
$\delta_{m}\to 0^{+}$. For each $m$ we define a subsequence $u_{m,n}$ of
$u_{n}$ recursively as follows:
* •
$u_{1,n}=u_{n}$ for all $n$.
* •
For $m\geq 2$, we take $\\{u_{m,n}\\}_{n}$ to be a subsequence of
$\\{u_{m-1,n}\\}_{n}$ that converges in $L^{p}(\Omega_{m})$ to some
$v_{m}:\Omega_{m}\to\\{-1,1\\}$ with
$\operatorname*{essVar}_{\Omega_{m}}v_{m}\leq C_{3}$, which exists by the
claim.
Let $w_{n}=u_{n,n}$. We claim that for some $u\in L^{p}(\Omega;\\{-1,1\\})$ we
have that $w_{n}\to u$ in $L^{p}(\Omega_{m})$ for every $m$, so that in
particular we have $w_{n}\to u$ in $L^{p}_{\text{loc}}(\Omega)$.
To see this, consider $m_{1},m_{2}$ with $m_{2}>m_{1}$, so that
$\\{u_{m_{2},n}\\}_{n}$ is a subsequence of $\\{u_{m_{1},n}\\}_{n}$. Then we
may write $u_{m_{2},k}=u_{m_{1},n_{k}}$ for some sequence $n_{k}$, and so
$\|v_{m_{2}}-v_{m_{1}}\|_{L^{p}(\Omega_{m_{1}})}\leq\|v_{m_{2}}-u_{m_{2},k}\|_{L^{p}(\Omega_{m_{1}})}+\|u_{m_{1},n_{k}}-v_{m_{1}}\|_{L^{p}(\Omega_{m_{1}})}.$
Sending $k\to+\infty$ we deduce that $v_{m_{2}}=v_{m_{1}}$ almost everywhere.
over $\Omega_{m_{1}}$, so in general $v_{m_{2}}$ is an extension of
$v_{m_{1}}$ for all $m_{2}$ and $m_{1}$ with $m_{2}>m_{1}$. Take $u$ to be the
maximal such extension, defined over all of $\Omega$. Then for any $m$, we
have that
$\|w_{n}-u\|_{L^{p}(\Omega_{m})}=\|u_{n,n}-v_{m}\|_{L^{p}(\Omega_{m})}\xrightarrow{n\to+\infty}0$
because $\\{u_{n,n}\\}_{n=m}^{\infty}$ is a subsequence of
$\\{u_{m,n}\\}_{n}$, which converges to $v_{m}$ in $L^{p}(\Omega_{m})$.
This proves the claim. In particular, we see that $w_{n}\to u$ in measure.
Moreover the family $\\{|w_{n}|^{p}\\}_{n}$ is uniformly integrable by Lemma
3.3. Thus by Vitali, we get that in fact, $w_{n}\to u$ in $L^{p}(\Omega)$.
Lastly, to see that $u\in BPV(\Omega)$, note that
$\operatorname*{essVar}_{\Omega}u=\lim_{m\to\infty}\operatorname*{essVar}_{\Omega_{m}}u=\lim_{m\to\infty}\operatorname*{essVar}_{\Omega_{m}}v_{m}\leq
C_{3},$
so we are done. ∎
We now find the Gamma-limit under $L^{p}(\Omega)$ convergence.
###### Theorem 3.5.
Let $\Omega=(a,b)$ be a non-empty open interval, and let $1\leq p<\infty$. For
each $\varepsilon>0$, define a functional
$F_{\varepsilon}:L^{p}(\Omega)\to\overline{\mathbb{R}}$ as in (10). Let
$\varepsilon_{n}$ be a sequence of positive reals with $\varepsilon_{n}\to
0^{+}$. For all $u\in L^{p}(\Omega)$, we have
$\Gamma\text{-}\lim_{n\to+\infty}F_{\varepsilon_{n}}(u)=\begin{cases}\alpha\operatorname*{essVar}_{\Omega}u,&u\in
BPV(\Omega;\pm 1)\\\ +\infty,&\text{otherwise}\end{cases}$
under $L^{p}(\Omega)$ convergence, where $\alpha$ is defined as in (8).
We begin with a useful interpolation result.
###### Lemma 3.6.
Let $T>0$ and $A,m\in\mathbb{R}$ with $m\neq 0$. Then there exists
$f:[0,T]\to\mathbb{R}$ satisfying the following properties:
1. (i)
$f\in C^{\infty}([0,T])$,
2. (ii)
$f(0)=A$, $f^{\prime}(0)=m$, $f(T)=0$, and $f^{\prime}(T)=0$,
3. (iii)
$\int_{0}^{T}|f^{(k)}(x)|^{2}\,dx\leq C_{k}(A^{2}+m^{2}T^{2})T^{1-2k}$ for a
constant $C_{k}$ depending only on $k$,
4. (iv)
$\sup_{[0,T]}|f|\leq|A|+\frac{1}{2}|m|\cdot|T|$.
###### Proof.
We take $g:\mathbb{R}\to\mathbb{R}$ with
$g(x)=\begin{cases}mx+A,&-T/2<x<T/2\\\ 0,&\text{otherwise}\end{cases}.$
Now let $\varphi\in C^{\infty}_{c}((-1,1))$ be a non-negative and symmetric
mollifier, so that $\varphi(x)=\varphi(-x)$ for all $x\in(-1,1)$. For each
$\varepsilon>0$ we define
$\varphi_{\varepsilon}(x):=\frac{1}{\varepsilon}\varphi\left(\frac{x}{\varepsilon}\right)$
and the mollification
$g_{\varepsilon}(x):=\int_{x-\varepsilon}^{x+\varepsilon}g(y)\varphi_{\varepsilon}(x-y)\,dy.$
For $\varepsilon<T/2$ we recover the properties $g_{\varepsilon}(0)=A$,
$g^{\prime}_{\varepsilon}(0)=m$, $g_{\varepsilon}(T)=0$, and
$g_{\varepsilon}^{\prime}(T)=0$, because by symmetry of
$\varphi_{\varepsilon}$ we have that $g_{\varepsilon}(x)=mx+A$ for $x$ near
$0$ and $g_{\varepsilon}(x)=0$ for $x$ near $T$. Note that for each $k$, we
have that
$|g(y)\varphi_{\varepsilon}^{(k)}(x-y)|\leq\varepsilon^{-k}\|\varphi^{(k)}\|_{\infty}|g(y)|$
for all $x,y\in\mathbb{R}$. As $g$ is integrable, we may use dominated
convergence to obtain
$\displaystyle g^{(k)}_{\varepsilon}(x)$
$\displaystyle=\frac{d^{k}}{{dx}^{k}}\int_{x-\varepsilon}^{x+\varepsilon}g(y)\varphi_{\varepsilon}(x-y)\,dy=\frac{d^{k}}{{dx}^{k}}\int_{\mathbb{R}}g(y)\varphi_{\varepsilon}(x-y)\,dy$
$\displaystyle=\int_{\mathbb{R}}g(y)\varphi_{\varepsilon}^{(k)}(x-y)\,dy=\int_{x-\varepsilon}^{x+\varepsilon}g(y)\varphi_{\varepsilon}^{(k)}(x-y)\,dy.$
Now we integrate in $x$ and apply Minkowski’s inequality for integrals to get
$\displaystyle\int_{0}^{T}g_{\varepsilon}^{(k)}(x)^{2}\,dx$
$\displaystyle=\int_{0}^{T}\left(\int_{x-\varepsilon}^{x+\varepsilon}g(y)\varphi_{\varepsilon}^{(k)}(x-y)\,dy\right)^{2}\,dx$
$\displaystyle\leq\left(\int_{-\varepsilon}^{T+\varepsilon}\left(\int_{(y-\varepsilon,y+\varepsilon)\cap[0,T]}|g(y)|^{2}|\varphi_{\varepsilon}^{(k)}(x-y)|^{2}\,dx\right)^{1/2}\,dy\right)^{2}$
$\displaystyle\leq\left(\int_{-\varepsilon}^{T+\varepsilon}|g(y)|\left(\int_{y-\varepsilon}^{y+\varepsilon}|\varphi_{\varepsilon}^{(k)}(x-y)|^{2}\,dx\right)^{1/2}\,dy\right)^{2}$
$\displaystyle=\left(\int_{-\varepsilon}^{T+\varepsilon}|g(y)|\,dy\right)^{2}\int_{-\varepsilon}^{\varepsilon}|\varphi_{\varepsilon}^{(k)}(z)|^{2}\,dz.$
Handling the first integral is a matter of computation, which gets us the
bound
$\int_{-\varepsilon}^{T+\varepsilon}|g(y)|\,dy\leq\int_{-\varepsilon}^{T+\varepsilon}|m|\cdot|y|+|A|\,dy=\frac{|m|}{2}\cdot\left((T+\varepsilon)^{2}+\varepsilon^{2}\right)+(T+2\varepsilon)|A|.$
As for the second integral, we may write
$\varphi_{\varepsilon}^{(k)}(x)=\frac{d^{k}}{{dx}^{k}}\frac{1}{\varepsilon}\varphi\left(\frac{x}{\varepsilon}\right)=\frac{1}{\varepsilon^{k+1}}\varphi^{(k)}\left(\frac{x}{\varepsilon}\right)$,
so that
$\int_{-\varepsilon}^{\varepsilon}|\varphi_{\varepsilon}^{(k)}(z)|^{2}\,dz=\int_{-\varepsilon}^{\varepsilon}\frac{1}{\varepsilon^{2k+2}}|\varphi^{(k)}(z/\varepsilon)|^{2}\,dz=\frac{1}{\varepsilon^{2k+1}}\int_{-1}^{1}|\varphi^{(k)}(z)|^{2}\,dz.$
Altogether, we get that
$\int_{0}^{T}g_{\varepsilon}^{(k)}(x)^{2}\,dx\leq\frac{\left(\frac{|m|}{2}\cdot\left((T+\varepsilon)^{2}+\varepsilon^{2}\right)+(T+2\varepsilon)|A|\right)^{2}}{\varepsilon^{2k+1}}\cdot\|\varphi^{(k)}\|_{L^{2}(\mathbb{R})}^{2}.$
Choosing $\varepsilon=T/4$, and applying the inequality $(a+b)^{2}\leq
2a^{2}+2b^{2}$, we see that
$\int_{0}^{T}g_{T/4}^{(k)}(x)^{2}\,dx\leq\frac{|m|^{2}T^{4}+|A|^{2}T^{2}}{T^{2k+1}}\cdot
C_{k}=C_{k}(|A|^{2}+|m|^{2}T^{2})T^{1-2k}$
for a constant $C_{k}$ depending only on $k$.
Lastly, since $|g(x)|\leq|m|\cdot|x|+|A|\leq|m|\cdot\frac{|T|}{2}+|A|$ for all
$x$, we must have
$|g_{T/4}(x)|\leq\int_{x-\varepsilon}^{x+\varepsilon}|g(y)|\varphi_{\varepsilon}(x-y)\,dy\leq\left(|m|\cdot\frac{|T|}{2}+|A|\right)\int_{x-\varepsilon}^{x+\varepsilon}\varphi_{\varepsilon}(x-y)\,dy=|m|\cdot\frac{|T|}{2}+|A|$
for all $x$, so that in particular
$\sup_{[0,T]}|g_{T/4}(x)|\leq|m|\cdot\frac{|T|}{2}+|A|$. We conclude that
taking $f\equiv g_{T/4}$ satisfies the required properties. ∎
Now we prove the theorem.
###### Proof.
Step 1
If $u\not\in L^{p}(\Omega;\pm 1)$ then for some $\varepsilon>0$, the set
$E=\\{x\in\Omega:||u(x)|-1|\geq\varepsilon\\}$ has positive measure. We now
claim that the inequality $(z^{2}-1)^{2}\geq\varepsilon^{2-p}||z|-1|^{p}$
holds for all $z$ with $||z|-1|\geq\varepsilon$. To see this, note that for
all such $z$ with $z\geq 0$ we have
$|z+1|^{2}\cdot|z-1|^{2-p}\geq\varepsilon^{2-p}$
which rearranges to the claimed inequality, and similarly for $z\leq 0$. Hence
we may use this and Minkowski’s inequality to write
$\displaystyle\int_{E}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}\,dx$
$\displaystyle\geq\int_{E}\varepsilon_{n}^{-1}\varepsilon^{2-p}||u_{n}|-1|^{p}\,dx$
$\displaystyle\geq\varepsilon_{n}^{-1}\varepsilon^{2-p}\left(\left(\int_{E}||u|-1|^{p}\,dx\right)^{1/p}-\left(\int_{E}\big{|}|u|-|u_{n}|\big{|}^{p}\,dx\right)^{1/p}\right)^{p}$
$\displaystyle\geq\varepsilon_{n}^{-1}\varepsilon^{2-p}\left(\varepsilon\mathcal{L}^{1}(E)^{1/p}-\left(\int_{E}|u-u_{n}|^{p}\,dx\right)^{1/p}\right)^{p}.$
Now it is clear that $F_{\varepsilon_{n}}(u_{n})\to+\infty$ because
$\|u-u_{n}\|_{L^{p}(E)}\to 0$ and $\varepsilon_{n}^{-1}\to+\infty$. Hence
$\Gamma\text{-}\lim_{n\to\infty}F_{\varepsilon_{n}}(u)=+\infty$.
Step 2
We now consider the case $u\in BPV(\Omega;\pm 1)$.
Suppose $u$ jumps between $-1$ and $1$ or vice versa at the $J$ points
$x_{1},x_{2},\cdots,x_{J}\in\Omega$. Find $\delta_{1},\cdots,\delta_{J}>0$ so
small that the intervals $(a_{i},b_{i})$ are disjoint, where
$(a_{i},b_{i}):=(x_{i}-\delta_{i},x_{i}+\delta_{i})$.
Let $u_{n}\to u$ in $L^{p}(\Omega)$. Suppose
$L=\displaystyle\liminf_{n\to\infty}F_{\varepsilon_{n}}(u_{n})$. For the
liminf inequality, we need to prove that $L\geq\alpha\operatorname*{essVar}u$.
By extraction of a subsequence, assume that actually
$L=\displaystyle\lim_{n\to\infty}F_{\varepsilon_{n}}(u_{n})$. Fix an arbitrary
$\eta\in(0,1)$. Then we may find $N_{\eta}$ so large that
$L+\eta\geq F_{\varepsilon_{n}}(u_{n})$
for all $n\geq N_{\eta}$.
Pick some $x_{i}$ and, for simplicity, assume without loss of generality that
$u$ jumps from $-1$ to $1$ at $x_{i}$.
The key claim that we shall prove in this step is that, for all $n$ large
enough, we may modify $u_{n}$ in $(a_{i},b_{i})$ by “anchoring" it to $-1$ at
$a_{i}$ and to $1$ at $b_{i}$ with a derivative of $0$ at both points, without
spending more than $C\eta$ potential energy for a constant $C$ not depending
on $\eta$. More concretely, there exists $v_{n,i}:[a_{i},b_{i}]\to\mathbb{R}$
such that:
* •
$v_{n,i}\in W^{2,p}((a_{i},b_{i}))$,
* •
$v_{n,i}(a_{i})=-1$, $v_{n,i}(b_{i})=1$,
* •
$v_{n,i}^{\prime}(a_{i})=v_{n,i}^{\prime}(b_{i})=0$,
* •
$\displaystyle\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(v_{n,i}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}|^{2}\,dx\leq\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx+C\eta$.
To do this, we consider only $n\geq N_{\eta}$, and it suffices to modify
$u_{n}$ in $(x_{i},b_{i})$ by affixing to $1$ in the manner described. First,
observe that since $u_{n}\to u$ in $L^{p}(\Omega)$, we have in particular that
$u_{n}\to 1$ in $L^{p}((x_{i},b_{i}))$, so
$\|u_{n}-1\|_{L^{p}((x_{i},b_{i}))}\to 0$ as $n\to+\infty$. Putting this
aside, note that we also have
$\int_{x_{i}}^{b_{i}}\frac{|u_{n}-1|^{p}}{\|u_{n}-1\|_{L^{p}((x_{i},b_{i}))}^{p}}\,dx=1$.
Combining this with $F_{\varepsilon_{n}}(u_{n})\leq L+\eta$, it follows that
$\int_{x_{i}}^{b_{i}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}+\frac{|u_{n}-1|^{p}}{\|u_{n}-1\|_{L^{p}((x_{i},b_{i}))}^{p}}\,dx\leq
L+\eta+1.$
Now let $K_{n}=\lceil\varepsilon_{n}^{-1}\rceil$ and subdivide the interval
$(x_{i},b_{i})$ into $K_{n}$ same-length intervals $(y_{j-1},y_{j})$, so that:
$x_{i}=y_{0}<y_{1}<\cdots<y_{K_{n}}=b_{i}$
By a “pigeonhole principle-like argument", there exists $1\leq j\leq K_{n}$
such that
$\int_{y_{j-1}}^{y_{j}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}+\frac{|u_{n}-1|^{p}}{\|u_{n}-1\|_{L^{p}((x_{i},b_{i}))}^{p}}\,dx\leq\frac{L+\eta+1}{K_{n}}\leq\varepsilon_{n}(L+\eta+1).$
(22)
A particular consequence is the bound
$\int_{y_{j-1}}^{y_{j}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx\leq\frac{L+\eta+1}{K_{n}}\leq\varepsilon_{n}(L+\eta+1)$,
which implies by Theorem 3.2 (applied with the intervals in the inclusion
$(\frac{2}{3}y_{j-1}+\frac{1}{3}y_{j},\frac{1}{3}y_{j-1}+\frac{2}{3}y_{j})\Subset(y_{j-1},y_{j})$)
that
$\int_{\frac{2}{3}y_{j-1}+\frac{1}{3}y_{j}}^{\frac{1}{3}y_{j-1}+\frac{2}{3}y_{j}}\varepsilon_{n}|u_{n}^{\prime}|^{2}\,dx\leq
c\int_{y_{j-1}}^{y_{j}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx\leq\varepsilon_{n}c(L+\eta+1)$
for all sufficiently large $n$, for a universal constant $c$. Combining this
with (22), we obtain
$\int_{\frac{2}{3}y_{j-1}+\frac{1}{3}y_{j}}^{\frac{1}{3}y_{j-1}+\frac{2}{3}y_{j}}\varepsilon_{n}|u_{n}^{\prime}|^{2}+\frac{|u_{n}-1|^{p}}{\|u_{n}-1\|_{L^{p}((x_{i},b_{i}))}^{p}}\,dx\leq\varepsilon_{n}(c+1)(L+\eta+1).$
Let the integrand
$\varepsilon_{n}|u_{n}^{\prime}|^{2}+\frac{|u_{n}-1|^{p}}{\|u_{n}-1\|_{L^{p}((x_{i},b_{i}))}^{p}}$
be $H$. If there does not exist
$x\in(\frac{2}{3}y_{j-1}+\frac{1}{3}y_{j},\frac{1}{3}y_{j-1}+\frac{2}{3}y_{j})$
for which $H(x)\leq\frac{6K_{n}\varepsilon_{n}(c+1)(L+\eta+1)}{b_{i}-x_{i}}$,
then
$\varepsilon_{n}(c+1)(L+\eta+1)\geq\int_{\frac{2}{3}y_{j-1}+\frac{1}{3}y_{j}}^{\frac{1}{3}y_{j-1}+\frac{2}{3}y_{j}}H\,dx\geq\frac{6K_{n}\varepsilon_{n}(c+1)(L+\eta+1)}{b_{i}-x_{i}}\cdot\frac{y_{j}-y_{j-1}}{3}$
$=\frac{6K_{n}\varepsilon_{n}(c+1)(L+\eta+1)}{b_{i}-x_{i}}\cdot\frac{b_{i}-x_{i}}{3K_{n}}=2\varepsilon_{n}(c+1)(L+\eta+1),$
which is a contradiction. We conclude that for all large enough $n$, there
exists
$x_{0}\in(\frac{2}{3}y_{j-1}+\frac{1}{3}y_{j},\frac{1}{3}y_{j-1}+\frac{2}{3}y_{j})$,
depending on $n$, such that
$H(x_{0})\leq(C_{1}/2)K_{n}\varepsilon_{n}\leq(C_{1}/2)\cdot\frac{\varepsilon_{n}+1}{\varepsilon_{n}}\cdot\varepsilon_{n}=(C_{1}/2)(\varepsilon_{n}+1)\leq
C_{1}$ for a constant $C_{1}$ that does not depend on $n$. In particular, we
now know that such an $x_{0}$ satisfies:
1. (1)
$\varepsilon_{n}^{2}|u_{n}^{\prime}(x_{0})|^{2}\leq\varepsilon_{n}C_{1}$,
2. (2)
$|u_{n}(x_{0})-1|\leq C_{1}^{1/p}\|u_{n}-1\|_{L^{p}((x_{i},b_{i}))}$.
These properties, combined with the fact that $u_{n}\to 1$ in
$L^{p}(x_{i},b_{i})$, imply that, for all large enough $n$, we may obtain
$\varepsilon_{n}^{2}|u_{n}^{\prime}(x_{0})|^{2}+|u_{n}(x_{0})-1|^{2}\leq\eta$.
Let $A=u_{n}(x_{0})-1$, $m=u_{n}^{\prime}(x_{0})$, and $T=y_{j}-x_{0}$. Using
these constants, we may find a smooth $f:[0,T]\to\mathbb{R}$ as described in
Lemma 3.6. We are now ready to define $v_{n,i}$ over $(x_{i},b_{i})$ as
$v_{n,i}(x):=\begin{cases}u_{n}(x),&x_{i}<x\leq x_{0}\\\
f(x-x_{0})+1,&x_{0}<x\leq y_{j}\\\ 1,&y_{j}<x<b_{i}\end{cases}.$
By virtue of $f$ being a smooth connector, we must have $v_{n,i}\in
W^{2,p}(x_{i},b_{i})$. Moreover, by property (iii) in Lemma 3.6, we have the
bounds
$\int_{x_{i}}^{b_{i}}(v_{n,i}-1)^{2}\,dx=\int_{x_{0}}^{y_{j}}|f(x-x_{0})|^{2}\,dx\leq
C_{0}(A^{2}+m^{2}T^{2})T$ (23)
and
$\int_{x_{i}}^{b_{i}}|v_{n,i}^{\prime\prime}|^{2}\,dx=\int_{x_{0}}^{y_{j}}|f^{\prime\prime}(x-x_{0})|^{2}\,dx\leq
C_{2}(A^{2}+m^{2}T^{2})T^{-3}$ (24)
for universal constants $C_{0},C_{2}>0$.
Since (23) is not quite a bound on the integral of $(v_{n,i}^{2}-1)^{2}$, we
will need to prove the inequality
$\int_{x_{i}}^{b_{i}}(v_{n,i}^{2}-1)^{2}\,dx\leq
C^{\prime}\int_{x_{i}}^{b_{i}}(v_{n,i}-1)^{2}\,dx$ (25)
for all $n$ large enough, for some constant $C^{\prime}>0$. Indeed, observe
that for all $x\in(x_{i},b_{i})$, we have by property (iv) of Lemma 3.6 that
$|v_{n,i}(x)+1|\leq 2+|v_{n,i}-1|\leq 2+\sup_{[0,T]}|f|\leq
2+|A|+\frac{1}{2}|m|\cdot|T|.$
Now, $T\leq\varepsilon_{n}$, and from
$\varepsilon_{n}^{2}|u_{n}^{\prime}(x_{0})|^{2}+|u_{n}(x_{0})-1|^{2}\leq\eta\leq
1$ we have that $|A|=|u_{n}(x_{0})-1|\leq 1$ and
$|m|\cdot|T|=|u_{n}^{\prime}(x_{0})|\cdot(y_{j}-x_{0})\leq\varepsilon_{n}|u_{n}^{\prime}(x_{0})|\leq
1$. We hence obtain $|v_{n,i}(x)+1|\leq 4$ for all $x\in(x_{i},b_{i})$, and so
taking $C^{\prime}=4^{2}$ we get
$\int_{x_{i}}^{b_{i}}(v_{n,i}^{2}-1)^{2}\,dx=\int_{x_{i}}^{b_{i}}(v_{n,i}-1)^{2}(v_{n,i}+1)^{2}\,dx\leq
C^{\prime}\int_{x_{i}}^{b_{i}}(v_{n,i}-1)^{2},$
as we wanted.
We may now add (23) to (24) and apply (25) to obtain
$\displaystyle\phantom{{}={}}\int_{x_{0}}^{b_{i}}\varepsilon_{n}^{-1}(v_{n,i}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}|^{2}\,dx$
$\displaystyle\leq\max(C_{0},C_{2})\max(1,C^{\prime})\left[\frac{T}{\varepsilon_{n}}(A^{2}+m^{2}T^{2})+\frac{\varepsilon_{n}^{3}}{T^{3}}(A^{2}+m^{2}T^{2})\right]$
$\displaystyle=C_{3}(A^{2}+m^{2}T^{2})\left(\frac{T}{\varepsilon_{n}}+\frac{\varepsilon_{n}^{3}}{T^{3}}\right)$
for some constant $C_{3}>0$.
Since
$\frac{2}{3}y_{j-1}+\frac{1}{3}y_{j}<x_{0}<\frac{1}{3}y_{j-1}+\frac{2}{3}y_{j}$,
we have that
$(b_{i}-x_{i})\varepsilon_{n}\geq\frac{b_{i}-x_{i}}{K_{n}}=y_{j}-y_{j-1}>y_{j}-x_{0}=T>y_{j}-(\frac{1}{3}y_{j-1}+\frac{2}{3}y_{j})=\frac{1}{3}(y_{j}-y_{j-1})=\frac{b_{i}-x_{i}}{3K_{n}}\geq\frac{b_{i}-x_{i}}{3}\cdot\frac{\varepsilon_{n}}{1+\varepsilon_{n}}$.
So we may use this to write
$\displaystyle\phantom{{}={}}\int_{x_{0}}^{b_{i}}\varepsilon_{n}^{-1}(v_{n,i}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}|^{2}\,dx$
$\displaystyle\leq
C_{3}((u_{n}(x_{0})-1)^{2}+|u_{n}^{\prime}(x_{0})|^{2}T^{2})\left(\frac{T}{\varepsilon_{n}}+\frac{\varepsilon_{n}^{3}}{T^{3}}\right)$
$\displaystyle\leq
C_{3}((u_{n}(x_{0})-1)^{2}+|u_{n}^{\prime}(x_{0})|^{2}(b_{i}-x_{i})^{2}\varepsilon_{n}^{2})\left(\frac{(b_{i}-x_{i})\varepsilon_{n}}{\varepsilon_{n}}+\varepsilon_{n}^{3}\cdot\frac{27(1+\varepsilon_{n})^{3}}{(b_{i}-x_{i})^{3}\varepsilon_{n}^{3}}\right)$
$\displaystyle\leq
C_{3}((u_{n}(x_{0})-1)^{2}+|u_{n}^{\prime}(x_{0})|^{2}(b_{i}-x_{i})^{2}\varepsilon_{n}^{2})\left(b_{i}-x_{i}+\frac{27\cdot
8}{(b_{i}-x_{i})^{3}}\right)$ $\displaystyle\leq
C_{4}((u_{n}(x_{0})-1)^{2}+|u_{n}^{\prime}(x_{0})|^{2}\varepsilon_{n}^{2})\leq
C_{4}\eta,$
Where $C_{4}>0$ is a constant with no dependence on $n$, and we have applied
our choice of $x_{0}$. At last, it follows that
$\displaystyle\phantom{{}={}}\int_{x_{i}}^{b_{i}}\varepsilon_{n}^{-1}(v_{n,i}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}|^{2}\,dx$
$\displaystyle=\int_{x_{i}}^{x_{0}}\varepsilon_{n}^{-1}(v_{n,i}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}|^{2}\,dx+\int_{x_{0}}^{b_{i}}\varepsilon_{n}^{-1}(v_{n,i}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}|^{2}\,dx$
$\displaystyle=\int_{x_{i}}^{x_{0}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx+\int_{x_{0}}^{b_{i}}\varepsilon_{n}^{-1}(v_{n,i}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}|^{2}\,dx$
$\displaystyle=\int_{x_{i}}^{x_{0}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx+C_{4}\eta$
$\displaystyle\leq\int_{x_{i}}^{b_{i}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx+C_{4}\eta.$
Doing the same thing for the interval $(a_{i},x_{i})$, we finally obtain
$\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(v_{n,i}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}|^{2}\,dx\leq\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx+2C_{4}\eta.$
(26)
Thus the key claim has been proven.
Step 3
We may now complete the liminf argument. We recall the definitions of the
family $\mathscr{J}$ and the constant $\alpha$ from $\eqref{eq:famJ}$ and
$\eqref{eq:alpha}$ respectively.
Let us first write
$L+\eta\geq
F_{\varepsilon_{n}}(u_{n})\geq\sum_{i=1}^{J}\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx$
for all large enough $n$. We now apply the key claim (26) to every interval
$(a_{i},b_{i})$ to get
$L+\eta\geq-2C_{4}J\eta+\sum_{i=1}^{J}\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(v_{n,i}(x)^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n,i}^{\prime\prime}(x)|^{2}\,dx.$
We now apply the change of variables $y=\frac{x-a_{i}}{b_{i}-a_{i}}$. Defining
$w_{n,i}:[0,1]\to\mathbb{R}$ via
$w_{n,i}(y)=v_{n,i}\left((b_{i}-a_{i})y+a_{i}\right)$, we have that
$w_{n,i}^{\prime\prime}(y)=(b_{i}-a_{i})^{2}v_{n,i}^{\prime\prime}\left((b_{i}-a_{i})y+a_{i}\right)$,
so that we get
$\displaystyle L+\eta$
$\displaystyle\geq-2C_{4}J\eta+\sum_{i=1}^{J}(b_{i}-a_{i})\int_{0}^{1}\varepsilon_{n}^{-1}(w_{n,i}(y)^{2}-1)^{2}+\frac{\varepsilon_{n}^{3}}{(b_{i}-a_{i})^{4}}|w_{n,i}^{\prime\prime}(y)|^{2}\,dy$
$\displaystyle=-2C_{4}J\eta+\sum_{i=1}^{J}\frac{b_{i}-a_{i}}{\varepsilon_{n}}\int_{0}^{1}(w_{n,i}(y)^{2}-1)^{2}\,dy+\frac{\varepsilon_{n}^{3}}{(b_{i}-a_{i})^{3}}\int_{0}^{1}|w_{n,i}^{\prime\prime}(y)|^{2}\,dy,$
and by applying the weighted AM-GM inequality we may go down again to get
$\displaystyle L+\eta$
$\displaystyle\geq-2C_{4}J\eta+\sum_{i=1}^{J}\frac{4}{3^{3/4}}\left(\int_{0}^{1}(w_{n,i}(y)^{2}-1)^{2}\,dy\right)^{3/4}\left(\int_{0}^{1}|w_{n,i}^{\prime\prime}(y)|^{2}\,dy\right)^{1/4}$
$\displaystyle=-2C_{4}J\eta+\sum_{i=1}^{J}\frac{4}{3^{3/4}}\Phi(w_{n,i}).$
Lastly, as $w_{n,i}\in\mathscr{J}$ for all $i$, we have that
$\frac{4}{3^{3/4}}\Phi(w_{n,i})\geq\frac{4}{3^{3/4}}\inf_{w\in\mathscr{J}}\Phi(w)=2\alpha$,
thus
$\displaystyle L+\eta$ $\displaystyle\geq-2C_{4}J\eta+\sum_{i=1}^{J}2\alpha$
$\displaystyle=-2C_{4}J\eta+2J\alpha=-2C_{4}J\eta+\alpha\operatorname*{essVar}_{\Omega}u.$
Hence $L+\eta\geq-2C_{4}J\eta+\alpha\operatorname*{essVar}_{\Omega}u$. As
$\eta$ was arbitrary, we conclude that
$L\geq\alpha\operatorname*{essVar}_{\Omega}u$, as needed. This proves the
liminf inequality for $u\in BPV(\Omega;\pm 1)$.
Step 4
In this step, we briefly resolve the case $u\in L^{2}(\Omega;\pm 1)\setminus
BPV(\Omega;\pm 1)$.
Showing that $F_{\varepsilon_{n}}(u_{n})\to+\infty$ in this case (which is
sufficient to conclude the Gamma limit) requires modification of the argument
in Steps 2 and 3. We sketch the proof: Find $2N$ Lebesgue points
$a<x_{1}<y_{1}<\cdots<x_{N}<y_{N}<b$ for which $u(x_{i})=-1$ and $u(y_{i})=1$.
For each $i$ we may find an interval $(x_{i},x_{i}+\delta)$ for which $u=-1$
in most of $(x_{i},\delta)$ and an interval $(y_{i}-\delta,y_{i})$ for which
$u=1$ in most of $(y_{i}-\delta,y_{i})$. We can then apply the pigeonhole
argument in these intervals by using the fact that
$\|u_{n}+1\|_{L^{p}((x_{i},x_{i}+\delta)\cap\\{u=-1\\}}\to 0$ and
$\|u_{n}-1\|_{L^{p}((y_{i}-\delta,y_{i})\cap\\{u=1\\}}\to 0$. In the end, we
obtain a lower bound for $F_{\varepsilon_{n}}(u_{n})$ that tends to $+\infty$
when we send $N\to+\infty$.
Step 5
The remaining thing to prove is the limsup inequality for $u\in BPV(\Omega;\pm
1)$. This entails finding a sequence $u_{n}\in W^{2,p}(\Omega)$ for which
$u_{n}\to u$ in $L^{p}(\Omega)$ and
$\limsup_{n\to\infty}F_{\varepsilon_{n}}(u_{n})\leq\alpha\operatorname*{Var}_{\Omega}u$.
First, recall the definition of $\Phi$ as in (5) and the family $\mathscr{J}$
as in (6). Define the subfamily
$\mathscr{J}_{n}:=\left\\{h\in\mathscr{J}:\frac{\int_{0}^{1}|h^{\prime\prime}|^{2}\,dx}{\int_{0}^{1}(h^{2}-1)^{2}\,dx}\leq\frac{1}{\varepsilon_{n}}\right\\}.$
Since $\varepsilon_{n}\to 0^{+}$, it is clear that
$\bigcup_{n=1}^{\infty}\mathscr{J}_{n}=\mathscr{J}$. Thus if we define the
sets
$S_{n}:=\left\\{\Phi(h):h\in\mathscr{J}_{n}\right\\},$
$S:=\left\\{\Phi(h):h\in\mathscr{J}\right\\},$
then we have $\bigcup_{n=1}^{\infty}S_{n}=S$, and it is not hard to show that
$\lim_{n\to\infty}\inf S_{n}=\inf S$. We note also that $\alpha$, as defined
in (8), may be written as $\alpha=\frac{2}{3^{3/4}}\inf S$.
For all $n$, find $h_{n}\in\mathscr{J}_{n}$ for which
$\inf S_{n}\leq\Phi(h_{n})\leq\frac{1}{n}+\inf S_{n}.$ (27)
As in Step 2, suppose $u$ “jumps" at the points
$x_{1},x_{2},\cdots,x_{J}\in\Omega$, and find pairwise disjoint intervals
$(a_{i},b_{i})\subseteq\Omega$ with $x_{i}\in(a_{i},b_{i})$. Consider some
$x_{i}$, and without loss of generality assume that $u$ jumps from $-1$ to $1$
at $x_{i}$. Let $\zeta_{n}\to 0$ be a sequence of positive reals that we shall
choose later. We now define $u_{n}$ over $(a_{i},b_{i})$ as
$u_{n}(x):=\begin{cases}-1,&a_{i}<x<x_{i}-\zeta_{n}/2\\\
h_{n}\left(\frac{x-x_{i}+\zeta_{n}/2}{\zeta_{n}}\right),&x_{i}-\zeta_{n}/2\leq
x\leq x_{i}+\zeta_{n}/2\\\ 1,&x_{i}+\zeta_{n}/2<x<b_{i}\end{cases}.$
We define $u_{n}$ similarly over all other intervals $(a_{j},b_{j})$, and in
$\Omega\setminus\bigcup_{j=1}^{J}(a_{j},b_{j})$ we let $u_{n}$ agree with $u$.
Since $h_{n}(0_{+})=-1$, $h_{n}(1_{-})=1$, and
$h_{n}^{\prime}(0_{+})=h_{n}^{\prime}(1_{-})=0$, we have that $u_{n}\in
W^{2,p}(\Omega)$. Moreover, $u_{n}\to u$ almost everywhere as a consequence of
$\zeta_{n}\to 0^{+}$, and $\\{|u_{n}|\\}_{n}$ is uniformly integrable, so by
Vitali we have $u_{n}\to u$ in $L^{p}(\Omega)$. It remains to verify the
limsup inequality. We may compute
$\displaystyle F_{\varepsilon_{n}}(u_{n})$
$\displaystyle=\sum_{i=1}^{J}\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx$
$\displaystyle=\sum_{i=1}^{J}\int_{a_{i}}^{b_{i}}\varepsilon_{n}^{-1}\left(h_{n}\left(\frac{x-x_{i}+\zeta_{n}/2}{\zeta_{n}}\right)^{2}-1\right)^{2}+\frac{\varepsilon_{n}^{3}}{\zeta_{n}^{4}}\left|h_{n}^{\prime\prime}\left(\frac{x-x_{i}+\zeta_{n}/2}{\zeta_{n}}\right)\right|^{2}\,dx,$
and after changing variables we get
$\displaystyle F_{\varepsilon_{n}}(u_{n})$
$\displaystyle=\sum_{i=1}^{J}\zeta_{n}\int_{0}^{1}\varepsilon_{n}^{-1}\left(h_{n}\left(y\right)^{2}-1\right)^{2}+\frac{\varepsilon_{n}^{3}}{\zeta_{n}^{4}}\left|h_{n}^{\prime\prime}\left(y\right)\right|^{2}\,dy$
$\displaystyle=\sum_{i=1}^{J}\frac{\zeta_{n}}{\varepsilon_{n}}\int_{0}^{1}\left(h_{n}\left(y\right)^{2}-1\right)^{2}\,dy+\frac{\varepsilon_{n}^{3}}{\zeta_{n}^{3}}\int_{0}^{1}\left|h_{n}^{\prime\prime}\left(y\right)\right|^{2}\,dy.$
(28)
We now choose $\zeta_{n}$ so that we obtain the equality case in the AM-GM
inequality. Specifically, we choose
$\zeta_{n}=\varepsilon_{n}\left(\frac{3\displaystyle\int_{0}^{1}\left|h_{n}^{\prime\prime}\left(y\right)\right|^{2}\,dy}{\displaystyle\int_{0}^{1}\left(h_{n}\left(y\right)^{2}-1\right)^{2}\,dy}\right)^{1/4}.$
Showing that this choice is valid for all sufficiently large $n$ amounts to
proving that $\zeta_{n}\to 0^{+}$. Fortunately, since
$h_{n}\in\mathscr{J}_{n}$, we have that
$\zeta_{n}\leq\varepsilon_{n}\left(\frac{3}{\varepsilon_{n}}\right)^{1/4}=3^{1/4}\varepsilon_{n}^{3/4}\to
0^{+}.$
With the choice of $\zeta_{n}$ justified, we may now continue the computation
in (28) by using the choice of $\zeta_{n}$ and (27) to write
$F_{\varepsilon_{n}}(u_{n})=\sum_{i=1}^{J}\frac{4}{3^{3/4}}\left(\int_{0}^{1}(h_{n}^{2}-1)^{2}\,dy\right)^{3/4}\left(\int_{0}^{1}|h_{n}^{\prime\prime}|^{2}\,dy\right)^{1/4}\leq\frac{4J}{3^{3/4}}\left(\frac{1}{n}+\inf
S_{n}\right).$
Taking the limsup, we obtain
$\limsup_{n\to\infty}F_{\varepsilon_{n}}(u_{n})\leq\limsup_{n\to\infty}\frac{4J}{3^{3/4}}\left(\frac{1}{n}+\inf
S_{n}\right)=\frac{2\operatorname*{Var}_{\Omega}u}{3^{3/4}}\left(\inf
S\right)=\alpha\operatorname*{Var}_{\Omega}u.$
This completes the proof. ∎
## 4\. Boundary Conditions
The Gamma limit will change upon restricting to the boundary conditions
$u(a_{+})=a_{\varepsilon}$ and $u(b_{-})=b_{\varepsilon}$ for
$a_{\varepsilon}\to-1$ and $b_{\varepsilon}\to 1$ as $\varepsilon\to 0^{+}$. A
portion of the work needed to account for the boundary conditions has already
been done in the proof of Theorem 3.5, but there is still much to be done.
Intuitively, every jump in the interior of $(a,b)$ induces a factor of
$\alpha$, whereas a jump at the boundary induces a factor of $\beta(t)$
depending on the height of the jump. Recall that $\beta(t)$ is defined as in
the statement of Theorem 1.1, and we will again reference the family
$\mathscr{J}^{\prime}(t)$ and the functional $\Phi$.
We first introduce a new family of functions $\mathscr{J}^{\prime}(t)$,
defined as
$\mathscr{J}^{\prime}_{\infty}(t):=\left\\{u\in
W^{2,\infty}(-\infty,0):u(0)=t\text{ and }u(x)=-1\ \forall x\leq-L\text{ for
some }L>0\right\\}$
for a parameter $t\in\mathbb{R}$. We associate with each
$u\in\mathscr{J}^{\prime}_{\infty}(t)$ the constant $L_{u}$, where $-L_{u}$ is
the first time that $u$ reaches $-1$ and remains at this value indefinitely.
That is,
$L_{u}:=\inf\left\\{L>0:u(x)=-1\ \forall x\leq-L\right\\}.$
We also define a new functional $\Psi:W^{2,\infty}(-\infty,0)\to\mathbb{R}$
via
$\Psi(u):=\int_{-\infty}^{0}(u^{2}-1)^{2}+|u^{\prime\prime}|^{2}\,dx.$
The relevance of these constructs is as follows.
###### Lemma 4.1.
$\beta(t)=\frac{4}{3^{3/4}}\inf_{u\in\mathscr{J}^{\prime}(t)}\Phi(u)=\inf_{v\in\mathscr{J}^{\prime}_{\infty}(t)}\Psi(v).$
###### Proof.
We first note that if $t=-1$ then clearly
$\inf_{u\in\mathscr{J}^{\prime}(t)}\Phi(u)=0$ and
$\inf_{u\in\mathscr{J}^{\prime}_{\infty}(t)}\Psi(u)=0$, so we may assume that
$t\neq-1$. Now consider $u\in\mathscr{J}^{\prime}(t)$. Then take $L>0$
depending only on $u$ that we shall choose later, and define
$v\in\mathscr{J}^{\prime}_{\infty}(t)$ via
$v(x)=\begin{cases}-1,&x\leq-L\\\ u(x/L+1),&-L<x<0\end{cases}.$
Then
$\displaystyle\Psi(v)$
$\displaystyle=\int_{-L}^{0}(v(x)^{2}-1)^{2}+|v^{\prime\prime}(x)|^{2}\,dx$
$\displaystyle=\int_{-L}^{0}(u(x/L+1)^{2}-1)^{2}+\frac{1}{L^{4}}|u^{\prime\prime}(x/L+1)|^{2}\,dx$
$\displaystyle=\int_{0}^{1}L(u(y)^{2}-1)^{2}+\frac{1}{L^{3}}|u^{\prime\prime}(y)|^{2}\,dy.$
(29)
By examination of the equality case of AM-GM, this is precisely
$\frac{4}{3^{3/4}}\Phi(u)$ for a proper choice of $L$. Specifically, one may
take
$L=\left(\frac{3\int_{0}^{1}|u^{\prime\prime}|^{2}\,dx}{\int_{0}^{1}(u^{2}-1)^{2}\,dx}\right)^{1/4}$,
which is well-defined from the assumption that $t\neq-1$. This gives
$\frac{4}{3^{3/4}}\Phi(u)=\Psi(v)\geq\inf_{v\in\mathscr{J}_{\infty}^{\prime}(t)}\Psi(v)$,
and taking the infimum gives
$\inf_{u\in\mathscr{J}^{\prime}(t)}\Phi(u)\geq\inf_{v\in\mathscr{J}_{\infty}^{\prime}(t)}\Psi(v)$.
On the other hand, if $v\in\mathscr{J}^{\prime}_{\infty}(t)$, then from taking
$u(x):=v\left(L_{v}(x-1)\right)$ we get from using the same computations done
in (29) and applying AM-GM that
$\displaystyle\Psi(v)$
$\displaystyle=\int_{0}^{1}\frac{L_{v}}{3}(u(y)^{2}-1)^{2}+\frac{1}{L_{v}^{3}}|u^{\prime\prime}(y)|^{2}\,dy$
$\displaystyle\geq\frac{4}{3^{3/4}}\left(\int_{0}^{1}(u(y)^{2}-1)^{2}\,dy\right)^{3/4}\left(\int_{0}^{1}|u^{\prime\prime}(y)|^{2}\,dy\right)^{1/4}$
$\displaystyle=\frac{4}{3^{3/4}}\Phi(u),$
so that
$\Psi(v)\geq\frac{4}{3^{3/4}}\inf_{u\in\mathscr{J}^{\prime}(t)}\Phi(u)$. Thus
$\inf_{v\in\mathscr{J}^{\prime}_{\infty}(t)}\Psi(v)\geq\frac{4}{3^{3/4}}\inf_{u\in\mathscr{J}^{\prime}(t)}\Phi(u)$,
finishing the proof. ∎
We will now prove several crucial results concerning the family
$\mathscr{J}^{\prime}_{\infty}(t)$ and the functional $\Psi$. The first is a
compactness result.
###### Lemma 4.2.
Let $\Omega\subseteq\mathbb{R}$ be a non-empty bounded open set and $u_{n}\in
w^{2,\infty}(\Omega)$ such that
$M:=\sup_{n\in\mathbb{N}}\int_{\Omega}(u_{n}^{2}-1)^{2}+|u_{n}^{\prime\prime}|^{2}\,dx<\infty.$
There exists a subsequence $u_{n_{k}}$ and $u\in W^{2,2}(\Omega)$ such that
1. (1)
$u_{n_{k}}\rightharpoonup u$ in $W^{2,2}(\Omega)$,
2. (2)
$u_{n_{k}}^{\prime}\to u^{\prime}$ almost everywhere and in $L^{2}(\Omega)$,
and
3. (3)
$u_{n_{k}}\to u$ almost everywhere and in $L^{2}(\Omega)$
###### Proof.
Find $A>0$ large enough so that $z^{2}\leq(z^{2}-1)^{2}$ for all $|z|\geq A$.
Then
$\int_{\Omega}u_{n}^{2}\,dx\leq\int_{\\{|u_{n}|<A\\}}u_{n}^{2}\,dx+\int_{\\{|u_{n}|\geq
A\\}}u_{n}^{2}\,dx\leq
A^{2}\mathcal{L}^{1}(\Omega)+A\int_{\Omega}(u_{n}^{2}-1)^{2}\leq
A^{2}\mathcal{L}^{1}(\Omega)+AM,$
so $\\{u_{n}\\}_{n}$ is uniformly bounded in $L^{2}(\Omega)$. Since
$\\{u_{n}^{\prime\prime}\\}_{n}$ is uniformly bounded in $L^{2}(\Omega)$ as
well, we may apply Theorem 2.10 with some $l<\mathcal{L}^{1}(\Omega)$ to
deduce that $\\{u_{n}^{\prime}\\}_{n}$ is uniformly bounded in
$L^{2}(\Omega)$. Hence $\\{u_{n}\\}_{n}$ is uniformly bounded in
$W^{2,2}(\Omega)$.
By Theorem 2.14, there exists a subsequence $u_{n_{k}}$ and $u\in
W^{2,2}(0,T)$ such that $u_{n_{k}}\rightharpoonup u$ in $W^{2,2}(0,T)$.
Moreover, by 2.13, we have that $u_{n_{k}}\to u$ strongly in $L^{2}(0,T)$ and
$u_{n_{k}}^{\prime}\to u^{\prime}$ strongly in $L^{2}(0,T)$. By extracting a
subsequence, we may also get $u_{n_{k}}\to u$ and $u_{n_{k}}^{\prime}\to
u^{\prime}$ almost everywhere. ∎
Next, we show that we may “force" a bound on $L_{u}$ for functions
$u\in\mathscr{J}^{\prime}_{\infty}(t)$.
###### Lemma 4.3.
Let $M,\eta>0$. Then there exists a constant $L_{M,\eta}>0$ depending only on
$M$ and $\eta$ such that for every $t\in\mathbb{R}$ and
$u\in\mathscr{J}^{\prime}_{\infty}(t)$ with $\Psi(u)\leq M<\infty$, there
exists $v\in\mathscr{J}^{\prime}_{\infty}(t)$ such that
$\Psi(v)\leq\Psi(u)+O(\eta)$ and $v(x)=-1$ for all $x\leq-L_{M,\eta}$. That
is, $L_{v}\leq L_{M,\eta}$.
###### Proof.
We begin by constructing a function $v_{1}\in\mathscr{J}^{\prime}_{\infty}(t)$
such that $\Psi(v_{1})\leq\Psi(u)+O(\eta)$ and
$\mathcal{L}^{1}(v_{1}^{-1}((0,2)))\leq C$ for a constant $C$ depending only
on $M$ and $\eta$.
Let $K=\frac{M}{\eta}$. We will take $C=2K+2$. Let $E:=u^{-1}((0,2))$. If we
have that $\mathcal{L}^{1}(E)\leq 2K+2$, then we simply take $v_{1}=u$.
Otherwise, we may let $E_{1}:=(-\infty,y)\cap E$, $F:=[y,z]\cap E$, and
$E_{2}:=(z,0)\cap E$, where $y,z$ are chosen such that
$\mathcal{L}^{1}(E_{1})=\mathcal{L}^{1}(E_{2})=K$ and
$\mathcal{L}^{1}(F)=\mathcal{L}^{1}(E)-2K>2$.
Since $u>0$ over $E_{1}$, we may write
$M\geq\int_{E_{1}}(u^{2}-1)^{2}+|u^{\prime\prime}|^{2}\,dx\geq\int_{E_{1}}(u-1)^{2}+|u^{\prime\prime}|^{2}\,dx,$
so there exists $x_{1}\in E_{1}$ such that
$(u(x_{1})-1)^{2}+u^{\prime\prime}(x_{1})^{2}\leq\frac{M}{\mathcal{L}^{1}(E_{1})}=\frac{M}{K}=\eta$.
It follows by Lemma 3.6 that there exists $\tilde{u}_{1}\in
C^{\infty}([x_{1},x_{1}+1])$ such that $\tilde{u}_{1}(x_{1})=u(x_{1})$,
$\tilde{u}_{1}^{\prime}(x_{1})=u^{\prime}(x_{1})$, $\tilde{u}_{1}(x_{1}+1)=1$,
and $\tilde{u}_{1}^{\prime}(x_{1}+1)=0$, with
$\displaystyle\int_{x_{1}}^{x_{1}+1}(\tilde{u}_{1}^{2}-1)^{2}+|\tilde{u}^{\prime\prime}|^{2}\,dx$
$\displaystyle\leq\int_{x_{1}}^{x_{1}+1}9(\tilde{u}_{1}-1)^{2}\,dx+\int_{x_{1}}^{x_{1}+1}|\tilde{u}^{\prime\prime}|^{2}\,dx$
$\displaystyle\leq
9C^{\prime}(u(x_{1})-1)^{2}+C^{\prime}|u^{\prime}(x_{1})|^{2}$
$\displaystyle\leq 9C^{\prime}\eta=O(\eta)$
for some universal constant $C^{\prime}>0$. We construct $x_{2}\in E_{2}$ and
$\tilde{u}_{2}\in C^{\infty}([x_{2}-1,x_{2}])$ in a similar manner so that
$\int_{x_{2}-1}^{x_{2}}(\tilde{u}_{2}-1)^{2}+|\tilde{u}_{2}^{\prime\prime}|^{2}\,dx=O(\eta)$.
Now define $\tilde{u}$ as
$\tilde{u}(x):=\begin{cases}u(x),&-\infty<x<x_{1}\\\
\tilde{u}_{1}(x),&x_{1}\leq x\leq x_{1}+1\\\ 1,&x_{1}+1<x<x_{2}-1\\\
\tilde{u}_{2}(x),&x_{2}-1\leq x\leq x_{2}\\\ u(x),&x_{2}<x<0\end{cases}.$
Note that this is well-defined in the sense that $x_{1}+1<x_{2}-1$ because
$x_{2}-x_{1}>z-y\geq\mathcal{L}^{1}(F)\geq 2$. Moreover our choices for
$\tilde{u}_{1}$ and $\tilde{u}_{2}$ ensure that $u\in
W^{2,\infty}(-\infty,0)$, and
$\displaystyle\Psi(\tilde{u})$
$\displaystyle=\int_{(-\infty,x_{1})\cup(x_{2},0)}(\tilde{u}^{2}-1)^{2}+|\tilde{u}^{\prime\prime}|^{2}\,dx+\int_{[x_{1},x_{1}+1]\cup[x_{2}-1,x_{2}]}(\tilde{u}^{2}-1)^{2}+|\tilde{u}^{\prime\prime}|^{2}\,dx$
$\displaystyle\leq\int_{(-\infty,x_{1})\cup(x_{2},0)}(u^{2}-1)^{2}+|\tilde{u}^{\prime\prime}|^{2}\,dx+\int_{[x_{1},x_{1}+1]\cup[x_{2}-1,x_{2}]}(u^{2}-1)^{2}+|u^{\prime\prime}|^{2}\,dx+O(\eta)$
$\displaystyle\leq\Psi(u)+O(\eta).$
We may now take $v_{1}$ to be
$v_{1}(x):=\begin{cases}\tilde{u}(x-x_{2}+x_{1}+2),&-\infty<x<x_{2}-1\\\
\tilde{u}(x),&x_{2}-1<x<0\end{cases}.$
In essence, we have “deleted" an interval in which $\tilde{u}=1$. We still
have $v\in W^{2,\infty}(-\infty,0)$ and $v(0_{+})=u(0_{+})=t$, with
$\lim_{x\to-\infty}v(x)=-1$, so that $v\in\mathscr{J}^{\prime}_{\infty}(t)$.
Furthermore, $\Psi(v)=\Psi(\tilde{u})\leq\Psi(u)+O(\eta)$. Lastly, we see that
$\displaystyle\mathcal{L}^{1}(\\{x<0:0<v_{1}(x)<2\\})$
$\displaystyle\leq\mathcal{L}^{1}(\\{x\in(-\infty,x_{1}+1)\cup(x_{2}-1,0):0<\tilde{u}(x)<2\\})$
$\displaystyle\leq\mathcal{L}^{1}(\\{x\in E_{1}\cup E_{2}:0<u(x)<2\\})+2$
$\displaystyle=2K+2=C,$
which was our goal.
Now we can construct $v$. Observe that
$M\geq\int_{\\{\min(|v-1|,|v+1|)>1\\}}(v_{1}^{2}-1)^{2}+|v_{1}^{\prime\prime}|^{2}\,dx\geq\mathcal{L}^{1}(\\{x<0:\min(|v(x)-1|,|v(x)+1|)>1\\}).$
Thus, if we take $L_{M,\eta}=C+M+K+1$, then the set
$G:=\\{-L_{M,\eta}+1<x<0:-2<v(x)<0\\}$
satisfies $\mathcal{L}^{1}(G)\geq K$. Since
$\int_{G}(v_{1}+1)^{2}+|v^{\prime\prime}|^{2}\,dx\leq\int_{G}(v_{1}^{2}-1)^{2}+|v^{\prime\prime}|^{2}\,dx\leq
M$, there exists $x_{3}\in G$ such that
$(v_{1}(x_{3})+1)^{2}+|v_{1}^{\prime\prime}(x_{3})|^{2}\leq\frac{M}{\mathcal{L}^{1}(G)}\leq
M/K=\eta.$
Hence, as we did before to $u$, we may use Lemma 3.6 to modify $v_{1}$ in the
interval $(x_{3}-1,x_{3})$ and hence obtain a function $v$ for which $v(x)=-1$
for all $x<x_{3}-1$, $v(x)=v_{1}(x)$ for $x_{3}<x<0$, and
$\int_{x_{3}-1}^{x_{3}}(v_{1}^{2}-1)^{2}+|v_{1}^{\prime\prime}|^{2}\,dx\leq
O(\eta)$.
This function $v$ satisfies
$\Psi(v)\leq\Psi(v_{1})+O(\eta)\leq\Psi(u)+O(\eta)$ and, since
$-L_{M,\eta}+1<x_{3}$, we have that $v(x)=-1$ for all $x<-L_{M,\eta}$, where
$L_{M,\eta}$ depends only on $M$ and $\eta$, as desired. ∎
Now we arrive at our first major result.
###### Lemma 4.4.
Let $t_{0}\in\mathbb{R}$. Then we have that
$\lim_{t\to t_{0}}\beta(t)=\beta(t_{0}),$
where $\beta(t)$ is defined as in (9). That is, $\beta$ is continuous.
###### Proof.
Step 1
We claim that $\beta(t)$ is decreasing over $t\leq-1$ and increasing over
$t\geq-1$.
First note that $\beta(t)\geq 0$ for all $t$ and $\beta(-1)=0$. Now suppose
$s,t\in\mathbb{R}$ satisfy either $-1<s\leq t$ or $t\leq s<-1$. We show that
$\beta(s)\leq\beta(t)$ by proving that for all $v\in\mathscr{J}^{\prime}(t)$
we may find $u\in\mathscr{J}^{\prime}(s)$ for which $\Phi(u)\leq\Phi(v)$.
Indeed, since $v(1_{-})=t$ and $v(0_{+})=-1$, there must exist $T\in(0,1)$ for
which $v(T)=s$. Now take $u(x):=v(Tx)$. Evidently,
$u\in\mathscr{J}^{\prime}(s)$. Moreover we have
$\int_{0}^{1}(u^{2}-1)^{2}\,dx=\int_{0}^{T}T^{-1}(v^{2}-1)^{2}\,dx\leq\int_{0}^{1}T^{-1}(v^{2}-1)^{2}\,dx$
and
$\int_{0}^{1}|u^{\prime\prime}|^{2}\,dx=\int_{0}^{T}T^{3}|v^{\prime\prime}|^{2}\,dx\leq\int_{0}^{1}T^{3}|v^{\prime\prime}|^{2}\,dx.$
Thus
$\displaystyle\Phi(u)$
$\displaystyle=\left(\int_{0}^{1}(u^{2}-1)^{2}\,dx\right)^{3/4}\left(\int_{0}^{1}|u^{\prime\prime}|^{2}\,dx\right)^{1/4}$
$\displaystyle\leq\left(\int_{0}^{1}T^{-1}(v^{2}-1)^{2}\,dx\right)^{3/4}\left(\int_{0}^{1}T^{3}|v^{\prime\prime}|^{2}\,dx\right)^{1/4}$
$\displaystyle=\left(\int_{0}^{1}(v^{2}-1)^{2}\,dx\right)^{3/4}\left(\int_{0}^{1}|v^{\prime\prime}|^{2}\,dx\right)^{1/4}$
$\displaystyle=\Phi(v),$
as needed.
Step 2
Take $t_{0}\in\mathbb{R}$. We show that $\limsup_{t\to
t_{0}}\beta(t)\leq\beta(t_{0})$. It is sufficient to prove that for any
$v\in\mathscr{J}^{\prime}(t_{0})$, we may pick
$u_{t}\in\mathscr{J}^{\prime}(t)$ for each $t\in\mathbb{R}$ such that
$\lim_{t\to t_{0}}\Phi(u_{t})=\Phi(v)$. This is because if we fix $\eta>0$,
then we may take $v\in\mathscr{J}^{\prime}(t_{0})$ such that
$\beta(t_{0})\leq\frac{4}{3^{3/4}}\Phi(v)\leq\beta(t_{0})+\eta$, take $u_{t}$
for each $t\in\mathbb{R}$ as above, and then choose $\delta>0$ so small that
$|\Phi(v)-\Phi(u_{t})|<\eta$ for all $t$ with $|t-t_{0}|<\delta$, so that
$\beta(t)\leq\frac{4}{3^{3/4}}\Phi(u_{t})\leq\frac{4}{3^{3/4}}(\Phi(v)+\eta)\leq\beta(t_{0})+\eta+\frac{4}{3^{3/4}}\eta,$
which is enough.
For $v\in\mathscr{J}^{\prime}(t_{0})$, we take
$u_{t}\in\mathscr{J}^{\prime}(t)$ to be $u_{t}:=\frac{1+t}{1+t_{0}}(v+1)-1$.
Then
$\int_{0}^{1}(u_{t}^{2}-1)^{2}\,dx=\left(\frac{1+t}{1+t_{0}}\right)^{4}\int_{0}^{1}(v+1)^{2}\left(v+1-2\cdot\frac{1+t_{0}}{1+t}\right)^{2}\,dx.$
Taking the limit, we obtain
$\lim_{t\to t_{0}}\int_{0}^{1}(u_{t}^{2}-1)^{2}\,dx=\lim_{t\to
t_{0}}\int_{0}^{1}(v+1)^{2}\left(v+1-2\cdot\frac{1+t_{0}}{1+t}\right)^{2}\,dx=\int_{0}^{1}(v^{2}-1)^{2}\,dx$
because $(v+1)^{2}\left(v+1-2\cdot\frac{1+t_{0}}{1+t}\right)^{2}$ converges to
$(v^{2}-1)^{2}$ pointwise as $t\to t_{0}$ and, since $v$ is bounded, we have
that $(v+1)^{2}\left(v+1-2\cdot\frac{1+t_{0}}{1+t}\right)^{2}$ is uniformly
bounded for all $t$ sufficiently near $t$, which is enough to apply dominated
convergence.
We also have
$\lim_{t\to t_{0}}\int_{0}^{1}|u_{t}^{\prime\prime}|^{2}\,dx=\lim_{t\to
t_{0}}\left(\frac{1+t}{1+t_{0}}\right)^{2}\int_{0}^{1}|v^{\prime\prime}|^{2}\,dx=\int_{0}^{1}|v^{\prime\prime}|^{2}\,dx.$
Altogether, we see that
$\displaystyle\lim_{t\to t_{0}}\Phi(u_{t})$ $\displaystyle=\left(\lim_{t\to
t_{0}}\int_{0}^{1}(u_{t}^{2}-1)^{2}\,dx\right)^{3/4}\left(\lim_{t\to
t_{0}}\int_{0}^{1}|u_{t}^{\prime\prime}|^{2}\,dx\right)^{1/4}$
$\displaystyle=\left(\int_{0}^{1}(v^{2}-1)^{2}\,dx\right)^{3/4}\left(\int_{0}^{1}|v^{\prime\prime}|^{2}\,dx\right)^{1/4}$
$\displaystyle=\Phi(v).$
This completes the proof that $\beta$ is upper semi-continuous.
Step 3
We now define yet another family $\overline{\mathscr{J}^{\prime}_{\infty}(t)}$
via
$\overline{\mathscr{J}^{\prime}_{\infty}(t)}:=\left\\{u\in
W^{2,\infty}(-\infty,0):u(0_{-})=t\text{ and
}\lim_{x\to-\infty}u(x)=-1\right\\}.$
We claim that
$\beta(t)=\inf_{u\in\mathscr{J}^{\prime}_{\infty}(t)}\Psi(u)=\inf_{u\in\overline{\mathscr{J}^{\prime}_{\infty}(t)}}\Psi(u).$
Since
$\mathscr{J}^{\prime}_{\infty}(t)\subseteq\overline{\mathscr{J}^{\prime}_{\infty}(t)}$,
it is clear that
$\inf_{u\in\mathscr{J}^{\prime}_{\infty}(t)}\Psi(u)\geq\inf_{u\in\overline{\mathscr{J}^{\prime}_{\infty}(t)}}\Psi(u)$.
To show that
$\inf_{u\in\mathscr{J}^{\prime}_{\infty}(t)}\Psi(u)\leq\inf_{u\in\overline{\mathscr{J}^{\prime}_{\infty}(t)}}\Psi(u)$,
we fix $\varepsilon>0$ and $u\in\overline{\mathscr{J}^{\prime}_{\infty}(t)}$.
It is then sufficient to find
$v\in\overline{\mathscr{J}^{\prime}_{\infty}(t)}$ for which
$\Psi(v)\leq\Psi(u)+O(\varepsilon)$.
Find $M$ so large that
$\int_{-\infty}^{-M}|u^{\prime\prime}|^{2}\,dx<\varepsilon^{2}$ and
$|u(x)+1|<\varepsilon$ for all $x\leq-M$. Then
$\int_{-M-1}^{-M}(u+1)^{2}\,dx\leq\varepsilon^{2},$
and so by Theorem 2.10 applied to $u+1$ with $l=1$, we see that
$\left(\int_{-M-1}^{-M}|u^{\prime}|^{2}\,dx\right)^{1/2}\leq
c\left(\int_{-M-1}^{-M}(u+1)^{2}\,dx\right)^{1/2}+c\left(\int_{-M-1}^{-M}|u^{\prime\prime}|^{2}\,dx\right)^{1/2}\leq
2c\varepsilon$
for a universal constant $c>0$, so that
$\int_{-M-1}^{-M}|u^{\prime}|^{2}\,dx\leq 4c^{2}\varepsilon^{2}$. It follows
that there exists $T\in(M,M+1)$ such that $|u^{\prime}(-T)|\leq
2c\varepsilon$. Since $|u(-T)+1|<\varepsilon$, we may apply Lemma 3.6 to find
$f\in C^{\infty}([-T-1,-T])$ such that
1. (i)
$f(-T-1)=f^{\prime}(-T-1)=0$, $f(-T)=u(-T)$, $f^{\prime}(-T)=u^{\prime}(-T)$,
2. (ii)
for a universal constant $C_{0}$ we have
$\int_{-T-1}^{-T}|f+1|^{2}\,dx\leq
C_{0}(|u(-T)+1|^{2}+|u^{\prime}(-T)|^{2})=C_{0}(\varepsilon^{2}+4c^{2}\varepsilon^{2}),$
3. (iii)
for a universal constant $C_{2}$ we have
$\int_{-T-1}^{-T}|f^{\prime\prime}|^{2}\,dx\leq
C_{2}(|u(-T)+1|^{2}+|u^{\prime}(-T)|^{2})=C_{2}(\varepsilon^{2}+4c^{2}\varepsilon^{2}),$
and
4. (iv)
$\sup_{[-T-1,-T]}|f|\leq|u(-T)+1|+\frac{1}{2}|u^{\prime}(-T)|=(1+2c)\varepsilon$.
Naturally we now take
$v(x):=\begin{cases}0,&x<-T-1\\\ f(x),&-T-1\leq x\leq-T\\\
u(x),&-T<x<0\end{cases}.$
By item (i), we see that $v\in W^{2,\infty}(-\infty,0)$, and so
$v\in\mathscr{J}^{\prime}_{\infty}(t)$.
For $\varepsilon$ sufficiently small, we have $(1+2c)\varepsilon<1$, which
ensures that $|f(x)-1|\leq 2$ for all $x\in[-T-1,-T]$ by item (iv). This bound
and item (ii) gives
$\displaystyle\int_{-\infty}^{0}(v^{2}-1)^{2}\,dx$
$\displaystyle\leq\int_{-\infty}^{0}(u^{2}-1)^{2}\,dx+\int_{-T-1}^{-T}(f^{2}-1)^{2}\,dx$
$\displaystyle\leq\int_{-\infty}^{0}(u^{2}-1)^{2}\,dx+4\int_{-T-1}^{-T}(f+1)^{2}\,dx$
$\displaystyle\leq\int_{-\infty}^{0}(u^{2}-1)^{2}\,dx+4C_{0}(\varepsilon^{2}+4c^{2}\varepsilon^{2})\,dx,$
(30)
and item (iii) gives
$\displaystyle\int_{-\infty}^{0}|v^{\prime\prime}|^{2}\,dx$
$\displaystyle\leq\int_{-\infty}^{0}|u^{\prime\prime}|^{2}\,dx+\int_{-T-1}^{-T}|f^{\prime\prime}|^{2}\,dx$
$\displaystyle\leq\int_{-\infty}^{0}|u^{\prime\prime}|^{2}\,dx+C_{2}(\varepsilon^{2}+4c^{2}\varepsilon^{2}).$
(31)
Adding (30) and (31) gives
$\Psi(v)\leq\Psi(u)+(4C_{0}+C_{2})(\varepsilon^{2}+4c^{2}\varepsilon^{2})=O(\varepsilon),$
which is enough. This proves the claim.
Step 4
In this step, we show that if $u\in W^{2,\infty}(-\infty,0)$ such that
$\Psi(u)<\infty$, then $\lim_{x\to-\infty}u(x)\in\\{-1,1\\}$.
It is easy to see that this is true provided that the limit
$\lim_{x\to-\infty}u(x)$ exists. To show that the limit exists, let
$l_{1}:=\liminf_{x\to-\infty}u(x)$ and $l_{2}:=\limsup_{x\to-\infty}u(x)$, and
suppose for contradiction that $l_{1}<l_{2}$. Then we may find $k_{1},k_{2}$
with $l_{1}<k_{1}<k_{2}<l_{2}$ such that $k_{1},k_{2}\not\in\\{-1,1\\}$. The
continuity of $u$ ensures that $u^{-1}(k_{1})$ and $u^{-1}(k_{2})$ are
unbounded.
Now consider the quantity
$\gamma:=\inf\left\\{\int_{0}^{T}(v^{2}-1)^{2}+|v^{\prime\prime}|^{2}\,dx:v\in
W^{2,\infty}(0,T),T>1,v(0_{+})=k_{1},v(T_{-})=k_{2}\right\\}.$
We claim that $\gamma=0$. Indeed, we may find sequences $x_{n},y_{n}$ such
that $u(-x_{n})=k_{1}$, $u(-y_{n})=k_{2}$, and $x_{n}+1<y_{n}<x_{n+1}$ for all
$n$. But now
$\infty>\Psi(u)\geq\sum_{n=1}^{\infty}\int_{-y_{n}}^{-x_{n}}(u^{2}-1)^{2}+|u^{\prime\prime}|^{2}\,dx\geq\sum_{n=1}^{\infty}\gamma,$
which can only be possible if $\gamma=0$.
Now find sequences $T_{n}>1$ and $v_{n}\in W^{2,\infty}(0,T_{n})$ for which
$v_{n}(0_{+})=k_{1}$, $v_{n}({T_{n}}_{-})=k_{2}$, and
$\int_{0}^{T_{n}}(v_{n}^{2}-1)^{2}+|v_{n}^{\prime\prime}|^{2}\,dx\to 0$.
Since $T_{n}>1$, we may obtain the uniform bound
$\int_{0}^{1}(v_{n}^{2}-1)^{2}+|v_{n}^{\prime\prime}|^{2}\,dx\leq M$ for some
$M>0$. Also,
$0=\lim_{n\to\infty}\int_{0}^{1}(v_{n}^{2}-1)^{2}+|v_{n}^{\prime\prime}|^{2}\,dx\geq\lim_{n\to\infty}\int_{0}^{1}|v_{n}^{\prime\prime}|^{2}\,dx,$
so we have that $v_{n}^{\prime\prime}\to 0$ in $L^{2}(0,1)$. Moreover, by
Lemma 4.2, there exists a subsequence $v_{n_{k}}$ and $v\in W^{2,2}(0,1)$ such
that $v_{n_{k}}^{\prime\prime}\rightharpoonup v^{\prime\prime}$ in
$L^{2}(0,1)$, $v_{n_{k}}^{\prime\prime}\to 0$ in $L^{2}(0,1)$, and
$v_{n_{k}}\to v$ pointwise almost everywhere. Now, notice that both
$v_{n_{k}}^{\prime\prime}\rightharpoonup 0$ and
$v_{n_{k}}^{\prime\prime}\rightharpoonup v^{\prime\prime}$ in $L^{2}(0,1)$, so
$v^{\prime\prime}=0$ almost everywhere. It follows that $v$ is affine, so that
$v(x)=mx+b$ for some $m,b\in\mathbb{R}$.
Now, Fatou’s Lemma gives that
$0=\lim_{n\to\infty}\int_{0}^{1}(v_{n}^{2}-1)^{2}+|v_{n}^{\prime\prime}|^{2}\,dx\geq\lim_{n\to\infty}\int_{0}^{1}(v_{n}^{2}-1)^{2}\,dx\geq\int_{0}^{1}(v^{2}-1)^{2}\,dx,$
which implies that $m=0$ and $b\in\\{-1,1\\}$.
However, we claim that $v$ must satisfy $v(0_{+})=k_{1}$, which would imply
that $k_{1}\in\\{-1,1\\}$, resulting in a contradiction. To see this, use the
fact that $v_{n}(0_{+})=k_{1}$ for all $n$, the Fundamental Theorem of
Calculus, and Hölder’s inequality to write
$|v_{n}(x)-k_{1}|\leq\int_{0}^{x}|v_{n}^{\prime}(t)|\,dt\leq\left(\int_{0}^{x}|v_{n}^{\prime}(t)|^{2}\,dt\right)^{1/2}\sqrt{x}\leq\left(\int_{0}^{1}|v_{n}^{\prime}(t)|^{2}\,dt\right)^{1/2}\sqrt{x}.$
Then since $v_{n}^{\prime}$ converges in $L^{p}(0,1)$ we must have that
$\int_{0}^{1}|v_{n}^{\prime}(t)|^{2}\,dt$ is bounded, so $|v_{n}(x)-k_{1}|\leq
M_{1}\sqrt{x}$ for a constant $M_{1}>0$. Sending $n\to+\infty$ we then have
$|v(x)-k_{1}|\leq M_{1}\sqrt{x}$ for almost every $x\in(0,1)$, so we may send
$x\to 0^{+}$ along an appropriate sequence to deduce that
$|v(0_{+})-k_{1}|=0$, as needed.
Step 5
We are now ready to show that $\liminf_{t\to t_{0}}\beta(t)\geq\beta(t_{0})$.
Letting $t_{n}\to t_{0}$ be arbitrary, we just need to show that
$\liminf_{n\to\infty}\beta(t_{n})\geq\beta(t_{0})$.
By the monotone properties that we have proven about $\beta$ in Step 1, we see
that $\\{\beta(t_{n}):n\in\mathbb{N}\\}$ is bounded by a constant $M$.
Specifically, we may take
$M=\max(\beta(\sup_{n\in\mathbb{N}}t_{n}),\beta(\inf_{n\in\mathbb{N}}t_{n})).$
Next, by extraction of a subsequence, let us assume that there exists the
limit $L:=\lim_{n\to\infty}\beta(t_{n})$.
Now fix $\eta>0$. Select
$\tilde{u}_{n}\in\mathscr{J}^{\prime}_{\infty}(t_{n})$ such that
$\beta(t_{n})\leq\Psi(\tilde{u}_{n})\leq\beta(t_{n})+\frac{1}{n}$. Since
$\Psi(\tilde{u}_{n})\leq M+1$ for all $n$, we may use Step 6 to find
$u_{n}\in\mathscr{J}^{\prime}_{\infty}(t_{n})$ such that
$\mathcal{L}^{1}(\\{x<0:0<u_{n}(x)<2\\})\leq C_{M,\eta}$ where $C_{M,\eta}$
depends only on $M$ and $\eta$, and
$\Psi(u_{n})\leq\Psi(\tilde{u}_{n})+O(\eta)$, so that
$\beta(t_{n})\leq\Psi(u_{n})\leq\beta(t_{n})+\frac{1}{n}+O(\eta).$ (32)
Note that $\\{u_{n}\\}$ is uniformly bounded in $\Psi$. In particular,
$\sup_{n\in\mathbb{N}}\int_{-m}^{0}|u_{n}^{\prime\prime}|^{2}\,dx<\infty$ for
each $m\in\mathbb{N}$.
Consider $m=2$. By Lemma 4.2 we may extract a subsequence $\\{u_{2,n}\\}_{n}$
of $\\{u_{n}\\}_{n}$ for which $u_{2,n}\rightharpoonup v_{2}$ in
$W^{2,2}(-2,0)$ for some $v_{2}\in W^{2,2}(-2,0)$. and $u_{2,n}\to v_{2}$
almost everywhere.
Inductively, for $m\geq 3$ we take $\\{u_{m,n}\\}_{n}$ to be a subsequence of
$\\{u_{m-1,n}\\}_{n}$ such that $u_{m,n}\rightharpoonup v_{m}$ in
$W^{2,2}(-m,0)$ for some $v_{m}\in W^{2,2}(-m,0)$ and such that $u_{m,n}\to
v_{m}$ almost everywhere.
We claim that $v_{m}$ extends $v_{m-1}$ for all $m\geq 3$. Indeed,
$u_{m-1,n}\to v_{m-1}$ almost everywhere, and since $\\{u_{m,n}\\}_{n}$ is a
subsequence of $\\{u_{m-1,n}\\}_{n}$, we have that $u_{m,n}\to v_{m-1}$ almost
everywhere over $(-m+1,0)$. Since $u_{m,n}\to v_{m}$ almost everywhere, it
follows that $v_{m-1}$ and $v_{m}$ agree over $(-m+1,0)$ as needed.
It follows that there exists a unique $u:(-\infty,0)\to\mathbb{R}$ extending
each $v_{m}$. Now consider $\\{u_{n,n}\\}$. For each $m$, $\\{u_{n,n}\\}_{n}$
may be viewed as a subsequence of $\\{u_{m,n}\\}$ for $n$ large enough, so
$u_{n,n}\to v_{n}=u$ almost everywhere in $(-m,0)$. Thus $u_{n,n}\to v$ over
$(-\infty,0)$ almost everywhere. Similarly, we see that
$u_{n,n}\rightharpoonup v$ in $W^{2,2}(-m,0)$ for all $m>0$. In particular,
$u_{n,n}\rightharpoonup v$ in $W^{2,2}_{\text{loc}}(-m,0)$.
Since $\\{u_{n,n}\\}_{n}$ is a subsequence of $\\{u_{n}\\}_{n}$, we let
$n_{k}$ be such that $\\{u_{n_{k}}\\}_{k}=\\{u_{n,n}\\}_{n}$.
To finish we use the fact that $u_{n_{k}}\to u$ almost everywhere in
$(-\infty,0)$ to obtain the inequality
$\int_{-\infty}^{0}(u^{2}-1)^{2}\,dx\leq\liminf_{k\to\infty}\int_{0}^{T}(u_{n_{k}}^{2}-1)^{2}\,dx$
by Fatou’s Lemma. Next, we use the property that
$u_{n_{k}}^{\prime\prime}\rightharpoonup u^{\prime\prime}$ in
$L^{2}_{\text{loc}}(0,T)$ and the fact that $z\mapsto z^{2}$ is convex,
together with Theorem 2.15, to conclude that
$\int_{-\infty}^{0}|u^{\prime\prime}|^{2}\,dx\leq\liminf_{k\to\infty}\int_{0}^{T}|u_{n_{k}}^{\prime\prime}|^{2}\,dx.$
Combining these two inequalities gives
$\Psi(u)\leq\liminf_{k\to\infty}\Psi(u_{n_{k}})$
Now, on one hand, we have from (32) that
$\Psi(u_{n_{k}})\leq\beta(t_{n_{k}})+\frac{1}{n_{k}}+O(\eta)$, and taking the
liminf we have
$L+O(\eta)=\liminf_{k\to\infty}\beta(t_{n_{k}})+O(\eta)\geq\liminf_{k\to\infty}\Psi(u_{n_{k}}).$
On the other hand, we claim that $\Psi(u)\geq\beta(t_{0})$. It suffices to
prove that $u\in\overline{\mathscr{J}^{\prime}_{\infty}(t_{0})}$.
Since $u_{n_{k}}\to u$ almost everywhere, and $u_{n_{k}}(0_{-})=t_{n_{k}}$
with $t_{n_{k}}\to t_{0}$, we must have $u(0_{-})=t_{0}$. Moreover, since
$\Psi(u)<\infty$, we have by Step 5 that the limit $\lim_{x\to-\infty}u(x)$
exists, and is either $1$ or $-1$. However,
$\mathcal{L}^{1}(\\{u>0\\})=\mathcal{L}^{1}\left(\liminf_{k\to\infty}\\{u_{n_{k}}>0\\}\right)\leq\liminf_{k\to\infty}\mathcal{L}^{1}(\\{u_{n_{k}}>0\\})\leq
C_{M,\eta}<\infty,$
which eliminates the possibility that the limit is 1. Thus
$\lim_{x\to-\infty}u(x)=-1$, so we may conclude that
$u\in\overline{\mathscr{J}^{\prime}_{\infty}(t_{0})}$.
With $\Psi(u)\geq\beta(t_{0})$, we have that
$L+O(\eta)\geq\liminf_{k\to\infty}\Psi(u_{n_{k}})\geq\Psi(u)\geq\beta(t_{0}).$
But $\eta>0$ was arbitrary, so $L\geq\beta(t_{0})$, which is what we wanted to
show. ∎
We now prove our main result, Theorem 1.1.
###### Proof.
The cases in which $u\not\in L^{p}(\Omega,\\{-1,1\\})$ or $u\in
L^{p}(\Omega,\\{-1,1\\})\setminus BPV(\Omega;\\{-1,1\\})$ are handled as in
Steps 1 and 4 of the proof of Theorem 3.5, so let us assume that $u\in
BPV(\Omega;\\{-1,1\\})$.
For the liminf inequality, let $u_{n}\to u$ in $L^{p}(\Omega)$. We may assume
that $u_{n}(a_{+})=a_{\varepsilon_{n}}$ and $u_{n}(b_{-})=b_{\varepsilon_{n}}$
for all $n$.
Fix $\eta>0$, and suppose that $u(b_{-})=-1$. Then we may find an interval
$(b-\delta,b)$ over which $u=-1$, and now for all large enough $n$ we follow
Step 2 of the proof of Theorem 3.5 in which we “modify" $u_{n}$ in
$(b,b-\delta)$ so that it becomes “affixed" to $-1$ at $b-\delta$. That is, we
find $v_{n}:(b-\delta,b)\to\mathbb{R}$ such that $v_{n}\in
W^{2,p}(b-\delta,b)$, $v_{n}(b-\delta_{-})=-1$,
$v_{n}^{\prime}(b-\delta_{-})=0$,
$v_{n}(b_{-})=u_{n}(b_{-})=b_{\varepsilon_{n}}$, and
$\int_{b-\delta}^{b}\varepsilon_{n}^{-1}(v_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n}^{\prime\prime}|^{2}\,dx\leq\int_{b-\delta}^{b}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx+\eta.$
Now, as in Step 3 of the proof of Theorem 3.5, we may change variables and
apply the AM-GM inequality to eventually obtain the bound
$\int_{b-\delta}^{b}\varepsilon_{n}^{-1}(v_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|v_{n}^{\prime\prime}|^{2}\,dx\geq\frac{4}{3^{3/4}}\inf_{u\in\mathscr{J}^{\prime}(b_{\varepsilon_{n}})}\Phi(u)=2\beta(b_{\varepsilon_{n}}).$
If instead $u(b_{-})=1$, then by a symmetrical argument, we instead obtain the
term $2\beta(-b_{\varepsilon_{n}})$. Both of these terms may be written as
$2\beta(-\operatorname{sgn}(u(b_{-}))b_{\varepsilon_{n}})$.
Using the same argument, we may obtain the term
$2\beta(-\operatorname{sgn}(u(a_{+}))a_{\varepsilon_{n}})$, and we recover the
term $\alpha\operatorname*{essVar}_{\Omega}u$ as in Steps 2 and 3 of the proof
of Theorem 3.5.
Overall, we see that the inequality
$\liminf_{n\to\infty}G_{\varepsilon_{n}}(u_{n})\geq\alpha\operatorname*{essVar}_{\Omega}u+\liminf_{n\to\infty}2\beta(-\operatorname{sgn}(u(a_{+}))a_{\varepsilon_{n}})+2\beta(-\operatorname{sgn}(u(b_{-}))b_{\varepsilon_{n}})$
may be obtained, and so we get
$\liminf_{n\to\infty}G_{\varepsilon_{n}}(u_{n})\geq\alpha\operatorname*{essVar}_{\Omega}u+2\beta(-\operatorname{sgn}(u(a_{+}))a_{0})+2\beta(-\operatorname{sgn}(u(b_{-}))b_{0})$
by continuity of $\beta$.
For the limsup inequality, we use a construction similar to that done in Step
5 of the proof of Theorem 3.5. We begin by strengthening the continuity result
on $\beta(t)$. Define the constant
$\beta_{\varepsilon}(t):=\inf\left\\{\Psi(v):v\in\mathscr{J}^{\prime}_{\infty}(t),L_{v}\leq\frac{1}{\sqrt{\varepsilon}}\right\\}$
and let $t_{0}\in\mathbb{R}$ with $t_{n}\to t_{0}$. We claim that
$\lim_{n\to\infty}\beta_{\varepsilon_{n}}(t_{n})=\beta(t_{0})$. To see this,
fix $\eta>0$ and for each $n\in\mathbb{N}$ take
$v_{n}\in\mathscr{J}^{\prime}_{\infty}(t_{n})$ for which
$\beta(t_{n})\leq\Psi(v_{n})\leq\beta(t_{n})+\frac{1}{n}$. The continuity of
$\beta$ ensures that $\\{\Psi(v_{n})\\}_{n}$ is bounded by a constant $M>0$.
By Lemma 4.3 we may construct $\tilde{v}_{n}$ for which
$\Psi(\tilde{v}_{n})\leq\Psi(v_{n})+O(\eta)\leq\beta(t_{n})+\frac{1}{n}+O(\eta)$
and $L_{\tilde{v}_{n}}\leq L_{M,\eta}$ where $L_{M,\eta}$ depends only on $M$
and $\eta$. Particularly $L_{M,\eta}$ has no dependence on $n$, thus
$L_{\tilde{n}_{n}}\leq\frac{1}{\sqrt{\varepsilon_{n}}}$ for all $n$ large
enough. For all such $n$ we may write
$\beta_{\varepsilon_{n}}(t_{n})\leq\Psi(\tilde{v}_{n})\leq\beta(t_{n})+\frac{1}{n}+O(\eta),$
and so by taking the limsup we may obtain
$\limsup_{n\to\infty}\beta_{\varepsilon_{n}}(t_{n})\leq\beta(t_{0})+O(\eta)$
by continuity of $\beta$. As $\eta>0$ was arbitrary, we get
$\limsup_{n\to\infty}\beta_{\varepsilon_{n}}(t_{n})\leq\beta(t_{0})$. But we
clearly also have $\beta_{\varepsilon_{n}}(t_{n})\geq\beta(t_{n})$, and taking
the liminf finishes the proof of the claim.
We now turn to the proof of the limsup inequality. As we did for the liminf
inequality, assume the case $u(b_{-})=-1$ and find $\delta>0$ small enough so
that $u=-1$ over the interval $(b-\delta,b)$. We will define $u_{n}$ over
$(b-\delta,b)$. Take
$v_{n}\in\mathscr{J}^{\prime}_{\infty}(b_{\varepsilon_{n}})$ satisfying
$L_{v_{n}}\leq\frac{1}{\sqrt{\varepsilon_{n}}}$ such that
$\beta_{\varepsilon_{n}}(b_{\varepsilon_{n}})\leq\Psi(v_{n})\leq\beta_{\varepsilon_{n}}(b_{\varepsilon_{n}})+\frac{1}{n}.$
(33)
Since $\varepsilon_{n}L_{v_{n}}\leq\sqrt{\varepsilon_{n}}\to 0$, we have
$\varepsilon_{n}L_{v_{n}}<\delta$ for all $n$ large enough. For all such $n$,
we define
$u_{n}(x):=\begin{cases}-1,&b-\delta<x\leq b-\varepsilon_{n}L_{v_{n}}\\\
v_{n}\left(\frac{x-b}{\varepsilon_{n}}\right),&b-\varepsilon_{n}L_{v_{n}}<x<b\end{cases}.$
This satisfies $u_{n}(b-\delta)=-1$, $u_{n}^{\prime}(b-\delta)=0$, and the
boundary condition $u_{n}(b_{-})=b_{\varepsilon_{n}}$. Moreover
$\displaystyle\int_{b-\delta}^{b}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}$
$\displaystyle+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx$
$\displaystyle=\int_{b-\varepsilon_{n}L_{v_{n}}}^{b}\varepsilon_{n}^{-1}\left(v_{n}\left(\frac{x-b}{\varepsilon_{n}}\right)^{2}-1\right)^{2}+\frac{\varepsilon_{n}^{3}}{\varepsilon_{n}^{4}}v_{n}^{\prime\prime}\left(\frac{x-b}{\varepsilon_{n}}\right)^{2}\,dx$
$\displaystyle=\int_{-L_{v_{n}}}^{0}(v_{n}^{2}-1)^{2}+|v_{n}^{\prime\prime}|^{2}\,dx$
$\displaystyle=\Psi(v_{n})\leq\beta_{\varepsilon_{n}}(b_{\varepsilon_{n}})+\frac{1}{n},$
thus
$\limsup_{n\to\infty}\int_{b-\delta}^{b}\varepsilon_{n}^{-1}(u_{n}^{2}-1)^{2}+\varepsilon_{n}^{3}|u_{n}^{\prime\prime}|^{2}\,dx\leq\limsup_{n\to\infty}\beta_{\varepsilon_{n}}(b_{\varepsilon_{n}})+\frac{1}{n}=\beta(b_{0})$
by the claim. It remains to show that $u_{n}\to-1$ in $L^{p}(b-\delta,b)$.
Since $\varepsilon_{n}L_{v_{n}}\to 0$, we have that $u_{n}\to-1$ almost
everywhere in $(b-\delta,b)$. By Vitali, it suffices to prove that
$\\{|u_{n}|^{p}\cdot 1_{(b-\delta,b)}\\}_{n}$ is uniformly integrable. Indeed,
we have
$\int_{b-\delta}^{b}(u_{n}^{2}-1)^{2}\,dx=\varepsilon_{n}\int_{-L_{v_{n}}}^{0}(v_{n}^{2}-1)^{2}\,dx=\varepsilon_{n}\Psi(v_{n}),$
and since
$\limsup_{n\to\infty}\Psi(v_{n})\leq\lim_{n\to\infty}\beta_{\varepsilon_{n}}(b_{\varepsilon_{n}})+\frac{1}{n}<\infty$,
we have that $\lim_{n\to\infty}\int_{b-\delta}^{b}(u_{n}^{2}-1)^{2}\,dx=0$, so
uniform integrability follows from Lemma 3.3.
Our construction for $u_{n}$ over $(b-\delta,b)$ provides us with the term
$\beta(b_{0})$ for the Gamma limit in the case that $u(b_{-})=-1$. If instead
$u(b_{-})=1$, we may negate our construction to instead acquire the term
$\beta(-b_{0})$. The term in both cases is equal to
$\beta(b_{0}\operatorname{sgn}u(b_{-}))$. A symmetrical construction near the
endpoint $a$ yields the term $\beta(a_{0})\operatorname{sgn}u(a_{+}))$. Lastly
we may define $u_{n}$ away from the endpoints as in Step 5 of the proof of
Theorem 3.5 in order to recover the term
$\alpha\operatorname*{essVar}_{\Omega}u$, completing the proof. ∎
## 5\. Acknowledgments
This research was partially supported by the National Science Foundation under
grants No. DMS 2108784 and DMS 1714098. I am also indebted to my advisor
Giovanni Leoni, whose guidance and feedback made this paper possible.
## References
* [1] A. Braides. Gamma-convergence for Beginners. Oxford University Press, Great Clarendon Street, Oxford 0X2 6DP, 2002\.
* [2] M. Chermisi, G. D. Maso, I. Fonseca, and G. Leoni. Singular perturbation models in phase transitions for second-order materials. Indiana University Mathematics Journal, 60(2):367–409, 2011.
* [3] M. Cicalese, E. N. Spadaro, and C. I. Zeppieri. Asymptotic analysis of a second-order singular perturbation model for phase transitions. Calculus of Variations and Partial Differential Equations, 41(5):127–150, 2011.
* [4] R. Cristoferi and G. Gravina. Sharp interface limit of a multi-phase transitions model under nonisothermal conditions. To appear in Calculus of Variations and Partial Differential Equations.
* [5] I. Fonseca and G. Leoni. Modern Methods in the Calculus of Variations: Lp Spaces. Springer Science+Business Media, 233 Spring Street, New York, NY, 10013, USA, 2007.
* [6] I. Fonseca and C. Mantegazza. Second order singular perturbation models for phase transitions. SIAM Journal on Mathematical Analysis, 31(5):1121–1143, May 2000\.
* [7] J. E. H. John W. Cahn. Free energy of a nonuniform system. i. interfacial free energy. The Journal of Chemical Physics, 28(2):258–267, 1958.
* [8] T. Lam. Masters thesis. Carnegie Mellon University. In Preparation.
* [9] G. Leoni. Additional material for the book. https://www.ams.org/publications/authors/books/postpub/gsm-181, 2017\. Accessed: 2022-08-18.
* [10] G. Leoni. A First Course in Sobolev Spaces, Second Edition. American Mathematical Society, Providence, Rhode Island, 2017.
* [11] L. Modica. The gradient theory of phase transitions and the minimal interface criterion. Archive for Rational Mechanics and Analysis, 98:123–142, 1987.
* [12] N. C. Owen, J. Rubinstein, and P. Sternberg. Minimizers and gradient flows for singularly perturbed bi-stable potentials with a dirichlet condition. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 429(1877):505–542, 1988.
* [13] P. Sternberg. The effect of a singular perturbation on nonconvex variational problems. Archive for Rational Mechanics and Analysis, 101:209–260, 1988\.
|
# Decepticons: Corrupted Transformers Breach
Privacy in Federated Learning for Language Models
Liam Fowl$\ \ ,\ \ $ Jonas Geiping∗
University of Maryland
{lfowl<EMAIL_ADDRESS>
&Steven Reich
University of Maryland
&Yuxin Wen
University of Maryland Wojtek Czaja
University of Maryland &Micah Goldblum
New York University &Tom Goldstein
University of Maryland
<EMAIL_ADDRESS>Authors contributed equally. Order chosen randomly.
###### Abstract
Privacy is a central tenet of Federated learning (FL), in which a central
server trains models without centralizing user data. However, gradient updates
used in FL can leak user information. While the most industrial uses of FL are
for text applications (e.g. keystroke prediction), the majority of attacks on
user privacy in FL have focused on simple image classifiers and threat models
that assume honest execution of the FL protocol from the server. We propose a
novel attack that reveals private user text by deploying malicious parameter
vectors, and which succeeds even with mini-batches, multiple users, and long
sequences. Unlike previous attacks on FL, the attack exploits characteristics
of both the Transformer architecture and the token embedding, separately
extracting tokens and positional embeddings to retrieve high-fidelity text. We
argue that the threat model of malicious server states is highly relevant from
a user-centric perspective, and show that in this scenario, text applications
using transformer models are much more vulnerable than previously thought.
## 1 Introduction
Federated learning (FL) has recently emerged as a central paradigm for
decentralized training. Where previously, training data had to be collected
and accumulated on a central server, the data can now be kept locally and only
model updates, such as parameter gradients, are shared and aggregated by a
central party. The central tenet of federated learning is that these protocols
enable privacy for users (McMahan & Ramage, 2017; Google Research, 2019). This
is appealing to industrial interests, as user data can be leveraged to train
machine learning models without user concerns for privacy, app permissions or
privacy regulations, such as GDPR (Veale et al., 2018; Truong et al., 2021).
However, in reality, these federated learning protocols walk a tightrope
between actual privacy and the appearance of privacy. Attacks that invert
model updates sent by users can recover private information in several
scenarios Phong et al. (2017); Wang et al. (2018) if no measures are taken to
safe-guard user privacy. Optimization-based inversion attacks have
demonstrated the vulnerability of image data when only a few datapoints are
used to calculate updates (Zhu et al., 2019; Geiping et al., 2020; Yin et al.,
2021). To stymie these attacks, user data can be aggregated securely before
being sent to the server as in Bonawitz et al. (2017), but this incurs
additional communication overhead, and as such requires an estimation of the
threat posed by inversion attacks against specific levels of aggregation,
model architecture, and setting.
Most of the work on gradient inversion attacks so far has focused on image
classification problems. Conversely, the most successful industrial
applications of federated learning have been in language tasks. There,
federated learning is not just a promising idea, it has been deployed to
consumers in production, for example to improve keystroke prediction (Hard et
al., 2019; Ramaswamy et al., 2019) and settings search on the Google Pixel
(Bonawitz et al., 2019). However, attacks in this area have so far succeeded
only on limited examples of sequences with few ($<25$) tokens (Deng et al.,
2021; Zhu et al., 2019; Dimitrov et al., 2022), even for massive models such
as BERT (with worse recovery for smaller models). This leaves the impression
that these models are already hard to invert, and limited aggregation is
already sufficient to protect user privacy, without the necessity to employ
stronger defenses such as local or distributed differential privacy (Dwork &
Roth, 2013).
In this work, we revisit the privacy of transformer models. We focus on the
realistic threat model where the server-side behavior is untrusted by the
user, and show that a malicious update sent by the server can completely
corrupt the behavior of user-side models, coercing them to spill significant
amounts of user data. The server can then collect the original words and
sentences entered by the user with straightforward statistical evaluations and
assignment problems. We show for the first time that recovery of all tokens
and most of their absolute positions is feasible even on the order of several
thousand tokens and even when applied to small models only 10% the size of
BERT discussed for FL use in Wang et al. (2021). Furthermore, instead of
previous work which only discuss attacks for updates aggregated over few
users, this attack breaks privacy even when updates are aggregated over more
than 100 users. We hope that these observations can contribute to re-
evaluation of privacy risks in FL applications for language.
Figure 1: An example reconstruction from a small GPT-2 model using a
Decepticon attack, showing the first $20$ tokens reconstructed from a randomly
selected user for different combinations of sequence length and batch size on
a challenging text fragment. Highlighted text represents _exact_ matches for
position and token.
## 2 Motivation and Threat Model
At first glance, gradients from Transformer architectures might not appear to
leak significant amounts of user data. Both the attention mechanisms and the
linear components learn operations that act individually on tokens, so that
their gradients are naturally averaged over the entire length of the sequence
(e.g. 512 tokens). Despite most architectures featuring large linear layers,
the mixing of information reduces the utility of their content to an attacker.
In fact, the only operation that “sees" the entire sequence, the scaled dot
product attention, is non-learned and does not leak separate gradients for
each entry in the sequence. If one were to draw intuition from vision-based
attacks, gradients whose components are averaged over 512 images are
impossible to invert even for state-of-the-art attacks (Yin et al., 2021).
On the other hand, recovering text appears much more constrained than
recovering images. The attacker knows from the beginning that only tokens that
exist in the vocabulary are possible solutions and it is only necessary to
find their location from a limited list of known positions and identify such
tokens to reconstruct the input sequence perfectly.
##### Previous Attacks
Previous gradient inversion attacks in FL have been described for text in Deng
et al. (2021); Zhu et al. (2019); Dimitrov et al. (2022) and have focused on
optimization-based reconstruction in the honest-but-curious server model.
Recently, Gupta et al. (2022) use a language model prior to improve attack
success via beam-search that is guided by the user gradient. Our work is
further related to investigations about the unintended memorization abilities
of fully trained, but benign, models as described in Carlini et al. (2021);
Inan et al. (2021); Brown et al. (2021), which can extract up to $1\%$ of
training data in some cases (Carlini et al., 2022). In contrast, we
investigate attacks which directly attempt to retrieve a large fraction of the
training data utilized in a single model update. This is the worst-case for a
privacy leak from successive benign model snapshots discussed in Zanella-
Béguelin et al. (2020).
##### Threat Model
We consider the threat model of an untrusted server that is interested in
recovering private user data from user updates in FL. Both user and server are
bound by secure implementations to follow a known federated learning protocol,
and the model architecture is compiled into a fixed state and verified by the
same implementation to be a standard Transformer architecture. The user
downloads a single malicious server state and returns their model updates
according to protocol. The server then recovers user data from this update.
This threat model is the most natural setting from a user-centric perspective.
Stronger threat models would, for example, allow the server to execute
arbitrary code on user devices, but such threats are solvable by software
solutions such as secure sandboxing (Frey, 2021). Still other work
investigates malicious model architectures (Fowl et al., 2021; Boenisch et
al., 2021), but such malicious modifications have to be present at the
conception of the machine learning application, before the architecture is
fixed and compiled for production (Bonawitz et al., 2019), making those
attacks most feasible for actors in the “analyst" role (Kairouz et al.,
2021b). Ideally, user and server communicate through a verified protocol
implementation that only allows pre-defined and vetted model architectures.
However, none of these precautions stop an adversarial server from sending
malicious updates (Wang et al., 2021). Such attacks are naturally ephemeral -
the server can send benign server states nearly all of the time to all users,
then switch to a malicious update for single round and group of users (or
single secure aggregator), collect user information, and return to benign
updates immediately after. The malicious update is quickly overwritten and not
detectable after the fact. Such an attack can be launched by any actor with
temporary access to the server states sent to users, including, aside from the
server owner, temporary breaches of server infrastructure, MITM attacks or
single disgruntled employees. Overall we argue that server states sent to
users, especially for very expressive models such as transformers, should be
treated by user applications as analogue to untrusted code that operates on
user data and should be handled with the appropriate care.
## 3 Malicious Parameters for Data Extraction
Our attack leverages several interacting strategies that we present in the
sub-sections below.
For simplicity, we assume in this section that the FL protocol is fedSGD, i.e.
model update returned by the users is the average gradient of the model
parameters sent by the server computed over all local user data. If the server
is malicious, then this protocol can either be selected directly or another
protocol such as federated averaging (fedAVG), which includes multiple local
update steps (McMahan et al., 2017), could be modified maliciously to run with
either only a single local step or a large enough batch size. Both variants
effectively convert the protocol to fedSGD. The local user data is tokenized
text fed into the transformer model, although the attack we describe does not
make use of text priors and can read out an arbitrary stream of tokens.
We separate the exposition into two parts. First, we discuss how the server
constructs a malicious parameter vector for the target architecture that is
sent to users. Second, we then discuss how the server can reconstruct private
information from this update.
Figure 2: Recovering each token and its positions from the inputs to the first
transformer layer. We find the correlation matrix between recovered input
embeddings to known positions and tokens, and solve a sum assignment problem
to determine which position is most likely for each input.
We first discuss parameter modifications for a single sequence of tokens, and
then move to modifications that allow recovery of multiple sequences. For the
following segment, we assume that the targeted model is based on a standard
transformer architecture, containing first an embedding component that, for
each token, adds token embedding and positional embedding, and second, a
series of transformer layers, each containing an attention block and a feed-
forward block.
##### Modifications to Recover Token Ids and Positions:
In principle, the goal of the attacker is simple: If the attacker can
reconstruct the inputs to the first transformer layer, which are token
embeddings plus positional embeddings, then a simple matching problem can
identify the token id and position id by correlating each input to the known
token and position embedding matrices, see Figure 2. However, what the user
returns is not a token but a gradient, representing changes to the model
relative to the user’s data. This gradient is also an average over all tokens
in the sequence. As a first step to improve attack success, the attacker can
disable all attention layers (setting their output transformation to 0) and
mostly disable outputs of each feed-forward block. The outputs of each feed-
forward layer cannot be entirely disabled (otherwise all gradients would be
zero), so we set all outputs except the last entry to zero. A transformer
modified in this way does not mix token information and the inputs to each
layer are the same (except for the last entry in the hidden dimension) and
equal to token embeddings plus positional embeddings.
##### Unmixing Gradient Signals:
Let’s investigate the resulting gradient of the first linear layer in each
feed-forward block. We define the input embeddings that the attacker intends
to recover as $\\{u_{k}\\}_{k=1}^{S}$ coming from a sequence of text
containing $S$ elements. The operation of the first linear layer in each
($i^{th}$) feed-forward block is $y_{k}=W_{i}u_{k}+b_{i}$. The gradient
averaged over all tokens in this sequence is:
$\displaystyle\sum_{k=1}^{S}\nabla_{W_{i}^{(j)}}\mathcal{L}_{k}$
$\displaystyle=\sum_{k=1}^{S}\bigg{(}\frac{\partial\mathcal{L}_{k}}{\partial
y_{k}^{(j)}}\cdot\nabla_{W_{i}^{(j)}}y_{k}^{(j)}\bigg{)},\qquad$ (1)
$\displaystyle\sum_{k=1}^{S}\frac{\partial\mathcal{L}_{k}}{\partial
b_{i}^{(j)}}$
$\displaystyle=\sum_{k=1}^{S}\frac{\partial\mathcal{L}_{k}}{\partial
y_{k}^{(j)}}\cdot\frac{\partial y_{k}^{(j)}}{\partial b_{i}},$ (2)
For each of the $j$ entries, rows of the bias and weight, respectively. This
can be simplified further to:
$\sum_{k=1}^{S}\frac{\partial\mathcal{L}_{k}}{\partial y_{k}^{(j)}}\cdot
u_{k}$ for weight and $\sum_{k=1}^{S}\frac{\partial\mathcal{L}_{k}}{\partial
y_{k}^{(j)}}$ for bias (Geiping et al., 2020). Even after this reduction, it
seems quite difficult to recover $u_{k}$ from these sums. Yet, a malicious
attacker can modify the weights of $W_{i}$ and of the subsequent matrix to
allow for a closed-form solution.
To this end, we leverage recent strategies for separating gradient signals
(Fowl et al., 2021). In essence, this strategy encodes a measurement function
into each weight row $W_{i}^{(j)}$ of a linear layer $i$ and an increasing
offset into each entry of the bias vectors $b_{i}$. For linear layers that are
followed by activations which cut off smaller input values (such as ReLU or
GELU), the gradient of the weights in this linear layer will then encode a
cumulative sum of the values of all inputs up to some offset value. This
structures the gradients observed in Equation 1 and original inputs $u_{k}$
can then be recovered by subtraction of adjacent rows. This is visualized in
Figure 3.
Figure 3: How embeddings are extracted from linear layers in the transformer
blocks. We initialize the rows of the first linear layer in each Transformer
block to a randomly sampled Gaussian vector (all rows being the same). The
bottom portion of the figure depicts the internals of a forward pass through
this FC layer. When an incoming embedding vector $u_{k}$ enters the FC layer,
the inner products $\langle m,u_{k}\rangle$ fall into a distribution which the
attacker partitions into bins. If $u_{k}$ is the only token whose inner
product lies in a given bin, the ReLU activation will encode information about
this single token, and (3) can be used to directly recover the token.
To go further into detail into this modification, the rows of all weight
matrices $W_{i}$ are replaced by a measurement vector $m$ and the biases over
all layers are modified to be sequentially ascending. As a result, all linear
layers now compute $y_{k}^{(j)}=\langle m,u_{k}\rangle+b^{(j)}$. And $y_{k}$
is ordered so that $y_{k}^{(j)}\geq y_{k}^{(j^{\prime})},\forall
j<j^{\prime}$. The subsequent ReLU unit will threshold $y_{k}^{(j)}$, so that
$y_{k}^{(j)}=0$ directly implies $\langle m,u_{k}\rangle<-b^{(j)}$. This
ordering can now be exploited by subtracting the $(j+1)^{th}$ entry in the
gradient from the $j^{th}$ entry to recover:
$\displaystyle\sum_{k=1}^{S}\nabla_{W_{i}^{(j)}}\mathcal{L}_{k}-\sum_{k=1}^{S}\nabla_{W_{i}^{(j+1)}}\mathcal{L}_{k}$
$\displaystyle=\lambda_{i}^{(j)}\cdot
u_{k},\qquad\sum_{k=1}^{S}\frac{\partial\mathcal{L}_{k}}{\partial
b^{(j)}}-\sum_{k=1}^{S}\frac{\partial\mathcal{L}_{k}}{\partial b^{(j+1)}}$
$\displaystyle=\lambda_{i}^{(j)}$ (3)
for scalars $\lambda_{i}^{(j)}$. Assuming the number of embeddings that
fulfill this condition is either one or zero, we recover the input vector
$u_{k}$ simply by dividing the term of the left of Equation 3 by the term on
the right. All embeddings that collide and fall into the same interval are not
easily recoverable.
We minimize the probability of a collision by first sampling a Gaussian vector
$m\sim\mathcal{N}(\vec{0},\mathbb{1}_{d})$ where $d=d_{model}.$ We then set
the biases to $b_{i}=[c_{i\cdot k},\dots,c_{(i+1)\cdot k-1}]$, where $k$ is
the width of each linear layer and $c_{l}=-\Phi^{-1}(\frac{l}{M}),$ where
$\Phi^{-1}$ is the inverse of a Gaussian CDF and $M$ is the sum over the
widths of all included Transformer blocks. This uniformly distributes the
values of $\langle m,u_{k}\rangle$ over the bins created by the bias vector.
If the width of the first linear layer in each block is larger than the number
of tokens, collisions will be unlikely. We can estimate the rough mean and
variance of {$\langle m,u_{k}\rangle$} from either a small volume of public
text or a random stream of tokens, see Appendix D.
##### Disambiguating Multiple Sequences
Recovering multiple sequences – either from user data comprising multiple
separate sentences of tokens, or aggregates of multiple users – presents the
most difficult task for the attacker. Naively using the strategy described so
far can only recover a partially ordered set of tokens. For example, if a
model update consists of five user sequences, then the attacker recovers five
tokens at position 0, five tokens at position 1, and so on, because positional
embeddings repeat in each sequence. Re-grouping these tokens into salient
sequences quickly becomes intractable as the number of possible groupings
grows as $n^{l}$ for $n$ sequences of length $l$. Further complicating matters
for the attacker is that no learned parameters (and thus no parameters
returning gradients) operate on the entire length of a sequence in the
Transformer model. In fact, the only interaction between embeddings from the
same sequence comes in the scaled dot product attention mechanism.
Figure 4: A high-level schematic of MHA manipulation. The MHA block attends to
the first word identically for every input sequence, encoding a part of the
first token for each embedding in the entire sequence.
Thus, if any malicious modification can be made to encode sequence information
into embeddings, it must utilize the attention mechanism of Transformers. We
describe a mechanism by which an attacker can disambiguate user sequences,
even when model updates are aggregated over a large number of separate
sequences. At a high level, we introduce a strategy that encodes unique
information about each sequence into a predefined location in the embedding of
each token in the sequence. Then, we can read out a unique quantity for each
sequence and thus assign tokens to separate sequences. This behavior is
illustrated in Figure 4.
Let $W_{Q},W_{K},W_{V}$ represent the query, key, and value weight matrices
respectively, and let $b_{Q},b_{K},b_{V}$ represent their biases. For
simplicity of presentation, we explain this attack on a single head, but it is
easily adapted to multi-head attention. We first set the $W_{K}$ matrix to the
identity ($\mathbb{1}_{d_{model}}$), and $b_{K}=\vec{0}$. This leaves incoming
embeddings unaltered. Then, we set $W_{Q}=\vec{0}$, and
$b_{Q}=\gamma\cdot\vec{p}_{0}$ where $\vec{p}_{0}$ is the first positional
encoding. Here we choose the first position vector for simplicity, but there
are many potential choices for the identifying vector. This query matrix then
transforms each embedding identically to be a scaled version of the first
positional encoding. We then set $W_{V}=\mathbb{1}_{d^{\prime}}$ to be a
partial identity (identity in the first $d^{\prime}$ entries where
$d^{\prime}\leq d$ is a hyperparameter that the server controls). Finally, we
set $b_{v}=\vec{0}$.
Now, we investigate how these changes transform an incoming sequence of
embeddings. Let $\\{u_{k}\\}_{i=0}^{l-1}$ be embeddings for a sequence of
length $l$ that enters the attention mechanism. $u_{0}$ is the embedding
corresponding to the first token in the sequence, so $u_{0}$ is made up of the
token embedding for the first token in the sequence, and the first positional
encoding. $W_{K}$ produces keys $K$ that exactly correspond to the incoming
embeddings, however, $W_{Q},b_{Q}$ collapses the embeddings to produce $Q$,
consisting of $l$ identical copies of a single vector, the first positional
encoding. Then, when the attention weights are calculated as:
$\operatorname*{softmax}\bigg{(}\frac{QK^{T}}{\sqrt{d_{k}}}\bigg{)},$
the attacker finds that the first embedding dominates the attention weights
for all the embeddings, as the query vectors all correlate with the first
embedding the most. In fact, the $\gamma$ parameter can effectively turn the
attention weights to a delta function on the first position. Finally, when the
attention weights are used to combine the values $V$ with the embeddings, by
construction, a part of the embedding for the first word in the sequence is
identically added to each other word in that sequence. So the embeddings are
transformed as
$\\{x_{i}\\}_{i=0}^{l-1}\rightarrow\\{x_{i}+x_{0,d^{\prime}}\\}_{i=0}^{l-1}$
where $x_{0,d^{\prime}}$ is a vector where the first $d^{\prime}$ entries are
the first $d^{\prime}$ entries of $x_{0}$, and the other
$d_{model}-d^{\prime}$ entries are identically $0$.
If the attacker chooses to perform this modification on the first Transformer
block, this means that embeddings $u_{k}$ that the attacker recovers from each
of the linear layers now also contain unique information about the first token
of the sequence from which they came. The attacker can then calculate
correlations between first-position embeddings and later-position embeddings
and group each embedding into a sequence with other embeddings.
### 3.1 Extracting User Data
Now that we have introduced the main mechanisms by which an attacker prepares
a malicious parameter vector, we summarize how to extract user data. The
attack begins after a user or aggregate group of users has computed their
gradient using the corrupted parameters. Then, the server retrieves their
update and begins the inversion procedure. We summarize the entire attack in
Algorithm 1.
##### Getting a Bag-of-Words Containing All Tokens
Figure 5: Left: A high-level schematic of token leaking. The token embedding
layer leaks tokens and frequencies solely through its sparse gradient entries.
Right: Distribution and Histogram of log of norms of all token embedding
gradients for GPT-2 for $13824$ tokens. In this case, gradient entries are
non-sparse due to the tied encoder-decoder structure of GPT, but the
embeddings of true user data (red) are clearly separable from the mass of all
embeddings (blue) by a cutoff at 1.5 standard deviations (marked in black).
Even without malicious parameters, an attacker can retrieve the bag-of-words
(unordered list of tokens) of the user from the gradient of the token
embedding. Melis et al. (2019) identified that the embedding layer gradient is
only non-zero at locations corresponding to tokens that were used for
training, as visualized in the left plot of Figure 5. Furthermore, the
frequency of all words can be extracted by analyzing the bias gradient of the
embedding or decoding (last linear) layer, as the magnitude of the update of
each row of a random embedding matrix is proportional to the frequency of word
usage, allowing for a greedy estimation by adapting the strategy of Wainakh et
al. (2021) which was proposed for simpler classification tasks. We use this
finding as a first step to restrict the number of possible tokens that should
be matched. More details can be found in Appendix C.
Algorithm 1 Decepticon Data Readout Overview
1: Input: Sequence Length $S$, number of expected sequences $N$. Gradients
$\nabla W_{i},\nabla b_{i}$ of the weight and bias of the first linear layer
in each FFN block. Positional embeddings $P$, token embeddings $T$.
2: $t_{v}$ $\leftarrow$ Token Embeddings of estimated bag-of-words of leaked
tokens or all tokens
3: $p_{k}$ $\leftarrow$ Known positional embeddings
4: $u_{h}$ $\leftarrow$
$\frac{\nabla_{W_{ij}}-\nabla_{W_{i,j+1}}}{\nabla_{b^{i}_{j}}-\nabla_{b^{i}_{j+1}}}$
for linear layers $i=1,..L$ and rows $j=1,...,r$.
5: $L_{\text{batch}}$ $\leftarrow$ Cluster index for each $u_{h}$ clustering
the first $d^{\prime}$ entries of each embedding.
6: for n in $0...N$ do
7: $u_{hn}$ $\leftarrow$ Entries of cluster $n$ in $L_{\text{batch}}$
8: $u^{n}_{k}$ $\leftarrow$ Match($P_{k}$, $u_{hn}$) on all entries
$d>d^{\prime}$
9: end for
10: $u_{nk}$ $\leftarrow$ concatenate $\\{u^{n}_{k}\\}_{n=1}^{N}$
11: $t^{\text{final tokens}}_{nk}$ $\leftarrow$ Indices of
Match($u_{nk}-p_{k}$, $t_{v}$) on all entries $d>d^{\prime}$
Recovering Input Embeddings: As a first step, we recover all input embeddings
$u_{h}$ using the divided difference method described in Equation 3 for all
rows where $\nabla_{b^{i}_{j}}-\nabla_{b^{i}_{j+1}}$ is non-zero, i.e. at
least one embedding exist with $-b_{j}<\langle m,u_{k}\rangle<-b_{j+1}$. This
results in a list of vectors $u_{h}$ with $h<SN$, for maximal sequence length
$S$ and maximal number of expected sequences $N$, that have to be matched to
separate sequences, positions and tokens.
Recovering Sequence Clusters: The first $d^{\prime}$ entries of each embedding
encode the sequence identity as described in Section 3. The server can hence
group the embeddings $u_{h}$ uniquely into their sequences by a clustering
algorithm. We utilize constrained K-Means as described in (Bradley et al.,
2000), as the maximal sequence length $S$ and hence cluster size is known.
This results in a label for each vector $u_{h}$, sorting them into separate
sequences.
Assigning Positions in each Sequence: Now, the matching problem has been
reduced to separately matching each sequence, based on all entries
$d>d^{\prime}$. For a given sequence with index $n$ first match the embedding
$u_{hn}$ to the available positional embeddings $P_{k}$. To do so, we
construct the matrix of correlations between all $u_{hn}$ and $p_{k}$ and
solve a rectangular linear sum assignment to find the optimal position $p_{k}$
for each $u_{hn}$. For the assignment problem, we use the solver proposed in
Crouse (2016) which is a modification of the shortest augmenting path strategy
originally proposed in Jonker & Volgenant (1987). Due to collisions, some
positions will possibly not be filled after this initial matching step. We
thus iteratively fill up un-matched positions with the best-matching
embeddings from $u_{hn}$, even if they have been used before in other
positions, until each position is assigned a match.
Converting Embeddings to Tokens: Finally, we match all $u_{nk}$ to all token
embeddings $t_{v}$ in either our bag of possible tokens or all tokens in the
vocabulary. For this, we first subtract or decorrelate the positional encoding
$p_{k}$ from $u_{nk}$ to remove interference, and then again solve a linear
sum assignment problem as described. The indices of the solution to this
assignment problem will finally be the token ids $t_{nk}$ for each position
$k$ in sequence $n$.
In general, this process works very well to recover batches containing
multiple long sequences. In the limit of many tokens, the quality of recovery
eventually degrades as collisions become more common, making the
identification of positions and token id less certain. In the regime of a
massive number of tokens, only a subset of positions are accurately recovered,
and the remaining tokens are inaccurate, appearing in wrong positions. This
process is random, and even for a massive number of tokens there are often
some sequences that can be recovered very well.
## 4 Empirical Evaluation of the Attack
We evaluate the proposed attack using a range of Transformers. Our main focus
is the application to next-word prediction as emphasized in practical use-
cases (Hard et al., 2019; Ramaswamy et al., 2019). We consider three
architectures of differing sizes: the small 3-layer Transformer discussed as a
template for federated learning scenarios in Wang et al. (2021) (11M
parameters), the BERT-Base model (Devlin et al., 2019) attacked in previous
work (Deng et al., 2021; Zhu et al., 2019) (110M parameters), and the smallest
GPT-2 variation (Radford et al., 2019) (124M parameters). We train the small
Transformer and GPT-2 as causal language models and BERT as a masked language
model. Predictably, the attack performance degrades most quickly for the
3-layer Transformer as this has the fewest parameters and thus the least
capacity to uniquely capture a large number of user embeddings.
Figure 6: Total Average Accuracy. Total accuracy (i.e. percentage of tokens
recovered correctly in their correct position) for all considered and varying
batch sizes for sequence length 32 from wikitext(left) and sequence length 512
from stackoverflow (right).
Figure 7: Most-vulnerable Accuracy. Accuracy on the most-leaked sentence for
sequence length 32 from wikitext(left) and sequence length 512 from
stackoverflow (right). Note the size difference between BERT/GPT and the small
transformer depicted here.
We test this attack on several datasets, wikitext (Merity et al., 2016)
shakespeare (Caldas et al., 2019) and stackoverflow (Wang et al., 2021). For
wikitext, We partition into separate “users” by article, while for the other
datasets user partitions are given. We tokenize using the GPT-2 (BPE)
tokenizer for the small Transformer and GPT-2, and the original BERT
(WordPiece) tokenizer for BERT. We always report average scores over the first
100 users, which are have enough data to fill batch size $\times$ sequence
length tokens, skipping users with less data who would be more vulnerable. We
focus on fedSGD, i.e. single gradient updates from all users, but note the
option of a malicious server to convert another protocol to fedSGD discussed
in Section 3.
We evaluate using _total_ average accuracy, BLEU (Papineni et al., 2002) and
ROUGE-L (Lin, 2004). Our total accuracy is stricter than the token (i.e., bag-
of-words) accuracy described previously Deng et al. (2021); we only count
success if both token id _and_ position are correct. Figure 1 shows partial
reconstructions for a randomly selected, challenging sequence as batch size
and sequence length increase. We find that even for more difficult scenarios
with more tokens, a vast majority of tokens are recovered, with a majority in
their _exact_ correct position.
Average Accuracy: Our multi-head attention strategy allows an attacker to
reconstruct larger batches of sequences, as seen in Figure 6 (We include
single sequences in Figure 16). This applies to an update from a single user
with more than one sequence, and also multiple users aggregating model
updates. To the best of our knowledge, this is the first approach to
explicitly attack multi-user updates with strong average accuracy. We find
that for a sequence length of $32$, the proposed attack can recover almost all
tokens used in the user updates, and $>50\%$ of tokens at their _exact_
position for $>100$ users. Even for a sequence length of $512$, the attack
remains effective for large batches.
Most-vulnerable Accuracy: We further chart the accuracy on the most-recovered
sequence in Figure 7, finding that even at larger aggregation rates, some
sequences remain vulnerable to attack and are almost perfectly recovered by an
attacker.
Figure 8: Comparing attacks on several data sources. Left: Total accuracy
(i.e. percentage of tokens recovered correctly in their correct position) for
GPT-2 and varying batch sizes with sequence length 32. Right: Accuracy on the
most-leaked sentence for all data sources.
Comparison to other threat models.
Figure 9: Comparison between the malicious server threat model with our attack
and TAG (honest-but-curious) for the 3-layer transformer described in Wang et
al. (2021), variable seq. length (batch size $1$) on wikitext.
Several works approached the topic of reconstructing text from FL updates,
under a “honest-but-curious" threat model where benign parameters are sent to
the user, and the FL protocol is performed normally. We compare by fixing an
architecture – in this case the Transformer-3 model described earlier. We then
consider the setting with batch size $1$ and evaluate our proposed method
against the TAG attack Deng et al. (2021) which improves upon previous results
in Zhu et al. (2019). We experiment for varying sequence lengths in Figure 9.
The performance of TAG very quickly degrades for sequences of even moderate
length. For a single sequence, our proposed attack maintains high total
accuracy for sequences exceeding $1000$ tokens, whereas the total accuracy for
TAG soon approaches $0\%$. Overall, we find that the severity of the threat
posed by our attack is orders of magnitude greater than in the “honest-but-
curious" threat model, and argue for the re-evaluation of the amount of
privacy leaked by FL applications using transformers.
##### Accuracy on other Datasets:
The attack is largely invariant to the source of user text, as detailed in
Figure 8, where diverse sources of data are almost equally vulnerable.
Shakespeare data is most vulnerable due to the use of short and simple
sentences which are consistently well-recovered. Random text is noticeably
more vulnerable, which is relevant from a security perspective, as an attacker
often be interested in text fragments that are not usual text, e.g. passwords
or social security numbers.
##### Attack Confidence
In contrast to existing attacks, the attacker can verify recovered tokens.
After recovering positions and tokens for each breach $u_{h}$, one can cross-
check whether the assigned position $p_{k}$ and token embedding $t_{v}$
perfectly explain $u_{h}$. Tokens that fulfill this condition are perfectly
recovered and allow the attacker to certify which parts of their recovered
text accurately reflect user data.
## 5 Conclusions
In this work we re-evaluate the threat posed by attacks against privacy for FL
for Transformer-based language models. We argue that from a user-perspective,
the most natural threat model is not to trust the server state. We show that a
malicious server can send updates that encode large amounts of private user
data into the update sent by users. This attack and threat model significantly
lower the bar for possible attacks, recovering up to several thousand tokens
of user data. Our results underline the necessity of adoption of strong
privacy guarantees for users in federated learning.
#### Ethics Statement and Mitigations
Several known case studies of FL models deployed in production in Hard et al.
(2019); Hartmann (2021) rely only on aggregation to preserve privacy in FL for
text. However, the attack we describe in this work shows that aggregation
levels considered safe based on previous work may not be enough to protect
user privacy in several use cases: The setting deployed to production in Hard
et al. (2019) runs FL on users updates with $400$ sentences with $4$ words per
average message sent to the server without further aggregation, well within
the range of the investigated attack if trained with a transformer model. We
thus briefly discuss other mitigation strategies for attacks like this. We
roughly group them into two categories: parameter inspection and differential
privacy based defenses.
Parameter inspection and verification of the server state is currently not
implemented in any major industrial FL framework (Paulik et al., 2021;
Dimitriadis et al., 2022; Li et al., 2019; Bonawitz et al., 2019), but after
the publication of this attack, a rule could be designed to mitigate it (which
we would encourage!). However, we caution that the design space for attacks as
described here seems too large to be defended by inspecting parameters for a
list of known attacks.
Based on these concerns, general differential privacy (DP) based defenses
continue to be the more promising route. Local or distributed differential
privacy, controlled directly by the user and added directly on-device (instead
of at the server level as in McMahan et al. (2018)) allows for general
protection for users without the need to trust the update sent by the server
(Kairouz et al., 2021a; Bhowmick et al., 2019). Local DP does come at a cost
in utility, observed in theoretical (Kairouz et al., 2021b; Kasiviswanathan et
al., 2011; Ullman, 2021) as well as practical investigations (Jayaraman &
Evans, 2019) when a model is pretrained from scratch (Li et al., 2022; Noble
et al., 2022). DP currently also comes at some cost in fairness, as model
utility is reduced most for underrepresented groups (Bagdasaryan et al.,
2019). Yet, as any attack on privacy will break down when faced with
sufficient differential privacy, with this work we advocate for strong
differential privacy on the user side, as incorporated for example into recent
system designs in Paulik et al. (2021).
We include further discussion about mitigations in practice in Appendix G,
including limited experimentation with gradient clipping/noising, as well as
parameter inspection adaptations.
Attacks like the one presented in this paper present a privacy risk to users
and their data. As such, the publication of the possibility of attacks like
the one described in this work has potential negative consequences for the
privacy of deployed user systems that do not apply mitigation strategies as
described above. As FL becomes more integrated into privacy-critical
applications, it is of paramount importance that the community is aware of
worst-case threats to user privacy, so that mitigating measures can be taken.
If left undefended, an attack like this could compromise large amounts of user
data, including sensitive text messages and emails. Because previous research
on privacy attacks against language models in FL has been relatively scarce,
some might believe that simple defenses such as aggregation are sufficient to
ensure user privacy. This work shows that this belief is unfounded and further
measures must be taken. We hope that this attack raises awareness of the
vulnerability of FL models.
#### Reproducibility Statement
We provide technical details in Appendix A, and additional pseudocode for
mentioned algorithms in Appendix B and provide code in conjunction with this
submission to reproduce all attacks and settings discussed in this work, which
can be found at github.com/JonasGeiping/breaching. We provide command-line
interfaces and jupyter notebooks to evaluate the attacks and interactively
vary parameters. All attacks discussed in this work can be cheaply reproduced
using Laptop CPUs, as no extensive computations on GPUs are necessary to run
the attack.
#### Acknowledgements
This work was supported by the Office of Naval Research (#N000142112557), the
AFOSR MURI program, DARPA GARD (HR00112020007), the National Science
Foundation (IIS-2212182), and Capital One Bank.
## References
* Bagdasaryan et al. (2019) Eugene Bagdasaryan, Omid Poursaeed, and Vitaly Shmatikov. Differential Privacy Has Disparate Impact on Model Accuracy. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d\textquotesingle Alché-Buc, E. Fox, and R. Garnett (eds.), _Advances in Neural Information Processing Systems 32_ , pp. 15479–15488. Curran Associates, Inc., 2019. URL http://papers.nips.cc/paper/9681-differential-privacy-has-disparate-impact-on-model-accuracy.pdf.
* Bhowmick et al. (2019) Abhishek Bhowmick, John Duchi, Julien Freudiger, Gaurav Kapoor, and Ryan Rogers. Protection Against Reconstruction and Its Applications in Private Federated Learning. _arXiv:1812.00984 [cs, stat]_ , June 2019. URL http://arxiv.org/abs/1812.00984.
* Boenisch et al. (2021) Franziska Boenisch, Adam Dziedzic, Roei Schuster, Ali Shahin Shamsabadi, Ilia Shumailov, and Nicolas Papernot. When the Curious Abandon Honesty: Federated Learning Is Not Private. _arXiv:2112.02918 [cs]_ , December 2021. URL http://arxiv.org/abs/2112.02918.
* Bonawitz et al. (2017) Kallista Bonawitz, Vladimir Ivanov, Ben Kreuter, Antonio Marcedone, H. Brendan McMahan, Sarvar Patel, Daniel Ramage, Aaron Segal, and Karn Seth. Practical Secure Aggregation for Privacy Preserving Machine Learning. Technical Report 281, 2017. URL http://eprint.iacr.org/2017/281.
* Bonawitz et al. (2019) Kallista Bonawitz, Hubert Eichner, Wolfgang Grieskamp, Dzmitry Huba, Alex Ingerman, Vladimir Ivanov, Chloe Kiddon, Jakub Konečný, Stefano Mazzocchi, H. Brendan McMahan, Timon Van Overveldt, David Petrou, Daniel Ramage, and Jason Roselander. Towards Federated Learning at Scale: System Design. _arXiv:1902.01046 [cs, stat]_ , March 2019. URL http://arxiv.org/abs/1902.01046.
* Bradley et al. (2000) Paul S. Bradley, Kristin P. Bennett, and Ayhan Demiriz. Constrained k-means clustering. _Microsoft Research, Redmond_ , 20(0):0, 2000\.
* Brown et al. (2021) Gavin Brown, Mark Bun, Vitaly Feldman, Adam Smith, and Kunal Talwar. When is Memorization of Irrelevant Training Data Necessary for High-Accuracy Learning? In _Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing_ , pp. 123–132, June 2021. doi: 10.1145/3406325.3451131. URL http://arxiv.org/abs/2012.06421.
* Caldas et al. (2019) Sebastian Caldas, Sai Meher Karthik Duddu, Peter Wu, Tian Li, Jakub Konečný, H. Brendan McMahan, Virginia Smith, and Ameet Talwalkar. LEAF: A Benchmark for Federated Settings. _arxiv:1812.01097[cs, stat]_ , December 2019. doi: 10.48550/arXiv.1812.01097. URL http://arxiv.org/abs/1812.01097.
* Carlini et al. (2021) Nicholas Carlini, Florian Tramèr, Eric Wallace, Matthew Jagielski, Ariel Herbert-Voss, Katherine Lee, Adam Roberts, Tom Brown, Dawn Song, Úlfar Erlingsson, Alina Oprea, and Colin Raffel. Extracting Training Data from Large Language Models. In _30th USENIX Security Symposium (USENIX Security 21)_ , pp. 2633–2650, 2021. ISBN 978-1-939133-24-3. URL https://www.usenix.org/conference/usenixsecurity21/presentation/carlini-extracting.
* Carlini et al. (2022) Nicholas Carlini, Daphne Ippolito, Matthew Jagielski, Katherine Lee, Florian Tramer, and Chiyuan Zhang. Quantifying Memorization Across Neural Language Models. _arxiv:2202.07646[cs]_ , February 2022. URL http://arxiv.org/abs/2202.07646.
* Crouse (2016) David F. Crouse. On implementing 2D rectangular assignment algorithms. _IEEE Transactions on Aerospace and Electronic Systems_ , 52(4):1679–1696, August 2016. ISSN 1557-9603. doi: 10.1109/TAES.2016.140952.
* Deng et al. (2021) Jieren Deng, Yijue Wang, Ji Li, Chao Shang, Hang Liu, Sanguthevar Rajasekaran, and Caiwen Ding. TAG: Gradient Attack on Transformer-based Language Models. _arXiv:2103.06819 [cs]_ , September 2021. URL http://arxiv.org/abs/2103.06819.
* Deng et al. (2020) Yuntian Deng, Anton Bakhtin, Myle Ott, Arthur Szlam, and Marc’Aurelio Ranzato. Residual Energy-Based Models for Text Generation. _arXiv:2004.11714 [cs]_ , April 2020. URL http://arxiv.org/abs/2004.11714.
* Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. _arXiv:1810.04805 [cs]_ , May 2019. URL http://arxiv.org/abs/1810.04805.
* Dimitriadis et al. (2022) Dimitrios Dimitriadis, Mirian Hipolito Garcia, Daniel Madrigal Diaz, Andre Manoel, and Robert Sim. FLUTE: A Scalable, Extensible Framework for High-Performance Federated Learning Simulations. _arxiv:2203.13789[cs]_ , March 2022. URL http://arxiv.org/abs/2203.13789.
* Dimitrov et al. (2022) Dimitar I. Dimitrov, Mislav Balunović, Nikola Jovanović, and Martin Vechev. LAMP: Extracting Text from Gradients with Language Model Priors. February 2022. URL https://arxiv.org/abs/2202.08827v1.
* Dwork & Roth (2013) Cynthia Dwork and Aaron Roth. The Algorithmic Foundations of Differential Privacy. _Foundations and Trends® in Theoretical Computer Science_ , 9(3-4):211–407, 2013. ISSN 1551-305X, 1551-3068. doi: 10.1561/0400000042. URL http://www.nowpublishers.com/articles/foundations-and-trends-in-theoretical-computer-science/TCS-042.
* Fowl et al. (2021) Liam H. Fowl, Jonas Geiping, Wojciech Czaja, Micah Goldblum, and Tom Goldstein. Robbing the Fed: Directly Obtaining Private Data in Federated Learning with Modified Models. In _International Conference on Learning Representations_ , September 2021. URL https://openreview.net/forum?id=fwzUgo0FM9v.
* Frey (2021) Suzanne Frey. Introducing Android’s Private Compute Services, September 2021\. URL https://security.googleblog.com/2021/09/introducing-androids-private-compute.html.
* Geiping et al. (2020) Jonas Geiping, Hartmut Bauermeister, Hannah Dröge, and Michael Moeller. Inverting Gradients - How easy is it to break privacy in federated learning? In _Advances in Neural Information Processing Systems_ , volume 33, December 2020. URL https://proceedings.neurips.cc//paper_files/paper/2020/hash/c4ede56bbd98819ae6112b20ac6bf145-Abstract.html.
* Google Research (2019) Team Google Research. Federated Learning, May 2019. URL https://federated.withgoogle.com.
* Gupta et al. (2022) Samyak Gupta, Yangsibo Huang, Zexuan Zhong, Tianyu Gao, Kai Li, and Danqi Chen. Recovering Private Text in Federated Learning of Language Models. _arxiv:2205.08514[cs]_ , May 2022. doi: 10.48550/arXiv.2205.08514. URL http://arxiv.org/abs/2205.08514.
* Hard et al. (2019) Andrew Hard, Kanishka Rao, Rajiv Mathews, Swaroop Ramaswamy, Françoise Beaufays, Sean Augenstein, Hubert Eichner, Chloé Kiddon, and Daniel Ramage. Federated Learning for Mobile Keyboard Prediction. _arXiv:1811.03604 [cs]_ , February 2019. URL http://arxiv.org/abs/1811.03604.
* Hartmann (2021) Florian Hartmann. Predicting Text Selections with Federated Learning, November 2021\. URL http://ai.googleblog.com/2021/11/predicting-text-selections-with.html.
* Hendrycks & Gimpel (2016) Dan Hendrycks and Kevin Gimpel. Gaussian error linear units (gelus). _arXiv preprint arXiv:1606.08415_ , 2016.
* Inan et al. (2021) Huseyin A. Inan, Osman Ramadan, Lukas Wutschitz, Daniel Jones, Victor Rühle, James Withers, and Robert Sim. Training Data Leakage Analysis in Language Models. _arxiv:2101.05405[cs]_ , February 2021. doi: 10.48550/arXiv.2101.05405. URL http://arxiv.org/abs/2101.05405.
* Jayaraman & Evans (2019) Bargav Jayaraman and David Evans. Evaluating Differentially Private Machine Learning in Practice. In _28th{}USENIX{} Security Symposium ({}USENIX{} Security 19)_, pp. 1895–1912, 2019. ISBN 978-1-939133-06-9. URL https://www.usenix.org/conference/usenixsecurity19/presentation/jayaraman.
* Jonker & Volgenant (1987) R. Jonker and A. Volgenant. A shortest augmenting path algorithm for dense and sparse linear assignment problems. _Computing_ , 38(4):325–340, December 1987. ISSN 1436-5057. doi: 10.1007/BF02278710. URL https://doi.org/10.1007/BF02278710.
* Kairouz et al. (2021a) Peter Kairouz, Ziyu Liu, and Thomas Steinke. The Distributed Discrete Gaussian Mechanism for Federated Learning with Secure Aggregation. In _Proceedings of the 38th International Conference on Machine Learning_ , pp. 5201–5212. PMLR, July 2021a. URL https://proceedings.mlr.press/v139/kairouz21a.html.
* Kairouz et al. (2021b) Peter Kairouz, H. Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Kallista Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, Rafael G. L. D’Oliveira, Hubert Eichner, Salim El Rouayheb, David Evans, Josh Gardner, Zachary Garrett, Adrià Gascón, Badih Ghazi, Phillip B. Gibbons, Marco Gruteser, Zaid Harchaoui, Chaoyang He, Lie He, Zhouyuan Huo, Ben Hutchinson, Justin Hsu, Martin Jaggi, Tara Javidi, Gauri Joshi, Mikhail Khodak, Jakub Konečný, Aleksandra Korolova, Farinaz Koushanfar, Sanmi Koyejo, Tancrède Lepoint, Yang Liu, Prateek Mittal, Mehryar Mohri, Richard Nock, Ayfer Özgür, Rasmus Pagh, Mariana Raykova, Hang Qi, Daniel Ramage, Ramesh Raskar, Dawn Song, Weikang Song, Sebastian U. Stich, Ziteng Sun, Ananda Theertha Suresh, Florian Tramèr, Praneeth Vepakomma, Jianyu Wang, Li Xiong, Zheng Xu, Qiang Yang, Felix X. Yu, Han Yu, and Sen Zhao. Advances and Open Problems in Federated Learning. _arXiv:1912.04977 [cs, stat]_ , March 2021b. URL http://arxiv.org/abs/1912.04977.
* Kasiviswanathan et al. (2011) Shiva Prasad Kasiviswanathan, Homin K. Lee, Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. What Can We Learn Privately? _SIAM Journal on Computing_ , 40(3):793–826, January 2011. ISSN 0097-5397. doi: 10.1137/090756090. URL https://epubs.siam.org/doi/10.1137/090756090.
* Li et al. (2019) Wenqi Li, Fausto Milletarì, Daguang Xu, Nicola Rieke, Jonny Hancox, Wentao Zhu, Maximilian Baust, Yan Cheng, Sébastien Ourselin, M. Jorge Cardoso, and Andrew Feng. Privacy-Preserving Federated Brain Tumour Segmentation. In Heung-Il Suk, Mingxia Liu, Pingkun Yan, and Chunfeng Lian (eds.), _Machine Learning in Medical Imaging_ , Lecture Notes in Computer Science, pp. 133–141, Cham, 2019. Springer International Publishing. ISBN 978-3-030-32692-0. doi: 10.1007/978-3-030-32692-0_16.
* Li et al. (2022) Xuechen Li, Florian Tramer, Percy Liang, and Tatsunori Hashimoto. Large Language Models Can Be Strong Differentially Private Learners. In _International Conference on Learning Representations_ , March 2022. URL https://openreview.net/forum?id=bVuP3ltATMz.
* Lin (2004) Chin-Yew Lin. ROUGE: A Package for Automatic Evaluation of Summaries. In _Text Summarization Branches Out_ , pp. 74–81, Barcelona, Spain, July 2004. Association for Computational Linguistics. URL https://aclanthology.org/W04-1013.
* Mallat & Zhang (1993) S.G. Mallat and Zhifeng Zhang. Matching pursuits with time-frequency dictionaries. _IEEE Transactions on Signal Processing_ , 41(12):3397–3415, December 1993. ISSN 1941-0476. doi: 10.1109/78.258082.
* McMahan & Ramage (2017) Brendan McMahan and Daniel Ramage. Federated Learning: Collaborative Machine Learning without Centralized Training Data, April 2017. URL http://ai.googleblog.com/2017/04/federated-learning-collaborative.html.
* McMahan et al. (2017) H. Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Agüera y Arcas. Communication-Efficient Learning of Deep Networks from Decentralized Data. _arXiv:1602.05629 [cs]_ , February 2017. URL http://arxiv.org/abs/1602.05629.
* McMahan et al. (2018) H. Brendan McMahan, Daniel Ramage, Kunal Talwar, and Li Zhang. Learning Differentially Private Recurrent Language Models. _arXiv:1710.06963 [cs]_ , February 2018. URL http://arxiv.org/abs/1710.06963.
* Melis et al. (2019) Luca Melis, Congzheng Song, Emiliano De Cristofaro, and Vitaly Shmatikov. Exploiting Unintended Feature Leakage in Collaborative Learning. In _2019 IEEE Symposium on Security and Privacy (SP)_ , pp. 691–706, May 2019. doi: 10.1109/SP.2019.00029.
* Merity et al. (2016) Stephen Merity, Caiming Xiong, James Bradbury, and Richard Socher. Pointer Sentinel Mixture Models. November 2016. URL https://openreview.net/forum?id=Byj72udxe.
* Noble et al. (2022) Maxence Noble, Aurélien Bellet, and Aymeric Dieuleveut. Differentially Private Federated Learning on Heterogeneous Data. _arxiv:2111.09278[cs, math, stat]_ , February 2022. URL http://arxiv.org/abs/2111.09278.
* ONNX (2022) Documentation ONNX. ONNX Operator Schemas. Open Neural Network Exchange, January 2022. URL https://github.com/onnx/onnx/blob/50a1981dc3d50af13075cec33b08b4c87fb0e41f/docs/Operators.md.
* Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: A Method for Automatic Evaluation of Machine Translation. In _Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics_ , pp. 311–318, Philadelphia, Pennsylvania, USA, July 2002. Association for Computational Linguistics. doi: 10.3115/1073083.1073135. URL https://aclanthology.org/P02-1040.
* Paszke et al. (2017) Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch. In _NIPS 2017 Autodiff Workshop_ , Long Beach, CA, 2017\. URL https://openreview.net/forum?id=BJJsrmfCZ.
* Paulik et al. (2021) Matthias Paulik, Matt Seigel, Henry Mason, Dominic Telaar, Joris Kluivers, Rogier van Dalen, Chi Wai Lau, Luke Carlson, Filip Granqvist, Chris Vandevelde, Sudeep Agarwal, Julien Freudiger, Andrew Byde, Abhishek Bhowmick, Gaurav Kapoor, Si Beaumont, Áine Cahill, Dominic Hughes, Omid Javidbakht, Fei Dong, Rehan Rishi, and Stanley Hung. Federated Evaluation and Tuning for On-Device Personalization: System Design & Applications. _arXiv.2102.08503_ , February 2021. doi: 10.48550/arXiv.2102.08503. URL https://arxiv.org/abs/2102.08503v1.
* Phong et al. (2017) Le Trieu Phong, Yoshinori Aono, Takuya Hayashi, Lihua Wang, and Shiho Moriai. Privacy-Preserving Deep Learning via Additively Homomorphic Encryption. Technical Report 715, 2017. URL http://eprint.iacr.org/2017/715.
* Radford et al. (2019) Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language Models are Unsupervised Multitask Learners. _OpenAI_ , pp. 24, 2019.
* Ramaswamy et al. (2019) Swaroop Ramaswamy, Rajiv Mathews, Kanishka Rao, and Françoise Beaufays. Federated Learning for Emoji Prediction in a Mobile Keyboard. _arXiv:1906.04329 [cs]_ , June 2019. URL http://arxiv.org/abs/1906.04329.
* Rubinstein et al. (2008) Ron Rubinstein, Michael Zibulevsky, and Michael Elad. Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit. Technical report, Computer Science Department, Technion, 2008.
* Somepalli et al. (2021) Gowthami Somepalli, Micah Goldblum, Avi Schwarzschild, C Bayan Bruss, and Tom Goldstein. Saint: Improved neural networks for tabular data via row attention and contrastive pre-training. _arXiv preprint arXiv:2106.01342_ , 2021.
* Truong et al. (2021) Nguyen Truong, Kai Sun, Siyao Wang, Florian Guitton, and YiKe Guo. Privacy preservation in federated learning: An insightful survey from the GDPR perspective. _Computers & Security_, 110:102402, November 2021. ISSN 0167-4048. doi: 10.1016/j.cose.2021.102402. URL https://www.sciencedirect.com/science/article/pii/S0167404821002261.
* Ullman (2021) Jonathan Ullman. Tight Lower Bounds for Locally Differentially Private Selection. _arXiv:1802.02638 [cs]_ , May 2021. URL http://arxiv.org/abs/1802.02638.
* Veale et al. (2018) Michael Veale, Reuben Binns, and Lilian Edwards. Algorithms that remember: Model inversion attacks and data protection law. _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , 376(2133):20180083, November 2018. doi: 10.1098/rsta.2018.0083. URL https://royalsocietypublishing.org/doi/full/10.1098/rsta.2018.0083.
* Wainakh et al. (2021) Aidmar Wainakh, Fabrizio Ventola, Till Müßig, Jens Keim, Carlos Garcia Cordero, Ephraim Zimmer, Tim Grube, Kristian Kersting, and Max Mühlhäuser. User Label Leakage from Gradients in Federated Learning. _arXiv:2105.09369 [cs]_ , June 2021. URL http://arxiv.org/abs/2105.09369.
* Wang et al. (2021) Jianyu Wang, Zachary Charles, Zheng Xu, Gauri Joshi, H. Brendan McMahan, Blaise Aguera y Arcas, Maruan Al-Shedivat, Galen Andrew, Salman Avestimehr, Katharine Daly, Deepesh Data, Suhas Diggavi, Hubert Eichner, Advait Gadhikar, Zachary Garrett, Antonious M. Girgis, Filip Hanzely, Andrew Hard, Chaoyang He, Samuel Horvath, Zhouyuan Huo, Alex Ingerman, Martin Jaggi, Tara Javidi, Peter Kairouz, Satyen Kale, Sai Praneeth Karimireddy, Jakub Konecny, Sanmi Koyejo, Tian Li, Luyang Liu, Mehryar Mohri, Hang Qi, Sashank J. Reddi, Peter Richtarik, Karan Singhal, Virginia Smith, Mahdi Soltanolkotabi, Weikang Song, Ananda Theertha Suresh, Sebastian U. Stich, Ameet Talwalkar, Hongyi Wang, Blake Woodworth, Shanshan Wu, Felix X. Yu, Honglin Yuan, Manzil Zaheer, Mi Zhang, Tong Zhang, Chunxiang Zheng, Chen Zhu, and Wennan Zhu. A Field Guide to Federated Optimization. _arXiv:2107.06917 [cs]_ , July 2021. URL http://arxiv.org/abs/2107.06917.
* Wang et al. (2018) Zhibo Wang, Mengkai Song, Zhifei Zhang, Yang Song, Qian Wang, and Hairong Qi. Beyond Inferring Class Representatives: User-Level Privacy Leakage From Federated Learning. _arXiv:1812.00535 [cs]_ , December 2018. URL http://arxiv.org/abs/1812.00535.
* Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. HuggingFace’s Transformers: State-of-the-art Natural Language Processing. _arXiv:1910.03771 [cs]_ , July 2020. URL http://arxiv.org/abs/1910.03771.
* Yin et al. (2021) Hongxu Yin, Arun Mallya, Arash Vahdat, Jose M. Alvarez, Jan Kautz, and Pavlo Molchanov. See Through Gradients: Image Batch Recovery via GradInversion. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 16337–16346, 2021. URL https://openaccess.thecvf.com/content/CVPR2021/html/Yin_See_Through_Gradients_Image_Batch_Recovery_via_GradInversion_CVPR_2021_paper.html.
* Zanella-Béguelin et al. (2020) Santiago Zanella-Béguelin, Lukas Wutschitz, Shruti Tople, Victor Rühle, Andrew Paverd, Olga Ohrimenko, Boris Köpf, and Marc Brockschmidt. Analyzing Information Leakage of Updates to Natural Language Models. In _Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security_ , pp. 363–375. Association for Computing Machinery, New York, NY, USA, October 2020. ISBN 978-1-4503-7089-9. URL https://doi.org/10.1145/3372297.3417880.
* Zhu et al. (2019) Ligeng Zhu, Zhijian Liu, and Song Han. Deep Leakage from Gradients. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d\textquotesingle Alché-Buc, E. Fox, and R. Garnett (eds.), _Advances in Neural Information Processing Systems 32_ , pp. 14774–14784. Curran Associates, Inc., 2019. URL http://papers.nips.cc/paper/9617-deep-leakage-from-gradients.pdf.
## Appendix A Technical Details
We implement all attacks in an extensible PyTorch framework (Paszke et al.,
2017) for this type of attack which allows for a full reproduction of the
attack and which we attach to this submission. We utilize huggingface models
and data extensions for the text data (Wolf et al., 2020).The attack is
separated into two parts. The first part is the malicious modification of the
server parameters, which is triggered immediately before the server sends out
their model update payload to users.
We implement the malicious parameter modifications as described in Sec. 3. We
initialize from a random model initialization and reserve the first $6$
entries of the embedding layer for the sentence encoding for the Transformer-3
model and the first $32$ entries for the larger BERT and GPT-2 models. The
corresponding entries in the positional and token embeddings are reset to $0$.
In the MHA modification, we choose a softmax skewing of $1e8$, although this
value can be significantly lower in practice as well. We reserve the last
entry of the embedding layer for gradient flow to each linear layer, so that
the attack is able to utilize all layers as described. This entry is scaled by
$\varepsilon=1e-6$ for the smaller Transformer-3 model and $\varepsilon=1e-8$
for the larger models, so that the layer normalization is not skewed by large
values arising from the last embedding entry.
For the attack part, we retrieve the gradient update on the server and run all
recovery computations in single float precision. We first run a token recovery
as discussed in Sec 3.1, which we additionally summarize in Sec. 3.4 and then
proceed with the attack following Algorithm 1. For the sentence labeling we
cluster using constrained K-means as described in Bradley et al. (2000). For
all assignment problems we utilize the linear sum assignment solver proposed
in Crouse (2016) which is a modification of the shortest augmenting path
strategy originally proposed in Jonker & Volgenant (1987).
During the quantitative evaluation, we trial 100 users each with a request for
updates of size sequence length $\times$ batches. We skip all users which do
not own enough data and do not pad user data with [PAD] tokens, which we think
would skew results as it would include a large number of easy tokens to
recover. All measurements are hence done on non-trivial sentences, which are
concatenated to reach the desired sequence length. Each user’s data is
completely separated, representing different wikipedia articles as described
in the main body. Overall, the attack is highly successful over a range of
users even with very different article content.
To compute metrics, we first resort the recovered batches toward the ground
truth order by assigning to ground truth batches based on total token accuracy
per sentence. This leads to a potential underestimation of the true accuracy
and ROUGE metrics of the recovery as sentence are potentially mismatched. We
compute Rouge (Lin, 2004) scores based on the sorted batches and BLEU scores
(Papineni et al., 2002) (with the default huggingface implementation) by
giving all batches as references. We measure total accuracy as the number of
tokens that are perfectly identified and in the correct position, assigning no
partial credit for either token identity or position. This is notably in
contrast to the naming of metrics in Deng et al. (2021) where accuracy refers
to only overall token accuracy. We refer to that metric as "token accuracy",
measuring only the overlap between reference and reconstruction sentence bag-
of-words, but report only the total accuracy, given that token accuracy is
already near-perfect after the attack on the bag-of-words described in Sec
3.1.
## Appendix B Algorithm Details
We detail sub-algorithms as additional material in this section. Algorithm 2
and Algorithm 3 detail the token recovery for transformers with decoder bias
and for transformer with a tied embedding. These roughly follow the principles
of greedy label recovery strategy proposed in Wainakh et al. (2021) and we
reproduce them here for completeness, incorporating additional considerations
necessary for token retrieval.
Algorithm 2 Token Recovery
(Decoder Bias)
1: Input: Decoder bias gradient $g_{b}$, embedding gradient $g_{e}$, sequence
length $s$, number of sequences $n$.
2: $v_{\text{tokens}}$ $\leftarrow$ all indices where $g_{b}<0$
3: $v_{e}$ $\leftarrow$ all indices where $g_{e}<0$
4: $v_{\text{tokens}}$ append $v_{\text{tokens}}\setminus v_{e}$
5: $m_{\text{impact}}=\frac{1}{sn}\sum_{i\in v_{\text{tokens}}}{g_{b}}_{i}$
6: $g_{b}[v_{\text{tokens}}]$ $\leftarrow$
$g_{b}[v_{\text{tokens}}]-m_{\text{impact}}$
7: while Length of $v_{\text{tokens}}<sn$ do
8: $j=\operatorname*{arg\,min}_{i}{g_{b}}_{i}$
9: ${g_{b}}_{j}$ $\leftarrow$ ${g_{b}}_{j}-m_{\text{impact}}$
10: $v_{\text{tokens}}$ append $j$
11: end while
Algorithm 3 Token Recovery
(Tied encoder Embedding)
1: Input: Embedding weight gradient $g_{e}$, sequence length $s$, number of
expected sequences $n$, cutoff factor $f$
2: ${n_{e}}_{j}=||{g_{e}}_{j}||_{2}$ for all $j=1,...N$ embedding rows
3: $\mu,\sigma=\text{Mean}(\log{n_{e}}),\text{Std}(\log{n_{e}})$
4: $c=\mu+f\sigma$
5: $v_{\text{tokens}}$ $\leftarrow$ all indices where $n_{e}>c$
6: $m_{\text{impact}}=\frac{1}{sn}\sum_{i\in v_{\text{tokens}}}{n_{e}}_{i}$
7: $n_{e}[v_{\text{tokens}}]$ $\leftarrow$
$n_{e}[v_{\text{tokens}}]-m_{\text{impact}}$
8: while Length of $v_{\text{tokens}}<sn$ do
9: $j=\operatorname*{arg\,min}_{i\in v_{\text{tokens}}}{n_{e}}_{i}$
10: ${n_{e}}_{j}$ $\leftarrow$ ${n_{e}}_{j}-m_{\text{impact}}$
11: $v_{\text{tokens}}$ append $j$
12: end while
## Appendix C Token Bag-of-Words Recovery
Even without any malicious parameter modifications, an attacker can
immediately retrieve the bag-of-words (the unordered list of all tokens and
their assorted frequencies) of the user data from the gradient of the token
embedding. Melis et al. (2019) previously identified that unique tokens can be
recovered due to the sparsity of rows in the token embedding gradient, as
visualized in the left plot of Figure 5. But, perhaps surprisingly, we find
that even the frequency of all words can be extracted. Depending on the model
architecture, this frequency estimation can either be triggered by analyzing
either the bias gradient of the decoding (last linear) layer, or the norm of
the embedding matrix. In both cases, the magnitude of the update of each row
of a random embedding matrix is proportional to the frequency of word usage,
allowing for a greedy estimation by adapting the strategy of Wainakh et al.
(2021) which was originally proposed to extract label information in
classification tasks.
For models which contain a decoder bias parameter, this estimation directly
leads to a $>99\%$ accurate estimate of word frequencies. For models without
decoder bias, such as GPT (Radford et al., 2019), the same strategy can again
be employed based on the magnitude of embedding rows. We can further
accurately predict token frequency even for models which tie their embedding
and decoding weights, and as such do not feature naturally sparse embedding
gradients. However, examining the distributions of embedding row norms in log-
space for tokens contained in the other updates and all tokens, as visualized
in the right plot of Figure 5, we find that unused tokens can be removed by a
simple thresholding. The resulting token frequency estimation is inexact due
to the cut-off, but still reaches a bag-of-words accuracy of $93.1\%$ accuracy
for GPT-2, even for an update containing 13824 tokens.
This first insight already appears to have been overlooked in optimization-
based attacks (Deng et al., 2021; Dimitrov et al., 2022) and does not require
any malicious modifications of the embedding sent by the server. However, for
a model update averaged over multiple users and thousands of tokens, this
recovery alone is less of a threat to privacy. Naively, the ways to order
recovered tokens scale as $n!$, and any given ordering of a large amount of
tokens may not reveal sensitive information about any constituent user
involved in the model update.
Figure 10: Left: A high-level schematic of a simple case of token leaking. The
token embedding layer can leak tokens and frequency solely through its sparse
gradient entries. Right: Distribution and Histogram of log of norms of all
token embedding gradients for GPT-2 for $13824$ tokens. In this case, gradient
entries are non-sparse due to the tied encoder-decoder structure of GPT, but
the embeddings of true user data (red) are clearly separable from the mass of
all embeddings (blue) by a simple cutoff at 1.5 standard deviations (marked in
black).
Figure 11: Bag-of-words Accuracy for from embedding gradients out of 13824
tokens from wikitext. Left: The small transformer model where tokens frequency
can be estimated directly from the decoder bias. Token frequency estimation is
$96.7\%$ accurate, unique token recovery are $99.8\%$ accurate. Right: The
(small) GPT-2 variant where tokens are estimated from the norms of the
embedding rows. Token frequency estimation is $93.1\%$ accurate, unique tokens
are $97.1\%$ accurate due to the cutoff at $\frac{3\sigma}{2}$.
## Appendix D Measurement Normalization
In order to retrieve positionally encoded features from the first linear layer
in the feed-forward part of each Transformer block, we create “bins" which
partition the possible outcomes of taking the scalar product of an embedding
with a random Gaussian vector. To do this, we estimate the mean and variance
of such a measurement, and create bins according to partitions of equal mass
for a Gaussian with this mean and variance. A natural question arises: how
well can we estimate this quantity? If the server does not have much data from
the user distribution, will this present a problem for recovering user data?
To this end, we demonstrate that the server can estimate these measurement
quantities well using a surrogate corpus of data. In Figure 12, we see that
the Gaussian fit using the Wikitext dataset is strikingly similar to the one
the server would have found, even on a very different dataset (Shakespeare).
We further note that an ambitious server could also directly use model updates
retrieved from users in a first round of federated learning, to directly glean
the distribution of these linear layers, given that the lowest bin records the
average of all activations of this layer. However, given the ubiquity of
public text data, this step appears almost unnecessary - but possibly relevant
in specialized text domains or when using Transformers for other data
modalities such as tabular data (Somepalli et al., 2021).
However, for language data, these quantities can also simply be estimated
using a small number of batches (100) filled with random token ids. We show in
Figure 13 that this leads to essentially the same performance in average total
accuracy and to only minor differences in peak accuracy.
Figure 12: Comparison of Gaussian fit to the measurements of Shakespeare
dataset, and a Gaussian fit to the measurements (against the same vector) for
the Wikitext dataset.
Figure 13: Using random data to estimate the measurement distribution. Left:
Total accuracy (i.e. percentage of tokens recovered correctly in their correct
position) for GPT-2 and varying batch sizes with sequence length 32. Right:
Average token accuracy on the most-leaked sentence. Random tokens can be an
effective substitute for public data to estimate the measurement mean and
standard deviation.
### D.1 Random token vulnerability
Note that in Figure 8, we see that sequences generated randomly from the
vocabulary are more vulnerable than other datasets. This is because randomly
generated sequences on average result in fewer collisions as all possible
tokens from the vocabulary are used with equal probability. Zipf’s law for the
distribution of words in English lends credence to this interpretation. More
common words, and associated positions for that matter, will result in more
collisions and degrade the attacker’s ability to reconstruct sequences.
## Appendix E Variants and Details
### E.1 Masked Language Modelling
For problems with masked language modelling, the loss is sparse over the
sequence length, as only masked tokens compute loss. This impacts the strategy
proposed in Sec 3.2, as with disabled attention mechanisms in all but the
first layer, no gradient information flows to unmasked entries in the
sequence. However, this can be quickly solved by reactivating the last
attention layer, so that it equally attends to all elements in the sequence
with a minor weight increment which we set to $10$. This way, gradient flow is
re-enabled and all computations can proceed as designed. For BERT, we further
disable the default huggingface initialiazation of the token embedding, which
we reset to a random normal initialization.
Masked language modelling further inhibits the decoder bias strategy discussed
in Sec 3.1, as only masked tokens lead to a non-positive decoder bias
gradient. However, we can proceed for masked language models by recovering
tokens from the embedding layer as discussed for models without decoder bias.
The mask tokens extracted from the bias can later be used to fill masked
positions in the input.
### E.2 GELU
A minor stumbling for the attacker occurs if the pre-defined model uses GELU
(Hendrycks & Gimpel, 2016) activation instead of ReLU. This is because GELU
does not threshold activations in the same was as ReLU, and transmits gradient
signal when the activation $<0$. However, a simple workaround for the attacker
is to increase the size of the measurement vector and, by doing so, push
activations away from 0, and thus more toward standard ReLU behavior. For the
main-body experiments, we use ReLU activation for simplicity of presentation,
but using the workaround described above, we find in Figure 14, Figure 15 that
the performance of the attack against a GELU network is comparable to its
level against a ReLU network.
Figure 14: Comparison of GELU and ReLU activation with magnified measurement
vector (sequence length $256$). Figure 15: Comparison of GELU and ReLU
activation with magnified measurement vector (batch size $1$). Figure 16:
Baseline results for our method for different popular architectures and
metrics for variable length sequence input (batch size $1$). Note BERT’s
positional encoding caps out at sequence length $512$, so we end our
experiments for BERT there. Figure 17: Results for our method for different
popular architectures and metrics for variable length batch size (sequence
length $32$).
### E.3 Dropout
All experiments in the main body, including comparisons, have assumed that
dropout has been turned off for all models under consideration. In standard
applications, dropout can be modified from the server-side, even for compiled
and otherwise fixed models (ONNX, 2022). In our threat model, dropout hence
falls into the category of a server-side parameter that a malicious update
could turn off. We also investigated the performance of the attack if the
attacker cannot alter standard dropout parameters, finding that dropout
decreased total accuracy of the attack by about 10%, e.g. from $93.36\%$ for a
sequence of length 512 on GPT-2 to $81.25\%$ with dropout.
## Appendix F Additional Background Material
The attack TAG in Deng et al. (2021) approaches gradient inversion attacks
against transformer models from the direct optimization-based angle. This was
first suggested in Zhu et al. (2019), who provide some preliminary experiments
on recovery of short sequences from BERT. Basically, the attack works to
recover user data $(x^{*},y^{*})$ from the measurement of the gradient on this
data $g$ by solving the optimization problem of
$\min_{x,y}||\mathcal{L}(n(x,\theta),y)-g||^{2},$ (4)
where $x$ and $y$ are the inputs and labels to be recovered, $\mathcal{L}$ is
the loss and $n(\cdot,\theta)$ is the language model with parameters $\theta$.
This optimization approach does succeed for short sequences, but the
optimization becomes quickly stuck in local minima and cannot recover the
original input data. Zhu et al. (2019) originally propose the usage of an
L-BFGS solver to optimize this problem, but this optimizer can often get stuck
in local minima. Deng et al. (2021) instead utilize the "BERT" Adam optimizer
with hyperparameters as in BERT training (Devlin et al., 2019). They further
improve the objective by adding an additional term that especially penalizes
gradients of the first layers, which we also implement when running their
attack. A major problem for the attacks of Zhu et al. (2019) and Deng et al.
(2021), however, is the large label space for next-word prediction. In
comparison to binary classification tasks as mainly investigated in Deng et
al. (2020), the large label space leads to significant uncertainty in the
optimization of labels $y$, which leads their attack to reduce in performance
as vocabulary size increases.
We further note that Boenisch et al. (2021) also propose malicious model
modifications (as opposed to our more realistic malicious parameter
modifications) to breach privacy in FL for text, however the proposed model in
their work is a toy two-layer fully connected model that is not a Transformer-
based model. In fact, the strategy employed in their attack cannot be deployed
against modern language models that do not construct a linear layer of the
length of the sequence, aside from handling of facets of modern language
models like positional encodings, or attention mechanisms.
## Appendix G Mitigation Strategies
As stated in the main body, a central message of our paper is that aggregation
alone may not be sufficient to defend against malicious parameter attacks in
FL. Further defenses may be needed. Such defenses could include parameter
inspection wherein a defender could look for identifiable signatures in the
shared parameter vector. For example, the given attack duplicates a
measurement vector, leading to low rank linear layers. To the best of our
knowledge, there do not exist systematic parameter inspection based defenses
to malicious models in FL. The attacker could, however, easily adapt to some
obvious parameter inspection defenses. For example, simply adding a small
amount of noise to the measurement vector, and other conspicuous parts of the
attack, does not affect the success of the attack significantly, but does
immediately make a strategy like rank inspection null and void. We confirm
this adaptation by adding a small amount of Gaussian noise to each row of the
measurement vector in the linear layers (for the Transformer-3 model, $8$
users, $32$ sequence length) and find that numerically, these layers do indeed
become full rank, and the success of the attack remains largely unchanged
(accuracy $90.62\%$). Because such simple adaptations are easily made, we do
not recommend this line of defense.
Instead, we advocate for strong differential privacy on the user side. Such
measures could include gradient clipping/noising (c.f. .
The immediate impact of differential privacy mechanisms on the attack
described in this work is that the measured weight and bias become noisy, so
that the estimation in Equation 3 becomes subsequently noisy. However, an
attacker can counteract this simply by noticing that the number of jumps in
the cumulative sum is sparse, i.e only number-of-tokens many jumps exist,
independent from the number of bins. As such, as long as the number of tokens
is smaller than the number of bins, the attacker can denoise this measurement,
for example using tools from sparse signal recovery such orthogonal matching
pursuit (Mallat & Zhang, 1993). We use the implementation via K-SVD from
Rubinstein et al. (2008) to do so. This way, the attacker reduces the
immediate effect of gradient noise, utilizing the redundancy of the
measurements of the cumulative sum of embeddings in every bin. Overall, with
the above strategy, we find the attack can still succeed in the presence of
gradient clipping and noising, although the success does predictably degrade
at higher noise levels. (see Figure 18).
Furthermore, the imprint strategy, as found in Fowl et al. (2021), has been
shown to be robust to certain amounts of gradient noise, and an attacker could
further attempt to implement strategies found in that attack to resist this
defense.
##### Mitigations through local loss audits
Moreover, federated learning often takes place in the background, for example,
at night when the device is charging (see Hard et al. (2019)) and can be a
regular occurrence on participating devices. Even if user applications were
modified to make loss values on local data observable to the user, a user
could not know whether a model was malicious, or whether a new model was
simply in an early round of its training run. As such, auditing and exposition
of local model loss values to the user is not a strong defense against the
proposed attack.
Figure 18: Attack success rate against different levels of added Laplacian
noise. In each experiment, the gradient was also clipped to a value of $1$. We
see that the attack can remain successful even in the presence of added noise.
In this experiment, a single sequence of length $32$ was considered.
## Appendix H Further Results
Figure 19: Baseline results for our method for different popular architectures
and metrics for variable batch size (sequence length $512$). Figure 20:
Comparison to TAG attack for variable batch size (sequence length $32$).
In Figure 3, we illustrate the embedding recovery process that we adapt from
Fowl et al. (2021). We opt to initialize the rows of the first linear layer in
each Transformer block to a randomly sampled Gaussian vector (all rows being
the same vector). Then, following the calculation of Fowl et al. (2021), when
an incoming embedding vector, for example, corresponding to the word "pepper",
is propagated through this layer, the ascending biases threshold values of the
inner product between the Gaussian vector, $m$, and the embedding vector, $x$.
Because of the ReLU activation, this means that individual embedding vectors
can be directly encoded into gradient entries for rows of this linear layer.
### H.1 Sensitivity to threshold in token estimation
We include an evaluation of the threshold parameter for the embedding-norm
token estimation in Figure 21. This value is set to 1.5 standard deviations in
all other experiments.
Figure 21: Sensitivity to token estimation threshold. Total accuracy (i.e.
percentage of tokens recovered correctly in their correct position) for GPT-2
and varying batch sizes with sequence length 32. We evaluate the threshold
parameter for the token estimation. The default value in the remainder of all
experiments is 1.5, although we find the estimation to be robust to a range of
similar cutoffs.
### H.2 Total Amount of Token Leaked
Finally, in Figure 22 we include the total amount of tokens leaked correctly
through the attack for a variety of sequence lengths, batch sizes and models.
We see that the total amount of tokens leaked with GPT-2 has not yet reached a
peak, and larger sequences could leak more tokens. For the smaller 3-layer
transformer we see that the number of leaked tokens peaks around a batch size
of 128 for a sequence length of 32, and 8 for a sequence length of 512, i.e.
around 4096 tokens. This is related to the number of bins in this
architecture, which in turn is given by the width of all linear layers,
leading to a bin size of $4608$ for the 3-layer transformer. For the small
GPT-2 the number of bins is 36864.
Figure 22: Total number of tokens leaked in their exact location for various
settings (instead of percentages as shown elsewhere). Top row: Total number of
token for GPT-2 (left) and the 3-layer transformer (right) for a sequence
length of 32 and various batch sizes. Bottom row: total number of token for
GPT-2 (left) and the 3-layer transformer (right) for a sequence length of 512
and various batch sizes.
|
# Joint CFO, Gridless Channel Estimation and Data Detection for Underwater
Acoustic OFDM Systems
Lei Wan, Jiang Zhu, En Cheng and Zhiwei Xu Lei Wan and En Cheng are with
Department of Information and Communication Engineering and Key Laboratory of
Underwater Acoustic Communication and Marine Information Technology (Xiamen
University), Ministry of Education, China (email: {leiwan,
chengen}@xmu.edu.cn). Jiang Zhu and Zhiwei Xu are with the engineering
research center of oceanic sensing technology and equipment, Ministry of
Education, Ocean College, Zhejiang University, and also with the key
laboratory of ocean observation-imaging tested of Zhejiang Province, No.1
Zheda Road, Zhoushan, 316021, China (email: {jiangzhu16, xuzw}@zju.edu.cn).
The corresponding author is Jiang Zhu (email: jiangzhu16@zju.edu.cn).
###### Abstract
In this paper, we propose an iterative receiver based on gridless variational
Bayesian line spectra estimation (VALSE) named JCCD-VALSE that _j_ ointly
estimates the _c_ arrier frequency offset (CFO), the _c_ hannel with high
resolution and carries out _d_ ata decoding. Based on a modularized point of
view and motivated by the high resolution and low complexity gridless VALSE
algorithm, three modules named the VALSE module, the minimum mean squared
error (MMSE) module and the decoder module are built. Soft information is
exchanged between the modules to progressively improve the channel estimation
and data decoding accuracy. Since the delays of multipaths of the channel are
treated as continuous parameters, instead of on a grid, the leakage effect is
avoided. Besides, the proposed approach is a more complete Bayesian approach
as all the nuisance parameters such as the noise variance, the parameters of
the prior distribution of the channel, the number of paths are automatically
estimated. Numerical simulations and sea test data are utilized to demonstrate
that the proposed approach performs significantly better than the existing
grid-based generalized approximate message passing (GAMP) based _j_ oint _c_
hannel and _d_ ata decoding approach (JCD-GAMP). Furthermore, it is also
verified that joint processing including CFO estimation provides performance
gain.
keywords: Iterative receivers, message passing, expectation propagation, grid-
less, OFDM channel estimation
## I Introduction
Due to the propagation characteristics, underwater acoustic communications
typically suffer from strong multipath and Doppler effect [1]. Since the
propagation speed is low (around 1500 m/s), the reflection of acoustic waves
from surface and bottom of the ocean, along with the refraction caused by
nonuniformity of sound speed in water, cause significant multipath delay
spread. The typical delay spread of underwater acoustic communication channels
is from several milliseconds to tens of milliseconds, and could be hundreds of
milliseconds and more in extreme cases. On the other hand, due to the movement
of transmitter and receiver platforms, as well as water media, Doppler effect
is almost inevitable in underwater acoustic communications. Considering the
wide band nature of underwater acoustic communication systems, the Doppler
effect usually exhibits itself as signal dilation or compression [2].
With the advantages of low complexity frequency domain multipath channel
equalization and high bandwidth efficiency, orthogonal frequency division
multiplexing (OFDM) technology was introduced into underwater acoustic
communications, and soon became a popular choice especially for high speed
communications [3, 4, 5]. However, Doppler effect in underwater acoustic
channels could destroy the orthogonality between the subcarriers in OFDM and
hence deteriorates its performance. In [4], a two-step approach was proposed
to compensate the Doppler distortion, in which a resampling operation is
carried out first to convert the wide band Doppler distortion into narrow band
carrier frequency offset (CFO). In the second step, the CFO is estimated and
compensated. Experimental results in [4] demonstrated that the proposed scheme
can effectively mitigate the Doppler distortion.
In coherent underwater acoustic communication systems, accurate channel
estimation is essential. Since compressed sensing based channel estimation
methods exploit the structure of the channel and require less number of
measurements, they have been the most popular choices for underwater acoustic
OFDM systems [6, 7]. Among compressed sensing algorithms, orthogonal matching
pursuit (OMP) algorithm [8] features lower computational complexity than basis
pursuit methods, and it is widely adopted in channel estimations [9, 10]. On
the other hand, as an approximation for sum-product algorithm (SPA),
approximate message passing (AMP) has been utilized for sparse linear
estimation problems [11], and hence is also employed for sparse channel
estimations [12].
To further improve the performance of wireless communication systems in
challenging channel conditions, joint receivers in which modules such as
channel estimation, channel equalization interactively work together are
proposed, both for radio and underwater acoustic communication systems. In
[13], a joint channel estimation, equalization and data detection system based
on expectation-maximization (EM) algorithm is proposed for OFDM systems with
high mobility. In the proposed scheme, the time-varying channel is represented
by discrete cosine orthogonal basis functions and the equalized symbols are
quantized into nearest data symbol constellation point in each iteration. A
low complexity generalized AMP (GAMP) based joint channel estimation and data
decoding approach for OFDM system is proposed [14]. Later, a joint channel
estimation and symbol detection scheme based on parametric bilinear GAMP
algorithm is proposed for single-carrier systems [15], which supports the
sparse characteristic of wireless communication channels and soft-input soft-
output (SISO) equalizer. Recently, based on Markov chain Monte Carlo (MC)
algorithm with Gibbs sampling and convolutional coding plus differential
coding, a joint phase shift keying (PSK) data detection and channel estimation
scheme utilizing channel sparsity is proposed in [16].
For underwater acoustic communications, the concept of joint processing has
been widely pursued. In the early work [17], an iterative receiver based on
message passing techniques is proposed for single carrier underwater acoustic
communication systems, in which equalization and decoding are performed
multiple times in a turbo manner. However, channel estimation is carried out
based on training symbols, utilizing maximum-likelihood approach. In [18, 19,
20], iterative receivers were developed for single carrier multiple-input
multiple-output (MIMO) underwater acoustic communication systems, for both
time-domain and frequency domain equalization. In the proposed iterative
receivers, soft equalization outputs are feedback for interference cancelation
and channel estimation. For OFDM underwater acoustic communication systems in
doubly spread channels, an iterative sparse channel estimation and data
detection receiver using partial interval demodulation is proposed in [21]. In
[22], an iterative receiver is proposed for inter-frequency interference (IFI)
cancelation for single-carrier underwater acoustic communication systems
adopting frequency domain equalization. Channel impulse responses, explicit
and residual IFI are updated in an iterative fashion based on the soft symbol
estimates in the proposed receiver. Considering the impulse noise in
underwater acoustic communication channels, a joint channel estimation and
impulsive noise mitigation receiver utilizing sparse Bayesian learning (SBL)
algorithm is proposed [23], in which the information on data symbols are
employed to carry out joint time-varying channel estimation and data
detection, to further improve the system performance. The performance of the
proposed algorithm is verified through both numerical simulations and real
data. More recently, a joint channel estimation and equalization scheme with
superimposed training is proposed for single carrier underwater acoustic
communication systems [24]. A message-passing-based bidirectional channel
estimation algorithm is adopted in the proposed scheme, utilizing the
correlation between consecutive channels.
It is worth noting that all the above compressed sensing based approaches
utilize the fixed dictionary and suffer grid mismatch [25], and the gridless
parametric channel estimation approach indeed improves both the channel
estimation and data decoding performance over the grid based approaches [26,
27]. Moreover, existing iterative receivers seldom incorporate the CFO
estimation and compensation into the design. In this paper, according to
message passing and from the modularized point of view [28], an iterative
receiver based on gridless variational Bayesian line spectra estimation
(VALSE) [29] which _j_ ointly estimates the residual _C_ FO and _c_ hannel,
meanwhile also achieves _d_ ata decoding named JCCD-VALSE is proposed for
underwater acoustic OFDM systems. The JCCD-VALSE consists of three modules
named as high resolution gridless channel estimation, minimum mean squared
error (MMSE) estimation, and low-density parity-check code (LDPC) decoder.
Through the exchange of soft information between modules, the channel
estimation and data decoding accuracy progressively improves. Compared with
existing receivers, the joint processing including residual CFO estimation and
compensation, the gridless channel estimation, and the feature of automatical
estimation of nuisance parameters are the advantages of the proposed scheme.
Hence, the contributions of this paper include:
1. 1.
Targeting at underwater acoustic OFDM systems in channels with an
approximately common Doppler scaling factor, the system model for received
signal after resampling is derived, and the corresponding receiver JCCD-VALSE
which jointly estimates residual CFO, channel, and finishes data decoding is
proposed.
2. 2.
In JCCD-VALSE, all the nuisance parameters such as the number of paths, the
noise variance are automatically estimated. This reduces the reliance on
priori information, and also improves the estimation accuracy.
3. 3.
In JCCD-VALSE, gridless channel estimation based on VALSE is adopted, which
improves the channel estimation and data decoding performance, compared with
the existing channel estimation and data decoding methods. Besides, it is also
shown that taking residual CFO into consideration benefits both the channel
estimation and data decoding performances.
4. 4.
Sea trial data of mobile acoustic communication experiment (MACE10) is
utilized to further demonstrate the effectiveness of the proposed scheme.
In addition, it is also worth noting that our proposed approach is very
flexible and can deal with both uniform and nonuniform pilots.
The rest of this paper is organized as follow: Section II introduces the
system model. The probabilistic formulation of the channel, the data symbol
and the normalized residual CFO are presented in Section III. The ensuing
Section IV proposes the joint CFO, channel estimation and data decoding
algorithm. Section V presents the performance metrics in terms of the coded
bit error rate (BER) and the normalized mean squared error (NMSE) of the
various algorithms. The sea trial data decoding results are introduced in
Section VI. Finally, we conclude the paper in Section VII.
Notation: For a matrix $\mathbf{A}$, let $[{\mathbf{A}}]_{i,j}$ or $A_{ij}$
denote the $(i,j)$th element of ${\mathbf{A}}$, and ${\rm diag}(\mathbf{A})$
returns a vector with elements being the diagonal elements of $\mathbf{A}$.
Let $\Re\\{\cdot\\}$ and $\Im\\{\cdot\\}$ denote the real and imaginary part
operator, respectively. For random vectors $\mathbf{x}$ and $\mathbf{y}$ with
joint probability density function (PDF) $p({\mathbf{x}},{\mathbf{y}})$, the
marginal PDF $p({\mathbf{x}})=\int p({\mathbf{x}},{\mathbf{y}}){\rm
d}{\mathbf{y}}$ with ${\rm d}{\mathbf{y}}$ being ${\rm
d}\Re\\{\mathbf{y}\\}{\rm d}\Im\\{\mathbf{y}\\}$. Let $\odot$ denote the
Hadamard product operator. For a vector ${\mathbf{a}}$, let ${\rm
diag}({\mathbf{a}})$ return a diagonal matrix with diagonal elements being
$\mathbf{a}$. Let ${\mathcal{S}}\subset\\{1,\cdots,N\\}$ be a subset of
indices. For a square matrix ${\mathbf{A}}\in{\mathbb{C}}^{N\times N}$, let
$\mathbf{A}_{{\mathcal{S}},{\mathcal{S}}}$ denote the submatrix by choosing
both the rows and columns of $\mathbf{A}$ indexed by $\mathcal{S}$. Let
${(\cdot)}^{*}_{\mathcal{S}}$, ${(\cdot)}^{\rm T}_{\mathcal{S}}$ and
${(\cdot)}^{\rm H}_{\mathcal{S}}$ be the conjugate, transpose and Hermitian
transpose operator of ${(\cdot)}_{\mathcal{S}}$, respectively. For two vectors
${\mathbf{a}}\in{\mathbb{C}}^{n}$ and ${\mathbf{b}}\in{\mathbb{C}}^{n}$,
${\mathbf{a}}/{\mathbf{b}}$ denotes the componentwise division.
## II System model
This section establishes the OFDM model similar to [4], and the assumption
that all paths have a similar Doppler scaling factor is adopted, which seems
to be justified as long as the dominant Doppler effect is caused by the direct
transmitter/receiver motion. For completeness, the detailed derivations are
presented.
Consider the cyclic prefix (CP) OFDM system, and let $T$ and $T_{\rm cp}$
denote the OFDM symbol duration and the CP duration. The number of subcarriers
is $N$ and the frequency spacing is $\Delta f=1/T$. With carrier frequency
$f_{c}$, the frequency of the $k$th subcarrier is
$\displaystyle f_{k}=f_{c}+k\Delta f,~{}k=-N/2,\dots,N/2-1,$
and the bandwidth is $B=N\Delta f$.
Let $d[k]$ denote the information symbol to be transmitted on the $k$th
subcarrier. The transmitted continuous time OFDM block in passband is
$\displaystyle
s(t)=\frac{2}{\sqrt{N}}\Re\left\\{\sum\limits_{k=-N/2}^{N/2-1}d[k]{\rm
e}^{{\rm j}2\pi f_{k}t}g(t)\right\\},$
where $g(t)$ is
$\displaystyle g(t)=\begin{cases}1,\quad t\in[-T_{\rm cp},T],\\\ 0,\quad{\rm
otherwise}.\end{cases}$
A multipath underwater channel where all paths have a similar Doppler scaling
factor $a$ is considered, and the channel impulse response is
$\displaystyle
c(\tau,t)=\sum\limits_{p=1}^{L}A_{p}\delta(\tau-\tau_{p}(t))=\sum\limits_{p=1}^{L}A_{p}\delta(\tau-(\tau_{p}-at)),$
where $A_{p}$ and $\tau_{p}$ denote the gain and delay of the $p$th path, $L$
is the total number of paths. The received continuous time signal in passband
$\tilde{y}(t)$ is
$\displaystyle\tilde{y}(t)=s(t)\ast c(\tau,t)$
$\displaystyle=\sum\limits_{p=1}^{L}A_{p}s(t-\tau_{p}+at)$
$\displaystyle=\frac{2}{\sqrt{N}}\Re\left\\{\sum\limits_{k=-N/2}^{N/2-1}d[k]\sum\limits_{p=1}^{L}A_{p}{\rm
e}^{{\rm j}2\pi f_{k}(t-(\tau_{p}-at))}g((1+a)t-\tau_{p})\right\\}.$
After downconverting and low pass filtering, the baseband signal ${y}(t)$ can
be obtained as
$\displaystyle{y}(t)$ $\displaystyle={\rm LPF}\left[\tilde{y}(t){\rm e}^{-{\rm
j}2\pi f_{c}t}\right]$
$\displaystyle\approx\frac{1}{\sqrt{K}}\sum\limits_{k=-N/2}^{N/2-1}d[k]{\rm
e}^{{\rm j}2\pi k\Delta ft}{\rm e}^{{\rm j}2\pi
af_{k}t}\left[\sum\limits_{p=1}^{L}A_{p}{\rm e}^{-{\rm j}2\pi
f_{k}\tau_{p}}g((1+a)t-\tau_{p})\right]+n(t),$
where $n(t)$ is the additive noise in baseband. Note that each subcarrier
experiences a Doppler-induced frequency shift ${\rm e}^{{\rm j}2\pi af_{k}t}$.
Provided that the bandwidth of the OFDM signal is comparable to the center
frequency, the Doppler-induced frequency shift differs considerably on
different subcarriers, which introduces the intercarrier interference.
In order to mitigate the Doppler effect and convert the wideband OFDM system
into a narrowband system, a resampling approach is carried out first [4].
Suppose the Doppler factor $a$ is estimated as $\hat{a}$. Resample ${y}(t){\rm
e}^{-{\rm j}2\pi\hat{a}f_{c}t}$ with a resampling factor $\hat{a}$ to yield
$\displaystyle{z^{\prime}}(t)$ $\displaystyle={y}(\frac{t}{1+\hat{a}}){\rm
e}^{-{\rm j}2\pi\hat{a}f_{c}\frac{t}{1+\hat{a}}}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum\limits_{k=-N/2}^{N/2-1}d[k]{\rm e}^{{\rm
j}2\pi\frac{1+a}{1+\hat{a}}k\Delta ft}{\rm e}^{{\rm
j}2\pi\frac{a-\hat{a}}{1+\hat{a}}f_{c}t}\left[\sum\limits_{p=1}^{L}A_{p}{\rm
e}^{-{\rm j}2\pi
f_{k}\tau_{p}}g(\frac{1+a}{1+\hat{a}}t-\tau_{p})\right]+n(t){\rm e}^{{\rm
j}2\pi\frac{\hat{a}}{1+\hat{a}}f_{c}t},$ (1)
The resampling factor $\hat{a}$ is chosen to make $\frac{1+a}{1+\hat{a}}$ as
close to one as possible. As a result, (II) can be approximated as
$\displaystyle{z^{\prime}}(t)\approx\frac{1}{\sqrt{N}}{\rm e}^{{\rm
j}2\pi\frac{a-\hat{a}}{1+\hat{a}}f_{c}t}\sum\limits_{k=-N/2}^{N/2-1}d[k]{\rm
e}^{{\rm j}2\pi k\Delta ft}\left[\sum\limits_{p=1}^{L}A_{p}{\rm e}^{-{\rm
j}2\pi f_{k}\tau_{p}}g\left(t-\tau_{p}\right)\right]+v(t).$ (2)
From (2), usually CFO estimation and compensation are carried out as the
second step to further remove the impact of Doppler [4]. Let ${r}(t)$ denote
the output of the CFO compensation, sampling $r(t)$ at $(n-1)T_{s}=(n-1)/B$,
$n=1,\dots,N$ yields 111In engineering application, fftshift operator ${\rm
fftshift}(\cdot)$ is usually introduced and model (3) can also be formulated
as ${\mathbf{r}}={\mathbf{e}}(\omega)\odot\left({\mathbf{F}}^{\rm H}{\rm
fftshift}({\mathbf{h}}\odot{\mathbf{d}})\right)+{\mathbf{w}}$.
$\displaystyle{\rm e}^{{\rm
j}\pi(n-1)}{\mathbf{r}}={\mathbf{e}}(\omega)\odot\left({\mathbf{F}}^{\rm
H}({\mathbf{h}}\odot{\mathbf{d}})\right)+{\mathbf{w}},$ (3)
where ${r}_{n}=r(t)|_{t=(n-1)T_{s}}$; the $n$th entry $d_{n}$ of
${\mathbf{d}}$ is $d_{n}=d[n-1-N/2]$; ${\mathbf{F}}$ denotes the normalized
DFT matrix with the $(m,n)$th entry being $\frac{1}{\sqrt{N}}{\rm e}^{-{\rm
j}2\pi(n-1)(m-1)/N}$; $\omega$ denotes the normalized residual CFO which is
usually very small and close to $0$, ${\mathbf{e}}(\omega)=[1,{\rm e}^{{\rm
j}\omega},\cdots,{\rm e}^{{\rm j}(N-1)\omega}]^{\rm T}$; $\mathbf{h}$ is the
frequency-domain channel response vector and can be viewed as a line spectra,
which can be represented as
$\displaystyle{\mathbf{h}}={\mathbf{A}}({\bm{\theta}}){\bm{\beta}},$ (4)
where ${{\bm{\beta}}\in{\mathbb{C}}^{L}}$ denotes the complex coefficient
vector related to the path gains and delays of the channel and
$\displaystyle\beta_{p}=A_{p}{\rm e}^{-{\rm j}2\pi f_{-\frac{N}{2}}\tau_{p}},$
${{\bm{\theta}}\in{\mathbb{R}}^{L}}$ denotes the frequency component related
to the path time delay of the channel and
$\displaystyle\theta_{p}=-2\pi\Delta f\tau_{p},$
$L$ denotes the number of paths, ${\mathbf{A}}({\bm{\theta}})$ is
$\displaystyle{\mathbf{A}}({\bm{\theta}})=[{\mathbf{a}}(\theta_{1}),{\mathbf{a}}(\theta_{2}),\cdots,{\mathbf{a}}(\theta_{L})],$
and ${\mathbf{a}}(\theta)=[1,{\rm e}^{{\rm j}\theta},\cdots,{\rm e}^{{\rm
j}(N-1)\theta}]^{\rm T}$; $\mathbf{w}$ denotes the noise and approximation
error which is assumed to follow the Gaussian distribution, i.e.,
${\mathbf{w}}\sim{\mathcal{CN}}({\mathbf{0}},\sigma^{2}{\mathbf{I}}_{N})$ with
$\sigma^{2}$ being the unknown variance.
Define
$\displaystyle{y}_{n}={\rm e}^{{\rm j}\pi(n-1)}{z}_{n}.$
Then, (3) is simplified as
$\displaystyle{\mathbf{y}}={\mathbf{e}}(\omega)\odot\left({\mathbf{F}}^{\rm
H}({\mathbf{h}}\odot{\mathbf{d}})\right)+{\mathbf{w}}.$ (5)
Assume that the symbol vector $\mathbf{d}$ are partitioned into three parts
corresponding to the pilot, null, and data, which are represented as
${\mathbf{d}}_{{\mathcal{P}}}$, ${\mathbf{d}}_{{\mathcal{N}}}$ and
${\mathbf{d}}_{{\mathcal{D}}}$ and
${\mathcal{P}}\cup{\mathcal{N}}\cup{\mathcal{D}}=\\{1,2,\cdots,N\\}$. Then the
goal is to jointly estimate the data symbol ${\mathbf{d}}_{{\mathcal{D}}}$,
the channel vector $\mathbf{h}$, the normalized residual CFO $\omega$, and the
nuisance parameters such as $\sigma^{2}$ by fully exploiting the channel
structure $\mathbf{h}$. In the following, we design an approximate Bayesian
algorithm.
## III Probabilistic Formulation
This section describes the probabilistic formulation of the channel
$\mathbf{h}$, the data symbol $\mathbf{d}$ and the normalized residual CFO
$\omega$.
### III-A Parametric Channel Model
For the channel $\mathbf{h}$, the number of path $L$ is usually unknown. Here
the number of paths is assumed to be $L_{\rm max}$ and $L<L_{\rm max}$ [29],
i.e.,
$\displaystyle{\mathbf{h}}=\sum\limits_{l=1}^{L_{\rm
max}}{\beta}_{l}{\mathbf{a}}({\theta}_{l})\triangleq{\mathbf{A}}({\bm{\theta}}){\bm{\beta}},$
(6)
where
${\mathbf{A}}({\bm{\theta}})=[{\mathbf{a}}({\theta}_{1}),\cdots,{\mathbf{a}}({\theta}_{L_{\rm
max}})]$. Since the number of path is supposed to be $L_{\rm max}$, the binary
hidden variables ${\mathbf{s}}=[s_{1},...,s_{L_{\rm max}}]^{\rm T}$ are
introduced, where $s_{l}=1$ means that the $l$th path is active, otherwise
deactive ($\beta_{l}=0$). Let $\rho$ denote the prior probability of each path
being active, i.e.,
$\displaystyle p(s_{l})=\rho^{s_{l}}(1-\rho)^{(1-s_{l})},\quad
s_{l}\in\\{0,1\\}.$ (7)
Then the probability mass function (PMF) of ${\mathbf{s}}$ is
$p({\mathbf{s}})=\prod\limits_{l=1}^{L_{\rm max}}p(s_{l})$. Given that
$s_{l}=1$, we assume that ${\beta}_{l}\sim{\mathcal{CN}}({\beta}_{l};0,\nu)$,
where $\nu$ denotes the priori variance of the path being active. Thus
$(s_{l},{\beta}_{l})$ follows a Bernoulli-Gaussian distribution, that is
$\displaystyle
p({\beta}_{l}|s_{l};\nu)=(1-s_{l}){\delta}({\beta}_{l})+s_{l}{\mathcal{CN}}({\beta}_{l};0,\nu).$
(8)
In addition, the distribution of $\bm{\beta}$ conditioned on $\mathbf{s}$ is
$p({\bm{\beta}}|{\mathbf{s}})=\prod\limits_{l=1}^{L_{\rm
max}}p({\beta}_{l}|s_{l};\nu)$.
From (7) and (8), it can be seen that the parameter $\rho$ denotes the
probability of the $l$th component being active and $\nu$ is a variance
parameter. The variable ${\bm{\theta}}=[\theta_{1},...,\theta_{L_{\rm
max}}]^{\rm T}$ has the prior PDF
$p({\bm{\theta}})=\begin{matrix}\prod_{l=1}^{L_{\rm
max}}p(\theta_{l})\end{matrix}$. Generally, $p(\theta_{l})$ is encoded through
the von Mises distribution [30, p. 36]
$\displaystyle
p(\theta_{l})={\mathcal{VM}}(\theta_{l};\mu_{0,l},\kappa_{0,l})=\frac{1}{2\pi{I_{0}}(\kappa_{0,l})}{\rm
e}^{\kappa_{0,l}{\cos(\theta-\mu_{0,l})}},$ (9)
where $\mu_{0,l}$ and $\kappa_{0,l}$ are the mean direction and concentration
parameters of the prior of the $l$th frequency $\theta_{l}$, $I_{p}(\cdot)$ is
the modified Bessel function of the first kind and the order $p$ [30, p. 348].
For a given von Mises distribution, ${\rm arg}({\rm
E}_{{\mathcal{VM}}(\theta;\mu,\kappa)}[{\rm e}^{{\rm j}\theta}])={\rm
arg}\left({\rm e}^{{\rm
j}\mu}\frac{I_{1}(\kappa)}{I_{0}(\kappa)}\right)=\mu={\rm
E}_{{{\mathcal{VM}}(\theta;\mu,\kappa)}}[\theta]$. In addition, ${\rm E}[{\rm
e}^{{\rm j}m\theta}]={\rm e}^{{\rm j}m\mu}(I_{m}(\kappa)/I_{0}(\kappa))$ [30,
pp. 26]. Without any knowledge of the frequency $\theta_{l}$, the
uninformative prior distribution $p(\theta_{l})={1}/({2\pi})$ is used. For
more details please refer to [29].
### III-B OFDM System
A single-input single-output OFDM system with $N$ subcarriers is considered.
Since the structure of the consecutive OFDM symbols is not exploited, a single
OFDM system is modeled 222As you see, the proposed approach outputs the PDFs
of the amplitudes and delays of all the paths, which is very suitable for
channel tracking and will be left for future work.. Of the $N$ subcarriers,
$N_{\mathcal{P}}$ and $N_{\mathcal{N}}$ are dedicated as pilots and nulls,
respectively, and the remaining
$N_{\mathcal{D}}=N-N_{\mathcal{P}}-N_{\mathcal{N}}$ are used to transmit the
coded/interleaved data bits. The sets $\mathcal{N}$, $\mathcal{P}$,
$\mathcal{D}$ denote the indices of the null, pilot and data subcarriers,
respectively. It follows that
${\mathcal{P}}\cup{\mathcal{N}}\cup{\mathcal{D}}=\\{1,2,\cdots,N\\}$ and
${\mathcal{P}}\cap{\mathcal{N}}=\emptyset$,
${\mathcal{P}}\cap{\mathcal{D}}=\emptyset$ and
${\mathcal{N}}\cap{\mathcal{D}}=\emptyset$. Besides, the null subcarriers are
not necessary to delicate and can be set as the empty set
${\mathcal{N}}=\emptyset$ to improve the data transmission rate.
Let the vector ${\mathbf{u}}\in\\{0,1\\}^{K}$ be the transmitted (equi-
probable) information bits. These bits are coded by a rate-$R$ encoder and
interleaved to get the length-$K/R$ vector $c={\mathcal{C}}({\mathbf{u}})$,
where ${\mathcal{C}}:\\{0,1\\}^{K}\rightarrow\\{0,1\\}^{K/R}$ is the
interleaving and coding function such as a turbo code and a LDPC code. The
coded/interleaved bits are partioned into subvectors
${\mathbf{c}}^{(i)}\in\\{0,1\\}^{Q}$, $i\in{\mathcal{D}}$, which are then
mapped to the $i$th subcarrier. Let
${\mathcal{M}}:\\{0,1\\}^{Q}\rightarrow{\mathbb{A}}_{D}\subset{\mathbb{C}}$
denote the $2^{Q}$-ary mapping, where ${\mathbb{A}}_{D}$ is the symbol
alphabet. The pilots are drawn from the pilot symbol alphabet
${\mathbb{A}}_{P}\subset{\mathbb{C}}$. In OFDM, ${\mathbb{A}}_{D}$ is
typically a $2^{Q}$-ary quadrature amplitude modulation (QAM) alphabet and
${\mathbb{A}}_{P}$ is a quadrature phase shift keying (QPSK) alphabet. The
nulls, pilots and data symbols are stacked in vector $\mathbf{d}$. Vectors
${\mathbf{d}}_{\mathcal{N}}$, ${\mathbf{d}}_{\mathcal{P}}$ and
${\mathbf{d}}_{\mathcal{D}}$ contain the nulls, pilot symbol and data symbols,
respectively.
For the OFDM symbol, the information bits ${\mathbf{u}}$ has the following PDF
$\displaystyle
p({\mathbf{u}})=\prod\limits_{k\in{\mathcal{K}}}p(u_{k})=\prod\limits_{k\in{\mathcal{K}}}\left(0.5\times{1}_{[u_{k}\in\\{0,1\\}]}\right),$
where ${1}_{[\cdot]}$ denotes the indicator function, ${\mathcal{K}}$ denotes
the index set of the information bits. For the interleaving and coding
function, the PMFs $p({\mathbf{c}}|{\mathbf{u}})$ of subvectors ${\mathbf{c}}$
conditioned on $\mathbf{u}$ are
$\displaystyle
p({\mathbf{c}}|{\mathbf{u}})={1}_{[{\mathbf{c}}=C({\mathbf{u}})]}.$
Note that $p({\mathbf{c}}|{\mathbf{u}})$ describe the structure of the channel
code and interleaver, whose details are not provided. Conditioned on
$\mathbf{c}$, the PMF of $\mathbf{d}$ is $p({\mathbf{d}}|{\mathbf{c}})$.
The normalized residual CFO $\omega$ is treated as an unknown deterministic
parameter and $\omega\in[-\pi,\pi]$.
Figure 1: Factor graph of the joint PDF (10) describing the parametric channel
and OFDM system. Here the circle denotes the variable (vector) node, and the
square denotes the factor node. The nulls are not plotted here for simplicity,
as it can only be used for noise variance estimation.
According to the above probabilistic model, the factor graph representation
describing the channel and OFDM system model is given in Fig. 1. The joint PDF
is
$\displaystyle
p({\mathbf{y}},{\mathbf{d}}_{{\mathcal{D}}},{\mathbf{c}},{\mathbf{u}},{\bm{\beta}},{\bm{\theta}},{\mathbf{s}},;\omega,\nu,\rho,\sigma^{2})=p({\mathbf{s}};\rho)p({\bm{\beta}}|{\mathbf{s}};\nu)p({\bm{\theta}})p({\mathbf{u}})p({\mathbf{c}}|{\mathbf{u}})p({\mathbf{d}}|{\mathbf{c}})p({\mathbf{y}}|{\bm{\theta}},{\bm{\beta}},{\mathbf{d}}_{{\mathcal{D}}};\omega,\sigma^{2}).$
(10)
Firstly, the maximum likelihood estimation of the nuisance parameters are
$\displaystyle(\omega_{\rm{ML}},\nu_{\rm{ML}},\rho_{\rm{ML}},\sigma_{\rm{ML}}^{2})$
$\displaystyle=\underset{\omega,\nu,\rho,\sigma^{2}}{\operatorname{argmax}}\quad
p({\mathbf{y}};\omega,\nu,\rho,\sigma^{2})$ $\displaystyle=\int
p({\mathbf{y}},{\mathbf{d}}_{{\mathcal{D}}},{\mathbf{c}},{\mathbf{u}},{\bm{\beta}},{\bm{\theta}},{\mathbf{s}},;\omega,\nu,\rho,\sigma^{2}){\rm
d}{\mathbf{d}}_{{\mathcal{D}}}{\rm d}{\bm{\beta}}{\rm d}{\mathbf{u}}{\rm
d}{\mathbf{c}}{\rm d}{\bm{\theta}}{\rm d}{\mathbf{s}}.$ (11)
Then the BER optimal receiver, i.e., maximum a posterior (MAP) estimate is
$\displaystyle\hat{u}_{k}=\underset{u_{k}\in\\{0,1\\}}{\operatorname{argmax}}~{}p(u_{k}|{\mathbf{y}};\omega_{\rm{ML}},\nu_{\rm{ML}},\rho_{\rm{ML}},\sigma_{\rm{ML}}^{2}),$
(12)
where
$p(u_{k}|{\mathbf{y}};\omega_{\rm{ML}},\nu_{\rm{ML}},\rho_{\rm{ML}},\sigma_{\rm{ML}}^{2})\propto
p(u_{k},{\mathbf{y}};\omega_{\rm{ML}},\nu_{\rm{ML}},\rho_{\rm{ML}},\sigma_{\rm{ML}}^{2})$
can be obtained via marginalizing all the random variables but $u_{k}$ in the
joint PDF
$p({\mathbf{y}},{\mathbf{d}}_{{\mathcal{D}}},{\mathbf{c}},{\mathbf{u}},{\bm{\beta}},{\bm{\theta}},{\mathbf{s}},;\omega_{\rm{ML}},\nu_{\rm{ML}},\rho_{\rm{ML}},\sigma_{\rm{ML}}^{2})$.
Note that solving either (III-B) or (12) is computationally intractable.
Therefore, we resort to approximate Bayesian methods in the ensuing section.
## IV Algorithm
This section utilizes expectation propagation (EP) [34] to develop the JCCD-
VALSE algorithm through novelly combining the gridless VALSE, the MMSE module
and the LDPC decoder module.
Figure 2: Factor graph of the joint PDF (10) with an additional hidden
variable node $\mathbf{h}$ and the module of the JCCD-VALSE algorithm. This
factor graph is equivalent to the factor graph shown in Fig. 1.
It is worth noting that in conventional underwater OFDM systems, null
subcarriers are beneficial for both Doppler and noise estimation. While in
this work, the pilot and data can be used for both Doppler and noise
estimation. As a consequence, the null subcarriers can be replaced with the
data subcarriers, which improves the spectrum efficiency. Define the ordered
set ${\mathcal{M}}={\mathcal{P}}\cup{\mathcal{D}}$.
Before designing the algorithm, an important point is emphasized. Note that
$\displaystyle p({\mathbf{y}}|{\mathbf{h}},{\mathbf{d}};\omega,\sigma^{2})$
$\displaystyle=\frac{1}{(\pi\sigma^{2})^{N}}{\rm
e}^{-\frac{\left({\mathbf{y}}-{\mathbf{e}}(\omega)\odot\left({\mathbf{F}}^{\rm
H}({\mathbf{h}}\odot{\mathbf{d}})\right)\right)^{\rm
H}\left({\mathbf{y}}-{\mathbf{e}}(\omega)\odot\left({\mathbf{F}}^{\rm
H}({\mathbf{h}}\odot{\mathbf{d}})\right)\right)}{\sigma^{2}}}$
$\displaystyle=\frac{1}{(\pi\sigma^{2})^{N}}{\rm
e}^{-\frac{\|\tilde{\mathbf{y}}(\omega)-{\mathbf{h}}\odot{\mathbf{d}}\|_{2}^{2}}{\sigma^{2}}}$
$\displaystyle=\prod\limits_{i=1}^{N}p(\tilde{y}_{i}(\omega)|{h}_{i},{d}_{i};\omega,\sigma^{2}),$
(13)
where $\tilde{\mathbf{y}}(\omega)$ is
$\displaystyle\tilde{\mathbf{y}}(\omega)={\mathbf{F}}\left({\mathbf{y}}\odot{\mathbf{e}}(-\omega)\right).$
(14)
This means that
$\displaystyle\tilde{\mathbf{y}}(\omega)={\mathbf{h}}\odot{\mathbf{d}}+\tilde{\mathbf{w}}$
(15)
is equivalent to the original observation model (5), where
$\tilde{\mathbf{w}}\sim{\mathcal{CN}}({\mathbf{0}},\sigma^{2}{\mathbf{I}})$.
It is beneficial to introduce an additional hidden variable $\mathbf{h}$
defined in (4), and an equivalent factor shown in Fig. 2 (a) can be obtained.
Compared to the original factor graph in Fig. 1, a delta factor node [28] is
introduced, which is the key to decompose the original problem into
subproblems to be solved by designing the respective modules. According to
Fig. 2 (b), the JCCD-VALSE can be summarized as follows: Firstly, initialize
the extrinsic message $m_{{\rm A}\rightarrow{\rm
B}}({\mathbf{h}}_{\mathcal{M}})$ from module A to module B as
$\displaystyle m_{{\rm A}\rightarrow{\rm
B}}({\mathbf{h}}_{{\mathcal{M}}})={\mathcal{CN}}({\mathbf{h}}_{{\mathcal{M}}};{\mathbf{h}}_{{\rm
A},{\mathcal{M}}}^{\rm ext},{\rm diag}({\mathbf{v}}_{{\rm
A},{\mathbf{h}}_{{\mathcal{M}}}}^{\rm ext})),$ (16)
and calculate the extrinsic message $m_{{\rm B}\rightarrow{\rm
C}}({\mathbf{d}}_{\mathcal{D}})$ from module B to module C. Secondly, running
the LDPC decoder algorithm in module C to obtain the posterior means and
variances of ${\mathbf{d}}_{\mathcal{D}}$. Thirdly, update the residual CFO
and noise variance estimate in module B. Fourthly, calculate the extrinsic
message $m_{{\rm B}\rightarrow{\rm A}}({\mathbf{h}}_{\mathcal{M}})$ from
module B to module A. Finally, calculate the extrinsic message $m_{{\rm
A}\rightarrow{\rm B}}({\mathbf{h}}_{\mathcal{M}})$ from module A to module B,
which closes the algorithm. The details of the above steps are described
below.
### IV-A Calculate the Message $m_{{\rm B}\rightarrow{\rm
C}}({\mathbf{d}}_{\mathcal{D}})$
According to the belief propagation (BP), the message $m_{{\rm
B}\rightarrow{\rm C}}({\mathbf{d}}_{\mathcal{D}})$ is
$\displaystyle m_{{\rm B}\rightarrow{\rm C}}({\mathbf{d}}_{\mathcal{D}})$
$\displaystyle\propto\int m_{{\rm A}\rightarrow{\rm
B}}({\mathbf{h}})p({\mathbf{y}}|{\mathbf{h}},{\mathbf{d}}_{\mathcal{D}};\omega,\sigma^{2}){\rm
d}{\mathbf{h}}$
$\displaystyle=p({\mathbf{y}}|{\mathbf{d}}_{\mathcal{D}};\omega,\sigma^{2})=\prod\limits_{i\in{\mathcal{D}}}{\mathcal{CN}}(\tilde{y}_{i}(\omega);h_{{\rm
A},i}^{\rm ext}d_{i},|d_{i}|^{2}v_{{\rm A},{h}_{i}}^{\rm ext}).$ (17)
For each $i\in\mathcal{D}$, $d_{i}$ is drawn from a finite alphabet set, and
the PMF $m_{{\rm B}\rightarrow{\rm C}}({\mathbf{d}}_{\mathcal{D}})$ can be
obtained by evaluating the above Gaussian PDF at the points of the symbol
alphabet ${\mathbb{A}}_{D}$ followed by normalization.
### IV-B Calculate the Posterior Means and Variances of
${\mathbf{d}}_{\mathcal{D}}$ in LDPC Module
The message $m_{{\rm B}\rightarrow{\rm C}}({\mathbf{d}}_{\mathcal{D}})$ is
input to the LDPC module, where the SPA is applied. After several iterations
where the change of the loglikelihood ratio of the bits is less than a
threshold or the number of iterations exceeds the setting maximum, the beliefs
of the data symbols are obtained. These beliefs are further used to calculate
the posterior means $d_{{\rm C},i}^{\rm post}$ and variances $v_{{\rm
C},d_{i}}^{\rm post}$ of the data.
### IV-C Residual CFO and Noise Variance Estimation
The type II approach [31] and the EM algorithm are used to estimate the
normalized residual CFO $\omega$ and noise variance $\sigma^{2}$,
respectively. The type-II method is also termed as type-II maximum likelihood
(ML) method [32], which aims to to maximize the marginal likelihood in
Bayesian models. To estimate the residual CFO via type II approach, the
marginal likelihood $p(\tilde{\mathbf{y}}(\omega);\omega)$ is obtained. Define
$\displaystyle{\mathbf{z}}={\mathbf{h}}\odot{\mathbf{d}}.$ (18)
and model (19) can be formulated as
$\displaystyle\tilde{\mathbf{y}}(\omega)={\mathbf{z}}+\tilde{\mathbf{w}},$
(19)
where $\tilde{\mathbf{w}}\triangleq{\mathbf{F}}^{\rm
H}({\mathbf{w}}\odot{\mathbf{e}}(-\omega))\sim{\mathcal{CN}}({\mathbf{0}},\sigma^{2}{\mathbf{I}})$.
Straightforward calculation shows that the posterior means ${\mathbf{z}}_{\rm
B}^{\rm post}$ and variances ${\mathbf{v}}_{{\rm B},{\mathbf{z}}}^{\rm post}$
of $\mathbf{z}$ are
$\displaystyle{\mathbf{z}}_{\rm B}^{\rm post}$
$\displaystyle={\mathbf{h}}_{\rm A}^{\rm post}\odot{\mathbf{d}}_{\rm C}^{\rm
post},$ (20a) $\displaystyle{\mathbf{v}}_{{\rm B},{\mathbf{z}}}^{\rm post}$
$\displaystyle=|{\mathbf{d}}_{\rm C}^{\rm post}|^{2}\odot{\mathbf{v}}_{{\rm
A},{\mathbf{h}}}^{\rm post}+|{\mathbf{h}}_{\rm A}^{\rm
post}|^{2}\odot{\mathbf{v}}_{{\rm C},{\mathbf{d}}}^{\rm
post}+{\mathbf{v}}_{{\rm A},{\mathbf{h}}}^{\rm post}\odot{\mathbf{v}}_{{\rm
C},{\mathbf{d}}}^{\rm post},$ (20b)
where the posterior means and variances of $\mathbf{h}$ are obtained in module
A, i.e., the VALSE module, see Subsection IV-E (eq. (53a) and eq. (53b)).
Suppose that $\mathbf{z}$ follows Gaussian distribution with diagonal
covariance matrix and independent of $\tilde{\mathbf{w}}$, i.e.,
${\mathbf{z}}\sim{\mathcal{CN}}({\mathbf{z}};{\mathbf{z}}_{\rm B}^{\rm
post},{\rm diag}({\mathbf{v}}_{{\rm B},{\mathbf{z}}}^{\rm post}))$. For the
pseudo measurement $\tilde{\mathbf{y}}(\omega)$, it follows
$\tilde{\mathbf{y}}(\omega)\sim{\mathcal{CN}}(\tilde{\mathbf{y}}(\omega);{\mathbf{z}}_{\rm
B}^{\rm post},{\rm diag}({\mathbf{v}}_{{\rm B},{\mathbf{z}}}^{\rm
post})+\sigma^{2}{\mathbf{I}})$. The type II ML estimation problem can be
formulated as
$\displaystyle\hat{\omega}=\underset{\omega}{\operatorname{argmin}}~{}(\tilde{\mathbf{y}}(\omega)-{\mathbf{z}}_{\rm
B}^{\rm post})^{\rm H}({\rm diag}({\mathbf{v}}_{{\rm B},{\mathbf{z}}}^{\rm
post})+\sigma^{2}{\mathbf{I}})^{-1}(\tilde{\mathbf{y}}(\omega)-{\mathbf{z}}_{\rm
B}^{\rm post})\triangleq g(\omega).$ (21)
One can apply the Newston step to refine the previous estimate
$\hat{\omega}_{g}^{{}^{\prime}}$ as
$\displaystyle\hat{\omega}_{g}=\hat{\omega}_{g}^{{}^{\prime}}-\frac{\partial
g(\omega)}{\partial\omega}/\frac{\partial^{2}g(\omega)}{\partial\omega^{2}}\Big{|}_{\omega=\hat{\omega}_{g}^{{}^{\prime}}},$
(22)
where
$\displaystyle\frac{\partial
g(\omega)}{\partial\omega}=2\Re\left\\{\left(\frac{\partial\tilde{\mathbf{y}}(\omega)}{\partial\omega}\right)^{\rm
H}({\rm diag}({\mathbf{v}}_{{\rm B},{\mathbf{z}}}^{\rm
post})+\sigma^{2}{\mathbf{I}})^{-1}(\tilde{\mathbf{y}}(\omega)-{\mathbf{z}}_{\rm
B}^{\rm post})\right\\}$ (23)
and
$\displaystyle\frac{\partial^{2}g(\omega)}{\partial\omega^{2}}$
$\displaystyle=2\Re\left\\{\left(\frac{\partial^{2}\tilde{\mathbf{y}}(\omega)}{\partial\omega^{2}}\right)^{\rm
H}({\rm diag}({\mathbf{v}}_{{\rm B},{\mathbf{z}}}^{\rm
post})+\sigma^{2}{\mathbf{I}})^{-1}(\tilde{\mathbf{y}}(\omega)-{\mathbf{z}}_{\rm
B}^{\rm post})\right\\}$
$\displaystyle+2\left(\frac{\partial\tilde{\mathbf{y}}(\omega)}{\partial\omega}\right)^{\rm
H}({\rm diag}({\mathbf{v}}_{{\rm B},{\mathbf{z}}}^{\rm
post})+\sigma^{2}{\mathbf{I}})^{-1}\frac{\partial\tilde{\mathbf{y}}(\omega)}{\partial\omega},$
(24)
where
$\displaystyle\frac{\partial\tilde{\mathbf{y}}(\omega)}{\partial\omega}={\mathbf{F}}\left({\mathbf{y}}\odot{\bm{\chi}}\odot{\mathbf{e}}(-\omega)\right)$
(25)
and
$\displaystyle\frac{\partial^{2}\tilde{\mathbf{y}}(\omega)}{\partial\omega^{2}}={\mathbf{F}}\left({\mathbf{y}}\odot{\bm{\chi}}^{2}\odot{\mathbf{e}}(-\omega)\right),$
(26)
where ${\bm{\chi}}=-{\rm j}[0,1,\cdots,N-1]^{\rm T}$. It is worth noting that
the residual CFO estimation approach (21) is very general and generalize the
null subcarriers based approach in [4]. By letting ${\mathbf{v}}_{{\rm
B},z_{i}}^{\rm post}\rightarrow\infty$, $i\in{\mathcal{P}}\cup{\mathcal{D}}$,
i.e., the posterior variances of $\mathbf{z}$ corresponding to pilot and data
subcarriers tend to infinity, (21) reduces to the null subcarriers based
approach [4]. Since we utilize more data to estimate the residual CFO, it
makes sense that our approach estimates the residual CFO more accurately,
leading to better channel estimation and data decoding performance, as
illustrated in the numerical experiments.
Once the residual CFO $\omega$ is updated, the pseudo measurements
$\tilde{\mathbf{y}}(\omega)$ are refined according to (14).
The EM approach is adopted to estimate the noise variance, which is
$\displaystyle\hat{\sigma}^{2}=\underset{{\sigma}^{2}}{\operatorname{argmax}}~{}{\rm
E}_{{\mathbf{z}}}\left[\ln
p(\tilde{\mathbf{y}}(\omega);{\mathbf{z}},\sigma^{2})\right]=-\frac{{\rm
E}_{{\mathbf{z}}}\|\tilde{\mathbf{y}}(\omega)-{\mathbf{z}}\|_{2}^{2}}{\sigma^{2}}-N\ln\sigma^{2}+{\rm
const},$ (27)
where the expectation is taken with respect to the posterior PDF of
${\mathbf{z}}$.
Substituting (20) in (27), one obtains
$\displaystyle\hat{\sigma}^{2}=\underset{{\sigma}^{2}}{\operatorname{argmax}}~{}-\frac{\|\tilde{\mathbf{y}}(\omega)-{\mathbf{z}}_{\rm
B}^{\rm post}\|_{2}^{2}+{\mathbf{1}}^{\rm T}{\mathbf{v}}_{{\rm
B},{\mathbf{z}}}^{\rm post}}{\sigma^{2}}-N\ln\sigma^{2}.$ (28)
Setting the objective function of (28) with respect to $\sigma^{2}$ to zero,
the noise variance is estimated as
$\displaystyle\hat{\sigma}^{2}=\frac{\|\tilde{\mathbf{y}}(\hat{\omega}_{g})-{\mathbf{z}}_{\rm
B}^{\rm post}\|^{2}+{\mathbf{1}}^{\rm T}{\mathbf{v}}_{{\rm
B},{\mathbf{z}}}^{\rm post}}{N}.$ (29)
### IV-D Calculate the Extrinsic Message $m_{{\rm B}\rightarrow{\rm
A}}({\mathbf{h}}_{\mathcal{M}})$ from Module B to Module A
According to variational message passing [33], the message $m_{{\rm
B}\rightarrow{\rm A}}({\mathbf{h}}_{\mathcal{M}})$ can be calculated as
$\displaystyle\ln m_{{\rm B}\rightarrow{\rm A}}({\mathbf{h}}_{\mathcal{M}})$
$\displaystyle={\rm E}_{\mathbf{d}}\left[\ln
p({\mathbf{y}}|{\mathbf{h}},{\mathbf{d}})\right]+{\rm const}$
$\displaystyle={\rm E}_{\mathbf{d}}\left[\ln
p(\tilde{\mathbf{y}}(\omega)|{\mathbf{h}},{\mathbf{d}})\right]+{\rm const}$
$\displaystyle={\rm
E}_{\mathbf{d}}\left[-\frac{|\tilde{\mathbf{y}}(\omega)-{\mathbf{h}}\odot{\mathbf{d}}|^{2}}{\sigma^{2}}\right]+{\rm
const},$ (30)
where ${\rm const}$ is to ensure that the PDF $m_{{\rm B}\rightarrow{\rm
A}}({\mathbf{h}}_{\mathcal{M}})$ is normalized and ${\rm
E}_{\mathbf{d}}[\cdot]$ denotes the expectation of $\mathbf{d}$ with respect
to the posterior PDF. It can be seen that $m_{{\rm B}\rightarrow{\rm
A}}({\mathbf{h}}_{\mathcal{M}})$ is Gaussian distributed.
From (IV-D), one has $m_{{\rm B}\rightarrow{\rm
A}}({\mathbf{h}}_{\mathcal{M}})=\prod\limits_{i\in{\mathcal{M}}}m_{{\rm
B}\rightarrow{\rm A}}({h}_{i})$. Now we calculate the message $m_{{\rm
B}\rightarrow{\rm A}}({h}_{i})$ corresponding to the pilots and data
subcarriers, respectively.
For $i\in{\mathcal{P}}$, straightforward calculation shows that
$\displaystyle m_{{\rm B}\rightarrow{\rm
A}}({h}_{i})={\mathcal{CN}}(h_{i};h_{{\rm B},i}^{\rm ext},v_{{\rm
B},h_{i}}^{\rm ext}),$ (31)
where
$\displaystyle h_{{\rm B},i}^{\rm ext}=\tilde{y}_{i}(\omega)/d_{i},\quad
v_{{\rm B},h_{i}}^{\rm ext}=\sigma^{2}/|{d}_{i}|^{2},i\in{\mathcal{P}}.$ (32)
For the data symbols where $i\in{\mathcal{D}}$,
$\displaystyle\ln m_{{\rm B}\rightarrow{\rm A}}({h}_{i})={\rm
E}_{d_{i}}\left[\ln
p(\tilde{\mathbf{y}}(\omega)|{\mathbf{h}},{\mathbf{d}})\right]+{\rm const}$
$\displaystyle={\rm
E}_{\mathbf{d}}\left[-\frac{|\tilde{y}_{i}(\omega)-{h_{i}}d_{i}|^{2}}{\sigma^{2}}\right]+{\rm
const}=-{\rm
E}[|d_{i}|^{2}]\frac{|h|_{i}^{2}}{\sigma^{2}}+2\frac{\Re\\{\tilde{y}_{i}^{*}(\omega)h_{i}{\rm
E}[d_{i}]\\}}{\sigma^{2}}+{\rm const}$ $\displaystyle=-\frac{(|d_{{\rm
C},i}^{\rm post}|^{2}+v_{{\rm C},d_{i}}^{\rm
post})|h|_{i}^{2}}{\sigma^{2}}+2\frac{\Re\\{\tilde{y}_{i}^{*}(\omega)h_{i}d_{{\rm
C},i}^{\rm post}\\}}{\sigma^{2}}+{\rm const}.$ (33)
As a consequence, one has
$\displaystyle h_{{\rm B},i}^{\rm ext}$
$\displaystyle=\frac{\tilde{y}_{i}(\omega)d_{i}^{*}}{|d_{{\rm C},i}^{\rm
post}|^{2}+v_{{\rm C},d_{i}}^{\rm post}},$ (34a) $\displaystyle v_{{\rm
B},h_{i}}^{\rm ext}$ $\displaystyle=\sigma^{2}/(|d_{{\rm C},i}^{\rm
post}|^{2}+v_{{\rm C},d_{i}}^{\rm post}),i\in{\mathcal{D}}.$ (34b)
Note that if $v_{{\rm C},d_{i}}^{\rm post}=0$, (34) is consistent with (32).
### IV-E Calculate the message $m_{{\rm A}\rightarrow{\rm
B}}({\mathbf{h}}_{\mathcal{M}})$
Define ${\mathbf{A}}_{\mathcal{M}}$ as the matrix chosen from the rows of
$\mathbf{A}$ indexed by ${\mathcal{M}}$. The pseudo linear measurement model
under heterogenous noise is
$\displaystyle\widetilde{\mathbf{h}}_{\mathcal{M}}={\mathbf{A}}_{\mathcal{M}}({\bm{\theta}}){\bm{\beta}}+\widetilde{\bm{\epsilon}}_{\mathcal{M}},$
(35)
where $\widetilde{\mathbf{h}}_{\mathcal{M}}={\mathbf{h}}_{{\rm
B},{\mathcal{M}}}^{\rm ext}$,
$\widetilde{\bm{\epsilon}}_{\mathcal{M}}\sim{\mathcal{CN}}({\mathbf{0}},{\rm
diag}(\tilde{\bm{\sigma}}_{\mathcal{M}}^{2}))$ and
$\widetilde{\bm{\sigma}}_{\mathcal{M}}^{2}={\mathbf{v}}_{{\rm
B},{\mathbf{h}}_{\mathcal{M}}}^{\rm ext}$.
The VALSE algorithm tries to construct a structured PDF
$q({\bm{\theta}},{\bm{\beta}},{\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})$
to approximate the true PDF
$p({\bm{\theta}},{\bm{\beta}},{\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})$
by minimizing their Kullback-Leibler (KL) divergence
${\rm{KL}}\left(q({\bm{\theta}},{\bm{\beta}},{\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})||p({\bm{\theta}},{\bm{\beta}},{\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})\right)$.
For
$q({\bm{\theta}},{\bm{\beta}},{\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})$,
it is supposed to be factored as
$\displaystyle
q({\bm{\theta}},{\bm{\beta}},{\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})=\prod_{i=1}^{L_{\rm
max}}q(\theta_{i}|\widetilde{\mathbf{h}}_{\mathcal{M}})q({\bm{\beta}}|\widetilde{\mathbf{h}}_{\mathcal{M}},{\mathbf{s}})q({\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}}),$
(36)
where
$q({\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})=\delta({\mathbf{s}}-\hat{\mathbf{s}})$
is restricted to a point estimate. Given that VALSE outputs the approximated
PDF
$q({\bm{\theta}},{\bm{\beta}},{\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})$,
according to EP [34], the extrinsic message $m_{{\rm A}\rightarrow{\rm
B}}({\mathbf{h}}_{\mathcal{M}})={\mathcal{CN}}({\mathbf{h}}_{\mathcal{M}};{\mathbf{h}}_{{\rm
A},{\mathcal{M}}}^{\rm ext},{\rm diag}({\mathbf{v}}_{{\rm
A},{\mathbf{h}}_{\mathcal{M}}}^{\rm ext}))$ of ${\mathbf{h}}_{\mathcal{M}}$
from module A to module B is
$\displaystyle m_{{\rm A}\rightarrow{\rm B}}({\mathbf{h}}_{\mathcal{M}})$
$\displaystyle\propto\frac{{\rm Proj}[\int
q({\bm{\beta}}_{\hat{S}}|\widetilde{\mathbf{h}}_{\mathcal{M}})\delta({\mathbf{h}}_{\mathcal{M}}-{\mathbf{A}}_{{\mathcal{M}},\hat{S}}({\bm{\theta}}){\bm{\beta}}_{\hat{S}})q({\bm{\theta}}|\widetilde{\mathbf{h}}_{\mathcal{M}}){\rm
d}{\mathbf{w}}_{\hat{S}}{\rm d}{\bm{\theta}}]}{m_{{\rm B}\rightarrow{\rm
A}}({\mathbf{h}}_{\mathcal{M}})}$ $\displaystyle\triangleq\frac{{\rm
Proj}[q_{\rm A}({\mathbf{h}}_{\mathcal{M}})]}{m_{{\rm B}\rightarrow{\rm
A}}({\mathbf{h}}_{\mathcal{M}})},$ (37)
where ${\rm Proj}[q({\mathbf{x}})]$ denotes the projection operation which
approximates the target distribution $q({\mathbf{x}})$ as Gaussian
distribution with moment matching, i.e., the means and variances of
$q({\mathbf{x}})$ matches with that of the approximated Gaussian distribution.
Let ${\mathbf{h}}_{{\rm A},{\mathcal{M}}}^{\rm post}$ and ${\mathbf{v}}_{{\rm
B},{\mathbf{h}}_{\mathcal{M}}}^{\rm post}$ denote the posterior means and
variances of ${\mathbf{h}}_{\mathcal{M}}$ with respect to $q_{\rm
A}({\mathbf{h}}_{\mathcal{M}})$, then
$\displaystyle{\rm Proj}[q_{\rm
A}({\mathbf{h}}_{\mathcal{M}})]={\mathcal{CN}}({\mathbf{h}}_{\mathcal{M}};{\mathbf{h}}_{{\rm
A},{\mathcal{M}}}^{\rm post},{\rm diag}({\mathbf{v}}_{{\rm
A},{\mathbf{h}}_{\mathcal{M}}}^{\rm post})).$ (38)
From (IV-E), ${\mathbf{h}}_{{\rm A},{\mathcal{M}}}^{\rm ext}$ and
${\mathbf{v}}_{{\rm A},{\mathbf{h}}_{\mathcal{M}}}^{\rm ext}$ can be
calculated as
$\displaystyle{\mathbf{v}}_{{\rm A},{\mathbf{h}}_{\mathcal{M}}}^{\rm
ext}=\left(\frac{1}{{\mathbf{v}}_{{\rm A},{\mathbf{h}}_{\mathcal{M}}}^{\rm
post}}-\frac{1}{{\mathbf{v}}_{{\rm A},{\mathbf{h}}_{\mathcal{M}}}^{\rm
ext}}\right)^{-1},$ (39a) $\displaystyle{\mathbf{h}}_{{\rm
A},{\mathcal{M}}}^{\rm ext}={\mathbf{v}}_{{\rm
A},{\mathbf{h}}_{\mathcal{M}}}^{\rm ext}\odot\left(\frac{{\mathbf{h}}_{{\rm
A},{\mathcal{M}}}^{\rm post}}{{\mathbf{v}}_{{\rm
A},{\mathbf{h}}_{\mathcal{M}}}^{\rm post}}-\frac{{\mathbf{h}}_{{\rm
B},{\mathcal{M}}}^{\rm ext}}{{\mathbf{v}}_{{\rm
B},{\mathbf{h}}_{\mathcal{M}}}^{\rm ext}}\right),$ (39b)
where $\odot$ denotes the Hadarmard product. In the follow, the details of
obtaining the approximated posterior PDFs through VALSE is briefly described
333For further details, please refer to [29, 35]..
The VALSE proceeds as follows: Firstly, the frequencies
$\\{\theta_{i}\\},i\in\hat{S}$ are inferred and the von Mises approximations
$q(\theta_{i}|\widetilde{\mathbf{h}}_{\mathcal{M}}),i\in\hat{S}$ of the PDFs
are obtained. Secondly, the weights $\bm{\beta}$ and support $\mathbf{s}$ are
inferred and the posterior PDFs
$q({\bm{\beta}}|\widetilde{\mathbf{h}}_{\mathcal{M}})$ of the weight and the
PMF $q({\mathbf{s}}|\widetilde{\mathbf{h}}_{\mathcal{M}})$ of the support are
obtained. Finally, the model parameters $\rho$ and $\nu$ are estimated. In the
following, we detail the procedures.
#### IV-E1 Inferring the frequencies
Let $\mathcal{S}$ be the set of indices of the non-zero components of
$\mathbf{s}$, i.e.,
$\displaystyle\mathcal{S}=\\{i|1\leq i\leq L_{\rm max},s_{i}=1\\}.$
Analogously, define $\widehat{\mathcal{S}}$ based on $\widehat{\mathbf{s}}$.
For $i\notin{\mathcal{S}}$, we have
$q(\theta_{i}|\widetilde{\mathbf{h}}_{\mathcal{M}})=p(\theta_{i})$. For
$i\in{\mathcal{S}}$, $q(\theta_{i}|\widetilde{\mathbf{h}}_{\mathcal{M}})$ is
$\displaystyle q(\theta_{i}|\widetilde{\mathbf{h}}_{\mathcal{M}})\propto
p(\theta_{i}){\rm exp}(\Re\\{{\bm{\eta}}_{i}^{\rm
H}{\mathbf{a}}(\theta_{i})\\})\triangleq\exp\\{f(\theta_{i})\\},$ (40)
where the complex vector $\bm{\eta}_{i}$ is given by
$\displaystyle{\bm{\eta}}_{i}=2\left[\left(\widetilde{\mathbf{h}}_{\mathcal{M}}-\sum_{l\in\widehat{\mathcal{S}}\backslash\\{i\\}}{\widehat{\mathbf{a}}}_{{\mathcal{M}},l}{\widehat{\beta}}_{l}\right){\widehat{\beta}}^{*}_{i}-\sum_{l\in\widehat{\mathcal{S}}\backslash\\{i\\}}{\widehat{\mathbf{a}}}_{{\mathcal{M}},l}{\widehat{C}}_{l,i}\right]/\tilde{\bm{\sigma}}_{\mathcal{M}}^{2},$
(41)
where ${\widehat{\mathbf{a}}}_{{\mathcal{M}},l}$ denotes the $l$th column of
${\widehat{\mathbf{A}}}_{{\mathcal{M}}}$, $``\backslash i"$ denotes the
indices $\widehat{\mathcal{S}}$ excluding $i$,
$\hat{\bm{\beta}}_{\widehat{\mathcal{S}}}$ denotes the subvector of
$\hat{\bm{\beta}}$ by choosing the $\widehat{\mathcal{S}}$ rows of
${\bm{\beta}}$. Since it is hard to obtain the analytical results of the
expected value of ${\mathbf{a}}_{\mathcal{M}}(\theta_{i})$ for the PDF (40),
$q(\theta_{i}|\tilde{\mathbf{h}})$ is approximated as a von Mises distribution
via the following two steps 444For further details, please refer to [29,
Algorithm 2: Heurestic 2].: Firstly, search for the most dominant component of
(40) via Heuristic 2 [29, Algorithm 2]. Secondly, use second-order Taylor
approximation to approximate the dominant component as a single von Mises PDF.
It is worth noting that, the number of subcarriers $N$ is usually very large,
for example $N=1024$, thus the peak of the single von Mises PDF should be well
matched to that of (40) in very high accuracy. Let $\bar{\theta}_{i}$ denote
the mean of the dominant component. In contrast with [29] where a single
Newston refinement is adopted, we apply a two step procedure to improve the
robustness. First, apply the Newston step to refine the estimate
$\displaystyle\hat{\mu}_{i}=\bar{\theta}_{i}-\frac{f^{{}^{\prime}}(\bar{\theta}_{i})}{f^{{}^{\prime\prime}}(\bar{\theta}_{i})},$
(42)
where $f(\theta_{i})$ is defined in (40). Since the Newston step may be too
large to across the peak, we set
$\displaystyle\bar{\mu}_{i}=\frac{\hat{\mu}_{i}+\bar{\theta}_{i}}{2}.$ (43)
Then we continue to apply the Newston step to obtain the final estimate as
$\displaystyle\hat{\theta}_{i}=\bar{\mu}_{i}-\frac{f^{{}^{\prime}}(\bar{\mu}_{i})}{f^{{}^{\prime\prime}}(\bar{\mu}_{i})}.$
(44)
Numerical experiments demonstrate that such a two step approach improves the
robustness of the VALSE significantly, compared to the original implementation
[29]. Besides, the concentration parameter $\hat{\kappa}_{i}$ is approximated
as
$\displaystyle\hat{\kappa}_{i}=A^{-1}\left({\rm
e}^{0.5/f^{{}^{\prime\prime}}(\hat{\theta}_{i})}\right),$ (45)
where $A^{-1}(\cdot)$ is the inverse function of
$A(\cdot)=I_{1}(\cdot)/I_{0}(\cdot)$. The expected value of
${\mathbf{a}}_{\mathcal{M}}(\theta_{i})$ is
$\displaystyle\hat{\mathbf{a}}_{\mathcal{M}}(\theta_{i})=\frac{I_{\mathcal{M}}(\hat{\kappa}_{i})}{I_{0}(\hat{\kappa}_{i})}\odot{\mathbf{a}}_{\mathcal{M}}(\hat{\theta}_{i}).$
(46)
It is found that calculating the most dominate component via Heurestic 2 [29,
Algorithm 2] in each iteration is very time consuming. Consequently, we adopt
the warm start strategy where we only implement Heuristic 2 in the
initialization stage. Then we use the previous estimates to perform
refinement, which is very efficiently.
#### IV-E2 Inferring the weights and support
Next by fixing the posterior PDFs
$q(\theta_{i}|\widetilde{\mathbf{h}}_{\mathcal{M}}),i=1,...,L_{\rm max}$, the
posterior PDF $q({\bm{\beta}},{\mathbf{s}}|\tilde{\mathbf{h}}_{\mathcal{M}})$
is obtained. Define the matrices $\mathbf{J}$ and $\mathbf{u}$ as
$\displaystyle{J}_{ij}=\begin{cases}{\mathbf{1}}_{\mathcal{M}}^{\rm
T}\left(\frac{1}{\tilde{\bm{\sigma}}_{\mathcal{M}}^{2}}\right),&i=j\\\
{\widehat{\mathbf{a}}}^{\rm H}_{{\mathcal{M}},i}{\rm
diag}\left(\frac{1}{\tilde{\bm{\sigma}}_{\mathcal{M}}^{2}}\right){\widehat{\mathbf{a}}}_{{\mathcal{M}},j},&i\neq{j}\end{cases},\quad
i,j\in\\{1,2,\cdots,L_{\rm max}\\},$ (47a)
$\displaystyle{\mathbf{u}}=\widehat{\mathbf{A}}_{\mathcal{M}}^{\rm
H}\left(\tilde{\mathbf{h}}_{\mathcal{M}}/\tilde{\bm{\sigma}}_{\mathcal{M}}^{2}\right).$
(47b)
According to (36), the posterior approximation
$q({\bm{\beta}},{\mathbf{s}}|\tilde{\mathbf{h}}_{\mathcal{M}})$ is factored as
the product of $q({\bm{\beta}}|\tilde{\mathbf{h}}_{\mathcal{M}},{\mathbf{s}})$
and $\delta({\mathbf{s}}-{\widehat{\mathbf{s}}})$. For a given
$\widehat{\mathbf{s}}$,
$q({\bm{\beta}}_{\widehat{\mathcal{S}}}|\tilde{\mathbf{h}}_{\mathcal{M}})$ is
a complex Gaussian distribution, and
$q({\bm{\beta}}|\tilde{\mathbf{h}}_{\mathcal{M}};\widehat{\mathbf{s}})$ is
$\displaystyle
q({\bm{\beta}}|\tilde{\mathbf{h}};\widehat{\mathbf{s}})={\mathcal{CN}}({\bm{\beta}}_{\widehat{\mathcal{S}}};\widehat{\bm{\beta}}_{\widehat{\mathcal{S}}},\widehat{\mathbf{C}}_{\widehat{\mathcal{S}}})\prod_{i\not\in\widehat{\mathcal{S}}}\delta(\beta_{i}),$
(48)
where
$\displaystyle\widehat{\mathbf{C}}_{{\mathcal{S}}}=\left({{\mathbf{J}}_{{\mathcal{S}}}}+\frac{{\mathbf{I}}_{|{\mathcal{S}}|}}{\nu}\right)^{-1},$
(49a)
$\displaystyle\widehat{\bm{\beta}}_{{\mathcal{S}}}=\widehat{\mathbf{C}}_{{\mathcal{S}}}{\mathbf{u}}_{{\mathcal{S}}}.$
(49b)
Although (49a) involves a matrix inversion, it can be avoided by utilizing the
rank one update formula.
Then we need to find $\hat{\mathbf{s}}$ which maximizes $\ln Z(\mathbf{s})$,
i.e.,
$\displaystyle\hat{\mathbf{s}}=\underset{\mathbf{s}}{\operatorname{argmax}}\ln
Z(\mathbf{s}),$ (50)
where $\ln Z(\mathbf{s})$ is
$\displaystyle\ln Z(\mathbf{s})\triangleq$
$\displaystyle-\ln\det\left({\mathbf{J}}_{\mathcal{S}}+\frac{1}{\nu}{\mathbf{I}}_{|\mathcal{S}|}\right)+{\mathbf{u}}_{\mathcal{S}}^{\rm
H}\left(\mathbf{J}_{\mathcal{S}}+\frac{1}{{\nu}}\mathbf{I}_{|\mathcal{S}|}\right)^{-1}{\mathbf{u}}_{\mathcal{S}}$
$\displaystyle+||\mathbf{s}||_{0}\ln\frac{\rho}{1-\rho}+||\mathbf{s}||_{0}\ln\frac{1}{{\nu}}+{\rm
const}.$ (51)
Problem (50) is solved via a greedy algorithm, and local optimum is
guaranteed.
#### IV-E3 Inferring the model parameters
After updating the PDFs of the frequencies and weights, the model parameters
$\\{\rho,~{}\nu\\}$ are updated as
$\displaystyle\widehat{\rho}=$
$\displaystyle\frac{||\widehat{\mathbf{s}}||_{0}}{{L_{\rm max}}},$
$\displaystyle\widehat{{\nu}}=$ $\displaystyle\frac{\widehat{\bm{\beta}}^{\rm
H}_{\widehat{\mathcal{S}}}\widehat{\bm{\beta}}_{\widehat{\mathcal{S}}}+{\rm
tr}(\widehat{{\mathbf{C}}}_{\widehat{\mathcal{S}}})}{||\widehat{\mathbf{s}}||_{0}}.$
(52)
Now the the posterior mean and variances of ${\mathbf{h}}$ can be obtained as
$\displaystyle{\mathbf{h}}_{{\rm A},{\mathcal{M}}}^{\rm
post}=\hat{\mathbf{A}}_{{\mathcal{M}},\hat{\mathcal{S}}}\hat{\bm{\beta}}_{\hat{\mathcal{S}}},$
(53a) $\displaystyle{\mathbf{v}}_{{\rm A},{\mathbf{h}}_{\mathcal{M}}}^{\rm
post}={\rm
diag}(\hat{\mathbf{A}}_{{\mathcal{M}},\hat{\mathcal{S}}}\hat{\mathbf{C}}_{\hat{\mathcal{S}}}\hat{\mathbf{A}}_{{\mathcal{M}},\hat{\mathcal{S}}}^{\rm
H})+\left(\hat{\bm{\beta}}_{\hat{\mathcal{S}}}^{\rm
H}\hat{\bm{\beta}}_{\hat{\mathcal{S}}}{\mathbf{1}}_{|{\mathcal{M}}|}-|\hat{\mathbf{A}}_{{\mathcal{M}},\hat{\mathcal{S}}}|^{2}|\hat{\bm{\beta}}_{\hat{\mathcal{S}}}|^{2}\right)$
$\displaystyle+\left[{\rm
tr}(\hat{\mathbf{C}}_{\hat{\mathcal{S}}}){\mathbf{1}}_{|{\mathcal{M}}|}-|\hat{\mathbf{A}}_{{\mathcal{M}},\hat{\mathcal{S}}}|^{2}{\rm
diag}(\hat{\mathbf{C}}_{\hat{\mathcal{S}}})\right],$ (53b)
respectively. Then, the extrinsic message $m_{{\rm A}\rightarrow{\rm
B}}({\mathbf{h}}_{\mathcal{M}})={\mathcal{CN}}({\mathbf{h}}_{\mathcal{M}};{\mathbf{h}}_{{\rm
A},{\mathcal{M}}}^{\rm ext},{\rm diag}({\mathbf{v}}_{{\rm
A},{\mathbf{h}}_{\mathcal{M}}}^{\rm ext}))$ from module A to module B can be
obtained from (39b) and (39a). Remember that we have initialized the extrinsic
message $m_{{\rm A}\rightarrow{\rm B}}({\mathbf{h}}_{\mathcal{M}})$ from
module A to module B at the beginning of Section IV, the algorithm is closed.
### IV-F Initialization
Performing the joint estimation of the channel, residual CFO and data
detection is a very challenging task, and initialization is a very important
step to have good performance. Since the residual CFO $\omega$ is close to
zero, we initialize it as zero. To obtain a good estimate of the channel, we
run the pilot based VALSE algorithm to obtain the posterior means and
variances of the channel, which are provided as the initialization of the
extrinsic message $m_{{\rm A}\rightarrow{\rm B}}({\mathbf{h}}_{\mathcal{M}})$
from module A to module B. Besides, we also initialize the nuisance parameters
$\sigma^{2}$ and ${\nu}$ via the pilot based VALSE algorithm. The whole
algorithm is summarized as Algorithm 1.
Algorithm 1 JCCD-VALSE algorithm
1: Initialize the extrinsic message $m_{{\rm A}\rightarrow{\rm
B}}({\mathbf{h}}_{\mathcal{M}})$ from module A to module B, the nuisance
parameters $\sigma^{2}$ and $\nu$ via the pilot based VALSE algorithm, as
described in Subsection IV-F; Set the number of iterations $T_{\rm max}$, the
maximum number of paths $L_{\rm max}$;
2: Calculate the message $m_{{\rm B}\rightarrow{\rm
C}}({\mathbf{d}}_{\mathcal{D}})$ (IV-A) and input it to the LDPC decoder
module to obtain the posterior means $d_{{\rm C},i}^{\rm post}$ and variances
$v_{{\rm C},d_{i}}^{\rm post}$ of the data, as described in Subsection IV-B.
3: Calculate the extrinsic message $m_{{\rm B}\rightarrow{\rm
A}}({\mathbf{h}}_{\mathcal{M}})$ through (32) and (IV-D) from module B to
module A.
4: Initialize $q(\theta_{i}|\widetilde{\mathbf{h}}_{\mathcal{M}})$,
$i=1,2,\cdots,L_{\rm max}$; compute
$\hat{\mathbf{a}}_{\mathcal{M}}(\theta_{i})$, $\mathbf{J}$ (47a) and
$\mathbf{u}$ (47b).
5: for $t=1,\cdots,T_{\rm max}$ do
6: Update $\widehat{\mathbf{s}}$ (50),
$\widehat{\bm{\beta}}_{\widehat{\mathcal{S}}}~{}{\rm
and}~{}\widehat{\mathbf{C}}_{\widehat{\mathcal{S}}}$ (49).
7: Update $\widehat{\rho}$, $\widehat{\nu}$ (IV-E3) and Update $\bm{\eta}_{i}$
and $\widehat{\mathbf{a}}_{i}$ for all $i\in\widehat{\mathcal{S}}$.
8: Compute the posterior means and variances of $\mathbf{h}$ as
${\mathbf{h}}_{\rm A}^{\rm post}$ (53a), ${\mathbf{v}}_{{\rm A},{\rm h}}^{\rm
post}$ (53b).
9: Calculate the extrinsic message $m_{{\rm A}\rightarrow{\rm
B}}({\mathbf{h}}_{\mathcal{M}})$ from module B to module A.
10: Calculate the message $m_{{\rm B}\rightarrow{\rm
C}}({\mathbf{d}}_{\mathcal{D}})$ (IV-A) and input it to the LDPC decoder
module to obtain the posterior means $d_{{\rm C},i}^{\rm post}$ and variances
$v_{{\rm C},d_{i}}^{\rm post}$ of the data.
11: Compute the posterior means and variances of $\mathbf{z}$ as
${\mathbf{z}}_{\rm B}^{\rm post}$ (20a) and ${\mathbf{v}}_{{\rm
B},{\mathbf{z}}}^{\rm post}$ (20b), respectively. Implement the Newston step
to update the residual CFO and the EM step to update the noise variance via
(22) and (29), respectively.
12: Calculate the extrinsic message $m_{{\rm B}\rightarrow{\rm
A}}({\mathbf{h}}_{\mathcal{M}})$ through (32) and (IV-D) from module B to
module A.
13: Update $\mathbf{J}$ (47a) and $\mathbf{u}$ (47b).
14: end for
15: Return belief function of data bits $u_{k}$, $k\in{\mathcal{K}}$, channel
estimate $\hat{\mathbf{h}}$.
## V Numerical Simulation Results
In this section, numerical simulations are conducted to demonstrate the
advantages of the proposed joint processing scheme. A typical underwater
acoustic channel setting is adopted in the simulation: there are $15$ paths,
the arrival time between two adjacent paths follows exponential distribution
with mean value $1$ ms, and the amplitude of each path follows Rayleigh
distribution with mean power decreasing by $20$ dB in $30$ ms delay spread
[7]. The simulation parameters of the OFDM system are listed in Table I. In
addition, a common Doppler scaling factor exists in all paths, which follows
uniform distribution in the range $\left[-2\times 10^{-3},2\times
10^{-3}\right]$. In our simulation, null subcarriers are utilized to estimate
the Doppler scaling factor and the residual CFO [36], based on the results of
which the resampling and CFO compensation are implemented. Performance metrics
of decoding BER and NMSE of channel estimation defined as
$\displaystyle{\rm
NMSE}(\hat{\mathbf{h}})=10\log_{10}\left(\frac{\left\|\hat{\mathbf{h}}-\mathbf{h}_{0}\right\|^{2}}{\left\|\mathbf{h}_{0}\right\|^{2}}\right)$
are compared in the simulation, where $\mathbf{h}_{0}$, $\hat{\mathbf{h}}$ are
the ground truth and estimated value of the frequency-domain channel response
vector, respectively. All the results are averaged over $500$ MC trials.
TABLE I: CP-OFDM parameter settings Bandwidth | $B$ | $4882.8~{}\text{Hz}$
---|---|---
Carrier frequency | $f_{c}$ | $13000~{}\text{Hz}$
Sampling frequency | $f_{s}$ | $39062.5~{}\text{Hz}$
No. subcarriers | $N$ | $1024$
No. data subcarriers | $N_{\mathcal{D}}$ | $672$
No. pilot subcarriers | $N_{\mathcal{P}}$ | $256$
No. null subcarriers | $N_{\mathcal{N}}$ | $96$
Symbol duration | $T$ | $209.72~{}\text{ms}$
Cyclic-prefix length | $T_{\text{cp}}$ | $40.35~{}\text{ms}$
### V-A Benchmark Algorithms
Several approaches classified as pilot-only based and joint channel and data
decoding based are implemented to make performance comparison. The pilot-only
based are listed as follows:
* •
The pilot based OMP approach [7]. The dictionary is constructed by $\times 8$
oversampling for time delay search, and the Bayesian information stopping
criterion [37] is adopted, and the algorithm is termed as OMP.
* •
The pilot based AMP approach [38]. It is numerically found that AMP diverges
with the oversampling factor greater than $2$. We construct the dictionary by
$\times 2$ oversampling. Besides, the Bernoulli Gaussian prior is adopted for
the coefficients and the EM algorithm [38, 39] is incorporated to jointly
estimate the nuisance parameters of the prior distribution and the noise
variance. The algorithm is termed as AMP.
* •
The pilot based VALSE approach [29]. The observations corresponding to pilots
are directly input to the VALSE algorithm to obtain the path delays and
amplitudes, which are then used to reconstruct the whole channel. The
algorithm is termed as VALSE.
Note that after the whole channel is reconstructed, the estimates (including
means and variances) of the data are input to the single input single output
(SISO) equalizer, i.e, iterative decoding is conducted to obtain the BER.
The GAMP based _j_ oint _c_ hannel estimation and _d_ ata decoding approach
(JCD) named JCD-GAMP is also conducted [14]. Note that the original approach
uses the Markov chain prior and learn the nuisance parameters of the prior via
fitting the simulated channel data. Besides, it uses the true noise variance.
Here we use the Bernoulli Gaussian prior and use the EM to jointly estimate
the nuisance parameters of the prior distribution and the noise variance
instead.
To show the benefits of estimating the residual CFO, we also implement the
VALSE based JCD approach without estimating the CFO, and we name the approach
as JCD-VALSE.
To further show the effectiveness of the proposed algorithm, we also evaluate
the coded BER under the perfect channel state information (PCSI) and the NMSE
of the JCCD-VALSE with data aware, which are termed as PCSI and JCCD-VALSE
(data aware), respectively. Note that the coded BER of PCSI and the JCCD-VALSE
(data aware) will be the lower bounds of the coded BER and the NMSE,
respectively.
### V-B QPSK Results
Figure 3: Performance metrics evaluated by the various algorithms for QPSK.
(a) coded BER, (b) NMSE.
Fig. 3 presents the performance metrics evaluated by the various algorithms
for QPSK modulation. For the coded BER shown in Fig. 3, PCSI works best,
followed by JCCD-VALSE, JCD-VALSE, JCD-GAMP, VALSE, OMP and AMP. For ${\rm
BER}=10^{-3}$, the SNR needed for the PCSI, JCCD-VALSE, JCD-VALSE, JCD-GAMP,
VALSE, and OMP are $4.21$ dB, $4.36$ dB, $4.47$ dB, $4.81$ dB, $5.11$ dB,
$6.61$ dB. The SNR gap between the JCCD-VALSE and the PCSI method (oracle) is
$4.36-4.21=0.15$ dB, which is very close. Compared to JCD-GAMP, VALSE and AMP
approaches, our JCCD method has SNR gains $4.81-4.36=0.45$ dB,
$5.11-4.36=0.75$ dB, $6.61-4.36=1.75$ dB, respectively. As for the NMSE of the
channel estimation, Fig. 3 shows that JCCD-VALSE (data aware) works best,
followed by JCCD-VALSE, JCD-VALSE (without estimating the residual CFO), JCD-
GAMP, VALSE, OMP and AMP. Besides, JCCD-VALSE asymptotically approaches to
JCCD-VALSE (data aware). To sum up, the proposed approach achieves excellent
data decoding and channel estimation performance for QPSK modulated symbols.
### V-C $16$ QAM Results
The performance metrics evaluated by the various algorithms for $16$ QAM
modulation are shown in Fig. 4. Note that the results are similar to the QPSK
modulation setting except the following two differences. The first difference
is that because the constellation becomes larger, the SNR needed to achieve
the same coded BER and the NMSE increases (from about $4.36$ dB to $9.7$ dB of
JCCD-VALSE). The second difference is that the gain of the CFO estimation for
the JCCD-VALSE is marginal, compared to JCD-VALSE (without estimating the
residual CFO). The reason is probably that the Doppler scaling factor
estimation accuracy is better in the higher SNR region, which leads to smaller
residual CFO. Still, we conclude that the proposed approach has performance
gains compared to the other approaches for $16$ QAM modulated symbols.
Figure 4: Performance metrics evaluated by the various algorithms for $16$
QAM. (a) coded BER, (b) NMSE.
## VI Sea Trial Data Decoding Results
To further verify the performance of the proposed JCCD-VALSE scheme in real
underwater acoustic communication channels, sea test data from MACE10 is
adopted for decoding. MACE10 experiment was carried out off the coast of
Marthas Vineyard, Massachusetts, at Jun. 2010. The depth of water was about 80
meters. The source was towed from the ship and towed back at relatively slow
speeds (1-2 m/s). The relative distance between the transmitter and the
receiver changed from 500 m to 7 km. In total 31 transmissions were carried
out during the experiment, each containing 20 OFDM symbols in QPSK and 20
symbols in $16$ QAM, respectively. The transmitted OFDM signal adopts the same
parameters shown in Table I. For data decoding, one transmission data file
recorded during the turn of the source which has quite low SNR is excluded. In
Fig. 5, the estimated typical channel impulse response and the estimated
Doppler speed corresponding to the 30 transmissions in MACE10 are presented.
(a) Estimated channel impulse response
(b) Estimated Doppler speed
Figure 5: Channel condition in MACE10 [40].
### VI-A QPSK Decoding Results
In total 600 OFDM symbols in QPSK modulation were collected during the sea
experiment. Since all the original received data can be decoded successfully,
additional noise in different level are added in order to show the differences
between different algorithms. Denote the original noise variance in the
received signal to be $\delta^{2}$, the decoding BER results corresponding to
different additional noise levels are shown in Table II. It can be observed
that the proposed JCCD-VALSE algorithm always achieves the lowest BER in all
additional noise levels.
TABLE II: MACE10 decoding BER: QPSK Algorithm | OMP | AMP | VALSE | JCCD-VALSE | JCD-VALSE | JCD-GAMP
---|---|---|---|---|---|---
$0.8\delta^{2}$ | 0.0015 | 0.0021 | 0.0010 | 0 | 0 | 0.0001
$1.6\delta^{2}$ | 0.0085 | 0.0100 | 0.0078 | 0.0056 | 0.0058 | 0.0070
$2.4\delta^{2}$ | 0.0182 | 0.0194 | 0.0153 | 0.0118 | 0.0118 | 0.0125
$3.2\delta^{2}$ | 0.0338 | 0.0382 | 0.0281 | 0.0204 | 0.0214 | 0.0235
$4.0\delta^{2}$ | 0.0511 | 0.0568 | 0.0460 | 0.0358 | 0.0363 | 0.0397
$4.8\delta^{2}$ | 0.0662 | 0.0712 | 0.0616 | 0.0531 | 0.0535 | 0.0578
### VI-B $16$ QAM decoding results
For 16 QAM modulation, 523 complete OFDM symbols were collected during the
experiment. A direct decoding of the received signals with different
algorithms leads to the results shown in Table III, which clearly shows that
the proposed JCCD-VALSE scheme achieves the best BER performance.
TABLE III: MACE10 decoding BER: 16 QAM Algorithm | OMP | AMP | VALSE | JCCD-VALSE | JCD-VALSE | JCD-GAMP
---|---|---|---|---|---|---
BER | 0.0152 | 0.0144 | 0.0130 | 0.0081 | 0.0084 | 0.0131
## VII Conclusion
This paper proposes the joint CFO, gridless channel estimation and data
decoding algorithm named as JCCD-VALSE, which automatically estimates the
number of paths, the nuisance parameters of the prior distribution and the
noise variance. The algorithm is decomposed as three modules named the VALSE
module, the minimum mean squared error (MMSE) module and the decoder module,
and by iteratively exchange extrinsic information between modules, the channel
estimation and data decoding performances gradually improve. Numerical
experiments and real data are used to demonstrate the superior performance of
the proposed approach.
## References
* [1] M. Stojanovic and J. Preisig, “Underwater acoustic communication channels: Propagation models and statistical characterization,” _IEEE Commun. Mag._ , vol. 47, no. 1, pp. 84-89, Jan. 2009.
* [2] B. S. Sharif, J. Neasham, O. R. Hinton and A. E. Adams, “A computationlly efficient Doppler compensation system for underwater acoustic communications,” _IEEE J. Ocean. Eng._ , vol. 25, no. 1, pp. 52-61, Jan. 2000.
* [3] M. Stojanovic, “Low complexity OFDM detector for underwater channels,” _in Proc. of MTS/IEEE OCEANS Conference_ , Boston, MA, Sept. 18-21, 2006.
* [4] B. Li, S. Zhou, M. Stojanovic, L. Freitag and P. Willett, “Multicarrier communication over underwater acoustic channels with nonuniform Doppler shifts,” _IEEE J. Ocean. Eng._ , vol. 33, no. 2, pp. 198-209, April 2009.
* [5] S. Zhao, S. Yan and J. Xi, “Adaptive turbo equalization for differential OFDM systems in underwater acoustic communications,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 11, pp. 13937-13941, 2020.
* [6] W. Li and J. C. Preisig, “Estimation of rapidly time-varying sparse channels,” _IEEE J. Ocean. Eng._ , vol. 32, no. 4, pp. 927-939, Oct. 2007.
* [7] C. R. Berger, S. Zhou, J. C. Preisig, and P. Willett, “Sparse channel estimation for multicarrier underwater acoustic communication: From subspace methods to compressed sensing,” _IEEE Trans. Signal Process._ , vol. 58, no. 3, pp. 1708-1721, 2010.
* [8] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” _IEEE Trans. Inform. Theory_ , vol. 53, no. 12, pp. 4655-4666, 2007.
* [9] B. Peng, P. S. Rossi, H. Dong and K. Kansanen, “Time-domain oversampled OFDM communication in doubly-selective underwater acoustic channels,” _IEEE Commun. Lett._ , vol. 19, no. 6, pp. 1081-1084, June 2015.
* [10] E. Panayirci, M. T. Altabbaa, M. Uysal, and H. V. Poor, “Sparse channel estimation for OFDM-based underwater acoustic systems in Rician fading with a new OMP-MAP algorithm,” _IEEE Trans. Signal Process._ , vol. 67, no. 6, pp. 1550-1565, 2019.
* [11] D. L. Donoho, A. Maleki, and A. Montanari, “Message passing algorithms for compressed sensing: I. Motivation and construction,” _in Proc. Inf. Theory Workshop_ , Cairo, Egypt, Jan. 2010, pp. 1-5.
* [12] C. Wei, H. Liu, Z. Zhang, J. Dang and L. Wu, “Near-optimum sparse channel estimation based on least squares and approximate message passing,” _IEEE Wireless Commun. Lett._ , vol. 6, no. 6, pp. 754-757, Dec. 2017.
* [13] E. Panayirci, H. Senol and H. V. Poor, “Joint channel Estimation, equalization, and data detection for OFDM systems in the presence of very high mobility,” _IEEE Trans. Signal Process._ , vol. 58, no. 8, pp. 4225-4238, Aug. 2010.
* [14] P. Schniter, “A message-passing receiver for BICM-OFDM over unknown clustered-sparse channels,” _IEEE J. Sel. Topics Signal Process._ , vol. 5, no. 8, pp. 1462-1474, Dec. 2011.
* [15] P. Sun, Z. Wang and P. Schniter, “Joint channel-estimation and equalization of single-carrier systems via bilinear AMP,” _IEEE Trans. Signal Process._ , vol. 66, no. 10, pp. 2772-2785, May 2018.
* [16] Z. Jiang, X. Shen, H. Wang and Z. Ding, “Joint PSK data detection and channel estimation under frequency selective sparse multipath channels,” _IEEE Trans. Commun._ , vol. 68, no. 5, pp. 2726-2739, May 2020.
* [17] J. F. Sifferlen, H. C. Song, W. S. Hodgkiss, W. A. Kuperman and J. M. Stevenson, “An iterative equalization and decoding approach for underwater acoustic communication,” _IEEE J. Ocean. Eng._ , vol. 33, no. 2, pp. 182-197, April 2008.
* [18] Z. Yang and Y. R. Zheng, “Iterative channel estimation and Turbo equalization for multiple-input multiple-output underwater acoustic communications,” _IEEE J. Ocean. Eng._ , vol. 41, no. 1, pp. 232-242, Jan. 2016.
* [19] Z. Chen, J. Wang and Y. R. Zheng, “Frequency-domain Turbo equalization with iterative channel estimation for MIMO underwater acoustic communications,” _IEEE J. Ocean. Eng._ , vol. 42, no. 3, pp. 711-721, July 2017.
* [20] X. Qin, F. Qu and Y. R. Zheng, “Bayesian iterative channel estimation and Turbo equalization for multiple-input multiple-output underwater acoustic communications,” _IEEE J. Ocean. Eng._ , vol. 46, no. 1, pp. 326-337, Jan. 2021.
* [21] K. P. Arunkumar and C. R. Murthy, “Iterative sparse channel estimation and data detection for underwater acoustic communications using partial interval demodulation,” _IEEE Trans. Signal Process._ , vol. 66, no. 19, pp. 5041-5055, Oct., 2018.
* [22] X. Tu, X. Xu and A. Song, “Frequency-domain decision feedback equalization for single-carrier transmissions in fast time-varying underwater acoustic channels,” _IEEE J. Ocean. Eng._ , vol. 46, no. 2, pp. 704-716, April 2021.
* [23] S. Wang, Z. He, K. Niu, P. Chen and Y. Rong, “New results on joint channel and impulsive noise estimation and tracking in underwater acoustic OFDM systems,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 4, pp. 2601-2612, April 2020.
* [24] G. Yang, Q. Guo, H. Ding, Q. Yan and D. D. Huang, “Joint message-passing-based bidirectional channel estimation and equalization with superimposed training for underwater acoustic communications,” _IEEE J. Ocean. Eng._ , early access, 2021.
* [25] Y. Chi, L. L. Scharf, A. Pezeshki and R. Calderbank, “Sensitivity of basis mismatch to compressed sensing,” _IEEE Trans. on Signal Process._ , vol. 59, pp. 2182 - 2195, 2011.
* [26] T. L. Hansen, B. H. Fleury and B. D. Rao, “Superfast line spectral estimation,” _IEEE Trans. Signal Process._ , vol. 66, no. 10, pp. 2511-2526, 2018.
* [27] T. L. Hansen, P. B. Jørgensen, M. A. Badiu and B. H. Fleury, “An iterative receiver for OFDM with sparsity-based parametric channel estimation,” _IEEE Trans. Signal Process._ , vol. 66, no. 20, pp. 5454-5469, 2018.
* [28] X. Meng, S. Wu and J. Zhu, “A unified Bayesian inference framework for generalized linear models,” _IEEE Signal Process. Lett._ , vol. 25, no. 3, pp. 398-402, 2018.
* [29] M. A. Badiu, T. L. Hansen and B. H. Fleury, “Variational Bayesian inference of line spectra,” _IEEE Trans. Signal Process._ , vol. 65, no. 9, pp. 2247-2261, 2017.
* [30] K. V. Mardia and P. E. Jupp, _Directional Statistics_. New York, NY, USA: Wiley, 2000.
* [31] D. J. C. MacKay, “Bayesian interpolation,” _Neural Computation_ , vol. 4, no. 3, pp. 415-447, 1992.
* [32] J. O. Berger, _Statistical Decision Theory and Bayesian Analysis_. Springer, second edition, 1985.
* [33] J. Winn and C. M. Bishop, “Variational message passing,” _J. Mach. Learn. Res._ , vol. 6, pp. 661-694, 2005.
* [34] T. Minka, “A family of algorithms for approximate Bayesian inference,” Ph.D. dissertation, Department of Electrical Engineering and Computer Science, Mass. Inst. Technol., Cambridge, MA, USA, 2001.
* [35] J. Zhu, Q. Zhang, and X. Meng, “Grid-less variational Bayesian inference of line spectral from quantized samples,” 2019, arXiv:1811.05680, to appear in _China Communications_. [Online]. Available: https://arxiv.org/abs/1811.05680
* [36] L. Wan, Z.-H. Wang, S. Zhou, T. C. Yang, and Z. Shi, “Performance comparison of Doppler scale estimation methods for underwater acoustic OFDM,” _Journal of Electrical and Computer Engineering_ , vol. 2012, Article ID 703243, 2012.
* [37] N. Eldarov, G. Tan, and T. Herfet, “Delay-Doppler search for matching pursuit algorithms in time-variant channels,” _in Proc. IEEE Int. Symp. Broadband Multimedia Syst. Broadcast._ , Jun. 2015, pp. 1-5.
* [38] J. P. Vila and P. Schniter, “Expectation-Maximization Bernoulli-Gaussian approximate message passing,” _Proc. Asilomar Conf. on Signals, Systems, and Computers_ , Pacific Grove, CA, Nov. 2011.
* [39] J. P. Vila and P. Schniter, “Expectation-Maximization Gaussian-mixture approximate message passing,” _IEEE Trans. Signal Process._ , vol. 61, no. 19, pp. 4658-4672, Oct. 2013.
* [40] L. Wan, H. Jia, F. Zhou, M. Muzzammil, T. Li and Y. Huang, “Fine Doppler scale estimations for an underwater acoustic CP-OFDM system,” _Signal Process._ , vol. 170, May 2020.
|
# Roadmap to Thermal Dark Matter Beyond the WIMP Unitarity Bound
Ronny Frumkin<EMAIL_ADDRESS>Eric Kuflik
<EMAIL_ADDRESS>Itay Lavie<EMAIL_ADDRESS>Tal
Silverwater<EMAIL_ADDRESS>Racah Institute of Physics,
Hebrew University of Jerusalem, Jerusalem 91904, Israel
(August 27, 2024)
###### Abstract
We study the general properties of the freezeout of a thermal relic. We give
analytic estimates of the relic abundance for an arbitrary freezeout process,
showing when instantaneous freezeout is appropriate and how it can be
corrected when freezeout is slow. This is used to generalize the relationship
between the dark mater mass and coupling that matches the observed abundance.
The result encompasses well-studied particular examples, such as WIMPs, SIMPs,
coannihilation, coscattering, inverse decays, and forbidden channels, and
generalizes beyond them. In turn, this gives an approximate perturbative
unitarity bound on the dark matter mass for an arbitrary thermal freezeout
process. We show that going beyond the maximal masses allowed for freezeout
via dark matter self-annihilations (WIMP-like, $m_{\rm
DM}\gg\mathcal{O}(100\leavevmode\nobreak\ \rm TeV)$) predicts that there are
nearly degenerate states with the dark matter and that the dark matter is
generically metastable. We show how freezeout of a thermal relic may allow for
dark matter masses up to the Planck scale.
## I Introduction and General Idea
The identity of dark matter (DM) remains one of the biggest mysteries of the
universe. Although the evidence for its existence is overwhelming, we still do
not know much about it. All we know about DM’s existence and properties is
inferred from its gravitational effects. We still do not know its particle
nature: its mass, which forces it interacts with, the number of DM particles,
if it is elementary or composite, and perhaps most importantly, how we may
discover it.
It has long been known that the current abundance of DM can be related to its
microscopic properties, such as its mass and interaction cross-sections. This
is analogous, for instance, to how the abundance of the light nuclear elements
are determined Alpher:1948ve ; Cyburt:2015mya . There are many different types
of interactions and processes in the early universe that can determine a
particle’s abundance today, but only a small number of those can give rise to
the DM we observe.
The possibility that DM is a thermal relic—namely, that its interactions
brought it into thermal and chemical equilibrium with the big bang bath, and
that its abundance is determined by those interactions when it finally departs
equilibrium—is an exciting possibility Lee:1977ua ; Scherrer:1985zt . The
abundance today is then insensitive to initial conditions because thermal
equilibrium keeps no memory about the past. Additionally, since the relic
abundance would determine the size of the interactions, thermal relics are
very predictive. Some recent ideas for new thermal processes controlling the
DM abundance beyond ordinary DM-DM annihilations, include DEramo:2010keq ;
Hochberg:2014dra ; Kuflik:2015isi ; Cline:2017tka ; DAgnolo:2017dbv ;
Garny:2017rxs ; Smirnov:2020zwf ; Kramer:2020sbb ; Fitzpatrick:2020vba ;
Frumkin:2021zng .
The common lore on thermal relics within a standard cosmological history has
been that an elementary DM candidate has an upper bound on its mass
Griest:1989wd (for heavy dark matter in other scenarios see the review in
Carney:2022gse and references within). This result comes from considering
WIMP-like DM, whose abundance is determined by the freezeout of annihilations
of DM pairs. The rate for a particle $\chi$, representing a DM candidate, to
self-annihilate is
$\Gamma=n_{\chi}\left<\sigma v\right>_{\chi\chi}\,,$ (1)
where $n_{\chi}$ is the DM density and $\left<\sigma v\right>_{\chi\chi}$ is
the thermally averaged cross section. We assume that this process is fast
enough to keep the DM in chemical equilibrium with the bath prior to
freezeout. As the temperature drops below the mass of the dark matter
$m_{\chi}$, the density of DM becomes exponentially Boltzmann suppressed,
$n_{\chi}=n_{\chi}^{\rm eq}\propto e^{-m_{\chi}/T}\,.$ (2)
This implies that the rate of annihilation—the process which keeps the DM in
equilibrium and annihilates away its abundance—is also becoming exponentially
slower. Eventually the process and the number of DM particles freezes out,
leaving a relic abundance of DM particles. Additionally, since cross sections
are (generally) bounded above by perturbative unitarity, there is a limit on
the amount of time the DM can stay in equilibrium and reduce its abundance.
This results in a lower bound on the DM number density at freezeout. Since the
energy density (mass$\times$density) of DM is fixed to the observed value,
this gives an upper bound on the DM mass. This number is around
$m_{\rm WIMP}\lesssim\left({m_{\rm pl}T_{\rm eq}}\right)^{1/2}\sim
100\leavevmode\nobreak\ {\rm TeV}\,,$ (3)
where $T_{\rm eq}=0.8$ eV is the temperature at matter-radiation equality and
$m_{\rm pl}=2\times 10^{18}$ GeV is the reduced Planck mass Griest:1989wd .
Another way to state the WIMP unitarity bound is that if a stable relic has
mass above the bound, it will be overabundant if the freezeout process is
self-annihilations.
However, the unitarity bound can be weakened if DM can more efficiently
annihilate away and stay in equilibrium longer. The strong bound can be
traced, in part, to the exponentially decaying rates from Eqs. (1) and (2). If
there is less exponential suppression, or it is removed entirely, then the DM
density can be reduced more and the mass limit increased. For a given
exponential dependence on the annihilation rate,
$\Gamma\propto e^{-b\,m_{\chi}/T},$ (4)
for some value of $b$, one can recalculate the perturbative unitarity bound on
the DM mass. Provided that $b$ is not too close to zero, the unitarity bound
becomes
$m_{\chi}\lesssim\left({m_{\rm pl}T_{\rm eq}^{b}}\right)^{\frac{1}{1+b}}\,.$
(5)
We will derive the exact form of the above bound, and the case where $b\sim
0$, later in the text.
At the minimum, this would require that the DM annihilate with a particle
other than itself, and lighter in mass, in order for the rate to be less
exponentially suppressed. For instance, if the DM, $\chi$, annihilates with a
particle $\psi$ (via a process $\chi+\psi\to\rm\phi_{1}+\phi_{2}$), then the
rate of annihilation scales as
$\Gamma\propto e^{-m_{\psi}/T}=e^{-(m_{\psi}/m_{\chi})m_{\chi}/T}$ (6)
and $b=m_{\psi}/m_{\chi}<1$ implies that $m_{\psi}<m_{\chi}$. Additionally, to
go beyond the WIMP unitarity bound (Eq. (3)), the total mass in the final
state in the annihilation process should not be larger than the initial
particles’ masses (except for special circumstances; see inverse decays
later). Otherwise, the process may enter a forbidden regime Griest:1990kh ;
DAgnolo:2015ujb , where additional exponential suppression in the thermally
averaged cross section can come about, since only DM particles on the
exponential Boltzmann tail can annihilate.
Moreover, the DM should be heavier than the sum of all other particles in the
process, or else it will be too unstable to be DM (in the above process Eq.
(6) this corresponds to $m_{\chi}<m_{\psi}+m_{\phi_{1}}+m_{\phi_{2}}$).
Together, these constraints suggest that in order to go beyond the WIMP
unitarity bound, there will be particles degenerate with the DM. We quantify
this in the paper and show that, in fact,
$m_{\chi}\leq\sum_{j}m_{j}<3m_{\chi},$ (7)
where $\sum_{j}m_{j}$ is the sum of the mass of all other particles (initial
and final) in the annihilation process. Therefore, if the DM mass is beyond
the WIMP unitarity bound, there is another particle close to mass with it,
which we will call $\chi^{\prime}$. However, $\chi^{\prime}$ must have
abundance less than the DM (since it’s not the DM), but its mass is above the
WIMP unitarity bound. Its abundance will be too large, unless it has some
process that efficiently depletes its abundance, or it decays away. If it is
unstable, then it potentially causes the DM itself to be unstable. If it is
stable, then it must annihilated away via a non-WIMP like process, and Eq. (7)
applies to $\chi^{\prime}$. Then there is another particle close to mass with
it, $\chi^{\prime\prime}$, and the problem continues, leading to a chain of DM
particles and interactions. This can end when we reach a particle below the
WIMP unitarity bound or one that decays. Such cases are exemplified in
scenarios within a standard cosmological history Kim:2019udq ; Kramer:2020sbb
and within an early matter dominated scenario Berlin:2017ife .
This paper is organized as follows. In Section II we study thermal freezeout
for a general process, giving a relationship between the DM mass and coupling
that matches the observed abundance. We show when instantaneous freezeout is
appropriate and how it is corrected when freezeout is slow. Section III
derives the unitarity bound for an arbitrary process and shows what is needed
to in order to go beyond the WIMP unitarity bound. Explicit examples of DM
chains with thermal DM well above the WIMP unitarity bound are given in
Section IV. We conclude with a discussion in Section V.
## II Thermal Freezeout
| Process | Masses | $b$ | $\beta-1$ | Unitarity bound (GeV)
---|---|---|---|---|---
WIMP | $\chi\chi\to\phi\phi$ | $m_{\phi}\ll m_{\chi}$ | 1 | $\frac{1}{x_{d}}\log(1+x_{d})$ | $\mathcal{O}(10^{5})$
SIMP | $\chi\chi\chi\to\chi\chi$ | | 2 | $\frac{1}{2x_{d}}\log(1+x_{d}/2)$ | $\mathcal{O}(1)$
Forbidden | $\chi\chi\to\psi\psi$ | $m_{\psi}=m_{\chi}(1+\Delta)$ | $2\Delta\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\frac{1}{x_{d}}\log(1+x_{d})$ | $\mathcal{O}(10^{5-14\frac{\Delta}{1+\Delta}})$
Coannihilations | $\begin{array}[]{l}\psi\psi\to\phi\phi\\\ \chi\chi\leftrightarrow\psi\psi\end{array}$ | $m_{\psi}=m_{\chi}(1+\Delta)$ | $1+\Delta$ | $\frac{1}{x_{d}}\log(1+x_{d})$ | $\mathcal{O}(10^{5-7\frac{\Delta}{1+\Delta/2}})$
Coscattering | $\chi\phi\to\psi\phi$ | $m_{\phi}\ll m_{\psi}=m_{\chi}(1+\Delta)$ | $\Delta$ | $\begin{cases}\frac{1}{\Delta x_{d}}&{\rm for\leavevmode\nobreak\ }c=1{\rm\leavevmode\nobreak\ or\leavevmode\nobreak\ }\Delta\gg 0\\\ \frac{1}{c-2}&{\rm for\leavevmode\nobreak\ }c>2{\rm\leavevmode\nobreak\ and\leavevmode\nobreak\ }\Delta=0\\\ \end{cases}$ | $\mathcal{O}(10^{9\frac{2-\Delta}{1+\Delta}})$
Zombie | $\chi\psi\to\psi\psi$ | $m_{\psi}=m_{\chi}(1-\Delta)$ | $1-\Delta$ | $\frac{1}{(1-\Delta)x_{d}}$ | $\mathcal{O}(10^{5+7\frac{\Delta}{1-\Delta/2}})$
Inverse decays | $\chi\phi\to\psi$ | $m_{\phi}\ll m_{\psi}=m_{\chi}(1+\Delta)$ | $\Delta$ | $\frac{1}{\Delta x_{d}}\left(1+\frac{1}{\Delta x_{d}}\right)$ | $\mathcal{O}(10^{9\frac{2-\Delta}{1+\Delta}})$
Table 1: Parameters and the maximum mass allowed by perturbative unitarity for
different freezeout processes. Here $\chi$ is the DM, $\psi$ is particle close
to mass with $\chi$, and $\phi$ is a particle much lighter than both $\chi$
and $\psi$. Both $\psi$ and $\phi$ are assumed to be in thermal and chemical
equilibrium throughout the freezeout process. In the $2\to 2$ freezeout cases,
we assumed s-wave annihilations for calculating $\beta$. For coannihilations,
the process $\psi\psi\to\phi\phi$ controls the abundance, where
$\chi\chi\leftrightarrow\psi\psi$ is assumed to be in equilibrium throughout
freezeout.
Consider a DM candidate in thermal equilibrium with the bath and in chemical
equilibrium via the $n\to m$ process of the form
$\underbrace{\chi\,\chi\,\cdots\chi}_{k}\,i_{1}\,i_{2}\cdots
i_{n-k}\to\underbrace{\chi\,\chi\,\cdots\chi}_{\ell}\,f_{1}\,f_{2}\cdots\,f_{m-\ell}.$
(8)
The general form of the Boltzmann equation for the $\chi$ number density is
$\dot{n}_{\chi}+3Hn_{\chi}=-\langle\sigma v\rangle(n_{i_{1}}^{\rm eq}\cdots
n_{i_{n}}^{\rm
eq})\left(n_{\chi}^{k}-n_{\chi}^{k-\ell}\left(n_{\chi}^{eq}\right)^{k}\right),$
(9)
neglecting Pauli-blocking and stimulated emission. Here $\langle\sigma
v\rangle$ is the thermally averaged cross-section of the $n\to m$ process in
Eq. (8). On dimensional grounds, this thermally averaged cross section will
have the form
$\langle\sigma v\rangle\equiv\frac{\alpha_{\rm
eff}^{m+n-2}}{m_{\chi}^{3n-4}}\times\frac{e^{-b^{\prime}x}}{x^{c^{\prime}}}\,.$
(10)
for some coupling $\alpha_{\rm eff}$, and parameters $b^{\prime}$ and
$c^{\prime}$. Here we define the dimensionless temperature,
$x={m_{\chi}}/{T}$. The power of $\alpha_{\rm eff}$ is chosen to be the form
expected if the amplitude is determined solely by cubic interactions, such as
gauge or Yukawa interactions, but this need not be the case. The thermally
averaged cross section can also have temperature dependence. Usually the
temperature dependence can be well approximated at freezeout by a power law
(for instance $c^{\prime}=1$ for p-wave annihilation) or exponential
dependence on the temperature, which occurs for instance for forbidden
channels.
Defining the yield $Y={n}/{s}$, the Boltzmann equation takes the familiar form
$\frac{\partial Y_{\chi}}{\partial x}=-\frac{\langle\sigma v\rangle
s^{k-1}}{xH}(n_{i_{1}}^{\rm eq}\cdots n_{i_{n}}^{\rm
eq})\left(Y_{\chi}^{k}-Y_{\chi}^{k-\ell}\left(Y_{\chi}^{eq}\right)^{k}\right)\,.$
(11)
Typically, this is the form of the Boltzmann equation that is solved
numerically for the asymptotic value of the DM abundance, $Y_{\chi}(\infty)$.
An estimate for the relic abundance of $\chi$ can be obtained by the
instantaneous freezeout approximation. In this approximation, it is assumed
that the DM density departs from equilibrium and instantly stops annihilating,
where its co-moving density freezes to the value when it departed equilibrium.
Departure from equilibrium, which defines the temperature $x_{d}$, occurs
roughly when
$\frac{\Gamma(x_{d})}{H(x_{d})x_{d}}=1.$ (12)
The annihilation rate is given as:
${\Gamma(x)}={(n_{i_{1}}^{\rm eq}\cdots n_{i_{n}}^{\rm eq})(n^{\rm
eq}_{\chi})^{k-1}}{}\langle\sigma v\rangle\equiv\alpha_{\rm
eff}^{m+n-2}m_{\chi}\frac{ae^{-bx}}{x^{c}}$ (13)
for some constants, $a$, $b$, and $c$, which are defined by this equation. The
values $b$ in different freezeout processes studied in the literature can be
seen in Table 1.
Requiring that the co-moving density at the time of freezeout matches the one
observed today by Planck Aghanim:2018eyx yields the constraint
$Y_{\chi}(\infty)=0.55\times\frac{T_{\rm eq}}{m_{\chi}}.$ (14)
Assuming instantaneous freezeout, namely that the DM density abruptly changes
from an equilibrium density to a fixed a co-moving density at $x=x_{d}$, then
${Y_{\chi}(\infty)=Y^{\rm
eq}_{\chi}(x_{d})}=\frac{45g_{\chi}}{2^{5/2}\pi^{7/2}g_{\star
s}}x_{d}^{3/2}e^{-x_{d}}$ (15)
for a DM particles freezing out while non-relativistic. Using Eqs. (12) –
(15), one finds that the mass-coupling relationship that gives the observed
abundance is
$m_{\chi}\simeq\left(\frac{\alpha^{n+m-2}_{\rm eff}m_{\rm
pl}T_{eq}^{b}}{x_{d}^{c+\frac{3}{2}b-1}}\right)^{\frac{1}{1+b}}\,.$ (16)
where $x_{d}$ is the solution to
$x_{d}\simeq\frac{1}{b+1}\log\left(\frac{\alpha_{\rm eff}^{m+n-2}m_{\rm
pl}}{x_{d}^{c-\frac{5}{2}}T_{\rm eq}}\right).$ (17)
Overall numerical factors are left out for simplicity. These factors can be
recovered here and later with the replacement
$\displaystyle m_{\rm pl}$ $\displaystyle\to$ $\displaystyle
a\sqrt{\frac{90}{\pi^{2}g_{\star}(x_{d})}}m_{\rm pl}$ $\displaystyle T_{\rm
eq}$ $\displaystyle\to$ $\displaystyle{\frac{\pi^{2}(2\pi)^{3/2}g_{\star
s}(x_{d})}{30(1+\Omega_{\rm b}/\Omega_{\rm dm})g_{\chi}}}T_{\rm eq}\,.$ (18)
The factor is typically an $\mathcal{O}(1-10)$ correction to the mass-coupling
relationship.
The relic abundance is mostly determined by the mass, coupling, and the degree
of exponential suppression of the annihilation rate, $b$. From Eq. (16) one
can obtain as particular examples the mass-coupling relationships for well-
studied scenarios such as WIMPs Lee:1977ua ; Scherrer:1985zt , SIMPs
Hochberg:2014dra , forbidden Griest:1990kh ; DAgnolo:2015ujb , co-
annihilations Griest:1990kh , co-scattering DAgnolo:2017dbv ; Kim:2019udq ,
zombie Berlin:2017ife ; Kramer:2020sbb and inverse decay processes
Garny:2017rxs ; Frumkin:2021zng , which are given in Table 1.
Figure 1: Left: $m_{\chi}Y_{\chi}$ vs. $x$ for DM freezeout. The orange
curves shows the solution of the Boltzmann equation for a $m_{\chi}=1$ GeV
WIMP. The red point shows the value $x_{d}$, given by Eqs (17) and (24). After
decoupling, the abundance continues to reduce by a factor
$e^{-x_{d}(\beta-1)}$ until it completely freezes out to a final yield,
$m_{\chi}Y_{\chi}(\infty)=0.55T_{\rm eq}$. In gray we also show the abundance
curves for for a $m_{\chi}=1$ GeV DM candidate whose relic abundance is
determined by a coscattering or $3\to 2$ (SIMP) freezeout process. For
coscattering, decoupling happens very early, freezeout is slow and $\beta$ is
quite large. On the other hand, SIMP DM freezes out very fast and instanaeous
freezeout works as a better approximation of the final relic abundance. Right:
Mass vs. coupling relationship that matches the observed abundance. The solid
line shows the results from the numerical simulation, while the dashed line
shows the analytical solution in Eq. (23). The definitions of $\Delta$ for
zombie and forbidden annihilations are given in Table 1.
Instantaneous freezeout is a good approximation in many scenarios, because the
annihilation rate is dropping exponentially with time (or temperature).
However, when exponential suppression is weaker ($b<1$) instantaneous
freezeout may no longer be a good approximation for estimating the relic
abundance. This will often be the case when going beyond the WIMP unitarity
bound, which requires $b<1$. Indeed, it was shown in Ref. Kim:2019udq ;
Kramer:2020sbb that freezeout can be very slow.
We define a parameter $\beta$ as a measure of the deviation from instantaneous
freezeout as
$Y_{\chi}(\infty)\equiv Y^{\rm eq}_{\chi}(x_{d})e^{-x_{d}(\beta-1)}$ (19)
For $\beta=1$, the instantaneous freezeout approximation is perfect. For $b\ll
1$, the correction can be large. Large deviations from instantaneous freezeout
are expected when the DM is annihilating with a lighter particle than itself.
Therefore, we consider here corrections to instantaneous freezeout when there
is only one DM particle in the initial state ($k=1$), although the method here
can be applied to any scenario. The Boltzmann equation (11) for $k=1$, using
the definitions above, can be written as
$\frac{\partial Y_{\chi}}{\partial x}=-\frac{\alpha_{\rm
eff}^{m+n-2}\frac{m_{\chi}}{x^{c}}e^{-bx}}{xH}\left(Y_{\chi}-Y_{\chi}^{eq}\right)\,.$
(20)
There exists an integral solution to this Boltzmann equation. However, an easy
way to get a good estimate for the correction to Eq. (16) is to evolve the
equation from the moment $\chi$ departs equilibrium, which we approximate as
occurring at $x=x_{d}$. For $x>x_{d}$ the inverse reaction can be dropped. The
solution is
$Y_{\chi}(\infty)=Y^{\rm
eq}_{\chi}(x_{d})\exp\left(-x_{d}\int_{1}^{\infty}x^{-c+1}e^{-bx_{d}(x-1)}\,dx\right)\,.$
(21)
Thus we see an additional possible exponential suppression compared to the
instantaneous freezeout case given in Eq. (15), where
$\beta=1+\int_{1}^{\infty}dx\,x^{-c+1}e^{-bx_{d}(x-1)}.$ (22)
In this form, the mass-coupling relationship in Eq. (16), can easily be found
as before, but using Eq. (19) instead of Eq. (15). One finds
$m_{\chi}\simeq\left(\frac{\alpha^{n+m-2}_{\rm eff}m_{\rm
pl}T_{eq}^{b/\beta}}{x_{d}^{c+\frac{3}{2}\frac{b}{\beta}-1}}\right)^{\frac{1}{1+b/\beta}}\,.$
(23)
where
$x_{d}\simeq\frac{1}{b+\beta}\log\left(\frac{\alpha_{\rm eff}^{m+n-2}m_{\rm
pl}}{x_{d}^{c-\frac{5}{2}}T_{\rm eq}}\right).$ (24)
Again, overall numerical factors can be restored with the replacement Eq.
(18). Notice that Eq. (22) and Eq. (24) still provide two equations needed to
solve for $x_{d}$ and $\beta$. However, the integral $\beta$ can usually be
done semi-analytically. For $b\gtrsim 1$, where instantaneous freezeout it
expected to be good, one finds that $\beta\simeq 1+1/(bx_{d})$. For $bx_{d}\ll
1$ (and further more for $c\to 0$), $\beta$ can be quite large. In the extreme
$b\to 0$ case, $\beta\simeq 1+1/(c-2)$. Note that for $b=0$ and $c\leq 2$,
freezeout never occurs, because the annihilation rate is decreasing slower
relative to the Hubble rate; this occurs for decay processes and can also
occur for co-scattering with a massless mediator. The value of $\beta$ for co-
scattering, inverse decays, and zombie processes are given in Table 1.111We
also include values of $\beta$ for WIMPs and SIMPs and forbidden, still
defined by Eq. (19). Here $\beta$ is calculated in the same way, but starting
with the Boltzmann equation Eq. (11), and dropping the inverse annihilation
term for $x>x_{d}$.
In left panel of Fig. 1 we show a schematic of freezeout, exemplifying the
meaning of the different parameters defined in this section. In the right
panel of Fig. 1 we show the mass vs coupling relationship that matches the
observed abundance. The solid line indicates the results from the numerical
simulation, while the dashed line indicates the analytical solution in Eq.
(23). The numerical and analytical results are in excellent agreement across
many different freezeout processes.
## III Unitary bound
From Eq. (16), the perturbative unitarity bound on WIMP DM is immediately
apparent. For DM-DM annihilations into two light bath particles, we have $b=1$
and $m=n=2$, which gives
$m_{\rm WIMP}\simeq\alpha_{\rm eff}\left(m_{\rm pl}T_{\rm
eq}\right)^{1/2}\simeq\alpha_{\rm eff}\times 30\leavevmode\nobreak\ \rm TeV.$
(25)
For perturbative couplings, one sees that the mass of a WIMP DM candidate is
never greater than a few hundred TeV. However, going beyond this mass is still
possible for a thermal relic, provided that $b<1$—in other words, that the
annihilation rate is less exponentially suppressed as the DM density is
annihilating away.222There are another two interesting possibilities to get an
exponentially fast rate, but they require species to be out of chemical
equilibrium. One is to consider enhancement via stimulated emission if the DM
annihilates to a boson with large occupation number. Another possibility is if
DM annihilates with a particle that has already frozen out and therefore has a
large chemical potential. See Refs. Garny:2017rxs ; Dror:2016rxc ;
Kramer:2020sbb ; DAgnolo:2017dbv for freezeout mechanisms that involve
particles chemically decoupled from the bath. The unitarity bound for an
arbitrary interaction is given by
$m_{\chi}\lesssim m_{\rm pl}\left(\frac{T_{\rm eq}}{m_{\rm
pl}}\right)^{\frac{1}{1+\beta/b}}{x_{d}^{-\frac{c-1}{1+b/\beta}}}.$ (26)
Thus for sufficiently small $b$ and $c\leq 1$, one can expect that there might
be a perturbative thermal candidate for DM all the way up to the Planck scale.
As we will now show, going above the perturbative WIMP unitarity bound
generically implies that there are additional nearly degenerate particles to
the DM, and that the DM is metastable. To go beyond WIMP unitarity, first
consider the process
$\chi+i_{1}+\cdots i_{n}\to f_{1}+\cdots f_{m}\ $ (27)
setting the abundance.333We assume that this is the dominant process, which
itself can impose hierarchy on the masses of the particles involved. We need
only consider the case with one DM in the initial state, otherwise $b\geq 1$.
The rate for this process scales as
$\Gamma\propto\exp\left[-\begin{cases}{\frac{\sum_{i}m_{i}}{m_{\chi}}x}&\scriptstyle
m_{\chi}+\sum\limits_{i}m_{i}>\sum\limits_{f}m_{f}\\\
{\frac{\sum_{f}m_{f}-m_{\chi}}{m_{\chi}}x}&\scriptstyle
m_{\chi}+\sum\limits_{i}m_{i}\leq\sum\limits_{f}m_{f}\end{cases}\right]$ (28)
where the second case is for forbidden DM. Requiring $b<1$ imposes
$\sum_{i}m_{i}<m_{\chi}\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\
m_{\chi}+\sum_{i}m_{i}>\sum_{f}m_{f}$ (29)
or
$\sum_{i}m_{f}<2m_{\chi}\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\
m_{\chi}+\sum_{i}m_{i}\leq\sum_{f}m_{f}.$ (30)
Finally, we require that the DM is stable from the decay induced by moving all
initial particles to the final state
$\sum_{i}m_{i}+\sum_{f}m_{f}\geq m_{\chi}.$ (31)
Combining these conditions gives one of the main results of this paper,
$m_{\chi}\leq\sum_{i}m_{i}+\sum_{f}m_{f}<3m_{\chi}.$ (32)
Therefore, one of the particles, which we will call a partner, will always
have mass close to the DM particle.444It is possible that there is a very
large number of particles in the process, possibly avoiding a degeneracy.
However, this would give a large phase space suppression in the cross-section
which would ultimately reduce the DM mass. However, since the partner is close
to the mass of the DM, it must have a process with $b<1$ in order to reduce
its abundance for the same reasons. Therefore this degeneracy repeats itself,
leading to a potential chain of DM reactions. The chain will end either when
the mass of a partner goes below the WIMP unitarity bound, or when it reaches
a particle that decays in equilibrium with the bath.
Eq. (32) gives a constraint on the masses of the particles involved in the
freezeout process, when we require that the DM mass is above the WIMP
unitarity bound ($b=1$). A stronger constraint can be obtained if we require
that the DM mass be even heavier. For instance, we can impose that the DM is
heavier than the mass given by the unitarity limit in Eq. (26) for arbitrary
$b$. We repeat the same steps as above, but instead of requiring that the rate
in Eq. (28) is faster than the WIMP, we require it to be faster than freezeout
with an exponential suppression ${\Gamma\propto e^{-bx}}$. Eqs. (29) and (30)
become
$\sum_{i}m_{i}<bm_{\chi}\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\
m_{\chi}+\sum_{i}m_{i}>\sum_{f}m_{f}$ (33)
or
$\sum_{i}m_{f}<(1+b)m_{\chi}\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\
m_{\chi}+\sum_{i}m_{i}\leq\sum_{f}m_{f}.$ (34)
Therefore, the constraint on the masses for DM above the unitarity bound value
found in Eq. (26) is
$m_{\chi}\leq\sum_{i}m_{i}+\sum_{f}m_{f}<(1+2b)m_{\chi}.$ (35)
## IV Freezeout chains
Having established a roadmap to going beyond the WIMP unitarity limit for
thermal DM, we now turn to new mechanisms that predict super heavy DM. We will
focus on two chain mechanisms allowing for orders of magnitude larger mass
than the WIMP unitarity bound. First we discuss a zombie chain, in which
absolutely stable DM is possible well beyond the WIMP unitarity bound with
only a small chain. Next we study an inverse decay chain, where DM is
metastable, but whose mass may be much larger.
Figure 2: Thermal evolution of the comoving energy density
$m_{\chi_{i}}Y_{\chi_{i}}$ vs. temperature $x=m_{\chi_{i}}/T$ for the zombie
chain (left) and inverse decay chain (right). The solid lines shows the
numerical abundance for each particle, while the dashed lines represent
equilibrium abundance curves. In each case, we indicate the DM mass, mass-
splitting, effective coupling and number of particles in the chain on the
plot.
### IV.1 The Zombie Chain
Next we study a chain based on zombie type-interactions. It was shown in
Kramer:2020sbb that a zombie process can allow for a heavy DM candidate
without a chain, but with an unstable DM particle. Here we introduce a chain
of zombie interactions, and show that it supports exponentially larger mass
for each additional particle in the chain, up to the bound set by Eq. (23),
while allowing for completely stable DM.
The zombie chain consists of $N$ DM particles, $\chi_{i}$ ($i=1...N$) with
zombie-type nearest neighbor interactions
$\chi_{i}+\chi_{i+1}\leftrightarrow\chi_{i+1}+\chi_{i+1}\,.$ (36)
For simplicity, we will assume that each chain interaction has the same
strength and that the mass hierarchy is constant going from the heaviest to
lightest. Namely, we take $\left<\sigma
v\right>_{i,{i+1}\to{i+1},{i+1}}\equiv\alpha^{2}/m_{\chi_{i}}^{2}$ and
$m_{i+1}/m_{i}$ independent of $i$. The last particle in the chain,
$\chi_{N}$, is assumed to also have direct annihilations into the SM bath
particles
$\chi_{N}+\chi_{N}\leftrightarrow{\rm SM}+{\rm SM}\,.$ (37)
Since the abundance of $\chi_{N}$ is determined by standard self-
annihilations, and is assumed to be stable, its mass must be less that the
WIMP unitarity bound, i.e., $m_{\chi_{N}}\lesssim 100$ TeV.
For the zombie interactions, the annihilation rate is
$\Gamma_{i,{i+1}\to{i+1},{i+1}}=n_{{i+1}}^{\rm eq}\left<\sigma
v\right>_{i,{i+1}\to{i+1},{i+1}}\propto
e^{-\frac{m_{{i+1}}}{m_{i}}\frac{m_{{i}}}{T}}$ (38)
and here $b_{\rm zombie}=m_{i+1}/m_{i}$. Following Eq. (32) and Eq. (35), we
see that in order to go beyond the WIMP unitarity bound, each link in the
chain must satisfy
$m_{i}\leq 3m_{i+1}=3b\,m_{i}$ (39)
The highest mass is obtained for the smallest allowed value $b=1/3$; below
this value, the DM is unstable. Furthermore, for the interactions above, each
$\chi_{i}$ is absolutely stable, although each will have an exponentially
smaller relic abundance than the previous.
Precise solutions to the relic abundance can be obtained by solving the
coupled Boltzmann equations. However, the freezeout occurs sequentially: first
$\chi_{1}$ freezes out from $\chi_{2}$, then $\chi_{2}$ freezes out from
$\chi_{3}$, and so on. At each stage each particle freezes out with its
neighbors still in chemical equilibrium. For each $\chi_{i}$, we can simply
use Eqs. (16) and (17) to determine the abundance.
For $b=1/3$, the maximum allowed mass for $\chi_{1}$ is given by Eq. (26),
which gives
$m_{\chi_{1}}\lesssim 10^{11}\leavevmode\nobreak\ \rm GeV$ (40)
(for $\alpha_{\rm eff}=10$). In terms of the last particle in the chain, the
${\chi_{1}}$ mass is $m_{\chi_{1}}=3^{N-1}m_{\chi_{N}}$. At least $N=13$
particles are required in the chain to achieve $m_{\chi_{1}}\simeq
10^{11}\leavevmode\nobreak\ \rm GeV$, if $m_{\chi_{N}}\simeq
100\leavevmode\nobreak\ \rm TeV$.
In the left panel of Fig. 2, we show the numerical solution for $N=18$,
$b=0.337$, $\alpha_{\rm eff}=1$, and ${m_{\chi_{1}}=3.6\times 10^{9}}$ GeV,
which exceeds the WIMP unitarity bound Eq. (3). The thermal evolution of the
comoving density is shown for each $\chi_{i}$. The abundances of $\chi_{1}$
through $\chi_{N-1}$ freezeout via zombie processes and have decreasing relic
abundance $m_{\chi_{i}}Y_{\chi_{i}}/m_{\chi_{i+1}}Y_{\chi_{i+1}}\simeq 70$.
The abundance of $\chi_{N}$ freezes out via self annihilations Eq. (37), and
therefore has a relatively large relic abundance, but much smaller than the
${\chi_{1}}$ abundance.
### IV.2 The Inverse Decay Chain
Consider freezeout of $\chi$ via inverse decay (INDY) process Garny:2017rxs ;
Frumkin:2021zng
$\chi+\gamma^{\prime}\to\psi,$ (41)
where $\gamma^{\prime}$ is a light particle in equilibrium with the bath. It
is easy to see from detailed balance that the rate for depleting $\chi$ is
$\Gamma_{\chi+\gamma^{\prime}\to\psi}=\frac{n_{\psi}^{\rm eq}}{n_{\chi}^{\rm
eq}}\Gamma_{\psi\to\chi+\gamma^{\prime}}\propto
e^{-\frac{m_{\psi}-m_{\chi}}{m_{\chi}}x}$ (42)
where $\Gamma_{\psi\to\chi+\gamma^{\prime}}$ is the partial width of $\psi$.
For inverse decays, in terms of the definition of the rate in Eq. (13),
$b=\frac{m_{\psi}-m_{\chi}}{m_{\chi}}\equiv\Delta,\qquad c=0,$ (43)
and
$\alpha_{\rm
eff}=\frac{\Gamma_{\psi\to\chi+\gamma^{\prime}}}{m_{\chi}}(1+\Delta)^{5/2}\,.$
(44)
The suppression of the rate in temperature is only due to the mass-splitting
between $\psi$ and $\chi$. Therefore, the rate only decreases when the mass
splitting is of order the temperature, i.e. $x\Delta>1$. For sufficiently
small $\Delta$, decoupling and freezeout can occur arbitrarily late. For these
reasons it is expected that arbitrarily high DM masses are possible.
However, for the reasons discussed in the previous section, if $m_{\chi}$ is
well above the WIMP unitarity bound, then there needs to be a way for $\psi$
to deplete in order to not be over abundant. If it is unstable while decaying
in equilibrium, then certainly $\chi$ will be too short lived to be DM. Thus,
we need to consider $\psi$ depleting by inverse decays as well. To this end,
we consider a chain of inverse decays in order to achieve arbitrarily high DM
mass. The chain must be long enough so that the phase space in the $\chi$
decay is small enough to stabilize the DM on cosmological scales.
Consider the chain of inverse decay processes
$\chi_{i}+\gamma^{\prime}\to\chi_{i+1},$ (45)
for $i=1\ldots N$, and the decay for $\chi_{N}$ directly into the Standard
Model bath
$\chi_{N}\to{\rm sm}+{\rm sm}.$ (46)
The decay rate $\chi_{N}$ is taken to be fast, so that it is an equilibrium
process, i.e., $\Gamma_{\chi_{N}\to{\rm sm}+{\rm sm}}>H(m_{\chi_{N}})$. For
simplicity we consider the case where the decay width for each process in the
chain is constant
${\Gamma\equiv\Gamma_{\chi_{i+1}\to\chi_{i}+\gamma^{\prime}}}$ and a constant
mass splitting ${m_{\chi_{i+1}}=m_{\chi_{i}}(1+\Delta)}$.
The relic abundance can be found by solving the $N$ coupled Boltzmann
equations:
$\displaystyle xH\frac{\partial Y_{1}}{\partial x}$ $\displaystyle=$
$\displaystyle{\Gamma}{}\left(Y_{2}-Y_{1}\frac{n_{{2}}^{\rm eq}}{n_{1}^{\rm
eq}}\right)$ $\displaystyle\vdots$ $\displaystyle xH\frac{\partial
Y_{i}}{\partial x}$ $\displaystyle=$
$\displaystyle{\Gamma}{}\left(Y_{{i+1}}-Y_{i}\frac{n_{{i+1}}^{\rm
eq}}{n_{i}^{\rm eq}}\right)+{\Gamma}{}\left(Y_{{i-1}}\frac{n_{{i}}^{\rm
eq}}{n_{{i-1}}^{\rm eq}}-Y_{{i}}\right)$ $\displaystyle\vdots$ $\displaystyle
xH\frac{\partial Y_{N}}{\partial x}$ $\displaystyle=$
$\displaystyle{\Gamma}{}\left(Y_{{N-1}}\frac{n_{{N}}^{\rm eq}}{n_{{N-1}}^{\rm
eq}}-Y_{{N}}\right)-{\Gamma_{N\to{\rm sm}}}\left(Y_{N}-Y_{N}^{\rm
eq}\right)\,,$
where $x=m_{\chi_{1}}/T$.
We show a numerical solution in the right panel of Fig. 2 for $N=30$,
$\Delta=0.01$, $\alpha_{\rm eff}=1$, and $m_{\chi_{1}}=7.35\times 10^{11}$
GeV, which exceeds the WIMP unitarity bound, Eq. (3). One can see that the
analytic mass-coupling relationship derived in Eq. (23) does not well-describe
the results here. Similar to what was observed for the coscattering chain in
Ref. Kim:2019udq , the chain can weaken the strength of the DM depletion. For
scattering, it was shown that the effective rate for annihilation is
suppressed by $N^{2}$. In other words, one should replace $\alpha_{\rm
eff}\to\alpha_{\rm eff}/N^{2}$ in order to obtain an estimate for correct
coupling needed to match the observed abundance. Exact analytic solutions to
the coupled set of equations are difficult to obtain, and we leave this to
future work inversechain .
Finally, we comment on the stability of the DM for the example shown. The DM
candidate, $\chi_{1}$, can decay to $N$ SM particles via $N-1$ off-shell
$\chi_{i}$ particles. We estimate the phase-space and choose $N=30$ as a safe
value that guarantees that the phase space in the $\chi$ decay is small enough
to stabilize the DM on cosmological scales.
## V Outlook
In this paper we have derived general formula for the thermal relic abundance
of dark matter for an arbitrary freezeout process, and have outlined the
roadmap to obtain a thermal relic with mass above the standard WIMP
perturbative unitarity bound of a few hundred TeV.
Thermal freezeout provides a well-motivated framework for studying the relic
abundance of DM. The relic abundance can be determined in the instantaneous
freezeout approximation for most cases with Eq. (16), or with Eq. (23) which
corrects for when freezeout is slow. Depending on the process controlling
freezeout, there is a different bound on the dark matter mass from
perturbative unitarity, which can be substantially higher than that of the
WIMP standard lore. To go beyond the standard WIMP unitarity bound, we find
that there are particles degenerate with the DM, possibly constituting a long
chain of DM interactions.
Much work remains to be done to better understand heavy dark matter in context
of a chain and to populate the model parameter space that achieves such heavy
thermal relics. We leave this to future work.
Acknowledgments. We thank Hyun Min Lee and Josh Ruderman for useful
discussions. We are especially grateful to Yonit Hochberg and Nadav
Outmezguine for useful discussions and comments on the manuscript. The works
of RF, EK, IL, and TS are supported by the US-Israeli Binational Science
Foundation (grants 2016153 and 2020220), the Israel Science Foundation (grant
No. 1111/17), and by the I-CORE Program of the Planning and Budgeting
Committee (grant No. 1937/12).
## References
* (1) R.A. Alpher, H. Bethe and G. Gamow, _The origin of chemical elements_ , _Phys. Rev._ 73 (1948) 803.
* (2) R.H. Cyburt, B.D. Fields, K.A. Olive and T.-H. Yeh, _Big Bang Nucleosynthesis: 2015_ , _Rev. Mod. Phys._ 88 (2016) 015004 [1505.01076].
* (3) B.W. Lee and S. Weinberg, _Cosmological Lower Bound on Heavy Neutrino Masses_ , _Phys. Rev. Lett._ 39 (1977) 165.
* (4) R.J. Scherrer and M.S. Turner, _On the Relic, Cosmic Abundance of Stable Weakly Interacting Massive Particles_ , _Phys. Rev. D_ 33 (1986) 1585.
* (5) F. D’Eramo and J. Thaler, _Semi-Annihilation of Dark Matter_ , _JHEP_ 06 (2010) 109 [1003.5912].
* (6) Y. Hochberg, E. Kuflik, T. Volansky and J.G. Wacker, _Mechanism for Thermal Relic Dark Matter of Strongly Interacting Massive Particles_ , _Phys. Rev. Lett._ 113 (2014) 171301 [1402.5143].
* (7) E. Kuflik, M. Perelstein, N.R.-L. Lorier and Y.-D. Tsai, _Elastically Decoupling Dark Matter_ , _Phys. Rev. Lett._ 116 (2016) 221302 [1512.04545].
* (8) J.M. Cline, H. Liu, T. Slatyer and W. Xue, _Enabling Forbidden Dark Matter_ , _Phys. Rev. D_ 96 (2017) 083521 [1702.07716].
* (9) R.T. D’Agnolo, D. Pappadopulo and J.T. Ruderman, _Fourth Exception in the Calculation of Relic Abundances_ , _Phys. Rev. Lett._ 119 (2017) 061102 [1705.08450].
* (10) M. Garny, J. Heisig, B. Lülf and S. Vogl, _Coannihilation without chemical equilibrium_ , _Phys. Rev. D_ 96 (2017) 103521 [1705.09292].
* (11) J. Smirnov and J.F. Beacom, _New Freezeout Mechanism for Strongly Interacting Dark Matter_ , _Phys. Rev. Lett._ 125 (2020) 131301 [2002.04038].
* (12) E.D. Kramer, E. Kuflik, N. Levi, N.J. Outmezguine and J.T. Ruderman, _Heavy Thermal Dark Matter from a New Collision Mechanism_ , _Phys. Rev. Lett._ 126 (2021) 081802 [2003.04900].
* (13) P.J. Fitzpatrick, H. Liu, T.R. Slatyer and Y.-D. Tsai, _New Pathways to the Relic Abundance of Vector-Portal Dark Matter_ , 2011.01240.
* (14) R. Frumkin, Y. Hochberg, E. Kuflik and H. Murayama, _Thermal Dark Matter from Freezeout of Inverse Decays_ , 2111.14857.
* (15) K. Griest and M. Kamionkowski, _Unitarity Limits on the Mass and Radius of Dark Matter Particles_ , _Phys. Rev. Lett._ 64 (1990) 615.
* (16) D. Carney et al., _Snowmass2021 Cosmic Frontier White Paper: Ultraheavy particle dark matter_ , 2203.06508.
* (17) K. Griest and D. Seckel, _Three exceptions in the calculation of relic abundances_ , _Phys. Rev._ D43 (1991) 3191.
* (18) R.T. D’Agnolo and J.T. Ruderman, _Light Dark Matter from Forbidden Channels_ , _Phys. Rev. Lett._ 115 (2015) 061301 [1505.07107].
* (19) H. Kim and E. Kuflik, _Superheavy Thermal Dark Matter_ , _Phys. Rev. Lett._ 123 (2019) 191801 [1906.00981].
* (20) A. Berlin, _Wimps with Guts: Dark Matter Coannihilation with a Lighter Species_ , _Phys. Rev. Lett._ 119 (2017) 121801 [1704.08256].
* (21) Planck collaboration, _Planck 2018 Results. Vi. Cosmological Parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
* (22) J.A. Dror, E. Kuflik and W.H. Ng, _Codecaying Dark Matter_ , _Phys. Rev. Lett._ 117 (2016) 211801 [1607.03110].
* (23) R. Frumkin, Y. Hochberg, E. Kuflik, I. Lavie and T. Silverwater work in progress.
|
11institutetext: School of Physical Science and Technology, Xinjiang
University, Urumqi 830046, PR China
11email<EMAIL_ADDRESS>22institutetext: Center for Theoretical
Physics, Xinjiang University, Urumqi 830046, PR China
33institutetext: College of Mechanical and Electronic Engineering, Tarim
University, Alar 843300, PR China
44institutetext: CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
55institutetext: School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100101, China
# Li-rich and super Li-rich giants produced by element diffusion††thanks: The
full version of Table LABEL:tab1 only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
Jun Gao 1122 Chunhua Zhu 1122 Jinlong Yu 33 Helei Liu 1122 Xizhen Lu 1122
Jianrong Shi 4455 Guoliang Lü 112*2*
###### Abstract
Context. About 0.2-2% of giant stars are Li-rich, whose lithium abundance
(A(Li)) is higher than 1.5 dex. Among them, near 6% are super Li-rich with
A(Li) exceeding 3.2 dex. Meanwhile, the formation mechanism of these Li-rich
and super Li-rich giants is still under debate.
Aims. Considering the compact He core of red giants, attention is paid to the
effect of element diffusion on A(Li). In particular, when the He core flash
occurs, the element diffusion makes the thermohaline mixing zone extend inward
and connect to the inner convection region of stars. Then, a large amount of
7Be produced by the He flash can be transferred to stellar surface, finally
turning into 7Li. Thus, the goal of this work is to propose the mechanism of
A(Li) enrichment and achieve the consistency between the theoretical and
observation data.
Methods. Using the Modules for Experiments in Stellar Astrophysics (MESA), we
simulate the evolution of low-mass stars, with considering the effects of
element diffusion on the Li abundances. The timescale ratio of Li-rich giants
to normal giants is estimated by population synthesis method. Then we get the
theoretical value of A(Li) and make a comparison with observations.
Results. Considering the influence of element diffusion in the model results
in the increase of lithium abundance up to about 1.8 dex, which can reveal Li-
rich giants. Simultaneously, introducing high constant diffusive mixing
coefficients ($D_{\rm mix}$) with the values from $10^{11}$ to $10^{15}$cm2
s-1 in the model allows A(Li) to increase from 2.4 to 4.5 dex, which can
explain the most of Li-rich and super Li-rich giant stars. The population
synthesis method reveals that the amount of Li-rich giants among giants is
about 0.2-2%, which is consistent with observation estimated levels.
Conclusions. In our model, the element diffusion mainly triggered by the
gravity field changes the mean molecular weight at the junction zone between
the stellar envelope and the He core, which makes the thermohaline mixing
region expanding to the inner convection region of stars. A transport channel,
efficiently transporting 7Be in the hydrogen burning region of the star to the
convective envelope where 7Be decays into 7Li, is formed. Combing a high
constant diffusive mixing coefficients, the transport channel can explain the
origin of Li-rich and super Li-rich giant, even the most super Li-rich giants.
###### Key Words.:
stars: evolution; standards; stars: low-mass; diffusion; stars: abundances
## 1 Introduction
Lithium (Li) is one of the important elements to study the origin of the
universe. In the evolution of low-mass stars, Li begins to deplete in the main
sequence (MS) stage. This process experienced the first dredge-up and some
deep mixing, and most of the Li will be consumed. The phenomenon has been
predicted by a standard staller evolution model(Deepak & Reddy 2019), and
confirmed by numerous observations (Brown et al. 1989; Lind et al. 2009; Liu
et al. 2014; Kirby et al. 2016).
Since the first giant star with a high Li abundance was discovered by
Wallerstein & Sneden (1982), the development of the related research field
challenged the traditional stellar evolution mechanism. These stars were
called Li-rich giants, with a classic definition of A(Li) $\geq$ 1.5 dex(Iben
1967a, b). 111Here, A(Li) is the Li abundance expressed as A(Li) = log [n (Li)
/ n (H)] + 12, where n (Li) and n (H) is the number density of Li and
hydrogen, respectively. It is noteworthy that Li-rich giants are extremely
rare, accounting about 1%-2% or even less among all normal giants(Brown et al.
1989; Kumar et al. 2011; Liu et al. 2014; Casey et al. 2016; Kirby et al.
2016; Monaco et al. 2011; Li et al. 2018; Gao et al. 2019). In particular, the
ratios estimated from some large survey programs are $\sim$0.9% from Gaia-ESO
survey (Casey et al. 2016; Smiljanic et al. 2018), $\sim$0.8% from RAVE
(Ruchti et al. 2011) and $\sim$0.2%-0.3% from SDSS and GALAH data (Martell &
Shetrone 2013; Deepak & Reddy 2019).
In the past 40 years, various Li-rich giant stars have been successively
identified. Based on the Large Sky Area Multi-Object Fiber Spectroscopic
Telescope (LAMOST) low resolution spectra acquired in China: in 2018, a star
with the highest Li abundance was found,with A(Li) $\sim$ 4.5 dex(Yan et al.
2018). The Li-rich giants with A(Li) higher than 3.2 dex are called super Li-
rich giants, whose amount among all Li-rich giants is about 6%(Singh et al.
2021). In the period from October 2011 to June 2019, a total of 10,535 Li-rich
giants with A(Li) $\geq$ 1.5 dex were screened from the LAMOST low-resolution
spectra, which allowed one to expand the existing observation sample database
by about 5 times and greatly enriches research samples.
There are two main hypotheses about the origin of Li-rich giant stars: one is
that Li comes from the outside of stars, such as planetary engulfment or
pollution of binary companion stars (Stephan et al. 2018; Lodders 2019;
Holanda et al. 2020). The other is through
3He($\alpha$,$\gamma$)7Be(e,$\nu$)7Li, also known as Cameron-Fowler (CF)
mechanism (Cameron 1955; Cameron & Fowler 1971). In the latter process, 7Be
produced at a high-temperature region of H burning has to be quickly carried
away to a low-temperature region, such as the convective envelope, where it
will decay into 7Li, and this fresh 7Li will survive. As is predicted by
Schwab (2020) that there is enough 7Be in the H-burning shell prior to the
first helium (He) subflash.
Combining the Kepler and spectroscopy information, Yan et al. (2021) confirmed
that most Li-rich stars are red clump (RC), while a few of them are red giant
branch (RGB). In order to explain the Li enhancement in RC stars, Mori et al.
(2021) introduces a neutrino magnetic moment (NMM), which shows that 7Be
production becomes more active owing to the fact that the delay of He flash
makes themohaline mixing more effective when the NMM is excited. Thus Li was
produced at the tip of the red giant branch (TRGB). However, they did not
attempt to explain the origin of super Li-rich giants. At the same time, Kumar
has also used the GALAH DR2(Buder et al. 2018) and Gaia DR2(Gaia Collaboration
2018) data to confirm that the Li abundance of RC stars is 40 times higher
than those at TRGB(Kumar et al. 2020). Perhaps, this abnormality of Li
abundance might be due to a complex interaction between TRGB and RC phase. The
most significant event between these two stages is the occurrence of a He
flash, especially the first, strongest He subflash.
Recently, Schwab (2020) has reported that the He flash induced mixing links
the H-burning shell and the convective zone, and plenty of 7Be circulate
toward the cooler convective zone to turn into 7Li through the CF mechanism.
The model proposed by Schwab (2020) implies there is the possibility of very
high Li abundances (see Fig.4), although the author notes that these high Li
abundances are quickly depleted in their model. Therefore, searching for a
physical mechanism of Li enrichment (including super Li-rich giant stars) to
obtain the consistency between observation and theory is of great significance
for fundamental and applied research. It is well known that the surface
chemical abundance of a star can be affected by many factors such as
convection, thermohaline mixing or element diffusion(e. g., Kippenhahn et al.
1980; Dupuis et al. 1992; Zhu et al. 2021). The effects of convection and
thermohaline mixing on Li abundances of RGB stars have been investigated by
Yan et al. (2018) and Martell et al. (2021). However, the element diffusion is
seldom considered. Element diffusion is a dynamic process that changes the
distribution of chemical elements in stars. It is mainly the result of the
joint action of pressure, temperature, material concentration, and other
factors. Element diffusion plays a very important role in the stellar
evolution, especially in the chemical element distribution on the stellar
surface(Semenova et al. 2020).
In this paper, we consider the effects of the element diffusion impact on the
Li abundance for stars in the RC phase. In particular, Sect. 2 describes the
details of the stellar model and element diffusion. Sect. 3 presents the Li
abundances predicted by our model and the comparative analysis of the
observation and theoretical results. The summary is given in Sect. 4.
## 2 Models
We use Modules for Experiments in Stellar Astrophysics (MESA, [rev. 12778];
Paxton et al. (2011, 2013, 2015, 2018, 2019)) to construct one-dimensional
low-mass stellar models. MESA adopts the equation of state of Rogers &
Nayfonov (2002) and Timmes & Swesty (2000) and the opacity of Iglesias &
Rogers (1996, 1993) and Ferguson et al. (2005).
Our model use the standard MESA $pp\\_and\\_cno\\_extras$ nuclear network,
which includes 25 species and the reactions covering the pp-chains and CNO-
cycles. We adopt nuclear reaction rates compiled by JINA REACLIB (Cyburt et
al. 2010). Treatment of electron screening is based on Alastuey & Jancovici
(1978) and Itoh et al. (1979). The mass loss formula in Reimers (1975) is
adopted. We use the electron-capture rate on 7Be from Simonucci et al. (2013),
as made available in machine-readable form by Vescovi et al. (2019). Models
are initialized on the pre-main sequence with the Asplund et al. (2009) solar
abundance pattern and Z = 0.014. This initializes Li to the meteoritic
abundance A(Li) = 3.26 dex.
The size of convective zone depends on mixing length parameter. We adopt a
mixing length of 1.8 times the pressure scale height. These models include
thermohaline mixing($\alpha_{\rm th}$) using the Kippenhahn et al. (1980)
prescription with an efficiency of $\alpha_{\rm th}$ = 100. This gives the
deep mixing necessary to destroy 7Li on the first ascent giant branch.
Elemental diffusion in stars is mainly driven by a combination of pressure
gradients (or gravity), temperature gradients, compositional gradients, and
radiation pressures. The main driving factor of element diffusion is gravity
sedimentation(Paxton et al. 2015, 2018). In the model, we input a mixing
diffusive coefficient $D_{\rm mix}$$\sim$($\Delta$R)2/($\Delta$t) to show the
efficiency of element diffusion in different regions. Standard stellar
evolution theory points out that the H-burning core in the MS stage
continuously generates helium elements, which are deposited into the stellar
interior, so as to form a helium core. The He core reaches a certain mass, it
will begin to burn. We speculate that the influence of element diffusion will
affect the element abundance on the star surface, which will affect the
formation process of helium core, the occurrence time of the first He flash,
and then stimulate the thermohline mixing to make the mixing process more
sufficient.
By solving the Burgers equation (Burgers 1969), Thoul et al. (1994) proposed a
general method to arrange the whole set of equations into a single matrix
equation, that is to input Burgers equation into the matrix structure without
readjusting any number. There is no approximation of the relative
concentrations of various species, nor is there any limitation on the number
of elements to be considered. Therefore, this method is suitable for a wide
variety of astrophysical problems. Using the method of Thoul et al. (1994),
MESA can calculate the diffusion of chemical elements in stellar interior
(Paxton et al. 2015, 2018).
The inputs provided by the MESA model are the number densities $n_{\rm s}$,
temperature $T$, the gradients of these quantities $d$ ln $n_{\rm s}$/$dr$ and
$d$ ln $T$/$dr$, species mass in atomic units $A_{\rm s}$, species mean charge
as an average ionization state $\overline{Z}_{\rm s}$, and the resistance
coefficients $K_{\rm st}$, $z_{\rm st}$, $z_{\rm st}$′ and $z_{\rm st}$′′, as
defined by Equation (86) in Paxton et al. (2015). In our model, diffusion
coefficients $D_{\rm mix}$ derived from Paquette et al. (1986) and updated by
Stanton & Murillo (2016). By calculating the characteristic duration of the
first He flash and the characteristic length scale of the mixing region, this
suggest a effective mixing diffusion coefficient requires $D_{\rm mix}$ > 1010
cm 2 s-1(Schwab 2020). Together with the mean ionization states, these are key
parts of the input physics that determine the diffusion of all ions. The
additional acceleration terms $g_{\rm rad,s}$ of radiation suspension is set
to zero by default.
## 3 Result
Figure 1: The evolution of Li on the stellar surface for 1 $M_{\odot}$ model.
The x axis is the Li abundance. The y axis represents the logarithm of
luminosity. The lines indicates the evolution process of Li abundance.
According to Yan et al. (2021), the Li-rich RGB and RC stars have different
mass distributions with the peaks at about 1.7 $M_{\odot}$ and 1.2
$M_{\odot}$, respectively. In general, when the stellar mass at a zero age
main sequence (ZAMS) is lower than 0.9 $M_{\odot}$, the star hardly evolves
into a giant phase. Simultaneously, when the stellar mass is larger than 1.8
$M_{\odot}$, the temperature in the envelope of the star in the giant phase is
sufficiently high. As a result, beryllium (Be) elements produced by the
H-burning shell are quickly destroyed and cannot be brought to the surface of
the star. Therefore, we calculated the evolution of Li abundance for the
models with masses of 0.9, 1.0, 1.2, 1.4, 1.6, and 1.8 $M_{\odot}$. For
simplicity, a 1.0 $M_{\odot}$ model was taken as a sample.
### 3.1 Effect of element diffusion
The solid line in Fig. 1 shows the standard evolutionary track. It shows that
low mass star samples usually start from the MS turnoff with A(Li) $\sim$ 3.2
dex and suffer depletion through the first dredge-up. Then the star reaches
the RGB bump with a luminosity of 101.6 $L_{\odot}$, where it begins to
consume Li rapidly again(Iben 1967a; Charbonnel & Zahn 2007; Lattanzio et al.
2015). The black solid line represents the stellar evolution without element
diffusion at Stage II. It shows the luminosity drops sharply to the level of
RC, whilst maintaining the RGB tip A(Li)(Kumar et al. 2020). The black dotted
line denotes the model with element diffusion. A(Li) successfully increases
from -1 to 1.8 dex in Stage II. As a result, the element diffusion improves
the efficiency of thermohaline mixing, and its activity range gets connected
with the convective zone of stellar interior. Also, the 7Be in the H-burning
shell get transferred to the convective envelope, where they decay into 7Li
through the CF mechanism. This figure reveals that the increase of Li
abundance up to 1.8 dex can be realized by considering the element diffusion.
Almost all explanations proposed for Li-rich giants involve the He flash(Kumar
et al. 2020; Mori et al. 2021; Schwab 2020). Based on the standard model of
stellar structure and evolution, the He-core burning occurs when the mass of
the He core increases to $M_{\rm He}$ $\approx$ 0.45 $M_{\odot}$(Thomas 1967;
Bildsten et al. 2012). Therefore, the stellar evolution in this work was
divided into two stages: Stage I is from the MS stage to $M_{\rm He}$ = 0.45
$M_{\odot}$. The next range until the RC stage is called Stage II.
Figure 1 displays the evolution trajectory of Li abundance on the surface of a
1$M_{\odot}$ star without and with the element diffusion. As shown by the
dark-blue solid line, the evolution of Li abundance in both models was similar
during Stage I. In the MS phase, the Li abundance is kept constant. When a
low-mass star evolves into a red giant, it undergoes the first dredge-up.
After the first dredge-up, Li starts to rapidly decrease because the Li
elements are mixed up the stellar interior material by the envelope deepen
process and then are diluted. This depletion is due to dilution as the
envelope deepens and mixes up interior material heavily depleted in Li. Before
the RGB tip, the Li abundance continued to drop due to the thermohaline
mixing. From the ZAMS to the RGB tip, the Li abundance on the stellar surface
decreased by about 4 orders of magnitudes, which was consistent with the
results of Kumar et al. (2020).
Figure 2: Profiles of element abundances (top left corner) and diffusion
coefficient (top right corner) ($D_{\rm mix}$) on the first He flash. The
solid lines are for model with element diffusion, and the dashed lines are for
models without element diffusion. The left and right panels at the bottom show
the relative changes between the mean molecular weights ($\mu_{\rm diff}$ and
$\mu$) with and without element diffusion, and between the mean molecular
weight gradients ($\nabla\mu_{\rm diff}$ and $\nabla\mu$) with and without
element diffusion, respectively.
The He flash emerges on the RGB tip. The Li abundance in the model without
element diffusion was falling during Stage II. However, A(Li) in the case of
element diffusion increased from -1 to 1.8 dex. It indicates that element
diffusion can enhance the Li abundance on the stellar surface. The main reason
for this phenomenon is depicted in Fig. 2. Especialy, a compact He core is
formed on the RGB tip, whose strong gravity can produce the efficient element
diffusion, resulting in the expansion of the thermohaline mixing zone. As
shown in the top-right panel of Fig. 2, for the model with element diffusion,
the thermohaline mixing zone connected to the convection zone in the stellar
interior in which the He flash occurred, that is, it extended to the deeper
interior of the hydrogen burning region. Therefore, the large amounts of 7Be,
produced by the H-burning shell, could be transferred to the stellar surface,
finally turning into 7Li. On the other hand, for the model without element
diffusion, 7Be could not or rarely be brought to the stellar envelope due to
the disconnection of the stellar interior convection zone and the thermohaline
mixing zone.
Thermohaline convection is a turbulent mixing process that can occur in
stellar radiative regions whenever the mean molecular weight increases with
radius. In some cases, it can have a significant observable impact on stellar
structure and evolution(Ulrich 1972; Kippenhahn et al. 1980; Brown et al.
2013; Garaud 2018). The left and right panels at the bottom of the Fig. 2 show
the relative changes of mean molecular weights ($\mu_{\rm diff}$ and $\mu$)
and mean molecular weight gradients ($\nabla\mu_{\rm diff}$ and $\nabla\mu$)
respectively at the junction zone between the stellar envelope and the He
core. It can be seen that due to the influence of element diffusion, the
$\mu_{\rm diff}$ and $\nabla\mu_{\rm diff}$ have decreased significantly. The
local decrease of mean molecular weight can drive a more efficient
thermohaline mixing. It expands to the inner convection region of stars, which
is shown by pink lines in the top-right panel. These indicate that element
diffusion can greatly affect the mean molecular weight and the mean molecular
weight gradient, which leads to the expansion of the mixing region of
thermohaline convection.
The essence of this phenomenon is that the element diffusion mainly triggered
by the gravity field suppress the the mean molecular weight and changes the
element concentration gradient at the junction zone between the stellar
envelope and the He core. It is one of the most important factors which affect
the thermohaline mixing. It is well known that the mean molecular weight
greatly affects the efficiency and range of thermohaline mixing, and make the
thermohaline mixing region expanding to the inner convection region of stars.
That is, the thermohaline mixing can occur in the internal area of hydrogen
burning, and bring products and by-products of nuclear reactions to the
surface. Thus, a transport channel, efficiently transporting 7Be in the
hydrogen burning region of the star to the convective envelope where 7Be
decays into 7Li, is formed.
### 3.2 Effect of element diffusion with constant diffusive coefficients
Although A(Li) on the surface of the star predicted by the model with element
diffusion can increase up to about 1.8 dex, it cannot explain the formation of
the super Li-rich giants.
Figure 3: Similar to Fig. 1, but for the evolutional tracks of Li abundances of four models. |
---|---
|
Figure 4: Similar to Fiure 1, but for the models with different masses and
constant diffusive mixing coefficient of $D_{\rm mix}$ to $10^{11}$ cm2 s-1,
$10^{12}$ cm2 s-1, $10^{13}$ cm2 s-1 and $10^{15}$ cm2 s-1. Different values
are displayed in the upper left corner of each subgraph. The solid and dashed
lines show the evolutional tracks at Stages I and II, respectively. The red
and blue circles are the Li-rich giant stars and RC stars listed in Table 1,
respectively. The star represent the most Li-rich giant star, TYC 429-2097-1
observed by Yan et al. (2018). Figure 5: The evolution of A(Li) after the
first He flash with time. The solid lines represents the increase produced
with element diffusion and different constant diffusive coefficient( $D_{\rm
mix}$). The dotted line indicates the A(Li) $\sim$ 1.5 dex and A(Li) $\sim$
3.2 dex.
According to Fig. 2, the diffusive coefficient is a very important factor for
the formation of the Li-rich giant. Very recently, in order to produce the Li-
rich giant, Schwab (2020) has considered that turbulent convective motions can
excite internal gravity waves, and a chemical mixing occurs when the
luminosity of the He flash ($L_{\rm He}$) is higher than $10^{4}$ L⊙. In
particular, the effective diffusive mixing coefficient was estimated to be
about $10^{11}$ cm2 s-1. Given the constant diffusive mixing coefficients
$D_{\rm mix}$ of $10^{10}$, $10^{12}$ and $10^{14}$ cm2 s-1 from different
models, the maximum value of A(Li) calculated by Schwab (2020) was about 3.6
dex. Meanwhile, the model proposed by Schwab (2020) fails to explain the
origin of super Li-rich giants.
In this section, combining the element diffusion and constant diffusive mixing
coefficients, we calculate the evolution of A(Li) on the stellar surface.
Following Schwab (2020), we assume a constant diffusive mixing coefficient for
the mixing when $L_{\rm He}$ ¿ $10^{4}$ L⊙. According to Fig. 2, $D_{\rm mix}$
during the He flash could reach $10^{15}$ cm2 s-1. Therefore, in this work,
$D_{\rm mix}=10^{11}$, $10^{12}$, $10^{13}$ and $10^{15}$ cm2 s-1 were adopted
in different models.
Figure 3 displays the A(Li) evolution on the surface of a star with the mass
of 1 $M_{\odot}$. Using the model with a constant diffusive mixing coefficient
of $10^{12}$ cm2 s-1 but without element diffusion could enhance the Li
abundance up to about 1.0 dex, which agreed with the result of Schwab (2020).
In the model with element diffusion, the Li abundance could be increased to
1.8 dex. At this time, the inner convection region was connected with the
thermohaline mixing zone to form a channel for transporting 7Be elements,
which greatly increased the 7Li on the stellar surface. Surprisingly, the
model combining the element diffusion and the constant diffusive mixing
coefficient exhibited an increase of A(Li) up to 3.4 dex. The reason for this
is that the diffusive mixing coefficient improves the mixing efficiency of the
channel excited by the element diffusion. Therefore, A(Li) in the model with
the element diffusion and the constant diffusive mixing coefficient can keep a
constant value above 3.0 dex.
### 3.3 Li-rich giants and super Li-rich giants
In recent years, many large survey programs have revealed the existence of
numerous Li-rich giants. Combining the astrometric data from the Gaia
satellite (Gaia Collaboration et al. 2016) with spectroscopic abundance
surveys (such as GALAH survey, LAMOST survey), twenty Li-rich abundances could
be identified. Based on GALAH DR2 and DR3 surveys, Deepak & Reddy (2019),
Deepak et al. (2020), Kumar et al. (2020) and Martell et al. (2021) measured
the Li abundances of 1872 giant stars. According to LAMOST survey, Singh et
al. (2019) and Yan et al. (2021) explored the Li abundances of 456 giant stars
which are in the Kepler field. In order to compare with the theoretical
results with observation samples in this work, we selected 351 published Li-
rich giants with precise values of luminosity, temperature and Li abundance as
our samples (see Table LABEL:tab1).
Figure 4 depicts the observed data on Li-rich giants and the theoretical
results for the models with different $D_{\rm mix}$s and masses. In this
study, an increase in $D_{\rm mix}$ from $10^{11}$ to $10^{15}$ cm2 s-1 led to
a rise in A(Li) from 2.4 to 4.5 dex. For the models with element diffusion and
$D_{\rm mix}>10^{12}$ cm2 s-1, the evolutionary tracks passed through most of
observed samples of super Li-rich giants (A(Li) $\geq$ 3.2 dex), Li-rich
giants (A(Li $\geq$ 1.5 dex) to the normal giants. Especially, the value of
A(Li), calculated in the model with element diffusion and $D_{\rm
mix}=10^{15}$ cm2 s-1, reached 4.5 dex, which could explain the Li abundance
of the most super Li-rich giants.
### 3.4 Population synthesis for Li-rich giants
As mentioned in the Introduction, the Li-rich giants among giants are scarce
(about 0.2-2%). Based on the models in this work, we estimate the theoretical
ratio by the population synthesis method which is used in the previous
investigations by our group(Lü et al. 2006, 2009, 2013, 2020; Yu et al. 2019,
2021; Zhu et al. 2021).
In the population synthesis method for single-star systems, the initial mass
function (IMF) is the most important input parameter. The IMF used in the
present research was derived from the stellar distribution toward both
Galactic poles as well as that within 5.2 pc of the Sun by Kroupa et al.
(1993). Based on this IMF, 106 stars were produced by Monte Carlo calculation.
In order to estimate their percentage, the lifetimes of Li-rich giants and
giants were afterward estimated.
Figure 5 displays the evolution of A(Li) in all models. After the first He
flash, the element diffusion firstly initiates and enhances the Li abundance.
According to Schwab (2020), the constant diffusive mixing coefficient can work
when $L_{\rm He}>$ 104 $L_{\odot}$. The Li abundance greatly increased after
about 0.2 Myr of the first He flash, and remained constant within several Myr
due to a constant $D_{\rm mix}$. Of course, A(Li) decreases when the stellar
luminosity is lower than 104 $L_{\odot}$. When the mass is greater than about
1.9 M⊙, the element diffusion and constant diffusive mixing coefficient are
not excited because the temperature within the stellar envelope is too high.
Simultaneously, the lifetimes of the Li-rich giants decrease with the increase
of $D_{\rm mix}$. The main reason for this is that the high diffusive
coefficient accelerates the circulation process of elements, so that Be or Li
elements can be quickly carried to a high-temperature zone and destroyed.
In this study, MESA was applied to calculate the evolution of stars with
initial masses of 0.9 M⊙, 1.0 M⊙, 1.2 M⊙, 1.4 M⊙, 1.6 M⊙ and 1.8 M⊙. Through a
linear interpolation method, the lifetimes of Li-rich giants and giants for
these 106 stars were estimated. Then, the percentage of Li-rich giants among
giants was assessed. In particular, the ratio values of models with $D_{\rm
mix}=10^{11}$, $10^{12}$, $10^{13}$ and $10^{15}$ cm2 s-1 were 0.5, 1.2, 1.1
and 0.2%, respectively, which was consistent with the observational estimates.
Very recently, Zhang et al. (2021) has shown that the deterioration of Li in
the RC stage is not so obvious and the low ratio (such as 0.2%) may be an
anomaly. If it is true, the high diffusive coefficient (such as $10^{15}$ cm2
s-1) may be undesirable. It means that the model proposed in this study fails
to produce all super Li-rich giants, especially the most Li-rich giant star
TYC 429-2097-1. Thus, the binary merging model proposed by Zhang et al. (2020)
may be competitive.
## 4 Conclusions
Considering the element diffusion, we used MESA to calculate the evolution of
Li abundance. The element diffusion mainly triggered by the gravity field
suppress the the mean molecular weight and changes the element concentration
gradient at the junction zone between the stellar envelope and the He core.
The local decrease of mean molecular weight greatly affects the efficiency and
range of thermohaline mixing, and make the thermohaline mixing region
expanding to the inner convection region of stars. A transport channel,
efficiently transporting 7Be in the hydrogen burning region of the star to the
convective envelope where 7Be decays into 7Li, is formed. Therefore, a large
amount of 7Be, produced by the He flash, could be transferred to the stellar
surface, finally turning into 7Li. However, the value of A(Li) could be
increased up to only 1.8 dex, which was insufficient to produce super Li-rich
giant stars.
In turn, combining the element diffusion and constant diffusive mixing
coefficients enabled one to increase the theoretical A(Li) values from 2.4 to
4.5 dex by increasing $D_{\rm mix}$ from $10^{11}$ to $10^{15}$ cm2 s-1. This
means that the element diffusion in the proposed model can result in the
extension of the thermohaline mixing zone and its connection with the stellar
interior convection zone. Then, 7Be produced by He burning can be mixed in the
stellar envelope. The high diffusive mixing coefficient can improve the
efficiency of 7Be transfer to the stellar surface. Therefore, our model can
produce the Li-rich giants, and even the most of super Li-rich giants. The
results provided by population synthesis method were also consistent with the
observations, which confirmed the feasibility of this mechanism. However, the
accuracy of the results in the model under consideration may be affected by
the uncertain input parameter, $D_{\rm mix}$. Since calculating an accurate
$D_{\rm mix}$ is beyond the scope of this work, attention is rather paid to
the diffusive mixing coefficients.
###### Acknowledgements.
We are grateful to anonymous referee for careful reading of the paper and
constructive criticism. This work received the generous support of the
National Natural Science Foundation of China, project Nos. U2031204, 11863005,
12163005, and 12090044, the science research grants from the China Manned
Space Project with NO. CMS-CSST-2021-A10, and the Natural Science Foundation
of Xinjiang No.2021D01C075.
Table 1: From about 11000 observational samples, the 351 Li-rich giant stars whose luminosities are measured are selected in this work. The observational data come from the references listed in the last column. (The full version of this Table is available at the CDS.) Object ID | Teff (K) | log g (dex) | [Fe/H] (dex) | log(L/$L_{\odot}$) | A(Li) | Reference
---|---|---|---|---|---|---
HD 8676 | 4860 | 2.95 | 0.02 | 1.68 | 3.55 | Kumar et al.(2011)
HD 10437 | 4830 | 2.85 | 0.1 | 1.77 | 3.48 | Kumar et al.(2011)
HD 12203 | 4870 | 2.65 | -0.27 | 1.69 | 2.08 | Kumar et al.(2011)
HD 37719 | 4650 | 2.4 | 0.09 | 1.76 | 2.71 | Kumar et al.(2011)
HD 40168 | 4800 | 2.5 | 0.1 | 2.1 | 1.7 | Kumar et al.(2011)
HD 51367 | 4650 | 2.55 | 0.2 | 1.59 | 2.6 | Kumar et al.(2011)
HD 77361 | 4580 | 2.35 | -0.02 | 1.66 | 3.8 | Kumar et al.(2011)
HD 88476 | 5100 | 3.1 | -0.01 | 1.87 | 2.21 | Kumar et al.(2011)
HD 107484 | 4640 | 2.5 | 0.18 | 1.78 | 2.14 | Kumar et al.(2011)
HD 118319 | 4700 | 2.2 | -0.25 | 1.68 | 2.02 | Kumar et al.(2011)
HD 133086 | 4940 | 2.98 | 0.02 | 1.7 | 2.14 | Kumar et al.(2011)
HD 145457 | 4850 | 2.75 | -0.08 | 1.61 | 2.49 | Kumar et al.(2011)
HD 150902 | 4690 | 2.55 | 0.09 | 1.83 | 2.65 | Kumar et al.(2011)
HD 167304 | 4860 | 2.95 | 0.18 | 1.93 | 2.85 | Kumar et al.(2011)
HD 170527 | 4810 | 2.85 | -0.1 | 1.69 | 3.12 | Kumar et al.(2011)
TYC 429-2097-1 | 4696 | 2.25 | -0.36 | 1.95 | 4.51 | Yan et al. (2018)
Gaia DR2 6423511482552457344 | 4828.68 | 2.84 | 0.18 | 1.56 | 3.54 | Deepak and Reddy(2019)
Gaia DR2 6216747182780840576 | 4773.08 | 2.69 | 0.12 | 1.54 | 3.41 | Deepak and Reddy(2019)
Gaia DR2 3080569351805501824 | 4995.53 | 2.6 | 0.03 | 1.71 | 3.41 | Deepak and Reddy(2019)
Gaia DR2 5920543908525756800 | 4815.52 | 2.68 | 0.14 | 1.54 | 3.39 | Deepak and Reddy(2019)
Gaia DR2 5676420200792553600 | 4854.1 | 2.31 | -0.11 | 1.83 | 3.38 | Deepak and Reddy(2019)
Gaia DR2 6721793108675117440 | 4911.04 | 2.45 | -0.04 | 1.64 | 3.33 | Deepak and Reddy(2019)
Gaia DR2 4488063566731544960 | 4778.94 | 2.37 | -0.02 | 1.52 | 3.27 | Deepak and Reddy(2019)
Gaia DR2 2939800046333110272 | 4985.2 | 2.56 | -0.13 | 1.55 | 3.26 | Deepak and Reddy(2019)
Gaia DR2 4168437628181576192 | 4749.88 | 2.71 | 0.19 | 1.21 | 3.26 | Deepak and Reddy(2019)
Gaia DR2 5229729170925959552 | 5038.38 | 2.79 | -0.15 | 1.65 | 3.24 | Deepak and Reddy(2019)
Gaia DR2 5293680581122445184 | 4832.25 | 2.5 | -0.01 | 1.65 | 3.23 | Deepak and Reddy(2019)
Gaia DR2 5242382659974594688 | 4786.39 | 2.8 | 0.28 | 1.6 | 3.23 | Deepak and Reddy(2019)
Gaia DR2 5628302754467688576 | 4868.94 | 2.37 | -0.2 | 1.57 | 3.21 | Deepak and Reddy(2019)
Gaia DR2 3202012502737830784 | 4906.32 | 2.42 | -0.14 | 1.77 | 3.21 | Deepak and Reddy(2019)
Gaia DR2 5460011229840058880 | 4813.14 | 2.63 | 0.16 | 1.7 | 3.21 | Deepak and Reddy(2019)
Gaia DR2 5452473905831060480 | 4711.6 | 2.17 | -0.3 | 1.84 | 3.2 | Deepak and Reddy(2019)
Gaia DR2 6162898261508964992 | 4835.99 | 2.58 | 0.01 | 1.7 | 3.2 | Deepak and Reddy(2019)
Gaia DR2 6779302244026689920 | 4541.3 | 2.17 | -0.48 | 1.99 | 3.2 | Deepak and Reddy(2019)
Gaia DR2 3496188144418768640 | 4776.14 | 2.59 | 0.04 | 1.63 | 3.2 | Deepak and Reddy(2019)
… | … | … | … | … | … | …
## References
* Alastuey & Jancovici (1978) Alastuey, A. & Jancovici, B. 1978, ApJ, 226, 1034
* Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
* Bildsten et al. (2012) Bildsten, L., Paxton, B., Moore, K., & Macias, P. J. 2012, ApJ, 744, L6
* Brown et al. (1989) Brown, J. A., Sneden, C., Lambert, D. L., & Dutchover, Edward, J. 1989, ApJS, 71, 293
* Brown et al. (2013) Brown, J. M., Garaud, P., & Stellmach, S. 2013, The Astrophysical Journal, 768, 34
* Buder et al. (2018) Buder, S., Asplund, M., Duong, L., et al. 2018, MNRAS, 478, 4513
* Burgers (1969) Burgers, J. M. 1969, Flow Equations for Composite Gases
* Cameron (1955) Cameron, A. G. W. 1955, ApJ, 121, 144
* Cameron & Fowler (1971) Cameron, A. G. W. & Fowler, W. A. 1971, ApJ, 164, 111
* Casey et al. (2016) Casey, A. R., Ruchti, G., Masseron, T., et al. 2016, MNRAS, 461, 3336
* Charbonnel & Zahn (2007) Charbonnel, C. & Zahn, J. P. 2007, A&A, 467, L15
* Cyburt et al. (2010) Cyburt, R. H., Amthor, A. M., Ferguson, R., et al. 2010, ApJS, 189, 240
* Deepak et al. (2020) Deepak, Lambert, D. L., & Reddy, B. E. 2020, MNRAS, 494, 1348
* Deepak & Reddy (2019) Deepak & Reddy, B. E. 2019, Monthly Notices of the Royal Astronomical Society, 484, 2000
* Dupuis et al. (1992) Dupuis, J., Fontaine, G., Pelletier, C., & Wesemael, F. 1992, ApJS, 82, 505
* Ferguson et al. (2005) Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005, ApJ, 623, 585
* Gaia Collaboration (2018) Gaia Collaboration. 2018, VizieR Online Data Catalog, I/345
* Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., Brown, A. G. A., & Vallenari, A. 2016, A&A, 595, A1
* Gao et al. (2019) Gao, Q., Shi, J.-R., Yan, H.-L., et al. 2019, The Astrophysical Journal Supplement Series, 245, 33
* Garaud (2018) Garaud, P. 2018, Annual Review of Fluid Mechanics, 50, 275
* Holanda et al. (2020) Holanda, N., Drake, N. A., & Pereira, C. B. 2020, AJ, 159, 9
* Iben (1967a) Iben, Icko, J. 1967a, ApJ, 147, 650
* Iben (1967b) Iben, Icko, J. 1967b, ApJ, 147, 624
* Iglesias & Rogers (1993) Iglesias, C. A. & Rogers, F. J. 1993, ApJ, 412, 752
* Iglesias & Rogers (1996) Iglesias, C. A. & Rogers, F. J. 1996, ApJ, 464, 943
* Itoh et al. (1979) Itoh, N., Totsuji, H., Ichimaru, S., & Dewitt, H. E. 1979, ApJ, 234, 1079
* Kippenhahn et al. (1980) Kippenhahn, R., Ruschenplatt, G., & Thomas, H. C. 1980, A&A, 91, 175
* Kirby et al. (2016) Kirby, E. N., Guhathakurta, P., Zhang, A. J., et al. 2016, VizieR Online Data Catalog, J/ApJ/819/135
* Kroupa et al. (1993) Kroupa, P., Tout, C. A., & Gilmore, G. 1993, MNRAS, 262, 545
* Kumar et al. (2020) Kumar, Y. B., Reddy, B. E., Campbell, S. W., et al. 2020, Nature Astronomy, 4, 1059
* Kumar et al. (2011) Kumar, Y. B., Reddy, B. E., & Lambert, D. L. 2011, ApJ, 730, L12
* Lattanzio et al. (2015) Lattanzio, J. C., Siess, L., Church, R. P., et al. 2015, MNRAS, 446, 2673
* Li et al. (2018) Li, H., Aoki, W., Matsuno, T., et al. 2018, ApJ, 852, L31
* Lind et al. (2009) Lind, K., Primas, F., Charbonnel, C., Grundahl, F., & Asplund, M. 2009, A&A, 503, 545
* Liu et al. (2014) Liu, Y. J., Tan, K. F., Wang, L., et al. 2014, ApJ, 785, 94
* Lodders (2019) Lodders, K. 2019, in Nuclei in the Cosmos XV, Vol. 219, 165–170
* Lü et al. (2006) Lü, G., Yungelson, L., & Han, Z. 2006, MNRAS, 372, 1389
* Lü et al. (2013) Lü, G., Zhu, C., & Podsiadlowski, P. 2013, ApJ, 768, 193
* Lü et al. (2020) Lü, G., Zhu, C., Wang, Z., et al. 2020, ApJ, 890, 69
* Lü et al. (2009) Lü, G., Zhu, C., Wang, Z., & Wang, N. 2009, MNRAS, 396, 1086
* Martell & Shetrone (2013) Martell, S. L. & Shetrone, M. D. 2013, MNRAS, 430, 611
* Martell et al. (2021) Martell, S. L., Simpson, J. D., Balasubramaniam, A. G., et al. 2021, MNRAS, 505, 5340
* Monaco et al. (2011) Monaco, L., Villanova, S., Moni Bidin, C., et al. 2011, A&A, 529, A90
* Mori et al. (2021) Mori, K., Kusakabe, M., Balantekin, A. B., Kajino, T., & Famiano, M. A. 2021, MNRAS, 503, 2746
* Paquette et al. (1986) Paquette, C., Pelletier, C., Fontaine, G., & Michaud, G. 1986, ApJS, 61, 177
* Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3
* Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4
* Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15
* Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34
* Paxton et al. (2019) Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10
* Reimers (1975) Reimers, D. 1975, Memoires of the Societe Royale des Sciences de Liege, 8, 369
* Rogers & Nayfonov (2002) Rogers, F. J. & Nayfonov, A. 2002, ApJ, 576, 1064
* Ruchti et al. (2011) Ruchti, G. R., Fulbright, J. P., Wyse, R. F. G., et al. 2011, ApJ, 743, 107
* Schwab (2020) Schwab, J. 2020, ApJ, 901, L18
* Semenova et al. (2020) Semenova, E., Bergemann, M., Deal, M., et al. 2020, A&A, 643, A164
* Simonucci et al. (2013) Simonucci, S., Taioli, S., Palmerini, S., & Busso, M. 2013, ApJ, 764, 118
* Singh et al. (2019) Singh, R., Reddy, B. E., Bharat Kumar, Y., & Antia, H. M. 2019, ApJ, 878, L21
* Singh et al. (2021) Singh, R., Reddy, B. E., Campbell, S. W., Kumar, Y. B., & Vrard, M. 2021, ApJ, 913, L4
* Smiljanic et al. (2018) Smiljanic, R., Franciosini, E., Bragaglia, A., et al. 2018, A&A, 617, A4
* Stanton & Murillo (2016) Stanton, L. G. & Murillo, M. S. 2016, Phys. Rev. E, 93, 043203
* Stephan et al. (2018) Stephan, A. P., Naoz, S., & Gaudi, B. S. 2018, AJ, 156, 128
* Thomas (1967) Thomas, H. C. 1967, in Late-Type Stars, ed. M. Hack, 395
* Thoul et al. (1994) Thoul, A. A., Bahcall, J. N., & Loeb, A. 1994, ApJ, 421, 828
* Timmes & Swesty (2000) Timmes, F. X. & Swesty, F. D. 2000, ApJS, 126, 501
* Ulrich (1972) Ulrich, R. K. 1972, The Astrophysical Journal, 172, 165
* Vescovi et al. (2019) Vescovi, D., Piersanti, L., Cristallo, S., et al. 2019, A&A, 623, A126
* Wallerstein & Sneden (1982) Wallerstein, G. & Sneden, C. 1982, ApJ, 255, 577
* Yan et al. (2018) Yan, H.-L., Shi, J.-R., Zhou, Y.-T., et al. 2018, Nature Astronomy, 2, 790
* Yan et al. (2021) Yan, H.-L., Zhou, Y.-T., Zhang, X., et al. 2021, Nature Astronomy, 5, 86
* Yu et al. (2019) Yu, J., Li, Z., Zhu, C., et al. 2019, ApJ, 885, 20
* Yu et al. (2021) Yu, J., Zhang, X., & Lü, G. 2021, MNRAS, 504, 2670
* Zhang et al. (2021) Zhang, J., Shi, J.-R., Yan, H.-L., et al. 2021, ApJ, 919, L3
* Zhang et al. (2020) Zhang, X., Jeffery, C. S., Li, Y., & Bi, S. 2020, ApJ, 889, 33
* Zhu et al. (2021) Zhu, C., Liu, H., Wang, Z., & Lü, G. 2021, A&A, 654, A57
|
# Measurement methods of radial flow in relativistic heavy-ion collisions
Peng Yang Key Laboratory of Quark and Lepton Physics (MOE), Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China Lin Li
School of Science, Wuhan University of Technology, Wuhan 430070, China Yu
Zhou Department of Mathematics, University of California, Los Angeles,
California 90095, USA Zhiming Li Key Laboratory of Quark and Lepton Physics
(MOE), Institute of Particle Physics, Central China Normal University, Wuhan
430079, China Mingmei Xu Key Laboratory of Quark and Lepton Physics (MOE),
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
China Yeyin Zhao Key Laboratory of Quark and Lepton Physics (MOE), Institute
of Particle Physics, Central China Normal University, Wuhan 430079, China
Yuanfang Wu<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics
(MOE), Institute of Particle Physics, Central China Normal University, Wuhan
430079, China
###### Abstract
Radial flow can be directly extracted from the azimuthal distribution of mean
transverse rapidity. We apply the event-plane method and the two-particle
correlation method to estimate the anisotropic Fourier coefficient of the
azimuthal distribution of mean transverse rapidity. Using the event sample
generated by a multiphase transport model with string melting, we show that
both methods are effective. For the two-particle correlation method to be
reliable, the mean number of particles in an azimuthal bin must be above a
certain threshold. Using these two methods, anisotropic radial flow can be
estimated in a model-independent way in relativistic heavy-ion collisions.
###### pacs:
25.75.Nq, 25.75.Dw, 25.75.Ld
## I Introduction
Collectivity, or flow, is one of the main characteristics of the newly formed
quark-gluon plasma in relativistic heavy-ion collisions lab1 ; lab2 ; lab3 ;
lab4 . Traditionally, we have observed radial flow, direct flow, elliptic
flow, triangular flow, and so on. Radial flow and elliptic flow are the two
prominent classes among them.
Elliptic flow is defined as the second Fourier component of the azimuthal
multiplicity distribution. It is generated by the initial geometric asymmetry
of noncentral collisions where the overlap of two incident nuclei is of an
almond shape in the transverse coordinate plane. The minor axis of the overlap
is on the reaction plane, which is spanned by the vector of impact parameter
and beam direction lab1 ; lab4 ; lab5 . This initial geometric asymmetry leads
to a larger density gradient along the in-plane direction, i.e., the
anisotropic distribution of final-state particles in momentum space.
Therefore, elliptic flow provides information about initial conditions and
system properties lab2 ; lab6 .
Radial flow is originally deduced from an analysis of transverse momentum
spectra in central collisions lab1 ; lab4 ; lab7 . Later, it is generalized
into two parameters: isotropic radial velocity and anisotropic radial velocity
lab8 ; lab9 ; lab10 ; lab11 ; lab12 . The isotropic radial velocity features
the isotropic transverse expansion of the source at kinetic freeze-out. The
anisotropic radial velocity presents the difference of the radial flow between
the in-plane direction and the out-of-plane direction and arises in noncentral
collisions.
The interplay of radial expansion and elliptic flow results in what we observe
as the particle mass splitting of the differential elliptic flow lab3 ; lab4 ;
lab6 . Namely, the heavier particles show smaller elliptic flow values. This
mass ordering of elliptic flow has been well understood by hydrodynamics with
a set of kinetic freeze-out constraints, i.e., radial flow, temperature, and
source deformation lab13 ; lab14 .
In addition, momentum transfer due to viscosity is proportional to the first
derivative of the velocity in hydrodynamics lab15 ; lab16 . Bulk viscosity is
associated with isotropic transverse velocity, whereas shear viscosity is
associated with the anisotropic transverse velocity. The proportion constants
are equal to the bulk and shear viscosities, respectively. Therefore, the
determination of anisotropic radial flow is essential for hydrodynamic
calculations and for measuring the shear viscosity in relativistic heavy-ion
collisions lab17 ; lab18 ; lab19 .
Conventionally, anisotropic radial flow is extracted by fitting the transverse
momentum spectrum of particles with the blast-wave parametrization lab20 ;
lab21 . To minimize $\chi^{2}$ of the fitting, the spectra of all particle
species are fitted simultaneously. For the relativistic heavy-ion collisions
at the Brookhaven National Laboratory (BNL) Alternating Gradient Synchrotron,
the BNL Relativistic Heavy Ion Collider (RHIC), and the CERN Large Hadron
Collider energies, the fitting results have all been presented lab22 ; lab23 ;
lab24 ; lab25 . However, these results are obviously model dependent.
Therefore, a model-independent measurement is called for.
In order to measure radial flow, we had introduced the mean-transverse
rapidity (MTR) and its azimuthal distribution, replacing velocity with
rapidity in the original definition of radial flow lab12 ; lab15 ; lab26 . The
MTR is averaged over the number of particles in an azimuthal angle bin. It
presents the raw kinetic expansion in a specified azimuthal direction.
The azimuthal distribution of the MTR presents the transverse expansion of the
source at kinetic freeze-out lab15 ; lab27 ; lab28 . The second Fourier
coefficient of this distribution features the anisotropic kinetic expansion
and is consistent with the anisotropic radial flow extracted from the blast-
wave parametrization lab12 .
The aim of this paper is to estimate the anisotropic radial flow in
relativistic heavy-ion collisions. This estimation is similar to how we
estimate elliptic flow where the reaction plane cannot be measured directly
due to random fluctuations of the impact parameter vector and is instead
estimated from the reconstructed particles. This estimation method is the so-
called event-plane method (EPM) lab29 ; lab30 ; lab31 . Later, it is found
that the two-particle correlation method (TPCM) can also be used in place of
the EPM to estimate the elliptic flow lab32 ; lab33 ; lab34 ; lab35 ; lab36 .
In this paper, we first describe the EPM and the TPCM that are designed to
estimate the anisotropic coefficients of azimuthal distributions for both
total-transverse rapidity (TTR) and MTR in Sec. II. Then, we demonstrate the
effectivity of both methods using the event sample generated by the AMPT model
with string melting in Sec. III. The limitation of the TPCM is discussed in
Sec. IV. Finally, summary and conclusions are presented in Sec. V.
## II Measurement methods
Transverse rapidity is defined as lab2 ; lab12 ; lab15 ; lab26
$y_{T}=\ln\left(\frac{m_{T}+p_{T}}{m_{0}}\right),$ (1)
where $m_{0}$ is the particle mass in the rest frame, $p_{T}$ is transverse
momentum, and $m_{T}=\sqrt{m_{0}^{2}+p_{T}^{2}}$ is the transverse mass.
The TTR of the _m_ th azimuthal bin is the summation of all particles’
transverse rapidities in an event, and its average is lab12 ; lab15 ; lab26
as follows:
$\langle Y_{T}(\phi_{m}-\Psi_{r})\rangle=\frac{1}{N_{\rm
event}}\sum_{k=1}^{N_{\rm
event}}\sum_{i=1}^{N_{m}^{k}}y_{T,i}^{k}(\phi_{m}-\Psi_{r}).$ (2)
It measures total transverse expansion in a specified azimuthal bin
$(\phi_{m}-\Psi_{r})$, where $\Psi_{r}$ is the reaction plane angle.
$\phi_{m}$ is the _m_ th azimuthal angle bin. The bin width is $2\pi/N_{\rm
bin}$. $N_{\rm bin}$ is the total number of bins.
$y_{T,i}^{k}(\phi_{m}-\Psi_{r})$ is the transverse rapidity of the _i_ th
particle in the _k_ th event and _m_ th azimuthal angle bin. $N_{m}^{k}$ is
the total number of particles in the _k_ th event and the _m_ th azimuthal
angle bin. $N_{\rm event}$ is the total number of events.
The MTR of the _m_ th azimuthal bin is defined as
$\langle\langle y_{{\rm T}}(\phi_{m}-\Psi_{r})\rangle\rangle=\frac{1}{N_{\rm
event}}\sum_{k=1}^{N_{\rm
event}}\frac{1}{N_{m}^{k}}\sum_{i=1}^{N_{m}^{k}}y_{T,i}^{k}(\phi_{m}-\Psi_{r}),$
(3)
where $\langle\langle\cdots\rangle\rangle$ is first averaged over the number
of particles in the _k_ th event and the _m_ th azimuthal angle bin
$N_{m}^{k}$, and then over the total number of events. It measures the raw
kinetic expansion in a specified azimuthal direction. The contribution of the
number of particles is removed by the first average.
The Fourier expansions of azimuthal TTR and MTR distributions are as follows:
$\begin{split}&\frac{d\langle
Y_{T}(\phi-\Psi_{r})\rangle}{d(\phi-\Psi_{r})}\\\ &\
=v_{0}(Y_{T})\left(1+\sum_{n=1}^{\infty}2v_{n}(Y_{{\rm
T}})\cos\left[n\left(\phi-\Psi_{r}\right)\right]\right),\end{split}$ (4)
and
$\begin{split}&\frac{d\langle\langle
y_{T}(\phi-\Psi_{r})\rangle\rangle}{d(\phi-\Psi_{r})}\\\ &\
=v_{0}(y_{T})\left(1+\sum_{n=1}^{\infty}2v_{n}(y_{T})\cos\left[n\left(\phi-\Psi_{r}\right)\right]\right),\end{split}$
(5)
where azimuthal-angle-independent $v_{0}(Y_{T})$ and $v_{0}(y_{T})$ are
isotropic TTR flow and radial flow, respectively. The second Fourier
coefficients $v_{2}(Y_{T})$ and $v_{2}(y_{T})$ are defined as anisotropic TTR
flow and radial flow, respectively lab12 ; lab15 ; lab26 .
They are similar to the mean multiplicity in the _m_ th azimuthal angle bin,
i.e.,
$\langle N(\phi_{m}-\Psi_{r})\rangle=\frac{1}{N_{\rm event}}\sum_{k=1}^{N_{\rm
event}}N^{k}(\phi_{m}-\Psi_{r}),$ (6)
where $N^{k}(\phi_{m}-\Psi_{r})$ is the number of particles in the _k_ th
event and the $(\phi_{m}-\Psi_{r})$ direction. Its azimuthal distribution in
Fourier expansion is as follows:
$\begin{split}&\frac{d\langle N(\phi-\Psi_{r})\rangle}{d(\phi-\Psi_{r})}\\\ &\
=v_{0}(N)\left(1+\sum_{n=1}^{\infty}2v_{n}(N)\cos\left[n\left(\phi-\Psi_{r}\right)\right]\right),\end{split}$
(7)
where $v_{0}(N)$ is azimuthal-angle independent. $v_{2}(N)$ is the elliptic
flow, which is estimated, in practice, by the EPM and the TPCM. In the
following, we apply these two methods to estimate $v_{2}(Y_{T})$ and
$v_{2}(y_{T})$.
### II.1 The event-plane method
In nuclear-nuclear collisions, the azimuthal angle of the final-state particle
is measured with respect to the reaction plane. However, the angle of reaction
plane is unknown in experiments and fluctuates from event to event. Usually, a
reconstructed event plane is considered as a substitute lab29 , i.e.,
$\Psi_{n}$ of the _n_ th coefficient is given by
$\Psi_{n}=\left.\left(\tan^{-1}\frac{Q_{n,y}}{Q_{n,x}}\right)\middle/n\right.,$
(8)
where $Q_{n}$ is the flow vector and defined as,
$Q_{n,x}=\sum_{i=1}^{M}\omega_{i}\cos(n\phi_{i})=Q_{n}\cos(n\Psi_{n}),$ (9)
$Q_{n,y}=\sum_{i=1}^{M}\omega_{i}\sin(n\phi_{i})=Q_{n}\sin(n\Psi_{n}).$ (10)
_M_ is the number of particles used in the event-plane determination.
$\phi_{i}$ and $\omega_{i}$ are the azimuthal angle and the weight of the _i_
th particle.
A general Fourier expansion with respect to the event-plane ($\Psi_{n}$) can
be expressed as
$\begin{split}&\frac{d\langle\omega
N(\phi-\Psi_{n})\rangle}{d(\phi-\Psi_{n})}\\\ &\ =v_{0}(\omega
N)\left(1+\sum_{n=1}^{\infty}2v_{n}^{\rm obs}(\omega
N)\cos\left[n\left(\phi-\Psi_{n}\right)\right]\right),\end{split}$ (11)
where weight $\omega$ changes with the observable lab29 , such as different
kinds of flow defined in Eqs. (4), (5), and (7).
For multiplicity, the weight is unity, and Eq. (11) corresponds to Eq. (7).
Its $v_{2}^{\rm obs}(N)$ is as follows:
$v_{2}^{\rm obs}\left(N\right)=\frac{1}{N_{\rm event}}\sum_{k=1}^{N_{\rm
event}}\left(\frac{1}{N^{k}}\sum_{i=1}^{N^{k}}\cos\left[2\left(\phi_{i}^{k}-\Psi_{2}\right)\right]\right),$
(12)
where $\Psi_{2}$ is the event-plane angle of the second harmonic and $N^{k}$
is the total number of particles in the _k_ th event.
For the TTR and MTR, their weights are $\omega_{i}=y_{T,i}$ and
$\omega_{i}=\left.y_{T,i,m}\middle/N_{m}\right.$, respectively. Their
$v_{2}^{\rm obs}(Y_{T})$ and $v_{2}^{\rm obs}(y_{T})$ are as follows:
$\begin{split}v_{2}^{\rm obs}\left(Y_{T}\right)=&\frac{1}{N_{\rm
event}}\sum_{k=1}^{N_{\rm event}}\\\
&\left(\frac{1}{\sum_{i=1}^{N^{k}}y_{T,i}^{k}}\sum_{i=1}^{N^{k}}y_{T,i}^{k}\cos\left[2\left(\phi_{i}^{k}-\Psi_{2}\right)\right]\right),\end{split}$
(13)
and
$\begin{split}v_{2}^{\rm obs}\left(y_{T}\right)=&\frac{1}{N_{\rm
event}}\sum_{k=1}^{N_{\rm event}}\left(\frac{1}{\sum_{m=1}^{N_{\rm
bin}}\frac{1}{N_{m}^{k}}\sum_{k=1}^{N_{m}^{k}}y_{T,i,m}^{k}}\right.\\\
&\left.\sum_{m=1}^{N_{\rm
bin}}\frac{1}{N_{m}^{k}}\sum_{k=1}^{N_{m}^{k}}y_{T,i,m}^{k}\cos\left[2\left(\phi_{i,m}^{k}-\Psi_{2}\right)\right]\right).\end{split}$
(14)
With finite multiplicity, $v_{2}^{\rm obs}(N)$ has to be corrected for the
event-plane resolution lab4 ; lab29 , i.e.,
$\rm Re_{2}=\left\langle\cos\left[2(\Psi_{2}-\Psi_{r})\right]\right\rangle,$
(15)
which is obtained by the iteration of sub-event-plane resolution. The subevent
is usually constructed in a longitudinal rapidity window that is different
from the particles of interest lab23 ; lab37 . Therefore, the elliptic flow is
as follows:
$v_{2}(N)=\left.v_{2}^{\rm obs}(N)\middle/\rm Re_{2}\right..$ (16)
After the same corrections for both $v_{2}^{\rm obs}(Y_{T})$ and $v_{2}^{\rm
obs}(y_{T})$, $v_{2}(Y_{T})$ and $v_{2}(y_{T})$ are as follows:
$v_{2}(Y_{T})=\left.v_{2}^{\rm obs}(Y_{T})\middle/\rm Re_{2}\right.,$ (17)
and
$v_{2}(y_{T})=\left.v_{2}^{\rm obs}(y_{T})\middle/\rm Re_{2}\right..$ (18)
### II.2 The two-particle correlation method
The coefficients of Fourier expansion Eq. (7) can also be estimated by the
TPCM lab30 , i.e.,
$\left\langle\cos\left[n\left(\phi_{1}-\phi_{2}\right)\right]\right\rangle=\left\langle
e^{in\left(\phi_{1}-\phi_{2}\right)}\right\rangle=\left\langle
v_{n}^{2}\right\rangle+\delta_{n}.$ (19)
Here a cross term that depends both on $v_{n}$ and on $\delta_{n}$ is
neglected as it is very small in most cases of interest. $\delta_{2}$
represents the so-called nonflow contribution and is irrelevant to initial
geometry. All correlations are first averaged over all particle pairs in a
given event and then over all events. The latter average involves weight
depending on event. For convenience, single-event average two-particle
azimuthal correlations is defined as
$\left\langle 2\right\rangle^{k}\equiv\frac{1}{\sum_{i,j=1,i\neq
j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}}\sum_{i,j=1,i\neq
j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}e^{in\left(\phi_{i}^{k}-\phi_{j}^{k}\right)}.$
(20)
Its average over all events is as follows:
$\begin{split}\left\langle\left\langle
2\right\rangle\right\rangle&\equiv\frac{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j=1,i\neq j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}\left\langle
2\right\rangle^{k}}{\sum_{k=1}^{N_{\rm event}}\sum_{i,j=1,i\neq
j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}}\\\ &=\frac{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j=1,i\neq
j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}e^{in\left(\phi_{i}^{k}-\phi_{j}^{k}\right)}}{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j=1,i\neq j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}}.\end{split}$
(21)
For multiplicity and TTR, their weights are $\omega_{i}=1$ and
$\omega_{i}=y_{T,i}$, respectively. So their $\left\langle\left\langle
2\right\rangle\right\rangle$ are
$\left\langle\left\langle
2\right\rangle\right\rangle_{N}=\frac{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j,i\neq
j}^{N^{k}}e^{in\left(\phi_{i}^{k}-\phi_{j}^{k}\right)}}{\sum_{k=1}^{N_{\rm
event}}N^{k}\left(N^{k}-1\right)},$ (22)
and
$\left\langle\left\langle
2\right\rangle\right\rangle_{Y_{T}}=\frac{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j,i\neq
j}^{N^{k}}y_{T,i}^{k}y_{T,j}^{k}e^{in\left(\phi_{i}^{k}-\phi_{j}^{k}\right)}}{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j,i\neq j}^{N^{k}}y_{T,i}^{k}y_{T,j}^{k}}.$ (23)
Elliptic flow can be estimated from two-particle correlations by lab30 ; lab31
$v_{2}(N)=\sqrt{\left\langle\left\langle 2\right\rangle\right\rangle_{N}}.$
(24)
Similarly, corresponding anisotropic coefficient of TTR can be estimated by,
$v_{2}(Y_{T})=\sqrt{\left\langle\left\langle
2\right\rangle\right\rangle_{Y_{T}}}.$ (25)
Equation (25) is obtained in the same way as Eq. (24) except for the nonunit
weight factor.
For MTR, its weight is $\omega_{i}=\left.y_{T,i,m}\middle/N_{m}\right.$. The
two correlated particles could come from the same azimuthal angle bin or from
different bins. If they come from different bins, the single-event average
two-particle azimuthal correlations is as follows:
$\begin{split}\langle 2\rangle_{\rm d}^{k}=&\sum_{m,n=1,m\neq n}^{N_{\rm
bin}}\left(\frac{1}{N_{m}^{k}N_{n}^{k}}\right.\\\
&\left.\sum_{i=1}^{N_{m}^{k}}\sum_{j=1}^{N_{n}^{k}}y_{T,i,m}^{k}y_{T,j,n}^{k}e^{in(\phi_{i,m}^{k}-\phi_{j,n}^{k})}\right).\end{split}$
(26)
Here $N_{m}^{k}$ and $N_{n}^{k}$ are the numbers of particles in the _m_ th
and _n_ th bin, respectively. If they come from the same bin, the single-event
average two-particle azimuthal correlations are as follows:
$\begin{split}\langle 2\rangle_{\rm s}^{k}=&\sum_{m=1}^{N_{\rm
bin}}\left(\frac{1}{N_{m}^{k}(N_{m}^{k}-1)}\right.\\\ &\left.\sum_{i,j=1,i\neq
j}^{N_{m}^{k}}y_{T,i,m}^{k}y_{T,j,m}^{k}e^{in\left(\phi_{i,m}^{k}-\phi_{j,m}^{k}\right)}\right).\end{split}$
(27)
Therefore, $\left\langle\left\langle 2\right\rangle\right\rangle$ of the
azimuthal distribution of MTR is as follows:
$\langle\left\langle 2\rangle\right\rangle_{y_{T}}=\frac{\sum_{k=1}^{N_{\rm
event}}\langle 2\rangle^{k}_{\rm d}+\langle 2\rangle^{k}_{\rm
s}}{\sum_{k=1}^{N_{\rm event}}W^{k}},$ (28)
where $W^{k}$ is the event weight, i.e.,
$\begin{split}W^{k}=&\sum_{m,n=1,m\neq n}^{N_{\rm
bin}}\frac{1}{N_{m}^{k}N_{n}^{k}}\sum_{i=1}^{N_{m}^{k}}\sum_{j=1}^{N_{n}^{k}}y_{T,i,m}^{k}y_{T,j,n}^{k}\\\
\ &+\sum_{m=1}^{N_{\rm
bin}}\frac{1}{N_{m}^{k}\left(N_{m}^{k}-1\right)}\sum_{i,j=1,i\neq
j}^{N_{m}^{k}}y_{T,i,m}^{k}y_{T,j,m}^{k}.\end{split}$ (29)
The corresponding anisotropic radial flow is as follows:
$v_{2}(y_{T})=\sqrt{\left\langle\left\langle
2\right\rangle\right\rangle_{y_{T}}}.$ (30)
Up to now, we derive the anisotropic coefficients of azimuthal multiplicity,
TTR and MTR distributions by both the EPM and the TPCM. In order to check how
these two methods work in practice, we apply them to the event sample
generated by Monte Carlo simulation in the following section.
## III Application
Figure 1: Centrality dependence of elliptic flow (solid black dots),
anisotropic radial flow (solid red stars), anisotropic TTR flow (solid blue
triangles), and the summation of elliptic flow and anisotropic radial flow
(solid purple squares). (a) is the true results. (b) and (c) are the results
estimated by the EPM and the TPCM, respectively. The error of each point is
smaller than the symbol size.
The AMPT model with string melting lab38 ; lab39 can reproduce the observed
elliptic flow at the RHIC lab32 ; lab33 ; lab34 ; lab35 . The event sample
generated by this model has the basic characteristics of flows, in general,
and can be used to test the effectivity of the two aforementioned methods.
With this model, we generate $4\times 10^{6}$ events for Au + Au collisions at
200 GeV. The analysis is presented in the kinetic ranges $|\eta|\leq 2.5$, and
$p_{T}\in\left[0.15,2\right]\rm GeV/\mathit{c}$ which is the same as the
RHIC/STAR Collaboration lab33 ; lab35 . Nine centrality bins are defined by
the $N_{\rm part}$ corresponding to the nine multiplicity-ranges, consistent
with those at the RHIC/STAR Collaboration lab35 .
For the EPM, two subevents [(a) and (b)] for resolution corrections are
constructed in two longitudinal symmetry windows
$|\eta|\in\left[3.3,4.5\right]$, similar to those at the RHIC/STAR
Collaboration lab23 . We also use a random method to make the number of
charged particles in each subevent even lab29 ; lab33 .
Since the azimuthal angle of the true reaction plane $\Psi_{r}$ is known and
set to zero in this model, $v_{2}(Y_{T}),v_{2}(N)$, and $v_{2}(y_{T})$ can be
directly obtained by the corresponding azimuthal distributions. These directly
obtained $v_{2}(Y_{T}),v_{2}(N)$, and $v_{2}(y_{T})$ can be regarded as their
true values. Their centrality dependence are represented in Fig. 1(a) by solid
blue triangles, black dots, and red stars, respectively.
Figure 2: Centrality dependence of (a) $v_{2}(N)$, (b) $v_{2}(Y_{T})$, and
(c) $v_{2}(y_{T})$ where solid black dots are true values, solid red stars,
and solid blue triangles are estimated by the EPM and the TPCM, respectively.
The error of each point is smaller than the symbol size.
In Fig. 1(a), three kinds of flows have similar centrality dependence,
consistent with those obtained from UrQMD lab27 ; lab28 ; lab40 . In each of
the centrality interval, $v_{2}(Y_{T})$ is the largest, $v_{2}(N)$ is medial,
and $v_{2}(y_{T})$ is the smallest. It is no surprise that $v_{2}(N)$ and
$v_{2}(y_{T})$ are both smaller than $v_{2}(Y_{T})$. From the definitions of
Eqs. (2), (3), and (6), TTR contains the contributions from both the number of
particles (multiplicity) and the transverse movement. MTR is raw kinetics, and
the influence of multiplicity is excluded. Therefore, $v_{2}(y_{T})$ and
$v_{2}(N)$ are both smaller than $v_{2}(Y_{T})$.
It is interesting that $v_{2}(y_{T})+v_{2}(N)$ (purple squares) overlaps with
$v_{2}(Y_{T})$ (blue triangles) at each of the nine centralities. This
relationship shows that the anisotropic radial flow and elliptic flow are
additive, although MTR and multiplicity are not simply related to TTR.
In order to compare the anisotropic coefficients estimated by the two methods
with their true values of $v_{2}(N),v_{2}(Y_{T})$, and $v_{2}(y_{T})$ are
presented in Figs. 2(a)-2(c), respectively, where solid black dots are the
true values, solid red stars, and blue triangles are estimated by the EPM and
the TPCM.
For $v_{2}(N)$ as shown in Fig. 2(a), in each centrality interval, the black
dot is the lowest, the red star is in the middle, and the blue triangle is the
highest. This shows that the estimation given by TPCM is higher that given by
the EPM, and the estimations given these two methods are both higher than the
true value. The excess parts are the contributions of nonflow and flow
fluctuations lab41 ; lab42 . Nevertheless, the difference is roughly an
overall shift. The general trends of three data sets are consistent with each
other and with the expectation that the larger elliptic flow appears in the
midcentral collisions and small elliptic flow in peripheral and central
collisions. Therefore, both methods can be considered as a valid estimation of
true elliptic flow.
Similarly, for $v_{2}(Y_{T})$ as shown in Fig. 2(b), in each centrality
interval, three kinds of points have the same position ordering as shown in
Fig. 2(a), i.e., the lowest black dot, the middle red star, and the highest
blue triangle. By the same reasoning, these results demonstrate that the both
methods are effective in estimating $v_{2}(Y_{T})$.
For $v_{2}(y_{T})$ as shown in Fig. 2(c), three kinds of points also display
the same ordering as shown in Figs. 2(a) and 2(b), i.e., the lowest black
dots, the middle red stars, and the highest blue triangles for the seven
middle and central collisions. These results are not really surprising. As
even in some toy Monte Carlo studies where there is only flow fluctuations and
no systematic biases due to nonflow, the elliptic flows given by the EPM and
the TPCM are still different lab42 .
The centrality dependence of elliptic flow, anisotropic TTR flow, and radial
flow estimated by the EPM and the TPCM are presented in Figs. 1(b) and 1(c),
respectively, in comparison to their true values in Fig. 1(a). The summations
of elliptic flow $v_{2}(N)$ and anisotropic radial flow $v_{2}(y_{T})$ (purple
squares) estimated by these two methods also overlap with corresponding
anisotropic TTR flow $v_{2}(Y_{T})$ (blue triangles), consistent with the case
in Fig. 1(a). Therefore, the elliptic flow and anisotropic radial flow
estimated by these two methods are also additive, just like their true values.
This demonstrates that the two methods are equally effective.
However, the TPCM is ineffective in estimating $v_{2}(y_{T})$ for the two
peripheral-collision bins. We will show in the following section why and how
the TPCM fails.
## IV The limitation of the TPCM
$v_{2}(y_{T})$ given by the TPCM in Eq. (30) is the square root of
$\langle\langle 2\rangle\rangle_{y_{T}}$. For the two peripheral-collision
bins, $\langle\langle 2\rangle\rangle_{y_{T}}$ becomes negative. This is why
the TPCM fails.
We can understand the cause of this problem from the original definition of
MTR. MTR is defined to be averaged over $N_{m}$ (the number of particles in
the _m_ th bin). If all particles are uniformly distributed in the whole
azimuthal region $N_{m}\sim N/N_{\rm bin}$ ($N$ is multiplicity, and $N_{\rm
bin}$ is the total number of bins). Now, assuming $\langle\langle
2\rangle\rangle_{y_{T}}$ is well defined, we should have selected a proper
$N_{\rm bin}$ for which $N_{m}$ is large enough.
To examine the proper range of $N_{\rm bin},\langle\langle
2\rangle\rangle_{y_{T}}$ versus $N_{\rm bin}$ at the nine centralities are
presented in Fig. 3. For midcentral collisions and central collisions,
$\langle\langle 2\rangle\rangle_{y_{T}}$ decreases with $N_{\rm bin}$ when
$N_{\rm bin}\leq 20$, and then becomes independent of $N_{\rm bin}$ when
$N_{\rm bin}\geq 20$. There is a wide range of $N_{\rm bin}$, where
$\langle\langle 2\rangle\rangle_{y_{T}}$ stays flat. This range guarantees
that $\langle\langle 2\rangle\rangle_{y_{T}}$ is independent of $N_{\rm bin}$.
Therefore, we choose $N_{\rm bin}=25$ for all the aforementioned analysis.
Figure 3: $\langle\langle 2\rangle\rangle_{y_{T}}$ versus $N_{\rm bin}$ at
the nine centralities where the error of each point is smaller than the symbol
size.
However, for the two peripheral-collision bins, $\langle\langle
2\rangle\rangle_{y_{T}}$ rapidly decreases to negative as $N_{\rm bin}$
increases. Obviously, this is not caused by the change in $N_{\rm bin}$ but
instead by the decrease of multiplicity $N$ from central to peripheral
collisions. When $N_{\rm bin}$ is fixed, $N_{m}$ is solely proportional to
$N$. Therefore, $N_{m}$ also decreases from central to peripheral collisions.
To find the lower threshold of $\langle N_{m}\rangle$ in general, we randomly
drop some particles in each event. Using this method, we can show how
$\langle\langle 2\rangle\rangle_{y_{T}}$ changes with $\langle N_{m}\rangle$,
and where in each centrality interval the TPCM becomes invalid.
$\langle\langle 2\rangle\rangle_{y_{T}}$ versus $\langle N_{m}\rangle$ for
four cases are presented in Figs. 4(a)-4(d). For the two upper subfigures, we
choose the pseudorapidity range $|\eta|\leq 2.5$ and $N_{\rm bin}=25$ for Fig.
4(a) and $N_{\rm bin}=35$ for Fig. 4(b). In Fig. 4(a), with a decrease in
$\langle N_{m}\rangle,\langle\langle 2\rangle\rangle_{y_{T}}$ stays flat and
positive. This implies that $\langle\langle 2\rangle\rangle_{y_{T}}$ can still
be estimated correctly, even though some particles in each of the events are
dropped. When $\langle N_{m}\rangle$ decreases even lower, $\langle\langle
2\rangle\rangle_{y_{T}}$ rapidly drops to negative. This implies that the
correlations between the two particles decrease significantly, and the method
becomes invalid. Meanwhile, Fig. 4(b) also shows the same $\langle
N_{m}\rangle$ which makes $\langle\langle 2\rangle\rangle_{y_{T}}$ rapidly
drop to negative. Therefore, independent of $N_{\rm bin}$, as long as the mean
number of particles in a bin is less than 10, the TPCM will fail.
Figure 4: $\langle\langle 2\rangle\rangle_{y_{T}}$ versus $\langle
N_{m}\rangle$ at the nine centralities where the error of each point is
smaller than the symbol size.
For the two lower subfigures, we choose the pseudorapidity range $|\eta|\leq
1$ and $N_{\rm bin}=25$ for Fig. 4(c) and $N_{\rm bin}=35$ for Fig. 4(d). In
contrast with the two upper row subfigures, narrow pseudorapidity range makes
the total multiplicity smaller. The general trends in Figs. 4(c) and 4(d) are
consistent with each other and with the two upper subfigures. This means that
$\langle\langle 2\rangle\rangle_{y_{T}}$ is independent of the total mean
multiplicity $\langle N\rangle$ and $N_{\rm bin}$ but is dependent on $\langle
N_{m}\rangle$. Both Figs. 4(c) and 4(d) show negative $\langle\langle
2\rangle\rangle_{y_{T}}$ when $\langle N_{m}\rangle<10$. Therefore, the lower
threshold of $\langle N_{m}\rangle$ is ten for the TPCM.
In addition, one may ask why there is no such lower threshold in the EPM. This
is because $v_{2}(y_{T})$ given by Eq. (30) is of higher order than that given
by Eq. (18). The right side of Eq. (30) is the square root of two-particle
correlations. Therefore, higher precision is required for the TPCM.
Moreover, as shown in Fig. 1, anisotropic radial flow is the smallest one
among three. Its measurement is more difficult than anisotropic TTR flow, or
elliptic flow. Therefore, a larger number of two-particle pairs is required.
## V Summary and conclusions
Anisotropic radial flow can be directly extracted from the azimuthal
distribution of the MTR. First, we apply the EPM and the TPCM to estimate the
anisotropic coefficients of the azimuthal distributions of TTR and MTR in
relativistic heavy-ion collisions.
Then, using the event sample of Au + Au collisions at 200 GeV generated by the
AMPT model with string melting, we show that the EPM and the TPCM are both
effective in estimating the anisotropic TTR flow. The EPM is also effective in
estimating anisotropic radial flow.
For central collisions and midcentral collisions, the TPCM is effective in
estimating anisotropic radial flow. However, for the two peripheral-collision
bins, this method fails when the mean number of particles in a bin becomes
less than ten.
Therefore, anisotropic radial flow can be estimated in a model-independent way
by the EPM and the TPCM in relativistic heavy-ion collisions. For the TPCM to
be reliable, the mean number of particles in an azimuthal bin must be above a
certain threshold.
We find that the summation of the true elliptic flow and true anisotropic
radial flow is consistent with the true anisotropic TTR flow. This
relationship also holds for the corresponding coefficients estimated by the
EPM and the TPCM. Therefore, elliptic flow and anisotropic radial flow are
additive.
## VI Acknowledgement
We are very grateful to Dr. G. Wang for his valuable comments and suggestions.
We thank Dr. A. Tang, F. Wang, Y. Zhou, and Z. Lin for helpful discussions.
This work was supported, in part, by the Ministry of Science and Technology
(MoST) under Grant No. 2016YFE0104800 and the Fundamental Research Funds for
the Central Universities under Grant No. CCNU19ZN019.
## References
* (1) J. Y. Ollitrault, Nucl. Phys. A 638, 195c (1998).
* (2) P. Braun-Munzinger, J. Stachel, Nature (London) 448, 302 (2007).
* (3) S. A. Voloshin, Phys. Rev. C 55, R1630(R) (1997).
* (4) S. A. Voloshin, A. M. Poskanzer, and R. Snellings, in _Landolt-Boernstein, Relativistic Heavy Ion Physics_ , Vol. 1/23 (Springer-Verlag, 2010), pp. 5–54.
* (5) T. Hirano, arXiv:nucl-th/9904082.
* (6) P. Danielewicz, Phys. Rev. C 51, 716 (1995).
* (7) C. M. Hung and E. Shuryak, Phys. Rev. C 57, 1891 (1998).
* (8) P. Huovinen and P. V. Ruuskanen, Annu. Rev. Nucl. Part. Sci. 56, 163 (2006).
* (9) M. Shao, L. Yi, Z. B. Tang, H. F. Chen _et al._ , J. Phys. G: Nucl. Part. Phys. 37, 085104 (2010).
* (10) Z. B. Tang, L. Yi, L. J. Ruan _et al._ , Chin. Phys. Lett. 30, 031201 (2013).
* (11) J. Chen, J. Deng, Z. Tang, Z. Xu, and L. Yi, arXiv: 2012.02986.
* (12) L. Li, N. Li, and Y. F. Wu, J. Phys. G: Nucl. Part. Phys. 40, 075104 (2013).
* (13) P. Huovinen, P. F. Koblb, U. W. Heinz, P. V. Ruuskanen, and S. A. Voloshin, Phys. Lett. B 503, 58 (2001).
* (14) N. Borghini and J. Y. Ollitrault, Phys. Lett. B 642, 227 (2006).
* (15) L. Li, N. Li, and Y. F. Wu, Chin. Phys. C 36, 423 (2012).
* (16) L. D. Landau and E. M. Lifshitz, _Fluid Mechanics_ , 2nd. ed., Vol. 6, Course of Theoretical Physics (Pergamon, Oxford, 1987), p.44.
* (17) C. Shen and U. Heinz, Phys. Rev. C 85, 054902 (2012).
* (18) Iu. A. Karpenko, P. Huovinen, H. Petersen, and M. Bleicher, Phys. Rev. C 91, 064901 (2015).
* (19) P. Liu and R. A. Lacey, Phys. Rev. C 98, 021902 (2018).
* (20) E. Schnedermann, J. Sollfrank, and U. W. Heinz, Phys. Rev. C 48, 2462 (1993).
* (21) Y. Oh, Z. W. Lin, and C. M. Ko, Phys. Rev. C 80, 064902 (2009).
* (22) J. Barrette _et al._ (E877 Collaboration), Phys. Rev. C 62, 024901 (2000).
* (23) L. Adamczyk _et al._ (STAR Collaboration), Phys. Rev. C 93, 014907 (2016).
* (24) L. Adamczyk _et al._ (STAR Collaboration), Phys. Rev. C 96, 044904 (2017).
* (25) B. Abelev _et al._ (ALICE Collaboration), Phys. Rev. C 88, 044910 (2013).
* (26) P. Yang, L. Li, and Y. F. Wu, Beijing: Sciencepaper Online, 201404-454 (2014) [arXiv:1405.0686].
* (27) S. Sarkar, P. Mali, and A. Mukhopadhyay, Phys. Rev. C 95, 014908 (2017).
* (28) S. Sarkar, P. Mali, S. Ghosh, and A. Mukhopadhyay, Adv. High Energy Phys. 2018, 7453752 (2018).
* (29) A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 58, 1671 (1998).
* (30) A. Bilandzic, R. Snellings, and S. A. Voloshin, Phys. Rev. C 83, 044913 (2011).
* (31) N. Borghini, P. M. Dinh, and J. Y. Ollitrault, Phys. Rev. C 64, 054901 (2001).
* (32) C. Adler _et al._ (STAR Collaboration), Phys. Rev. C 66, 034904 (2002).
* (33) B. I. Abelev _et al._ (STAR Collaboration), Phys. Rev. C 77, 054901 (2008).
* (34) L. Adamczyk _et al._ (STAR Collaboration), Phys. Rev. C 86, 054908 (2012).
* (35) J. Adams _et al._ (STAR Collaboration), Phys. Rev. C 72, 014904 (2005).
* (36) S. Wang, Y. Z. Jiang, Y. M. Liu, D. Keane, D. Beavis, S. Y. Chu, S. Y. Fung, M. Vient, C. Hartnack, and H. Stöcker, Phys. Rev. C 44, 1091 (1991).
* (37) J. Y. Ollitrault, Phys. Rev. D 48, 1132 (1993).
* (38) Z. W. Lin and C. M. Ko, Phys. Rev. C 65, 034904 (2002).
* (39) Z. W. Lin, C. M. Ko, B. A. Li, B. Zhang and S. Pal, Phys. Rev. C 72, 064901 (2005).
* (40) S. Sarkar and A. Mukhopadhyay, Proc. DAE-BRNS Symp. Nucl. Phys. 60, 752 (2015).
* (41) J. Y. Ollitrault, A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 80, 014904 (2009).
* (42) M. Luzum and J. Y. Ollitrault, Phys. Rev. C 87, 044907 (2013).
|
# On description of blow-up behavior for ODEs with normally hyperbolic nature
in dynamics at infinity
Kaname Matsue , 111International Institute for Carbon-Neutral Energy Research
(WPI-I2CNER), Kyushu University, Fukuoka 819-0395, Japan Institute of
Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan
<EMAIL_ADDRESS>
###### Abstract
We describe blow-up behavior for ODEs by means of dynamics at infinity with
complex asymptotic behavior in autonomous systems, as well as in nonautonomous
systems. Based on preceding studies, a variant of closed embeddings of phase
spaces and the time-scale transformation determined by the structure of vector
fields at infinity reduce our description of blow-ups to unravel the shadowing
property of (pre)compact trajectories on the horizon, the geometric object
expressing the infinity, with the specific convergence rates. Geometrically,
this description is organized by asymptotic phase of invariant sets on the
horizon. Blow-up solutions in nonautonomous systems can be described in a
similar way. As a corollary, normally, or partially hyperbolic invariant
manifolds on the horizon possessing asymptotic phase are shown to induce blow-
ups.
Keywords: blow-up solutions, embeddings, desingularization, normally
hyperbolic invariant manifolds, asymptotic phase, nonautonomous differential
equations
AMS subject classifications: 34A26, 34C08, 34C45, 35B44, 37C60, 37D10, 58K55
## 1 Introduction
Blow-up behavior of solutions in differential equations is widely studied in
decades from many aspects such as mathematical, numerical and physical ones.
Fundamental questions in blow-up phenomena are whether or not a solution blows
up at a finite time and, if it does, when, where and how it blows up. While
numerous studies involving blow-ups and related finite-time singularities such
as finite-time extinction, collapse and quenching are provided from various
viewpoints, the author and his collaborators have provided characterizations
of finite-time singularities involving blow-ups by means of dynamics at
infinity based on compactifications of phase spaces and appropriate time-scale
desingularizations [38, 39, 2, 30]. These machineries reduce the problem for
finding and characterizing blow-up solutions to that for characterizing
(center-)stable manifolds of equilibria/periodic orbits on the horizon, the
geometric object expressing infinity as the boundaries of compactified
manifolds or their upper-tangent spaces, for the transformed vector fields
called desingularized vector fields so that “dynamics at infinity” makes
sense. In particular, asymptotic theory in dynamical systems provides various
characterizations of blow-up solutions under appropriate setting and transform
in vector fields. Note that this approach is applied to studying features of
solutions of PDEs with blow-up and/or quenching behavior (e.g., [23, 24, 25,
26]), while the approach itself is also used to study bounded objects in
dynamical systems (e.g., [7, 8, 17, 22, 31]). Moreover, the proposed machinery
also provides various achievements in computer-assisted proofs of blow-up
solutions with well-established concepts in dynamical systems [35, 40, 41,
52].
In the previous studies, blow-up solutions are considered only for autonomous
vector fields ${\bf y}^{\prime}=f({\bf y})$ admitting, as already mentioned,
equilibria and/or periodic orbits at infinity. While various generalization
will be proposed, we shall pay our attention to the following generalizations
here:
* •
blow-up behavior with more complex nature at infinity, and
* •
characterization of blow-ups in nonautonomous systems.
One of appropriate candidates in the first direction would be blow-ups
shadowing normally hyperbolic invariant manifolds (NHIMs for short) at
infinity (cf. [34] for characterizing unbounded geometry by means of NHIMs).
In the case of nonautonomous systems
${\bf y}^{\prime}=f(t,{\bf y})$ (1.1)
under a suitable setting to $f$, an appropriate treatment of time variable $t$
is required. One of typical treatments is to regard the time variable $t$ as
an additional phase variable and the original system as the extended
autonomous system
$\frac{d{\bf y}}{d\eta}=f(t,{\bf y}),\quad\frac{dt}{d\eta}=1,$ (1.2)
in which case treatments of invariant sets becomes different from general
autonomous systems222 For example, equilibria do not make sense in (1.2). .
While useful machineries such as compactifications in the time variable (e.g.,
[53]) are established for studying global-in-time solutions in nonautonomous
systems, we will see that the above extended autonomous system (1.2) with an
appropriate rule of scaling in $t$ is suitable for characterizing blow-ups in
nonautonomous systems in the sense that machineries developed for autonomous
systems can be applied.
Our main aims here are to see that the following stuff essentially
characterize blow-ups both in autonomous and nonautonomous systems:
* •
shadowing property of trajectories at infinity;
* •
specific (and sufficiently fast) “converge rate” to them.
Geometrically, the first requirement is described by means of asymptotic phase
of invariant sets (e.g., [37] and references therein), which is manifested in
invariant manifolds with partial / normal hyperbolicity. An example is
invariant foliations of stable manifolds of NHIMs. The second requirement is
essentially related to “convergence rate”, which is stronger than usual
characterizations of the above hyperbolicity. In a special case, we refer to
the theory in linear differential systems, e.g., [45, 46, 47] (see also [29,
51]), where dichotomy spectrum characterize stability or asymptotic behavior
of solutions. We shall see that a special structure of the spectrum provides
our requirement of “convergence rate”, which includes the case of the presence
of hyperbolic equilibria/periodic orbits at infinity ([38]).
The rest of the present paper is organized as follows. In Section 2, we review
a fundamental concept of NHIMs and put a remark on asymptotic phase. In
Section 3, we summarize fundamental settings of vector fields we shall treat
throughout the present paper and machineries to characterize dynamics at
infinity including blow-up solutions; embeddings of phase spaces and time-
scale desingularizations. In Section 4, we pay our attention to autonomous
systems, and provide a characterization of blow-up solutions shadowing
trajectories at infinity. This characterization generalizes preceding works
where blow-up solutions shadowing hyperbolic equilibria and periodic orbits at
infinity are characterized (e.g., [38, 39]). We further describe blow-up
solutions with asymptotic rates automatically determined by the quasi-
homogeneity of vector fields, which are often referred to as type-I blow-ups,
under the specific “convergence rate” to shadowed trajectories. In Section 5,
we move to nonautonomous systems and derive a criterion of the existence of
blow-ups with their asymptotic behavior. In addition to a fundamental
description, a geometric treatment of NHIMs with boundaries towards our aims
is also presented to geometrically describe the blow-up criterion. In Section
6, we show several examples of blow-up solutions in nonautonomous systems. Our
examples shown there revisit to preceding works in various backgrounds to
characterize finite-time singularities, and will be shown that blow-ups are
described in a unified way.
Several technical details are collected in Appendix. In Appendix A, another
embedding frequently applied in practical problems is reviewed. In Appendix B,
a sufficient condition to satisfy the rate requirement discussed in Section 4
is provided by means of spectral theory in linear differential systems. In
Appendix C.3, a geometric technique to provide a sufficient condition for
blow-up rates of blow-up solutions in nonautonomous systems discussed in
Section 5 is presented.
###### Remark 1.1 (Notation on vector bundles. e.g., [19, 45]).
Let $\xi=(\pi,E,X)$ be an $n$-dimensional vector bundle over the base
(topological) space $X$, that is, the triple of $X$, the total space
$E=\bigcup_{{\bf x}\in X}F_{\bf x}$ with the fiber $F=F_{\bf
x}\cong\mathbb{R}^{n}$, and the continuous projection $\pi:E\to X$. For any
subset $M\subset X$ of the base space, let $\xi|_{M}=(\pi,E(M),M)$ with
$E(M)=\bigcup_{{\bf x}\in M}F_{\bf x}$ be the restriction of $\xi$ over $M$.
## 2 Preliminaries 1: normally hyperbolic invariant manifolds
Here we briefly summarize a fundamental concept in dynamical systems we shall
use later, normally hyperbolic invariant manifolds (NHIMs) followed by [10,
20], as well as a remark on a property which NHIMs admit, asymptotic phase.
Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a $C^{r}$-vector field with $r\geq
1$ and $\varphi=\varphi_{f}$ be the generated flow. For any $t\in\mathbb{R}$,
the associated time-$t$ map is denoted by $\varphi^{t}=\varphi_{f}^{t}$. For
any subset $M\subset\mathbb{R}^{n}$, let $T_{M}\mathbb{R}^{n}:=\pi^{-1}(M)$
associated with the vector bundle $(\pi,T\mathbb{R}^{n},\mathbb{R}^{n})$.
###### Remark 2.1.
Arguments in this subsection are valid for $C^{r}$-vector fields $f:Q\to TQ$,
where $Q$ is a $C^{r}$ Riemannian manifold. Nevertheless we pay our attention
to the case $Q=\mathbb{R}^{n}$ for simplicity. Indeed, our interests here are
submanifolds of $\mathbb{R}^{n}$ for some $n$ and associated vector fields.
###### Definition 2.2 (Normal hyperbolicity, cf. [20, 10]).
Let $M\subset\mathbb{R}^{n}$ be a compact, connected manifold with $\partial
M=\emptyset$ which is invariant for $\varphi$. We say that $M$ is $r$-normally
hyperbolic333 In [20], the above definition is referred to as $r$-eventually
relatively normal hyperbolicity. This property is sufficient in our purpose.
if there is a continuous splitting
$T_{M}\mathbb{R}^{n}=TM\oplus E^{u}\oplus E^{s}$ (2.1)
of $T_{M}\mathbb{R}^{n}$ such that, for any $t$, all subbundles $TM$, $E^{u}$
and $E^{s}$ are $D\varphi^{t}$-invariant and that there are constants $C>0$
and $\lambda_{s}<0<\lambda_{u}$ satisfying
$\|D\varphi^{t}|_{E^{s}_{p}}\|\leq
Ce^{\lambda_{s}t}m\left(D\varphi^{t}|_{T_{p}M}\right)^{i},\quad
m\left(D\varphi^{t}|_{E^{u}_{p}}\right)\geq\frac{e^{\lambda_{u}t}}{C}\|D\varphi^{t}|_{T_{p}M}\|^{i}$
(2.2)
for all $p\in M$, $t\geq 0$, and $0\leq i\leq r$. We shall call $M$ an
($r$-)normally hyperbolic invariant manifold, or ($r$-)NHIM for short. Now
$\oplus$ in (2.1) denotes the Whitney sum of vector bundles and
$m(A):=\inf\left\\{\|Av\|\mid\|v\|=1\right\\}$
is the minimum norm444 If $A$ in invertible, then $m(A)=\|A^{-1}\|^{-1}$. of
a matrix $A$. We particularly say a NHIM $M$ being normally attracting ($M$
being a NAIM for short) if $E^{u}$ is the zero bundle; $E^{u}=\\{0\\}$.
Denote $n_{T},n_{u},n_{s}$ with $n=n_{T}+n_{u}+n_{s}$ by the rank ($=$
dimension of fibers) of $TM$, $E^{u}$ and $E^{s}$, respectively. In
particular, $TM$, $E^{u}$ and $E^{s}$ are vector bundles with the base space
$M$ and $n_{T}$-, $n_{u}$\- and $n_{s}$-dimensional fibers. We shall call
$E^{u}$ and $E^{s}$ the unstable bundle and the stable bundle of $M$,
respectively, while $TM$ is the tangent bundle of $M$. When $M$ is a NAIM,
then $n_{u}=0$. The vector bundle structure of $E^{s}$ is written by
$\xi^{s}=(\pi_{s},E^{s},F^{s})$ with
$E^{s}=\bigcup_{p\in M}F^{s}_{p},\quad F^{s}\cong
F^{s}_{p}\cong\mathbb{R}^{n_{s}}$
for all $p\in M$. A similar description holds for $E^{u}$. Several properties
which NHIMs possess are summarized below.
###### Proposition 2.3 (e.g., [20, 43, 44, 54]).
Let $M\subset\mathbb{R}^{n}$ be a boundaryless, compact NHIM for
$\varphi_{f}$.
1. 1.
(Invariant foliation). There is a local stable manifold $W^{s}_{\rm loc}(M)$
tangent to the Whitney sum $TM\oplus E^{s}$ at $M$, which is a
$C^{r}$-embedded submanifold of $\mathbb{R}^{n}$ and invariantly fibered by
embedded ($n_{s}$-dimensional) disks $D_{p}^{s}$, denoted by $W^{s}_{\rm
loc}(p)$ for each $p\in M$, constructing fibers of the local stable invariant
foliation
$W^{s}_{\rm loc}(M)(=W^{s}_{\rm loc}(M;\varphi))=\bigcup_{p\in M}W^{s}_{\rm
loc}(p).$
In particular, $W^{s}_{\rm loc}(p)$ is an invariant manifold satisfying
$T_{p}W^{s}_{\rm loc}(p)=E_{p}^{s}$ for each $p\in M$.
2. 2.
(Exponential decay along fibers). There exist $C_{s}>0$ and $\lambda_{s}<0$
such that, for each $p\in M$, the following holds for all $t\geq 0$ whenever
$q\in W^{s}_{\rm loc}(p)$:
$\|\varphi_{f}(t,q)-\varphi_{f}(t,p)\|<C_{s}e^{\lambda_{s}t}.$ (2.3)
3. 3.
(Linearization). $\varphi_{f}$ is topologically conjugate to $N\varphi_{f}$
near $M$, namely there are neighborhoods $U,U^{\prime}$ of $M$ and the zero
section $Z(M)$ of the bundle $\mathcal{N}=E^{u}\oplus E^{s}$, respectively in
$\mathbb{R}^{n}$, and a homeomorphism $h:U\to U^{\prime}$ such that
$N\varphi_{f}^{t}\circ h=h\circ\varphi_{f}^{t}$ in $U$ for all
$t\in\mathbb{R}$ if trajectories are inside corresponding neighborhoods, where
$N\varphi_{f}^{t}=(\varphi_{f}^{t}|_{Z(M)},D\varphi_{f}^{t}|_{E^{u}\oplus
E^{s}})$ denotes the (linear skew-product) flow555 The definition is left to
Appendix B because it is argued only when technical details in our objectives
are discussed. on $\mathcal{N}$. The flow $\varphi_{f}^{t}|_{Z(M)}$ is
identified with $\varphi_{f}^{t}|_{M}$ through the identity $Z(M)\cong M$.
Because $M$ is assumed to be compact, taking $U$ and $U^{\prime}$ smaller if
necessary, both $h$ and $h^{-1}$ may be assumed to be bounded on $U$ and
$U^{\prime}$, respectively.
The theory of NHIMs is also established for manifolds with nonempty boundary.
We collect basic notions of normal hyperbolicity in such a case.
###### Definition 2.4 (Inflowing/overflowing invariant manifolds, e.g.,
[54]).
Suppose that $M$ is a compact manifold in an ambient manifold $Q$ possibly
with nonempty boundary: $\partial M\not=\emptyset$. We say that $M$ is
inflowing invariant for the vector field $f$ if $\varphi_{f}^{t}(M)\subset M$
for all $t\geq 0$ and if $f$ points strictly inward at $\partial M$.
Similarly, we say that $M$ is overflowing invariant for the vector field $f$
if $\varphi_{f}^{t}(M)\subset M$ for all $t\leq 0$ and if $f$ points strictly
outward at $\partial M$. If $M$ is inflowing/overflowing invariant with the
structures (2.1) and (2.2), we call $M$ inflowing/overflowing NHIM,
respectively666 If $\partial M=\emptyset$ and $M$ is invariant, then $M$ is
both overflowing and inflowing invariant. But if $\partial M\not=\emptyset$,
$M$ being invariant does not imply that $M$ is both overflowing and inflowing
invariant. .
It is proved that inflowing invariant manifolds admit invariant foliations of
local stable manifolds, while overflowing invariant manifolds admit invariant
foliations of local unstable manifolds (e.g., [10, 54]).
Finally we shall make a remark on invariant foliations; shadowing property for
NHIMs. Even if an invariant manifold is not normally hyperbolic, there are
several cases which it admits the similar property.
###### Remark 2.5 (Asymptotic phase, cf. [36, 37]).
Let $X=(X,d)$ be a metric space and $\varphi$ be a flow on $X$. Suppose that
there is an attracting invariant set $A$ with a basin $B$:
$\lim_{t\to\infty}d(\varphi^{t}({\bf x}),A)=0,\quad\forall{\bf x}\in B.$
It is said that a mapping $t\mapsto\varphi^{t}({\bf x})$, ${\bf x}\in B$, has
an asymptotic phase if there is ${\bf x}_{\ast}\in A$ such that
$\lim_{t\to\infty}d(\varphi^{t}({\bf x}),\varphi^{t}({\bf x}_{\ast}))=0.$
Stable invariant foliation for a NHIM $M$ is a special case of asymptotic
phase: $A=M$ and $B=W^{s}_{\rm loc}(M)$. Asymptotic phase beyond normal
hyperbolicity is constructed in several studies. See e.g., [36, 37] for a
brief summary of historical backgrounds in asymptotic phase and its existence
in a generalized setting.
## 3 Preliminaries 2: dynamics at infinity
In this section we provide tools and methodologies to consider divergent
solutions in terms of global-in-time trajectories converging to invariant sets
for appropriately transformed vector fields as general as possible.
### 3.1 Asymptotically quasi-homogeneous systems
First of all, we collect our present settings precisely, which are based on
[38].
###### Definition 3.1 (Homogeneity index and admissible domain).
Let $\alpha=(\alpha_{1},\cdots,\alpha_{n})$ be a set of nonnegative integers.
We say the index set $I_{\alpha}=\\{i\in\\{1,\cdots,n\\}\mid\alpha_{i}>0\\}$
the set of homogeneity indices associated with
$\alpha=(\alpha_{1},\cdots,\alpha_{n})$. Let $U\subset\mathbb{R}^{n}$. We say
the domain $U\subset\mathbb{R}^{n}$ admissible with respect to the sequence
$\alpha$ if
$U=\left\\{{\bf x}=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}\mid
x_{i}\in\mathbb{R}\text{ if }i\in I_{\alpha},\
(x_{j_{1}},\cdots,x_{j_{n-l}})\in\tilde{U}\right\\},$
where $\\{j_{1},\cdots,j_{n-l}\\}=\\{1,\cdots,n\\}\setminus I_{\alpha}$ and
$\tilde{U}$ is an open set in $\mathbb{R}^{n-l}$ spanning
$(x_{j_{1}},\cdots,x_{j_{n-l}})$ with
$\\{j_{1},\cdots,j_{n-l}\\}=\\{1,\cdots,n\\}\setminus I_{\alpha}$.
###### Definition 3.2 (Asymptotically quasi-homogeneous vector fields, cf.
[6, 38]).
Let $U$ be an admissible set $U\subset\mathbb{R}^{n}$ with respect to
$\alpha$. Also, let $f_{0}:U\to\mathbb{R}$ be a function. Let
$\alpha_{1},\ldots,\alpha_{n}$ be nonnegative integers with
$(\alpha_{1},\ldots,\alpha_{n})\not=(0,\ldots,0)$ and $k>0$. We say that
$f_{0}$ is a quasi-homogeneous function777 In preceding studies, all
$\alpha_{i}$’s and $k$ are typically assumed to be natural numbers. In the
present study, on the other hand, the above generalization is valid. of type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and order $k$ if
$f_{0}(s^{\Lambda_{\alpha}}{\bf x})=s^{k}f_{0}({\bf x})\quad\text{ for all
}{\bf x}=(x_{1},\ldots,x_{n})^{T}\in U\text{ and }s>0,$
where888 Throughout the rest of this paper, the power of real positive numbers
or functions to matrices is described in the similar manner.
$\Lambda_{\alpha}={\rm diag}\left(\alpha_{1},\ldots,\alpha_{n}\right),\quad
s^{\Lambda_{\alpha}}{\bf
x}=(s^{\alpha_{1}}x_{1},\ldots,s^{\alpha_{n}}x_{n})^{T}.$
Next, let $X=\sum_{i=1}^{n}f_{i}({\bf x})\frac{\partial}{\partial x_{i}}$ be a
continuous vector field defined on $U$. We say that $X$, or simply
$f=(f_{1},\ldots,f_{n})^{T}$ is a quasi-homogeneous vector field of type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and order $k+1$ if each component
$f_{i}$ is a quasi-homogeneous function of type $\alpha$ and order
$k+\alpha_{i}$.
Finally, we say that $X=\sum_{i=1}^{n}f_{i}({\bf x})\frac{\partial}{\partial
x_{i}}$, or simply $f:U\to\mathbb{R}^{n}$ is an asymptotically quasi-
homogeneous vector field of type $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and
order $k+1$ (at infinity) if there is a quasi-homogeneous vector field
$f_{\alpha,k}=(f_{i;\alpha,k})_{i=1}^{n}$ of type $\alpha$ and order $k+1$
such that
$f_{i}(s^{\Lambda_{\alpha}}{\bf x})-s^{k+\alpha_{i}}f_{i;\alpha,k}({\bf
x})=o(s^{k+\alpha_{i}}),\quad i\in\\{1,\ldots,n\\},$ (3.1)
as $s\to+\infty$ uniformly on $\left\\{{\bf x}\in U\mid\sum_{i\in
I_{\alpha}}x_{i}^{2}=1,(x_{j_{1}},\cdots,x_{j_{n-l}})\in\tilde{K}\right\\}$
for any compact subset $K\subset\tilde{U}$.
###### Remark 3.3.
In the above definition, non-polynomial-like functions such as $\sin x$ are
not included to characterize quasi-homogeneity. Indeed, such functions are
allowed to exist only in the residual terms characterized by the asymptotic
quasi-homogeneity (3.1). On the other hand, (3.1) is required for all
$i\in\\{1,\ldots,n\\}$.
A fundamental property of quasi-homogeneous functions and vector fields is
reviewed in e.g. [2]. Throughout the rest of this section, consider an
(autonomous) $C^{r}$ vector field (1.1) with $r\geq 1$, where
$f:U\to\mathbb{R}^{n}$ is asymptotically quasi-homogeneous of type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and order $k+1$ at infinity defined on
an admissible set $U\subset\mathbb{R}^{n}$ with respect to $\alpha$.
### 3.2 Embedding of phase spaces and dynamics at infinity
Our process to describe blow-up solutions from the viewpoint of dynamical
systems is summarized below.
1. 1.
For given (asymptotically quasi-homogeneous) vector field $f$, determine its
type $\alpha$ and order $k+1$. See Section 5 for the treatment of $t$ in
nonautonomous systems.
2. 2.
Apply an embedding of phase spaces with the type $\alpha$, precisely defined
below, and transform $f$ through this embedding.
3. 3.
Introduce the time-scale transformation determined by the order $k+1$ of $f$
to desingularize the transformed vector field at infinity.
The above process is widely used in autonomous systems (cf. [21, 22, 23, 24,
25, 26, 35, 38, 39, 40, 52]) and we follow this strategy in the present
argument as well as nonautonomous systems later.
###### Remark 3.4.
In preceding studies, embeddings we would review below were referred to as
compactifications. On the other hand, because the embedded manifolds involving
our interests here are typically non-compact, including the case of
nonautonomous systems, we shall use the terminology “embeddings” instead of
“compactifications”.
In the following arguments in this section, we fix a type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{Z}_{\geq
0}^{n}\setminus\\{(0,\ldots,0)\\}$, and let $I_{\alpha}$ be the associated set
of homogeneity indices. Also, let $U\subset\mathbb{R}^{n}$ be an admissible
set with respect to $\alpha$. A “global-type” embedding and the associated
vector field suitable for describing dynamics at infinity are introduced here.
Another example of embeddings is shown in Appendix A.
###### Definition 3.5 (Quasi-parabolic embedding, cf. [40]).
Let $\\{\beta_{i}\\}_{i\in I_{\alpha}}$ be the collection of natural numbers
so that
$\alpha_{i}\beta_{i}\equiv c\in\mathbb{N},\quad i\in I_{\alpha}$ (3.2)
is the least common multiplier. In particular, $\\{\beta_{i}\\}_{i\in
I_{\alpha}}$ is chosen to be the smallest among possible collections. Let
$p({\bf y})$ be a functional given by
$p({\bf y})\equiv\left(\sum_{i\in
I_{\alpha}}y_{i}^{2\beta_{i}}\right)^{1/2c}.$ (3.3)
Define the mapping $T_{{\rm para};\alpha}:U\to\mathbb{R}^{n}$ as the inverse
of
$S_{{\rm para};\alpha}({\bf x})={\bf y},\quad
y_{j}=\kappa^{\alpha_{j}}x_{j},\quad j=1,\ldots,n,$ (3.4)
where
$\kappa=\kappa({\bf x})=(1-p({\bf x})^{2c})^{-1}\equiv\left(1-\sum_{j\in
I_{\alpha}}x_{j}^{2\beta_{j}}\right)^{-1}.$
We say the mapping $T_{{\rm para};\alpha}$ the quasi-parabolic embedding (with
type $\alpha$).
###### Remark 3.6.
The functional $\kappa=\tilde{\kappa}({\bf y})$ as a functional determined by
${\bf y}$ is implicitly determined by $p({\bf y})$. Details of such a
characterization of $\kappa$ in terms of ${\bf y}$, and the bijectivity and
smoothness of $T_{{\rm para};\alpha}$ are shown in [40] with a general class
of embeddings including quasi-parabolic compactifications (embeddings).
The above definition is originally introduced in the case
$I_{\alpha}=\\{1,\ldots,n\\}$. It is nevertheless valid for the general case,
$I_{\alpha}\not=\\{1,\ldots,n\\}$, to characterize infinity suitable for the
behavior of $f$. It should be however noted that $T_{{\rm para};\alpha}$ and
its inverse $S_{{\rm para};\alpha}$ are defined for all
$j\in\\{1,\ldots,n\\}$, not only $j\in I_{\alpha}$.
As proved in [40], $T_{{\rm para};\alpha}$ maps $U$ one-to-one onto the set
$\mathcal{D}\equiv\\{{\bf x}\in U\mid p({\bf x})<1\\}$. Infinity in the
original coordinate then corresponds to a point on the level set of $p$:
$\mathcal{E}=\\{{\bf x}\in U\mid p({\bf x})=1\\}.$
###### Definition 3.7.
We call the set $\mathcal{E}$ the horizon.
Once we fix an embedding associated with the type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ of the vector field $f$ with order
$k+1$, we can derive the vector field which makes sense including the horizon.
Then the dynamics at infinity makes sense through the appropriately
transformed vector field called the desingularized vector field, denoted by
$g$. The common approach is twofold. Firstly, we rewrite the vector field
(1.1) with respect to the new variable defined by embeddings. Secondly, we
introduce the time-scale transformation of the form $d\tau=q({\bf
x})\kappa({\bf x}(t))^{k}dt$ for some function $q({\bf x})$ which is bounded
including the horizon. We then obtain the vector field with respect to the new
time variable $\tau$, which is continuous, including the horizon.
###### Remark 3.8.
Continuity of the desingularized vector field $g$ including the horizon is
guaranteed by the smoothness of $f$ and asymptotic quasi-homogeneity ([38]).
In the case of parabolic-type embeddings introduced here, $g$ inherits the
smoothness of $f$ including the horizon, which is not always the case of other
embeddings in general. Details are discussed in [38].
###### Definition 3.9 (Time-scale desingularization).
Define the new time variable $\tau$ by
$d\tau=q({\bf x})^{-1}(1-p({\bf x})^{2c})^{-k}dt,\quad q({\bf
x})=1-\frac{2c-1}{2c}(1-p({\bf x})^{2c}),$ (3.5)
equivalently
$t-t_{0}=\int_{\tau_{0}}^{\tau}q({\bf x}(\tau))(1-p({\bf
x}(\tau))^{2c})^{k}d\tau,$
where $\tau_{0}$ and $t_{0}$ denote the correspondence of initial times, ${\bf
x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ and ${\bf y}(\tau)$ is a
solution ${\bf y}(t)$ under the parameter $\tau$. We shall call (3.5) the
time-scale desingularization of order $k+1$.
The change of coordinate and the above desingularization yield the following
vector field $g=(g_{1},\ldots,g_{n})^{T}$, which is continuous on
$\widetilde{\mathcal{D}}\equiv\mathcal{D}\cup\mathcal{E}$:
$\displaystyle\dot{x}_{i}\equiv\frac{dx_{i}}{d\tau}=g_{i}({\bf x})=q({\bf
x})\left\\{\tilde{f}_{i}({\bf
x})-\alpha_{i}x_{i}\sum_{j=1}^{n}(\nabla\kappa)_{j}\kappa^{\alpha_{j}-1}\tilde{f}_{j}({\bf
x})\right\\},$
where
$\tilde{f}_{j}({\bf
x}):=\kappa^{-(k+\alpha_{j})}f_{j}(\kappa^{\Lambda_{\alpha}}{\bf x}),\quad
j=1,\ldots,n,$ (3.6)
and $\nabla\kappa=\nabla_{\bf x}\kappa=((\nabla_{\bf
x}\kappa)_{1},\ldots,(\nabla_{\bf x}\kappa)_{n})^{T}$ is
$(\nabla_{\bf
x}\kappa)_{j}=\begin{cases}\frac{\kappa^{1-\alpha_{j}}x_{j}^{2\beta_{j}-1}}{\alpha_{j}q({\bf
x})}&j\in I_{\alpha},\\\ 0&\text{otherwise},\end{cases}$
as derived in [40]. In particular, the vector field $g$ and the differential
equation is simply written as follows:
$\displaystyle\dot{\bf x}=g({\bf x})=q({\bf x})\tilde{f}({\bf x})-G({\bf
x})\Lambda_{\alpha}{\bf x},$ (3.7)
where $\tilde{f}=(\tilde{f}_{1},\ldots,\tilde{f}_{n})^{T}$ and
$\displaystyle G({\bf x})$ $\displaystyle\equiv\sum_{j\in
I_{\alpha}}\frac{x_{j}^{2\beta_{j}-1}}{\alpha_{j}}\tilde{f}_{j}({\bf x}).$
(3.8)
Smoothness of $f$ and its asymptotic quasi-homogeneity guarantee the
smoothness of the right-hand side $g$ of (3.7) including the horizon
$\mathcal{E}\equiv\\{p({\bf x})=1\\}$. In particular, dynamics at infinity,
such as divergence of solutions to specific directions, is characterized
through dynamics generated by (3.7) around the horizon. See [38, 40] for
details.
###### Remark 3.10 (Invariant structure).
The horizon $\mathcal{E}$ is invariant in $\widetilde{\mathcal{D}}$. Indeed,
direct calculations yield that
$\left.\frac{d}{d\tau}p({\bf x}(\tau))^{2c}\right|_{\tau=0}=0\quad\text{
whenever }\quad{\bf x}(0)\in\mathcal{E}.$
In particular, $\mathcal{E}$ is a codimension-one invariant submanifold of
$\widetilde{\mathcal{D}}$ if $I_{\alpha}=\\{1,\ldots,n\\}$. See e.g. [38],
where detailed calculations are shown in a similar type of global embeddings.
We shall apply this invariant structure to extracting the detailed blow-up
structure later.
## 4 Normally hyperbolic blow-up
In the present and the next sections, we state the main results: description
of blow-up solutions by means of solutions shadowing “trajectories at
infinity”. Key features of blow-up description we shall show here are that,
for the desingularized vector field, the convergence to a trajectory on the
horizon at a suitably fast rate provides finite-time blow-up, and that the
asymptotic blow-up profile is described by the shadowed trajectory, which are
extensions of blow-up characterization derived in preceding works (e.g.,
[38]). The shadowing property is geometrically described by asymptotic phase,
which is observed in invariant manifolds with “hyperbolic” property. A
suitable class of such manifolds is NHIMs, which yields that NHIMs on the
horizon induce blow-up solutions.
Another issue in blow-up description is the magnitude of blow-ups referred to
as blow-up rates. In the present argument, we assume convergence to shadowed
trajectories with specific rates, which is stronger than typical convergence
properties observed in asymptotic phase. These convergence rates can relate to
spectral properties of associated linearized systems, which are left to
Appendix B, where the correspondence by means of dichotomy spectrum
established in the theory of linear differential systems (e.g., [45, 46]) is
discussed.
Now let $f:U\to\mathbb{R}^{n}$ be a smooth (at least $C^{1}$) asymptotically
quasi-homogeneous vector field of type $\alpha=(\alpha_{1},\ldots,\alpha_{n})$
associated with the set of homogeneity indices $I_{\alpha}$ with an admissible
domain $U\subset\mathbb{R}^{n}$, and order $k+1$. Also, let $g$ be the
desingularized vector field (3.7) associated with $f$ and the quasi-parabolic
embedding $T_{{\rm para};\alpha}$ of type $\alpha$, and $\varphi_{g}$ be the
flow generated by $g$. The result is also valid for directional embeddings
summarized in Appendix A, and possibly other ones with suitable assumptions999
As far as we understand, other possible choices of compactificaitons are
Poincaré-type ones (e.g., [38]) and Poincaré-Lyapunov disks (e.g., [7]).
Roughly speaking, the former is the embedding of $\mathbb{R}^{n}$ into the
hemisphere in $\mathbb{R}^{n+1}$, and the latter is the change of variables
with quasi-polar coordinates. .
First we provide a general description of blow-up solutions by means of
solutions shadowing “trajectories at infinity”.
###### Theorem 4.1 (Blow-up: a general description).
Suppose that $g$ admits a trajectory $\gamma=\\{{\bf
x}_{\gamma}(\tau)\\}_{\tau\in\mathbb{R}}\subset\mathcal{E}$ such that, for
some ${\bf y}_{0}\in U$ and the solution ${\bf y}(\tau)$ through ${\bf y}_{0}$
at $\tau=0$,
$\displaystyle\|T_{{\rm para};\alpha}({\bf y}(\tau))-{\bf
x}_{\gamma}(\tau)\|<Ce^{-\lambda\tau}$ (4.1)
for constants $C>0$ and $\lambda>0$. Then $t_{\max}<\infty$ holds; namely,
${\bf y}(t)$ is a blow-up solution.
If we further assume
$\lim_{\tau\to+\infty}e^{(\lambda+\epsilon)\tau}\|T_{{\rm para};\alpha}({\bf
y}(\tau))-{\bf x}_{M}(\tau)\|=\begin{cases}+\infty&\epsilon>0\\\
0&\epsilon<0\end{cases},$ (4.2)
then
$\displaystyle p({\bf y}(t))\sim
C_{0}(-\ln(t_{\max}-t))(t_{\max}-t)^{-1/k}\quad\text{ as }\quad t\to
t_{\max}-0$
for some function $C_{0}(-\ln(t_{\max}-t))$ satisfying
$C_{0}(\tau)=o(e^{\epsilon\tau}),\quad C_{0}(\tau)^{-1}=o(e^{\epsilon\tau})$
for any $\epsilon>0$ as $\tau\to\infty$, equivalently
$-\ln(t_{\max}-t)\to+\infty$. Moreover, we have
$y_{i}(t)\sim
C_{0}(-\ln(t_{\max}-t))^{\alpha_{i}}x_{\gamma,i}(-\bar{c}\ln(t_{\max}-t))(t_{\max}-t)^{-\alpha_{i}/k}\quad\text{
as }\quad t\to t_{\max}-0$
for some constant $\bar{c}>0$, provided $x_{\gamma,i}(\tau)\not\to 0$ as
$\tau\to\infty$.
###### Remark 4.2.
The asymptotic behavior (4.2) covers the cases
$\lim_{\tau\to+\infty}\frac{\|T_{{\rm para};\alpha}({\bf y}(\tau))-{\bf
x}_{M}(\tau)\|}{e^{-\lambda\tau}\tau^{m}}=c_{0}>0,$
for some integer $m$, or
$0\leq c_{1}\leq\frac{\|T_{{\rm para};\alpha}({\bf y}(\tau))-{\bf
x}_{M}(\tau)\|}{e^{-\lambda\tau}\tau^{m}}<c_{2}$
for all sufficiently large $\tau>0$.
###### Proof.
Let ${\bf x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ in the
$\tau$-timescale. The maximal existence time $t_{\max}$ for ${\bf y}(t)$ in
the $t$-timescale is then estimated through (3.5) as follows:
$\displaystyle t_{\max}$ $\displaystyle\equiv\int_{0}^{\infty}q({\bf
x}(\tau))(1-p({\bf x}(\tau))^{2c})^{k}d\tau\leq\int_{0}^{\infty}\\{1-p({\bf
x}(\tau))^{2c}\\}^{k}d\tau$
from the fact that $q({\bf x})\leq 1$. On the other hand, because ${\bf
x}(\tau)$ shadows ${\bf x}_{\gamma}(\tau)$ in the sense of (4.1), the quantity
$1-p({\bf x}(\tau))^{2c}$ can be rewritten as
$\displaystyle 1-p({\bf x}(\tau))^{2c}$ $\displaystyle=1-\sum_{i\in
I_{\alpha}}x_{i}(\tau)^{2\beta_{i}}$ $\displaystyle=\sum_{i\in
I_{\alpha}}x_{\gamma,i}(\tau)^{2\beta_{i}}-\sum_{i\in
I_{\alpha}}x_{i}(\tau)^{2\beta_{i}}$ $\displaystyle=\sum_{i\in
I_{\alpha}}(x_{\gamma,i}(\tau)^{2}-x_{i}(\tau)^{2})\left\\{\sum_{j=0}^{\beta_{i}-1}x_{\gamma,i}(\tau)^{2(\beta_{i}-1-j)}x_{i}(\tau)^{2j}\right\\}$
$\displaystyle=\sum_{i\in
I_{\alpha}}(x_{\gamma,i}(\tau)-x_{i}(\tau))(x_{\gamma,i}(\tau)+x_{i}(\tau))\left\\{\sum_{j=0}^{\beta_{i}-1}x_{\gamma,i}(\tau)^{2(\beta_{i}-1-j)}x_{i}(\tau)^{2j}\right\\}$
(4.3)
and, using the inequality
$\left\|(x_{\gamma,i}(\tau)-x_{i}(\tau))(x_{\gamma,i}(\tau)+x_{i}(\tau))\left\\{\sum_{j=0}^{\beta_{i}-1}x_{\gamma,i}(\tau)^{2(\beta_{i}-1-j)}x_{i}(\tau)^{2j}\right\\}\right\|\leq
C_{i}e^{\lambda\tau}$
obtained from (4.1) for some constant101010 Because only components $x_{i}$
and $x_{\gamma,i}$ with $i\in I_{\alpha}$ are considered, we have
$|x_{i}(\tau)|,|x_{\gamma,i}(\tau)|\leq 1$ for all $\tau\geq 0$. $C_{i}>0$,
we have
$\displaystyle t_{\max}$ $\displaystyle\leq\int_{0}^{\infty}\\{1-p({\bf
x}(\tau))^{2c}\\}^{k}d\tau\leq\tilde{C}\int_{0}^{\infty}e^{k\lambda\tau}d\tau<\infty$
for some constant $\tilde{C}>0$. In particular, ${\bf y}(t)$ in the original
$t$-timescale is a blow-up solution.
Next we assume (4.2). For $t$ less than but sufficiently close to $t_{\max}$,
we have
$t_{\max}-t=\int_{\tau}^{\infty}q({\bf x}(\tilde{\tau}))(1-p({\bf
x}(\tilde{\tau}))^{2c})^{k}d\tilde{\tau}$
with sufficiently large $\tau$, and the asymptotic expression of the right-
hand side is our interest here. Here estimate the functional
$\kappa^{-1}=1-p({\bf x}(\tau))^{2c}$. Using (4.3), the functional is written
as follows:
$\displaystyle 1-p({\bf x}(\tau))^{2c}$ $\displaystyle=\sum_{i\in
I_{\alpha}}(x_{\gamma,i}(\tau)-x_{i}(\tau))(x_{\gamma,i}(\tau)+x_{i}(\tau))\left\\{\sum_{j=0}^{\beta_{i}-1}x_{\gamma,i}(\tau)^{2(\beta_{i}-1-j)}x_{i}(\tau)^{2j}\right\\}$
$\displaystyle=e^{-\lambda\tau}\bar{F}(\tau),$
where $\bar{F}(\tau)=o(e^{\epsilon\tau})$ and
$\bar{F}(\tau)^{-1}=o(e^{\epsilon\tau})$ for any $\epsilon>0$ as
$\tau\to\infty$. Therefore we have
$\frac{1}{k}\ln(t_{\max}-t)=-\lambda\tau+\tilde{F}(\tau)$
as $\tau\to\infty$, where $\tilde{F}(\tau)=o(\tau)$ and
$\tilde{F}(\tau)^{-1}=o(\tau)$ as $\tau\to\infty$. The above estimates are
combined to obtain
$p({\bf y}(t))=\left\\{1-\sum_{i\in
I_{\alpha}}x_{i}(t)^{2\beta_{i}}\right\\}^{-1}\left(\sum_{i\in
I_{\alpha}}x_{i}(t)^{2\beta_{i}}\right)^{1/2c}\sim
C_{0}(-\ln(t_{\max}-t))(t_{\max}-t)^{-1/k}$
as $t\to t_{\max}$, and the proof involving the asymptotic behavior of $p({\bf
y}(t))$ is completed. The last statement follows from the definition of
$T_{{\rm para};\alpha}$ (Definition 3.5) and asymptotic relations we have
obtained. ∎
A usual way to verify assumptions, in particular (4.1), is to construct a
collection of trajectories on the horizon guaranteeing the shadowing property
in advance. One specific characterization of such a property is asymptotic
phase. Once one can construct invariant sets admitting asymptotic phase with
appropriate convergence rate, assumptions involving (4.1) are satisfied and
blow-up solutions can be described by means of invariant manifolds for $g$ on
the horizon. For example, when NHIMs on the horizon are constructed, our
description is summarized as follows, which shall be called normally
hyperbolic blow-up. Note that other characterizations of asymptotic phase (cf.
Remark 2.5) yield the similar results.
###### Corollary 4.3 (Normally hyperbolic blow-up).
Suppose that $g$ admits a compact connected NHIM $M\subset\mathcal{E}$ with
$\partial M=\emptyset$. If the solution ${\bf y}(t)$ of (1.1) with a bounded
initial point ${\bf y}_{0}\in\mathbb{R}^{n}$ whose image ${\bf
x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ is on $W_{\rm
loc}^{s}(M;\varphi_{g})$, then $t_{\max}<\infty$ holds; namely, ${\bf y}(t)$
is a blow-up solution. Moreover, there is a trajectory $\gamma=\\{{\bf
x}_{\gamma}(\tau)\\}_{\tau\geq 0}$ for (3.7) on $M\subset\mathcal{E}$ such
that (4.1) holds.
###### Proof.
Thanks to the stable foliation of $W_{\rm loc}^{s}(M)\equiv W_{\rm
loc}^{s}(M;\varphi_{g})$, there is a trajectory $\gamma=\\{{\bf
x}_{\gamma}(\tau)\\}_{\tau\geq 0}\subset M$ such that ${\bf x}(0)=T_{{\rm
para};\alpha}({\bf y}_{0})\in W_{\rm loc}^{s}({\bf x}_{\gamma}(0))$ and ${\bf
x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))\in W_{\rm loc}^{s}({\bf
x}_{\gamma}(\tau))$ for all $\tau\geq 0$. The decay estimate (4.1) is the
direct consequence of (2.3). ∎
As mentioned in the beginning, description of blow-up rates requires more
precise information about convergence rates. Details are left to Appendix B.
## 5 Nonautonomous blow-up
Next we describe blow-up solutions in the nonautonomous system
${\bf y}^{\prime}=f(t,{\bf y}),\quad{\bf y}(t_{0})={\bf y}_{0}\in U$ (5.1)
with $t_{0}\in\mathbb{R}$, an open set $U\subset\mathbb{R}^{n}$ and a
$C^{r}$-mapping $f:\mathbb{R}\times U\to\mathbb{R}^{n}$ with $r\geq 1$. While
the essential idea to describe blow-up solutions is the same as the autonomous
case, namely embeddings of phase spaces, we need to treat nonautonomous terms
appropriately. First, we have to consider a “natural scaling of time $t$” so
that dynamics at infinity can be considered in the similar way to autonomous
cases. Second, a natural determination of “invariant sets at infinity” for
nonautonomous systems is necessary so that the similar ideas to autonomous
systems can be applied. We shall see later that NHIMs will be an appropriate
class to describe blow-up solutions in nonautonomous systems.
In the present section, we first propose a natural treatment of the time
variable $t$ in our consideration of dynamics at infinity based on extended
autonomous systems, and provide a description of blow-up solutions in a
general setting then. To apply the description to e.g., NHIMs, a geometric
technique is introduced so that invariant manifolds with boundary can be
treated within the framework of compact, boundaryless invariant manifolds, in
particular NHIMs. Details of the technique are stated in Section 5.3 and
Appendix C.3. With the help of this technique, blow-ups in the extended
autonomous systems can be characterized in the similar way to Section 4.
### 5.1 Desingularized vector fields for nonautonomous systems
One natural treatment to consider (5.1) is to regard the time variable $t$ as
another independent variable depending on an extra variable $\eta$, namely
regard (5.1) as the extended autonomous system:
$\frac{d{\bf y}}{d\eta}=f(t,{\bf
y}),\quad\frac{dt}{d\eta}=1,\quad(t(\eta_{0}),{\bf y}(\eta_{0}))=(t_{0},{\bf
y}_{0})\in\mathbb{R}\times U.$ (5.2)
We pay attention to the scaling of the “time” variable $t$ so that the
extended system is regarded as an asymptotically quasi-homogenenous system.
Because the motion of time does not originally change as solution evolve, it
is natural to regard the scaling of $t$ as
$t=\kappa^{0}\tilde{t}=\tilde{t},$ (5.3)
indicating that the time-variable is not scaled depending on the location of
trajectories. Once this regulation is determined, the machinery provided in
Section 3 is applied to (5.2) towards a blow-up description in a similar way
to autonomous systems. We shall describe the desingularized vector field
associated with (5.1) under the assertion (5.3) with the quasi-parabolic
embedding. Application of directional embeddings (Appendix A) is derived in
the similar way.
Throughout the rest of this section, given a system (5.1), assume that the
extended vector field is asymptotically quasi-homogeneous of type
$\alpha=(0,\alpha_{1},\ldots,\alpha_{n})$ and order $k+1$ such that the set
$\mathbb{R}\times U$ is admissible with respect to $\alpha$. Here note that
the exponent $0$ in the first component of $\alpha$ stems from the scaling
(5.3). In particular, the associated set of homogeneity indices $I_{\alpha}$
is necessarily a proper subset of $\\{0,1,\ldots,n\\}$. We then apply the
quasi-parabolic embedding in Definition 3.5. In the present case, the
functional $p$ in (3.3), written as $p(t,{\bf y})$, is independent of $t$ and
we shall identify $p(t,{\bf y})$ with $p({\bf y})$. Using this identification,
derive the desingularized vector field introducing
$d\tau=(1-p({\bf x})^{2c})^{-k}q({\bf x})^{-1}d\eta$ (5.4)
in a usual manner, where $q({\bf x})$ is given in (3.5). We then obtain the
corresponding desingularized vector field, which can be written by
$\frac{d}{d\tau}\begin{pmatrix}t\\\ {\bf x}\end{pmatrix}=g(t,{\bf x})\equiv
q({\bf x})\bar{f}(t,{\bf x})-G(t,{\bf
x})\bar{\Lambda}_{\alpha}\begin{pmatrix}t\\\ {\bf x}\end{pmatrix}$ (5.5)
with the following notations, which are consistent with the general derivation
of desingularized vector fields in autonomous systems:
$\displaystyle\bar{f}(t,{\bf x})$
$\displaystyle=\begin{pmatrix}\tilde{f}_{0},\tilde{f}_{1},\ldots,\tilde{f}_{n}\end{pmatrix}^{T}\quad\text{
with }\quad f_{0}(t,{\bf y})=1,$ $\displaystyle G(t,{\bf x})$
$\displaystyle\equiv\sum_{j\in
I_{\alpha}}\frac{x_{j}^{2\beta_{j}-1}}{\alpha_{j}}\tilde{f}_{j}(t,{\bf
x}),\quad\bar{\Lambda}_{\alpha}={\rm diag}(0,\alpha_{1},\ldots,\alpha_{n}).$
Note that, in the above notation, $\tilde{f}_{0}(t,{\bf x})=(1-p({\bf
x})^{2c})^{k}$ via (3.6). The above identifications are also consistent with
the evolution of $t$ followed by the time-scale desingularization (5.4).
### 5.2 Fundamental description of blow-ups
Like autonomous cases, geometric description of blow-ups for (5.2) is based on
the construction of invariant manifolds with “hyperbolic” structure on the
horizon for the desingularized vector field (5.5). On the other hand,
$dt/d\tau=0$ on the horizon $\mathcal{E}=\\{p({\bf x})=1\\}$, and invariant
sets on $\mathcal{E}$ in the present case will consist of slices of invariant
sets parameterized by the fixed variable $t$, the first component of phase
variables $(t,{\bf x})$. Therefore, even in the simplest case such as
“equilibria” on the horizon, they are not isolated. In this case, “hyperbolic
structure” will not be expected for invariant sets on the horizon. Instead,
normal, or partial hyperbolicity can be naturally considered regarding the
evolution in $t$-variable as the tangential direction. In particular, the
concept of normally hyperbolic blow-up stated in Section 4 will be naturally
applied in practical problems.
As in the previous section, we only consider quasi-parabolic embeddings. Fix
the corresponding embedding $T_{{\rm para};\alpha}$ associated with $f$. Let
$g$ be the desingularized vector field (5.5) associated with $f$ and $T_{{\rm
para};\alpha}$. Also, let $\varphi_{g}$ be the flow generated by $g$. Finally,
let $\pi_{\nu}$ be the projection onto the $\nu$-component, $\nu=t,{\bf x}$.
Here we describe blow-up solutions in a general setting. As seen in Section 4,
our blow-up description; existence and blow-up rates, relies on the existence
of asymptotic phase and precise exponential decay behavior of trajectory
shadowing invariant sets on the horizon for $g$. Because the desingularized
vector field $g(t,{\bf x})$ in (5.5) essentially has the same structure as $g$
in autonomous cases, (3.7), the similar calculations and estimates yield the
corresponding description of blow-ups.
###### Theorem 5.1 (Nonautonomous blow-up: a general description).
Suppose that $g$ admits a precompact trajectory $\gamma=\\{{\bf
x}_{\gamma}(\tau)\\}_{\tau\in\mathbb{R}}\subset\mathcal{E}$ such that, for
some $(t_{0},{\bf y}_{0})\in U$ and the solution ${\bf y}(\tau)$ through
$(t_{0},{\bf y}_{0})$ at $\tau=0$ corresponding $\tau=0$ to $t=t_{0}$,
$\displaystyle\lim_{\tau\to+\infty}\|\pi_{\bf x}T_{{\rm
para};\alpha}(t(\tau),{\bf y}(\tau))-{\bf x}_{\gamma}(\tau)\|=0.$ (5.6)
Then $t_{\max}<\infty$ holds; namely, ${\bf y}(t)$ is a blow-up solution.
If we further assume
$\displaystyle\lim_{\tau\to+\infty}e^{(\lambda+\epsilon)\tau}\|\pi_{\bf
x}T_{{\rm para};\alpha}(t(\tau),{\bf y}(\tau))-{\bf
x}_{\gamma}(\tau)\|=\begin{cases}+\infty&\epsilon>0\\\
0&\epsilon<0\end{cases}$ (5.7)
for a positive constant $\lambda>0$, then
$\displaystyle p({\bf y}(t))\sim
C_{0}(-\ln(t_{\max}-t))(t_{\max}-t)^{-1/k}\quad\text{ as }\quad t\to
t_{\max}-0$
for some function $C_{0}(-\ln(t_{\max}-t))$ satisfying
$C_{0}(\tau)=o(e^{\epsilon\tau}),\quad C_{0}(\tau)^{-1}=o(e^{\epsilon\tau})$
for any $\epsilon>0$ as $\tau\to\infty$, equivalently
$-\ln(t_{\max}-t)\to+\infty$. Moreover, we have
$y_{i}(t)\sim
C_{0}(-\ln(t_{\max}-t))^{\alpha_{i}}x_{\gamma,i}(-\bar{c}\ln(t_{\max}-t))(t_{\max}-t)^{-\alpha_{i}/k}\quad\text{
as }\quad t\to t_{\max}-0$
for some constant $\bar{c}>0$, provided $x_{\gamma,i}(\tau)\not\to 0$ as
$\tau\to\infty$.
###### Proof.
Observe that $dt/d\tau=0$ on the horizon $\mathcal{E}$. From the compactness
of the closure ${\rm cl}(\gamma)$ in $\mathcal{E}$, the trajectory $\gamma$ is
bounded. Hence $\gamma$ is included in $\mathcal{E}\cap\\{t=\bar{t}\\}$ for
some $\bar{t}$. Combining with the convergence (4.1), the maximal existence
time $t_{\max}$ of ${\bf y}(t)$ with ${\bf y}(t_{0})={\bf y}_{0}$ must be
equal to $\bar{t}<\infty$, in particular ${\bf y}(t)$ is a blow-up solution.
The remaining results follow from the same arguments as Theorem 4.1. ∎
In the nonautonomous setting, convergence rate of trajectories to shadowed
ones is not required because finiteness of $t_{\max}$ is automatically
achieved from the boundedness of $\gamma$ and the fact $dt/d\tau=0$ on
$\mathcal{E}$. The remaining requirements to describe blow-ups are similar to
those stated in e.g., Proposition B.7.
### 5.3 Blow-up descriptions for concrete problems
Here we address an intrinsic difficulty so that the first part of our result,
Theorem 5.1, can be applied and a technique to overcome it, which ensures the
application of the result to a wide range of nonautonomous systems. Theorem
5.1 essentially relies on the existence of asymptotic phase, which are
typically ensured for compact, boundaryless invariant manifolds. On the other
hand, as seen in simple cases (e.g., Section 6.1), constructed invariant
manifolds on $\mathcal{E}$ for the extended system is not compact in general.
Although several generalizations to noncompact invariant manifolds can be
applied (e.g., [9]), we notice that our attention is blow-up behavior occuring
in finite range of time $t$ (not $\tau$ !), and hence the restriction of
invariant manifolds to a compact segment of invariant manifolds is sufficient
to consider the concrete behavior of solutions.
We shall pay attention to NHIMs for simplicity, in which case a well-known
vector field modification technique on the boundaries can be applied to
constructing inflowing/overflowing invariant manifolds (see Definition 2.4 and
comments below). Its direct application provides invariant foliations,
yielding the existence of blow-up solutions in our problems. As provided
below, we can easily construct a modification to obtain both compact
inflowing/overflowing invariant manifolds depending on our requirements. We
shall only show a modification for constructing inflowing invariant manifolds.
Modification for constructing overflowing invariant manifolds is similar111111
Overflowing invariant manifolds on the horizon are applied to detecting blow-
up behavior in the reverse time direction. .
For any set $M\subset\widetilde{D}\equiv\mathcal{D}\cup\mathcal{E}$,
$\bar{t}\in\mathbb{R}$ and an interval $I\subset\mathbb{R}$, let
$M_{\bar{t}}\equiv M\cap\\{t=\bar{t}\\},\quad M_{I}\equiv M\cap\\{t\in I\\}$
(5.8)
be the slice and the tube of $M$ on $I$, respectively. First observe that,
when $M\subset\mathcal{E}$ is a compact invariant manifold for $g$ admitting
the normally hyperbolic structure, the slice $M_{\bar{t}}$ is a compact NHIM
in $\mathcal{E}\cap\\{t=\bar{t}\\}$ because the $dt/d\tau$ always vanishes on
$\mathcal{E}$ and hence $t$-evolution plays a role in (non-essential)
tangential evolution for $g$. Here we choose a collection of compact intervals
$I,I^{\prime}$ satisfying
$t_{0}\in I\subset{\rm int}\,I^{\prime}\subset I^{\prime},$ (5.9)
which are used to modify manifolds over a small interval in
$I^{\prime}\setminus I$ so that dynamics over $I$ is unchanged.
###### Assumption 5.2.
Fix an initial time $t_{0}\in\mathbb{R}$. There is a collection of compact
intervals $I,I^{\prime}$ satisfying (5.9) and an invariant manifold
$M\subset\mathcal{E}$, not necessarily $\partial M\not=\emptyset$, for $g$ in
(5.5) satisfying the following assertions:
* •
$M_{I^{\prime}}$ admits the normally hyperbolic structure for $\varphi_{g}$,
in particular, there is a continuous splitting
$T_{M_{I^{\prime}}}\mathbb{R}^{n+1}=TM_{I^{\prime}}\oplus
E^{u}(M_{I^{\prime}})\oplus E^{s}(M_{I^{\prime}})$
and, for some constants $C\geq 1$ and $\mu<0<\lambda$,
$m(D\varphi_{g}|_{E^{u}(M_{I^{\prime}})})\geq\frac{e^{\lambda
t}}{C}\|D\varphi_{g}|_{TM_{I^{\prime}}}\|^{i},\quad\|D\varphi_{g}|_{E^{s}(M_{I^{\prime}})}\|\leq
Ce^{\mu t}m(D\varphi_{g}|_{TM_{I^{\prime}}})^{i}$
hold for all $t\geq 0$ and $0\leq i\leq r$.
* •
For any $\bar{t}\in I^{\prime}$, the slice $M_{\bar{t}}$ is nonempty, compact,
connected and boundaryless. In particular, such a slice $M_{\bar{t}}$ is a
NHIM.
###### Remark 5.3.
Without the loss of generality, we may assume that $t_{0}\in I$ in the above
assumption. Indeed, our interest is asymptotic behavior of solutions ${\bf
u}(\tau)=(t(\tau),{\bf x}(\tau))$ for $g$ approaching to $\mathcal{E}$ as
$\tau\to\infty$. If $t_{0}=t(\tau_{0})\not\in I$, evolve the solution until
$\tau_{0}+\tilde{\tau}$ so that $t(\tau_{0}+\bar{\tau})\in I$ and replacing
$t_{0}$ by $t(\tau_{0}+\bar{\tau})$, which is the case of our assumption.
In Assumption 5.2, we have a compact (normally hyperbolic) invariant manifold
$M_{I^{\prime}}\subset\mathcal{E}$ with the (nontrivial) invariant boundary.
Modification of vector fields on the boundary (cf. [54]) is well used to
guarantee persistence and/or invariant foliations of invariant manifolds. In
our case, let $\rho_{I,I^{\prime}}:\mathbb{R}\to\mathbb{R}$ be a smooth,
nonnegative bump-type function such that
$\rho_{I,I^{\prime}}(t)=\begin{cases}1&t\not\in I^{\prime},\\\ 0&t\in
I.\end{cases}$
We then modify the desingularized vector field (5.5) to the following:
$\frac{d}{d\tau}\begin{pmatrix}t\\\ {\bf x}\end{pmatrix}=\tilde{g}(t,{\bf
x})\equiv g(t,{\bf
x})+\delta\rho_{I,I^{\prime}}(t)\begin{pmatrix}\nu_{I^{\prime}}\\\
0\end{pmatrix},$ (5.10)
where $\delta>0$, and $\nu_{I^{\prime}}$ is the inward unit normal
vector121212 For constructing overflowing invariant manifolds, modify
$\nu_{I^{\prime}}$ to be the outward unit normal vector, which ensures the
invariant foliation of unstable manifolds ([54]). of $\partial I^{\prime}$.
Then we know that $\tilde{g}=g$ on $M_{I}$ and that trajectories on
$\mathcal{E}$ approach to $M_{I}$ in $t$-direction. In other words, for any
interval $\tilde{I}$ satisfying $I\subset{\rm int}\tilde{I}\subset
I^{\prime}$, the tube $M_{\tilde{I}}$ is inflowing invariant for the modified
vector field $\tilde{g}$. Therefore, according to the general theory of NHIMs,
the inflowing invariant manifold $M_{\tilde{I}}$ for $\tilde{g}$ admits the
local stable manifold $W^{s}_{\rm loc}(M_{\tilde{I}};\tilde{g})$ and its
invariant foliation (e.g., [54]) such that the restriction of the base space
of $W^{s}_{\rm loc}(M_{\tilde{I}};\tilde{g})$ to $M_{I}$ provides the
invariant foliation of $W^{s}_{\rm loc}(M_{I};g)$; for the original vector
field $g$ in $\\{t\in I\\}$. As a result, blow-up solutions for (5.2) stated
in Theorem 5.1 can be described by means of $M_{I}$.
###### Corollary 5.4 (Nonautonomous blow-up: the existence by means of
invariant manifolds).
Fix an initial time $t_{0}\in\mathbb{R}$. Suppose that $g$ admits an invariant
manifold $M\subset\mathcal{E}$ satisfying all requirements in Assumption 5.2
with compact intervals $I,I^{\prime}\subset\mathbb{R}$. Also assume that the
solution ${\bf y}(t)$ of (5.2) with a bounded initial point ${\bf y}_{0}(={\bf
y}(t_{0}))\in\mathbb{R}^{n}$ whose image ${\bf x}(\tau)=T_{{\rm
para};\alpha}({\bf y}(\tau))$ for $T_{{\rm para};\alpha}$ is on $W_{\rm
loc}^{s}(M_{I};g)$, where $\tau_{0}$ is the corresponding time of $t_{0}$ for
the time-scale desingularization. Then $t_{\max}<\infty$ and (5.6) holds with
a trajectory $\gamma\subset M_{I}$.
We leave the treatment of blow-up rates in Appendix C because several
additional techniques stemming from differential topology are required, in
addition to techniques discussed in Appendix B. Here we only provide a special
case, which can be simply applied to various nonautonomous systems such as
examples in Section 6.
###### Corollary 5.5 (Blow-up rates for nonautonomous blow-up: a special
case).
In addition to settings in Assumption 5.2, suppose that
$\displaystyle M_{I^{\prime}}=\\{(t,{\bf x}_{\ast}(t))\mid t\in
I^{\prime},\,{\bf x}_{\ast}(t)\text{ is an equilibrium satisfying (\ref{NH-
spec-nonaut-stationary}) below}\\}$ $\displaystyle\sharp\\{{\rm
Spec}(Dg(t,{\bf x}_{\ast}(t)))\cap i\mathbb{R}\\}=1\quad\text{ for all }\quad
t\in I^{\prime}$ (5.11)
where ${\rm Spec}(A)$ denotes the set of eigenvalues of a squared matrix $A$,
that is, $M_{I^{\prime}}$ is a curve of hyperbolic equilibria parameterized by
$t$. Then the asymptotic estimate (5.7) holds true with ${\bf
x}_{\gamma}(\tau)\equiv{\bf x}_{\ast}(t)$ for each $t\in I$. In particular,
asymptotic behavior for blow-up solutions stated in Theorem 5.1 is obtained.
###### Proof.
See Section C.2. ∎
## 6 Examples
Examples showing the applicability of our results are collected. In addition
to the applicability, we aim at seeing that the present methodology unifies
the approach to finite-time singularities in nonautonomous problems. The
examples we shall show are highlighted as follows:
* •
Section 6.1 addresses the first Painlevé equation, which the existence of
singular solutions is well-known in the treatment of regular singular points.
* •
Section 6.2 relates to a regularized system for exhibiting singular shocks in
(non-)hyperbolic systems of conservation laws.
* •
Section 6.3 addresses self-similar profiles of nonlinear diffusion equations
with finite-time singularity.
* •
Section 6.4 addresses radially symmetric profiles of a reaction-diffusion
equation with negative power nonlinearity.
All examples were investigated individually in preceding studies, and
independent techniques were required to unravel the comprehensive morphology
of solutions. When we pay our attention to blow-ups, the proposed methodology
unravels all these features in a systematic way, indicating that a series of
machineries involving embeddings of phase spaces as well as time-scale
desingularizations can unify descriptions of finite-time singularities.
###### Remark 6.1.
In all examples, invariant manifolds in our interest are collections of
equilibria. Therefore Corollary 5.5 are mainly applied.
###### Remark 6.2.
As we shall see in Section 6.1, as well as commented in [35], arguments
through global (parabolic-type or Poincaré-type, say [38]) embeddings require
lengthy and tedious calculations, although we do not suffer from any change of
charts131313 Advantages that quasi-parabolic embeddings are used as our
central machinery are discussed in [30, 35]. . This is the reason why we apply
another embeddings, called directional embeddings briefly summarized in
Appendix A, to the remaining examples. Note that all results in Sections 4 and
5 are essentially valid. See e.g. [38] for the correspondence of arguments.
### 6.1 The first Painlevé equation
First we consider the first Painlevé equation
$u^{\prime\prime}=6u^{2}+t,\quad^{\prime}=\frac{d}{dt}$ (6.1)
from the viewpoint of blow-up description. It is well-known that (6.1)
possesses the following solution: for fixed $t_{0}\in\mathbb{R}$,
$u(t)\sim 6(t_{\max}-t)^{-2}\quad\text{ as }\quad t\to t_{\max}-0$ (6.2)
for some $t_{\max}=t_{\max}(t_{0})>t_{0}$ and sufficiently large initial point
$u_{0}=u(t_{0})$. Although this asymptotic behavior can be derived via
substitution of a formal (Frobenius-type) power series solution into the
equation (e.g., [33]), we shall derive the existence of such a solution
through dynamics at infinity.
Rewrite (6.1) as the system of the first order ODEs:
$\begin{cases}\chi^{\prime}=1,&\\\ u^{\prime}=v,&\\\
v^{\prime}=6u^{2}+\chi,&\end{cases}\quad^{\prime}=\frac{d}{dt}.$ (6.3)
We immediately have the following property.
###### Lemma 6.3.
The system (6.3) is asymptotically quasi-homogeneous of type $(0,2,3)$ and
order $k+1=2$.
In the present example, we apply the quasi-parabolic embedding
$\chi=\chi,\quad u=\frac{x_{1}}{(1-p({\bf x})^{6})^{2}},\quad
v=\frac{x_{2}}{(1-p({\bf x})^{6})^{3}},$
where
${\bf x}=(x_{1},x_{2})^{T},\quad p({\bf x})^{6}=x_{1}^{6}+x_{2}^{4}$
with $(\beta_{1},\beta_{2})=(3,2)$ and $c=6$. Following the general derivation
(Section 3), let
$\displaystyle\kappa:=(1-p({\bf x})^{6})^{-1},\quad\tilde{f}_{1}(\chi,{\bf
x}):=x_{2},\quad\tilde{f}_{2}(\chi,{\bf x}):=6x_{1}^{2}+\kappa^{-4}\chi,$
$\displaystyle G(\chi,{\bf x}):=\frac{x_{1}^{5}}{2}\tilde{f}_{1}(\chi,{\bf
x})+\frac{x_{2}^{3}}{3}\tilde{f}_{2}(\chi,{\bf
x})=\frac{x_{1}^{5}}{2}x_{2}+\frac{x_{2}^{3}}{3}(6x_{1}^{2}+\kappa^{-4}\chi).$
Under the time-scale desingularization
$\frac{d\tau}{dt}=\kappa^{-1},$
we have the desingularized vector field (5.5) with $c=6$, $k=1$ and
$\bar{\Lambda}_{\alpha}={\rm diag}(0,2,3)$.
The next step is to find invariant sets on the horizon $\mathcal{E}=\\{p({\bf
x})=1\\}$. On $\mathcal{E}$, equilibria satisfy $g_{1}(\chi,{\bf
x})=g_{2}(\chi,{\bf x})=0$, where
$g_{1}(\chi,{\bf x})=x_{2}(1-x_{1}^{6}-4x_{1}^{3}x_{2}^{2}),\quad
g_{2}(\chi,{\bf x})=\frac{3}{2}x_{1}^{2}(4-x_{1}^{3}x_{2}^{2}-4x_{2}^{4})$
are the second and the third components of the desingularized vector field,
respectively. Note that $\dot{\chi}=0$ is automatically satisfied on
$\mathcal{E}$. We find, for each $\chi\in\mathbb{R}$, the points $(\chi,{\bf
x}_{\ast})$, where
$\displaystyle{\bf
x}_{\ast}=(x_{1;+},x_{2;+}),\quad(x_{1;+},x_{2;-}),\quad(x_{1;-},x_{2;+}),\,\text{
or }\,(x_{1;-},x_{2;-}),$ $\displaystyle x_{1;\pm}=\pm\frac{1}{17^{1/6}},\quad
x_{2;\pm}=\pm\frac{2}{17^{1/4}}$
are equilibria on $\mathcal{E}$.
Now we pay attention to the family of equilibria
$M_{++}=\\{(\chi,x_{1;+},x_{2;+})\\}$. The Jacobian matrix of (5.5) at each
point on $M_{++}$ is
$\begin{pmatrix}0&\ast\\\
O&\tilde{A}\end{pmatrix},\quad\tilde{A}=\begin{pmatrix}\frac{11}{2}x_{1}^{5}\tilde{f}_{1}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{1}}\right|_{\mathcal{E}}-2\left.\left(G+x_{1}\frac{\partial G}{\partial
x_{1}}\right)\right|_{\mathcal{E}}&\frac{11}{3}x_{2}^{3}\tilde{f}_{1}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{2}}\right|_{\mathcal{E}}-\left.2x_{1}\frac{\partial G}{\partial
x_{2}}\right|_{\mathcal{E}}\\\
\frac{11}{2}x_{1}^{5}\tilde{f}_{2}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{1}}\right|_{\mathcal{E}}-\left.3x_{2}\frac{\partial G}{\partial
x_{1}}\right|_{\mathcal{E}}&\frac{11}{3}x_{2}^{3}\tilde{f}_{2}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{2}}\right|_{\mathcal{E}}-3\left.\left(G+x_{2}\frac{\partial G}{\partial
x_{2}}\right)\right|_{\mathcal{E}}\end{pmatrix},$
where
$\displaystyle\tilde{f}_{1}|_{\mathcal{E}}=x_{2}|_{\mathcal{E}},\quad\tilde{f}_{2}|_{\mathcal{E}}=6x_{1}^{2}|_{\mathcal{E}},\quad\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{1}}\right|_{\mathcal{E}}=0,\quad\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{2}}\right|_{\mathcal{E}}=1,\quad\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{1}}\right|_{\mathcal{E}}=12x_{1},\quad\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{2}}\right|_{\mathcal{E}}=0,$ $\displaystyle
G|_{\mathcal{E}}=\frac{x_{1}^{5}}{2}\tilde{f}_{1}|_{\mathcal{E}}+\frac{x_{2}^{3}}{3}\tilde{f}_{2}|_{\mathcal{E}}=\left.\left(\frac{1}{2}x_{1}^{5}x_{2}+2x_{1}^{2}x_{2}^{3}\right)\right|_{\mathcal{E}},$
$\displaystyle\left.\frac{\partial G}{\partial
x_{1}}\right|_{\mathcal{E}}=\frac{5}{2}x_{1}^{4}\tilde{f}_{1}|_{\mathcal{E}}+\frac{x_{1}^{5}}{2}\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{1}}\right|_{\mathcal{E}}+\frac{x_{2}^{3}}{3}\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{1}}\right|_{\mathcal{E}}=\left.\left(\frac{5}{2}x_{1}^{4}x_{2}+4x_{1}x_{2}^{3}\right)\right|_{\mathcal{E}},$
$\displaystyle\left.\frac{\partial G}{\partial
x_{2}}\right|_{\mathcal{E}}=\frac{x_{1}^{5}}{2}\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{2}}\right|_{\mathcal{E}}+x_{2}^{2}\tilde{f}_{2}|_{\mathcal{E}}+\frac{x_{2}^{3}}{3}\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{2}}\right|_{\mathcal{E}}=\left.\left(\frac{1}{2}x_{1}^{5}+6x_{1}^{2}x_{2}^{2}\right)\right|_{\mathcal{E}},$
and several terms being identically zero on $\mathcal{E}$ are omitted. Note
that the above matrix is independent of $\chi$, namely the flow and its
linearization are autonomous on $\mathcal{E}$. Substituting
$(x_{1},x_{2})=(x_{1;+},x_{2;+})$ into the above matrix, we have
$\displaystyle\frac{11}{2}x_{1}^{5}\tilde{f}_{1}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{1}}\right|_{\mathcal{E}}-2\left.\left(G+x_{1}\frac{\partial G}{\partial
x_{1}}\right)\right|_{\mathcal{E}}$
$\displaystyle=\frac{11}{17^{13/12}}-\frac{108}{17^{13/12}}=-\frac{97}{17^{13/12}},$
$\displaystyle\frac{11}{3}x_{2}^{3}\tilde{f}_{1}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{2}}\right|_{\mathcal{E}}-\left.2x_{1}\frac{\partial G}{\partial
x_{2}}\right|_{\mathcal{E}}$ $\displaystyle=\frac{11\cdot
16}{51}+1-\frac{49}{17}=\frac{80}{51},$
$\displaystyle\frac{11}{2}x_{1}^{5}\tilde{f}_{2}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{1}}\right|_{\mathcal{E}}-\left.3x_{2}\frac{\partial G}{\partial
x_{1}}\right|_{\mathcal{E}}$ $\displaystyle=\frac{33}{17^{7/6}}+\frac{12\cdot
17}{17^{7/6}}-\frac{6}{17^{7/6}}\left(5+32\right)=\frac{15}{17^{7/6}},$
$\displaystyle\frac{11}{3}x_{2}^{3}\tilde{f}_{2}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{2}}\right|_{\mathcal{E}}-3\left.\left(G+x_{2}\frac{\partial G}{\partial
x_{2}}\right)\right|_{\mathcal{E}}=\frac{176}{17^{13/12}}-\frac{3}{17^{13/12}}\left(1+16+1+48\right)=-\frac{22}{17^{13/12}}$
and hence the matrix $\tilde{A}$ is
$\tilde{A}=\begin{pmatrix}-97/17^{13/12}&80/51\\\
15/17^{7/6}&-22/17^{13/12}\end{pmatrix}\equiv\begin{pmatrix}a&b\\\
c&d\end{pmatrix}.$
Now
$\displaystyle
a+d=-\frac{119}{17^{13/12}},\quad(a+d)^{2}=\frac{119^{2}}{17^{13/6}},\quad ad-
bc=\frac{97\cdot 22-400}{17^{13/6}},$ $\displaystyle(a+d)^{2}-4(ad-
bc)=\frac{119^{2}-4(97\cdot 22-400)}{17^{13/6}}=\frac{4025}{17^{13/6}}$
and hence the eigenvalues of $\tilde{A}$ are
$\lambda=\frac{1}{2}\left\\{(a+d)\pm\sqrt{(a+d)^{2}-4(ad-
bc)}\right\\}=\frac{-119\pm\sqrt{7225}}{2\cdot 17^{13/12}}=\frac{-119\pm
85}{2\cdot 17^{13/12}}=\frac{-1}{17^{1/12}},\quad\frac{-6}{17^{1/12}},$
indicating that $(x_{1;+},x_{2;+})$ is attracting (sink-type), and hence the
invariant manifold $M_{++}$ is normally attracting. In particular, for any
compact interval $I\subset\mathbb{R}$, the restriction $M_{++;I}\equiv
M_{++}\cap\\{t\in I\\}$ is a NAIM satisfying all requirement in Corollary 5.5.
We therefore see that, for particular choice of an interval
$I\subset\mathbb{R}$, solutions ${\bf x}(\tau)\subset W^{s}_{\rm
loc}(M_{++;I})$ characterize blow-up solutions of (6.3) with the asymptotic
behavior $O((t_{\max}-t)^{-2})$ as $t\to t_{\max}-0$. This result confirms and
guarantees of the existence of blow-up solutions of the form (6.2) without any
algebraic trial-and-error arguments.
###### Theorem 6.4 (cf. [33]).
Consider (6.1) with the initial time $t_{0}\in\mathbb{R}$ and
$u(t_{0})=u_{0}\in\mathbb{R}$. If $u_{0}$ is sufficiently large, the solution
$u(t)=u(t;t_{0},u_{0})$ with $u(t_{0})=u_{0}$ blows up at a time
$t_{\max}=t_{\max}(t_{0})<\infty$ with
$u(t)\sim c_{u}(t_{\max}-t)^{-2}\quad\text{ as }\quad t\to t_{\max}-0$
for some constant $c_{u}>0$.
### 6.2 A problem in Dafermos profiles for singular shocks
The next example is the following system with $0\leq\epsilon\ll 1$ as a
parameter controlling the multi-scale structure:
$\begin{cases}\chi^{\prime}=\epsilon,&\\\ u_{1}^{\prime}=u_{1}^{2}-u_{2}-\chi
u_{1}-w_{1},&\\\ u_{2}^{\prime}=\frac{1}{3}u_{1}^{3}-u_{1}-\chi
u_{2}-w_{2},&\\\ w_{1}^{\prime}=-\epsilon u_{1},&\\\ w_{2}^{\prime}=-\epsilon
u_{2},&\\\ \end{cases}\quad^{\prime}=\frac{d}{dt}.$ (6.4)
###### Remark 6.5.
The origin of (6.4) is the system of conservation laws
$(u_{1})_{t}+\left(u_{1}^{2}-u_{2}\right)_{x}=0,\quad(u_{2})_{t}+\left(\frac{1}{3}u_{1}^{3}-u_{1}\right)_{x}=0\quad\Leftrightarrow\quad
U_{t}+F(U)_{x}=0,$ (6.5)
proposed by Keyfitz and Kranser (e.g., [28, 32, 27]. See also [48, 50]) with
identification
$U(t,x)=(u_{1}(t,x),u_{2}(t,x))^{T},\quad
F(U)=\left(u_{1}^{2}-u_{2},\frac{1}{3}u_{1}^{3}-u_{1}\right)^{T}.$
Applying the Dafermos regularization (e.g., [49]) to (6.5), we have
$U_{t}+F(U)_{x}=\epsilon tU_{xx},$ (6.6)
and applying the ansatz of self-similar profiles, called the Dafermos
profiles: $U(t,x)=U^{\epsilon}(x/t)$, we obtain the extended autonomous system
$\chi^{\prime}=\epsilon,\quad U^{\prime}=V,\quad V^{\prime}=\\{DF(U)-\chi
I\\}V,\quad^{\prime}=\frac{d}{dt},$
where $U=(u_{1},u_{2})^{T}$ and $V=(v_{1},v_{2})^{T}$. An alternative form is
derived through the change of variables (e.g., [49])
$(U,V,\chi)\mapsto(U,W,\chi)=(U,F(U)-\chi U-V,\chi),$
which is exactly (6.4), where $W=(w_{1},w_{2})^{T}$. In [49], this system is
considered to construct the Dafermos profile of singular shocks, namely shock
fronts with Dirac’s delta-type singularity on the front. A novel observation
in singular shocks is that these shocks can be constructed as the singular
limit of singularly perturbed solutions admitting arbitrarily large
magnitudes. The corresponding trajectories are constructed by the connecting
orbits of equilibria at infinity under technical change of coordinates. The
present example as well as preceding works by the author (e.g., [38]) is
inspired by reconstructing such singular objects within standard and unified
arguments of dynamics at infinity.
Based on arguments in preceding works, we have the following observation.
###### Lemma 6.6 (cf. [38, 49]).
The system (6.4) is asymptotically quasi-homogeneous of type $(0,1,2,1,2)$ and
order $k+1=2$.
We apply the directional embedding
$T_{d;\alpha}:(\chi,u_{1},u_{2},w_{1},w_{2})\mapsto(\chi,x_{1},s,x_{3},x_{4})$
defined by
$\chi=\chi,\quad u_{1}=\frac{x_{1}}{s},\quad u_{2}=\frac{1}{s^{2}},\quad
w_{1}=\frac{x_{3}}{s},\quad w_{2}=\frac{x_{4}}{s^{2}}.$ (6.7)
###### Remark 6.7.
The present choice of the localizing direction, namely $u_{2}=s^{-2}$, stems
from the previous arguments for deriving blow-up behavior of dynamical systems
with nonlinearity extracted from (6.5) (cf. [35, 38, 40]):
$u_{1}^{\prime}=u_{1}^{2}-u_{2},\quad
u_{2}^{\prime}=\frac{1}{3}u_{1}^{3}-u_{1}.$
It is observed in the above references that the corresponding desingularized
vector fields through global embeddings like quasi-parabolic ones admit
totally four equilibria on the horizon, all of which correspond to blow-up
solutions (in the positive or negative time direction) such that $v$ goes to
“positive” infinity. See references mentioned for details. This is the reason
why we have chosen the embedding in the direction specified as (6.7).
Using the change of coordinates (6.7) as well as the time-scale
desingularization
$\frac{d\tau}{dt}=s^{k}=s,$
we obtain the associated desingularized vector field:
$\begin{cases}\dot{\chi}=\epsilon s,&\\\
\dot{x}_{1}=x_{1}\tilde{g}_{0}+(x_{1}^{2}-1)-s(\chi x_{1}+x_{3}),\\\
\dot{s}=s\tilde{g}_{0},&\\\ \dot{x}_{3}=x_{3}\tilde{g}_{0}-\epsilon x_{1}s,\\\
\dot{x}_{4}=2x_{4}\tilde{g}_{0}-\epsilon s,\\\
\end{cases}\quad\dot{}=\frac{d}{d\tau},\quad\tilde{g}_{0}=-\frac{1}{6}x_{1}^{3}+\frac{x_{1}}{2}s^{2}+\frac{s}{2}(\chi+x_{4}).$
(6.8)
We easily see that the horizon $\\{s=0\\}$ admits four families of equilibria
$(\chi,x_{1},s,x_{3},x_{4})=\left(\chi,\pm\sqrt{3\pm\sqrt{3}},0,0,0\right).$
Let
$p_{1,\pm}:=\pm\sqrt{3+\sqrt{3}},\quad p_{2,\pm}:=\pm\sqrt{3-\sqrt{3}}.$ (6.9)
The Jacobian matrices on the above family of equilibria are
$\begin{pmatrix}0&0&\epsilon&0&0\\\
0&-\frac{2}{3}p_{i,\pm}^{3}+2p_{i,\pm}&-\frac{1}{2}\chi p_{i,\pm}&0&0\\\
0&0&-\frac{1}{6}p_{i,\pm}^{3}&0&0\\\ 0&0&-\epsilon
p_{i,\pm}&-\frac{1}{6}p_{i,\pm}^{3}&0\\\
0&0&-\epsilon&0&-\frac{1}{6}p_{i,\pm}^{3}\\\ \end{pmatrix}$
For simplicity, we further assume that $\epsilon=0$ for a while. Then, in the
singular fast limit system, namely (6.8) with $\epsilon=0$, four families of
equilibria are invariant with normally hyperbolic structure. Among four points
(6.9), $p_{1,+}$ and $p_{2,+}$ are those where the flow is attracting towards
the horizon. Moreover, $p_{1,+}$ is a saddle, which is our main issue141414
According to the mechanism of singular shocks mentioned in Remark 6.5, saddle-
like invariant sets would be considered to play a key role in constructing
these shocks. . From our construction and the above observation, the family
$M_{1,+}=\\{(\chi,p_{1,+},0,0,0)\mid\chi\in\mathbb{R}\\}$
determines a (noncompact) NHIM. Because $\chi$ is just a a parameter on the
horizon $\\{s=0\\}$, for any compact interval $I\subset\mathbb{R}$, the
restriction
$M_{1,+;I}=\\{(\chi,p_{1,+},0,0,0)\mid\chi\in I\\}$
becomes a compact NHIM (with boundary). In particular, all requirements in
Corollary 5.5 are satisfied. By the geometric singular perturbation theory
(cf. [14]) (with boundary modification), the above conclusion also holds for
all $0\leq\epsilon\ll 1$.
We therefore conclude that, for sufficiently large $I$ depending on the
initial time $t_{0}$, the local stable manifold $W_{\rm loc}^{s}(M_{1,+;I})$
induce a blow-up solution. More precisely, there is an equilibrium
$p_{\ast}=(\chi_{\ast},p_{1,+},0,0,0)\in M_{1,+;I}$ for some
$\chi_{\ast}=\chi_{\ast}(t_{0})$ such that the trajectory on $W_{\rm
loc}^{s}(p_{\ast})$ induces a blow-up solution with the asymptotic behavior
$u_{1}(t)=\frac{x_{1}(t)}{s(t)}\sim c(t_{\max}-t)^{-1}\quad\text{ as }\quad
t\to t_{\max}-0$
for some constant $c>0$.
###### Theorem 6.8 (cf. [28, 49]).
Consider (6.4) and a solution $(\chi,u_{1},u_{2},w_{1},w_{2})(t)$ with
$0\leq\epsilon\ll 1$. Let $g$ be the desingularized vector field (6.8). Assume
that, under the directional embedding $T_{d;\alpha}$ given in (6.7),
$T_{d;\alpha}(\chi,u_{1},u_{2},w_{1},w_{2})(\bar{\tau})\in W_{\rm
loc}^{s}(p_{\ast})$ holds for some $\bar{\tau}$. Then
$(\chi,u_{1},u_{2},w_{1},w_{2})(t)$ is a blow-up solution with
$u_{1}(t)\sim c_{1}(t_{\max}-t)^{-1},\quad u_{2}(t)\sim
c_{2}(t_{\max}-t)^{-2}\quad\text{ as }\quad t\to t_{\max}-0$
for some constants $c_{1},c_{2}>0$.
In the above theorem we do not mention the asymptotic behavior of $w_{1}$ and
$w_{2}$ because the corresponding $x$-component of $p_{\ast}$ is $0$, in which
case further terms have to be calculated for these asymptotic behavior within
the present approach (cf. [2]).
### 6.3 A system associated with self-similarity
The next example we shall consider is the following system:
$(u^{m-1}u^{\prime})^{\prime}+\beta\chi u^{\prime}+\alpha
u=0,\quad^{\prime}=\frac{d}{d\chi},\quad\chi\in\mathbb{R},$ (6.10)
where $\alpha,\beta\in\mathbb{R}$ are parameters. The system (6.10) originates
from the diffusion equation (e.g., [15])
$U_{t}=(U^{m-1}U_{x})_{x}.$
The parameter $m$ controls the strength of nonlinear diffusion. Paying our
attention to self-similar solutions of the form151515 The ansatz (6.11)
represents the backward self-similarity. Although there are another types of
self-similarity; forward and exponential-type ones, the governing equation
becomes the same one, (6.10).
$U(t,x)=(T-t)^{\alpha}u(x(T-t)^{\beta}),\quad t<T$ (6.11)
for some $T>0$, the system is reduced to (6.10). We concentrate on the very
fast diffusion case, $m<0$, and assume that $\beta<0$. In what follows we
investigate blow-up solutions following our proposed machinery. First rewrite
(6.10) as the first order extended autonomous system:
$\begin{cases}\chi^{\prime}=1,&\\\ u^{\prime}=u^{1-m}v,&\\\
v^{\prime}=-\beta\chi u^{1-m}v-\alpha u,&\\\
\end{cases}\quad^{\prime}=\frac{d}{dt}.$ (6.12)
Direct calculations yield the following observation.
###### Lemma 6.9.
The system (6.12) is asymptotically quasi-homogeneous of type $(0,1,1)$ and
order $k+1=2-m$.
We then apply the directional embedding of type $(0,1,1)$:
$\chi=\chi,\quad u=\frac{1}{s},\quad v=\frac{x}{s}.$ (6.13)
Using this change of coordinates as well as the time-scale desingularization
$\frac{d\tau}{dt}=s^{1-m},$
we obtain the associated desingularized vector field:
$\begin{cases}\dot{\chi}=s^{1-m},&\\\ \dot{s}=-sx,&\\\
\dot{x}=-x^{2}-\beta\chi x-\alpha s^{1-m},&\\\
\end{cases}\quad\dot{}=\frac{d}{d\tau}.$ (6.14)
We easily see that the horizon $\\{s=0\\}$ admits a family of equilibria
$\\{(\chi,0,0)\\},\quad\text{ and }\quad\\{(\chi,0,-\beta\chi)\\}.$
We further investigate the dynamics around the invariant manifold
$M_{\geq\chi_{0}}=\\{(\chi,0,-\beta\chi)\in\mathbb{R}^{3}\mid\chi\geq\chi_{0}\\}$
with $\chi_{0}>0$. Thanks to the assumption $m<0$, the linearized matrix at
each point on $M_{\geq\chi_{0}}$ becomes very simple:
$\begin{pmatrix}0&0&0\\\ 0&\beta\chi&0\\\
\beta^{2}\chi&0&\beta\chi\end{pmatrix},$
which indicates that, for any fixed $\chi_{1}\in(\chi_{0},\infty)$, the
invariant manifold $M_{[\chi_{0},\chi_{1}]}\equiv
M_{\geq\chi_{0}}\cap\\{\chi\in[\chi_{0},\chi_{1}]\\}$ is invariant with
normally attracting structure. Therefore Corollary 5.5 can be applied to
(6.12), yielding the following result. The rate we have obtained coincides
with the preceding result shown in [15].
###### Theorem 6.10 (cf. [15]).
Assume that $m<0$ and $\beta<0$. Then the solution $(\chi,u(\chi),v(\chi))$
through the initial point $(\chi_{0},u_{0},v_{0})$ with $\chi_{0}>0$ whose
image under the directional embedding (6.13) is on the (local) stable manifold
of $M_{\geq\chi_{0}}$ for (6.14) is a blow-up solution with the blow-up rate
$u(\chi)\sim c_{u}(\chi_{\max}-\chi)^{-1/(1-m)}$ and $v(\chi)\sim
c_{v}(\chi_{\max}-\chi)^{-1/(1-m)}$ for some constants $c_{u},c_{v}>0$.
###### Remark 6.11.
In [15], dynamics of self-similar profiles is studied under a technical change
of variables, which changes (6.10) to an autonomous system. More precisely,
(6.10) with $u(0)=1$ and $u^{\prime}(0)=a\in\mathbb{R}$ is transformed via
$\chi=e^{r},\quad X(r)=\frac{\chi u^{\prime}}{u},\quad Y(r)=\chi^{2}u^{1-m}$
into
$\begin{cases}\dot{X}=X-mX^{2}-(\alpha+\beta X)Y,&\\\
\dot{Y}=(2+(1-m)X)Y,&\end{cases}\quad\dot{X}=dX/dr.$
In our methodology, on the other hand, no additional techniques are introduced
to obtain the same result when we pay our attention to blow-up solutions.
We have assumed $\beta<0$ in the present study. We can easily check that the
family of equilibria $\\{(\chi,0,-\beta\chi)\\}$ for (6.14) on the horizon
undergoes a transcritical bifurcation at $\beta=0$. In particular, when
$\beta>0$, the invariant manifold $M_{\geq\chi_{0}}$ becomes normally
repelling and hence blow-up associated with $M_{\geq\chi_{0}}$ never occurs,
which is consistent with observations in [15]. One of the original aims in
[15] was to construct self-similar profiles $U$ mentioned in (6.11). Another
parameter $\alpha$ should have the constraint for the backward self-
similarity: $2\beta=(1-m)\alpha-1$, which can be determined once $\beta$ is
fixed, and $\alpha$ is not explicitly appeared in characterization of blow-up
profiles in Theorem 6.10.
###### Remark 6.12.
Calculations of the Jacobian matrices imply that another branch of equilibria
on the horizon; $\\{(\chi,0,0)\mid\chi\in I\\}$ for any compact interval
$I\subset\mathbb{R}$, is not normally hyperbolic. Indeed, this branch contains
a center direction in the normal component. In this case, the center manifold
reduction in the corresponding direction will be required to determine the
precise blow-up behavior (cf. [21, 23, 24, 39]).
### 6.4 Radially symmetric solutions in a MEMS-type equation
The final example is the following system:
$u^{\prime\prime}+\frac{n-1}{r}u^{\prime}-r^{q}u^{-p}=0,\quad^{\prime}=\frac{d}{dr},$
(6.15)
where $n\in\mathbb{N}$, $p\in\mathbb{N}$, $q\in\mathbb{R}$ and
$r\in(0,\infty)$.
###### Remark 6.13.
The equation (6.15) originates from the following PDE, which is known as a
MEMS (micro-electro mechanical system) type equation:
$U_{t}=\Delta U-|x|^{q}U^{-p},\quad t>0,\quad x\in\mathbb{R}^{n}.$ (6.16)
Restricting our attention to radially symmetric stationary solutions
$U(t,x)=u(r)$ with $r=|x|$, then $u$ must satisfy (6.15). In [25], radially
symmetric stationary solutions $u$ with compact supports and derivatives
diverging as $r\to r_{+}-0$ and/or $r\to r_{-}+0$ for some
$0<r_{-}<r_{+}<\infty$, namely quenching solutions, are constructed. In
particular, all solutions (with $p\in 2\mathbb{N}$) $u(r)$ are negative on
these supports. This work reflects our selection of the directional embedding
in (6.19) below.
Keeping assertions in Remark 6.13 in mind, we pay attention to $w=u^{-1}$
instead of $u$ itself:
$2w^{-1}(w^{\prime})^{2}-w^{\prime\prime}-\frac{n-1}{r}w^{\prime}-r^{q}w^{p+2}=0,\quad^{\prime}=\frac{d}{dr},$
(6.17)
equivalently
$\frac{d}{d\rho}\begin{pmatrix}r\\\ w\\\ v\end{pmatrix}=\begin{pmatrix}1\\\
v\\\ -\frac{n-1}{r}v-r^{q}w^{p+2}+2w^{-1}v^{2}\end{pmatrix}.$ (6.18)
###### Lemma 6.14.
The system (6.18) is asymptotically quasi-homogeneous of type $(0,2,p+3)$ and
order $p+2$ when $p$ is even, while type $(0,1,(p+3)/2)$ and order $(p+3)/2$
when $p$ is odd.
In the case of $p$ being even, introducing the directional embedding
$r=r,\quad w=-\frac{1}{s^{2}},\quad v=\frac{x}{s^{p+3}}$ (6.19)
and the time-scale desingularization
$\frac{d\tau}{d\rho}=s^{-(p+1)},$
we have the following desingularized vector field:
$\frac{d}{d\tau}\begin{pmatrix}r\\\ s\\\
x\end{pmatrix}=\begin{pmatrix}s^{p+1}\\\ \frac{1}{2}sx\\\
\frac{p-1}{2}x^{2}-\frac{n-1}{r}s^{p+1}x-r^{q}\end{pmatrix}.$ (6.20)
Equilibria on the horizon $\mathcal{E}=\\{s=0\\}$ exist because $p$ is
even161616 When $p$ is odd, the equation requiring the existence of equilibria
on $\mathcal{E}$ is essentially the same as the one associated with (6.20),
which yields that there are no equilibria on $\mathcal{E}$, because the sign
of $r^{q}$ in (6.20) becomes “$+$”. , in which case these are
$p_{\pm}(r)=\left(r,0,\pm\frac{r^{q/2}}{\sqrt{\frac{p-1}{2}}}\right),$
parameterized by $r>0$. The Jacobian matrix on the horizon with $r>0$ is
$\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}x&0\\\
-qr^{q-1}&0&2x\sqrt{\frac{p-1}{2}}\end{pmatrix},$
implying that, for any fixed $r_{0}>0$, the invariant set
$M_{+}=\\{p_{+}(r)\mid r\geq r_{0}\\}$ is normally repelling, while
$M_{-}=\\{p_{-}(r)\mid r\geq r_{0}\\}$ is normally attracting. In particular,
$M_{-}$ satisfies all requirements in Corollary 5.5. As a consequence, we have
the following theorem.
###### Theorem 6.15.
Consider (6.17) with $p\in 2\mathbb{N}$. Then the solution $(r,w(r),v(r))$
through the initial point $(r_{0},w_{0},v_{0})$ with $r_{0}>0$ whose image
under the directional embedding (6.19) is on the (local) stable manifold of
$M_{\geq r_{0}}$ for (6.20) is a blow-up solution with the blow-up rate
$w(r)\sim-c_{w}(r_{\max}-r)^{-2/(p+1)},\quad v(r)\sim-
c_{v}(r_{\max}-r)^{-(p+3)/(p+1)}\quad\text{ as }\quad r\to r_{\max}-0$
for some constants $r_{\max}>r_{0}$ and $c_{w},c_{v}>0$. The corresponding
solution $u(r)$ of (6.15) has the following asymptotic behavior:
$u(r)\sim-c_{u}(r_{\max}-r)^{2/(p+1)},\quad
u^{\prime}(r)\sim\tilde{c}_{u}(r_{\max}-r)^{-(p-1)/(p+1)}\quad\text{ as }\quad
r\to r_{\max}-0$
for some constants $c_{u},\tilde{c}_{u}>0$. In particular, $u(r)$ is a
quenching solution of (6.15).
The family of blow-up solutions we have obtained is different from those
discussed in [25]. This can be because dominant terms of the vector field at
infinity we pay attention to, namely the basic scalings, are different.
## Conclusion
In this paper, we have developed methodologies to characterize blow-up
solutions shadowing “trajectories at infinity”, and those in nonautonomous
systems. A unified description of blow-up solutions is organized by means of
shadowing trajectories, or geometrically asymptotic phase of invariant
manifolds, and specific exponential decay rates, which generalizes
methodologies developed in preceding works [38, 39]. Indeed, as investigated
in examples (Section 6), various finite-time singularities involving blow-ups
are extracted by a common machinery. We strongly believe that the present
investigation will be the basis on characterizing blow-up solutions from the
viewpoint of dynamical systems. We end this paper by providing several
directions as future works.
### Asymptotic expansions and applications to computer-assisted proofs
One direction of future works would be multi-order asymptotic expansions of
blow-ups, as developed for stationary blow-ups, namely the case where blow-ups
are described by equilibria on the horizon when the original vector field is
autonomous [2, 30]. A natural question in this theme towards the
generalization is the development of the similar methodology to blow-ups
induced by e.g., periodic orbits, NHIMs or others such as chaotic invariant
sets on the horizon, and in nonautonomous systems. In [2, 30], not only a
systematic methodology to calculate asymptotic expansions of blow-up solutions
in arbitrary orders, but also a natural correspondence between the essential
information characterizing asymptotic expansions and linearized information
around equilibria on the horizon are shown. The latter correspondence enables
us to extract blow-up structure in a simple way. Indeed, as seen in Section
6.1, globally defined compactifications or embeddings require tedious
calculations towards the conclusion, which affect the success of applications,
such as computer-assisted proofs, equivalently rigorous numerics (e.g., [35,
40, 41, 52]). In general, global embeddings drastically raise the complexity
of vector fields, such as types and orders of polynomial, or general
asymptotically quasi-homogeneous vector fields, and require quite large
computational costs, even for typical numerical computations of blow-ups. The
achievement of the correspondence to asymptotic expansions can reduce the
computational costs for blow-ups, in particular for them induced by periodic
orbits and NHIMs on the horizon and for nonautonomous systems once the
corresponding derivation for these setting is achieved, because the systems
involving asymptotic expansions are much simpler than those involving
embeddings. Because applications involving computer-assisted proofs of
periodic orbits and NHIMs themselves are widely achieved (e.g., [18]),
combination with the present and future machineries will immediately
contribute to develop computer-assisted proofs of blow-ups with the above
asymptotic behavior at infinity and their global extensions established in
[35] for stationary blow-ups.
### Blow-up rates, spectral intervals and unique ergodicity
Another natural direction in studies of blow-up solutions is to determine
blow-up rates, the principal terms of blow-up solutions, where the leading
term is our interest unlike the multi-order asymptotic expansions. This is one
of central issues in blow-up studies for decades, in particular for PDEs,
while all blow-ups mentioned in the present paper are referred to as type-I,
namely the case where the blow-up rates are automatically and uniquely
determined by type and order of asymptotically quasi-homogeneous vector
fields. Note that examples exhibiting rates other than type-I are collected in
[39], where nonhyperbolicity of invariant sets on the horizon plays a central
role in determining special blow-up rates, including ones referred to as type-
II in the field of PDEs. Although investigations of such blow-up rates with
complex asymptotic behavior and in nonautonomous systems are ones of natural
direction of future investigations (cf. [34]), we would like to leave another
direction involving blow-up rates.
One essential condition to determine blow-up rates in the present arguments is
the precise exponential rate of convergence to trajectories on the horizon. It
will be shown in Appendix B that discrete distribution of dichotomy spectra of
linear skew-product flows provides the required convergence rate in the case
of NHIMs. In special cases mentioned in Proposition B.4, dichotomy spectra (or
specially eigenvalues and Floquet exponents for constant and periodic cases,
respectively) determine the unique exponential decay rates. It would be also
true from comments in [47], that unique ergodicity of flows on the base
spaces; the hulls of trajectories on NHIMs being minimal sets, could yield the
same conclusion. On the other hand, there are several examples that dichotomy
spectra include continuous spectral intervals, namely intervals with positive
Lebesgue measure in the spectra, which are observed when the flow on the base
space is non-uniquely ergodic. Our question here is the following:
* •
Is the blow-up rate uniquely determined even when the associated base sets in
NHIMs admit more than one ergodic measures, or (4.2) is violated ?
If uniquely determined, its determination with the mechanism is a central
issue. Otherwise, the concrete description of blow-up rate in such a case
would be the issue.
## Acknowledgements
KM was partially supported by World Premier International Research Center
Initiative (WPI), Ministry of Education, Culture, Sports, Science and
Technology (MEXT), Japan, and JSPS Grant-in-Aid for Scientists (B) (No.
JP23K20813). He would like also to thank Professors Jordi-Lluís Figueras and
Akitoshi Takayasu for giving him inspiring suggestions to the present study.
## References
* [1] L. Adrianova. Introduction to linear systems of differential equations. Translations of Mathematical Monographs, 1995.
* [2] T. Asai, H. Kodani, K. Matsue, H. Ochiai, and T. Sasaki. Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. I - Method and Justification. arXiv:2211.06865, 2022.
* [3] L. Dieci and E.S. Van Vleck. Lyapunov and other spectra: a survey. Collected lectures on the preservation of stability under discretization (Fort Collins, CO, 2001), 197:218, 2002.
* [4] L. Dieci and E.S. Van Vleck. Lyapunov spectral intervals: theory and computation. SIAM Journal on Numerical Analysis, 40(2):516–542, 2002.
* [5] L. Dieci and E.S. Van Vleck. Lyapunov and Sacker-Sell spectral intervals. Journal of dynamics and differential equations, 19:265–293, 2007\.
* [6] F. Dumortier. Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations. In Bifurcations and Periodic Orbits of Vector Fields, pages 19–73. Springer, 1993.
* [7] F. Dumortier and C. Herssens. Polynomial Liénard equations near infinity. Journal of Differential Equations, 153(1):1–29, 1999.
* [8] F. Dumortier, J. Llibre, and J.C. Artés. Qualitative theory of planar differential systems. Springer, 2006.
* [9] J. Eldering. Normally hyperbolic invariant manifolds: the noncompact case, volume 2. Springer, 2013.
* [10] J. Eldering, M. Kvalheim, and S. Revzen. Global linearization and fiber bundle structure of invariant manifolds. Nonlinearity, 31(9):4202–4245, 2018.
* [11] N. Fenichel. Persistence and smoothness of invariant manifolds for flows. Indiana University Mathematics Journal, 21(3):193–226, 1971.
* [12] N. Fenichel. Asymptotic stability with rate conditions. Indiana University Mathematics Journal, 23(12):1109–1137, 1974.
* [13] N. Fenichel. Asymptotic stability with rate conditions, II . Indiana University Mathematics Journal, 26(1):81–93, 1977\.
* [14] N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations, 31(1):53–98, 1979.
* [15] R. Ferreira and J.L. Vázquez. Study of self-similarity for the fast-diffusion equation. Advances in Differential Equations, 8(9):1125–1152, 2003.
* [16] A.M. Fink. Almost periodic differential equations, volume 377 of Springer Lecture Notes in Mathematics. Springer, Berlin, Heidelberg, 1974.
* [17] A. Giraldo, B. Krauskopf, and H.M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. arXiv:2206.12031, 2022.
* [18] A. Haro, M. Canadell, J.-L. Figueras, A. Luque, and J.-M. Mondelo. The parameterization method for invariant manifolds : from rigorous results to effective computations, volume 195. Springer, 2016.
* [19] M.W. Hirsch. Differential Topology, volume 33 of Graduate Texts in Mathematics. Springer New York, 1994.
* [20] M.W. Hirsch, C.C. Pugh, and M. Shub. Invariant manifolds, volume 583 of Springer Lecture Notes in Mathematics. Springer, Berlin, Heidelberg, 1977.
* [21] Y. Ichida. Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations. Discrete and Continuous Dynamical Systems-B, 28(2):1116–1132, 2023.
* [22] Y. Ichida. Radially symmetric stationary solutions for certain chemotaxis systems in higher dimensions: A geometric approach. Discrete and Continuous Dynamical Systems, 43(5):1975–2001, 2023.
* [23] Y. Ichida, K. Matsue, and T.O. Sakamoto. A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations. JSIAM Letters, 12:65–68, 2020.
* [24] Y. Ichida and T.O. Sakamoto. Quasi traveling waves with quenching in a reaction-diffusion equation in the presence of negative powers nonlinearity. Proc. Japan Acad. Ser. A Math Sci., 96:1–6, 2020.
* [25] Y. Ichida and T.O. Sakamoto. Radial symmetric stationary solutions for a MEMS type reaction-diffusion equation with spatially dependent nonlinearity. Japan Journal of Industrial and Applied Mathematics, 38(1):297–322, 2021.
* [26] Y. Ichida and T.O. Sakamoto. Radially symmetric stationary solutions for a MEMS type reaction-diffusion equation with fringing field. Nonlinearity, 36(1):71, 2022.
* [27] B.L. Keyfitz. Singular shocks: retrospective and prospective. Confluentes Mathematici, 3(03):445–470, 2011.
* [28] B.L. Keyfitz and H.C. Kranzer. Spaces of weighted measures for conservation laws with singular shock solutions. Journal of Differential Equations, 118(2):420–451, 1995.
* [29] P.E. Kloeden and M. Rasmussen. Nonautonomous dynamical systems, volume 176 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2011.
* [30] H. Kodani, K. Matsue, H. Ochiai, and A. Takayasu. Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. II - Dynamical Correspondence. arXiv:2211.06868, 2022.
* [31] H. Kokubu and R. Roussarie. Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: Part I. Journal of Dynamics and Differential Equations, 16(2):513–557, 2004\.
* [32] H.C. Kranzer and B.L. Keyfitz. A strictly hyperbolic system of conservation laws admitting singular shocks. In Nonlinear evolution equations that change type, pages 107–125. Springer, 1990.
* [33] M.D. Kruskal, N. Joshi, and R. Halburd. Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property. Integrability of nonlinear systems, pages 171–205, 1997.
* [34] C. Kuehn. Normal hyperbolicity and unbounded critical manifolds. Nonlinearity, 27(6):1351–1366, 2014.
* [35] J.-P. Lessard, K. Matsue, and A. Takayasu. Saddle-type blow-up solutions with computer-assisted proofs: Validation and extraction of global nature. Journal of Nonlinear Science, pages 1–75, 2023.
* [36] A. Luchko and I. Parasyuk. On asymptotic phase of dynamical system hyperbolic along attracting invariant manifold. arXiv:1810.00268, 2018.
* [37] A. Luchko and I. Parasyuk. Asymptotic phase for flows with exponentially stable partially hyperbolic invariant manifolds. Electronic Journal of Qualitative Theory of Differential Equations, 2021(36):1–28, 2021.
* [38] K. Matsue. On blow-up solutions of differential equations with Poincaré-type compactifications. SIAM Journal on Applied Dynamical Systems, 17(3):2249–2288, 2018\.
* [39] K. Matsue. Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs. Journal of Differential Equations, 267(12):7313–7368, 2019.
* [40] K. Matsue and A. Takayasu. Numerical validation of blow-up solutions with quasi-homogeneous compactifications. Numerische Mathematik, 145:605–654, 2020.
* [41] K. Matsue and A. Takayasu. Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity. Journal of Computational and Applied Mathematics, 374:112607, 2020\.
* [42] J.C. Oxtoby. Ergodic sets. Bulletin of the American Mathematical Society, 58(2):116–136, 1952.
* [43] J. Palis and F. Takens. Topological equivalence of normally hyperbolic dynamical systems. Topology, 16(4):335–345, 1977.
* [44] C. Pugh and M. Shub. Linearization of normally hyperbolic diffeomorphisms and flows. Inventiones mathematicae, 10(3):187–198, 1970.
* [45] R.J. Sacker and G.R. Sell. Existence of dichotomies and invariant splittings for linear differential systems I. Journal of Differential Equations, 15(3):429–458, 1974.
* [46] R.J. Sacker and G.R. Sell. A spectral theory for linear differential systems. Journal of Differential Equations, 27(3):320–358, 1978.
* [47] R.J.J Sacker and G.R. Sell. The spectrum of an invariant submanifold. Journal of Differential Equations, 38(2):135–160, 1980.
* [48] D.G. Schaeffer, S. Schecter, and M. Shearer. Nonstrictly hyperbolic conservation laws with a parabolic line. Journal of differential equations, 103(1):94–126, 1993.
* [49] S. Schecter. Existence of Dafermos profiles for singular shocks. Journal of Differential Equations, 205(1):185–210, 2004.
* [50] S. Schecter and P. Szmolyan. Composite waves in the Dafermos regularization. Journal of Dynamics and Differential Equations, 16(3):847–867, 2004\.
* [51] G.R. Sell. Topological dynamics and ordinary differential equations, volume 33. Van Nostrand-Reinhold, 1971.
* [52] A. Takayasu, K. Matsue, T. Sasaki, K. Tanaka, M. Mizuguchi, and S. Oishi. Numerical validation of blow-up solutions for ordinary differential equations. Journal of Computational and Applied Mathematics, 314:10–29, 2017\.
* [53] S. Wieczorek, C. Xie, and C.K.R.T. Jones. Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds. Nonlinearity, 34(5):2970, 2021.
* [54] S. Wiggins. Normally hyperbolic invariant manifolds in dynamical systems, volume 105. Springer Science & Business Media, 2013.
## Appendix A Directional embeddings
Here a locally defined embedding is introduced, which shall be called a
directional embedding.
###### Definition A.1 (Directional embedding, cf. [8, 38]).
A directional embedding of type $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ is
defined as
$\displaystyle y=(y_{1},\ldots,y_{n})\mapsto
T_{d;\alpha}(y)=(s,\hat{x})\equiv(s,\hat{x}_{1},\ldots,\hat{x}_{i_{0}-1},\hat{x}_{i_{0}+1},\ldots,\hat{x}_{n}),$
$\displaystyle
y_{i}:=\frac{\hat{x}_{i}}{s^{\alpha_{i}}}\quad(i\not=i_{0}),\quad
y_{i_{0}}:=\pm\frac{1}{s^{\alpha_{i_{0}}}}$ (A.1)
with given direction $i_{0}$ and the signature $\pm$, provided $i_{0}\in
I_{\alpha}$. This embedding is bijective in $U\cap\\{\pm y_{i_{0}}>0\\}$, in
which sense directional embeddings are local ones. In particular, this
embedding is available when we are interested in solutions of (1.1) such that
the $i_{0}$-th component has the identical sign during time evolution. The
image of $T_{d;\alpha}$ is
$\mathcal{D}=\\{(s,\hat{x}_{1},\ldots,\hat{x}_{i_{0}-1},\hat{x}_{i_{0}+1},\ldots,\hat{x}_{n})\mid
s>0,\,\hat{x}_{i}\in\mathbb{R}\,(i\in
I_{\alpha}\setminus\\{i_{0}\\}),\,(\hat{x}_{j_{1}},\ldots,\hat{x}_{j_{n-l}})\in\tilde{U}\\},$
(A.2)
where $\\{j_{1},\cdots,j_{n-l}\\}=\\{1,\cdots,n\\}\setminus I_{\alpha}$. The
set $\mathcal{E}=\\{s=0\\}$ corresponds to the infinity in the original
coordinate, which shall be called the horizon.
For simplicity, fix $i_{0}=1$ in (A.1) in the following arguments. Next
transform (1.1) via (A.1), which is straightforward:
$\displaystyle\frac{ds}{dt}$
$\displaystyle=-\frac{1}{\alpha_{1}}s^{-k+1}\hat{f}_{1}(s,\hat{x}_{2},\ldots,\hat{x}_{n}),$
$\displaystyle\frac{d\hat{x}_{i}}{dt}$
$\displaystyle=s^{-k}\left\\{\hat{f}_{i}(s,\hat{x}_{2},\ldots,\hat{x}_{n})-\frac{\alpha_{i}}{\alpha_{1}}x_{i}\hat{f}_{1}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\right\\}\quad(i=2,\ldots,n),$
where
$\hat{f}_{i}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\equiv
s^{k+\alpha_{i}}f_{i}(s^{-\alpha_{1}},s^{-\alpha_{2}}\hat{x}_{2},\ldots,s^{-\alpha_{n}}\hat{x}_{n}),\quad
i=1,\ldots,n.$ (A.3)
The resulting vector field is still singular near the horizon, but it turns
out that the order of divergence of vector field as $s\to+0$ is $O(s^{-k})$,
and hence the following time-scale transformation is available.
###### Definition A.2 (Time-variable desingularization: the directional
version).
Define the new time variable $\tau_{d}$ by
$d\tau_{d}=s(t)^{-k}dt$ (A.4)
equivalently,
$t=t_{0}+\int_{\tau_{0}}^{\tau}s(\tau_{d})^{k}d\tau_{d},$ (A.5)
where $\tau_{0}$ and $t_{0}$ denote the correspondence of initial times, and
$s(\tau_{d})$ is the solution trajectory $s(t)$ under the parameter
$\tau_{d}$. We shall call (A.4) the time-variable desingularization (of order
$k+1$).
The vector field $g=g_{d}$ in $\tau_{d}$-time-scale is171717 The existence of
$B$ follows by cyclic permutations and the fact that $\alpha_{1}>0$. See also
[38].
$\begin{pmatrix}\frac{ds}{d\tau_{d}}\\\ \frac{dx_{2}}{d\tau_{d}}\\\ \vdots\\\
\frac{dx_{n}}{d\tau_{d}}\end{pmatrix}=g_{d}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\equiv\begin{pmatrix}-s&0&\cdots&0\\\
0&1&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&1\end{pmatrix}B\begin{pmatrix}\hat{f}_{1}\\\ \hat{f}_{2}\\\
\vdots\\\ \hat{f}_{n}\end{pmatrix},\quad
B=\begin{pmatrix}\alpha_{1}&0&\cdots&0&0\\\
\alpha_{2}\hat{x}_{2}&1&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
\alpha_{n-1}\hat{x}_{n-1}&0&\cdots&1&0\\\
\alpha_{n}\hat{x}_{n}&0&\cdots&0&1\end{pmatrix}.$ (A.6)
The componentwise expression is
$\displaystyle\frac{ds}{d\tau_{d}}$
$\displaystyle=g_{d,1}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\equiv-\frac{1}{\alpha_{1}}s\hat{f}_{1}(s,\hat{x}_{2},\ldots,\hat{x}_{n}),$
$\displaystyle\frac{d\hat{x}_{i}}{d\tau_{d}}$
$\displaystyle=g_{d,i}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\equiv\hat{f}_{i}(s,\hat{x}_{2},\ldots,\hat{x}_{n})-\frac{\alpha_{i}}{\alpha_{1}}x_{i}\hat{f}_{1}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\quad(i=2,\ldots,n).$
Note that the above derivation makes sense even if
$I_{\alpha}\not=\\{1,\ldots,n\\}$ provided $1\in I_{\alpha}$, namely
$\alpha_{1}>0$. The resulting vector field is as smooth as $f$ including $s=0$
and hence dynamics at infinity makes sense through dynamics generated by (A.6)
around the horizon $\mathcal{E}=\\{s=0\\}$. Finally note that a correspondence
between global/directional embeddings is provided in [38].
## Appendix B Discussion 1: Blow-up rates and spectral properties of
invariant manifolds
Here we discuss the precise nature of blow-up behavior stated in Theorem 4.1
by means of spectral theory for linear differential systems (e.g., [45, 46, 3,
4, 5]).
An essence to describe (type-I) blow-up rates is the exponential decay with
the uniquely determined rate (4.2). One of the next issue will be when (4.2)
is achieved. Because the usual decay estimates in characterizing normal
hyperbolicity as shown in Proposition 2.3, or general hyperbolicity describing
asymptotic phase, provide only upper bounds of decay rates, which are not
sufficient to obtain (4.2) in general. Fortunately, compact boundaryless NHIMs
admits a local topological conjugacy to the linearized flow and the decay
rates are characterized by spectral properties of the linearized systems on
NHIMs. In particular, our interests are reduced to the asymptotic behavior in
systems of the form
${\bf v}^{\prime}=A(t){\bf v},$ (B.1)
where $A:\mathbb{R}\to\mathbb{R}^{n\times n}$ denotes a bounded, continuous
matrix-valued function. The spectral theory for (B.1) is back to Sacker and
Sell (e.g., [45]). Recall that the linear skew-product flow $\theta$ on the
vector bundle $\xi=(\pi,E,X)$ is the pair of continuous maps
$\theta=(\varphi,\Phi)$ given by $\theta(t,{\bf v},{\bf x})=(\varphi(t,{\bf
x}),\Phi(t,{\bf x}){\bf v})$, where $\varphi$ is a flow on the base space $X$
of $\xi$, and $\Phi(t,{\bf x})$ is linear on the fiber $F_{\varphi(t,{\bf
x})}$.
Let $\theta=(\varphi,\Phi)$ be a linear skew-product flow on a vector bundle
$\xi=(\pi,E,X)$ with a compact Hausdorff base space $X$. For each
$\lambda\in\mathbb{R}$, define the linear skew-product flow $\theta_{\lambda}$
by
$\theta_{\lambda}(t,{\bf v},{\bf x})\equiv(\varphi(t,{\bf
x}),\Phi_{\lambda}(t,{\bf x}){\bf v}):=(\varphi(t,{\bf x}),e^{-\lambda
t}\Phi(t,{\bf x}){\bf v}).$
Stability of solutions for linear skew-product flows is discussed in terms of
dichotomy given below, which is a generalization of hyperbolicity for
equilibria and periodic orbits.
###### Definition B.1 (Exponential dichotomy. e.g., [46]).
Let $M$ be a subset of the base space $X$. We say that the linear skew-product
flow $\theta_{\lambda}$ admits an exponential dichotomy over $M$ if there is a
projector $P:E(M)\to E(M)$ and positive constants $K,a>0$ such that
$\displaystyle\left|\Phi_{\lambda}(t,{\bf x})P({\bf
x})\Phi_{\lambda}^{-1}(s,{\bf x})\right|$ $\displaystyle\leq
Ke^{-a(t-s)},\quad s\leq t,$ $\displaystyle\left|\Phi_{\lambda}(t,{\bf
x})[I-P({\bf x})]\Phi_{\lambda}^{-1}(s,{\bf x})\right|$ $\displaystyle\leq
Ke^{-a(s-t)},\quad t\leq s,$
for all ${\bf x}\in M$.
The dichotomy determines the spectrum for linear skew-product flows on vector
bundles. Let $(\xi,\theta)$ be the pair of a vector bundle $\xi$ and a linear
skew-product flow $\theta$ on $\xi$.
###### Definition B.2 (Dichotomy spectrum, cf. [45, 46]).
For a subset $M\subset X$, define
$\displaystyle\rho({\bf x})$
$\displaystyle:=\\{\lambda\in\mathbb{R}\mid\theta_{\lambda}\text{ has the
exponential dichotomy}\\}\quad({\bf x}\in X),$ $\displaystyle\rho(M)$
$\displaystyle:=\bigcap_{{\bf x}\in M}\rho({\bf
x}),\quad\Sigma(M):=\mathbb{R}\setminus\rho(M).$
We call $\Sigma(M)$ the dichotomy spectrum of $(\xi,\theta)$ over $M$. By
$\Sigma({\bf x})$ we mean $\Sigma(\\{{\bf x}\\})$ for each ${\bf x}\in X$.
One of benefits to apply dichotomy spectra is that decomposition of solutions
to (B.1) into invariant subbundles is applied. It is known that the dichotomy
spectrum $\Sigma(M)$ for a compact invariant set $M$ consists of at most $n$
mutually disjoint compact intervals, which is partially characterized as
follows.
###### Lemma B.3 (cf. [46], Lemma 6).
Under the setting in Definition B.2, let $M$ be a compact invariant set in
$X$. Then the dichotomy spectrum $\Sigma(M)$ is compact. More precisely, we
have
$\Sigma(M)=[a_{1},b_{1}]\cup\cdots\cup[a_{k},b_{k}]$ (B.2)
with181818 The notation in the present subsection of $k$ is independent of the
order of asymptotically quasi-homogeneous vector fields. $k\leq n=\dim E(M)$
and $a_{1}\leq b_{1}<a_{2}\leq\cdots\leq b_{k-1}<a_{k}\leq b_{k}$.
###### Proposition B.4 (Invariant subbundles, e.g., [46]).
Let $M$ be a compact invariant set in $X$ and $\Sigma(M)$ be the dichotomy
spectrum of $(\xi|_{M},\theta)$ given by (B.2). Then the vector bundle $E(M)$
over $M$ is decomposed into
$E(M)=E_{1}\oplus\cdots\oplus E_{k}$
as the Whitney sum, such that each subbundle $E_{i}$ over $M$ is associated
with the spectral interval $[a_{i},b_{i}]$ and invariant under $\theta$. When
$\lambda\in(b_{i},a_{i+1})$ for some $i\in\\{1,\ldots,k-1\\}$, we have
$\displaystyle\lim_{t\to+\infty}\left\|e^{-\lambda t}\Phi(t,{\bf x}){\bf
v}\right\|\to 0\quad\text{ if }\quad({\bf v},{\bf x})\in
E_{1}\oplus\cdots\oplus E_{i},$
$\displaystyle\lim_{t\to-\infty}\left\|e^{-\lambda t}\Phi(t,{\bf x}){\bf
v}\right\|\to 0\quad\text{ if }\quad({\bf v},{\bf x})\in
E_{i+1}\oplus\cdots\oplus E_{k}.$
Once we detect the location of trajectories for the linear skew-product flow
on $(\xi|_{M},\theta)$, its spectral property is extracted by means of
invariant spectral subbundles, which can vary in time-evolution within the
specified spectral intervals. The simplest case will be the discrete
distribution of spectrum.
###### Proposition B.5 (Discrete distribution of spectrum. e.g., [46]).
For a pair $(\xi,\theta)$ mentioned with $\dim\xi=n$, let $X$ be the compact
Hausdorff base space of $\xi$ and $\varphi$ be the flow on $X$ determining
$\theta$. If one of the following for the trajectory ${\bf x}(t)$ on $X$ for
$\varphi$ holds true, then $\Sigma(M)$ consists of finite points, were
$M=\overline{\\{{\bf x}(t)\\}}\equiv H({\bf x}(t))$ denotes the hull of ${\bf
x}(t)$;
* •
${\bf x}(t)$ is an equilibrium;
* •
${\bf x}(t)$ is a time-periodic solution;
* •
${\bf x}(t)$ is almost periodic (cf. [51, 16]) and the spectrum $\Sigma(M)$
consists of $n$-mutually disjoint closed intervals
It is also mentioned in [46] that this claim is also true for linear skew-
product flows on the vector bundle $\xi$ over the base space $X$ being
compact, minimal and a uniquely ergodic measure $\mu$ (unique ergodicity of
sets). Indeed, [46] indicates that it is possible to have a nontrivial
dichotomy spectral interval when one does not have unique ergodicity. We shall
refer to e.g., [42, 51] for ergodicity, because we will not discuss ergodicity
of dynamical systems in the present paper, while it is left to future works.
See e.g., [51] for detailed arguments of minimal sets.
###### Remark B.6 (Lyapunov exponents and dichotomy spectrum).
Lyapunov exponents are another well-known quantities to characterize
exponential behavior of solutions to (B.1). See e.g., [1] for details.
Lyapunov exponents determine the Lyapunov spectrum, the collection of
intervals between upper and lower Lyapunov exponents. It is well known that
Lyapunov spectrum is generally a proper subset of dichotomy spectrum,
indicating that Lyapunov exponents are not suitable for characterizing
decaying/expanding properties of solutions to (B.1). See e.g., [3, 4, 5]. On
the other hand, it is shown in e.g., [46] that, if a spectral interval
$[a_{i},b_{i}]$ in the dichotomy spectrum consists of one point;
$a_{i}=b_{i}$, then the quantity coincides with the corresponding Lyapunov
exponent, namely the exponent included in $[a_{i},b_{i}]$.
###### Proposition B.7.
Under the setting of Corollary 4.3, let $N\varphi_{g;{\bf x}_{\gamma}}$ be the
linear skew-product flow on the algebraic normal bundle
$\mathcal{N}=E^{u}(V_{{\bf x}_{\gamma}})\oplus E^{s}(V_{{\bf x}_{\gamma}})$
over the hull $V_{{\bf x}_{\gamma}}=H({\bf x}_{\gamma}(\tau))$ induced by the
normally hyperbolic structure of $M$ for $g$. Assume that the dichotomy
spectrum $\Sigma_{E^{s}}(V_{{\bf x}_{\gamma}})$ of $(E^{s}(V_{{\bf
x}_{\gamma}}),N\varphi_{g;{\bf x}_{\gamma}})$ restricted to the stable
subbundle $E^{s}(V_{{\bf x}_{\gamma}})$ is discrete:
$\Sigma_{E^{s}}(V_{{\bf
x}_{\gamma}})=\\{\lambda_{1}^{s},\ldots,\lambda_{m_{s}}^{s}\\}\subset(-\infty,0),$
where $m_{s}\leq n_{s}$. Then the exponential estimate (4.2) holds.
###### Proof.
Let ${\bf x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ and denote
$E^{s}=E^{s}(V_{{\bf x}_{\gamma}})$ for simplicity. Our assumption implies
that the bundle $\mathcal{N}=E^{s}\oplus E^{u}$ admits the following Whitney
sum decomposition into spectral invariant subbundles (Proposition B.4):
$E^{s}=E_{1}^{s}\oplus\ldots\oplus E_{m_{s}}^{s},$ (B.3)
$E_{i}^{s}$ is the spectral invariant subbundle associated with the point
spectrum $\\{\lambda_{i}^{s}\\}$, $i=1,\ldots,m_{s}$. Invariant structure of
the spectral decomposition indicates that the fiber component ${\bf v}(\tau)$
of a trajectory $({\bf v}(\tau),{\bf x}_{\gamma}(\tau))=h({\bf x}(\tau))$ for
the linear skew-product flow $N\varphi_{g;{\bf x}_{\gamma}}$ with ${\bf
x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ and the topological conjugacy
$h$ (cf. Proposition 2.3) is located on the subbundle
$E^{s}_{i_{1}}\oplus\cdots\oplus E^{s}_{i_{p}}\quad\text{ with }\quad
i_{1},\ldots,i_{p}\in\\{1,\ldots,m_{s}\\}\quad\text{ and }\quad
i_{1}<\cdots<i_{p}$ (B.4)
for all $\tau\in\mathbb{R}$. In particular, ${\bf v}(\tau)$ is uniquely
decomposed into
$\displaystyle{\bf
v}(\tau)=v_{i_{1}}(\tau)+\cdots+v_{i_{p}}(\tau),\quad\text{where}\quad(v_{i_{l}}(\tau),{\bf
x}_{\gamma}(\tau))\in E_{i_{l}}^{s}\,\text{ with }\quad
v_{i_{l}}(\tau)\not=0\,\text{ for all }\quad l=1,\ldots,p.$
Without the loss of generality, we may assume that
$\lambda_{1}<\cdots<\lambda_{m_{s}}<0$. By the characterization of exponential
dichotomy, each component $v_{i_{l}}(\tau)$ satisfies
$e^{(-\lambda_{i_{l}}+\epsilon)\tau}|v_{i_{l}}(\tau)|=\begin{cases}+\infty&\epsilon>0\\\
0&\epsilon<0\end{cases}\quad\text{ as }\quad\tau\to+\infty.$
The original ${\bf x}(\tau)$ as the image of such a trajectory through the
conjugacy $h$ is written by
${\bf x}(\tau)\equiv h^{-1}({\bf v}(\tau),{\bf x}_{\gamma}(\tau))={\bf
x}_{\gamma}(\tau)+{\bf v}(\tau)+\tilde{h}^{-1}({\bf v}(\tau),{\bf
x}_{\gamma}(\tau))$
via the identity $\tilde{h}^{-1}({\bf x})=h^{-1}({\bf x})-{\bf x}$ being
continuous and bounded (choosing a neighborhood of $M$ in $W_{\rm loc}^{s}(M)$
smaller so that it is included in a compact neighborhood of $M$ in $W_{\rm
loc}^{s}(M)$ ensuring the existence of $h$). Because the conjugacy $h$
preserves $M$, namely $h({\bf x})={\bf x}$ for all ${\bf x}\in M$, exponential
decay of ${\bf v}(\tau)$, continuity of $h$ with the mentioned property as
well as the estimate (2.3) yield
$\displaystyle e^{(\lambda_{i}+\epsilon)\tau}\left|(h^{-1})_{i}({\bf
v}(\tau),{\bf x}_{\gamma}(\tau))-x_{i;\gamma}(\tau)\right|$
$\displaystyle=e^{(\lambda_{i}+\epsilon)\tau}\left|v_{i}(\tau)+(\tilde{h}^{-1})_{i}({\bf
v}(\tau),{\bf x}_{\gamma}(\tau))\right|$
$\displaystyle=\begin{cases}+\infty&\epsilon>0\\\
0&\epsilon<0\end{cases}\quad\text{ as }\quad\tau\to+\infty$
with some $-\lambda_{i}\leq\lambda_{s}<0$ for all $i=1,\ldots,n$, where
$\lambda_{s}\in\\{\lambda_{i}\\}_{i=1}^{m_{s}}$ is uniformly determined. ∎
As a consequence, for NHIMs admitting the spectral information mentioned in
the proposition, Theorem 4.1 can be applied. In particular, combining
Proposition B.5, we can rephrase results in the preceding paper [38].
###### Corollary B.8 (Stationary and periodic blow-ups, [38]).
In Theorem 4.3, assume that $M$ is either a hyperbolic equilibrium or a
hyperbolic periodic orbit. Then all statements in Theorem 4.1 hold.
Blow-ups in backward time direction can be described in the similar way, in
which case trajectories on the local unstable manifold $W_{\rm
loc}^{u}(M;\varphi_{g})$ determine the behavior.
## Appendix C Discussion 2: Blow-up rates in nonautonomous blow-ups
This appendix addresses several practical ideas to apply Theorem 5.1 involving
blow-up rates in concrete nonautonomous systems, and their validity to prove
the concrete blow-up behavior.
### C.1 Blow-up rates
One approach is to use the idea stated in Proposition B.7 in autonomous cases;
the spectral information of $M_{I}$, the tube of an invariant manifold over
$I\subset\mathbb{R}$ given in (5.8), for the linearized systems. In
Proposition B.7, the spectral information is applied through the local
topological conjugacy, which is generally admitted for boundaryless NHIMs (cf.
[20, 44]). Because our invariant manifolds (for the modified vector field
$\tilde{g}$ given in (5.10)) of the form $M_{I^{\prime}}$ have nontrivial
boundary, some modifications are necessary if we apply the above conjugacy.
Here we refer to a result in [10], where global linearization of (normally
attracting) invariant manifolds is provided, no matter when invariant
manifolds have boundaries. The technique relies on several boundaryless
treatments in differential topology (e.g., [19]) and the uniformity
lemma191919 If we try to provide the similar result to invariant manifolds
which are not normally hyperbolic, the corresponding estimate will be
necessary. stemming from normally hyperbolic structure of manifolds (e.g.,
[11, 12, 54]) to ensure persistence of normally hyperbolic structure in
perturbed vector fields.
###### Proposition C.1.
Consider the desingularized vector field $g$ in (5.5) satisfying requirements
in Assumption 5.2. Then there are a Riemannian manifold $\hat{Q}$ and a vector
field $\hat{g}$ on $\hat{Q}$ such that
* •
$M_{I}$ is embedded into a compact, boundaryless, normally attracting
invariant manifold $\hat{N}\subset\hat{Q}$,
* •
$M_{I}$ is normally attracting for $\hat{g}$ as the submanifold of $Q$, and
that
* •
$\hat{g}=g$ on $W^{s}(M_{I};\varphi_{\hat{g}})$ as the submanifold of
$W^{s}(\hat{N};\varphi_{\hat{g}})$, where $\varphi_{\hat{g}}$ is the flow
generated by $\hat{g}$. In particular, the stable foliation of
$W^{s}(M_{I};\varphi_{\hat{g}})$ is constructed, which is consequently
regarded as the foliation of $W^{s}(M_{I};\varphi_{g})$.
###### Proof.
Let $\tilde{g}$ be the modified desingularized vector field given in (5.10)
admitting a sequence of normally hyperbolic manifolds $M_{I}\subset
M_{\tilde{I}}\subset M_{I^{\prime}}$ such that $M_{I}$ is invariant, while
$M_{\tilde{I}}$ and $M_{I^{\prime}}$ are inflowing invariant. Let
$Q:=W^{s}(M_{I^{\prime}};\varphi_{\tilde{g}})$, the global stable manifold of
$M_{I^{\prime}}$ for $\tilde{g}$. We then apply Proposition C.4 below to
$N=M_{\tilde{I}}$ and $M=M_{\hat{I}}$, which directly yields the result. ∎
Once the treatment of invariant manifolds on $\mathcal{E}$ admitting normally
hyperbolic structure is understood, blow-up rates are evaluated in the similar
way to autonomous cases.
###### Corollary C.2 (Nonautonomous blow-up: blow-up rates).
Under the setting in Corollary 5.4, if all assumptions in Proposition B.7 are
satisfied for the hull $V_{{\bf x}_{\gamma}}\equiv{\rm cl}(\gamma)$, then the
blow-up solution ${\bf y}(t)$ admits the asymptotic behavior
$\displaystyle p({\bf y}(t))\sim
C_{0}(-\ln(t_{\max}-t))(t_{\max}-t)^{-1/k}\quad\text{ as }\quad t\to
t_{\max}-0$
for some function $C_{0}(-\ln(t_{\max}-t))$ satisfying
$C_{0}(\tau)=o(e^{\epsilon\tau}),\quad C_{0}(\tau)^{-1}=o(e^{\epsilon\tau})$
for any $\epsilon>0$ as $\tau\to\infty$, equivalently
$-\ln(t_{\max}-t)\to+\infty$, and
$y_{i}(t)\sim
C_{0}(-\ln(t_{\max}-t))^{\alpha_{i}}x_{\gamma,i}(-\bar{c}\ln(t_{\max}-t))(t_{\max}-t)^{-\alpha_{i}/k}\quad\text{
as }\quad t\to t_{\max}-0$
for some constant $\bar{c}>0$, provided $x_{M,i}(\tau)\not\to 0$ as
$\tau\to\infty$.
###### Proof.
It follows from Proposition C.1 that $M_{I}$ as the submanifold of
$Q=W^{s}(M_{I^{\prime}};\varphi_{g})$ admits the stable foliation of $W_{\rm
loc}^{s}(M_{I};\varphi_{\hat{g}})=W_{\rm loc}^{s}(M_{I};\varphi_{g})$. The
statement then follows from the distribution of $M_{I}\subset\mathcal{E}$, in
particular of $\pi_{t}M_{I}$ in $\mathbb{R}$. Results involving asymptotic
behavior of $p({\bf y}(t))$ and $y_{i}(t)$ are the direct consequence of
Proposition B.7 replacing $M$ with $\hat{N}$ stated in Proposition C.1. ∎
###### Remark C.3 (Beyond normal hyperbolicity).
Here we have restricted our attention to NHIMs because our geometric
description of blow-ups relies on embeddings of compact invariant manifolds
with boundary to compact “boundaryless” manifolds with the same normal
hyperbolicity restricted to the original manifolds. This technique is based on
[10] and its validity beyond normal hyperbolicity (like situations in [37])
remains nontrivial. If the similar embedding technique is constructed to
general invariant manifolds (admitting asymptotic phase), the same conclusion
for blow-up rates will be provided.
### C.2 Proof of Corollary 5.5
The proof is a direct consequence of combinations of Proposition B.5 and
Corollary C.2.
First, thanks to Proposition C.1, the manifold $M_{I}$ is embedded into a
compact, boundaryless NAIMs in a manifold $\hat{Q}$ (mentioned in Proposition
C.4 for $Q=W^{s}(M_{I^{\prime}};\varphi_{\tilde{g}})$), and invariant
foliation of $W^{s}_{\rm loc}(M_{I};g)$ is admitted. In particular, $g$ admits
a local topological conjugacy on $M_{I}$ to $Dg|_{\hat{Q}}$, where we have
used the identification of $g$ and the modified vector field $\hat{g}$ on
$W^{s}_{\rm loc}(M_{I};g)$. Next, because $M_{I}$ is assumed to be a
collection of equilibria, the dichotomy spectrum of $Dg$ on each hull $V_{{\bf
x}_{\gamma}}$ of trajectories on $M_{I}$, which is exactly a point ${\bf
x}(t)\in M_{I}$ itself for each $t\in I$, are discrete by Proposition B.5.
Therefore, Corollary C.2 can be applied to obtaining the conclusion.
### C.3 Reduction to boundaryless inflowing NAIMs
Finally we review a technique to modify inflowing invariant manifolds with
normally hyperbolic structure with non-trivial boundary to boundaryless ones
used in the argument of Proposition C.1. This was provided in [10], where the
reduction of inflowing normally attracting invariant manifolds (NAIMs) with
boundaries to boundaryless ones was discussed. The main aim of the discussion
was to apply global linearization theorem for boundaryless NHIMs stated in
preceding works (e.g. [20, 44]) to inflowing NAIMs. See e.g., [19] for
fundamental concepts in differential geometry and differential topology. Let
$Q$ be a $C^{r\geq 1}$ Riemannian manifold.
###### Proposition C.4 (Reduction to boundaryless NAIM, [10], Proposition
B.1).
Let $M,N\subset Q$ be compact inflowing $r$-NAIMs, with $M\subset{\rm
int}_{Q}N$, for the $C^{r\geq 1}$ flow $\varphi$ generated by the $C^{r\geq
1}$ vector field $f$ on $Q$. Let $U_{0}$ be an arbitrarily small tubular
neighborhood of $\partial N$, having smooth boundary $\partial U_{0}$ and
disjoint from $W^{s}_{\rm loc}(M;\varphi)$. Define $\hat{Q}$ to be the double
of $Q\setminus U_{0}$.
Then there is a $C^{\infty}$ differential structure on $\hat{Q}$ and a $C^{r}$
vector field $\hat{f}:\hat{Q}\to T\hat{Q}$ satisfying the following
properties:
1. 1.
$\hat{f}$ is equal to $f$ on each copy of $Q\setminus U_{0}$ except on an
arbitrarily small neighborhood of $\partial U_{0}$.
2. 2.
There is a compact and boundaryless $r$-NAIM $\hat{N}$ for $\hat{f}$, with
$\hat{N}$ equal to $N$ on each copy of $Q\setminus U_{0}$, except on an
arbitrarily small neighborhood of $\partial U_{0}$.
3. 3.
The global stable foliation of $M$ for $f$ does not intersect $U_{0}$, and it
coincides with the global stable foliation of $M$ for $\hat{f}$, where $M$ and
$W^{s}(M)$ are identified via inclusion with subset of a copy of $Q\setminus
U_{0}$ in $\hat{Q}$.
Furthermore, let $\hat{\varphi}$ be the $C^{r}$-flow generated by $\hat{f}$,
and let $\hat{E}^{s}$ be the $D\hat{\varphi}^{t}|_{\hat{N}}$-invariant stable
vector bundle for the NAIM $\hat{N}$. If, additionally, there are constants
$C>0$ and $\alpha<0$ such that, for all $m\in M$, $t\geq 0$ and $0\leq i\leq
k$, the $k$-center bunching condition202020 This condition implies that
$W^{s}(M;\varphi)$ and $E^{s}$ as fiber or vector bundles are $C^{k}$,
according to [13].
$\|D\varphi^{t}|_{T_{m}M}\|^{i}\|D\varphi^{t}|_{E^{s}_{m}}\|\leq Ce^{\alpha
t}m\left(D\varphi^{t}|_{T_{m}M}\right)$ (C.1)
is satisfied for the original system on $Q$, then (C.1) is also satisfied
replacing $M$, $E^{s}$ and $\varphi^{t}$ with $\hat{N}$, $\hat{E}^{s}$ and
$\hat{\varphi}^{t}$, respectively, and $\alpha$ with some different constant
$\hat{\alpha}<0$.
Similarly, if additionally there are constants $0<\delta<-\alpha<-\beta$ and
$C\geq 1$ such that, for all $t\geq 0$,
$\begin{cases}C^{-1}e^{-\delta t}\leq
m\left(D\varphi^{t}|_{TM}\right)\leq\|D\varphi^{t}|_{TM}\|\leq Ce^{\delta
t},&\\\ C^{-1}e^{-\delta t}\leq
m\left(\left(D\varphi^{t}|_{TM}\right)^{-1}\right)\leq\|\left(D\varphi^{t}|_{TM}\right)^{-1}\|\leq
Ce^{\delta t},&\\\ C^{-1}e^{\beta t}\leq
m\left(D\varphi^{t}|_{E^{s}}\right)\leq\|D\varphi^{t}|_{E^{s}}\|\leq
Ce^{\alpha t}&\end{cases}$ (C.2)
uniformly on $TM$ and $E^{s}$, then we can choose $\hat{f}$ appropriately such
that the same conclusion holds true for $\hat{\varphi}^{t}$, $T\hat{N}$ and
$\hat{E}^{s}$ with modified constants
$0<\hat{\delta}<-\hat{\alpha}<-\hat{\beta}$ arbitrarily close to
$\delta,\alpha,\beta$, respectively.
The essence of this result is that the differential structure of manifolds and
vector fields are modified only in a neighborhood of $\partial U_{0}$,
although the ambient manifold $Q$ is implicitly assumed to be boundaryless.
Even when $Q$ has a nontrivial boundary $\partial Q$, the above technique can
be applied as long as $(Q,\partial Q)$ is a Riemannian manifold and $\partial
Q$ is invariant for the vector field $f:Q\to TQ$. In this result, normally
attracting, or normally hyperbolic properties of $N$ and $M$ is assumed. In
this situation, the uniformity lemma212121 If we try to provide the similar
result to invariant manifolds which are not normally hyperbolic, the
corresponding estimate will be necessary. can be applied (e.g., [11, 12, 54])
to making uniform estimates in (C.2).
In Appendix C.1, this proposition is used to provide linearization of dynamics
at infinity for nonautonomous systems. More precisely, this technique is used
to characterize spectral properties for invariant manifolds in nonautonomous
systems by means of compact, boundaryless invariant manifolds, where local
topological conjugacy can be applied. Although our interests are NHIMs, all
arguments involving NAIMs can be applied by restricting our attention to the
(global) stable manifold: $Q=W^{s}(M;\varphi)$ with a compact, connected
(inflowing) invariant manifold $M$ with a normally hyperbolic structure, then
$M$ is normally attracting for the flow restricted to $Q$, and hence the
theory of NAIMs can be directly applied222222 This is possible because the
stable manifold $W^{s}(M)$ of an invariant manifold $M$ for a given flow is
also invariant. .
|
# Supplemental material
Observation of interatomic Coulombic decay induced
by double excitation of helium in nanodroplets
B. Bastian Wilhelm Ostwald Institute for Physical and Theoretical Chemistry,
University of Leipzig, Linnéstraße 2, 04103 Leipzig, Germany J. D. Asmussen
L. Ben Ltaief H. B. Pedersen Department of Physics and Astronomy, Aarhus
University, 8000 Aarhus C, Denmark K. Sishodia S. De Indian Institute of
Technology Madras, Chennai 600036, India S. R. Krishnan Department of
Physics and QuCenDiEM-group, Indian Institute of Technology Madras, Chennai
600036, India C. Medina Institute of Physics, University of Freiburg, 79104
Freiburg, Germany N. Pal R. Richter Elettra-Sincrotrone Trieste, 34149
Basovizza, Trieste, Italy N. Sisourat Sorbonne Université, CNRS, Laboratoire
de Chimie Physique Matière et Rayonnement, UMR 7614, F-75005 Paris, France M.
Mudrich<EMAIL_ADDRESS>Department of Physics and Astronomy, Aarhus
University, 8000 Aarhus C, Denmark
(September 1, 2024)
Figure 1: Interactions at a double excitation (Fano) resonance in a helium
nanodroplet. An incident photon $h\nu$ can (1) doubly excite a helium atom or
(2) directly ionize. A doubly excited helium atom will quickly autoionize (1a)
or undergo single deexcitation (1b). The autoionization (1a) and direct
ionization (2) pathways are indistinguishable and give rise to the Fano
resonance profile when varying the photon energy. The resulting photoelectron
may escape the droplet with $35.8\text{\,}\mathrm{e}\mathrm{V}$ kinetic energy
or excite a neighbor atom by inelastic scattering (dashed) and escape as a
slow $14.6\text{\,}\mathrm{e}\mathrm{V}$ electron. In the single deexcitation
pathway (1b), instead the doubly excited atom decays to a singly excited one
while a neigboring atom is ionized by energy transfer and a
$14.6\text{\,}\mathrm{e}\mathrm{V}$ ICD electron is ejected. The aim of this
work is to disentangle the inelastic scattering and ICD contributions to the
formation of slow electrons.
## I Disentanglement of ICD and inelastically scattered electrons
To spectroscopically disentangle the ICD and inelastic scattering
contributions to electron-energy loss (EEL) electrons around
$E_{e}=$15\text{\,}\mathrm{e}\mathrm{V}$$ electron kinetic energy, the ratio
of EEL to total electron counts is divided by its off-resonant
($E_{h\nu}\leq$59.9\text{\,}\mathrm{e}\mathrm{V}$$) average which gives the
relative count ratio
$r(E_{e})=\frac{c_{\text{\tiny
EEL}}(E_{e})}{c_{\text{total}}(E_{e})}\left<\frac{c_{\text{\tiny
EEL}}}{c_{\text{total}}}\right>_{\text{\\!off}}^{-1}$ (1)
where $c$ denotes absolute counts and $\langle\leavevmode\nobreak\
\rangle_{\text{\\!off}}$ the average over the off-resonant photon energies. If
only inelastic scattering contributed to EEL, $c_{\text{\tiny EEL}}$ would be
proportional to $c_{\text{total}}$ because the inelastic scattering cross
sections are nearly constant across the resonance. Consequently, the ratio
would be constant and $r(E_{e})=1$ across the tuning range of the photon
energy studied here. The ICD channel adds additional counts $c_{\text{\tiny
EEL}}(E_{e})$ which results in values $r(E_{e})>1$ on the resonance. The ratio
of ICD to total EEL electrons is then $\left(r-1\right)/r$ which gives the
number of ICD electrons as a function of EEL or total electron counts
$c_{\text{\tiny ICD}}(E_{e})=\frac{r(E_{e})-1}{r(E_{e})}\,c_{\text{\tiny
EEL}}(E_{e})=\left(r(E_{e})-1\right)\,\left<\frac{c_{\text{\tiny
EEL}}}{c_{\text{total}}}\right>_{\text{\\!off}}\,c_{\text{total}}(E_{e})$ (2)
and the probability of an electron to originate from the ICD process
$p_{\text{\tiny ICD}}(E_{e})=\frac{c_{\text{\tiny
ICD}}(E_{e})}{c_{\text{total}}(E_{e})}=\frac{r(E_{e})-1}{r(E_{e})}\,\frac{c_{\text{\tiny
EEL}}(E_{e})}{c_{\text{total}}(E_{e})}=\left(r(E_{e})-1\right)\,\left<\frac{c_{\text{\tiny
EEL}}}{c_{\text{total}}}\right>_{\text{\\!off}}\;.$ (3)
Uncertainties are determined by Gaussian error propagation assuming $\sqrt{c}$
errors for the EEL and total electron counts and using the standard deviation
of the off-resonant $c_{\text{\tiny EEL}}/c_{\text{total}}$ values for the
error of their average. To evaluate the hemispherical electron analyzer (HEA)
data for which the total electrons were not measured, the ICD probability is
factorized into
$p_{\text{\tiny ICD}}(E_{e})=\frac{c_{\text{\tiny
ICD}}(E_{e})}{c_{\text{total}}(E_{e})}=\frac{c_{\text{\tiny
ICD}}(E_{e})}{c_{\text{\tiny EEL}}(E_{e})}\frac{c_{\text{\tiny
EEL}}(E_{e})}{c_{\text{total}}(E_{e})}=\frac{c_{\text{\tiny
ICD}}(E_{e})}{c_{\text{\tiny EEL}}(E_{e})}\,p_{\text{\tiny EEL}}(E_{e})$ (4)
and the second factor ($p_{\text{\tiny EEL}}$) is taken from VMI data with a
similar average droplet size.
## II Baseline correction for count ratios from the HEA data
As photoelectrons were not detected in the HEA measurement, measured total
electron scans from a previous beamtime with the same droplet source but a
different detector were used [1]. The resulting skewed baseline was corrected
with a constant slope which is justified by comparing to the present VMI data
where the ratios stem from a single measurement, see Fig. 2 and Fig. 2 (d) in
the main text. The derived normalized EEL to total electron ratios and total
ICD counts are presented Fig. 3 (b) and Fig. 4 (b) in the main text,
respectively.
Figure 2: Linear correction of the count ratio of low kinetic energy
($18\text{\,}\mathrm{e}\mathrm{V}23\text{\,}\mathrm{e}\mathrm{V}$) to total
electrons. VMI and HEA data are shown for the He nanodroplet (HND) expansion
conditions specified in the legend. The HEA data are corrected by a linear fit
to the off-resonant photon energy range around the double excitation
resonance. The agreement of the corrected data with the result from velocity
map imaging (VMI) data justifies this approach.
Figure 3: (a) Average HEA high-resolution EEL spectrum with fit residuals
(gray crosses, right axis) and individual Gaussians (see labels in the legend)
of the multi-peak fit that are not shown in Fig. 4 (a) in the main text. Black
circles are averaged over 39 spectra recorded in the range $h\nu=58.4$ – 62.2
eV. Vertical dashed lines show atomic excitations with values from NIST [2].
Shaded areas correspond to He∗He+ states with boundaries computed for ICD
decay of the He∗∗ droplet excitation into the He${}_{2}^{+*}$ states shown in
(c). (b) Photon energy dependence of the normalized peak area to total
electron ratios for different states with colors corresponding to (a). In two
cases (magenta and gray), the sum of two peak area is considered. Those
components assigned to molecular excitations display pronounced maxima in the
range of the Fano resonance. The ratios for the molecular 4–5
${}^{2}\Sigma/^{1}\Pi$ excitations are given by the second $y$-axis on the
right. (c) Estimate of the total ICD counts in the electron impact excitation
(EEL) and electron impact ionization (EII) range of the electron kinetic
energy. Integration ranges of the electron energy loss are specified in the
legend. The 1s3p/1s4p range is excluded because it overlaps with the EII
region. (d) Photon energy dependence of ICD counts at different levels of
molecular excitation in the 21 – 23 eV range. Higher excitation is
preferentially observed at higher photon energies. (e) and (f) Ratios of EEL
(18 – 23 eV e- kinetic energy loss) to photoelectron counts ($-$2 – 2 eV loss)
as a function of photon energy at different HND sizes specified in the legend.
Total photoelectrons at the $n=2$ and $n=3$ Fano resonances (assigned
according to Ref. [1] Fig. 1) are shown as solid red lines (right axes). The
lower photon energies (e) feature the discussed ICD resonance around 60.5 eV
photon energy. Assuming a similar ICD efficiency for the higher photon
energies (f) would give an increase of the order of 2 % which is in the noise
level.
## III Count ratios from Fano and Lorentzian fits
To evaluate the ICD probability in (3) or (4) and related quantities, we fit
the measured electron counts as a function of photon energy with the Fano
profile [3]
$c(h\nu)=s_{d}+s_{r}\,\frac{\left(q+\frac{h\nu-
E_{r}}{\Gamma}\right)^{2}}{1+\left(\frac{h\nu-E_{r}}{\Gamma}\right)^{2}}$ (5)
with photon energy $h\nu$, asymmetry (Fano) parameter $q$, resonance energy
and width $E_{r}$ and $\Gamma$, the resonant amplitude $s_{r}$ for double
excitation and the direct ionization amplitude $s_{d}$. In this case, $q$ is
negative and the maximum, resonance and minimum points are
$\displaystyle E_{r}+\Gamma/q,$ $\displaystyle E_{r}$ and $\displaystyle
E_{r}-\Gamma\,q$ (6) with the amplitudes $\displaystyle
s_{d}+\left(q^{2}+1\right)s_{r},$ $\displaystyle s_{d}+q^{2}s_{r}$ and
$\displaystyle s_{d}\,.$ (7)
In case of ICD electrons only the resonant pathway is involved and the shape
is found to reduce to a symmetric Lorentzian profile with $q=0$,
$s_{d}+s_{r}\approx 0$ and the maximum amplitude $s_{d}$ is found at the
resonance energy. To compute count ratios (relative probabilities), we
evaluate all amplitudes at the resonance energies obtained from the fit which
gives $s_{d}+q^{2}s_{r}$. The resonance energy of all fits to the total, EEL
and ICD electrons is close to 60.4 eV. All fit results are summarized in Table
III.
Table 1: Fit results from Fano and Lorentzian fits. The photon energy range
from 59.8 eV to 60.2 eV was excluded for the fits because of the sharply
peaked photoline from an atomic He fraction in the droplet beam. The He+
coincidences are most strongly affected so the fits for the total coincidence
electrons are not reliable and impossible at 16 K. Counts ($s_{d}$, $s_{r}$
and $s_{d}+q^{2}s_{r}$) are tabulated in units of $10^{6}$ for all electrons
(all $e^{-}$) and $10^{4}$ for He${}_{n}^{+}$–$\,e^{-}$ coincidences and HEA
data.
Det. | T/K | p/bar | Data | Electrons | $q$ | $E_{r}$/eV | $\Gamma$/eV | $s_{d}$ | $s_{r}$ | $s_{d}+q^{2}s_{r}$
---|---|---|---|---|---|---|---|---|---|---
VMI | 10 | 30 | all $e^{-}$ | total | $-8.21\pm 1.21$ | $60.349\pm 0.006$ | $0.217\pm 0.005$ | $0.800\pm 0.015$ | $0.016\pm 0.004$ | $1.9\pm 0.4$
VMI | 10 | 30 | all $e^{-}$ | EEL | $-8.63\pm 1.32$ | $60.354\pm 0.006$ | $0.220\pm 0.005$ | $0.201\pm 0.004$ | $0.004\pm 0.001$ | $0.50\pm 0.12$
VMI | 10 | 30 | all $e^{-}$ | ICD | $0$ | $60.415\pm 0.003$ | $0.220\pm 0.009$ | $0.011\pm 0.000$ | $-0.022\pm 0.001$ | $0.0110\pm 0.0003$
VMI | 12 | 30 | all $e^{-}$ | total | $-2.36\pm 0.07$ | $60.402\pm 0.005$ | $0.233\pm 0.005$ | $1.361\pm 0.011$ | $0.272\pm 0.015$ | $2.88\pm 0.12$
VMI | 12 | 30 | all $e^{-}$ | EEL | $-3.10\pm 0.14$ | $60.409\pm 0.006$ | $0.220\pm 0.006$ | $0.161\pm 0.002$ | $0.023\pm 0.002$ | $0.38\pm 0.03$
VMI | 12 | 30 | all $e^{-}$ | ICD | $0$ | $60.428\pm 0.006$ | $0.183\pm 0.012$ | $0.056\pm 0.002$ | $-0.059\pm 0.002$ | $0.056\pm 0.002$
VMI | 14 | 30 | all $e^{-}$ | total | $-3.03\pm 0.18$ | $60.365\pm 0.009$ | $0.245\pm 0.010$ | $4.441\pm 0.060$ | $0.567\pm 0.064$ | $9.6\pm 0.9$
VMI | 14 | 30 | all $e^{-}$ | EEL | $-3.91\pm 0.27$ | $60.401\pm 0.007$ | $0.219\pm 0.008$ | $0.413\pm 0.006$ | $0.040\pm 0.005$ | $1.02\pm 0.11$
VMI | 14 | 30 | all $e^{-}$ | ICD | $0$ | $60.439\pm 0.010$ | $0.173\pm 0.019$ | $0.196\pm 0.013$ | $-0.210\pm 0.013$ | $0.196\pm 0.013$
VMI | 16 | 30 | all $e^{-}$ | total | $-2.34\pm 0.13$ | $60.391\pm 0.014$ | $0.276\pm 0.013$ | $0.522\pm 0.008$ | $0.121\pm 0.011$ | $1.18\pm 0.09$
VMI | 16 | 30 | all $e^{-}$ | EEL | $-2.41\pm 0.12$ | $60.491\pm 0.011$ | $0.243\pm 0.012$ | $0.040\pm 0.001$ | $0.011\pm 0.001$ | $0.104\pm 0.009$
VMI | 16 | 30 | all $e^{-}$ | ICD | $0$ | $60.483\pm 0.014$ | $0.147\pm 0.025$ | $0.028\pm 0.003$ | $-0.030\pm 0.003$ | $0.028\pm 0.003$
VMI & 10 30 He+ coinc. total $-3.39\pm 1.96$ $60.300\pm 0.043$ $0.344\pm
0.033$ $0.118\pm 0.005$ $0.007\pm 0.007$ $0.19\pm 0.12$ VMI 10 30 He+ coinc.
EEL $-24.56\pm 59.89$ $60.335\pm 0.032$ $0.263\pm 0.020$ $0.032\pm 0.004$
$0.000\pm 0.000$ $0.032\pm 0.004$ VMI 10 30 He+ coinc. ICD $0$ $60.418\pm
0.002$ $0.235\pm 0.008$ $0.002\pm 0.000$ $-0.004\pm 0.000$ $0.00182\pm
0.00004$ VMI 12 30 He+ coinc. total $-2.04\pm 0.12$ $60.435\pm 0.013$
$0.277\pm 0.015$ $1.130\pm 0.013$ $0.174\pm 0.018$ $1.85\pm 0.11$ VMI 12 30
He+ coinc. EEL $-4.07\pm 0.32$ $60.398\pm 0.009$ $0.255\pm 0.012$ $0.277\pm
0.006$ $0.029\pm 0.004$ $0.76\pm 0.10$ VMI 12 30 He+ coinc. ICD $0$ $60.420\pm
0.004$ $0.205\pm 0.009$ $0.108\pm 0.002$ $-0.113\pm 0.003$ $0.108\pm 0.002$
VMI 14 30 He+ coinc. total $-2.12\pm 0.33$ $60.457\pm 0.035$ $0.300\pm 0.038$
$2.001\pm 0.019$ $0.097\pm 0.025$ $2.44\pm 0.18$ VMI 14 30 He+ coinc. EEL
$-6.04\pm 1.09$ $60.376\pm 0.014$ $0.236\pm 0.018$ $0.482\pm 0.013$ $0.018\pm
0.006$ $1.1\pm 0.3$ VMI 14 30 He+ coinc. ICD $0$ $60.429\pm 0.006$ $0.204\pm
0.013$ $0.207\pm 0.007$ $-0.219\pm 0.007$ $0.207\pm 0.007$ VMI 16 30 He+
coinc. total $-3.41\pm 0.47$ $60.154\pm 0.010$ $0.040\pm 0.006$ $1.021\pm
0.097$ $0.298\pm 0.104$ $4.5\pm 1.5$ VMI 16 30 He+ coinc. EEL $-3.51\pm 0.30$
$60.453\pm 0.019$ $0.289\pm 0.021$ $0.123\pm 0.003$ $0.017\pm 0.003$ $0.33\pm
0.05$ VMI 16 30 He+ coinc. ICD $0$ $60.459\pm 0.013$ $0.200\pm 0.026$
$0.084\pm 0.005$ $-0.089\pm 0.005$ $0.084\pm 0.005$ VMI 10 30 He${}_{2}^{+}$
coinc. total $-10.11\pm 1.70$ $60.299\pm 0.005$ $0.242\pm 0.004$ $0.380\pm
0.005$ $0.004\pm 0.001$ $0.79\pm 0.17$ VMI 10 30 He${}_{2}^{+}$ coinc. EEL
$-10.76\pm 6.73$ $60.312\pm 0.020$ $0.258\pm 0.015$ $0.054\pm 0.003$ $0.001\pm
0.001$ $0.17\pm 0.19$ VMI 10 30 He${}_{2}^{+}$ coinc. ICD $0$ $60.417\pm
0.003$ $0.247\pm 0.012$ $0.003\pm 0.000$ $-0.006\pm 0.000$ $0.0025\pm 0.0001$
VMI 12 30 He${}_{2}^{+}$ coinc. total $-2.39\pm 0.10$ $60.384\pm 0.007$
$0.232\pm 0.007$ $4.210\pm 0.042$ $0.757\pm 0.056$ $8.5\pm 0.5$ VMI 12 30
He${}_{2}^{+}$ coinc. EEL $-3.19\pm 0.21$ $60.400\pm 0.009$ $0.240\pm 0.009$
$0.486\pm 0.007$ $0.060\pm 0.007$ $1.10\pm 0.11$ VMI 12 30 He${}_{2}^{+}$
coinc. ICD $0$ $60.432\pm 0.007$ $0.191\pm 0.013$ $0.158\pm 0.006$ $-0.166\pm
0.006$ $0.158\pm 0.006$ VMI 14 30 He${}_{2}^{+}$ coinc. total $-3.01\pm 0.26$
$60.359\pm 0.011$ $0.214\pm 0.012$ $6.410\pm 0.081$ $0.586\pm 0.095$ $11.7\pm
1.3$ VMI 14 30 He${}_{2}^{+}$ coinc. EEL $-4.54\pm 0.84$ $60.379\pm 0.018$
$0.236\pm 0.020$ $0.644\pm 0.015$ $0.030\pm 0.011$ $1.3\pm 0.3$ VMI 14 30
He${}_{2}^{+}$ coinc. ICD $0$ $60.455\pm 0.011$ $0.187\pm 0.022$ $0.237\pm
0.016$ $-0.256\pm 0.016$ $0.237\pm 0.016$ VMI 16 30 He${}_{2}^{+}$ coinc.
total $-2.06\pm 0.08$ $60.456\pm 0.010$ $0.242\pm 0.010$ $2.175\pm 0.027$
$0.528\pm 0.037$ $4.4\pm 0.2$ VMI 16 30 He${}_{2}^{+}$ coinc. EEL $-2.86\pm
0.23$ $60.478\pm 0.015$ $0.214\pm 0.016$ $0.202\pm 0.005$ $0.034\pm 0.005$
$0.48\pm 0.06$ VMI 16 30 He${}_{2}^{+}$ coinc. ICD $0$ $60.480\pm 0.014$
$0.152\pm 0.025$ $0.127\pm 0.013$ $-0.135\pm 0.012$ $0.127\pm 0.013$ VMI 10 30
He${}_{3}^{+}$ coinc. total $-4.23\pm 0.59$ $60.361\pm 0.011$ $0.261\pm 0.008$
$0.042\pm 0.001$ $0.002\pm 0.001$ $0.08\pm 0.02$ VMI 10 30 He${}_{3}^{+}$
coinc. EEL $-10.82\pm 20.34$ $60.340\pm 0.073$ $0.341\pm 0.060$ $0.003\pm
0.001$ $0.000\pm 0.000$ $0.003\pm 0.001$ VMI 10 30 He${}_{3}^{+}$ coinc. ICD
$0$ $60.416\pm 0.003$ $0.225\pm 0.011$ $0.000\pm 0.000$ $-0.000\pm 0.000$
$0.00023\pm 0.00001$ VMI 12 30 He${}_{3}^{+}$ coinc. total $-2.34\pm 0.11$
$60.412\pm 0.007$ $0.228\pm 0.008$ $0.569\pm 0.007$ $0.105\pm 0.009$ $1.14\pm
0.07$ VMI 12 30 He${}_{3}^{+}$ coinc. EEL $-2.29\pm 0.29$ $60.468\pm 0.021$
$0.239\pm 0.022$ $0.053\pm 0.002$ $0.010\pm 0.002$ $0.105\pm 0.017$ VMI 12 30
He${}_{3}^{+}$ coinc. ICD $0$ $60.440\pm 0.007$ $0.191\pm 0.013$ $0.017\pm
0.001$ $-0.018\pm 0.001$ $0.017\pm 0.001$ VMI 14 30 He${}_{3}^{+}$ coinc.
total $-2.93\pm 0.28$ $60.397\pm 0.013$ $0.219\pm 0.014$ $0.775\pm 0.011$
$0.073\pm 0.013$ $1.40\pm 0.16$ VMI 14 30 He${}_{3}^{+}$ coinc. EEL $-6.36\pm
3.73$ $60.412\pm 0.046$ $0.266\pm 0.050$ $0.066\pm 0.004$ $0.002\pm 0.002$
$0.15\pm 0.12$ VMI 14 30 He${}_{3}^{+}$ coinc. ICD $0$ $60.466\pm 0.012$
$0.191\pm 0.024$ $0.026\pm 0.002$ $-0.028\pm 0.002$ $0.026\pm 0.002$ VMI 16 30
He${}_{3}^{+}$ coinc. total $-1.95\pm 0.09$ $60.488\pm 0.012$ $0.249\pm 0.012$
$0.261\pm 0.004$ $0.073\pm 0.006$ $0.54\pm 0.03$ VMI 16 30 He${}_{3}^{+}$
coinc. EEL $-1.61\pm 0.17$ $60.588\pm 0.025$ $0.205\pm 0.026$ $0.018\pm 0.001$
$0.008\pm 0.001$ $0.039\pm 0.005$ VMI 16 30 He${}_{3}^{+}$ coinc. ICD $0$
$60.492\pm 0.014$ $0.151\pm 0.025$ $0.013\pm 0.001$ $-0.013\pm 0.001$
$0.013\pm 0.001$ HEA 14 50 all $e^{-}$ EEL $-2.94\pm 0.15$ $60.405\pm 0.010$
$0.223\pm 0.011$ $0.356\pm 0.006$ $0.063\pm 0.006$ $0.90\pm 0.08$ HEA 14 50
all $e^{-}$ ICD $0.01\pm 0.02$ $60.381\pm 0.015$ $0.24\pm 0.03$ $0.133\pm
0.008$ $-0.141\pm 0.007$ $0.133\pm 0.008$
## IV Electron trapping probability
Trapping of slow ICD electrons in helium droplets with increasing size affects
the probability of their observation [4, 5]. It is modelled as a product of
the size independent ICD probability $p_{0}$ and the likelihood for electron
ejection before trapping (see fit in Fig. 3 in the main text),
$\displaystyle p^{\rm obs}_{\rm ICD}$
$\displaystyle=p_{0}\exp\left(-\frac{\bar{d}}{l_{\rm
slow}}\right)=p_{0}\exp\left(-\frac{3R}{4l_{\rm slow}}\right)$ (8)
where $\bar{d}=\frac{3}{4}R$ is the average travelling distance to the surface
in a droplet with radius $R$ and $l_{\rm slow}$ is the mean free path of the
electron. We have thus neglected the angular deflection by elastic and
inelastic collisions that slow down the electron and roughly estimate the
ejection probability alone from the average distance $\bar{d}$ from the
surface. The droplet radius is computed from the number of atoms $N_{\rm He}$
in the droplet by the expression
$\displaystyle R/\text{nm}$ $\displaystyle=0.22\sqrt[3]{N_{\rm He}}$ (9)
by Gomez et al. [6]. The expression for the mean distance $\bar{d}$ is derived
as follows. From Fig. 4 we find the distance $d$ of an electron from the
surface of a spherical droplet with radius $R$,
$\displaystyle d$
$\displaystyle=\sqrt{R^{2}-a^{2}\sin^{2}(\theta)}-a\cos(\theta)$ (10)
where $a$ is the distance from the electron to the center of the droplet. The
isotropic likelihood that the electron originates from a distance $a$ from the
center and moves toward a surface element $[\theta,\theta+{\rm
d}\theta]\times[\phi,\phi+{\rm d}\phi]$ is given by
$\displaystyle\frac{3a^{2}}{R^{3}}\,{\rm d}a\leavevmode\nobreak\
\times\leavevmode\nobreak\ \frac{1}{4\pi}\sin(\theta)\,{\rm d}\theta\,{\rm
d}\phi$ (11)
and we obtain the distance averaged over all positions and directions by
integration,
$\displaystyle\bar{d}$ $\displaystyle=\frac{3}{4\pi
R^{3}}\int_{0}^{R}\int_{0}^{\pi}\int_{0}^{2\pi}a^{2}\left(\sqrt{R^{2}-a^{2}\sin^{2}(\theta)}-a\cos(\theta)\right)\sin(\theta)\,{\rm
d}\phi\,{\rm d}\theta\,{\rm d}a$ (12)
$\displaystyle=\frac{3}{2R^{3}}\int_{0}^{R}\int_{0}^{\pi}\left(\sqrt{R^{2}-a^{2}\sin^{2}(\theta)}-a\cos(\theta)\right)a^{2}\sin(\theta)\,{\rm
d}\theta\,{\rm d}a\quad\text{(the $\cos(\theta)\sin(\theta)$ term vanishes by
symmetry)}$
$\displaystyle=\frac{3R}{2}\int_{0}^{1}\int_{0}^{\pi}\sqrt{1-k^{2}\sin^{2}(\theta)}\;k^{2}\sin(\theta)\,{\rm
d}\theta\,{\rm
d}k=\frac{3R}{2}\int_{0}^{1}\int_{-1}^{1}\sqrt{1-k^{2}\left(1-\mu^{2}\right)}\;k^{2}\,{\rm
d}\mu\,{\rm d}k$
$\displaystyle=\frac{3R}{2}\int_{0}^{1}k\int_{-k}^{k}\sqrt{x^{2}+1-k^{2}}\;\,{\rm
d}x\,{\rm
d}k=\frac{3R}{2}\int_{0}^{1}\frac{k}{2}\left[x\sqrt{x^{2}+1-k^{2}}+(1-k^{2})\ln\left(x+\sqrt{x^{2}+1-k^{2}}\right)\right]_{-k}^{k}\,{\rm
d}k$
$\displaystyle=\frac{3R}{2}\int_{0}^{1}\frac{k}{2}\left(k-(-k)+(1-k^{2})\ln\left(\frac{1+k}{1-k}\right)\right)\,{\rm
d}k=\frac{3R}{4}\int_{0}^{1}\left(2k^{2}+\left(k-k^{3}\right)\ln\left(\frac{1+k}{1-k}\right)\right)\,{\rm
d}k=\frac{3R}{4}\;.$
$R$$d$$a$$a\sin\theta$$a\cos\theta$$\theta$ Figure 4: Geometrical construction
of the distance $d$ of a point in a sphere from a point on its surface. The
distance of the inner point from the center of the sphere with radius $R$ is
denoted by $a$. One finds
$\left(d+a\cos\left(\theta\right)\right)^{2}=R^{2}-a^{2}\sin^{2}\left(\theta\right)$.
## V Electron impact excitation probability
The loss of fast electrons is predominated by electron impact excitation (EIE)
and ionization (EII) that we describe in a simple model by the average
travelling distance $\bar{d}$ to the surface that has been derived in the
previous section and a mean free path $l_{\rm fast}$ in analogy to Eq. (8).
The EIE probability is then given by
$\displaystyle P_{\rm EIE}/\text{\small\%}$ $\displaystyle=p_{\rm
EIE}\left(1-\exp\left(-\frac{\bar{d}}{l_{\rm fast}}\right)\right)=p_{\rm
EIE}\left(1-\exp\left(-\frac{3R}{4l_{\rm fast}}\right)\right)$ (13)
where $p_{\rm EIE}$ is the fraction of EIE from the total inelastic scattering
cross section (EIE = EEL $-$ ICD). The fit in Fig. 5 (a) gives the mean free
path for inelastic scattering $l_{\rm fast}=(17.6\pm 1.4)$ nm and a $(40\pm
2)$ % fraction of EIE of the 1s2s/1s2p states relative to the total inelastic
cross section in close agreement with $17\text{\,}\mathrm{n}\mathrm{m}$ and
$38\text{\,}\mathrm{\char 37\relax}$ in Sec. VI derived from the cross
sections in Ref. [7]. Surprisingly, the loss of slow electrons thus seems to
be negligible in the investigated droplet size range, in opposition to the
size dependence of the ICD probability that indicates suppressed ejection of
slow electrons. This discrepancy is only partially explained by the fact that
fast electrons typically travel a substantial distance to the droplet surface
before losing kinetic energy by inelastic scattering. A consistent explanation
of the observed size dependences would thus require an improved model
considering the different pathways for electrons propagating in a droplet
instead of using a single average distance from the surface.
Figure 5: Droplet-size dependence for total electrons and coincidences with
He${}_{n}^{+}$ cations, respectively. The points for $n=2,3$ are somewhat
shifted to lower and higher droplet sizes for better visibility. (a) EIE
probability relative to total electrons (all e-) or coincidences
(He${}_{n}^{+}$, $n=2,3$) with the EIE model fitted to the total electron
data. (b) Relative ratio of ICD and EIE (electron impact excitation of
1s2s/1s2p states) counts. The reen solid line connects the total electron
points to guide the eye.
The ratios of EIE coincidences to total coincidences in Fig. 5 (a) do increase
with growing EEL probability at larger droplet sizes, but the growth is
significantly weaker than for all electrons because the cations are more
easily trapped in the droplets than electrons [4]. At all sizes, the ICD/EIE
ratio in Fig. 5 (b) is within uncertainties identical for total electrons and
e-–He${}_{n}^{+}$, $n=1,2,3$, coincidences. There is consequently no evidence
that the ICD process at short distances in the He∗He+ potential energy curves
affects the ion formation and ejection dynamics.
## VI Electron impact cross section
To estimate the mean free path $l_{\rm fast}$ for inelastic scattering, we sum
up the electron impact ionization and excitation cross sections from the
initial 1 1S state at $E_{e}=$35.8\text{\,}\mathrm{e}\mathrm{V}$$ electron
energy (corresponding to $E_{h\nu}=$60.4\text{\,}\mathrm{e}\mathrm{V}$$ photon
energy) according to graphs 4 – 6 in Ref. [7] which gives the cross section
$\sigma_{\rm total}\approx$0.27\text{\,}\AA^{2}$$. With the helium density
$n_{\rm He}=N_{\rm He}/(4/3\pi
R^{3})\approx$22.42\text{\,}\mathrm{n}\mathrm{m}^{-3}$$ from Eq. (9) we obtain
the corresponding mean free path
$\displaystyle l_{\rm fast}=\frac{1}{\sigma_{\rm exc}\,n_{\rm
He}}\approx$17\text{\,}\mathrm{n}\mathrm{m}$.$ (14)
In the main text, EEL denotes all electrons with 18 to 23 eV energy loss,
excluding excitation of 1s3p and 1s4p states that overlap with electron impact
ionization. Subtracting the ICD contribution gives the EIE yield that
consequently only includes excitation of 1s2s and 1s2p states with the cross
section $\sigma_{\rm 1s2s/1s2p}\approx$0.10\text{\,}\AA^{2}$$ in Ref. [7]. Its
ratio to the total inelastic cross section is
$\displaystyle p_{\rm EIE}=\frac{\sigma_{\rm 1s2s/1s2p}}{\sigma_{\rm
total}}\approx 0.38\,.$ (15)
## VII Theoretical estimate of the ICD probability
Total and partial decay widths of the He(2s2p)–He(1s2) ${}^{1}\Sigma_{u}$ and
${}^{1}\Pi_{u}$ states have been computed as described in the main text at
five different distances. The resulting autoionization (AI) probabilities are
summarized in Table 2. The differences of 100 % and the AI probabilities are
the total probabilities for the decay into different ICD channels as a
function of distance which is plotted in Fig. 6. We are not aware of available
data on the relative excitation of the ${}^{1}\Sigma_{u}$ and ${}^{1}\Pi_{u}$
states and the respective ICD probabilities are not differing much.
Particularly at larger distances, they are close to zero such that the
computational method can only give an estimate with limited accuracy.
Therefore, we average the probabilities at each distance with equal weight and
the underlying assumption of similar cross sections has only a small effect on
the resulting average.
Table 2: Likelihood of AI for ${}^{1}\Sigma_{u}$ and ${}^{1}\Pi_{u}$ states at different pair distances $r$. r (a. u.) | 6 | 5.5 | 5 | 4.5 | 4
---|---|---|---|---|---
r (Å) | 3.18 | 2.91 | 2.65 | 2.38 | 2.12
He(2s2p) – He(1s2) ${}^{1}\Sigma_{u}$ $\rightarrow$ AI (%) | 98 | 97 | 94 | 92 | 85
He(2s2p) – He(1s2) ${}^{1}\Pi_{u}$ $\rightarrow$ AI (%) | 98 | 96 | 99 | 95 | 91
Figure 6: Computed ICD probabilities for the He(2s2p) – He(1s2)
${}^{1}\Sigma_{u}$ and ${}^{1}\Pi_{u}$ states with an exponential fit to their
average (magenta squares). Green circles represent the pair-distance
distribution function $g(r)$ [8, 9].
To derive the overall ICD probability, the distance dependent average
probabilities are fitted with an exponential curve. This curve is weighted
with the pair-distance distribution function $g(r)$ [8, 9] to obtain the total
ICD probability relative to the resonant double excitation cross section (AI
plus ICD) which is $(3.0\pm 2.8)\,\%$. The large uncertainty is due to the
standard deviations of the fit parameters $a=670\pm 380$, $b=1.9\pm 0.3$,
$p(r)=a\exp\left(-br\right)$. The ICD probability relative to the total
ionization cross section is obtained by multiplying this result with the ratio
$f_{\text{res}}$ of the resonant cross section to the total electron counts
$c(h\nu)$ in Eq. (5) evaluated at the double excitation resonance
($h\nu=E_{r}$),
$f_{\text{res}}=\frac{s_{r}q^{2}}{s_{d}+s_{r}q^{2}}\left(1+\frac{2}{\pi
q^{2}}\right)^{-1}$ (16)
which is a product of two ratios. According to Ref. [10], we identify a
constant background contribution $s_{d}$ and the unperturbed continuum cross
section $s_{r}$ in the Fano profile in Eq. (5). Then the total cross section
$c(h\nu)$ is a sum of the background $s_{d}$ and a transition probability —
which we denote as $c_{r}(h\nu)$ — that stems from the Fano interference of
the continuum with the discrete level (double excitation leading to AI or
ICD). At the resonance, the latter simplifies to $s_{r}q^{2}$ and its ratio to
the total cross section, $c_{r}(E_{r})/c(E_{r})$, is the first factor in Eq.
(16). For the second factor, we use the identification of $\frac{1}{2}\pi
q^{2}$ in Ref. [11] as the ratio of the transition probabilities to the
modified discrete level ($A$) and to a band width of unperturbed continuum
states ($B$). Assuming that the sum $A+B$ approximates the transition
probability $c_{r}(h\nu)$, its fraction of the resonant pathway is
$\frac{A}{c_{r}(h\nu)}\approx\frac{A}{A+B}=\frac{A/B}{A/B+1}=\frac{\frac{1}{2}\pi
q^{2}}{\frac{1}{2}\pi q^{2}+1}=\frac{1}{1+\frac{2}{\pi q^{2}}}$ (17)
which is the second factor needed to compute $f_{\text{res}}$. With the fit
results for total electrons in Table III we obtain the ratios in Table 3 for
four different nozzle temperatures. The values do not significantly differ
from each other and we use the average $f_{\text{res}}=0.51\pm 0.03$ as final
result.
Table 3: Fractions of resonant to total ionization according to Eq. 16 for VMI data at different nozzle temperatures. Nozzle temperature | 10 K | 12 K | 14 K | 16 K
---|---|---|---|---
$f_{\text{res}}$ | $0.57\pm 0.09$ | $0.47\pm 0.02$ | $0.50\pm 0.04$ | $0.49\pm 0.04$
## References
* LaForge _et al._ [2016] A. C. LaForge, D. Regina, G. Jabbari, K. Gokhberg, N. V. Kryzhevoi, S. R. Krishnan, M. Hess, P. O’Keeffe, A. Ciavardini, K. C. Prince, R. Richter, F. Stienkemeier, L. S. Cederbaum, T. Pfeifer, R. Moshammer, and M. Mudrich, Phys. Rev. A 93, 050502 (2016).
* Kramida _et al._ [2013] A. Kramida, Y. Ralchenko, J. Reader, and N. A. Team, NIST Atomic Spectra Database (version 5.1) (National Institute of Standards and Technology, Gaithersburg, MD) (2013).
* Hertel and Schulz [2008] I. V. Hertel and C.-P. Schulz, _Atoms, molecules and optical physics 1. Atomic physics and foundations of spectroscopy_ (Springer-Verlag Berlin Heidelberg, 2008).
* Asmussen _et al._ [2023a] J. D. Asmussen, L. Ben Ltaief, K. Sishodia, A. R. Abid, B. Bastian, S. Krishnan, H. B. Pedersen, and M. Mudrich, J. Chem. Phys. 159, 034301 (2023a).
* Asmussen _et al._ [2023b] J. D. Asmussen, K. Sishodia, B. Bastian, A. R. Abid, L. Ben Ltaief, H. B. Pedersen, S. De, C. Medina, N. Pal, R. Richter, T. Fennel, S. Krishnan, and M. Mudrich, Nanoscale 15, 14025 (2023b).
* Gomez _et al._ [2011] L. F. Gomez, E. Loginov, R. Sliter, and A. F. Vilesov, J. Chem. Phys. 135, 154201 (2011).
* Ralchenko _et al._ [2008] Y. Ralchenko, R. Janev, T. Kato, D. Fursa, I. Bray, and F. De Heer, At. Data Nucl. Data Tables 94, 603 (2008).
* Chin and Krotscheck [1995] S. A. Chin and E. Krotscheck, Phys. Rev. B 52, 10405 (1995).
* Peterka _et al._ [2007] D. S. Peterka, J. H. Kim, C. C. Wang, L. Poisson, and D. M. Neumark, J. Phys. Chem. A 111, 7449 (2007).
* Mezei _et al._ [2015] J. Mezei, I. F. Schneider, and C. Jungen, EPJ Web of Conferences 84, 04005 (2015).
* Fano [1961] U. Fano, Phys. Rev. 124, 1866 (1961).
|
[]Tamás G. Kovács
# High temperature $U(1)_{A}$ breaking in the chiral limit
###### Abstract
We solve the long-standing problem concerning the fate of the chiral
$U(1)_{A}$ symmetry in QCD-like theories at high temperature in the chiral
limit. We introduce a simple instanton based random matrix model that
precisely reproduces the properties of the lowest part of the lattice overlap
Dirac spectrum. We show that in the chiral limit the instanton gas splits into
a free gas component with a density proportional to $m^{N_{f}}$ and a gas of
instanton-antiinstanton molecules. While the latter do not influence the
chiral properties, for any nonzero quark mass the free gas component produces
a singular spectral peak at zero that dominates Banks-Casher type spectral
sums. By calculating these we show that the difference of the pion and delta
susceptibility vanishes only for three or more massless flavors, however, the
chiral condensate is zero already for two massless flavors.
## 1 Introduction
The approximate $SU(2)_{A}\times U(1)_{A}$ chiral symmetry of light quarks is
an essential feature of quantum-chromodynamics (QCD). While the $SU(2)_{A}$
part, spontaneously broken at low temperatures, is expected to be restored
above the crossover temperature $T_{c}$, the flavor singlet $U(1)_{A}$ part is
anomalous due to the presence of instantons. The fate of the flavor singlet
part is the subject of a long debate that was started by the seminal paper of
Pisarski and Wilczek, who discussed possible scenarios for the finite
temperature chiral transition in the chiral limit [1]. Their results were
based on the $\epsilon$-expansion, and there has been an ongoing effort to
check it using lattice QCD.
Figure 1: The distribution of the lowest Dirac eigenvalue on configurations
containing exactly one instanton-antiinstanton pair on the lattice and in the
random matrix model in a spatial volume of $32^{3}$ in lattice units (left
panel). The same comparison for the second lowest eigenvalue on a larger
volume of $40^{3}$ (right panel). For comparison we also show the distribution
used for the fit, the one in the previous figure.
In the chiral limit the order parameter of the spontaneously broken chiral
symmetry, $\langle\bar{\psi}\psi\rangle$ can be written in terms of the
spectral density $\rho(\lambda)$ of the quark Dirac operator as
$\lim_{m\rightarrow 0}\langle\bar{\psi}\psi\rangle\propto\rho(0).$
This is the Banks-Casher relation [2], showing that the low lying spectrum of
the Dirac operator is intimately related to how chiral symmetry is realized.
Indeed, for some time the standard lore was that at the transition temperature
to the quark-gluon plasma, the spontaneously broken chiral symmetry is
restored, and $\rho(0)$ vanishes, up to small explicit breaking effects due to
the finite, but small light quark masses.
However, this view was challenged when instead of a small $\rho(0)$, a sharp
rise was found in the spectral density of the overlap Dirac operator at zero
[3]. The reason this spike in the spectral density went unnoticed before, was
that unlike the overlap, the staggered and Wilson lattice Dirac operators,
used exclusively previously, did not respect chiral symmetry, and could not
properly resolve the smallest Dirac eigenvalues, the ones that make up the
spike of the density. For some time this finding remained largely ignored,
mostly because it was considered to be a quenched artifact, the result of
ignoring the quark determinant in the path integral. Indeed, the determinant
disfavors eigenvalues of small magnitude, and is expected to suppress the
spike in the spectral density at zero. Later on, the spike was found to
persist even in the presence of light dynamical quarks [4, 6, 5, 7]. However,
some doubts could still remain, as the lattice fermions used in these works
were not chirally symmetric, and their poor resolution of the small Dirac
eigenvalues might have lead to an improper suppression of the spike. Indeed,
results by the JLQCD collaboration using chiral lattice fermions suggest that
toward the chiral limit the spectral spike completely disappears already at a
nonzero quark mass [9, 8]. Another possibility, the one we advocate in the
present paper, is that indeed, chiral fermions are needed for a proper
suppression of the spike, but the spike is still present at any finite quark
mass, however, it becomes undetectably small with the currently used lattice
volumes and statistics.
It seems to us that using the currently available lattice technology it is not
possible to explore the fate of the spike in the spectral density and also the
fate of the $U(1)_{A}$ symmetry for light dynamical quarks. In the present
work we suggest a different approach, based on the finding that in the
quenched theory the statistics of the eigenvalues in the spectral spike are
consistent with mixing zero modes of a free instanton gas, a proposal already
put forward in Ref. [3] and recently confirmed in more detail [10].
## 2 Random matrix model
We propose a random matrix model for the description of the mixing zero modes,
the zero mode zone (ZMZ) of the free instanton gas in quenched QCD. In a free
instanton gas the number distribution of instantons and antiinstantons are
independent and identical Poisson distributions with mean $\chi_{0}V/2$, where
$\chi_{0}$ is the topological susceptibility. A random matrix is constructed
by first drawing the number of instantons $n_{\mbox{\scriptsize i}}$ and
antiinstantons $n_{\mbox{\scriptsize a}}$ from the Poisson distributions. The
size of the matrix is $(n_{\mbox{\scriptsize i}}+n_{\mbox{\scriptsize
a}})\times(n_{\mbox{\scriptsize i}}+n_{\mbox{\scriptsize a}})$. Since zero
modes of like charge objects are protected from mixing by the index theorem,
the matrix has two diagonal blocks of zeros of size $n_{\mbox{\scriptsize
i}}\times n_{\mbox{\scriptsize i}}$ and $n_{\mbox{\scriptsize a}}\times
n_{\mbox{\scriptsize a}}$. At high temperatures the instanton zero mode wave
functions decay exponentially with exponent $\pi T$ [11], and we assume that
the mixing matrix elements of instanton and antiinstanton zero modes also
follow this exponential decay with the distance of the given instanton and
antiinstanton. In a noninteracting gas, the location of the topological
objects are chosen randomly within a three-dimensional111At high temperature
the instantons typically occupy the whole available space in the temporal
direction, so we ignore the temporal location of the instantons. box of size
$L$. In this way the remaining off-diagonal blocks of the matrix are filled
with elements of the form $w_{kl}=A\cdot\exp(-\pi Tr_{kl})$, where $r_{kl}$ is
the distance of the randomly placed instanton $k$ and antiinstanton $l$. This
completes the construction of one random matrix, and we can easily generate
ensembles of such random matrices.
Figure 2: The distribution of the lowest Dirac eigenvalue on the lattice
ensemble described in the text, compared to that in the matrix model with the
best fit parameter $A$ (left panel). In both cases we used only the
configurations with exactly one instanton-antiinstanton pair , and to better
resolve the spike in the spectral density, we plotted the distribution of the
natural log of the eigenvalues. The right panel shows a similar comparison,
but with the second lowest eigenvalue, and on a larger lattice with $L=40$.
The model has two parameters, the topological susceptibility $\chi_{0}$ and
the prefactor $A$. To determine these parameters we consider an ensemble of
20k quenched lattice configurations generated with the Wilson action at
$T=1.11T_{c}$ on lattices of size $32^{3}\times 8$. Computing the lowest
eigenvalues of the overlap Dirac operator on this ensemble and counting the
exact zero eigenvalues yields the topological susceptibility. For fitting the
single remaining parameter $A$, we consider the distribution of the lowest
nonzero Dirac eigenvalue on those configurations that have exactly two complex
conjugate eigenvalues in the ZMZ, corresponding to an instanton-antiinstanton
pair. In Fig. 2 (left panel) we compare the distribution of the lattice
eigenvalues with that of the random matrix model with the best fit parameter
$A$. It is already remarkable that the whole distribution can be fitted with
just this one parameter, but now that the model is fixed we can compare
different properties of the Dirac spectrum on the lattice and in the matrix
model. The comparison can be made also for the distribution of the second
lowest eigenvalue or the full spectral density, and also on different volumes.
As an illustration, in Fig. 2 (right panel) we show the comparison for the
second lowest eigenvalue on a larger volume. These tests demonstrate that the
random matrix model properly describes the details of the lattice overlap
spectrum. Simulations of the random matrix model on larger volumes, not
accessible to direct lattice simulations, indicate the in the thermodynamic
limit the spectral density is singular at zero [12].
## 3 Including dynamical quarks
On the lattice, including dynamical quarks means that in addition to the
quenched Boltzmann weights, each configuration gets another weight factor
proportional to the quark Dirac determinant $\det(D+m)^{N_{f}}$. For
simplicity we assume $N_{f}$ degenerate quark flavors. The determinant of the
lattice Dirac operator can be split into the contribution of the zero mode
zone and that of the bulk as
$\det(D+m)=\prod_{i\in ZMZ}(\lambda_{i}+m)\cdot\prod_{i\in\mbox{\tiny
bulk}}(\lambda_{i}+m).$ (1)
As can be seen in Fig. 1, at high temperatures the ZMZ and the bulk are well
separated, therefore eigenvalues in the bulk are not expected to be correlated
with the ones in the ZMZ. Our main concern here is to study how the
determinant suppresses the eigenvalues in the ZMZ. It is thus a good
approximation to ignore the bulk contribution to the determinant, as that will
only give a trivial factor in the path integral for quantities depending on
the ZMZ. Especially for small quark mass, the suppression of the eigenvalues
in the ZMZ will be driven by the contribution of those small eigenvalues to
the determinant. This is exactly the part of the Dirac spectrum that is
faithfully represented by the random matrix model, so including the
determinant of the random matrices in the statistical weight will properly
describe the suppression of the spectral spike by dynamical quarks. Thus the
random matrix model we propose for the ZMZ of QCD with dynamical quarks is the
one described in the previous section, with the additional weight
$\det(D+m)^{N_{f}}$ for each instanton configuration, where $D$ is the random
matrix corresponding to the given instanton configuration.
The numerical simulation of this model reveals that the topological
susceptibility is suppressed by light dynamical quarks as
$\chi(m)=\chi_{0}m^{N_{f}},$ (2)
where $\chi_{0}$ is the quenched susceptibility, the one obtained without
including the quark determinant. To understand this behavior we note that
throughout the simulations we found that the eigenvalues in the spectral spike
always satisfied $|\lambda|\ll m$, even for the smallest quark mass of
$m=0.01$ (in lattice units). The reason for this is that as the quarks become
lighter, the determinant suppresses the number of instantons, the typical
instanton-antiinstanton distance grows, and the matrix elements become
exponentially small, resulting in ever smaller eigenvalues. As a result, in
the chiral limit the magnitude of the eigenvalues in the ZMZ decreases much
faster than the quark mass, and they will always remain smaller than the quark
mass. If the eigenvalues are much smaller than the quark mass then to a very
good approximation the determinant of a matrix with $n_{\mbox{\scriptsize i}}$
instantons and $n_{\mbox{\scriptsize a}}$ antiinstantons can be written as
$\det(D+m)=\prod_{i}(\lambda_{i}+m)\approx m^{n_{\mbox{\scriptsize
i}}+n_{\mbox{\scriptsize a}}}.$ (3)
Figure 3: The spectral density of the random matrix model with two degenerate
dynamical quark flavors of mass $m=0.05$ (in lattice units) compared to the
spectral density of the matrix model of the free instanton gas with the same
topological susceptibility (left panel). A comparison of the density of
nearest opposite charged topological objects between the quenched and the
dynamical random matrix ensemble (right panel).
With $N_{f}$ quark flavors, each (anti)instanton contributes a suppression
factor $m^{N_{f}}$ to the determinant, and the distribution of (anti)instanton
numbers is still Poissonian, but with a susceptibility suppressed as
$\chi_{0}\rightarrow\chi_{0}m^{N_{f}}$. This explains the quark mass
dependence of the susceptibility in Eq. (2). The fact that the Poisson
distribution of the number of topological objects is preserved also implies
that even in the presence of light dynamical quarks, the lowest part of the
Dirac spectrum can still be understood as the zero mode zone of a free
instanton gas. To demonstrate this, in the left panel of Fig. 3 we compare the
spectral density of the random matrix model with two degenerate dynamical
quark flavors of mass $m=0.05$ (in lattice units) with that of the matrix
model of the free instanton gas (without the determinant) with the same
topological susceptibility. The two curves exactly agree, except for the
largest eigenvalues, where the model with dynamical quarks shows an excess of
eigenvalues. Large eigenvalues in the matrix model indicate that there might
be large matrix elements which in turn would imply that in the dynamical case
there are nearby instanton-antiinstanton pairs. To check that, in the right
panel of Fig. 3 we compare the density of the nearest opposite charged objects
as a function of their distance for the dynamical and quenched matrix
ensembles of the left panel of the same figure. Indeed, in the dynamical case
we find an excess of tightly bound instanton-antiinstanton pairs at a distance
smaller than the instanton size $1/T$. These instanton-antiinstanton
“molecules” are held together by the attractive force due to light dynamical
quarks.
## 4 Chiral condensate and $U(1)_{A}$ breaking susceptibility
We have seen that with light dynamical quarks the instanton gas has two
components. There is a dilute gas of free instantons, responsible for the
small Dirac eigenvalues, i.e. the spectral spike. Besides that, there is a
component of tightly bound instanton-antiinstanton molecules. Our simulations
reveal that in the chiral limit the eigenvalues corresponding to these two
components behave differently. While the free instanton eigenvalues in the
spike become smaller as the gas becomes more dilute, the eigenvalues
corresponding to the molecules remain at a fixed scale in the spectrum. This
is because the size of the molecules does not change with the quark mass. An
important consequence of this is that in the chiral limit any nonvanishing
contribution to Banks-Casher type sums can only come from the free instanton
generated part of the spectrum.
The simplest of these sums is the one providing the chiral condensate. In the
chiral limit the chiral condensate can be written in terms of the Dirac
spectrum as
$\langle\bar{\psi}\psi\rangle\approx\langle\sum_{i}\frac{m}{m^{2}+\lambda_{i}^{2}}\rangle\approx\underbrace{\left(\mbox{\parbox{43.05542pt}{\tiny
avg.\ number of instantons in free
gas}}\right)}_{\chi_{0}m^{N_{\mbox{\scriptsize
f}}}V}\cdot\frac{1}{m}\;=\;m^{N_{\mbox{\scriptsize
f}}-1}\chi_{\mbox{\scriptsize 0}}V.$ (4)
Here we used the fact that the eigenvalues corresponding to the molecular
component of the instanton gas remain at a fixed scale, and in the chiral
limit they do not contribute to the sum. In contrast, the magnitude of
eigenvalues generated by the free gas component becomes smaller in the chiral
limit so rapidly that $\lim_{m\rightarrow 0}\frac{\lambda}{m}=0$, and all
these eigenvalues will contribute a term $1/m$ to the sum. So the chiral limit
of the condensate is given by the contribution of the free instanton gas. This
result shows that for two and more flavors of vanishing mass, the chiral
condensate goes to zero, and the spontaneously broken chiral symmetry is
restored, as expected. We also note that Eq. (4) is consistent with the
expansion of the QCD free energy density in terms of the quark mass around the
chiral point [13], and also consistent with the resulting quasi-instanton
picture of [14]. In fact, our instanton-based matrix model provides a physical
explanation for the quasi-instanton picture, and shows that it is valid not
only in the small quark mass limit.
Another quantity of interest in the chiral limit is the susceptibility
$\chi_{\pi}-\chi_{\delta}$, a nonzero value of which signals the breaking of
the $U(1)_{A}$ symmetry [6, 5, 9]. Similarly to the chiral condensate, in the
chiral limit this quantity can also be written as a sum over the Dirac
spectrum as
$\chi_{\pi}-\chi_{\delta}\approx\langle\sum_{i}\frac{m^{2}}{(m^{2}+\lambda_{i}^{2})^{2}}\rangle\approx
m^{N_{\mbox{\scriptsize f}}}\chi_{\mbox{\scriptsize
0}}V\cdot\frac{1}{m^{2}}\;=\;m^{N_{\mbox{\scriptsize
f}}-2}\chi_{\mbox{\scriptsize 0}}V,$ (5)
showing that even for two flavors of vanishing mass, the symmetry remains
effectively broken.
## Acknowledgments
This work was supported by Hungarian National Research, Development and
Innovation Office NKFIH Grant No. KKP126769 and NKFIH excellence grant
TKP2021-NKTA-64.
## References
* [1] R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338-341 (1984) doi:10.1103/PhysRevD.29.338
* [2] T. Banks and A. Casher, Nucl. Phys. B 169, 103-125 (1980) doi:10.1016/0550-3213(80)90255-2
* [3] R. G. Edwards, U. M. Heller, J. E. Kiskis and R. Narayanan, Phys. Rev. D 61, 074504 (2000) doi:10.1103/PhysRevD.61.074504.
* [4] A. Alexandru and I. Horváth, Phys. Rev. D 92, no.4, 045038 (2015) doi:10.1103/PhysRevD.92.045038.
* [5] H. T. Ding, S. T. Li, S. Mukherjee, A. Tomiya, X. D. Wang and Y. Zhang, Phys. Rev. Lett. 126, no.8, 082001 (2021) doi:10.1103/PhysRevLett.126.082001.
* [6] O. Kaczmarek, L. Mazur and S. Sharma, Phys. Rev. D 104, no.9, 094518 (2021) doi:10.1103/PhysRevD.104.094518.
* [7] X. L. Meng et al. [$\chi$QCD and CLQCD], [arXiv:2305.09459 [hep-lat]]. [8]
* [8] S. Aoki et al. [JLQCD], PTEP 2022, no.2, 023B05 (2022) doi:10.1093/ptep/ptac001
* [9] S. Aoki et al. [JLQCD], Phys. Rev. D 103, no.7, 074506 (2021) doi:10.1103/PhysRevD.103.074506.
* [10] R. A. Vig and T. G. Kovacs, Phys. Rev. D 103, no.11, 114510 (2021) doi:10.1103/PhysRevD.103.114510.
* [11] D. J. Gross, R. D. Pisarski and L. G. Yaffe, Rev. Mod. Phys. 53, 43 (1981) doi:10.1103/RevModPhys.53.43.
* [12] T. G. Kovacs, [arXiv:2311.04208 [hep-lat]].
* [13] T. Kanazawa and N. Yamamoto, JHEP 01, 141 (2016) doi:10.1007/JHEP01(2016)141.
* [14] T. Kanazawa and N. Yamamoto, Phys. Rev. D 91, 105015 (2015) doi:10.1103/PhysRevD.91.105015.
|
# CBDC-AquaSphere: Interoperable Central Bank Digital Currency Built on
Trusted Computing and Blockchain
Ivan Homoliak,† Martin Perešíni,† Patrik Holop,† Jakub Handzuš,† Fran
Casino$\ast$ †Brno University of Technology, Faculty of Information
Technology, Czech Republic∗Universitat Rovira i Virgili, Tarragona, Spain
###### Abstract.
The adoption of decentralized, tamper-proof ledger systems is paving the way
for new applications and opportunities in different contexts. While most
research aims to improve their scalability, privacy, and governance issues,
interoperability has received less attention. Executing transactions across
various blockchains is notably instrumental in unlocking the potential of
novel applications, particularly in the financial sector, where their
potential would otherwise be significantly diminished. Therefore,
interoperable ledgers are crucial to ensure the expansion and further adoption
of such a technology in various contexts.
In this paper, we present a protocol that uses a combination of trusted
execution environment (TEE) and blockchains to enable interoperability over
independent semi-centralized CBDC ledgers, guaranteeing the atomicity of
inter-bank transfers. Our interoperability protocol uses a custom adaptation
of atomic swap protocol and is executed by any pair of CBDC instances to
realize a one-way transfer. It ensures features such as atomicity,
verifiability, correctness, censorship resistance, and privacy while offering
high scalability in terms of the number of CBDC instances. Our approach
enables to possible deployment scenarios that can be combined: (1) CBDC
instances represent central banks of multiple countries, and (2) CBDC
instances represent the set of retail banks and a paramount central bank of a
single country. We provide a detailed description of our protocol as well as
an extensive analysis of its benefits, features, and security.
In this WIP paper, we made a proof-of-concept implementation and made a
partial evaluation, while the more extensive evaluation will be made in our
future work.
Blockchain, Interoperability, Central Bank Digital Currency (CBDC), Trusted
Execution Environment, Cross-chain Protocol, Privacy, Censorship.
## 1\. Introduction
Blockchain technology is becoming the backbone of a myriad of applications
since it provides features such as decentralization, immutability,
availability, and transparency. More recently, along with the increasing
adoption and maturity of such a technology (AlShamsi et al., 2022), central
banks all over the world are accelerating the process of Central Bank Digital
Currency (CBDC) development (Zhang and Huang, 2021). CBDC has received
increasing attention in the past few years. More than 85% of central banks are
actively researching the potential for CBDCs, and according to BIS survey
(Boar and Wehrli, 2021) conducted in 2021 central banks covering 20% of the
world’s population are likely to launch retail CBDCs before 2025. Some of the
reasons behind this new paradigm are the digitization of the economy, the
level of development of the financial sector, and a strong decline in the use
of cash (Náñez Alonso et al., 2020).
Despite the generalized will to improve the worldwide financial system by
utilizing blockchain technology (Ehrenfeld, 2022) in centralized environments,
there is still a road ahead for the realization of fast secure blockchain
payment systems. Nevertheless, some features are essential to enable financial
solutions to reach an operational level, making interoperability a crucial
requirement in this context. Note that a few cross-chain solutions and
protocols (Zhang and Huang, 2021; Lan et al., 2021; Bellavista et al., 2021)
that leverage the necessary level of interoperability for execution of inter-
bank financial transactions have been proposed in the literature. In this
regard, technologies such as Trusted Execution Environments (TEE) in a
potential combination with blockchains can efficiently enforce the required
security and privacy levels of centralized environments of banks, and thus
provide a high level of trustworthiness for the end users.
##### Motivation
CBDC legislation and adoption goes in hand with privacy and security concerns.
The centralized nature of banks implies that transactions are recorded in
private ledgers managed by banks, contrary to the very nature of public
decentralized cryptocurrencies. While this may prevent some potential
malicious scenarios, users are forced into trusting a single authority and its
corresponding regulations. Aiming at increasing decentralization and trust,
several authors have proposed the use of TEE to leverage verifiable protocols
enabling interoperability of multiple centralized isolated environments (Wang
et al., 2023; Lacoste and Lefebvre, 2023).
While achieving blockchain interoperability is challenging regardless of its
flavor (i.e., between centralized, decentralized or hybrid blockchain
structures) additional features such as scalability, confidentiality, and
censorship resistance are necessary to guarantee for practical scenarios.
Digital Euro Association released on October 2022 the CBDC manifesto (Digital
Euro Association, 2022), in which they highlight important features of CBDC,
such as strong value proposition for the end users, the highest degree of
privacy, and interoperability. Our work is inline with this manifesto and
adheres to its features while it provides even additional features that are
interesting for the users and the whole ecosystem assuming that the CBDC-
equipped bank might potentially be an untrusted entity.
##### Contributions
In this paper, we present a practical blockchain interoperability protocol
that integrates such features. On top of the above mentioned features, to the
best of our knowledge, our work represents the first TEE-based interoperable
CBDC approach that provides the proof-of-censorship. Our main contributions
are summarized as follows:
1. (1)
We specify requirements for an instance of CBDC that is controlled by a single
bank111As we will see it can be a central bank or even a retail bank,
depending on the deployment scenario described in Section 4.2.1. Therefore,
depending on the scenario, we well use the term CBDC instance even for a
retail bank. and forms an isolated environment. These requirements include
high processing performance, transparent token issuance, correctness of intra-
bank transfers, immutability of historical data, non-equivocation, privacy,
and the indisputable proofs of censorship.
2. (2)
We investigate state-of-the-art approaches applicable for a CBDC instance
assuming our requirements and identify the most convenient one, Aquareum
(Homoliak and Szalachowski, 2020), that we further base on. The Aquareum-based
CBDC ledger ensures immutability, non-equivocation, privacy, and the
indisputable proofs of transaction censorship by utilizing a permissionless
blockchain (e.g., Ethereum). Next, by using TEE (e.g., Intel SGX), it ensures
the correctness of any transaction execution.
3. (3)
Our main contribution resides in a design and implementation of a protocol
resolving interoperability over multiple instances of semi-centralized CBDC,
which guarantees the atomicity of inter-bank transfers.
4. (4)
We provide a security analysis, to prove the the properties of our approach.
##### Organization
The remainder of the article is organized as follows. In Section 2, we provide
a background on blockchain, atomic swap, and trusted computing. We define the
problem in Section 3, where we describe the attacker model and required
features of a single CBDC instance as well as the environment of multiple
interoperable CBDC instances. Section 4 provides a description of the proposed
interoperability protocol and its deployment scenarios. Next, the
implementation details of the designed protocol and its partial evaluation are
described in Section 5. We make a security analysis of our approach in Section
6. Section 7 reviews the state of the art of CBDC approaches and TEE-based
blockchain solutions. Section 8 discusses the benefits and limitations of our
approach. We conclude the paper in Section 9 with some final remarks.
## 2\. Background
This section provides the reader with the essential context needed to
understand the topics that will be discussed in this article.
### 2.1. Blockchain
Blockchain is a tamper-resistant data structure, in which data records (i.e.,
blocks) are linked using a cryptographic hash function, and each new block has
to be agreed upon by participants (a.k.a., miners) running a consensus
protocol (i.e., consensus nodes). Each block may contain data records
representing orders that transfer tokens, application codes written in a
platform-supported language, and the execution orders of such application
codes. These application codes are referred to as smart contracts, and they
encode arbitrary processing logic written in a supported language of a smart
contract platform. Interactions between clients and the smart contract
platform are based on messages called transactions.
### 2.2. Trusted Execution Environment
Trusted Execution Environment (TEE) is a hardware-based component that enables
secure (remote) execution (Subramanyan et al., 2017) of a pre-defined code
(i.e., enclave) in an isolated environment. TEE uses cryptographic primitives
and hardware-embedded secrets that protect data confidentiality and the
integrity of computations. In particular, the adversary model of TEE involves
the operating system (OS) that may compromise user-space applications but not
TEE-protected applications. An enclave process cannot execute system calls but
can read and write memory outside the enclave. Thus isolated execution in TEE
may be viewed as an ideal model in which a process is guaranteed to be
executed correctly with ideal confidentiality, while it might run on a
potentially malicious OS.
##### Intel SGX
While there exist multiple instances of TEE, in the context of this work we
will focus on Intel SGX (Software Guard Extensions) (Anati et al., 2013;
McKeen et al., 2013; Hoekstra et al., 2013). Intel SGX allows a local process
or a remote system to securely communicate with the enclave as well as execute
verification of the integrity of the enclave’s code. When an enclave is
created, the CPU outputs a report of its initial state, also referred to as a
measurement, which is signed by the private key of TEE and encrypted by a
public key of Intel Attestation Service (IAS). The hardware-protected
signature serves as the proof that the measured code is running in an SGX-
protected enclave, while the encryption by IAS public key ensures that the
SGX-equipped CPU is genuine and was manufactured by Intel. This proof is also
known as a quote or attestation, and it can be verified by a local process or
by a remote system. The enclave-provided public key can be used by a verifier
to establish a secure remote channel with the enclave or to verify the
signature during the attestation.
### 2.3. CBDC
CBDC is often defined as a digital liability backed and issued by a central
bank that is widely available to the general public. CBDC encompasses many
potential benefits such as efficiency and resiliency, flexible monetary
policies, and enables enhanced control of tax evasion and money laundering
(Kiff et al., 2020). However, regulations, privacy and identity management
issues, as well as design vulnerabilities are potential risks that are shared
with cryptocurrencies. Many blockchain-based CBDC projects rely on using some
sort of stable coins adapting permissioned blockchains due to their
scalability and the capability to establish specific privacy policies, as
compared to public blockchains (Sethaput and Innet, 2021; Zhang and Huang,
2021). Therefore, the level of decentralization and coin volatility are two
main differences between blockchain-based CBDCs and common cryptocurrencies.
These CBDCs are often based on permissioned blockchain projects such as Corda
(Brown et al., 2016), variants of Hyperledger (Hyperledger Foundation, 2022),
and Quorum (Espel et al., 2017).
CDBC solutions are often designed as multi-layer projects (Jin and Xia, 2022).
Wholesale CBDC targets communication of financial institutions and inter-bank
settlements. Retail CBDC includes accessibility to the general public or their
customers.
### 2.4. Atomic Swap
A basic atomic swap assumes two parties $\mathbb{A}$ and $\mathbb{B}$ owning
crypto-tokens in two different blockchains. $\mathbb{A}$ and $\mathbb{B}$ wish
to execute cross-chain exchange atomically and thus achieve a fairness
property, i.e., either both of the parties receive the agreed amount of
crypto-tokens or neither of them. First, this process involves an agreement on
the amount and exchange rate, and second, the execution of the exchange
itself.
In a centralized scenario (Micali, 2003), the approach is to utilize a trusted
third party for the execution of the exchange. In contrast to the centralized
scenario, blockchains allow us to execute such an exchange without a
requirement of the trusted party. The atomic swap protocol (Bitcoin Wiki,
2018) enables conditional redemption of the funds in the first blockchain to
$\mathbb{B}$ upon revealing of the hash pre-image (i.e., secret) that redeems
the funds on the second blockchain to $\mathbb{A}$. The atomic swap protocol
is based on two Hashed Time-Lock Contracts (HTLC) that are deployed by both
parties in both blockchains.
Although HTLCs can be implemented by Turing-incomplete smart contracts with
support for hash-locks and time-locks, for clarity, we provide a description
assuming Turing-complete smart contracts, requiring four transactions:
1. (1)
$\mathbb{A}$ chooses a random string $x$ (i.e., a secret) and computes its
hash $h(x)$. Using $h(x)$, $\mathbb{A}$ deploys $HTLC_{\mathbb{A}}$ on the
first blockchain and sends the agreed amount to it, which later enables
anybody to do a conditional transfer of that amount to $\mathbb{B}$ upon
calling a particular method of $HTLC_{\mathbb{A}}$ with $x=h(x)$ as an
argument (i.e., hash-lock). Moreover, $\mathbb{A}$ defines a time-lock, which,
when expired, allows $\mathbb{A}$ to recover funds into her address by calling
a dedicated method: this is to prevent aborting of the protocol by another
party.
2. (2)
When $\mathbb{B}$ notices that $HTLC_{\mathbb{A}}$ has been already deployed,
she deploys $HTLC_{\mathbb{B}}$ on the second blockchain and sends the agreed
amount there, enabling a conditional transfer of that amount to $\mathbb{A}$
upon revealing the correct pre-image of $h(x)$ ($h(x)$ is visible from already
deployed $HTLC_{\mathbb{A}}$). $\mathbb{B}$ also defines a time-lock in
$HTLC_{\mathbb{B}}$ to handle abortion by $\mathbb{A}$.
3. (3)
Once $\mathbb{A}$ notices deployed $HTLC_{\mathbb{B}}$, she calls a method of
$HTLC_{\mathbb{B}}$ with revealed $x$, and in turn, she obtains the funds on
the second blockchain.
4. (4)
Once $\mathbb{B}$ notices that $x$ was revealed by $\mathbb{A}$ on the second
blockchain, she calls a method of $HTLC_{\mathbb{A}}$ with $x$ as an argument,
and in turn, she obtains the funds on the first blockchain.
If any of the parties aborts, the counter-party waits until the time-lock
expires and redeems the funds.
### 2.5. Merkle Tree
A Merkle tree (Merkle, 1989) is a data structure based on the binary tree in
which each leaf node contains a hash of a single data block, while each non-
leaf node contains a hash of its concatenated children. Hence, the root node
provides a tamper-evident integrity snapshot of the tree contents. A Merkle
tree enables efficient membership verification (with logarithmic time/space
complexity) using the Merkle proof. To enable a membership verification of
element $x_{i}$ in the list $X$, the Merkle tree supports the following
operations:
:
$\mathbf{MkRoot(X)\rightarrow Root}$: an aggregation of all elements of the
list $X$ by a Merkle tree, providing a single value $Root$.
:
$\mathbf{MkProof(x_{i},X)\rightarrow\pi^{mk}}$: a Merkle proof generation for
the $i$th element $x_{i}$ present in the list of all elements $X$.
:
$\mathbf{\pi^{mk}.Verify(x_{i},Root)\rightarrow\\{T,F\\}}$: verification of
the Merkle proof $\pi^{mk}$, witnessing that $x_{i}$ is included in the list
$X$ that is aggregated by the Merkle tree with the root hash $Root$.
### 2.6. History Tree
A Merkle tree has been primarily used for proving membership. However, Crosby
and Wallach (Crosby and Wallach, 2009) extended its application for an append-
only tamper-evident log, named a history tree. In history tree, leaf nodes are
added in an append-only fashion while it enables to produce incremental proofs
witnessing that arbitrary two versions of the tree are consistent. The history
tree brings a versioned computation of hashes over the Merkle tree, enabling
to prove that different versions (i.e., commitments) of a log, with distinct
root hashes, make consistent claims about the past. The history tree $L$
supports the following operations:
$\mathbf{L.add(x)\rightarrow C_{j}}$::
appending of the record $x$ to $L$, returning a new commitment $C_{j}$ that
represents the most recent value of the root hash of the history tree.
$\mathbf{L.IncProof(C_{i},C_{j})\rightarrow\pi^{inc}}$::
an incremental proof generation between two commitments $C_{i}$ and $C_{j}$,
where $i\leq j$.
$\mathbf{L.MemProof(i,C_{j})\rightarrow\pi^{mem}}$::
a membership proof generation for $x_{i}$ from the commitment $C_{j}$, where
$i\leq j$.
$\mathbf{\pi^{inc}.Verify(C_{i},C_{j})\rightarrow\\{T,F\\}}$::
verification of the incremental proof $\pi^{inc}$, witnessing that the
commitment $C_{j}$ contains the same history of records
$x_{k},k\in\\{0,\ldots,i\\}$ as the commitment $C_{i}$, where $i\leq j$.
$\mathbf{\pi^{mem}.Verify(i,x_{i},C_{j})\rightarrow\\{T,F\\}}$::
verification of the membership proof $\pi^{mem}$, witnessing that $x_{i}$ is
the $i$th record in the $j$th version of $L$, fixed by the commitment $C_{j}$,
$i\leq j$.
$\mathbf{\pi^{inc}.ReduceRoot()\rightarrow C_{j}}$::
a reduction of the commitment $C_{j}$ from the incremental proof $\pi^{inc}$
that was generated by $L.Inc\\-Proof(C_{i},C_{j})$.
### 2.7. Aquareum
Aquareum (Homoliak and Szalachowski, 2020) is a centralized ledger that is
based on a combination of a trusted execution environment (TEE) with a public
blockchain platform (see our other submission with ID $\\#83$). It provides a
publicly verifiable non-equivocating censorship-evident private ledger.
Aquareum is integrated with a Turing-complete virtual machine (instantiated by
eEVM (Microsoft, 2020)), allowing arbitrary transaction processing logic, such
as transfers or client-specified smart contracts. In other words, Aquareum
provides most of the blockchain features while being lightweight and cheap in
contrast to them. Nevertheless, Aquareum does not provide extremely high
availability (such as blockchains) due to its centralized nature, which is,
however, common and acceptable for the environment of CBDC.
Figure 1. Architecture of Aquareum with our modifications in red.
The overview of Aquareum is depicted in Figure 1 (where parts in red are our
modifications and are irrelevant for the current description). In Aquareum,
clients $\mathbb{C}$s submit transactions to operator $\mathbb{O}$ (1), who
executes them in protected TEE enclave $\mathbb{E}$ (4) upon fetching a few
data of the ledger $L$ with the partial state containing only concerned
accounts (2). $\mathbb{E}$ outputs updated state of affected client accounts
with execution receipts and a version transition pair of $L$ (5) that is
periodically submitted to the smart contract $\mathbb{S}$ deployed on a public
blockchain (7). $\mathbb{S}$ verifies $\mathbb{E}$’s signature and the
consistency of the previous version of $L$ with $\mathbb{S}$’s local snapshot
(8) before updating the snapshot to a new version. Note that snapshot is
represented by the root hash (i.e., $LRoot$ ) of the history tree of $L$.
## 3\. Problem Definition
Our goal is to propose a CBDC approach that respects the features proposed in
DEA manifesto (Digital Euro Association, 2022) released in 2022, while on top
of it, we assume other features that might bring more benefits and guarantees.
First, we start with a specification of the desired features related to a
single instance of CBDC that we assume is operated by a single entity (further
a bank or its operator) that maintains its ledger. Later, we describe desired
features related to multiple instances of CBDC that co-exist in the ecosystem
of wholesale and/or retail CBDC.222Note that we will propose two deployment
scenarios (see Section 4.2.1), one for the wholesale environment and the
second one for the retail environment of multiple retail banks interacting
with a single central bank. In both cases, we assume that a central bank might
not be a trusted entity. All features that respect this assumption are marked
with asterisk ∗ and are considered as requirements for such an attacker model.
### 3.1. Single Instance of CBDC
When assuming a basic building block of CBDC – a single bank’s CBDC working in
an isolated environment from the other banks – we specify the desired features
of CBDC as follows:
1. Correctness of Operation Execution∗:
The clients who are involved in a monetary operation (such as a transfer)
should be guaranteed with a correct execution of their operation.
2. Integrity∗:
The effect of all executed operations made over the client accounts should be
irreversible, and no “quiet” tampering of the data by a bank should be
possible. Also, no conflicting transactions can be (executed and) stored by
the CBDC instance in its ledger.
3. Verifiability∗:
This feature extends integrity and enables the clients of CBDC to obtain
easily verifiable evidence that the ledger they interact with is internally
correct and consistent. In particular, it means that none of the previously
inserted transactions was neither modified nor deleted.
4. Non-Equivocation∗:
From the perspective of the client’s security, the bank should not be able to
present at least two inconsistent views on its ledger to (at least) two
distinct clients who would accept such views as valid.
5. Censorship Evidence∗:
The bank should not be able to censor a client’s request without leaving any
public audit trails proving the censorship occurrence.
6. Transparent Token Issuance∗:
Every CBDC-issued token should be publicly visible (and thus audit-able) to
ensure that a bank is not secretly creating token value “out-of-nothing,” and
thus causing uncontrolled inflation. The transparency also holds for burning
of existing tokens.
7. High Performance:
A CBDC instance should be capable of processing a huge number of transactions
per second since it is intended for daily usage by thousands to millions of
people.
8. Privacy:
All transfers between clients as well as information about the clients of CBDC
should remain private for the public and all other clients that are not
involved in particular transfers. However, a bank can access this kind of
information and potentially provide it to legal bodies, if requested.
### 3.2. Multiple Instances of CBDC
In the case of multiple CBDC instances that can co-exist in a common
environment, we extend the features described in the previous listing by
features that are all requirements:
1. Interoperability∗:
As a necessary prerequisite for co-existence of multiple CBDC instances, we
require them to be mutually interoperable, which means that tokens issued by
one bank can be transferred to any other bank. For simplicity, we assume that
all the CBDC instances are using the unit token of the same value within its
ecosystem.333On the other hand, conversions of disparate CBDC-backed tokens
would be possible by following trusted oracles or oracle networks. At the
hearth of interoperability lies atomicity of supported operations. Atomic
interoperability, however, requires means for accountable coping with
censorship and recovery from stalling. We specify these features in the
following.
1. Atomicity∗:
Any operation (e.g., transfer) between two interoperable CBDC instances must
be either executed completely or not executed at all. As a consequence, no new
tokens can be created out-of-nothing and no tokens can be lost in an inter-
bank operation. Note that even if this would be possible, the state of both
involved instances of CBDC would remain internally consistent; therefore,
consistency of particular instances (Section 3.1) is not a sufficient feature
to ensure atomicity within multiple interoperable CBDC instances. This
requirement is especially important due to trustless assumption about
particular banks, who might act in their benefits even for the cost of
imposing the extreme inflation to the whole system.444For example, if
atomicity is not enforced, one bank might send the tokens to another bank,
while not decreasing its supply due to pretended operation abortion.
2. Inter-CBDC Censorship Evidence∗:
Having multiple instances of CBDC enables a different way of censorship, where
one CBDC (and its clients) might be censored within some inter-CBDC operation
with another CBDC instance, precluding them to finish the operation.
Therefore, there should exist a means how to accountably detect this kind of
censorship as well.
3. Inter-CBDC Censorship Recovery∗:
If the permanent censorship happens and is indisputably proven, it must not
impact other instances of CBDC, including the ones that the inter-CBDC
operations are undergoing. Therefore, the interoperable CBDC environment
should provide a means to recover from inter-CBDC censorship of unfinished
operations.
2. Identity Management of CBDC Instances∗:
Since we assume that CBDC instances are trustless, in theory, there might
emerge a fake CBDC instance, pretentding to act as a valid one. To avoid this
kind of situation, it is important for the ecosystem of wholesale CBDC to
manage identities of particular valid CBDC instances in a secure manner.
### 3.3. Adversary Model
The attacker can be represented by the operator of a bank or the client of a
bank, and their intention is to break functionalities that are provided by the
features described above. Next, we assume that the adversary cannot undermine
the cryptographic primitives used, the blockchain platform, and the TEE
platform deployed.
## 4\. Proposed CBDC Approach
Figure 2. Overview of our CBDC architecture supporting interoperability among
multiple CBDC instances (i.e., banks). The schema depicts two instances, where
each of them has its own centralized ledger $L$ modified in a secure way
through TEE of $\mathbb{E}$, while its integrity is ensured by periodic
integrity snapshots to the integrity preserving smart contract
($\mathbb{IPSC}$) in a public blockchain $PB$. Each CBDC instance is
registered in the identity management smart contract $\mathbb{IMSC}$ of a
public blockchain, serving as a global registry of bank instances. A client
who makes an inter-bank transfer communicates with her bank and the counter-
party bank utilizing interoperability micro contracts ($\mathbb{IOMC}$),
running in the TEE. Any censored request of a client is resolved by
$\mathbb{IPSC}$ of a particular bank and can be initiated by its client or a
counter-party client.
We propose a holistic approach for the ecosystem of wholesale and/or retail
CBDC, which aims at meeting the features described in Section 3. To accomplish
these features, we leverage interesting properties stemming from a combination
of a public blockchain (with smart contract platform) and TEE. Such a
combination was proposed for various purposes in related work (see Section 7),
out of which the use case of generic centralized ledger Aquareum (Homoliak and
Szalachowski, 2020) is most convenient to build on. Therefore, we utilize
Aquareum as a building block for a single instance of CBDC, and we make a few
CBDC-specific modifications to it, enhancing its transparency and
functionality. Our modifications are outlined in Figure 1 by red color, while
the details of them (especially changes in programs of smart contract and
enclave) will be described in this section. First, we start by a description
of a single CBDC instance and then we extend it to a fully interoperable
environment consisting of multiple CBDC instances.
Note that in this paper, we focus solely on the transfer of tokens operation
within the context of CBDC interoperability. However, our approach could be
extended to different operations, involving inter-CBDC smart contract
invocations. Also, note that to distinguish between smart contracts on a
public blockchains and smart contracts running in TEE, we will denote latter
as micro contracts (or $\mu$-contracts). Similarly, we denote transactions
sent to TEE as micro transactions (or $\mu$-transactions) and blocks created
in the ledger of CBDC instance as micro blocks (or $\mu$-blocks).
### 4.1. A CBDC Instance
Alike in Aquareum, the primary entity of each CBDC instance is its operator
$\mathbb{O}$ (i.e., a bank), who is responsible for (1) maintaining the ledger
$L$, (2) running the TEE enclave $\mathbb{E}$, (3) synchronization of the
$L$’s snapshot to a public blockchain with smart contract $\mathbb{IPSC}$
($\mathbb{I}$ntegrity $\mathbb{P}$reserving $\mathbb{S}$mart
$\mathbb{C}$ontract), (4) resolving censorship requests, and (5) a
communication with clients $\mathbb{C}$s.
#### 4.1.1. Token Issuance
On top of Aquareum’s $\mathbb{S}$, our $\mathbb{IPSC}$ contains snapshotting
of the total issued tokens $t_{i}$ by the current CBDC instance and the total
supply $t_{s}$ available at the instance for the purpose of transparency in
token issuance (and potentially even burning). Therefore, we extend the
$\mathbb{E}$-signed version transition pair periodically submitted to
$\mathbb{IPSC}$ by these two fields that are relayed to $\mathbb{IPSC}$ upon
snapshotting $L$ (see red text in Figure 1). Notice that $t_{i}=t_{s}$ in the
case of a single instance since the environment of the instance is isolated.
#### An Inflation Bound
Although snapshotting the total tokens in circulation is useful for the
transparency of token issuance, $\mathbb{O}$ might still hyper-inflate the
CBDC instance. Therefore, we require $\mathbb{O}$ to guarantee a maximal
inflation rate $i_{r}$ per year, which can be enforced by $\mathbb{IPSC}$ as
well as $\mathbb{E}$ since the code of both is publicly visible and
attestable. The $i_{r}$ should be adjusted to a constant value by $\mathbb{O}$
at the initialization of $\mathbb{IPSC}$ and verified every time the new
version of $L$ is posted to $\mathbb{IPSC}$; in the case of not meeting the
constrain, the new version would not be accepted at $\mathbb{IPSC}$. However,
another possible option is that the majority vote of $\mathbb{C}$s can change
$i_{r}$ even after initialization. Besides, $\mathbb{E}$ also enforces $i_{r}$
on $t_{i}$ and does not allow $\mathbb{O}$ to issue yearly more tokens than
defined by $i_{r}$. Nevertheless, we put the inflation rate logic also into
$\mathbb{IPSC}$ for the purpose of transparency.
#### 4.1.2. Initialization
First, $\mathbb{E}$ with program $prog^{\mathbb{E}}$ (see Algorithm 5 of
Appendix) generates and stores two key pairs, one under $\Sigma_{pb}$ (i.e.,
$SK_{\mathbb{E}}^{pb}$, $PK_{\mathbb{E}}^{pb}$) and one under $\Sigma_{tee}$
(i.e., $SK_{\mathbb{E}}^{tee}$, $PK_{\mathbb{E}}^{tee}$). Then, $\mathbb{O}$
generates one key pair under $\Sigma_{pb}$ (i.e., $SK_{\mathbb{O}}^{pb}$,
$PK_{\mathbb{O}}^{pb}$), which is then used as the sender of a transaction
deploying $\mathbb{IPSC}$ with program $prog^{\mathbb{IPSC}}$ (see Algorithm 6
of Appendix) at public blockchain with parameters $PK_{\mathbb{E}}^{pb}$,
$PK_{\mathbb{E}}^{tee}$, $PK_{\mathbb{O}}^{pb}$, $t_{i}$, and $i_{r}$. Then,
$\mathbb{IPSC}$ stores the keys in parameters, sets the initial version of $L$
by putting $LRoot_{pb}\leftarrow~{}\perp$, and sets the initial total issued
tokens and the total supply, both to $t_{i}$.555Among these parameters, a
constructor of $\mathbb{IPSC}$ also accepts the indication whether an instance
is allowed to issue tokens. This is, however, implicit for the single
instance, while restrictions are reasonable in the case of multiple instances.
#### Client Registration
A client $\mathbb{C}$ registers with $\mathbb{O}$, who performs know your
customer (KYC) checks and submits her public key $PK_{pb}^{\mathbb{C}}$ to
$\mathbb{E}$. Then, $\mathbb{E}$ outputs an execution receipt about the
successful registration of $\mathbb{C}$ as well as her access ticket
$t^{\mathbb{C}}$ that will serve for potential communication with
$\mathbb{IPSC}$ and its purpose is to avoid spamming $\mathbb{IPSC}$ by
invalid requests. In detail, $t^{\mathbb{C}}$ is the $\mathbb{E}$-signed tuple
that contains $PK_{pb}^{\mathbb{C}}$ and optionally other fields such as the
account expiration timestamp. Next, $\mathbb{C}$ verifies whether her
registration (proved by the receipt) was already snapshotted by $\mathbb{O}$
at $\mathbb{IPSC}$.
#### 4.1.3. Normal Operation
$\mathbb{C}$s send $\mu$-transactions (writing to $L$) and queries (reading
from $L$) to $\mathbb{O}$, who validates them and relays them to $\mathbb{E}$,
which processes them within its virtual machine (Aquareum uses eEVM
(Microsoft, 2020)). Therefore, $L$ and its state are modified in a trusted
code of $\mathbb{E}$, creating a new version of $L$, which is represented by
the root hash $LRoot$ of the history tree. Note that program
$prog^{\mathbb{E}}$ is public and can be remotely attested by $\mathbb{C}$s
(or anybody). $\mathbb{O}$ is responsible for a periodic synchronization of
the most recent root hash $LRoot_{cur}$ (i.e., snapshotting the current
version of $L$ ) to $\mathbb{IPSC}$, running on a public blockchain $PB$.
Besides, $\mathbb{C}$s use this smart contract to resolve censored
transactions and queries, while preserving the privacy of data.
#### 4.1.4. Censorship Resolution
$\mathbb{O}$ might potentially censor some write transactions or read queries
of $\mathbb{C}s$. However, these can be resolved by Aquareum’s mechanism as
follows. If $\mathbb{C}$’s $\mu$-transaction $\mu$-tx is censored by
$\mathbb{O}$, $\mathbb{C}$ first creates $PK_{\mathbb{E}}^{tee}$-encrypted
$\mu$-$etx$ (to ensure privacy in $PB$), and then she creates and signs a
transaction containing $\mathbb{C}^{\prime}s$ access ticket $t^{\mathbb{C}}$
and $\mu$-$etx$. $\mathbb{C}$ sends this transaction to $\mathbb{IPSC}$, which
verifies $t^{\mathbb{C}}$ and stores $\mu$-$etx$, which is now visible to
$\mathbb{O}$ and the public. Therefore, $\mathbb{O}$ might relay $\mu$-$etx$
to $\mathbb{E}$ for processing and then provide $\mathbb{E}$-signed execution
receipt to $\mathbb{IPSC}$ that publicly resolves this censorship request. On
the other hand, if $\mathbb{O}$ were not to do it, $\mathbb{IPSC}$ would
contain an indisputable proof of censorship by $\mathbb{O}$ on a client
$\mathbb{C}$.
### 4.2. Multiple CBDC Instances
The conceptual model of our interoperable CBDC architecture is depicted in
Figure 2. It consists of multiple CBDC instances (i.e., at least two), whose
$\mathbb{C}s$ communicate in three different ways: (1) directly with each
other, (2) in the instance-to-instance fashion through the infrastructure of
their $\mathbb{O}$ as well as counterpart’s $\mathbb{O}$, (3) through $PB$
with $\mathbb{IPSC}$ of both $\mathbb{O}$s and a global registry
$\mathbb{IMSC}$ managing identities of instances.
For simplified description, in the following we assume the transfer operation
where a local CBDC instance in Figure 2 is A (i.e., the sender of tokens) and
the external one is B (i.e., the receiver of tokens). To ensure
interoperability, we require a communication channel of local clients
$\mathbb{C}s_{A}$ to external clients $\mathbb{C}s_{B}$ (the green arrow), the
local operator $\mathbb{O}_{A}$ (the black arrow), and the external operator
$\mathbb{O}_{B}$ (the black dashed arrow). In our interoperability protocol
$\Pi^{T}$ (described later in Section 4.3), external $\mathbb{C}s_{B}$ use the
channel with the local operator $\mathbb{O}_{A}$ only for obtaining
incremental proofs of $L_{A}$’s history tree to verify inclusion of some
$\mu$-transactions in $L_{A}$. However, there might arise a situation in which
$\mathbb{O}_{A}$ might censor such queries, therefore, we need to address it
by another communication channel – i.e., the public blockchain $PB$.
#### Censorship of External Clients
We allow external clients $\mathbb{C}s_{B}$ to use the same means of
censorship resolution as internal clients of a single CBDC instance (see
Section 4.1.4). To request a resolution of a censored query, the external
$\mathbb{C}_{B}$ uses the access ticket $t^{\mathbb{C}_{B}}$ at
$\mathbb{IPSC}_{A}$, which is issued by $\mathbb{E}_{A}$ in the first phase of
$\Pi^{T}$.
#### Identification of Client Accounts
To uniquely identify $\mathbb{C}$’s account at a particular CBDC instance,
first it is necessary to specify the globally unique identifier of the CBDC
instance. The best candidate is the blockchain address of the $\mathbb{IPSC}$
in $PB$ since it is publicly visible and unique in $PB$ (and we denote it by
$\mathbb{IPSC}$). Then, the identification of $\mathbb{C}$’s relevant account
is a pair $\mathbb{C}^{ID}=\\{PK_{pb}^{\mathbb{C}}||~{}\mathbb{IPSC}\\}$. Note
that $\mathbb{C}$ might use the same $PK_{pb}^{\mathbb{C}}$ for the
registration at multiple CBDC instances (i.e., equivalent of having accounts
in multiple banks); however, to preserve better privacy, making linkage of
$\mathbb{C}$’s instances more difficult, we recommend $\mathbb{C}s$ to have
dedicated key pair for each instance.
#### 4.2.1. Identity Management of CBDC Instances
To manage identities of all CBDC instances in the system, we need a global
registry of their identifiers – $\mathbb{IPSC}$ addresses. For this purpose,
we use the $\mathbb{I}$dentity $\mathbb{M}$anagement $\mathbb{S}$mart
$\mathbb{C}$ontract ($\mathbb{IMSC}$) deployed in $PB$ (see program
$prog^{\mathbb{IMSC}}$ in Algorithm 1). We propose $\mathbb{IMSC}$ to be
managed in either decentralized or centralized fashion, depending on the
deployment scenario described below.
### Deployment scenarios
#### Decentralized Scheme
In the decentralized scheme, the enrollment of a new CBDC instance must be
approved by a majority vote of the already existing instances. This might be
convenient for interconnecting central banks from various countries/regions.
The enrollment requires creating a request entry at $\mathbb{IMSC}$ (i.e.,
$newJoinRequest()$) by a new instance specifying the address of its
$\mathbb{IPSC}_{new}$ and $PK_{PB}^{\mathbb{O}_{new}}$. Then, the request has
to be approved by voting of existing instances. Prior to voting (i.e.,
$approveJoinRequest()$), the existing instances should first verify a new
instance by certain legal processes as well as by technical means: do the
remote attestation of $prog^{\mathbb{E}}_{new}$, verify the inflation rate
$i_{r}$ and the initial value of total issued tokens $t_{i}$ in
$\mathbb{IPSC}$, etc. Removing of the existing instance also requires the
majority of all instances, who should verify legal conditions prior to voting.
#### Centralized Scheme
So far, we were assuming that CBDC instances are equal, which might be
convenient for interconnection of central banks from different countries.
However, from the single-country point-of-view, there usually exist only one
central bank, which might not be interested in decentralization of its
competences (e.g., issuing tokens, setting inflation rate) among multiple
commercial banks. We respect this and enable our approach to be utilized for
such a use case, while the necessary changes are made to $\mathbb{IMSC}_{c}$
(see Algorithm 2), allowing to have only one CBDC authority that can add or
delete instances of (commercial) banks, upon their verification (as outlined
above). The new instances can be adjusted even with token issuance capability
and constraints on inflation, which is enforced within the code of
$\mathbb{E}$ as well as $\mathbb{IPSC}$.
$\triangleright$ Declaration of types and variables:
$msg$: a current transaction that called $\mathbb{IMSC}$,
struct InstanceInfo {
$operator$ : $PK_{\mathbb{O}}^{PB}$ of the instance’s $\mathbb{O}$,
$isApproved$: admission status of the instance,
$approvals\leftarrow[]$ : $\mathbb{O}$s who have approved the instance
creation (or deletion),
}
$instances[]$: a mapping of $\mathbb{IPSC}$ to InstanceInfo,
$\triangleright$ Declaration of functions:
function _$Init$($\mathbb{IPSC}s[],\mathbb{O}s[]$) public $\triangleright$
Initial instances are implicitly approved. _
assert $|\mathbb{IPSC}s|=|\mathbb{O}s|$ ;
for _$i\leftarrow 0;\ i\leq|\mathbb{O}s|;\ i\leftarrow i+1$_ do
$instances[\mathbb{IPSC}s[i]]\leftarrow\textbf{InstanceInfo}(\mathbb{O}s[i],True,[])$;
function _$newJoinRequest$($\mathbb{IPSC}$) public _
assert $instances[\mathbb{IPSC}]=~{}\perp$; $\triangleright$ The instance must
not exist yet.
$instances[\mathbb{IPSC}]\leftarrow\textbf{InstanceInfo}(msg.sender,False,[])$;
function _$approveJoinRequest$($\mathbb{IPSC}_{my},\mathbb{IPSC}_{new}$)
public _
assert $instances[\mathbb{IPSC}_{my}].operator=msg.sender$; $\triangleright$
Sender’s check.
assert $instances[\mathbb{IPSC}_{my}].isApproved$; $\triangleright$ The
sending $\mathbb{O}$ has valid instance.
assert $!instances[\mathbb{IPSC}_{new}].isApproved$; $\triangleright$ The new
instance is not approved.
$r\leftarrow instances[\mathbb{IPSC}_{new}]$;
$r.approvals[msg.sender]\leftarrow True$; $\triangleright$ The sender
acknowledges the request.
if _$|r.approvals| >\lfloor|instances|/2\rfloor$_ then
$r.isApproved\leftarrow True$; $\triangleright$ Majority vote applies.
$r.approvals\leftarrow[]$; $\triangleright$ Switch this field for a potential
deletion.
function _$approveDelete$($\mathbb{IPSC}_{my},\mathbb{IPSC}_{del}$) public _
assert $instances[\mathbb{IPSC}_{my}].operator=msg.sender$; $\triangleright$
Sender’s check.
assert $instances[\mathbb{IPSC}_{my}].isApproved$; $\triangleright$ The
sending $\mathbb{O}$ has valid instance.
assert $instances[\mathbb{IPSC}_{del}].isApproved$; $\triangleright$ An
instance to delete must be approved.
$r\leftarrow instances[\mathbb{IPSC}_{del}]$;
$r.approvals[msg.sender]\leftarrow True$; $\triangleright$ The sender
acknowledges the request.
if _$|r.approvals| >\lfloor|instances|/2\rfloor$_ then
delete $r$;
Algorithm 1 $prog^{\mathbb{IMSC}}_{d}$ of decentralized $\mathbb{IMSC}$
$\triangleright$ Declaration of types and variables:
$msg$: a current transaction that called $\mathbb{IMSC}$,
$authority$: $\mathbb{IPSC}$ of the authority bank,
$authority^{\mathbb{O}}$: $PK^{\mathbb{O}}_{pb}$ of $\mathbb{O}$ at authority
bank,
$instances[]$: a mapping of $\mathbb{IPSC}$ to $PK_{\mathbb{O}}^{PB}$,
$\triangleright$ Declaration of functions:
function _$Init$($\mathbb{IPSC}$) public $\triangleright$ Initial instances
are implicitly approved. _
$authority^{\mathbb{O}}\leftarrow msg.sender$;
$authority\leftarrow\mathbb{IPSC}$;
function _$addInstance$($\mathbb{IPSC}_{new},~{}\mathbb{O}_{new}$) public _
assert $msg.sender=authority^{\mathbb{O}}$ ; $\triangleright$ Only the
authority can add instances.
assert $instances[\mathbb{IPSC}_{new}]=~{}\perp$; $\triangleright$ The
instance must not exist yet.
$instances[\mathbb{IPSC}_{new}]\leftarrow\mathbb{O}_{new}$;
function _$delInstance$($\mathbb{IPSC}_{del}$) public _
assert $msg.sender=authority^{\mathbb{O}}$ ; $\triangleright$ Only the
authority can delete instances.
delete $instances[\mathbb{IPSC}_{del}]$;
Algorithm 2 $prog^{\mathbb{IMSC}}_{c}$ of centralized $\mathbb{IMSC}$
#### 4.2.2. Token Issuance
With multiple CBDC instances, $\mathbb{C}$s and the public can obtain the
total value of issued tokens in the ecosystem of CBDC and compare it to the
total value of token supply of all instances. Nevertheless, assuming only two
instances A and B, the value of $t_{s}$ snapshotted by $\mathbb{IPSC}_{A}$
might not reflect the recently executed transfers to instance B that might
have already made the snapshot of its actual $L_{B}$ version to
$\mathbb{IPSC}_{B}$, accounting for the transfers. As a consequence, given a
set of instances, the value of the aggregated $t_{s}$ should always be greater
or equal than the corresponding sum of $t_{i}$:
(1) $\displaystyle t_{i}^{A}+t_{i}^{B}$ $\displaystyle\leq$ $\displaystyle
t_{s}^{A}+t_{s}^{B}.$
We can generalize it for $N$ instances known by $\mathbb{IMSC}$ as follows:
(2) $\displaystyle\sum_{\forall X~{}\in~{}\mathbb{IMSC}}t_{i}^{X}$
$\displaystyle\leq$ $\displaystyle\sum_{\forall
X~{}\in~{}\mathbb{IMSC}}t_{s}^{X}.$
#### 4.2.3. Inflation Rate
In contrast to a single CBDC instance, multiple independent instances must
provide certain guarantees about inflation not only to their clients, but also
to each other. For this purpose, the parameter inflation rate $i_{r}$ is
adjusted to a constant value in the initialization of $\mathbb{IPSC}$ and
checked before the instance is approved at $\mathbb{IMSC}$.
If one would like to enable the update of $i_{r}$ at CBDC instances, a
majority vote at $\mathbb{IMSC}$ on a new value could be utilized (or just the
vote of authority in the case of centralized scenario). Nevertheless, to
support even fairer properties, $\mathbb{C}$s of a particular instance might
vote on the value of $i_{r}$ upon its acceptance by $\mathbb{IOMC}$ and before
it is propagated to $\mathbb{IPSC}$ of an instance. Then, based on the new
value of $\mathbb{IPSC}.i_{r}$, $\mathbb{E}.i_{r}$ can be adjusted as well
(i.e., upon the validation by the light client of $\mathbb{E}$). However, the
application of such a mechanism might depend on the use case, and we state it
only as a possible option that can be enabled in our approach.
$\triangleright$ Declaration of types and variables:
$\mathbb{E}$, $\triangleright$ The reference to $\mathbb{E}_{A}$ of sending
party.
$msg$, $\triangleright$ The current $\mu$-transaction that called
$\mathbb{IOMC}^{S}$.
struct LockedTransfer {
$sender$, $\triangleright$ Sending client $\mathbb{C}_{A}$.
$receiver$, $\triangleright$ Receiving client $\mathbb{C}_{B}$.
$receiver\mathbb{IPSC}$, $\triangleright$ The $\mathbb{IPSC}$ contract address
of the receiver’s instance.
$amount$, $\triangleright$ Amount of tokens sent.
$hashlock$, $\triangleright$ Hash of the secret of the sending
$\mathbb{C}_{A}$.
$timelock$, $\triangleright$ A timestamp defining the end of validity of the
transfer.
$isCompleted$, $\triangleright$ Indicates whether the transfer has been
completed.
$isReverted$, $\triangleright$ Indicates whether the transfer has been
canceled.
},
$transfers\leftarrow[]$, $\triangleright$ Initiated outgoing transfers (i.e.,
LockedTransfer).
const $timeout^{HTLC}\leftarrow 24h$, $\triangleright$ Set the time lock for
e.g., 24 hours.
$\triangleright$ Declaration of functions:
function _$sendInit$($receiver,receiver\mathbb{IPSC},hashlock$) public
payable _
assert $msg.value>0$; $\triangleright$ Checks the amount of tokens.
$timelock\leftarrow timestamp.now()+timeout^{HTLC}$;
$t\leftarrow\textbf{LockedTransfer}(msg.sender,receiver,receiver\mathbb{IPSC},$
$msg.value$, $hashlock,timelock,False,False)$; $\triangleright$ A new
receiving transfer.
$transfers.append(t)$;
Output $("sendInitialized",transferID\leftarrow|transfers|-1))$;
function _$sendCommit$($transferID,secret,extTransferID$) public _
assert $transfers[transferID]\neq\perp$; $\triangleright$ Check the existence
of locked transfer.
$t\leftarrow transfers[transferID]$;
assert $t.hashlock=h(secret)$; $\triangleright$ Check the secret.
assert $!t.isCompleted~{}\wedge~{}!t.isReverted$; $\triangleright$ Test if the
transfer is still pending.
$t.isCompleted\leftarrow True$;
burn t.amount; $\triangleright$ Burn tokens.
$\mathbb{E}.t_{s}\leftarrow\mathbb{E}.t_{s}-t.amount$; $\triangleright$
Decrease the total supply of the instance.
Output $("sendCommitted",transferID,$ $extTransferID,$ $t.receiver,$
$t.receiver\mathbb{IPSC},$ $t.amount)$;
function _$sendRevert$($transferID$) public _
assert $transfers[transferID]\neq\perp$; $\triangleright$ Check the existence
of locked transfer.
$t\leftarrow transfers[transferID]$;
assert $!t.isCompleted~{}\wedge~{}!t.isReverted$; $\triangleright$ Test the
transfer is still pending.
assert $t.timelock\leq timestamp.now()$; $\triangleright$ Check the HTLC
expiration.
$transfer(t.amount,t.sender)$; $\triangleright$ Returning tokens back to the
sender.
$t.isReverted\leftarrow True$;
Output$("sendReverted",transferID)$;
Algorithm 3 $prog^{\mathbb{IOMC}^{S}}$ of sending $\mathbb{IOMC}^{S}$
#### 4.2.4. Interoperability
The interoperability logic itself is provided by our protocol $\Pi^{T}$ that
utilizes $\mathbb{I}$nter$\mathbb{O}$perability $\mathbb{M}$icro
$\mathbb{C}$ontracts $\mathbb{IOMC}^{S}$ and $\mathbb{IOMC}^{R}$, which serve
for sending and receiving tokens, respectively. Therefore, in the context of
$\mathbb{E}$-isolated environment these $\mu$-contracts allow to mint and burn
tokens, reflecting the changes in $t_{s}$ after sending or receiving tokens
between CBDC instances. Both $\mu$-contracts are deployed in $\mathbb{E}$ by
each $\mathbb{O}$ as soon as the instance is created, while $\mathbb{E}$
records their addresses that can be obtained and attested by $\mathbb{C}$s. We
briefly review these contracts in the following, and we will demonstrate their
usage in Section 4.3.
#### The Sending $\mathbb{IOMC}^{S}$
The sending $\mathbb{IOMC}^{S}$ (see Algorithm 3) is based on Hash Time LoCks
(HTLC), thus upon initialization of transfer by $hashlock$ provided by
$\mathbb{C}_{A}$ (i.e., $hashlock\leftarrow h(secret)$) and calling
$sendInit(hash\\-lock,\ldots)$, $\mathbb{IOMC}^{S}$ locks transferred tokens
for the timeout required to complete the transfer by
$send\\-Commit\\-(secret,\ldots)$. If tokens are not successfully transferred
to the recipient of the external instance during the timeout, they can be
recovered by the sender (i.e., $sendRevert()$).666Note that setting a short
timeout might prevent the completion of the protocol. If tokens were sent
successfully from $\mathbb{C}_{A}$ to $\mathbb{C}_{B}$, then instance A burns
them within $sendCommit()$ of $\mathbb{IOMC}^{S}$ and deducts them from
$t_{s}$. Note that deducting $t_{s}$ is a special operation that cannot be
executed within standard $\mu$-contracts, but $\mathbb{IOMC}$ contracts are
exceptions and can access some variables of $\mathbb{E}$.
#### The Receiving $\mathbb{IOMC}^{R}$
The receiving $\mathbb{IOMC}^{R}$ (see Algorithm 4) is based on Hashlocks
(referred to as HLC) and works pairwise with sending $\mathbb{IOMC}^{S}$ to
facilitate four phases of our interoperable transfer protocol $\Pi_{T}$
(described below). After calling $\mathbb{IOMC^{S}}.sendInit()$, incoming
initiated transfer is recorded at $\mathbb{IOMC}^{R}$ by
$receiveInit(hash\\-lock,\ldots)$. Similarly, after executing token deduction
at instance A (i.e., $\mathbb{IOMC^{S}}.send\\-Commit\\-(secret,\ldots)$),
incoming transfer is executed at $\mathbb{IOMC}^{R}$ by
$receiveCommit(secret,\ldots)$ that mints tokens to $\mathbb{C}_{B}$ and
increases $t_{s}$. Similar to $\mathbb{IOMC}^{S}$, minting tokens and
increasing $t_{s}$ are special operations requiring access to $\mathbb{E}$,
which is exceptional for $\mathbb{IOMC}$. The overview of $\Pi_{T}$ is
depicted in Figure 3.
$\triangleright$ Declaration of types and variables:
$\mathbb{E}$, $\triangleright$ The reference to $\mathbb{E}_{B}$ of receiving
party.
struct LockedTransfer {
$sender$, $\triangleright$ Sending client $\mathbb{C}_{A}$.
$sender\mathbb{IPSC}$, $\triangleright$ The IPSC contract address of the
sender’s instance.
$receiver$, $\triangleright$ Receiving client $\mathbb{C}_{B}$.
$amount$, $\triangleright$ Amount of transferred tokens.
$hashlock$, $\triangleright$ Hash of the secret of the sending
$\mathbb{C}_{A}$.
$isCompleted$, $\triangleright$ Indicates whether the transfer has been
completed.
},
$transfers\leftarrow[]$, $\triangleright$ Initiated incoming transfers (i.e.,
LockedTransfer).
$\triangleright$ Declaration of functions:
function _$receiveInit$($sender,sender\mathbb{IPSC},hashlock,amount$) public
_
assert $amount>0$;
$t\leftarrow\textbf{LockedTransfer}(sender,sender\mathbb{IPSC},msg.sender,amount,$
$hashlock,False)$; $\triangleright$ Make a new receiving transfer entry.
$transfers.append(t)$;
Output$("receiveInitialized",transferID\leftarrow|transfers|-1)$;
function _$receiveCommit$($transferID,secret$) public _
assert $transfers[transferID]~{}\neq~{}\perp$; $\triangleright$ Check the
existence of transfer entry.
$t\leftarrow transfers[transferID]$;
assert $t.hashlock=h(secret)$; $\triangleright$ Check the secret.
assert $!t.isCompleted$; $\triangleright$ Check whether the transfer is
pending.
$\mathbb{E}.mint(this,t.amount)$; $\triangleright$ Call $\mathbb{E}$ to mint
tokens on $\mathbb{IOMC}_{R}$.
$\mathbb{E}.t_{s}\leftarrow\mathbb{E}.t_{s}+t.amount$ ; $\triangleright$
Increase the total supply of the instance.
$transfer(t.amount,t.receiver)$; $\triangleright$ Credit tokens to the
recipient.
$t.isCompleted\leftarrow True$;
Output$("receiveCommited",transferID)$;
Algorithm 4 $prog^{\mathbb{IOMC}^{R}}$ of receiving $\mathbb{IOMC}^{R}$ Figure
3. Overview of the protocol $\Pi^{T}$, consisting of 4 phases.
### 4.3. Interoperable Transfer Protocol $\mathbf{\Pi^{T}}$
In this section we outline our instance-to-instance interoperable transfer
protocol $\Pi^{T}$ for inter-CBDC transfer operation, which is inspired by the
atomic swap protocol (see Section 2.4), but in contrast to the exchange-
oriented approach of atomic swap, $\Pi^{T}$ focuses only on one-way atomic
transfer between instances of the custodial environment of CBDC, where four
parties are involved in each transfer – a sending $\mathbb{C}_{A}$ and
$\mathbb{O}_{A}$ versus a receiving $\mathbb{C}_{B}$ and $\mathbb{O}_{B}$. The
goal of $\Pi^{T}$ is to eliminate any dishonest behavior by $\mathbb{C}s$ or
$\mathbb{O}$s that would incur token duplication or the loss of tokens.
Figure 4. The details of the proposed interoperability protocol $\Pi^{T}$.
To execute $\Pi^{T}$ it is necessary to inter-connect $\mathbb{E}$s of two
instances involved in a transfer. However, $\mathbb{E}$ does not allow direct
communication with the outside world, and therefore it is necessary to use an
intermediary. One solution is to involve $\mathbb{O}$s but they might be
overwhelmed with other activities, updating the ledger by executing
$\mu$-transactions, and moreover, they might not have direct incentives to
execute inter-CBDC transfers. Therefore, we argue that in contrast to the
above option, involving $\mathbb{C}s$ as intermediaries has two advantages:
(1) elimination of the synchronous communication overhead on $\mathbb{O}$s and
(2) enabling $\mathbb{C}$s to have a transparent view about the status of the
transfer and take action if required. In the following, we describe phases of
$\Pi^{T}$ in detail (see also Figure 4).
### Phase 1 – Client $\mathbb{C}_{A}$ Initiates the Protocol
The client $\mathbb{C}_{A}$ creates a $\mu$-tx1 with the amount being sent,
which invokes the sendInit() of $\mathbb{IOMC}_{A}$ with arguments containing
the address of the external client $\mathbb{C}_{B}$, the address of
$\mathbb{IPSC}_{B}$ (denoted as $\mathbb{S}_{B}$ in Figure 4 for brevity), and
the hash of the secret that is created by $\mathbb{C}_{A}$. $\mathbb{C}_{A}$
sends signed $\mu$-tx1 to $\mathbb{O}_{A}$ who forwards it to the
$\mathbb{E}_{A}$. Before executing the $\mu$-tx1, $\mathbb{E}_{A}$ ensures
that the external recipient (i.e., $\mathbb{C}_{B}$) has the access ticket
already issued and valid, enabling her to post censorship resolution requests
to $\mathbb{IPSC}_{A}$ (if needed). The access ticket should be valid for at
least the entire period defined by the HTLC of $\mathbb{IOMC}_{A}$. In the
next step, a $\mu$-tx1 is executed by $\mathbb{E}_{A}$, creating a new
transfer record with $transferId$ in $\mathbb{IOMC}_{A}$. During the
execution, $\mathbb{C}_{A}$’s tokens are transferred (and thus locked) to the
$\mathbb{IOMC}_{A}$’s address. $\mathbb{C}_{A}$ waits until the new version of
$L_{A}$ is snapshotted to $\mathbb{IPSC}_{A}$, and then obtains $LRoot^{A}$
from it. Then $\mathbb{C}_{A}$ asks $\mathbb{O}_{A}$ for the execution receipt
$rcp_{1}$ of $\mu$-tx1 that also contains a set of proofs ($\pi_{hdr}^{mem}$,
$\pi_{rcp_{1}}^{mk}$) and the header of the $\mu$-block that includes
$\mu$-tx1. In detail, $\pi_{hdr}^{mem}$ is the inclusion proof of the
$\mu$-block b in the current version of $L_{A}$; $\pi_{rcp_{1}}^{mk}$ is the
Merkle proof proving that $rcp_{1}$ is included in b (while $rcp_{1}$ proves
that $\mu$-tx1 was executed correctly). The mentioned proofs and the receipt
are provided to $\mathbb{C}_{B}$, who verifies that $\mu$-tx1 was executed and
included in the $L_{A}$’s version that is already snapshotted to
$\mathbb{IPSC}_{A}$, thus irreversible (see below).
### Phase 2 – $\mathbb{C}_{B}$ Initiates Receive
First, $\mathbb{C}_{B}$ validates an access ticket to $\mathbb{IPSC}_{A}$
using the enclave $\mathbb{E}_{A}$’s public key accessible in that smart
contract. Next, $\mathbb{C}_{B}$ obtains the root hash $LRoot_{pb}^{A}$ of
$L_{A}$ to ensure that $\mathbb{C}_{B}$’s received state has been already
published in $\mathbb{IPSC}_{A}$, and thus contains $\mu$-tx1. After obtaining
$LRoot_{pb}^{A}$, $\mathbb{C}_{B}$ forwards it along with the root $LRoot^{A}$
obtained from $\mathbb{C}_{A}$ to $\mathbb{O}_{A}$, who creates an incremental
proof $\pi^{inc}$ of $\langle LRoot^{A},LRoot_{pb}^{A}\rangle$. Once the proof
$\pi^{inc}$ has been obtained and validated, the protocol can proceed to
validate the remaining proofs sent by the client $\mathbb{C}_{A}$ along with
verifying that the receiving address belongs to $\mathbb{C}_{B}$. Next,
$\mathbb{C}_{B}$ creates $\mu$-$tx_{2}$, invoking the method receiveInit()
with the arguments: the address of $\mathbb{C}_{A}$ obtained from
$\mu$-$tx_{1}$,777Note that we assume that the address is extractable from the
signature. the address $\mathbb{IPSC}_{A}.addr$ of $\mathbb{C}_{A}$’s
instance, the hash value of the secret, and the amount of crypto-tokens being
sent. $\mathbb{C}_{B}$ sends $\mu$-$tx_{2}$ to $\mathbb{O}_{B}$, who forwards
it to $\mathbb{E}_{B}$ for processing. During processing of $\mu$-$tx_{2}$,
$\mathbb{E}_{B}$ determines whether the external client (from its point of
view – i.e., $\mathbb{C}_{A}$) has an access ticket issued with a sufficiently
long validity period; if not, one is created. Subsequently, $\mathbb{E}_{B}$
creates a new record in $\mathbb{IOMC}_{B}$ with $extTransferId$. Afterward,
$\mathbb{C}_{B}$ retrieves the $LRoot^{B}$ from $L_{B}$ and requests the
execution receipt $rcp_{2}$ from $\mathbb{O}_{B}$, acknowledging that the
$\mu$-$tx_{2}$ has been executed. Finally, $\mathbb{C}_{B}$ sends a message
$\mathbb{C}_{A}$ with $\mu$-$tx_{2}$ and cryptographic proofs
$\pi_{hdr}^{mem}$, $\pi_{rcp_{2}}^{mk}$, the execution receipt of
$\mu$-$tx_{2}$, the block header $b$ in which the $\mu$-$tx_{2}$ was included,
$LRoot^{B}$ (i.e., the root value of $L_{B}$ after $\mu$-$tx_{2}$ was
executed), and the valid client access ticket for $\mathbb{C}_{A}$.
### Phase 3 – Confirmation of Transfer by $\mathbb{C}_{A}$
First, $\mathbb{C}_{A}$ validates the received access ticket to
$\mathbb{IPSC}_{B}$. Next, $\mathbb{C}_{A}$ obtains the snapshotted root hash
$LRoot_{pb}^{B}$ of $L_{B}$ from $\mathbb{IPSC}_{B}$. As in the previous
phases, it is necessary to verify that the version of $L_{B}$ that includes
$\mu$-$tx_{2}$ is represented by $LRoot_{pb}^{B}$ (thus is irreversible).
Next, both root hashes ($LRoot^{B}$ and $LRoot_{pb}^{B}$) are sent to the
external operator $\mathbb{O}_{B}$, which produces the incremental proof
$\pi^{inc}$ from them. Next, $\mathbb{C}_{A}$ creates $\mu$-$tx_{3}$ that
consists of invoking the sendCommit() method at $\mathbb{E}_{A}$ with the
arguments containing the published secret (i.e., $preimage$) and the record
identifier of the transfer at local instance (i.e., $transferId$) as well as
the external one (i.e., $extTransferId$). Along with the invocation of
sendCommit(), $\mu$-$tx_{3}$ also wraps $\pi^{inc}$ with its versions
($LRoot_{pb}^{B}$ and $LRoot^{B}$), $\mu$-$tx_{2}$, its execution receipt
$rcp_{2}$ with its Merkle proof $\pi_{rcp_{2}}^{mk}$, $b.hdr$ – the header of
the block that included $\mu$-$tx_{2}$, and its membership proof
$\pi^{mem}_{hdr}$ of $L_{B}$. Next, $\mathbb{C}_{A}$ sends $\mu$-$tx_{3}$ to
$\mathbb{E}_{A}$ through $\mathbb{O}_{A}$. During the execution of
$\mu$-$tx_{3}$, $\mathbb{E}_{A}$ validates the provided proofs and the
equality of transfer IDs from both sides of the protocol. Note that to verify
$\pi^{mem}_{hdr}$, $\mathbb{E}_{A}$ uses its light client to $L_{B}$.
$\mathbb{E}_{A}$ then validates whether $\mathbb{C}_{A}$’s provided secret
corresponds to the hashlock recorded in the 1st phase of the protocol, and if
so, it burns the sent balance of the transfer.
Next, $\mathbb{C}_{A}$ waits until the new version of $L_{A}$ is snapshotted
to $\mathbb{IPSC}_{A}$, and then obtains $LRoot^{A}$ from it. Then
$\mathbb{C}_{A}$ asks $\mathbb{O}_{A}$ for the execution receipt $rcp_{3}$ of
$\mu$-tx3 that also contains a set of proofs ($\pi_{hdr}^{mem}$,
$\pi_{rcp_{3}}^{mk}$) and the header of the $\mu$-block that includes
$\mu$-tx3. The proofs have the same interpretation as in the end of the 1st
phase. The mentioned proofs and the receipt are provided to $\mathbb{C}_{B}$,
who verifies that $\mu$-tx1 was executed and included in the $L_{A}$’s version
that is already snapshotted to $\mathbb{IPSC}_{A}$, thus irreversible.
### Phase 4 – Acceptance of Tokens by $\mathbb{C}_{B}$
After receiving a message from client $\mathbb{C}_{A}$, the client
$\mathbb{C}_{B}$ obtains $LRoot_{pb}^{A}$ from $\mathbb{IPSC}_{A}$ and then
requests the incremental proof between versions $\langle
LRoot^{A},LRoot_{pb}^{A}\rangle$ from $\mathbb{O}_{A}$. Then, $\mathbb{C}_{B}$
creates $\mu$-$tx_{4}$ invoking the receiveClaim() function at
$\mathbb{E}_{B}$ with $transferId$ and the disclosed secret by
$\mathbb{C}_{A}$ as the arguments. Moreover, $\mu$-$tx_{4}$ contains remaining
items received from $\mathbb{C}_{A}$. Then, $\mu$-$tx_{4}$ is sent to
$\mathbb{O}_{B}$, who forwards it to $\mathbb{E}_{B}$. During the execution of
$\mu$-$tx_{4}$, $\mathbb{E}_{B}$ verifies the provided proofs, the equality of
transfer IDs from both sides of the protocol, the amount being sent, and the
receiver of the transfer (i.e., $\mathbb{C}_{B}$ —— $\mathbb{IPSC}_{B}$). Note
that to verify $\pi^{mem}_{hdr}$, $\mathbb{E}_{B}$ uses its light client to
$L_{A}$. $\mathbb{E}_{A}$ then validates whether $\mathbb{C}_{A}$’s provided
secret corresponds to the hashlock recorded in the 2nd phase of the protocol,
and if so, it mints the sent balance of the transfer on the receiver’s account
$\mathbb{C}_{B}$. Finally, $\mathbb{C}_{B}$ verifies that $\mu$-$tx_{4}$ was
snapshotted at $\mathbb{IPSC}_{B}$, thus is irreversible. In detail, first
$\mathbb{C}_{B}$ obtains $LRoot_{pb}^{B}$ from $\mathbb{IPSC}_{B}$ and then
asks $\mathbb{O}_{B}$ to provide her with the execution receipt $rcp_{4}$ of
$\mu$-$tx_{4}$ in the version of $L_{B}$ that is equal or newer than
$LRoot_{pb}^{B}$. Then, $\mathbb{C}_{B}$ verifies $rcp_{4}$, which completes
the protocol.
## 5\. Implementation & Evaluation
The work is built on a proof-of-concept implementation of the decentralized
smart contract platform Aquareum in C++ and Intel SGX technology for enclave
instantiation. The IPSC contract on the public blockchain is constructed using
the Solidity programming language and is prepared for deployment on the
Ethereum network. The enclave employs the OpenEnclave SDK development
tool,888https://openenclave.io/sdk/, which is compatible with several TEE
technologies and OS systems. Aquareum incorporates the Ethereum virtual
machine – EVM, in its stripped-down, minimalist version eEVM.999Microsoft’s
Enclave EVM is available at https://github.com/microsoft/eEVM.
#### 5.0.1. Implementation Details
The C++ written client application enables the clients to execute internal and
external (i.e., between two instance) transfer operations as well as invoking
internal and external functions of micro contracts. The operator component is
represented by the C++ written server implemented as a concurrent non-blocking
application that processes messages from clients. So far, the PoC of the
server enables to process three types of messages: transaction execution,
client registration, query for IOMC contract addresses.
### 5.1. Evaluation
We used Ganache101010https://github.com/trufflesuite/ganache-cli and
Truffle,111111https://github.com/trufflesuite/truffle to develop
$\mathbb{IOMC}$, $\mathbb{IPSC}$, and $\mathbb{IMSC}$ contracts. In addition,
using the Pexpect121212https://github.com/pexpect/pexpect tool, we tested the
intercommunication of the implemented components and validated the correctness
of the implemented interoperability protocol. The tool enabled the parallel
execution and control of numerous programs (in this case, multiple Aquareum
instances and client programs) to check the correctness of the expected
output.
The computational cost of executing the operations defined in $\mathbb{IOMC}$
and $\mathbb{IMSC}^{X}$ contracts is presented in Table 1, Table 2, and Table
3.131313Note that we do not provide the gas measurements for $\mathbb{IPSC}$
since these are almost the same as in Aquareum (Homoliak and Szalachowski,
2020). We optimized our implementation to minimize the storage requirements of
smart contract platform. On the other hand, it is important to highlight that
$\mathbb{IOMC}^{X}$ $\mu$-contracts are executed on a private ledger
corresponding to the instance of CBDC, where the cost of gas is minimal or
negligible as compared to a public blockchain. Other experiments are the
subject of our future work.
Function | constructor | sendInitialize | sendCommit | sendRevert
---|---|---|---|---
Deployment | 901 509 | 160 698 | 64 629 | 60 923
Execution | 653 689 | 134 498 | 42 717 | 39 523
Table 1. The cost of deployment and invocation of functions in the sending $\mathbb{IOMC}^{S}$ $\mu$-contract in gas units (CBDC private ledger). Function | constructor | receiveInit | receiveClaim | fund
---|---|---|---|---
Deployment | 716 330 | 139 218 | 61 245 | 23 168
Execution | 509 366 | 112 762 | 39 653 | 1 896
Table 2. The cost of deployment and invocation of functions in the receiving $\mathbb{IOMC}^{R}$ $\mu$-contract in units of gas (CBDC private ledger). Function | constructor | newJoinRequest | approveRequest | isApproved
---|---|---|---|---
Deployment | 830 074 | 48 629 | 69 642 | 0
Execution | 567 838 | 25 949 | 45 554 | 0
Table 3. The invocation cost of functions in $\mathbb{IMSC}$ smart contract in
units of gas (Ethereum public blockchain).
## 6\. Security Analysis
In this section, we analyze our approach in terms of security-oriented
features and requirements specified in Section 3. In particular, we focus on
resilience analysis of our approach against adversarial actions that the
malicious CBDC instance (i.e., its operator $\mathcal{O}$) or malicious client
(i.e., $\mathcal{C}$) can perform to violate the security requirements.
### 6.1. Single Instance of CBDC
###### Theorem 1.
(Correctness of Operation Execution) $\mathcal{O}$ is unable to modify the
full state of $L$ in a way that does not respect the semantics of VM deployed
in $\mathbb{E}$ of CBDC instance.
###### Justification.
The update of the $L$’s state is performed exclusively in $\mathbb{E}$. Since
$\mathbb{E}$ contains trusted code that is publicly known and remotely
attested by $\mathbb{C}$s, $\mathcal{O}$ cannot tamper with this code. ∎
###### Theorem 2.
(Integrity) $\mathcal{O}$ is unable to modify the past records of $L$, and no
conflicting transactions can be stored in $L$.
###### Justification.
All extensions of $L$ are performed within trusted code of $\mathbb{E}$ (see
Theorem 1), while utilizing the history tree (Crosby and Wallach, 2009) as a
tamper evident data structure, which enables us to make only such incremental
extensions of $L$ that are consistent with $L$’s past. ∎
###### Theorem 3.
(Verifiability) $\mathcal{O}$ is unable to unnoticeably modify or delete a
transaction $tx$ that was previously inserted to $L$, if sync with
$\mathbb{IPSC}$ was executed anytime afterward.
###### Justification.
Since $tx$ was correctly executed (Theorem 1) as a part of the block $b_{i}$
in a trusted code of $\mathbb{E}$, $\mathbb{E}$ produced a signed version
transition pair $\\{h(L_{i-1}),h(L_{i}),t_{i},t_{s}\\}_{\mathbb{E}}$ of $L$
from the version $i-1$ to the new version $i$ that corresponds to $L$ with
$b_{i}$ included. $\mathcal{O}$ could either sync $L$ with $\mathbb{IPSC}$
immediately after $b_{i}$ was appended or she could do it $n$ versions later.
In the first case, $\mathcal{O}$ published
$\\{h(L_{i-1}),h(L_{i}),t_{i},t_{s}\\}_{\mathbb{E}}$ to $\mathbb{IPSC}$, which
updated its current version of $L$ to $i$ by storing $h(L_{i})$ into
$\mathbb{IPSC}.LRoot_{pb}$. In the second case, $n$ blocks were appended to
$L$, obtaining its $(i+n)$th version. $\mathbb{E}$ executed all transactions
from versions $(i+1),\ldots,(i+n)$ of $L$, while preserving correctness
(Theorem 1) and integrity (Theorem 2). Then $\mathbb{E}$ generated a version
transition pair $\\{h(L_{i-1}),h(L_{i+n}),t_{i},t_{s}\\}_{\mathbb{E}}$ and
$\mathcal{O}$ posted it to $\mathbb{IPSC}$, where the current version of $L$
was updated to $i+n$ by storing $h(L_{i+n})$ into $\mathbb{IPSC}.LRoot_{pb}$.
When any $\mathbb{C}$ requests $tx$ and its proofs from $\mathcal{O}$ with
regard to publicly visible $\mathbb{IPSC}.LRoot_{pb}$, she might obtain a
modified $tx^{\prime}$ with a valid membership proof $\pi^{mem}_{hdr_{i}}$ of
the block $b_{i}$ but an invalid Merkle proof $\pi^{mk}_{tx^{\prime}}$, which
cannot be forged. $\Box$ In the case of $tx$ deletion, $\mathcal{O}$ provides
$\mathbb{C}$ with the tampered full block $b_{i}^{\prime}$ (maliciously
excluding $tx$) whose membership proof $\pi^{mem}_{hdr_{i}^{\prime}}$ is
invalid – it cannot be forged. ∎
###### Theorem 4.
(Non-Equivocation) Assuming $L$ synced with $\mathbb{IPSC}$: $\mathcal{O}$ is
unable to provide two distinct $\mathbb{C}$s with two distinct valid views on
$L$.
###### Justification.
Since $L$ is periodically synced with publicly accessible $\mathbb{IPSC}$, and
$\mathbb{IPSC}$ stores only a single current version of $L$ (i.e.,
$\mathbb{IPSC}.LRoot_{pb}$), all $\mathbb{C}s$ share the same view on $L$. ∎
###### Theorem 5.
(Censorship Evidence) $\mathcal{O}$ is unable to censor any request
(transaction or query) from $\mathbb{C}$ while staying unnoticeable.
###### Justification.
If $\mathbb{C}$’s request is censored by CBDC’s operator $\mathcal{O}$,
$\mathbb{C}$ can ask for a resolution of the request through public
$\mathbb{IPSC}$. $\mathcal{O}$ observing the request might either ignore it
and leave the indisputable proof of censorship at $\mathbb{IPSC}$ or she might
submit the request to $\mathbb{E}$ and obtain an enclave-signed proof
witnessing that a request was processed (hence have not remained censored) –
this proof is submitted to $\mathbb{IPSC}$, whereby publicly resolving the
request. ∎
###### Theorem 6.
(Privacy) $\mathcal{C}$ is unable to obtain plain text of $\mu$-transactions
of other $\mathbb{C}$s even during the censorship resolution.
###### Justification.
$\mu$-transactions are sent to $\mathbb{O}$ in TLS-encrypted messages. In the
case of censorship resolution, submitted $\mu$-transactions by $\mathbb{C}$ to
public $\mathbb{IPSC}$ are encrypted by $\mathbb{E}$’s public key
$PK_{\mathbb{E}}^{tee}$. ∎
###### Theorem 7.
(Transparent Token Issuance) $\mathcal{O}$ is unable to issue or burn any
tokens without leaving a publicly visible evidence.
###### Justification.
All issued tokens of CBDC are publicly visible at $\mathbb{IPSC}$ since each
transaction posting a new version transition pair also contains
$\mathbb{E}$-signed information about the current total issued tokens $t_{i}$
and total supply of the instance $t_{s}$,141414Note that in the case of single
CBDC instance $t_{i}=t_{s}$ while $t_{i}$ was updated within the trusted code
of $\mathbb{E}$. The information about $t_{i}$ is updated at $\mathbb{IPSC}$
along with the new version of $L$. Note that the history of changes in total
issued tokens $t_{i}$ can be parsed from all transactions updating version of
$L$ published by $\mathcal{O}$ to $PB$. ∎
### 6.2. Multiple Instances of CBDC
In the following, we assume two CBDC instances A and B.
###### Theorem 8.
(Atomic Interoperability I) Neither $\mathcal{O}_{A}$ (operating $A$) nor
$\mathcal{O}_{B}$ (operating $B$) is unable to steal any tokens during the
inter-bank CBDC transfer.
###### Justification.
Atomic interoperability is ensured in our approach by adaptation of atomic
swap protocol for all inter-bank transfers, which enables us to preserve the
wholesale environment of CBDC in a consistent state (respecting Equation 2).
In detail, the transferred tokens from CBDC instance $A$ to instance $B$ are
not credited to $B$ until $A$ does not provide the indisputable proof that
tokens were deducted from a relevant $A$’s account. This proof confirms
irreversible inclusion of $tx_{3}$ (i.e., $\mathbb{E}_{A}.sendCommit()$ that
deducts account of $A$’s client) in $A$’s ledger and it is verified in 4th
stage of our protocol by the trusted code of $\mathbb{E}_{B}$.
In the case that $\mathcal{O}_{A}$ would like to present $B$ with integrity
snapshot of $L_{A}$ that was not synced to $\mathbb{IPSC}_{A}$ yet, B will not
accept it since the 4th phase of our protocol requires $\mathcal{O}_{B}$ to
fetch the recent $\mathbb{IPSC}_{A}.LRoot_{pb}$ and verify its consistency
with A-provided $LRoot$ as well as inclusion proof in $PB$; all
executed/verified within trusted code of $\mathbb{E}_{B}$. ∎
###### Theorem 9.
(Atomic Interoperability II) Colluding clients $\mathcal{C}_{A}$ and
$\mathcal{C}_{B}$ of two CBDC instances cannot steal any tokens form the
system during the transfer operation of our protocol.
###### Justification.
If the first two phases of our protocol have been executed, $\mathcal{C}_{A}$
might potentially reveal the $preimage$ to $\mathcal{C}_{B}$ without running
the 3rd phase with the intention to credit the tokens at $B$ while deduction
at $A$ had not been executed yet. However, this is prevented since the trusted
code of $\mathbb{E}_{B}$ verifies that the deduction was performed at $A$
before crediting the tokens to $\mathcal{C}_{B}$ – as described in Theorem 8.
∎
###### Theorem 10.
(Inter-CBDC Censorship Evidence) $\mathcal{O}_{A}$ is unable to unnoticeably
censor any request (transaction or query) from $\mathbb{C}_{B}$.
###### Justification.
If $\mathbb{C}_{B}$’s request is censored by $\mathcal{O}_{A}$,
$\mathbb{C}_{B}$ can ask for a resolution of the request through public
$\mathbb{IPSC}_{A}$ since $\mathbb{C}_{B}$ already has the access ticket to
instance $A$. The access ticket is signed by $\mathbb{E}_{A}$ and thus can be
verified at $\mathbb{IPSC}_{A}$. Hence, the censorship resolution/evidence is
the same as in Theorem 5 of a single CBDC instance. ∎
###### Theorem 11.
(Inter-CBDC Censorship Recovery) A permanent inter-CBDC censorship by
$\mathcal{O}_{A}$ does not cause an inconsistent state or permanently frozen
funds of undergoing transfer operations at any other CBDC instance – all
initiated and not finished transfer operations can be recovered from.
###### Justification.
If $\mathcal{O}_{A}$ were to censor $\mathbb{C}_{B}$ in the 2nd phase of our
protocol, no changes at ledger $L_{B}$ would be made. If $\mathcal{O}_{A}$
were to censor $\mathbb{C}_{B}$ in the 4th phase of our protocol, $L_{B}$
would contain an initiated transfer entry, which has not any impact on the
consistency of the ledger since it does not contain any locked tokens. $\Box$
If $\mathcal{O}_{B}$ were to censor $\mathbb{C}_{A}$ in the 3rd phase of our
protocol, $A$ would contain some frozen funds of the initiated transfer.
However, these funds can be recovered back to $\mathbb{C}_{A}$ upon a recovery
call of $\mathbb{E}_{A}$ after a recovery timeout has passed. Note that after
tokens of $\mathbb{C}_{A}$ have been recovered and synced to
$\mathbb{IPSC}_{A}$ in $PB$, it is not possible to finish the 4th stage of our
protocol since it requires providing the proof that tokens were deducted at
$A$ and such a proof cannot be constructed anymore. The same holds in the
situation where the sync to $\mathbb{IPSC}_{A}$ at $PB$ has not been made yet
– after recovery of tokens, $\mathbb{E}_{A}$ does not allow to deduct the same
tokens due to its correct execution (see Theorem 1). ∎
###### Theorem 12.
(Identity Management of CBDC Instances I) A new (potentially fake) CBDC
instance cannot enter the ecosystem of wholesale CBDC upon its decision.
###### Justification.
To extend the list of valid CBDC instances (stored in IMSC contract), the
majority vote of all existing CBDC instances must be achieved through public
voting on IMSC. ∎
###### Theorem 13.
(Identity Management of CBDC Instances II) Any CBDC instance (that e.g., does
not respect certain rules for issuance of tokens) might be removed from the
ecosystem of CBDC by majority vote.
###### Justification.
A publicly visible voting about removal of a CBDC instance from the ecosystem
is realized by IMSC contract that resides in $PB$, while each existing
instance has a single vote. ∎
### 6.3. Security of TEE
We assume that its TEE platform employed is secure. However, previous research
indicated that this might not be the case in practical implementations of TEE,
such as SGX that was vulnerable to memory corruption attacks (Biondo et al.,
2018) as well as side channel attacks (Brasser et al., 2017; Van Bulck et al.,
2018; Lipp et al., 2021; Murdock et al., 2020). A number of software-based
defense and mitigation techniques have been proposed (Shih et al., 2017; Gruss
et al., 2017; Chen et al., 2017; Brasser et al., 2017; Seo et al., 2017) and
some vulnerabilities were patched by Intel at the hardware level (Intel,
2018). Nevertheless, we note that our approach is TEE-agnostic thus can be
integrated with other TEEs such as ARM TrustZone or RISC-V architectures
(using Keystone-enclave (Enclave, 2019) or Sanctum (Costan et al., 2016)).
Another class of SGX vulnerabilities was presented by Cloosters et al.
(Cloosters et al., 2020) and involved incorrect application designs enabling
arbitrary reads and writes of protected memory. Since the authors did not
provide public with their tool (and moreover it does not support Open-enclave
SDK), we did manual inspection of our code and did not find any of the
concerned vulnerabilities. Another work was done by Borrello et al. (Borrello
et al., 2022) and involves more serious micro-architectural flaws in chip
design. Intel has already released microcode and SGX SDK updates to fix the
issue.
### 6.4. Public Blockchain & Finality
Many blockchain platforms suffer from accidental forks (i.e., availability-
favored blockchains in terms of CAP theorem), which temporarily create
parallel inconsistent blockchain views. To mitigate this phenomenon, it is
recommended to wait a certain number of block confirmations after a given
block is created before considering it irreversible with overwhelming
probability. This waiting time (a.k.a., time to finality) influences the non-
equivocation property of our approach, inheriting it from the underlying
blockchain platform. Most availability-favored blockchains have a long time to
finality, e.g., $\sim$3mins in Bitcoin (Nakamoto, 2008), $\sim$3mins in
Ethereum (wood2014ethereum), $\sim$2mins in Cardano (Kiayias et al., 2017).
However, consistency-favored blockchains in terms of the CAP theorem have a
short time to finality, e.g., HoneyBadgerBFT (Miller et al., 2016), Algorand
(Gilad et al., 2017), Hyperledger Besu (Hyperledger Foundation, 2022). The
selection of the underlying blockchain platform should respect low time to
finality in the critical environment of CBDC, and thus employ a consistency-
favored public blockchain.
## 7\. Related Work
In this section, we first review various approaches to interoperability and
CBDC. Moreover, since our protocol is designed using a combination of TEE and
the blockchain, we revise the most relevant solutions and stress the novelty
of our approach, which combines several unique features.
### 7.1. Blockchain Interoperability
Cross-chain interoperability is one of the most desirable yet challenging
features to be designed and developed in blockchains, affecting the impact and
usability of the solution (Wang et al., 2023; Belchior et al., 2021; Mohanty
et al., 2022; Qasse et al., 2019).
Cross-chain communication protocols define the process of synchronization
between different chains of the same blockchain, e.g., by the use of
sidechains. Additionally, cross-blockchain communication protocols, such as
Interledger Protocol (Siris et al., 2019), allow interaction of different
blockchains. While the cross-chain solutions can be employed by the native
constructs such as atomic swap, the cross-blockchain protocols require
adoption of the solution. Blockchain interoperability solutions can be
categorized into three groups according to the principle they are based on and
the type of chains that are supported (Belchior et al., 2021).
##### Public connectors.
Public connectors are a set of approaches that focuses on cryptocurrency
systems and their transactions. This includes the sidechains, relays, notary
schemes, and hash time locks (Musungate et al., 2019; Mohanty et al., 2022;
Dcunha et al., 2021).
##### Blockchain of blockchains.
Blockchain of blockchains focuses on application specific-solutions. The
example is Polkadot (Belchior et al., 2021; Burdges et al., 2020) – a network
for cross-blockchain interoperability. In Polkadot network, multiple
parallelized globally-coherent chains (parachains) are connected via bridges
that represent a specific type of parachain. Bridges also serve as a gateway
for communication with external networks, such as Bitcoin.
##### Hybrid Connectors
Hybrid solutions create an abstraction layer over the blockchain ecosystem and
provide a unified API for interaction between blockchain and applications
(Madine et al., 2021). Examples are trusted relays or blockchain migrators.
The interoperability requires validators present in both the source and target
blockchains. The validators collect cross-chain transactions and ensure that
they are delivered (Wang and Nixon, 2021).
The proposed solution in this paper contain a custom one-way atomic swap
protocol that utilizes hash time lock contracts. Such swaps are settled on
public blockchain $PB$. It is also expected that $PB$ used for the
synchronization of clients and CBDC instances deploys a single blockchain
technology. The usability of the proposed solution targets the financial
institutions such as banks, leveraging its potential use in CBDC projects. The
protocol does not specify a middleware layer providing API or the use of
gateway chains. Therefore, it can be categorized as a public connector that
augments and combines the features provided by individual solutions in the
same category.
### 7.2. CBDC Projects
While most CBDC projects are still in their early stages, some well-known
proposals are reaching maturity level (Zhang and Huang, 2021). For instance,
Project Jasper (Chapman et al., 2017) was one of the initial prototypes for
inter-bank payments using blockchain technology. Project Ubin (Monetary
Authority of Singapore, 2016) appeared with the aim of clearing and settling
of payments and securities efficiently by using several blockchain
technologies and smart contracts. Project E-krona (Armelius et al., 2020) was
designed, among others, to enable fast transactions between domestic and
cross-border entities. Stella (Kishi, 2019) is another well-known project that
uses permissioned blockchain technology to enable cross border operations as
well as confidentiality protection. The mBridge project (BIS innovation hub,
2021) (initially named Inthanon-LionRock) prototype is built by ConsenSys on
Hyperledger Besu. The prototype encompasses several jurisdictions and aims at
creating a cross-border payment infrastructure that improves on key pain
points, including high cost, low speed, and operational complexities. Finally,
Project Khokha (South African Reserve Bank, 2018) was designed for efficient,
confidential inter-bank transactions.
Despite the maturity of some projects, research on CBDC technology is still in
its infancy. In addition, the road to creating a native interoperable protocol
that can be used regardless of the underlying blockchain technology still
requires further exploration and is one of the main objectives of this
article. Compared to other CBDC projects, our approach is the first protocol
combining TEE and blockchain to bring interesting security and privacy
features, accompanied by external interoperability. In detail, our protocol
guarantees a set of features such as integrity, non-equivocation (i.e., we
provide snapshots to public blockchain to avoid reverts and forks of the local
CBDC ledgers), correctness (i.e., the EVM is executed in an enclave which can
be remotely attested), and censorship evidence. Since the designed protocol
addresses inter-bank communication and payment settlements, it can be
potentially integrated as a part of the above-mentioned wholesale CBDC
projects. The advantages of the retail CBDC approach towards individual
clients are also preserved by the privacy support, censorship evidence and
mitigation of malicious approach described in Section 6. The general approach
is also invariant towards token differences introduced by different projects
with regard to the public blockchain.
### 7.3. Combining Blockchain and TEE
The combination of Trusted Execution Environment (TEE) technologies and
blockchain has gained increased attention in the past few years. Hybridchain
(Wang et al., 2020) is an architecture for confidentiality-preserving in
permissioned blockchain. Such architecture extends the enclave memory of TEE
that allows blockchain applications running in TEE to securely store
transaction records outside of TEE. Ekiden (Cheng et al., 2018) is a
blockchain-agnostic solution that offloads smart contract execution to TEE
enclaves. Teechain (Lind et al., 2017) focuses on the Bitcoin network and
enables the secure execution of transactions in TEE, enhancing the scalability
of the network. Fastkitten enables extended functionality in the Bitcoin
network by using Turing-complete smart contracts executed via TEE-enabled
operators (Das et al., 2019).
However, solutions combining interoperability with TEE-based blockchains are
still in their infancy. Only a few authors have explored this such as
Bellavista et al. (Bellavista et al., 2021), and Lan et al. (Lan et al.,
2021), which are the works most similar to ours. More concretely, Bellavista
et al. (Bellavista et al., 2021) explore the use of a relay scheme based on
TEE to provide blockchain interoperability in the context of collaborative
manufacturing and supply chains. Lan et al. (Lan et al., 2021) aim to preserve
confidentiality in interoperable cross-chain platforms and propose a protocol
to ensure privacy-preserving communications among them. Nevertheless, our
approach is the first one designing a functional protocol for interoperable
CBDC, considering features such as the ones mentioned in Section 3.
## 8\. Discussion
As seen in Section 7, this is the first blockchain TEE-based interoperable
protocol that operates in the context of CBDC. However, our protocol allows
modifications if additional requirements were to be fulfilled (i.e.,
considering the ones defined in Section 3). The latter enables a certain
degree of dynamism when adapting the protocol to specific application
contexts.
Following the interest of countries in CBDC (Atlantic Council, [n. d.]),
research on CBDCs and their potential challenges has also been receiving
increasing attention in the last years (i.e., the number of contributions has
been doubling yearly since 2020 according to Scopus, using the query TITLE-
ABS-KEY ( ( ( central AND bank AND digital AND currency ) OR CBDC ) AND
challenges ). While a profound analysis of state of the art is out of the
scope of this paper, we found that authors typically follow two strategies to
discuss CBDC and its challenges, namely considering a local perspective (i.e.,
at a jurisdiction or national level) and adopting a global challenge
abstraction. Overall, we considered the most recent reviews and surveys
analyzing CBDC and its challenges (Catalini et al., 2022; Alwago, 2022;
Koziuk, 2021; Sebastião et al., 2021) and other grey literature, such as the
Digital Euro Association (Digital Euro Association, [n. d.]), or the US
federal reserve (US Federal Reserve, [n. d.]), to extract the challenges and
represent them according to a high-level hierarchical abstraction. Since one
of the aims of our proposal is to provide solutions to as many challenges as
possible, we describe, for each challenge, the benefits and features that our
proposal provides in Table 4.
Topic | Main concerns | Our proposal’s contribution
---|---|---
Technology | The design, implementation and maintenance of CBDC’s as well as their scalability, resiliency and compatibility with the current financial structure. | Our system is scalable and compatible with current financial system
Monetary Policy | Monetary policy transmission, including interest rates, the value of money, or other tools, should not be hindered by CBDC. | Our system relies on smart contracts to enforce specific policies if required, such as token expiration or token usability.
Financial Stability | The potentially disrupting impact of CBDC on the existing financial system could create new financial vulnerabilities, uncontrolled disintermediation or illicit activities. | The use of blockchain and the policies translated into the system should be audited and verified. Our system is compatible with the latter and other policies in the above layers.
Legal Framework | The legal framework for CBDC needs to comply with existing laws and regulations, including consumer protection, anti-money laundering, and countering the financing of terrorism. | The system is compatible with auditability layers compliant with current legal and regulations
Interoperability | Ensuring interoperability by guaranteeing that CBDCs are compatible with other countries monetary policies and promoting cross-border cooperation and standardisation. | Our protocol is interoperable by design and ensures the system remains in monetary equilibrium since no new tokens are created. The potential use of oracles enables further operations with different currencies beyond current ones, promoting cross-border cooperation and additional capabilities.
Security and Privacy | CBDC needs to ensure robustness to prevent cyberattacks and unauthorised access to data by guaranteeing privacy-preserving mechanisms of transactions and personal information. | Our proposal is robust and preserves the privacy of transactions since all the transactions are encrypted. We provide various security properties, such as atomicity, verifiability, integrity, non-equivocation, correctness of execution, censorship evidence, and others.
User Adoption and Inclusion | CBDCs will need to provide access to banking services to different populations. Users will require a behaviour change, acceptance and trust. | The use of our system is transparent to other layers, so it does not introduce any burden. TEE technologies enable trustable platforms, and our protocol allows verifiable censorship resolution.
Table 4. High-level abstraction of CBDC’s challenges and how our proposal
contributes to them. In some cases, our proposal slightly interferes with
these challenges since many only apply to other CBDC ecosystem layers.
## 9\. Conclusion
Although the controversy surrounding the coexistence of privacy and CBDC
(Lagarde, 2022), the latter promises a series of benefits, such as transaction
efficiency (e.g., by reducing costs and decreasing its finality at the
national or international level) and countering financial crime. Moreover,
CBDC complements current financial services by offering broader opportunities.
Nevertheless, the corresponding regulations should carefully manage these new
opportunities, ensuring they do not restrict citizens’ rights. Note that novel
functionalities enforced in financial transactions, such as token expiration
dates, negative interest rates for token holders (i.e., in an attempt to
stimulate the economy in recession periods) or tokens whose validity is tied
to a specific subset of goods (e.g., enforcing that part of the salary is
spent on energy or healthcare), could either be applied for the sustainability
of the society or state control in the context of authoritarian regimes.
Given the above circumstances, we provide the design and implementation of the
protocol that uses a custom adaptation of atomic swap and is executed by any
pair of CBDC instances to realize a one-way transfer, resolving
interoperability over multiple instances of semi-centralized CBDC. Our
protocol guarantees a series of properties such as verifiability, atomicity of
inter-bank transfers, censorship resistance, and privacy. Our contributions
result in a step forward toward enriching the capabilities of CBDC and their
practical deployment.
Future work will closer study token issuance management through protocol
directives, perform more extensive evaluation, and propose interoperable
execution of smart contracts between CBDC instances.
## Acknowledgments
This work was supported by the FIT BUT internal project FIT-S-23-8151. This
work was also supported by the European Commission under the Horizon Europe
Programme, as part of the LAZARUS project (http://lazarus-he.eu/ Grant
Agreement no. 101070303). Fran Casino was supported by the Government of
Catalonia with the Beatriu de Pinós programme (Grant No. 2020 BP 00035), and
by AGAUR with the 2021-SGR-00111 project. The content of this article does not
reflect the official opinion of the European Union. Responsibility for the
information and views expressed therein lies entirely with the authors.
## References
* (1)
* AlShamsi et al. (2022) Mohammed AlShamsi, Mostafa Al-Emran, and Khaled Shaalan. 2022\. A Systematic Review on Blockchain Adoption. _Applied Sciences_ 12, 9 (2022), 4245.
* Alwago (2022) W.O. Alwago. 2022\. Is the Renminbi a Global Currency in the Making? Globalization of Digital yuan. 67, 4 (2022), 553–566.
* Anati et al. (2013) Ittai Anati, Shay Gueron, Simon Johnson, and Vincent Scarlata. 2013\. Innovative technology for CPU based attestation and sealing. In _Proceedings of the 2nd international workshop on hardware and architectural support for security and privacy_ , Vol. 13. ACM New York, NY, USA.
* Armelius et al. (2020) Hanna Armelius, Gabriela Guibourg, Stig Johansson, and Johan Schmalholz. 2020. E-krona design models: pros, cons and trade-offs. _Sveriges Riksbank Economic Review_ 2 (2020), 80–96.
* Atlantic Council ([n. d.]) Atlantic Council. [n. d.]. Central Bank Digital Currency Tracker. https://www.atlanticcouncil.org/cbdctracker/.
* Belchior et al. (2021) Rafael Belchior, André Vasconcelos, Sérgio Guerreiro, and Miguel Correia. 2021. A survey on blockchain interoperability: Past, present, and future trends. _ACM Computing Surveys (CSUR)_ 54, 8 (2021), 1–41.
* Bellavista et al. (2021) Paolo Bellavista, Christian Esposito, Luca Foschini, Carlo Giannelli, Nicola Mazzocca, and Rebecca Montanari. 2021. Interoperable Blockchains for Highly-Integrated Supply Chains in Collaborative Manufacturing. _Sensors_ 21, 15 (2021), 4955.
* Biondo et al. (2018) Andrea Biondo, Mauro Conti, Lucas Davi, Tommaso Frassetto, and Ahmad-Reza Sadeghi. 2018. The Guard’s Dilemma: Efficient Code-Reuse Attacks Against Intel $\\{$SGX$\\}$. In _27th $\\{$USENIX$\\}$ Security Symposium ($\\{$USENIX$\\}$ Security 18)_. 1213–1227.
* BIS innovation hub (2021) BIS innovation hub. 2021\. Inthanon-LionRock to mBridge: Building a multi CBDC platform for international payments. https://www.bis.org/publ/othp40.htm
* Bitcoin Wiki (2018) Bitcoin Wiki. 2018\. Atomic Swap. {https://en.bitcoinwiki.org/wiki/Atomic_Swap}
* Boar and Wehrli (2021) Codruta Boar and Andreas Wehrli. 2021. Ready, steady, go?-Results of the third BIS survey on central bank digital currency. (2021).
* Borrello et al. (2022) Pietro Borrello, Andreas Kogler, Martin Schwarzl, Moritz Lipp, Daniel Gruss, and Michael Schwarz. 2022\. ÆPIC Leak: Architecturally Leaking Uninitialized Data from the Microarchitecture. In _31st USENIX Security Symposium (USENIX Security 22)_.
* Brasser et al. (2017) Ferdinand Brasser, Srdjan Capkun, Alexandra Dmitrienko, Tommaso Frassetto, Kari Kostiainen, Urs Müller, and Ahmad-Reza Sadeghi. 2017\. DR. SGX: hardening SGX enclaves against cache attacks with data location randomization. _arXiv preprint arXiv:1709.09917_ (2017).
* Brown et al. (2016) Richard Gendal Brown, James Carlyle, Ian Grigg, and Mike Hearn. 2016. Corda: an introduction. _R3 CEV, August_ 1, 15 (2016), 14.
* Burdges et al. (2020) Jeff Burdges, Alfonso Cevallos, Peter Czaban, Rob Habermeier, Syed Hosseini, Fabio Lama, Handan Kilinc Alper, Ximin Luo, Fatemeh Shirazi, Alistair Stewart, and Gavin Wood. 2020. Overview of Polkadot and its Design Considerations. https://doi.org/10.48550/ARXIV.2005.13456
* Catalini et al. (2022) C. Catalini, A. De Gortari, and N. Shah. 2022\. Some Simple Economics of Stablecoins. 14 (2022), 117–135.
* Chapman et al. (2017) James Chapman, Rodney Garratt, Scott Hendry, Andrew McCormack, and Wade McMahon. 2017. Project Jasper: Are distributed wholesale payment systems feasible yet. _Financial System_ 59 (2017).
* Chen et al. (2017) Sanchuan Chen, Xiaokuan Zhang, Michael K Reiter, and Yinqian Zhang. 2017. Detecting privileged side-channel attacks in shielded execution with Déjá Vu. In _Proceedings of the 2017 ACM on Asia Conference on Computer and Communications Security_. 7–18.
* Cheng et al. (2018) Raymond Cheng, Fan Zhang, Jernej Kos, Warren He, Nicholas Hynes, Noah Johnson, Ari Juels, Andrew Miller, and Dawn Song. 2018. Ekiden: A platform for confidentiality-preserving, trustworthy, and performant smart contract execution. _arXiv preprint arXiv:1804.05141_ (2018).
* Cloosters et al. (2020) Tobias Cloosters, Michael Rodler, and Lucas Davi. 2020\. TeeRex: Discovery and Exploitation of Memory Corruption Vulnerabilities in $\\{$SGX$\\}$ Enclaves. In _29th $\\{$USENIX$\\}$ Security Symposium ($\\{$USENIX$\\}$ Security 20)_. 841–858.
* Costan et al. (2016) Victor Costan, Ilia Lebedev, and Srinivas Devadas. 2016\. Sanctum: Minimal hardware extensions for strong software isolation. In _25th $\\{$USENIX$\\}$ Security Symposium ($\\{$USENIX$\\}$ Security 16)_. 857–874.
* Crosby and Wallach (2009) Scott A Crosby and Dan S Wallach. 2009. Efficient Data Structures For Tamper-Evident Logging.. In _USENIX Security Symposium_. 317–334.
* Das et al. (2019) Poulami Das, Lisa Eckey, Tommaso Frassetto, David Gens, Kristina Hostáková, Patrick Jauernig, Sebastian Faust, and Ahmad-Reza Sadeghi. 2019. $\\{$FastKitten$\\}$: Practical Smart Contracts on Bitcoin. In _28th USENIX Security Symposium (USENIX Security 19)_. 801–818.
* Dcunha et al. (2021) Snoviya Dcunha, Srushti Patel, Shravani Sawant, Varsha Kulkarni, and Mahesh Shirole. 2021. Blockchain Interoperability Using Hash Time Locks. In _Proceeding of Fifth International Conference on Microelectronics, Computing and Communication Systems_ , Vijay Nath and J. K. Mandal (Eds.). Springer Singapore, Singapore, 475–487.
* Digital Euro Association ([n. d.]) Digital Euro Association. [n. d.]. Central Bank Digital Currency. https://home.digital-euro-association.de/cbdc/en.
* Digital Euro Association (2022) Digital Euro Association. 2022\. The CBDC Manifesto. https://cbdcmanifesto.com/
* Ehrenfeld (2022) Jonathan Ehrenfeld. 2022\. SWIFT partners with Chainlink for cross-chain crypto transfer project. https://blog.chain.link/smartcon-2022-recap/
* Enclave (2019) Keystone Enclave. 2019\. Keystone: An Open Framework for Architecting Trusted Execution Environments. https://keystone-enclave.github.io/.
* Espel et al. (2017) Thomas Espel, Laurent Katz, and Guillaume Robin. 2017\. Proposal for Protocol on a Quorum Blockchain with Zero Knowledge. 2017 (2017).
* Gilad et al. (2017) Yossi Gilad, Rotem Hemo, Silvio Micali, Georgios Vlachos, and Nickolai Zeldovich. 2017. Algorand: Scaling byzantine agreements for cryptocurrencies. In _SOSP_.
* Gruss et al. (2017) Daniel Gruss, Julian Lettner, Felix Schuster, Olya Ohrimenko, Istvan Haller, and Manuel Costa. 2017\. Strong and efficient cache side-channel protection using hardware transactional memory. In _26th $\\{$USENIX$\\}$ Security Symposium ($\\{$USENIX$\\}$ Security 17)_. 217–233.
* Hoekstra et al. (2013) Matthew Hoekstra, Reshma Lal, Pradeep Pappachan, Vinay Phegade, and Juan Del Cuvillo. 2013. Using innovative instructions to create trustworthy software solutions. _HASP@ ISCA_ 11 (2013).
* Homoliak and Szalachowski (2020) Ivan Homoliak and Pawel Szalachowski. 2020. Aquareum: A Centralized Ledger Enhanced with Blockchain and Trusted Computing. arXiv:2005.13339 [cs.CR]
* Hyperledger Foundation (2022) Hyperledger Foundation. 2022\. Hyperledger. https://github.com/hyperledger.
* Intel (2018) Intel. 2018. Resources and Response to Side Channel L1 Terminal Fault. https://www.intel.com/content/www/us/en/architecture-and-technology/l1tf.html
* Jin and Xia (2022) Si Yuan Jin and Yong Xia. 2022. CEV Framework: A Central Bank Digital Currency Evaluation and Verification Framework With a Focus on Consensus Algorithms and Operating Architectures. _IEEE Access_ 10 (2022), 63698–63714. https://doi.org/10.1109/ACCESS.2022.3183092
* Kiayias et al. (2017) Aggelos Kiayias, Al‘exander Russell, Bernardo David, and Roman Oliynykov. 2017. Ouroboros: A provably secure proof-of-stake blockchain protocol. In _CRYPTO’17_.
* Kiff et al. (2020) Mr John Kiff, Jihad Alwazir, Sonja Davidovic, Aquiles Farias, Mr Ashraf Khan, Mr Tanai Khiaonarong, Majid Malaika, Mr Hunter K Monroe, Nobu Sugimoto, Hervé Tourpe, et al. 2020\. A survey of research on retail central bank digital currency. (2020).
* Kishi (2019) Michinobu Kishi. 2019\. Project Stella and the impacts of fintech on financial infrastructures in Japan.
* Koziuk (2021) V. Koziuk. 2021\. Confidence in digital money: Are central banks more trusted than age is matter? 18, 1 (2021), 12–32.
* Lacoste and Lefebvre (2023) Marc Lacoste and Vincent Lefebvre. 2023. Trusted Execution Environments for Telecoms: Strengths, Weaknesses, Opportunities, and Threats. _IEEE Security & Privacy_ (2023).
* Lagarde (2022) Christine Lagarde. 2022\. High level conference: Towards a legislative framework enabling a digital euro for citizens and businesses. https://www.ecb.europa.eu/press/key/date/2022/html/ecb.sp221107~dcc0cd8ed9.en.html
* Lan et al. (2021) Ying Lan et al. 2021\. TrustCross: Enabling Confidential Interoperability across Blockchains Using Trusted Hardware. In _2021 4th International Conference on Blockchain Technology and Applications_ (Xi’an, China) _(ICBTA 2021)_. Association for Computing Machinery, New York, NY, USA, 17–23. https://doi.org/10.1145/3510487.3510491
* Lind et al. (2017) Joshua Lind, Ittay Eyal, Florian Kelbert, Oded Naor, Peter Pietzuch, and Emin Gün Sirer. 2017. Teechain: Scalable blockchain payments using trusted execution environments. _arXiv preprint arXiv:1707.05454_ (2017).
* Lipp et al. (2021) Moritz Lipp, Andreas Kogler, David Oswald, Michael Schwarz, Catherine Easdon, Claudio Canella, and Daniel Gruss. 2021. PLATYPUS: Software-based Power Side-Channel Attacks on x86. In _2021 IEEE Symposium on Security and Privacy (SP)_. IEEE.
* Madine et al. (2021) Mohammad Madine, Khaled Salah, Raja Jayaraman, Yousof Al-Hammadi, Junaid Arshad, and Ibrar Yaqoob. 2021. appXchain: Application-Level Interoperability for Blockchain Networks. _IEEE Access_ 9 (2021), 87777–87791. https://doi.org/10.1109/ACCESS.2021.3089603
* McKeen et al. (2013) Frank McKeen, Ilya Alexandrovich, Alex Berenzon, Carlos V Rozas, Hisham Shafi, Vedvyas Shanbhogue, and Uday R Savagaonkar. 2013. Innovative instructions and software model for isolated execution. _Hasp@isca_ 10, 1 (2013).
* Merkle (1989) Ralph C Merkle. 1989\. A certified digital signature. In _Conference on the Theory and Application of Cryptology_. Springer, 218–238.
* Micali (2003) Silvio Micali. 2003\. Simple and fast optimistic protocols for fair electronic exchange. In _Proceedings of the twenty-second annual symposium on Principles of distributed computing_. ACM, 12–19.
* Microsoft (2020) Microsoft. 2020\. Enclave EVM. https://github.com/microsoft/eEVM
* Miller et al. (2016) Andrew Miller, Yu Xia, Kyle Croman, Elaine Shi, and Dawn Song. 2016. The honeybadger of BFT protocols. In _ACM CCS_.
* Mohanty et al. (2022) Debasis Mohanty, Divya Anand, Hani Moaiteq Aljahdali, and Santos Gracia Villar. 2022. Blockchain Interoperability: Towards a Sustainable Payment System. _Sustainability_ 14, 2 (2022), 913.
* Monetary Authority of Singapore (2016) Monetary Authority of Singapore. 2016\. Project Ubin. https://www.mas.gov.sg/schemes-and-initiatives/Project-Ubin
* Murdock et al. (2020) Kit Murdock, David Oswald, Flavio D. Garcia, Jo Van Bulck, Daniel Gruss, and Frank Piessens. 2020\. Plundervolt: Software-based Fault Injection Attacks against Intel SGX. In _Proceedings of the 41st IEEE Symposium on Security and Privacy (S &P’20)_.
* Musungate et al. (2019) Blessing Musungate, Busra Candan, Umut Cabuk, and Gokhan Dalkilic. 2019. Sidechains: Highlights and Challenges. https://doi.org/10.1109/ASYU48272.2019.8946384
* Nakamoto (2008) Satoshi Nakamoto. 2008\. Bitcoin: A peer-to-peer electronic cash system.
* Náñez Alonso et al. (2020) Sergio Luis Náñez Alonso, Miguel Ángel Echarte Fernández, David Sanz Bas, and Jarosław Kaczmarek. 2020. Reasons fostering or discouraging the implementation of central bank-backed digital currency: A review. _Economies_ 8, 2 (2020), 41.
* Qasse et al. (2019) Ilham A Qasse, Manar Abu Talib, and Qassim Nasir. 2019\. Inter blockchain communication: A survey. In _Proceedings of the ArabWIC 6th Annual International Conference Research Track_. 1–6.
* Sebastião et al. (2021) H.M.C.V. Sebastião, P.J.O.R. da Cunha, and P.M.C. Godinho. 2021\. Cryptocurrencies and blockchain. Overview and future perspectives. 21, 3 (2021), 305–342.
* Seo et al. (2017) Jaebaek Seo, Byoungyoung Lee, Seong Min Kim, Ming-Wei Shih, Insik Shin, Dongsu Han, and Taesoo Kim. 2017. SGX-Shield: Enabling Address Space Layout Randomization for SGX Programs.. In _NDSS_.
* Sethaput and Innet (2021) Vijak Sethaput and Supachate Innet. 2021. Blockchain Application for Central Bank Digital Currencies (CBDC). In _2021 Third International Conference on Blockchain Computing and Applications (BCCA)_. IEEE, 3–10.
* Shih et al. (2017) Ming-Wei Shih, Sangho Lee, Taesoo Kim, and Marcus Peinado. 2017\. T-SGX: Eradicating Controlled-Channel Attacks Against Enclave Programs.. In _NDSS_.
* Siris et al. (2019) Vasilios Siris, Pekka Nikander, Spyros Voulgaris, Nikos Fotiou, Dmitrij Lagutin, and George Polyzos. 2019. Interledger Approaches. _IEEE Access_ PP (07 2019), 1–1. https://doi.org/10.1109/ACCESS.2019.2926880
* South African Reserve Bank (2018) South African Reserve Bank. 2018\. Project Khokha: Blockchain Case Study for Central Banking in South Africa . https://consensys.net/blockchain-use-cases/finance/project-khokha/
* Subramanyan et al. (2017) Pramod Subramanyan, Rohit Sinha, Ilia Lebedev, Srinivas Devadas, and Sanjit A Seshia. 2017. A formal foundation for secure remote execution of enclaves. In _Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security_. 2435–2450.
* US Federal Reserve ([n. d.]) US Federal Reserve. [n. d.]. Central Bank Digital Currency. https://www.federalreserve.gov/central-bank-digital-currency.htm.
* Van Bulck et al. (2018) Jo Van Bulck, Marina Minkin, Ofir Weisse, Daniel Genkin, Baris Kasikci, Frank Piessens, Mark Silberstein, Thomas F Wenisch, Yuval Yarom, and Raoul Strackx. 2018\. Foreshadow: Extracting the keys to the intel $\\{$SGX$\\}$ kingdom with transient out-of-order execution. In _27th $\\{$USENIX$\\}$ Security Symposium ($\\{$USENIX$\\}$ Security 18)_. 991–1008.
* Wang and Nixon (2021) Gang Wang and Mark Nixon. 2021. InterTrust: Towards an Efficient Blockchain Interoperability Architecture with Trusted Services. In _2021 IEEE International Conference on Blockchain (Blockchain)_. 150–159. https://doi.org/10.1109/Blockchain53845.2021.00029
* Wang et al. (2023) Gang Wang, Qin Wang, and Shiping Chen. 2023. Exploring Blockchains Interoperability: A Systematic Survey. _Comput. Surveys_ (2023).
* Wang et al. (2020) Yong Wang, June Li, Siyu Zhao, and Fajiang Yu. 2020\. Hybridchain: A novel architecture for confidentiality-preserving and performant permissioned blockchain using trusted execution environment. _IEEE Access_ 8 (2020), 190652–190662.
* Zhang and Huang (2021) Tao Zhang and Zhigang Huang. 2021. Blockchain and central bank digital currency. _ICT Express_ (2021).
$\triangleright$ Declaration of types and functions:
Header { $ID$, $txsRoot$, $rcpRoot$, $stRoot$};
$\\#(r)\rightarrow v$: denotes the version $v$ of $L$ having $LRoot$ $=$ $r$,
$\triangleright$ Variables of TEE:
$SK_{\mathbb{E}}^{tee},PK_{\mathbb{E}}^{tee}$: keypair of $\mathbb{E}$ under
$\Sigma_{tee}$,
$SK_{\mathbb{E}}^{pb},PK_{\mathbb{E}}^{pb}$: keypair of $\mathbb{E}$ under
$\Sigma_{pb}$,
$hdr_{last}\leftarrow\perp$: the last header created by $\mathbb{E}$,
$LRoot_{pb}\leftarrow\perp$: the last root of $L$ flushed to PB’s
$\mathbb{IPSC}$,
$LRoot_{cur}\leftarrow\perp$: the root of $L\cup blks_{p}$ (not flushed to
PB),
$ID_{cur}\leftarrow 1$: the current version of $L$ (not flushed to PB),
$FH_{cur}\leftarrow[]$: the frozen hashes cache of the current $L$’s history
tree.
$\triangleright$ Declaration of functions:
function _$Init$() public_
($SK_{\mathbb{E}}^{pb}$,
$PK_{\mathbb{E}}^{pb}$)$\leftarrow\Sigma_{pb}.Keygen()$;
($SK_{\mathbb{E}}^{tee}$,
$PK_{\mathbb{E}}^{tee}$)$\leftarrow\Sigma_{tee}.Keygen()$;
Output($PK_{\mathbb{E}}^{tee},PK_{\mathbb{E}}^{pb}$);
function _$Exec$($txs[],\partial st^{old}$) public_
assert $\partial st^{old}.root=hdr_{last}.stRoot$;
$\partial
st^{new},rcps,txs_{er}\leftarrow~{}~{}\\_\\_processTxs(txs,~{}\partial
st^{old})$;
$\sigma\leftarrow\Sigma_{pb}.sign(SK_{\mathbb{E}}^{pb},(LRoot_{pb},LRoot_{cur}))$;
Output($LRoot_{pb},LRoot_{cur},\partial st^{new},hdr_{last},rcps$, $txs_{er}$,
$\sigma$);
function _$Flush$() public_
$LRoot_{pb}\leftarrow LRoot_{cur}$; $\triangleright$ Shift the version of $L$
synchronized with PB.
function _$\\_\\_processTxs$($txs[],\partial st^{old}$) private_
$\partial st^{new},rcps[],txs_{er}\leftarrow$ runVM($txs$, $\partial
st^{old}$); $\triangleright$ Run $\mu$-$txs$ in VM.
$txs\leftarrow txs\setminus txs_{er}$; $\triangleright$ Filter out parsing
errors/wrong signatures.
$hdr\leftarrow~{}~{}\mathbf{Header}(ID_{cur},MkRoot(txs),MkRoot(rcps),\partial
st^{new}.root))$;
$hdr_{last}\leftarrow hdr$;
$LRoot_{cur}\leftarrow\\_\\_newLRoot(hdr)$;
$ID_{cur}\leftarrow ID_{cur}+1$;
return $\partial st^{new}$, $rcps$, $txs_{er}$;
function _$\\_\\_newLRoot(hdr)$ private_
$\\_\\_udpateFH(h(hdr));$
return $FH_{cur}.ReduceRoot()$;
$\triangleright$ Since $FH_{cur}=\pi^{inc}_{next}$, inc. proof. for 1 element
commitment.
function _$\\_\\_updateFH(hdrH)$ private_
$FH_{cur}.add(hdrH);$
$l\leftarrow\lfloor log_{2}(ID_{cur})\rfloor$;
for _$i\leftarrow 2;\ i\leq 2^{l};\ i\leftarrow 2i$_ do
if _$0=ID_{cur}\bmod i$_ then
$FH_{cur}[\text{-}2]\leftarrow h(FH_{cur}[\text{-}2]\ ||\
FH_{cur}[\text{-}1])$
$\mathbf{delete}~{}~{}FH_{cur}[\text{-1}]$; $\triangleright$ Remove the last
element.
Algorithm 5 The program $prog^{\mathbb{E}}$ of enclave $\mathbb{E}$
$\triangleright$ Declaration of types and constants:
CensInfo { $\mu$-$etx,\mu$-$equery,status,edata$ },
$msg$: a current transaction that called $\mathbb{IPSC}$,
$\triangleright$ Declaration of functions:
function
_$Init$($PK_{\mathbb{E}}^{pb},PK_{\mathbb{E}}^{tee},PK_{\mathbb{O}},~{}{\color[rgb]{1,0,0}\\_i_{r}},~{}{\color[rgb]{1,0,0}[ia\leftarrow\textbf{T}]}$)
public _
$PK_{\mathbb{E}}^{tee}[].add(PK_{\mathbb{E}}^{tee})$; $\triangleright$ PK of
enclave $\mathbb{E}$ under $\Sigma_{tee}$.
$PK_{\mathbb{E}}^{pb}[].add(PK_{\mathbb{E}}^{pb})$; $\triangleright$ PK of
enclave $\mathbb{E}$ under $\Sigma_{pb}$.
$PK_{\mathbb{O}}^{pb}\leftarrow PK_{\mathbb{O}}$; $\triangleright$ PK of
operator $\mathbb{O}$ under $\Sigma_{pb}$.
$LRoot_{pb}\leftarrow\perp$; $\triangleright$ The most recent root hash of $L$
synchronized with $\mathbb{IPSC}$.
$censReqs\leftarrow[]$; $\triangleright$ Request that $\mathbb{C}$s wants to
resolve publicly.
$t_{s}\leftarrow 0$; $\triangleright$ The total supply of the instance.
$t_{i}\leftarrow 0$; $\triangleright$ The total issued tokens by the instance.
const $issueAuthority\leftarrow ia$; $\triangleright$ Token issuance
capability of the instance.
const $i_{r}\leftarrow\\_i_{r}$; $\triangleright$ Max. yearly inflation of the
instance.
const $createdAt\leftarrow timestamp()$; $\triangleright$ The timestamp of
creation a CBDC instance.
function
_$snapshotLedger$($root_{A},root_{B},{\color[rgb]{1,0,0}\\_t_{i},\\_t_{s},}~{}\sigma$)
public _
$\triangleright$ Verify whether msg was signed by $\mathbb{E}$.
assert
$\Sigma_{pb}.verify((\sigma,PK_{\mathbb{E}}^{pb}[\text{-}1]),(root_{A},root_{B},{\color[rgb]{1,0,0}\\_t_{i},\\_t_{s}}))$;
$\triangleright$ Snapshot issued tokens and total supply.
if _$issueAuthority$_ then
assert $\\_\\_meetsInflationRate(\\_t_{i})$; $\triangleright$ The code is
trivial, and we omit it.
$t_{i}\leftarrow\\_t_{i}$;
else
assert $t_{i}=\\_t_{i}$;
$\triangleright$ Verify whether a version transition extends the last one.
if _$LRoot_{pb}=root_{A}$_ then
$LRoot_{pb}\leftarrow root_{B}$; $\triangleright$ Do a version transition of
$L$.
function _$SubmitCensTx$($\mu$-$etx,\sigma_{msg}$) public _
$\triangleright$ Called by $\mathbb{C}$ in the case her $\mu$-tx is censored.
$\mathbb{C}$ encrypts it by $PK^{tee}_{\mathbb{E}}$.
accessControl($\sigma_{msg},msg.PK_{\mathbb{C}}^{pb}$);
$censReqs$.add(CensInfo($\mu$-$etx,\perp,\perp,\perp$));
function _$ResolveCensTx(idx_{req},status,\sigma$) public _
$\triangleright$ Called by $\mathbb{O}$ to prove that $\mathbb{C}$’s $\mu$-tx
was processed.
assert $idx_{req}<|censReqs|$;
$r\leftarrow censReqs[idx_{req}]$;
assert
$\Sigma_{pb}.verify((\sigma,PK_{\mathbb{E}}^{pb}[\text{-}1]),~{}(h(r.\mu$-$etx),status))$;
$r.status\leftarrow status$;
function _$SubmitCensQry$($\mu$-$equery,\sigma_{msg}$) public _
$\triangleright$ Called by $\mathbb{C}$ in the case its read query is
censored. $\mathbb{C}$ encrypts it by $PK^{tee}_{\mathbb{E}}$.
accessControl($msg$, $\sigma_{msg},msg.PK_{\mathbb{C}}^{pb}$);
$censReqs$.add(CensInfo($\perp,\mu$-$equery,\perp,\perp$));
function _$ResolveCensQry(idx_{req},status,edata,\sigma$) public _
$\triangleright$ Called by $\mathbb{O}$ as a response to the $\mathbb{C}$’s
censored read query.
assert $idx_{req}<|censReqs|$;
$r\leftarrow censReqs[idx_{req}]$;
assert
$\Sigma_{pb}.verify((\sigma,PK_{\mathbb{E}}^{pb}[\text{-}1]),(h(r.\mu$-$equery),status,h(edata)))$;
$r.\\{edata\leftarrow edata,status\leftarrow status\\}$;
function
_$ReplaceEnc$($PKN_{\mathbb{E}}^{pb},PKN_{\mathbb{E}}^{tee},r_{A},r_{B},~{}{\color[rgb]{1,0,0}\\_t_{i},\\_t_{s},}~{}\sigma,\sigma_{msg}$)
public _
$\triangleright$ Called by $\mathbb{O}$ in the case of enclave failure.
assert $\Sigma_{pb}.verify((\sigma_{msg},PK_{\mathbb{O}}^{pb}),msg)$;
$\triangleright$ Avoiding MiTM attack.
$snapshotLedger(r_{A},r_{B},~{}{\color[rgb]{1,0,0}\\_t_{i},\\_t_{s}},~{}\sigma)$
; $\triangleright$ Do a version transition.
$PK_{\mathbb{E}}^{tee}.add(PKN_{\mathbb{E}}^{tee})$; $\triangleright$ Upon
change, $\mathbb{C}s$ make remote attestation.
$PK_{\mathbb{E}}^{pb}.add(PKN_{\mathbb{E}}^{pb})$;
Algorithm 6 The program $prog^{\mathbb{IPSC}}$ of $\mathbb{IPSC}$.
|
# Enhanced anomalous Nernst effects in ferromagnetic materials driven by Weyl
nodes
Ilias Samathrakis<EMAIL_ADDRESS>Teng Long Zeying Zhang
Harish K. Singh Hongbin Zhang Institute of Materials Science, TU Darmstadt,
64287 Darmstadt, Germany
###### Abstract
Based on high-throughput first-principles calculations, we evaluated the
anomalous Hall and anomalous Nernst conductivities of 266 transition-metal-
based ferromagnetic compounds. Detailed analysis based on the symmetries and
Berry curvatures reveals that the origin of singular-like behaviour of
anomalous Hall/Nernst conductivities can be mostly attributed to the
appearance of Weyl nodes or nodal lines located in the proximity of the Fermi
energy, which can be further tailored by external stimuli such as biaxial
strains and magnetic fields. Moreover, such calculations are enabled by the
automated construction of Wannier functions with a success rate of 92%, which
paves the way to perform accurate high-throughput evaluation of the physical
properties such as the transport properties using the Wannier interpolation.
## I Introduction
In the last decades, materials of the nontrivial topological nature have
attracted quite intensive attention of the community, leading to many
interesting physical properties such as ultra high mobility [1], protected
surface states [2], anomalous magnetoresistance [3, 1] and exotic optical
properties [4], promising for future applications. One particularly
interesting class of materials are those with long range magnetic orderings at
finite temperatures, including topological insulators, Dirac (Weyl) semimetals
(DSM,WSM) and nodal line semimetals. These materials are characterized by
gapless surface Dirac fermions with a spin-momentum-locked energy dispersion,
a spin-(non-)degenerate band touching point and lines respectively. [5, 6]
These special energy dispersions are responsible the generation of interesting
physical phenomena. For instance, for ferromagnetic materials, the anomalous
Hall effect (AHE), which describes the generation of a transversal current
perpendicular to the electric current in the absence of an external magnetic
field [7, 8], can be induced. The same can be achieved using the thermal
gradient as the external stimulus, resulting in the anomalous Nernst effect
(ANE) [9, 10]. Both AHE and ANE are caused by the nonzero Berry curvature of
the Fermi sea states, [8] which acts as a fictitious magnetic field [11]
giving rise to these effects that lead to promising applications such as data
storage and data transfer. [12, 13] Note that like the other linear response
properties, the anomalous Hall conductivities (AHC) can also be equally
evaluated by integrating the Fermi surfaces [14], arising one interesting
aspect whether AHC can be tuned by external stimuli.
Recently, motivated by the emergent antiferromagnetic spintronics [15], it is
demonstrated that significant AHCs and ANCs can be obtained in magnetic
materials with noncollinear antiferromagnetic ordering [16, 17]. The most
representative materials include the Mn3X with X=Ge,Ga,Sn,Rh,In,Pt family [18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28], the Mn3XN with X=Ga,Zn,Ag,Ni family
[29, 30, 31], Mn5Sn3 [32, 33] and Pr2Ir2O7 [34]. Besides AHC, the triangular
noncollinear antiferromagnetic structure of the Mn3X family is responsible for
the generation of a spin current in the absence of external field, dubbed Spin
Hall effect (SHE) [35] and the Magneto-optical Kerr effect (MOKE) [36] which
is the rotation of the plane of polarization of a light beam reflected from
the surface in the presence of a magnetic field. Such peculiar properties can
be traced back to the existence of Weyl nodes in the electronic structure.
[22, 37] WSM are topological materials where the electronic structure is
dominated by non-accidental linear touching and historically Y2Ir2O7 was the
first material to be realized as a WSM [38]. Recently, Co3Sn2S2 and several
magnetic Weyl Heusler compounds were classified as WSM which also exhibit
touching points with linear dispersion close to the Fermi level [39, 40],
giving rise to large AHC. Therefore, there is a strong impetus to search for
more magnetic WSM and to investigate the resulting physical properties.
In this work, after collecting the known ferromagnetic materials, we performed
high-throughput (HTP) calculations on AHC and ANC of 266 existing transition-
metal-based ferromagnetic compounds, where the magnetization directions and
biaxial strains are considered as perturbations to tune such effects. It is
observed that the appearance of linearly degenerate states such as Weyl nodes
and nodal lines will lead to singular-like behavior of AHC and hence enhanced
ANC. Nevertheless, there exist also materials with significant ANC but without
hot-spot contributions to AHC. Therefore, it is suspected that further HTP
calculations are required in order to identify promising candidates with
enhanced ANC for transversal thermoelectric applications, where automated
construction of Wannier functions can be valuable.
## II Numerical details
Out of 5487 experimentally known ferromagnets, reported in AtomWork Adv
database [41], the crystal structure of 3956 was found and collected using
ICSD, [42] Materials Project, [43] and AtomWork databases. Compounds being
either non-stoichiometric or containing elements with partial occupation were
excluded, leading to 1827 compounds. Additionally, rare earths, oxides and
compounds including more than 30 atoms in their unit cell were screened out as
well since they increase the complexity and the time needed for the
calculation. These constraints further reduced the number of available
ferromagnetic compounds for calculations to 335.
The transport properties of the ferromagnetic compounds are computed
automatically using an in-house-developed scheme, written in Python, linking
VASP, Wannier90 and Wanniertools software. The first step involves the self
consistent first-principles calculation of each compound using the projected
augmented wave method (PAW), implemented in VASP. [44] The exchange
correlation functional is approximated in the general gradient approximation
(GGA), as parameterized by Perdew-Burke-Ernzerhof [45] and the spin orbit
coupling is included in all calculations. A $\Gamma$-centered kmesh of density
50, in respect to the lattice parameters, as well as an energy cutoff of
500$eV$ are selected. Subsequently, the Bloch wave functions are projected
onto maximally localised wannier functions (MLWF), using Wannier90, following
Ref. 46. The number of MLWF as well as the disentanglement and frozen windows
of each compound are automatically computed following Ref. 47. The AHC is then
evaluated by integrating the Berry curvature, according to the formula:
$\displaystyle\sigma_{\alpha\beta}=-\frac{e^{2}}{\hbar}\int\frac{d\mathbf{k}}{\left(2\pi\right)^{3}}\sum_{n}f\big{[}\epsilon\left(\mathbf{k}\right)-\mu\big{]}\Omega_{n,\alpha\beta}\left(\mathbf{k}\right)$
(1)
$\displaystyle\Omega_{n,\alpha\beta}\left(\mathbf{k}\right)=-2Im\sum_{m\neq
n}\frac{\braket{\mathbf{k}n}{v_{\alpha}}{\mathbf{k}m}\braket{\mathbf{k}m}{v_{\beta}}{\mathbf{k}n}}{\big{[}\epsilon_{m}\left(\mathbf{k}\right)-\epsilon_{n}\left(\mathbf{k}\right)\big{]}^{2}},$
(2)
where $f\big{[}\epsilon\left(\mathbf{k}\right)-\mu\big{]}$ denotes the Fermi
distribution function, $\mu$ the Fermi energy, n (m) the occupied (empty)
Bloch band,
$\epsilon_{n}\left(\mathbf{k}\right)\left(\epsilon_{m}\left(\mathbf{k}\right)\right)$
their energy eigenvalues and $v_{\alpha}$ $\left(v_{\beta}\right)$ the
velocity operator. The integration is performed on 400 dense kmesh in respect
to the lattice parameters, using Wanniertools. [48] The ANC is defined
according to the formula:
$a_{\alpha\beta}=-\frac{1}{e}\int d\epsilon\frac{\partial
f}{\partial\mu}\sigma_{\alpha\beta}\left(\epsilon\right)\frac{\epsilon-\mu}{T},$
(3)
where $\epsilon$ is the energy point of the energy grid, e the electron’s
charge, T the temperature and $\mu$ the Fermi level. Regarding the ANC, an in
house developed Python script in an energy grid of 1000 points at a range
$\big{[}-0.5,0.5\big{]}$ eV in respect to the Fermi level is used.
## III Results and Discussion
As stated above, starting from 335 transition-metal-based ferromagnetic
materials, we managed to converge the DFT calculations for 289 compounds for
both magnetization directions, i.e., the [001] and [100]. MLWFs can be
successfully constructed for 266 of them, giving rise to a success rate of
92%. In comparison, a success rate of 93% and 97% is achieved in Ref. 49 and
Ref. 50 respectively. Our results provide an alternative to the automatic
construction of Wannier functions and enable us to perform further HTP
calculations to evaluate the desired physical properties, in particular making
use of the Wannier interpolation technique [51].
Figure. 1 shows the z-component ($\sigma_{z}=\sigma_{xy}$) of AHC and ANC for
266 FM intermetallic compounds with the magnetization aligned along the [001]
direction. Obviously, the magnitude of AHC ranges between -2051$S/cm$ for
Ni3Pt and 2040$S/cm$ for CrPt3, where there are 11 compounds with the absolute
value of AHC exceeding 1000$S/cm$. The largest magnitude of AHC is larger than
the already reported -1862$S/cm$ for Rh2MnGa and -1723$S/cm$ for Rh2MnAl, [52]
arising the question whether there is an upper limit and opens up the
possibility to engineer materials with more significant magnitudes.
Concomitant with AHC, the z-component ($\alpha_{z}=\alpha_{xy}$) of ANC varies
between -7.29$A/(m\cdot K)$ for Ni3Pt and 5.83$A/(m\cdot K)$ for BCo4Y, and
exceeds 3$A/(m\cdot K)$ in 16 compounds.
Figure 1: $\sigma_{z}$ (at T=0K) and $\alpha_{z}$ (at T=300K) for selected
ferromagnets with the magnetization direction parallel to the [001] axis. Our
results are indicated with black and compared to the ones available in
literature in red.
In general, our results match reasonably well with the existing literature for
the magnetization direction being parallel to z-axis. More specifically the
reported value of 1130$S/cm$ for Co3Sn2S2 [53] is comparable with the
988$S/cm$ obtained from our calculations. Moreover, good agreement is achieved
in Heusler compounds where our values of 140$S/cm$ and 200$S/cm$ for Co2VGa
and Co2MnSn agree with the reported cases of 137$S/cm$ in Ref. 54 and
118$S/cm$ in Ref. 55 respectively. Comparable results are also observed in the
XPt3 family with $X=\left(Cr,Mn\right)$ where the reported cases of 2040$S/cm$
and 1400$S/cm$ [56] are reproduced by our values of 2000$S/cm$ and 1471$S/cm$
respectively. Regarding ANC, the calculated value of 4.58$A/(m\cdot K)$ for
MnPt3 is consistent with the reported value of 4$A/(m\cdot K)$ in Ref. [56].
It is important to mention that the ANC is extremely sensitive to the density
of points used for the calculation as well as the temperature that might
explain several differences between the reported and the calculated values.
The shape of the AHC and ANC tensors is the same and can be determined by the
magnetic space group of the compound. For the AHC tensor given by Eq. 1, it
primarily depends on the Berry curvature which behaves as a pseudovector under
symmetry operations, yielding:
$s\mathbf{v\left(r\right)}=\pm
det\left(\mathbf{D}\left(R\right)\right)\mathbf{D}\left(R\right)\mathbf{v}\left(s^{-1}\mathbf{r}\right)$
(4)
where $\mathbf{v}\left(r\right)$ denotes the pseudovector Berry curvature,
$\mathbf{D}\left(R\right)$ the three-dimensional representation of a symmetry
operation without the translation part and $s$ an arbitrary symmetry
operation. Taking SiMnY as an example, we observe that its crystal structure
belongs to the space group $P4/nmm$ (129) and the magnetization direction
along the [100]-direction renders it to the magnetic space group is
$Pmm^{\prime}n^{\prime}$ (BNS: 59.410). Correspondingly, the Berry curvature
(in the Cartesian basis) behaves as an odd pseudovector under the application
of $m_{100}$ and obeys:
$\displaystyle\Omega_{x}\left(-k_{x},k_{y},k_{z}\right)=\Omega_{x}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{y}\left(-k_{x},k_{y},k_{z}\right)=-\Omega_{y}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{z}\left(-k_{x},k_{y},k_{z}\right)=-\Omega_{z}\left(k_{x},k_{y},k_{z}\right).$
Thus, the summation over the whole Brillouin zone forces
$\sigma_{y}=\sigma_{xz}$ and $\sigma_{z}=\sigma_{xy}$ to vanish. On the other
hand, the magnetic space group of SiMnY changes to $P4/nm^{\prime}m^{\prime}$
(BNS: 129.417) for the magnetization direction parallel to [001] axis. In this
case, the Berry curvature behaves as an odd pseudovector under the application
of $2_{100}$ and obeys
$\displaystyle\Omega_{x}\left(-k_{x},-k_{y},k_{z}\right)=-\Omega_{x}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{y}\left(-k_{x},-k_{y},k_{z}\right)=-\Omega_{y}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{z}\left(-k_{x},-k_{y},k_{z}\right)=\Omega_{z}\left(k_{x},k_{y},k_{z}\right).$
It is obvious that the summation over the whole Brillouin zone forces
$\sigma_{x}=\sigma_{yz}$ and $\sigma_{y}=\sigma_{xz}$ to vanish, leading to
the conclusion that for the high-symmetric magnetization directions, the
direction of the AHC tensor is always aligned with the magnetization
direction. It is further noted that the absence of symmetries dictating
specific components of the Berry curvature to be zero does not necessarily
guarantee finite AHC of the respective component, since AHC depends on the
distribution of the Berry curvature in the whole BZ. For instance, it is
calculated negligible AHC value for all components in Co2NbAl
($Im^{\prime}m^{\prime}m$, BNS:71.536) despite the absence of symmetry forcing
$\sigma_{x}=0$.
For magnetic materials without time-reversal symmetry, the actual symmetry and
the actual electronic structure depend on the magnetization directions after
considering SOC, leading to in general anisotropic responses. In order to
quantify the changes in AHC and ANC, we performed calculations on two
different magnetization directions, i.e., [100] and [001]. The resulting
anisotropy can be expressed as the ratio of the so obtained values for the two
magnetization directions, i.e.,
$\sigma_{x,M\parallel[100]}/\sigma_{z,M\parallel[001]}$ or
$\alpha_{x,M\parallel[100]}/\alpha_{z,M\parallel[001]}$. It is clear that
“large” values signify large changes in favor of the [100] direction, as it
happens for AHC and ANC in Rh2MnSb and Fe3Se4, where anisotropy of 53.32 and
-33.39 is calculated respectively. On the other hand, small values (excluding
the cases where both AHC values are lower than 10$S/cm$ and both ANC values
lower than 0.05$A/(m\cdot K)$ and therefore considered negligible) are in
favor of the [001] magnetization direction, such as -0.01 and 0.004, that are
present in PFe and Fe2Ge for AHC and ANC respectively. The full list of the
values obtained for AHC, ANC and anisotropy for all ferromagnetic compounds
are available in Table S1 of the supplementary[59]. Compounds with AHC larger
than or equal to 1000$S/cm$ (irrespective of the direction), ANC larger than
or equal to 3$A/(m\cdot K)$ and anisotropy larger than 4 or smaller than 0.25
are classified as extreme cases and they are highlighted.
Figure 2: (a) Crystal structure of MnZn. (b) The z-component of the AHC
($\sigma_{z}$) (blue) and the AHC contribution originating from the nodal
lines (red) as a function of energy for MnZn with magnetization direction
parallel to [001] axis. (c) The z-component of the ANC ($\alpha_{z}$)
evaluated at T=300K (blue) and the ANC contribution originating from the nodal
lines (red) as a function of energy for MnZn with magnetization direction
parallel to [001] axis. (d) Symmetry related nodal lines of MnZn, with
magnetiation direction parallel to [001] axis, contributing to the total AHC
value.
Recent works have shown that Weyl nodes and nodal lines, being bands touching
points, behave either as sinks or as sources of the Berry curvature [8, 57,
58] and hence they are expected to contribute to the total AHC, the origin of
which demonstrates the validity of the Mott relation. To further elucidate the
origin of singular-like AHC with resulting magnificent ANC, detailed analysis
is done on the energy-dependent AHC for MnZn (BNS: 123.345) (for crystal
structure see Fig. 2(a)), as shown in Fig. 2(b) with the magnetization
direction along the [001] axis. There exists a sharp peak of AHC about
1082$S/cm$ located about 4$meV$ below the Fermi energy. Explicit band
structure analysis reveals the presence of a circularly shaped nodal line at
$k_{y}=0.5$ plane (red part in Fig. 2(d)), which further appears at the planes
$k_{x}=\pm 0.5$, $k_{z}=\pm 0.5$ and $k_{y}=-0.5$ due to the presence of the
symmetries of the compound (black and blue parts). Importantly, we found that
about 73% of the total AHC (782$S/cm$ out of 1082$S/cm$) can be attributed to
the contribution of the nodal lines (Fig. 2(b)), similarly to the role of Weyl
nodes in Mn3PdN [57]. Exactly due to the presence of such nodal lines and
their contribution to the singular-like AHC, the behaviour of ANC around EF is
dominated by the contributions from the nodal lines (Fig. 2(c)). Therefore,
the presence of Weyl nodes and nodal lines close to the Fermi energy can lead
to an anomalous energy dependence of AHC and hence enhanced ANC.
A note about the role of symmetry of the nodal lines and their contributions
to AHC is in order. As discussed above, the symmetry of the Berry Curvature in
the Brillouin zone is essential to understand the origin of AHC. For MnZn with
the magnetization direction along the z-axis, the corresponding Magnetic Laue
group $4/mm^{\prime}m^{\prime}$ indicates the presence of the six closed-loop-
nodal lines of Fig. 2(d). However, not all of these symmetry equivalent nodal
lines contribute equally to the same component of AHC, though their geometry
are dictated by the energy eigenvalues and thus the symmetry. The underlined
Laue group for the compound includes the $m_{z}$ mirror plane symmetry that
flips the sign of $x$ and $y$ components of Berry curvature while it leaves
$z$ component unchanged, according to:
$\displaystyle\Omega_{x}\left(k_{x},k_{y},-k_{z}\right)=-\Omega_{x}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{y}\left(k_{x},k_{y},-k_{z}\right)=-\Omega_{y}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{z}\left(k_{x},k_{y},-k_{z}\right)=\Omega_{z}\left(k_{x},k_{y},k_{z}\right).$
Hence, the contribution from the top black nodal ring as well as from the
upper half of the red and the blue will exactly cancel out the one from the
bottom black and the lower half of the red and the blue respectively for $x$
and $y$ components. A complete list of Berry curvature transformations with
respect to each symmetry operation of the magnetic Laue group
$4/mm^{\prime}m^{\prime}$ is found in Table S2 of the supplementary [59].
Figure 3: (a) Band gap (in eV) at a slice of Brillouin Zone ($k_{z}=-0.479$)
for Ni3Pt. (b) The x-component of the AHC evaluated in 216 Brillouin Zone
cubes for Ni3Pt with magnetization direction parallel to [100] axis.
Enhanced ANC values may originate from AHC contributions other than isolated
Weyl nodes and nodal lines. The presence of nodal rings are responsible for
almost 3/4 of the total AHC of MnZn, as discussed above. However, there are
cases with more complicated behaviour, such as Ni3Pt that hosts a surprisingly
large AHC of -2051$S/cm$ that renders it as the largest calculated AHC. In
order to investigate its origin, we split the BZ
($k_{i}\in\left[-0.500,0.500\right)$) in $6\times 6\times 6=216$ cubes and
calculate the AHC within each of these, as illustrated in Fig. 3(b)). Our
results show that each part of the BZ contributes to the total AHC,
demonstrating the presence of numerous Weyl nodes, nodal lines as well as
extended small band gap areas. Such areas are illustrated in the region
$\left(k_{x},k_{y}\right)\in\left[0.33,0.50\right]$ of the band gap plot of
Fig. 3(a). Despite the absence of isolated Weyl nodes, there is a giant ANC of
-7.29$A/(m\cdot K)$ (at the Fermi energy) calculated, originating from the
whole BZ, showing at the same time that large ANC values are possible even in
the absence of isolated hot-spot contributions to the Berry curvature.
Figure 4: (a) Crystal structure of CoSe4. (b) The AHC components as a function
of energy for CoSe4 with magnetization direction parallel to [100] axis. (c)
The ANC components evaluated at T=300K as a function of energy for CoSe4 with
magnetization direction parallel to [100] axis. (d) The x-component of the AHC
of CoSe4 with magnetization direction parallel to [100] axis for different
values of applied strain
As the Fermi level can be tuned by several mechanisms including doping and
mechanical strain, it is fruitful to discuss the possibility of inducing
larger AHC and ANC values away from the charge neutral point. In order to
investigate the impact of such modifications to the transport properties, we
consider the hexagonal compound CoSe4 (BNS: 61.436) (for crystal structure see
Fig. 4(a). Interestingly, the compound exhibits an AHC of 202$S/cm$ at the
Fermi energy and moreover a sharp peak at almost 500$S/cm$, 5 meV above (see
Fig. 4(b)) that can be tuned by doping. Since, in general, compounds that
exhibit single-valued AHC peaks close to the Fermi energy offer great
opportunities for modifying AHC and and hence ANC values by means of doping,
some promising candidates include VAu4, CoS2 and Co2CrAl that exhibit AHC
peaks of 1955$S/cm$, -1048$S/cm$ and 1140$S/cm$ at 0.051$eV$, -0.024$eV$ and
0.027$eV$ respectively. AHC and ANC can also be modified by applying biaxial
strain. That is, the tensile strain of 0.5% reduces the size of the peak to
306$S/cm$ and moves it at 0.01$eV$ in respect to the Fermi energy. On the
other hand, the applied compressive strain of 1.0% has the opposite effect and
hence the size of the peak is increased to 683$S/cm$ and its position lies at
-0.001$eV$ (see Fig. 4(d)).
## IV Conclusion
Based on high throughput first-principles calculations, we evaluated the
anomalous Hall conductivity as well as the anomalous Nernst conductivity of
266 transition-metal based ferromagnets. We report that the absolute value of
AHC (ANC) of 11 (16) compounds is larger than 1000$S/cm$ (3$A/(m\cdot K)$)
with the largest being equal to 2060$S/cm$ for CrPt3 (-7.24$A/(m\cdot K)$ for
Ni3Pt). Moreover, we find that the AHC and ANC values are by 3/4 originating
from linear degenerate states such as Weyl nodes and nodal lines in MnZn and
they can further be enhanced by a factor of 43% by applying external stimuli,
such as 1.5% compressive strain in CoSe4. However, large AHC and ANC values
can arise even without the presence of isolated singular hot spots in the
Berry curvature but instead from uniform extended small band gap areas from
the whole BZ in Ni3Pt.
ACKNOWLEDGMENTS
This work was financially supported by the Deutsche Forschungsgemeinschaft
(DFG) via the priority programme SPP 1666 and the calculations were conducted
on the Lichtenberg high performance computer of the TU Darmstadt.
## References
* Liang _et al._ [2015] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. Cava, and N. Ong, Ultrahigh mobility and giant magnetoresistance in the dirac semimetal cd 3 as 2, Nature materials 14, 280 (2015).
* Xu _et al._ [2015] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, _et al._ , Discovery of a weyl fermion semimetal and topological fermi arcs, Science 349, 613 (2015).
* Huang _et al._ [2015] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, _et al._ , Observation of the chiral-anomaly-induced negative magnetoresistance in 3d weyl semimetal taas, Physical Review X 5, 031023 (2015).
* Wu _et al._ [2017] L. Wu, S. Patankar, T. Morimoto, N. L. Nair, E. Thewalt, A. Little, J. G. Analytis, J. E. Moore, and J. Orenstein, Giant anisotropic nonlinear optical response in transition metal monopnictide weyl semimetals, Nature Physics 13, 350 (2017).
* Zhang [2020] H. Zhang, High-throughput Design of Magnetic Materials, Electronic Structure 10.1088/2516-1075/abbb25 (2020).
* Sato and Ando [2017] M. Sato and Y. Ando, Topological superconductors: a review, Reports on Progress in Physics 80, 076501 (2017).
* Žutić _et al._ [2004] I. Žutić, J. Fabian, and S. D. Sarma, Spintronics: Fundamentals and applications, Reviews of modern physics 76, 323 (2004).
* Nagaosa _et al._ [2010] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall effect, Reviews of modern physics 82, 1539 (2010).
* Behnia and Aubin [2016] K. Behnia and H. Aubin, Nernst effect in metals and superconductors: a review of concepts and experiments, Reports on Progress in Physics 79, 046502 (2016).
* Lee _et al._ [2004] W.-L. Lee, S. Watauchi, V. Miller, R. Cava, and N. Ong, Anomalous hall heat current and nernst effect in the c u c r 2 s e 4- x b r x ferromagnet, Physical review letters 93, 226601 (2004).
* Xiao _et al._ [2010] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Reviews of modern physics 82, 1959 (2010).
* Hasegawa _et al._ [2015] K. Hasegawa, M. Mizuguchi, Y. Sakuraba, T. Kamada, T. Kojima, T. Kubota, S. Mizukami, T. Miyazaki, and K. Takanashi, Material dependence of anomalous nernst effect in perpendicularly magnetized ordered-alloy thin films, Applied Physics Letters 106, 252405 (2015).
* Huang _et al._ [2011] S. Huang, W. Wang, S. Lee, J. Kwo, and C. Chien, Intrinsic spin-dependent thermal transport, Physical review letters 107, 216604 (2011).
* Wang _et al._ [2007] X. Wang, D. Vanderbilt, J. R. Yates, and I. Souza, Fermi-surface calculation of the anomalous hall conductivity, Physical Review B 76, 195109 (2007).
* Baltz _et al._ [2018] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Antiferromagnetic spintronics, Reviews of Modern Physics 90, 015005 (2018).
* Suzuki _et al._ [2017] M.-T. Suzuki, T. Koretsune, M. Ochi, and R. Arita, Cluster multipole theory for anomalous hall effect in antiferromagnets, Phys. Rev. B 95, 094406 (2017).
* Seemann _et al._ [2015] M. Seemann, D. Ködderitzsch, S. Wimmer, and H. Ebert, Symmetry-imposed shape of linear response tensors, Physical Review B 92, 155138 (2015).
* Chen _et al._ [2014] H. Chen, Q. Niu, and A. H. MacDonald, Anomalous hall effect arising from noncollinear antiferromagnetism, Phys. Rev. Lett. 112, 017205 (2014).
* Zhang _et al._ [2018a] Y. Zhang, J. Železný, Y. Sun, J. van den Brink, and B. Yan, Spin hall effect emerging from a noncollinear magnetic lattice without spin–orbit coupling, New Journal of Physics 20, 073028 (2018a).
* Nayak _et al._ [2016] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle, J. Kübler, C. Felser, and S. S. P. Parkin, Large anomalous hall effect driven by a nonvanishing berry curvature in the noncolinear antiferromagnet mn3ge, Science Advances 2, 10.1126/sciadv.1501870 (2016), https://advances.sciencemag.org/content/2/4/e1501870.full.pdf .
* Ganguly _et al._ [2011] S. Ganguly, M. Costa, A. B. Klautau, A. Bergman, B. Sanyal, A. Mookerjee, and O. Eriksson, Augmented space recursion formulation of the study of disordered alloys with noncollinear magnetism and spin-orbit coupling: Application to mnpt and mn 3 rh, Physical Review B 83, 094407 (2011).
* Yang _et al._ [2017] H. Yang, Y. Sun, Y. Zhang, W.-J. Shi, S. S. Parkin, and B. Yan, Topological weyl semimetals in the chiral antiferromagnetic materials mn3ge and mn3sn, New Journal of Physics 19, 015008 (2017).
* Kübler and Felser [2014] J. Kübler and C. Felser, Non-collinear antiferromagnets and the anomalous hall effect, EPL (Europhysics Letters) 108, 67001 (2014).
* Li _et al._ [2017] X. Li, L. Xu, L. Ding, J. Wang, M. Shen, X. Lu, Z. Zhu, and K. Behnia, Anomalous nernst and righi-leduc effects in mn 3 sn: Berry curvature and entropy flow, Physical Review Letters 119, 056601 (2017).
* Guo and Wang [2017] G.-Y. Guo and T.-C. Wang, Large anomalous nernst and spin nernst effects in the noncollinear antiferromagnets ${\mathrm{mn}}_{3}x$ ($x=\mathrm{Sn},\mathrm{Ge},\mathrm{Ga}$), Phys. Rev. B 96, 224415 (2017).
* Kiyohara _et al._ [2016] N. Kiyohara, T. Tomita, and S. Nakatsuji, Giant anomalous hall effect in the chiral antiferromagnet ${\mathrm{mn}}_{3}\mathrm{Ge}$, Phys. Rev. Applied 5, 064009 (2016).
* Ikhlas _et al._ [2017] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki, D. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakatsuji, Large anomalous nernst effect at room temperature in a chiral antiferromagnet, Nature Physics 13, 1085 (2017).
* Železný _et al._ [2017] J. Železný, Y. Zhang, C. Felser, and B. Yan, Spin-polarized current in noncollinear antiferromagnets, Phys. Rev. Lett. 119, 187204 (2017).
* Zhou _et al._ [2020] X. Zhou, J.-P. Hanke, W. Feng, S. Blügel, Y. Mokrousov, and Y. Yao, Giant anomalous nernst effect in noncollinear antiferromagnetic mn-based antiperovskite nitrides, Physical Review Materials 4, 024408 (2020).
* Gurung _et al._ [2019] G. Gurung, D.-F. Shao, T. R. Paudel, and E. Y. Tsymbal, Anomalous hall conductivity of noncollinear magnetic antiperovskites, Phys. Rev. Materials 3, 044409 (2019).
* Huyen _et al._ [2019] V. T. N. Huyen, M.-T. Suzuki, K. Yamauchi, and T. Oguchi, Topology analysis for anomalous hall effect in the noncollinear antiferromagnetic states of mn 3 a n (a= ni, cu, zn, ga, ge, pd, in, sn, ir, pt), Physical Review B 100, 094426 (2019).
* Sürgers _et al._ [2017] C. Sürgers, T. Wolf, P. Adelmann, W. Kittler, G. Fischer, and H. v. Löhneysen, Switching of a large anomalous hall effect between metamagnetic phases of a non-collinear antiferromagnet, Scientific Reports 7, 42982 (2017), article.
* Sürgers [2018] C. Sürgers, Electrical switching of the anomalous hall effect, Nature Electronics 1, 154 (2018).
* Machida _et al._ [2007] Y. Machida, S. Nakatsuji, Y. Maeno, T. Tayama, T. Sakakibara, and S. Onoda, Unconventional anomalous hall effect enhanced by a noncoplanar spin texture in the frustrated kondo lattice ${\mathrm{pr}}_{2}{\mathrm{ir}}_{2}{\mathrm{o}}_{7}$, Phys. Rev. Lett. 98, 057203 (2007).
* Zhang _et al._ [2017] Y. Zhang, Y. Sun, H. Yang, J. Železný, S. P. P. Parkin, C. Felser, and B. Yan, Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn 3 X ( X = Ge , Sn, Ga, Ir, Rh, and Pt), Physical Review B 95, 075128 (2017).
* Higo _et al._ [2018] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune, O. M. J. van ’t Erve, Y. P. Kabanov, D. Rees, Y. Li, M.-T. Suzuki, S. Patankar, M. Ikhlas, C. L. Chien, R. Arita, R. D. Shull, J. Orenstein, and S. Nakatsuji, Large magneto-optical Kerr effect and imaging of magnetic octupole domains in an antiferromagnetic metal, Nature Photonics 12, 73 (2018).
* Šmejkal _et al._ [2018] L. Šmejkal, Y. Mokrousov, B. Yan, and A. H. MacDonald, Topological antiferromagnetic spintronics, Nature physics 14, 242 (2018).
* Wan _et al._ [2011] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates, Physical Review B 83, 205101 (2011).
* Armitage _et al._ [2018] N. Armitage, E. Mele, and A. Vishwanath, Weyl and dirac semimetals in three-dimensional solids, Reviews of Modern Physics 90, 015001 (2018).
* Yan and Felser [2017] B. Yan and C. Felser, Topological materials: Weyl semimetals, Annual Review of Condensed Matter Physics 8, 337 (2017).
* Xu _et al._ [2011] Y. Xu, M. Yamazaki, and P. Villars, Inorganic materials database for exploring the nature of material, Japanese Journal of Applied Physics 50, 11RH02 (2011).
* Hellenbrandt [2004] M. Hellenbrandt, The inorganic crystal structure database (icsd)—present and future, Crystallography Reviews 10, 17 (2004).
* Jain _et al._ [2013] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, _et al._ , Commentary: The materials project: A materials genome approach to accelerating materials innovation, Apl Materials 1, 011002 (2013).
* Kresse and Hafner [1993] G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Physical Review B 47, 558 (1993).
* Perdew _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical review letters 77, 3865 (1996).
* Mostofi _et al._ [2008] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, wannier90: A tool for obtaining maximally-localised wannier functions, Computer Physics Communications 178, 685 (2008).
* Zhang _et al._ [2018b] Z. Zhang, R.-W. Zhang, X. Li, K. Koepernik, Y. Yao, and H. Zhang, High-throughput screening and automated processing toward novel topological insulators, The journal of physical chemistry letters 9, 6224 (2018b).
* Wu _et al._ [2018] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, Wanniertools: An open-source software package for novel topological materials, Computer Physics Communications 224, 405 (2018).
* Garrity and Choudhary [2021] K. F. Garrity and K. Choudhary, Database of wannier tight-binding hamiltonians using high-throughput density functional theory, Scientific data 8, 1 (2021).
* Vitale _et al._ [2020] V. Vitale, G. Pizzi, A. Marrazzo, J. R. Yates, N. Marzari, and A. A. Mostofi, Automated high-throughput wannierisation, npj Computational Materials 6, 1 (2020).
* Marzari _et al._ [2012] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Reviews of Modern Physics 84, 1419 (2012), publisher: American Physical Society.
* Noky _et al._ [2020] J. Noky, Y. Zhang, J. Gooth, C. Felser, and Y. Sun, Giant anomalous hall and nernst effect in magnetic cubic heusler compounds, npj Computational Materials 6, 1 (2020).
* Liu _et al._ [2018] E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y. Yang, D. Liu, A. Liang, Q. Xu, _et al._ , Giant anomalous hall effect in a ferromagnetic kagome-lattice semimetal, Nature physics 14, 1125 (2018).
* Manna _et al._ [2018] K. Manna, L. Muechler, T.-H. Kao, R. Stinshoff, Y. Zhang, J. Gooth, N. Kumar, G. Kreiner, K. Koepernik, R. Car, _et al._ , From colossal to zero: controlling the anomalous hall effect in magnetic heusler compounds via berry curvature design, Physical Review X 8, 041045 (2018).
* Kübler and Felser [2012] J. Kübler and C. Felser, Berry curvature and the anomalous hall effect in heusler compounds, Physical Review B 85, 012405 (2012).
* Markou _et al._ [2020] A. Markou, J. Gayles, E. Derunova, P. Swekis, J. Noky, L. Zhang, M. N. Ali, Y. Sun, and C. Felser, Hard magnetic topological semimetals in xpt3: Harmony of berry curvature, arXiv preprint arXiv:2008.11914 (2020).
* Singh _et al._ [2020] H. K. Singh, I. Samathrakis, N. M. Fortunato, J. Zemen, C. Shen, O. Gutfleisch, and H. Zhang, Multifunctional antiperovskites driven by strong magnetostructural coupling, arXiv preprint arXiv:2009.06440 (2020).
* Xu _et al._ [2020] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. Vergniory, N. Regnault, Y. Chen, C. Felser, and B. A. Bernevig, High-throughput calculations of magnetic topological materials, Nature 586, 702 (2020).
* [59] See Supplemental Material for Table S1 summarizing all calculated AHC, ANC, anisotropy and reported literature values for all 266 compounds and for Table S2 illustrating the the Berry curvature transformation under all symmetry operations of Magnetic Laue group $4/mm^{\prime}m^{\prime}$..
|
2020 082 19 May
$p_{\rm T}$ and ESE dependence of D-meson $v_{\mathrm{n}}$ harmonics
ALICE Collaboration††thanks: See Appendix A for the list of collaboration
members ALICE Collaboration
The elliptic and triangular flow coefficients $v_{2}$ and $v_{3}$ of prompt
${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ mesons were measured at
midrapidity ($|y|<0.8$) in Pb–Pb collisions at the centre-of-mass energy per
nucleon pair of $\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\
\mathrm{TeV}$ with the ALICE detector at the LHC. The D mesons were
reconstructed via their hadronic decays in the transverse momentum interval
$1<p_{\rm T}<36\leavevmode\nobreak\ \mathrm{GeV}/c$ in central (0–10%) and
semi-central (30–50%) collisions. Compared to pions, protons, and ${\rm
J}/\psi$ mesons, the average D-meson $v_{\mathrm{n}}$ harmonics are compatible
within uncertainties with a mass hierarchy for $p_{\rm T}\lesssim
3\leavevmode\nobreak\ \mathrm{GeV}/c$, and are similar to those of charged
pions for higher $p_{\rm T}$. The coupling of the charm quark to the light
quarks in the underlying medium is further investigated with the application
of the event-shape engineering (ESE) technique to the D-meson $v_{2}$ and
$p_{\rm T}$-differential yields. The D-meson $v_{2}$ is correlated with
average bulk elliptic flow in both central and semi-central collisions. Within
the current precision, the ratios of per-event D-meson yields in the ESE-
selected and unbiased samples are found to be compatible with unity. All the
measurements are found to be reasonably well described by theoretical
calculations including the effects of charm-quark transport and the
recombination of charm quarks with light quarks in a hydrodynamically
expanding medium.
## 1 Introduction
The formation of a strongly coupled colour-deconfined medium in ultra-
relativistic heavy-ion collisions, called quark–gluon plasma (QGP), has been
established both at RHIC and LHC energies [1, 2]. The QGP behaves as a near-
perfect fluid with small shear viscosity over entropy density ratio, $\eta/s$,
undergoing an expansion that can be described by relativistic hydrodynamics
[3].
In heavy-ion collisions, heavy quarks (charm and beauty) are predominantly
produced via hard-scattering processes on a time scale shorter than the QGP
formation time [4, 5], and therefore they experience all the stages of the
system evolution, interacting with the medium constituents via both elastic
(collisional) [6] and inelastic (gluon radiation) [7, 8, 9] processes. The
measurement of the suppression of the yield of heavy-flavour hadrons in
central nucleus–nucleus collisions relative to pp collisions scaled by the
number of nucleon–nucleon collisions at both RHIC [10, 11, 12, 13, 14] and LHC
energies [15, 16, 17, 18, 19, 20, 21] provides compelling evidence of heavy-
quark energy loss in deconfined strongly interacting matter.
Additional insights into the QGP properties can be obtained by measuring the
azimuthal anisotropy of heavy-flavour hadrons. In non-central nucleus–nucleus
collisions the initial spatial anisotropy of the overlap region is converted
via multiple interactions into an azimuthally anisotropic distribution in the
momentum space of the produced particles [22, 23]. This anisotropy is
characterised in terms of the Fourier coefficients
$v_{\mathrm{n}}=\langle\cos[\mathrm{n}(\varphi-\Psi_{\mathrm{n}})]\rangle$,
where $\varphi$ is the azimuthal angle of the particle and $\Psi_{\mathrm{n}}$
is the azimuthal angle of the symmetry plane for the nth-order harmonic [23,
24]. The values of the Fourier coefficients depend on the geometry of the
collision, the fluctuations in the distributions of nucleons and gluons within
the nuclei [25], and the dynamics of the expansion. The second order flow
coefficient $v_{\mathrm{2}}$, called elliptic flow, is related to the almond-
shaped geometry of the overlap region between the colliding nuclei and,
consequently, is the largest contribution to the anisotropy in non-central
collisions. The third harmonic coefficient $v_{\mathrm{3}}$, named triangular
flow, originates from event-by-event fluctuations in the initial distribution
of nucleons and gluons in the overlap region [26]. In particular, the
measurement of the azimuthal anisotropy of heavy-flavour hadrons at low
$p_{\rm T}$ can help quantify the extent to which charm and beauty quarks
participate in the collective expansion of the medium [27], as well as the
fraction of heavy-flavour hadrons hadronising via recombination with flowing
light quarks [28, 29]. At high $p_{\rm T}$, instead, the charm hadron
azimuthal anisotropy can constrain the path-length dependence of heavy-quark
in-medium energy loss [30, 31]. Precise measurements of heavy-flavour
$v_{\mathrm{n}}$ coefficients are useful to constrain the parameters of models
that implement the heavy-quark transport in the QGP. In this context, the
heavy-quark spatial diffusion coefficient $D_{s}$ in the QGP is particularly
interesting, since it is related to the relaxation (equilibration) time of
heavy quarks $\tau_{\mathrm{Q}}=(m_{\mathrm{Q}}/T)D_{s}$, where
$m_{\mathrm{Q}}$ is the quark mass and $T$ is the medium temperature [32].
Further investigation into the dynamics of heavy quarks in the medium can be
performed with the event-shape engineering (ESE) technique [33], which allows
for selection of events with the same centrality but different initial
geometry on the basis of the magnitude of the average bulk flow. In fact,
hydrodynamic calculations show that the average flow of the bulk of soft
hadrons is proportional to the initial-state eccentricities [34] for small
values of $\eta/s$ [3, 35, 36]. By classifying the events with the ESE
technique it is possible to investigate the correlation between the flow
coefficients of D mesons and soft hadrons. According to the available
calculations [37, 38, 34], the initial system ellipticity is converted into
parton flow with a similar efficiency for bulk and charm quarks, despite the
different production mechanisms, dynamics, and hadronisation of heavy quarks
and light partons forming the bulk of the medium. Moreover, the measurement of
the D-meson spectra in events with different average eccentricity provides
information about the possible correlation between the radial and elliptic
flows at low-intermediate $p_{\rm T}$, and the charm-quark energy loss and the
elliptic flow at high $p_{\rm T}$. The correlation with the radial flow is
expected to be present from the observation of the scaling of the flow
harmonics with the particle mass [39], while the correlation with the in-
medium energy loss would be motivated by the different path traversed by the
charm quark in the medium in the case of an isotropic or an eccentric system.
A positive D-meson $v_{2}$ is observed at both RHIC [10, 40, 41] and the LHC
[42, 43, 44, 45, 46, 47, 48]. The comparison of the D-meson $v_{2}$ with the
charged-pion $v_{2}$ and with theoretical models [49, 50, 51, 52, 53, 54, 55,
56, 57] indicates that charm quarks participate in the collective expansion of
the medium and that both collisional processes and the recombination of charm
and light quarks contribute to the observed elliptic flow. Furthermore, a
positive ${\rm D^{0}}$-meson $v_{3}$ was measured by the CMS Collaboration
[47]. The $p_{\rm T}$-differential yields and $v_{2}$ of D mesons were
measured by the ALICE Collaboration in samples of events selected on the basis
of the average bulk elliptic flow with the ESE technique [48]. A correlation
between the D-meson $v_{2}$ and the $v_{2}$ of the bulk of light hadrons was
observed, while the ratio of the $p_{\rm T}$-differential yields in ESE-
selected samples to the yields measured without any ESE selection was found to
be compatible with unity within the large uncertainties.
In this Letter, the measurement of the non-strange D-meson flow harmonics
performed on a large sample of Pb–Pb collisions at
$\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02$ TeV collected by ALICE in 2018 is
reported. With this data sample, the D-meson $v_{2}$ is measured with the
Scalar Product (SP) method in an extended $p_{\rm T}$ interval and with
smaller uncertainties with respect to the previous results obtained with the
Event Plane (EP) method described in [46, 48] in the 30–50% centrality class.
The results obtained with the SP and the EP method were found to be compatible
between each other, as reported in previous publications [43]. The measurement
of the D-meson $v_{2}$ coefficient in the 0–10% centrality class and $v_{3}$
coefficient in the 0–10% and 30–50% centrality classes are also presented. In
addition, the measurement of the $v_{2}$ and the modification of the $p_{\rm
T}$ distributions in the ESE-selected samples is reported in narrower classes
of the average event flow with respect to [48]. The measurements are compared
to theoretical calculations in order to assess information about the
participation of the charm quark in the collective motion of the system and
its interactions with the QGP constituents.
## 2 Detector and data sample
A detailed description of the ALICE apparatus and data acquisition framework
can be found in [58, 59]. The main detectors used for this analysis are the
Inner Tracking System (ITS) [60], the Time Projection Chamber (TPC) [61], and
the Time-Of-Flight (TOF) detector [62]. The ITS is a six-layer silicon
detector which provides the event selection, the reconstruction of primary and
secondary vertices, and the tracking of charged particles. The TPC detector is
used for the track reconstruction and the particle identification (PID) via
the measurement of the specific energy loss ${\rm d}E/{\rm d}x$, while the TOF
detector provides PID via the measurement of the flight time of the particles.
These detectors are located inside a solenoid providing a uniform magnetic
field of 0.5 T parallel to the LHC beam direction and cover the pseudorapidity
interval $|\eta|<$ 0.9. A minimum-bias interaction trigger was provided by the
coincidence of signals in the two scintillator arrays of the V0 detector [63],
covering the full azimuth in the pseudorapidity regions $-3.7<\eta<-1.7$ (V0C)
and $2.8<\eta<5.1$ (V0A). An online selection based on the V0 signal
amplitudes was applied in order to enhance the sample of central and mid-
central collisions through two separate trigger classes. Background events
from beam–gas interactions were removed offline using the time information
provided by the V0 and the neutron Zero-Degree Calorimeters (ZDC) [64]. Only
events with a primary vertex reconstructed within $\pm 10$ cm from the centre
of the detector along the beam line were considered in the analysis.
Events were divided into centrality classes, defined in terms of percentiles
of the hadronic Pb–Pb cross section, using the amplitudes of the signals in
the V0 arrays. The number of events in each centrality class considered for
this analysis (0–10% and 30–50%) is about $100\times 10^{6}$ and $85\times
10^{6}$, corresponding to an integrated luminosity of $\simeq
130\leavevmode\nobreak\ \mu\mathrm{b}^{-1}$ and $\simeq 56\leavevmode\nobreak\
\mu\mathrm{b}^{-1}$, respectively [65]. In order to apply the ESE technique,
the events in each centrality class were further divided into samples with
different average elliptic anisotropy of final-state particles, selected
according to the magnitude of the second-order harmonic reduced flow vector
$q_{2}$ [36], defined as
$q_{2}=|\boldsymbol{Q}_{2}|/\sqrt{M},$ (1)
where $M$ is the number of tracks used in the $|\boldsymbol{Q}_{2}|$
calculation selected as described below, and
$\boldsymbol{Q}_{2}=\sum_{k=1}^{M}e^{i2\varphi_{k}}$ (2)
is a vector built from the azimuthal angles ($\varphi_{k}$) of the considered
particles. The $\boldsymbol{Q}_{2}$ vector was measured using charged tracks
reconstructed in the TPC with $|\eta|<0.8$ and $0.2<p_{\rm
T}<5\leavevmode\nobreak\ \mathrm{GeV}/c$ to exploit the $\varphi$ resolution
of the TPC and the large multiplicity at midrapidity, which are crucial to
maximise the selectivity of $q_{2}$ with respect to the final state flow
eccentricity [66, 48]. The denominator in Eq. 1 is introduced to remove the
dependence of $|\boldsymbol{Q}_{2}|$ on $\sqrt{M}$ in the absence of flow
[36]. The tracks used to form the D-meson candidates were excluded from the
computation of $q_{2}$ to partially remove autocorrelations between D mesons
and $q_{2}$. The effect of residual autocorrelations between the D mesons and
$q_{2}$ was studied in [48] by computing $q_{2}$ from the azimuthal
distribution of the energy deposition measured in the V0A detector, and hence
introducing a pseudorapidity gap of two units between the D mesons and
$q_{2}$. The $v_{\mathrm{2}}$ values obtained with the $q_{2}$ calculated with
TPC tracks and using the V0 detector were found to be compatible with a
reduction of the eccentricity discriminating power of the two detectors
without allowing for a firm conclusion on the magnitude of non-flow
contamination. The same study was repeated for the data sample used for this
analysis, leading to the same conclusions.
The selection of the events according to the average elliptic flow of the
event was performed by defining $q_{2}$ percentiles in 1%-wide centrality
intervals as described in [48] and [67] to avoid the introduction of biases in
the centrality (multiplicity) distribution of the selected events. The ESE-
selected classes defined for the analyses presented in this paper correspond
to the 20% of events with smallest and largest $q_{2}$, respectively, and will
be indicated as “small-$q_{2}$” and “large-$q_{2}$”. In case of no ESE
selection, the term “unbiased” will be used.
## 3 Analysis technique
The charmed mesons were reconstructed at midrapidity via the decay channels
${\rm D}^{0}\to{\rm K}^{-}\pi^{+}$ (with branching ratio, $\mathrm{BR}=3.89\pm
0.04\%$), ${\rm D}^{+}\to{\rm K}^{-}\pi^{+}\pi^{+}$ ($\mathrm{BR}=8.98\pm
0.28\%$), and ${\rm D}^{*+}\to{\rm D}^{0}\pi^{+}$ ($\mathrm{BR}=67.7\pm
0.5\%$) and their charge conjugates [68]. ${\rm D^{0}}$ and ${\rm D^{+}}$
candidates were built combining pairs and triplets of tracks with the proper
charge, $p_{\rm T}>0.4\leavevmode\nobreak\ \mathrm{GeV}/c$, $|\eta|<$ 0.8, a
fit quality $\chi^{2}$/ndf $<$ 2 in the TPC (where ndf is the number of
degrees of freedom involved in the track fit procedure), at least 70 (out of
159) associated space points in the TPC, and a minimum number of two hits in
the ITS, with at least one in the two innermost layers. ${\rm D^{*+}}$
candidates were formed by combining ${\rm D^{0}}$ candidates with low-$p_{\rm
T}$ tracks, referred to here as “soft pions”, which were required to have
$p_{\rm T}>$ 0.1 $\mathrm{GeV}/c$, $|\eta|<$ 0.8, and at least three
associated hits in the ITS. These selections limit the D-meson acceptance in
rapidity, which drops to zero for $|y|>0.6$ for $p_{\rm
T}=1\leavevmode\nobreak\ \mathrm{GeV}/c$ and $|y|>0.8$ for $p_{\rm
T}>5\leavevmode\nobreak\ \mathrm{GeV}/c$. A $p_{\rm T}$-dependent fiducial
acceptance cut, $|y_{\mathrm{D}}|<y_{\rm fid}(p_{\rm T})$, was therefore
applied, defined as a second-order polynomial function increasing from 0.6 to
0.8 in the range $1<p_{\rm T}<5\leavevmode\nobreak\ \mathrm{GeV}/c$, and fixed
to 0.8 for $p_{\rm T}>5\leavevmode\nobreak\ \mathrm{GeV}/c$.
The D-meson candidate selection approach adopted to reduce the combinatorial
background is similar to that used in previous analyses [43, 46]. The analysis
procedure searches for decay vertices displaced from the primary vertex,
exploiting the mean proper decay lengths of about 123 and 312 $\mathrm{\mu m}$
for ${\rm D^{0}}$ and ${\rm D^{+}}$ mesons, respectively [68]. The variables
mainly used to distinguish between signal and background candidates are based
on the separation between the primary and decay vertices, the displacement of
the tracks from the primary vertex, and the pointing angle of the
reconstructed D-meson momentum to the primary vertex, and are the same as
described in [69, 21]. In the strong decay of the ${\rm D^{*+}}$ meson the
primary vertex cannot be differentiated from the secondary vertex. Therefore
the geometrical selections were applied on the secondary vertex topology of
the produced ${\rm D^{0}}$ mesons. The optimisation of the selection criteria
for each D-meson species was performed as a function of $p_{\rm T}$ and
independently for the two centrality classes. Further reduction of the
combinatorial background was obtained by applying PID for the daughter tracks
with the TPC and TOF detectors. A selection in units of resolution ($\pm
3\,\sigma$) was applied on the difference between the measured and expected
signals of pions and kaons for both ${\rm d}E/{\rm d}x$ and time-of-flight.
The same selections are applied both for the unbiased and the ESE-selected
measurements.
The D-meson elliptic and triangular flow coefficients, $v_{2}$ and $v_{3}$,
were measured using the Scalar Product (SP) method [36, 70, 71]. For each
D-meson candidate, the $v_{\mathrm{n}}$ coefficients can be expressed in terms
of the $Q_{\mathrm{n}}$ vectors, introduced in Sec. 2, as
$v_{\mathrm{n}}\\{\mathrm{SP}\\}=\langle\langle\boldsymbol{u}_{\mathrm{n}}\cdot\frac{\boldsymbol{Q}_{\mathrm{n}}^{\rm
A*}}{M^{\rm
A}}\rangle\rangle\bigg{/}\sqrt{\frac{\langle\frac{\boldsymbol{Q}_{\mathrm{n}}^{\rm
A}}{M^{\rm A}}\cdot\frac{\boldsymbol{Q}_{\mathrm{n}}^{\rm B*}}{M^{\rm
B}}\rangle\langle\frac{\boldsymbol{Q}_{\mathrm{n}}^{\rm A}}{M^{\rm
A}}\cdot\frac{\boldsymbol{Q}_{\mathrm{n}}^{\rm C*}}{M^{\rm
C}}\rangle}{\langle\frac{\boldsymbol{Q}_{\mathrm{n}}^{\rm B}}{M^{\rm
B}}\cdot\frac{\boldsymbol{Q}_{\mathrm{n}}^{\rm C*}}{M^{\rm C}}\rangle}},$ (3)
where u${}_{\mathrm{n}}=e^{i\mathrm{n}\varphi_{\mathrm{D}}}$ is the unit flow
vector of the D-meson candidate with azimuthal angle $\varphi_{\mathrm{D}}$
and the symbol "*" denotes the complex conjugation. The denominator is called
the resolution ($R_{\rm n}$) and is calculated with the formula introduced in
[36], where the three subevents, indicated as A, B, and C, are defined by the
particles measured in the V0C, V0A, and TPC detectors, respectively.
$\boldsymbol{Q}^{k}_{\mathrm{n}}$ is the subevent flow vector corresponding to
the nth-order harmonic for the subevent $k$, and $M^{k}$ represents the
subevent multiplicity. This is defined as the sum of the amplitudes measured
in each channel for the V0A and the V0C. For the V0A and V0C detectors, the
$Q_{\mathrm{n}}$ vectors were calculated from the azimuthal distribution of
the energy deposition, and their components are given by
$Q_{\mathrm{n},x}^{\text{V0A or V0C}}=\sum_{k=1}^{N_{\rm
sectors}}w_{k}\cos(\mathrm{n}\varphi_{k}),\quad Q_{\mathrm{n},y}^{\text{V0A or
V0C}}=\sum_{k=1}^{N_{\rm sectors}}w_{k}\sin(\mathrm{n}\varphi_{k}),$ (4)
where the sum runs over the 32 sectors ($N_{\rm sectors}$) of the V0A or V0C
detector, $\varphi_{k}$ is the azimuthal angle of the centre of the sector
$k$, and $w_{k}$ is the amplitude measured in sector $k$, once the gain of the
single channels is equalised and the recentering is applied to correct effects
of non-uniform acceptance [72]. For the TPC detector, the $Q_{\mathrm{n}}$
vectors were computed using Eq. 2. The single bracket $\langle\rangle$ in Eq.
3 refers to an average over all the events, while the double brackets
$\langle\langle\rangle\rangle$ denote the average over all particles in the
considered $p_{\rm T}$ interval and all events. The $R_{\rm n}$ is obtained as
a function of the collision centrality.
The $v_{\mathrm{n}}$ of the D mesons cannot be directly measured using Eq. 3
as ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ cannot be identified on a
particle-by-particle basis. Therefore, a simultaneous fit to the invariant-
mass spectrum and the $v^{\mathrm{tot}}_{\mathrm{n}}$ distribution as a
function of the invariant mass ($M_{\mathrm{D}}$) was performed in each
$p_{\rm T}$ interval for ${\rm D^{0}}$ and ${\rm D^{+}}$ candidates separately
in order to measure the raw yields and the flow coefficients. For the ${\rm
D^{*+}}$ case the distributions are studied as a function of the mass
difference $\Delta M=M(\mathrm{K}\pi\pi)-(\mathrm{K}\pi)$. The measured
anisotropic flow coefficient, $v^{\mathrm{tot}}_{\mathrm{n}}$, can be written
as a weighted sum of the $v_{\mathrm{n}}$ of the D-meson candidate,
$v^{\mathrm{sig}}_{\mathrm{n}}$, and that of background,
$v^{\mathrm{bkg}}_{\mathrm{n}}$ [73] as
$v^{\mathrm{tot}}_{\mathrm{n}}(M_{\mathrm{D}})=v^{\mathrm{sig}}_{\mathrm{n}}\frac{N^{\mathrm{sig}}}{N^{\mathrm{sig}}+N^{\mathrm{bkg}}}(M_{\mathrm{D}})+v^{\mathrm{bkg}}_{\mathrm{n}}(M_{\mathrm{D}})\frac{N^{\mathrm{bkg}}}{N^{\mathrm{sig}}+N^{\mathrm{bkg}}}(M_{\mathrm{D}}),$
(5)
where $N^{\mathrm{sig}}$ and $N^{\mathrm{bkg}}$ are the raw signal and
background yields, respectively. The fit function for the invariant-mass
distributions was composed of a Gaussian term to describe the signal and an
exponential distribution for the background for ${\rm D^{0}}$ and ${\rm
D^{+}}$ candidates, while for the ${\rm D^{*+}}$ candidates the background was
described by the function $a\sqrt{\Delta M-m_{\pi}}e^{b(\Delta M-m_{\pi})}$,
where $a$ and $b$ are free parameters. In the case of the ${\rm D^{0}}$
invariant mass the contribution of signal candidates with the reflected
K-$\pi$ mass assignment was taken into account with an additional term. Its
invariant-mass distribution was parameterised with a double-Gaussian
distribution based on Monte Carlo (MC) simulations [74, 48, 69, 46, 43]. In
the MC simulation, the underlying Pb–Pb events at
$\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\ \mathrm{TeV}$
were simulated using the HIJING v1.383 generator [75] and $\rm c\overline{c}$
or $\rm b\overline{b}$ pairs were added with the PYTHIA 6.4.25 generator [76]
with Perugia-2011 tune [77]. In the simultaneous fit, the $v_{\mathrm{n}}$
parameter for the candidates with wrong K-$\pi$ mass assignment was set to be
equal to $v^{\mathrm{sig}}_{\mathrm{n}}$, provided that the origin of these
candidates are real ${\rm D^{0}}$ mesons. The $v^{\mathrm{sig}}_{\mathrm{n}}$
is measured from the fit to the $v^{\mathrm{tot}}_{\mathrm{n}}$ distribution
with the function of Eq. 5, where $v^{\mathrm{bkg}}_{\mathrm{n}}$ is a linear
function for ${\rm D^{+}}$ and ${\rm D^{*+}}$ mesons, and ${\rm D^{0}}$ mesons
with $p_{\rm T}>4\leavevmode\nobreak\ \mathrm{GeV}/c$. For the ${\rm D^{0}}$
candidates with $p_{\rm T}<4\leavevmode\nobreak\ \mathrm{GeV}/c$, a second-
order polynomial function was used instead. Figure 1 shows the simultaneous
fit to the invariant-mass spectrum and $v_{2}^{\mathrm{tot}}(M_{\mathrm{D}})$
in the $p_{\rm T}$ intervals 3–4 $\mathrm{GeV}/c$ for ${\rm D^{0}}$, 5–6
$\mathrm{GeV}/c$ for ${\rm D^{+}}$, and 8–10 $\mathrm{GeV}/c$ for ${\rm
D^{*+}}$ in the 30–50% centrality class.
Figure 1: Simultaneous fits to the invariant-mass spectrum and $v_{2}$
($M_{\rm{D}}$) of ${\rm D^{0}}$ (left panel), ${\rm D^{+}}$ (middle panel),
and ${\rm D^{*+}}$ (right panel) meson candidates in the $3<p_{\rm
T}<4\leavevmode\nobreak\ \mathrm{GeV}/c$, $5<p_{\rm T}<6\leavevmode\nobreak\
\mathrm{GeV}/c$, and $8<p_{\rm T}<10\leavevmode\nobreak\ \mathrm{GeV}/c$
intervals, respectively, for the 30–50% centrality class. The solid blue and
the dotted red curves represent the total and the combinatorial-background fit
functions, respectively. For the ${\rm D^{0}}$ candidates, the green dashed
curve represents the contribution of the reflected signal.
The reconstructed D-meson signal is a mixture of prompt D mesons from
$c$-quark hadronisation or strong decays of excited charmonium or open-charm
states, and D mesons from beauty-hadron decays, called “feed-down" in the
following. The $v^{\mathrm{sig}}_{\mathrm{n}}$ is therefore a linear
combination of prompt ($v^{\mathrm{prompt}}_{\mathrm{n}}$) and feed-down
($v^{\mathrm{feed\text{-}down}}_{\mathrm{n}}$) contributions, and can be
expressed as
$v^{\mathrm{sig}}_{\mathrm{n}}=f_{\mathrm{prompt}}v^{\mathrm{prompt}}_{\mathrm{n}}+(1-f_{\mathrm{prompt}})v^{\mathrm{feed\text{-}down}}_{\mathrm{n}},$
(6)
where $f_{\mathrm{prompt}}$ is the fraction of promptly produced D mesons
estimated as a function of $p_{\rm T}$ with the theory-driven method described
in [21]. This method uses (i) FONLL calculations [78, 79] for the production
cross section of beauty hadrons, (ii) the beauty-hadron decay kinematics from
the EvtGen package [80], (iii) the product of efficiency and acceptance $({\rm
Acc}\times\varepsilon)$ from Monte Carlo simulations, and (iv) a hypothesis on
the nuclear modification factor of feed-down D mesons.
The anisotropic flow coefficients of promptly produced D mesons were obtained
assuming
$v^{\mathrm{feed\text{-}down}}_{\mathrm{n}}=v^{\mathrm{prompt}}_{\mathrm{n}}/2$.
The hypothesis is based on the measurement of the non-prompt ${\rm J}/\psi$
performed by CMS [19] and on the available model calculations [49, 81, 82],
that indicate
$0<v^{\mathrm{feed\text{-}down}}_{\mathrm{n}}<v^{\mathrm{prompt}}_{\mathrm{n}}$.
For the measurement of the modification of the $p_{\rm T}$-differential
distributions of D mesons in the ESE-selected samples compared to the unbiased
sample, the raw yields were extracted via fits to the invariant-mass
distributions of ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ candidates
and normalised to the corresponding number of events in the corresponding ESE-
selected sample. The same functions adopted in the simultaneous fits for the
invariant-mass distributions were used. The extracted raw yields were not
corrected for the efficiency in the ratio calculation, under the assumption
that the reconstruction and selection efficiencies do not depend on $q_{2}$.
This assumption was previously verified in [48].
## 4 Systematic uncertainties
The D-meson $v_{\mathrm{n}}$ coefficients are affected by the systematic
uncertainties due to (i) the signal extraction from the invariant-mass and
$v^{\mathrm{tot}}_{\mathrm{n}}$ distributions, (ii) the beauty feed-down
contribution, (iii) on the selection of the centrality interval in which
$R_{\rm n}$ is calculated, and (iv), for the ESE-selected samples, the
uncertainties due to possible bias caused by the presence of auto-correlation
effects between the subevents used for $R_{\rm n}$ and $q_{2}$ calculations.
The uncertainty due to the simultaneous fit was estimated by repeating the fit
several times with different configurations. In particular, the lower and
upper limits of the fit range, the bin width, and the background fit functions
used for the invariant-mass and $v^{\mathrm{tot}}_{\mathrm{n}}$ distributions
were varied. For each configuration the D-meson $v_{\mathrm{n}}$ was
calculated and the absolute systematic uncertainty for each $p_{\rm T}$
interval was assigned as the r.m.s. of the $v_{\mathrm{n}}$ distribution
obtained from the different trials. The absolute systematic uncertainty values
on the $v_{\mathrm{n}}$ are reported in Table 1 and they depend on the D-meson
species, the $p_{\rm T}$ interval and the ESE-selected class. This source of
uncertainty was considered as uncorrelated among the $p_{\rm T}$ intervals and
the centrality classes for the two harmonics. The correlation between the
small-$q_{2}$/large-$q_{2}$ and the unbiased case was investigated and the
outcome indicated that this uncertainty source is uncorrelated between the
different $q_{2}$-selected samples.
For the $p_{\rm T}$-differential yield ratios in ESE-selected samples, the
uncertainty for the signal extraction was estimated using the same approach
described above, directly on the ratio of the yields in the ESE-selected and
unbiased samples, leading to a systematic uncertainty value from 0.7$\%$ to
5$\%$, depending on the $p_{\rm T}$ and the D-meson species.
Table 1: Summary of systematic uncertainties on the measurement of the D-meson $v_{\mathrm{2}}$, in the unbiased and ESE-selected samples, and $v_{\mathrm{3}}$ in Pb–Pb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\ \mathrm{TeV}$. The range of the uncertainties on the fitting procedure and feed-down subtraction are quoted as absolute uncertainties, while those on the $R_{\rm n}$ as relative uncertainty. Systematic uncertainty source | $v_{\mathrm{2}}$ | $v_{\mathrm{3}}$ | $v_{\mathrm{2}}$ small-$q_{2}$ | $v_{\mathrm{2}}$ large-$q_{2}$
---|---|---|---|---
| 0–10%
$M$ and $v_{\mathrm{n}}$ fits | 0.005–0.03 | 0.006–0.03 | 0.006–0.01 | 0.006–0.01
Feed-down | 0.002–0.01 | 0.0007–0.01 | 0.0003–0.006 | 0.003–0.016
$R_{\rm n}$ determination | 3.5$\%$ | negl. | 3.5$\%$ | 3.5$\%$
Autocorrelations on $R_{\rm 2}$ and $q_{2}$ | - | - | 3.5$\%$ | 1$\%$
| 30–50%
$M$ and $v_{\mathrm{n}}$ fits | 0.006–0.025 | 0.01–0.05 | 0.006–0.015 | 0.004–0.015
Feed-down | 0.0004–0.02 | 0.003–0.018 | 0.003–0.01 | 0.004–0.029
$R_{\rm n}$ determination | 0.5$\%$ | 0.5$\%$ | 0.5$\%$ | 0.5$\%$
Autocorrelations on $R_{\rm 2}$ and $q_{2}$ | - | - | 0.5$\%$ | 0.5$\%$
The systematic uncertainty source related to the beauty feed-down correction
has two main contributions. The first is due to the $f_{\mathrm{prompt}}$
calculation and it was studied by varying the quark mass and the
renormalisation and factorisation scales in the FONLL calculations, the
$R_{\rm AA}^{\mathrm{feed\text{-}down}}$ hypothesis as reported in [21]. The
second contribution is due to the assumption of
$v^{\mathrm{feed\text{-}down}}_{\mathrm{n}}$ =
$v^{\mathrm{prompt}}_{\mathrm{n}}$/2, previously described in Sec. 3, and was
estimated by assuming a flat distribution of
$v^{\mathrm{feed\text{-}down}}_{\mathrm{n}}$ between 0 and
$v^{\mathrm{prompt}}_{\mathrm{n}}$ and by varying the central value of
$v^{\mathrm{feed\text{-}down}}_{\mathrm{n}}$ by $\pm
v^{\mathrm{prompt}}_{\mathrm{n}}$/$\sqrt{12}$. The values of the absolute
systematic uncertainty from the beauty feed-down correction are reported in
Table 1 and they depend on the D-meson species, the $p_{\rm T}$ interval and
the ESE-selected class. The uncertainty due to the beauty feed-down correction
was assumed to be fully correlated among the $p_{\rm T}$ bins for the measured
$v_{\mathrm{n}}$ coefficients in the same centrality class.
The non-flow effects are naturally suppressed because of the pseudorapidity
gap of at least 0.9 units between the pseudorapidity interval used for the
D-meson reconstruction, and the V0C used for the $Q_{\mathrm{n}}$-vector
determination. Furthermore, the auto-correlation effect due to the usage of
the TPC tracks for the $q_{2}$ estimate has been discussed in Sec. 2 and the
related systematic uncertainty was found to be negligible, as described in
[48].
The contribution of the $R_{\rm n}$ to the systematic uncertainty is due to
the centrality dependence. The central value of $R_{\rm n}$ was estimated
using the three subevent formula, as described in Sec. 3, averaged over the
events in the 0–10% and 30–50% intervals. The uncertainty was evaluated as the
difference of the centrality integrated $R_{\rm n}$ values with those obtained
as weighted averages of $R_{\rm n}$ values in narrow centrality intervals
using the D-meson yields as weights. A systematic uncertainty of 3.5$\%$ and
0.5$\%$ was assigned on $R_{\rm 2}$ in the 0–10% and 30–50% centrality classes
and for all ESE-selected samples. For the $R_{3}$, an uncertainty of 0.5$\%$
was assigned in the 30–50% interval while it was found to be negligible for
the 0–10% class. The uncertainty associated with the resolution factor is
smaller for the third harmonic than for the second harmonic, due to the milder
centrality dependence of $R_{3}$ compared with that of $R_{2}$.
For the ESE-selected samples an additional source of systematic uncertainty on
the resolution originates from auto-correlations due to the usage of the TPC
tracks both for $q_{2}$ and $R_{\rm 2}$ determination. This potential bias is
assessed by replacing the ratio $\langle\boldsymbol{Q}_{\mathrm{n}}^{\rm
V0C}/M^{\rm V0C}\cdot\boldsymbol{Q}_{\mathrm{n}}^{\rm TPC*}/M^{\rm
TPC}\rangle$/$\langle\boldsymbol{Q}_{\mathrm{n}}^{\rm V0A}/M^{\rm
V0A}\cdot\boldsymbol{Q}_{\mathrm{n}}^{\rm TPC*}/M^{\rm TPC}\rangle$ in Eq. 3
with the one from the $q_{2}$-integrated analysis, following the same approach
used for the ${\rm J}/\psi$ azimuthal anisotropy measurement [83]. In this
case, the systematic uncertainty was estimated to be $3.5\%$ for the
small-$q_{2}$ and $1\%$ for the large-$q_{2}$ samples in the 0–10% centrality
class, and $0.5\%$ for both $q_{2}$-selected classes in the 30–50% centrality
class, as reported in Table 1. The last two sources of systematic uncertainty,
related to the resolution, are considered to be fully correlated among the
different $p_{\rm T}$ intervals.
For the analysis of the $p_{\rm T}$-differential yield ratios in ESE-selected
and unbiased samples the reconstruction efficiency was verified to be
independent of $q_{2}$. Consequently, it cancels out in the ratio of the two
ESE-selected classes.
## 5 Results
### 5.1 Unbiased flow harmonics
Figure 2 shows the average $v_{2}$ (top panels) and $v_{3}$ (bottom panels)
coefficients of prompt ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ mesons
measured in the unbiased sample as a function of $p_{\rm T}$ in the 0–10%
(left panels) and 30–50% (right panels) centrality classes. The average
$v_{\mathrm{n}}$ of prompt ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$
mesons was computed by using the inverse squared absolute statistical
uncertainties as weights, after having compared their compatibility [67]. The
systematic uncertainties were propagated to the average by considering the
contributions from the centrality dependence of the $R_{\rm n}$ resolution and
the correction for the beauty feed-down component in the D-meson yields as
correlated among the D-meson species. The D-meson $v_{\mathrm{n}}$ harmonics
are compared to the corresponding coefficients measured for charged pions and
protons at midrapidity ($|y|<0.5$) [39] as well as to inclusive ${\rm J}/\psi$
mesons at forward rapidity ($2.5<y<4$) [84].
Figure 2: Average $v_{2}$ (top panels) and $v_{3}$ (bottom panels)
coefficients of prompt ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ mesons
as a function of $p_{\rm T}$ for Pb–Pb collisions at
$\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\ \mathrm{TeV}$ in
the 0–10% (left panels) and 30–50% (right panels) centrality classes. The
$v_{2}$ and $v_{3}$ of $\pi^{\pm}$, $\mathrm{p+\bar{p}}$ [39] and inclusive
${\rm J}/\psi$ mesons [84] measured at the same centre-of-mass energy and in
the same centrality classes are shown for comparison.
The D-meson elliptic flow increases significantly from central to semi-central
collisions, as expected from the increasing eccentricity of the interaction
region. Conversely, the triangular flow is compatible in the two centrality
classes within the large uncertainties, following the milder centrality
dependence of the third flow harmonic observed for light-flavour particles
[39]. For $p_{\rm T}<3\text{--}4\leavevmode\nobreak\ \mathrm{GeV}/c$ ($p_{\rm
T}<4\text{--}5\leavevmode\nobreak\ \mathrm{GeV}/c$) the measured D-meson
$v_{2}$ ($v_{3}$) is lower than that of pions and protons. This observation is
consistent within uncertainties with the hypothesis of a mass hierarchy,
$v_{\mathrm{n}}(\mathrm{D})<v_{\mathrm{n}}(\mathrm{p})<v_{\mathrm{n}}(\pi)$,
in the low $p_{\rm T}$ region ($p_{\rm T}\lesssim 3\leavevmode\nobreak\
\mathrm{GeV}/c$). In semi-central events the $v_{\mathrm{n}}$ coefficients of
${\rm J}/\psi$ mesons seem to follow the mass hierarchy ($v_{\mathrm{n}}({\rm
J}/\psi)<v_{\mathrm{n}}(\mathrm{D})$). In central events the data suggests a
similar behaviour, however within the current uncertainties no firm
conclusions can be drawn. This observation can be explained by the interplay
between the anisotropic flow and the isotropic expansion of the system (radial
flow), which imposes an equal velocity boost to all particles. For $4\lesssim
p_{\rm T}\lesssim 6\text{--}8\leavevmode\nobreak\ \mathrm{GeV}/c$, the D-meson
$v_{\mathrm{n}}$ coefficients are similar to those of charged pions and lower
than those of protons. This observation is consistent with a scaling of the
$v_{\mathrm{n}}$ coefficients with the number of constituent quarks, which
supports the hypothesis of particle production via quark coalescence [85]. In
the same $p_{\rm T}$ interval, for the 30–50% centrality class the larger
values of $v_{\mathrm{n}}$ for D mesons compared to ${\rm J}/\psi$ mesons can
be explained by (i) the hadronisation via coalescence together with the larger
flow coefficients of up and down quarks compared to that of charm quarks [28]
and (ii) the fraction of ${\rm J}/\psi$ mesons coming from beauty-hadron
decays [86, 87], which are expected to have lower $v_{2}$ and $v_{3}$ than
charmed mesons [88, 26]. In the 0–10% centrality class, the current
experimental uncertainties do not allow for firm conclusions on the expected
difference for ${\rm J}/\psi$ and D mesons. The measured $v_{\mathrm{n}}$
coefficients for all the hadron species are compatible within uncertainties
for $p_{\rm T}\gtrsim 8\leavevmode\nobreak\ \mathrm{GeV}/c$. Similar values of
$v_{\mathrm{n}}$ coefficients are expected, because in this kinematic range
the charm-quark mass is small compared to the momentum, and because the path-
length dependence of the in-medium parton energy loss is similar for
high-$p_{\rm T}$ charm quarks and gluons.
Figure 3: Average $v_{2}$ (top panels) and $v_{3}$ (bottom panels)
coefficients of prompt ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ mesons
as a function of $p_{\rm T}$ for Pb–Pb collisions at
$\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\ \mathrm{TeV}$ in
the 0–10% (left panels) and 30–50% (right panels) centrality classes compared
with model calculations [89, 56, 34, 49, 55, 53, 26, 90, 57, 91, 92, 37, 93,
94, 95].
In Fig. 3, the average D-meson $v_{\mathrm{n}}$ coefficients are compared to
theoretical calculations that include the charm-quark transport in a
hydrodynamically expanding medium. The theoretical uncertainties, where
available, are displayed with a coloured band. In the TAMU [89], POWLANG HTL
[56, 34], PHSD [53], Catania [94, 95], and BAMPSel [49] calculations the
interactions between the charm quarks and the medium constituents are modelled
with collisional processes, while the MC@sHQ+EPOS2 [26], LBT [90, 57], LIDO
[91, 92], BAMPSel+rad [55], DAB-MOD(M&T) [37, 93], and LGR [96] models include
also radiative processes. The difference in the variants of the BAMPS model
indicates that in this model elastic collisions are the dominant process that
imparts a positive D-meson $v_{2}$ in the low and intermediate $p_{\rm T}$
region. All the models except for BAMPS include the hadronisation of the charm
quark via coalescence, in addition to the fragmentation mechanism. Initial-
state event-by-event fluctuations are included in the POWLANG HTL, LIDO, PHSD,
MC@sHQ+EPOS2, LBT, and DAB-MOD(M&T) models, which are therefore the only ones
that provide predictions for the triangular flow. Although the models differ
in several aspects related to the interactions both in the QGP and in the
hadronic phase as well as to the medium expansion, most of them provide a fair
description of the measured $v_{\mathrm{n}}$ harmonics. The largest difference
is observed in the $2<p_{\rm T}<6\leavevmode\nobreak\ \mathrm{GeV}/c$ interval
for the $v_{2}$ in the 30–50% centrality class, where most of the models
provide a prediction lower than the measured points. This is more evident for
the LIDO model, which shows a deviation of $5.4\,\sigma$, and BAMPSel+rad,
which underestimates the measured $v_{2}$ by about a factor two with more than
$10\,\sigma$ significance. In contrast to this, BAMPSel overestimates the
measurement by about $3\,\sigma$. The underestimation of the data by the
BAMPSel+rad model can be eventually due to the missing implementation of the
charm-quark coalescence with light quarks from the medium, which seems to be
necessary in the description of the measured $v_{2}$. In the same $p_{\rm T}$
range, the DAB-MOD model overestimates the measured $v_{2}$ in the 0–10%
centrality class by $3.7\,\sigma$. These discrepancies expressed in number of
standard deviations were computed combining the probability to observe a
deviation from the null hypothesis (i.e. the model prediction) for all the
measured points in the $2<p_{\rm T}<6\leavevmode\nobreak\ \mathrm{GeV}/c$
interval, considering both the experimental (statistical and systematic) and
the theoretical uncertainties, when available.
Table 2: Summary of $\chi^{2}/\mathrm{ndf}$ values obtained for the different model predictions compared with the measured D-meson $v_{\mathrm{n}}$ harmonics. Model | $p_{\rm T}$ ($\mathrm{GeV}/c)$ | $\chi^{2}/\mathrm{ndf}$
---|---|---
| $v_{\mathrm{2}}$ | | $v_{\mathrm{3}}$ | global
| 0–10% | 30–50% | | 0–10% | 30–50% |
BAMPSel [49] | [1–24] | - | 31.7/11 | | - | - | -
BAMPSel+rad [55] | [1–24] | - | 203.6/11 | | - | - | -
Catania [94, 95] | [1–12] | 3.1/7 | 14.0/8 | | 15.1/7 | 8.1/4 | 40.3/26
DAB-MOD(M&T) [37, 93] | [1–8] | 24.6/7 | 9.8/6 | | 16.1/7 | 7.1/3 | 57.6/23
LBT [90, 57] | [1–36] | 18.2/11 | 15.8/12 | | 24.9/11 | 8.4/7 | 67.4/41
LIDO [91, 92] | [1–24] | 10.7/10 | 62.0/11 | | 17.8/10 | 12.5/6 | 102.9/37
LGR [96] | [1–24] | - | 15.5/11 | | - | - | -
MC@sHQ+EPOS2 [26] | [1–36] | - | 15.7/12 | | - | - | -
PHSD [53] | [1–24] | 13.2/10 | 19.6/11 | | 7.9/10 | 8.6/6 | 48.9/37
POWLANG HTL [56, 34] | [1–12] | 9.6/7 | 13.5/8 | | 14.6/7 | 8.3/4 | 45.9/26
TAMU [89] | [1–12] | - | 8.15/9 | | - | - | -
The global agreement between the data and the theoretical models was evaluated
by computing the $\chi^{2}/\mathrm{ndf}$, as done in [46]. The values are
reported in Table 2. All the centrality classes and $v_{\mathrm{n}}$ harmonics
were considered when the model predictions were available. Compared to the
results in [46], for almost all the models the $\chi^{2}/\mathrm{ndf}$ is
found to be higher than unity, most likely because of the improved precision
of the measurement. The models that describe the data with
$\chi^{2}/\mathrm{ndf}<2$ are MC@sHQ+EPOS2, LBT, LGR, PHSD, POWLANG, Catania,
and TAMU which is more in agreement with the data compared to [46], thanks to
the improved description of the charm-quark coalescence in its latest version
[89]. These models use a value of heavy-quark spatial diffusion coefficient in
the range $1.5<2\pi D_{s}T_{\mathrm{c}}<7$ at the critical temperature
$T_{\mathrm{c}}=155\leavevmode\nobreak\ \mathrm{MeV}$ [97], which is
consistent with the interval obtained in [46]. It is however important to
consider that not all the theoretical models provide predictions for all the
$v_{\mathrm{n}}$ harmonics in all the centrality classes reported in this
article, hence the global interpretation of these comparisons could not be
conclusive.
### 5.2 Event-shape engineered flow harmonics and $\boldsymbol{p_{\rm
T}}$-differential yields
The average $v_{2}$ of prompt ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$
mesons measured in the ESE-selected samples is shown in Fig. 4 for the 0–10%
(top row) and the 30–50% (bottom row) centrality classes. The measurements in
the small-$q_{2}$ sample are reported in the left column, those in the
large-$q_{2}$ sample in the right column, while the measurements in the
unbiased samples recomputed in the same $p_{\rm T}$ intervals of the ESE
analysis are in the middle column. A reduced $p_{\rm T}$ range ($2<p_{\rm
T}<16\leavevmode\nobreak\ \mathrm{GeV}/c$) and wider $p_{\rm T}$ intervals
compared to the unbiased $v_{2}$ measurement were adopted due to the limited
size of the ESE-selected samples. The average $v_{2}$ among the three D-meson
species was computed as described in Sec. 5.1. In Fig. 5 the ratio between the
average D-meson $v_{2}$ measured in the ESE-selected samples with respect to
that in the unbiased sample is depicted. The statistical uncertainties of the
ratio were calculated taking into consideration the degree of correlation
between the measurements in the ESE-selected and unbiased samples. The
systematic uncertainties arising from the centrality dependence of $R_{\rm
n}$, the non-flow contaminations among sub-events, and the correction for the
beauty feed-down contribution were considered as fully correlated.
Figure 4: Average of prompt ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$
meson $v_{2}$ as a function of $p_{\rm T}$ in Pb–Pb collisions at
$\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\ \mathrm{TeV}$ in
the small-$q_{2}$, large-$q_{2}$ (see text for details), and unbiased samples,
for the 0–10% (top panels) and 30–50% (bottom panels) centrality classes,
compared to model calculations [34, 91, 37, 93, 94, 95]. In the LIDO, DAB-MOD,
and Catania predictions, the ESE selection is performed with a $q_{2}$
estimator, while in the POWLANG model the elliptic eccentricity $\epsilon_{2}$
is used. Figure 5: Ratio of the average prompt ${\rm D^{0}}$, ${\rm D^{+}}$,
and ${\rm D^{*+}}$ meson $v_{2}$ coefficients measured in the small-$q_{2}$
(left panels) and large-$q_{2}$ (right panels) selected samples with respect
to that of the unbiased sample as a function of $p_{\rm T}$ in Pb–Pb
collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\
\mathrm{TeV}$ for the 0–10% (top panels) and 30–50% (bottom panels) centrality
classes, compared to model calculations [34, 91, 37, 93, 95].
The D-meson $v_{2}$ was found to be on average about 50% higher (lower) in the
20% of the events with largest (smallest) $q_{2}$ in both the 0–10% and 30–50%
centrality classes. No significant centrality dependence was found within the
current uncertainties. The corresponding variation of the average $q_{2}$ in
the small-$q_{2}$ (large-$q_{2}$) sample with respect to the unbiased one was
found to be about 65% (75%) and 60% (65%) for the 0–10% and 30–50% centrality
class, respectively. This confirms the correlation between the D-meson
azimuthal anisotropy and the collective expansion of the bulk matter already
observed in [48]. This modification of the $v_{2}$ coefficient was found to be
independent of $p_{\rm T}$ within uncertainties, which might suggest that the
ESE selection is related to a global property of the events (i.e. a property
that is independent of the measured particle and is related to the entire
event). A similar trend was also observed for light-flavour particles [66].
Figures 4 and 5 also compare the measured $v_{2}$ and $v_{2}$ ratios between
ESE-selected and unbiased samples to the POWLANG, LIDO, DAB-MOD, and Catania
theoretical predictions. For the POWLANG model, both the predictions obtained
with the transport coefficients from weak coupling (Hard Thermal Loop, HTL
[98]) and from lattice QCD calculations (lQCD [99]) are reported. For the DAB-
MOD model, a version based on the heavy-quark transport (M&T [32]) and a
parametric model for the heavy-quark energy loss ($E_{\mathrm{loss}}$ [100])
were considered. In the LIDO, DAB-MOD, and Catania models the ESE selection is
performed with a $q_{2}$ estimator computed starting from generated quantities
[91, 93, 95], while in the POWLANG model the elliptic eccentricity
$\epsilon_{2}$ is directly used [34]. The $v_{2}$ measured in the
small-$q_{2}$ sample is described by all the available models within the
uncertainties. On the contrary, in the 30–50% centrality class the LIDO, DAB-
MOD, and Catania models underestimate the measurement in the large-$q_{2}$
sample, which is instead well described by the POWLANG HTL prediction. In the
case of POWLANG lQCD, the theoretical prediction is compatible with the
measured $v_{2}$ for $p_{\rm T}<4\leavevmode\nobreak\ \mathrm{GeV}/c$ and
lower for higher $p_{\rm T}$. The DAB-MOD calculations give a better
description of the experimental data with the M&T approach for $p_{\rm
T}<5\leavevmode\nobreak\ \mathrm{GeV}/c$ and in the $E_{\mathrm{loss}}$ case
for $p_{\rm T}>5\leavevmode\nobreak\ \mathrm{GeV}/c$. When the ratios between
the $v_{2}$ in the ESE-selected and the unbiased samples are considered, the
models seem to better describe the measured values, owing to similar
discrepancies between theoretical predictions and experimental data in the
ESE-selected and unbiased samples which lead to similar ratio values in the
different models. In the small-$q_{2}$ samples the model predictions are more
similar to each other and the discrepancies are less significant, also due to
the larger experimental uncertainties. Interestingly, different
implementations of the same model with the studied transport parameterisations
(i.e. POWLANG HTL vs. POWLANG lQCD, and DAB-MOD(M&T) vs. DAB-MOD($E_{\rm
loss}$)) give similar predictions, suggesting that the effect of the ESE
selection is more related to the initial geometry and the underlying
hydrodynamic expansion rather than the dynamic evolution of the heavy quarks
in the medium.
Figure 6: Average of the ratio of $p_{\rm T}$-differential ${\rm D^{0}}$,
${\rm D^{+}}$, and ${\rm D^{*+}}$ yields measured in the ESE-selected samples
to those in the unbiased sample in Pb–Pb collisions at
$\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\ \mathrm{TeV}$
for the 0–10% (top panels) and 30–50% (bottom panels) centrality classes,
compared to the POWLANG [34] and LIDO [91, 92] predictions.
To study a possible interplay between the azimuthal anisotropy of the event
and the charm-quark radial flow (at low/intermediate $p_{\rm T}$) and in-
medium energy loss (at high $p_{\rm T}$), the ratio of the measured per-event
yields of prompt ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ mesons in
the ESE-selected and unbiased samples has been calculated as a function of
$p_{\rm T}$ in the range $2<p_{\rm T}<24\leavevmode\nobreak\ \mathrm{GeV}/c$.
The average D-meson ratios, computed by using the inverse of the squared
relative statistical uncertainties as weights, are compared to the POWLANG and
LIDO models in Fig. 6. The POWLANG model predicts a hardening (softening) of
the $p_{\rm T}$ distributions in the large (small)-$q_{2}$ class of events due
to an interplay between the radial and elliptic flows, while no significant
modification is predicted by the LIDO model. Within the current precision, the
measured per-event yield ratios and are found compatible with unity, and hence
to the LIDO model predictions, and with the POWLANG model in the case of lQCD,
while the measured effect seems to be lower than the effect predicted with HTL
transport coefficients.
## 6 Conclusions
The elliptic and triangular flow of ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm
D^{*+}}$ mesons was measured with the SP method at midrapidity ($|y|<0.8$) as
a function of $p_{\rm T}$ in central (0–10%) and semi-central (30–50%) Pb–Pb
collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=5.02\leavevmode\nobreak\
\mathrm{TeV}$.
Compared to other particle species, the average D-meson $v_{\mathrm{n}}$
harmonics were found to be compatible with the hypothesis of a mass hierarchy
for $p_{\rm T}\lesssim 3\leavevmode\nobreak\ \mathrm{GeV}/c$ as observed for
light-flavour hadrons [39]. At intermediate $p_{\rm T}$, the D-meson
$v_{\mathrm{n}}$ is similar to those of charged pions, lower than those of
protons, and higher than those of ${\rm J}/\psi$ mesons, supporting the
hypothesis of charm-quark hadronisation via coalescence. Moreover, the
contribution to the hadronisation of charm quarks from coalescence with light
quarks from the medium seems to be necessary in the theoretical models to
quantitatively reproduce the measured D-meson $v_{\mathrm{n}}$. For $p_{\rm
T}\gtrsim 8\leavevmode\nobreak\ \mathrm{GeV}/c$, the D-meson $v_{2}$ and
$v_{3}$ are compatible within uncertainties with the values measured for the
other particle species, indicating a similar path-length dependence of the
energy loss of high-$p_{\rm T}$ charm quarks and gluons. The comparison of the
measured D-meson $v_{\mathrm{n}}$ with theoretical calculations suggests that
the interactions with the hydrodynamically expanding medium impart a positive
$v_{2}$ and $v_{3}$ to the charm quarks.
The elliptic flow and the modification of the $p_{\rm T}$ distributions of
${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ mesons were also investigated
with the event-shape engineering technique. The D-meson $v_{2}$ was found to
be larger (smaller) in events with larger (smaller) $q_{2}$, confirming the
correlation with average bulk elliptic flow. The ratios of the $p_{\rm
T}$-differential yields measured in the ESE-selected samples and the unbiased
sample were found to be compatible with unity. The measurements in the ESE-
selected samples are qualitatively described by theoretical calculations and
provide new constraints to models based on charm-quark transport in a
hydrodynamically expanding medium and charm-quark energy loss in the QGP.
## Acknowledgements
The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS), Armenia; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of Communications and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de Estudos e Projetos (Finep), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Universidade Federal do Rio Grande do Sul (UFRGS), Brazil; Ministry of Education of China (MOEC) , Ministry of Science & Technology of China (MSTC) and National Natural Science Foundation of China (NSFC), China; Ministry of Science and Education and Croatian Science Foundation, Croatia; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research | Natural Sciences, the VILLUM FONDEN and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat à l’Energie Atomique (CEA) and Institut National de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre National de la Recherche Scientifique (CNRS), France; Bundesministerium für Bildung und Forschung (BMBF) and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy Government of India (DAE), Department of Science and Technology, Government of India (DST), University Grants Commission, Government of India (UGC) and Council of Scientific and Industrial Research (CSIR), India; Indonesian Institute of Science, Indonesia; Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi and Istituto Nazionale di Fisica Nucleare (INFN), Italy; Institute for Innovative Science and Technology , Nagasaki Institute of Applied Science (IIST), Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway, Norway; Commission on Science and Technology for Sustainable Development in the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru; Ministry of Science and Higher Education, National Science Centre and WUT ID-UB, Poland; Korea Institute of Science and Technology Information and National Research Foundation of Korea (NRF), Republic of Korea; Ministry of Education and Scientific Research, Institute of Atomic Physics and Ministry of Research and Innovation and Institute of Atomic Physics, Romania; Joint Institute for Nuclear Research (JINR), Ministry of Education and Science of the Russian Federation, National Research Centre Kurchatov Institute, Russian Science Foundation and Russian Foundation for Basic Research, Russia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; Suranaree University of Technology (SUT), National Science and Technology Development Agency (NSDTA) and Office of the Higher Education Commission under NRU project of Thailand, Thailand; Turkish Atomic Energy Agency (TAEK), Turkey; National Academy of Sciences of Ukraine, Ukraine; Science and Technology Facilities Council (STFC), United Kingdom; National Science Foundation of the United States of America (NSF) and United States Department of Energy, Office of Nuclear Physics (DOE NP), United States of America.
## References
* [1] B. Müller and J. L. Nagle, “Results from the relativistic heavy ion collider,” Ann. Rev. Nucl. Part. Sci. 56 (2006) 93–135, arXiv:nucl-th/0602029 [nucl-th].
* [2] G. Roland, K. Safarik, and P. Steinberg, “Heavy-ion collisions at the LHC,” Prog. Part. Nucl. Phys. 77 (2014) 70–127.
* [3] U. Heinz and R. Snellings, “Collective flow and viscosity in relativistic heavy-ion collisions,” Ann. Rev. Nucl. Part. Sci. 63 (2013) 123–151, arXiv:1301.2826 [nucl-th].
* [4] F.-M. Liu and S.-X. Liu, “Quark-gluon plasma formation time and direct photons from heavy ion collisions,” Phys. Rev. C89 no. 3, (2014) 034906, arXiv:1212.6587 [nucl-th].
* [5] A. Andronic et al., “Heavy-flavour and quarkonium production in the LHC era: from protonproton to heavy-ion collisions,” Eur. Phys. J. C76 no. 3, (2016) 107, arXiv:1506.03981 [nucl-ex].
* [6] E. Braaten and M. H. Thoma, “Energy loss of a heavy quark in the quark - gluon plasma,” Phys. Rev. D44 no. 9, (1991) R2625.
* [7] M. Gyulassy and M. Plümer, “Jet Quenching in Dense Matter,” Phys. Lett. B243 (1990) 432–438.
* [8] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff, “Radiative energy loss and $p_{\perp}$-broadening of high-energy partons in nuclei,” Nucl. Phys. B484 (1997) 265–282, arXiv:hep-ph/9608322 [hep-ph].
* [9] F. Prino and R. Rapp, “Open Heavy Flavor in QCD Matter and in Nuclear Collisions,” J. Phys. G43 no. 9, (2016) 093002, arXiv:1603.00529 [nucl-ex].
* [10] PHENIX Collaboration, A. Adare et al., “Heavy Quark Production in $p+p$ and Energy Loss and Flow of Heavy Quarks in Au+Au Collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 200 GeV,” Phys. Rev. C84 (2011) 044905, arXiv:1005.1627 [nucl-ex].
* [11] STAR Collaboration, B. Abelev et al., “Transverse momentum and centrality dependence of high-$p_{T}$ non-photonic electron suppression in Au+Au collisions at $\sqrt{s_{NN}}=200$\,GeV,” Phys. Rev. Lett. 98 (2007) 192301, arXiv:nucl-ex/0607012. [Erratum: Phys.Rev.Lett. 106, 159902 (2011)].
* [12] STAR Collaboration, L. Adamczyk et al., “Observation of $D^{0}$ Meson Nuclear Modifications in Au+Au Collisions at $\sqrt{s_{NN}}=200$ GeV,” Phys. Rev. Lett. 113 no. 14, (2014) 142301, arXiv:1404.6185 [nucl-ex]. [Erratum: Phys.Rev.Lett. 121, 229901 (2018)].
* [13] PHENIX Collaboration, S. S. Adler et al., “Nuclear modification of electron spectra and implications for heavy quark energy loss in Au+Au collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 200 GeV,” Phys. Rev. Lett. 96 (2006) 032301, arXiv:nucl-ex/0510047 [nucl-ex].
* [14] PHENIX Collaboration, A. Adare et al., “Single electron yields from semileptonic charm and bottom hadron decays in Au$+$Au collisions at $\sqrt{s_{NN}}=200$ GeV,” Phys. Rev. C93 no. 3, (2016) 034904, arXiv:1509.04662 [nucl-ex].
* [15] ALICE Collaboration, J. Adam et al., “Transverse momentum dependence of D-meson production in Pb–Pb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 2.76 TeV,” JHEP 03 (2016) 081, arXiv:1509.06888 [nucl-ex].
* [16] ALICE Collaboration, B. Abelev et al., “Production of muons from heavy flavour decays at forward rapidity in pp and Pb–Pb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 2.76 TeV,” Phys. Rev. Lett. 109 (2012) 112301, arXiv:1205.6443 [hep-ex].
* [17] ALICE Collaboration, J. Adam et al., “Measurement of the production of high-$p_{\rm T}$ electrons from heavy-flavour hadron decays in Pb–Pb collisions at $\mathbf{\sqrt{\it s_{\rm{NN}}}}$ = 2.76 TeV,” Phys. Lett. B771 (2017) 467–481, arXiv:1609.07104 [nucl-ex].
* [18] ALICE Collaboration, J. Adam et al., “Measurement of electrons from beauty-hadron decays in p-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV and Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV,” JHEP 07 (2017) 052, arXiv:1609.03898 [nucl-ex].
* [19] CMS Collaboration, V. Khachatryan et al., “Suppression and azimuthal anisotropy of prompt and nonprompt ${\mathrm{J}}/\psi$ production in PbPb collisions at $\sqrt{{s_{{}_{\text{NN}}}}}=2.76$ $\,\mathrm{TeV}$,” Eur. Phys. J. C77 no. 4, (2017) 252, arXiv:1610.00613 [nucl-ex].
* [20] CMS Collaboration, A. M. Sirunyan et al., “Nuclear modification factor of D0 mesons in PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV,” Phys. Lett. B782 (2018) 474–496, arXiv:1708.04962 [nucl-ex].
* [21] ALICE Collaboration, S. Acharya et al., “Measurement of D0, D+, D∗+ and D${}_{s}^{+}$ production in Pb–Pb collisions at $\sqrt{{\mathrm{s}}_{\mathrm{NN}}}=5.02$ TeV,” JHEP 10 (2018) 174, arXiv:1804.09083 [nucl-ex].
* [22] J.-Y. Ollitrault, “Anisotropy as a signature of transverse collective flow,” Phys. Rev. D46 (1992) 229–245.
* [23] S. Voloshin and Y. Zhang, “Flow study in relativistic nuclear collisions by Fourier expansion of Azimuthal particle distributions,” Z. Phys. C70 (1996) 665–672, arXiv:hep-ph/9407282 [hep-ph].
* [24] A. M. Poskanzer and S. A. Voloshin, “Methods for analyzing anisotropic flow in relativistic nuclear collisions,” Phys. Rev. C58 (1998) 1671–1678, arXiv:nucl-ex/9805001 [nucl-ex].
* [25] G.-Y. Qin, H. Petersen, S. A. Bass, and B. Müller, “Translation of collision geometry fluctuations into momentum anisotropies in relativistic heavy-ion collisions,” Phys. Rev. C82 (2010) 064903, arXiv:1009.1847 [nucl-th].
* [26] M. Nahrgang, J. Aichelin, S. Bass, P. B. Gossiaux, and K. Werner, “Elliptic and triangular flow of heavy flavor in heavy-ion collisions,” Phys. Rev. C91 no. 1, (2015) 014904, arXiv:1410.5396 [hep-ph].
* [27] S. Batsouli, S. Kelly, M. Gyulassy, and J. L. Nagle, “Does the charm flow at RHIC?,” Phys. Lett. B557 (2003) 26–32, arXiv:nucl-th/0212068 [nucl-th].
* [28] D. Molnar, “Charm elliptic flow from quark coalescence dynamics,” J. Phys. G31 (2005) S421–S428, arXiv:nucl-th/0410041 [nucl-th].
* [29] V. Greco, C. M. Ko, and R. Rapp, “Quark coalescence for charmed mesons in ultrarelativistic heavy ion collisions,” Phys. Lett. B595 (2004) 202–208, arXiv:nucl-th/0312100 [nucl-th].
* [30] M. Gyulassy, I. Vitev, and X. N. Wang, “High $p_{\rm T}$ azimuthal asymmetry in noncentral A+A at RHIC,” Phys. Rev. Lett. 86 (2001) 2537–2540, arXiv:nucl-th/0012092 [nucl-th].
* [31] E. V. Shuryak, “The Azimuthal asymmetry at large $p_{\rm T}$ seem to be too large for a ‘jet quenching’,” Phys. Rev. C66 (2002) 027902, arXiv:nucl-th/0112042 [nucl-th].
* [32] G. D. Moore and D. Teaney, “How much do heavy quarks thermalize in a heavy ion collision?,” Phys. Rev. C71 (2005) 064904, arXiv:hep-ph/0412346 [hep-ph].
* [33] J. Schukraft, A. Timmins, and S. A. Voloshin, “Ultra-relativistic nuclear collisions: event shape engineering,” Phys. Lett. B719 (2013) 394–398, arXiv:1208.4563 [nucl-ex].
* [34] A. Beraudo, A. De Pace, M. Monteno, M. Nardi, and F. Prino, “Event-shape engineering and heavy-flavour observables in relativistic heavy-ion collisions,” Eur. Phys. J. C79 no. 6, (2019) 494, arXiv:1812.08337 [physics.data-an].
* [35] F. G. Gardim, F. Grassi, M. Luzum, and J.-Y. Ollitrault, “Characterizing the hydrodynamic response to the initial conditions,” Nucl. Phys. A904-905 (2013) 503c–506c, arXiv:1210.8422 [nucl-th].
* [36] S. A. Voloshin, A. M. Poskanzer, and R. Snellings, “Collective phenomena in non-central nuclear collisions,” Landolt-Bornstein 23 (2010) 293–333, arXiv:0809.2949 [nucl-ex].
* [37] C. A. G. Prado, J. Noronha-Hostler, R. Katz, A. A. P. Suaide, J. Noronha, M. G. Munhoz, and M. R. Cosentino, “Event-by-event correlations between soft hadrons and $D^{0}$ mesons in 5.02 TeV PbPb collisions at the CERN Large Hadron Collider,” Phys. Rev. C96 no. 6, (2017) 064903, arXiv:1611.02965 [nucl-th].
* [38] P. B. Gossiaux, J. Aichelin, M. Nahrgang, V. Ozvenchuk, and K. Werner, “Global view on coupled dynamics of heavy and light flavor observables from EPOSHQ,” Nucl. Phys. A967 (2017) 672–675, arXiv:1705.02271 [hep-ph].
* [39] ALICE Collaboration, S. Acharya et al., “Anisotropic flow of identified particles in Pb–Pb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 5.02 TeV,” JHEP 09 (2018) 006, arXiv:1805.04390 [nucl-ex].
* [40] STAR Collaboration, L. Adamczyk et al., “Elliptic flow of electrons from heavy-flavor hadron decays in Au + Au collisions at $\sqrt{s_{\rm NN}}=$ 200, 62.4, and 39 GeV,” Phys. Rev. C95 no. 3, (2017) 034907, arXiv:1405.6348 [hep-ex].
* [41] STAR Collaboration, L. Adamczyk et al., “Measurement of ${\rm D^{0}}$ Azimuthal Anisotropy at Midrapidity in Au+Au Collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 200 GeV,” Phys. Rev. Lett. 118 no. 21, (2017) 212301, arXiv:1701.06060 [nucl-ex].
* [42] ALICE Collaboration, B. Abelev et al., “D meson elliptic flow in non-central Pb–Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76TeV,” Phys. Rev. Lett. 111 (2013) 102301, arXiv:1305.2707 [nucl-ex].
* [43] ALICE Collaboration, B. Abelev et al., “Azimuthal anisotropy of D meson production in Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV,” Phys. Rev. C90 no. 3, (2014) 034904, arXiv:1405.2001 [nucl-ex].
* [44] ALICE Collaboration, J. Adam et al., “Elliptic flow of electrons from heavy-flavour hadron decays at mid-rapidity in Pb–Pb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 2.76 TeV,” JHEP 09 (2016) 028, arXiv:1606.00321 [nucl-ex].
* [45] ALICE Collaboration, J. Adam et al., “Elliptic flow of muons from heavy-flavour hadron decays at forward rapidity in Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV,” Phys. Lett. B753 (2016) 41–56, arXiv:1507.03134 [nucl-ex].
* [46] ALICE Collaboration, S. Acharya et al., “D-meson azimuthal anisotropy in midcentral Pb–Pb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 5.02 TeV,” Phys. Rev. Lett. 120 no. 10, (2018) 102301, arXiv:1707.01005 [nucl-ex].
* [47] CMS Collaboration, A. M. Sirunyan et al., “Measurement of prompt $D^{0}$ meson azimuthal anisotropy in Pb–Pb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 5.02 TeV,” Phys. Rev. Lett. 120 no. 20, (2018) 202301, arXiv:1708.03497 [nucl-ex].
* [48] ALICE Collaboration, S. Acharya et al., “Event-shape engineering for the D-meson elliptic flow in mid-central Pb–Pb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 5.02 TeV,” JHEP 02 (2019) 150, arXiv:1809.09371 [nucl-ex].
* [49] J. Uphoff, O. Fochler, Z. Xu, and C. Greiner, “Open Heavy Flavor in Pb+Pb Collisions at $\sqrt{s}=2.76$ TeV within a Transport Model,” Phys. Lett. B717 (2012) 430–435, arXiv:1205.4945 [hep-ph].
* [50] M. He, R. J. Fries, and R. Rapp, “Heavy Flavor at the Large Hadron Collider in a Strong Coupling Approach,” Phys. Lett. B735 (2014) 445–450, arXiv:1401.3817 [nucl-th].
* [51] M. Monteno, W. M. Alberico, A. Beraudo, A. De Pace, A. Molinari, M. Nardi, and F. Prino, “Heavy-flavor dynamics in nucleus-nucleus collisions: from RHIC to LHC,” J. Phys. G38 (2011) 124144, arXiv:1107.0256 [hep-ph].
* [52] S. Cao, G.-Y. Qin, and S. A. Bass, “Heavy-quark dynamics and hadronization in ultrarelativistic heavy-ion collisions: Collisional versus radiative energy loss,” Phys. Rev. C88 (2013) 044907, arXiv:1308.0617 [nucl-th].
* [53] T. Song, H. Berrehrah, D. Cabrera, W. Cassing, and E. Bratkovskaya, “Charm production in Pb + Pb collisions at energies available at the CERN Large Hadron Collider,” Phys. Rev. C93 no. 3, (2016) 034906, arXiv:1512.00891 [nucl-th].
* [54] M. Nahrgang, J. Aichelin, P. B. Gossiaux, and K. Werner, “Influence of hadronic bound states above $T_{c}$ on heavy-quark observables in Pb + Pb collisions at the CERN Large Hadron Collider,” Phys. Rev. C89 no. 1, (2014) 014905, arXiv:1305.6544 [hep-ph].
* [55] J. Uphoff, O. Fochler, Z. Xu, and C. Greiner, “Elastic and radiative heavy quark interactions in ultra-relativistic heavy-ion collisions,” J. Phys. G42 no. 11, (2015) 115106, arXiv:1408.2964 [hep-ph].
* [56] A. Beraudo, A. De Pace, M. Monteno, M. Nardi, and F. Prino, “Heavy flavors in heavy-ion collisions: quenching, flow and correlations,” Eur. Phys. J. C75 no. 3, (2015) 121, arXiv:1410.6082 [hep-ph].
* [57] S. Cao, T. Luo, G.-Y. Qin, and X.-N. Wang, “Heavy and light flavor jet quenching at RHIC and LHC energies,” Phys. Lett. B777 (2018) 255–259, arXiv:1703.00822 [nucl-th].
* [58] ALICE Collaboration, K. Aamodt et al., “The ALICE experiment at the CERN LHC,” JINST 3 (2008) S08002.
* [59] ALICE Collaboration, B. Abelev et al., “Performance of the ALICE Experiment at the CERN LHC,” Int. J. Mod. Phys. A29 (2014) 1430044, arXiv:1402.4476 [nucl-ex].
* [60] ALICE Collaboration, K. Aamodt et al., “Alignment of the ALICE Inner Tracking System with cosmic-ray tracks,” JINST 5 (2010) P03003, arXiv:1001.0502 [physics.ins-det].
* [61] J. Alme et al., “The ALICE TPC, a large 3-dimensional tracking device with fast readout for ultra-high multiplicity events,” Nucl. Instrum. Meth. A622 (2010) 316–367, arXiv:1001.1950 [physics.ins-det].
* [62] A. Akindinov et al., “Performance of the ALICE Time-Of-Flight detector at the LHC,” Eur. Phys. J. Plus 128 (2013) 44.
* [63] ALICE Collaboration, E. Abbas et al., “Performance of the ALICE VZERO system,” JINST 8 (2013) P10016, arXiv:1306.3130 [nucl-ex].
* [64] R. Arnaldi et al., “The Zero degree calorimeters for the ALICE experiment,” Nucl. Instrum. Meth. A 581 (2007) 397–401. [Erratum: Nucl.Instrum.Meth.A 604, 765 (2009)].
* [65] C. Loizides, J. Kamin, and D. d’Enterria, “Improved monte carlo glauber predictions at present and future nuclear colliders,” Phys. Rev. C97 (2018) 054910, arXiv:1710.07098.
* [66] ALICE Collaboration, J. Adam et al., “Event shape engineering for inclusive spectra and elliptic flow in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV,” Phys. Rev. C93 no. 3, (2016) 034916, arXiv:1507.06194 [nucl-ex].
* [67] ALICE Collaboration Collaboration, “Supplemental figures: "Transverse-momentum and event-shape dependence of D-meson flow harmonics in Pb-Pb collisions at ${\sqrt{s_{\rm{NN}}}=5.02}$ TeV",”. http://cds.cern.ch/record/2719037.
* [68] Particle Data Group Collaboration, M. Tanabashi et al., “Review of Particle Physics,” Phys. Rev. D98 no. 3, (2018) 030001.
* [69] ALICE Collaboration, S. Acharya et al., “Measurement of D-meson production at mid-rapidity in pp collisions at ${\sqrt{s}=7}$ TeV,” Eur. Phys. J. C77 no. 8, (2017) 550, arXiv:1702.00766 [hep-ex].
* [70] M. Luzum and J.-Y. Ollitrault, “Eliminating experimental bias in anisotropic-flow measurements of high-energy nuclear collisions,” Phys. Rev. C87 no. 4, (2013) 044907, arXiv:1209.2323 [nucl-ex].
* [71] STAR Collaboration, C. Adler et al., “Elliptic flow from two and four particle correlations in Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}=130$-GeV,” Phys. Rev. C 66 (2002) 034904, arXiv:nucl-ex/0206001.
* [72] I. Selyuzhenkov and S. Voloshin, “Effects of non-uniform acceptance in anisotropic flow measurement,” Phys.Rev.C 77 (2008) 034904, arXiv:0707.4672 [nucl-th].
* [73] N. Borghini and J. Y. Ollitrault, “Azimuthally sensitive correlations in nucleus-nucleus collisions,” Phys. Rev. C70 (2004) 064905, arXiv:nucl-th/0407041 [nucl-th].
* [74] ALICE Collaboration, S. Acharya et al., “Measurement of ${{\mathrm{D}}^{0}}$, ${{\mathrm{D}}^{+}}$, ${{\mathrm{D}}^{*+}}$ and ${{\mathrm{D}}^{+}_{\mathrm{s}}}$ production in pp collisions at ${\sqrt{{\textit{s}}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 5.02\leavevmode\nobreak\ {\text{TeV}}}$ with ALICE,” Eur. Phys. J. C79 no. 5, (2019) 388, arXiv:1901.07979 [nucl-ex].
* [75] X.-N. Wang and M. Gyulassy, “HIJING: A Monte Carlo model for multiple jet production in p p, p A and A A collisions,” Phys. Rev. D44 (1991) 3501–3516.
* [76] T. Sjostrand, S. Mrenna, and P. Z. Skands, “PYTHIA 6.4 Physics and Manual,” JHEP 05 (2006) 026, arXiv:hep-ph/0603175 [hep-ph].
* [77] P. Z. Skands, “Tuning Monte Carlo Generators: The Perugia Tunes,” Phys. Rev. D82 (2010) 074018, arXiv:1005.3457 [hep-ph].
* [78] M. Cacciari, M. Greco, and P. Nason, “The $p_{\rm T}$ spectrum in heavy flavor hadroproduction,” JHEP 05 (1998) 007, arXiv:hep-ph/9803400 [hep-ph].
* [79] M. Cacciari, S. Frixione, and P. Nason, “The $p_{\rm T}$ spectrum in heavy flavor photoproduction,” JHEP 03 (2001) 006, arXiv:hep-ph/0102134 [hep-ph].
* [80] D. J. Lange, “The EvtGen particle decay simulation package,” Nucl. Instrum. Meth. A462 (2001) 152–155.
* [81] J. Aichelin, P. B. Gossiaux, and T. Gousset, “Radiative and Collisional Energy Loss of Heavy Quarks in Deconfined Matter,” Acta Phys. Polon. B43 (2012) 655–662, arXiv:1201.4192 [nucl-th].
* [82] V. Greco, H. van Hees, and R. Rapp, “Heavy-quark kinetics at RHIC and LHC,” in Nuclear physics. Proceedings, 23rd International Conference, INPC 2007, Tokyo, Japan, June 3-8, 2007. 2007\. arXiv:0709.4452 [hep-ph].
* [83] ALICE Collaboration, S. Acharya et al., “Study of J/$\psi$ azimuthal anisotropy at forward rapidity in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV,” JHEP 02 (2019) 012, arXiv:1811.12727 [nucl-ex].
* [84] ALICE Collaboration, S. Acharya et al., “J/$\psi$ elliptic and triangular flow in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV,” arXiv:2005.14518 [nucl-ex].
* [85] D. Molnar and S. A. Voloshin, “Elliptic flow at large transverse momenta from quark coalescence,” Phys. Rev. Lett. 91 (2003) 092301, arXiv:nucl-th/0302014 [nucl-th].
* [86] ALICE Collaboration, J. Adam et al., “Inclusive, prompt and non-prompt J/$\psi$ production at mid-rapidity in Pb–Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV,” JHEP 07 (2015) 051, arXiv:1504.07151 [nucl-ex].
* [87] CMS Collaboration, S. Chatrchyan et al., “Suppression of non-prompt $J/\psi$, prompt $J/\psi$, and Y(1S) in PbPb collisions at $\sqrt{s_{\rm\scriptscriptstyle NN}}=$ 2.76 TeV,” JHEP 05 (2012) 063, arXiv:1201.5069 [nucl-ex].
* [88] A. Beraudo, A. De Pace, M. Monteno, M. Nardi, and F. Prino, “Development of heavy-flavour flow-harmonics in high-energy nuclear collisions,” JHEP 02 (2018) 043, arXiv:1712.00588 [hep-ph].
* [89] M. He and R. Rapp, “Hadronization and Charm-Hadron Ratios in Heavy-Ion Collisions,” Phys. Rev. Lett. 124 no. 4, (2020) 042301, arXiv:1905.09216 [nucl-th].
* [90] S. Cao, T. Luo, G.-Y. Qin, and X.-N. Wang, “Linearized Boltzmann transport model for jet propagation in the quark-gluon plasma: Heavy quark evolution,” Phys. Rev. C94 no. 1, (2016) 014909, arXiv:1605.06447 [nucl-th].
* [91] W. Ke, Y. Xu, and S. A. Bass, “Linearized Boltzmann-Langevin model for heavy quark transport in hot and dense QCD matter,” Phys. Rev. C98 no. 6, (2018) 064901, arXiv:1806.08848 [nucl-th].
* [92] W. Ke, Y. Xu, and S. A. Bass, “Modified Boltzmann approach for modeling the splitting vertices induced by the hot QCD medium in the deep Landau-Pomeranchuk-Migdal region,” Phys. Rev. C 100 no. 6, (2019) 064911, arXiv:1810.08177 [nucl-th].
* [93] R. Katz, C. A. G. Prado, J. Noronha-Hostler, J. Noronha, and A. A. P. Suaide, “DAB-MOD sensitivity study of heavy flavor $R_{\rm{AA}}$ and azimuthal anisotropies based on beam energy, initial conditions, hadronization, and suppression mechanisms,” arXiv:1906.10768 [nucl-th].
* [94] F. Scardina, S. K. Das, V. Minissale, S. Plumari, and V. Greco, “Estimating the charm quark diffusion coefficient and thermalization time from D meson spectra at energies available at the BNL Relativistic Heavy Ion Collider and the CERN Large Hadron Collider,” Phys. Rev. C 96 no. 4, (2017) 044905, arXiv:1707.05452 [nucl-th].
* [95] S. Plumari, G. Coci, V. Minissale, S. K. Das, Y. Sun, and V. Greco, “Heavy - light flavor correlations of anisotropic flows at LHC energies within event-by-event transport approach,” Phys. Lett. B 805 (2020) 135460, arXiv:1912.09350 [hep-ph].
* [96] S. Li and J. Liao, “Data-driven extraction of heavy quark diffusion in quark-gluon plasma,” arXiv:1912.08965 [hep-ph].
* [97] Wuppertal-Budapest Collaboration, S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, “Is there still any T_c mystery in lattice QCD? Results with physical masses in the continuum limit III,” JHEP 09 (2010) 073, arXiv:1005.3508 [hep-lat].
* [98] W. M. Alberico, A. Beraudo, A. De Pace, A. Molinari, M. Monteno, M. Nardi, F. Prino, and M. Sitta, “Heavy flavors in $AA$ collisions: production, transport and final spectra,” Eur. Phys. J. C73 (2013) 2481, arXiv:1305.7421 [hep-ph].
* [99] D. Banerjee, S. Datta, R. Gavai, and P. Majumdar, “Heavy Quark Momentum Diffusion Coefficient from Lattice QCD,” Phys. Rev. D85 (2012) 014510, arXiv:1109.5738 [hep-lat].
* [100] B. Betz and M. Gyulassy, “Constraints on the Path-Length Dependence of Jet Quenching in Nuclear Collisions at RHIC and LHC,” JHEP 08 (2014) 090, arXiv:1404.6378 [hep-ph]. [Erratum: JHEP 10, 043 (2014)].
## Appendix A The ALICE Collaboration
S. Acharyaorg141&D. Adamováorg95&A. Adlerorg74&J. Adolfssonorg81&M.M.
Aggarwalorg100&G. Aglieri Rinellaorg34&M. Agnelloorg30&N.
Agrawalorg10,org54&Z. Ahammedorg141&S. Ahmadorg16&S.U. Ahnorg76&Z.
Akbarorg51&A. Akindinovorg92&M. Al-Turanyorg107&S.N. Alamorg40,org141&D.S.D.
Albuquerqueorg122&D. Aleksandrovorg88&B. Alessandroorg59&H.M. Alfandaorg6&R.
Alfaro Molinaorg71&B. Aliorg16&Y. Aliorg14&A. Aliciorg10,org26,org54&N.
Alizadehvandchaliorg125&A. Alkinorg2,org34&J. Almeorg21&T. Altorg68&L.
Altenkamperorg21&I. Altsybeevorg113&M.N. Anaamorg6&C. Andreiorg48&D.
Andreouorg34&A. Andronicorg144&M. Angelettiorg34&V. Anguelovorg104&C.
Ansonorg15&T. Antičićorg108&F. Antinoriorg57&P. Antonioliorg54&N.
Apadulaorg80&L. Aphecetcheorg115&H. Appelshäuserorg68&S. Arcelliorg26&R.
Arnaldiorg59&M. Arratiaorg80&I.C. Arseneorg20&M. Arslandokorg104&A.
Augustinusorg34&R. Averbeckorg107&S. Azizorg78&M.D. Azmiorg16&A.
Badalàorg56&Y.W. Baekorg41&S. Bagnascoorg59&X. Baiorg107&R. Bailhacheorg68&R.
Balaorg101&A. Balbinoorg30&A. Baldisseriorg137&M. Ballorg43&S.
Balouzaorg105&D. Banerjeeorg3&R. Barberaorg27&L. Barioglioorg25&G.G.
Barnaföldiorg145&L.S. Barnbyorg94&V. Barretorg134&P. Bartaliniorg6&C.
Bartelsorg127&K. Barthorg34&E. Bartschorg68&F. Baruffaldiorg28&N.
Bastidorg134&S. Basuorg143&G. Batigneorg115&B. Batyunyaorg75&D.
Bauriorg49&J.L. Bazo Albaorg112&I.G. Beardenorg89&C. Beattieorg146&C.
Beddaorg63&N.K. Beheraorg61&I. Belikovorg136&A.D.C. Bell Hechavarriaorg144&F.
Belliniorg34&R. Bellwiedorg125&V. Belyaevorg93&G. Bencediorg145&S.
Beoleorg25&A. Bercuciorg48&Y. Berdnikovorg98&D. Berenyiorg145&R.A.
Bertensorg130&D. Berzanoorg59&M.G. Besoiuorg67&L. Betevorg34&A.
Bhasinorg101&I.R. Bhatorg101&M.A. Bhatorg3&H. Bhattorg49&B.
Bhattacharjeeorg42&A. Bianchiorg25&L. Bianchiorg25&N. Bianchiorg52&J.
Bielčíkorg37&J. Bielčíkováorg95&A. Bilandzicorg105&G. Biroorg145&R.
Biswasorg3&S. Biswasorg3&J.T. Blairorg119&D. Blauorg88&C. Blumeorg68&G.
Bocaorg139&F. Bockorg96&A. Bogdanovorg93&S. Boiorg23&J. Bokorg61&L.
Boldizsárorg145&A. Bolozdynyaorg93&M. Bombaraorg38&G. Bonomiorg140&H.
Borelorg137&A. Borissovorg93&H. Bossiorg146&E. Bottaorg25&L. Bratrudorg68&P.
Braun-Munzingerorg107&M. Bregantorg121&M. Brozorg37&E. Brunaorg59&G.E.
Brunoorg106&M.D. Bucklandorg127&D. Budnikovorg109&H. Bueschingorg68&S.
Bufalinoorg30&O. Bugnonorg115&P. Buhlerorg114&P. Buncicorg34&Z.
Butheleziorg72,org131&J.B. Buttorg14&S.A. Bysiakorg118&D. Caffarriorg90&M.
Caiorg6&A. Calivaorg107&E. Calvo Villarorg112&J.M.M. Camachoorg120&R.S.
Camachoorg45&P. Cameriniorg24&F.D.M. Canedoorg121&A.A. Caponorg114&F.
Carnesecchiorg26&R. Caronorg137&J. Castillo Castellanosorg137&A.J.
Castroorg130&E.A.R. Casulaorg55&F. Catalanoorg30&C. Ceballos Sanchezorg75&P.
Chakrabortyorg49&S. Chandraorg141&W. Changorg6&S. Chapelandorg34&M.
Chartierorg127&S. Chattopadhyayorg141&S. Chattopadhyayorg110&A.
Chauvinorg23&C. Cheshkovorg135&B. Cheynisorg135&V. Chibante Barrosoorg34&D.D.
Chinellatoorg122&S. Choorg61&P. Chochulaorg34&T. Chowdhuryorg134&P.
Christakoglouorg90&C.H. Christensenorg89&P. Christiansenorg81&T.
Chujoorg133&C. Cicaloorg55&L. Cifarelliorg10,org26&L.D. Cilladiorg25&F.
Cindoloorg54&M.R. Ciupekorg107&G. Claiorg54orgI&J. Cleymansorg124&F.
Colamariaorg53&D. Colellaorg53&A. Colluorg80&M. Colocciorg26&M.
Concasorg59orgII&G. Conesa Balbastreorg79&Z. Conesa del Valleorg78&G.
Continorg24,org60&J.G. Contrerasorg37&T.M. Cormierorg96&Y. Corrales
Moralesorg25&P. Corteseorg31&M.R. Cosentinoorg123&F. Costaorg34&S.
Costanzaorg139&P. Crochetorg134&E. Cuautleorg69&P. Cuiorg6&L.
Cunqueiroorg96&D. Dabrowskiorg142&T. Dahmsorg105&A. Daineseorg57&F.P.A.
Damasorg115,org137&M.C. Danischorg104&A. Danuorg67&D. Dasorg110&I.
Dasorg110&P. Dasorg86&P. Dasorg3&S. Dasorg3&A. Dashorg86&S. Dashorg49&S.
Deorg86&A. De Caroorg29&G. de Cataldoorg53&J. de Cuvelandorg39&A. De
Falcoorg23&D. De Gruttolaorg10&N. De Marcoorg59&S. De Pasqualeorg29&S.
Deborg50&H.F. Degenhardtorg121&K.R. Dejaorg142&A. Delofforg85&S.
Delsantoorg25,org131&W. Dengorg6&P. Dhankherorg49&D. Di Bariorg33&A. Di
Mauroorg34&R.A. Diazorg8&T. Dietelorg124&P. Dillensegerorg68&Y. Dingorg6&R.
Diviàorg34&D.U. Dixitorg19&Ø. Djuvslandorg21&U. Dmitrievaorg62&A.
Dobrinorg67&B. Dönigusorg68&O. Dordicorg20&A.K. Dubeyorg141&A.
Dublaorg90,org107&S. Dudiorg100&M. Dukhishyamorg86&P. Dupieuxorg134&R.J.
Ehlersorg96&V.N. Eikelandorg21&D. Eliaorg53&B. Erazmusorg115&F.
Erhardtorg99&A. Erokhinorg113&M.R. Ersdalorg21&B. Espagnonorg78&G.
Eulisseorg34&D. Evansorg111&S. Evdokimovorg91&L. Fabbiettiorg105&M.
Fagginorg28&J. Faivreorg79&F. Fanorg6&A. Fantoniorg52&M. Faselorg96&P.
Fecchioorg30&A. Felicielloorg59&G. Feofilovorg113&A. Fernández Téllezorg45&A.
Ferreroorg137&A. Ferrettiorg25&A. Festantiorg34&V.J.G. Feuillardorg104&J.
Figielorg118&S. Filchaginorg109&D. Finogeevorg62&F.M. Fiondaorg21&G.
Fiorenzaorg53&F. Flororg125&A.N. Floresorg119&S. Foertschorg72&P.
Fokaorg107&S. Fokinorg88&E. Fragiacomoorg60&U. Frankenfeldorg107&U.
Fuchsorg34&C. Furgetorg79&A. Fursorg62&M. Fusco Girardorg29&J.J.
Gaardhøjeorg89&M. Gagliardiorg25&A.M. Gagoorg112&A. Galorg136&C.D.
Galvanorg120&P. Ganotiorg84&C. Garabatosorg107&J.R.A. Garciaorg45&E. Garcia-
Solisorg11&K. Gargorg115&C. Gargiuloorg34&A. Garibliorg87&K. Garnerorg144&P.
Gasikorg105,org107&E.F. Gaugerorg119&M.B. Gay Ducatiorg70&M. Germainorg115&J.
Ghoshorg110&P. Ghoshorg141&S.K. Ghoshorg3&M. Giacaloneorg26&P.
Gianottiorg52&P. Giubellinoorg59,org107&P. Giubilatoorg28&A.M.C.
Glaenzerorg137&P. Glässelorg104&A. Gomez Ramirezorg74&V.
Gonzalezorg107,org143&L.H. González-Truebaorg71&S. Gorbunovorg39&L.
Görlichorg118&A. Goswamiorg49&S. Gotovacorg35&V. Grabskiorg71&L.K.
Graczykowskiorg142&K.L. Grahamorg111&L. Greinerorg80&A. Grelliorg63&C.
Grigorasorg34&V. Grigorievorg93&A. Grigoryanorg1&S. Grigoryanorg75&O.S.
Groettvikorg21&F. Grosaorg30,org59&J.F. Grosse-Oetringhausorg34&R.
Grossoorg107&R. Guernaneorg79&M. Guittiereorg115&K. Gulbrandsenorg89&T.
Gunjiorg132&A. Guptaorg101&R. Guptaorg101&I.B. Guzmanorg45&R. Haakeorg146&M.K.
Habiborg107&C. Hadjidakisorg78&H. Hamagakiorg82&G. Hamarorg145&M. Hamidorg6&R.
Hanniganorg119&M.R. Haqueorg63,org86&A. Harlenderovaorg107&J.W.
Harrisorg146&A. Hartonorg11&J.A. Hasenbichlerorg34&H. Hassanorg96&Q.U.
Hassanorg14&D. Hatzifotiadouorg10,org54&P. Hauerorg43&L.B. Havenerorg146&S.
Hayashiorg132&S.T. Heckelorg105&E. Hellbärorg68&H. Helstruporg36&A.
Herghelegiuorg48&T. Hermanorg37&E.G. Hernandezorg45&G. Herrera Corralorg9&F.
Herrmannorg144&K.F. Hetlandorg36&H. Hillemannsorg34&C. Hillsorg127&B.
Hippolyteorg136&B. Hohlwegerorg105&J. Honermannorg144&D. Horakorg37&A.
Hornungorg68&S. Hornungorg107&R. Hosokawaorg15,org133&P. Hristovorg34&C.
Huangorg78&C. Hughesorg130&P. Huhnorg68&T.J. Humanicorg97&H.
Hushnudorg110&L.A. Husovaorg144&N. Hussainorg42&S.A. Hussainorg14&D.
Hutterorg39&J.P. Iddonorg34,org127&R. Ilkaevorg109&H. Ilyasorg14&M.
Inabaorg133&G.M. Innocentiorg34&M. Ippolitovorg88&A. Isakovorg95&M.S.
Islamorg110&M. Ivanovorg107&V. Ivanovorg98&V. Izucheevorg91&B. Jacakorg80&N.
Jacazioorg34,org54&P.M. Jacobsorg80&S. Jadlovskaorg117&J. Jadlovskyorg117&S.
Jaelaniorg63&C. Jahnkeorg121&M.J. Jakubowskaorg142&M.A. Janikorg142&T.
Jansonorg74&M. Jercicorg99&O. Jevonsorg111&M. Jinorg125&F.
Jonasorg96,org144&P.G. Jonesorg111&J. Jungorg68&M. Jungorg68&A. Juskoorg111&P.
Kalinakorg64&A. Kalweitorg34&V. Kaplinorg93&S. Karorg6&A. Karasu Uysalorg77&D.
Karatovicorg99&O. Karavichevorg62&T. Karavichevaorg62&P. Karczmarczykorg142&E.
Karpechevorg62&A. Kazantsevorg88&U. Kebschullorg74&R. Keidelorg47&M.
Keilorg34&B. Ketzerorg43&Z. Khabanovaorg90&A.M. Khanorg6&S. Khanorg16&A.
Khanzadeevorg98&Y. Kharlovorg91&A. Khatunorg16&A. Khuntiaorg118&B.
Kilengorg36&B. Kimorg61&B. Kimorg133&D. Kimorg147&D.J. Kimorg126&E.J.
Kimorg73&H. Kimorg17&J. Kimorg147&J.S. Kimorg41&J. Kimorg104&J. Kimorg147&J.
Kimorg73&M. Kimorg104&S. Kimorg18&T. Kimorg147&T. Kimorg147&S. Kirschorg68&I.
Kiselorg39&S. Kiselevorg92&A. Kisielorg142&J.L. Klayorg5&C. Kleinorg68&J.
Kleinorg34,org59&S. Kleinorg80&C. Klein-Bösingorg144&M. Kleinerorg68&A.
Klugeorg34&M.L. Knichelorg34&A.G. Knospeorg125&C. Kobdajorg116&M.K.
Köhlerorg104&T. Kolleggerorg107&A. Kondratyevorg75&N. Kondratyevaorg93&E.
Kondratyukorg91&J. Konigorg68&S.A. Konigstorferorg105&P.J. Konopkaorg34&G.
Kornakovorg142&L. Koskaorg117&O. Kovalenkoorg85&V. Kovalenkoorg113&M.
Kowalskiorg118&I. Králikorg64&A. Kravčákováorg38&L. Kreisorg107&M.
Krivdaorg64,org111&F. Krizekorg95&K. Krizkova Gajdosovaorg37&M. Krügerorg68&E.
Kryshenorg98&M. Krzewickiorg39&A.M. Kuberaorg97&V. Kučeraorg34,org61&C.
Kuhnorg136&P.G. Kuijerorg90&L. Kumarorg100&S. Kunduorg86&P. Kurashviliorg85&A.
Kurepinorg62&A.B. Kurepinorg62&A. Kuryakinorg109&S. Kushpilorg95&J.
Kvapilorg111&M.J. Kweonorg61&J.Y. Kwonorg61&Y. Kwonorg147&S.L. La
Pointeorg39&P. La Roccaorg27&Y.S. Laiorg80&M. Lamannaorg34&R. Langoyorg129&K.
Lapidusorg34&A. Lardeuxorg20&P. Larionovorg52&E. Laudiorg34&R. Lavickaorg37&T.
Lazarevaorg113&R. Leaorg24&L. Leardiniorg104&J. Leeorg133&S. Leeorg147&S.
Lehnerorg114&J. Lehrbachorg39&R.C. Lemmonorg94&I. León Monzónorg120&E.D.
Lesserorg19&M. Lettrichorg34&P. Lévaiorg145&X. Liorg12&X.L. Liorg6&J.
Lienorg129&R. Lietavaorg111&B. Limorg17&V. Lindenstruthorg39&A.
Lindnerorg48&C. Lippmannorg107&M.A. Lisaorg97&A. Liuorg19&J. Liuorg127&S.
Liuorg97&W.J. Llopeorg143&I.M. Lofnesorg21&V. Loginovorg93&C. Loizidesorg96&P.
Loncarorg35&J.A. Lopezorg104&X. Lopezorg134&E. López Torresorg8&J.R.
Luhderorg144&M. Lunardonorg28&G. Luparelloorg60&Y.G. Maorg40&A.
Maevskayaorg62&M. Magerorg34&S.M. Mahmoodorg20&T. Mahmoudorg43&A.
Maireorg136&R.D. Majkaorg146org*&M. Malaevorg98&Q.W. Malikorg20&L.
Malininaorg75orgIII&D. Mal’Kevichorg92&P. Malzacherorg107&G.
Mandaglioorg32,org56&V. Mankoorg88&F. Mansoorg134&V. Manzariorg53&Y.
Maoorg6&M. Marchisoneorg135&J. Marešorg66&G.V. Margagliottiorg24&A.
Margottiorg54&J. Marguttiorg63&A. Marínorg107&C. Markertorg119&M.
Marquardorg68&C.D. Martinorg24&N.A. Martinorg104&P. Martinengoorg34&J.L.
Martinezorg125&M.I. Martínezorg45&G. Martínez Garcíaorg115&S.
Masciocchiorg107&M. Maseraorg25&A. Masoniorg55&L. Massacrierorg78&E.
Massonorg115&A. Mastroserioorg53,org138&A.M. Mathisorg105&O.
Matonohaorg81&P.F.T. Matuokaorg121&A. Matyjaorg118&C. Mayerorg118&F.
Mazzaschiorg25&M. Mazzilliorg53&M.A. Mazzoniorg58&A.F. Mechlerorg68&F.
Meddiorg22&Y. Melikyanorg62,org93&A. Menchaca-Rochaorg71&E.
Meninnoorg29,org114&A.S. Menonorg125&M. Meresorg13&S. Mhlangaorg124&Y.
Miakeorg133&L. Michelettiorg25&L.C. Migliorinorg135&D.L. Mihaylovorg105&K.
Mikhaylovorg75,org92&A.N. Mishraorg69&D. Miśkowiecorg107&A. Modakorg3&N.
Mohammadiorg34&A.P. Mohantyorg63&B. Mohantyorg86&M. Mohisin Khanorg16orgIV&Z.
Moravcovaorg89&C. Mordasiniorg105&D.A. Moreira De Godoyorg144&L.A.P.
Morenoorg45&I. Morozovorg62&A. Morschorg34&T. Mrnjavacorg34&V.
Mucciforaorg52&E. Mudnicorg35&D. Mühlheimorg144&S. Muhuriorg141&J.D.
Mulliganorg80&A. Mulliriorg23,org55&M.G. Munhozorg121&R.H. Munzerorg68&H.
Murakamiorg132&S. Murrayorg124&L. Musaorg34&J. Musinskyorg64&C.J.
Myersorg125&J.W. Myrchaorg142&B. Naikorg49&R. Nairorg85&B.K. Nandiorg49&R.
Naniaorg10,org54&E. Nappiorg53&M.U. Naruorg14&A.F. Nassirpourorg81&C.
Nattrassorg130&R. Nayakorg49&T.K. Nayakorg86&S. Nazarenkoorg109&A.
Neaguorg20&R.A. Negrao De Oliveiraorg68&L. Nellenorg69&S.V. Nesboorg36&G.
Neskovicorg39&D. Nesterovorg113&L.T. Neumannorg142&B.S. Nielsenorg89&S.
Nikolaevorg88&S. Nikulinorg88&V. Nikulinorg98&F. Noferiniorg10,org54&P.
Nomokonovorg75&J. Normanorg79,org127&N. Novitzkyorg133&P. Nowakowskiorg142&A.
Nyaninorg88&J. Nystrandorg21&M. Oginoorg82&A. Ohlsonorg81,org104&J.
Oleniaczorg142&A.C. Oliveira Da Silvaorg130&M.H. Oliverorg146&C.
Oppedisanoorg59&A. Ortiz Velasquezorg69&A. Oskarssonorg81&J.
Otwinowskiorg118&K. Oyamaorg82&Y. Pachmayerorg104&V. Pacikorg89&S.
Padhanorg49&D. Paganoorg140&G. Paićorg69&J. Panorg143&S. Panebiancoorg137&P.
Pareekorg50,org141&J. Parkorg61&J.E. Parkkilaorg126&S. Parmarorg100&S.P.
Pathakorg125&B. Paulorg23&J. Pazziniorg140&H. Peiorg6&T. Peitzmannorg63&X.
Pengorg6&L.G. Pereiraorg70&H. Pereira Da Costaorg137&D. Peresunkoorg88&G.M.
Perezorg8&S. Perrinorg137&Y. Pestovorg4&V. Petráčekorg37&M.
Petroviciorg48&R.P. Pezziorg70&S. Pianoorg60&M. Piknaorg13&P. Pillotorg115&O.
Pinazzaorg34,org54&L. Pinskyorg125&C. Pintoorg27&S. Pisanoorg10,org52&D.
Pistoneorg56&M. Płoskońorg80&M. Planinicorg99&F. Pliquettorg68&M.G.
Poghosyanorg96&B. Polichtchoukorg91&N. Poljakorg99&A. Poporg48&S. Porteboeuf-
Houssaisorg134&V. Pozdniakovorg75&S.K. Prasadorg3&R. Preghenellaorg54&F.
Prinoorg59&C.A. Pruneauorg143&I. Pshenichnovorg62&M. Puccioorg34&J.
Putschkeorg143&S. Qiuorg90&L. Quagliaorg25&R.E. Quishpeorg125&S.
Ragoniorg111&S. Rahaorg3&S. Rajputorg101&J. Rakorg126&A.
Rakotozafindrabeorg137&L. Ramelloorg31&F. Ramiorg136&S.A.R. Ramirezorg45&R.
Raniwalaorg102&S. Raniwalaorg102&S.S. Räsänenorg44&R. Rathorg50&V.
Ratzaorg43&I. Ravasengaorg90&K.F. Readorg96,org130&A.R. Redelbachorg39&K.
Redlichorg85orgV&A. Rehmanorg21&P. Reicheltorg68&F. Reidtorg34&X. Renorg6&R.
Renfordtorg68&Z. Rescakovaorg38&K. Reygersorg104&A. Riabovorg98&V.
Riabovorg98&T. Richertorg81,org89&M. Richterorg20&P. Riedlerorg34&W.
Rieglerorg34&F. Riggiorg27&C. Risteaorg67&S.P. Rodeorg50&M. Rodríguez
Cahuantziorg45&K. Røedorg20&R. Rogalevorg91&E. Rogochayaorg75&D. Rohrorg34&D.
Röhrichorg21&P.F. Rojasorg45&P.S. Rokitaorg142&F. Ronchettiorg52&A.
Rosanoorg56&E.D. Rosasorg69&K. Roslonorg142&A. Rossiorg28,org57&A.
Rotondiorg139&A. Royorg50&P. Royorg110&O.V. Ruedaorg81&R. Ruiorg24&B.
Rumyantsevorg75&A. Rustamovorg87&E. Ryabinkinorg88&Y. Ryabovorg98&A.
Rybickiorg118&H. Rytkonenorg126&O.A.M. Saarimakiorg44&S. Sadhuorg141&S.
Sadovskyorg91&K. Šafaříkorg37&S.K. Sahaorg141&B. Sahooorg49&P. Sahooorg49&R.
Sahooorg50&S. Sahooorg65&P.K. Sahuorg65&J. Sainiorg141&S. Sakaiorg133&S.
Sambyalorg101&V. Samsonovorg93,org98&D. Sarkarorg143&N. Sarkarorg141&P.
Sarmaorg42&V.M. Sartiorg105&M.H.P. Sasorg63&E. Scapparoneorg54&J.
Schambachorg119&H.S. Scheidorg68&C. Schiauaorg48&R. Schickerorg104&A.
Schmahorg104&C. Schmidtorg107&H.R. Schmidtorg103&M.O. Schmidtorg104&M.
Schmidtorg103&N.V. Schmidtorg68,org96&A.R. Schmierorg130&J. Schukraftorg89&Y.
Schutzorg136&K. Schwarzorg107&K. Schwedaorg107&G. Scioliorg26&E.
Scomparinorg59&J.E. Segerorg15&Y. Sekiguchiorg132&D. Sekihataorg132&I.
Selyuzhenkovorg93,org107&S. Senyukovorg136&D. Serebryakovorg62&A.
Sevcencoorg67&A. Shabanovorg62&A. Shabetaiorg115&R. Shahoyanorg34&W.
Shaikhorg110&A. Shangaraevorg91&A. Sharmaorg100&A. Sharmaorg101&H.
Sharmaorg118&M. Sharmaorg101&N. Sharmaorg100&S. Sharmaorg101&O.
Sheibaniorg125&K. Shigakiorg46&M. Shimomuraorg83&S. Shirinkinorg92&Q.
Shouorg40&Y. Sibiriakorg88&S. Siddhantaorg55&T. Siemiarczukorg85&D.
Silvermyrorg81&G. Simatovicorg90&G. Simonettiorg34&B. Singhorg105&R.
Singhorg86&R. Singhorg101&R. Singhorg50&V.K. Singhorg141&V. Singhalorg141&T.
Sinhaorg110&B. Sitarorg13&M. Sittaorg31&T.B. Skaaliorg20&M. Slupeckiorg44&N.
Smirnovorg146&R.J.M. Snellingsorg63&C. Sonccoorg112&J. Songorg125&A.
Songmoolnakorg116&F. Soramelorg28&S. Sorensenorg130&I. Sputowskaorg118&J.
Stachelorg104&I. Stanorg67&P.J. Steffanicorg130&E. Stenlundorg81&S.F.
Stiefelmaierorg104&D. Stoccoorg115&M.M. Storetvedtorg36&L.D.
Strittoorg29&A.A.P. Suaideorg121&T. Sugitateorg46&C. Suireorg78&M.
Suleymanovorg14&M. Suljicorg34&R. Sultanovorg92&M. Šumberaorg95&V.
Sumberiaorg101&S. Sumowidagdoorg51&S. Swainorg65&A. Szaboorg13&I.
Szarkaorg13&U. Tabassamorg14&S.F. Taghaviorg105&G. Taillepiedorg134&J.
Takahashiorg122&G.J. Tambaveorg21&S. Tangorg6,org134&M. Tarhiniorg115&M.G.
Tarzilaorg48&A. Tauroorg34&G. Tejeda Muñozorg45&A. Telescaorg34&L.
Terlizziorg25&C. Terrevoliorg125&D. Thakurorg50&S. Thakurorg141&D.
Thomasorg119&F. Thoresenorg89&R. Tieulentorg135&A. Tikhonovorg62&A.R.
Timminsorg125&A. Toiaorg68&N. Topilskayaorg62&M. Toppiorg52&F. Torales-
Acostaorg19&S.R. Torresorg37&A. Trifiróorg32,org56&S. Tripathyorg50,org69&T.
Tripathyorg49&S. Trogoloorg28&G. Trombettaorg33&L. Tropporg38&V.
Trubnikovorg2&W.H. Trzaskaorg126&T.P. Trzcinskiorg142&B.A.
Trzeciakorg37,org63&A. Tumkinorg109&R. Turrisiorg57&T.S. Tveterorg20&K.
Ullalandorg21&E.N. Umakaorg125&A. Urasorg135&G.L. Usaiorg23&M. Valaorg38&N.
Valleorg139&S. Valleroorg59&N. van der Kolkorg63&L.V.R. van Doremalenorg63&M.
van Leeuwenorg63&P. Vande Vyvreorg34&D. Vargaorg145&Z. Vargaorg145&M. Varga-
Kofaragoorg145&A. Vargasorg45&M. Vasileiouorg84&A. Vasilievorg88&O. Vázquez
Doceorg105&V. Vecherninorg113&E. Vercellinorg25&S. Vergara Limónorg45&L.
Vermuntorg63&R. Vernetorg7&R. Vértesiorg145&L. Vickovicorg35&Z.
Vilakaziorg131&O. Villalobos Baillieorg111&G. Vinoorg53&A. Vinogradovorg88&T.
Virgiliorg29&V. Vislaviciusorg89&A. Vodopyanovorg75&B. Volkelorg34&M.A.
Völklorg103&K. Voloshinorg92&S.A. Voloshinorg143&G. Volpeorg33&B. von
Hallerorg34&I. Vorobyevorg105&D. Voscekorg117&J. Vrlákováorg38&B.
Wagnerorg21&M. Weberorg114&S.G. Weberorg144&A. Wegrzynekorg34&S.C.
Wenzelorg34&J.P. Wesselsorg144&J. Wiechulaorg68&J. Wikneorg20&G. Wilkorg85&J.
Wilkinsonorg10,org54&G.A. Willemsorg144&E. Willsherorg111&B.
Windelbandorg104&M. Winnorg137&W.E. Wittorg130&J.R. Wrightorg119&Y.
Wuorg128&R. Xuorg6&S. Yalcinorg77&Y. Yamaguchiorg46&K. Yamakawaorg46&S.
Yangorg21&S. Yanoorg137&Z. Yinorg6&H. Yokoyamaorg63&I.-K. Yooorg17&J.H.
Yoonorg61&S. Yuanorg21&A. Yuncuorg104&V. Yurchenkoorg2&V. Zaccoloorg24&A.
Zamanorg14&C. Zampolliorg34&H.J.C. Zanoliorg63&N. Zardoshtiorg34&A.
Zarochentsevorg113&P. Závadaorg66&N. Zaviyalovorg109&H. Zbroszczykorg142&M.
Zhalovorg98&S. Zhangorg40&X. Zhangorg6&Z. Zhangorg6&V. Zherebchevskiiorg113&D.
Zhouorg6&Y. Zhouorg89&Z. Zhouorg21&J. Zhuorg6,org107&Y. Zhuorg6&A.
Zichichiorg10,org26&G. Zinovjevorg2&N. Zurloorg140&
## Affiliation notes
org*Deceased
orgIItalian National Agency for New Technologies, Energy and Sustainable
Economic Development (ENEA), Bologna, Italy
orgIIDipartimento DET del Politecnico di Torino, Turin, Italy
orgIIIM.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of
Nuclear, Physics, Moscow, Russia
orgIVDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India
orgVInstitute of Theoretical Physics, University of Wroclaw, Poland
## Collaboration Institutes
org1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia
org2Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine
org3Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India
org4Budker Institute for Nuclear Physics, Novosibirsk, Russia
org5California Polytechnic State University, San Luis Obispo, California,
United States
org6Central China Normal University, Wuhan, China
org7Centre de Calcul de l’IN2P3, Villeurbanne, Lyon, France
org8Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba
org9Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City
and Mérida, Mexico
org10Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi’, Rome, Italy
org11Chicago State University, Chicago, Illinois, United States
org12China Institute of Atomic Energy, Beijing, China
org13Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovakia
org14COMSATS University Islamabad, Islamabad, Pakistan
org15Creighton University, Omaha, Nebraska, United States
org16Department of Physics, Aligarh Muslim University, Aligarh, India
org17Department of Physics, Pusan National University, Pusan, Republic of
Korea
org18Department of Physics, Sejong University, Seoul, Republic of Korea
org19Department of Physics, University of California, Berkeley, California,
United States
org20Department of Physics, University of Oslo, Oslo, Norway
org21Department of Physics and Technology, University of Bergen, Bergen,
Norway
org22Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN,
Rome, Italy
org23Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy
org24Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
org25Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy
org26Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy
org27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy
org28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy
org29Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy
org30Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy
org31Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy
org32Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy
org33Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy
org34European Organization for Nuclear Research (CERN), Geneva, Switzerland
org35Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia
org36Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway
org37Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic
org38Faculty of Science, P.J. Šafárik University, Košice, Slovakia
org39Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany
org40Fudan University, Shanghai, China
org41Gangneung-Wonju National University, Gangneung, Republic of Korea
org42Gauhati University, Department of Physics, Guwahati, India
org43Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany
org44Helsinki Institute of Physics (HIP), Helsinki, Finland
org45High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico
org46Hiroshima University, Hiroshima, Japan
org47Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation
(ZTT), Worms, Germany
org48Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania
org49Indian Institute of Technology Bombay (IIT), Mumbai, India
org50Indian Institute of Technology Indore, Indore, India
org51Indonesian Institute of Sciences, Jakarta, Indonesia
org52INFN, Laboratori Nazionali di Frascati, Frascati, Italy
org53INFN, Sezione di Bari, Bari, Italy
org54INFN, Sezione di Bologna, Bologna, Italy
org55INFN, Sezione di Cagliari, Cagliari, Italy
org56INFN, Sezione di Catania, Catania, Italy
org57INFN, Sezione di Padova, Padova, Italy
org58INFN, Sezione di Roma, Rome, Italy
org59INFN, Sezione di Torino, Turin, Italy
org60INFN, Sezione di Trieste, Trieste, Italy
org61Inha University, Incheon, Republic of Korea
org62Institute for Nuclear Research, Academy of Sciences, Moscow, Russia
org63Institute for Subatomic Physics, Utrecht University/Nikhef, Utrecht,
Netherlands
org64Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia
org65Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India
org66Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic
org67Institute of Space Science (ISS), Bucharest, Romania
org68Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
org69Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico
org70Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil
org71Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico
org72iThemba LABS, National Research Foundation, Somerset West, South Africa
org73Jeonbuk National University, Jeonju, Republic of Korea
org74Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany
org75Joint Institute for Nuclear Research (JINR), Dubna, Russia
org76Korea Institute of Science and Technology Information, Daejeon, Republic
of Korea
org77KTO Karatay University, Konya, Turkey
org78Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France
org79Laboratoire de Physique Subatomique et de Cosmologie, Université
Grenoble-Alpes, CNRS-IN2P3, Grenoble, France
org80Lawrence Berkeley National Laboratory, Berkeley, California, United
States
org81Lund University Department of Physics, Division of Particle Physics,
Lund, Sweden
org82Nagasaki Institute of Applied Science, Nagasaki, Japan
org83Nara Women’s University (NWU), Nara, Japan
org84National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece
org85National Centre for Nuclear Research, Warsaw, Poland
org86National Institute of Science Education and Research, Homi Bhabha
National Institute, Jatni, India
org87National Nuclear Research Center, Baku, Azerbaijan
org88National Research Centre Kurchatov Institute, Moscow, Russia
org89Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
org90Nikhef, National institute for subatomic physics, Amsterdam, Netherlands
org91NRC Kurchatov Institute IHEP, Protvino, Russia
org92NRC «Kurchatov» Institute - ITEP, Moscow, Russia
org93NRNU Moscow Engineering Physics Institute, Moscow, Russia
org94Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United
Kingdom
org95Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy,
Czech Republic
org96Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
org97Ohio State University, Columbus, Ohio, United States
org98Petersburg Nuclear Physics Institute, Gatchina, Russia
org99Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia
org100Physics Department, Panjab University, Chandigarh, India
org101Physics Department, University of Jammu, Jammu, India
org102Physics Department, University of Rajasthan, Jaipur, India
org103Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany
org104Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany
org105Physik Department, Technische Universität München, Munich, Germany
org106Politecnico di Bari, Bari, Italy
org107Research Division and ExtreMe Matter Institute EMMI, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany
org108Rudjer Bošković Institute, Zagreb, Croatia
org109Russian Federal Nuclear Center (VNIIEF), Sarov, Russia
org110Saha Institute of Nuclear Physics, Homi Bhabha National Institute,
Kolkata, India
org111School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
org112Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru
org113St. Petersburg State University, St. Petersburg, Russia
org114Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria
org115SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes,
France
org116Suranaree University of Technology, Nakhon Ratchasima, Thailand
org117Technical University of Košice, Košice, Slovakia
org118The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland
org119The University of Texas at Austin, Austin, Texas, United States
org120Universidad Autónoma de Sinaloa, Culiacán, Mexico
org121Universidade de São Paulo (USP), São Paulo, Brazil
org122Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
org123Universidade Federal do ABC, Santo Andre, Brazil
org124University of Cape Town, Cape Town, South Africa
org125University of Houston, Houston, Texas, United States
org126University of Jyväskylä, Jyväskylä, Finland
org127University of Liverpool, Liverpool, United Kingdom
org128University of Science and Technology of China, Hefei, China
org129University of South-Eastern Norway, Tonsberg, Norway
org130University of Tennessee, Knoxville, Tennessee, United States
org131University of the Witwatersrand, Johannesburg, South Africa
org132University of Tokyo, Tokyo, Japan
org133University of Tsukuba, Tsukuba, Japan
org134Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
org135Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, Lyon, France
org136Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg,
France, Strasbourg, France
org137Université Paris-Saclay Centre d’Etudes de Saclay (CEA), IRFU,
Départment de Physique Nucléaire (DPhN), Saclay, France
org138Università degli Studi di Foggia, Foggia, Italy
org139Università degli Studi di Pavia, Pavia, Italy
org140Università di Brescia, Brescia, Italy
org141Variable Energy Cyclotron Centre, Homi Bhabha National Institute,
Kolkata, India
org142Warsaw University of Technology, Warsaw, Poland
org143Wayne State University, Detroit, Michigan, United States
org144Westfälische Wilhelms-Universität Münster, Institut für Kernphysik,
Münster, Germany
org145Wigner Research Centre for Physics, Budapest, Hungary
org146Yale University, New Haven, Connecticut, United States
org147Yonsei University, Seoul, Republic of Korea
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# Surface modified mesoporous g-$C_{3}N_{4}$@FeN$i_{3}$ as prompt and
proficient magnetic adsorbent for crude oil recovery
Meenakshi Talukdar, Sushant Kumar Behera, Kakoli Bhattacharya and Pritam Deb
Corresponding Author<EMAIL_ADDRESS>Advanced Functional Material
Laboratory, Department of Physics, Tezpur University (Central University),
Tezpur-784028, India.
###### Abstract
Efficient oil adsorption and recovery is a generous universal importance for
future energy demand and environmental protection. Adsorbents based on 2D
flatland with engineered surfaces can overcome the limitations of conventional
methods for selective oil absorption. Here, we report magnetic
hydrophobic/oleophilic graphitic $C_{3}N_{4}$ nanosheets that exhibit
excellent oil sorption performance and ready removal of adsorbed oil using
magnetic field. Combining porous and nanosheet structure along with magnetic
FeN$i_{3}$ and fatty acid surface functionalization make the system an ideal
adsorbent for adsorbing and separating viscous crude oil from water. The
nanocomposite can be further recycled and reused in an ecofriendly manner for
oil adsorption and recovery. The graphitic sheets selectively absorb a wide
range of drilled oils with enhancement of thickness upto 9 folds than the
pristine one. Oil can be collected and recovered with high efficiencies once,
it gets adsorbed by the adsorbent.
keywords
graphitic carbon nitride . hydrophobicity and oleophilicity . magnetic
nanocomposite .
oil-water sorbent . oil separation.
Crude oil is important and predominant energy resource in present human
societies. The availability of crude oil needs to be increased to match the
rapidly growing consumption. Hence, the proficient recovery of spilled oil
from oily-water is an importance concern. During crude oil exploration, a vast
amount of oil in mud is generated and wasted causing soil pollution L. Lee
(2010). The wastage of oil during drilling not only leads to compromisation on
future fuel consumption, but also creates significant threats to environment
and human health. On the other side, oil spills often causes immediate and
long term environmental damage Jernelov (2010). The oil spillage can be
cleaned traditionally by mechanical collection Keller and Broje (2006),
absorbent materials Choi and Cloud (1992), chemical dispersants E. B.
Kujawinski (2011a), bioremediations H. A. Aziz (2010), in situ burning I.
Buist (1999), dispersants E. B. Kujawinski (2011b), solidifiers A. Vidyasagar
and Sureshan (2011), skimmers Broje and Keller (2006), etc. These conventional
oil-removal technologies fail to meet the required efficiency without
affecting the ecosystem. In order to sustain the crude oil sources for future
generations, it is required to adopt corrective means for maximum utilization
and minimum wastage.
While developing an effective treatment method, it is essential to choose an
appropriate adsorbing material to yield oil from water surface. Worldwide
efforts are underway for developing sorption material with optimum water
repelling property (i.e. hydrophobicity) as well as oil adsorption capacity
(i.e. oleophilicity) for practical utilization. Oil adsorption generally
follows three primary steps, like the diffusion of oil molecules into the
sorbent surface, capillary entrapment of the same molecules in the sorbent
structure and oil droplet agglomeration in the porous structure of the sorbent
D. L. Tian (2011). In this regard, few of the synthetic and natural sorption
materials, with both hydrophobic and oleophilic properties, have been
investigated and tested for oil-removal purpose H. Sun (2013). An active area
of research in this regard is to optimize adsorption performance, where the
surface area plays major role Y. Si (2014). Primarily, they should be
developed with high specific surface area and large proportion of meso-pores.
Figure 1: Surface atomic configuration of (a) g-$C_{3}N_{4}$ layer structure
with a interplanner spacing of 12 $\AA{}$, (b) composite of g-$C_{3}N_{4}$ and
FeN$i_{3}$ where the FeN$i_{3}$ is stabilized on surface of the layer
structure, monolayers of (c) g-$C_{3}N_{4}$ and (d) composite. Color codes of
the respective atoms are marked in the figure. Dotted red circle shows the
presence of FeN$i_{3}$ on g-$C_{3}N_{4}$ surface.
In such situation, two dimensional (2D) materials have great potential towards
adsorption processes due to its high surface to volume ratio unlike, its bulk
form S. Barg (2014). Besides, for decent adsorption outcome, high content of
carbon and nitrogen are preferred to present as a prerequisite in the
adsorbent moiety. The strategy here is to employ graphitic $C_{3}N_{4}$, which
provides high surface area owing to its 2D sheet structure. Nontoxic iron
nickel (FeN$i_{3}$) magnetic nanoparticles (MNPs) have been mixed with
graphitic carbon nitride (g-$C_{3}N_{4}$) to form the composite material
(shown in toc Fig. 1). The layers in the g-$C_{3}N_{4}$ sheet are connected
via tertiary amines in a stacked fashion and separated by weak van der Waals
(vdW) forces. The vdW force hardly regulate the uniform delocalization (i.e.
the distribution of magnetic spheres) of FeN$i_{3}$ MNPs on the sheet surface
enhancing the sorption capability of the host flakes. Besides, the
nanocomposite exhibits hydrophobic and oleophilic properties after surface
engineering with stearic acid $(CH_{3}(CH_{2})_{16}$COOH), which makes the oil
extraction easy and effective. Moreover, the nanocomposite efficiently removes
variety of oils from water surface under an external magnet.
The minimum energy based structures of the composite system are fixed using
geometry optimization via Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm W.
Lei (2013). The geometry optimization is fixed via affecting the organizing
atoms in the supercell to minimum total energy position for stable geometry
(Figure 1) with its electronic structure to verify its stability.
Figure 2: (a)TEM micrograph of surface modified g-$C_{3}N_{4}@FeNi_{3}$
nanocomposite before oil absorbtion with SAED pattern embedded at the left
top, AFM 2D images of the surface modified nanocomposite (b) before oil
absortion and (c) after oil absorption embeded with their respective 3D image
on the left top. (d) hydrophilic/oleophobic surface of nanocomposite showing
contact angle below 9$0^{0}$ and (e) surface modified to
hydrophobic/oleophilic showing contact angle above 9$0^{0}$. (f) The final
surface topography of the material is shown after oil absorption corroborating
AFM image.
A representative TEM image of the composite system before surface modification
was shown in Figure 2 (a) with its SAED pattern and schematic structure with
contact angle (Figure 2 (d)). The image revealed uniformity in shape and size
of MNPs, with a diameter around 70 nm over the sheet surface in the composite
system. A stacked graphitic structure was noticed with the dispersion of MNPs
on the carbon nitride sheets impeding the formation of composite system. SAED
pattern indicated that the nanoparticles are polycrystalline in nature with
(002), (111) and (100) planes of both FeN$i_{3}$ and g-$C_{3}N_{4}$ identified
in the composite. AFM images of the system were shown in Figure 2 (b) and (c))
to visualize the surface topography before and after oil removal, respectively
along with their schematic structure (Figure 2 (e) and (f)) for easy
visualization. AFM images provided convincing evidence regarding post-oil
absorption enhancement in MNP size indicating the formation of finite number
of nanoparticle clusters over the sheet surface. Also, the images revealed the
uniform increase of nine times in thickness of the nanosheets after oil
absorption (Figure 2 (c)) compared to the pristine nanosheets (Figure 2(b)).
Micro-structural study of the composite was carried out through x-ray
diffraction (Figure S2-4). The diffraction peaks observed at 2θ=27.40, 44.10,
51.30 and 75.70 in the composite system indexed to (002), (111), (200) and
(220) lattice planes respectively. The above results endorsed the formation of
g-$C_{3}N_{4}@FeNi_{3}$ composite. However, the peak intensity became weaker
and the width of the diffraction peak became broader indicating interactions
between MNPs and host sheet due to presence of Fe in the composite system.
Figure 3: (a) GC-MS plot of the surface modified composite before and after
oil absorptions. (b) Nitrogen adsorption-desorption isotherm, where the pore
size distribution of the same is in the inset. (c) Light microscope image
showing the oil presence on the surface of the composite.
We first used a laser with 514 nm wavelength with hardly detectable Raman
peaks. Fluorescence background of the composite suppressed the Raman peaks at
lower wavelength excitation source of 514 nm A. D. Liao (2011). Lastly, 699
c$m^{-1}$ was due to bending and stretching mode on ring atoms of NCN and CN
(shown in Table S1). Due to breaking of the cyano group, no further Raman
modes were obtained beyond 1650 cm-1 (Figure S6). With rise in temperature
from 45$0^{0}$ C to 55$0^{0}$ C, the crystalline phase transforms to complete
amorphous state corroborating our XRD results T. Rojalin (2016).
The GC-MS characterization (Figure 3(a)) was performed to confirm the
adsorption of the viscous oil and its removal efficacy. It was observed that
several chromatographic peaks were present revealing the existence of crude
oil in the oil-water mixture before oil removal. The water, left-over in the
Petri dish after oil adsorption, was tested again and found no such peaks. The
absence of aforementioned peaks indicated the absence of oil in the oil water
mixture showing efficient oil removal by the composite.
The FTIR spectra of g-$C_{3}N_{4}$ sheet were plotted for 45$0^{0}$ C and
55$0^{0}$ C and 50$0^{0}$ C for comparison with the composite system. The
adsorption band of composite system at 1605 c$m^{-1}$ were assigned to C=N.
Similarly, the absorption bands at 1257, 1311 and 1424 c$m^{-1}$ were assigned
to aromatic C-N stretching heterocycles. The presence of MNPs suppressed the
peak intensity in case of composite suggesting weak interaction between the
other carbon and nitrogen bonds X. Gao (2016). The textural properties of
carbon nitride based catalysts were studied by nitrogen sorption. Figure 3(b)
displayed the absorption-desorption isotherm curves of g-$C_{3}N_{4}$ sheet
showing a characteristic of type IV isotherm pattern. The surface area was
found to be 41 $m^{2}g^{−1}$ for g-$C_{3}N_{4}$ from the Brunauer-Emmett-
Teller (BET) curve. Barrett-Joyner-Halenda (BJH) curve indicated the pore-size
distribution of the flake reflecting the existence of mesopore in the flake
structure (inset of Figure 3(b)). The pore-size distribution curve showed the
mesoporous structure with the pore size in the range of 4-40 nm. The textural
property of the g-$C_{3}N_{4}@FeNi_{3}$ is basically consistent with that of
g-$C_{3}N_{4}$ support, suggesting that the FeN$i_{3}$ nanoparticles do not
block the pore distribution on the sheet surface. Figure 3(c) gives the light
microscopic image of the oil absorbed g-C3N4@FeNi3 surface. The spotted green
colour dotted line reveals the presence of crude oil on nanocomposite surface.
Figure 4: Particle size distribution pattern of (a) surface modified
g-$C_{3}N_{4}@FeNi_{3}$ nanocomposite (i) before oil absorbtion embe dded
(right hand top position of the image) with contact angle and (ii) after oil
absorption inset with contact angle showing hydrophobic and oleophilic
behaviour. (b) Removal process of crude oil ((i)-(iv)) from water surface by
nanocomposite under magnetic field.
The surface engineering enabled in wettability transition of the
nanocomposite, which was ensured by contact angle measurements before and
after oil absorption (shown inset of Figure 4a (i) and (ii) respectively). The
increase in contact angle upto 140° implied the hydrophobic and oleophilic
nature of the prepared nanocomposite material. The shape of the water droplet
above the sample was uniform and indicating the surface to be highly
hydrophobic. The contact angle showed hydrophobic behavior after reusing the
composite for oil removal for the second time (shown inset of Figure 4a (ii)).
The nanocomposite retained its hydrophobic and oleophilic behavior after its
first use which showed the transition in wettability nature due to good
stearic acid based surface engineering of the nanocomposite. The dynamic light
scattering (DLS) measurements are shown in Figure 4a (i) and (ii), which
evaluated the hydrodynamic diameters. The hydrodynamic diameter was obtained
to be almost 1.2 $\mu$m before oil extraction, while this diameter value
increased significantly after oil adsorption. The hydrodynamic diameter,
measured by DLS, was affected by the viscosity and concentration of the
medium. As a result, the value was larger than that the value obtained from
TEM image of the same system (Figure 2a).
The highly hydrophobic and oleophilic nanocomposite exhibited a selective
absorbance for oil recovery. When brought into contact with a layer of crude
oil in oil-water mixture, the nanocomposite adsorbed the oil instantaneously
repelling the water. Interestingly, the oil-adsorbed nanocomposite could be
removed from oil-water mixture with an external magnet with a very quick
response. The process of oil absorption and removal is shown in Figure 4b (i)
to (iv) in stepwise manner.
The outcome efficacy is measured using the formula C. Li (2017)
$n=(m_{2}-m_{1})/c_{0}$
where $m_{1}$ is the weight of nanocomposite before extraction and $m_{2}$ is
the weight of nanocomposite after oil extraction. $c_{0}$ is the weight of the
added crude oil. The outcome efficiency of crude oil extraction is found to be
92 % with the prepared magnetic nanocomposite. Apart from crude oil
separation, other oils such as Mobil, Petrol and Mustard oil were also used in
extraction process. It has been found that surface modified 2D
g-$C_{3}N_{4}@FeNi_{3}$ can also be used to separate these variety of oils.
Apart from crude oil separation, it showed noted removal efficiency in case of
other oils.
The nanocomposite can be further recycled and reused for oil adsorption and
separation in an ecofriendly manner as shown in Figure 5. The graphitic
magnetic nanocomposite once used can be recycled and reuse again. The system
can be further reused easily by washing the composite with ethanol and then
drying at a temperature of 6$0^{0}$ C. The removed oil can be recovered
successfully, apart from its reusability. Figure 5(c) shows the recovery of
oil after separation.
Figure 5: The collected surface modified g-$C_{3}N_{4}@FeNi_{3}$ nanocomposite
(a) is washed (b) and oil is collected (c).
In summary, we develop a novel and smart surface engineered sorbent material
to realize magnetic separation of oil from oily water mixtures. The stearic
acid based surface functionalized porous graphitic flakes exhibit not only
wettability transition from hydrophilicity to hydrophobicity, but also
selective oleophilicity in oil removal from water body. 2D graphitic flatland
with precise surface area and high proportion of mesopore achieves near-
absolute adsorption efficiency of crude oil. As a result, the sheet thickness
enhanced upto 9 folds compared to the pristine one. The developed system
efficiently recovers crude oil from water surface under an external magnet
with a quick response. Apart from the crude oil, the composite system
selectively adsorbs a wide range of oils. Moreover, we present density
functional theory DFT calculations to validate the stability and sorption
activity of the composite system. Thus, the prepared g-$C_{3}N_{4}@FeNi_{3}$
nanocomposite acts as promising absorbent material for efficient oil recovery
and hence provides feasible solution towards the upcoming oil consumption.
###### Acknowledgements.
PD would like to acknowledge UGC research award grant. MT acknowledges SAIC,
Tezpur University and SAIF, NEHU, Shillong for few characterization analyses.
MT acknowledges Tezpur University for providing financial support. SKB
acknowledges CSE, Tezpur University for providing HPCC facility and DST, Govt.
of India for INSPIRE Fellowship. MT gratefully acknowledges Dr. Pabitra Nath
and Prof. Ashok Kumar for their valuable inputs.
## References
* L. Lee (2010) W. X. Z. B. H. . W. J. H. L. Lee, Q. Kalb, Science 328, 76 (2010).
* Jernelov (2010) A. Jernelov, Nature 466, 182 (2010).
* Keller and Broje (2006) A. A. Keller and V. Broje, Environ. Sci. Technol. 40, 7914 (2006).
* Choi and Cloud (1992) H. M. Choi and R. M. Cloud, Environ. Sci. Technol. 26, 772 (1992).
* E. B. Kujawinski (2011a) D. L. V. A. K. B. K. L. M. C. R. E. B. Kujawinski, M. C. K. Soule, Environ. Sci. Technol. 45, 1298 (2011a).
* H. A. Aziz (2010) M. H. I. L. M. S. M. H. A. Aziz, M. A. Zahed, Bioresour. Technol. 101, 9455 (2010).
* I. Buist (1999) S. P. S. R. K. T. I. Buist, J. McCourt, Pure Appl. Chem. 71, 43 (1999).
* E. B. Kujawinski (2011b) . L. V. A. K. B. . . L. M. C. R. E. B. Kujawinski, M. C. K. Soule, Environ. Sci. Technol. 45, 1298 (2011b).
* A. Vidyasagar and Sureshan (2011) K. H. A. Vidyasagar and K. M. Sureshan, Angew. Chem. Int. Ed. 50, 8021 (2011).
* Broje and Keller (2006) V. Broje and A. A. Keller, Environ. Sci. Technol. 40, 7914 (2006).
* D. L. Tian (2011) X. W. J. Z. L. J. D. L. Tian, X. F. Zhang, Phys. Chem. Chem. Phys. 13, 14606 (2011).
* H. Sun (2013) C. G. H. Sun, Z. Xu, Adv. Mater. 25, 2554 (2013).
* Y. Si (2014) X. T. J. G. B. D. Y. Si, J. Yu, Nat. Commun. 5, 5802 (2014).
* S. Barg (2014) N. N. P. d. V. P. R. C. M. E. G.-T. S. E. S. A. C. M. E. S. S. Barg, F. M. Perez, Nat. Commun. 5, 4328 (2014).
* W. Lei (2013) D. L. S. Q. Y. C. W. Lei, D. Portehault, Nat. Commun. 4, 1777 (2013).
* A. D. Liao (2011) X. W. K. T. D. J. H. D. E. P. A. D. Liao, J. Z. Wu, Phys. Rev. Lett. 106, 256801 (2011).
* T. Rojalin (2016) T. L. T. . J. . K. C. . L. S. .-H. C. J. . M. T. Rojalin, L. Kurki, Anal. Bioanal. Chem. 408, 761 (2016).
* X. Gao (2016) R. D. Z. X. S. D. R. L. Z. L. J. Z. X. Gao, J. Zhou, Adv. Mater. 28, 168 (2016).
* C. Li (2017) C. Y. Z. D. L. J. C. Li, L. Wu , Angew. Chem. 129, 3811 (2017).
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# Transformations of Stabilizer States in Quantum Networks
Matthias Englbrecht Institute for Theoretical Physics, University of
Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria Tristan Kraft
Institute for Theoretical Physics, University of Innsbruck, Technikerstraße
21A, 6020 Innsbruck, Austria Barbara Kraus Institute for Theoretical
Physics, University of Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria
###### Abstract
Stabilizer states and graph states find application in quantum error
correction, measurement-based quantum computation and various other concepts
in quantum information theory. In this work, we study party-local Clifford
(PLC) transformations among stabilizer states. These transformations arise as
a physically motivated extension of local operations in quantum networks with
access to bipartite entanglement between some of the nodes of the network.
First, we show that PLC transformations among graph states are equivalent to a
generalization of the well-known local complementation, which describes local
Clifford transformations among graph states. Then, we introduce a mathematical
framework to study PLC equivalence of stabilizer states, relating it to the
classification of tuples of bilinear forms. This framework allows us to study
decompositions of stabilizer states into tensor products of indecomposable
ones, that is, decompositions into states from the entanglement generating set
(EGS). While the EGS is finite up to $3$ parties [Bravyi et al., J. Math.
Phys. 47, 062106 (2006)], we show that for $4$ and more parties it is an
infinite set, even when considering party-local unitary transformations.
Moreover, we explicitly compute the EGS for $4$ parties up to $10$ qubits.
Finally, we generalize the framework to qudit stabilizer states in prime
dimensions not equal to $2$, which allows us to show that the decomposition of
qudit stabilizer states into states from the EGS is unique.
## 1 Introduction
The pursuit of building large-scale quantum networks to implement a quantum
internet [1, 2] has recently attracted much interest from the experimental and
the theoretical perspective [3, 4, 5]. In particular, several important
building blocks of quantum networks have been demonstrated experimentally [6,
7]. Such networks open interesting possibilities for advanced quantum
information processing tasks, which include distributed quantum computing [8],
quantum metrology [9], and, most prominently, long-distance quantum
communication [10]. Beyond future applications, the strong correlations in
quantum networks also open new possibilities for fundamental tests of quantum
mechanics (cf. Ref. [11]). As technology progresses it becomes more and more
relevant to understand quantum networks from the theoretical point of view.
The fundamental building block of these networks are entangled quantum
systems. Entanglement itself is quite well understood in the bipartite case,
but understanding multipartite entanglement, despite many efforts, still
remains to be an extremely challenging task [12, 13, 14, 15]. In particular,
our understanding of network entanglement is still quite limited, even in the
case of a few qubits only, which is due to its complicated structure. Indeed,
recent results demonstrate that its characterization requires new mathematical
tools to be explored [16, 17, 18].
In this regard, so-called _stabilizer states_ constitute an interesting
subclass of multipartite entangled pure states. These states are defined as
the unique simultaneous eigenstate of a maximal set of commuting Pauli
observables, the so-called _stabilizer_ [20, 19]. The stabilizer formalism was
originally introduced by Gottesman in the realm of quantum error correction
codes [21]. Nowadays, stabilizer states are known to be important to many
aspects of quantum information processing such as, e.g., measurement based
quantum computing [22] or the classical simulation of quantum circuits [23].
_Graph states_ [20, 19] admit a simple description in terms of mathematical
graphs 111In the context of quantum error correction graph states were first
considered in Ref. [24].. It is known that any stabilizer state is equivalent
to a graph state under a certain class of local operations, the so-called
local _Clifford operations_ , which map the set of stabilizer states onto
itself. Due to their relevance, as outlined above, gaining a deeper
understanding of entanglement in stabilizer states is an important goal.
In this paper we study stabilizer states in quantum networks. Creating
bipartite entanglement in quantum networks is considerably less demanding than
the distribution of large entangled states. In particular, well connected
nodes can share a large amount of bipartite entanglement at basically no cost.
This bipartite entanglement can be used, for instance, to implement non-local
transformations of stabilizer states via gate teleportation [25]. Here,
Clifford operations turn out to be particularly important as they can be
implemented deterministically. Thus, well connected nodes having access to the
additional bipartite entanglement can be viewed as a single party. We call the
operations that can be applied on those parties _party-local Clifford_ (PLC)
operations. In case each party only holds a single qubit it is known that two
stabilizer states are equivalent under local Cliffords if and only if their
corresponding graph states are connected by a series of _local
complementations_ [26].
If all qubits of a stabilizer state are grouped into two parties the bipartite
stabilizer state can be converted into multiple independent copies of the
maximally entangled state $\ket{\phi^{+}}$, and locally separable states
$\ket{0}$ using PLC transformations [27]. For tripartite stabilizer states it
was shown in Ref. [28] that any state can be converted to multiple independent
copies of the Greenberger-Horne-Zeilinger (GHZ) state $\ket{GHZ}$ between the
three parties, the Bell state $\ket{\phi^{+}}$ between pairs of parties, and
locally separable states $\ket{0}$, see Figure 1. The fact that these
decompositions are unique, even under more general party local unitary (PLU)
transformations, motivated the authors of Ref. [28] to call these sets
_entanglement generating sets_. These results were later generalized to
tripartite qudit graph states in Ref. [29].
Figure 1: Top: A stabilizer state is distributed over two parties. Under party
local unitary transformations the state can be converted to multiple
independent copies of the Bell state, and locally separable states. Bottom:
The same state, when grouped into three parties, can be decomposed into a GHZ
state, a Bell state and locally separable states.
Furthermore, Ref. [28] provides criteria for the PLC equivalence of stabilizer
states. In Ref. [30] homological invariants of stabilizer states under local
Clifford operations are studied, and were linked to the GHZ extraction
condition of Ref. [28]. The extraction of GHZ states from random stabilizer
states was investigated in Refs. [31, 32, 33]. However, a systematic approach
to study the extraction of arbitrary stabilizer states in the multipartite
scenario was missing. Putting forward such an approach that allows us to gain
a deeper insight into the structure of entanglement in multipartite stabilizer
states is precisely the aim of this work.
In this paper we study PLC transformations and state extraction of stabilizer
states. First, we introduce notations and definitions relevant in the context
of PLC transformations of stabilizer states in Section 2.1. Then we study how
PLC transformations between graph states alter the corresponding graphs in
Section 2.3, generalizing the concept of local complementation. As any
stabilizer state is LC equivalent to a graph state this characterizes the
action of any PLC operations on an arbitrary stabilizer state. In Section 2.4
we put forward a complete characterization of PLC equivalence classes based on
PLC invariant polynomials which seems computationally not very practical. Then
we recall previous works on PLC transformations of stabilizer states (Section
2.5). We mainly focus on the results of Ref. [28] which solves the
classification of PLC classes for the $3$-partite case and introduces the
notion of the entanglement generating set. In Section 3, we introduce one of
our main results, namely, a new mathematical formalism, that we call the
_commutation matrix formalism_ , that allows us to systematically study PLC
transformations and state extraction of stabilizer states shared between an
arbitrary number of parties. In this formalism we associate a tuple of
alternating bilinear forms to every stabilizer state. We show that the
classification of PLC classes is equivalent to the classification of
commutation matrices up to simultaneous congruence. This type of problem was
studied in linear algebra and we demonstrate how these results can be utilized
in the commutation matrix formalism in Sections 3.1 and 3.2. In particular
this leads to necessary and sufficient conditions for a state to be contained
in the EGS. In Section 3.3, using the commutation matrix formalism, we derive
another main result by showing that the EGSM is an infinite set for $M\geq 4$,
even with respect to party-local unitary transformations (PLU). Moreover, the
new formalism allows us to compute the EGS4 up to $10$ qubits. Going beyond
the EGS4 we provide evidence that the structure of the EGS becomes rapidly
more complex. In Section 4 we show that the commutation matrix formalism
extends to qudit stabilizer states of prime dimension. As the commutation
matrices in this case are defined over the field $\mathbb{Z}_{d}$ for $d\geq
3$, the results of Ref. [34] imply that every qudit stabilizer states admits a
unique decomposition into a tensor product of states in the EGS. Whether or
not the uniqueness of the decomposition extends to all qubit stabilizer states
is unclear. Finally, in case one considers more general states or operations
it happens that the decomposition is not unique anymore, which we discuss in
Section 4.2.
## 2 Party-local operations
### 2.1 Preliminaries
Let us introduce some notation and recall some important definitions relevant
in the context of PLC transformations of stabilizer states. We denote by
$X,Y,Z$ the single-qubit Pauli operators
$X=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\ \ Y=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\ \ Z=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$ (1)
and by $\mathcal{P}_{n}$ the $n$-qubit Pauli group, i.e., the group generated
by $n$-fold tensor products of single-qubit Pauli operators. A stabilizer
$\mathcal{S}$ is an abelian subgroup of the Pauli group $\mathcal{P}_{n}$,
which does not contain $-\mathds{1}$. A maximal abelian subgroup of
$\mathcal{P}_{n}$ is generated by $n$ independent commuting Pauli operators.
The operators of a stabilizer forming a maximal abelian subgroup admit a
unique, common $+1$ eigenstate, which is called a stabilizer state. Examples
of stabilizer states are the Bell state
$\ket{\phi^{+}}=1/\sqrt{2}(\ket{00}+\ket{11})$ and the $n$-qubit GHZ state
$\ket{GHZ}=1/\sqrt{n}(\ket{0\ldots 0}+\ket{1\ldots 1})$.
It follows from the definition of stabilizer states that the density matrix
$\outerproduct{\psi}{\psi}$ of a $n$-qubit stabilizer state $\ket{\psi}$ with
stabilizer $\mathcal{S}$ can we written as
$\rho=\outerproduct{\psi}{\psi}=\frac{1}{2^{n}}\sum_{s\in\mathcal{S}}s.$ (2)
Using this relation together with the fact that Pauli operators are traceless,
one finds that the reduced state on a subset of qubits $\alpha$ is given by
$\rho_{\alpha}=\frac{1}{2^{|\alpha|}}\sum_{s\in\mathcal{S}_{\alpha}}s,$ (3)
where $\mathcal{S}_{\alpha}$ contains those elements of $\mathcal{S}$ which
act trivially on all qubits not in $\alpha$. Here, and in the following,
$|\alpha|$ denotes the cardinality of the set $\alpha$. The rank of the
reduced state $\varrho_{\alpha}$ depends on how many qubits in $\alpha$ are
entangled with qubits outside of $\alpha$, e.g., if all qubits are entangled
$\mathcal{S}_{\alpha}$ only contains the identity operator and
$\varrho_{\alpha}$ has full rank.
The unitary normalizer (modulo $U(1)$) of the Pauli group $\mathcal{P}_{n}$ is
called the $n$-qubit Clifford group $\mathcal{C}_{n}$, i.e.,
$\mathcal{C}_{n}=\\{U\in U(2^{n})|UPU^{\dagger}\in\mathcal{P}_{n}\forall
P\in\mathcal{P}_{n}\\}\backslash U(1)$. For any $n$, the Clifford group
$\mathcal{C}_{n}$ is generated by single and two qubit gates, e.g.,
$\begin{split}H&=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix},\
P=\begin{pmatrix}1&0\\\ 0&i\end{pmatrix},\\\ CZ&=\begin{pmatrix}1&0\\\
0&0\end{pmatrix}\otimes\mathds{1}+\begin{pmatrix}0&0\\\
0&1\end{pmatrix}\otimes Z.\end{split}$ (4)
A subgroup of the Clifford group $\mathcal{C}_{n}$ is the local Clifford group
$\mathcal{C}_{n}^{L}$, which consists of those elements of $\mathcal{C}_{n}$
which can be written as a tensor product of elements of $\mathcal{C}_{1}$.
Every stabilizer state is local Clifford (LC) equivalent to a graph state
[26]. A graph state $\ket{G}$ is the unique stabilizer state associated to a
simple (no loops, undirected) graph $G=(V,E)$ with vertices
$V=\\{1,2,\ldots,n\\}$ and edges $E\subset\\{\\{i,j\\}|i,j\in V\\}$. Its
stabilizer $\mathcal{S}_{G}=\left<g_{1},\ldots,g_{n}\right>$ is generated by
the _canonical generators_
$g_{i}=X_{i}\bigotimes_{j\in N_{i}}Z_{j},$ (5)
where $N_{i}=\\{k\in V|\\{k,i\\}\in E\\}$ is the neighbourhood of vertex $i$
in $G$. A local Clifford transformation which maps a graph state to another
graph state corresponds to a sequence of local complementations on the level
of the graphs [26]. A local complementation with respect to (w.r.t.) the
vertex $j\in E$ removes all edges between neighbours of vertex $j$ and adds
all edges between neighbours of vertex $j$, which where previously not present
in the graph. That is, it removes the edges in $\\{\\{k,l\\}\in E|k,l\in
N_{j}\\}$ and adds all edges in $\\{\\{k,l\\}\not\in E|k,l\in N_{j}\\}$ to the
graph $G$. This corresponds to the application of the operator
$\exp(-i\pi/4X)$ on qubit $j$, and $\exp(i\pi/4Z)$ on each of its neighbours.
Let us mention here that counting the number of local Clifford equivalent
graph states is #P-Complete [35]. Note that the graphical description of local
Clifford transformations between graph states has also been studied for qudit
systems [36, 37]. A particularly useful way of representing an $n$-qubit graph
state $\ket{G}$ is by its _adjacency matrix_ $\Gamma_{G}\in M_{n\times n}$,
defined by
$(\Gamma_{G})_{i,j}=\begin{cases}1,&\text{if $i$ and $j$ are connected}\\\
0,&\,\text{otherwise.}\end{cases}$ (6)
For a graph state $\ket{G}$ any element of its stabilizer $\mathcal{S}$ is a
local symmetry. However, the state $\ket{G}$ can have more local symmetries
than those in $\mathcal{S}$ (see Ref. [38]). For example, consider a graph
state $\ket{G}$ where qubit $1$ is only connected to qubit $2$. Then qubit $1$
is called a _leaf_ , qubit $2$ is called its _parent_ and for any
$\alpha\in\mathbb{C}$ the operator
$e^{i\alpha X}\otimes e^{-i\alpha Z}\otimes\mathds{1}$ (7)
is a local symmetry of $\ket{G}$ (see Ref. [39]). In Ref. [38] some of the
authors of the current work studied, among other things, all local symmetries
of stabilizer states. While preparing the current work, we became aware of
Ref. [40] (and Ref. [41]), which studies equivalence of graphs under local
complementation. In this context, Ref. [40] also derives the local Clifford
symmetries of graph states.
In this work we are interested in a superset of local Clifford
transformations, so called party-local Clifford transformations (PLC). To
define these transformations, consider a partition $P(M,n)$ of the set
$[n]:=\\{1,\ldots n\\}$ into $M$ subsets. In the following, we refer to the
elements of $P(M,n)$ as parties. We call operators of the form
$\bigotimes_{\alpha\in P(M,n)}C_{\alpha}\ \text{ with }\
C_{\alpha}\in\mathcal{C}_{|\alpha|},$ (8)
party-local Clifford operators and we denote the subgroup generated by these
operators by $\mathcal{C}^{P(M,n)}$. Note that
$\mathcal{C}^{P(M,n)}\subset\mathcal{C}_{n}$ and equality only holds for
$M=1$. We say that an $n$-qubit stabilizer state is $M$-partite if its qubits
are distributed among $M$ parties. In the following we denote by
$\text{Stab}(P(M,n))$ the set of $n$-qubit stabilizer states that are
distributed according to the partition $P(M,n)$.
As we will discuss in Section 2.1, the Pauli group can be represented as a
vector space over a finite field. We denote by $\mathbb{Z}_{d}$ the finite
field of order $d$ for any prime number $d$. Over this field addition and
multiplication are carried out modulo $d$.
### 2.2 Motivation
In this section we explain our motivation for studying PLC transformation of
stabilizer states in quantum networks. We argue that PLC transformations are a
physically motivated extension of local operations.
As experimental techniques progress, the distribution of bipartite
entanglement between well connected nodes in a quantum network will no longer
be an expensive resource. For instance, if two nodes are spatially close they
might be considered as well connected. Nevertheless, two nodes can be not
spatially close and well connected if they can establish a large amount of
entanglement over long distances, e.g., via a satellite [42]. One can use this
bipartite entanglement to implement non-local gates. This setting effectively
breaks the network down into several parties, within which non-local
operations are possible.
Figure 2: Local complementation (LC) on the parties highlighted in green
compared to local complementation and edge removal/addition (LCE) on the
parties highlighted in grey.
Clifford operations play a special role in such a setting as they can be
implemented deterministically via gate teleportation (see Refs. [25, 43]).
Indeed, to apply a non-local gate $U\in\mathcal{C}_{2}$ to two qubits held by
spatially separated parties, we can distribute the Choi state
$U\otimes\mathds{1}\ket{\phi^{+}}^{\otimes 2}$ among the parties. After
performing a Bell measurement in each of the parties (each involving one qubit
of the Choi state and the one originally held by the party), the state of the
two unmeasured qubits is
$UP\varrho P^{\dagger}U^{\dagger}=UPU^{\dagger}(U\varrho
U^{\dagger})UP^{\dagger}U^{\dagger},$ (9)
where $P\in\mathcal{P}_{n}$ depends on the measurement result and $\rho$
denotes the initial state of the two qubits originally held by the parties. As
Clifford operators map the Pauli group onto itself, local Pauli corrections
suffice to invert $UPU^{\dagger}$. As the Clifford group is generated by
single and two-qubit gates (see Section 2.1), the above well-known
construction allows to implement any non-local Clifford gate. For other non-
local gates, which are not Clifford gates, the correction $UPU^{\dagger}$ will
not be local in general and thus a deterministic local correction is not
possible.
In this work we only consider PLC transformations among special types of
states, namely stabilizer states. This is, on the one hand, due to the
outstanding importance of these states for quantum information protocols. On
the other hand, Clifford operations map stabilizer states to stabilizer states
and thus these states are a natural candidate to study in the setting we
consider. Additionally, the experimental requirements to implement party-local
Clifford gates on stabilizer states are actually significantly less than what
we described above. For stabilizer states, the distribution of Bell states,
local $Z$ basis measurements and the application of Clifford gates, which act
locally on the nodes of the network, suffice to implement any party-local
Clifford gate. Indeed, it is easy to see that after distributing the Bell
state between two parties both parties can apply a $CZ$ between the qubits of
the stabilizer state and the Bell state, perform local complementation on the
qubits of the Bell state and then measure them in the $Z$ basis to implement a
non-local $CZ$ gate up to local Pauli corrections.
Motivated by the above, it is the goal of this work to approach a complete
characterization of PLC equivalence classes of stabilizer states.
### 2.3 Generalized local complementation
In this section, we study how party-local Clifford transformations from graph
states to graph states alter the corresponding graphs. We mentioned in Section
2.1 that local Clifford transformations which map a graph state to another
graph state correspond to a sequence of local complementations on the level of
the graphs [26]. We analyze how these results generalize to party-local
Clifford transformations in the following theorem.
###### Theorem 1.
Let $\ket{G}$ and $\ket{G^{\prime}}$ be $n$-qubit graph states distributed
among the parties $P(M,n)$. Then $\ket{G}$ is PLC equivalent to
$\ket{G^{\prime}}$ if and only if the graphs $G$ and $G^{\prime}$ are related
via a sequence of local complementations and edge additions/removals within
parties (LCE).
Let us compare Theorem 1 to the results of [26] for local Clifford operations.
We see that the only additional power party-local Clifford transformations
have over local Clifford transformations is the possibility of adding and
removing edges within parties. However, note that adding and removing edges
within parties and local complementations are non-commuting operations and
thus LCE generates larger orbits. The proof of Theorem 1 makes use of Lemma 2,
which is stated and proven in Appendix A . Here, we state only an informal
version of it, as this is sufficient to understand the reason for the validity
of Theorem 1.
###### Lemma 2 (informal version).
The action of any two-qubit Clifford gate on a graph state is equivalent to
the action of a suitable sequence of LCE up to local Clifford corrections.
Using Lemma 2 and the fact that $\mathcal{C}_{n}$ is generated by single and
two-qubit Clifford operators (see Section 2.1) we can then show Theorem 1. The
proof can be found in Appendix A.
### 2.4 Classification of PLC classes via invariant polynomials
The characterization of equivalence classes of states under certain operations
is a common task in quantum information theory. In the context of stochastic
local operations assisted by classical communication (SLOCC), invariant
polynomials proved to be a fruitful approach [44]. Motivated by this, we study
the use of invariant polynomials in the classification of PLC classes of
stabilizer states.
Consider a stabilizer state $\ket{\psi}\in\text{Stab}(P(M,n))$. We want to
identify all states in the PLC class of $\ket{\psi}$. To this end, let us
choose a set of generators $g_{1}^{\psi},\ldots,g_{n}^{\psi}$ for the
stabilizer $\mathcal{S}$ of $\ket{\psi}$ and define the polynomial
$I_{\ket{\psi}}(\ket{\phi})=\sum_{C\in\mathcal{C}_{n}^{P(M,n)}}\prod_{j=1}^{n}\innerproduct*{\phi}{C^{\dagger}g_{j}^{\psi}C|\phi}^{2}.$
(10)
Observe that this polynomial is obviously PLC invariant, i.e., for all
$C\in\mathcal{C}^{P(M,n)}$ it holds that
$I_{\ket{\psi}}(C\ket{\phi})=I_{\ket{\psi}}(\ket{\phi})$, and that
$I_{\ket{\psi}}=I_{C\ket{\psi}}$. Moreover, for any stabilizer state
$\ket{\phi}$ this polynomial is nonzero if and only if $\ket{\phi}$ is PLC
equivalent to $\ket{\psi}$, as we show in the following. Using the fact that
any stabilizer state can be written in terms of its stabilizer as in Equation
(2), and that every $P\in\mathcal{P}_{n}$ not equal to the identity is
traceless, a term in the polynomial in Equation (10) is nonzero if and only if
$\\{C^{\dagger}g_{i}^{\psi}C\\}_{i=1}^{n}$ generate the stabilizer of
$\ket{\phi}$ for some $C\in\mathcal{C}^{P(M,n)}$. Hence, the polynomial is
nonzero if and only if $\ket{\psi}$ and $\ket{\phi}$ are PLC equivalent. We
will see in Section 2.5 that it follows from the results of Ref. [28] that the
polynomial $I_{\ket{\psi}}$ only depends on the commutation relations of the
generators $g_{1}^{\psi},\ldots,g_{n}^{\psi}$ on the individual parties.
While each PLC equivalence class can be characterized by a polynomial as in
Equation (10), this characterization is not very practical. To our knowledge,
the only way to compute the value of such a polynomial (with the possible
exception of a small number of parties) for a given stabilizer state is to
explicitly perform the sum over all terms. If that is the case, computing the
polynomial is not more efficient than directly checking PLC equivalence by
computing the full PLC orbit.
Let us remark that invariants have been used previously to study PLC
equivalence of stabilizer states. Ref. [45] introduces a finite set of
invariants that completely characterize local Clifford equivalence of
stabilizer states. That is, two states are local Clifford equivalent if and
only if they agree in these invariants. Ref. [30] takes a similar approach,
defining so called homological invariants of stabilizer states, which apply to
arbitrary party sizes, and even qudit stabilizer states. These are used to
rederive the results on the extraction of GHZ states from Ref. [28] (see
Section 2.5).
### 2.5 Previous results
In this section we summarize the main results obtained in the context of PLC
transformations of stabilizer states, focusing on those that we use in the
following.
The classification of $3$-partite PLC equivalence classes of stabilizer states
was completely solved in Ref. [28]. There, it is shown that every $3$-partite
stabilizer state is PLC equivalent to a tensor product of a unique number of
copies of the states $\ket{0}$, $\ket{\phi^{+}}$ and $\ket{GHZ}$, see Figure 1
for an example. That is, two stabilizer states are PLC equivalent if and only
if their unique decompositions coincide. In the following we make use of some
of the techniques developed in Ref. [28]. Therefore, let us briefly introduce
some of the notation that was being used and discuss the content of the paper
in more detail. First, recall that the Pauli group $\mathcal{P}_{n}$ modulo
phases is isomorphic to the vector space $\mathbb{Z}_{2}^{2n}$. To see that,
one identifies the single-qubit Pauli operators and the identity operator with
elements of $\mathbb{Z}_{2}^{2}$ as
$\sigma_{0,0}=\mathds{1},\ \sigma_{1,0}=X,\ \sigma_{1,1}=Y,\ \sigma_{0,1}=Z.$
(11)
Then, one can define a map
$\sigma:\mathbb{Z}_{2}^{2n}\rightarrow\mathcal{P}_{n}$ via
$\sigma((a_{1},b_{1},\ldots,a_{n},b_{n}))=\sigma_{a_{1},b_{1}}\otimes\ldots\otimes\sigma_{a_{n},b_{n}}.$
(12)
From this relation one sees that the multiplication of Pauli operators
corresponds to the addition of their corresponding vectors in
$\mathbb{Z}_{2}^{2n}$, i.e.,
$\sigma(\bm{f})\sigma(\bm{g})\propto\sigma(\bm{f}+\bm{g})\ \ \
\bm{f},\bm{g}\in\mathbb{Z}_{2}^{2n}.$ (13)
The commutation relations of the Pauli operators define a symplectic bilinear
form $\omega:\mathbb{Z}_{2}^{2n}\rightarrow\mathbb{Z}_{2}$ by
$\sigma(\bm{f})\sigma(\bm{g})=e^{i\pi\omega(\bm{f},\bm{g})}\sigma(\bm{g})\sigma(\bm{f}).$
(14)
Indeed, writing $\bm{f}=(a_{1},b_{1}\ldots,a_{n},b_{n})$ and
$\bm{g}=(a_{1}^{\prime},b_{1}^{\prime},\ldots,a_{n}^{\prime},b_{n}^{\prime})$
the above definition implies that
$\displaystyle\begin{split}&\omega(\bm{f},\bm{g})\\\
&=(a_{1},\ldots,a_{n},b_{1},\ldots b_{n})\begin{pmatrix}0&\mathds{1}\\\
\mathds{1}&0\end{pmatrix}(a_{1}^{\prime},\ldots,a_{n}^{\prime},b_{1}^{\prime},\ldots
b_{n}^{\prime})^{T}.\end{split}$ (15)
Thus, the bilinear form $\omega$ satisfies $\omega(\bm{f},\bm{f})=0$, and is
non-degenerate, i.e., $\omega(\bm{f},\bm{g})=0$ for all $\bm{g}$ implies that
$\bm{f}=0$. We conclude that $\mathbb{Z}^{2n}_{2}$ together with $\omega$ is a
symplectic vector space. A brief introduction to symplectic vector spaces can
be found in Appendix B.
Every stabilizer $\mathcal{S}\subset\mathcal{P}_{n}$ can be written as
$\mathcal{S}=\\{\epsilon(\bm{f})\sigma(\bm{f})|\bm{f}\in V^{\mathcal{S}}\\}$
where $V^{\mathcal{S}}$ is a maximally isotropic subspace of
$\mathbb{Z}^{2n}_{2}$ and $\epsilon:V^{\mathcal{S}}\rightarrow\\{\pm 1\\}$ is
a suitable sign choice. A maximally isotropic subspace is a subspace which is
equal to its orthogonal complement defined via the symplectic form $\omega$
(for details see Appendix B). A set of generators of $\mathcal{S}$ corresponds
to a basis of $V^{\mathcal{S}}$ via the map $\sigma$. Choosing a different set
of generators of $\mathcal{S}$ is equivalent to a basis change in
$V^{\mathcal{S}}$ via Equation (13) and it is described by an invertible
matrix. Note that subspaces which are isotropic but not maximally isotropic,
i.e., which are contained in their orthogonal complement but not necessarily
equal to it, correspond to stabilizer codes.
Any Clifford operator $C\in\mathcal{C}_{n}$ defines a linear map
$u:\mathbb{Z}_{2}^{2n}\rightarrow\mathbb{Z}_{2}^{2n}$ via
$C\sigma(\bm{f})C^{\dagger}\propto\sigma(u(\bm{f})).$ (16)
As $C$ is unitary, $u$ is invertible and preserves the symplectic form
$\omega$, i.e., it is an isometry. In fact, one can show that any isometry on
$\mathbb{Z}_{2}^{2n}$ corresponds to a $C\in\mathcal{C}_{n}$ via the above
relation. This correspondence is unique up to phases.
Consider a set of parties $P(M,n)$. For every $\bm{f}\in\mathbb{Z}_{2}^{2n}$
and $\alpha\in P(M,n)$ we denote by
$\bm{f}_{\alpha}\in\mathbb{Z}_{2}^{2|\alpha|}$ the restriction of $\bm{f}$ to
party $\alpha$. For every party $\alpha\in P(M,n)$, we consider two subspaces,
the local subspace
$V^{\mathcal{S}}_{\alpha}=\\{\bm{f}\in V^{\mathcal{S}}|\bm{f}_{\beta}=0\
\forall\beta\in M,\beta\neq\alpha\\}$ (17)
and the colocal subspace
$V_{\hat{\alpha}}^{\mathcal{S}}=\\{\bm{f}\in
V^{\mathcal{S}}|\bm{f}_{\alpha}=0\\}.$ (18)
That is, while $V^{\mathcal{S}}_{\alpha}$ contains those vectors of
$V^{\mathcal{S}}$ that vanish on every party but $\alpha$, the subspace
$V^{\mathcal{S}}_{\alpha}$ contains those vectors that vanish on party
$\alpha$. Note that the dimension of $V_{\alpha}^{\mathcal{S}}$ is related to
rank of the reduced state via ${\rm rk}(\rho_{\alpha})=2^{|\alpha|-{\rm
dim}(V_{\alpha}^{\mathcal{S}})}$ (see Equation (3)). In this context, it is
helpful to define the support of a vector
$\bm{f}=(a_{1},b_{1},\ldots,a_{n},b_{n})\in\mathbb{Z}_{2}^{2n}$ as
$\text{supp}(\bm{f})=\\{i\in[n]|a_{i}=1\lor b_{i}=1\\}.$ (19)
Similarly, the support of a Pauli operator $P\propto\sigma(\bm{f})$ is given
$\text{supp}(P)=\text{supp}(\bm{f})$, i.e., the support of $P$ is where it
acts non-trivially.
With all the definitions in place let us recall the findings of Ref. [28] that
we will use here. There are two types of results in Ref. [28]. On the one
hand, the authors provide necessary and sufficient conditions for PLC
equivalence of stabilizer states. On the other hand, they use those conditions
to find the exact number of $M$-qubit, $M$-partite GHZ states one can extract
from an $M$-partite stabilizer state. Using this result, as well as the
conditions for PLC equivalence, they provide a complete solution to the
problem of PLC equivalence for $3$-partite stabilizer states. We start by
summarizing the necessary and sufficient conditions for PLC equivalence
derived in Ref. [28].
It is shown in Ref. [28] that the PLC equivalence of two stabilizer states
solely depends on how elements of their stabilizers commute on the individual
parties, as stated in the following theorem.
###### Theorem 3 ([28]).
Let $\ket{\psi},\ket{\phi}\in\text{Stab}(P(M,n))$. Then, $\ket{\psi}$ and
$\ket{\phi}$ are PLC equivalent if and only if there exists a basis
$\bm{g}_{1},\ldots,\bm{g}_{n}$ of $V^{{\mathcal{S}}_{\psi}}$ and a basis
$\bm{f}_{1},\ldots,\bm{f}_{n}$ of $V^{{\mathcal{S}}_{\phi}}$ such that
$\omega((\bm{f}_{i})_{\alpha},(\bm{f}_{j})_{\alpha})=\omega((\bm{g}_{i})_{\alpha},(\bm{g}_{j})_{\alpha})$
(20)
for all $\alpha\in P(M,n)$ and $i,j\in[n]$.
That is, two stabilizer states are PLC equivalent iff there exist sets of
generators for $\mathcal{S}_{\psi}$ and $\mathcal{S}_{\phi}$ whose respective
commutation relations on all parties agree. In Ref. [28], this theorem is only
stated for stabilizer states of maximal local rank, i.e., states for which
$\text{dim}(V_{\alpha}^{\mathcal{S}})=0$ for all $\alpha\in P(M,n)$. However,
the authors mention that it extends to general stabilizer states. In Appendix
G we provide a proof for the general statement in the case of qudit systems of
prime dimension, as we use it in Section 4 where we study qudit systems of
prime dimension.
Moreover, Ref. [28] provides a theorem which gives necessary and sufficient
conditions for when a stabilizer state $\ket{\phi}$ can be extracted from a
stabilizer state $\ket{\psi}$ via PLC. We say a state
$\ket{\phi}\in\text{Stab}(P(M,m))$ can be extracted from a state
$\ket{\psi}\in\text{Stab}(P(M,n))$ with $n\geq m$ if there exists a PLC
transformation $C\in\mathcal{C}^{P(M,n)}$ and a $(n-m)$-qubit stabilizer state
$\ket{\phi^{\prime}}$ such that
$C\ket{\psi}=\ket{\phi}\otimes\ket{\phi^{\prime}}$. In this case, we say
$\ket{\psi}$ decomposes into $\ket{\phi}$ and $\ket{\phi^{\prime}}$.
###### Theorem 4 ([28]).
Let $\ket{\psi}\in\text{Stab}(P(M,n))$ and $\ket{\phi}\in\text{Stab}(P(M,m))$
with $n\geq m$. The state $\ket{\phi}$ can be extracted from $\ket{\psi}$ if
and only if there exists a linear injective map
$T:V^{\mathcal{S}_{\phi}}\rightarrow V^{\mathcal{S}_{\psi}}$ such that
1. (i)
$\omega(T(\bm{f})_{\alpha},T(\bm{g})_{\alpha})=\omega(\bm{f}_{\alpha},\bm{g}_{\alpha})\
\forall\bm{f},\bm{g}\in V^{\mathcal{S}_{\phi}}\ \alpha\in P(M,n)$
2. (ii)
$[T(V^{\mathcal{S}_{\phi}})]_{\hat{\alpha}}=T(V^{\mathcal{S}_{\phi}}_{\hat{\alpha}})$
for all $\alpha\in P(M,n)$.
Condition $(i)$ in Theorem 4 guarantees that the map $T$ preserves commutation
relations on every party. Condition $(ii)$ enforces that the party support of
$f$ and $T(f)$ is the same for all $\bm{f}\in V^{\mathcal{S}_{\phi}}$. This
condition is necessary as PLC transformations cannot change the party support
of Pauli operators. Note that Theorem 3 considers an extremal case of Theorem
4 where $n=m$. One can show that condition $(ii)$ in Theorem 4 becomes
obsolete in this case.
A simple case of extractability occurs if there exists a party $\alpha\in
P(M,n)$ such that $\text{dim}(V^{\mathcal{S}}_{\alpha})=d_{\alpha}>0$. It
follows directly from Theorem 4 that then $\ket{\psi}$ is PLC equivalent to a
state $\ket{\phi}\otimes\ket{0}^{\otimes d_{\alpha}}$. Note that an analogous
reasoning also applies to the qudit case (see Appendix G).
Using, in particular, Theorem 4 one can determine the maximum number of
$M$-qubit GHZ states shared among all parties that can be extracted from a
state $\ket{\psi}\in\text{Stab}(P(M,n))$ via PLC. This number is given by the
minimum number of generators of $\mathcal{S}_{\psi}$ which are truly
$M$-partite. To be more precise, let us consider the subspace
$V_{loc}=\text{span}\bigcup_{\alpha\in P(M,n)}V^{S}_{\hat{\alpha}}.$ (21)
It is shown in Ref. [28] that the maximum number of $M$-qubit GHZ states
distributed among all $M$ parties that can be extracted from $\ket{\psi}$ via
PLC is given by $\Delta=n-\text{dim}(V_{loc})$.
In the $3$-partite case, this result in combination with Theorem 4 leads to a
complete characterization of all PLC equivalence classes of stabilizer states
of arbitrary qubit number. As we have mentioned before, it is shown in Ref.
[28], that every $3$-partite stabilizer state decomposes into a tensor product
of $k_{1}$ copies of the $\ket{0}$ state, $k_{2}$ copies of the
$\ket{\phi^{+}}$ state, and $k_{3}$ copies of the $\ket{GHZ}$ state under PLC.
The number of copies of each of those states is unique and can be computed
efficiently. Two stabilizer states are PLC equivalent if and only if their
decompositions coincide, i.e., if the numbers of copies in the decompositions
of each of the states coincides. This statement still holds if one considers
party-local unitary operations (PLU) instead of PLC operations. Conversely,
any tensor product of the states $\ket{0},\ket{\phi^{+}}$ and $\ket{GHZ}$
uniquely characterizes a PLC class (of three parties), and also a PLU class
(Ref. [28]).
The authors of Ref. [28] therefore suggest to call
$\\{\ket{0},\ket{\phi^{+}},\ket{GHZ}\\}$ the entanglement generating set (EGS)
for $3$-partite stabilizer states. In this work, we use the EGS w.r.t. PLC
operations to characterize PLC equivalence classes for an arbitrary number of
parties. The entanglement generating set (EGS) for $M$-partite stabilizer
states under PLC is a minimal set EGSM of indecomposable $M$-partite
stabilizer states such that every $M$-partite stabilizer state decomposes
under PLC into a tensor product of the states contained in EGSM. It follows
directly from its definition that the EGS is unique up to PLC transformations
of the individual states it contains. In particular, it contains one
representative of every PLC class, which consists of indecomposable stabilizer
states. Note that there are stabilizer states which are LU equivalent but not
LC equivalent (see Ref. [46]). Therefore, the EGS, as we define it, can be
overcomplete regarding PLU transformations. Moreover, the EGSM is defined for
a fixed number of parties $M$ but an unbounded number of qubits. Thus, it is
possible that it contains infinitely many states.
The results of Ref. [28] imply that $3$-partite stabilizer states decompose
uniquely into states from the EGS3. Note that the results of [28] were also
partially extended to $2$\- and $3$-partite qudit systems, with $d$ such that
its prime decomposition only contains primes of power $0$ or $1$ in Ref. [29].
More precisely, the authors show that any $2$\- and $3$-partite qudit
stabilizer state, with $d$ as defined above, decomposes into a tensor product
of the states $\ket{0}$,
$\ket{\phi_{d}^{+}}=\frac{1}{\sqrt{d}}\sum_{j}\ket{jj}$ and
$\ket{GHZ^{d}}=\frac{1}{\sqrt{d}}\sum_{j}\ket{jjj}$.
In the light of the above summary let us briefly outline the rest of the
paper. In the following sections we introduce the commutation matrix formalism
as a powerful tool to study PLC equivalence of stabilizer states. This allows
us to use results from the classification theory of bilinear forms to study
the structure of the EGSM for qubits and qudits. Particularly, we derive
necessary and sufficient conditions for when a state is decomposable. While we
do not know whether or not qubit stabilizer state decompose uniquely into
states from the EGS, we prove that for qudit stabilizer states of prime
dimension the decomposition is indeed unique.
## 3 PLC transformations of stabilizer states and the commutation matrix
formalism
As we saw in the previous section, whether or not two stabilizer states are
PLC equivalent solely depends on the local commutation relations of operators
in their stabilizers (Theorem 3 shown in [28]). In this section, we introduce
a new formalism to exploit these findings in studying the PLC transformations
of stabilizer states.
Let $\ket{\psi}\in\text{Stab}(P(M,n))$ with stabilizer $\mathcal{S}$ and let
$\\{\bm{f}_{1},\ldots,\bm{f}_{n}\\}$ be a basis of its corresponding maximally
isotropic subspace $V^{\mathcal{S}}$. As we are only interested in the
subspace $V^{\mathcal{S}}$, we define the linear map
$R:V^{\mathcal{S}}\rightarrow\mathbb{Z}^{n}_{2}$ via $R\bm{f}_{i}=\bm{e}_{i}$
for all $i\in[n]$, where $\\{\bm{e}_{i}\\}$ denotes the standard basis of
$\mathbb{Z}_{2}^{n}$ 222This representation of the maximally isotropic
subspace $V^{\mathcal{S}}$ is equivalent to the representation of any element
$g\in\mathcal{S}$ as $g=g_{1}^{x_{1}}g_{2}^{x_{2}}\cdots g_{n}^{x_{n}}$, where
$\bm{x}\in\mathbb{Z}_{2}^{n}$.. The map $R$ is invertible and therefore
describes an isomorphism. For every party $\alpha\in P(M,n)$ we define a
bilinear form
$C_{\alpha}:\mathbb{Z}_{2}^{n}\times\mathbb{Z}_{2}^{n}\rightarrow\mathbb{Z}_{2}$
via
$C_{\alpha}(\bm{k},\bm{l})=\omega((R^{-1}\bm{k})_{\alpha},(R^{-1}\bm{l})_{\alpha}).$
(22)
As $\omega$ is a symplectic form, the bilinear forms $(C_{\alpha})_{\alpha}$
are _alternating_ , that is,
$C_{\alpha}(\bm{k},\bm{l})=-C_{\alpha}(\bm{l},\bm{k})$, and
$C_{\alpha}(\bm{k},\bm{k})=0$ for all $\bm{k},\bm{l}\in\mathbb{Z}_{2}^{n}$. As
we operate over the field $\mathbb{Z}_{2}$ we also have that
$C_{\alpha}(\bm{k},\bm{l})=C_{\alpha}(\bm{l},\bm{k})$. Note that they are not
symplectic in general as they can be degenerate, i.e., if
$C_{\alpha}(\bm{k},\bm{l})=0$ for all $\bm{l}\in\mathbb{Z}_{2}^{n}$, then this
does not imply that $\bm{k}=0$.
In slight abuse of notation we use $C_{\alpha}$ also to denote the matrix
representation of the respective bilinear form w.r.t. to the basis
$\\{\bm{e}_{i}\\}_{i\in[n]}$. Let us now make the following definition.
###### Definition 5.
Let $\ket{\psi}$ be an $M$-partite stabilizer state. We call the matrices
$(C_{\alpha})_{\alpha\in P(M,n)}$, describing the bilinear forms defined in
Equation (22) w.r.t. to the basis choice $\\{\bm{f}_{i}\\}$ of
$V^{\mathcal{S}}$, commutation matrices.
A commutation matrix $C_{\alpha}$ contains all information about the
commutation relations of the Pauli operators $\sigma(\bm{f}_{j})$ on the party
$\alpha$. More precisely, $(C_{\alpha})_{ij}=0$ iff $\sigma(\bm{f}_{i})$ and
$\sigma(\bm{f}_{j})$ commute on party $\alpha$, i.e.,
$[\sigma(\bm{f}_{i})_{\alpha},\sigma(\bm{f}_{j})_{\alpha}]=0$. Moreover, by
replacing the field $\mathbb{Z}_{2}$ with a finite field $\mathbb{Z}_{d}$ for
$d\geq 3$, commutation matrices can also be defined for qudit systems of prime
dimension, which we discuss in Section 4.
Let us discuss some properties of commutation matrices. Commutation matrices
are alternating as they describe alternating forms. Moreover they satisfy
$\sum_{\alpha\in P(M,n)}C_{\alpha}=0,$ (23)
where addition is performed modulo $2$. This condition is a direct consequence
of the fact that all generators globally commute, i.e., that $V^{\mathcal{S}}$
is isotropic. Indeed, as $[\sigma(\bm{f}_{i}),\sigma(\bm{f}_{j})]=0$ for all
$i,j$, the operators $\sigma(\bm{f}_{i})$ and $\sigma(\bm{f}_{j})$ anti-
commute on an even number of parties. Therefore
$\sum_{\alpha}(C_{\alpha})_{ij}=0$ for all $i,j$. Moreover, for two parties
$\alpha_{i},\alpha_{j}\in P(M,n)$ one finds that
$C_{\alpha_{1}}+C_{\alpha_{2}}=C_{\alpha_{1}\cup\alpha_{2}}.$ (24)
As noted in the definition, the commutation matrices depend on the chosen
basis $\\{\bm{f}_{1},\ldots,\bm{f}_{n}\\}$. A basis change in
$V^{\mathcal{S}}$, which is equivalent to choosing a different set of
generators for $\mathcal{S}$, is described by an invertible matrix $Q\in
M_{n\times n}(\mathbb{Z}_{2})$ on $\mathbb{Z}_{2}^{n}$. As commutation
matrices represent bilinear forms, the commutation matrices with respect to
the new basis are given by
$(Q^{T}C_{\alpha}Q)_{\alpha\in M}.$ (25)
In order to illustrate the formalism, let us present some examples of
commutation matrices that are relevant in the following. For the state
$\ket{\phi^{+}}$ distributed among the parties $P(2,2)=\\{\\{1\\},\\{2\\}\\}$
the commutation matrices w.r.t the canonical generators $g_{1}=X_{1}Z_{2}$ and
$g_{2}=Z_{1}X_{2}$, defined by Equation (5), are
$C_{\\{1\\}}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,\text{and}\,C_{\\{2\\}}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$ (26)
For the state $\ket{GHZ}$ distributed among the parties
$P(3,3)=\\{\\{1\\},\\{2\\},\\{3\\}\\}$ the commutation matrices w.r.t the
canonical generators $g_{1}=X_{1}Z_{2}$, $g_{2}=Z_{1}X_{2}Z_{3}$ and
$g_{3}=Z_{2}X_{3}$ are
$\displaystyle C_{\\{1\\}}$ $\displaystyle=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},C_{\\{2\\}}=\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\end{pmatrix},$ $\displaystyle C_{\\{3\\}}$
$\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}.$ (27)
It can be easily verified that they satisfy all the requirements listed above,
i.e., they are alternating, and sum up to zero.
If we instead choose the generators $g_{1},g_{2},g_{2}g_{3}$, the associated
tuple of commutation matrices is
$\displaystyle\tilde{C}_{\\{1\\}}$ $\displaystyle=\begin{pmatrix}0&1&1\\\
1&0&0\\\ 1&0&0\end{pmatrix},\tilde{C}_{\\{2\\}}=\begin{pmatrix}0&1&1\\\
1&0&1\\\ 1&1&0\end{pmatrix},$ $\displaystyle\tilde{C}_{\\{3\\}}$
$\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}.$ (28)
Again, it is straightforward to verify that these commutation matrices result
from the original ones by a congruence transformation with the invertible
matrix
$Q=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&1&1\\\ \end{pmatrix}.$ (29)
In fact, the states that we have just discussed are graph states, as the
generators of their stabilizers correspond to the canonical generators in
Equation 5. For these states there exists a simple way to compute a tuple of
commutation matrices from the adjacency matrix $\Gamma_{G}$. One finds that if
$\Gamma_{G}(i)$ denotes the matrix that agrees with $\Gamma_{G}$ in row and
column $i$ and is zero everywhere else, then
$C_{\alpha}=\sum_{i\in\alpha}\Gamma_{G}(i).$ (30)
However, not every tuple of alternating matrices which satisfies Equation (23)
corresponds to a stabilizer state, i.e., a maximally isotropic subspace. As we
show in the following Theorem, the ranks of the matrices in tuples
corresponding to stabilizer states are constrained.
###### Theorem 6.
Let $(C_{\alpha})_{\alpha\in P(M,n)}$, $C_{\alpha}\in M_{n\times
n}(\mathbb{Z}_{2})$ be a tuple of alternating matrices such that
$\sum_{\alpha}C_{\alpha}=0$ for some partition $P(M,n)$. Then there exists a
$\ket{\psi}\in\text{Stab}(P(M,n))$ such that $(C_{\alpha})_{\alpha}$ are the
corresponding commutation matrices if and only if
$2\,{\rm rk}([C_{\alpha}]_{\alpha})=\sum_{\alpha}{\rm rk}(C_{\alpha}).$ (31)
Here, $[C_{\alpha}]_{\alpha}$ denotes the $n\times(nm)$ matrix obtained by
concatenating all matrices $C_{\alpha}$, and the rank is taken over the field
$\mathbb{Z}_{2}$.
To explain this condition let us note that the rank of a commutation matrix is
related to the rank of the reduced stabilizer state as
$2^{\text{rk}(C_{\alpha})/2}=\text{rk}(\rho_{\alpha})$. Therefore,
$\text{rk}(C_{\alpha})/2$ coincides with the number of qubits in party
$\alpha$ that are effectively entangled to another party (see discussion in
Appendix C). Observe that ${\rm rk}([C_{\alpha}]_{\alpha})={\rm
rk}([QC_{\alpha}Q^{T}]_{\alpha})$ for any invertible matrix $Q$. Therefore,
$n-{\rm rk}([C_{\alpha}]_{\alpha})$ is the maximum number of generators that
locally commute with all other generators of $\mathcal{S}$. As $\mathcal{S}$
is a maximally abelian subgroup it follows that those generators can be chosen
to have support only on one party, and thus correspond to extractable qubits
in the state $\ket{0}$. Thus, the rank condition in Equation (31) states the
following. The sum over the number of qubits in each party which are
effectively entangled to another one outside the party has to be equal to the
difference between $n$ and the total number of qubits in the state $\ket{0}$
that can be extract from $\ket{\psi}$. The latter number is precisely given by
$n-{\rm rk}([C_{\alpha}]_{\alpha})$.
For general alternating matrices $(C_{\alpha})_{\alpha\in P(M,n)}$ which sum
up to zero, Equation (31) becomes an inequality
$2\,{\rm rk}([C_{\alpha}]_{\alpha})\leq\sum_{i=1}^{M}{\rm rk}(C_{i}).$ (32)
Commutation matrices are precisely those sets of matrices that saturate this
bound. For the details of the proof of this inequality and for the proof of
Theorem 6 see Appendix C and the discussion below. Note that one can show that
tuples of alternating matrices which do not satisfy Equation (31) are
associated to stabilizer codes. Thus, one can also use this formalism to study
PLC equivalence of stabilizer codes. However, in this case the condition for
PLC equivalence is more complicated. Addressing this task is beyond the scope
of this paper.
Given a tuple of commutation matrices one can construct a corresponding
stabilizer state, as we show in the following. The idea is to go through all
parties and transform the respective commutation matrix to a form where it is
easy to choose a correct form for the generators. Finally, the transformations
are reversed. More precisely, one proceeds as follows. Let us first consider
first the case ${\rm rk}([C_{\alpha}]_{\alpha})=n$ and then deal with the case
${\rm rk}([C_{\alpha}]_{\alpha})<n$.
* –
Initialize the operators $g_{1},\ldots,g_{n}$ to $\mathds{1}$.
* –
For each party $\alpha$ proceed as follows:
* $\rightarrow$
Assign ${\rm rk}(C_{\alpha})/2$ qubits to party $\alpha$. For this iteration
of the loop, these qubits are labeled from $1$ to ${\rm rk}(C_{\alpha})/2$.
* $\rightarrow$
Find an invertible matrix $P\in M_{n\times n}(\mathbb{Z}_{2})$ such that
$PC_{\alpha}P^{T}$ is of the form described in Theorem 18, i.e., a direct sum
of the blocks
$\begin{pmatrix}0\end{pmatrix},\text{ and }\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$
* $\rightarrow$
Let $\tilde{g}_{1},\ldots,\tilde{g}_{n}$ be this new set of generators
(obtained as products of the old generators). For all $j\in[n-1]$, if
$(PC_{\alpha}P^{T})_{j,j+1}=1$, set $(\tilde{g}_{j})_{\alpha}=X_{(j+1)/2}$ and
$(\tilde{g}_{j+1})_{\alpha}=Z_{(j+1)/2}$.
* $\rightarrow$
Then undo the change of generators according to $P^{-1}$.
* –
After determining all operators $g_{j}$ check whether $g_{j}^{2}=+\mathds{1}$
or $-\mathds{1}$. In the latter case replace $g_{j}$ by $ig_{j}$.
First, observe that the rank condition in Eq. (31) ensures that the algorithm
defines generators acting on $n$ qubits. It is clear that implementing this
algorithm for every party, the commutation relations of the operators
$g_{1},\ldots,g_{n}$ are described by $(C_{\alpha})_{\alpha}$. Thus, the
$g_{1},\ldots,g_{n}$ commute. Moreover, they are independent as ${\rm
rk}([C_{\alpha}]_{\alpha})=n$ and, due to the last step of the algorithm,
$g_{i}^{2}=\mathds{1}$ for all $i\in[n]$. Therefore, any element of the
generated subgroup is a product of powers zero or one of each generator. If
any of these powers is one, the generated operator is not proportional to
$\mathds{1}$ as the generators are independent. If all powers are zero, then
the operator is equal to $\mathds{1}$ due to the last step of the algorithm.
We conclude that $g_{1},\ldots,g_{n}$ generate an $n$-qubit stabilizer
corresponding to a stabilizer state.
In case ${\rm rk}([C_{\alpha}]_{\alpha})<n$, the construction has to be
slightly modified. First, one finds an invertible matrix $R$ such that the
last $w=n-\text{rk}([C_{\alpha}]_{\alpha})$ rows of
$[RC_{\alpha}R^{T}]_{\alpha}$ are zero. Then, one can assign
$\text{rk}(C_{\alpha})/2$ qubits to party $\alpha$ for every $\alpha\in
P(M,n)$ and determine the generators $g_{1},\ldots,g_{n-w}$ as above. The
remaining $n-{\rm rk}([C_{\alpha}]_{\alpha})$ qubits, which are in the state
$\ket{0}$, can be distributed arbitrarily among the parties. Let us number the
qubits in the state such that those appear last. Then we define $g_{j}=Z_{j}$
for all $j\in\\{n-w+1,\ldots,n\\}$. Finally, we undo the change of generators
according to $R^{-1}$. It is straightforward to verify that the operators
obtained this way define a stabilizer corresponding to a stabilizer state.
Again Eq. (31) ensures that the algorithm defines generators acting on $n$
qubits.
Let us discuss the relevance of commutation matrices in the context of PLC
transformations. Theorem 3 ([28]) states that two stabilizer states are PLC
equivalent if their stabilizers admit sets of generators with the same
commutation relations on each party. In terms of commutation matrices this
Theorem reads as follows.
###### Theorem 7.
Let $\ket{\psi},\ket{\phi}\in\text{Stab}(P(M,n))$. Then $\ket{\psi}$ is PLC
equivalent to $\ket{\phi}$ if and only if they admit the same tuple of
commutation matrices.
As all sets of commutation matrices of a state are related via a congruence
transformation, this theorem is equivalent to the following: $\ket{\psi}$ and
$\ket{\phi}$ are PLC equivalent if any two tuples of commutation matrices
$(C_{\alpha}^{\psi})_{\alpha}$, and $(C_{\alpha}^{\phi})_{\alpha}$ are
congruent to each other. Consequently, a set of commutation matrices and every
congruent set uniquely characterizes a PLC equivalence class. Thus, in order
to characterize all PLC classes we have to find a maximal set of non-congruent
tuples of commutation matrices.
### 3.1 The EGS and commutation matrices
We argued in the previous section that the classification of PLC classes is
equivalent to the classification of commutation matrices up to congruence. A
common approach to classify tuples of matrices under certain operations is to
first establish that these tuples decompose uniquely into indecomposable
blocks (up to reordering and equivalence of the blocks). Then, the
classification task is solved by presenting a complete set of indecomposable
tuples. We show in this section that these two tasks are equivalent to the
ones we want to solve for stabilizer states. Namely, we show that stabilizer
states decompose uniquely into indecomposable states if and only if this is
true for commutation matrices. Moreover, finding the EGS is equivalent to
finding a maximal set of non-congruent indecomposable commutation matrices.
Let us start by making more precise the notion of decomposability of
stabilizer states and their corresponding tuples of commutation matrices. To
that end, consider the tensor product of two stabilizer states
$\ket{\psi}\otimes\ket{\varphi}$, and let $\\{g_{i}\\}_{i}$ be the generators
of $\mathcal{S}_{\psi}$ and $\\{h_{i}\\}_{i}$ the generators of
$\mathcal{S}_{\varphi}$ respectively. Such a state is clearly decomposable.
Observe that the two sets of generators are mutually commuting, globally but
also locally, as they are acting on different qubits. Thus, their commutation
matrices are block-diagonal, where one block corresponds to the generators of
$\ket{\psi}$ and the other to the generators of $\ket{\varphi}$, i.e.,
$C_{\alpha}(\ket{\psi}\otimes\ket{\varphi})=C_{\alpha}(\ket{\psi})\oplus
C_{\alpha}(\ket{\varphi}).$ (33)
To see that the converse is also true the following theorem establishes the
connection between decomposability of stabilizer states and the block-
diagonalizability of their commutation matrices.
###### Theorem 8.
Let $\ket{\psi}\in\text{Stab}(P(M,n))$ and let $(C_{\alpha})_{\alpha\in
P(M,n)}$ be a tuple of commutation matrices. Then $\ket{\psi}$ is decomposable
if and only if there exists an invertible matrix $Q\in M_{n\times
n}(\mathbb{Z}_{2})$, and matrices $B_{\alpha}^{1}\in M_{n_{1}\times
n_{1}}(\mathbb{Z}_{2})$, $B_{\alpha}^{2}\in M_{n_{2}\times
n_{2}}(\mathbb{Z}_{2})$ such that $(QC_{\alpha}Q^{T})_{\alpha\in
M}=(B_{\alpha}^{1}\oplus B_{\alpha}^{2})_{\alpha\in M}$.
###### Proof.
The proof of the only if part follows directly from Theorem 3 and the fact
that all tuples of commutation matrices associated to same state are
congruent.
Next, we prove the if part. Suppose there exists a an invertible matrix $Q\in
M_{n\times n}(\mathbb{Z}_{2})$ such that $(QC_{\alpha}Q^{T})_{\alpha\in
P(M,n)}=(B_{\alpha}^{1}\oplus B_{\alpha}^{2})_{\alpha}$. As
$(B_{\alpha}^{1}\oplus B_{\alpha}^{2})_{\alpha}$ is a tuple of commutation
matrices, it satisfies Equation (31). That is,
$2\,{\rm rk}([B_{\alpha}^{1}+B_{\alpha}^{2}])=\sum_{\alpha}{\rm
rk}(B_{\alpha}^{1}\oplus B_{\alpha}^{2})$ (34)
or equivalently,
$2\,{\rm rk}([B_{\alpha}^{1}])+2\,{\rm rk}([B_{\alpha}^{2}])=\sum_{\alpha}{\rm
rk}(B_{\alpha}^{1})+\sum_{\alpha}{\rm rk}(B_{\alpha}^{2}).$ (35)
As $2\,{\rm rk}([B_{\alpha}^{i}])\leq\sum_{\alpha}{\rm rk}(B_{\alpha}^{i})$
(see Inequality (32)) Equation (35) can hold only if $2\,{\rm
rk}([B_{\alpha}^{i}])=\sum_{\alpha}{\rm rk}(B_{\alpha}^{i})$. Thus,
$(B_{\alpha}^{1})$ and $(B_{\alpha}^{2})$ are tuples of commutation matrices.
Let $\ket{\phi_{1}}$ and $\ket{\phi_{2}}$ be the corresponding $M$-partite
stabilizer states. It follows from Theorem 3 that $\ket{\psi}$ is PLC
equivalent to $\ket{\phi_{1}}\otimes\ket{\phi_{2}}$ and is therefore
decomposable. ∎
This theorem establishes that decomposing stabilizer states into other
stabilizer states is equivalent to decomposing tuples of commutation matrices
into blocks. Therefore, to identify states in the EGS we have to identify
indecomposable tuples of commutation matrices. The question whether or not
stabilizer states decompose uniquely into indecomposable ones is equivalent to
whether or not tuples of commutation matrices uniquely decompose into
indecomposable blocks.
These questions were studied for general tuples of alternating matrices in
linear algebra in the last century. We summarize the main results in the
following. For a more detailed description we refer the reader to Appendix D.
Let us stress that we are only interested in the classification of a special
subset of these tuples, namely those which satisfy Equation (31).
How much is known about the problem very much depends on the field one
considers. For the field $\mathbb{Z}_{2}$, up to our knowledge, both questions
have only been answered for tuples of length $1$ and $2$. The solution for a
single matrix is given by Theorem 18 (see Appendix B), which establishes that
the only indecomposable blocks are given by
$\begin{pmatrix}0\end{pmatrix},\text{ and }\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$ (36)
Theorem 18 also implies that the decomposition into these blocks is unique.
This follows from the simple fact that a congruence transformation cannot
change the rank of a matrix and therefore the number of blocks of each type is
fixed. Comparing the above result to the commutation matrices of the state
$\ket{\phi^{+}}$ in Equation (26) one sees that this implies that any
bipartite stabilizer state decomposes uniquely into a tensor product of copies
of the state $\ket{\phi^{+}}$ and the state $\ket{0}$.
For tuples of length $2$ Ref. [47] establishes the uniqueness of the
decomposition and states a complete set of indecomposable pairs of alternating
forms. To obtain these results the author shows that the classification of
pairs of alternating forms is equivalent to the classification of the better
known matrix pencils. Due to Equation (23), these results directly apply to
the characterization of the EGS for $3$-partite stabilizer states. In fact,
they precisely reproduce the results of Ref. [28] regarding PLC equivalence
classes. To see this, one starts from the general classification of Ref. [47]
and considers only those indecomposable tuples which satisfy the rank
condition in Equation (31). For a more detailed discussion we refer the reader
to Appendix D.
Up to our knowledge, the solution for tuples of length $\geq 3$ for the field
$\mathbb{Z}_{2}$ is not known. However, for algebraically closed fields of
characteristic $\neq 2$ it is shown in Ref. [48] that the classification of
triples of alternating forms is a _wild_ problem, i.e., it is at least as hard
as classifying pairs of matrices up to simultaneous similarity. The latter
task is considered to be very challenging. This suggests that also in the case
of the field $\mathbb{Z}_{2}$, classifying general triples of alternating
forms is difficult. However, due to Equation (31) we are only interested in
very special triples of matrices and thus a complete characterization of
indecomposables tuples might be possible beyond pairs of matrices.
For the field $\mathbb{Z}_{d}$ with $d\geq 3$ the situation is different.
Here, Ref. [34] establishes the uniqueness of the decomposition for arbitrary
tuples of alternating matrices (cf. Theorem 2 in Ref. [34], see also Appendix
D). As mentioned earlier, the commutation matrix formalism generalizes to
qudit stabilizer states for prime dimension. In this case, the commutation
matrices are defined over the field $\mathbb{Z}_{d}$. Therefore, by
establishing the equivalence between the decomposition of commutation matrices
and the decomposition of stabilizer states (see Section 4) we show via the
results of Ref. [34] that qudit stabilizer states decompose uniquely into
indecomposable ones under PLC (see Section 4).
We comment in Appendix H on our approach to show the uniqueness of the
decomposition for qubit systems. We prove that stabilizer states without PLC
symmetries decompose uniquely. However, these symmetries seem to be common, at
least for four parties (see the states in Figure 4) as many states in the EGS4
up to $10$ qubits contain leafs, which yield additional symmetries.
### 3.2 Necessary and sufficient conditions for decomposability
In this section we introduce useful tools for studying decompositions of
stabilizer states. In particular, we focus on the application of the
commutation matrix formalism. We discuss Fitting’s Lemma (see, e.g., Ref.
[34]), which provides necessary and sufficient conditions for decomposability
of tuples of alternating matrices, and in particular of commutation matrices.
First, let us present a lemma which gives sufficient conditions for the
extraction of the state $\ket*{GHZ_{M-1}}$ or $\ket*{GHZ_{M}}$ from an
$M$-partite stabilizer state.
###### Lemma 9.
Let $P(M,n)=\\{\alpha_{1},\ldots,\alpha_{M}\\}$ be a partition and let
$\ket{\psi}\in\text{Stab}(P(M,n))$ have full local ranks, i.e.,
$\text{dim}(V^{\mathcal{S}}_{\alpha})=0$ for all $\alpha\in P(M,n)$. Then one
can extract either $\ket*{GHZ_{M-1}}$ or $\ket{GHZ_{M}}$ from $\ket{\psi}$ if
and only if there exist elements $f_{1},\ldots,f_{M-2}$ in $\mathcal{S}$ such
that $\text{supp}(f_{j})=\alpha_{1}\cup\alpha_{j+1}$ and
$(f_{j})_{\alpha_{1}}=(f_{k})_{\alpha_{1}}$ for all $j,k\in[M-2]$.
The proof of this lemma can be found in Appendix E. It makes use of the fact
that $\mathcal{S}$ is a maximally isotropic subspace. This fact in combination
with the special operators $f_{j}$ it has to contain is enough to show that
the conditions of Theorem 4 are satisfied and an extraction is possible
333Note that Ref. [30] also contains conditions under which one can extract a
GHZ state from a given stabilizer states, even between less than $M-1$
parties..
Next, let us discuss how one can analyze decompositions of tuples of
commutations (and thereby decompositions of the corresponding stabilizer
states). It is a well-known fact that to study decompositions of a tuples of
alternating matrices into blocks one can equivalently study the ring of self-
adjoint endomorphisms of the corresponding tuple. A self-adjoint endomorphism
of a tuple of alternating matrices $(C_{\alpha})_{\alpha\in P(M,n)}$,
$C_{\alpha}\in M_{n\times n}(\mathbb{Z}_{2})$ is a matrix $E\in M_{n\times
n}(\mathbb{Z}_{2})$ such that
$C_{\alpha}E=E^{T}C_{\alpha}$ (37)
holds for all $\alpha\in P(M,n)$. Self-adjoint endomorphisms enjoy many useful
properties that we will frequently make use of in what follows. It directly
follows from the defining Equation (37) that the set of self-adjoint
endomorphisms is closed under products and additions and therefore forms a
ring. In particular, whenever $E$ is a self-adjoint endomorphism, so are its
powers $E^{m}$. It is straightforward to verify that for any $m\in\mathbb{N}$
it holds that $\text{ran}(E^{m})\supseteq\text{ran}(E^{m+1})$, and
$\text{ker}(E^{m})\subseteq\text{ker}(E^{m+1})$. For invertible endomorphisms
the range coincides with the full vector space for any power of $E$, and for
nilpotent endomorphisms one finds $k\in\mathbb{N}$ such that $E^{k}=0$. For
finite dimensional vector spaces, and in particular over finite fields, one
can always find an $l\in\mathbb{N}$ for which
$\text{ran}(E^{l})=\text{ran}(E^{l+1})$, and
$\text{ker}(E^{l})=\text{ker}(E^{l+1})$, that is $E^{l}$ acts as a bijection
on its range. Over finite fields, $\text{ran}(E^{l})$ is a finite set and,
therefore, $E^{l}$ is a permutation on it. Therefore, there exists a power
$p\geq l$ for which $E^{p}$ acts as the identity on its range and thus
fulfills $(E^{p})^{2}=E^{p}$, i.e., $E^{p}$ is _idempotent_. The existence of
idempotent elements that are neither invertible nor nilpotent plays a crucial
role in deciding if a set of commutation matrices is decomposable. The
following lemma, known as Fitting’s lemma, explains how properties of self-
adjoint endomorphisms relate to the decomposition of $(C_{\alpha})_{\alpha\in
P(M,n)}$ into blocks.
###### Lemma 10 (Fitting’s lemma [34]).
Let $F$ be any field, let $I\subset\mathbb{N}$ be a finite index set and let
$(A_{\alpha})_{\alpha\in I}$, $A_{\alpha}\in M_{n\times n}(F)$ be a tuple of
alternating matrices. Then, $(A_{\alpha})_{\alpha\in I}$ is indecomposable
under congruence if and only if all self-adjoint endomorphisms of
$(A_{\alpha})_{\alpha\in I}$ are either nilpotent or invertible.
Let us include a proof of the lemma for finite fields as it illustrates its
application in the context of commutation matrices.
###### Proof.
Let $(A_{i})_{i}$ be a tuple of alternating $n\times n$ matrices. As $F$ is
finite, for any endomorphism of $(A_{i})_{i}$ there exists a $k\in\mathbb{N}$
such that $E^{2k}=E^{k}$ (see the discussion above Lemma 10). Thus, there
exists an invertible matrix $R$ such that $E^{k}=R^{-1}(\mathds{1}\oplus 0)R$.
It is straightforward to verify that $\mathds{1}\oplus 0$ is a self-adjoint
endomorphism of $(RA_{i}R^{T})$. We conclude that if $\text{ker}(E^{k})\neq
F^{n}$ or $\text{R}(E^{k})\neq F^{n}$, then $(RA_{i}R^{T})_{i}$ is
blockdiagonal. ∎
Hence, they are simultaneously decomposable if and only if there exists an
idempotent endomorphism which is neither invertible nor nilpotent 444Note that
Fitting’s lemma is usually stated for decompositions into blocks under
isomorphisms, i.e., under transformations of the form $QC_{\alpha}R$, where
$Q,R$ are invertible (see for instance Ref. [49]). However, for our purposes
the version presented in Ref. [34] is the relevant one, as it is formulated
for congruence.
Lemma 10 is particularly useful if one wants to numerically decide whether or
not a stabilizer state is decomposable. As Equation (37) is linear in the
entries of $E$, it is efficient to compute a basis of the endomorphism ring.
The endomorphism ring generated by this basis is finite as we work over the
finite field $\mathbb{Z}_{2}$ and there are only finitely many matrices over
$\mathbb{Z}_{2}$. Thus, if the ring is not too large one can check whether
every element is either nilpotent or invertible, or equivalently, if there
exists a idempotent element that is neither invertible nor nilpotent. As we
will see later, this insight is very useful for numerically computing the EGS.
### 3.3 The EGSM for $M\geq 4$
We explained in the previous sections that to characterize the PLC classes of
$M$-partite stabilizer states we can characterize the EGSM and establish that
$M$-partite stabilizer states decompose uniquely into indecomposable ones. In
this section we are concerned with the structure of the EGSM for $M\geq 4$. We
show that the EGS4 and therefore any EGSM for $M\geq 4$ is an infinite set.
This is even the case if we consider PLU instead of PLC transformations.
Let us show that the EGS4 is an infinite set w.r.t. PLC transformations.
Consider the spiral graph states $\ket{G_{n}}$ defined for any
$n\in\mathbb{N}$, $n\geq 4$, in Figure 3. These states are distributed among
the parties $P(4,n)=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$.
Figure 3: Sequence of the spiral graph states $\ket{G_{n}}$ and their
distribution among four parties
$P(4,n)=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$; all of these graph
states are indecomposable.
###### Theorem 11.
For any $n\in\mathbb{N}$ the state $\ket{G_{n}}$ is indecomposable.
We show this Theorem in Appendix F. The proof makes use of the fact that if a
tuple of linear combinations of commutation matrices is indecomposable, then
so is the original tuple. Using the complete classification of indecomposable
pairs of alternating forms from Ref. [47] it is straightforward to verify that
if
$(C_{\alpha_{1}}^{(n)},C_{\alpha_{2}}^{(n)},C_{\alpha_{3}}^{(n)},C_{\alpha_{4}}^{(n)})$
are the commutation matrices of the spiral graph state $\ket{G_{n}}$, then
$(C_{\alpha_{1}}^{(n)}+C_{\alpha_{2}}^{(n)},C_{\alpha_{2}}^{(n)}+C_{\alpha_{3}}^{(n)})$
is indecomposable for any $n$. As the EGS4 contains one representative of
every PLC class which only consists of indecomposable states,
$\ket{G_{n}}\in\text{EGS}_{4}$ for any $n$ and thus EGS4 contains infinitely
many states. Moreover, as $\text{EGS}_{4}\subset\text{EGS}_{M}$ for any $M\geq
4$, we have the following corollary.
###### Corollary 12.
The EGSM is infinite for any $M\geq 4$.
Thus, by simply increasing the number of parties from $3$ to $4$ the EGS
changes from a finite set of three states (up to permutation) to an infinite
set.
Let us address the question whether the EGS is infinite only due to
considering PLC operations instead of more general PLU operations. Indeed, it
could be that the EGS is finite when considering PLU operations. The following
discussion, however, shows that this is not the case. By studying which
additional transformations of the spiral graph states $\ket{G_{4n}}$ are
enabled by PLU operations compared to PLC operations one realizes that they
are equally powerful, as stated in the following theorem. A proof can be found
in Appendix F.
###### Theorem 13.
Let $U$ be a PLU transformation such that $U\ket{G_{4n}}=\ket{H_{4n}}$ is a
stabilizer state. Then, there exists a PLC transformation
$C\in\mathcal{C}_{4n}^{P(4,4n)}$ such that $C\ket{G_{4n}}=\ket{H_{4n}}$.
The theorem relies on properties of the PLU symmetry group of the spiral graph
states $\ket{G_{4n}}$. Note that any PLU transformation between stabilizer
states defines a bijection between their respective PLU symmetry groups.
Theorem 13 implies that the spiral graph states $\ket{G_{4n}}$ are
indecomposable under PLU operations and therefore are elements of the EGSM
w.r.t. PLU operations for any $M\geq 4$. Thus, stabilizer states on $M\geq 4$
parties contain infinitely many different types of stabilizer state
entanglement.
### 3.4 The EGS4 up to $10$ qubits
Figure 4: EGS4 for up to $10$ qubits. The first three states generate all
stabilizer states on two- and three-partite qubit stabilizer states [27, 28].
The EGS4 on four qubits contains the GHZ state as well as the linear cluster
state (up to permutations) with different positions of its leaf parent pairs.
For five, six and seven qubits one finds that only the PLC classes of spiral
graphs are indecomposable. For eight qubits, the spiral graph appears three
times with different positions of its leaf parent pairs, while the other two
graphs have a structure different from the spiral graph. Particularly the last
graph for eight qubits has an interesting structure, since it contains three
leaf parent pairs. Finally, for nine and ten qubits only the PLC classes of
the spiral graphs are indecomposable.
The results from the previous section suggest that a complete characterization
of all indecomposable stabilizer states for an arbitrary number of parties is
a difficult task. Nevertheless, let us turn to the simplest non-solved case,
namely EGS4, and present some results on the states it contains. Intuitively,
we expect that not every distribution of qubits among the $4$ parties allows
for the existence of an indecomposable state. Indeed, it can be shown that any
state, where one party contains more than twice the number of qubits of the
party with the smallest number of qubits, is decomposable, as stated by the
following observation.
###### Observation 14.
Let $P(4,n)=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$ be a partition
such that $|\alpha_{1}|\geq|\alpha_{2}|\geq|\alpha_{3}|\geq|\alpha_{4}|$ and
$\ket{\psi}\in\text{Stab}(P(M,n))$. Then $\ket{\psi}$ is decomposable if
$|\alpha_{1}|>2|\alpha_{4}|$.
The bound is simple consequence of Lemma 9, which gives sufficient conditions
for when one can extract GHZ states from a stabilizer state. Note that this
bound is not tight, as we will see later, i.e., there exist qubit
configurations for which all states are decomposable which is not implied by
the above observation. This is not surprising as Observation 14 does not take
into account whether $4$-partite stabilizer states other than the GHZ state
can be extracted. Now let us compute the EGS4 up to $10$ qubits. We explain in
detail how we utilize the commutation matrix formalism in this computation
555The code is written for Python 3 and can be made available from the authors
upon reasonable request..
From Observation 14 it is clear that for up to $9$ qubits only qubits
distributed among parties specified by the tuples $(1,1,1,1)$, $(2,1,1,1)$,
$(2,2,1,1)$, $(2,2,2,1)$, $(2,2,2,2)$, and $(3,2,2,2)$ can lead to
indecomposable stabilizer states. To find the maximal set of indecomposable
PLC inequivalent set of stabilizer states we proceed as follows. First, for
each configuration containing up to $7$ qubits we generate all possible graph
states and remove the decomposable ones. To this end, we compute a tuple of
commutation matrices for each state and then check decomposability by
computing a basis of the endomorphism ring and checking whether or not there
exists an idempotent element that is neither nilpotent nor invertible.
According to Fitting’s Lemma 10 the existence of such an element is equivalent
to the set of commutation matrices, and hence, the state, being decomposable.
It remains to identify one representative of each PLC equivalence class from
the obtained list of indecomposable graph states. We select a state from the
list and then remove all states which are PLC equivalent to the chosen state.
To check PLC equivalence we proceed as follows. Recall that according to
Theorem 7 two states are PLC equivalent if they admit congruent tuples of
commutation matrices. Thus, given two graph states $\ket{G^{\prime}}$ and
$\ket{G^{\prime}}$ with commutation matrices $(C_{\alpha})_{\alpha}$ and
$(D_{\alpha})_{\alpha}$ respectively, we have to decide whether
$(C_{\alpha})_{\alpha}$ and $(D_{\alpha})_{\alpha}$ are congruent to each
other or not. This can be done by solving the linear system of equations
$(QC_{\alpha})_{\alpha}=(D_{\alpha}P)_{\alpha}$ (38)
for the entries of $Q$ and $P$. Then, we check if any solution satisfies
$Q=(P^{-1})^{T}$. The runtime for both, verifying indecomposability and
removing congruent tuples, can in principle be exponentially large in the
number of qubits. This is due to the fact that the solution space of the
respective linear equation system can have a basis of $O(n^{2})$ elements, and
hence the solution space contains $O(2^{n^{2}})$ elements. Nevertheless, the
algorithm seems to be fast for most graph states in practice.
For the cases $(2,2,2,2)$ and $(3,2,2,2)$ containing eight and nine qubits we
have to follow a different approach as the number of graph states of eight
qubits is already too large ($\approx 268$ million). Therefore, we start from
a complete list of local Clifford inequivalent states of eight and nine qubits
(see [50]) and consider all possible distributions of these states among $4$
parties resulting in the configuration $(2,2,2,2)$, and $(3,2,2,2)$
respectively. Then, we proceed as before by removing all decomposable states
and determining a state for each PLC class.
For $10$ qubits we proceed in a similar way as for eight and nine qubits but
now we have to consider qubits distributed as $(3,3,2,2)$ and $(4,2,2,2)$.
Note, that the latter distribution is not excluded to contain indecomposable
states by Observation 14. Nevertheless, we find that it contains only
decomposable states. For the distribution $(3,3,2,2)$ only the PLC class of
the ten qubit spiral graph state is indecomposable.
In Figure 4 we display all states in the EGS4 up to $10$ qubits up to
relabeling of parties. Remarkably, up to $7$ qubits, the only additions to the
EGS4 besides the $4$-qubit GHZ state are the spiral graph states. For $8$
qubits more complex indecomposable graph states exist. In particular, the last
graph for eight qubits in Figure 4 is different, since it contains three leaf
parent pairs. Finally, for $9$ and $10$ qubits the only additional states are
again the spiral graphs.
Going beyond the EGS4 we find that for $6$ qubits on 5 parties there exist
$19$ different PLC classes (or $10$ different classes up to permutations of
the parties holding a single qubit). This suggests that beyond the EGS4 the
number of different PLC classes increases very quickly, already in the case
where the EGS4 for $5$ qubits still has a very simple structure, i.e., where
it contains only a single PLC class (see Figure 4).
## 4 Qudit systems
In this section, we generalize the commutation matrix formalism to qudit
stabilizer states of prime dimension. Similar to the qubit case, non-local
qudit Clifford operations can be implemented deterministically using gate
teleportation. Consequently, it is again interesting to study the same task of
characterizing PLC equivalence classes of qudit stabilizer states.
We first recall the definition of the Pauli group and stabilizer states for
qudits. We will closely follow Refs. [51, 52]. In the following,
$d\in\mathbb{N}$ is prime and $d\geq 3$. We consider the common generalization
of the Pauli operators to qudit systems as
$\displaystyle X$ $\displaystyle=\sum_{k=0}^{d-1}\ket{k+1}\bra{k},$ (39)
$\displaystyle Z$ $\displaystyle=\sum_{k=0}^{d-1}\eta^{k}\ket{k}\bra{k},$ (40)
where $\eta=\exp(2\pi i/d)$ so that $X^{d}=Z^{d}=\mathds{1}$. In analogy to
the single-qubit Pauli group the single-qudit Pauli group is defined as
$\mathcal{P}^{d}=\\{\eta^{m}\sigma_{a,b}|a,b,m\in\mathbb{Z}_{d}\\},$ (41)
where $\sigma_{a,b}=X^{a}Z^{b}$. As $d$ is prime, all elements in the single-
qudit Pauli group have the same order, namely $d$, and the same set of non-
degenerate eigenvalues. The $n$-qudit Pauli group $\mathcal{P}_{n}^{d}$ is the
group generated by all $n$-fold tensor products of elements of
$\mathcal{P}^{d}$. In analogy to the qubit case, there is a correspondence
between the Pauli group and the vector space $\mathbb{Z}_{d}^{2n}$ with a
symplectic form $\omega$. Let us again define the map
$\sigma:\mathbb{Z}_{d}^{2n}\rightarrow\mathcal{P}_{n}^{d}$ via
$\sigma(a_{1},b_{1},\ldots,a_{n},b_{n})=\sigma_{a_{1},b_{1}}\otimes\ldots\otimes\sigma_{a_{n},b_{n}}$.
Under this map, the group structure of the Pauli group is preserved. More
precisely, from the relation
$\sigma(\bm{f})\sigma(\bm{g})\propto\sigma(\bm{f}+\bm{g})$ (42)
we see that elements of the Pauli group modulo phases can be represented as
vectors of $\mathbb{Z}_{d}^{2n}$, and the multiplication of Pauli operators
corresponds to addition of their corresponding vectors in
$\mathbb{Z}_{d}^{2n}$. The generalized Pauli operators obey the following
commutation relations
$\sigma(\bm{f})\sigma(\bm{g})=\eta^{\omega(\bm{f},\bm{g})}\sigma(\bm{g})\sigma(\bm{f})\
\ \bm{f},\bm{g}\in\mathbb{Z}_{d}^{2n},$ (43)
with the symplectic form
$\displaystyle\omega(\bm{f},\bm{g})=(\bm{a},\bm{b})\begin{pmatrix}0&\mathds{1}\\\
-\mathds{1}&0\end{pmatrix}(\bm{a^{\prime}},\bm{b^{\prime}})^{T}.$ (44)
The Clifford group $\mathcal{C}_{n}(d)$ is defined as the unitary normalizer
of the Pauli group up to phases, i.e., $\mathcal{C}_{n}(d)=\\{U\in
SU(d^{n})|UPU^{\dagger}\in\mathcal{P}_{n}^{d}\ \forall
P\in\mathcal{P}_{n}^{d}\\}/U(1)$. Every $U\in\mathcal{C}_{n}(d)$ defines an
isometry on the symplectic space $\mathbb{Z}_{d}^{2n}$ via Equation (16) and
vice versa.
A stabilizer state is the unique $+1$ eigenstate to a maximal abelian subgroup
of $\mathcal{P}_{n}^{d}$ which does not contain any nontrivial multiples of
the identity. Such a subgroup contains $d^{n}$ elements and is generated by
$n$ independent elements. An abelian subgroup of $\mathcal{P}_{n}^{d}$ which
does not contain any nontrivial multiples of the identity is called a
stabilizer. In complete analogy to the qubit case, a stabilizer associated to
a stabilizer state corresponds to a maximally isotropic subspace of
$\mathbb{Z}_{d}^{2n}$ via the map $\sigma$. Moreover, every qudit stabilizer
state is local Clifford equivalent to a qudit graph state [53]. A qudit graph
state is the unique stabilizer state associated to a multigraph $G$ with
vertices $V$ and edges $E$. In contrast to a simple graph, multigraphs allow
for an arbitrary number of edges between two vertices. Similar to the qubit
case, edges correspond to controlled-$Z$ interactions between qudits, which
are represented by the vertices of the graph. The existence of distinct non-
trivial powers of the gate $CZ=\sum_{k=0}^{d-1}\outerproduct{k}{k}\otimes
Z^{k}$ gives rise to the different numbers of edges between vertices. The
number of edges $m$ between two vertices is called the _multiplicity_ , and
the corresponding interaction is given by $CZ^{m}$, where $0\leq m\leq d-1$.
Given a multigraph the canonical generators of the corresponding stabilizer
are the operators
$g_{i}=X_{i}\bigotimes_{j\in N_{i}}Z_{j}^{m_{ij}},$ (45)
where $m_{ij}$ is the multiplicity of the edges connecting vertex $i$ and $j$.
Finally, the adjacency matrix of a graph state $\ket{G}$ is defined by
$(\Gamma_{G})_{ij}=m_{ij}.$ (46)
In the following we denote by $\text{Stab}_{d}(P(M,n))$ the set of $n$-qudit
stabilizer states for qudits of dimension $d$, which are distributed among $M$
parties according to the partition $P(M,n)$.
The $n$-qudit Pauli group $\mathcal{P}_{n}^{d}$ has an analogous relation to
the symplectic vector space $(\mathbb{Z}_{d}^{2n},\omega)$ as
$\mathcal{P}_{n}$ has to the symplectic vector space
$(\mathbb{Z}_{2}^{2n},\omega)$. It is therefore straightforward to define
commutation matrices via the same construction as in Section 3 for qudit
stabilizer states. Commutation matrices are now tuples of matrices
$(C_{\alpha})_{\alpha\in P(M,n)}$ such that $C_{\alpha}\in\ M_{n\times
n}(\mathbb{Z}_{d})$. The matrices are again alternating due to the fact that
$\omega(\bm{f},\bm{g})=-\omega(\bm{g},\bm{f})$ and satisfy Equation (23),
i.e., they sum up to zero modulo $d$.
Before discussing the other properties of commutation matrices, let us
consider an example. Consider a four qutrit state distributed over $M=4$
parties and let the generator of its stabilizer be $g_{1}=X_{1}Z_{2}Z_{4}$,
$g_{2}=Z_{1}X_{2}$, $g_{3}=X_{3}Z_{4}^{2}$, and $g_{4}=Z_{1}Z_{3}^{2}X_{4}$.
One can then verify that it admits the following set of commutation matrices.
$\begin{split}C_{1}&=\begin{pmatrix}0&2&0&2\\\ 1&0&0&0\\\ 0&0&0&0\\\
1&0&0&0\end{pmatrix}C_{2}=\begin{pmatrix}0&1&0&0\\\ 2&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix}\\\ C_{3}&=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&1\\\
0&0&2&0\end{pmatrix}C_{4}=\begin{pmatrix}0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&2\\\
2&0&1&0\end{pmatrix}.\end{split}$ (47)
These matrices are alternating as $-1=2$ over $\mathbb{Z}_{3}$, and they sum
up to zero modulo $3$. In fact, this state is again a graph state with edges
$\\{\\{1,2\\}_{1},\\{1,4\\}_{1},\\{3,4\\}_{2}\\}$, where the subscript denotes
the multiplicity of its edges. One can again verify that the commutation
matrices can be obtained from the adjacency matrix $\Gamma_{G}$ via
$C_{\alpha}=\sum_{i\in\alpha}\Gamma_{G}(i),$ (48)
where $\Gamma_{G}(i)$ is the matrix that contains the $i$-th column and $(-1)$
times the $i$-th row of $\Gamma_{G}$, and zero everywhere else. E.g., for the
the above example one has
$\Gamma_{G}=\begin{pmatrix}0&1&0&1\\\ 1&0&0&0\\\ 0&0&0&2\\\
1&0&2&0\end{pmatrix},$ (49)
from which the commutation matrices in Equation (47) directly follow.
Let us continue our discussion on properties of commutation matrices. As
stabilizers again correspond to maximally isotropic subspaces of
$\mathbb{Z}_{d}^{2n}$, it is straightforward to verify that the rank condition
in Theorem 6 still holds. Moreover, commutation matrices still have the same
interpretation concerning PLC equivalence of stabilizer states. In fact, a
theorem similar to Theorem 7 in the qubit case still holds in the case of
qudits.
###### Theorem 15.
Let $\ket{\psi},\ket{\phi}\in\text{Stab}_{d}(P(M,n))$. Then $\ket{\psi}$ is
PLC equivalent to $\ket{\phi}$ if and only if they admit the same tuple of
commutation matrices.
The proof closely follows Ref. [28] and can be found in Appendix G.
Figure 5: States in the EGS4 for qutrits. Highlighted in green is the complete
EGS4 for up to five qutrits. The first three states generate all two- and
three-partite qutrit stabilizer states [29]. An exhaustive search through all
four-partite graph states up to 5 qutrits reveals six more states from the
EGS4. Beyond $5$ qutrits we have identified some states that are contained in
the EGS4 for qutrits, however, this list might be incomplete. The EGS4 for
qutrits already has a much richer structure, due to the appearance of graphs
that contain edges of higher multiplicities.
It directly follows that a qudit stabilizer state is decomposable if and only
if its commutation matrices can be block-diagonalized under congruence (cf.
Theorem 8). This is due to the fact that Equation (31) and Inequality (61)
still hold. Thus, we again have that finding states in the EGS is equivalent
to finding indecomposable tuples of commutation matrices. Stabilizer states
decompose uniquely into indecomposable ones if commutation matrices decompose
uniquely into indecomposable blocks. In contrast to qubit systems, the
commutation matrices are now defined over the finite field $\mathbb{Z}_{d}$
with $d\geq 3$, i.e., a field with characteristic not equal to two. Theorem
$2$ of Ref. [34] shows that tuples of alternating matrices over such fields
indeed decompose uniquely into indecomposable blocks. Thus, we have shown the
following theorem.
###### Theorem 16.
All states in $\text{Stab}_{d}(P(M,n))$ decompose uniquely into indecomposable
states in the EGSM, where $d\geq 3$ is a prime.
Due to this theorem, any PLC class is uniquely characterized by a tensor
product of states from the EGS. Therefore, the remaining task is to determine
the EGS.
First, however, some remarks are in order. Let us stress that (qubit and
qudit) stabilizer states distributed among parties (where each party can hold
more than one qubit) can be described by a subset of higher-dimensional qudit
stabilizer states with a single qudit per party and a composite dimension (cf.
Ref. [54]). PLC operations then form a subset of qudit Clifford
transformations. On the other hand, viewing qudit stabilizer states of non-
prime dimension as lower-dimensional weighted graph states [54] also reveals
that for qudit graph states of non-prime dimension the decompositions under
PLU are not necessarily unique. Consider a four ququart state on two parties
that decomposes into a maximally entangled state $\ket{\phi^{+}_{4}}$,
containing 2 ebits of entanglement, and two $\ket{0}$ states. Using some of
the results of Ref. [54] one can see that this decomposition is PLU equivalent
to a decomposition consisting of two entangled pairs of ququarts that each
contain only one ebit of entanglement. Thus, there exist to decompositions
that are PLU equivalent, but the states that occur in these decompositions are
PLU inequivalent. Note that qudit stabilizer states and qudit Clifford
transformations were also investigated in Refs. [36, 37].
Moreover, let us remark that, similar to the qubit case, the results of
Scharlau [47] cannot lead to anything else than the EGS3 for prime dimensions.
In particular, they cannot be used to construct the EGS4 since choosing two
commutation matrices to be indecomposable but not equal to the
$\ket{\phi^{+}_{3}}$ or the $\ket{GHZ_{3}^{3}}$ one cannot find two additional
commutation matrices so that the overall rank constraint is fulfilled.
### 4.1 States in the EGS4 for qutrits
Using the same techniques as in the case of qubits in Section 3.2, i.e.,
Fitting’s Lemma 10, we can compute the EGS4 for up to $5$ qutrits by
performing an exhaustive search through all $4$-partite graph states up to $5$
qutrits. First, one recovers the states $\ket{0}$, $\ket{\phi^{+}_{3}}$, and
$\ket{GHZ_{3}}$ that where already proven in Ref. [29] to generate all three-
partite stabilizer states. This can again be derived from the complete set of
indecomposable pairs of alternating forms in Ref. [47], as these results hold
for any field. Additionally, we identify the $6$ additional states from the
EGS4 up to $5$ qutrits, see Figure 5.
Beyond $5$ qutrits the exhaustive search through all graph states becomes
computationally intractable due to the increasing number of graphs but also
due to the increasing sizes of the endomorphism rings. Nevertheless, one can
gain some additional insight into the EGS4 for more than $5$ qutrits. To that
end, let us first see if the states that we found to be in the EGS4 for qubits
also appear in the case of qutrits. Again employing Lemma 10 (Fitting’s
Lemma), one can show that the spiral graphs containing only edges of
multiplicities $1$ on $5$ to $10$ qubits are also indecomposable and thus in
the EGS4 for qutrits. This raises the question if other spiral graph states
containing edges of other multiplicities are also contained in the EGS4. In
the following we argue that this is not the case, as for many states edges of
higher multiplicities can be removed by local Clifford operations.
To this end, let us recall how the action of the local Clifford group can be
described on the level of graphs. The action of the local Clifford group on
qudit graph states was completely characterized in Ref. [53]. In particular,
it was shown that there exists a local Clifford operation that, when applied
to any vertex $v$, multiplies the multiplicities of all edges connected to $v$
by a constant factor $0\neq b\in\mathbb{Z}_{d}$. Thus, for qutrit graph
states, there exists a local Clifford operation that converts all edges of
multiplicity $2$ to edges of multiplicity $1$ and vice versa. Recall, that a
_tree graph_ is a graph that does not contain any cycle. Then, we have the
following observation.
###### Observation 17.
Any PLC orbit that contains a tree graph, also contains the same tree graph,
where all edges are of multiplicity $1$.
This observation simply follows from the fact that in a tree graph all edges
of higher multiplicities can be moved to a leaf and can then be locally
converted to edges of multiplicities $1$. Notably, this means that any qutrit
spiral graph state, regardless of the multiplicities of its edges, is always
PLC equivalent to the spiral graph states with all multiplicities being equal
to $1$.
Moreover, the graph state containing three leafs is indecomposable and edges
of higher multiplicities play no role by the arguments above. The only state
where edges of higher multiplicities play a role is the one containing cycles.
It is again indecomposable for qutrits and appears in four PLC inequivalent
forms with different numbers of edges with multiplicity $2$.
Finally, as in the case of four qutrits, the closed loop graph containing a
single edge of multiplicity two is indecomposable. From these few states one
can already deduce, that the EGS4 for qutrits already has a much richer
structure, due to the appearance of graphs that contain edges of higher
multiplicities, see Figure 5.
### 4.2 Beyond PLC transformations
Let us finally briefly comment on how (stabilizer) states transform under more
general types of transformations, also constrained by locality.
First, let us consider general PLU transformations (not just PLC). In Ref.
[55] it was shown that already in the case of two parties decompositions of
arbitrary states into indecomposable states is not unique. More precisely, let
$\ket{\psi_{1}}$ ($\ket{\psi_{2}}$) be a state of two qudits of dimension
$d_{1}$ ($d_{2}$). Ref. [55] shows that for any $d_{1},d_{2}\geq 4$ there
exist states of the form $\ket{\psi_{1}}\otimes\ket{\psi_{2}}$ shared among
two parties and a PLU transformation $U$ such that
$U\ket{\psi_{1}}\otimes\ket{\psi_{2}}=\ket{\psi_{1}^{\prime}}\otimes\ket{\psi_{2}^{\prime}}$
(50)
and such that $\ket{\psi_{1}}$ (and $\ket{\psi_{2}}$) is not PLU equivalent to
any $\ket{\psi_{i}^{\prime}}$. If we consider quasilocal unitary operations
instead of PLU, it follows from the results of Ref. [56] that also
decompositions of stabilizer are not unique.
If we allow for measurements in addition to PLC transformations,
transformations between PLC inequivalent decompositions become possible .
Consider for instance two Bell states $\ket{\phi^{+}}^{\otimes 2}$ shared
between parties $\alpha_{1}=\\{1\\}$, $\alpha_{2}=\\{2,3\\}$ and
$\alpha_{3}=\\{4\\}$. Then, applying $CZ_{2,3}$ followed by local
complementation w.r.t. qubit $3$ and a $Z$ basis measurement of qubit $3$
results in the state $\ket{GHZ_{3}}$ shared between all three parties and
$\ket{0}$ at party $\alpha_{2}$ (up to local Pauli corrections). It is clear
that $\ket{\phi^{+}}^{\otimes 2}$ and $\ket{GHZ_{3}}\otimes\ket{0}$ are PLU
inequivalent.
## 5 Conclusion and Outlook
In this work, we studied PLC transformations of stabilizer states. PLC
transformations are a physically motivated extension of local Clifford
operations that naturally arise in the context of quantum networks that can
provide large amounts of bipartite entanglement between well connected nodes.
Moreover, the fact that these operations can be implemented deterministically
makes them particularly interesting to study.
First, we addressed the question of how PLC transformations between graph
states change the corresponding graph. It was shown in Ref. [26] that local
Clifford transformations between graph states correspond to a sequence of
local complementations on the respective graphs. We found that any PLC
transformation between graph states can be realized by local complementations
supplemented by the addition and removal of edges within parties.
Then, we studied the classification of PLC equivalence classes of stabilizer
states. We discussed an approach based on invariant polynomials and showed
that this approach achieves a complete classification. However, since we are
not aware of an efficient way to evaluate those polynomial, the approach seems
unpractical.
Motivated by the results of Ref. [28], we introduced a new mathematical tool,
the commutation matrix formalism, to study PLC equivalence classes of
stabilizer states. More precisely, we related the problem of classifying PLC
equivalence classes to the classification problem of tuples of alternating
forms, which is a well studied problem in linear algebra.
We showed that two stabilizer states are PLC equivalent if they admit the same
set of commutation matrices, and moreover, states can be decomposed into
tensor products of smaller stabilizer states under PLC if their commutation
matrices can be block-diagonalized. This approach allowed us to gain several
insights into the entanglement structure of multipartite stabilzier states. We
showed, that in contrast to the $3$-partite case, $M\geq 4$-partite stabilizer
states contain infinitely many different types of entanglement under PLC
transformations and even under PLU transformations. We derived necessary and
sufficient conditions to decide whether or not a given stabilizer state is
decomposable. To demonstrate the power of our approach we numerically computed
the EGS4 for up to $10$ qubits. We furthermore showed that the decomposition
of states into indecomposable states is unique in case the states do not admit
additional PLC symmetries.
Finally, we generalized the commutation matrix formalism to qudit systems of
prime dimension $d$. As the commutation matrices are now defined over the
field $\mathbb{Z}_{d}$, the results of Ref. [34] imply that qudit stabilizer
states decompose uniquely into tensor products of indecomposable ones. Again
we employed the commutation matrix formalism to compute the EGS4 up to $5$
qutrits.
An interesting topic for future research is to resolve the question whether or
not qubit stabilizer states decompose uniquely into indecomposable ones.
Furthermore, the structure of the EGS4 up to $10$ qubits suggests that maybe a
complete description of the stabilizer states in this set is possible.
Considering PLU transformations, it would be interesting to understand whether
stabilizer states which are indecomposable under PLC can decompose under PLU
into states which are not PLU equivalent to stabilizer states. Note that these
states would have to be locally maximally entangleable (LME) states [56]. In
case such decompositions exist, one can study the question how the EGS for
stabilizer states looks like if it can contain states which are not PLU
equivalent to stabilizer states.
## 6 Acknowledgements
We thank Vladimir V. Sergeĭchuk for helpful correspondence. We acknowledge
financial support from the Austrian Science Fund (FWF): W1259-N27, P32273-N27,
FG5-L, and the SFB BeyondC (Grant No. F7107-N38).
## References
* [1] H. J. Kimble, ”The quantum internet.”, Nature 453, 1023 (2008).
* [2] S. Wehner, D. Elkouss, and R. Hanson, ”Quantum internet: A vision for the road ahead.”, Science 362, 9288 (2018).
* [3] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, ”Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network.”, Phys. Rev. Lett. 78, 3221 (1997).
* [4] S. Muralidharan, L. Li, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, ”Efficient long distance quantum communication.”, Sci. Rep. 6, 20463 (2016).
* [5] S. Perseguers, G. J. Lapeyre Jr, D. Cavalcanti, M. Lewenstein, and A. Acín, ”Distribution of entanglement in large-scale quantum networks.”, Rep. Prog. Phys. 76, 096001 (2013).
* [6] L.-M. Duan, and C. Monroe, ”Colloquium: Quantum networks with trapped ions.”, Rev. Mod. Phys. 82, 1209 (2010).
* [7] A. Reiserer, and G. Rempe, ”Cavity-based quantum networks with single atoms and optical photons.”, Rev. Mod. Phys. 87,1379 (2015).
* [8] J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, ”Distributed quantum computation over noisy channels.”, Phys. Rev. A 59, 4249 (1999).
* [9] T. J. Proctor, P. A. Knott, and J. A. Dunningham, ”Multiparameter Estimation in Networked Quantum Sensors.”, Phys. Rev. Lett. 120, 080501 (2018).
* [10] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, ”Long-distance quantum communication with atomic ensembles and linear optics.”, Nature 414, 413 (2001).
* [11] A. Tavakoli, A. Pozas-Kerstjens, M. Luo, and M.-O. Renou, ”Bell nonlocality in networks.”, Rep. Prog. Phys. (2021).
* [12] O. Gühne, and G. Tóth, ”Entanglement detection.”, Phys. Rep. 474, 1 (2009).
* [13] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, ”Quantum entanglement.”, Rev. Mod. Phys. 81, 865 (2009).
* [14] I. Bengtsson, and K. Zyczkowski, ”Geometry of Quantum States.”, (Cambridge University Press, 2006)
* [15] D. Sauerwein, N. R. Wallach, G. Gour, and B. Kraus, ”Transformations among Pure Multipartite Entangled States via Local Operations are Almost Never Possible.”, Phys. Rev. X 8, 031020 (2018).
* [16] T. Kraft, S. Designolle, C. Ritz, N. Brunner, O. Gühne, and M. Huber, ”Quantum entanglement in the triangle network.”, Phys. Rev. A 103, L060401 (2021).
* [17] M. Navascués, E. Wolfe, D. Rosset, and A. Pozas-Kerstjens, ”Genuine Network Multipartite Entanglement.”, Phys. Rev. Lett. 125, 240505 (2020).
* [18] K. Hansenne, Z.-P. Xu, T. Kraft, and O. Gühne, ”Symmetries in quantum networks lead to no-go theorems for entanglement distribution and to verification techniques.”, Nat. Commun. 13, 496 (2022).
* [19] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H.-J. Briegel, ”Entanglement in Graph States and its Applications.”, in _Quantum Computers, Algorithms and Chaos_, edited by G. Casati, D. L. Shepelyansky, P. Zoller, and G. Benenti (IOS Press, Amsterdam, 2006).
* [20] M. Hein, J. Eisert, and H. J. Briegel, ”Multiparty entanglement in graph states.”, Phys. Rev. A 69, 062311 (2004).
* [21] D. Gottesman, ”Stabilizer Codes and Quantum Error Correction.”, _PhD thesis_ , California Institute of Technology Pasadena, California (1997).
* [22] R. Raussendorf and H.J. Briegel, ”A One-Way Quantum Computer.”, Phys. Rev. Lett. 86, 5188 (2001).
* [23] D. Gottesman, ”The Heisenberg Representation of Quantum Computers.”, in _Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics_ , edited by S. P. Corneyet al. (International Press, Cambridge, MA 1999).
* [24] D. Schlingemann and R. F. Werner, ”Quantum error-correcting codes associated with graphs.”, Phys. Rev. A 65, 012308 (2001).
* [25] D. Gottesman, and I. L. Chuang, ”Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations.”, Nature 402, 390 (1999).
* [26] M. Van den Nest, J. Dehaene, and B. De Moor, ”Graphical description of the action of local Clifford transformations on graph states.”, Phys. Rev. A 69, 022316 (2004).
* [27] D. Fattal, T. S. Cubitt, Y. Yamamoto, S. Bravyi, & I. L. Chuang, ”Entanglement in the stabilizer formalism.”, arXiv:quant-ph/0406168 (2004).
* [28] S. Bravyi, D. Fattal, and D. Gottesman, ”GHZ extraction yield for multipartite stabilizer states.”, J. Math. Phys. 47, 062106 (2006).
* [29] S. Y. Looi, and R. B. Griffiths, ”Tripartite entanglement in qudit stabilizer states and application in quantum error correction.”, Phys. Rev. A 84, 052306 (2011).
* [30] K. Wirthmüller, ”Homological invariants of stabilizer states.” Quantum Inf. Comput. 8, 595 (2008).
* [31] G. Smith, and D. Leung, ”Typical entanglement of stabilizer states.”, Phys. Rev. A 74, 062314 (2006).
* [32] K. Wirthmüller, ”Homology of Generic Stabilizer States.”, arXiv:0809.3346 (2008).
* [33] S. Nezami, and M. Walter, ”Multipartite Entanglement in Stabilizer Tensor Networks.”, Phys. Rev. Lett. 125, 241602 (2020).
* [34] V. V. Sergeĭchuk, ”Classification problems for systems of forms and linear mappings.”, Math. USSR Izv. 31, 481 (1988).
* [35] A. Dahlberg, J. Helsen, and S. Wehner, ”Counting single-qubit Clifford equivalent graph states is #P-Complete.”, J. Math. Phys. 61, 022202 (2020).
* [36] F. E. S. Steinhoff, C. Ritz, N. Miklin, and O. Gühne, ”Qudit Hypergraph States.”, Phys. Rev. A 95, 052340 (2017).
* [37] C. Ritz, ”Characterizing the structure of multiparticle entanglement in high-dimensional systems.”, PhD thesis, University of Siegen 2018.
* [38] M. Englbrecht, and B. Kraus, ”Symmetries and entanglement of stabilizer states.”, Phys. Rev. A 101, 062302 (2020).
* [39] D. H. Zhang, H. Fan, and D. L. Zhou, ”Stabilizer dimension of graph states.”, Phys. Rev. A 79, 042318 (2009).
* [40] A. Bouchet, ”An efficient algorithm to recognize locally equivalent graphs.”, Combinatorica 11, 315 (1991).
* [41] A. Bouchet, ”Recognizing locally equivalent graphs.” Discrete Math. 114, 75 (1993).
* [42] S.-K. Liao et al., ”Satellite-Relayed Intercontinental Quantum Network.” Phys. Rev. Lett. 120, 030501 (2018).
* [43] J. I. Cirac, W. Dür, B. Kraus, and M Lewenstein, ”Entangling operations and their implementation using a small amount of entanglement.” Phys. Rev. Lett. 86, 544 (2001).
* [44] G. Gour, and N. R. Wallach, ”Classification of multipartite entanglement of all finite dimensionality.”, Phys. Rev. Lett. 111, 060502 (2013).
* [45] M. Van den Nest, J. Dehaene, and B. De Moor, ”Finite set of invariants to characterize local Clifford equivalence of stabilizer states.”, Phys. Rev. A 72, 014307 (2005).
* [46] Z. Ji, J. Chen, Z. Wei, and M. Ying, ”The LU-LC conjecture is false.”, Quantum Inf. Comput. 10, 97-108 (2010).
* [47] R. Scharlau, ”Paare alternierender Formen.”, Math. Z. 147, 13-19 (1976).
* [48] G. Belitskii, R. Lipyanski, and V. V. Sergeichuk, ”Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild.”, Linear Algebra Its Appl. 407, 249-262 (2005).
* [49] M. Barot, ”Representations of quivers.”, available at https://www.matem.unam.mx/ barot/articles/
notes_ictp.pdf.
* [50] L. E. Danielsen, Database of Entanglement in Graph States (2011), available at http://www.ii.uib.no/l̃arsed/entanglement/.
* [51] E. Hostens, J. Dehaene, and B. De Moor, ”Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic.”, Phys. Rev. A 71, 042315 (2005).
* [52] J. M. Farinholt, ”An ideal characterization of the Clifford operators.” J. Phys. A 47, 305303 (2014).
* [53] M. Bahramgiri, and S. Beigi, ”Graph states under the action of local Clifford group in non-binary case.”, arXiv:quant-ph/0610267 (2006).
* [54] T. Kraft, C. Ritz, N. Brunner, M. Huber, and O. Gühne, ”Characterizing Genuine Multilevel Entanglement.”, Phys. Rev. Lett. 120, 060502 (2018).
* [55] A. Neven, D. K. Gunn, M. Hebenstreit, and B. Kraus, ”Local transformations of multiple multipartite states.”, SciPost Phys., 11, 042 (2021).
* [56] C. Kruszynska, and B. Kraus. ”Local entanglability and multipartite entanglement.” Phys. Rev. A 79, 052304 (2009).
* [57] M. Aschbacher, Finite group theory, (Cambridge University Press, Cambridge, MA, 2000).
* [58] S. Lang, Algebra, (Springer, New York, NY, 2002).
* [59] D. Gross, ”Hudson’s theorem for finite-dimensional quantum systems.” J. Math. Phys. 47, 122107 (2006).
* [60] F. J. MacWilliams, and N. J. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company (1977).
* [61] G. Marsaglia, ”Bounds for the Rank of the Sum of Two Matrices.” Mathematical Note, Boeing Scientific Research Labs, Seattle, WA, (1964).
* [62] C. G. Bartolone, and M. A. Vaccaro. ”The action of the symplectic group associated with a quadratic extension of fields.” J. Algebra 220, 115 (1999).
* [63] G. Falcone, M. A. Vaccaro, ”Kronecker modules and reductions of a pair of bilinear forms.”, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 43, 55 (2004).
* [64] J. Dieudonné, ”Sur la réduction canonique des couples de matrices.” Bull. Soc. Math. Fr. 74, 130 (1946).
* [65] Y. A. Drozd, and A. I. Plakosh, ”On nilpotent Chernikov 2-groups with elementary tops.”, Algebra Discrete Math. 22, 201 (2016).
## Appendix A Generalized local complementation
In this section we provide a proof of Theorem 1. To that end, we first prove
Lemma 2, which shows that the action of any $2$-qubit Clifford operator on a
graph state can be implemented by $CZ$ gates and local complementation (LCE)
on the respective qubits up to local Clifford operations. Let us state the
lemma in its formal version.
###### Lemma 2 (formal version).
Let $\ket{G}$ be a $n\geq 2$ qubit graph state. Then for any
$C\in\mathcal{C}_{2}$ acting on qubits 1 and 2 there exists a local Clifford
operator $L_{1}\in\mathcal{C}_{n}^{L}$ and a LCE transformation with respect
to vertex $1$ and $2$ described by the operator $L_{2}\otimes L_{3}$, where
$L_{2}\in\mathcal{C}_{2}$ acts on qubits 1 and 2, and
$L_{3}\in\mathcal{C}_{n-2}^{L}$ acts on qubits $\\{3,\ldots n\\}$, such that
$C\otimes\mathds{1}\ket{G}=L_{1}(L_{2}\otimes L_{3})\ket{G}.$ (51)
Note that the 2-qubit Clifford group has $|\mathcal{C}_{2}|=11520$ elements
and its subgroup, the local Clifford group, has $|\mathcal{C}_{2}^{L}|=24^{2}$
elements. Thus, $\mathcal{C}_{2}$ partitions into $11520/24^{2}=20$ different
right cosets
$\mathcal{C}_{2}^{L}g=\\{fg|f\in\mathcal{C}_{2}^{L},g\in\mathcal{C}_{2}\\}$ of
$\mathcal{C}_{2}^{L}$, i.e.,
$\mathcal{C}_{2}=\bigcup_{i=1}^{20}\mathcal{C}^{L}_{2}g_{i}$, with
$g_{i}\in\mathcal{C}_{2}^{L}$. With these insights let us show Lemma 2.
###### Proof.
Observe, that Lemma 2 is equivalent to following statement. Clifford operators
corresponding to LCE with respect to vertices $1$ and $2$ can generate (up to
local Clifford operations) an element of every right coset
$\mathcal{C}_{2}^{L}g$, with $g\in\mathcal{C}_{2}$ acting on qubits $1$ and
$2$. We show this in the following by generating an element of each coset via
LCE.
To this end we consider the operators that correspond to LCE. Adding and
removing the edge between vertex $1$ and $2$ corresponds to a controlled phase
gate $CZ_{12}=\outerproduct{0}{0}\otimes\mathds{1}+\outerproduct{1}{1}\otimes
Z$ between qubits $1$ and $2$. Local complementation at qubit $1$ ($2$ qubit)
corresponds to the operator $\exp(-i\pi/4X)\otimes\exp(i\pi/4Z)$
($\exp(i\pi/4Z)\otimes\exp(-i\pi/4X)$) on qubits $1$ and $2$ if qubit $1$ and
$2$ are connected. Otherwise, it corresponds to the operator
$\exp(-i\pi/4X)\otimes\mathds{1}$ ($\mathds{1}\otimes\exp(-i\pi/4X)$).
Moreover, local complementation creates additional local Clifford gates on
other neighbours of qubit $1$ and $2$. Let us write all possible operators on
qubit $1$ and $2$ as an (ordered) tuple
$\left(CZ_{12},e^{-i\frac{\pi}{4}X}\otimes\mathds{1},\mathds{1}\otimes
e^{-i\frac{\pi}{4}X},e^{-i\frac{\pi}{4}X}\otimes
e^{i\frac{\pi}{4}Z},e^{i\frac{\pi}{4}Z}\otimes e^{-i\frac{\pi}{4}X}\right).$
(52)
Figure 6 shows a representative of each right coset of $\mathcal{C}_{2}^{L}$
in $\mathcal{C}_{2}$ and a sequence of the above operators generating it. The
sequence respects that the operators for local complementation depend on
whether qubit $1$ and $2$ are connected. Figure 6 considers the case where
qubit $1$ and $2$ are initially disconnected. Let $\\{C_{j}\\}$ be the
operators listed in the table. Clearly, if qubit $1$ and $2$ are initially
connected, then $\\{C_{j}CZ_{1,2}\\}$ are representatives of all different
cosets. ∎
Next let us show Theorem 1, which we state here again.
###### Theorem 1.
Let $\ket{G}$ and $\ket{G^{\prime}}$ be $n$-qubit graph states distributed
among $P(M,n)$ parties. Then $\ket{G}$ is PLC equivalent to $\ket{G^{\prime}}$
if and only if the graphs $G$ and $G^{\prime}$ are related via a sequence of
local complementations and edge additions/removals within parties (LCE).
###### Proof of Theorem 1.
The if part is trivial. To prove the only if part let $\ket{G}$ and
$\ket{G^{\prime}}$ be $M$-partite graph states. Suppose there exists a PLC
operator $\bigotimes_{\alpha\in P(M,n)}C_{\alpha}$ such that
$\ket{G^{\prime}}=\bigotimes_{\alpha}C_{\alpha}\ket{G}$. Then, for every party
$\beta\in P(M,n)$ we do the following. As $\mathcal{C}_{|\beta|}$ is generated
by single- and two-qubit Clifford operators (see also Section 2.1) we can
write $C_{\beta}=\prod_{j=1}^{k}C_{\beta}^{j}$, where
$C_{\beta}^{j}\in\mathcal{C}_{2}$ for all $j\in[k]$. Lemma 2 allows to replace
the action of $C_{\beta}^{k}$ by the action of a local Clifford operator
$L_{1}$ on the state $L_{2}\otimes L_{3}\ket{G}$. The state $L_{2}\otimes
L_{3}\ket{G}$ is a again a graph state as the operator $L_{2}\otimes L_{3}$
describes a sequence of LCE. The local operator $L_{1}$ can then be commuted
to the end of the circuit by possibly changing the remaining operators
$C_{\beta}^{j}$ to some other $2$-qubit Clifford operators and possibly the
operators $C_{\alpha}$ for $\alpha\neq\beta$ to some other PLC operators. We
now continue like this for the operators $C_{\beta}^{j}$ for $j\in[k-1]$. It
is straightforward to verify that, after applying this reasoning to every
party, we end up with an equation of the form
$\ket{G^{\prime}}=L\ket{G^{\prime\prime}}$, where $\ket{G^{\prime\prime}}$ is
obtained from $\ket{G}$ via LCE. It remains to show that the operator $L$
corresponds to local complementation. In Ref. [26] it is shown that two graph
states are related by a local Clifford operator if and only if their graphs
are related via a sequence of local complementations. We conclude that
$G^{\prime\prime}$ and $G^{\prime}$ are related via a sequence of local
complementations and therefore $G^{\prime}$ and $G$ by a sequence of LCE. ∎
$B$ | generation | $B$ | generation | $B$ | generation | $B$ | generation
---|---|---|---|---|---|---|---
$\begin{pmatrix}1&1&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}$ | $[1]$ | $\begin{pmatrix}1&1&0&1\\\ 0&1&0&0\\\ 0&1&1&1\\\ 0&0&0&1\end{pmatrix}$ | $[0,3,0,2,0,3,0]$ | $\begin{pmatrix}1&0&1&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&1&0&1\end{pmatrix}$ | $[1,0,3,0]$ | $\begin{pmatrix}1&0&0&0\\\ 0&1&0&1\\\ 1&0&1&1\\\ 0&0&0&1\end{pmatrix}$ | $[0,4,0]$
$\begin{pmatrix}0&0&1&1\\\ 0&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\end{pmatrix}$ | $[0,4,3,0,2,1,0,4,0]$ | $\begin{pmatrix}0&1&1&0\\\ 0&0&0&1\\\ 1&0&0&1\\\ 0&1&0&0\end{pmatrix}$ | $[0,4,3,4,0]$ | $\begin{pmatrix}1&1&1&1\\\ 0&1&0&0\\\ 0&0&1&1\\\ 0&1&0&1\end{pmatrix}$ | $[2,0,3,0]$ | $\begin{pmatrix}1&0&0&0\\\ 0&1&1&0\\\ 0&0&1&0\\\ 1&0&0&1\end{pmatrix}$ | $[0]$
$\begin{pmatrix}0&1&1&1\\\ 0&1&0&1\\\ 1&1&1&0\\\ 0&1&0&0\end{pmatrix}$ | $[1,2,0,4,3,0]$ | $\begin{pmatrix}0&1&1&0\\\ 0&1&0&1\\\ 1&1&1&1\\\ 0&1&0&0\end{pmatrix}$ | $[1,0,4,3,0]$ | $\begin{pmatrix}1&0&0&0\\\ 0&1&1&1\\\ 0&0&1&1\\\ 1&0&0&1\end{pmatrix}$ | $[2,0]$ | $\begin{pmatrix}1&0&1&0\\\ 0&0&0&1\\\ 1&1&0&1\\\ 0&1&0&1\end{pmatrix}$ | $[1,0,3,4,0]$
$\begin{pmatrix}1&0&1&1\\\ 0&0&1&0\\\ 0&1&1&0\\\ 1&1&1&0\end{pmatrix}$ | $[2,1,0,3,0,2,0]$ | $\begin{pmatrix}1&0&1&0\\\ 0&0&1&1\\\ 0&1&1&1\\\ 1&1&1&1\end{pmatrix}$ | $[0,3,0,2,1,0]$ | $\begin{pmatrix}1&1&1&0\\\ 0&0&0&1\\\ 1&0&0&1\\\ 0&1&0&1\end{pmatrix}$ | $[0,3,4,0]$ | $\begin{pmatrix}1&1&0&0\\\ 0&1&0&1\\\ 1&1&1&1\\\ 0&0&0&1\end{pmatrix}$ | $[1,0,4,0]$
$\begin{pmatrix}1&1&1&1\\\ 0&0&1&0\\\ 0&1&1&0\\\ 1&0&1&0\end{pmatrix}$ | $[2,0,3,0,2,0]$ | $\begin{pmatrix}1&1&0&0\\\ 0&1&1&0\\\ 0&0&1&0\\\ 1&1&0&1\end{pmatrix}$ | $[1,0]$ | $\begin{pmatrix}1&1&0&0\\\ 0&1&1&1\\\ 0&0&1&1\\\ 1&1&0&1\end{pmatrix}$ | $[2,1,0]$ | $\begin{pmatrix}1&1&0&1\\\ 1&1&1&0\\\ 1&0&1&1\\\ 1&1&0&0\end{pmatrix}$ | $[0,4,0,1,0]$
Figure 6: This table contains an element $B$ of each right coset of
$\mathcal{C}_{2}^{L}$ in $\mathcal{C}_{2}$ and a sequence of operators from
the tuple in Eq. (52) generating it. Each sequence is specified by a list of
indices of the respective elements in the tuple (starting from $0$) and
assuming that qubit $1$ and $2$ are initially not connected; the left most
entry is the index of the operator applied first; the order of the generators
is compatible with the fact the operators for local complementation depend on
whether qubit $1$ and $2$ are connected.
## Appendix B Symplectic vector spaces and their connection to stabilizers
There is a close connection between stabilizer states and symplectic vector
spaces. In this section we recall some important results on symplectic vector
spaces, which have been studied extensively in the literature. For more
details we refer the reader to, e.g., Refs. [57, 59, 58].
Let $V$ be a finite-dimensional vector space over any field $F$. A bilinear
form is a map $\omega:V\times V\rightarrow F$ which is linear in both
arguments. After fixing a basis $\\{\bm{e}_{i}\\}$ for $V$, we can associate
to the bilinear form $\omega$ the matrix $B_{\omega}\in
M_{\text{dim}(V)\times\text{dim}(V)}(F)$ whose entries are given by
$(B_{\omega})_{ij}=\omega(\bm{e}_{i},\bm{e}_{j})$. The bilinear form may then
be written as
$\omega(\bm{f},\bm{g})=\bm{f}^{T}B_{\omega}\bm{g}.$ (53)
A bilinear form is called _alternating_ if $\omega(\bm{f},\bm{f})=0$ for all
$\bm{f}\in V$. Note that any alternating form is also skew-symmetric, i.e.,
$\omega(\bm{x},\bm{y})=-\omega(\bm{y},\bm{x})$, which follows from
$\omega(\bm{x}+\bm{y},\bm{x}+\bm{y})=0$ together with the bilinearity of
$\omega$. Consequently, any matrix representation of an alternating bilinear
form satisfies $(B_{\omega})_{ij}=-(B_{\omega})_{ji}$ and
$(B_{\omega})_{ii}=0$ for all $i,j\in[\text{dim}(V)]$. Any matrix with these
properties is called alternating.
Given an alternating form and a subspace $W\subseteq V$, we define the dual
space as $W^{\perp}=\\{\bm{f}\in V|\omega(\bm{f},\bm{g})=0\ \forall\bm{g}\in
W\\}$. An important type of subspace which appears in this context is a
_hyperbolic plane_. A hyperbolic plane is a $2$-dimensional subspace spanned
by a hyperbolic pair, which is a pair of vectors $\bm{f},\bm{g}\in V$ such
that $\omega(\bm{f},\bm{g})=1$. Given a vector space and an alternating form,
the vector space decomposes into a orthogonal direct sum of hyperbolic planes
and a subspace on which the form is $0$. This is stated in the following
theorem, a proof of which can for instance be found in Ref. [57].
###### Theorem 18.
Let $V$ be a vector space and let $\omega:V\times V\rightarrow F$ be an
alternating form. Then $V=V_{1}\oplus\ldots\oplus V_{m}\oplus V^{\perp}$ where
$V_{1},\ldots,V_{m}$ are hyperbolic planes for some $m\in\mathbb{N}$ and
$\omega(\bm{f},\bm{g})=0$ if $\bm{f}\in V^{\perp}$ or $\bm{g}\in V^{\perp}$.
This Theorem is equivalent to the following statement. For every alternating
form $\omega$ on a vector space $V$ there exists a basis of $V$ w.r.t. which
the matrix representation of $\omega$ is a direct sum of the blocks
$\begin{pmatrix}0\end{pmatrix},\text{ and }\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}.$ (54)
Stated differently, there always exists a basis change in $V$ described by the
invertible matrix $R$ such that $R^{T}B_{\omega}R$ is block-diagonal, with the
blocks as specified above. Note that for $V=\mathbb{Z}_{2}^{n}$ Theorem 18 is
also known as _Dickson’s Theorem_ (cf. Chapter 15 in Ref. [60]). We state it
here for general fields as, when we treat qudit stabilizer states with qudits
of prime dimension $d$, we need the Theorem for the field $\mathbb{Z}_{d}$.
Let us continue with some definitions. A bilinear form is called _non-
degenerate_ if the conditions $\omega(\bm{f},\bm{g})=0$ for all $\bm{g}\in V$
imply that $\bm{f}=0$. An alternating bilinear form which is non-degenerate is
called _symplectic_. A symplectic vector space is a vector space $V$ together
with symplectic bilinear form $\omega$. Symplectic vector spaces have even
dimension as follows for instance from Theorem 18. A subspace $W\subseteq V$
of a symplectic vector space $V$ is called _isotropic_ if $W\subseteq
W^{\perp}$ and _maximally isotropic_ if $W^{\perp}=W$. Moreover, for any
subspace $W\subseteq V$ it holds that
$\text{dim}(W)+\text{dim}(W^{\perp})=\text{dim}(V)$.
An invertible linear map $U:V\rightarrow V$ is referred to as _isometry_ for a
symplectic form $\omega$ if it preserves $\omega$, i.e., if
$\omega(U\bm{f},U\bm{g})=\omega(\bm{f},\bm{g})$ for all $\bm{f},\bm{g}\in V$,
or equivalently, if $U^{T}B_{\omega}U=B_{\omega}$. Such transformations are
also called _symplectic_ transformations. Symplectic vector spaces allow for
the extension of any isometry between subspaces to isometries on the whole
vector space, as stated in the following lemma which can be found in Ref.
[57].
###### Lemma 19 (Witt’s Lemma).
Let $V$ be a symplectic space with symplectic form $\omega:V\times
V\rightarrow\mathcal{F}$. Let $A,B$ be subspaces of $V$ and let
$U:A\rightarrow B$ be an invertible linear map which satisfies
$\omega(U\bm{f},U\bm{g})=\omega(\bm{f},\bm{g})$. Then $U$ can be extended to
an isometry of $V$.
Let us see how the above results connect to the Pauli group and stabilizer
states. As described in Section 2.5, the Pauli group $\mathcal{P}_{n}$ modulo
phases is isomorphic to the vector space $\mathbb{Z}_{2}^{2n}$. The
commutation relations of the Pauli operators define a symplectic form on
$\mathbb{Z}_{2}^{2n}$, i.e., the Pauli group modulo phases is isomorphic to a
symplectic vector space. This correspondence also exists for the qudit Pauli
group and $\mathbb{Z}_{d}^{2n}$ (see Section 4). A stabilizer corresponding to
a stabilizer state is a maximal abelian subgroup of $\mathcal{P}_{n}$ which
does not contain $-\mathds{1}$. Via the above isomorphism it corresponds to a
maximally isotropic subspace of $\mathbb{Z}_{2}^{2n}$.
For instance, consider the Bell state $\ket{\phi^{+}}$, whose stabilizer is
$\mathcal{S}=\\{\mathds{1},X_{1}Z_{2},Z_{1}X_{2},-Y_{1}Y_{2}\\}$. It is
straightforward to verify that any operator in $\mathcal{P}_{2}$ which
commutes with every element of $\mathcal{S}$ is already an element of
$\mathcal{S}$ up to a phase. The stabilizer $\mathcal{S}$ corresponds to the
subspace
$V^{\mathcal{S}}=\\{(0,0,0,0),(1,0,0,1),(0,1,1,0),(1,1,1,1)\\}\subset\mathbb{Z}_{2}^{4}$
via the homomorphism $\sigma:\mathbb{Z}_{2}^{4}\rightarrow\mathcal{P}_{n}$
defined by
$\displaystyle\sigma((1,0,0,0))$ $\displaystyle=X_{1}$ (55)
$\displaystyle\sigma((0,1,0,0))$ $\displaystyle=Z_{1}$ (56)
$\displaystyle\sigma((0,0,1,0))$ $\displaystyle=X_{2}$ (57)
$\displaystyle\sigma((0,0,0,1))$ $\displaystyle=Z_{2}.$ (58)
This subspace is maximally isotropic, i.e., if
$\bm{f}=(a_{1},b_{1},a_{2},b_{2})\in\mathbb{Z}_{2}^{4}$ such that
$\omega(\bm{f},\bm{g})=0$ for all
$\bm{g}=(a_{1}^{\prime},b_{1}^{\prime},a_{2}^{\prime},b_{2}^{\prime})\in
V^{\mathcal{S}}$ with respect to the symplectic form
$\omega(\bm{f},\bm{g})=(a_{1},a_{2},b_{1},b_{2})^{T}\begin{pmatrix}0&\mathds{1}_{2}\\\
\mathds{1}_{2}&0\end{pmatrix}(a_{1}^{\prime},a_{2}^{\prime},b_{1}^{\prime},b_{2}^{\prime}),$
(59)
then $\bm{f}\in V^{\mathcal{S}}$. It is straightforward to see that $\omega$
describes the commutation relations of the respective Pauli operators, for
example $\omega((1,0,0,0),(0,1,0,0))=1$ as $X_{1}Z_{1}=-Z_{1}X_{1}$ but
$\omega((1,0,0,0),(0,0,1,0))=0$ as $X_{1}X_{2}=X_{2}X_{1}$.
Clifford operators map Pauli operators to Pauli operators. As they preserve
the commutation relations of the Pauli operators Clifford operators correspond
to isometries on the symplectic vector space $(\mathbb{Z}_{2}^{2n},\omega)$.
On the contrary, any homomorphism between subgroups of the Pauli group which
preserves commutation relations and is invertible can be extended to an
isometry due to Lemma 19 and can be implemented by a Clifford operator (cf.
Equation (16)).
## Appendix C Properties of commutation matrices
In this section we are concerned with the rank condition (cf. Equation (31)),
which is necessary for commutation matrices. First, we prove Theorem 6 and
then show that the equality in Equation (31) is an inequality for general
tuples of alternating matrices. We note that the theorem also holds for any
finite field, and the proof follows simply by replacing $\mathbb{Z}_{2}$ with
$\mathbb{Z}_{d}$.
For convenience, let us restate Theorem 6.
###### Theorem 6.
Let $(C_{\alpha})_{\alpha\in P(M,n)}$, $C_{\alpha}\in M_{n\times
n}(\mathbb{Z}_{2})$ be a tuple of alternating matrices such that
$\sum_{\alpha}C_{\alpha}=0$ for some partition $P(M,n)$. Then there exists a
$\ket{\psi}\in\text{Stab}(P(M,n))$ such that $(C_{\alpha})_{\alpha}$ are the
corresponding commutation matrices if and only if
$2\,{\rm rk}([C_{\alpha}]_{\alpha})=\sum_{\alpha}{\rm rk}(C_{\alpha}).$ (60)
Here, $[C_{\alpha}]_{\alpha}$ is the $n\times(nm)$ matrix obtained by
concatenating all matrices $C_{\alpha}$, and the rank is taken over the field
$\mathbb{Z}_{2}$.
###### Proof.
First, we prove the only if part of the statement. Suppose that
$(C_{\alpha})_{\alpha}$, with $C_{\alpha}\in M_{n\times n}(\mathbb{Z}_{2})$,
correspond to an $n$ qubit stabilizer state. We want to prove that Equation
(60) holds. To begin with, let us consider the l.h.s. of Equation (60).
Observe that ${\rm rk}([C_{\alpha}]_{\alpha})={\rm
rk}([QC_{\alpha}Q^{T}]_{\alpha})$ for any invertible $Q\in M_{n\times
n}(\mathbb{Z}_{2})$. Consider a $Q$ such that $[QC_{\alpha}Q^{T}]_{\alpha}$
has the maximum number of zero rows. The number of zero rows is equal to the
number of generators of $\mathcal{S}$ that locally commute with all other
elements in the stabilizer. These generators correspond to qubits that can be
locally extracted, as they are not entangled with any other qubit outside of
their corresponding parties (see discussion above Theorem 8). For convenience,
let us now assume that those states have been extracted, so that ${\rm
rk}([C_{\alpha}]_{\alpha})=N\leq n$ is of full rank, i.e., equal to the number
$N$ of qubits that are entangled with at least one other qubit outside of
their own party. Next, let us consider the r.h.s of Equation (60), and let us
establish the connection between the ranks of the commutation matrices and the
ranks of the corresponding reduced states. Recall, that for stabilizer states
the reduced states are given by Equation (3), i.e., by the sum over all
elements $s\in\mathcal{S}$ with their support fully contained in $\alpha$.
Consider the matrix $C_{\alpha}$. Due to Theorem 18 there exists an invertible
matrix $Q$ such that $QC_{\alpha}Q^{T}$ is a direct sum of blocks of the form
as in Equation (54). We observe that whenever a generator anti-commutes with
another generator on party $\alpha$ they also have to anti-commute on at least
one other party (as the generators have to commute as a whole), and hence,
every nonzero block increases the rank of the reduced state by a factor of
two, and thus, $\text{rk}(\varrho_{\alpha})=2^{\text{rk}(C_{\alpha})/2}$. From
this equation it follows that $\text{rk}(C_{\alpha})/2$ is the number of
qubits in $\alpha$ that are entangled with at least one other qubit in another
party. Thus, $\sum_{\alpha}\text{rk}(C_{\alpha})/2$ is equal to the total
number $N$ of entangled qubits, and thus, Equation (60) holds.
The if part of the statement follows directly form the construction below
Theorem 6 in the main text. ∎
Next, let us show that Equation (31) is an inequality for general sets of
alternating matrices. We show the following theorem over any field $F$ as we
use it later not only for $\mathbb{Z}_{2}$ but any $\mathbb{Z}_{d}$ with $d$ a
prime number.
###### Theorem 20.
Let $\\{A_{\alpha}\\}_{\alpha\in[M]}\subset M_{n\times n}(F)$ with
$M\in\mathbb{N}$ be a finite set of alternating matrices such that
$\sum_{\alpha}A_{\alpha}=0$. Then
$2\,{\rm rk}([A_{\alpha}]_{\alpha\in[M]})\leq\sum_{\alpha\in[M]}{\rm
rk}(A_{\alpha})$ (61)
where $[A_{\alpha}]_{\alpha\in[M]}\in M_{n\times(kn)}(F)$ is the matrix
obtained by concatenating the matrices $\\{A_{\alpha}\\}_{\alpha\in[M]}$.
To prove Theorem 20 we use the following Lemma shown in Ref. [61].
###### Lemma 21.
Let $A,B\in M_{n\times m}(F)$. Then
${\rm rk}([A,B])+{\rm rk}\begin{pmatrix}A\\\ B\end{pmatrix}\leq{\rm
rk}(A)+{\rm rk}(B)+{\rm rk}(A+B).$ (62)
###### Proof of Theorem 20.
Following Ref. [61], we denote in the following for any matrix $A\in
M_{k\times l}(F)$, $k,l\in\mathbb{N}$ by $\overline{A}\in M_{k\times
k}(\mathbb{Z}_{2})$ the orthogonal projector on the columnspace of $A$.
We prove the statement via induction. For $M=3$ the statement is equivalent to
Lemma 21 and holds. Suppose the statement holds for all $M$ from $3$ to $N$.
Let us show that then it also holds for $M=N+1$.
Let $\\{A_{\alpha}\\}_{\alpha\in[N+1]}\subset M_{n\times n}(F)$ be a set of
alternating matrices such that $\sum_{\alpha}A_{\alpha}=0$. We want to show
that Inequality (61) holds. To this end, let us first show another inequality
that will be helpful in the proof. In the following we write $C_{X}$ for the
column space of the matrix $X$. Then, we find that
$\displaystyle{\rm
rk}((\mathds{1}-\overline{[A_{\alpha}]_{\alpha\in[N-1]}})A_{N})+\text{dim}(C_{\sum_{\alpha=1}^{N-1}A_{\alpha}}\cap
C_{A_{N}})$ (63) $\displaystyle\leq$ $\displaystyle{\rm
rk}((\mathds{1}-\overline{[A_{\alpha}]_{\alpha\in[N-1]}})A_{N})+\text{dim}(C_{[A_{\alpha}]_{\alpha\in[N-1]}}\cap
C_{A_{N}})={\rm rk}(A_{N}).$ (64)
The inequality follows from the fact that
$C_{\sum_{\alpha=1}^{N-1}A_{\alpha}}\subseteq
C_{[A_{\alpha}]_{\alpha\in[N-1]}}$. The last equality can be seen as follows.
First observe that the equality holds if and only if it holds for $A_{N}R$,
where $R$ is an invertible matrix. Let $f_{j}=f_{j}^{\perp}\oplus f_{j}^{||}$
for any $j$ be the columns of $A_{N}R$ where
$(\mathds{1}-\overline{[A_{\alpha}]_{\alpha\in[N-1]}})f_{j}^{\perp}=f_{j}^{\perp}$
and $(\mathds{1}-\overline{[A_{\alpha}]_{\alpha\in[N-1]}})f_{j}^{||}=0$. We
choose $R$ such that there exists a $k\leq n$ such that $f_{j}^{\perp}=0$ for
all $j>k$ and $f_{1}^{\perp},\ldots,f_{k}^{\perp}$ are independent. Then, |
# Guiding GANs: How to control non-conditional pre-trained GANs for
conditional image generation
Manel Mateos, Alejandro González111Corresponding author:
<EMAIL_ADDRESS>Xavier Sevillano GTM - Grup de Recerca en
Tecnologies Mèdia. La Salle - Universitat Ramon Llull
###### Abstract
Generative Adversarial Networks (GANs) are an arrange of two neural networks
–the generator and the discriminator– that are jointly trained to generate
artificial data, such as images, from random inputs. The quality of these
generated images has recently reached such levels that can often lead both
machines and humans into mistaking fake for real examples. However, the
process performed by the generator of the GAN has some limitations when we
want to condition the network to generate images from subcategories of a
specific class. Some recent approaches tackle this conditional generation by
introducing extra information prior to the training process, such as image
semantic segmentation or textual descriptions. While successful, these
techniques still require defining beforehand the desired subcategories and
collecting large labeled image datasets representing them to train the GAN
from scratch. In this paper we present a novel and alternative method for
guiding generic non-conditional GANs to behave as conditional GANs. Instead of
re-training the GAN, our approach adds into the mix an encoder network to
generate the high-dimensional random input vectors that are fed to the
generator network of a non-conditional GAN to make it generate images from a
specific subcategory. In our experiments, when compared to training a
conditional GAN from scratch, our guided GAN is able to generate artificial
images of perceived quality comparable to that of non-conditional GANs after
training the encoder on just a few hundreds of images, which substantially
accelerates the process and enables adding new subcategories seamlessly.
###### keywords:
Neural Network , Generative Adversarial Networks , Conditional image
generation , Guiding process , Encoder Networks
††journal: Neural Networks
## 1 Introduction
The generation of artificial data that follows real distributions has
encouraged the computer science community to develop generative algorithms
that aim to create data as indistinguishable as possible from real data.
Applications range from the generation of missing data for incomplete datasets
[1] to coherent text generation [2], among many others.
Recently, the computer vision community has focused on the generation of real-
looking artificial images, and a specific type of neural networks called
generative adversarial networks (GANs) have attained remarkable performance in
this area [3]. In GANs, models are built with two neural networks: the
generator, which is a convolutional neural network (CNN) trained to generate
images that mimic the distribution of the training dataset, and the
discriminator, which tries to distinguish between real images and fake images
generated by the generator. The process of the generator trying to fool the
discriminator leads to a joint learning process that allows to effectively
generate fake real-looking images similar to those in the training set.
Despite their success, giving users total control on the characteristics of
the images generated by a GAN is still an issue to be solved. Traditionally,
researchers have used additional information as inputs in the generator
network training to condition the generation process [4]. This gives rise to
conditional GANs, which are able to generate artificial images with certain
specific, desired characteristics. Many authors in recent years have explored
different ways of including this conditional information, be it through image
textual descriptions [5], semantic segmentations [6] or image category
definition [7], among others. However, all these approaches set the number and
definition of characteristics before the GAN training process and cannot be
changed later. Thus, for any addition or variation, a new labeled dataset must
be collected, and the GAN must be re-trained, making the whole process complex
and time-consuming.
In the face of these limitations of current conditional GANs, in this paper we
present a novel and efficient way of guiding pre-trained non-conditional GANs
for generating images belonging to specific subcategories avoiding burdensome
re-training processes. In Figure 1, the general scheme of the proposed method
is depicted.
Figure 1: Guiding GANs process: the user collects a small sample of images
corresponding to the specific subcategory. Based on this sample, the encoder
network generates the subcategory prototype vector that represents the
distribution of the sample. This vector is then randomly sampled to create
random vectors that are fed to the pre-trained non-conditional GAN to generate
images of the desired subcategory.
The key element of our proposal is an encoder network, which is first trained
to learn the opposite transformation of the pre-trained non-conditional GAN.
To do so, pairs of random vectors and the corresponding images generated by
the GAN are employed.
Then, the guiding process starts with the user collecting a small sample of
non-annotated real images as examples of the specific image subcategory he/she
wants the pre-trained non-conditional GAN to generate. These images are fed to
the encoder network, which estimates the distribution of the selected
subcategory, and embeds it in the so-called subcategory prototype vector. By
randomly sampling the prototype vector, we create multiple random vectors
which are fed to the pre-trained generator network of the GAN to produce
artificial images similar to the previously selected ones. By proceeding this
way, we enable users to obtain real-looking artificial images at a reduced
computational cost.
This paper is organized as follows: in Section 2 the theoretical background of
our proposal, as well as relevant previous work is reviewed. Then, Section 3
describes the proposed method. Next, in Section 4, our proposal is evaluated
in a series of experiments. Finally, Section 5 discusses the obtained results,
highlights the advantages of our method and outlines future research lines.
## 2 State of the art
The image generation problem has pushed researchers to find the most optimal
frameworks and models to produce real-looking artificial images.
In addition to GANs [8], other relevant approaches include adversarial auto-
encoders (AAEs) [9], variational auto-encoders (VAEs) [10] and auto-regression
models (ARMs) (e.g. PixelRNN [11]).
As GANs are probably the most widely employed technique for the conditional
generation of high resolution artificial images, this section is focused on
reviewing the basic concepts of GANs (Section 2.1), as well as common
approaches to GAN-based increased resolution image generation (Section 2.2)
and proposals on conditional generation image generation via GANs (Section
2.3).
### 2.1 Artificial image generation via GANs
GANs are a subset of implicit density generative models that focus on
transforming a random input into an image which aims to be part of the true
distribution of the training data set [8]. As mentioned earlier, the GAN
models are built through the interaction of two neural networks, the generator
and the discriminator, that together learn how to generate new real-looking
images effectively by trying to confuse one another. On the one hand, the
generator network (G) is a CNN that takes a random vector from a distribution
and maps it through the model to an output with the desired sample size. Its
objective is to produce images that look as similar as possible to the
training examples. On the other hand, the discriminator (D) is a binary
classification CNN that predicts if a given image is a real one (i.e. a
training sample) or a fake one (generated by G). In this sense, the GAN
training process can be understood as a two-player game where G tries to fool
D by generating real-looking images and D tries to distinguish between real
and fake images, as shown in Figure 2.
Figure 2: Generic GAN architecture. Generator, discriminator and their
connections.
These two models are trained jointly minimizing the possible loss for a worst-
case scenario by using as the objective function:
$\min\limits_{G}\max\limits_{D}V(D,G)=E_{x\sim p_{data}(x)}[logD(x)]+E_{z\sim
p_{z}(z)}[log(1-D(G(z)))]$, where $D(x)$ is the discriminator output for the
real data sample $x$, and $D(G(z))$ is the discriminator output for the
artificial data generated by the generator with $z$ as the random input
vector.
During the training process, the discriminator tries to maximize the objective
function by making $D(x)$ close to 1 and $D(G(z))$ close to 0, while the
generator focuses on minimizing it such that $D(G(z))$ is close to 1, fooling
the discriminator.
### 2.2 Increasing artificial image resolution via progressive growing GANs
The generation of high resolution and complex images via GANs requires up-
scaling their architecture. In this sense, higher resolutions imply some
additional challenges, such as i) gradients during training become useless,
generating poor images easily identifiable as fake, and therefore making the
training process fail, ii) memory constraints increase, forcing researchers to
reduce batch sizes, which compromises the stability of the training process,
and iii) better hardware acceleration is needed to train these bigger models
and handle them efficiently.
Figure 3: Progressive GANs training process. The training process for G and D
starts at a low-resolution ($4\times 4$ pixels) and is gradually adapted to
higher resolutions by adding layers to G and D. At each training step all
existing layers remain trainable. Added layers are convolutional layers
operating on $N\times N$ spatial resolution.
These problems forced the research community to formulate new approaches to
scale up the resolution of the generated images successfully. Some authors
like Salimans et al. [12] and Gulrajani et al. [13] presented improvements on
the GAN training process, and others like Berthelot et al. [14] and Kodali et
al. [15] proposed new GAN architectures.
An alternative and interesting approach are Progressive Growing GANs (PG-GANs)
[16], which consist in gradually training GANs and iteratively adapting them
to higher resolution images on each step of the training. The authors propose
starting the training process on low-resolution images and then gradually
increasing the resolution by adding layers to both G and D (see Figure 3).
This procedure implies that G and D must be symmetrical and grow
synchronously. The method is based on the premise that CNNs are capable of
first learning large general features, generalizing the training images, and
the addition of more layers allows to move into the finer details. According
to Karras et al., the use of this progressive growing approach i) reduces GANs
training time, ii) improves convergence, as low-resolution neural networks are
stable and the progressive increase of image resolution allows starting the
higher resolution training process with a stable pre-trained network, and iii)
introduces an adaptation degree in terms of resolution, allowing the control
of the training process for obtaining artificial images of a given desired
resolution. Due to this flexibility and reduced training time, we adopt PG-
GANs in our work to generate artificial images.
### 2.3 Conditional image generation with GANs
Another issue worth considering is how to control the generation process
beyond the training dataset, a process that goes by the name of conditional
generation.
In this context, authors have developed different approaches based on training
the GANs not only with uncategorized images but also introducing categorical
information of the images included in the training dataset. For instance,
works like [5] used the categories of the training images as extra features in
the generation and discrimination processes.
Other authors addressed the problem by using both categorical information of
the training images and also their semantic segmentation, as in [6, 7, 17,
18]. In those works, the authors train the discriminator to distinguish real
and fake images and at the same time, to match the objective pixel wise
semantic information given.
An example of a use case for conditional GANs consists on generating human
body images simulating specific body poses. In this context, some authors
proposed new architectures for the training of GANs, which receive the body
poses of the training images as an extra feature. [19, 20, 21], once the GAN
is trained users may generate images where the pose is freely chosen.
The following section describes our proposal, which guides non-conditional PG-
GANs for generating images of a specific subcategory within the training set,
at will and without the need of retraining the GAN models.
## 3 Guiding non-conditional pre-trained GANs
Our proposal is based on considering the $d$-dimensional input space of random
vectors that feed the generator network G of a non-conditional GAN once it is
trained to generate images of a specific category $C$, which we refer to as
$\mathrm{G}_{C}$. Without loss of generality, any category $C$ is
intrinsically composed of multiple (say $m$) subcategories $SC_{i}$, that is
$C=\displaystyle\bigcup_{i=1}^{m}{SC_{i}}$.
In response to these random input vectors, $\mathrm{G}_{C}$ generates images
corresponding to the category represented in the training set, but no control
mechanism is available to “tell” $\mathrm{G}_{C}$ to generate images of a
specific subcategory $SC_{k}$. In our method, we propose hand-picking the
random vector input to $\mathrm{G}_{C}$ to produce images belonging to the
desired subcategory, thus giving the user total control over the artificial
image generation process.
Our method is described step by step in the following paragraphs. Please refer
to Figure 1 for a graphical reference.
Figure 4: Encoder training.
Step 1) Encoder training: an encoder network is trained to learn the opposite
transformation from the one carried out by the trained generator
$\mathrm{G}_{C}$. To that end, the encoder is trained using pairs of
$d$-dimensional random input vectors and their correspondent generated images
directly extracted from the non-conditional generator network
$\mathrm{G}_{C}$, therefore making the training data supply virtually endless
(see Figure 4). As a result, we obtain an encoder model that given an image
generated by $\mathrm{G}_{C}$ returns the $d$-dimensional input vector which
would have created that image. After finishing the training process, the
encoder is capable of successfully returning input vectors from random images
not produced by the generator.
Step 2) Subcategory random vectors generation: The user collects real images
of the desired subcategory $SC_{k}$ and feeds the trained encoder with them.
This image collection process can be fully automated, as described in Section
4. In response, the encoder returns a $d$-dimensional random vector
$\vec{x}_{i}^{SC_{k}}$ corresponding to each input image. Notice that the
larger the number of collected images (referred to as $N$), the more accurate
the estimation of the distribution of the desired subcategory. Moreover, it is
also to note that the user can decide to add a new subcategory, and the
obtainment of the corresponding vectors through the encoder can be started at
any point.
Step 3) Subcategory prototype vector creation and sampling: the mean value and
standard deviation of each of the $d$ components of the vectors
$\vec{x}_{i}^{SC_{k}}$ ($\forall i=1...N$) output by the encoder in response
to the images of the desired subcategory $SC_{k}$ are computed and embedded in
the subcategory prototype vector $\vec{p}^{SC_{k}}$. Next, this prototype
vector is used to generate as many random vectors as desired by sampling $d$
normal random variables $X_{j}\sim
N\left(\mu_{j},\alpha\cdot\sigma_{j}\right)$ (with $\forall j=1..d$), where
$\mu_{j}$ and $\sigma_{j}$ are the mean value and the standard deviation of
the $j$th component of the vectors $\vec{x}_{i}^{SC_{k}}$ ($\forall i=1...N$),
and $\alpha$ is a scalar parameter. In our experiments, we heuristically tuned
the value of this parameter to 2.5. These random vectors follow the
distribution of the desired subcategory $SC_{k}$, so they will make the pre-
trained generator network $\mathrm{G}_{C}$ generate images belonging to the
specific subcategory of choice.
Figure 5: Training real examples of the “mountains” category used to train the
non-conditional progressive growing GAN
## 4 Experiments and results
The experiments described in this section aim to evaluate our method to guide
a non-conditional progressive growing GAN.
We start by describing the data employed in our experiments. Subsequently, we
present the architecture of the PG-GAN, and an experiment involving a
subjective quality evaluation test to assess the quality of the images it
generates.
In the final experiment, we guide the non-conditional PG-GAN to generate
images from specific subcategories of choice. We describe the architecture of
the encoder network employed in the experiments, and then evaluate i) the
ability of the non-conditional network to effectively generate images that
correspond to the chosen subcategories, and ii) the perceived quality of the
generated images.
### 4.1 Dataset
The non-conditional PG-GAN was trained to generate images of the category $C=$
“mountains”. To that end, a total of 19.765 images of mountains were
downloaded from the Flickr image hosting service and used to train the GAN.
Some example images from the training dataset are presented in Figure 5.
On the other hand, to train the encoder network, we created 500.000 random
vectors, fed them to the pre-trained non-generic PG-GAN, and collected the
corresponding images.
The images used for guiding the non-conditional GAN to generate images from a
specific subcategory were obtained using the Flickr API, downloading $N$
images that were tagged as one of the following selected subcategories
$SC_{k}=\\{$“mountains + snow”, “mountains + sunset”, “mountains + trees”,
“mountains + night” and “mountains + rocks”$\\}$.
### 4.2 Non-conditional progressive growing GAN
#### 4.2.1 Architecture
The PG-GAN used in these experiments follows the architecture presented by
Karras et al. in [16] and shown in Figure 3.
In a nutshell, the model starts training at a resolution of 4x4 and progresses
until reaching a final resolution of 128x128.
The architecture of both the generator and the discriminator are based on
strided convolutions with leakyReLU activations and constrain the signal
magnitude and competition during training through pixel wise feature
normalization and equalizing the learning. The whole model has over 45 million
parameters and was trained on Google Colab for 200 epochs.
Figure 6 shows several examples of the artificial images of the mountain
category generated by the GAN. All examples portray high fidelity and
variance, successfully capturing the true distribution of the images provided
during the training of the model.
Figure 6: Examples of the “mountain” category artificial images generated by
the non-conditional progressive GAN Figure 7: Normalized histogram of the
scores given by the participants in the subjective quality evaluation test of
the non-conditional progressive GAN
#### 4.2.2 Artificial image quality evaluation
To evaluate the quality of the images generated by the non-conditional PG-GAN,
we carried out a subjective quality evaluation test, in which 50 participants
were asked to evaluate the degree of realism of 20 artificial images using a
rating scale from 0 to 10 (the greater the score, the greater the realism).
The normalized histogram of the obtained ratings is depicted in Figure 7. The
left skewed distribution of scores reveals that the participants judged most
of the images as quite realistic, obtaining an mean opinion score of 6.3.
### 4.3 Guiding the non-conditional GAN
The architecture of the encoder network used to guide the non-conditional GAN
is presented in Figure 8.
Figure 8: Architecture of the encoder network
As mentioned earlier, the encoder was trained on 500.000 pairs of random
vectors and the corresponding artificial images generated by the non-
conditional PG-GAN. The training took 4 epochs to converge.
The computation of the subcategory prototype vector was made after
programmatically downloading a variable number $N$ of images from Flickr
corresponding to the desired subcategories that were described in section 4.1.
The experiments to evaluate the quality of the images generated by the guided
non-conditional PG-GAN are presented next.
#### 4.3.1 Effect of $N$ on the perceived quality of the images
First, we evaluated how the number of images fed to the encoder to create the
subcategory prototype vector affects the quality of the images that are
subsequently generated by the guided non-conditional PG-GAN.
To that end, we presented 50 participants with images generated when the
subcategory prototype vector was computed after feeding the encoder network
with $N=\\{64,128\,\mathrm{and}\,256\\}$ images.
Figure 9: Examples of images of the “mountains+snow”, “mountains+sunset”,
“mountains+trees”, “mountains+night” and “mountains+rocks” subcategories
generated by the guided non-conditional “mountain” GAN
The mean opinion score for these configurations was 5.9, 6.2 and 6.4,
respectively. Taking into account that the subjective quality evaluation of
the images created by the non-conditional progressive growing GAN yielded a
mean opinion score of 6.3, these results prove that using a few hundreds of
images corresponding to the desired subcategory suffices to generate images of
that subcategory with an equivalent level of perceived quality.
To illustrate this fact, Figure 9 presents images generated by the guided GAN
when asked to create images of the subcategories mentioned earlier with
$N=256$. It can be observed that the network succeeds in generating images of
the specific subcategory.
#### 4.3.2 Subcategory identification
In this experiment, we evaluate whether the participants in the subjective
evaluation test were able to correctly identify the subcategory of the images
generated by the guided GAN.
The experiment consisted of presenting the participants with 20 images that
had to be classified in the (“mountains+”) “snow”, “sunset”, “trees”, “night”
or “rocks” subcategories.
In average, the participants successfully chose the correct subcategory with a
85.2% accuracy. The confusion matrix corresponding to this experiment is
presented in Table 1. Notice that the “snow” and “sunset” subcategories are
identified close to perfection, while the “tree” subcategory is identified
with a 56.6% accuracy.
| | Predicted class
---|---|---
| | Snow | Sunset | Trees | Rocks | Night
Actual class | Snow | 99.5 | 0 | 0 | 0 | 0.5
Sunset | 0 | 99.5 | 0.5 | 0 | 0
Trees | 1 | 0.5 | 56.5 | 42 | 0
Rocks | 2 | 0.5 | 7 | 91 | 0
Night | 1 | 8 | 9 | 2.5 | 79.5
Table 1: Confusion matrix of the subcategory identification experiment (values
in %).
## 5 Conclusions
This work has introduced a novel method that gives users control over the
specific type of images generated by GANs. Our proposal enables the generation
of artificial images from a user-defined subcategory, guiding a non-
conditional GAN thanks to a new architecture that includes an encoder network
to feed the GAN.
This novel process transforms the conditional image generation problem into a
simpler task for general users, reaching a flexibility level that cannot be
reached by a non-conditional GAN.
Our proposal allows to considerably reduce the time needed to perform
conditional image generation, while maintaining similar results in terms of
artificial image quality. Additionally, since only a small set of images of
the desired subcategory is needed to guide the GAN, the process can be fully
automated. Moreover, the proposed method enables the user to select the
desired image subcategory on the go, which allows new ideas to be tested in
minutes, much faster than the time that would be required to train a new
regular, non-generic GAN from scratch.
Moving forward, we believe the subcategory prototype vector creation process
described in Section 3 could be further improved to better represent the
subcategory’s distribution, which would help the generator network yield more
variance between the images belonging to a single subcategory. Additionally,
studying how input vectors are transformed throughout the generator process,
and specifically trying to understand how dependent the perceived subcategory
is to each step of the network, could help better guide the model by not only
feeding it the right vector, but also further ”steering” the generation
process into the desired direction.
## References
* Hartley [1958] H. O. Hartley, Maximum likelihood estimation from incomplete data, Biometrics 14 (1958) 174–194.
* Roh and Lee [2003] J. Roh, J.-H. Lee, Coherent text generation using entity-based coherence measures, in: Advances in Computation of Oriental Languages–Proceedings of the 20th International Conference on Computer Processing of Oriental Languages, 2003.
* Alqahtani et al. [2019] H. Alqahtani, M. Kavakli-Thorne, G. Kumar, Applications of generative adversarial networks (gans): An updated review, Archives of Computational Methods in Engineering (2019) 1–28.
* Dai et al. [2017] B. Dai, S. Fidler, R. Urtasun, D. Lin, Towards diverse and natural image descriptions via a conditional gan, in: Proceedings of the IEEE International Conference on Computer Vision, 2017, pp. 2970–2979.
* Chang et al. [2019] C.-H. Chang, C.-H. Yu, S.-Y. Chen, E. Y. Chang, Kg-gan: Knowledge-guided generative adversarial networks, arXiv preprint arXiv:1905.12261 (2019).
* Tang et al. [2020] H. Tang, X. Qi, D. Xu, P. H. Torr, N. Sebe, Edge guided gans with semantic preserving for semantic image synthesis, arXiv preprint arXiv:2003.13898 (2020).
* Qi et al. [2018] X. Qi, Q. Chen, J. Jia, V. Koltun, Semi-parametric image synthesis, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 8808–8816.
* Goodfellow et al. [2014] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, Y. Bengio, Generative adversarial nets (2014) 2672–2680.
* Makhzani et al. [2015] A. Makhzani, J. Shlens, N. Jaitly, I. Goodfellow, B. Frey, Adversarial autoencoders, arXiv preprint arXiv:1511.05644 (2015).
* Kingma and Welling [2013] D. P. Kingma, M. Welling, Auto-encoding variational bayes, arXiv preprint arXiv:1312.6114 (2013).
* Oord et al. [2016] A. v. d. Oord, N. Kalchbrenner, K. Kavukcuoglu, Pixel recurrent neural networks, arXiv preprint arXiv:1601.06759 (2016).
* Salimans et al. [2016] T. Salimans, I. Goodfellow, W. Zaremba, V. Cheung, A. Radford, X. Chen, Improved techniques for training gans, in: Advances in neural information processing systems, 2016, pp. 2234–2242.
* Gulrajani et al. [2017] I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, A. C. Courville, Improved training of wasserstein gans, in: Advances in neural information processing systems, 2017, pp. 5767–5777.
* Berthelot et al. [2017] D. Berthelot, T. Schumm, L. Metz, Began: Boundary equilibrium generative adversarial networks, arXiv preprint arXiv:1703.10717 (2017).
* Kodali et al. [2017] N. Kodali, J. Abernethy, J. Hays, Z. Kira, How to train your dragan, arXiv preprint arXiv:1705.07215 2 (2017).
* Karras et al. [2017] T. Karras, T. Aila, S. Laine, J. Lehtinen, Progressive growing of gans for improved quality, stability, and variation, arXiv preprint arXiv:1710.10196 (2017).
* Bau et al. [2019] D. Bau, J.-Y. Zhu, J. Wulff, W. Peebles, H. Strobelt, B. Zhou, A. Torralba, Seeing what a gan cannot generate, in: Proceedings of the IEEE International Conference on Computer Vision, 2019, pp. 4502–4511.
* Park et al. [2019] T. Park, M.-Y. Liu, T.-C. Wang, J.-Y. Zhu, Semantic image synthesis with spatially-adaptive normalization, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2019, pp. 2337–2346.
* Tang et al. [2019] H. Tang, D. Xu, G. Liu, W. Wang, N. Sebe, Y. Yan, Cycle in cycle generative adversarial networks for keypoint-guided image generation, in: Proceedings of the 27th ACM International Conference on Multimedia, 2019, pp. 2052–2060.
* Dong et al. [2018] H. Dong, X. Liang, K. Gong, H. Lai, J. Zhu, J. Yin, Soft-gated warping-gan for pose-guided person image synthesis, in: Advances in neural information processing systems, 2018, pp. 474–484.
* Ma et al. [2018] L. Ma, Q. Sun, S. Georgoulis, L. Van Gool, B. Schiele, M. Fritz, Disentangled person image generation, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 99–108.
|
# The Role of SARS-CoV-2 Testing on Hospitalizations in California
J. Cricelio Montesinos-López Maria L. Daza–Torres Yury E. García Luis A.
Barboza Fabio Sanchez Alec J. Schmidt Brad H. Pollock Miriam Nuño
###### Abstract
The rapid spread of the new SARS-CoV-2 virus triggered a global health crisis
disproportionately impacting people with pre-existing health conditions and
particular demographic and socioeconomic characteristics. One of the main
concerns of governments has been to avoid the overwhelm of health systems. For
this reason, they have implemented a series of non-pharmaceutical measures to
control the spread of the virus, with mass tests being one of the most
effective control. To date, public health officials continue to promote some
of these measures, mainly due to delays in mass vaccination and the emergence
of new virus strains. In this study, we studied the association between
COVID-19 positivity rate and hospitalization rates at the county level in
California using a mixed linear model. The analysis was performed in the three
waves of confirmed COVID-19 cases registered in the state to September 2021.
Our findings suggests that test positivity rate is consistently associated
with hospitalization rates at the county level for all waves of study.
Demographic factors that seem to be related with higher hospitalization rates
changed over time, as the profile of the pandemic impacted different fractions
of the population in counties across California.
## 1 Introduction
The SARS-CoV-2 virus, responsible for the novel coronavirus disease
(COVID-19), was identified in late December 2019 in Wuhan, China [1], and
spread rapidly, causing a global health crisis. As of October 5, 2021, more
than 235 million cases and 4,812,221 deaths have been confirmed worldwide [2].
As the pandemic spread across the globe, governments started to enforce public
policies to suppress SARS-CoV-2 transmission, including social distancing,
contact tracing, stay-at-home orders, school closings, limits public space
utilization, and border closures [3, 4]. To date, public health officials
continue to promote some of these non-pharmaceutical measures, mainly due to
delays in mass vaccination and the growing number of new COVID-19 variants
[5]. Mass surveillance testing, efforts of isolation, quarantine, and contact
tracing became essential control measures for curtailing the burden of the
COVID-19 pandemic [6]. The successful epidemic control measures taken by
countries such as Korea, Taiwan, Japan, China, New Zealand, and the Czech
Republic, which emphasized high testing rates during the initial stages of the
pandemic, supported the proposal that mass surveillance testing could help
limit viral transmission when properly leveraged [7, 8, 9, 10, 11]. However,
it remains unknown which testing strategies are the best and whether different
approaches show significant and measurable effects on viral spread in general
and the rates of severe or deadly cases in particular [12]. Although
population-scale testing is proven to reduce SARS-CoV-2 [13], it appears to
become less effective as viral prevalence decreases and is insufficient to
eliminate viral transmission on its own [14, 15].
Public health officials commonly use the test positivity rate to infer the
adequacy of population-level testing and the rate of COVID-19 transmission in
a population [16]. A low test positivity rate indicates low viral prevalence
and a testing program with sufficient surveillance capacity. In contrast, a
high test positivity rate suggests that the amount of testing is insufficient
and that many infected people go unnoticed, especially when test positivity
rates are higher than the expected prevalence [17]. Implementing mass testing
may also lead to fewer hospitalizations by reducing new infections by offering
interventions for symptomatic and asymptomatic cases discovered early [18, 19,
6]. Hospitalization is also influenced by the demographic structure of the
population and health care system factors. In theory, a public health system
that is better prepared to identify and support the isolation of cases
discovered by surveillance testing and treat those who require medical care
should result in lower hospitalizations rates.
On January 26, 2020, the first documented case of COVID-19 in California
occurred in Orange County [20]. Since then, the state government has
implemented a variety of strategies to contain the spread of the virus [21].
On March 4, 2020, California declared a state of emergency, followed by a
mandatory statewide stay-at-home order on March 19, 2020. On June 18, 2021, a
statewide mask mandate was ordered due to the rising number of cases and
deaths. These mandates were in force until June 15, 2021, when California
started reopening the economy [22], with $70\%$ of eligible with at least one
dose of the COVID-19 vaccine and more than $40\%$ of the population fully
vaccinated [23]. As of September 22, 2021, California has had three COVID-19
case waves. The first peak occurred in mid-July 2020, reaching an average of
10,000 new cases per day (first wave, May-Sep 2020) [24]. During this first
wave, most infections were geographically concentrated in the Central Valley,
primarily dominated by agriculture, manufacturing services, and retail,
meaning few residents could make the transition to working from home [25]. In
Autumn 2020, COVID-19 cases spiked again, to a peak of 40,000 new cases per
day at the end of December (second wave Nov-Jan 2021). During this wave, Los
Angeles was one of the main epicenters of the pandemic [26, 27]. The third
wave associated with the SARS-CoV-2 delta variant started in mid-June 2021
after the lifting of the statewide stay-at-home order. By mid-September, the
number of reported daily COVID-19 infections were decreasing, and as of
September 20, 2021, California reported the lowest coronavirus state
incidences case rate in the U.S. [28, 29]
In this paper, we aim to provide an exploratory data analysis to verify how
demographics and positivity rate correlate with COVID-19 hospital admission in
California. The analysis was performed in each of the three waves, using a
mixed linear model and data related to hospitalizations for COVID-19, age,
race, ethnicity, poverty, and mobility. The paper is organized as follows:
Section 2 describes the data and the methods implemented for the analysis,
section 3 describes the results, and section 4 highlights the main results and
conclusions.
## 2 Materials and Methods
The main goal of this analysis is to describe the effect of surveillance
testing on hospitalizations for COVID-19. We performed a comparative analysis
using a mixed linear model to study the relationship between hospitalization
rates for COVID-19 and positive cases, diagnostic tests, mobility, age, race
and ethnic group, poverty, and education across the counties of California.
Sixteen of the fifty-eight counties were excluded from this analysis:
Calaveras, Colusa, Del Norte, Glenn, Inyo, Lassen, Mariposa, Modoc, Mono,
Plumas, San Benito, and Siskiyou, due to low quality of hospital and mobility
data; and rural counties like Alpine, Sierra, Sutter, and Trinity because they
do not have hospital wards, so patients from those counties would go to
neighboring counties for COVID-19 medical care.
We analyzed three waves according to the three primary outbreaks reported in
California [30]. We defined the first wave period from April 21, 2020, to
September 30, 2020; the second wave starts on October 1 and ends on February
28, 2021, and the third starts on March 1 and ends on September five, 2021
(Figure 1). We implemented a change-point analysis to select the different
waves intervals.
Figure 1: Confirmed cases (7-day moving average) and the number of patients
hospitalized in an inpatient bed who have laboratory-confirmed COVID-19 in
California.
### 2.1 Data Sources
Several data sources were used for this study. We used publicly available
epidemiological data for COVID-19 daily reported cases and hospitalization
admissions at the county level from the official website of the California
Department of Public Health (CDPH) [31]. Data from the American Community
Survey (ACS) [32] estimates characteristics at the county level for age and
race or ethnic group. We used the Healthy Places Index (HPI) to account for
community-level factors contributing to social vulnerability. The HPI is
produced by the Public Health Alliance of Southern California, which combines
twenty-five community characteristics (e.g., the number of people living below
the poverty line, the number of people with lower levels of education, areas
with more renters and fewer homeowners, among others) into a single index
value to account for the level of poverty, education, and life expectancy in a
particular community [33]. The degree of intra-community mobility was produced
from Google’s Community Mobility Reports [34]. Six Google-specific data
streams (grocery and pharmacy, parks, residential, retail and recreation,
transit stations, and workplaces) were combined to obtain a single mobility
measure for the county using principal component analysis (PCA) (see
Supplementary Material for details). All data that changed over time were
analyzed weekly to minimize fluctuations observed at the daily level. We
considered 7-day averages for daily test positivity rate, intra-community
mobility, and hospitalization rate (Figure: S2-S5), given that this is likely
to be less volatile.
### 2.2 Exposure and Outcome
The number of tests done and the number of positive cases discovered is not
meaningful without extra information. The number of confirmed cases on a given
day is related to the actual prevalence, the average duration of disease, and
the gross number of tests performed, such that an increase in the number of
tests can reveal more existing infections and a change in estimates of the
prevalence. Test positivity rate incorporates both the number of tests done
and the number of positive cases discovered, frequently used for monitoring
the progression of the COVID-19 pandemic [35, 36], and its correlation with
hospitalization rates has been shown in previous studies [37, 38] consistent
with our use here. We calculated the average positivity rate at the county
level by dividing the 7-day average of daily confirmed cases by the 7-day
average of daily tests. The hospitalization rate was conceptualized as the
average weekly hospital admission rate for laboratory-confirmed COVID-19 per
10,000 county residents. The weekly average positivity and hospitalization
rates were log-transformed to capture the effect of detected infections and
testing on COVID-19 hospitalizations.
### 2.3 Model
Hospitalization data are made up of repeated measurements. The first, second,
and third waves represent 24, 22, and 27 measurements of hospitalization rate,
respectively, corresponding to the number of weeks in each wave. The
traditional linear regression model is not appropriate for studying data with
multiple repeated measures [39]. Therefore, we employed a linear mixed-effects
model that incorporates repeated observations at the county level.
Let $\textbf{Y}_{j}$ be the $I\times 1$ dependent variable corresponding to
the log of the rate of hospital admissions for COVID-19 per 10,000 inhabitants
at the county $j$. The subscripts $j=1,2,...,J$ and $i=1,2,...,I$ represent
the 42 counties in California and the number of weeks in the wave data
collected, respectively. $\textbf{X}_{j}$ is the $I\times p$ fixed-effects
design matrix; $\boldsymbol{\beta}$ is the $p\times 1$ fixed-effects vector;
$\textbf{Z}_{j}$ correspond to $I\times q$ matrix of random-effects design
matrix; $\textbf{u}_{j}$ represents the $q\times 1$ vector of random effects
and $\boldsymbol{\varepsilon_{j}}$ is the $I\times 1$ vector of residuals.
$\textbf{u}_{j}$ is independent of $\boldsymbol{\varepsilon_{j}}$. G is the
$q\times q$ covariance matrix for the random effects, and $\textbf{R}_{j}$ is
the $I\times I$ covariance matrix for the residuals. The model we considered
includes a random intercept and a random slope concerning the positivity rate
($q=2)$ since we hypothesize that each county has a different baseline
positivity rate and that the effect of the positivity rate on hospitalization
differs between counties.
We define the general form of the mixed linear regression model as follows:
$\begin{split}\textbf{Y}_{j}&=\textbf{X}_{j}\boldsymbol{\beta}+\textbf{Z}_{j}\textbf{u}_{j}+\boldsymbol{\varepsilon_{j}}\\\
\textbf{u}_{j}&\sim N\left(0,\textbf{G}\right)\\\
\boldsymbol{\boldsymbol{\varepsilon}_{j}}&\sim
N\left(0,\textbf{R}_{j}\right).\end{split}$ (1)
The term $\textbf{X}_{j}\boldsymbol{\beta}$ corresponds to the fixed effect(s)
component (a standard general linear model) and $\textbf{Z}_{j}\textbf{u}_{j}$
to the random effects. The model was fitted using the lmer function in the
lme4 package for R [40].
Since only the hospitalization rate and the positivity rate were log-
transformed, we interpret the coefficient ($\beta_{r}$) for the log positivity
rate as the percent increase in the hospitalization rate for every $1\%$
increase in the positivity rate. The estimation for all other coefficients
($\beta_{p}$’s) requires transformation via $100\times(\exp(\beta_{p})-1)$,
which gives the percent increase (or decrease) in the hospitalizations rate
for every one-unit increase in the independent variable.
## 3 Results
The coefficient estimates and the $95\%$ confidence intervals (CI) for the
linear mixed model are presented in Table 1. The $\beta$ value represents the
effect that each variable has on the hospitalizations rate. Variables with a
p-value $<0.05$ were considered statistically significant. Results show that
significant variables changed over time, but the positivity rate consistently
remained significant across all three waves with a coefficient $\beta_{r}$
close to one. Regarding hospitalization rates for different racial and ethnic
groups, counties with a higher population percentage of non-White race or
ethnic groups had higher hospitalization rates in the first and second waves
1. In the first wave of infections, counties saw an average $7.4\%$ increase
in hospitalization rate for every $1\%$ of the population identified as
Hispanic or Latino, and a $16.6\%$ increase in hospitalization rate for every
$1\%$ of the population that identifies as African American. In the second
wave, counties with high proportions of Hispanic or Latino and African
American populations were not significantly different, but a $3.4\%$ increase
in hospitalization rates was associated with every $1\%$ of the population
that identifies as Asian.
HPI was significant and positive in the first wave, meaning that counties with
more significant economic, social, and healthcare resources reported increased
hospitalization rates compared to counties with fewer resources. Higher intra-
community mobility was associated with higher hospitalization rates; however,
in the second wave, we found that higher mobility was negatively associated
with hospitalization rates.
Table 1: Association between hospitalization rates and independent variables
at the county level.
| 1st wave | 2nd wave | 3rd wave
---|---|---|---
Variable | Estimate* (95% CI) | p-value | Estimate* (95% CI) | p-value | Estimate* (95% CI) | p-value
Positivity rate | 0.9 (0.6, 1.4) | $<$0.001 | 0.9 (0.8, 0.9) | $<$0.001 | 1.1 (0.9, 1.2) | $<$0.001
Pop over 65 | -13.2 (-30.6, 8.5) | 0.246 | 3.1 (-3.8, 10.6) | 0.416 | -7.7 (-15.1, -0.1) | 0.060
Asian | 0.8 (-7.2, 9.4) | 0.861 | 3.4 (0.7, 6.1) | 0.020 | -0.5 (-3.9, 2.9) | 0.763
Hispanic/Latino | 7.4 (2.1, 12.9) | 0.010 | 0.9 (-0.6, 2.5) | 0.259 | -0.6 (-2.3, 1.3) | 0.535
African American | 16.6 (0.1, 35.9) | 0.070 | 0.1 (-4.7, 5.1) | 0.982 | 1.8 (-4.3, 8.2) | 0.582
HPI | 4.9 (2.4, 7.4) | $<$0.001 | -0.4 (-1.2, 0.3) | 0.266 | 0.2 (-0.7, 1.2) | 0.686
Mobility | 4.9 (1.9, 7.9) | $<$0.001 | -2.8 (-3.4, -2.1) | $<$0.001 | 0.1 (-1.7, 1.8) | 0.950
*A $1\%$ increase in the positivity rate coefficient consistently corresponds to around a $1\%$ percent increase in the hospitalization rate. Interpretation for the other independent variables is a one-unit increment corresponding to a percent change, namely positive or negative, depending on the coefficient sign.
Table 2 displays the coefficient value related to the log positivity rate for
each county in the three waves. These values are equal to
($\beta_{r}+u_{r_{j}}$), where $\beta_{r}$ correspond to the general
coefficient for the log positivity rate (Table 1) and $u_{r_{j}}$ is the
random coefficient for the j-th county, $j=1,2,...,J$. Counties with higher
coefficient values had stronger associations between test positivity rate and
hospitalization rate.
Table 2: Estimates of the association between test positivity rate and
hospitalization rate for each wave of study.
County | Wave 1 | Wave 2 | Wave 3 | County | Wave 1 | Wave 2 | Wave 3
---|---|---|---|---|---|---|---
Yolo | 2.99 | 0.70 | 1.10 | Santa Clara | 0.95 | 1.08 | 0.82
Madera | 1.64 | 0.77 | 1.05 | Stanislaus | 0.92 | 0.80 | 0.78
El Dorado | 1.52 | 1.34 | 1.53 | Nevada | 0.92 | 0.64 | 0.94
Imperial | 1.34 | 1.02 | 0.92 | Tuolumne | 0.91 | 0.55 | 0.80
Los Angeles | 1.23 | 0.95 | 0.78 | Butte | 0.90 | 0.82 | 1.10
Orange | 1.22 | 1.02 | 0.94 | Merced | 0.90 | 0.86 | 1.30
Alameda | 1.18 | 0.82 | 0.98 | San Francisco | 0.82 | 1.06 | 0.87
Tulare | 1.15 | 0.86 | 1.00 | Marin | 0.79 | 0.79 | 0.98
Fresno | 1.14 | 0.88 | 0.87 | Sacramento | 0.75 | 0.85 | 1.16
Contra Costa | 1.09 | 0.99 | 1.21 | Napa | 0.73 | 0.95 | 1.04
San Bernardino | 1.06 | 1.01 | 1.02 | Lake | 0.71 | 1.06 | 2.62
Kern | 1.06 | 0.84 | 1.05 | Amador | 0.71 | 0.90 | 1.01
San Mateo | 1.04 | 1.03 | 0.91 | Shasta | 0.70 | 1.02 | 0.91
Ventura | 1.04 | 1.05 | 0.91 | Tehama | 0.69 | 0.76 | 1.70
Santa Barbara | 1.04 | 0.93 | 0.97 | Sonoma | 0.67 | 0.97 | 1.03
San Joaquin | 1.01 | 1.01 | 1.06 | Santa Cruz | 0.66 | 0.79 | 1.12
Kings | 1.00 | 0.77 | 0.89 | San Luis Obispo | 0.55 | 0.95 | 1.15
San Diego | 0.99 | 0.92 | 0.97 | Yuba | 0.52 | 0.95 | 1.02
Solano | 0.99 | 0.90 | 1.06 | Placer | 0.29 | 0.89 | 0.77
Riverside | 0.97 | 1.02 | 1.09 | Mendocino | 0.18 | 1.10 | 1.17
Monterey | 0.95 | 0.81 | 1.05 | Humboldt | 0.12 | 0.38 | 1.14
## 4 Discussion
A mixed linear model was used between the COVID-19 hospitalization rate and
factors such as age, ethnicity, race, poverty index, and intra-community
mobility. Our primary interest was studying the impact of testing rates on
county-level hospitalization rates, as county health departments were usually
responsible for public testing administration. We found that the test
positivity rate was consistently significant and positively associated with
the hospitalization rate during all three waves of COVID-19. Hospitalization
rate increased at an almost 1:1 basis with a positivity rate. While other
possible predictors of hospitalization rate, including the density of
different race or ethnic groups, social vulnerability, and intra-community
mobility, had pronounced effects at differing times during the pandemic, none
were consistent predictors of hospitalization rate for all three waves of
infection.
The actual local prevalence and the number of tests administered both affect
the positivity rate value. Generally, the higher the true prevalence, the
higher the positivity rate will be; as more tests are deployed, the positivity
rate will converge with the true prevalence. The nature of diagnostic testing
on a first-come-first-serve basis frequently leads to positivity rates more
than the actual prevalence if testing rates are insufficient to sample the
mild or asymptomatic cases. In other words, if the number of tests is a
limiting factor, and they are used primarily to confirm likely cases more
frequently than a random surveillance sampling of the population, positivity
rates will be biased upwards compared to the actual prevalence. This assumes
that those who suspect they have the disease or suspect exposure are more
likely to seek a test than those who have no such suspicion. Thus, high test
positivity rates are likely a mix of biased sampling and high prevalence, but
clarifying which is dominant during a specific time frame requires high-
quality auxiliary data that may not exist. Our results suggest that actions
that reduce the test positivity rate are likely to reduce the hospitalization
rate by a similar magnitude. Simply increasing the number of tests will only
significantly reduce the positivity rate if sampling bias is the dominant
reason for a high positivity rate. Determining the effect on hospitalization
rate of reducing test positivity rate in bias-dominant versus prevalence-
dominant systems is beyond the scope of this paper, but remains an important
question.
The response following detection is essential. Theoretically, early detection
of a new case, symptomatic or asymptomatic, and rapid isolation will prevent
further potential hospitalizations. Extrapolating from our results, we expect
that the counties that more regularly tested a more significant proportion of
their population- from asymptomatic surveillance or robust testing
requirements for essential workers- experienced lower hospitalization rates
than the counterfactual scenario. However, care must be taken extending this
reasoning too far: large-scale population testing can theoretically lead to
reduced hospitalizations, but the effect will always be indirect. The
resources and infrastructure must support proper mass testing and preparation
to respond to the information garnered from the testing program, which no two
counties will have done identically, hence why each county reported here
maintained intercepts that varied from each other over time.
A low positivity rate due to a high amount of testing does not always imply
adequate pandemic control. Not only does the gap between testing rates among
suspected cases, known exposures, and the unexposed or asymptomatic matter,
but testing rates among different demographic groups demonstrably effect the
value of testing data. Suppose the mass testing systemically excludes people
with a high-risk profile (as could quickly occur where healthcare
accessibility is low). In that case, many infections could remain undiscovered
for long periods, leading to a growth in the hospitalization rate despite low
positivity rates. The pandemic has not affected everyone equally. Disparities
in coronavirus disease outcomes by racial and ethnicity as well as
socioeconomic status have been reported since the beginning of the pandemic
[41]. Our findings highlight that areas with larger relative populations of
Hispanic or Latinos and African Americans were significantly correlated with
higher hospitalization rates in the first wave and with Asians in the second
wave, consistent with previous studies [42, 43, 44]. The underlying causes of
health disparities in Latinos, African Americans, and poor communities are
related to social and structural determinants of health [45]. Implementing
social distancing, especially at the beginning of the pandemic, may have been
challenging because these communities, on average, live in more crowded
conditions and work more frequently in essential public-facing occupations. In
addition, their access to health services is systemically limited, so that
populations have a disproportionate burden of underlying comorbidities and
lack the possibility of accessing adequate and timely treatment when affected
by the SARS-CoV-2 virus [46], and possibly confounding the relationship
between test positivity rates and hospitalization rates, as discussed above.
The Healthy Places Index is correlated positively with the hospitalization
rate in the first wave, which implies that counties with higher socioeconomic
status had a higher probability of reporting hospitalizations. One of the
reasons may be the capacity and better availability of hospital facilities
attributed to economic resources. Mobility was another significant variable
that positively and negatively correlated with the hospitalization rate in the
first and second waves. A similar result was reported in [47] for COVID-19
transmission and mortality rates. Early in the pandemic, mobility patterns
were drastically affected by containment measures implemented to slow the
spread of the disease. Our results show a linear correlation between mobility
and the rate of hospitalization in the first wave, in agreement with previous
reports [48], which implies that an increase in the circulation of people
could cause an increase in infections and, consequently, in hospitalizations.
However, it is not clear how mobility affected the growth rates of the
COVID-19 infection once the lockdowns were lifted because other interventions
became more widely available and easier to adhere to, such as wearing face
masks and social distancing, patterns of both mobility and growth of
infections became non-linear [48]. One interpretation could be that areas with
lower infection rates allowed for greater freedom in summer activities,
negatively correlating positivity and hospitalization rates. Care must be
taken in attributing causation to relationships between these covariates and
hospitalization rates without further study.
This study has some limitations that are important to consider. First, it is
focused on county-level analysis and is intended to investigate population-
level risk; conclusions at the individual level are not appropriate and should
not be applied. Second, as discussed earlier, we did not attempt to address
whether a given data point on test positivity was produced during a bias-
dominant or prevalence-dominant period. Thus an unknown proportion of the
relationship between positivity rate and hospitalization rate is likely due to
natural increases in the prevalence. Third, the hospitalization rate is also
dependent on available hospital beds, which we did not consider as a factor
given the limited availability and reliability of such data at the county
level. Thus, some instances where hospitalization would have been an outcome
for a patient except for bed availability were not accounted for, which could
have led to point underestimates of our primary outcome measurements.
Knowing the factors that affect the spread of the virus and hospitalizations
allows local decision-makers to help identify areas at higher risk for severe
COVID-19 and guide resource allocation and implementation of prevention and
mitigation strategies. These findings highlight how the most significant
factors impacting hospitalizations have changed with the pandemic’s evolution.
The positivity rate is the only factor to prevail over time as a significant
and directly correlated with hospitalization rate.
## References
* [1] Zunyou Wu and Jennifer M McGoogan. Characteristics of and important lessons from the coronavirus disease 2019 (covid-19) outbreak in china: summary of a report of 72 314 cases from the chinese center for disease control and prevention. JAMA, 323(13):1239–1242, 2020.
* [2] Johns hopkins coronavirus resource center. covid-19 map. Available online: https://coronavirus.jhu.edu/map.html. (Accessed on 10/05/2021).
* [3] Maria Nicola, Catrin Sohrabi, Ginimol Mathew, Ahmed Kerwan, Ahmed Al-Jabir, Michelle Griffin, Maliha Agha, and Riaz Agha. Health policy and leadership models during the covid-19 pandemic-review article. Int. J. Surg., 81:122–129, 2020.
* [4] Shahul H Ebrahim, Qanta A Ahmed, Ernesto Gozzer, Patricia Schlagenhauf, and Ziad A Memish. Covid-19 and community mitigation strategies in a pandemic. BMJ, 368, 2020.
* [5] James Hadfield, Colin Megill, Sidney M Bell, John Huddleston, Barney Potter, Charlton Callender, Pavel Sagulenko, Trevor Bedford, and Richard A Neher. Nextstrain: real-time tracking of pathogen evolution. Bioinformatics, 34(23):4121–4123, 2018.
* [6] Maria Bonet, I Blanco, Argimon Pallàs JM, B Clotet, J Ara Del Rey, and et al. A cost-benefit analysis of the covid-19 asymptomatic mass testing strategy in the north metropolitan area of barcelona. Int. J. Environ, 18(13), 2021.
* [7] C Jason Wang, Chun Y Ng, and Robert H Brook. Response to covid-19 in taiwan: big data analytics, new technology, and proactive testing. JAMA, 323(14):1341–1342, 2020.
* [8] Shiyi Cao, Yong Gan, Chao Wang, Max Bachmann, Shanbo Wei, Jie Gong, Yuchai Huang, Tiantian Wang, Liqing Li, Kai Lu, and et al. Post-lockdown sars-cov-2 nucleic acid screening in nearly ten million residents of wuhan, china. Nat. Commun, 11(1):1–7, 2020.
* [9] Yaqing Fang. Large-scale national screening for covid-19 in china. J. Med. Virol., 92(11):2266–2268, 2020.
* [10] Talha Burki. Mass testing for covid-19. The Lancet. Microbe., 1(8):e317, 2020.
* [11] Janice K Louie, Hyman M Scott, Amie DuBois, Natalya Sturtz, Wendy Lu, Juliet Stoltey, Godfred Masinde, Stephanie Cohen, Darpun Sachdev, Susan Philip, and et al. Lessons from mass-testing for coronavirus disease 2019 in long-term care facilities for the elderly in san francisco. Clin. Infect. Dis, 72(11):2018–2020, 2021.
* [12] Flávia Bulegon Pilecco, Carolina Gomes Coelho, Qeren Hapuk Rodrigues Ferreira Fernandes, Ismael Henrique Silveira, Júlia Moreira Pescarini, Naiá Ortelan, Ligia Gabrielli, Estela ML Aquino, and Maurício Lima Barreto. The effect of laboratory testing on covid-19 monitoring indicators: an analysis of the 50 countries with the highest number of cases. Epidemiol Serv Saude, 30, 2021.
* [13] Solomon Hsiang, Daniel Allen, Sébastien Annan-Phan, Kendon Bell, Ian Bolliger, Trinetta Chong, Hannah Druckenmiller, Luna Yue Huang, Andrew Hultgren, Krasovich E, and et al. The effect of large-scale anti-contagion policies on the covid-19 pandemic. Nature, 584(7820):262–267, 2020.
* [14] Daniel B Larremore, Bryan Wilder, Evan Lester, Soraya Shehata, James M Burke, James A Hay, Tambe Milind, Michael J Mina, and Roy Parker. Test sensitivity is secondary to frequency and turnaround time for covid-19 surveillance. MedRxiv, 2020.
* [15] Martin Pavelka, Kevin Van-Zandvoort, Sam Abbott, Katharine Sherratt, Marek Majdan, CMMID COVID, Inštitút Zdravotnỳch Analỳz, Pavol Jarčuška, Marek Krajčí, Stefan Flasche, and et al. The impact of population-wide rapid antigen testing on sars-cov-2 prevalence in slovakia. Science, 372(6542):635–641, 2021.
* [16] Alexis C. Madrigal Robinson Meyer. How bad is the coronavirus outbreak? here’s a key number. the atlantic, 2020. Available online: https://www.theatlantic.com/technology/archive/2020/04/us-coronavirus-outbreak-out-control-test-positivity-rate/610132/. (Accessed on 10/05/2021).
* [17] Tim R Mercer and Marc Salit. Testing at scale during the covid-19 pandemic. Nat. Rev. Genet., 22(7):415–426, 2021.
* [18] Anne M Neilan, Elena Losina, Audrey C Bangs, Clare Flanagan, Christopher Panella, G Eskibozkurt, Amir Mohareb, Emily P Hyle, Justine A Scott, Milton C Weinstein, and et al. Clinical impact, costs, and cost-effectiveness of expanded severe acute respiratory syndrome coronavirus 2 testing in massachusetts. Clin. Infect. Dis., 2020.
* [19] C Wei, CC Lee, TC Hsu, WT Hsu, CC Chan, SC Chen, and CJ Chen. Correlation of population mortality of covid-19 and testing coverage: A comparison among 36 oecd countries. Epidemiol. Infect., 149:e1, 2021.
* [20] Wikipedia. covid-19 pandemic in california. available online:. https://en.wikipedia.org/wiki/COVID-19_pandemic_in_California. (Accessed on 10/05/2021).
* [21] State of california. california takes action to combat covid-19 — california governor. available online:. https://www.gov.ca.gov/california-takes-action-to-combat-covid-19/. (Accessed on 10/05/2021).
* [22] Office of governor gavin newsom. as california fully reopens, governor newsom announces plans to lift pandemic executive orders. june 11, 2021. Available online: https://www.gov.ca.gov/2021/06/11/as-california-fully-reopens-governor-newsom-announces-plans-to-lift-pandemic-executive-orders/. (Accessed on 09/22/2021).
* [23] Cheri Mossburg. Here’s what you need to know about california’s reopening. cnn. Available online: https://edition.cnn.com/us/live-news/coronavirus-pandemic-vaccine-updates-06-15-21/index.html. (Accessed on 10/05/2021).
* [24] USAFacts. California coronavirus cases and deaths. Available online: https://usafacts.org/visualizations/coronavirus-covid-19-spread-map/state/california. (Accessed on 10/05/2021).
* [25] Dustin Manduffie. Covid ravaging the central valley, california’s new virus epicenter. august 4, 2020. Available online: https://www.courthousenews.com/covid-ravaging-the-central-valley-californias-new-virus-epicenter/. (Accessed 10/05/2021).
* [26] Mary Stringini and KJ Hiramoto. Is los angeles county the new “epicenter” of the covid-19 pandemic?. december 21, 2020. Available online: https://www.foxla.com/news/is-los-angeles-county-the-new-epicenter-of-the-covid-19-pandemic. (Accessed 10/05/2021).
* [27] Susannah Cullinane Paul Vercammen, Melissa Alonso. La county records more than 1 million coronavirus cases. january 17, 2021\. Available online: https://edition.cnn.com/2021/01/16/health/los-angeles-county-million-coronavirus/index.html. (Accessed 10/05/2021).
* [28] abc-7news. california has lowest covid-19 case rate in us. Available online: https://abc7news.com/california-covid-ca-coronavirus-rate-pandemic/11033019/. (Accessed on 10/05/2021).
* [29] Cbs news. california reports lowest covid-19 case rate in the country. Available online: https://www.cbsnews.com/news/california-covid-lowest-positivity-rate/. (Accessed on 10/05/2021).
* [30] California goverment. tracking covid-19 in california - coronavirus covid-19 response. Available online: https://covid19.ca.gov/state-dashboard/. (Accessed on 09/16/2021).
* [31] Californial department of public health. covid-19 time-series metrics by county and state - datasets - california health and human services open data portal. Available online: https://data.chhs.ca.gov/dataset/covid-19-time-series-metrics-by-county-and-state. (Accessed on 09/16/2021).
* [32] Us census bureau. 2019 data profiles — american community survey. Available online: https://www.census.gov/acs/www/data/data-tables-and-tools/data-profiles/. (Accessed on 09/16/2021).
* [33] The california healthy places index (hpi). data & reports – california healthy places index. Available online: https://healthyplacesindex.org/data-reports/. (Accessed on 09/16/2021).
* [34] Covid-19 community mobility reports. google (2020). Available online: https://www.google.com/covid19/mobility/. (Accessed on 11/09/2021).
* [35] Yuki Furuse, Yura K Ko, Kota Ninomiya, Motoi Suzuki, and Hitoshi Oshitani. Relationship of test positivity rates with covid-19 epidemic dynamics. Int. J. Environ, 18(9):4655, 2021.
* [36] Livio Fenga and Mauro Gaspari. Predictive capacity of covid-19 test positivity rate. Sensors, 21(7):2435, 2021.
* [37] Ram C Kafle, Doo Young Kim, Martin E Malandro, and Melinda Miller Holt. Modeling covid-19 positivity rates and hospitalizations in texas. Model Assist. Stat. Appl., 16(1):53–58, 2021.
* [38] Ahmed Al Dallal, Usama AlDallal, and Jehad Al Dallal. Positivity rate: an indicator for the spread of covid-19. Curr. Med. Res., 0(0):1–10, 2021.
* [39] Nan M. Laird and James H. Ware. Random-effects models for longitudinal data. Biometrics, 38(4):963–974, 1982.
* [40] Douglas Bates, Martin Mächler, Ben Bolker, and Steve Walker. Fitting linear mixed-effects models using lme4. J. Stat. Softw., 67(1):1–48, 2015.
* [41] Merlin Chowkwanyun and Adolph L Reed Jr. Racial health disparities and covid-19—caution and context. N. Engl. J. Med., 383(3):201–203, 2020.
* [42] Gabriel J Escobar, Alyce S Adams, Vincent X Liu, Lauren Soltesz, Yi-Fen Irene Chen, Stephen M Parodi, G Thomas Ray, Laura C Myers, Charulata M Ramaprasad, Richard Dlott, and et al. Racial disparities in covid-19 testing and outcomes: retrospective cohort study in an integrated health system. Ann. Intern. Med., 174(6):786–793, 2021.
* [43] Gbenga Ogedegbe, Joseph Ravenell, Samrachana Adhikari, Mark Butler, Tiffany Cook, Fritz Francois, Eduardo Iturrate, Girardin Jean-Louis, Simon A Jones, and et al Onakomaiya, Deborah. Assessment of racial/ethnic disparities in hospitalization and mortality in patients with covid-19 in new york city. JAMA Netw. Open, 3(12):e2026881–e2026881, 2020.
* [44] Laura Jean Podewils, Tori L Burket, Christie Mettenbrink, Abigail Steiner, Allison Seidel, Kenneth Scott, Lilia Cervantes, and Romana Hasnain-Wynia. Disproportionate incidence of covid-19 infection, hospitalizations, and deaths among persons identifying as hispanic or latino—denver, colorado march–october 2020. MMWR, 69(48):1812–1816, 2020.
* [45] Monica Webb Hooper, Anna María Nápoles, and Eliseo J Pérez-Stable. Covid-19 and racial/ethnic disparities. JAMA, 323(24):2466–2467, 2020.
* [46] This time must be different: disparities during the covid-19 pandemic, 2020.
* [47] Tanmoy Bhowmik, Sudipta Dey Tirtha, Naveen Chandra Iraganaboina, and Naveen Eluru. A comprehensive analysis of covid-19 transmission and mortality rates at the county level in the united states considering socio-demographics, health indicators, mobility trends and health care infrastructure attributes. Plos one, 16(4):e0249133, 2021.
* [48] Raju Gottumukkala, Satya Katragadda, Ravi Teja Bhupatiraju, Md Kamal, Vijay Raghavan, Henry Chu, Ramesh Kolluru, and Ziad. Ashkar. Exploring the relationship between mobility and covid- 19 infection rates for the second peak in the united states using phase-wise association. BMC Public Health, 21(1):1–14, 2021.
* [49] California health and human services open data portal. covid-19 hospital data. Available online: https://data.chhs.ca.gov/dataset/covid-19-hospital-data. (Accessed on 06/09/2021).
* [50] Pierre Nouvellet, Sangeeta Bhatia, Anne Cori, Kylie EC Ainslie, Marc Baguelin, Samir Bhatt, Adhiratha Boonyasiri, Nicholas F Brazeau, Lorenzo Cattarino, Laura V Cooper, and et al. Reduction in mobility and COVID-19 transmission. Nat. Commun, 12(1):1–9, 2021.
* [51] Hamada S Badr, Hongru Du, Maximilian Marshall, Ensheng Dong, Marietta M Squire, and Lauren M Gardner. Association between mobility patterns and COVID-19 transmission in the USA: a mathematical modelling study. Lancet Infect Dis, 20(11):1247–1254, 2020.
* [52] Nittai K Bergman and Ram Fishman. Mobility levels and covid-19 transmission rates. medRxiv, 2020.
* [53] Oliver Gatalo, Katie Tseng, Alisa Hamilton, Gary Lin, and Eili Klein. Associations between phone mobility data and covid-19 cases. Lancet Infect Dis, 21(5):e111, 2021.
* [54] Apple. covid-19 mobility trends reports. Available online: https://covid19.apple.com/mobility.
* [55] Social distancing metrics. Safegraph (2020). Available online: https://docs.safegraph.com/docs/social-distancing-metrics.
## Supplementary material
### Maps
This maps displays the coefficient value related to the log positivity rate
for each county in the three waves.
Figure S1. Values in the county-level maps represent the effect of the
positivity rate on hospitalizations at each county for each wave. Top Left:
Wave 1. Top Right: Wave 2. Bottom: Wave 3.
### Hospitalizations, testings, and positivity rate.
Figure S2. This figure displays the number of patients hospitalized in an
inpatient bed who have laboratory-confirmed COVID of the California counties
[49] considered in the study and testing per 10K population from January 5,
2020, to September 6, 2021. Figure S3. Number of patients hospitalized in an
inpatient bed who have laboratory-confirmed COVID (red line) and positivity
rate (7-day moving average, green line) from March 28, 2020 to September 6,
2021.
### Variable description
We describe the co-variable used in the analysis, mobility, age, race,
ethnicity, poverty, and education in the California counties.
#### Demographic variables
Demographic characteristics such as age, ethnicity, and race by county was
obtained from the United State Census Bureau [32].
Table S1. Demographic variables considered in the mixed linear model.
Variable | Description
---|---
Over 65 | Percentage of population ages 65 and above
Black or African | Percentage of Black or African American population
American Indian /and Alaska Native | Percentage of American Indian Alaska Native population
Asian | Percentage of Asian population
Hispanic or Latino | Percentage of Hispanic or Latino population
HPI | Healthy place index
Figure S4. Demographic variables to county level are taken from the United
States Census Bureau [32]. All the values are in percentage on the population
except the HPI [33], which is a number between 0 to 100.
#### Correlation and multicollinearity
To choose the variables used in the model, we checked for the
multicollinearity issue across the independent variables as some county-level
features may be highly correlated. We plot a correlation matrix to highlight
the Pearson correlation coefficient between each independent variable in the
dataset. Figure S5 highlights the weaker correlation across most demographics
variables while finding high correlations of the comorbidities between them
and most of the demographic variables.
Figure S5. Correlation matrix of the demographics variables and comorbidities.
To measures the severity of multicollinearity in our regression analysis, we
estimate the variance inflation factor (VIF) for the previous variables. VIF
indicates the increase in the variance of a regression coefficient for each
variable as a result of correlated predictors. A VIF value of 1 for a variable
implies this variable is not correlated to the remaining ones. When VIF is
higher than 10 there is significant multicollinearity that needs to be
corrected. In Tab. S2, we observe very large values of the VIF. When removing
variable by variable we are left with only the demographic variables given in
the Tab. S1. Therefore we can safely conclude that our analysis was unaffected
by the multicollinearity issue. For this reason, disease prevalence variables
were not included in the analysis.
Table S2. Variance inflation factor. Variable | Variance inflation factor
---|---
Heart disease | 3846.96
Obesity | 235.79
COPD | 1648.60
diabetes | 1430.87
CKD | 3342.10
65+ | 299.79
HPI | 11.71
Black or African American | 6.50
Asian | 11.19
Hispanic or Latino | 73.89
#### Mobility trends
The existence of social contacts could be proxied by mobility data [50, 51,
52, 53], with frameworks such as Google’s Community Mobility Report (CMR)
[34], COVID‑19 - Mobility Trends Reports - Apple [54] and Safegraph [55] being
able to measure mobility, as it measures citizens’ mobility according to
different types.
Our analysis is based on 40 counties of California for which both
hospitalization and Google mobility data were available. Google mobility data
included six data-streams: “grocery and pharmacy,” “parks,” “residential,”
“retail and recreation,” “transit stations,” and “workplaces.” We combined all
Google-specific data streams to obtain a google county mobility measure. We
used an unsupervised machine learning method known as principal component
analysis (PCA) to construct the google mobility index using the six mobility
metrics.The first principal component explained more than 50% of the
variability in the data by each county, indicating a good dimension reduction
(Table S3).
A regression analysis was used to estimate the lag length. The results show
that mobility is correlated with COVID-19 hospitalizations in most counties
with lags of 3-4 weeks.
Table S3. The variance explained by the first principal component for the
Google’s Community Mobility Report (CMR).
County | Explained variance | County | Explained variance | County | Explained variance
---|---|---|---|---|---
Alameda | 0.62 | Mendocino | 0.60 | San Mateo | 0.65
Amador | 0.43 | Merced | 0.68 | Santa Barbara | 0.67
Butte | 0.56 | Monterey | 0.66 | Santa Clara | 0.60
Contra Costa | 0.63 | Napa | 0.56 | Santa Cruz | 0.60
El Dorado | 0.50 | Nevada | 0.58 | Shasta | 0.51
Fresno | 0.67 | Orange | 0.68 | Solano | 0.60
Humboldt | 0.60 | Placer | 0.50 | Sonoma | 0.66
Imperial | 0.67 | Riverside | 0.65 | Stanislaus | 0.62
Kern | 0.60 | Sacramento | 0.66 | Tehama | 0.4
Kings | 0.61 | San Bernardino | 0.62 | Tulare | 0.57
Lake | 0.65 | San Diego | 0.71 | Tuolumne | 0.47
Los Angeles | 0.73 | San Francisco | 0.73 | Ventura | 0.56
Madera | 0.51 | San Joaquin | 0.68 | Yolo | 0.66
Marin | 0.62 | San Luis Obispo | 0.65 | Yuba | 0.54
|
60.00009pt-\mathbb{E}_{\mu}\left[f((Y^{\mathrm{Q}}+E^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M,\,\|E^{\mathrm{Q}}\|\leq\widebar{M}]}\right]\bigg{|},$ $\displaystyle T_{2}$
$\displaystyle=\bigg{|}\mathbb{E}_{\mu}\left[f((Y^{\mathrm{Q}}+E^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M,\,\|E^{\mathrm{Q}}\|\leq\widebar{M}]}\right]-\mathbb{E}_{\mu}\left[f((Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M]}\right]\bigg{|},$ $\displaystyle T_{3}$
$\displaystyle=\bigg{|}\mathbb{E}_{\widehat{\mu}}\left[f((Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M]}\right]-\mathbb{E}_{\mu}\left[f((Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M]}\right]\bigg{|}.$
By (F.18), we can conclude that $T_{1}\xrightarrow{a.s}0$ and by (F.19), we
can conclude that $T_{3}\xrightarrow{a.s}0$. Next, by the properties of
multivariate Gaussian distributions and arguments similar to those leading to
(F.18), we can also show that, as $N\rightarrow\infty$,
$\displaystyle\bigg{|}f((Y^{\mathrm{Q}}+E^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M,\,\|E^{\mathrm{Q}}\|\leq\widebar{M}]}-f((Y^{\mathrm{Q}}+E^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\mu)\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M,\,\|E^{\mathrm{Q}}\|\leq\widebar{M}]}\bigg{|}\xrightarrow{a.s}0,$
$\displaystyle\hskip
60.00009pt\bigg{|}f((Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M]}-f((Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\mu)\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M]}\bigg{|}\xrightarrow{a.s}0.$
Finally, as $\|E^{\mathrm{Q}}\|\xrightarrow{a.s}0$, by the Dominated
Convergence theorem and Lemma 4.1, we conclude that
$\bigg{|}f((Y^{\mathrm{Q}}+E^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\mu)\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M,\,\|E^{\mathrm{Q}}\|\leq\widebar{M}]}-f((Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\mu)\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M]}\bigg{|}\xrightarrow{a.s}0.$
Therefore,
$\bigg{|}f((Y^{\mathrm{Q}}+E^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M,\,\|E^{\mathrm{Q}}\|\leq\widebar{M}]}-f((Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M]}\bigg{|}\xrightarrow{a.s}0.$
Furthermore, since the functions in the above equation are uniformly dominated
by $1$, by the Dominated Convergence Theorem, we have
$\displaystyle\hskip-25.00003pt\bigg{|}\mathbb{E}_{\mu}\left[f((Y^{\mathrm{Q}}+E^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M,\,\|E^{\mathrm{Q}}\|\leq\widebar{M}]}\right]-\mathbb{E}_{\mu}\left[f((Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}}),\mathcal{C}_{\alpha},\widehat{\mu})\mathbbm{1}_{[(Y^{\mathrm{Q}},\widetilde{Y}^{\mathrm{Q}})\in
M]}\right]\bigg{|}\rightarrow 0,$ (F.21)
almost surely, given
$\\{\bar{\bm{V}}_{T,1},\ldots,\bar{\bm{V}}_{T,m},\widehat{\bm{D}}_{1},\ldots,\widehat{\bm{D}}_{m},\bm{V}_{1},\ldots,\bm{V}_{m}\\}$.
In other words, $T_{2}\stackrel{{\scriptstyle a.s.}}{{\to}}0$ conditionally.
Since, $\varepsilon$ chosen in (F.14) is arbitrary, by (F.20), we can get
(F.15) conditioning on
$\\{\bar{\bm{V}}_{T,1},\ldots,\bar{\bm{V}}_{T,m},\widehat{\bm{D}}_{1},\ldots,\widehat{\bm{D}}_{m},\bm{V}_{1},\ldots,\bm{V}_{m}\\}.$
As the conclusion holds regardless of the conditioning event, (F.15) holds
marginally. Now, as $\mathcal{C}_{\alpha}\in\mathscr{C}_{r+\widetilde{r}}$,
Lemma 4.4 implies,
$\displaystyle|\mu(\mathcal{C}_{\alpha})-\widehat{\mu}(\mathcal{C}_{\alpha})|\xrightarrow{a.s}0,\quad\mbox{as
$N\rightarrow\infty$,}$ (F.22)
conditioned on
$\\{\bar{\bm{V}}_{T,1},\ldots,\bar{\bm{V}}_{T,m},\widehat{\bm{D}}_{1},\ldots,\widehat{\bm{D}}_{m},\bm{V}_{1},\ldots,\bm{V}_{m}\\}$.
Since the right-hand side of (F.22) is non-random, (F.22) also holds
unconditionally. Hence, we almost surely have
$\displaystyle\lim_{N\rightarrow\infty}\mathbb{P}_{\mu}\left[(U^{\mathrm{Q}}_{1},\ldots,U^{\mathrm{Q}}_{m},\widetilde{U}^{\mathrm{Q}}_{1},\ldots,\widetilde{U}^{\mathrm{Q}}_{\widetilde{m}})\in\mathcal{C}_{\alpha}\right]$
$\displaystyle=\lim_{N\rightarrow\infty}\mu(\mathcal{C}_{\alpha})\geq\lim_{N\rightarrow\infty}\widehat{\mu}(\mathcal{C}_{\alpha})+\lim_{N\rightarrow\infty}{\color[rgb]{0,0,0}|}\widehat{\mu}(\mathcal{C}_{\alpha})-\mu(\mathcal{C}_{\alpha}){\color[rgb]{0,0,0}|}\geq
1-\alpha.$
Again as,
$\mathbb{P}_{\mu}\left[(U^{\mathrm{Q}}_{1},\ldots,U^{\mathrm{Q}}_{m},\widetilde{U}^{\mathrm{Q}}_{1},\ldots,\widetilde{U}^{\mathrm{Q}}_{\widetilde{m}})\in\mathcal{C}_{\alpha}\right]$
is a non-random quantity, our result follows.
## Appendix G Choices of Classes of Priors
In this section, we provide examples of two classes of distributions that can
be chosen as $\mathcal{P}$ and $\mathcal{P}_{\nu_{h}}$.
### G.1 Gaussian Mixture Models
We can choose $\mathcal{P}$ in (2.14) and each $\mathcal{P}_{\nu_{h}}$ in
(2.16) to be the set of distributions on $\mathbb{R}^{r+\widetilde{r}}$ and
$\mathbb{R}^{r_{h}}$ respectively, which are absolutely continuous with
respect to the Lebesgue measure and have bounded density. From [17, Chapter
3], we know that the class of Gaussian mixture distributions with large number
of components can approximate any distribution up to negligible error.
Inspired by this fact, for implementation purposes, we can approximate
$\mathcal{P}$ and all of the $\mathcal{P}_{\nu_{h}}$’s using Gaussian mixture
distributions with a large number of components. The number of components can
be chosen according to the size of the data and its complexity. In other
words, our approximating class of priors can be
$\widehat{\mathcal{P}}=\left\\{\sum_{k=1}^{K}\pi_{k}\,N_{r+\widetilde{r}}(m_{k},\bm{\Sigma}_{k}):m_{k}\in\mathbb{R}^{r+\widetilde{r}},\,\bm{\Sigma}_{k}\in\mathbb{S}^{r+\widetilde{r}}_{+},\,\pi_{k}\in[0,1]\;\mbox{for
$k\in[K]$}\right\\}.$ (G.1)
Similarly, for each $h\in[m]$, we can approximate $\mathcal{P}_{\nu_{h}}$ by
$\widehat{\mathcal{P}}_{\nu_{h}}=\left\\{\sum_{k=1}^{K_{h}}\pi_{k,h}\,N_{r_{h}}(m_{k,h},\bm{\Sigma}_{k,h}):m_{k,h}\in\mathbb{R}^{r_{h}},\,\bm{\Sigma}_{k,h}\in\mathbb{S}^{r_{h}}_{+},\,\pi_{k,h}\in[0,1]\;\mbox{for
$k_{h}\in[K_{h}]$}\right\\}.$ (G.2)
By working with these prior classes $\widehat{\mathcal{P}}$ and
$\widehat{\mathcal{P}}_{\nu_{h}}$ for $h\in[m]$, estimating $\mu$ and each
$\nu_{h}$ reduces to estimating the mixing proportions and the component means
and variances in a Gaussian mixture.
If $r_{h}$ for most $h\in[m]$ are very large, we can model $\bm{\Sigma}_{k}$
and $\bm{\Sigma}_{k,h}$ by diagonal matrices. In such situations, it might be
essential to have more components in the Gaussian mixture to account for the
simplification in the covariance structure, but overall we may need to
estimate fewer parameters, and consequently, we may have greater estimation
accuracy.
### G.2 Non Parametric Maximum Likelihood Estimators
An alternative approach for estimating the priors $\mu$ and $\nu_{h}$ is to
consider their nonparametric maximum likelihood estimators. This approach is
advocated in [46]. It is important to observe that, this estimator can suffer
from the “curse of dimensionality” if $r+\widetilde{r}$ or any of the $r_{h}$
are large. However, if the signal strengths $(\bm{D}_{h})_{ii}$ for
$i\in[r_{h}]$ and $h\in[m]$ are extremely high, then such “curse of
dimensionality can be avoided. For constructing the NPMLE, we can approximate
$\mathcal{P}$ and all of the $\mathcal{P}_{\nu_{h}}$’s using discrete
distributions with an appropriately chosen support. In particular, we can set
the support of $\mu$, denoted by
$\mathcal{S}_{\mu}\subseteq\mathbb{R}^{r+\widetilde{r}}$, as
$\mathcal{S}_{\mu}:=\left\\{((\widehat{\bm{S}}^{L,\mathrm{pc}}_{0,1})^{-1}({\bm{U}}^{\mathrm{pc}}_{0,1})_{i*},\ldots,(\widehat{\bm{S}}^{L,\mathrm{pc}}_{0,m})^{-1}({\bm{U}}^{\mathrm{pc}}_{0,m})_{i*},\widehat{\bm{L}}^{-1}_{1}(\widetilde{\bm{X}}_{1})_{i*},\ldots,\widehat{\bm{L}}^{-1}_{\widetilde{m}}(\widetilde{\bm{X}}_{\widetilde{m}})_{i*}):i\in[N]\right\\}.$
In other words, each point in the support of $\mu$ can be chosen to be one row
in the column concatenated matrix
$\begin{bmatrix}(\widehat{\bm{S}}^{L,\mathrm{pc}}_{0,1})^{-1}{\bm{U}}^{\mathrm{pc}}_{0,1}&\ldots&(\widehat{\bm{S}}^{L,\mathrm{pc}}_{0,m})^{-1}{\bm{U}}^{\mathrm{pc}}_{0,m}&\widehat{\bm{L}}^{-1}_{1}\widetilde{\bm{X}}_{1}&\ldots&\widehat{\bm{L}}^{-1}_{\widetilde{m}}\widetilde{\bm{X}}_{\widetilde{m}}\end{bmatrix}\in\mathbb{R}^{n\times(r+\widetilde{r})}.$
Then, our approximating class of priors will be
$\widehat{\mathcal{P}}=\\{\mbox{all prior $\mu$ supported on
$\mathcal{S}_{\mu}$}\\}.$ (G.3)
Similarly, for each $h\in[m]$, the support of $\nu_{h}$, denoted by
$\mathcal{S}_{\nu_{h}}\subset\mathbb{R}^{r_{h}}$, can be set at
$\mathcal{S}_{\nu_{h}}:=\left\\{(\widehat{\bm{S}}^{R,\mathrm{pc}}_{0,h})^{-1}({\bm{V}}^{\mathrm{pc}}_{0,h})_{i*}:i\in[p_{h}]\right\\}.$
Thus, each point in the support of $\nu_{h}$ is a row in
$(\widehat{\bm{S}}^{R}_{0,h})^{-1}{\bm{V}}_{0,h}$, and we can approximate
$\mathcal{P}_{\nu_{h}}$ by
$\widehat{\mathcal{P}}_{\nu_{h}}=\\{\mbox{all prior $\nu_{h}$ supported on
$\mathcal{S}_{\nu_{h}}$}\\}.$ (G.4)
By working with these discrete priors with a fixed support, estimating $\mu$
and each $\nu_{h}$ reduces to estimating the mixing proportions in a mixture
of degenerate Gaussians where the number of components equals the number of
support points. |
\node (b') at (0,2) {$ L(e \times_{h} e') $};
\node (a') at (0,0) {$ Lb' $};
\node (wfz) at (3,4) {$ v $};
\node (v'e'y') at (3,2) {$ x \times_{z} x' $};
\node (u'd'z') at (3,0) {$ v' $};
\node (i) at (6,4) {$ Lc $};
\node (h') at (6,2) {$ L(f \times_{i} f') $};
\node (g') at (6,0) {$ Lc' $};
\draw [cd] (c) to (wfz);
\draw [cd] (i) to (wfz);
\draw [cd] (b') to (v'e'y');
\draw [cd] (h') to (v'e'y');
\draw [cd] (a') to (u'd'z');
\draw [cd] (g') to (u'd'z');
\draw [cd] (b') to node[left]{$ \iso $} (c);
\draw [cd] (b') to node[left]{$ \iso $} (a');
\draw [>->] (v'e'y') to (wfz);
\draw [>->] (v'e'y') to (u'd'z');
\draw [cd] (h') to node[right]{$ \iso $} (i);
\draw [cd] (h') to node[right]{$ \iso $} (g');
\end{scope}
\end{tikzpicture}
\]
Lemma <ref> ensures the
marked arrows are monic. The horizontal
composition of these is
\[
\begin{tikzpicture}
\node at (0,2) {
$ ( \alpha \vcirc \beta )
\hcirc
( \alpha' \vcirc \beta' )
= $};
\begin{scope}[shift={(3,0)}]
\node (a) at (0,4) {$ La $};
\node (bcb') at (0,2) {$ Ld \times_{Lg} Ld' $};
\node (a') at (0,0) {$ La' $};
\node (udx) at (4.5,4) {$ u +_{Lb} v $};
\node (middle) at (4.5,2)
{ $ ( w \times_{y} w' )
+_{L(e \times_{h} e')}
( x \times_{z} x') $ };
\node (u'd'z') at (4.5,0) {$ x' +_{Lb'} v' $};
\node (g) at (9,4) {$ Lc $};
\node (hih') at (9,2) {$ Lf +_{Li} Lf' $};
\node (g') at (9,0) {$ Lc' $};
\draw [cd] (a) to (udx);
\draw [cd] (g) to (udx);
\draw [cd] (bcb') to (middle);
\draw [cd] (hih') to (middle);
\draw [cd] (a') to (u'd'z');
\draw [cd] (g') to (u'd'z');
\draw [cd] (bcb') to node[left]{$ \iso $} (a);
\draw [cd] (bcb') to node[left]{$ \iso $} (a');
\draw [>->] (middle) to (udx);
\draw [>->] (middle) to (u'd'z');
\draw [cd] (hih') to node[right]{$ \iso $} (g);
\draw [cd] (hih') to node[right]{$ \iso $} (g');
\end{scope}
\end{tikzpicture}
\]
It follows that the proof of Theorem
<ref> comes down to finding an
\[
(w \times_{y} w') +_{L(e \times_h e')} (x \times_z x')
\to
(w +_{Le} x) \times_{(y +_{Lh} z)} (w' +_{Le'} x')
\]
To simplify our diagrams, we introduce new
notation. We write
\[
\begin{array}{ll}
p \coloneqq
( w \times_y w' ) + ( x \times_z x' ), &
p' \coloneqq
( w \times_y w' ) +_{L(e \times_h e')} ( x' \times_z x' ), \\
q \coloneqq
( w + x ) \times_{ y + z } ( w' + x' ), &
q' \coloneqq
( w +_{Lg} x ) \times_{ y +_{Lh} z } ( w' +_{Li} x' ). \\
\end{array}
\]
In this notation, the isomorphism we
seek is
\begin{equation}
\label{eq:interchange-isomorphism}
\theta' \from p' \to q'
\end{equation}
Also, because $ Lb $, $ Le $, $ Lh $, $ Le' $, $ Lb' $, and
therefore $ L(e \times_{h} e') $ are all isomorphic, we
simply write $ L\ast $ to mean any of these. Each are
interchangeable in the diagrams below, and adjusting this
notation will not cause any false reasoning. While we do lose
the ability to discern between these objects, context should
help the reader determine this. Despite losing this ability, we gain
a breezier exposition and a more readable proof.
Apply Lemma <ref> to
the diagram
\[
\begin{tikzpicture}
\node (B) at (-4,2) {$ w \times_y w'$};
\node (A) at (0,2) {$ L\ast $};
\node (C) at (4,2) {$ x \times_z x'$};
\node (B') at (-4,0) {$ y $};
\node (A') at (0,0) {$ L\ast $};
\node (C') at (4,0) {$ z $};
\draw [cd] (A) edge (B);
\draw [cd] (A) edge node[right] {\scriptsize{$=$}} (A');
\draw [cd] (A) edge (C);
\draw [cd] (A') edge (B');
\draw [cd] (A') edge (C');
\draw [>->] (B) edge (B');
\draw [>->] (C) edge (C');
\end{tikzpicture}
\]
to get the pushout
\[
\begin{tikzpicture}
\node (BC) at (0,2) {$p$};
\node (BAC) at (4,2) {$p'$};
\node (BC') at (0,0) {$ y + z $};
\node (BAC') at (4,0) {$ y +_{L\ast} z $};
\draw [cd] (BC) edge (BAC);
\draw [>->] (BC) edge node[right]{$\psi$} (BC');
\draw [>->] (BAC) edge node[right]{$\psi'$} (BAC');
\draw [cd] (BC') edge (BAC');
\end{tikzpicture}
\]
Similarly, we get pushouts
\[
\begin{tikzpicture}
\begin{scope}
\node (BC) at (0,2) {$p$};
\node (BAC) at (3,2) {$p'$};
\node (BC') at (0,0) {$w+x$};
\node (BAC') at (3,0) {$w+_{L\ast}x$};
\draw [cd] (BC) edge (BAC);
\draw [>->] (BC) edge node[right]{$\sigma$} (BC');
\draw [>->] (BAC) edge node[right]{$\sigma'$} (BAC');
\draw [cd] (BC') edge (BAC');
\end{scope}
\node () at (4,1) {and};
\begin{scope}[shift={(5,0)}]
\node (BC) at (0,2) {$ p $};
\node (BAC) at (3,2) {$ p' $};
\node (BC') at (0,0) {$ w' + x' $};
\node (BAC') at (3,0) {$ w'+_{L\ast} x'$};
\draw [cd] (BC) edge (BAC);
\draw [>->] (BC) edge node[right]{$ \phi $} (BC');
\draw [>->] (BAC) edge node[right]{$ \phi' $} (BAC');
\draw [cd] (BC') edge (BAC');
\end{scope}
\end{tikzpicture}
\]
Now, $p$ forms a cone over the cospan
$ w+x \to y+z \gets w'+x' $ via the maps $ \psi $,
$ \sigma $, and $ \phi $. And so, we get a
canonical map $ \theta \from p \to q $.
The commuting diagram
\[
\begin{tikzpicture}
\node (Lg) at (0,0) {$ Lg $};
\node (Lh) at (2,2) {$ L\ast $};
\node (t) at (2,0) {$ t $};
\node (Li) at (4,0) {$ Li $};
\draw [cd] (Lg) to (Lh);
\draw [cd] (Lg) to (t);
\draw [cd] (Li) to (Lh);
\draw [cd] (Li) to (t);
\draw [>->] (Lh) to (t);
\end{tikzpicture}
\]
induces a canonical isomorphism between $ Lg
\times_{L\ast} Li $ and $ Lg \times_{t} L\ast $.
Via the projection maps, $ Lg \times_{L\ast} Li $ forms a
cone over the cospan $ Lg \to t \gets Li $
and, also, $ Lg \times_{t} Li $ forms a cone over the
cospan $ Lg \to L\ast \gets Li $, though the
latter requires the monic
$ L\ast \rightarrowtail t $ to do so. Universality
implies that the induced maps are mutual
inverses and they are the only such pair.
The map $\theta \from p \to q $ is an
Because colimits are stable under pullback
we get an isomorphism
\[
\gamma \from
( w \times_{ y + z } w' ) +
( w \times_{ y + z } x' ) +
( x \times_{ y + z } w' ) +
( x \times_{ y + z } x' )
\to q.
\]
But $ w \times_{ y + z } x' $ and
$ w' \times_{ y + z } x $ are initial. To see
this, recall that in a topos, all maps to the
initial object are isomorphisms. Now, consider
the diagram
\[
\begin{tikzpicture}
\node (STpb) at (0,4) {$w \times_{y+z} x'$};
\node (T'') at (4,4) {$z'$};
\node (0) at (2,2) {$0$};
\node (T) at (4,2) {$z$};
\node (S') at (0,0) {$w$};
\node (S) at (2,0) {$y$};
\node (ST) at (4,0) {$y+z$};
\draw [cd] (STpb) edge (S');
\draw [cd] (STpb) edge (T'');
\draw [cd] (STpb) edge[dashed] (0);
\draw [cd] (0) edge (S);
\draw [cd] (0) edge (T);
\draw [cd] (S) edge (ST);
\draw [cd] (T) edge (ST);
\draw [cd] (S') edge (S);
\draw [cd] (T'') edge (T);
\end{tikzpicture}
\]
whose lower right square is a pullback because
coproducts are disjoint in topoi. Similarly,
$ x \times_{ y + z } w' $ is initial. Hence
we get a canonical isomorphism
\begin{equation} \label{eq:B second iso}
\gamma'
\from ( w \times_{ y + z } w' ) +
( x \times_{ y + z } x' ) \to q
\end{equation}
that factors through $\gamma$. But Lemma
<ref> gives unique
\[
w \times_{ y } w' \cong
w \times_{ y + z } w'
\text{ and }
x \times_{ z } x' \cong x
\times_{ y + z } x'.
\]
This produces a canonical isomorphism
\[
\gamma'' \from p \to
( w \times_{ y + z } w' ) +
( x \times_{ y + z } x' ).
\]
One can show that
$ \theta = \gamma' \circ \gamma'' $ using
universal properties.
Having shown that $ \theta \from p \to q $ is an
isomorphism, we can write $ p $ in place of
\[ (w+x)
\times_{(y+z)} ( w'+x' )\]
in the following diagram
\begin{equation} \label{diag.the big cube}
\begin{tikzpicture}
\node (A) at (2,3) {$p$};
\node (Ay) at (0,0) {$p'$};
\node (By) at (2,0) {$q'$};
\node (ST) at (8,3) {$y+z$};
\node (ST') at (6,3.75) {$w+x$};
\node (ST'') at (4,2.25) {$w'+x'$};
\node (SYT) at (8,0) {$y+_{L\ast}z$};
\node (SYT') at (6,0.75) {$w+_{Le} x$};
\node (SYT'') at (4,-0.75) {$w'+_{Le'} x'$};
\draw [->>] (A) edge (Ay);
\draw [font=\scriptsize,>->] (A) edge node[below] {$\phi$} (ST'');
\draw [font=\scriptsize,>->] (A) edge node[above] {$\sigma$} (ST');
\draw [font=\scriptsize,cd] (Ay) edge[bend left=20,pos=0.65] node[above] {$\sigma'$} (SYT');
\draw [font=\scriptsize,cd] (Ay) edge[dashed] node[above] {$\theta'$} (By);
\draw [font=\scriptsize,cd] (Ay) edge[bend right=20] node[below] {$\phi'$} (SYT'');
\draw [font=\scriptsize,>->] (By) edge node[above] {$\omega$} (SYT');
\draw [>->] (By) edge (SYT'');
\draw [->>] (ST') edge (SYT');
\draw [>->] (ST') edge (ST);
\draw [>->] (ST'') edge (ST);
\draw [->>] (ST) edge (SYT);
\draw [>->] (SYT') edge (SYT);
\draw [>->] (SYT'') edge (SYT);
\draw [cd] (A) edge[white,line width=4pt] (By);
\draw [font=\scriptsize,cd] (A) edge[dashed] node[right] {$\rho$} (By);
\draw [cd] (ST'') edge[white,line width=6pt] (ST);
\draw [>->] (ST'') edge (ST);
\draw [cd] (ST'') edge[white,line width=6pt] (SYT'');
\draw [->>] (ST'') edge (SYT'');
\end{tikzpicture}
\end{equation}
where $\theta'$ from Equation
(<ref>) finally appears. It and
$\rho$ are the canonical maps arising from the pullback on
the bottom. Observe that $\rho$ factors through $\theta'$
in the above diagram. This follows from the universal
property of pullbacks.
The map
$ \theta' \from p' \to q' $ is an
Because we are working in a topos, it suffices
to show that $\theta'$ is both monic and epic.
It is monic because $\sigma'$ is monic.
To see that $\theta'$ is epic, it suffices to
show that $\rho$ is epic. The front and rear
right faces of (<ref>) are
pushouts by Lemma
<ref>. Then
because the top and bottom squares of
(<ref>) are pullbacks
consisting of only monomorphisms, Lemma
<ref> implies that the front and
rear left faces are pushouts. However, as
pushouts over monomorphisms, Lemma <ref> tells us they are pullbacks. But
in a topos, regular epimorphisms are stable under
pullback, and so $\rho$ is epic.
It remains to show that $\theta'$ serves as an isomorphism
between fine rewrites. This amounts to showing that
\begin{equation} \label{diag.theta 2-cell iso}
\begin{tikzpicture}
\node (La) at (-1.5,2) {$ La $};
\node (X) at (-4,0) {$L(d \times_g d')$};
\node (La') at (-6.5,-2) {$ La $};
\node (LyR) at (2.5,2) {$u +_{L\ast} v$};
\node (Lc) at (6.5,2) {$ Lc $};
\node (Y) at (4,0) {$L(f \times_i f')$};
\node (Lc') at (1.5,-2) {$ Lc' $};
\node (LyR') at (-2.5,-2) {$x'+_{L\ast} v'$};
\node (Ay) at (0,1) {$p'$};
\node (By) at (0,-1) {$q'$};
\draw [cd] (X) edge[] node[above]{$ $} (La);
\draw [cd] (X) edge[] node[above]{$ $} (La');
\draw [cd] (X) edge node[above] {$g$} (Ay);
\draw [cd] (X) edge node[above] {$h$} (By);
\draw [cd] (La) edge[] node[above]{$j$} (LyR);
\draw [cd] (La') edge[] node[below]{$ $} (LyR');
\draw [cd] (Y) edge (Lc);
\draw [cd] (Y) edge (Lc');
\draw [cd] (Y) edge[] (By);
\draw [cd] (Lc) edge[] (LyR);
\draw [cd] (Lc') edge[] (LyR');
\draw [cd] (Ay) edge node[above] {$k$} (LyR);
\draw [cd] (Ay) edge node[right] {$\theta'$} (By);
\draw [font=\scriptsize,cd] (By) edge[pos=0.4] node[below] {$fp$} (LyR);
\draw [cd] (By) edge[] (LyR');
\draw [cd] (Y) edge[white,line width=3.5pt] (Ay);
\draw [cd] (Y) edge[] (Ay);
\draw [cd] (Ay) edge[white,line width=3.5pt] (LyR');
\draw [cd] (Ay) edge[] (LyR');
\end{tikzpicture}
\end{equation}
commutes. Here $g$ and $k$ are induced from
applying vertical composition before horizontal,
$h$ from applying horizontal composition before
vertical, $j$ is from composing in either order,
$f$ is from horizontal composition as given in
Definition <ref> and
$ \omega $ is from (<ref>). The
top and bottom face commute by construction.
The inner triangles of diagram (<ref>) commute. That is, we have
$ k = f \rho \theta' $ and $ h = \theta' g $.
To see that $ k = f \omega \theta' $, consider the
\[
\begin{tikzpicture}
\node (Yt) at (0,2) {$L\ast$};
\node (Yb) at (0,0) {$L\ast$};
\node (Spb) at (2.5,1.25) {$w \times_{y} w'$};
\node (S') at (2.5,-0.75) {$w$};
\node (Tpb) at (5,2.75) {$x \times_zx'$};
\node (T') at (5,0.75) {$x$};
\node (Ay) at (7.5,2) {$p'$};
\node (SyT) at (7.5,0) {$y+_{L\ast}z$};
\node (LyR) at (10,0) {$u+_{L\ast}v$};
\draw [cd] (Yt) edge[] (Tpb);
\draw [cd] (Yt) edge[] (Spb);
\draw [font=\scriptsize,cd] (Yt) edge node[right] {$\iso$} (Yb);
\draw [cd] (Yb) edge[] (T');
\draw [cd] (Yb) edge[] (S');
\draw [cd] (Tpb) edge[] (Ay);
\draw [>->] (Tpb) edge[] (T');
\draw [font=\scriptsize,cd] (Ay) edge[dashed] node[above] {$k$} (LyR);
\draw [cd] (S') edge[] (SyT);
\draw [font=\scriptsize,cd] (S') edge[bend right=20] node[above] {$\iota_us$} (LyR);
\draw [font=\scriptsize,cd] (T') edge[bend left=20] node[above,pos=.2] {$\iota_vt$} (LyR);
\draw [cd] (T') edge[] (SyT);
\draw [font=\scriptsize,cd] (SyT) edge[dashed] node[above] {$f$} (LyR);
\draw [cd] (Spb) edge[white,line width=4pt] (Ay);
\draw [cd] (Spb) edge[] (Ay);
\draw [>->] (Spb) edge[white,line width=4pt] (S');
\draw [>->] (Spb) edge[] (S');
\draw [cd] (Ay) edge[white,line width=4pt] (SyT);
\draw [font=\scriptsize,cd] (Ay) edge[dashed] node[pos=0.75,left] {$\sigma'$} (SyT);
\end{tikzpicture}
\]
The bottom face is exactly the pushout diagram
from which $f$ was obtained. Universality
implies that $ k = f \sigma' $ and, as seen in
(<ref>), $ \sigma' = \rho \theta' $.
That $ h = \theta' g $ follows from
\[
f \rho h = j = k g = f \rho \theta' g
\]
and the fact that $ f \rho $ is monic.
Of course, we have only shown that two of the four
inner triangles commute, but we can replicate our
arguments to show the remaining two commute as
well. This lemma was the last step in proving
Theorem <ref>, the interchange
§ A SYMMETRIC MONOIDAL STRUCTURE
The double category $ _L\FFFineRewrite $ can be equipped
with a symmetric monoidal structure lifted from the
cocartesian structure on $ \A $ and $ \X $. Proving this
amounts to checking the axioms of Definition
$ _L\FFFineRewrite $ is a symmetric monoidal double category.
We denote $ _L\FFFineRewrite $ by $ \RRR $ for
convenience. Let us first show that the category of
objects $\RRR_0$ and the category of arrows $\RRR_1$ are
symmetric monoidal categories.
We obtain the monoidal structure $ (\otimes_0, 0_{\A}) $ on $\RRR_{0}$ by lifting
the cocartesian structure on $\A$ to the objects and by
\[
(a \xgets{f} b \xto{g} c) \otimes_0 (a' \xgets{f'} b' \xto{g'} c')
\bydef
(a+a' \xgets{f+g} b+b' \xto{f'+g'} c+c')
\]
on morphisms. Universal properties provide the associator
and unitors as well as the coherence axioms. This
monoidal structure is clearly symmetric.
Next, we have the category $\RRR_1$ whose objects are the
structured cospans and morphisms are their fine rewrites.
We obtain a symmetric monoidal structure
\[ ( \otimes_1 , L0_{\A} \to L0_{\A} \gets L0_{\A} ) \]
the objects via
\[
(La \to x \gets La') \otimes_1 (Lb \to y \gets Lb')
\bydef
(L(a+b) \to x+y \gets L(a'+b'))
\]
and on the morphisms by
\[
\raisebox{-0.5\height}{
\begin{tikzpicture}
\node (A) at (0,4) {$La$};
\node (A') at (0,2) {$Lb$};
\node (A'') at (0,0) {$Lc$};
\node (B) at (2,4) {$v$};
\node (B') at (2,2) {$w$};
\node (B'') at (2,0) {$x$};
\node (C) at (4,4) {$La'$};
\node (C') at (4,2) {$Lb'$};
\node (C'') at (4,0) {$Lc'$};
\draw[cd]
% horizontal arrows
(A) edge node[above]{} (B)
(A') edge node[above]{} (B')
(A'') edge node[above]{} (B'')
(C) edge node[above]{} (B)
(C') edge node[above]{} (B')
(C'') edge node[above]{} (B'')
% vertical arrows
(A') edge node[left]{} (A)
(A') edge node[left]{} (A'')
(B') edge[>->] node[left]{} (B)
(B') edge[>->] node[left]{} (B'')
(C') edge node[left]{} (C)
(C') edge node[left]{} (C'');
\end{tikzpicture}
\quad \otimes_1 \quad
\raisebox{-0.5\height}{
\begin{tikzpicture}
\node (A) at (0,4) {$La''$};
\node (A') at (0,2) {$Lb''$};
\node (A'') at (0,0) {$Lc''$};
\node (B) at (2,4) {$v'$};
\node (B') at (2,2) {$w'$};
\node (B'') at (2,0) {$x'$};
\node (C) at (4,4) {$La'''$};
\node (C') at (4,2) {$Lb'''$};
\node (C'') at (4,0) {$Lc'''$};
\draw[cd]
% horizontal arrows
(A) edge node[above]{} (B)
(A') edge node[above]{} (B')
(A'') edge node[above]{} (B'')
(C) edge node[above]{} (B)
(C') edge node[above]{} (B')
(C'') edge node[above]{} (B'')
% vertical arrows
(A') edge node[left]{} (A)
(A') edge node[left]{} (A'')
(B') edge[>->] node[left]{} (B)
(B') edge[>->] node[left]{} (B'')
(C') edge node[left]{} (C)
(C') edge node[left]{} (C'');
\end{tikzpicture}
\quad \bydef \quad
\]
\[
\raisebox{-0.5\height}{
\begin{tikzpicture}
\node (A) at (0,4) {$L(a+a'')$};
\node (A') at (0,2) {$L(b+b'')$};
\node (A'') at (0,0) {$L(c+c'')$};
\node (B) at (3,4) {$v+v'$};
\node (B') at (3,2) {$w+w'$};
\node (B'') at (3,0) {$x+x'$};
\node (C) at (6,4) {$L(a'+a''')$};
\node (C') at (6,2) {$L(b'+b''')$};
\node (C'') at (6,0) {$\L(c'+c''')$};
\draw[cd]
% horizontal arrows
(A) edge node[above]{} (B)
(A') edge node[above]{} (B')
(A'') edge node[above]{} (B'')
(C) edge node[above]{} (B)
(C') edge node[above]{} (B')
(C'') edge node[above]{} (B'')
% vertical arrows
(A') edge node[left]{} (A)
(A') edge node[left]{} (A'')
(B') edge[>->] node[left]{} (B)
(B') edge[>->] node[left]{} (B'')
(C') edge node[left]{} (C)
(C') edge node[left]{} (C'');
\end{tikzpicture}
\]
Again, universal properties provide the associator,
unitors, and coherence axioms. Hence both $\RRR_0$
and $\RRR_1$ are symmetric monoidal categories.
It remains to find globular isomorphisms $\mathfrak{x}$
and $\mathfrak{u}$ and their coherence. To find
$\mathfrak{x}$, fix horizontal 1-morphisms
\begin{align*}
La & \to x \gets La', & La' & \to x' \gets La'', \\
Lb & \to y \gets Lb', & Lb' & \to y' \gets Lb''.
\end{align*}
The globular isomorphism $\mathfrak{x}$ is an invertible 2-morphism with domain
\[
L(a+b) \to (x+y) +_{L(a'+b')} (x'+y') \gets L(a''+b'')
\]
and codomain
\[
L(a+b) \to (x+_{La'} y) + (x' +_{Lb'} y') \gets L(a''+b'')
\]
This comes down to finding an isomorphism in $\X$
between the apexes of the above cospans. Such an
isomorphism exists, and is unique, because both apexes are
colimits of the non-connected diagram
\[
\begin{tikzpicture}
\node (a) at (0,0) {$La$};
\node (b) at (1,.5) {$x$};
\node (c) at (2,0) {$La'$};
\node (d) at (3,.5) {$x'$};
\node (e) at (4,0) {$La''$};
\node (v) at (5,0) {$Lb$};
\node (w) at (6,.5) {$y$};
\node (x) at (7,0) {$Lb'$};
\node (y) at (8,.5) {$y'$};
\node (z) at (9,0) {$Lb''$};
\path[cd,font=\scriptsize,>=angle 90]
(a) edge node[above]{$$} (b)
(c) edge node[above]{$$} (b)
(c) edge node[above]{$$} (d)
(e) edge node[above]{$$} (d)
(v) edge node[above]{$$} (w)
(x) edge node[above]{$$} (w)
(x) edge node[above]{$$} (y)
(z) edge node[above]{$$} (y);
\end{tikzpicture}
\]
Moreover, the resulting globular isomorphism is a fine
rewrite of structured cospans because the universal maps are
isomorphisms. The globular isomorphism $\mathfrak{u}$ is
Finally, we check that the coherence axioms, namely
(a)-(k) of Definition \ref{def:mndl-dbl-cats},
hold. These are straightforward, though tedious, to
verify. For instance, if we have
\[
\begin{tikzpicture}
\node (a) at (0,0) {$La$};
\node (b) at (1,.5) {$x$};
\node (c) at (2,0) {$La'$};
\node (M1) at (-1,.25) {$M_1 =$};
\node (M2) at (3,.25) {$M_2 =$};
\node (c2) at (4,0) {$La'$};
\node (d) at (5,.5) {$x'$};
\node (e) at (6,0) {$La''$};
\node (M3) at (7,.25) {$M_3 =$};
\node (e2) at (8,0) {$La''$};
\node (f) at (9,.5) {$x''$};
\node (g) at (10,0) {$La'''$};
\node (t) at (0,-2) {$Lb$};
\node (u) at (1,-1.5) {$y$};
\node (v) at (2,-2) {$Lb'$};
\node (N1) at (-1,-1.75) {$N_1 =$};
\node (N2) at (3,-1.75) {$N_2 =$};
\node (v2) at (4,-2) {$Lb'$};
\node (w) at (5,-1.5) {$y'$};
\node (x) at (6,-2) {$Lb''$};
\node (N3) at (7,-1.75) {$N_3 =$};
\node (x2) at (8,-2) {$Lb''$};
\node (y) at (9,-1.5) {$y''$};
\node (z) at (10,-2) {$Lb'''$};
\path[cd,font=\scriptsize,>=angle 90]
(a) edge node[above]{$$} (b)
(c) edge node[above]{$$} (b)
(c2) edge node[above]{$$} (d)
(e) edge node[above]{$$} (d)
(v2) edge node[above]{$$} (w)
(e2) edge node[above]{$$} (f)
(g) edge node[above]{$$} (f)
(t) edge node[above]{$$} (u)
(v) edge node[above]{$$} (u)
(x) edge node[above]{$$} (w)
(x2) edge node[above]{$$} (y)
(z) edge node[above]{$$} (y);
\end{tikzpicture}
\]
then following Diagram \eqref{diag:MonDblCat} around the
top right gives the sequence of cospans
\[
\begin{tikzpicture}
\node (a) at (-5,0) {$L(a+b)$};
\node (b) at (0,0.5) {
$((x+y) +_{ L( a' + b') } (x'+y'))
+_{ L (a'' + b'') } (x''+y'') $ };
\node (c) at (5,0) {$L(a''+b'')$};
\node (M1) at (-2,1.5) {
$((M_1 \otimes N_1) \odot
(M_2 \otimes N_2)) \odot
(M_3 \otimes N_3) = $ };
\path[cd,font=\scriptsize]
(a) edge[in=180,out=45] node[above]{$$} (b)
(c) edge[in=0,out=135] node[above]{$$} (b);
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\node (a) at (-5,0) {$L(a+b)$};
\node (b) at (0,0.5) {
$ ( (x +_{La'} x') +
( y +_{Lb'} y')) +_{ L(a''+b'') }
(x''+y'') $ };
\node (c) at (5,0) {$ L(a''+b'') $};
\node (M1) at (-2,1.5) {
$((M_1 \odot M_2) \otimes
(N_1 \odot N_2)) \odot
(M_3 \otimes N_3) = $ };
\path[cd,font=\scriptsize]
(a) edge[in=180,out=45] node[above]{$$} (b)
(c) edge[in=0,out=135] node[above]{$$} (b);
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\node (a) at (-5,0) { $ L(a+b) $ };
\node (b) at (0,0.5) {
$ ( ( x +_{La'} x') +_{La''} x'' ) +
( ( y +_{Lb'} y' ) +_{Lb''} y'' ) $ };
\node (c) at (5,0) {$ L(a''+b'') $};
\node (M1) at (-2,1.5) {
$((M_1 \odot M_2) \odot M_3) \otimes
((N_1 \odot N_2) \odot N_3) = $};
\path[cd,font=\scriptsize]
(a) edge[in=180,out=45] node[above]{$$} (b)
(c) edge[in=0,out=135] node[above]{$$} (b);
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\node (a) at (-5,0) {$L(a+b)$};
\node (b) at (0,0.5) {
$ ( x +_{La'} ( x' +_{La''} x'' )) +
( y +_{Lb'} ( y' +_{Lb''} y'' ) ) $ };
\node (c) at (5,0) {$ L(a''+b'') $};
\node (M1) at (-2,1.5) {
$( M_1 \odot (M_2 \odot M_3)) \otimes
(N_1 \odot (N_2 \odot N_3)) = $ };
\path[cd,font=\scriptsize]
(a) edge[in=180,out=45] node[above]{$$} (b)
(c) edge[in=0,out=135] node[above]{$$} (b);
\end{tikzpicture}
\]
Following the diagram \eqref{diag:MonDblCat} around the
bottom left gives another sequence of cospans
\[
\begin{tikzpicture}
\node (a) at (-5,0) {$ L(a + b) $};
\node (b) at (0,0.5) {
$ ( (x + y) +_{L (a'+b')} (x'+y'))
+_{L(a''+b'')} (x'' + y'' ) $ };
\node (c) at (5,0) {$ L(a'' + b'') $};
\node (M1) at (-2,1.5) {
$ ((M_1 \otimes N_1) \odot
(M_2 \otimes N_2)) \odot
(M_3 \otimes N_3) = $ };
\path[cd,font=\scriptsize]
(a) edge[in=180,out=45] node[above]{$$} (b)
(c) edge[in=0,out=135] node[above]{$$} (b);
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\node (a) at (-5,0) {$L(a+b)$};
\node (b) at (0,0.5) {
$ (x + y) +_{L(a'+b')}
( (x' + y') +_{ L(a''+b'') } (x'' + y'') )$};
\node (c) at (5,0) {$L(a'''+b''')$};
\node (M1) at (-2,1.5) {
$(M_1 \otimes N_1) \odot
((M_2 \otimes N_2) \odot
(M_3 \otimes N_3)) = $ };
\path[cd,font=\scriptsize]
(a) edge[in=180,out=45] node[above]{$$} (b)
(c) edge[in=0,out=135] node[above]{$$} (b);
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\node (a) at (-5,0) {$ L(a+b) $};
\node (b) at (0,0.5) {
$ (x+y) +_{L(a'+b')}
( (x' +_{La''} x'') + ( y' +_{Lb''} y'') ) $};
\node (c) at (5,0) {$ L(a'''+b''') $};
\node (M1) at (-2,1.5) {
$(M_1 \otimes N_1) \odot
((M_2 \odot M_3) \otimes
(N_2 \odot N_3)) = $};
\path[cd,font=\scriptsize]
(a) edge[in=180,out=45] node[above]{$$} (b)
(c) edge[in=0,out=135] node[above]{$$} (b);
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\node (a) at (-5,0) {$ L(a+b) $};
\node (b) at (0,.5) {
$ ( x +_{La'} ( x' +_{La''} x'' )) +
( y +_{Lb'} ( y' +_{Lb''} y'' ) )$};
\node (c) at (5,0) {$ L(a'''+b''') $};
\node (M1) at (-2,1.5) {
$(M_1 \odot (M_2 \odot M_3))
\otimes (N_1 \odot (N_2 \odot N_3)) = $};
\path[cd,font=\scriptsize]
(a) edge[in=180,out=45] node[above]{$$} (b)
(c) edge[in=0,out=135] node[above]{$$} (b);
\end{tikzpicture}
\]
Putting these together gives the following commutative diagram.
\[
\begin{tikzpicture}
\node (a) at (-6,0) {$ L(a+b) $};
\node (b) at (0,0) {
$ ( (x+y) +_{L(a'+b')} (x'+y')) +_{L(a''+b'')} (x''+y'')$};
\node (c) at (6,0) {$L(a'''+b''')$};
\node (a2) at (-6,1.5) {$L(a+b)$};
\node (b2) at (0,1.5) {
$ (( x +_{La'} x')+ ( y+_{Lb'} y')) +_{L(a''+b'')} (x''+y'')$};
\node (c2) at (6,1.5) {$L(a'''+b''')$};
\node (a3) at (-6,3) {$L(a+b)$};
\node (b3) at (0,3) {
$ (( x +_{La'} x') +_{La''} x'')+ (( y +_{Lb'} y') +_{Lb''} y'')$};
\node (c3) at (6,3) {$L(a'''+b''')$};
\node (a4) at (-6,4.5) {$L(a+b)$};
\node (b4) at (0,4.5) {
$ ( x +_{La'} (x'+_{La''} x'')) + ( y +_{Lb'} ( y'+_{Lb''} y''))$};
\node (c4) at (6,4.5) {$L(a'''+b''')$};
\node (a5) at (-6,-1.5) {$L(a+b)$};
\node (b5) at (0,-1.5) {
$ (x+y) +_{L(a'+b')} ( (x'+y') +_{L(a''+b'')} (x''+y''))$};
\node (c5) at (6,-1.5) {$L(a'''+b''')$};
\node (a6) at (-6,-3) {$L(a+b)$};
\node (b6) at (0,-3) {
$(x+y)+_{L(a'+b')} ( (x' +_{la''} x'') + ( y' +_{La''} y''))$};
\node (c6) at (6,-3) {$L(a'''+b''')$};
\node (a7) at (-6,-4.5) {$L(a+b)$};
\node (b7) at (0,-4.5) {
$( x +_{La'} (x' +_{La''} x'')) + ( y +_{Lb'} (y'+_{Lb''} y''))$};
\node (c7) at (6,-4.5) {$L(a'''+b''')$};
\path[cd,font=\scriptsize,>=angle 90]
(a) edge node[above]{$$} (b)
(c) edge node[above]{$$} (b)
(a2) edge node[above]{$$} (b2)
(c2) edge node[above]{$$} (b2)
(a) edge node[above]{$$} (a2)
(b) edge node[above]{$$} (b2)
(c) edge node[above]{$$} (c2)
(a3) edge node[above]{$$} (b3)
(c3) edge node[above]{$$} (b3)
(a2) edge node[above]{$$} (a3)
(b2) edge node[above]{$$} (b3)
(c2) edge node[above]{$$} (c3)
(a4) edge node[above]{$$} (b4)
(c4) edge node[above]{$$} (b4)
(a3) edge node[above]{$$} (a4)
(b3) edge node[above]{$$} (b4)
(c3) edge node[above]{$$} (c4)
(a5) edge node[above]{$$} (b5)
(c5) edge node[above]{$$} (b5)
(a) edge node[above]{$$} (a5)
(b) edge node[above]{$$} (b5)
(c) edge node[above]{$$} (c5)
(a6) edge node[above]{$$} (b6)
(c6) edge node[above]{$$} (b6)
(a5) edge node[above]{$$} (a6)
(b5) edge node[above]{$$} (b6)
(c5) edge node[above]{$$} (c6)
(a7) edge node[above]{$$} (b7)
(c7) edge node[above]{$$} (b7)
(a6) edge node[above]{$$} (a7)
(b6) edge node[above]{$$} (b7)
(c6) edge node[above]{$$} (c7);
\end{tikzpicture}
\]
The vertical 1-morphisms on the left and right are the the
respective identity spans on $L(a+b)$ and $L(a'''+b''')$. The
vertical 1-morphisms in the center are isomorphism classes
of monic spans where each leg is given by a universal map
between two colimits of the same diagram. The horizontal
1-morphisms are given by universal maps into coproducts
and pushouts. The top cospan is the same as the bottom
cospan, making a bracelet-like figure in which all faces
commute. The other diagrams witnessing coherence are
given in a similar fashion.
\end{proof}
% =====================================================
% CHAPTER: spans of cospans
% SECTION: a compact closed bicategory
% =====================================================
\section{A compact closed bicategory of spans of cospans}
\label{sec:compact-closed-bicategory-spans-of-cospans}
Double categories have many nice features yet are not as
established in the world of higher categories as
bicategories. For those who more comfortable with
bicategories, we write this section to discuss a bicategory
of fine rewrites of structured cospans. Intuitively, it is
straightforward to pass from the double category
$ _L\FFFineRewrite $ to a bicategory of fine rewrites. By
only accepting the squares of $ _L\FFFineRewrite $ that fix
the inputs and outputs, that is disallow permutations, then
the only vertical arrows left are identities. But a double
category with only identity vertical arrows is virtually a
bicategory. Care is needed, though, because to actually
remove a bicategory of fine rewrites from
$ _L\FFFineRewrite $ requires more rigor than simply picking
out only the vertical arrows that are the identity.
More than a bicategory, we can actually extract a compact closed
bicategory from the symmetric monoidal double category
$ _L\FFFineRewrite $. To obtain a symmetric monoidal
bicategory from $ _L\FFFineRewrite $, we use machinery
developed by Shulman [58]. To show
that this bicategory is also compact closed, we use work by
Stay [59].
First, let us extract the `horizontal bicategory' of
$ _L\FFFineRewrite $, so named because we remove the
vertical arrows.
\begin{definition}
Define $ _L\FFineRewrite $ to be the bicategory whose
objects are the objects of $ \A $, 1-arrows are structured
cospans, and 2-arrows are fine rewrite rules of form
\begin{center}
\begin{tikzpicture}
\node (La) at (0,4) {$ La $};
\node (x) at (2,4) {$ x $};
\node (Lb) at (4,4) {$ Lb $};
\node (La') at (0,2) {$ La $};
\node (y) at (2,2) {$ y $};
\node (Lb') at (4,2) {$ Lb $};
\node (La'') at (0,0) {$ La $};
\node (z) at (2,0) {$ z $};
\node (Lb'') at (4,0) {$ Lb $};
\draw [cd]
(La) edge[] (x)
(Lb) edge[] (x)
(La') edge[] (y)
(Lb') edge[] (y)
(La'') edge[] (z)
(Lb'') edge[] (z)
(La') edge[] node[left]{$ \id $} (La)
(La) edge[] node[left]{$ \id $} (La'')
(y) edge[>->] (x)
(y) edge[>->] (z)
(Lb') edge[] node[right]{$ \id $} (Lb)
(Lb') edge[] node[right]{$ \id $} (Lb'');
\end{tikzpicture}
\end{center}
\end{definition}
That this is a double category follows from Shulman's
construction mentioned in Definition
\ref{def:horiz-bicat}. Had we used the same notation as that
definition, we would let $ _L\FFineRewrite \bydef
\mathcal{H} ( _L\FFFineRewrite ) $.
Shulman's construction continues to be useful, as we use it
to show that the double category $ _L\FFineRewrite $ is
symmetric monoidal. The first step towards this is showing
that $ _L\FFFineRewrite $ is isofibrant (see Definition
\ref{def:Fibrant}).
% DOUBLE CATEGORIES ARE ISOFIBRANT
\begin{lemma}
\label{lem:SpanCospanIsofibrant}
The symmetric monoidal double category $ _L\FFFineRewrite $
is isofibrant.
\end{lemma}
\begin{proof}
The companion of a vertical 1-morphism
\[
f = (a \xgets{\theta} b \xto{\psi} c)
\]
is given by
\[
\widehat{f} =
( La \xto{ L\theta^{-1} } Lb \xgets{ L\psi^{-1} } Lc )
\]
The required 2-arrows are given by
\[
\raisebox{-0.5\height}{
\begin{tikzpicture}
\node (A) at (0,4) {$La$};
\node (A') at (0,2) {$Lb$};
\node (A'') at (0,0) {$Lc$};
\node (B) at (2,4) {$Lb$};
\node (B') at (2,2) {$Lc$};
\node (B'') at (2,0) {$Lc$};
\node (C) at (4,4) {$Lc$};
\node (C') at (4,2) {$Lc$};
\node (C'') at (4,0) {$Lc$};
\draw[cd]
% horizontal arrows
(A) edge node[above]{} (B)
(A') edge node[above]{} (B')
(A'') edge node[above]{} (B'')
(C) edge node[above]{} (B)
(C') edge node[above]{} (B')
(C'') edge node[above]{} (B'')
% vertical arrows
(A') edge node[left]{} (A)
(A') edge node[left]{} (A'')
(B') edge node[left]{} (B)
(B') edge node[left]{} (B'')
(C') edge node[left]{} (C)
(C') edge node[left]{} (C'');
\end{tikzpicture}
\quad \text{ and } \quad
\raisebox{-0.5\height}{
\begin{tikzpicture}
\node (A) at (0,4) {$La$};
\node (A') at (0,2) {$La$};
\node (A'') at (0,0) {$La$};
\node (B) at (2,4) {$La$};
\node (B') at (2,2) {$La$};
\node (B'') at (2,0) {$Lb$};
\node (C) at (4,4) {$La$};
\node (C') at (4,2) {$Lb$};
\node (C'') at (4,0) {$Lc$};
\path[cd,font=\scriptsize,>=angle 90]
% horizontal arrows
(A) edge node[above]{} (B)
(A') edge node[above]{} (B')
(A'') edge node[above]{} (B'')
(C) edge node[above]{} (B)
(C') edge node[above]{} (B')
(C'') edge node[above]{} (B'')
% vertical arrows
(A') edge node[left]{} (A)
(A') edge node[left]{} (A'')
(B') edge node[left]{} (B)
(B') edge node[left]{} (B'')
(C') edge node[left]{} (C)
(C') edge node[left]{} (C'');
\end{tikzpicture}
\]
The conjoint of $f$ is given by $\check{f} = \widehat{f}^{\text{op}}$.
\end{proof}
Because the symmetric monoidal double category
$ _L\FFFineRewrite $ is isofibrant, Theorem \ref{thm:horz-bicat}
extracts a symmetric monoidal bicategory $ _L\FFineRewrite $
comprised of the same objects, structured cospans as arrows,
and isomorphism classes of fine rewrites of structured
cospans with form
\[
\begin{tikzpicture}
\node (A) at (0,4) {$La$};
\node (A') at (0,2) {$La$};
\node (A'') at (0,0) {$La$};
\node (B) at (2,4) {$v$};
\node (B') at (2,2) {$w$};
\node (B'') at (2,0) {$x$};
\node (C) at (4,4) {$La'$};
\node (C') at (4,2) {$La'$};
\node (C'') at (4,0) {$La'$};
\path[cd,font=\scriptsize,>=angle 90]
% horizontal arrows
(A) edge node[above]{} (B)
(A') edge node[above]{} (B')
(A'') edge node[above]{} (B'')
(C) edge node[above]{} (B)
(C') edge node[above]{} (B')
(C'') edge node[above]{} (B'')
% vertical arrows
(A') edge node[left]{\scriptsize{$ \id $}} (A)
(A') edge node[left]{\scriptsize{$ \id $}} (A'')
(B') edge[>->] node[left]{} (B)
(B') edge[>->] node[left]{} (B'')
(C') edge node[right]{\scriptsize{$ \id $}} (C)
(C') edge node[right]{\scriptsize{$ \id $}} (C'');
\end{tikzpicture}
\]
The difference between these fine rewrites and the squares
of $ _L\FFFineRewrite $ is that the vertical arrows are
identities. This is necessary given that bicategories have
no vertical arrows. However, the isofibrancy condition
ensures that information carried by the vertical arrows is
encoded the horizontal arrows.
\begin{theorem}
\label{thm:SpansCospasAreSMBicat}
$ _L\FFineRewrite$ is a symmetric monoidal bicategory.
\end{theorem}
\begin{proof}
Lemma \ref{lem:SpanCospanIsofibrant} states that $
_L\FFFineRewrite $ is isofibrant. The result then follows
from Theorem \ref{thm:horz-bicat}.
\end{proof}
It remains to show that this bicategory is compact
closed. This structure of $ _L\FFineRewrite $ is another
benefit of bicategories over double categories. Currently,
there is no notion of compact closedness for double
categories. However, it is a nice feature to have in a
category that serves as the syntax for open systems with
inputs and outputs. Here, we mention again that the terms
`inputs' and `outputs' do not imply a causal structure.
Instead, they partition the interface of an open system into
two parts, the purpose of which manifests when composing a
pair of systems. If we connect an open system, considered as
an structured cospan $ La \to x \gets La' $, to another
system, then $ La $ is parts of the connection and $ La' $
is not or vice versa. That is, partitioning an interface
into inputs and outputs allows a portion of the interface to
be part of a connection and the remain portion to be left
out of the connection. Compact closedness formalizes the
viewpoint that \emph{how} an interface is partitioned is
arbitrary. Indeed, every possible partition of the
interface exists as an arrow in $ _L\FFineRewrite $. That
is, given a system $ x $ with interface $ i $, then for any
two subobjects $ a $, $ a' $ of $ i $ such that
$ a+a' \cong i $, there is an an arrow
$ La \to x \gets La' $ in $ _L\FFineRewrite $.
\begin{example}
Denote by $ x $ the graph
\[
\begin{tikzpicture}
\node (1) at (0,0) {$ \bullet_a $};
\node (2) at (0,2) {$ \bullet_b $};
\node (3) at (2,0) {$ \bullet_c $};
\node (4) at (2,2) {$ \bullet_d$};
\draw[graph]
(1) edge (2)
(1) edge (3)
(3) edge[loop right] (3)
(4) edge (2)
(1) edge (4);
\end{tikzpicture}
\]
with interface $ \{a,c,d\} $. Then $ x $ appears as
an arrow in $ _L\FFineRewrite $ where $ L $ is from
\[
\adjunction{\Set}{\RGraph}{L}{R}{4}
\]
as all of the following
\begin{align*}
\{ a,c,d \} & \to x \gets 0 &
\{ a,c \} & \to x \gets \{ d \} \\
\{ a,d \} & \to x \gets \{ c \} &
\{ c,d \} & \to x \gets \{ a \} \\
\{ a \} & \to x \gets \{ c,d \} &
\{ c \} & \to x \gets \{ a,d \} \\
\{ d \} & \to x \gets \{ a,c \} &
0 & \to x \gets \{ a,b,c\}
\end{align*}
\end{example}
The ability to change an input to an output and vice versa
comes from the compact closed structure. We take the
remainder of this section to show that $ _L\FFineRewrite $
is compact closed.
We start with the following lemma. For this lemma, we
introduce the notation $ \nabla \from a+a \to a $ for the
folding map, which arises from the coproduct diagram
\[
\begin{tikzpicture}
\node (x1) at (-3,2) {$ a $};
\node (xx) at (0,2) {$ a+a $};
\node (x2) at (3,2) {$ a $};
\node (x) at (0,0) {$ a $};
\draw [cd] (x1) to
node [above] {\scriptsize{$ \iota $}} (xx);
\draw [cd] (x2)
to node [above] {\scriptsize{$ \iota $}} (xx);
\draw [cd] (x1) to
node [left] {\scriptsize{$ \id $}} (x);
\draw [cd] (x2) to
node [right] {\scriptsize{$ \id $}} (x);
\draw [cd,dashed] (xx) to node [left] {\scriptsize{$ \nabla $}} (x);
\end{tikzpicture}
\]
\begin{lemma}
\label{lem:PushoutDiagram}
In a category with coproducts, the diagram
\[
\begin{tikzpicture}
\node (UL) at (0,2) {$a+a+a$};
\node (LL) at (0,0) {$a+a$};
\node (UR) at (4,2) {$a+a$};
\node (LR) at (4,0) {$a$};
\path[cd,font=\scriptsize,>=angle 90]
(UL) edge node[above] {$\id + \nabla$} (UR)
(UL) edge node[left] {$\nabla + \id$} (LL)
(UR) edge node[right] {$\nabla$} (LR)
(LL) edge node[above] {$\nabla$} (LR);
\end{tikzpicture}
\]
is a pushout square.
\end{lemma}
\begin{proof}
Suppose that we have two maps $ f,g \from a+a \to b$
forming a cocone over the span inside the above diagram.
Let the arrow $\iota_{\text{m}} \from a \to a+a+a$ include
$a$ into the middle copy. Observe that
$\iota_{\text{l}} \coloneqq (\nabla + a) \circ \iota_{\text{m}}
$ and $\iota_r \coloneqq (a + \nabla) \circ \iota_{\text{m}} $ are,
respectively, the left and right inclusions $a \to a+a$.
Then $f \circ \iota_{\text{l}} = g \circ \iota_{\text{r}}$ is a
map $a \to b$, which we claim is the unique map making
\begin{center}
\begin{tikzpicture}
\node (UL) at (0,2) {$a+a+a$};
\node (LL) at (0,0) {$a+a$};
\node (UR) at (4,2) {$a+a$};
\node (LR) at (4,0) {$a$};
\node (b) at (6,-2) {$ b $};
\draw [cd]
(LR) edge[dashed] (b)
(LL) edge[bend right] (b)
(UR) edge[bend left] (b)
(UL) edge node[above] {$\id + \nabla$} (UR)
(UL) edge node[left] {$\nabla + \id$} (LL)
(UR) edge node[right] {$\nabla$} (LR)
(LL) edge node[above] {$\nabla$} (LR);
\end{tikzpicture}
\end{center}
commute. Indeed, given $h \from a \to b$ such that
$f = h \circ \nabla = g$, then
$g \circ \iota_
{ \text{r} } = f \circ \iota_{ \text{l} } = h \circ
\nabla \circ \iota_{ \text{l} } = h$.
\end{proof}
In the following theorem, we will make a slight abuse of
notation by writing $ \nabla $ to mean
\[
L(a+a) \to La + La \xto{\nabla} La.
\]
Here, $ L(a+a) \to La+La $ is the structure map which is
invertible because, as a left adjoint, $ L $ preserves coproducts.
\begin{theorem}
\label{thm:SpansMapsAreCCBicat}
The symmetric monoidal bicategory $ _L\FFineRewrite $ is compact closed.
\end{theorem}
\begin{proof}
First we show that each object is its own dual.
For an object $a$, define the counit
$\epsilon \from a + a \to 0$ and unit
$\eta \from 0 \to a+a$ to be the following cospans:
\[
\epsilon \bydef (L(a+a) \xto{\nabla} La \gets 0),
\quad \quad
\eta \bydef (0 \to La \xgets{\nabla} L(a+a)).
\]
Next we define the cusp isomorphisms, $\alpha$
and $\beta$. Note that $\alpha$ is a 2-morphism
whose domain is the composite
\[
a \xto{ \iota_{ \text{l} } }
a+a \xgets{ \id +\nabla }
a+a+a \xto{ \nabla + \id }
a+a \xgets{ \iota_{ \text{r} } }
\]
and whose codomain is the identity cospan on $a$. From
Lemma \ref{lem:PushoutDiagram} we have the equations
$\nabla+\id = \iota_{\text{l}} \circ \nabla$ and
$\id + \nabla = \iota_{\text{r}} \circ \nabla$ from which it
follows that the domain of $\alpha$ is the identity cospan
on $a$, and the codomain of $\beta$ is also the identity
cospan on $a$ obtained as the composite
\[
a \xto{\iota_{\text{r}}}
a+a \xgets{\nabla+ \id}
a+a+a \xto{X+\nabla}
a+a \xgets{\iota_{\text{l}}}
\]
Take $\alpha$ and $\beta$ each to be the
isomorphism class determined by the identity
2-morphism on $a$, which in particular is a
monic span of cospans. Thus we have a dual pair
$(a,a,\epsilon,\eta,\alpha,\beta)$. By Theorem
\ref{thm:StrictingDualPairs}, there exists a
cusp isomorphism $\beta'$ such that the tuple
$(a,a,\epsilon,\eta,\alpha,\beta')$ is a coherent dual
pair, and thus $ _L \FFineRewrite $ is
compact closed.
\end{proof}
%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% CHAPTER %%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% BOLD REWRITING %%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%
\chapter{Bold rewriting and structured cospans}
\label{sec:bold-rewriting}
We contrast this section with the previous section on fine
rewriting with an example. In the fine rewriting of
structured cospans, we ask for rewrite rules with the monic
arrows as in the diagram
\[
\begin{tikzpicture}
\node (1) at (0,4) {$ La $};
\node (2) at (2,4) {$ x $};
\node (3) at (4,4) {$ La' $};
\node (4) at (0,2) {$ Lb $};
\node (5) at (2,2) {$ y $};
\node (6) at (4,2) {$ Lb' $};
\node (7) at (0,0) {$ Lc $};
\node (8) at (2,0) {$ z $};
\node (9) at (4,0) {$ Lc' $};
\path[cd,font=\scriptsize]
(1) edge node[]{$ $} (2)
(3) edge node[]{$ $} (2)
(4) edge node[]{$ $} (5)
(6) edge node[]{$ $} (5)
(7) edge node[]{$ $} (8)
(9) edge node[]{$ $} (8)
(4) edge node[left]{$ \iso $} (1)
(4) edge node[left]{$ \iso $} (7)
(6) edge node[right]{$ \iso $} (3)
(6) edge node[right]{$ \iso $} (9);
\path[>->,font=\scriptsize]
(5) edge node[]{$ $} (2)
(5) edge node[]{$ $} (8);
\end{tikzpicture}
\]
There are situations, however, where requiring those monic
arrows is untenable. Consider, for instance, the string calculi
so frequently use to reason in monoidal categories. For
this example, we permit ourselves to ignore details and
subtleties so that we do not muddy the point we mean to
illustrate. For a detailed and complete look at string
calculi, Selinger's survey [57] provides an
excellent overview.
Given a monoidal category $ ( \C ,\otimes, I ) $, objects
are represented by certain isotopy classes of strings and
arrows are represented by nodes. This is illustrated in
Figure \ref{fig:string-diagrams}. The diagrams read from
left to right. Now, to draw a string for an identity arrow,
we do not include the node, giving the diagram
\[
\begin{tikzpicture}
\node (1) at (0,0) {$ a $};
\node (2) at (2,0) {$ a $};
\draw [-] (1) to (2);
\end{tikzpicture}
\]
to represent $ \id \from a \to a $. Composing with $ \id $
another arrow should result in nothing changing, as captured
in this equation
\begin{center}
\begin{tikzpicture}
\node (1) at (0,0) {$ a $};
\node[circle,draw=black] (2) at (4,0) {$ f $};
\node (3) at (6,0) {$ b $};
\path[-,font=\scriptsize]
(1) edge[] (2)
(2) edge node[]{$ $} (3);
\node () at (7,0) {$ = $};
\node (1) at (8,0) {$ a $};
\node[circle,draw=black] (2) at (10,0) {$ f $};
\node (3) at (12,0) {$ b $};
\path[-,font=\scriptsize]
(1) edge node[]{$ $} (2)
(2) edge node[]{$ $} (3);
\end{tikzpicture}
\end{center}
From this, we observe that the length of the string does not
matter. This accords with defining strings up to
isotopy. In particular, we want to have a string be
equivalent to a point. In the parlance of this thesis, we
want to be able to rewrite a string, with two distinct
endpoints, into a single point. Yet, this is not possible
to do with a fine rewrite rule.
\begin{figure}[h]
\centering
\fbox{
\begin{minipage}{1.0\linewidth}
\[
\begin{tikzpicture}
\node[circle,draw=black] (1) at (0,0) {$ f $};
\node (2) at (-2,0) {$ a $};
\node (3) at (2,0) {$ b $};
\path[-,font=\scriptsize]
(2) edge node[]{$ $} (1)
(1) edge node[]{$ $} (3);
\node () at (0,-1) {\scriptsize{A string representing
an arrow $ f \from a \to b $}};
\end{tikzpicture}
\quad \quad
\begin{tikzpicture}
\node[circle,draw=black] (1) at (0,0) {$ g $};
\node (2) at (-2,0.5) {$ a $};
\node (3) at (-2,-0.5) {$ b $};
\node (4) at (2,0.5) {$ a $};
\node (5) at (2,-0.5) {$ b' $};
% \draw (0,0) circle (0.5cm);
\path[-,font=\scriptsize]
(2.0) edge [bend left=10] (1.150)
(3.0) edge [bend right=10] (1.210)
(1.30) edge [bend left=10] (4.180)
(1.-30) edge [bend right=10] (5.180);
\node () at (0,-1) {\scriptsize{
A string representing an arrow
$ g \from a \otimes b \to a' \otimes b' $}};
\end{tikzpicture}
\]
\end{minipage}
\caption{String diagrams}
\label{fig:string-diagrams}
\end{figure}
Indeed, suppose we are working with strings in some topos of
spaces and we want to finely rewrite a string into a
point. Such a rewrite rule would be a span
\begin{equation} \label{eq:string-to-point}
\begin{tikzpicture}
\begin{scope}
\node (1) at (0,1) {$ \bullet $};
\node (2) at (0,0) {$ \bullet $};
\draw (1.center) -- (2.center);
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (l) at (0.6,0.5) {};
\end{scope}
\begin{scope}[shift={(3,0)}]
\node () at (0,0.5) {?};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (ml) at (-0.6,0.5) {};
\node (mr) at (0.6,0.5) {};
\end{scope}
\begin{scope}[shift={(6,0)}]
\node (1) at (0,0.5) {$ \bullet $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (r) at (-0.6,0.5) {};
\end{scope}
\draw[cd]
(ml) edge[] (l)
(mr) edge[] (r);
\end{tikzpicture}
\end{equation}
with `?' replaced by a subobject of both the string on the
left and point on the right. Thus, `?' must either be empty
or a point. Choosing the empty string does not scale. A
simple counter example is
\[
\begin{tikzpicture}
\begin{scope}
\node (a) at (0,0) {$ \bullet $};
\node (b) at (1,0) {$ \bullet $};
\draw (a.center) -- (b.center);
\draw [rounded corners] (-0.5,-0.5) rectangle (2.5,0.5);
\node (01b) at (1,-0.6) {$ $};
\node (01r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(4,0)}]
\node (c) at (1,0) {$ 0 $};
\draw [rounded corners] (-0.5,-0.5) rectangle (2.5,0.5);
\node (11l) at (-0.6,0) {$ $};
\node (11b) at (1,-0.6) {$ $};
\node (11r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(8,0)}]
\node (c) at (1,0) {$ \bullet $};
\draw [rounded corners] (-0.5,-0.5) rectangle (2.5,0.5);
\node (21l) at (-0.6,0) {$ $};
\node (21b) at (1,-0.6) {$ $};
\end{scope}
\begin{scope}[shift={(0,-3)}]
\node (a) at (0,0) {$ \bullet $};
\node (b) at (1,0) {$ \bullet $};
\node (c) at (2,1) {$ \bullet $};
\node (d) at (2,-1) {$ \bullet $};
\draw
(a.center) -- (b.center)
(b.center) -- (c.center)
(b.center) -- (d.center);
\draw [rounded corners]
(-0.5,-1.5) rectangle (2.5,1.5);
\node (00l) at (-0.6,0) {$ $};
\node (00t) at (1,1.6) {$ $};
\node (00r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(4,-3)}]
\node () at (1,0) {?};
\draw [rounded corners]
(-0.5,-1.5) rectangle (2.5,1.5);
\node (10l) at (-0.6,0) {$ $};
\node (10t) at (1,1.6) {$ $};
\node (10r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(8,-3)}]
\node (b) at (1,0) {$ \bullet $};
\node (c) at (2,1) {$ \bullet $};
\node (d) at (2,-1) {$ \bullet $};
\draw
(b.center) -- (c.center)
(b.center) -- (d.center);
\draw [rounded corners]
(-0.5,-1.5) rectangle (2.5,1.5);
\node (20l) at (-0.6,0) {$ $};
\node (20t) at (1,1.6) {$ $};
\node (20r) at (2.6,0) {$ $};
\end{scope}
\draw[cd]
(11l) edge node[]{$ $} (01r)
(11r) edge node[]{$ $} (21l)
(10l) edge node[]{$ $} (00r)
(10r) edge node[]{$ $} (20l)
(01b) edge node[]{$ $} (00t)
(11b) edge node[]{$ $} (10t)
(21b) edge node[]{$ $} (20t);
\end{tikzpicture}
\]
To see this more clearly, we reframe the question to take
advantage of the fact that pushing out over $ 0 $ is
the same as taking a disjoint union. So we can ask whether
\begin{center}
\begin{tikzpicture}
\node (a) at (0,0) {$ \bullet $};
\node (b) at (1,0) {$ \bullet $};
\node (c) at (2,1) {$ \bullet $};
\node (d) at (2,-1) {$ \bullet $};
\draw
(a.center) -- (b.center)
(b.center) -- (c.center)
(b.center) -- (d.center);
\draw [rounded corners]
(-0.5,-1.5) rectangle (2.5,1.5);
\node (00l) at (-0.6,0) {$ $};
\node (00t) at (1,1.6) {$ $};
\node (00r) at (2.6,0) {$ $};
\end{tikzpicture}
\end{center}
is the disjoint union of
\begin{center}
\begin{tikzpicture}
\node (a) at (0,0) {$ \bullet $};
\node (b) at (1,0) {$ \bullet $};
\draw (a.center) -- (b.center);
\draw [rounded corners] (-0.5,-0.5) rectangle (2.5,0.5);
\node (01b) at (1,-0.6) {$ $};
\node (01r) at (2.6,0) {$ $};
\end{tikzpicture}
\end{center}
and something else. Of course, it is not.
But maybe the issue was pushing out over $ 0 $ in
the first place. What about replacing $ 0 $ with a
point? A simple counter example to illustrate the failure of
this idea is
\[
\begin{tikzpicture}
\begin{scope}
\node (a) at (0,0) {$ \bullet $};
\node (b) at (1,0) {$ \bullet $};
\draw (a.center) -- (b.center);
\draw [rounded corners]
(-1.5,-0.5) rectangle (2.5,0.5);
\node (01l) at (-1.6,0) {$ $};
\node (01b) at (0.5,-0.6) {$ $};
\node (01r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(5,0)}]
\node (c) at (0.5,0) {$ \bullet $};
\draw [rounded corners]
(-1.5,-0.5) rectangle (2.5,0.5);
\node (11l) at (-1.6,0) {$ $};
\node (11b) at (0.5,-0.6) {$ $};
\node (11r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(10,0)}]
\node (c) at (0.5,0) {$ \bullet $};
\draw [rounded corners]
(-1.5,-0.5) rectangle (2.5,0.5);
\node (21l) at (-1.6,0) {$ $};
\node (21b) at (0.5,-0.6) {$ $};
\node (21r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(0,-3)}]
\node (0) at (-1,1) {$ \bullet $};
\node (1) at (-1,-1) {$ \bullet $};
\node (a) at (0,0) {$ \bullet $};
\node (b) at (1,0) {$ \bullet $};
\node (c) at (2,1) {$ \bullet $};
\node (d) at (2,-1) {$ \bullet $};
\draw
(a.center) -- (b.center)
(b.center) -- (c.center)
(b.center) -- (d.center)
(0.center) -- (a.center)
(1.center) -- (a.center);
\draw [rounded corners]
(-1.5,-1.5) rectangle (2.5,1.5);
\node (00l) at (-1.6,0) {$ $};
\node (00t) at (0.5,1.6) {$ $};
\node (00r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(5,-3)}]
\node () at (0.5,0) {?};
\draw [rounded corners]
(-1.5,-1.5) rectangle (2.5,1.5);
\node (10l) at (-1.6,0) {$ $};
\node (10t) at (0.5,1.6) {$ $};
\node (10r) at (2.6,0) {$ $};
\end{scope}
\begin{scope}[shift={(10,-3)}]
\node (0) at (-0.5,1) {$ \bullet $};
\node (1) at (-0.5,-1) {$ \bullet $};
\node (b) at (0.5,0) {$ \bullet $};
\node (c) at (1.5,1) {$ \bullet $};
\node (d) at (1.5,-1) {$ \bullet $};
\draw (0.center) -- (b.center);
\draw (1.center) -- (b.center);
\draw (b.center) -- (c.center);
\draw (b.center) -- (d.center);
\draw [rounded corners]
(-1.5,-1.5) rectangle (2.5,1.5);
\node (20l) at (-1.6,0) {$ $};
\node (20t) at (0.5,1.6) {$ $};
\node (20r) at (2.6,0) {$ $};
\end{scope}
\draw[cd]
(11l) edge node[above]{$ \theta $} (01r)
(11r) edge node[]{$ $} (21l)
(10l) edge node[]{$ $} (00r)
(10r) edge node[]{$ $} (20l)
(01b) edge node[]{$ $} (00t)
(11b) edge node[]{$ $} (10t)
(21b) edge node[]{$ $} (20t);
\end{tikzpicture}
\]
where we define $ \theta $ to choose the left or the right
point; the failure will occur regardless of the choice.
Again, there is nothing that we can place into the center,
bottom square to give a double pushout diagram. To see why,
we use the fact that if we could fill in `?', we already
know what it must be. The right square must also be a
pushout. This forces us to fill the blank with the graph
\begin{center}
\begin{tikzpicture}
\node (0) at (-0.5,1) {$ \bullet $};
\node (1) at (-0.5,-1) {$ \bullet $};
\node (b) at (0.5,0) {$ \bullet $};
\node (c) at (1.5,1) {$ \bullet $};
\node (d) at (1.5,-1) {$ \bullet $};
\draw (0.center) -- (b.center);
\draw (1.center) -- (b.center);
\draw (b.center) -- (c.center);
\draw (b.center) -- (d.center);
\draw [rounded corners]
(-1.5,-1.5) rectangle (2.5,1.5);
\node (20l) at (-1.6,0) {$ $};
\node (20t) at (0.5,1.6) {$ $};
\node (20r) at (2.6,0) {$ $};
\end{tikzpicture}
\end{center}
But then the left square is not a pushout.
And so, fine rewriting can be insufficient. In this
chapter, we define bold rewriting of structured cospans
to handle situations like this one found in string
calculi. We see that, though it largely mirrors the fine
rewriting of structured cospans, it has its own character:
the bicategory we extract is a bicategory of relations. At
the end of the chapter, we illustrate bold rewriting using
the string calculus from quantum computer science known as
the ZX-calculus.
% ===========================================================
\section{A double category of bold rewrites of structured
\label{sec:bold-rewrites}
% ===========================================================
In this section, we define a double category
$ _L\BBBoldRewrite $ whose objects are interface types,
whose vertical arrows are spans of interface types with
invertible legs, whose horizontal arrows are structured
cospans, and whose squares are bold rewrites of structured
cospans. The only difference between the definitions of
$ _L\FFFineRewrite $ and $ _L\BBBoldRewrite $ is in the
squares. The objects, horizontal arrows, and vertical arrows
are the same in each case. This winds up having an
interesting effect on the horizontal bicategory of
$ _L\BBBoldRewrite $ which we explore in Section
\ref{sec:cartesian-bicategory-spans-of-cospans}. Before
turning to that, we need to properly define
$ _L\BBBoldRewrite $, Fortunately, most of the work has been
done when constructing $ _L\FFFineRewrite $, so we
begin by defining the squares.
Recall from Definition \ref{def:span-str-cospan} that a
morphism of spans of structured cospans is an arrow
$ \theta $ that fits into a commuting diagram
\[
\begin{tikzpicture}
\node (La) at (0,4) {$ La $};
\node (Lb) at (1,2) {$ Lb $};
\node (Lc) at (2,0) {$ Lc $};
\node (x) at (3,4) {$ x $};
\node (y) at (4,3.5) {$ y $};
\node (y') at (4,0.5) {$ y' $};
\node (z) at (5,0) {$ z $};
\node (La') at (6,4) {$ La' $};
\node (Lb') at (7,2) {$ Lb' $};
\node (Lc') at (8,0) {$ Lc' $};
\draw[cd]
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc')
(y) edge node[]{$ $} (x)
(y') edge node[]{$ $} (x)
(y') edge node[]{$ $} (z)
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y')
(Lb') edge node[]{$ $} (y');
\path[line width=0.5em,draw=white]
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(y) edge node[]{$ $} (z)
(y) edge node[]{$ $} (y');
\path[cd,font=\scriptsize]
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(y) edge node[]{$ $} (z)
(y) edge node[left] {$ \theta $} (y');
\end{tikzpicture}
\]
Using a morphism of structured cospans, we can define the
connected components of structured cospans. We first define
a relation $ \sim $ setting
\[
\begin{tikzpicture}
\begin{scope}
\node (La) at (0,4) {$ La $};
\node (x) at (2,4) {$ x $};
\node (La') at (4,4) {$ La' $};
\node (Lb) at (0,2) {$ Lb $};
\node (y) at (2,2) {$ y $};
\node (Lb') at (4,2) {$ Lb' $};
\node (Lc) at (0,0) {$ Lc $};
\node (z) at (2,0) {$ z $};
\node (Lc') at (4,0) {$ Lc' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\node () at (5,2) {$ \sim $};
\begin{scope}[shift={(6,0)}]
\node (La) at (0,4) {$ La $};
\node (x) at (2,4) {$ x $};
\node (La') at (4,4) {$ La' $};
\node (Lb) at (0,2) {$ Lb $};
\node (y) at (2,2) {$ y' $};
\node (Lb') at (4,2) {$ Lb' $};
\node (Lc) at (0,0) {$ Lc $};
\node (z) at (2,0) {$ z $};
\node (Lc') at (4,0) {$ Lc' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\end{tikzpicture}
\]
if there is a morphism from the rewriting on the left side
of $ \sim $ to that on the right. A \defn{connected component of
structured cospans} is an equivalence class generated by
$ \sim $. The coarseness of the classes of squares is the
most important distinction between fine and bold rewriting.
\begin{definition}[Bold rewrite]
\label{def:bold-rewrite-str-cospans}
A \defn{bold rewrite of structured cospans} is a connected
component of structured cospans whose representative has
the form
\[
\begin{tikzpicture}
\begin{scope}
\node (La) at (0,4) {$ La $};
\node (x) at (2,4) {$ x $};
\node (La') at (4,4) {$ La' $};
\node (Lb) at (0,2) {$ Lb $};
\node (y) at (2,2) {$ y $};
\node (Lb') at (4,2) {$ Lb' $};
\node (Lc) at (0,0) {$ Lc $};
\node (z) at (2,0) {$ z $};
\node (Lc') at (4,0) {$ Lc' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[left]{$ \iso $} (La)
(Lb) edge node[left]{$ \iso $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[right]{$ \iso $} (La')
(Lb') edge node[right]{$ \iso $} (Lc');
\end{scope}
\end{tikzpicture}
\]
\end{definition}
The horizontal and vertical compositions for bold rewrites
of structured cospans are defined in the same way as for
fine rewrites. The classes are different, but the operation
on the class representatives work in the same way.
\begin{definition}
\label{def:composition-bold-rewrites}
The \defn{horizontal composition} $ \hcirc $ of bold
rewrites of structured cospans are defined by
the operation
\[
\begin{tikzpicture}
\begin{scope}
\node (La) at (0,4) {$ La $};
\node (x) at (2,4) {$ x $};
\node (La') at (4,4) {$ La' $};
\node (Lb) at (0,2) {$ Lb $};
\node (y) at (2,2) {$ y $};
\node (Lb') at (4,2) {$ Lb' $};
\node (Lc) at (0,0) {$ Lc $};
\node (z) at (2,0) {$ z $};
\node (Lc') at (4,0) {$ Lc' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\node () at (5,2) {$ \hcirc $};
\begin{scope}[shift={(6,0)}]
\node (La) at (0,4) {$ La' $};
\node (x) at (2,4) {$ x' $};
\node (La') at (4,4) {$ La'' $};
\node (Lb) at (0,2) {$ Lb' $};
\node (y) at (2,2) {$ y' $};
\node (Lb') at (4,2) {$ Lb'' $};
\node (Lc) at (0,0) {$ Lc' $};
\node (z) at (2,0) {$ z' $};
\node (Lc') at (4,0) {$ Lc'' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\node () at (11,2) {$ \bydef $};
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{scope}[shift={(0,0)}]
\node (La) at (0,4) {$ La $};
\node (x) at (3,4) {$ x +_{La'} x' $};
\node (La') at (6,4) {$ La'' $};
\node (Lb) at (0,2) {$ Lb $};
\node (y) at (3,2) {$ y +_{Lb'} y' $};
\node (Lb') at (6,2) {$ Lb'' $};
\node (Lc) at (0,0) {$ Lc $};
\node (z) at (3,0) {$ z +_{Lc'} z' $};
\node (Lc') at (6,0) {$ Lc'' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\end{tikzpicture}
\]
The \defn{vertical composition} of bold rewrites of structured
cospans is defined by
\[
\begin{tikzpicture}
\begin{scope}
\node (La) at (0,4) {$ La $};
\node (x) at (2,4) {$ v $};
\node (La') at (4,4) {$ La' $};
\node (Lb) at (0,2) {$ Lb $};
\node (y) at (2,2) {$ w $};
\node (Lb') at (4,2) {$ Lb' $};
\node (Lc) at (0,0) {$ Lc $};
\node (z) at (2,0) {$ x $};
\node (Lc') at (4,0) {$ Lc' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\node () at (5,2) {$ \hcirc $};
\begin{scope}[shift={(6,0)}]
\node (La) at (0,4) {$ Lc $};
\node (x) at (2,4) {$ x $};
\node (La') at (4,4) {$ Lc' $};
\node (Lb) at (0,2) {$ Ld $};
\node (y) at (2,2) {$ y $};
\node (Lb') at (4,2) {$ Ld' $};
\node (Lc) at (0,0) {$ Le $};
\node (z) at (2,0) {$ z $};
\node (Lc') at (4,0) {$ Le' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\node () at (11,2) {$ \bydef $};
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\begin{scope}[shift={(0,0)}]
\node (La) at (0,4) {$ La $};
\node (x) at (3,4) {$ v $};
\node (La') at (6,4) {$ La' $};
\node (Lb) at (0,2) {$ Lb \times_{Lc} Ld $};
\node (y) at (3,2) {$ w \times_{x} y $};
\node (Lb') at (6,2) {$ Lb' \times_{Lc'} Ld' $};
\node (Lc) at (0,0) {$ Le $};
\node (z) at (3,0) {$ z $};
\node (Lc') at (6,0) {$ Le' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\end{tikzpicture}
\]
\end{definition}
Unlike for fine rewrites of structured cospans, the
interchange law is straightforward to prove. The coarser
classes of rewrites of structured cospans vastly simplifies
concocting the isomorphism.
\begin{lemma}
\label{thm:bold-interchange}
\[
\begin{tikzpicture}
\node () at (-1,2) {$ \alpha \bydef $};
\node (La) at (0,4) {$ La $};
\node (x) at (2,4) {$ v $};
\node (La') at (4,4) {$ La' $};
\node (Lb) at (0,2) {$ Lb $};
\node (y) at (2,2) {$ w $};
\node (Lb') at (4,2) {$ Lb' $};
\node (Lc) at (0,0) {$ Lc $};
\node (z) at (2,0) {$ x $};
\node (Lc') at (4,0) {$ Lc' $};
\draw [cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{tikzpicture}
\quad \quad \quad \quad
\begin{tikzpicture}
\node () at (-1,2) {$ \alpha' \bydef $};
\node (La) at (0,4) {$ La' $};
\node (x) at (2,4) {$ v' $};
\node (La') at (4,4) {$ La'' $};
\node (Lb) at (0,2) {$ Lb' $};
\node (y) at (2,2) {$ w' $};
\node (Lb') at (4,2) {$ Lb'' $};
\node (Lc) at (0,0) {$ Lc' $};
\node (z) at (2,0) {$ x' $};
\node (Lc') at (4,0) {$ Lc'' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\node () at (-1,2) {$ \beta \bydef $};
\node (La) at (0,4) {$ Lc $};
\node (x) at (2,4) {$ x $};
\node (La') at (4,4) {$ Lc' $};
\node (Lb) at (0,2) {$ Ld $};
\node (y) at (2,2) {$ y $};
\node (Lb') at (4,2) {$ Ld' $};
\node (Lc) at (0,0) {$ Le $};
\node (z) at (2,0) {$ z $};
\node (Lc') at (4,0) {$ Le' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{tikzpicture}
\quad \quad \quad \quad
\begin{tikzpicture}
\node () at (-1,2) {$ \beta' \bydef $};
\node (La) at (0,4) {$ Lc' $};
\node (x) at (2,4) {$ x' $};
\node (La') at (4,4) {$ Lc'' $};
\node (Lb) at (0,2) {$ Ld' $};
\node (y) at (2,2) {$ y' $};
\node (Lb') at (4,2) {$ Ld'' $};
\node (Lc) at (0,0) {$ Le' $};
\node (z) at (2,0) {$ z' $};
\node (Lc') at (4,0) {$ Le'' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{tikzpicture}
\]
be bold rewrites of structured cospans. Then
\[
( \alpha \hcirc \alpha' ) \vcirc ( \beta \hcirc \beta' )
( \alpha \vcirc \beta ) \hcirc ( \alpha' \vcirc \beta' ).
\]
That is, the interchange law holds.
\end{lemma}
\begin{proof}
The left hand side of the equation is the bold rewrite of
structured cospans
\[
\begin{tikzpicture}
\begin{scope}
\node (La) at (0,4) {$ La $};
\node (x) at (5,4) {$ v +_{La'} v' $};
\node (La') at (10,4) {$ La'' $};
\node (Lb) at (0,2) {$ Lb \times_{Lc} Ld $};
\node (y) at (5,2)
{$ ( w +_{Lb'} w' )
\times_{( x +_{Lc'} x' )}
( y \times_{Ld'} y') $};
\node (Lb') at (10,2) {$ Lb \times_{Lc'} Ld' $};
\node (Lc) at (0,0) {$ Le $};
\node (z) at (5,0) {$ z +_{Le'} z' $};
\node (Lc') at (10,0) {$ Le'' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\end{tikzpicture}
\]
while the right hand side is
\[
\begin{tikzpicture}
\begin{scope}
\node (La) at (0,4) {$ La $};
\node (x) at (5,4) {$ v +_{La'} v' $};
\node (La') at (10,4) {$ La'' $};
\node (Lb) at (0,2) {$ Lb \times_{Lc} Ld $};
\node (y) at (5,2)
{$ ( w \times_{x} y)
+_{(Lb' \times_{Lc'} Ld')}
( w' \times_{x'} y' ) $};
\node (Lb') at (10,2) {$ Lb \times_{Lc'} Ld' $};
\node (Lc) at (0,0) {$ Le $};
\node (z) at (5,0) {$ z +_{Le'} z' $};
\node (Lc') at (10,0) {$ Le'' $};
\draw[cd]
(La) edge node[]{$ $} (x)
(La') edge node[]{$ $} (x)
(Lb) edge node[]{$ $} (y)
(Lb') edge node[]{$ $} (y)
(Lc) edge node[]{$ $} (z)
(Lc') edge node[]{$ $} (z)
(Lb) edge node[]{$ $} (La)
(Lb) edge node[]{$ $} (Lc)
(y) edge node[]{$ $} (x)
(y) edge node[]{$ $} (z)
(Lb') edge node[]{$ $} (La')
(Lb') edge node[]{$ $} (Lc');
\end{scope}
\end{tikzpicture}
\]
To show that these are equal as bold rewrites of
structured cospans, it suffices to find a \emph{morphism} between
them. Precisely, we need a morphism
\[
( w\times_{x}y )
+_{( Lb'\times_{Lc'}Ld')}
(w'\times_{x'}y' )
\to
( w+_{Lb'}w' )
\times_{(x+_{Lc'}x')}
( y\times_{Ld'}y')
\]
We can obtain the two objects as follows. Let $ \C $ be
the walking cospan category $ \{ \bullet \to \bullet \gets
\bullet \} $ and let $ \S $ be the walking span category $
\{ \bullet \gets \bullet \to \bullet \} $. Then $ \C
\times \S $ is the walking cospan of spans category
\[
\begin{tikzpicture}
\begin{scope}
\node (1) at (0,4) {$ \bullet $};
\node (2) at (2,4) {$ \bullet $};
\node (3) at (4,4) {$ \bullet $};
\node (4) at (0,2) {$ \bullet $};
\node (5) at (2,2) {$ \bullet $};
\node (6) at (4,2) { $ \bullet $};
\node (7) at (0,0) {$ \bullet $};
\node (8) at (2,0) {$ \bullet $};
\node (9) at (4,0) {$ \bullet $};
\draw[cd]
(1) edge node[]{$ $} (4)
(7) edge node[]{$ $} (4)
(2) edge node[]{$ $} (5)
(8) edge node[]{$ $} (5)
(3) edge node[]{$ $} (6)
(9) edge node[]{$ $} (6)
(2) edge node[]{$ $} (1)
(2) edge node[]{$ $} (3)
(5) edge node[]{$ $} (4)
(5) edge node[]{$ $} (6)
(8) edge node[]{$ $} (7)
(8) edge node[]{$ $} (9);
\end{scope}
\draw [rounded corners]
(-1,-1) rectangle (5,5);
\end{tikzpicture}
\]
Let $ F \from \C \times \S \to \X $ be the functor that
returns the diagram
\[
\begin{tikzpicture}
\begin{scope}
\node (1) at (0,4) {$ w $};
\node (2) at (2,4) {$ Lb' $};
\node (3) at (4,4) {$ w' $};
\node (4) at (0,2) {$ x $};
\node (5) at (2,2) {$ Lc' $};
\node (6) at (4,2) { $ x' $};
\node (7) at (0,0) {$ y $};
\node (8) at (2,0) {$ Ld' $};
\node (9') at (4,0) {$ y' $};
\draw[cd]
(1) edge node[]{$ $} (4)
(7) edge node[]{$ $} (4)
(2) edge node[]{$ $} (5)
(8) edge node[]{$ $} (5)
(3) edge node[]{$ $} (6)
(9) edge node[]{$ $} (6)
(2) edge node[]{$ $} (1)
(2) edge node[]{$ $} (3)
(5) edge node[]{$ $} (4)
(5) edge node[]{$ $} (6)
(8) edge node[]{$ $} (7)
(8) edge node[]{$ $} (9);
\end{scope}
\end{tikzpicture}
\]
which is the middle of the diagram obtained by gluing $
\alpha$, $ \beta $, $ \alpha' $, and $ \beta' $ together
along their coinciding edges. There is a canonical
morphism of type
\[
\colim\limits_\S \lim_\C F \to
\lim_\C \colim\limits_\S F
\]
where the domain is the image of $ F $ under the composite functor
\[
\X^{\C \times \S} \xto{\iso}
( \X^\C )^\S \xto{\lim_\C}
\X^\S \xto{\colim_\S} \X
\]
and the domain is the image of $ F $ under the composite
\[
\X^{\C \times \S} \xto{\iso}
( \X^\S )^\C \xto{\colim_\S}
\X^\C \xto{\lim_\C} \X.
\]
One can check that this canonical morphism gives the
morphism of bold rewrites of structured cospans we need.
\end{proof}
% ===================================================
\section{A bicategory of relations for bold
rewriting of structured cospans}
\label{sec:cartesian-bicategory-spans-of-cospans}
% ==================================================
There are two philosophies in rewriting. One is that we care
about how one object is rewritten into another, and so we
keep track of certain data to describe the rewriting. The
other perspective is that we do not care about \emph{how} an
object is rewritten into another, only that the rewriting is
possible. Bold rewriting of structured cospans belongs to
the latter philosophy. This is realized explicitly through
the fact that the horizontal bicategory forms a bicategory
of relations, specifically that it is locally posetal.
Appendix \ref{sec:cartesian-bicategories} discusses the
theory of such bicategories.
The first goal of this section is to define the bicategory
in question. We take the same approach as finding the
horizontal bicategory of fine rewrites of structured cospans
in Section
\ref{sec:compact-closed-bicategory-spans-of-cospans}. After
extracting the bicategory, we show that it is a bicategory
of relations (see Definition \ref{def:bicat-relations}).
This next theorem is proved with virtually the same argument
as Lemma \ref{lem:SpanCospanSM}.
\begin{theorem}
$ _L \BBBoldRewrite $ is a symmetric monoidal double
\end{theorem}
From here, we prove a series of lemmas that, when put
together, prove that the horizontal bicategory
$ _L \BBoldRewrite $ of $ _L \BBBoldRewrite $ is a
bicategory of relations. The first lemma in this string is
proved by replicating the proof of Lemma
\ref{lem:SpanCospanIsofibrant} and the second follows
from Theorem \ref{thm:horz-bicat}.
\begin{lemma}
$ _L \BBBoldRewrite $ is isofibrant.
\end{lemma}
\begin{lemma}
$ _L \BBoldRewrite $ is a symmetric monoidal bicategory.
\end{lemma}
In the following lemma, we use
$ \nabla \bydef [ \id,\id] \from a + a \to a $ to
denote the codiagonal map and $ ! $ to denote a canonical
arrow from the initial object.
\begin{lemma}
For each object $ a $ of $ _L \BBoldRewrite $, define
\[
\Delta_a \from a \to a + a
\quad \text{and} \quad
\varepsilon_a \from a \to 0
\]
to be the structured cospans
\[
La \xto{\id} La \xgets{L\nabla_a} L(a+a)
\quad \text{and} \quad
La \xto{\id} La \xgets{!} L0
\]
respectively. Then $ (a , \Delta_a , \varepsilon_a ) $ is
a cocommutative comonoid.
\end{lemma}
\begin{proof}
Proving this amounts to showing that the
coassociativity, counitality, and
cocommutativity diagrams commute.
The coassociativity diagram
\[
\begin{tikzpicture}
\node (01) at (0,2) {$ a $};
\node (21) at (6,2) {$ a+a $};
\node (00) at (0,0) {$ a+a $};
\node (10) at (3,0) {$ a+(a+a) $};
\node (20) at (6,0) {$ (a+a)+a $};
\draw[cd]
(01) edge node[above]{$ \nabla $} (21)
(01) edge node[left]{$ \nabla $} (00)
(00) edge node[below]{$ \alpha $} (10)
(10) edge node[below]{$ \id \otimes \nabla $} (20)
(21) edge node[right]{$ \nabla \otimes \id $} (20);
\end{tikzpicture}
\]
commutes because the top path, which is the composite
\[
\begin{tikzpicture}
\node (00) at (0,0) {$ La $};
\node (10) at (4,0) {$ L(a+a) $};
\node (20) at (8,0) {$ L((a+a)+a) $};
\node (01) at (2,2) {$ La $};
\node (11) at (6,2) {$ L((a+a)+a $};
\node (12) at (4,4) {$ La $};
\draw[cd]
(00) edge node[above,left]{$ \id $} (01)
(10) edge node[above,right]{$ L\nabla $} (01)
(10) edge node[above,left]{$ \id $} (11)
(20) edge node[above,right]{$ L(\nabla + \id) $} (11)
(01) edge node[above,left]{$ \id $} (12)
(11) edge node[above,right]{$ L\nabla $} (12);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
equals the bottom path, which is the composite
\[
\begin{tikzpicture}
\node (00) at (0,0) {$ La $};
\node (10) at (4,0) {$ L(a+a) $};
\node (20) at (8,0) {$ L(a+(a+a)) $};
\node (30) at (12,0) {$ L((a+a)+a $};
\node (01) at (2,2) {$ La $};
\node (11) at (6,2) {$ L(a+a) $};
\node (21) at (10,2) {$ L(a+(a+a)) $};
\node (02) at (4,4) {$ La $};
\node (12) at (8,4) {$ L(a+a) $};
\node (03) at (6,6) {$ La $};
\draw[cd]
(00) edge node[above,left]{$ \id $} (01)
(10) edge node[above,right]{$ L\nabla $} (01)
(10) edge node[above,left]{$ \id $} (11)
(20) edge node[above,right]{$ L(\id + \nabla) $} (11)
(20) edge node[above,left]{$ \id $} (21)
(30) edge node[above,right]{$ L(\alpha) $} (21)
(01) edge node[above,left]{$ \id $} (02)
(11) edge node[above,right]{$ L\nabla $} (02)
(11) edge node[above,left]{$ \id $} (12)
(21) edge node[above,right]{$ L(\id + \nabla) $} (12)
(02) edge node[above,left]{$ \id $} (03)
(12) edge node[above,right]{$ L\nabla $} (03);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\draw (7.9,3.5) -- (8,3.4) -- (8.1,3.5);
\draw (5.9,5.5) -- (6,5.4) -- (6.1,5.5);
\end{tikzpicture}
\]
The counitality diagram
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ 0 \otimes a $};
\node (21) at (3,0) {$ a \otimes a $};
\node (31) at (6,0) {$ a \otimes 0 $};
\node (12) at (3,2) {$ a $};
\draw[cd]
(12) edge node[left]{$ \lambda $} (11)
(12) edge node[right]{$ \nabla $} (21)
(12) edge node[right]{$ \rho $} (31)
(21) edge node[below]{$ \epsilon \otimes \id $} (11)
(21) edge node[below]{$ \id \otimes \epsilon $} (31);
\end{tikzpicture}
\]
commutes because the composite
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ La $};
\node (21) at (4,0) {$ L(a+a) $};
\node (31) at (8,0) {$ La $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ L(a+a) $};
\node (13) at (4,4) {$ L(0+a) $};
\draw[cd]
(11) edge node[above,left]{$ \id $} (12)
(21) edge node[above,right]{$ L\nabla $} (12)
(21) edge node[above,left]{$ \id $} (22)
(31) edge node[above,right]{$ L(!+\id) $} (22)
(12) edge node[above,left]{$ \id $} (13)
(22) edge node[above,right]{$ L\nabla $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
is equal to the left unitor and
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ La $};
\node (21) at (4,0) {$ L(a+a) $};
\node (31) at (8,0) {$ La $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ L(a+a) $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(11) edge node[above,left]{$ \id $} (12)
(21) edge node[above,right]{$ L\nabla $} (12)
(21) edge node[above,left]{$ \id $} (22)
(31) edge node[above,right]{$ L(\id+!) $} (22)
(12) edge node[above,left]{$ \id $} (13)
(22) edge node[above,right]{$ L\nabla $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
is the right unitor. Finally, the cocommutative
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ a \otimes a $};
\node (21) at (4,0) {$ a \otimes a $};
\node (12) at (2,2) {$ a $};
\draw[cd]
(12) edge node[above,left]{$ \nabla $} (11)
(12) edge node[above,right]{$ \nabla $} (21)
(11) edge node[below]{$ \beta $} (21);
\end{tikzpicture}
\]
commutes because the composite $ \nabla \beta $
is given by
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ La $};
\node (21) at (4,0) {$ L(a+a) $};
\node (31) at (8,0) {$ L(a+a) $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ L(a+a) $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(11) edge node[above,left]{$ \id $} (12)
(21) edge node[above,right]{$ L\nabla $} (12)
(21) edge node[above,left]{$ \beta $} (22)
(31) edge node[above,right]{$ \id $} (22)
(12) edge node[above,left]{$ \id $} (13)
(22) edge node[above,right]{$ L\nabla $}
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
which is exactly the comuliplication.
\end{proof}
In the following lemma, we follow the convention of writing
$ f \leq g $ to represent a 2-arrow from $ f $ to $ g $ in a
locally posetal bicategory. This notation is faithful to
the fact that the hom-categories are actually hom-posets. This
is discussed further in Section
\ref{sec:cartesian-bicategories}.
\begin{lemma}
Let $ (a,\Delta_a,\varepsilon_a) $ and
$ ( b,\Delta_b,\varepsilon_b ) $ be cocommutative comonoid
objects in the double category $ _L \BBoldRewrite $. Every structured cospan $
La \to x \gets Lb $ in $ _L \BBoldRewrite $ is a lax comonoid
homomorphism. That is,
\[
\Delta_b x \leq (x + x) \Delta_a
\quad \text{and} \quad
\varepsilon_b x \leq \varepsilon a
\]
\end{lemma}
\begin{proof}
The first 2-arrow is
\[
\begin{tikzpicture}
\node (11) at (3,0) {$ La +_{L(a+a)} (x+x) $};
\node (12) at (0,2) {$ La $};
\node (22) at (3,2) {$ La +_{L(a+a)} (x+x) $};
\node (32) at (9,2) {$ L(b+b) $};
\node (13) at (3,4) {$ x $};
\node (3'2) at (6,2) {$ x+x $};
\draw[cd]
(12) edge node[above,left]{$ f $} (13)
(12) edge node[above]{$ \psi $} (22)
(12) edge node[left]{$ \psi $} (11)
(32) edge node[above]{$ \langle g,g \rangle $} (13)
(32) edge node[above]{$ g+g $} (3'2)
(3'2) edge node[above]{$ \theta $} (22)
(3'2) edge node[right]{$ \nabla $} (13)
(32) edge node[right]{$ \theta (g+g) $} (11)
(22) edge [dashed] node[left]{$ $} (13)
(22) edge node[left]{$ \id $} (11);
\end{tikzpicture}
\]
where the dashed line is the universal arrow
formed in reference to $ f $ and $ \nabla
$. The source of this 2-arrow is the composite
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ La $};
\node (21) at (4,0) {$ Lb $};
\node (31) at (8,0) {$ L(b+b) $};
\node (12) at (2,2) {$ x $};
\node (22) at (6,2) {$ Lb $};
\node (13) at (4,4) {$ x $};
\draw[cd]
(11) edge node[above,left]{$ f $} (12)
(21) edge node[above,right]{$ g $} (12)
(21) edge node[above,left]{$ \id $} (22)
(31) edge node[above,right]{$ L\nabla $} (22)
(12) edge node[above,left]{$ \id $} (13)
(22) edge node[above,right]{$ g $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
and the target is the composite
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ La $};
\node (21) at (4,0) {$ L(a+a) $};
\node (31) at (8,0) {$ L(b+b) $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ x+x $};
\node (13) at (4,4) {$ La +_{L(a+a)} (x+x) $};
\draw[cd]
(11) edge node[above,left]{$ \id $} (12)
(21) edge node[above,right]{$ L\nabla $} (12)
(21) edge node[above,left]{$ f+f $} (22)
(31) edge node[above,right]{$ g+g $} (22)
(12) edge node[above,left]{$ \psi $} (13)
(22) edge node[above,right]{$ \theta $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
The second is witnessed by the 2-arrow
\[
\begin{tikzpicture}
\node (11) at (2,0) {$ x $};
\node (12) at (0,2) {$ La $};
\node (22) at (2,2) {$ La $};
\node (32) at (4,2) {$ L0 $};
\node (13) at (2,4) {$ La $};
\draw[cd]
(12) edge node[above]{$ \id $} (13)
(12) edge node[above]{$ \id $} (22)
(12) edge node[below]{$ f $} (11)
(32) edge node[above]{$ ! $} (13)
(32) edge node[above]{$ ! $} (22)
(32) edge node[below]{$ ! $} (11)
(22) edge node[left]{$ \id $} (13)
(22) edge node[left]{$ f $} (11);
\end{tikzpicture}
\]
where target 2-arrow is the composite
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ La $};
\node (21) at (4,0) {$ Lb $};
\node (31) at (8,0) {$ L0 $};
\node (12) at (2,2) {$ x $};
\node (22) at (6,2) {$ Lb $};
\node (13) at (4,4) {$ x $};
\draw[cd]
(11) edge node[above]{$ f $} (12)
(21) edge node[above]{$ g $} (12)
(21) edge node[above]{$ \id $} (22)
(31) edge node[above]{$ ! $} (22)
(12) edge node[above]{$ \id $} (13)
(22) edge node[above]{$ g $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\qedhere
\]
\end{proof}
\begin{lemma}
For any object $ a $ in $ _L \BBoldRewrite $, each
cocommutative comonoid structure map
\[
\nabla \bydef
\left(
La \xto{\id} La \xgets{L\nabla_a} L(a+a)
\right)
\quad \text{ and } \quad
\epsilon \bydef
\left(
La \xto{\id} La \xgets{!} L0
\right)
\]
has a right adjoint (see Definition
\ref{def:adjoint-in-bicat}), respectively,
\[
\nabla^\ast \bydef
\left(
L(a+a) \xto{L\nabla_a} La \xgets{\id} La
\right)
\quad \text{ and } \quad
\epsilon^\ast \bydef
\left(
L0 \xto{!} La \xgets{\id} La \right).
\]
\end{lemma}
\begin{proof}
The unit of the adjunction
$ \nabla \dashv \nabla^\ast $ is
\[
\begin{tikzpicture}
\node (11) at (4,0) {$ La $};
\node (12) at (0,2) {$ La $};
\node (22) at (4,2) {$ La $};
\node (32) at (8,2) {$ La $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(12) edge node[above,left]{$ \id $} (13)
(12) edge node[above]{$ \id $} (22)
(12) edge node[below,left]{$ \id $} (11)
(32) edge node[above,right]{$ \id $} (13)
(32) edge node[above]{$ \id $} (22)
(32) edge node[below,left]{$ \id $} (11)
(22) edge node[left]{$ \id $} (13)
(22) edge node[left]{$ \id $} (11);
\end{tikzpicture}
\]
where the target is the composite 1-arrow
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ La $};
\node (21) at (4,0) {$ L(a+a) $};
\node (31) at (8,0) {$ La $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ La $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(11) edge node[above,left]{$ \id $} (12)
(21) edge node[above,right]{$ \nabla $} (12)
(21) edge node[above,left]{$ \nabla $} (22)
(31) edge node[above,right]{$ \id $} (22)
(12) edge node[above,left]{$ \id $} (13)
(22) edge node[above,right]{$ \id $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
The counit of $ \nabla \dashv \nabla^\ast $ is
the 2-arrow
\[
\begin{tikzpicture}
\node (11) at (4,0) {$ L(a+a) $};
\node (12) at (0,2) {$ L(a+a) $};
\node (22) at (4,2) {$ L(a+a) $};
\node (32) at (8,2) {$ L(a+a) $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(12) edge node[above,left]{$ \nabla $} (13)
(12) edge node[above]{$ \id $} (22)
(12) edge node[below,left]{$ \id $} (11)
(32) edge node[above,right]{$ \nabla $} (13)
(32) edge node[above]{$ \id $} (22)
(32) edge node[below,left]{$ \id $} (11)
(22) edge node[left]{$ \nabla $} (13)
(22) edge node[left]{$ \id $} (11);
\end{tikzpicture}
\]
where the source is the composite 1-arrow
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ L(a+a) $};
\node (21) at (4,0) {$ La $};
\node (31) at (8,0) {$ L(a+a) $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ La $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(11) edge node[above,left]{$ \nabla $} (12)
(21) edge node[above,right]{$ \id $} (12)
(21) edge node[above,left]{$ \id $} (22)
(31) edge node[above,right]{$ \nabla $} (22)
(12) edge node[above,left]{$ \id $} (13)
(22) edge node[above,right]{$ \id $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
Checking the triangle identities is
The unit of the adjunction $ \epsilon \dashv
\epsilon^\ast $ is the 2-arrow
\[
\begin{tikzpicture}
\node (11) at (4,0) {$ L(a+a) $};
\node (12) at (0,2) {$ La $};
\node (22) at (4,2) {$ L(a+a) $};
\node (32) at (8,2) {$ La $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(12) edge node[above,left]{$ \id $} (13)
(12) edge node[above]{$ \lambda $} (22)
(12) edge node[left]{$ \lambda $} (11)
(32) edge node[right]{$ \id $} (13)
(32) edge node[above]{$ \rho $} (22)
(32) edge node[right]{$ \rho $} (11)
(22) edge node[left]{$ \nabla $} (13)
(22) edge node[left]{$ \id $} (11);
\end{tikzpicture}
\]
where the target is the composite 1-arrow
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ La $};
\node (21) at (4,0) {$ L0 $};
\node (31) at (8,0) {$ La $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ La $};
\node (13) at (4,4) {$ L(a+a) $};
\draw[cd]
(11) edge node[above,left]{$ \id $} (12)
(21) edge node[above,right]{$ ! $} (12)
(21) edge node[above,left]{$ ! $} (22)
(31) edge node[above,right]{$ \id $} (22)
(12) edge node[above,left]{$ \lambda $} (13)
(22) edge node[above,right]{$ \rho $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
the counit of $ \epsilon \dashv \epsilon^\ast $
is the 2-arrow
\[
\begin{tikzpicture}
\node (11) at (4,0) {$ La $};
\node (12) at (0,2) {$ L0 $};
\node (22) at (4,2) {$ La $};
\node (32) at (8,2) {$ L0 $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(12) edge node[above]{$ ! $} (13)
(12) edge node[above]{$ ! $} (22)
(12) edge node[below]{$ ! $} (11)
(32) edge node[above]{$ ! $} (13)
(32) edge node[above]{$ ! $} (22)
(32) edge node[below]{$ ! $} (11)
(22) edge node[left]{$ \id $} (13)
(22) edge node[left]{$ \id $} (11);
\end{tikzpicture}
\]
where the source is the composite 1-arrow
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ L0 $};
\node (21) at (4,0) {$ La $};
\node (31) at (8,0) {$ L0 $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ La $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(11) edge node[above,left]{$ ! $} (12)
(21) edge node[above,right]{$ \id $} (12)
(21) edge node[above,left]{$ \id $} (22)
(31) edge node[above,right]{$ ! $} (22)
(12) edge node[above,left]{$ \id $} (13)
(22) edge node[above,right]{$ \id $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
Again, the triangle equations are straightforward to
\end{proof}
The following lemma refers to a `Frobenius monoid', a monoid
and comonoid that satisfy some nice properties that we spell
out in Definition \ref{def:frobenius-monoid}.
\begin{lemma} \label{thm:frobenius}
For any object $ a $ of $ _L \BBoldRewrite $,
$ ( a , \nabla^\ast , \epsilon^\ast , \nabla ,
\epsilon ) $ is a Frobenius monoid. In particular,
\begin{equation} \label{eq:frobenius}
\nabla \nabla^\ast =
\left( \nabla^\ast \otimes \id \right)
\left( \id \otimes \nabla \right)
\end{equation}
\end{lemma}
\begin{proof}
The left-hand side of Equation
\ref{eq:frobenius} is given by the composite
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ L(a+a) $};
\node (21) at (4,0) {$ La $};
\node (31) at (8,0) {$ L(a+a) $};
\node (12) at (2,2) {$ La $};
\node (22) at (6,2) {$ La $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(11) edge node[above]{$ \nabla $} (12)
(21) edge node[above]{$ \id $} (12)
(21) edge node[above]{$ \id $} (22)
(31) edge node[above]{$ \nabla $} (22)
(12) edge node[above]{$ \id $} (13)
(22) edge node[above]{$ \id $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
The right-hand side is given by
\[
\begin{tikzpicture}
\node (11) at (0,0) {$ L(a+a) $};
\node (21) at (4,0) {$ L(a+a+a) $};
\node (31) at (8,0) {$ L(a+a) $};
\node (12) at (2,2) {$ L(a+a) $};
\node (22) at (6,2) {$ L(a+a) $};
\node (13) at (4,4) {$ La $};
\draw[cd]
(11) edge node[above]{$ \id $} (12)
(21) edge node[above,right]{$ L(\id +\nabla ) $} (12)
(21) edge node[below,right]{$ L(\nabla + \id) $} (22)
(31) edge node[above]{$ \id $} (22)
(12) edge node[above]{$ L\nabla $} (13)
(22) edge node[above,right]{$ L\nabla $} (13);
\draw (3.9,3.5) -- (4,3.4) -- (4.1,3.5);
\end{tikzpicture}
\]
These both compose to $ L(a+a) \xto{L\nabla} La
\xgets{L\nabla} L(a+a) $.
\end{proof}
The following structure theorem follows from this
string of lemmas.
\begin{theorem} \label{thm:bold-rewrite-bicat-rels}
$ _L \BBoldRewrite $ is a bicategory of relations.
\end{theorem}
% ===================================================
\section{The ZX-calculus}
\label{sec:zx-calculus}
% ===================================================
Perhaps one of the most interesting features of quantum
mechanics is the incompatibility of observables. Roughly,
an observable is a measurable quantity of some system, for
instance the spin of a photon. In classical physics,
measureable quantities are comparable, meaning that we can
obtain arbitrarily precise values at the same time. For
example, given a Porsche speeding down the highway, we can
simultaneously measure its velocity and its mass with
arbitrary precision. Knowledge about its velocity does not
preclude us from obtain information about its mass. The
situation is quite different in quantum mechanics. Given two
measurable quantities, knowledge of one may prevent us from
obtaining knowledge about the other. This is illustrated by
the famous Heisenberg uncertainty principle which quantifies
the limits of precision to which one can simultaneously
measure the position and momentum of a particle. In general,
the strength of this restriction depends on the
situation. The most extreme case is that knowing one
quantity with total precision implies total uncertainty
about the other quantity. Such a pair of observables are
called \defn{complementary}.
Historically, a quantum physicist would reason about
observables, complementary or otherwise, using Hilbert
spaces. Given the rapid progress of quantum physics in the
twentieth century, this framework seems to have worked quite
well for scientists. Working with Hilbert spaces, however,
is challenging even for skilled researchers. But the
language of quantum physics is now relevant to a wider
audience since the dawn of quantum computing. Given the
challenge of working with Hilbert spaces, perhaps developing
a simpler language is worth pursuing.
Such a high-level language was invented by Coecke and Duncan
[24]. This language, called the
ZX-calculus, was immediately used to generalize both quantum
circuits [51] and the measurement
calculus [26]. Its
validity was further justified when Duncan and Perdrix
presented a non-trivial method of verifying
measurement-based quantum computations
[31]. At its core, the
ZX-calculus is an intuitive graphical language in which to
reason about complementary observables.
In this section, we illustrate our framework with the
ZX-calculus. The backstory of the ZX-calculus dates to
Penrose's tensor networks [54] and, more recently,
to the relationship between graphical languages and monoidal
categories [39, 57]. Abramsky and Coecke
capitalized on this relationship when inventing a
categorical framework for quantum physics
[1]. Soon after, Coecke and Duncan
introduced a diagrammatic language in which to reason about
complementary quantum observables
[19]. After a fruitful period
of development [20, 23, 30, 31, 29, 53], a full presentation of the
ZX-calculus was published [24].
The completeness of the ZX-calculus for stabilizer quantum
mechanics was later proved by Backens
The ZX-calculus begins with the five diagrams depicted in
Figure \ref{fig:zx-generators}. On each diagram, the
dangling wires on the left are \defn{inputs} and those on
the right are \defn{outputs}. By connecting inputs to
outputs, we can form larger diagrams, which we call
\defn{ZX-diagrams}. These diagrams generate the arrows of a
dagger compact category $\ZX$ whose objects, the
non-negative integers, count the inputs and outputs of a
diagram. Below, we give a presentation of $\ZX$ along with
a brief discussion on the origins of its generating arrows
(Figure \ref{fig:zx-generators}) and relations (Figure
\ref{fig:zx-equations}).
\begin{figure}
\fbox{%
\centering
\begin{minipage}{1\textwidth}
% ~~~~~~~~~~~~~
\begin{minipage}{0.2\linewidth}
\centering
\includegraphics{InclGrphx--generater--wire} \\
\vspace{0.5em}
\textsc{Wire}
\end{minipage}
\begin{minipage}{0.2\linewidth}
\centering
\includegraphics{InclGrphx--generater--green_spider} \\
\vspace{0em}
\textsc{Green Spider}
\end{minipage}%
\begin{minipage}{0.2\linewidth}
\centering
\includegraphics{InclGrphx--generater--red_spider} \\
\vspace{0em}
\textsc{Red Spider}
\end{minipage}%
\begin{minipage}{0.2\linewidth}
\centering
\includegraphics{InclGrphx--generater--hadamard} \\
\vspace{0.5em}
\textsc{Hadamard}
\end{minipage}%
\begin{minipage}{0.2\linewidth}
\centering
\includegraphics{InclGrphx--generater--diamond} \\
\vspace{0.5em}
\textsc{Diamond}
\end{minipage}%
\end{minipage}
\caption{Generators for the ZX-calculus diagrams}
\label{fig:zx-generators}
\end{figure}
Our goal with this example is to generate, using the
machinery laid out in this chapter, a bicategory of
relations $\ZZX$ to provide a syntax for the ZX-calculus. We
show that $ \ZZX $ extends $ \ZX $ in a way we make precise below.
The five \defn{basic diagrams} in the ZX-calculus
are depicted in Figure \ref{fig:zx-generators} and
are to be read from left to right. They are
\begin{itemize}
\item a \defn{wire} with a single input and
\item \defn{green spiders} with a non-negative
integer number of inputs and outputs and paired
with a phase $\alpha \in [-\pi,\pi)$,
\item \defn{red spiders} with a non-negative
integer number inputs and outputs and paired
with a phase $\beta \in [-\pi,\pi)$,
\item the \defn{Hadamard node} with a single input
and output, and
\item a \defn{diamond node} with no inputs or
\end{itemize}
The wire plays the role of an identity, much like a wire
without resistance in an electrical circuit, or straight
pipe in a plumbing system. The green and red spiders each
arise from a pair of complementary observables. In
categorical quantum mechanics [1],
observables correspond to certain commutative Frobenius
algebras $A$ living in a dagger symmetric monoidal category
$(\C , \otimes , I)$, the classic example
$ \C \bydef \FinHilb $ being the category of finite
dimensional Hilbert spaces and linear maps. A pair of
complementary observables gives a pair of Frobenius
algebras whose operations interact via laws like those of a
Hopf algebra
[21, 22].
This is particularly nice because Frobenius algebras have
beautiful string diagram representations. There is an
morphism $\C(I,A) \to \C(A,A)$ of commutative monoids that
gives rise to a group structure on $A$ known as the
\defn{phase group}, which Coecke and Duncan detail
\cite[Def.~7.5]{coecke-duncan_quant-full}. The phases on
the green and red spider diagrams arise from this group.
The Hadamard node embodies the Hadamard gate. The diamond
is a scalar obtained when connecting a green and red node
together. A deeper exploration of these notions goes beyond
the scope of this paper. For those interested, the original
paper on the topic
[24]
is an excellent place learn more.
In the spirit of compositionality, we present a category
$ \ZX $ whose arrows are generated by the five
basic diagrams. We sketched $ \ZX $ at the beginning of this
section, but we now detail the construction.
We start by allowing the basic ZX-diagrams from Figure
\ref{fig:zx-generators} generate the arrows of a free dagger
compact category whose objects are the non-negative
integers. We then subject the arrows (ZX-diagrams) to the
relations given in Figure \ref{fig:zx-equations}, to which
we add equations obtained by exchanging red and green nodes,
daggering, and taking diagrams up to ambient isotopy in
$4$-space. These listed relations are called \defn{basic}.
Spiders with no phase indicated have a phase of $0$.
This category, denoted as $ \ZX $, was introduced by Coecke
and Duncan
[24]
and further studied by Backens
To compose in $\ZX$, connect compatible diagrams
along a bijection between inputs and the outputs. For
\begin{center}
\begin{tikzpicture}
\node (1) at (0,2) {$ $};
\node (2) at (0,0) {$ $};
\node [zxgreen] (3) at (2,1) {$ $};
\node (4) at (4,1) {$ $};
\node ( ) at (5,1) {$ \circ $};
\node (5) at (6,1) {$ $};
\node [zxyellow] (6) at (8,1) {$ $};
\node (7) at (10,1) {$ $};
\node ( ) at (11,1) {$ = $};
\draw
(1.60) --(3.120)
(2.-60) -- (3.-120)
(3) -- (4)
(5) -- (6)
(6) -- (7);
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}
\node (1) at (0,2) {$ $};
\node (2) at (0,0) {$ $};
\node [zxgreen] (3) at (2,1) {$ $};
\node (4) at (4,1) {$ $};
\node [zxyellow] (5) at (6,1) {$ $};
\node (6) at (8,1) {$ $};
\draw
(1.60) -- (3.120)
(2.-60) -- (3.-120)
(3) -- (4.center)
(4.center) -- (5)
(5) -- (6);
\end{tikzpicture}
\end{center}
A monoidal structure is given by adding numbers and taking
the disjoint union of ZX-diagrams. The identity on $n$ is
the disjoint union of $n$ wires:
\begin{center}
\begin{tikzpicture}
\node (1) at (0,0) {$ $};
\node (2) at (0,0.25) {$ $};
\node ( ) at (1.5,0.6) {$ \vdots $};
\node (3) at (0,0.75) {$ $};
\node (1') at (3,0) {$ $};
\node (2') at (3,0.25) {$ $};
\node (3') at (3,0.75) {$ $};
\draw
(1) -- (1')
(2) -- (2')
(3) -- (3');
\end{tikzpicture}
\end{center}
The symmetry and compactness of the monoidal product provide
a braiding, evaluation, and coevaluation morphisms:
\[
\includegraphics{InclGrphx--morphism--zx_braiding}
\quad \quad \quad \quad
\raisebox{-0.25\height}{%
\includegraphics[scale=0.75]{InclGrphx--morphism--zx_evaluation}
\quad \quad \quad \quad
\raisebox{-0.25\height}{%
\includegraphics[scale=0.75]{InclGrphx--morphism--zx_coevaluation}
\]
The evaluation and coevalutation arrows are of type
$2n \to 0$ and $0 \to 2n$ for each object
$n \geq 1$ and the empty diagram for $n=0$. On
the spider diagrams, the dagger structure swaps
inputs and outputs then multiplies the phase by
\[
\includegraphics{InclGrphx--functor--dagger}
\]
The dagger acts trivially on the wire, Hadamard,
and diamond elements.
\begin{figure}
\fbox{
\begin{minipage}{\textwidth}
\centering
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics{InclGrphx--equation--spider} \\
\textsc{Spider}
\end{minipage}%
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics{InclGrphx--equation--bialgebra} \\
\textsc{Bialgebra}
\end{minipage}
\vspace{1em}
\linebreak
\begin{minipage}{0.3\linewidth}
\centering
\includegraphics{InclGrphx--equation--copy} \\
\textsc{Copy}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\centering
\includegraphics{InclGrphx--equation--pi_copy} \\
\textsc{$ \pi $-Copy}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\centering
\includegraphics{InclGrphx--equation--cup} \\
\textsc{Cup}
\end{minipage}
\vspace{1em}
\linebreak
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics{InclGrphx--equation--trivial_spider} \\
\textsc{Trivial Spider}
\end{minipage}%
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics{InclGrphx--equation--pi_commutation} \\
\textsc{$\pi$-Commutation}
\end{minipage}%
\linebreak
\begin{minipage}{0.3\linewidth}
\centering
\includegraphics{InclGrphx--equation--color_change} \\
\textsc{Color Change}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\centering
\includegraphics{InclGrphx--equation--loop} \\
\textsc{Loop}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\centering
\includegraphics{InclGrphx--equation--diamond} \\
\textsc{Diamond}
\end{minipage}%
\end{minipage}
\caption{Relations in the category $\ZX$}
\label{fig:zx-equations}
\end{figure}
A major advantage of using string diagrams, apart
from their intuitive nature, is that computations
are more easily programmed into computers.
Indeed, graphical proof assistants like
[10, 28]
and Globular
[10]
were made for such graphical reasoning. The logic
of these programs are encapsulated by double
pushout rewrite rules. However, the algebraic
structure of $\ZX $ and other graphical calculi do
not contain the rewrite rules as explicit
elements. On the other hand, the framework developed in this thesis
explicitly includes the rewrite rules.
To model the ZX-calculus using structured cospans, we need
an appropriate adjunction $ L \from \A \lrto \X \from R $.
Determining the correct pieces to fill in requires some
discussion. Before providing the details, we sketch the
process. Let $ \A \bydef \FinSet $ be the topos of finite
sets and functions. Let $ \X \bydef \FinGraphGamma $ be the
over-category where we chose a graph $ \Gamma $ to provide
the objects of $ \X \bydef \FinGraph / \Gamma $ with the
same type information as the ZX-diagrams. The functor $ L $
turns a finite set $ a $ into a certain discrete graph over
$ \Gamma $ so that $ La $ can serve as inputs or outputs. To
unpack what this all means, we start with the over-category.
\begin{definition}
Let $g$ be a graph. By a \defn{graph over $g$},
we mean a graph morphism $x \to g$. A morphism
between graphs over $g$ is a graph morphism
$x \to y$ such that
\[
\begin{tikzpicture}
\node (1) at (-1,2) {$x$};
\node (2) at (1,2) {$y$};
\node (3) at (0,0) {$g$};
\draw[cd]
(1) edge (2)
(1) edge (3)
(2) edge (3);
\end{tikzpicture}
\]
\end{definition}
One way to think of a graph over $ g $ is as
a $ g $-typed graph. Consider the following simple
\begin{example}
\label{ex:graph-over-g}
Let $ g $ be the graph
\[
\begin{tikzpicture}
\node (a) at (0,0) {$ A $};
\node (b) at (3,0) {$ B $};
\draw[graph]
(a) edge[bend left] node[above]{$F$} (b)
(b) edge[bend left] node[below]{$G$} (a);
\end{tikzpicture}
\]
Let $ x $ be the graph
\[
\begin{tikzpicture}
\node (a) at (0,0) {$ a $};
\node (b) at (4,0) {$ b $};
\node (c) at (2,2) {$ c $};
\node (d) at (2,-2) {$ d $};
\draw[graph]
(a) edge node[above,left]{$ e $} (c)
(a) edge node[below,left]{$ f $} (d)
(c) edge node[above,right]{$ g $} (b)
(d) edge node[below,right]{$ h $} (b);
\end{tikzpicture}
\]
that lies over $ g $ via the map
\begin{align*}
a,b & \mapsto A & e,f & \mapsto F \\
c,d & \mapsto B & g,h & \mapsto G
\end{align*}
If we think of the nodes and edges of $ g $ as
types, then these types are transported to $ x $
along the fibers of this map. Thus $ x $ is a
graph with the following type-assignment:
\begin{center}
\begin{tabular}{ccccc}
$ a : A $ && $ b : A $ && $ c : B $ \\
$ d : B $ && $ e : F $ && $ f : F $ \\
$ g : G $ && $ h : G $ && \\
\end{tabular}
\end{center}
where `$ : $' should be read `is type'. Any graph over
$ g $ can have two node types $ A,B $ and two edge types
$ F,G $. Edges can only go from an $ A $-type
node to a $ B $-type node or vice versa. Edges cannot
traverse nodes of the same type simply because there are
no looped edges in $ g $.
A compact way to draw a graph over $ g $ is to
label its nodes and edges with their types.
Thus, the over-graph $ x \to g $ can be drawn as
\[
\begin{tikzpicture}
\node (a) at (0,0) {$ (a,A) $};
\node (b) at (4,0) {$ (b,A) $};
\node (c) at (2,2) {$ (c,B) $};
\node (d) at (2,-2) {$ (d,B) $};
\draw[graph]
(a) edge node[above,left]{$ (e,F) $} (c)
(a) edge node[below,left]{$ (f,F) $} (d)
(c) edge node[above,right]{$ (g,G) $} (b)
(d) edge node[below,right]{$ (h,G) $} (b);
\end{tikzpicture}
\]
One might recognize the class of graphs over $ g $ as
something like a bipartite graph. The difference between
graphs over $ g $ and bipartite graphs is that bipartite
graphs are usually defined by graph theorists to satisfy
\emph{the property} that the nodes can be partitioned into
two classes and the source and target of each edge must
belong to different classes. On the other hand, graphs
over $ g $ are graphs equipped with extra structure,
namely the type information. This distinction does not
appear in the graphs themselves, so we look at their
A morphism of graphs over $ g $ must respect the type
information. So if $ x \to g $ and $ y \to g $ are graphs
over $ g $, then a morphism between them is a graph
morphism $ x \to y $ such that the diagram
\[
\begin{tikzpicture}
\node (x) at (0,0) {$ x $};
\node (y) at (4,0) {$ y $};
\node (g) at (2,-2) {$ g $};
\draw [cd]
(x) edge (y)
(x) edge (g)
(y) edge (g);
\end{tikzpicture}
\]
commutes. Suppose that $ x $ is a single node typed $ A $
and $ y $ is a single node typed $ B $. There is no
morphism between them because the node in $ x $ must be
sent to a node of type $ A $. However, any two bipartite
graphs with a single node and no edges are isomorphic. The
moral of this example is by adding the type information, we
added structure instead of imposing a property. We denote by
$ \Graph \downarrow g $ the category of graphs over $ g $
and their morphisms.
\end{example}
We exploit this method of defining
`typed graphs' to transform typical combinatorial
graphs into ZX-diagrams. The types needed to make
ZX-diagrams from graphs encoded into the graph $ \Gamma $
that we define now.
\begin{definition}
\label{ex:basic-graph-over-g}
Let $ \Gamma $ be the graph
\begin{equation}
\label{eq:gamma}
\begin{tikzpicture}
\node [zxwhite] (w) at (0,0) {$ $};
\node [zxgreen] (g) at (-1,1) {};
\node () at (-1.33,1) {\scriptsize{$\alpha$}};
\node [zxred] (r) at (1,1) {};
\node () at (1.33,1) {\scriptsize{$\beta$}};
\node [zxblack] (b) at (-1,-1) {$$};
\node [zxyellow] (y) at (1,-1) {$$};
\path[graph]
(w) edge [loop below] (w)
(w) edge [bend left=10] (g)
(w) edge [bend left=10] (r)
(w) edge [bend left=10] (b)
(w) edge [bend left=10] (y)
(g) edge [bend left=10] (w)
(r) edge [bend left=10] (w)
(b) edge [bend left=10] (w)
(y) edge [bend left=10] (w);
\node () at (3,0) {$ \alpha,\beta \in [ -\pi , \pi ) $};
\end{tikzpicture}.
\end{equation}
We have not drawn the entirety of $ \Gamma $. In
actuality, the green and red nodes run through
$ [ -\pi , \pi ) $ and each of them have a single
arrow to and from the white node
\end{definition}
Note that the graphs over $ \Gamma $ are completely
determined by the function's behavior on the nodes. This is
because there is at most one arrow between any two
nodes. When comparing the $ \Gamma $-types to the types
appearing in the basic ZX-diagrams of Figure
\ref{fig:zx-generators}, there is a clear correlation
except, perhaps, for the white node. To explain the white
node, first observe that ZX-diagrams have dangling wires on
either end. Dangling edges are not permitted in our
definition of graphs, so the white node anchors them.
To draw graphs over $ \Gamma $, we attach the type
information to the nodes by rendering the nodes as
red, greed, white, black, or yellow. This manner
of drawing is more economical than drawing a graph
and describing its map to $ \Gamma $. For
example, consider the graph
\[
\begin{tikzpicture}
\node (0) at (0,0) {$ \bullet_a $};
\node (1) at (2,0) {$ \bullet_b $};
\node (2) at (4,0) {$ \bullet_c $};
\draw[graph]
(0) edge (1)
(1) edge (2);
\end{tikzpicture}
\]
with the map to $ \Gamma $ determined by
\[
\begin{tikzpicture}
\node at (0,1) {$ a,c \mapsto $};
\node [zxwhite] at (1,1) {$ $};
\node at (0,0) {$ b \mapsto $};
\node [zxgreen] at (1,0) {$ \beta $};
\end{tikzpicture}
\]
We draw this as
\[
\begin{tikzpicture}
\node [zxwhite] (1) at (0,0) {$ $};
\node [zxgreen] (2) at (2,0) {$ \beta $};
\node [zxwhite] (3) at (4,0) {$ $};
\draw[graph]
(1) edge (2)
(2) edge (3);
\end{tikzpicture}
\]
In our adjunction $ L \from \A \lrto \X \from R $, we let
$ \X $ be $ \FinGraphGamma $. This is a topos by the
fundamental theorem of topos theory, which we present in
Theorem \ref{thm:fund-thm-topos}.
The most important objects in $ \FinGraphGamma $ are those
corresponding to the basic ZX-diagrams. These are displaying
in Figure \ref{fig:basic-diagrams-over-gamma}. To choose a
category $ \A $ of interface types, we want to faithfully
represent the fact that ZX-diagrams have a non-negative
integer number of inputs and outputs. Therefore, we let
$ \A $ be the topos $\FinSet $ of finite sets and functions.
\begin{figure}
\fbox{
\begin{minipage}{\textwidth}
\centering
\begin{minipage}{0.3\linewidth}
\centering
\[
\begin{tikzpicture}
\node [zxwhite] (a) at (0,0) {};
\node [zxwhite] (b) at (2,0) {};
\draw [graph]
(a) to (b);
\end{tikzpicture}
\]
\textsc{Wire}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\centering
\[
\begin{tikzpicture}
\node [zxwhite] (a) at (0,0) {$ $};
\node ( ) at (0,0.66) {$ \vdots $};
\node [zxwhite] (b) at (0,1) {$ $};
\node [zxgreen] (c) at (1,0.5) {$ $};
\node ( ) at (1,1.25) {$\alpha$};
\node [zxwhite] (d) at (2,0) {$ $};
\node ( ) at (2,0.66) {$ \vdots $};
\node [zxwhite] (e) at (2,1) {$ $};
\draw[graph]
(a) edge [bend right=10] (c)
(b) edge [bend left=10] (c)
(c) edge [bend right=10] (d)
(c) edge [bend left=10] (e);
\end{tikzpicture}
\]
\textsc{Green Spider}
\end{minipage}
\begin{minipage}{0.3\linewidth}
\centering
\[
\begin{tikzpicture}
\node [zxwhite] (a) at (0,0) {$ $};
\node ( ) at (0,0.66) {$ \vdots $};
\node [zxwhite] (b) at (0,1) {$ $};
\node [zxred] (c) at (1,0.5) {$ $};
\node ( ) at (1,1.25) {$\alpha$};
\node [zxwhite] (d) at (2,0) {$ $};
\node ( ) at (2,0.66) {$ \vdots $};
\node [zxwhite] (e) at (2,1) {$ $};
\draw[graph]
(a) edge [bend right=10] (c)
(b) edge [bend left=10] (c)
(c) edge [bend right=10] (d)
(c) edge [bend left=10] (e);
\end{tikzpicture}
\]
\textsc{Red Spider}
\end{minipage}
\vspace{1em}
\linebreak
\begin{minipage}{0.5\linewidth}
\centering
\[
\begin{tikzpicture}
\node [zxwhite] (a) at (0,0) {$$};
\node [zxyellow] (b) at (1,0) {$$};
\node [zxwhite] (c) at (2,0) {$$};
\draw[graph]
(a) edge (b)
(b) edge (c);
\end{tikzpicture}
\]
\textsc{Hadamard}
\end{minipage}%
\begin{minipage}{0.5\linewidth}
\centering
\[
\begin{tikzpicture}
\node [zxblack] () at (0,0) {$$};
\end{tikzpicture}
\]
\textsc{Diamond}
\end{minipage}%
\end{minipage}
\caption{Basic ZX-diagrams as graphs over $ \Gamma $}
\label{fig:basic-diagrams-over-gamma}
\end{figure}
We still need to define $ L $ and $ R $ in the adjunction
\[
\adjunction{\FinSet}{\FinGraphGamma}{L}{R}{4}
\]
\[
L \from \FinSet \to \FinGraphGamma
\]
by letting $ La $ be the edgeless graph with node
set $ a $ that is constant over the whites
node in $ \Gamma $. A function $ f \from a \to b $ of finite
sets becomes of morphism $ Lf $ of graphs over $
\Gamma $ that simply reinterprets the action of $
f $ on elements of a set to white nodes in a
graph. Define
\[
R \from \FinGraphGamma \to \FinSet
\]
by defining $ R ( x \to \Gamma ) $ as the fiber in
$ x $ of the white node. Given a morphism of
graphs over $ \Gamma $, $ R $ restricts it to the
function on only the white nodes.
\begin{lemma}
The functor pair
\[
\adjunction{\FinSet}{\FinGraphGamma}{L}{R}{4}
\]
forms an adjunction and $ L $ preserves pullbacks.
\end{lemma}
\begin{proof}
Observe that the composite $ RL $ is the
identity functor. So the unit
$ \eta \from a \to RLa $ is the identity which
is natural in $ a $. The
counit $ \epsilon \from LRx \to x $ is the
inclusion of the white nodes of $ x $ into
$ x $. Given an arrow $ f \from x \to y $ in $
\FinGraphGamma $, the diagram
\[
\begin{tikzpicture}
\node (LRx) at (0,2) {$ LRx $};
\node (x) at (2,2) {$ x $};
\node (LRy) at (0,0) {$ LRy $};
\node (y) at (2,0) {$ y $};
\draw[cd]
(LRx) edge node[above]{$ \epsilon_x $} (x)
(LRy) edge node[below]{$ \epsilon_y $} (y)
(LRx) edge node[left]{$ LRf $} (LRy)
(x) edge node[right]{$ f $} (y);
\end{tikzpicture}
\]
commutes since $ LRf $ is a restriction of $ f $. To show
that $ L $ preserves pullbacks, take a cospan
\[ a \to b \gets c \] in $ \Set $ with pullback
$ a \times_b c $ and apply $ L $ to get the diagram
\[
\begin{tikzpicture}
\node (La) at (-2,0) {$ La $};
\node (Lb) at (0,0) {$ Lb $};
\node (Lc) at (0,2) {$ Lc $};
\node (Gamma) at (2,-2) {$ \Gamma $};
\path [cd]
(La) edge (Lb)
(Lc) edge (Lb)
(La.-90) edge[bend right] (Gamma.180)
(Lb) edge (Gamma)
(Lc.0) edge[bend left] (Gamma.90);
\end{tikzpicture}
\]
comprised of edgeless graphs $ La $, $ Lb $, and $ Lc $
that are constant over the white node in $ \Gamma $. The
pullback of this diagram is
$ La \times_{Lb} Lc \to \Gamma $ which is constant over
the white node. This is isomorphic to $ L(a \times_b c)
\to \Gamma $ which is constant over the white node.
\end{proof}
With our adjunction established, we can define structured
cospans of graphs over $ \Gamma $ and therefore also the
symmetric monoidal double category of bold rewrites
$ _L \BBBoldRewrite $ for the functor
$ L \from \FinSet \to \FinGraphGamma $ defined above. This
double category has as objects the finite sets, as
horizontal 1-arrows the structured cospans of graphs over
$ \Gamma $, as vertical 1-arrows the spans of finite sets
with invertible legs, and as squares all possible
bold rewrites of structured cospans. Clearly,
$ _L \BBBoldRewrite $ is far bigger than the ZX-calculus
because it contains graphs over $ \Gamma $ with no
corresponding ZX-diagram. This does not mean, however, that
$ _L \BBBoldRewrite $ serves no purpose. It plays the role
of an ambient space in which we chisel out a sub-double
category that \emph{does} correspond to the ZX-calculus.
To begin the process of constructing this sub-double
category of $ _L \BBBoldRewrite $, we identify structured
cospans to capture the basic ZX-diagrams and identify bold
rewrites of structured cospans for the basic
ZX-relations. We also include some additional structured
cospans to give the desired structure. Figure
\ref{fig:basic-diagrams-as-str-csps}
depicts the basic ZX-diagrams as structured
\begin{figure}
\fbox{
\scalebox{0.9}{
\begin{minipage}{\textwidth}
\centering
\begin{minipage}{\textwidth}
\centering
\[
\begin{tikzpicture}
\begin{scope}
\node [zxwhite] (a) at (0,0) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,0.5);
\node (l) at (0.6,0) {};
\end{scope}
\begin{scope}[shift={(2,0)}]
\node [zxwhite] (a) at (0,0) {};
\node [zxwhite] (b) at (2,0) {};
\draw [graph]
(a) to (b);
\draw [rounded corners]
(-0.5,-0.5) rectangle (2.5,0.5);
\node (cr) at (2.6,0) {};
\node (cl) at (-0.6,0) {};
\node () at (1,-1) {\textsc{Wire}};
\end{scope}
\begin{scope}[shift={(6,0)}]
\node [zxwhite] (a) at (0,0) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,0.5);
\node (r) at (-0.6,0) {};
\end{scope}
\draw[cd]
(l) edge (cl)
(r) edge (cr);
\end{tikzpicture}
\]
\end{minipage}%
\vspace{1em}
\linebreak
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{minipage}{0.40\linewidth}
\[
\begin{tikzpicture}
\begin{scope}
\node [zxwhite] (a) at (0,0) {};
\node ( ) at (0,0.66) {$ \vdots $};
\node [zxwhite] (b) at (0,1) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (l) at (0.6,0.5) {};
\end{scope}
\begin{scope}[shift={(2,0)}]
\node [zxwhite] (a) at (0,0) {$ $};
\node ( ) at (0,0.66) {$ \vdots $};
\node [zxwhite] (b) at (0,1) {$ $};
\node [zxgreen] (c) at (1,0.5) {$ $};
\node ( ) at (1,1) {$\alpha$};
\node [zxwhite] (d) at (2,0) {$ $};
\node ( ) at (2,0.66) {$ \vdots $};
\node [zxwhite] (e) at (2,1) {$ $};
\draw[graph]
(a) edge[bend right=10] (c)
(b) edge[bend left=10] (c)
(c) edge[bend right=10] (d)
(c) edge[bend left=10] (e);
\draw [rounded corners]
(-0.5,-0.5) rectangle (2.5,1.5);
\node (cl) at (-0.6,0.5) {};
\node (cr) at (2.6,0.5) {};
\node () at (1,-1) {\textsc{Green Spider}};
\end{scope}
\begin{scope}[shift={(6,0)}]
\node [zxwhite] (a) at (0,0) {};
\node ( ) at (0,0.66) {$ \vdots $};
\node [zxwhite] (b) at (0,1) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (r) at (-0.6,0.5) {};
\end{scope}
\path[cd]
(l) edge (cl)
(r) edge (cr);
\end{tikzpicture}
\]
\end{minipage}
\hspace{3em}
\begin{minipage}{0.45\linewidth}
\[
\begin{tikzpicture}
\begin{scope}
\node [zxwhite] (a) at (0,0) {};
\node ( ) at (0,0.66) {$ \vdots $};
\node [zxwhite] (b) at (0,1) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (l) at (0.6,0.5) {};
\end{scope}
\begin{scope}[shift={(2,0)}]
\node [zxwhite] (a) at (0,0) {$ $};
\node ( ) at (0,0.66) {$ \vdots $};
\node [zxwhite] (b) at (0,1) {$ $};
\node [zxred] (c) at (1,0.5) {$ $};
\node ( ) at (1,1) {$\alpha$};
\node [zxwhite] (d) at (2,0) {$ $};
\node ( ) at (2,0.66) {$ \vdots $};
\node [zxwhite] (e) at (2,1) {$ $};
\draw[graph]
(a) edge[bend right=10] (c)
(b) edge[bend left=10] (c)
(c) edge[bend right=10] (d)
(c) edge[bend left=10] (e);
\draw [rounded corners]
(-0.5,-0.5) rectangle (2.5,1.5);
\node (cl) at (-0.6,0.5) {};
\node (cr) at (2.6,0.5) {};
\node () at (1,-1) {\textsc{Red Spider}};
\end{scope}
\begin{scope}[shift={(6,0)}]
\node [zxwhite] (a) at (0,0) {};
\node ( ) at (0,0.66) {$ \vdots $};
\node [zxwhite] (b) at (0,1) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (r) at (-0.6,0.5) {};
\end{scope}
\path[cd]
(l) edge (cl)
(r) edge (cr);
\end{tikzpicture}
\]
\end{minipage}
\vspace{1em}
\linebreak
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{minipage}{0.5\linewidth}
\centering
\[
\begin{tikzpicture}
\begin{scope}
\node [zxwhite] (a) at (0,0) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,0.5);
\node (l) at (0.6,0) {};
\end{scope}
\begin{scope}[shift={(2,0)}]
\node [zxwhite] (a) at (0,0) {$$};
\node [zxyellow] (b) at (1,0) {$$};
\node [zxwhite] (c) at (2,0) {$$};
\draw [graph]
(a) edge (b)
(b) edge (c);
\draw [rounded corners]
(-0.5,-0.5) rectangle (2.5,0.5);
\node (cl) at (-0.6,0) {};
\node (cr) at (2.6,0) {};
\node () at (1,-1) {\textsc{Hadamard}};
\end{scope}
\begin{scope}[shift={(6,0)}]
\node [zxwhite] (a) at (0,0) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,0.5);
\node (r) at (-0.6,0) {};
\end{scope}
\path[cd]
(l) edge (cl)
(r) edge (cr);
\end{tikzpicture}
\]
\end{minipage}%
\begin{minipage}{0.5\linewidth}
\centering
\[
\begin{tikzpicture}
\begin{scope}[shift={(0,0)}]
\node () at (0,0) {$ \emptyset $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,0.5);
\node (l) at (0.6,0) {};
\end{scope}
\begin{scope}[shift={(2,0)}]
\node [zxblack] () at (0,0) {$$};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,0.5);
\node (cl) at (-0.6,0) {};
\node (cr) at (0.6,0) {};
\node () at (0,-1) {\textsc{Diamond}};
\end{scope}
\begin{scope}[shift={(4,0)}]
\node (a) at (0,0) {$ \emptyset $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,0.5);
\node (r) at (-0.6,0) {};
\end{scope}
\draw[cd]
(l) edge (cl)
(r) edge (cr);
\end{tikzpicture}
\]
\end{minipage}
\vspace{1em}
\linebreak
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{minipage}{0.45\linewidth}
\centering
\[
\begin{tikzpicture} % CUP
\begin{scope} % LEFT
\node [zxwhite] (b) at (0,1) {$ $};
\node ( ) at (0,0.5) {$\vdots$};
\node [zxwhite] (d) at (0,0) {$ $};
\node (0r) at (0.6,0.5) {$ $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~
\begin{scope}[shift={(2,0)}] % MID
\node [zxwhite] (a) at (0,1) {$ $};
\node ( ) at (0,0.5) {$\vdots$};
\node [zxwhite] (b) at (0,0) {$ $};
\node [zxwhite] (c) at (1,0.5) {$ $};
\draw [graph]
(a) edge[bend left=10] (c)
(b) edge[bend right=10] (c);
\node (1l) at (-0.6,0.5) {$ $};
\node (1r) at (1.6,0.5) {$ $};
\node () at (0,-1) {\textsc{Cup}};
\draw [rounded corners]
(-0.5,-0.5) rectangle (1.5,1.5);
\end{scope}
% ~~~
\begin{scope}[shift={(5,0)}] % RIGHT
\node (a) at (0,0.5) {$ \emptyset $};
\node (2l) at (-0.6,0.5) {$ $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~~~~~ ARROWS
\draw[cd]
(0r) edge (1l)
(2l) edge (1r);
\end{tikzpicture}
\]
\end{minipage}
\begin{minipage}{0.45\linewidth} % CAP
\centering
\[
\begin{tikzpicture}
\begin{scope} % LEFT
\node (a) at (0,0.5) {$ \emptyset $};
\node (0r) at (0.6,0.5) {$ $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~
\begin{scope}[shift={(2,0)}] % MID
\node [zxwhite] (a) at (1,1) {$ $};
\node ( ) at (1,0.5) {$\vdots$};
\node [zxwhite] (b) at (1,0) {$ $};
\node [zxwhite] (c) at (0,0.5) {$ $};
\draw [graph]
(c) edge [bend left=10] (a)
(c) edge [bend right=10] (b);
\node (1l) at (-0.6,0.5) {$ $};
\node (1r) at (1.6,0.5) {$ $};
\node () at (0,-1) {\textsc{Cap}};
\draw [rounded corners]
(-0.5,-0.5) rectangle (1.5,1.5);
\end{scope}
% ~~~~~~~~
\begin{scope}[shift={(5,0)}] % RIGHT
\node [zxwhite] (b) at (0,1) {$ $};
\node ( ) at (0,0.5) {$\vdots$};
\node [zxwhite] (d) at (0,0) {$ $};
\node (2l) at (-0.6,0.5) {$ $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~~~~~
\path[cd]
(0r) edge (1l)
(2l) edge (1r);
\end{tikzpicture}
\]
\end{minipage}
\linebreak
\vspace{1em}
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{minipage}{0.45\linewidth}
\centering
\[
\begin{tikzpicture} % MULTIPLICATION
\begin{scope} % LEFT
\node [zxwhite] () at (0,1) {$ $};
\node [zxwhite] () at (0,0) {$ $};
\node (0r) at (0.6,0.5) {$ $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~
\begin{scope}[shift={(2,0)}] % MID
\node [zxwhite] (b) at (0.5,0.5) {$ $};
\node (1l) at (-0.6,0.5) {$ $};
\node (1r) at (1.6,0.5) {$ $};
\node () at (0,-1) {\textsc{Multiplication}};
\draw [rounded corners]
(-0.5,-0.5) rectangle (1.5,1.5);
\end{scope}
% ~~~
\begin{scope}[shift={(5,0)}] % RIGHT
\node [zxwhite] (a) at (0,0.5) {};
\node (2l) at (-0.6,0.5) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~~~~~ ARROWS
\draw[cd]
(0r) edge (1l)
(2l) edge (1r);
\end{tikzpicture}
\]
\end{minipage}
\begin{minipage}{0.45\linewidth} % COMULTIPLICATION
\centering
\[
\begin{tikzpicture}
\begin{scope} % LEFT
\node [zxwhite] () at (0,0.5) {};
\node (0r) at (0.6,0.5) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~
\begin{scope}[shift={(2,0)}] % MID
\node [zxwhite] (a) at (0.5,0.5) {};
\node (1l) at (-0.6,0.5) {$ $};
\node (1r) at (1.6,0.5) {$ $};
\node () at (0,-1) {\textsc{Comultiplication}};
\draw [rounded corners]
(-0.5,-0.5) rectangle (1.5,1.5);
\end{scope}
% ~~~~~~~~
\begin{scope}[shift={(5,0)}] % RIGHT
\node [zxwhite] () at (0,1) {$ $};
\node [zxwhite] () at (0,0) {$ $};
\node (2l) at (-0.6,0.5) {$ $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~~~~~
\path[cd]
(0r) edge (1l)
(2l) edge (1r);
\end{tikzpicture}
\]
\end{minipage}
\linebreak
\vspace{1em}
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{minipage}{0.45\linewidth}
\centering
\[
\begin{tikzpicture} % UNIT
\begin{scope} % LEFT
\node () at (0,0.5) {$ \emptyset $};
\node (0r) at (0.6,0.5) {$ $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~
\begin{scope}[shift={(2,0)}] % MID
\node [zxwhite] (b) at (0.5,0.5) {$ $};
\node (1l) at (-0.6,0.5) {$ $};
\node (1r) at (1.6,0.5) {$ $};
\node () at (0,-1) {\textsc{Unit}};
\draw [rounded corners]
(-0.5,-0.5) rectangle (1.5,1.5);
\end{scope}
% ~~~
\begin{scope}[shift={(5,0)}] % RIGHT
\node [zxwhite] (a) at (0,0.5) {};
\node (2l) at (-0.6,0.5) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~~~~~ ARROWS
\draw[cd]
(0r) edge (1l)
(2l) edge (1r);
\end{tikzpicture}
\]
\end{minipage}
\begin{minipage}{0.45\linewidth} % COUNIT
\centering
\[
\begin{tikzpicture}
\begin{scope} % LEFT
\node [zxwhite] () at (0,0.5) {};
\node (0r) at (0.6,0.5) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~
\begin{scope}[shift={(2,0)}] % MID
\node [zxwhite] (a) at (0.5,0.5) {};
\node (1l) at (-0.6,0.5) {$ $};
\node (1r) at (1.6,0.5) {$ $};
\node () at (0,-1) {\textsc{Counit}};
\draw [rounded corners]
(-0.5,-0.5) rectangle (1.5,1.5);
\end{scope}
% ~~~~~~~~
\begin{scope}[shift={(5,0)}] % RIGHT
\node () at (0,0.5) {$ \emptyset $};
\node (2l) at (-0.6,0.5) {$ $};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\end{scope}
% ~~~~~~~~
\path[cd]
(0r) edge (1l)
(2l) edge (1r);
\end{tikzpicture}
\]
\end{minipage}
\linebreak
\vspace{1em}
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{minipage}{1.0\linewidth}
\centering
\[
\begin{tikzpicture}
\begin{scope}[shift={(0,0)}]
\node [zxwhite] () at (0,0) {};
\node [zxwhite] () at (0,1) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (l) at (0.6,0.5) {};
\end{scope}
\begin{scope}[shift={(2,0)}]
\node [zxwhite] (00) at (0,0) {};
\node [zxwhite] (01) at (0,1) {};
\node [zxwhite] (10) at (1,0) {};
\node [zxwhite] (11) at (1,1) {};
\draw [graph]
(00) edge (11)
(01) edge[draw=white,line width=2pt] (10)
(01) edge (10);
\draw [rounded corners]
(-0.5,-0.5) rectangle (1.5,1.5);
\node (cl) at (-0.6,0.5) {};
\node (cr) at (1.6,0.5) {};
\node () at (0.5,-1) {\textsc{Braid}};
\end{scope}
\begin{scope}[shift={(5,0)}]
\node [zxwhite](a) at (0,0) {};
\node [zxwhite](a) at (0,1) {};
\draw [rounded corners]
(-0.5,-0.5) rectangle (0.5,1.5);
\node (r) at (-0.6,0.5) {};
\end{scope}
\draw[cd]
(l) edge (cl)
(r) edge (cr);
\end{tikzpicture}
\]
\end{minipage}
\end{minipage}
\caption{Basic ZX-diagrams as structured cospans}
\label{fig:basic-diagrams-as-str-csps}
\end{figure}
Translating the relations between ZX-diagrams to
structured cospans is quite straightforward. We
provide several examples.
\[
\begin{tikzpicture}
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# A Latent Variational Framework for Stochastic Optimization
Philippe Casgrain
Department of Statistical Sciences
University of Toronto
Toronto, ON, Canada
<EMAIL_ADDRESS>
###### Abstract
This paper provides a unifying theoretical framework for stochastic
optimization algorithms by means of a latent stochastic variational problem.
Using techniques from stochastic control, the solution to the variational
problem is shown to be equivalent to that of a Forward Backward Stochastic
Differential Equation (FBSDE). By solving these equations, we recover a
variety of existing adaptive stochastic gradient descent methods. This
framework establishes a direct connection between stochastic optimization
algorithms and a secondary latent inference problem on gradients, where a
prior measure on gradient observations determines the resulting algorithm.
## 1 Introduction
Stochastic optimization algorithms are tools which are crucial to solving
optimization problems arising in machine learning. The initial motivation for
these algorithms comes from the fact that computing the gradients of a target
loss function becomes increasingly difficult as the scale and dimension of an
optimization problem grows larger. In these large-scale optimization problems,
deterministic gradient-based optimization algorithms perform poorly due to the
computational load of repeatedly computing gradients. Stochastic optimization
algorithms remedy this issue by replacing exact gradients of the target loss
with a computationally cheap gradient estimator, trading off noise in gradient
estimates for computational efficiency at each step.
To illustrate this idea, consider the problem of minimizing a generic risk
function $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$, taking the form
$f(x)=\frac{1}{|{\mathfrak{N}}|}\sum_{z\in{\mathfrak{N}}}\ell(x;z)\;,$ (1)
where $\ell:\mathbb{R}^{d}\times\mathcal{Z}\rightarrow\mathbb{R}$, and where
we define the set ${\mathfrak{N}}:=\\{z_{i}\in\mathcal{Z}\;,\;i=1,\dots,N\\}$
to be a set of training points. In this definition, we interpret $\ell(x;z)$
as the model loss at a single training point $z\in{\mathfrak{N}}$ for the
parameters $x\in\mathbb{R}^{d}$.
When $N$ and $d$ are typically large, computing the gradients of $f$ can be
time-consuming. Knowing this, let us consider the path of an optimization
algorithm as given by $\\{x_{t}\\}_{t\in\mathbb{N}}$. Rather than computing
$\nabla f(x_{t})$ directly at each point of the optimization process, we may
instead collect noisy samples of gradients as
$g_{t}=\frac{1}{|{\mathfrak{N}}_{t}^{m}|}\sum_{z\in{\mathfrak{N}}_{t}^{m}}\nabla_{x}\ell(x_{t};z)\;,$
(2)
where for each $t$, ${\mathfrak{N}}_{t}^{m}\subseteq{\mathfrak{N}}$ is an
independent sample of size $m$ from the set of training points. We assume that
$m\ll N$ is chosen small enough so that $g_{t}$ can be computed at a
significantly lower cost than $\nabla f(x_{t})$. Using the collection of noisy
gradients $\\{g_{t}\\}_{t\in\mathbb{N}}$, stochastic optimization algorithms
construct an estimator $\widehat{\nabla f}(x_{t})$ of the gradient $\nabla
f(x_{t})$ in order to determine the next step $x_{t+1}$ of the optimizer.
This paper presents a theoretical framework which provides new perspectives on
stochastic optimization algorithms, and explores the implicit model
assumptions that are made by existing ones. We achieve this by extending the
approach taken by Wibisono et al. (2016) to stochastic algorithms. The key
step in our approach is to interpret the task of optimization with a
stochastic algorithm as a latent variational problem. As a result, we can
recover algorithms from this framework which have built-in online learning
properties. In particular, these algorithms use an online Bayesian filter on
the stream of noisy gradient samples, $g_{t}$, to compute estimates of $\nabla
f(x_{t})$. Under various model assumptions on $\nabla f$ and $g$, we recover a
number of common stochastic optimization algorithms.
### 1.1 Related Work
There is a rich literature on stochastic optimization algorithms as a
consequence of their effectiveness in machine learning applications. Each
algorithm introduces its own variation on the gradient estimator
${\widehat{\nabla f}(x_{t})}$ as well as other features which can improve the
speed of convergence to an optimum. Amongst the simplest of these is
_stochastic gradient descent_ and its variants Robbins and Monro (1951), which
use an estimator based on single gradient samples. Others, such as Lucas et
al. (2018); Nesterov , use momentum and acceleration as features to enhance
convergence, and can be interpreted as using exponentially weighted moving
averages as gradient estimators. Adaptive gradient descent methods such as
AdaGrad from Duchi et al. (2011) and Adam from Kingma and Ba (2014) use
similar moving average estimators, as well as dynamically updated
normalization factors. For a survey paper which covers many modern stochastic
optimization methods, see Ruder (2016).
There exist a number of theoretical interpretations of various aspects of
stochastic optimization. Cesa-Bianchi et al. (2004) have shown a parallel
between stochastic optimization and online learning. Some previous related
works, such as Gupta et al. (2017) provide a general model for adaptive
methods, generalizing the subgradient projection approach of Duchi et al.
(2011). Aitchison (2018) use a Bayesian model to explain the various features
of gradient estimators used in stochastic optimization algorithms . This paper
differs from these works by naturally generating stochastic algorithms from a
variational principle, rather than attempting to explain their individual
features. This work is most similar to that of Wibisono et al. (2016) who
provide a variational model for continuous deterministic optimization
algorithms.
There is a large body of research on continuous-time approximations to
deterministic optimization algorithms via dynamical systems (ODEs) (Su et al.
(2014); Krichene et al. (2015); Wilson et al. (2016); da Silva and Gazeau
(2018)), as well as approximations to stochastic optimization algorithms by
stochastic differential equations (SDEs) (Xu et al. (2018a, b); Raginsky and
Bouvrie (2012); Mertikopoulos and Staudigl (2018); Krichene and Bartlett
(2017)). In particular, the most similar of these works, Raginsky and Bouvrie
(2012); Xu et al. (2018a, b), study continuous approximations to stochastic
mirror descent by adding exogenous Brownian noise to the continuous dynamics
derived in Wibisono et al. (2016). This work differs by deriving continuous
stochastic dynamics for optimizers from a broader theoretical framework,
rather than positing the continuous dynamics as-is. Although the equations
studied in these papers may resemble some of the results derived in this one,
they differ in a number of ways. Firstly, this paper finds that the source of
randomness present in the optimizer dynamics obtained in this paper are not
generated by an exogenous source of noise, but are in fact an explicit
function of the randomness generated by observed stochastic gradients during
the optimization process. Another important difference is that the optimizer
dynamics presented in this paper make no use of the gradients of the objective
function, $\nabla f$ (which is inaccessible to a stochastic optimizer), and
are only a function of the stream of stochastic gradients $g_{t}$.
### 1.2 Contribution
To the author’s knowledge, this is the first paper to produce a theoretical
model for stochastic optimization based on a variational interpretation. This
paper extends the continuous variational framework Wibisono et al. (2016) to
model stochastic optimization. From this model, we derive optimality
conditions in the form of a system of forward-backward stochastic differential
equations (FBSDEs), and provide bounds on the expected rate of convergence of
the resulting optimization algorithm to the optimum. By discretizing solutions
of the continuous system of equations, we can recover a number of well-known
stochastic optimization algorithms, demonstrating that these algorithms can be
obtained as solutions of the variational model under various assumptions on
the loss function, $f(x)$, that is being minimized.
### 1.3 Paper Structure
In Section 2 we define a continuous-time surrogate model of stochastic
optimization. Section 3 uses this model to motivate a stochastic variational
problem over optimizers, in which we search for stochastic optimization
algorithms which achieve optimal average performance over a collection of
minimization problems. In Section 4 we show that the necessary and sufficient
conditions for optimality of the variational problem can be expressed as a
system of Forward-Backward Stochastic Differential Equations. Theorem 4.2
provides rates of convergence for the optimal algorithm to the optimum of the
minimization problem. Lastly, Section 5 recovers SGD, mirror descent,
momentum, and other optimization algorithms as discretizations of the
continuous optimality equations derived in Section 4 under various model
assumptions. The proofs of the mathematical results of this paper are found
within the appendices.
## 2 A Statistical Model for Stochastic Optimization
Over the course of the section, we present a variational model for stochastic
optimization. The ultimate objective will be to construct a framework for
measuring the average performance of an algorithm over a random collection of
optimization problems. We define random variables in an ambient probability
space
$\smash{(\Omega,{\mathbb{P}},\mathfrak{G}=\\{{\mathcal{G}}_{t}\\}_{t\in[0,T]})}$,
where ${\mathcal{G}}_{t}$ is a filtration which we will define at a later
point in this section. We assume that loss functions are drawn from a random
variable $f:\Omega\rightarrow C^{1}(\mathbb{R}^{d})$. Each draw from the
random variable satisfies $f(x)\in\mathbb{R}$ for fixed $x\in\mathbb{R}^{d}$,
and $f$ is assumed to be an almost-surely continuously differentiable in $x$.
In addition, we make the technical assumption that $\mathbb{E}\,\lVert\nabla
f(x)\rVert^{2}<\infty$ for all $x\in\mathbb{R}^{d}$.
We define an optimizer $X=(X_{t}^{\nu})_{t\geq 0}$ as a controlled process
satisfying $X_{t}^{\nu}\in\mathbb{R}^{d}$ for all $t\geq 0$, with initial
condition $X_{0}\in\mathbb{R}^{d}$. The paths of $X$ are assumed to be
continuously differentiable in time so that the dynamics of the optimizer may
be written as $dX_{t}^{\nu}=\nu_{t}\,dt$, where $\nu_{t}\in\mathbb{R}^{d}$
represents the control, where we use the superscript to express the explicit
dependence of $X^{\nu}$ on the control $\nu$. We may also write the optimizer
in its integral form as $X_{t}^{\nu}=X_{0}+\int_{0}^{t}\nu_{u}\,du$,
demonstrating that the optimizer is entirely characterized by a pair
$(\nu,X_{0})$ consisting of a control process $\nu$ and an initial condition
$X_{0}$. Using an explicit Euler discretization with step size $\epsilon>0$,
the optimizer can be approximately represented through the update rule
$X_{t+\epsilon}^{\nu}\approx X^{\nu}_{t}+\epsilon\,\nu_{t}$. This leads to the
interpretation of $\nu_{t}$ as the (infinitesimal) step the algorithm takes at
each point $t$ during the optimization process.
In order to capture the essence of stochastic optimization, we construct our
model so that optimizers have restricted access to the gradients of the loss
function $f$. Rather than being able to directly observe $\nabla f$ over the
path of $X_{t}^{\nu}$, we assume that the algorithm may only use a noisy
source of gradient samples, modeled by a càdlàg semi-martingale111A _càdlàg_
(continue à droite, limite à gauche) process is a continuous time process that
is almost-surely right-continuous with finite left limit at each point t. A
_semi-martingale_ is the sum of a process of finite variation and a local
martingale. For more information on continuous time stochastic processes and
these definitions, see the canonical text Jacod and Shiryaev (2013).
$g=\left(g_{t}\right)_{t\geq 0}$. As a simple motivating example, we can
consider the model $g_{t}=\nabla f(X_{t}^{\nu})+\xi_{t}$, where $\xi_{t}$ is a
white noise process. This particular model for the noisy gradient process can
be interpreted as consisting of observing $\nabla f(X_{t}^{\nu})$ plus an
independent source of noise. This concrete example will be useful to keep in
mind to make sense of the results which we present over the course of the
paper.
To make the concept of information restriction mathematically rigorous, we
restrict ourselves only to optimizers $X^{\nu}$ which are measurable with
respect to the information generated by the noisy gradient process $g$. To do
this, we first define the global filtration ${\mathcal{G}}$, as
${\mathcal{G}}_{t}=\sigma\left((g_{u})_{u\in[0,t]},f\right)$ as the sigma
algebra generated by the paths of $g$ as well as the realizations of the loss
surface $f$. The filtration ${\mathcal{G}}_{t}$ is defined so that it contains
the complete set of information generating the optimization problem until time
$t$.
Next, we define the coarser filtration
${\mathcal{F}}_{t}=\sigma(g_{u})_{u\in[0,t]}\subset{\mathcal{G}}_{t}$
generated strictly by the paths of the noisy gradient process. This filtration
represents the total set of information _available to the optimizer_ up until
time $t$. This allows us to formally restrict the flow of information to the
algorithm by restricting ourselves to optimizers which are adapted to
${\mathcal{F}}_{t}$. More precisely, we say that the optimizer’s control $\nu$
is admissible if
$\nu\in\mathcal{A}:=\left\\{\omega=\left(\omega_{t}\right)_{t\geq
0}\,:\;\omega\text{ is
${\mathcal{F}}$-adapted}\,,\;\mathbb{E}\int_{0}^{T}\,\lVert\omega_{t}\lVert^{2}+\lVert\nabla
f(X^{\omega}_{t})\lVert^{2}\,dt<\infty\right\\}\;.$ (3)
The set of optimizers generated by $\mathcal{A}$ can be interpreted as the set
of optimizers which may only use the source of noisy gradients, which have
bounded expected travel distance and have square-integrable gradients over
their path.
## 3 The Optimizer’s Variational Problem
Having defined the set of admissible optimization algorithms, we set out to
select those which are optimal in an appropriate sense. We proceed similarly
to Wibisono et al. (2016), by proposing an objective functional which measures
the performance of the optimizer over a finite time period.
The motivation for the optimizer’s performance metric comes from a physical
interpretation of the optimization process. We can think of our optimization
process as a particle traveling through a potential field define by the target
loss function $f$. As the particle travels through the potential field, it may
either gain or lose momentum depending on its location and velocity, which
will in turn affect the particle’s trajectory. Naturally, we may seek to find
the path of a particle which reaches the optimum of the loss function while
minimizing the total amount of kinetic and potential energy that is spent. We
therefore turn to the Lagrangian interpretation of classical mechanics, which
provides a framework for obtaining solutions to this problem. Over the
remainder of this section, we lay out the Lagrangian formalism for the
optimization problem we defined in Section 2.
To define a notion of energy in the optimization process, we provide a measure
of distance in the parameter space. We use the _Bregman Divergence_ as the
measure of distance within our parameter space, which can embed additional
information about the geometry of the optimization problem. The Bregman
divergence, $D_{h}$, is defined as
$D_{h}(y,x)=h(y)-h(x)-\langle\nabla h(x),y-x\rangle$ (4)
where $h:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a strictly convex function
satisfying $h\in C^{2}$. We assume here that the gradients of $h$ are
$L$-Lipschitz smooth for a fixed constant $L>0$. The choice of $h$ determines
the way we measure distance, and is typically chosen so that it mimics
features of the loss function $f$. In particular, this quantity plays a
central role in mirror descent and non-linear sub-gradient algorithms. For
more information on this connection and on Bregman Divergence, see Nemirovsky
and Yudin (1983) and Beck and Teboulle (2003).
We define the total energy in our problem as the kinetic energy, accumulated
through the movement of the optimizer, and the potential energy generated by
the loss function $f$. Under the assumption that $f$ almost surely admits a
global minimum $x^{\star}=\arg\min_{x\in\mathbb{R}^{d}}f(x)$, we may represent
the total energy via the Bregman Lagrangian as
$\mathcal{L}(t,X,\nu)=e^{\gamma_{t}}{(}\underbrace{\vphantom{\int}e^{\alpha_{t}}D_{h}\left(X+e^{-\alpha_{t}}\nu,X\right)}_{\mathclap{\text{Kinetic
Energy}}}-\underbrace{\vphantom{\int}e^{\beta_{t}}\left(f(X)-f(x^{\star})\right)}_{\mathclap{\text{Potential
Energy}}}{)}\;,$ (5)
for fixed inputs $(t,X,\nu)$, and where we assume that
$\gamma,\alpha,\beta:\mathbb{R}^{+}\rightarrow\mathbb{R}$ are deterministic,
and satisfy $\gamma,\alpha,\beta\in C^{1}$. The functions
$\gamma,\alpha,\beta$ can be interpreted as hyperparameters which tune the
energy present at any state of the optimization process. An important property
to note is that the Lagrangian is itself a random variable due to the
randomness introduced by the latent loss function $f$.
The objective is then to find an optimizer within the admissible set
$\mathcal{A}$ which can get close to the minimum
$x^{\star}=\min_{x\in\mathbb{R}^{d}}f(x)$, while simultaneously minimizing the
energy cost over a finite time period $[0,T]$. The approach taken in classical
mechanics and in Wibisono et al. (2016) fixes the endpoint of the optimizer at
$x^{\star}$. Since we assume that the function $f$ is not directly visible to
our optimizer, it is not possible to add a constraint of this type that will
hold almost surely. Instead, we introduce a soft constraint which penalizes
the algorithm’s endpoint in proportion to its distance to the global minimum,
$f(X_{T})-f(x^{\star})$. As such, we define the _expected action functional_
$\mathcal{J}:\mathcal{A}\rightarrow\mathbb{R}$ as
$\mathcal{J}(\nu)=\mathbb{E}{\Big{[}}\;\underbrace{\int_{0}^{T}\mathcal{L}(t,X_{t}^{\nu},\nu_{t})\,dt}_{\text{Total
Path
Energy}}+\underbrace{\vphantom{\int_{0}}e^{\delta_{T}}\left(\vphantom{\sum}f(X_{T}^{\nu})-f(x^{\star})\right)}_{\text{Soft
End Point Constraint}}{\Big{]}}\;,$ (6)
where $\delta_{T}\in C^{1}$ is assumed to be an additional model
hyperparameter, which controls the strength of the soft constraint.
With this definition in place, the objective will be to select amongst
admissible optimizers for those which minimize the expected action. Hence, we
seek optimizers which solve the stochastic variational problem
$\nu^{\ast}=\arg\min_{\nu\in\mathcal{A}}\mathcal{J}(\nu)\;.$ (7)
###### Remark 1.
Note that the variational problem (7) is identical to the one with Lagrangian
$\tilde{\mathcal{L}}(t,X,\nu)=e^{\gamma_{t}}(e^{\alpha_{t}}D_{h}\left(X+e^{-\alpha_{t}}\nu,X\right)-e^{\beta_{t}}f(X))$
(8)
and terminal penalty $e^{\delta_{T}}f(X_{T}^{\nu})$, since they differ by
constants independent of $\nu$. Because of this, the results presented in
Section 4 also hold the case where $x^{\star}$ and $f(x^{\star})$ do not exist
or are infinite.
## 4 Critical Points of the Expected Action Functional
In order to solve the variational problem (7), we make use techniques from the
calculus of variations and infinite dimensional convex analysis to provide
optimality conditions for the variational problem (7). To address issues of
information restriction, we rely on the stochastic control techniques
developed by Casgrain and Jaimungal (2018a, c, b).
The approach we take relies on the fact that a necessary condition for the
optimality of a Gâteaux differentiable functional $\mathcal{J}$ is that its
Gâteaux derivative vanishes in all directions. Computing the Gâteaux
derivative of $\mathcal{J}$, we find an equivalence between the Gâteaux
derivative vanishing and a system of Forward-Backward Stochastic Differential
Equations (FBSDEs), yielding a generalization of the Euler-Lagrange equations
to the context of our optimization problem. The precise result is stated in
Theorem 4.1 below.
###### Theorem 4.1 (Stochastic Euler-Lagrange Equation).
A control $\nu^{\ast}\in\mathcal{A}$ is a critical point of $\mathcal{J}$ if
and only if $((\frac{\partial\mathcal{L}}{\partial\nu}),{\mathcal{M}})$ is a
solution to the system of FBSDEs,
$d\left(\frac{\partial\mathcal{L}}{\partial\nu}\right)_{t}=\mathbb{E}\left[\left(\frac{\partial\mathcal{L}}{\partial
X}\right)_{t}{\Big{\lvert}}{\mathcal{F}}_{t}\right]\,dt+d{\mathcal{M}}_{t}\;\;\forall
t<T\,,\;\;\left(\frac{\partial\mathcal{L}}{\partial\nu}\right)_{T}=-e^{\delta_{T}}\,\mathbb{E}\left[\nabla
f(X_{T}){\Big{\lvert}}{\mathcal{F}}_{T}\right]\;,$ (9)
where we define the processes
$\displaystyle{\left(\frac{\partial\mathcal{L}}{\partial X}\right)_{t}}$
$\displaystyle=e^{\gamma_{t}+\alpha_{t}}{{(}}\,\nabla
h(X_{t}^{\nu^{\ast}}+e^{-\alpha_{t}}\nu^{\ast}_{t})-\nabla
h(X_{t}^{\nu^{\ast}})-e^{-\alpha_{t}}\nabla^{2}h(X_{t}^{\nu^{\ast}})\nu^{\ast}_{t}-e^{\beta_{t}}\nabla
f(X_{t}^{\nu^{\ast}})\,{{)}}$ (10)
$\displaystyle\vphantom{{{{\int}}}}\left(\frac{\partial\mathcal{L}}{\partial\nu}\right)_{t}$
$\displaystyle=e^{\gamma_{t}}\left(\nabla
h(X_{t}^{\nu^{\ast}}+e^{-\alpha_{t}}\nu^{\ast}_{t})-\nabla
h(X_{t}^{\nu^{\ast}})\right)\;,$ (11)
and where the process $\mathcal{M}=(\mathcal{M}_{t})_{t\in[0,T]}$ is an
${\mathcal{F}}$-adapted martingale. As a consequence, if the solution to this
FBSDE is unique, then it is the unique critical point of the functional
$\mathcal{J}$ up to null sets.
###### Proof.
See Appendix C ∎
Theorem 4.1 presents an analogue of the Euler-Lagrange equation with free
terminal boundary. Rather than obtaining an ODE as in the classical result, we
obtain an FBSDE222For a background on FBSDEs, we point readers to Pardoux and
Tang (1999); Ma et al. (1999); Carmona (2016). At a high level, the solution
to an FBSDE of the form (9) consists of a pair of processes
$(\nicefrac{{\partial\mathcal{L}}}{{\partial\nu}},\mathcal{M})$, which
simultaneously satisfy the dynamics and the boundary condition of (9).
Intuitively, the martingale part of the solution can be interpreted as a
random process which guides $(\nicefrac{{\partial\mathcal{L}}}{{\partial
X}})_{t}$ towards the boundary condition at time $T$. , with backwards process
$(\nicefrac{{\partial\mathcal{L}}}{{\partial\nu}})_{t}$, and forward state
processes $\mathbb{E}[(\nicefrac{{\partial\mathcal{L}}}{{\partial
X}})_{t}\lvert{\mathcal{F}}_{t}]$, $\int_{0}^{t}\left\|\nu_{u}\right\|\,du$
and $X_{t}^{\nu^{\ast}}$. We can also interpret the dynamics of equation (9)
as being the filtered optimal dynamics of (Wibisono et al., 2016, Equation
2.3), $\mathbb{E}[(\nicefrac{{\partial\mathcal{L}}}{{\partial
X}})_{t}\lvert{\mathcal{F}}_{t}]$, plus the increments of data-dependent
martingale ${\mathcal{M}}_{t}$, with mechanics similar to that of the
‘innovations process’ of filtering theory. This martingale term should not be
interpreted as a source of noise, but as an explicit function of the data, as
is evident from its explicit form
${\mathcal{M}}_{t}=\mathbb{E}\left[\int_{0}^{T}\left(\frac{\partial\mathcal{L}}{\partial
X}\right)_{u}\,du-e^{\delta_{T}}\,\nabla
f(X_{T}){\Big{\lvert}}{\mathcal{F}}_{t}\right]\;.$ (12)
A feature of equation (9), is that optimality relies on the projection of
$(\nicefrac{{\partial\mathcal{L}}}{{\partial X}})_{t}$ onto
${\mathcal{F}}_{t}$. Thus, the optimization algorithm makes use of past noisy
gradient observations in order to make local gradient predictions. Local
gradient predictions are updated using a Bayesian mechanism, where the prior
model for $\nabla f$ is conditioned with the noisy gradient information
contained in ${\mathcal{F}}_{t}$. This demonstrates that the solution depends
only on the gradients of $f$ along the path of $X_{t}$ and no higher order
properties.
### 4.1 Expected Rates of Convergence of the Continuous Algorithm
Using the dynamics (9) we obtain a bound on the rate of convergence of the
continuous optimization algorithm that is analogous to Wibisono et al. (2016,
Theorem 2.1). We introduce the Lyapunov energy functional
$\mathcal{E}_{t}=D_{h}(x^{\star},X^{\nu^{\ast}}_{t}+e^{-\alpha_{t}}\nu_{t})+e^{\beta_{t}}\left(f(X^{\nu^{\ast}}_{t})-f(x^{\star})\right)-[\nabla
h(X^{\nu^{\ast}}+e^{-\alpha_{t}}\nu),X^{\nu^{\ast}}+e^{-\alpha_{t}}\nu]_{t}\;,$
(13)
where we define $x^{\star}$ to be a global minimum of $f$. Under additional
model assumptions, and by showing that this quantity is a super-martingale
with respect to the filtration ${\mathcal{F}}$, we obtain an upper bound for
the expected rate of convergence from $X_{t}$ towards the minimum.
###### Theorem 4.2 (Convergence Rate).
Assume that the function $f$ is almost surely convex and that the _scaling
conditions_ $\dot{\gamma}_{t}=e^{\alpha_{t}}$ and $\dot{\beta}_{t}\leq
e^{\alpha_{t}}$ hold. Moreover, assume that in addition to $h$ having
$L$-Lipschitz smooth gradients, $h$ is also $\mu$-strongly-convex with
$\mu>0$. Define $x^{\star}=\arg\min_{x\in\mathbb{R}^{d}}f(x)$ to be a global
minimum of $f$. If $x^{\star}$ exists almost surely, the optimizer defined by
FBSDE (9) satisfies
$\mathbb{E}\left[f(X_{t})-f(x^{\star})\right]=O\left(e^{-\beta_{t}}\max\left\\{1\,,\mathbb{E}\left[\,[e^{-\gamma_{t}}{\mathcal{M}}]_{t}\right]\right\\}\right)\;,$
(14)
where $\left[e^{-\gamma_{t}}\mathcal{M}\right]_{t}$ represents the quadratic
variation of the process $e^{-\gamma_{t}}{\mathcal{M}}_{t}$, where
$\mathcal{M}$ is the martingale part of the solution defined in Theorem 4.1.
###### Proof.
See Appendix D. ∎
We may interpret the term
$\mathbb{E}\left[\,[e^{-\gamma_{t}}{\mathcal{M}}]_{t}\right]$ as a penalty on
the rate of convergence, which scales with the amount of noise present in our
gradient observations. To see this, note that if there is no noise in our
gradient observations, we obtain that ${\mathcal{F}}_{t}={\mathcal{G}}_{t}$,
and hence $\mathcal{M}_{t}\equiv 0$, which recovers the exact deterministic
dynamics of Wibisono et al. (2016) and the optimal convergence rate
$O(e^{-\beta_{t}})$. If the noise in our gradient estimates is large, we can
expect $\mathbb{E}\left[\,[e^{-\gamma}{\mathcal{M}}]_{t}\right]$ to grow at
quickly and to counteract the shrinking effects of $e^{-\beta_{t}}$. Thus, in
the case of a convex objective function $f$, any presence of gradient noise
will proportionally hurt rate of convergence to an optimum. We also point out,
that there will be a nontrivial dependence of
$\mathbb{E}\left[\,[e^{-\gamma}{\mathcal{M}}]_{t}\right]$ on all model
hyperparameters, the specific definition of the random variable $f$, and the
model for the noisy gradient stream, $(g_{t})_{t\geq 0}$.
###### Remark 2.
We do not assume that the conditions of Theorem 4.2 carry throughout the
remainder of the paper. In particular, Sections 5 study models which may not
guarantee almost-sure convexity of the latent loss function.
## 5 Recovering Discrete Optimization Algorithms
In this section, we use the optimality equations of Theorem 4.1 to produce
discrete stochastic optimization algorithms. The procedure we take is as
follows. We first define a model for the processes $(\nabla
f(X_{t}),g_{t})_{t\in[0,T]}$. Second, we solve the optimality FBSDE (9) in
closed form or approximate the solution via the first-order singular
perturbation (FOSP) technique, as described in Appendix A. Lastly, we
discretize the solutions with a simple Forward-Euler scheme in order to
recover discrete algorithms.
Over the course of Sections 5.1 and 5.2, we show that various simple models
for $(\nabla f(X_{t}),g_{t})_{t\in[0,T]}$ and different specifications of $h$
produce many well-known stochastic optimization algorithms. These establish
the conditions, in the context of the variational problem of Section 2, under
which each of these algorithms are optimal. As a consequence, this allows us
to understand the prior assumptions which these algorithms make on the
gradients of the objective function they are trying to minimize, and the way
noise is introduced in the sampling of stochastic gradients, $(g_{t})_{t\geq
0}$.
### 5.1 Stochastic Gradient Descent and Stochastic Mirror Descent
Here we propose a Gaussian model on gradients which loosely represents the
behavior of mini-batch stochastic gradient descent with a training set of size
$n$ and mini-batches of size $m$. By specifying a martingale model for $\nabla
f(X_{t})$, we recover the stochastic gradient descent and stochastic mirror
descent algorithms as solutions to the variational problem described in
Section 2.
Let us assume that $\nabla f(X_{t})=\sigma W_{t}^{f}$, where $\sigma>0$ and
$(W^{f}_{t})_{t\geq 0}$ is a Brownian motion. Next, assume that the noisy
gradients samples obtained from mini-batches over the course of the
optimization, evolve according to the model
$\smash{g}_{t}=\sigma(W_{t}^{f}+\rho W_{t}^{e})$, where
$\rho=\sqrt{\nicefrac{{(n-m)}}{{m}}}$ and $W^{e}$ is an independent copy of
$W_{t}^{f}$. Here, we choose $\rho$ so that
$\mathbb{V}[g_{t}]=(\nicefrac{{n}}{{m}})\mathbb{V}[\nabla
f(X_{t})]=O(m^{-1})$, which allows the variance to scale in $m$ and $n$ as it
does with mini-batches.
Using symmetry, we obtain the trivial solution to the gradient filter,
$\mathbb{E}[\nabla f(X_{t}){\lvert}{\mathcal{F}}_{t}]=(1+\rho^{2})^{-1}g_{t}$,
implying that the best estimate of the gradient at the point $X_{t}$ will be
the most recent mini-batch sample observed. re-scaled by a constant depending
on $n$ and $m$. Using this expression for the filter, we obtain the following
result.
###### Proposition 5.1.
The FOSP approximation to the solution of the optimality equations (9) can be
expressed as
$dX_{t}=e^{\alpha_{t}}\left(\nabla h^{\ast}\left(\nabla
h(X_{t})-\tilde{\Phi}_{t}(1+\rho^{2})^{-1}g_{t}\right)-X_{t}^{\nu^{\ast}}\right)\,dt\;,$
(15)
where $h^{\ast}$ is the convex dual of $h$ and where
$\tilde{\Phi}_{t}=e^{-\gamma_{t}}(\Phi_{0}+\int_{0}^{t}e^{\alpha_{u}+\beta_{u}+\gamma_{u}}\,du)$
is a deterministic learning rate with
$\smash{\Phi_{0}=e^{\delta_{T}}-\int_{0}^{T}e^{\alpha_{u}+\beta_{u}+\gamma_{u}}\,du}$.
When $h$ has the form $h(x)=x^{\intercal}Mx$ for a symmetric positive-definite
matrix $M$, the FOSP approximation is exact, and (15) is the exact solution to
the optimality FBSDE (9). The martingale portion of the solution to (9) can be
expressed as
${\mathcal{M}}_{t}={\mathcal{M}}_{0}-(1+\rho^{2})^{-1}\int_{0}^{t}e^{\alpha_{u}+\beta_{u}+\gamma_{u}}\,dg_{u}$.
###### Proof.
See Appendix E.1. ∎
To obtain a discrete optimization algorithm from the result of 5.1, we employ
a forward-Euler discretization of the ODE (15) on the finite mesh
$\mathcal{T}=\\{t_{0}=0\,,\;t_{k+1}=t_{k}+e^{-\alpha_{t_{k}}}:k\in\mathbb{N}\\}$.
This discretization results in the update rule
$X_{t_{k+1}}=\nabla h^{\ast}\left(\nabla
h(X_{t_{k}})-\tilde{\Phi}_{t_{k}}\,g_{t_{k}}\right)\;,$ (16)
corresponding exactly to mirror descent (e.g. see Beck and Teboulle (2003))
using the noisy mini-batch gradients $g_{t}$ and a time-varying learning rate
$\tilde{\Phi}_{t_{k}}$. Moreover, setting $h(x)=\frac{1}{2}\|x\|^{2}$, we
recover the update rule
$X_{t_{k+1}}-X_{t_{k}}=-\tilde{\Phi}_{t_{k}}\,g_{t_{k}}$, exactly
corresponding to the mini-batch SGD with a time-dependent learning rate.
This derivation demonstrates that the solution to the variational problem
described in Section 2, under the assumption of a Gaussian model for the
evolution of gradients, recovers mirror descent and SGD. In particular, the
martingale gradient model proposed in this section can be roughly interpreted
as assuming that gradients behave as random walks over the path of the
optimizer. Moreover, the optimal gradient filter $\mathbb{E}[\nabla
f(X_{t}){\lvert}{\mathcal{F}}_{t}]=(1+\rho^{2})^{-1}g_{t}$ shows that, for the
algorithm to be optimal, mini-batch gradients should be re-scaled in
proportion to $(1+\rho^{2})^{-1}=\nicefrac{{m}}{{n}}$.
### 5.2 Kalman Gradient Descent and Momentum Methods
Using a _linear state-space model_ for gradients, we can recover both the
Kalman Gradient Descent algorithm of Vuckovic (2018) and momentum-based
optimization methods of Polyak (1964). We assume that each component of
$\smash{\nabla f(X_{t})=(\nabla_{i}f(X_{t}))_{i=1}^{d}}$ is modeled
independently as a linear diffusive process. Specifically, we assume that
there exist processes $\smash{y_{i}=(y_{i,t})_{t\geq 0}}$ so that for each
$i$, $\smash{\nabla_{i}f(X_{t})=b^{\intercal}y_{i,t}}$, where
$\smash{y_{i,t}\in\mathbb{R}^{\tilde{d}}}$ is the solution to the linear SDE
$\smash{dy_{i,t}=-A\,y_{i,t}dt+L\,dW_{i,t}}$. In particular, we the notation
$\hat{y}_{i,j,t}$ to refer to element $(i,j)$ of
$\smash{\hat{y}\in\mathbb{R}^{d\times\tilde{d}}}$, and use the notation
$\smash{\hat{y}_{\cdot,j,t}=(\hat{y}_{i,j,t})_{i=1}^{d}}$. We assume here that
$\smash{A,L\in\mathbb{R}^{\tilde{d}\times\tilde{d}}}$ are positive definite
matrices and each of the $\smash{W_{i}=(W_{i,t})_{t\geq 0}}$ are independent
$\tilde{d}$-dimensional Brownian Motions.
Next, we assume that we may write each element of a noisy gradient process as
${g_{i,t}=b^{\intercal}y_{i,\cdot,t}+\sigma\xi_{i,t}}$, where $\sigma>0$ and
where $\xi_{i}=(\xi_{i,t})_{t\geq 0}$ are independent white noise processes.
Noting that
$\smash{\mathbb{E}[\,\nabla_{i}f(X_{t+h})\lvert{\mathcal{F}}_{t}]=b^{\intercal}e^{-Ah}y_{i,t}}$,
we find that this model implicitly assumes that gradients are expected
decrease in exponentially in magnitude as a function of time, at a rate
determined by the eigenvalues of the matrix $A$. The parameters $\sigma$ and
$L$ can be interpreted as controlling the scale of the noise within the
observation and signal processes.
Using this model, we obtain that the filter can be expressed as
$\smash{\mathbb{E}[\,\nabla_{i}f(X_{t}){\lvert}{\mathcal{F}}_{t}]=b^{\intercal}\hat{y}_{i,t}}$,
where $\hat{y}_{i,t}=\mathbb{E}[y_{i,t}\lvert{\mathcal{F}}_{t}]$. The process
$\hat{y}_{i,t}$ is expressed as the solution to the Kalman-Bucy333For
information on continuous time filtering and the Kalman-Bucy filter we refer
the reader to the text of Bensoussan (2004) or the lecture notes of Van Handel
(2007). filtering equations
$d\hat{y}_{i,t}=-A\hat{y}_{i,t}\,dt+\sigma^{-1}\bar{P}_{t}\,b\,d\hat{B}_{i,t}\;,\;\hskip
20.00003pt\dot{\bar{P}}=-A\bar{P}_{t}-\bar{P}_{t}^{\intercal}A-\sigma^{-2}\bar{P}_{t}b\,b^{\intercal}\bar{P}_{t}^{\intercal}+LL^{\intercal}\;,$
(17)
with the initial conditions $\hat{y}_{i,0}=0$ and
$\smash{\bar{P}_{0}=\mathbb{E}[y_{i,0}y_{i,0}^{\intercal}]}$, and where we
define innovations process
$d\hat{B}_{i,t}=\sigma^{-1}\left(g_{i,t}-b^{\intercal}\hat{y}_{i,t}\right)\,dt$
with the property that each $\hat{B}_{i}$ is an independent
${\mathcal{F}}$-adapted Brownian motion.
Inserting the linear state space model and its filter into the optimality
equations (9) we obtain the following result.
###### Proposition 5.2 (State-Space Model Solution to the FOSP).
Assume that the gradient state-space model described above holds. The FOSP
approximation to the solution of the optimality equations (9) can be expressed
as
$dX_{t}=e^{\alpha_{t}}{{{(}}}\nabla h^{\ast}(\nabla
h(X_{t})-{\textstyle\sum_{j=1}^{\tilde{d}}}\tilde{\Phi}_{j,t}\hat{y}_{\cdot,j,t})-X_{t}^{\nu^{\ast}}{{{)}}}\,dt\;,$
(18)
where
$\tilde{\Phi}_{t}=e^{-\gamma_{t}}(b^{\intercal}e^{-At}\Phi_{0}+\int_{0}^{t}e^{\alpha_{u}+\beta_{u}+\gamma_{u}}b^{\intercal}e^{-A(t-u)}\,du)\in\mathbb{R}^{\tilde{d}}$
is a deterministic learning rate, where $e^{A}$ represents the matrix
exponential, and where
$\Phi_{0}=e^{\delta_{T}}e^{AT}-\int_{0}^{T}e^{\alpha_{u}+\beta_{u}+\gamma_{u}}e^{Au}\,du$
can be chosen to have arbitrarily large eigenvalues by scaling $\delta_{T}$.
The martingale portion of the solution of (9) can be expressed as
${\mathcal{M}}_{t}={\mathcal{M}}_{0}-\sigma^{-1}\int_{0}^{t}e^{\alpha_{u}+\beta_{u}+\gamma_{u}}b^{\intercal}e^{-A(t-u)}\bar{P}_{u}b\,d\hat{B}_{u}$.
###### Proof.
See Appendix E.2 ∎
#### 5.2.1 Kalman Gradient Descent
In order to recover Kalman Gradient Descent, we discretize the processes
$X_{t}^{\nu^{\ast}}$ and $\hat{y}$ over the finite mesh $\mathcal{T}$, defined
in equation (18). Applying a Forward-Euler-Maruyama discretization of (18) and
the filtering equations (17), we obtain the discrete dynamics
$y_{i,t_{k+1}}=(I-e^{-\alpha_{t_{k}}}A)y_{i,t_{k}}+Le^{-\alpha_{t}}w_{i,k}\;,\hskip
30.00005ptg_{i,t_{k}}=b^{\intercal}y_{i,t_{k}}+\sigma
e^{-\alpha_{t}}\xi_{i,k}\;,$ (19)
where each of the $\xi_{i,k}$ and $w_{i,k}$ are standard Gaussian random
variables of appropriate size. The filter
$\smash{\hat{y}_{i,k}=\mathbb{E}[y_{t_{k}}{\lvert}\\{g_{t_{k^{\prime}}}\\}_{k^{\prime}=1}^{k}]}$
for the discrete equations can be written as the solution to the discrete
_Kalman filtering equations_ , provided in Appendix B. Discretizing the
process $X^{\nu^{\ast}}$ over $\mathcal{T}$ with the Forward-Euler scheme, we
obtain discrete dynamics for the optimizer in terms of the _Kalman Filter_
$\hat{y}$, as
$X_{t_{k+1}}=\nabla h^{\ast}\left(\nabla
h(X_{t_{k}})-{\textstyle\sum_{j=1}^{\tilde{d}}}\tilde{\Phi}_{j,t_{k}}\hat{y}_{\cdot,j,k}\right)\;,$
(20)
yielding a generalized version of Kalman gradient descent of Vuckovic (2018)
with $\tilde{d}$ states for each gradient element. Setting
$h(x)=\frac{1}{2}\|x\|^{2}$, $\tilde{d}=1$ and $b=1$ recovers the original
Kalman gradient descent algorithm with a time-varying learning rate.
Just as in Section 5.1, we interpret each $g_{t_{k}}$ as being a mini-batch
gradient, as with equation (2). The algorithm (20) computes a Kalman filter
from these noisy mini-batch observations and uses it to update the optimizer’s
position.
#### 5.2.2 Momentum and Generalized Momentum Methods
By considering the asymptotic behavior of the Kalman gradient descent method
described in Section 5.2.1, we recover a generalized version of momentum
gradient descent methods, which includes mirror descent behavior, as well as
multiple momentum states. Let us assume that $\alpha_{t}=\alpha_{0}$ remains
constant in time. Then, using the asymptotic update rule for the Kalman
filter, as shown in Proposition B.2, and equation (20), we obtain the update
rule
$X_{t_{k+1}}=\nabla h^{\ast}\left(\nabla
h(X_{t_{k}})-{\textstyle\sum_{j=1}^{\tilde{d}}}\tilde{\Phi}_{j,t_{k}}\hat{y}_{\cdot,j,k}\right)\;,\hskip
20.00003pt\hat{y}_{i,\cdot,k}=\left(\tilde{A}-K_{\infty}b^{\intercal}\tilde{A}\right)\hat{y}_{i,\cdot,k}+K_{\infty}g_{i,k}\;,$
(21)
where $\tilde{A}=I-e^{-\alpha_{0}}A$ and where
$K_{\infty}\in\mathbb{R}^{\tilde{d}}$ is defined in the statement of the
Proposition B.2. This yields a generalized momentum update rule where we keep
track of $\tilde{d}$ momentum states with
$(\hat{y}_{i,j,k})_{j=1}^{\tilde{d}}$, and update its position using a linear
update rule. This algorithm can be seen as being most similar to the
Aggregated Momentum technique of Lucas et al. (2018), which also keeps track
of multiple momentum states which decay at different rates.
Under the special case where $\tilde{d}=1$, $b=1$, and
$h=\frac{1}{2}\|x\|^{2}$ we recover the exact momentum algorithm update rule
of Polyak (1964) as
$X_{t_{k+1}}-X_{t_{k}}=-\tilde{\Phi}_{t_{k}}\hat{y}_{k}\;,\hskip
20.00003pt\hat{y}_{i,k}=p_{1}\,\hat{y}_{k}+p_{2}\,g_{t_{k}}\;,$ (22)
where we have a scalar learning rate $\tilde{\Phi}_{t_{k}}$, where
$p_{1}=\tilde{A}-K_{\infty}b^{\intercal}\tilde{A}$, $p_{2}=K_{\infty}$ are
positive scalars, and where $g_{t_{k}}$ are mini-batch draws from the gradient
as in equation 2.
The recovery of the momentum algorithm of Polyak (1964) has some interesting
consequences. Since $p_{1}$ and $p_{2}$ are functions of the model parameters
$\sigma,A$ and $\alpha_{0}$, we obtain a direct relationship between the
optimal choice for the momentum model parameters, the assumed scale of
gradient noise $\sigma,L>0$ and the assumed expected rate of decay of
gradients, as given by $e^{-At}$. This result gives insight as to how momentum
parameters should be chosen in terms of their prior beliefs on the
optimization problem.
## 6 Discussion and Future Research Directions
Over the course of the paper we present a variational framework on optimizers,
which interprets the task of stochastic optimization as an inference problem
on a latent surface that we wish to optimize. By solving a variational problem
over continuous optimizers with asymmetric information, we find that optimal
algorithms should satisfy a system of FBSDEs projected onto the filtration
${\mathcal{F}}$ generated by the noisy observations of the latent process.
By solving these FBSDEs and obtaining continuous-time optimizers, we find a
direct relationship between the measure assigned to the latent surface and its
relationship to how data is observed. In particular, assigning simple prior
models to the pair of processes $(\nabla f(X_{t}),g_{t})_{t\in[0,T]}$,
recovers a number of well-known and widely used optimization algorithms. The
fact that this framework can naturally recover these algorithms begs further
study. In particular, it is still an open question whether it is possible to
recover other stochastic algorithms via this framework, particularly those
with second-order scaling adjustments such as ADAM or AdaGrad.
From a more technical perspective, the intent is to further explore properties
of the optimization model presented here and the form of the algorithms it
suggests. In particular, the optimality FBSDE 9 is nonlinear, high-dimensional
and intractable in general, making it difficult to use existing FBSDE
approximation techniques, so new tools may need to be developed to understand
the full extent of its behavior.
Lastly, numerical work on the algorithms generated by this framework can
provide some insights as to which prior gradient models work well when
discretized. The extension of simplectic and quasi-simplectic stochastic
integrators applied to the BSDEs and SDEs that appear in this paper also has
the potential for interesting future work.
## References
* Aitchison (2018) Laurence Aitchison. A unified theory of adaptive stochastic gradient descent as bayesian filtering. _arXiv preprint arXiv:1807.07540_ , 2018.
* Beck and Teboulle (2003) Amir Beck and Marc Teboulle. Mirror descent and nonlinear projected subgradient methods for convex optimization. _Operations Research Letters_ , 31(3):167–175, 2003.
* Bensoussan (2004) Alain Bensoussan. _Stochastic control of partially observable systems_. Cambridge University Press, 2004.
* Carmona (2016) René Carmona. _Lectures on BSDEs, stochastic control, and stochastic differential games with financial applications_ , volume 1. SIAM, 2016.
* Casgrain and Jaimungal (2018a) Philippe Casgrain and Sebastian Jaimungal. Mean field games with partial information for algorithmic trading. _arXiv preprint arXiv:1803.04094_ , 2018a.
* Casgrain and Jaimungal (2018b) Philippe Casgrain and Sebastian Jaimungal. Mean-field games with differing beliefs for algorithmic trading. _arXiv preprint arXiv:1810.06101_ , 2018b.
* Casgrain and Jaimungal (2018c) Philippe Casgrain and Sebastian Jaimungal. Trading algorithms with learning in latent alpha models. _arXiv preprint arXiv:1806.04472_ , 2018c.
* Cesa-Bianchi et al. (2004) Nicolo Cesa-Bianchi, Alex Conconi, and Claudio Gentile. On the generalization ability of on-line learning algorithms. _IEEE Transactions on Information Theory_ , 50(9):2050–2057, 2004.
* da Silva and Gazeau (2018) André Belotto da Silva and Maxime Gazeau. A general system of differential equations to model first order adaptive algorithms. _arXiv preprint arXiv:1810.13108_ , 2018.
* Duchi et al. (2011) John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. _Journal of Machine Learning Research_ , 12(Jul):2121–2159, 2011.
* Gupta et al. (2017) Vineet Gupta, Tomer Koren, and Yoram Singer. A unified approach to adaptive regularization in online and stochastic optimization. _arXiv preprint arXiv:1706.06569_ , 2017.
* Jacod and Shiryaev (2013) Jean Jacod and Albert Shiryaev. _Limit theorems for stochastic processes_ , volume 288. Springer Science & Business Media, 2013.
* Janković et al. (2012) Svetlana Janković, Miljana Jovanović, and Jasmina Djordjević. Perturbed backward stochastic differential equations. _Mathematical and Computer Modelling_ , 55(5-6):1734–1745, 2012.
* Kingma and Ba (2014) Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. _arXiv preprint arXiv:1412.6980_ , 2014.
* Krichene and Bartlett (2017) Walid Krichene and Peter L Bartlett. Acceleration and averaging in stochastic descent dynamics. In _Advances in Neural Information Processing Systems_ , pages 6796–6806, 2017.
* Krichene et al. (2015) Walid Krichene, Alexandre Bayen, and Peter L Bartlett. Accelerated mirror descent in continuous and discrete time. In _Advances in neural information processing systems_ , pages 2845–2853, 2015.
* Lucas et al. (2018) James Lucas, Shengyang Sun, Richard Zemel, and Roger Grosse. Aggregated momentum: Stability through passive damping. _arXiv preprint arXiv:1804.00325_ , 2018.
* Ma et al. (1999) Jin Ma, J-M Morel, and Jiongmin Yong. _Forward-backward stochastic differential equations and their applications_. Number 1702. Springer Science & Business Media, 1999.
* Mertikopoulos and Staudigl (2018) Panayotis Mertikopoulos and Mathias Staudigl. On the convergence of gradient-like flows with noisy gradient input. _SIAM Journal on Optimization_ , 28(1):163–197, 2018.
* Nemirovsky and Yudin (1983) Arkadii Semenovich Nemirovsky and David Borisovich Yudin. Problem complexity and method efficiency in optimization. 1983\.
* (21) Yu Nesterov. A method of solving a convex programming problem with convergence rate o (1/k2). In _Sov. Math. Dokl_ , volume 27.
* Pardoux and Tang (1999) Etienne Pardoux and Shanjian Tang. Forward-backward stochastic differential equations and quasilinear parabolic pdes. _Probability Theory and Related Fields_ , 114(2):123–150, 1999.
* Polyak (1964) Boris T Polyak. Some methods of speeding up the convergence of iteration methods. _USSR Computational Mathematics and Mathematical Physics_ , 4(5):1–17, 1964.
* Raginsky and Bouvrie (2012) Maxim Raginsky and Jake Bouvrie. Continuous-time stochastic mirror descent on a network: Variance reduction, consensus, convergence. In _2012 IEEE 51st IEEE Conference on Decision and Control (CDC)_ , pages 6793–6800. IEEE, 2012.
* Robbins and Monro (1951) Herbert Robbins and Sutton Monro. A stochastic approximation method. _The annals of mathematical statistics_ , pages 400–407, 1951.
* Ruder (2016) Sebastian Ruder. An overview of gradient descent optimization algorithms. _arXiv preprint arXiv:1609.04747_ , 2016.
* Su et al. (2014) Weijie Su, Stephen Boyd, and Emmanuel Candes. A differential equation for modeling nesterov’s accelerated gradient method: Theory and insights. In _Advances in Neural Information Processing Systems_ , pages 2510–2518, 2014.
* Van Handel (2007) Ramon Van Handel. Stochastic calculus, filtering, and stochastic control. _Course notes., URL http://www. princeton. edu/~ rvan/acm217/ACM217. pdf_ , 2007.
* Vuckovic (2018) James Vuckovic. Kalman gradient descent: Adaptive variance reduction in stochastic optimization. _arXiv preprint arXiv:1810.12273_ , 2018.
* Walrand and Dimakis (2006) Jean Walrand and Antonis Dimakis. Random processes in systems - lecture notes. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley CA 94720, August 2006.
* Wibisono et al. (2016) Andre Wibisono, Ashia C Wilson, and Michael I Jordan. A variational perspective on accelerated methods in optimization. _Proceedings of the National Academy of Sciences_ , 113(47):E7351–E7358, 2016.
* Wilson et al. (2016) Ashia C Wilson, Benjamin Recht, and Michael I Jordan. A lyapunov analysis of momentum methods in optimization. _arXiv preprint arXiv:1611.02635_ , 2016.
* Xu et al. (2018a) Pan Xu, Tianhao Wang, and Quanquan Gu. Accelerated stochastic mirror descent: From continuous-time dynamics to discrete-time algorithms. In _International Conference on Artificial Intelligence and Statistics_ , pages 1087–1096, 2018a.
* Xu et al. (2018b) Pan Xu, Tianhao Wang, and Quanquan Gu. Continuous and discrete-time accelerated stochastic mirror descent for strongly convex functions. In _International Conference on Machine Learning_ , pages 5488–5497, 2018b.
## Appendix A Obtaining Solutions to the Optimality FBSDE
### A.1 A Momentum-Based Representation of the Optimizer Dynamics
Using a simple change of variables we may represent the dynamics of the FBSDE
(9) in a simpler fashion, which will aid us in obtaining solutions to this
system of equations. Let us define the momentum process
$p=(p_{t})_{t\in[0,T]}$ as
$p_{t}=\left(\frac{\partial\mathcal{L}}{\partial\nu}\right)_{t}=e^{\gamma_{t}}\left(\nabla
h(X_{t}^{\nu^{\ast}}+e^{-\alpha_{t}}\nu^{\ast})-\nabla
h(X_{t}^{\nu^{\ast}})\right)\;.$ (23)
Noting that since $h$ is convex, we have the property that $\nabla h^{\hskip
0.56003pt\mathclap{\ast}}(x)=(\nabla h)^{-1}(x)$, we may use equation (23) to
write $\nu^{\ast}$ in terms of the momentum process as
$\nu^{\ast}=e^{-\alpha_{t}}\left(\nabla h^{\hskip
0.56003pt\mathclap{\ast}}\left(\nabla
h(X_{t})+e^{-\gamma_{t}}p_{t}\right)-X_{t}\right)\;.$ (24)
The introduction of this process allows us to represent the solution to the
optimality FBSDE (9), and by extension the optimizer, in a much more tractable
way. Re-writing (9) in terms of $p_{t}$, we find that
$\left\\{\begin{aligned}
dp_{t}&=-\left\\{e^{\gamma_{t}+\alpha_{t}+\beta_{t}}\mathbb{E}\left[\nabla
f(X_{t}^{\nu^{\ast}}){\big{\lvert}}{\mathcal{F}}_{t}\right]+\left(e^{\gamma_{t}}\nabla^{2}h(X_{t})\,\nu^{\ast}_{t}-e^{\alpha_{t}}p_{t}\right)\right\\}\,dt+d\mathcal{M}_{t}\\\
p_{T}&=-e^{\delta_{T}}\mathbb{E}\left[\nabla
f(X_{T}^{\nu^{\ast}}){\big{\lvert}}{\mathcal{F}}_{T}\right]\end{aligned}\right.$
(25)
where the dynamics of the forward process $X^{\nu^{\ast}}$ can be expressed as
$dX_{t}^{\nu^{\ast}}=e^{\alpha_{t}}\left(\nabla h^{\hskip
0.56003pt\mathclap{\ast}}\left(\nabla
h(X_{t}^{\nu^{\ast}})+e^{-\gamma_{t}}p_{t}\right)-X_{t}^{\nu^{\ast}}\right)\,dt\;.$
(26)
This particular change of variables corresponds exactly to the Hamiltonian
representation of the optimizer’s dynamics, which we show in Appendix A.3.
Writing out the explicit solution to the FBSDE (25), we obtain a
representation for the optimizer’s dynamics as
$p_{t}=\mathbb{E}\left[\int_{t}^{T}e^{\gamma_{u}}\left\\{e^{\alpha_{u}+\beta_{u}}\nabla
f(X_{u}^{\nu^{\ast}})+\left(\nabla^{2}h(X_{u})\,\nu^{\ast}_{u}-e^{\alpha_{u}-\gamma_{u}}p_{u}\right)\right\\}\,du\,-e^{\delta_{T}}\nabla
f(X_{T}^{\nu^{\ast}})\;{\Big{\lvert}}{\mathcal{F}}_{t}\right]\;,$ (27)
showing that optimizer’s momentum can be represented as a time-weighted
average of the expected future gradients over the remainder of the
optimization and the term
$e^{\gamma_{t}}\nabla^{2}h(X_{t})\,\nu^{\ast}_{t}-e^{\alpha_{t}}p_{t}$, where
the weights are determined by the choice of hyperparameters $\alpha,\beta$ and
$\gamma$. Noting that
$\nabla^{2}h(X_{t})\,\nu^{\ast}_{t}-e^{\alpha_{t}-\gamma_{t}}p_{t}=\nabla^{2}h(X_{t})\nu^{\ast}_{t}-\left(\frac{\nabla
h(X_{t}+e^{-\alpha_{t}}\nu^{\ast}_{t})-\nabla
h(X_{t})}{e^{-\alpha_{t}}}\right)\;,$ (28)
we find that the additional correction term in (27) can be interpreted as the
remainder in the first-order Taylor expansion of the term $\nabla
h(X_{t}+e^{-\alpha_{t}}\nu^{\ast})$.
The representation (27) demonstrates optimizer does not only depend on the
instantaneous value of gradients at the point $X_{t}^{\nu^{\ast}}$. Rather, we
find that the algorithm’s behaviour depends on the expected value of all
future gradients that will be encountered over the remainder of the
optimization process, projected onto the set of accumulated gradient
information, ${\mathcal{F}}_{t}$. This is in stark contrast to most known
stochastic optimization algorithms which only make explicit use of local
gradient information in order to bring the optimizer towards an optimum.
### A.2 First-Order Singular Perturbation Approximation
When $h$ does not take the quadratic form $h(x)=\frac{1}{2}x^{\intercal}Mx$
for some positive-definite matrix $M$, the nonlinear dynamics of the FBSDE (9)
or in the equivalent momentum form (25) make it difficult to provive a
solution for general $h$. More precisely, the Taylor expansion term (28)
constitutes the main obstacle in obtaining solutions in general.
In cases where the scaling parameter $\alpha_{t}$ is sufficiently large, we
can assume that the Taylor expansion remainder term of equation (28) will
become negligibly small. Hence, we may approximate the optimality dynamics of
the FBSDE (25) by setting this term to zero. This can be interpreted as the
first-order term in a singular perturbation expansion of the solution to the
momentum FBSDE (25).
Under the assumption that the Taylor remainder term vanishes, we obtain the
approximation
$\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}=(\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}})_{t\in[0,T]}$
for the momentum, which we present in the following proposition.
###### Proposition A.1 (First-Order Singular Perturbation (FOSP)).
The linear FBSDE
$\left\\{\begin{aligned}
&d\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}_{t}=-e^{\gamma_{t}+\alpha_{t}+\beta_{t}}\,\mathbb{E}\left[\nabla
f\left(X_{t}\right)\lvert{\mathcal{F}}_{t}\right]\,dt+d\tilde{\mathcal{M}}^{\scaleto{\mathstrut(0)}{4.5pt}}_{t}\\\
&\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}_{T}=-e^{\delta_{T}}\mathbb{E}\left[\nabla
f(X_{T}^{\nu^{\ast}}){\big{\lvert}}{\mathcal{F}}_{T}\right]\end{aligned}\right.\;,$
(29)
admits a solution that can be expressed as
$\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}_{t}=\mathbb{E}\left[\left.\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\nabla
f\left(X_{u}\right)\,du-e^{\delta_{T}}\nabla
f(X_{T}^{\nu^{\ast}})\right\lvert{\mathcal{F}}_{t}\right]\;,$ (30)
provided that
$\mathbb{E}\left[\int_{0}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\lVert\nabla
f\left(X_{u}\right)\lVert\,du\right]<\infty$.
###### Proof.
Noting that the remainder term in the expression (28) vanishes, we get that
$\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}_{t}=\mathbb{E}\left[\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\nabla
f\left(X_{u}\right)\,du-e^{\delta_{T}}\,\nabla
f(X_{T}^{\nu^{\ast}}){\Big{\lvert}}{\mathcal{F}}_{u}\right]\;.$ (31)
Under the assumption that $\alpha,\beta,\delta,\gamma$ are continuous over
$[0,T]$ and that $\mathbb{E}\|f(x)\|^{2}\|<\infty$, the right part of (31) is
bounded. Now note that the integral on the left side of (31) is upper bounded
for all $T$ by the integral provided in the integrability condition of
Proposition A.1, and therefore this condition is a sufficient condition for
the expression (31) to be finite and well-defined.
∎
Although a general, model independent bound for the accuracy of such
approximations is beyond the scope of this paper, it can still serve as a
reasonable and computationally cheap alternative to attempting to solve the
original problem dynamics directly with a BSDE numerical scheme. For more
information on singular perturbation methods in the context of FBSDEs, see
Janković et al. (2012).
### A.3 Hamiltonian Representation of the Optimizer Dynamics
Just as in Hamiltonian classical mechanics, it is possible to express the
optimality FBSDE of Theorem (4.1) with Hamiltonian equations of motion. We
define the Hamiltonian $\mathcal{H}$ as the Legendre dual of $\mathcal{L}$ at,
which can be written as
$\mathcal{H}(t,X,p)=\left\langle
p\,,\nu^{\ast}\right\rangle-\mathcal{L}(t,X,\nu^{\ast})\;,$ (32)
where $p=\frac{\partial\mathcal{L}}{\partial X}$. Using the identity
$D_{h}(x,y)=D_{h^{\hskip 0.40002pt\mathclap{\ast}}}(\nabla h(x),\nabla h(y))$,
where $h^{\hskip 0.56003pt\mathclap{\ast}}$ is the Legendre dual of $h$, and
inverting the expression for $\frac{\partial\mathcal{L}}{\partial X}$ in terms
$p$, we may compute equation (32) as444See Wibisono et al. (2016)[Appendix
B.4] for the full details of the computation.
$\mathcal{H}(t,X,p)=e^{\alpha_{t}+\gamma_{t}}D_{h^{\hskip
0.40002pt\mathclap{\ast}}}\left(\nabla h(X)+e^{-\gamma_{t}}p\,,\,\nabla
h(X)\right)+e^{\gamma_{t}+\beta_{t}}f(X_{t})\;.$ (33)
Using this definition of $\mathcal{H}$, and using the FBSDE (9), we obtain the
following equivalent representation for the dynamics of the optimizer.
Using the simple substitution $p_{t}=\left(\frac{\partial\mathcal{L}}{\partial
X}\right)_{t}$ and noting from equations (10) and (11) that
$p_{t}=e^{\gamma_{t}}\left(\nabla
h(X_{t}+e^{-\alpha_{t}}\nu^{\ast}_{t})-\nabla h(X_{t})\right)\;,$ (34)
a straightforward computation applied to the definition of $\mathcal{H}$ shows
that the dynamics of the optimality FBSDE (9) admit the alternate Hamiltonian
representation of the optimizer dynamics
$dX_{t}=\left(\frac{\partial\mathcal{H}}{\partial p}\right)_{t}dt\hskip
4.0pt,\hskip
12.00002ptdp_{t}=-\mathbb{E}\left[\left(\frac{\partial\mathcal{H}}{\partial
X}\right)_{t}{\Big{\lvert}}{\mathcal{F}}_{t}\right]\,dt-d\mathcal{M}_{t}$ (35)
along with the boundary condition $p_{T}=0$.
## Appendix B The Discrete Kalman Filter
Here we present the reader to the Kalman Filtering equations used in Section
5.2. Consider the model presented in equations (19),
$y_{i,t_{k+1}}=\tilde{A}_{k}y_{i,t_{k}}+\tilde{L}_{k}w_{i,k}\;,\hskip
24.00005ptg_{i,t_{k}}=b^{\intercal}y_{i,t_{k}}+\sigma
e^{-\alpha_{t}}\xi_{i,k}\;,$ (36)
where we use the notation $\tilde{A}_{k}=(I-e^{-\alpha_{t_{k}}}A)$ and
$\tilde{L}_{k}=Le^{-\alpha_{t}}$, and where $w_{i,k}$ and $\xi_{i,k}$ are all
independent standard Gaussian random variables. We provide the Kalman
filtering equations for this model in the following proposition.
###### Proposition B.1 (Walrand and Dimakis (2006, Theorem 10.2)).
Let
$\hat{y}_{i,k}=\mathbb{E}[y_{t_{k}}{\lvert}\sigma(g_{t_{k^{\prime}}})_{k^{\prime}=1}^{k}]$.
Then $\hat{y}_{i,k}$ satisfies the recursive equation
$\hat{y}_{i,k}=\tilde{A}_{k}\hat{y}_{i,k}+K_{k}\left(g_{i,k}-b^{\intercal}\tilde{A}_{k}\hat{y}_{i,k}\right)\;,$
(37)
where the matrices $K_{k}$ are obtained via the independent recursive
equations
$\displaystyle{P}_{k\mid k-1}$ $\displaystyle=\tilde{A}_{k}{P}_{k-1\mid
k-1}\tilde{A}_{k}^{\intercal}+\tilde{L}_{k}^{\intercal}\tilde{L}_{k}\,,$ (38)
$\displaystyle{S}_{k}$ $\displaystyle=\sigma^{2}+b^{\intercal}{P}_{k\mid
k-1}b\,,$ (39) $\displaystyle{K}_{k}$ $\displaystyle={P}_{k\mid
k-1}b\,{S}_{k}^{-1}\,,$ (40) $\displaystyle{P}_{k|k}$
$\displaystyle=\left({I}-{K}_{k}b^{\intercal}\right){P}_{k|k-1}\,.$ (41)
For more information on the discrete Kalman filter, its derivation and for
asymptotic properties, we refer the reader to the lecture notes Walrand and
Dimakis (2006).
Next, we provide a result on the asumptotic properties of the Kalman filter in
the proposition that follows.
###### Proposition B.2 (Walrand and Dimakis (2006, Theorem 11.2)).
Assume that $\alpha_{t_{k}}=\alpha_{t_{0}}$ is constant, so that
$\tilde{A}_{k}=\tilde{A}$ and $\tilde{L}_{k}=\tilde{L}$ become constant, and
assume that there exists a positive-definite solution
$K_{\infty}\in\mathbb{R}^{\tilde{d}\times\tilde{d}}$ to the algebraic matrix
equation
$\tilde{K}=\tilde{A}\tilde{K}\tilde{A}^{\intercal}+\tilde{L}\tilde{L}^{\intercal}\;.$
(42)
Then, we may write the asymptotic dynamics of the filter $\hat{y}_{i}$ as
$\hat{y}_{i,k}=\tilde{A}\hat{y}_{i,k}+K_{\infty}\left(g_{i,k}-b^{\intercal}\tilde{A}\hat{y}_{i,k}\right)\;,$
(43)
where $K_{\infty}$ is the solution to the system of algebraic matrix equations
$K_{\infty}=(I-RC)S\,,\;\;R=Sb\left(b^{\intercal}Sb+\sigma^{2}\right)^{-1}\,,\;\;S=\tilde{A}K_{\infty}\tilde{A}^{\intercal}+\tilde{L}\tilde{L}^{\intercal}\,.$
(44)
For more information on the Kalman Filter, its derivation and theoretical
properties, see Walrand and Dimakis (2006).
## Appendix C Proofs Relating to Theorem 4.1
Before going forward with the main part of the proof, we first present a lemma
for the computation of the Gâteaux derivative of $\mathcal{J}$.
###### Lemma C.1.
The functional $\mathcal{J}$ is everywhere Gâteaux differentiable in
$\mathcal{A}$. The Gâteaux at a point $\nu\in\mathcal{A}$ in the direction
$\tilde{\omega}=\omega-\nu$ for $\omega\in\mathcal{A}$ takes the form
$\left\langle
D\mathcal{J}(\nu),\tilde{\omega}\right\rangle=\mathbb{E}\left[\int_{0}^{T}\left\langle\omega_{t}\,,\;\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}-\mathbb{E}\left[\int_{t}^{T}\frac{\partial\mathcal{L}\left(u,X_{u}^{\nu},\nu_{u}\right)}{\partial
X}\,du-e^{\delta_{T}}\,\nabla
f\left(X_{T}^{\nu}\right)\Big{\lvert}{\mathcal{F}}_{t}\right]\right\rangle\,dt\right]\;.$
(45)
###### Proof.
If we assume that the conditions of Leibniz’ rule hold, we may compute the
Gâteax derivative as
$\displaystyle\partial_{\rho}\mathcal{J}\left(\nu+\rho\,\tilde{\omega}\right)$
$\displaystyle=\partial_{\rho}\mathbb{E}\left[\int_{0}^{T}\mathcal{L}\left(t,X_{t}^{\nu+\rho\,\tilde{\omega}},\nu_{t}+\rho\,\tilde{\omega}_{t}\right)\,dt+e^{\delta_{T}}\left(f(X_{T}^{\nu+\rho\,\tilde{\omega}})-f(x^{\star})\right)\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\partial_{\rho}\mathcal{L}\left(t,X_{t}^{\nu+\rho\,\tilde{\omega}},\nu_{t}+\rho\,\tilde{\omega}_{t}\right)\,dt+e^{\delta_{T}}\partial_{\rho}f(X_{T}^{\nu+\rho\,\tilde{\omega}})\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\left\\{\left\langle\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial
X}\,,\;\int_{0}^{t}\tilde{\omega}_{u}\,du\right\rangle+\left\langle\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}\,,\;\tilde{\omega}_{t}\right\rangle\right\\}\,dt+\left\langle\int_{0}^{T}\tilde{\omega}_{u}\,du,\Phi\nabla
f(X_{T}^{\nu})\right\rangle\right]\;,$ (46)
where we have
$\displaystyle\frac{\partial\mathcal{L}\left(t,X,\nu\right)}{\partial X}$
$\displaystyle=e^{\gamma_{t}+\alpha_{t}}{{(}}\,\nabla
h(X+e^{-\alpha_{t}}\nu)-\nabla
h(X)-e^{-\alpha_{t}}\nabla^{2}h(X)\nu-e^{\beta_{t}}\nabla f(X)\,{{)}}$ (47)
$\displaystyle\frac{\partial\mathcal{L}\left(t,X,\nu\right)}{\partial\nu}$
$\displaystyle=e^{\gamma_{t}}\left(\nabla h(X+e^{-\alpha_{t}}\nu)-\nabla
h(X)\right)\;.$ (48)
Note here that the derivative in $f$ is path-wise for every fixed realization
of the function $f$. Since $f\in C^{1}$, we have that $\nabla f$ is also well-
defined for every realization of $f$.
To ensure that this computation is valid, and that the conditions of the
Leibniz rule are met, due to the continuity of (46) in $\tilde{\omega}$, is
sufficient for us to show that the integrals in equation (46) are bounded for
any $\tilde{\omega}$ and $\nu$. First, note that by the Young and Jensen
inequalities,
$\displaystyle\mathbb{E}\left[\left\langle\int_{0}^{T}\tilde{\omega}_{u}\,du,\Phi\nabla
f(X_{T}^{\nu})\right\rangle\right]$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\left[\int_{0}^{T}\|\tilde{\omega}_{u}\|^{2}\,du+\Phi\|\nabla
f(X_{T}^{\nu})\|^{2}\right]<\infty\;,$ (49)
where the boundedness holds from the fact that $\tilde{\omega}\in\mathcal{A}$
and that $\mathbb{E}\|f(x)\|^{2}<\infty$ for all $x\in\mathbb{R}^{d}$.
Next, we focus on the left part of equation (46). By the Cauchy-Schwarz and
Young inequalities, we have
$\displaystyle\left\lvert\left\langle\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial
X}\,,\;\int_{0}^{t}\tilde{\omega}_{u}\,du\right\rangle+\left\langle\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}\,,\;\tilde{\omega}_{t}\right\rangle\right\rvert$
$\displaystyle\leq\left\|\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial
X}\right\|\left\|\int_{0}^{t}\tilde{\omega}_{u}\,du\right\|+\left\|\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}\right\|\|\tilde{\omega}_{t}\|$
(50)
$\displaystyle\leq\frac{1}{2}\left\\{\left\|\int_{0}^{t}\tilde{\omega}_{u}\,du\right\|^{2}+\left\|\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial
X}\right\|^{2}+\|\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}\|^{2}+\|\tilde{\omega}_{t}\|^{2}\right\\}\;.$
(51)
Using the $L$-Lipschitz property of the gradients of $h$, we can also bound
the partial derivatives of the Lagrangian with the triangle inequality as
$\displaystyle\left\|\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial
X}\right\|$ $\displaystyle\leq e^{\gamma_{t}+\alpha_{t}}\left\|\nabla
h(X+e^{-\alpha_{t}}\nu)-\nabla
h(X)\right\|+e^{\gamma_{t}}\|\nabla^{2}h(X)\nu\|+e^{\beta_{t}+\gamma_{t}+\alpha_{t}}\|\nabla
f(X)\|$ $\displaystyle\leq
L(e^{\gamma_{t}+\alpha_{t}}+e^{\gamma_{t}})\|\nu\|+e^{\beta_{t}+\gamma_{t}+\alpha_{t}}\|f(X)\|$
$\displaystyle\leq C_{0}\left(\|\nu\|+\|\nabla f(X)\|\right)$
$\displaystyle\left\|\frac{\partial\mathcal{L}\left(t,X,\nu\right)}{\partial\nu}\right\|$
$\displaystyle\leq e^{\gamma_{t}}\left\|\nabla h(X+e^{-\alpha_{t}}\nu)-\nabla
h(X)\right\|$ $\displaystyle\leq e^{\gamma_{t}}L\left\|\nu\right\|$
$\displaystyle\leq C_{0}\,\left\|\nu\right\|\;,$
where
$C_{0}=\sup_{t\in[0,T]}\\{e^{\alpha_{t}+\gamma_{t}}+e^{\gamma_{t}}+e^{\alpha_{t}+\gamma_{t}+\beta_{t}}\\}$
is bounded by the assumption that $\alpha,\beta,\gamma$ are continuous in
$[0,T]$.
Using the above result, and applying Young’s inequality to the previous
result, we can upper bound equation (51) as
(51) $\displaystyle\leq
32\,(1+C)\,\left\\{1+\int_{0}^{T}\|\tilde{\omega}_{u}\|^{2}\,du+\|\nu_{t}\|^{2}+\|\tilde{\omega}_{t}\|^{2}+\|\nabla
f\left(X_{t}\right)\|^{2}\right\\}$ (52) $\displaystyle\leq
64\,(1+C)\,\left\\{1+\int_{0}^{T}\|\omega_{u}\|^{2}\,du+\int_{0}^{T}\|\nu_{u}\|^{2}\,du+\|\nu_{t}\|^{2}+\|\omega_{t}\|^{2}+\|\nabla
f\left(X_{t}\right)\|^{2}\right\\}\;,$ (53)
where the number 32 is chosen to be much larger than what is strictly
necessary by Young’s inequality. Notice here that by the definition of
$\mathcal{A}$, this forms an integrable upper bound to the left integral of
equation (46), validating our use of Leibniz’s rule, and showing that
$\mathcal{J}$ is indeed Gâteaux integrable.
Now that integrability concerns have been dealt with, we can proceed with the
computation of the Gâteaux derivative. By applying integration by parts to the
left side of equation (54) and moving the right hand side into the integral,
we obtain
$\partial_{\rho}\mathcal{J}\left(\nu+\rho\,\tilde{\omega}\right)=\mathbb{E}\left[\int_{0}^{T}\left\langle\tilde{\omega}_{t}\,,\;\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}-\int_{t}^{T}\frac{\partial\mathcal{L}\left(u,X_{u}^{\nu},\nu_{u}\right)}{\partial
X}\,du-e^{\delta_{T}}\nabla f(X_{T}^{\nu})\right\rangle\,dt\right]$
Using the tower property and Fubini’s theorem on the right, we get
$\left\langle
D\mathcal{J}(\nu),\tilde{\omega}\right\rangle=\mathbb{E}\left[\int_{0}^{T}\left\langle\tilde{\omega}_{t}\,,\;\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}-\mathbb{E}\left[\int_{t}^{T}\frac{\partial\mathcal{L}\left(u,X_{u}^{\nu},\nu_{u}\right)}{\partial
X}\,du+e^{\delta_{T}}\nabla
f(X_{T}^{\nu})\,\Big{\lvert}{\mathcal{F}}_{t}\right]\right\rangle\,dt\right]\;,$
(54)
as desired. ∎
### C.1 Proof of Theorem 4.1
Using the representation of the Gâteux derivative of $\mathcal{J}$ brought
forth by Lemma C.1, we may proceed with the proof of Theorem 4.1.
###### Proof of Theorem 4.1.
The goal is to show that the BSDE (9) is a necessary and sufficient condition
for $\nu^{\ast}$ to be a critical point of $\mathcal{J}$. For any Gâteaux
differentiable function $\mathcal{J}$, a necessary and sufficient condition
for a point $\nu^{\ast}\in\mathcal{A}$ to be a critical point is that its
Gâteaux derivative vanished in any valid direction. Lemma C.1 shows that the
Gâteaux derivative takes the form of equation (45). Therefore, all that
remains is to show that the FBSDE 9 is a necessary and sufficient condition
for equation (45) to vanish.
Sufficiency. We will show that equation (45) vanishes when the FBSDE (9)
holds. Assume that there exists a solution to the FBSDE (9) satisfying
$\nu^{\ast}\in\mathcal{A}$. We may then express the solution to the FBSDE
explicitly as
$\left(\frac{\partial\mathcal{L}}{\partial\nu}\right)_{t}=\mathbb{E}\left[\int_{t}^{T}\left(\frac{\partial\mathcal{L}}{\partial
X}\right)_{u}\,du\,-e^{\delta_{T}}\nabla
f(X_{T}^{\nu})\,\Big{\lvert}{\mathcal{F}}_{t}\right]\;.$
Inserting this into the right side of (45), we find that $\left\langle
D\mathcal{J}(\nu),\omega\right\rangle$ vanishes for all
$\omega\in\mathcal{A}$, demonstrating sufficiency.
Necessity. Conversely, let us assume that $\left\langle
D\mathcal{J}(\nu),\omega-\nu\right\rangle=0$ for all $\omega\in\mathcal{A}$
and for some $\nu\in\mathcal{A}$ for which the FBSDE (9) is not satisfied. We
will show by contradiction that this statement cannot hold by choosing a
direction in which the Gâteax derivative does not vanish. Consider the choice
$\omega_{t}^{\rho}=\nu_{t}+\rho\,\left(\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}-\mathbb{E}\left[\int_{t}^{T}\frac{\partial\mathcal{L}\left(u,X_{u}^{\nu},\nu_{u}\right)}{\partial
X}\,du-e^{\delta_{T}}\nabla
f(X_{T}^{\nu})\Big{\lvert}{\mathcal{F}}_{t}\right]\right)\;,$ (55)
for some sufficiently small $\rho>0$. We will first show that
$\omega^{\rho}\in\mathcal{A}$ for some $\rho>0$.
First, note that clearly $\omega^{\rho}$ must be ${\mathcal{F}}_{t}$-adapted,
and we have $\omega^{0}=\nu_{t}$. Moreover, note that since
$\nu\in\mathcal{A}$, we have that
$\mathbb{E}\int_{0}^{T}\,\lVert\nu_{t}\lVert^{2}+\lVert\nabla
f(X^{\nu})\lVert^{2}\,dt<\infty$, that $\omega^{0}=\nu$. Notice that by the
continuity of $\nabla f$ and the definition of $X$, the expression
$\mathbb{E}\int_{0}^{T}\,\lVert\omega^{\rho}_{t}\lVert^{2}+\lVert\nabla
f(X^{\omega^{\rho}})\lVert^{2}\,dt$ (56)
is continuous in $\rho$. Since (56) is bounded for $\rho=0$, by continuity
there exists some $\rho>0$ for which (56) is bounded and by extension where
$\omega^{\rho}\in\mathcal{A}$ for this same value of $\rho$.
Inserting (55) into the Gâteaux derivative (45), we get that
$\left\langle
D\mathcal{J}(\nu),\omega^{\rho}-\nu\right\rangle=\rho\;\mathbb{E}\left[\int_{0}^{T}\left\lVert\frac{\partial\mathcal{L}\left(t,X_{t}^{\nu},\nu_{t}\right)}{\partial\nu}-\mathbb{E}\left[\int_{t}^{T}\frac{\partial\mathcal{L}\left(u,X_{u}^{\nu},\nu_{u}\right)}{\partial
X}\,du-e^{\delta_{T}}\nabla
f(X_{T}^{\nu})\,\Big{\lvert}{\mathcal{F}}_{t}\right]\right\rVert^{2}\,dt\right]\;,$
(57)
which is strictly positive unless the FBSDE (9) is satisfied, thus forming a
contradiction and demonstrating that the condition is necessary. ∎
## Appendix D Proof of Theorem 4.2
###### Proof.
The proof of this theorem is broken up into multiple parts. The idea will be
to first show that the energy functional $\mathcal{E}$ is a super-martingale
with respect to ${\mathcal{F}}_{t}$, and then to use this property to bound
the expected distance to the optimum. Lastly, we bound a quadratic co-
variation term which appears within these equations to obtain the final
result.
Before delving into the proof, we introduce standard notation for semi-
martingale calculus. We use the noation $dY_{t}=dY_{t}^{c}+\Delta Y_{t}$ to
indicate the increments of the continuous part $Y^{c}$ of a process $Y$ and
its discontinuities $\Delta Y_{t}=Y_{t}-Y_{t-}$, where we use the notation
$t-$ to indicate the left limit of the process. We use the notation
$[Y,Z]_{t}$ to represent the quadratic co-variation of two processes $Y$ and
$Z$. This quadratic variation term can be decomposed into
$d[Y,Z]_{t}=d[Y,Z]_{t}^{c}+\langle\Delta Y_{t},\Delta Z_{t}\rangle$, where
$[Y,Z]_{t}^{c}$ represents the quadratic covariation between $Y^{c}$ and
$Z^{c}$, and where $\langle\Delta Y_{t},\Delta Z_{t}\rangle$ represents the
inner product of their discontinuities at $t$. For more information on semi-
martingale calculus and the associated notation, see Jacod and Shiryaev (Jacod
and Shiryaev, 2013, Sections 3-5).
Dynamics of the Bregman Divergence. The idea will now be to show that the
energy functional $\mathcal{E}$, defined in equation (13), is a super-
martingale with respect to the visible filtration ${\mathcal{F}}_{t}$.
Using Itô’s formula and Itô’s product rule for càdlàg semi-martingales Jacod
and Shiryaev (2013)[Theorem 4.57], as well as the short-hand notation
$Y_{t}=X_{t}+e^{-\alpha_{t}}\nu^{\ast}_{t}$, we obtain
$\displaystyle dD_{h}(x^{\star},Y_{t})$ $\displaystyle=-\left\\{\langle\nabla
h(Y_{t}),dY_{t}^{c}\rangle+\frac{1}{2}\sum_{i,j=1}^{d}\frac{\partial^{2}h(Y_{t})}{\partial
x_{i}\partial x_{i}}d\left[Y_{i},Y_{j}\right]_{t}^{c}+\Delta
h(Y_{t})\right\\}-\left\\{\vphantom{\sum_{i,j=1}^{d}}\langle d\nabla
h(Y_{t}),x^{\star}-Y_{t}\rangle-\langle\nabla
h(Y_{t}),dY_{t}\rangle-d\left[\nabla h(Y),Y\right]_{t}\right\\}$
$\displaystyle=-\left\\{\langle\nabla h(Y_{t}),-\Delta
Y_{t}\rangle+\frac{1}{2}\sum_{i,j=1}^{d}\frac{\partial^{2}h(Y_{t})}{\partial
x_{i}\partial x_{i}}d\left[Y_{i},Y_{j}\right]_{t}^{c}+\Delta
h(Y_{t})\right\\}-\left\\{\langle d\nabla
h(Y_{t}),x^{\star}-Y_{t}\rangle-\sum_{i,j=1}^{d}\frac{\partial^{2}h(Y_{t})}{\partial
x_{i}\partial
x_{i}}d\left[Y_{i},Y_{j}\right]_{t}^{c}-\langle\Delta\left(\nabla
h(Y_{t})\right),\Delta Y_{t}\rangle\right\\}$ $\displaystyle=-\left\\{\Delta
h(Y_{t})-\langle\nabla h(Y_{t}),\Delta Y_{t}\rangle\right\\}-\langle d\nabla
h(Y_{t}),x^{\star}-Y_{t}\rangle+\left\\{\frac{1}{2}\sum_{i,j=1}^{d}\frac{\partial^{2}h(Y_{t})}{\partial
x_{i}\partial
x_{i}}d\left[Y_{i},Y_{j}\right]_{t}^{c}+\langle\Delta\left(\nabla
h(Y_{t})\right),\Delta Y_{t}\rangle\right\\}\;,$
where from line 1 to 2, we use the identity $d[\nabla
g(Y),Y]_{t}=\sum_{i,j}\frac{\partial^{2}g(Y_{t})}{\partial x_{i}\partial
x_{j}}d[Y_{i},Y_{j}]_{t}^{c}+\langle\Delta(\nabla g(Y_{t})),\Delta
Y_{t}\rangle$ for any $C^{2}$ function $g$.
Note that since $h$ is convex, $\nabla^{2}h$ must have positive eigenvalues,
and hence $\frac{1}{2}\sum_{i,j=1}^{d}\frac{\partial^{2}h(Y_{t})}{\partial
x_{i}\partial x_{i}}d\left[Y_{i},Y_{j}\right]_{t}^{c}\geq 0$. The convexity of
$h$ also implies that $\langle\nabla h(x)-\nabla h(y),x-y\rangle\leq 0$, and
therefore we get $\langle\Delta\left(\nabla h(Y_{t})\right),\Delta
Y_{t}\rangle\geq 0$. The convexity of $h$ also implies that $\Delta
h(Y_{t})-\langle\nabla h(Y_{t}),\Delta Y_{t}\rangle\geq 0$. Combining these
observations, we find that
$\displaystyle dD_{h}(x^{\star},Y_{t})$ $\displaystyle\leq-\langle d\nabla
h(Y_{t}),x^{\star}-Y_{t}\rangle+\left\\{\sum_{i,j=1}^{d}\frac{\partial^{2}h(Y_{t})}{\partial
x_{i}\partial
x_{i}}d\left[Y_{i},Y_{j}\right]_{t}^{c}+\langle\Delta\left(\nabla
h(Y_{t})\right),\Delta Y_{t}\rangle\right\\}$ (58) $\displaystyle=-\langle
d\nabla h(Y_{t}),x^{\star}-Y_{t}\rangle+[\nabla h(Y),Y]_{t}\;.$ (59)
Super-martingale property of $\mathcal{E}$. Applying the scaling conditions
to the optimality FBSDE (9), we obtain the dynamics
$d\nabla
h(X_{t}^{\nu^{\ast}}+e^{-\alpha_{t}}\nu^{\ast})=-e^{\alpha_{t}+\beta_{t}}\mathbb{E}\left[\nabla
f(X_{t}^{\nu^{\ast}}){\big{\lvert}}{\mathcal{F}}_{t}\right]\,dt+d\tilde{\mathcal{M}}_{t}\;.$
(60)
Inserting this in to the dynamics of for the energy functional, and applying
the upper bound (59), we find that
$\displaystyle d\mathcal{E}_{t}\leq$ $\displaystyle-\langle d\nabla
h(Y_{t}),x^{\star}-Y_{t}\rangle+\dot{\beta}_{t}e^{\beta_{t}}\left(f(X_{t})-f(x^{\star})\right)\,dt+e^{\beta_{t}}\left\langle\nabla
f(X_{t}),\nu_{t}\right\rangle\,dt$ (61) $\displaystyle=$
$\displaystyle\left\langle e^{\alpha_{t}+\beta_{t}}\mathbb{E}[\nabla
f\left(X_{t}\right)\lvert{\mathcal{F}}_{t}]\,dt-d\mathcal{M}_{t}\;,\;x^{\star}-Y_{t}\right\rangle+\dot{\beta}_{t}e^{\beta_{t}}\left(f(X_{t})-f(x^{\star})\right)\,dt+e^{\beta_{t}}\left\langle\nabla
f(X_{t}),\nu_{t}\right\rangle\,dt$ (62) $\displaystyle=$
$\displaystyle-\left\\{D_{f}(x^{\star},Y_{t})+\left(e^{\alpha_{t}}-\dot{\beta}_{t}\right)e^{\beta_{t}}\left(f(X_{t})-f(x^{\star})\right)\right\\}\,dt+d\mathcal{M}_{t}^{\prime}\;,$
(63)
where we use the notation $\mathcal{M}^{\prime}_{t}$ to represent the
${\mathcal{F}}_{t}$-martingale defined as
$d\mathcal{M}_{t}^{\prime}=\left\langle
e^{\alpha_{t}+\beta_{t}}\left(\mathbb{E}[\nabla
f\left(X_{t}\right)\lvert{\mathcal{F}}_{t}]-f(X_{t})\right)\,dt-d\mathcal{M}_{t}\;,\;x^{\star}-Y_{t}\right\rangle\;.$
(64)
Now note that due to the assumed convexity of $f$, we have that
$D_{f}(x^{\star},Y_{t})$ is almost surely non-negative. Second, by the scaling
conditions, $e^{\alpha_{t}}-\dot{\beta}_{t}$ is positive. Hence, the drift in
equation (63) is almost surely negative, and $\mathcal{E}_{t}$ is a super-
martingale.
Using the super-martingale property, we find that
$\mathbb{E}\left[\mathcal{E}_{t}\right]\leq\mathbb{E}\left[\mathcal{E}_{0}\right]=\mathbb{E}\left[D_{h}(x^{\star},X_{0}+e^{-\alpha_{0}}\nu_{0})+e^{\beta_{0}}\left(f(X_{0})-f(x^{\star})\right)\right]=C_{0}\;,$
where $C_{0}\geq 0$. Using the definition of $\mathcal{E}$, and using the fact
that $D_{h}\geq 0$ if $h$ is convex, we obtain
$e^{\beta_{t}}\mathbb{E}\left[\left(f(X_{t})-f(x^{\star})\right)\right]\leq\mathbb{E}\left[D_{h}(x^{\star},X_{t}+e^{-\alpha_{t}}\nu_{t})+e^{\beta_{t}}\left(f(X_{t})-f(x^{\star})\right)\right]\leq
C_{0}+\mathbb{E}\left[\,[\nabla h(Y),Y]_{t}\,\right]\;.$ (65)
Upper bound on the Quadratic Co-variation. Now we upper bound the quadratic
co-variation term appearing on the right hand side of (65). Using the further
change of variable $Z_{t}=\nabla h(Y_{t})$, and noting that by the assumed
convexity of $h$ that $\nabla h^{\hskip 0.56003pt\mathclap{\ast}}(x)=(\nabla
h)^{-1}(x)$, we get $[\nabla h(Y),Y]_{t}=[Z,\nabla h^{\hskip
0.56003pt\mathclap{\ast}}(Z)]_{t}$.
Assuming that $\nabla h$ is $\mu$-strongly convex, we get that $\nabla
h^{\hskip 0.56003pt\mathclap{\ast}}$ must have $\mu^{-1}$-Lipschitz smooth
gradients. This implies that (i) the eigenvalues of $\nabla^{2}h^{\hskip
0.56003pt\mathclap{\ast}}$ must be bounded above by $\mu^{-1}$ (ii) from the
Cauchy-Schwarz inequality, we have $\langle\nabla h^{\hskip
0.56003pt\mathclap{\ast}}(x)-\nabla h^{\hskip
0.56003pt\mathclap{\ast}}(y),x-y\rangle\leq\mu^{-1}\|x-y\|^{2}$. Using these
two observations and writing out the expression for $[Z,\nabla h^{\hskip
0.56003pt\mathclap{\ast}}(Z)]_{t}$, we get
$\displaystyle[Z,\nabla h^{\hskip 0.56003pt\mathclap{\ast}}(Z)]_{t}$
$\displaystyle=\sum_{i,j=1}^{d}\frac{\partial^{2}h(Y_{t})}{\partial
x_{i}\partial x_{i}}d\left[Y_{i},Y_{j}\right]_{t}^{c}+\langle\Delta(\nabla
h^{\hskip 0.56003pt\mathclap{\ast}}(Z)),\Delta Z_{t}\rangle$ (66)
$\displaystyle\leq\mu^{-1}[Z]_{t}\;.$ (67)
Moreover, note that since $Z_{t}=\nabla
h(X_{t}^{\nu^{\ast}}+e^{-\alpha_{t}}\nu^{\ast}_{t})$ and since $\nabla
h(X_{t}^{\nu^{\ast}})$ is a process of finite variation, the optimality
dynamics (9) imply that
$[Z]_{t}=[e^{-\gamma_{t}}\mathcal{M}]_{t}]=e^{-\gamma_{t}}[\mathcal{M}]_{t}$
Inserting the quadratic co-variation bound into equation (65) and using the
super-martingale property, we obtain the final result
$\displaystyle\mathbb{E}\left[\left(f(X_{t})-f(x^{\star})\right)\right]$
$\displaystyle\leq
e^{-\beta_{t}}\left(C_{0}+\frac{1}{2}\mathbb{E}\left[\,[\nabla
h(X+e^{-\alpha_{t}}\nu),\nu]_{t}\right]\right)$ $\displaystyle\leq
e^{-\beta_{t}}\left(C_{0}+\frac{1}{2}e^{-2\gamma_{t}}\mathbb{E}\left[\,[{\mathcal{M}}]_{t}\right]\right)$
$\displaystyle\leq(C_{0}+\frac{1}{2})e^{-\beta_{t}}\max\left\\{1\,,\,e^{-2\gamma_{t}}\mathbb{E}\left[\,[{\mathcal{M}}]_{t}\right]\right\\}$
$\displaystyle=O\left(e^{-\beta_{t}}\max\left\\{1\,,\,e^{-\beta_{t}+2\gamma_{t}}\mathbb{E}\left[\,[{\mathcal{M}}]_{t}\right]\right\\}\right)\;,$
as desired. ∎
## Appendix E Proofs of Propositions 5.1 and Proposition 5.2
Both of the proofs contained in this sections are applications of the momentum
representation of the optimizer dynamics, and the FOSP approximation to the
solution of the optimality FBSDE (9).
### E.1 Proof of Proposition 5.1
###### Proof.
Using Proposition A.1, we find that the solution to the FOSP takes the form
$\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}_{t}=\mathbb{E}\left[\left.\int_{t}^{T}e^{\gamma_{t}+\alpha_{t}+\beta_{t}}\,\nabla
f\left(X_{u}\right)\,du-e^{\delta_{T}}\nabla
f(X_{T}^{\nu^{\ast}})\right\lvert{\mathcal{F}}_{t}\right]\;.$
Applying Fubini’s theorem, and the martingale property of
$\mathbb{E}\left[\nabla
f\left(X_{u}\right)\lvert{\mathcal{F}}_{u}\right]=\nicefrac{{g_{u}}}{{(1+\rho^{2})}}$,
we find that
$\displaystyle\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}_{t}$
$\displaystyle=\mathbb{E}\left[\left.\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\nabla
f\left(X_{u}\right)\,du-e^{\delta_{T}}\nabla
f(X_{T}^{\nu^{\ast}})\right\lvert{\mathcal{F}}_{t}\right]$
$\displaystyle=\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\mathbb{E}\left[\nabla
f(X_{u}^{\nu^{\ast}})\lvert{\mathcal{F}}_{t}\right]\,du-e^{\delta_{T}}\mathbb{E}\left[\nabla
f(X_{T}^{\nu^{\ast}})\lvert{\mathcal{F}}_{t}\right]$
$\displaystyle=\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\mathbb{E}\left[\nabla
f(X_{t}^{\nu^{\ast}})\lvert{\mathcal{F}}_{t}\right]\,du-e^{\delta_{T}}\mathbb{E}\left[\nabla
f(X_{t}^{\nu^{\ast}})\lvert{\mathcal{F}}_{t}\right]$
$\displaystyle=g_{t}(1+\rho^{2})^{-1}\left(\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,du-e^{\delta_{T}}\right)\;.$
Inserting expression above into equation (24), and re-arranging terms, we
obtain the desired result. ∎
### E.2 Proof of Proposition 5.2
###### Proof.
Using Proposition A.1, we find that the solution to the FOSP takes the form
$\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}_{t}=\mathbb{E}\left[\left.\int_{t}^{T}e^{\gamma_{t}+\alpha_{t}+\beta_{t}}\,\nabla
f\left(X_{u}\right)\,du-e^{\delta_{T}}\nabla
f(X_{T}^{\nu^{\ast}})\right\lvert{\mathcal{F}}_{t}\right]\;.$
Applying Fubini’s theorem, and noting that
$\mathbb{E}[\,\nabla_{i}f(X_{t+h})\lvert
y_{i,t}]=\sum_{j=1}^{\tilde{d}}(b^{\intercal}e^{-Ah})_{j}\,y_{\cdot,j,t}$, we
obtain
$\displaystyle\tilde{p}^{\scaleto{\mathstrut(0)}{4.5pt}}_{t}$
$\displaystyle=\mathbb{E}\left[\left.\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\nabla
f\left(X_{u}\right)\,du-e^{\delta_{T}}\nabla
f(X_{T}^{\nu^{\ast}})\right\lvert{\mathcal{F}}_{t}\right]$
$\displaystyle=\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\mathbb{E}\left[\nabla
f(X_{u}^{\nu^{\ast}})\lvert{\mathcal{F}}_{t}\right]\,du-e^{\delta_{T}}\mathbb{E}\left[\nabla
f(X_{T}^{\nu^{\ast}})\lvert{\mathcal{F}}_{t}\right]$
$\displaystyle=\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\left(\sum_{j=1}^{\tilde{d}}(b^{\intercal}e^{-A(u-t)})_{j}\,y_{\cdot,j,t}\right)\,du-e^{\delta_{T}}\left(\sum_{j=1}^{\tilde{d}}(b^{\intercal}e^{-A(T-t)})_{j}\,y_{\cdot,j,t}\right)$
$\displaystyle=\sum_{j=1}^{\tilde{d}}\left(\int_{t}^{T}e^{\gamma_{u}+\alpha_{u}+\beta_{u}}\,\left(b^{\intercal}e^{-A(u-t)}\right)_{j}\,du-e^{\delta_{T}}(b^{\intercal}e^{-A(T-t)})_{j}\right)y_{\cdot,j,t}$
Inserting expression above into equation (24), and re-arranging terms, we
obtain the desired result. ∎
|
# A positive feedback method based on F-measure value for Salient Object
Detection
Ailing Pan 1 Chao Dai 1 Chen Pan 1 Corresponding author. Email:
<EMAIL_ADDRESS>Dongping Zhang 1 This work was supported by Key Research and
Development Projects in Zhejiang Province of China (NO.2021C03192,
2023C01032). Yunchao Xu 1
1Department of Computer Engineering, China JiLiang University, Hangzhou,
Zhejiang, 310018, China
###### Abstract
The majority of current salient object detection (SOD) models are focused on
designing a series of decoders based on fully convolutional networks (FCNs) or
Transformer architectures and integrating them in a skillful manner. These
models have achieved remarkable high performance and made significant
contributions to the development of SOD. Their primary research objective is
to develop novel algorithms that can outperform state-of-the-art models, a
task that is extremely difficult and time-consuming. In contrast, this paper
proposes a positive feedback method based on F-measure value for SOD, aiming
to improve the accuracy of saliency prediction using existing methods.
Specifically, our proposed method takes an image to be detected and inputs it
into several existing models to obtain their respective prediction maps. These
prediction maps are then fed into our positive feedback method to generate the
final prediction result, without the need for careful decoder design or model
training. Through applying the positive feedback method for decision fusion of
multi-model perception results. Moreover, our method is adaptive and can be
implemented based on existing models without any restrictions. Experimental
results on five publicly available datasets show that our proposed positive
feedback method outperforms the latest 12 methods in five evaluation metrics
for saliency map prediction. Additionally, we conducted a robustness
experiment, which shows that when at least one good prediction result exists
in the selected existing model, our proposed approach can ensure that the
prediction result is not worse. Our approach achieves a prediction speed of 20
frames per second (FPS) when evaluated on a low configuration host and after
removing the prediction time overhead of inserted models. These results
highlight the effectiveness, efficiency, and robustness of our proposed
approach for salient object detection. Code and Saliency maps will be
available.
## 1 Introduction
Salient object detection (SOD) aims to mimic the human visual perception
system to capture the most prominent regions in given images or videos. It can
serve as a pre-processing step for other realated computer vision tasks,
including object tracking[1], action recognition[2], video segmentation[3],
and image captioning[4].
Existing SOD models can be mainly divided into traditional algorithm-based and
deep learning-based approaches. Traditional SOD methods rely on handcrafted
features and use these features to predict saliency maps in a bottom-up
manner. Common handcrafted features include center prior[5] and distance
transform[6], which only contain low-level clues and perform poorly in complex
scenes. With the emergence of deep learning techniques, SOD has made
remarkable progress. Recent SOD models are mainly implemented based on fully
convolutional networks (FCNs) and Transformer architectures, among which FCN
is still the mainstream SOD architecture. SOD FCNs-based models mainly design
a series of decoders around feature extraction, refinement or enhancement, and
multi-level feature fusion and set specific loss functions to assist model
training[7][8][9][10][11][12][13][14][15][16]. Inspired by the impressive
performance of the Transformer in the field of natural language processing
(NLP), researchers have also carried out some innovative work using the
Transformer architecture in the SOD research field and achieved impressive
performance improvements. These models mainly rely on the powerful long-
distance feature correlation capture ability of the Transformer architecture
and help the model generate more complete predictions by effectively
extracting global contextual clues[17][18]. In addition, a hybrid architecture
of FCN and Transformer has been proposed by integrating the advantages of both
models[19][20].
The main research content of the above work is to design different network
models carefully based on the characteristics of the SOD task and demonstrate
the performance beyond the previous proposed models, playing an extremely
important role in the rapid development process of the SOD research field.
However, the process of carrying out these works is usually challenging, time-
consuming, and requires difficultly obtaining higher performance than the
latest methods. Therefore, in contrast to the above work, this paper proposes
a positive feedback approach based on F-measure value for SOD, which is based
on existing methods to obtain more accurate prediction results. The approach
allows any existing method to be inserted, such as traditional algorithms and
various deep learning methods, and can achieve performance beyond the inserted
methods.
In summary, the main contributions of this paper are:
* •
Under the current SOD research background, we have explored how to utilize
existing methods rarely. This paper proposes a positive feedback approach
based on F-measure value for SOD, which allows any method to be inserted and
performs better than the inserted method in terms of performance. The
prediction process of the approach is training-free, adaptive weights, and
contains only one tunable hyperparameter.
* •
Based on the F-measure value, a positive feedback process is designed, which
does not require human involvement during the calculation process, and is a
completely self-updating process.
## 2 Related Work
Existing SOD models can be mainly divided into traditional algorithm-based and
deep learning-based approaches, with the latter being the current mainstream
SOD method.
### 2.1 FCNs-based network
FCNs-based SOD models mainly design a series of decoders around feature
extraction, refinement or enhancement, and multi-level feature fusion and set
specific loss functions to assist model training.
The successful performance of SOD models depends on the effective integration
of multi-level features. Several studies have highlighted the significant
differences between the low-level and high-level features extracted from
backbone networks. To address this issue, Chen et al.[21] proposed a
progressive context aware feature aggregation module, while Dai et al.[10]
developed a middle layer feature extraction module to achieve better feature
fusion results. Moreover, the integration of multiscale features has been
shown to improve SOD detection performance in various studies. For instance,
Zhang et al.[15] developed a neural structure based search unit to
automatically determine the multiscale features that need to be aggregated.
Fang et al.[9] proposed a densely nested-based network framework to utilize
multiscale high-level feature maps effectively. Meanwhile, Zhuge et al.[8]
used various multiscale feature extraction methods to obtain diverse
multiscale features and designed an integrity channel enhancement module to
highlight salient objects. Base on work[22], Wu et al. [14] proposed a dynamic
pyramid convolution to extract multiscale features instead of fixed size
convolutions. Effective supervision strategy have also been employed to help
models predict salient objects more accurately. For instance, Wei et al.[22]
proposed pixel position aware losss to guide networks to pay more attention to
local details, while Yang et al.[23] proposed progressive self-guided loss to
guide network learning to more complete salient regions. Additionally, Xu et
al.[16] designed a knowledge review network to roughly locate salient regions
first and then finely segment salient objects. Wu et al. [7] proposed a
decomposition and completion network to predict the saliency, edge, and
skeleton maps respectively, and then filled the saliency map using edge and
skeleton maps.
### 2.2 Transformer-based network
In the field of SOD, the incorporation of global context information is
extremely crucial for accurately and completely predicting salient regions.
Transformer-based network architectures, renowned for their ability to capture
long-range dependencies, have been successfully introduced in this domain.
To this end, Liu et al.[17] proposed a unified transformer-based model for
SOD, which facilitates the propagation of global context information among
image patches. Meanwhile, Zhang et al.[18] developed a generative vision
transformer network that generates a pixel-level uncertainty map, effectively
representing the significance confidence of salient objects. Additionally, Ren
et al.[24] proposed a simple yet effective deeply-transformer network that
preserves more unifying global-local representations to gradually restore
spatial details.
### 2.3 Hybrid framework-based network
To enhance the accuracy and completeness of prediction results in the field of
SOD, it is important to effectively extract and utilize both global context
from deep features and local context from shallow features. However, achieving
this in network frameworks can be challenging. Therefore, some hybrid network
frameworks have emerged as the times require.
Zhu et al.[25] proposed a deep supervised fusion transformer network,
extending the applicability of FCN to the transformer architecture for the
first time. They employed a transformer encoder to extract multiscale features
and designed a multiscale aggregation module to aggregate these features in a
coarse-to-fine manner. Similarly, Yun et al.[19] proposed a self-refined
transformer network that leverages the transformer encoder to capture long-
distance dependencies and designed a context refinement module. The module is
employed to integrate global context with decoder features and refine and
locate local details automatically. In contrast, Wang et al.[26] still used
FCN as an encoder and utilized a transformer module for multi-level feature
fusion to address the limited receptive field of FCN.
## 3 Method
### 3.1 Overview of the proposed Method
Figure 1: The overall pipeline of the proposed method.
Figure 1 shows structure of the proposed method, which includes a multi-branch
model structure and a positive feedback prediction structure. The former is
used to place existing SOD model sequences, including traditional algorithm
models, deep-learning supervised or unsupervised models. The positive feedback
prediction structure will iteratively calculate the respective weights
according to the outputs of all branches to obtain the final saliency maps,
where the weight calculation process is completely adaptive. In order to
verify our method more conveniently, we use the precious models in advance to
generate saliency maps, which are marked as $Smp_{i}(i=1,2,3,4)$.
### 3.2 Positive feedback prediction algorithm
The positive feedback prediction algorithm is summarized in Table 1, and its
conceptual principle is illustrated in Figure 2. Hereinafter, $smp_{n},N$
refers to the saliency maps generated by the multi-branch models and the
number of branch. $\varepsilon$ is a threshold, which is set to 0.95. The
$imbinarize$($\cdot$) is a binarization function and the $mat2grey$ is a
graying function. The $computerFmeasure$($\cdot$) is the function for
calculating $F$-$measure$[27].
The key steps of the algorithm:
Step1. Input: outputs ($smp_{n}$) of the multi-branch model structure are used
as inputs of the positive feedback prediction structure, and perform the
binarization function to get binary maps ($b\\_smp_{n}$).
Step2. Initialization: the above inputs are fused in the way of pixel-level
addition, and perform the binarization function to get binary maps
($B\\_Smp_{0}$). At this time, the weights of each branch are consistent.
Step3. Iteration: calculate $F$-$measure$ with the input ($b\\_smp_{n}$) of
each brach and the latest fusion result ($B\\_Smp_{i-1}$). Update the weights
of each branch, recalculate the fusion result ($Smp_{i}$) from the new weights
and perform the binarization function to get binary maps ($B\\_Smp_{i}$).
Step4. Judgement: calculate $F$-$measure$ with the latest and last fusion
result and compare it with $\varepsilon$. If it is greater than the threshold,
the two successive resluts are considered to be simmilar enough, and the
latest fusion result is taken as the final output; Otherwise, execute Step3
again.
Table 1: Positive feedback prediction algorithm. Input: |
---|---
| $smp_{n},N$;
Output: |
| $Smp_{i}$;
1: | $B\\_Smp_{i-1}=imbinarize(mat2grey(\sum_{n=1}^{N}smp_{n})),i=1$;
2: | $b\\_smp_{n}=imbinarize(smp_{n})$;
3: | while True do
4: | $F_{n}=computerFmeasure(b\\_smp_{n},B\\_Smp_{i-1})$;
5: | $F_{sum}=\sum_{n=1}^{N}F_{n}$; $\alpha_{n}={F_{n}}\setminus{F_{sum}}$;
6: | $Smp_{i}=\sum_{n=1}^{N}{\alpha_{n}}\cdot smp_{n};B\\_Smp_{i}=imbinarize(Smp_{i})$;
7: | $F=computerFmeasure(B\\_Smp_{i},B\\_Smp_{i-1})$;
8: | if $F\geq\varepsilon$ then
9: | $break$;
10: | else
11: | $i=i+1$;
12: | end if
13: | end while
14: | $Smp_{i}=mat2grey(Smp_{i})$;
15: | return $Smp_{i}$.
Figure 2: The conceptual principle of positive feedback prediction algorithm.
## 4 Experimental results and analysis
### 4.1 Datasets and Evaluation Metrics
To validate the efficacy of the proposed method in this paper, we performed a
series of experiments on five publicly available datasets. The datasets
utilized in this study are briefly described as follows: DUTS[28] comprises a
total of 10,000 images for training and 5,000 images for testing, with only
the test set being used in this study. DUT-OMRON[29] consists of 5,019 images
with complex structures and backgrounds. HKU-IS[30] contains 4,447 maps with
multiple salient objects. PASCAL-S[31] contains 850 natural images. ECSSD[32]
comprises a collection of 1,000 images obtained from the internet.
To quantitatively evaluate the proposed method, we adopt six evaluation
metrics as the performance measures, including mean absolute error
($MAE$)[33], maximum F-measure ($mF$) score, and S-measure ($Sm$) score[34],
precision-recall ($PR$) curves.
### 4.2 Implementation Details
To facilitate the testing of the proposed positive feedback method, we
implemented the following steps. Firstly, we generated prediction maps of a
multi-branch model using the model disclosed by the author and Python tools.
Secondly, we evaluated the positive feedback method using Matlab tools.
To evaluate the effectiveness and characteristics of our proposed method, we
conducted comparative, ablation, and robustness experiments. In the
comparative experiment, we compared the performance with other state-of-the
arts models. Results showed that the positive feedback mechanism improved the
model’s performance on multiple metrics. In the ablation experiment, we
compared the performance of the positive feedback mechanism with pixel-level
addition fusion on five datasets. In the robustness experiment, we manually
selected good and bad renderings from each model and input them into the
positive feedback mechanism to observe the visual comparison results.
In total, we conducted two sets of experiments. In the first set, we used four
branches, namely MSFNet (2021)[15], PAKRNet (2021)[16], ICONet (2022)[8], and
SelReformer (2022)[19]. The performance of the 2021 methods are weaker than
that of the 2022 methods. In the second set, we used two branches, namely
DPNet (2022)[14] and SelReformer. The performance of these two methods is
relatively close. For detailed experimental procedures and analysis, please
refer to each subsection.
Table 2: Quantitative evaluation. The mean absolute error (MAE, smaller is
better), maximum F-measure (mF, larger is better) and S-measure (Sm, larger is
better). The results for each saliency detection method are reported on five
different datasets, with the top three performing methods highlighted in red,
green, and blue. It is important to note that ”Our-SSSS” and ”Our-SS” denote
the results of the four-branch and two-branch models, respectively.
Methods | DUTS | ECSSD | HKU-IS | PASCAL-S | DUT-OMRON
---|---|---|---|---|---
mF | MAE | Sm | mF | MAE | Sm | mF | MAE | Sm | mF | MAE | Sm | mF | MAE | Sm
DCNet[7] | .894 | .035 | .895 | .952 | .032 | .928 | .939 | .027 | .922 | .872 | .062 | .861 | .823 | .051 | .845
BiconNet[35] | .888 | .038 | .890 | .949 | .034 | .927 | .939 | .029 | .923 | .877 | .063 | .863 | .817 | .053 | .842
DNTD[9] | .892 | .033 | .891 | .946 | .034 | .922 | .938 | .028 | .920 | .878 | .064 | .857 | .803 | .051 | .828
EDNet[12] | .895 | .035 | .892 | .951 | .032 | .927 | .941 | .026 | .924 | .886 | .062 | .865 | .828 | .049 | .850
EFNet[10] | .898 | .034 | .895 | .948 | .034 | .925 | .939 | .027 | .922 | .875 | .063 | .864 | .822 | .054 | .843
MRINet[11] | .899 | .035 | .894 | .950 | .032 | .927 | .941 | .027 | .922 | .877 | .060 | .864 | .829 | .054 | .848
RCSBNet[13] | .899 | .035 | .881 | .944 | .034 | .922 | .938 | .027 | .919 | .882 | .059 | .860 | .810 | .049 | .835
MSFNet[15] | .878 | .034 | .877 | .941 | .033 | .915 | .927 | .027 | .908 | .863 | .061 | .852 | .799 | .050 | .832
PAKRNet[16] | .907 | .033 | .900 | .953 | .032 | .928 | .943 | .027 | .924 | .873 | .067 | .858 | .834 | .050 | .853
ICONet[8] | .893 | .037 | .892 | .951 | .031 | .931 | .942 | .027 | .925 | .884 | .060 | .870 | .830 | .059 | .846
DPNet[14] | .917 | .028 | .912 | .954 | .031 | .931 | .950 | .023 | .934 | .894 | .054 | .877 | .834 | .049 | .853
SelReformer[19] | .916 | .027 | .911 | .958 | .027 | .936 | .947 | .024 | .931 | .894 | .051 | .881 | .837 | .043 | .861
Our-SSSS | .922 | .032 | .909 | .960 | .029 | .941 | .950 | .025 | .935 | .898 | .055 | .884 | .863 | .043 | .877
Our-SS | .931 | .028 | .919 | .962 | .028 | .943 | .954 | .023 | .939 | .899 | .052 | .889 | .859 | .046 | .869
### 4.3 Comparison with the State-of-the-arts
We compare the proposed method with 12 state-of-the-art methods, including
DCNet[7], BiconNet[35], DNTD[9], EDNet[12], EFNet[10], MRINet[11],
RCSBNet[13], MSFNet[15], PAKRNet[16], ICONet[8], DPNet[14], SelReformer[19].
To ensure a fair and objective comparison, we will utilize the saliency maps
provided by the authors and employ the same standardized evaluation function
to calculate each metric.
In Table 2, we present the quantitative comparison results based on MAE,
maximum F-measure, and S-measure. Our methods demonstrate the most
comprehensive performance across all five datasets, as evidenced by our
superior results across all three metrics. Notably, our method, Our-SS,
consistently outperforms the second-best result in terms of mF and Sm by 1.5%,
0.4%, 0.4%, 0.5%, and 0.9%, 0.8%, 0.6%, 0.9% on DUTS, ECSSD, HKU-IS, and
PASCAL-S datasets, respectively. Additionally, our method, Our-SSSS, performs
exceptionally well on ECSSD, HKU-IS, and PASCAL-S datasets, particularly on
the DUT-OMRON dataset, where it consistently outperforms SelReformer[19], the
strongest model among the four-branch structures. Moreover, in Figure 3, we
provide precision-recall curves (PR) for all five datasets. Our curves
demonstrate exceptional performance across most thresholds, particularly on
the DUT-OMRON and DUTS datasets.
Figure 3: Performance comparison with 15 state-of-the-art methods on five
saliency datasets. The red solid line and the blue solid line denote our
models.
### 4.4 Ablation experiment
Within this subsection, we have performed ablation experiments on two datasets
to demonstrate the efficacy and distinctive attributes of the proposed
approach. Specifically, we have replaced the proposed positive feedback fusion
with pixel-level addition and have presented the experimental outcomes in
Table 3. The results reveal that our proposed method yields superior overall
performance compared to the direct addition way. Notably, in the ablation
experiment pertaining to the four-branch architecture, our proposed approach
has demonstrated a considerably more substantial advantage, especially for the
mean absolute error (MAE) metric, which has reduced by 7.2%and 7.7%,
correspondingly.
Table 3: Ablation evaluation on two datasets.
Methods | PASCAL-S | DUT-OMRON
---|---|---
mF | MAE | Sm | mF | MAE | Sm
ablation-4S | .891 | .060 | .882 | .863 | .047 | .874
Our-SSSS | .898 | .055 | .884 | .863 | .043 | .877
ablation-2S | .897 | .053 | .888 | .858 | .046 | .869
Our-SS | .899 | .052 | .889 | .859 | .046 | .869
Based on the results of the two sets of comparative experiments on ablation,
we can derive several professional conclusions. Firstly, in cases where there
is a significant difference in performance between the selected methods and
their placement in a multi-branch structure, the proposed method exhibits
greater advantages over direct fusion. Conversely, when the performance
difference between the selected methods is minor, the proposed method can only
exhibit limited superiority over direct fusion. Additionally, it is important
to note that the performance synthesis level achieved through positive
feedback fusion is consistently superior to that of each individual model
within a multi-branch structure.
### 4.5 Robustness experiment
To assess the proposed method’s robustness against interference, we conducted
adversarial experiments on a four-branch structure. Despite the outstanding
detection performance of many existing methods, their capabilities are not
consistently strong, as depicted in Figure 4. To better showcase the
superiority of our approach, we manually selected several sets of input
predictions with both desirable and undesirable effects for the proposed
method, and presented visual comparisons in Figure 4. This not only explains
why the proposed method is superior to methods introduced into a multi-branch
structure, but also superior to direct fusion, as it allows for automatic
weight updating to achieve satisfactory results, instead of treating all
branches equally. Furthermore, it should be noted that even in the absence of
good predictions in the four-branch structural model, a favorable outcome may
still be yielded, as demonstrated in the seventh row.
Figure 4: Visual comparisons from the four-branch structure.
## Conclusion
Distinguished from most existing efforts in salient object detection (SOD),
this study focuses on enhancing the accuracy of previously established
algorithms. Specifically, we introduce a positive feedback approach based
F-measure value for SOD that comprises a multi-branch model structure and a
positive feedback prediction structure. The method involves feeding input
images into the multi-branch model structure to generate their corresponding
saliency maps, and then processing these maps through the positive feedback
prediction structure for obtaining the final result via positive feedback
calculation. Notably, our method requires no model training, entails minimal
hyperparameter tuning, and features automatic calculation. By integrating
multiple existing models, our method surpasses the performance of individual
models. Our proposed method achieves a prediction speed of 20 frames per
second (FPS) even on a low-end host, while removing the prediction time
overhead of inserted models. Our findings suggest that higher performance of
the inserted models leads to better overall results, and the initialization
stage is crucial due to the positive feedback mechanism. Selecting models with
insignificant performance differences weakly showcases the advantages of our
approach. We believe that our study is not only of significant research value
but also has practical applications, and thus, merits the attention and
scrutiny of future researchers.
## Conflict of interests
The authors declare that they have no conflict of interests.
## Acknowledgments
This work was supported by Key Research and Development Projects in Zhejiang
Province of China (NO.2021C03192, 2023C01032).
## Author contributions
Ailing Pan wrote the paper, conceived and designed the experiments;
Chao Dai assisted in revising paper, collected the data, performed the
experiments and recorded experimental results;
Chen Pan proposed the core idea of the paper and will act as the corresponding
author;
Dongping Zhang and Yunchao Xu are participants in the fund projects.
All authors agree to this submission.
## Code and data availability
Code and Saliency maps will be available at:
https://github.com/dc3234/PF/tree/main.
## References
* [1] Lee, Hyemin, and Daijin Kim. Salient region-based online object tracking. Proceedings of the IEEE Winter Conference on Applications of Computer Vision, 2018: 1170-1177.
* [2] Atrish, Abhay, et al. An automated hierarchical framework for player recognition in sports image. Proceedings of the international conference on video and image processing. 2017: 103-108.
* [3] Zhang, Dingwen, et al. SPFTN: A joint learning framework for localizing and segmenting objects in weakly labeled videos. IEEE transactions on pattern analysis and machine intelligence, 42(2), 2018: 475-489.
* [4] Fang, Hao, et al. From captions to visual concepts and back. Proceedings of the IEEE conference on computer vision and pattern recognition. 2015: 1473-1482.
* [5] Jiang, Zhuolin, and Larry S. Davis. Submodular salient region detection. Proceedings of the IEEE conference on computer vision and pattern recognition, 2013: 2043-2050.
* [6] Tu, Wei-Chih, et al. Real-time salient object detection with a minimum spanning tree. Proceedings of the IEEE conference on computer vision and pattern recognition, 2016: 2334-2342.
* [7] Wu, Zhe, Li Su, and Qingming Huang. Decomposition and completion network for salient object detection. IEEE Transactions on Image Processing, 30, 2021: 6226-6239.
* [8] Zhuge, Mingchen, et al. Salient object detection via integrity learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(3), 2023: 3738-3752.
* [9] Fang, Chaowei, et al. Densely nested top-down flows for salient object detection. Science China Information Sciences, 65(8), 2022: 1-13.
* [10] Dai, Chao, Chen Pan, and Wei He. Feature extraction and fusion network for salient object detection. Multimedia Tools and Applications, 81(23), 2022: 33955-33969.
* [11] Dai, Chao, et al. Multiple refinement and integration network for Salient Object Detection. AI Communications, 35(1), 2022: 31-44.
* [12] Wu, Yu-Huan, et al. EDN: Salient object detection via extremely-downsampled network. IEEE Transactions on Image Processing, 31, 2022: 3125-3136.
* [13] Ke, Yun Yi, and Takahiro Tsubono. Recursive contour-saliency blending network for accurate salient object detection. Proceedings of the IEEE Winter Conference on Applications of Computer Vision, 2022: 2940-2950.
* [14] Wu, Zhenyu, et al. Salient Object Detection via Dynamic Scale Routing. IEEE Transactions on Image Processing, 31, 2022: 6649-6663.
* [15] Zhang, Miao, et al. Auto-msfnet: Search multi-scale fusion network for salient object detection. Proceedings of the 29th ACM international conference on multimedia. 2021.
* [16] Xu, Binwei, et al. Locate globally, segment locally: A progressive architecture with knowledge review network for salient object detection. Proceedings of the AAAI Conference on Artificial Intelligence, 35(4), 2021: 3004-3012.
* [17] Liu, N., Zhang, N., Wan, K., Shao, L., Han, J. Visual saliency transformer. In Proceedings of the IEEE international conference on computer vision, 2021: 4722-4732.
* [18] Zhang, J., Xie, J., Barnes, N., Li, P. Learning generative vision transformer with energy-based latent space for saliency prediction. Advances in Neural Information Processing Systems, 34, 2021: 15448-15463.
* [19] Yun, Y. K., Lin, W. SelfReformer: Self-Refined Network with Transformer for Salient Object Detection. arXiv preprint arXiv:2205.11283, 2022.
* [20] Xie, C., Xia, C., Ma, M., Zhao, Z., Chen, X., Li, J. Pyramid grafting network for one-stage high resolution saliency detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2020: 11717-11726.
* [21] Chen, Z., Xu, Q., Cong, R., Huang, Q. Global context-aware progressive aggregation network for salient object detection. In Proceedings of the AAAI conference on artificial intelligence, 34(7), 2020: 10599-10606.
* [22] Wei, J., Wang, S., Huang, Q. F³Net: fusion, feedback and focus for salient object detection. In Proceedings of the AAAI conference on artificial intelligence, 34(7), 2020: 12321-12328.
* [23] Yang, Sheng, et al. Progressive self-guided loss for salient object detection. IEEE Transactions on Image Processing, 30, 2021: 8426-8438.
* [24] Ren, S., Wen, Q., Zhao, N., Han, G., He, S. Unifying global-local representations in salient object detection with transformer. arXiv preprint arXiv:2108.02759, 2021.
* [25] Zhu, H., Sun, X., Li, Y., Ma, K., Zhou, S., Zheng, Y. DFTR: Depth-supervised Fusion Transformer for Salient Object Detection. 2022.
* [26] Wang, Z., Zhang, Y., Liu, Y., Wang, Z., Coleman, S., Kerr, D. TF-SOD: a novel transformer framework for salient object detection. Neural Computing and Applications, 34(14), 2022: 11789-11806.
* [27] Achanta, R., Hemami, S., Estrada, F., Susstrunk, S. (2009, June). Frequency-tuned salient region detection. In 2009 IEEE conference on computer vision and pattern recognition (pp. 1597-1604). IEEE.
* [28] Wang, L., Lu, H., Wang, Y., Feng, M., Wang, D., Yin, B., Ruan, X. Learning to detect salient objects with image-level supervision. In Proceedings of the IEEE conference on computer vision and pattern recognition, 2013: 136-145.
* [29] Yang, C., Zhang, L., Lu, H., Ruan, X., Yang, M. H. Saliency detection via graph-based manifold ranking. In Proceedings of the IEEE conference on computer vision and pattern recognition, 2013: 3166-3173).
* [30] Li, G., Yu, Y. Visual saliency based on multiscale deep features. In Proceedings of the IEEE conference on computer vision and pattern recognition, 2015: 5455-5463.
* [31] Li, Y., Hou, X., Koch, C., Rehg, J. M., Yuille, A. L. The secrets of salient object segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, 2014: 280-287.
* [32] Shi, J., Yan, Q., Xu, L., Jia, J. Hierarchical image saliency detection on extended CSSD. IEEE transactions on pattern analysis and machine intelligence, 38(4), 2015: 717-729.
* [33] Perazzi, F., Krähenbühl, P., Pritch, Y., Hornung, A. Saliency filters: Contrast based filtering for salient region detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, 2012: 733-740.
* [34] Fan, D. P., Cheng, M. M., Liu, Y., Li, T., Borji, A. Structure-measure: A new way to evaluate foreground maps. In Proceedings of the IEEE international conference on computer vision, 2017: 4548-4557.
* [35] Yang, Ziyun, Somayyeh Soltanian-Zadeh, and Sina Farsiu. BiconNet: An edge-preserved connectivity-based approach for salient object detection. Pattern recognition, 121, 2022: 1-13.
|
# Quantifying the rise and fall of scientific fields
Chakresh Singh Université de Paris, INSERM U1284, Center for Research and
Interdisciplinarity (CRI), F-75006 Paris, France Emma Barme Université de
Paris, INSERM U1284, Center for Research and Interdisciplinarity (CRI),
F-75006 Paris, France Robert Ward School of Public Policy, Georgia Institute
of Technology, Atlanta, GA 30332 Liubov Tupikina Université de Paris, INSERM
U1284, Center for Research and Interdisciplinarity (CRI), F-75006 Paris,
France Nokia Bell labs, France Marc Santolini Université de Paris, INSERM
U1284, Center for Research and Interdisciplinarity (CRI), F-75006 Paris,
France Corresponding author<EMAIL_ADDRESS>
###### Abstract
Science advances by pushing the boundaries of the adjacent possible. While the
global scientific enterprise grows at an exponential pace, at the mesoscopic
level the exploration and exploitation of research ideas is reflected through
the rise and fall of research fields. The empirical literature has largely
studied such dynamics on a case-by-case basis, with a focus on explaining how
and why communities of knowledge production evolve. Although fields rise and
fall on different temporal and population scales, they are generally argued to
pass through a common set of evolutionary stages.To understand the social
processes that drive these stages beyond case studies, we need a way to
quantify and compare different fields on the same terms. In this paper we
develop techniques for identifying scale-invariant patterns in the evolution
of scientific fields, and demonstrate their usefulness using 1.5 million
preprints from the arXiv repository covering $175$ research fields spanning
Physics, Mathematics, Computer Science, Quantitative Biology and Quantitative
Finance. We show that fields consistently follows a rise and fall pattern
captured by a two parameters right-tailed Gumbel temporal distribution. We
introduce a field-specific rescaled time and explore the generic properties
shared by articles and authors at the creation, adoption, peak, and decay
evolutionary phases. We find that the early phase of a field is characterized
by the mixing of cognitively distant fields by small teams of
interdisciplinary authors, while late phases exhibit the role of specialized,
large teams building on the previous works in the field. This method provides
foundations to quantitatively explore the generic patterns underlying the
evolution of research fields in science, with general implications in
innovation studies.
## 1 Introduction
Quantifying the dynamics of scientific fields can help us understand the past
and design the future of scientific knowledge production. Several studies have
investigated the emergence and evolution of scientific fields, from the
discovery of new concepts to their adaptation and modification by the
scientific community [1, 2, 3, 4]. In particular, methods ranging from
bibliometric studies [5, 6] to network analyses [7, 8, 4] and natural language
processing [9, 10] have been implemented on large publication corpora to
monitor the propagation of concepts across articles [11, 12] and the social
interactions between researchers that are producing them [8, 13, 14].
The definition of research fields often relies on data-driven strategies using
self-reported keywords or content analysis. For example, the use of granular
author self-reported topic annotations from well-defined classification
schemes such as PACS (Physics and Astronomy Classification Scheme), MESH terms
and keywords has allowed to construct topic co-occurrence networks and extract
clusters corresponding to potential research fields [7, 9]. Beyond self-
reported annotations, other methods have exploited the citation network
between research articles to group articles by relatedness and map the
knowledge flow within and across research fields [12, 4], and inferential
methods have leveraged Natural Language Processing techniques to automatically
identify key topics of research [11, 10, 15] and their relations. These
various methods provide clusters of closely related topics corresponding to
putative research fields, allowing to monitor how the changing relations
between topics and ideas underlie the dynamic evolution and mutual
interactions between fields. Beyond topic-centric approaches, other methods
have leveraged the interactions between researchers to define research
communities with shared research interests. For example, the co-authorship
network between researchers has been shown to undergo a topological transition
during the emergence of a new field [13, 14]. Co-authorship relations also
influence the individual evolution of research interests and foster the
emergence of a consensus in a research community [16, 17, 18].
While fields rise and fall on different temporal and population scales, they
are generally argued to pass through a common set of evolutionary stages [19,
1]. These stages delineate how diverse actors and behaviors are involved in
successive phases. To study these temporal patterns, dynamical models were
introduced to characterize the evolution of research fields [20, 5] and the
spread of innovation [21, 22, 23]. Yet, we are still lacking a unified
framework to delineate stereotyped stages in the evolution of scientific
fields that can be validated over a large number of well-annotated research
fields.
Here, we address this gap by developing techniques for identifying scale-
invariant patterns in the evolution of fields. We demonstrate their usefulness
using a large corpus of 1.45 million articles from the arXiv repository with
self-reported field tags spanning 175 research fields in Physics, Computer
Science, Mathematics, Finance, and Biology. We show that the evolution of
fields follows a right-tailed distribution with two parameters characterizing
peak location and distribution width. This allows us to collapse the temporal
distributions onto a single rise-and-fall curve and delineate different
evolutionary stages of the fields: creation, adoption, peak, early decay, and
late decay. We then describe the characteristics of articles and authors
across these stages. We finish by discussing these results and their
implications for further work in science and innovation.
Figure 1: a Example of an article in arXiv, highlighting the metadata
extracted using the arXiv API. b The daily number of articles submitted to
arXiv since 1986 shows an exponential growth over time, with a doubling period
of 6 years. The data also shows strong seasonality with 10 times fewer
articles over the weekends. c Heatmap representing the share of articles in
each field (rows) over time (columns). Field are identified using the subject
tags within articles. The heatmap is row normalized for comparison across
fields. Rows are ordered in chronological order of their peak time. The right
side panel shows the total number of articles published in each field as
horizontal bars.
## 2 Results
### 2.1 Description of the data
Since its launch in $1991$, the arXiv repository has become a major venue for
community research, gaining considerable importance across the fields of
Physics, Mathematics and Computer Science. As an open and free contribution
platform, it provides an equal opportunity for publication to researchers
globally, and plays a dominant role in the diffusion of knowledge [24] and the
evolution of new ideas [25].
When submitting a contribution, authors declare the research fields that the
article is contributing to by selecting from a list of subject tags. Here we
collected information about authors, date of publication, and research fields
of 1,456,403 arXiv articles until 2018 (see Methods section and Fig 1a). The
number of articles and authors exhibit an exponential growth over time with a
doubling period of 6 years (see Fig 1b and S1). To control for this effect,
here we focus for each field $i$ on the yearly share of articles
$f_{i,y}=n_{i,y}/N_{y}$, where $n_{i,y}$ is the number of articles published
in the considered field at year $y$ and $N_{y}$ is the total number of
articles in arXiv in the same year. We represent the temporal distributions of
all fields in Fig 1c by chronological peak time. Over the past 30 years, the
research interests have shifted from high-energy physics to computer science
and more recently economy.
### 2.2 Quantifying the rise and fall of scientific fields
Despite differences in overall number of articles and eventual duration, we
observe a general rise-and-fall pattern across research fields (Fig 1c),
prompting us to explore if a simple model can capture their temporal
variation. Extreme value theory [26] predicts that under a broad range of
circumstances, temporal processes displaying periods of incubation (such as
incubation of ideas) or processes with multiple choice (such as the choice of
ideas or research fields) follow skewed right-tailed extreme value
distributions. Examples of such processes can be found in diverse areas, for
example when modeling the evolution of scientific citations [27] or disease
incubation periods [28, 29]. Following these insights, here we use the Gumbel
distribution (Eq. 1) as an ansatz to model the observed field temporal
distributions. Belonging to the general class of extreme value distributions
[30, 26], it provides interpretable parameters for the peak location $\alpha$
and distribution width $\beta$ (Fig. S3). Denoting by $t$ the time since the
first article was published in the field, the share of articles $G(t)$ follows
Eq. 1:
$G(t)=\frac{1}{\beta}e^{\frac{-(t-\alpha)}{\beta}}e^{-e^{\frac{-(t-\alpha)}{\beta}}}$
(1)
where $\alpha$ is the location parameter and $\beta$ the scale parameter.
Figure 2: a. Conditions for a field to be included in the analysis. b-d
Gumbel fits for the fields with the largest numbers of articles in Physics,
Mathematics and Computer Science: Mathematics - Combinatorics (b), Material
Science (c) and Computational Complexity (d). e. Evolution of the 72 studied
fields after temporal re-scaling from Eq. 2. The blue curve represents the
Gumbel fit, and red dots correspond to the empirical average over equal-sized
bins. Error bars indicate standard error.
In order to estimate the model fit, we consider fields that satisfy three
conditions: (i) longevity – having at least 10 years of activity to ensure a
sufficient observation period, (ii) unimodality – we exclude multimodal
distributions as it would require introducing a mixture model going beyond the
scope of this study and (iii) completeness – we require the peak of the
distribution to be at least 3 years away from the beginning and the end of the
collection period to ensure that we capture sufficient data on both sides of
the distribution. This reduces the number of fields to $72$, which we consider
in our analyses below.
Using a least-squares optimization fitting procedure (see Methods), we show
that 66 out of 72 fields ($91.6\%$) exhibit a significant goodness of fit
($k<0.3$ and $p>0.05$ under KS-test, see Fig S5). We show in Fig 2b-d the
temporal distributions and Gumbel fits for the fields with the largest total
numbers of articles in Physics, Mathematics and Computer Science. After
obtaining the location $\alpha$ and scale $\beta$ parameters from the fitting
procedure, we compute for each field the re-scaled time:
$t^{{}^{\prime}}=\frac{t-\alpha}{\beta}$ (2)
By re-normalizing fields with this standardized time, we observe that the
various temporal distributions align on a single curve, highlighting the
shared patterns of rise and fall across the fields studied (Fig 2e). In
particular, the Gumble distribution provides a more stringent fit of the
tails, as can be observed when comparing to a symmetric, Gaussian fit in a log
scale (Fig S5).
### 2.3 Characterizing the stages of research field evolution
Using the rescaled time from Eq. 2, we next explore the characteristics of
articles and researchers at different stages of a research field evolution. We
adopt hereafter the standard delineations of stages from the innovation
diffusion literature [21] and define 5 periods of research field evolution
(creation, adoption, peak, early decay, and late decay) delineated at the re-
scaled times corresponding respectively to the $2.5\%$, $16\%$, $50\%$ and
$84\%$ quantiles of the Gumbel distribution in Fig2e (blue curve). We then
group articles within these categories for each field and examine the
variation of their characteristics when averaging across all fields.
We consider characteristics of the articles submitted at various field stages,
and of the authors who submit them. For articles, we focus on the number of
fields reported (article multidisciplinarity), the number of authors (team
size), the number of references made to other arXiv articles, and the number
of citations received within arXiv (article impact). For authors, we consider
their career stage at the time of submitting the article (seniority), the
total number of articles submitted to arXiv (longevity), the number of fields
their articles span during their career and the number of fields per article
(author multidisciplinarity). We average these characteristics over the
article coauthors for which we have a unique identifier (ORCID). In the case
of career stage $s$, we use Eq. 3, where $N_{art}$ is the chronological rank
of the current article across the author’s publications and $N_{tot}$ is the
total number of articles:
$s=\frac{N_{art}-1}{N_{tot}-1}$ (3)
We show in Fig 3 the average values of these features for each stage across
the $72$ fields along with random expectations (see Methods 4.4). In the
context of article metrics (Fig 3a), we find that the early stages of research
fields are characterized by interdisciplinary articles ($2.36$ fields, vs.
$2.05$ for late decay) co-authored by small teams (2 authors vs 4.5). As
fields evolve, we observe a steady growth in the number of references to
earlier arXiv articles, indicating that the community builds on earlier works
in arXiv (Fig. 3 and Fig. S6a when restricting to the same field). Finally, we
find that article impact, measured by the number of citations within arXiv, is
maximal at the Adoption phase before the field has reached its peak. The
citation count observes a similar trend in the case of total citations within
arXiv shown in Fig. 3a as well as citations within arXiv received in the first
five years (Fig. S6b). For author metrics (Fig. 3b), we find that the early
stages of research fields are characterized by multidisciplinary authors (16.9
fields in career for creation vs 7.9 in career for late decay, and 2.13 fields
per article vs 1.91 fields per article), who tend to be in their early career
(8% of total duration vs 60%) with the longest longevity (55 papers vs 27).
Figure 3: Characteristics of articles and authors at different evolutionary
stages. The observed values are averaged over all fields (red bars). Gray bars
correspond to the average field-specific random expectation (see Methods 4.4).
Bottom plots represent the relative difference between observed and random
values. Error bars denote standard errors for observed values (red) and
standard deviation for random values (gray). a Article-centric features:
number of fields reported in the article (multidisciplinarity), number of
authors (team size), number of references made to other arXiv articles, and
number of citations received within arXiv (impact). b Author centric features:
career stage at the time of submitting the article (seniority), total number
of articles submitted to arXiv (longevity), number of fields their articles
span during their career and average number of fields per article
(multidisciplinarity).
### 2.4 Cognitive distance and early innovation
Figure 4: a Co-occurrence network of arXiv field tags. Nodes are colored based
on the major research area they belong to (Physics, Computer Science,
Mathematics, Statistics, Quantiative Finance, Quantitative Biology). Barplots
in b,c follow the same method than in Fig 3. b Average cognitive distance
across the field tags of articles. c Average cognitive distance across all the
field tags used by authors throughout their career.
The previous results show that works submitted in early phases of research
fields tend to mix a larger number of field tags. However, this measure does
not take into consideration the various levels of similarity between fields.
For example, publishing an article within sub-fields of physics is different
than publishing an article mixing quantitative biology, computer science, and
physics. This is rendered apparent when examining the co-occurrence network of
fields across arXiv articles (Fig. 4a). In the co-occurrence network, nodes
represent field tags, and edges represent their co-occurrence across articles.
To define edge weights, we first compute the number of co-occurrences between
two fields across the whole period. We then compute a hypergeometric p-value
that the two fields would have this number of co-occurrences given the number
of times they each have occurred. Lower p-values indicate stronger similarity.
We define the weight $W_{ij}$ between fields $i$ and $j$ as
$-log_{10}(p_{ij})$, where $p_{ij}$ is the hypergeometric p-value. Edges with
$p>0.01$ are finally filtered out. The network represents the landscape of
fields in the arXiv, with closely related fields clustering together into
communities corresponding to 6 broader categories: Physics (purple),
Quantitative biology (gray), Computer Science (green), Mathematics (blue),
Statistics (pink) and Quantitative Finance (orange).
Using this network embedding, we define the Cognitive distance $C_{i,j}$
between field tags $i$ and $j$ as the weighted shortest path
$C_{i,j}=\sum_{e}\frac{1}{W_{e}}$, where $e$ are the edges on the shortest
path between the two tags $i$ and $j$ and $W_{e}$ are their weights in the co-
occurrence network. This cognitive distance allows us to provide a weighted
proxy for interdisciplinarity. In particular, it allows to quantify the
distance between disconnected fields: an example of this is shown in Fig. S8a
where q-fin.ec (Economics) connects to hep-ph (High Energy Physics) by a path
length of $4$.
We use this measure to compute for each article with at least two field tags
the maximum cognitive distance between any pair of tags. We find that articles
published in the early stages of a research field have a significantly larger
cognitive distance, while the measure decays to the random level by peak stage
(Fig 4b). Similarly, for authors we find that in earlier stages authors
publish in cognitively distant fields, which narrows down to similar fields in
later stages (Fig. 4c). The relative difference with random at the creation
stage is more stringent that the previous measure using number of tags
(articles: 0.3 vs 0.1, authors: 0.8 vs 0.1), strengthening our previous
observation.
## 3 Discussion
In this study, we leverage the field annotation of 1.5M articles from the
arXiv preprint repository to explore the scale-invariant patterns in the
evolution of scientific fields and highlight the attributes of articles and
researchers across different evolutionary stages. We show that research fields
follow a right-tailed Gumbel temporal distribution, allowing to rescale their
evolution over a single curve. We demonstrate the usefulness of this approach
by highlighting characteristics shared by articles and authors across the
various stages of a field evolution. We observe that early stages are
characterized by articles written by small teams of early career,
interdisciplinary authors, while late stages exhibit the role of large, more
specialized teams. This supports the general finding that small teams disrupt
while large teams develop science and technology [31, 32]. We find that
maximum impact, measured by citations, is reached before the peak of the field
evolution during the Adoption stage. This may reflect foundational works
underlying the subsequent attractivity of the field and moving it to the
‘peak’ phase. In addition, we observe a steady increase in the within-field
references to earlier work as fields evolve. This suggests a consolidation of
the community over the particular body of work produced in the field, though
further work on the citation and collaboration networks would be needed to
investigate this aspect.
The main contribution of our work is to provide a method to rescale fields and
associate research patterns to standardized evolutionary stages. However, this
study has limitations. First, to capture sufficient data on the rise and fall
patterns of research fields, we limited ourselves to a subset of 72 fields out
of the 175 available. In particular, the choice of keeping only unimodal
fields could be overcome by implementing a simple extension of our approach by
using a mixture model, thereby capturing different “waves” of interest within
a research field. In addition, to avoid ambiguity in author names we focused
only on authors for which we could extract ORCID IDs, limiting the study to a
small and potentially biased subset of authors. Future work should extend such
analyses to larger databases with disambiguated authors and topic annotation
to gain in generality. Finally, while being an open repository, authors
submitting to the arXiv need to be invited by another existing member from the
main field of interest. These create social ‘chaperoning’ constraints [33]
that might influence the type of authors observed at various stages.
Overall, this study contributes to the Science of Science literature by
proposing a simple method to investigate the generic temporal properties of
research fields, and highlighting its use in the context of arXiv. Future work
should be conducted to provide mechanistic models recapitulating the observed
patterns, and extending these analyses to larger datasets. We expect these
insights to be helpful for researchers and policymakers interested in the
emergence and development of research fields and more broadly in the dynamics
of innovation [34].
## 4 Methods
### 4.1 Dataset extraction
We extracted the publication metadata from the arXiv website using the arXiv
API. The data spans years 1986 to 2018, with a total of 1,456,404 articles.
For each article we retrieved the following characteristics: a) the unique
article ID, b) the timestamp of article submission, c) the list of subjects
categories (field tags), d) the citations received within arXiv, e) the
references to other arXiv articles, and f) the list of last names of authors.
We show an example article in Fig. 1a. Furthermore, we extracted when possible
the ORCID IDs of the authors that declared it in arXiv. The number of unique
ORCID IDs was 50,402, allowing to disambiguate these authors’ names.
### 4.2 Fitting procedure
Uni-modality Test: For filtering multi-modal fields we use the $diptest$ R
library to compute the dip unimodality test. We remove fields that fail the
test ($p<0.05$).
Least square optimization: For the selected fields, we strip years before the
first publication to only consider years since first article. We then
constrain the mode of the fitted distribution to coincide with the empirical
one, and we fit the location and scale parameters using least-square
optimization.
### 4.3 Assigning articles and authors to evolutionary stages
We first collect for each field all articles containing the field tag. We
associate each article to the evolutionary stage corresponding to the re-
scaled time obtained for that particular field. We then assign the authors of
each article with an ORCID ID to the corresponding evolutionary stage. Note
that articles with multiple field tags can be assigned to different stages of
evolution corresponding to the re-scaled times of the different tags.
### 4.4 Randomization
The observed features in Fig 3 are compared with random expectation by
shuffling for each field the re-scaled times across articles. This procedure
is repeated 50 times for each field and we compute the average for each stage.
Finally, we compute the average and standard deviation across fields.
## 5 Acknowledgements
Thanks to the Bettencourt Schueller Foundation long term partnership, this
work was partly supported by the CRI Research Fellowship to Marc Santolini.
## References
* [1] Scott Frickel and Neil Gross “A general theory of scientific/intellectual movements” In _American sociological review_ 70.2 Sage Publications Sage CA: Los Angeles, CA, 2005, pp. 204–232
* [2] Uri Shwed and Peter S Bearman “The temporal structure of scientific consensus formation” In _American sociological review_ 75.6 Sage Publications Sage CA: Los Angeles, CA, 2010, pp. 817–840
* [3] Xiaoling Sun et al. “Social dynamics of science” In _Scientific reports_ 3 Nature Publishing Group, 2013, pp. 1069
* [4] David Jurgens et al. “Measuring the evolution of a scientific field through citation frames” In _Transactions of the Association for Computational Linguistics_ 6 MIT Press, 2018, pp. 391–406
* [5] Luís Bettencourt et al. “Population modeling of the emergence and development of scientific fields” In _Scientometrics_ 75.3 Akadémiai Kiadó, co-published with Springer Science+ Business Media BV …, 2008, pp. 495–518
* [6] Hongguang Dong et al. “Allometric scaling in scientific fields” In _Scientometrics_ 112.1 Springer, 2017, pp. 583–594
* [7] Mark Herrera, David C Roberts and Natali Gulbahce “Mapping the evolution of scientific fields” In _PloS one_ 5.5 Public Library of Science, 2010, pp. e10355
* [8] Xiaoling Sun, Kun Ding and Yuan Lin “Mapping the evolution of scientific fields based on cross-field authors” In _Journal of Informetrics_ 10.3 Elsevier, 2016, pp. 750–761
* [9] Christine Balili et al. “TermBall: Tracking and Predicting Evolution Types of Research Topics by Using Knowledge Structures in Scholarly Big Data” In _IEEE Access_ 8 IEEE, 2020, pp. 108514–108529
* [10] Laércio Dias, Martin Gerlach, Joachim Scharloth and Eduardo G Altmann “Using text analysis to quantify the similarity and evolution of scientific disciplines” In _Royal Society open science_ 5.1 The Royal Society Publishing, 2018, pp. 171545
* [11] David Chavalarias and Jean-Philippe Cointet “Phylomemetic patterns in science evolution—the rise and fall of scientific fields” In _PloS one_ 8.2 Public Library of Science, 2013, pp. e54847
* [12] Ye Sun and Vito Latora “The evolution of knowledge within and across fields in modern physics” In _Scientific Reports_ 10.1, 2020, pp. 12097 DOI: 10.1038/s41598-020-68774-w
* [13] Luís MA Bettencourt, David I Kaiser and Jasleen Kaur “Scientific discovery and topological transitions in collaboration networks” In _Journal of Informetrics_ 3.3 Elsevier, 2009, pp. 210–221
* [14] Luis Bettencourt and David I Kaiser “Formation of scientific fields as a universal topological transition” In _arXiv preprint arXiv:1504.00319_ , 2015
* [15] Mauro Dalle Lucca Tosi and Julio Cesar Reis “Understanding the evolution of a scientific field by clustering and visualizing knowledge graphs” In _Journal of Information Science_ SAGE Publications Sage UK: London, England, 2020, pp. 0165551520937915
* [16] Tao Jia, Dashun Wang and Boleslaw K Szymanski “Quantifying patterns of research-interest evolution” In _Nature Human Behaviour_ 1.4 Nature Publishing Group, 2017, pp. 1–7
* [17] An Zeng et al. “Increasing trend of scientists to switch between topics” In _Nature communications_ 10.1 Nature Publishing Group, 2019, pp. 1–11
* [18] Moreno Bonaventura, Vito Latora, Vincenzo Nicosia and Pietro Panzarasa “The advantages of interdisciplinarity in modern science” In _arXiv preprint arXiv:1712.07910_ , 2017
* [19] Thomas S Kuhn “The structure of scientific revolutions” University of Chicago press, 2012
* [20] Andrea Scharnhorst, Katy Börner and Peter Besselaar “Models of science dynamics: Encounters between complexity theory and information sciences” Springer Science & Business Media, 2012
* [21] Everett M Rogers “Diffusion of innovations” SimonSchuster, 2010
* [22] Thomas S Robertson “The process of innovation and the diffusion of innovation” In _Journal of marketing_ 31.1 SAGE Publications Sage CA: Los Angeles, CA, 1967, pp. 14–19
* [23] Elihu Katz, Martin L Levin and Herbert Hamilton “Traditions of research on the diffusion of innovation” In _American sociological review_ JSTOR, 1963, pp. 237–252
* [24] Vincent Larivière et al. “arXiv E-prints and the journal of record: An analysis of roles and relationships” In _Journal of the Association for Information Science and Technology_ 65.6 Wiley Online Library, 2014, pp. 1157–1169
* [25] Ye Sun and Vito Latora “The evolution of knowledge within and across fields in modern physics” In _arXiv preprint arXiv:2001.07199_ , 2020
* [26] Samuel Kotz and Saralees Nadarajah “Extreme value distributions: theory and applications” World Scientific, 2000
* [27] Roberta Sinatra et al. “Quantifying the evolution of individual scientific impact” In _Science_ 354.6312 American Association for the Advancement of Science, 2016
* [28] Ottino-Loffler Bertrand, Jacob G Scott and Steven H Strogatz “Evolutionary dynamics of incubation periods” In _eLife_ 6 eLife Sciences Publications Ltd., 2017
* [29] Aurélien Gautreau, Alain Barrat and Marc Barthelemy “Global disease spread: statistics and estimation of arrival times” In _Journal of theoretical biology_ 251.3 Elsevier, 2008, pp. 509–522
* [30] Emil Julius Gumbel “Statistics of extremes” Columbia university press, 1958
* [31] Staša Milojević “Quantifying the cognitive extent of science” In _Journal of Informetrics_ 9.4 Elsevier, 2015, pp. 962–973
* [32] Lingfei Wu, Dashun Wang and James A Evans “Large teams develop and small teams disrupt science and technology” In _Nature_ 566.7744 Nature Publishing Group, 2019, pp. 378–382
* [33] Vedran Sekara et al. “The chaperone effect in scientific publishing” In _Proceedings of the National Academy of Sciences_ 115.50 National Acad Sciences, 2018, pp. 12603–12607
* [34] Enrico Ubaldi, Raffaella Burioni, Vittorio Loreto and Francesca Tria “Emergence and evolution of social networks through exploration of the Adjacent Possible space” In _Communications Physics_ 4.1 Nature Publishing Group, 2021, pp. 1–12
* [35] Paul Ginsparg “ArXiv at 20” In _Nature_ 476.7359 Nature Publishing Group, 2011, pp. 145–147
## Appendix A Supplementary Information
### A.1 arXiv as a dataset
Paul Ginsparg created arXiv in 1991. It was initially designed for sharing
preprint articles with friends and colleagues [35]. The reasons why
researchers favor uploading their articles on arXiv are diverse. With a low
threshold in the review phase and a minimal time between submission and online
appearance, it provides a fast way for researchers to share their results with
the scientific community. This in turn helps them in getting feedback from the
larger ecosystem and gain intellectual precedence for their claims. The
management team of arXiv follows a strict and systematic procedure ensuring
accurate classification of an article to its subject domain (see Field tags
management). Though arXiv is lenient in its quality control as compared to a
stricter “peer-reviewed” system, an earlier study reports that $\sim 64\%$
arXiv articles end up publishing in WOS (Web Of Science) indexed journals and
many journals also have started accepting arXiv preprint for submissions [24],
supporting the credibility of arXiv articles.
Field tags management \- Users can choose appropriate field tags for their
articles from the existing ones. They, however, cannot create their tags. The
tags assigned by users are then reviewed by moderators of different subject
domains and changed if deemed necessary. New field tags can only be introduced
by the arXiv administration. They do consider proposals from researchers for
introducing new tags and only after considering multiple factors such as the
size of the research community, frequency of articles appearing in the field,
or its impact on arXiv. A recent example of this was the introduction of two
new tags in $2018$: econ.TH and econ.GN, corresponding to Economics Theory and
Economics General. This happened after a community of economists proposed it
to arXiv. However, most of the field tags appeared in the initial years (see
Fig. S2).
Growth rate \- To calculate the growth rate of the arXiv dataset, we consider
the growth function as defined in Eq4, with growth rate $r$:
$N(t)=N_{0}e^{rt}$ (4)
We then fit the cumulative number of articles and number of authors in the
dataset over time as shown in Fig. S1.
Figure S1: a Cumulative number of articles submitted to arXiv in time. b
Cumulative number of (unique) authors. Both number of articles and of authors
grow exponentially with a doubling period of $\sim$ 6 years.
The growth rates $r$ for articles and authors are respectively 0.117 and
0.119. Hence the doubling period i.e $\frac{ln2}{r}$ for articles and authors
is resp. 5.9 and 5.8 years.
Figure S2: Cumulative number of field tags across time, among the $72$
studied. After an initial growth in the early years, the number of unique tags
stays constant.
### A.2 Example of Gumbel distribution
To get more insights on the role of the $\alpha$ and $\beta$ parameters, we
show in Fig. S3 some examples of Gumbel distributions with varying parameters.
The location parameter $\alpha$ corresponds to the peak location, while the
scale parameter $\beta$ corresponds to the distribution width. Fields with a
low $\beta$ have a rapid rise followed by a rapid decay with a long tail.
These could be the fields promoted by sudden advances in science and
technologies or economics, for example, Pricing of Securities in Quantitative
finance (q-fin.pr) (Fig. S4a). On the other hand fields with a large $\beta$
have a gradual rise and fall with a long tail in the decay phase – for example
Condensed Matter Material Sciences (cond-mat.mtrl-sci) (Fig. S4b).
Figure S3: Examples of Gumbel distributions for different location and shape
parameter values. Figure S4: Empirical distribution and Gumbel fits for the
fields of a Quantitative Finance (q-fin.pr) and b Condensed matter material
sciences (cond-mat.mtrl-sci).
### A.3 Fitting the empirical data
Normalizing the Gumbel Distribution Function
Since for each field we only observe a finite sampling period of the full
distribution, we need to normalize the Gumbel distribution between times
$t_{1}$ and $t_{2}$ to improve the fit. Given the Gumbel function $G(x)$, we
find the normalizing constant such that:
$\displaystyle C\int_{t_{1}}^{t_{2}}G(x,\alpha,\beta)dx$ $\displaystyle=1$ (5)
$\displaystyle
C\int_{t_{1}}^{t_{2}}\frac{1}{\beta}e^{\frac{-(x-\alpha)}{\beta}}e^{-e^{\frac{-(x-\alpha)}{\beta}}}dx$
$\displaystyle=1$ (6)
Let $y=e^{-\frac{(x-\alpha)}{\beta}}$ $\implies$
$dy=-\frac{1}{\beta}e^{-\frac{(x-\alpha)}{\beta}}dx$. Replacing above in Eq.6
and adjusting limits we get:
$\displaystyle
C\int_{e^{-\frac{(t_{1}-\alpha)}{\beta}}}^{e^{-\frac{(t_{2}-\alpha)}{\beta}}}-e^{-y}dy$
$\displaystyle=1$ (7) $\displaystyle
Ce^{-y}\Bigr{|}_{e^{-\frac{(t_{1}-\alpha)}{\beta}}}^{e^{-\frac{(t_{2}-\alpha)}{\beta}}}$
$\displaystyle=1$ (8) $\displaystyle
C\left[e^{-{e^{-\frac{(t_{2}-\alpha)}{\beta}}}}-e^{-{e^{-\frac{(t_{1}-\alpha)}{\beta}}}}\right]$
$\displaystyle=1$ (9) $\displaystyle
C=\frac{1}{\left[e^{-{e^{-\frac{(t_{2}-\alpha)}{\beta}}}}-e^{-{e^{-\frac{(t_{1}-\alpha)}{\beta}}}}\right]}$
(10)
With the above $C$ value we can normalize the Gumbel distribution function for
any values of $t_{1}$ and $t_{2}$. Note that when $t_{1}\rightarrow-\infty$
and $t_{2}\rightarrow\infty$ the constant $C\rightarrow 1$.
Figure S5: a Kolmogorov-Smironov test KS values for Gumbel distribution fits.
Lower values indicate better fits. b Corresponding p-values for the KS-test.
Values of $p>0.05$ indicate a plausible fit. c Scatter plot of the fitted vs
empirical values of the temporal distributions for the 72 selected fields. The
Spearman correlation is $\rho=0.81$, with $p<1e-16$. d Same as Fig 2e, in a
log scale. We show both the Gumbel and Gaussian fits. The Gumbel fit provides
a better description of the tails. Figure S6: Same as Fig 3a, for references
(a) and citations (b) within the same field than the article. Citations are
limited to the 5 years following the article.
Figure S7: Article-centric and author-centric properties calculated with the
Gaussian distribution fits. Figure S8: Example of a shortest path linking the
distant fields of Quantitative finance and High Energy Physics in the field
co-occurrence network.
|
††footnotetext: Email addresses<EMAIL_ADDRESS>(John MacQuarrie),
<EMAIL_ADDRESS>(Marlon Stefano)
# Butler’s method applied to ${}_{p}[C_{p}\times C_{p}]$-permutation modules
John W. MacQuarrie Universidade Federal de Minas Gerais, Belo Horizonte, MG,
Brazil Marlon Stefano Universidade Federal de Minas Gerais, Belo Horizonte,
MG, Brazil
###### Abstract
Let $G$ be a finite $p$-group with normal subgroup $N$ of order $p$. The first
author and Zalesskii have previously given a characterization of permutation
modules for ${}_{p}G$ in terms of modules for $G/N$, but the necessity of
their conditions was not known. We apply a correspondence due to Butler to
demonstrate the necessity of the conditions, by exhibiting highly non-trivial
counterexamples to the claim that if both the $N$-invariants and the
$N$-coinvariants of a given lattice $U$ are permutation modules, then so is
$U$.
## 1 Introduction
Let $R$ be a complete discrete valuation ring whose residue field has
characteristic $p$ ($R$ will almost always be p or $\mathbb{F}_{p}$) and let
$G$ be a finite $p$-group. An $RG$-module is a _lattice_ if it is free of
finite $R$-rank. An $RG$-lattice is a _permutation module_ if it possesses an
$R$-basis that is set-wise preserved by multiplication from $G$. Given an
$RG$-module $U$ and a normal subgroup $N$ of $G$, the _$N$ -invariants_ of
$U$, $U^{N}$, is the largest submodule of $U$ on which $N$ acts trivially, and
the _$N$ -coinvariants_ of $U$, $U_{N}$, is the largest quotient module of $U$
on which $N$ acts trivially.
In [MZ20], extending the famous sufficient condition of Weiss [Wei88, Theorem
2] to a characterization, the first author and Pavel Zalesskii proved the
following:
###### Theorem 1.1.
Let $G$ be a finite $p$-group, $U$ a ${}_{p}G$-lattice and $N$ a normal
subgroup of $G$ of order $p$. Then $U$ is a permutation module if, and only if
1. 1.
$U^{N}$ and $U_{N}$ are permutation ${}_{p}[G/N]$-modules and
2. 2.
$(U/U^{N})_{N}$ is a permutation $\mathbb{F}_{p}[G/N]$-module.
Simple examples are given in [MZ20] showing that we cannot remove the
conditions of Part 1. It was however left open whether Condition 2 is
redundant. That is, whether $U$ is a permutation module if, and only if, both
$U^{N}$ and $U_{N}$ are permutation modules. We will show here that Condition
2 cannot be removed, by exhibiting examples of lattices $U$ for
${}_{p}[C_{p}\times C_{p}]$ ($p$ odd) and for ${}_{2}[C_{2}\times C_{2}\times
C_{2}]$ for which both $U^{N}$ and $U_{N}$ are permutation modules, but $U$ is
not. The examples seem to be highly non-trivial.
We believe that the methods used to construct these examples are interesting
in their own right, utilizing in a new way a wonderful correspondence due to
Butler [But74a, But74b], wherein he associates to a lattice what he calls a
“diagram”. For Butler, the idea was to analyze the _category_ of lattices for
a given abelian $p$-group, using the correspondence to compare it with a
category that is better understood. The novelty of our approach is to use
Butler’s correspondence to analyze the structure of a _given_ lattice, by
analyzing how prescribed properties of the lattice affect the corresponding
diagram. In particular, we will characterize the diagrams of those
${}_{p}[C_{p}\times C_{p}]$-lattices $U$ for which $U$, $U^{N}$ and $U_{N}$
are permutation modules. Using these characterizations, we find a diagram
whose corresponding lattice $U$ has the properties that both $U^{N}$ and
$U_{N}$ are permutation modules, but $U$ is not. In the future, we believe
that analyses of this type will yield further insights in the study of
permutation modules.
## 2 Preliminaries
Notations introduced here will be used throughout our discussion. In what
follows, $G$ is always a finite abelian $p$-group and $N$
is a subgroup of $G$ of order $p$.
Given a ${}_{p}G$-lattice $U$, the ${}_{p}[G/N]$-modules of $N$-invariants
$U^{N}$ and $N$-coinvariants $U_{N}$ are, respectively, the largest submodule
and the largest quotient module of $U$ on which $N$ acts trivially. Explicitly
$\displaystyle U^{N}$ $\displaystyle=\\{u\in U\,|\,nu=u\,\,\forall n\in N\\}$
$\displaystyle U_{N}$ $\displaystyle=U/I_{N}U$
where $I_{N}$, the augmentation ideal of ${}_{p}N$, is the ideal generated by
the kernel of the map ${}_{p}N\to{}_{p}$ sending $\sum_{n\in N}\lambda_{n}n$
to $\sum\lambda_{n}$.
By [HR62, Theorem 2.6] there are three isomorphism classes of indecomposable
${}_{p}N$-lattice, being the trivial module p, the free module ${}_{p}N$ and a
non-permutation module $S$ of p-rank $p-1$, which can be described in either
of the following equivalent ways:
$S=I_{N}={}_{p}N/({}_{p}N)^{N}.$
###### Lemma 2.1 ([MZ20, Lemma 8]).
Let $H$ be a cyclic group of order $p$ and $U$ a ${}_{p}H$-lattice. Then
$U_{H}$ is a lattice if, and only if, $U$ is a permutation module.
If ever $H$ is a subgroup of $G$, we use the symbol $\widehat{H}$ to denote
the element $\sum_{h\in H}h$ in ${}_{p}G$.
###### Lemma 2.2.
Let $G=C_{p}\times C_{p}$ and $H,K$ subgroups of $G$. In ${}_{p}G$ we have
$\widehat{H}\cdot\widehat{K}=|H\cap K|\cdot\widehat{HK}$.
###### Proof.
This follows by direct calculations.
∎
## 3 Butler’s Correspondence for ${}_{p}[C_{p}\times C_{p}]$
We describe the parts we require from [But74b] for the special case of
${}_{p}[C_{p}\times C_{p}]$, which is what we require to construct examples
for $p$ odd. The case of $p=2$ must be treated separately, but the methods are
similar and we only present the example, in Section 6.
In what follows, $G=C_{p}\times C_{p}=N\times C$ . We denote as usual by
${}_{p},\mathbb{F}_{p}$ the field of fractions and the residue field of the
$p$-adic integers p, respectively. The group algebra
${}_{p}G={}_{p}\otimes_{{}_{p}}{}_{p}G$ has a decomposition
${}_{p}G=e_{0}{}_{p}G\times\prod_{\begin{subarray}{c}H\leqslant G\\\
|H|=p\end{subarray}}{e_{H}{}_{p}G},$
where the $e_{i}$ are the $p+2$ primitive orthogonal idempotents of ${}_{p}G$,
given as
$e_{0}=\frac{1}{p^{2}}\widehat{G}\,,\quad
e_{H}=\frac{1}{p^{2}}(p\widehat{H}-\widehat{G}).$
Denote by $I$ the indexing set $\\{0,H_{1},\ldots,H_{p+1}\\}$ of the primitive
idempotents, where the $H_{i}$ are the subgroups of $G$ of order $p$. For each
$i\in I$ we denote by $\Lambda_{(i)}$ both the ${}_{p}G$-algebra
$e_{i}{}_{p}G$ and the free $e_{i}{}_{p}G$-module of rank 1iiiwe write
$\Lambda_{(i)}$ rather than $\Lambda_{i}$ to avoid confusion with
coinvariants.. Recall that ${}_{p}G$ is a local algebra, with maximal ideal
$I_{G}+p{{}_{p}G}$.
###### Lemma 3.1.
Let $H,K$ be distinct subgroups of order $p$.
1. 1.
$\widehat{H}\cdot e_{K}=0$,
2. 2.
$\Lambda_{(K)}$ has no non-zero $H$-fixed points,
3. 3.
$J(\Lambda_{(K)})=I_{H}\Lambda_{(K)}$.
###### Proof.
1. 1.
By Lemma 2.2,
$\widehat{H}\cdot
e_{K}=\frac{1}{p^{2}}(p\widehat{H}\widehat{K}-\widehat{H}\widehat{G})=\frac{1}{p^{2}}(p\widehat{G}-p\widehat{G})=0.$
2. 2.
$\widehat{H}$ acts as multiplication by $p$ on a fixed point of
$\Lambda_{(K)}$, and so ${\Lambda_{(K)}}^{H}=0$ by Part 1.
3. 3.
$p\Lambda_{(K)}\subseteq I_{H}\Lambda_{(K)}$ because by Part 1
$pe_{K}=pe_{K}-\widehat{H}e_{K}=\sum_{h\in H}(1-h)e_{K}\in
I_{H}\Lambda_{(K)}.$
∎
###### Definition 3.2.
The ${}_{p}G$-lattice $U$ is _reduced_ if it has neither free direct summands
nor summands isomorphic to any $\Lambda_{i}$.
###### Definition 3.3 ([But74b, §2.2]).
A _diagram_ for $C_{p}\times C_{p}$ is a $(p+3)$-tuple
$V_{*}=(V;V_{(i)}\,,\,i\in I)$, where
* •
$V$ is a finitely generated $\mathbb{F}_{p}[C_{p}\times C_{p}]$-module,
* •
the $V_{(i)}$ are $\mathbb{F}_{p}[C_{p}\times C_{p}]$-submodules of $V$,
* •
for each $j\in I$ we have $\sum_{i\neq j}V_{(i)}=V$,
* •
for each $i\in I$, the induced action of ${}_{p}[C_{p}\times C_{p}]$ on
$V_{(i)}$ factors through the canonical surjection ${}_{p}[C_{p}\times
C_{p}]\to\Lambda_{(i)}$ sending $1$ to $e_{i}$.
A _map of diagrams_ $V_{*}\to V^{\prime}_{*}$ is an
$\mathbb{F}_{p}[C_{p}\times C_{p}]$-module homomorphism $\rho:V\to V^{\prime}$
such that $\rho(V_{(i)})\subseteq V^{\prime}_{(i)}$ for each $i\in I$.
We make explicit how one obtains a diagram from a reduced lattice, and vice-
versa, fixing as we do some notation that will remain in force throughout the
rest of the article. As we follow Butler’s arguments in [But74b], simply
specialising to the case of ${}_{p}[C_{p}\times C_{p}]$ (which simplifies
things slightly), we don’t give any justifications.
* •
Getting a diagram from a reduced lattice: Let $U$ be a reduced lattice. We
regard $U$ as the sublattice $1\otimes U$ of the ${}_{p}G$-module
${}_{p}U={}_{p}\otimes_{{}_{p}}U$. Defining for each $i\in I$ the submodule
$U_{(i)}:=e_{i}U$ of ${}_{p}U$, the sum
$U_{*}:=\sum_{i\in I}U_{(i)}$
is direct, and since $U$ is reduced we have
$pU_{*}\subsetneq U\subsetneq U_{*}.$
Defining
$V=U_{*}/U\,,\quad V_{(i)}=(U_{(i)}+U)/U,$
the tuple $V_{*}=(V;V_{(i)}\,,\,i\in I)$ is the diagram corresponding to $U$.
* •
Getting a reduced lattice from a diagram: Let $V_{*}=(V;V_{(i)}\,,\,i\in I)$
be a diagram. For each $i\in I$, let $F_{(i)}$ be the direct sum of
$\textnormal{dim}_{\mathbb{F}_{p}}(V_{(i)}/\textnormal{Rad}(V_{(i)}))$ copies
of $\Lambda_{(i)}$ and denote by $F$ the direct sum $\bigoplus_{i\in
I}F_{(i)}$. For each $i$ let $f_{i}:F_{(i)}\to V_{(i)}$ be a surjective
homomorphism of $\Lambda_{(i)}$-modules inducing an isomorphism
$F_{(i)}/\textnormal{Rad}(F_{(i)})\to V_{(i)}/\textnormal{Rad}(V_{(i)})$. We
thus obtain for each $i$ a map
$F_{(i)}\xrightarrow{f_{i}}V_{(i)}\hookrightarrow V$ and hence by summing
these maps, a map $f:F\to V$. The reduced lattice corresponding to $V_{*}$ is
$\textnormal{Ker}(f)$.
These processes are mutually inverse in the sense that if $U^{\prime}$ is the
reduced lattice obtained from the diagram of $U$, then $U^{\prime}\cong U$ as
${}_{p}G$-lattices, and if $V^{\prime}_{*}$ is the diagram coming from the
reduced lattice associated to $V_{*}$, then $V^{\prime}_{*}\cong V_{*}$.
It follows that given a reduced lattice $U$, we may associate a unique (up to
isomorphism) diagram $V_{*}$ and from this, the modules $F_{(i)},F$, with $U$
being considered as a submodule of $F$. We will use these notations freely in
what follows.
## 4 Characterizations using diagrams
Throughout this section, $U$ is a reduced ${}_{p}G$-lattice with diagram
$V_{*}=(V;V_{(i)}\,,\,i\in I)$ and $F_{(i)},F$ are the corresponding modules
defined in the previous section. Given an element $x\in F$, we denote by
$\overline{x}$ its image in $V$. Throughout, the letters $H,K$ will always
refer to subgroups of order $p$. If an element of $\bigoplus F_{(i)}$ is
written like $x=\sum_{H\neq N}x_{H}$, this should be taken to mean that
$x\in\bigoplus_{\begin{subarray}{c}H\leqslant G\\\ |H|=p\\\ H\neq
N\end{subarray}}F_{(H)}$ with each $x_{H}\in F_{(H)}$, and similarly in $V$.
###### Theorem 4.1.
The reduced lattice $U$ is a permutation module if, and only if, $G$ acts
trivially on each $V_{(i)}$ and
$V=V_{(0)}=\bigoplus_{\begin{subarray}{c}H\leqslant G\\\
|H|=p\end{subarray}}V_{(H)}$.
###### Proof.
A direct calculation shows that the reduced permutation lattice
${}_{p}[G/H]=\langle u\rangle$ has diagram
$(\mathbb{F}_{p};\mathbb{F}_{p},0,\ldots,0,\mathbb{F}_{p},0,\ldots,0)$
where the last non-zero entry occurs in the position indexed by $H$: taking
$K$ to be any subgroup of order $p$ different from $H$ we have by Lemma 2.2:
$\displaystyle e_{0}\cdot u$
$\displaystyle=\frac{1}{p^{2}}\widehat{G}u=\frac{1}{p^{2}}\widehat{K}\widehat{H}u=\frac{1}{p}\widehat{K}u$
$\displaystyle e_{H}\cdot u$
$\displaystyle=\frac{1}{p^{2}}(p\widehat{H}-\widehat{K}\widehat{H})u=\frac{1}{p}(p\widehat{1}-\widehat{K})u$
$\displaystyle e_{K}\cdot u$
$\displaystyle=\frac{1}{p^{2}}(p\widehat{K}-\widehat{H}\widehat{K})u=\frac{1}{p^{2}}(p\widehat{1}-\widehat{H})\widehat{K}u=\frac{1}{p^{2}}(p\widehat{1}-p\widehat{1})\widehat{K}u=0.$
Since $e_{0}\cdot u+e_{H}\cdot u=u\in U$, $U_{*}/U$ has dimension 1 and the
claim follows. A reduced permutation module is a direct sum of modules
${}_{p}[G/H]$ for different $H$ of order $p$, and the diagram of a direct sum
is the direct sum of the diagrams. The result follows. ∎
###### Lemma 4.2.
1. 1.
$U^{N}=U\cap(F_{(0)}+F_{(N)})$
2. 2.
For any subgroup $H\neq N$ of order $p$ we have $I_{H}U^{N}\subseteq U\cap
F_{(N)}$, with equality if, and only if, $U^{N}$ is a permutation module.
###### Proof.
1. 1.
$N$ acts trivially on $F_{(0)},F_{(N)}$, while $F_{(H)}$ has no $N$-fixed
points for $H\neq N$ by Lemma 3.1. Part 1 now follows because $F$ is the
direct sum of the $F_{(i)}$.
2. 2.
The inclusion follows from Part 1 because $I_{H}F_{(0)}=0$. For any $u\in
U\cap F_{(N)}$ we have $\widehat{H}u=\widehat{H}e_{N}u=0$ by Lemma 3.1 and so
$pu=pu-\widehat{H}u=\sum_{h\in H}(1-h)u\in I_{H}U^{N}.$
If $U^{N}$ is a permutation module, then $U^{N}/I_{H}U^{N}$ is a lattice by
Lemma 2.1 and so $u\in I_{H}U^{N}$. On the other hand, if $I_{H}U^{N}=U\cap
F_{(N)}$ then
$U^{N}/I_{H}U^{N}=U^{N}/(U^{N}\cap F_{(N)}),$
which is a lattice since $F_{(N)}$ is a summand of $F$, and so $U^{N}$ is a
permutation module by Lemma 2.1 again.
∎
###### Theorem 4.3.
The lattice $U^{N}$ is a permutation module if, and only if,
${V_{(N)}}^{G}\subseteq V_{(0)}$.
###### Proof.
Suppose first that $U^{N}$ is a permutation module and fix
$\overline{x}\in{V_{(N)}}^{G}$, with lift $x\in F_{N}$. Since
$(1-c)\overline{x}=0$, we have $(1-c)x\in U\cap F_{(N)}$. But by Lemma 4.2,
$U\cap F_{(N)}=I_{C}U^{N}$ so by Part 1 of Lemma 4.2 there are $x_{0}\in
F_{(0)},x_{N}\in F_{(N)}$ with $x_{0}+x_{N}\in U$ and
$(1-c)x=(1-c)(x_{0}+x_{N})=(1-c)x_{N}$. As $(1-c)(x-x_{N})=0$ and $F_{(N)}$
has no non-zero $C$-fixed points, $x=x_{N}$, so that $x_{0}+x\in U$. Hence
$\overline{x}=-\overline{x_{0}}\in V_{(0)}$, as required.
Suppose now that ${V_{(N)}}^{G}\subseteq V_{(0)}$. To conclude that $U^{N}$ is
a permutation module, it is enough to check that $U\cap F_{(N)}\subseteq
I_{C}U^{N}$, by Lemma 4.2. So fix $y\in U\cap F_{(N)}$. Since $U$ is reduced,
$y\in\textnormal{Rad}(F_{(N)})=I_{C}F_{(N)}$, by Lemma 3.1, so we may write
$y=(1-c)x$ with $x\in F_{(N)}$. Since $(1-c)\overline{x}=0$,
$\overline{x}\in{V_{(N)}}^{G}$, so in $V_{(0)}$ by hypothesis. Thus there is
$x_{0}\in F_{(0)}$ with $\overline{x_{0}}=\overline{x}$, hence $x-x_{0}\in
U^{N}$ and so $y=(1-c)(x-x_{0})\in I_{C}U^{N}$ as required. ∎
###### Lemma 4.4.
We have
$I_{N}U\subseteq U\cap\Big{(}\bigoplus_{H\neq N}F_{(H)}\Big{)}$
with equality if, and only if, $U_{N}$ is a lattice.
###### Proof.
The inclusion is clear because $(1-n)(F_{(0)}+F_{(N)})=0$. Suppose that
$U_{N}$ is a lattice and take $x$ in the right hand set. Since
$\widehat{N}x=0$ by Lemma 3.1, $px=\sum_{n\in N}(1-n)x\in I_{N}U$ and hence
$x\in I_{N}U$ since $U_{N}$ is a lattice. On the other hand, the right hand
set is a pure submodule of $U$ because $\bigoplus_{H\neq N}F_{(H)}$ is a
summand of $F$, and hence if this set is equal to $I_{N}U$ then $U_{N}$ is a
lattice. ∎
We need a technical lemma:
###### Lemma 4.5.
Fix $a\in I$ and $B\subseteq I\backslash\\{a\\}$. The homomorphism
$\displaystyle\varphi:U\cap\bigoplus_{i\in B\cup\\{a\\}}F_{(i)}$
$\displaystyle\to V_{(a)}\cap\sum_{i\in B}V_{(i)}$ $\displaystyle\sum
x_{i}\quad$ $\displaystyle\mapsto\quad\overline{x_{a}}$
is well-defined and surjective.
###### Proof.
The element $\overline{x_{a}}$ is in $V_{(a)}\cap\sum_{i\in B}V_{(i)}$ because
$\sum x_{i}\in U$, and hence $\overline{x_{a}}=-\sum_{i\in
B}\overline{x_{i}}$. Given $v\in V_{(a)}\cap\sum_{i\in B}V_{(i)}$, write
$v=-\sum_{i\in B}v_{i}$ and lift $v$ to $x_{a}$ in $F_{(a)}$ and each $v_{i}$
to $x_{i}$ in $F_{(i)}$. The element $\sum_{i\in B\cup\\{a\\}}x_{i}\in
U\cap\bigoplus_{i\in B\cup\\{a\\}}F_{(i)}$ and maps to $v$ by $\varphi$. ∎
Recall that by Lemma 2.1, $U_{N}$ is a lattice if, and only if, the
restriction $U\downarrow_{{N}}$ of $U$ to $N$ is a permutation module. Thus
the following theorem is also a characterization of those reduced $U$ for
which $U\downarrow_{{N}}$ is a permutation module.
###### Theorem 4.6.
The module $U_{N}$ is a lattice if, and only if
1. 1.
For each $H\neq N$ of order $p$, $V_{(H)}\cap(\sum_{K\neq
N,H}V_{(K)})\subseteq I_{N}V_{(H)}$, and
2. 2.
$V^{N}\subseteq V_{(0)}+V_{(N)}$.
###### Proof.
Suppose first that $U_{N}$ is a lattice. We check the first condition, so fix
an element $v\in V_{(H)}\cap(\sum_{K\neq N,H}V_{(K)})$. By Lemma 4.5 there is
$u=\sum_{K\neq N}u_{K}\in U\cap\bigoplus_{K\neq N}F_{(K)}$ such that
$\overline{u_{H}}=v$. By Lemma 4.4, there is $u^{\prime}=\sum_{i\in
I}u^{\prime}_{i}\in U$ such that $u=(1-n)u^{\prime}=\sum(1-n)u^{\prime}_{i}$.
As the sum of the $F_{(i)}$ is direct, we have $u_{H}=(1-n)u_{H}^{\prime}$ and
hence
$v=\overline{u_{H}}=(1-n)\overline{u_{H}^{\prime}}\in I_{N}V_{(H)}.$
To show the second condition, it is enough to confirm that $(\sum_{H\neq
N}V_{(H)})^{N}\subseteq V_{(0)}+V_{(N)}$ so fix $v=\sum_{H\neq
N}\overline{x_{H}}\in V^{N}$ and a lift $\sum_{H\neq N}x_{H}$ of $v$. Then
$(1-n)\sum_{H\neq N}x_{H}\in U\cap\bigoplus_{H\neq N}F_{(H)}$, which is
$I_{N}U$ by Lemma 4.4 and so
$(1-n)\sum_{H\neq N}x_{H}=(1-n)\sum_{i\in I}y_{i}$
with $\sum y_{i}\in U$. As $F_{(H)}$ has no non-zero $N$-fixed points,
$x_{H}=y_{H}$ for each $H\neq N$. Hence
$V_{(0)}+V_{(N)}\ni-(\overline{y_{0}}+\overline{y_{N}})=\sum_{H\neq
N}\overline{y_{H}}=\sum_{H\neq N}\overline{x_{H}}=v.$
Suppose conversely that Conditions $1$ and $2$ hold for $U$. We will check the
conditions of Lemma 4.4. Given $u=\sum x_{H}\in U\cap\bigoplus_{H\neq
N}F_{(H)}$, Lemma 4.5 and Condition 1 imply that $\overline{x_{H}}\in
I_{N}V_{(H)}=\textnormal{Rad}(V_{(H)})$ for each $H\neq N$ and hence, since
$F_{(H)}/\textnormal{Rad}(F_{(H)})\to V_{(H)}/\textnormal{Rad}(V_{(H)})$ is an
isomorphism, $x_{H}\in\textnormal{Rad}(F_{(H)})=I_{N}F_{(H)}$. For each $H\neq
N$, write $x_{H}=(1-n)y_{H}$ for some $y_{H}\in F_{(H)}$. Then
$(1-n)\sum_{H\neq N}\overline{y_{H}}=0$ so that $\sum_{H\neq
N}\overline{y_{H}}\in V^{N}$ and hence
$\sum_{H\neq N}\overline{y_{H}}=-\overline{y_{0}}-\overline{y_{N}}\in
V_{(0)}+V_{(N)}$
for some $y_{0}\in F_{(0)}$ and $y_{N}\in F_{(N)}$, by Condition 2. Thus
$y=\sum_{i\in I}y_{i}\in U$ and
$I_{N}U\ni(1-n)y=\sum_{H\neq N}x_{H}=u.$
∎
###### Lemma 4.7.
Let $U$ be a reduced lattice such that $U_{N}$ is a lattice. Then
$I_{G}U\subseteq U\cap\Big{(}\bigoplus_{H}F_{(H)}\Big{)}$
with equality if, and only if, $U_{N}$ is a permutation module.
###### Proof.
The inclusion is clear because $(1-g)F_{(0)}=0$ for every $g\in G$. Since
$U_{N}$ is a lattice, Lemma 2.1 says that $(U_{N})_{C}=U/I_{G}U$ is a lattice
if, and only if, $U_{N}$ is a permutation module. Suppose that $U_{N}$ is a
permutation module and fix $u=\sum_{H}x_{H}\in U\cap\bigoplus_{H}F_{(H)}$. By
Lemma 2.2, $(p\cdot 1-\widehat{K})u=\sum_{H\neq K}px_{H}\in I_{G}U$, and
hence, summing over the subgroups of order $p$ we get that
$\sum_{H}p^{p}x_{H}=p^{p}u\in I_{G}U$, and hence $u\in I_{G}U$ since
$U/I_{G}U$ is a lattice. On the other hand, the right hand submodule is pure
in $U$ because $\bigoplus_{H}F_{(H)}$ is a summand of $F$, so if the sets are
equal then $I_{G}U$ is pure in $U$ and $U_{N}$ is a permutation module. ∎
###### Remark 4.8.
Lemma 4.7 is false without the condition that $U_{N}$ be a lattice: the given
equality applies for the ${}_{2}[C_{2}\times C_{2}]$-lattice $U$ whose diagram
has
$V=\mathbb{F}_{2}\oplus\mathbb{F}_{2},V_{(0)}=\langle(1,1)\rangle,V_{(N)}=0,V_{(C)}=\langle(1,0)\rangle$
and $V_{(\langle nc\rangle)}=\langle(0,1)\rangle$. Here $U_{N}$ is not a
lattice.
###### Theorem 4.9.
Let $U$ be a reduced lattice such that $U_{N}$ is a lattice. Then $U_{N}$ is a
permutation module if, and only if $V_{(H)}\cap\sum_{K\neq H}V_{(K)}\subseteq
I_{G}V_{(H)}$ for every subgroup $H$ of order $p$.
###### Proof.
Suppose that $U_{N}$ is a permutation module. Given an element
$\overline{x_{H}}=\sum_{K\neq H}\overline{x_{K}}$ in $V_{(H)}\cap\sum_{K\neq
H}V_{(K)}$, then $x_{H}-\sum_{K\neq H}x_{K}\in
U\cap\Big{(}\bigoplus_{H}F_{(H)}\Big{)}$, hence in $I_{G}U\subseteq I_{G}F$ by
Lemma 4.7. Thus $x_{H}\in I_{G}F_{(H)}$ and hence $\overline{x_{H}}\in
I_{G}V_{(H)}$.
Suppose now that $V_{(H)}\cap\sum_{K\neq H}V_{(K)}\subseteq I_{G}V_{(H)}$ for
every $H$. By Lemma 4.7, to prove that $U_{N}$ is a permutation module, it is
enough to consider an element $u\in U\cap\bigoplus_{K}F_{(K)}$ and check that
it is in $I_{G}U$. The given inclusion implies that
$U\cap(\bigoplus_{K}F_{(K)})\subseteq I_{G}(\bigoplus_{K}F_{(K)})$, so we can
write
$u=(1-c)y_{N}-(1-n)\sum_{K\neq N}y_{K},$
so that $(1-c)\overline{y_{N}}=(1-n)\sum_{K\neq N}\overline{y_{K}}$. But by
the definition of a diagram, we have $V=V_{(0)}+\sum_{K\neq N}V_{(K)}$ and so
we can write $\overline{y_{N}}=\overline{x_{0}}+\overline{x_{C}}+\sum_{K\neq
N,C}\overline{x_{K}}$, and hence
$(1-c)\overline{y_{N}}=(1-c)\sum_{K\neq N,C}\overline{x_{K}}.$
Thus the element $w:=(1-c)y_{N}-(1-c)\sum_{K\neq N,C}x_{K}$ is in $U$, and
indeed in $I_{C}U$ because $y_{N}-\sum_{K\neq N,C}x_{K}-x_{0}-x_{C}\in U$ and
$w=(1-c)(y_{N}-x_{0}-x_{C}-\sum_{K\neq N,C}x_{K}).$
On the other hand, since
$(1-n)\sum_{K\neq N}\overline{y_{K}}=(1-c)\overline{y_{N}}=(1-c)\sum_{K\neq
N,C}\overline{x_{K}},$
the element $w^{\prime}:=(1-n)\sum_{K\neq N}y_{K}-(1-c)\sum_{K\neq
N,C}x_{K}\in U\cap\bigoplus_{K\neq N}F_{(K)}$, so $w^{\prime}\in I_{N}U$ by
Lemma 4.4 since $U_{N}$ is a lattice. It follows that $u=w-w^{\prime}\in
I_{C}U+I_{N}U=I_{G}U$, as required. ∎
## 5 Examples for $C_{p}\times C_{p}$
If $p=2$, it follows from Theorem 1.1 that a ${}_{2}[C_{2}\times
C_{2}]$-lattice $U$ is a permutation module if, and only if, both $U^{N}$ and
$U_{N}$ are permutation modules, because every $\mathbb{F}_{2}[G/N]$-module is
a permutation module so that Condition 2 is automatically satisfied. But this
is not the case for odd primes. Here we present a counterexample for each odd
prime $p$, using the results from the previous section.
###### Example 5.1.
Fix an odd prime $p$ and let $G=C_{p}\times C_{p}=\langle
n\rangle\times\langle c\rangle$. We construct a diagram for ${}_{p}G$. Let $V$
be the $\mathbb{F}_{p}[C_{p}\times C_{p}]$-module with $\mathbb{F}_{p}$-basis
$v_{1},\ldots,v_{p+2}$ and action from $G$ given as follows:
$n\cdot v_{1}=v_{1},\quad n\cdot v_{2}=v_{2},\quad n\cdot
v_{j}=v_{1}+v_{j}\quad\forall j\geqslant 3,$ $c\cdot v_{1}=v_{1},\quad c\cdot
v_{2}=v_{1}+v_{2},\quad c\cdot v_{3}=v_{3},\quad c\cdot
v_{j}=-k_{j}v_{1}+v_{j}\quad\forall j\geqslant 4,$
where $k_{j}\in\\{1,\ldots,p-1\\}$ is such that $(j-3)k_{j}=1$ modulo $p$. The
submodules $V_{(i)}$ are defined as follows:
$\displaystyle V_{(0)}$ $\displaystyle=\langle
v_{1},\,v_{2}-v_{3}+v_{4},\,(k_{j+1}-k_{j})v_{2}-v_{j}+v_{j+1}\hbox{ for
}4\leqslant j\leqslant p+1\rangle,$ $\displaystyle V_{(N)}$
$\displaystyle=\langle v_{1},v_{2}\rangle,$ $\displaystyle V_{(C)}$
$\displaystyle=\langle v_{1},v_{3}\rangle,$ $\displaystyle V_{(\langle
nc^{a}\rangle)}$ $\displaystyle=\langle v_{1},v_{a+3}\rangle,\qquad
a\in\\{1,\ldots,p-1\\}$
The tuple $(V;V_{(i)}\,,\,i\in I)$ is a diagram, with corresponding reduced
lattice $U$. Then
* •
$U^{N}$ is a permutation module by Theorem 4.3, because ${V_{(N)}}^{G}=\langle
v_{1}\rangle\subseteq V_{(0)}$.
* •
$U_{N}$ is a lattice by Theorem 4.6: the first condition is satisfied because
$V_{(H)}\cap\sum_{K\neq N,H}V_{(K)}=\langle v_{1}\rangle$ and
$v_{1}=(n-1)v_{j}$ for any $j\geqslant 3$. The second condition is satisfied
because the element $\sum_{j=1}^{p+2}\lambda_{j}v_{j}\in V^{N}$ if, and only
if, $p\mid\sum_{j=3}^{p+2}\lambda_{j}$, and hence $V^{N}$ is generated by
$v_{1},v_{2}$ and the elements $v_{j}-v_{j+1}$ for $j\in\\{3,\ldots,p+1\\}$,
which are in $V_{(0)}+V_{(N)}$.
* •
$U_{N}$ is a permutation module by Theorem 4.9: $V_{(H)}\cap\sum_{K\neq
H}V_{(K)}=\langle v_{1}\rangle$ which is in $I_{N}V_{(H)}$ if $H\neq N$ and,
since $v_{1}=(c-1)v_{2}$, is in $I_{C}V_{(H)}$ if $H=N$.
* •
But $U$ is not a permutation module by Theorem 4.1, since $G$ does not act
trivially on $V$.
The lattice $U$ has p-rank $p^{2}+p-1$. In the smallest case, when $p=3$, it
has 3-rank $11$ and with respect to some 3-basis of $U$, the matrices of the
actions of $n$ and $c$ respectively, are
$\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&0&-1&-1&-1&-1&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&-2&0&0&-1&1&0&1&0&0\\\
0&0&0&-2&0&-1&-1&0&0&1&0\\\ 0&0&0&0&-2&-1&0&0&0&0&1\\\
0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0\\\
0&0&-3&0&0&-1&1&0&1&0&0\\\ 0&0&0&-3&0&-1&-1&0&0&1&0\\\
0&0&0&0&-3&-1&0&0&0&0&1\end{pmatrix}\hbox{ and
}\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&-1&0&1&-1&0&0&0&0&-1&0\\\
0&-2&0&0&0&0&-1&1&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&-1&0\\\
0&0&0&0&-2&-1&0&0&0&0&1\\\ 0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0\\\
0&-3&0&0&0&0&-1&1&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&3&0&1&1&0&0&-2&0\\\
0&0&0&0&-3&-1&0&0&0&0&1\end{pmatrix}.$
## 6 Example for $C_{2}\times C_{2}\times C_{2}$
A similar application of Butler’s Correspondence to the group $G=C_{2}\times
C_{2}\times C_{2}=\langle n\rangle\times\langle b\rangle\times\langle
c\rangle$ yields the following lattice $U$ for which $U^{N}$ and $U_{N}$ are
permutation modules, but $U$ is not. The matrices of $n,b,c$ respectively,
are:
$\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0\\\
0&1&0&0&1&0&0&1&1&1&1\\\ 0&1&0&0&0&1&0&1&1&1&1\\\
0&-1&0&0&0&0&1&-1&-1&-1&-1\\\ 0&-1&0&0&0&0&0&-1&0&0&0\\\
0&0&-1&1&0&0&0&0&-1&0&0\\\ 0&-1&1&0&0&0&0&0&0&-1&0\\\
0&0&0&-1&0&0&0&0&0&0&-1\end{pmatrix}\,,\,\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&1&0&1&0&0&0&1&1\\\ 0&1&-1&0&1&0&0&0&0&1&1\\\
1&0&1&-1&-1&-1&-1&0&0&-1&-1\\\ 0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&1&0&0\\\ 0&-1&1&0&0&0&0&0&0&-1&0\\\
0&0&0&-1&0&0&0&0&0&0&-1\end{pmatrix}\,,\,\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&1&0&0&1&-1&0&-1&0\\\ 1&1&0&0&-1&-1&-1&-1&0&-1&0\\\
0&0&0&-1&1&0&0&1&0&1&0\\\ 0&0&0&0&0&0&0&1&0&0&0\\\ 0&0&-1&1&0&0&0&0&-1&0&0\\\
0&0&0&0&0&0&0&0&0&1&0\\\ 0&0&0&-1&0&0&0&0&0&0&-1\end{pmatrix}.$
## References
* [But74a] M. C. R. Butler. The $2$-adic representations of Klein’s four group. In Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973), pages 197–203. Lecture Notes in Math., Vol. 372, 1974.
* [But74b] M. C. R. Butler. On the classification of local integral representations of finite abelian $p$-groups. In Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), Paper No. 6, pages 18 pp. Carleton Math. Lecture Notes, No. 9, 1974.
* [HR62] A. Heller and I. Reiner. Representations of cyclic groups in rings of integers. I. Ann. of Math. (2), 76:73–92, 1962.
* [MZ20] John W. MacQuarrie and Pavel A. Zalesskii. A characterization of permutation modules extending a theorem of Weiss. Doc. Math., 25:1159–1169, 2020.
* [Wei88] Alfred Weiss. Rigidity of $p$-adic $p$-torsion. Ann. of Math. (2), 127(2):317–332, 1988.
|
representable if and only if $G$ is cyclic.
###### Proof.
One direction is clear, so we assume $G$ is freely representable. So by the
above theorem, the order of the kernel $K$ of the action homomorphism
$B\to\mathrm{Aut}(A)$ must be divisible by exactly the primes that divide the
order of $B$. This means that $K=B$, so $G=AB=A\times B$ is cyclic. ∎
We can restate the above lemma in terms $\mu(G)$:
###### Corollary 77.
Let $G$ be a freely representable Sylow-cyclic group, and let $\mu(G)$ be the
MCC subgroup of $G$. Then every element of $G$ of prime order is in $\mu(G)$.
In particular, every prime dividing the order of $G$ must divide the order of
$\mu(G)$.
###### Proof.
Write $G$ as $AB=A\rtimes B$ where $A$ and $B$ are cyclic subgroups of $G$ of
relatively prime orders. Let $g=ab$ be an element of prime order where $a\in
A$ and $b\in B$. Under the projection $A\rtimes B\to B$ the element $g$ maps
to $b$, so $b$ is of prime order or is the trivial element. In either case, by
the above lemma, $b\in K$ where $K$ is the kernel of the action map
$B\to\mathrm{Aut}(A)$. So $g=ab\in AK$. But $AK=\mu(G)$ (Lemma 71) so
$g\in\mu(G)$. ∎
This gives a necessary condition for a Sylow-cyclic group to be freely
representable. In order to show it is sufficient we use induced
representations, but in a very basic manner. The following, which we take as
given, is all that we need to know about induced representations here:
###### Proposition 78.
Let $G$ be a finite group with subgroup $H$, and let $F$ be a field. Suppose
that $H$ acts linearly on an $F$-vector space $W$. Then there is a linear
action of $G$ on an $F$-vector space $V$ containing $W$ such that (1) the
action of $G$ on $V$ restricts to the given action of $H$ on $W$, (2) if
$g_{1}H,\ldots,g_{k}H$ are the distinct left cosets in $G/H$ then the vector
space $W$ is the direct sum of the spaces $g_{i}W$:
$V=\bigoplus g_{i}W.$
This representation, called the induced representation, is unique up to a
$F[G]$-module isomorphism fixing $W$.
###### Proposition 79.
Let $G$ be a finite group with a subgroup $H$ that contains all elements of
$G$ of prime order. Suppose $W$ is an $F$-vector space with a linear
representation of $H$ on $W$, and suppose $V$ is a $F$-vector space containing
$W$ with a representation of $G$ induced by the representation of $H$ on $W$.
If the linear representation of $H$ on $W$ is a free linear representation,
then the linear representation of $G$ on $V$ is also a free linear
representation.
###### Proof.
Let $g_{1}H,\ldots,g_{k}H$ be the distinct left cosets in $G/H$. Suppose
$\sigma\in G$ is not the identity and that $\sigma(v)=v$ where $v\in V$ is
equal to
$v=g_{1}w_{1}+\ldots+g_{k}w_{k}$
where $w_{i}\in W$. Let $m>1$ be the order of $\sigma$ and let $p$ be a prime
dividing $m$. Then $\tau=\sigma^{m/p}$ has order $p$ and also fixes $v$. By
assumption $\tau\in H$. Observe that the conjugate $\tau_{i}=g_{i}^{-1}\tau
g_{i}$ also has order $p$ so is in $H$. Since $\tau g_{i}=g_{i}\tau_{i}$
$\tau v=\tau g_{1}w_{1}+\ldots+\tau
g_{k}w_{k}=g_{1}(\tau_{1}w_{1})+\ldots+g_{k}(\tau_{k}w_{k}).$
Since $\tau v=v$,
$g_{1}(\tau_{1}w_{1})+\ldots+g_{k}(\tau_{k}w_{k})=g_{1}w_{1}+\ldots+g_{k}w_{k}.$
By the direct sum property of induced representations,
$g_{i}(\tau_{i}w_{i})=g_{i}w_{i}$
for each $1\leq i\leq k$. Multiplying by the inverse of $g_{i}$ gives
$\tau_{i}w_{i}=w_{i}$ for the induced representation. Since $\tau_{i}\in H$
and $w_{i}\in W$, we have $\tau_{i}w_{i}=w_{i}$ in the original representation
of $H$ on $W$. This representation is a free linear representation by
assumption, so $w_{i}=0$ for each $i$. This implies $v=0$, showing that the
induced representation is a free linear representation. ∎
###### Corollary 80.
Let $G$ be a finite group with freely representable subgroup $H$. If $H$
contains all elements of $G$ of prime order, then $G$ is also freely
representable.
Now we are ready for the main theorem:
###### Theorem 81.
Let $G$ be a Sylow-cyclic group and let $\mu(G)$ be its maximal characteristic
cyclic subgroup. The following are equivalent:
1. 1.
$G$ is freely representable.
2. 2.
Every prime dividing the order of $G$ also divides the order of $\mu(G)$.
3. 3.
For every prime $p$ dividing the order of $G$ there is a unique subgroup of
$G$ of order $p$.
###### Proof.
Suppose $G$ is freely representable. Then by Corollary 77 every prime dividing
the order of $G$ must divide the order of $\mu(G)$. So $(1)\implies(2)$.
Now assume $(2)$. If $p$ divides $|G|$ then, by assumption, $p$ divides
$|\mu(G)|$. Thus there is a unique subgroup of $G$ of order $p$ by Corollary
64. So $(2)\implies(3)$ holds.
Finally assume $(3)$. Let $g\in G$ be an element of prime $p$ order. By
assumption the subgroup $\left<g\right>$ is the unique subgroup of $G$ of
order $p$. This implies that $\left<g\right>$ is a characteristic cyclic
subgroup, and so $\left<g\right>$ is contained in $\mu(G)$. Since $\mu(G)$ is
cyclic, it is freely representable. Thus $G$ is freely representable by
Corollary 80. ∎
We can now strengthen Lemma 75.
###### Corollary 82.
Let $A$ and $B$ be cyclic subgroups of $G$ of relatively prime orders such
that $G=A\rtimes B$ (so $A$ is normal in $G$). Let $K$ be the kernel of the
action homomorphism $B\to{\mathrm{Aut}}(A)$ associated to the semidirect
product. Then $G$ is freely representable if and only if every prime dividing
$|B|$ divides $|K|$.
###### Proof.
Recall that $\mu(G)$ is $AK$ (Lemma 71).
If every prime dividing $|B|$ divides $|K|$ then every prime dividing the
order of $G$ must divide $|A|$ or $|K|$. Hence every prime dividing the order
of $G$ divides the order of $\mu(G)$, and so $G$ is freely representable by
the above theorem.
Conversely, if $G$ is freely representable, then every prime $p$ dividing the
order of $B$ must divide the order of $\mu(G)=AK\cong A\times K$ by the above
theorem. But $p$ does not divide the order of $A$, so $p$ divides the order of
$K$ as desired. ∎
###### Example 11.
Suppose $A$ a cyclic group of odd prime order $p$. Let $q$ be any prime
dividing $p-1$. Let $B$ be a cyclic group of order $q^{k}$ with $k>1$. We can
identify $\mathrm{Aut}(A)$ with ${\mathbb{F}}_{p}^{\times}$ which is cyclic of
order $p-1$. Let $B^{\prime}$ be the unique quotient of $B$ of size $q$, and
fix an injective homomorphism $B^{\prime}\to\mathrm{Aut}(A)$. Now have $B$ act
on $A$ by the composition
$B\to B^{\prime}\to\mathrm{Aut}(A)$
and let $G$ be the associated semidirect product $A\rtimes B$. The kernel $K$
of this action homomorphism has order $q^{k-1}$. By Lemma 71 we have that
$\mu(G)\cong A\times K$, and this has $pq^{k-1}$ elements. By the above
theorem, $G$ is a freely representable non-Abelian group of order $pq^{k}$. In
the case of order $p\cdot 2^{2}$ such groups arose already as binary dihedral
groups $2D_{p}$ in ${\mathbb{H}}^{\times}$. But if we take $q\neq 2$ we can
conclude the following: _There are an infinite number of odd orders such that
there exists non-Abelian Sylow-cyclic groups of that order that are freely
representable_ (for example, for a fixed $q$ take an infinite sequence of
primes $p\equiv 1\bmod q$). Note that such Sylow-cyclic groups cannot be
isomorphic to subgroups of ${\mathbb{H}}^{\times}$ since all finite non-
Abelian subgroups of ${\mathbb{H}}^{\times}$ have order divisible by $4$.
The smallest such order of this type of group of odd order is $7\cdot
3^{2}=63$. Note that if $G$ is a noncyclic freely representable group of order
$5\cdot 3^{2}$ then $G$ is $A\rtimes B$ where $A$ is cyclic subgroup of $G$ of
order $5$ and $B$ is a cyclic subgroup of $G$ of order $9$. Furthermore,
$\mu(G)$ has order $15$ or $45$. The second case cannot happen since $G$ is
not cyclic. The first case cannot happen either: the automorphism group of $A$
has order 4, so the kernel $K$ of the action of $B$ on $A$ must be all of $K$,
so $\mu(G)=AK=G$. We conclude that 63 is the smallest odd order possible for a
noncyclic freely representable group.
Here is another interesting application of Theorem 81.
###### Proposition 83.
Let $G$ be a freely representable Sylow-cyclic group and let $N$ be a normal
subgroup of $G$ of index $p$. If $p^{2}$ does not divide the order of $G$ then
$G=NC_{p}\cong N\times C_{p}$
where $C_{p}$ is a subgroup of $G$ of order $p$ (and is the unique subgroup of
order $p$).
###### Proof.
Since $G$ is freely representable, there is a unique subgroup $C_{p}$ of order
$p$, and so $C_{p}$ must be normal in $G$. Since $N\cap C_{p}=\\{1\\}$ we have
$NC_{p}\cong N\times C_{p}$. Finally, $G=NC_{p}$ since $[G:N]=p$. ∎
We cannot hope to generalize Theorem 81 to all freely representable groups.
For example, the binary tetrahedral group $2T$ is freely representable (as a
subgroup of ${\mathbb{H}}^{\times}$), but does not have a unique subgroup of
order 3. However, one implication holds in general:
###### Proposition 84.
Let $G$ be a finite group with the property that for each prime $p$ dividing
the order of $G$ there is a unique subgroup of order $p$. Then $G$ is freely
representable.
###### Proof.
Let $p_{1},\ldots,p_{k}$ be the primes dividing the order of $G$. Let
$C_{p_{i}}$ be the be the unique subgroup of order $p_{i}$. Then each
$C_{p_{i}}$ is normal and
$H\;\stackrel{{\scriptstyle\mathrm{def}}}{{\,=\,}}\;C_{p_{1}}\cdots
C_{p_{k}}\cong C_{p_{1}}\times\cdots\times C_{p_{k}}$
is a cyclic subgroup of $G$. So $H$ is freely representable. Now use Corollary
80. ∎
###### Remark.
The classification of Sylow-cyclic groups is enough to yield significant
applications to differential geometry. In fact, by a theorem of Vincent
(1947), every complete connected Riemannian manifold of constant positive
curvature of dimension not congruent to $3$ modulo $4$ has a fundamental group
that is Sylow-cyclic. From this Vincent was able to give a full classification
of such manifolds when the dimension is not congruent to $3$ modulo $4$. Wolf
[17] completed the classification to all dimensions by classifying freely
representable groups beyond the Sylow-cyclic groups.
### 2 Application to Automorphisms of Sylow-cyclic groups
Consider the automorphism group ${\mathrm{Aut}}(G)$ where $G$ is a Sylow-
cyclic group of odd order, and let $O({\mathrm{Aut}}(G))$ be the maximal odd
normal subgroup of ${\mathrm{Aut}}(G)$. Then we can use the above results to
show that ${\mathrm{Aut}}(G)/O({\mathrm{Aut}}(G))$ is an Abelian $2$-group.
This is clear if $G$ is cyclic since ${\mathrm{Aut}}(G)$ is an Abelian group.
The key to generalizing this is to relate this quotient to the corresponding
quotient for the cyclic subgroup $\mu(G)$. We start with a lemma.
###### Lemma 85.
Let $G$ be a Sylow-cyclic group of odd order and let $\phi$ be an automorphism
of $G$ such that $\phi^{2}$ is the identity map. If $\phi$ fixes the MCC
subgroup $\mu(G)$ then $\phi$ fixes all of $G$.
###### Proof.
Let $P$ be a nontrivial Sylow subgroup of $G$. We will show that $\phi$ acts
trivially on $P$. Since the Sylow subgroups of $G$ generate $G$, this gives
the result. Let $A$ be the commutator subgroup of $G$. Since $A$ and $G/A$
have relatively prime orders, either $P\subseteq A$ or $P\cap A=\\{1\\}$. In
the first case $\phi$ acts trivially on $P$ since $A\subseteq\mu(G)$. So from
now on we assume that $P\cap A=\\{1\\}$.
Observe that the image $\overline{P}$ of $P$ in $G/A$ is isomorphic to $P$.
Since $\overline{P}$ is a cyclic group of odd order, it has a unique
automorphism of order $2$. So $\phi$ acts on $\overline{P}$ either as
$x\mapsto x$ or as $x\mapsto x^{-1}$.
First suppose that $\phi$ acts on $\overline{P}$ as $x\mapsto x$. So if $b\in
P$ then $\phi(b)=ab$ for some $a\in A$. Observe then that
$b=\phi^{2}(b)=\phi(ab)=\phi(a)\phi(b)=a(ab)=a^{2}b.$
Thus $a^{2}=1$. Since $A$ is a cyclic group of odd order $a=1$, and
$\phi(b)=b$. We conclude that $\phi$ acts trivially on $P$.
Finally suppose that $\phi$ acts on $\overline{P}$ as $x\mapsto x^{-1}$ and
let $c\in P$ be a generator. Thus $\phi(c)=c^{-1}a_{0}$ for some $a_{0}\in A$.
Let $B$ be a complement of $A$ and observe that $P$ is conjugate to a Sylow
subgroup of $B$. So replacing $B$ with a complement of $B$ if necessary, we
can assume $P$ is a subgroup of $B$. Note that $\phi(c)\neq c$ since $c^{-1}$
and $c$ have distinct images in $\overline{P}$ (and $\overline{P}$ has odd
order greater than 1). Thus $c$ is not in the center $Z(G)$ since
$Z(G)\subseteq\mu(G)$. Since $c$ centralizes $B$, it cannot centralize $A$.
Let $a\in A$ be such that $cac^{-1}\neq a$. Note that $cac^{-1}\in A$ so
$cac^{-1}=\phi(cac^{-1})=\phi(c)a\phi(c)^{-1}=c^{-1}a_{0}aa_{0}^{-1}c=c^{-1}ac.$
So $c^{2}ac^{-2}=a$. However $c$ has odd order, so this implies that
$cac^{-1}=a$, a contradiction. So $\phi$ cannot act on $\overline{P}$ as
$x\mapsto x^{-1}$. ∎
###### Proposition 86.
Let $G$ be a Sylow-cyclic group of odd order. Let $\mu(G)$ be the MCC subgroup
of $G$, and let $O({\mathrm{Aut}}(G))$ be the maximal odd normal subgroup of
the automorphism group ${\mathrm{Aut}}(G)$. Then the quotient
${\mathrm{Aut}}(G)/O({\mathrm{Aut}}(G)).$
is isomorphic to a $2$-group inside ${\mathrm{Aut}}(\mu(G))$. In particular,
this quotient is an Abelian $2$-group.
###### Proof.
Observe that ${\mathrm{Aut}}(\mu(G))$ is Abelian since $\mu(G)$ is a cyclic
group. Recall that Abelian groups are the products of their Sylow subgroups,
and so
${\mathrm{Aut}}(\mu(G))=A_{1}A_{2}=A_{1}\times A_{2}$
where $A_{1}$ is the subgroup of ${\mathrm{Aut}}(\mu(G))$ consisting of
elements of odd order, and where $A_{2}$ is the $2$-Sylow subgroup of
${\mathrm{Aut}}(\mu(G))$. Since $\mu(G)$ is characteristic in $G$ we have a
homomorphism ${\mathrm{Aut}}(G)\to{\mathrm{Aut}}(\mu(G))$. We also have the
projection homomorphism ${\mathrm{Aut}}(\mu(G))=A_{1}A_{2}\to A_{2}$. Let $K$
be the kernel of the composition
${\mathrm{Aut}}(G)\to{\mathrm{Aut}}(\mu(G))\to A_{2}.$
Observe that $K$ contains $O({\mathrm{Aut}}(G))$.
Claim: $K$ contains only elements of odd order. Suppose otherwise that
$\psi\in K$ has order $2k$, and let $\phi=\psi^{k}$. Then $\phi$ has order $2$
and is in $K$. The image of $\phi$ in ${\mathrm{Aut}}(\mu(G))$ is just the
restriction $\phi|_{\mu(G)}$ and it is in the kernel of of the projection
${\mathrm{Aut}}(\mu(G))=A_{1}A_{2}\to A_{2}$. In other words,
$\phi|_{\mu(G)}\in A_{1}$ and so has odd order. Since $\phi$ has order $2$, we
conclude that $\phi|_{\mu(G)}$ has order 1. By the previous lemma $\phi$ is
the identity, a contradiction.
So the claim has been established. This means $K=O({\mathrm{Aut}}(G))$ and we
have an injection
${\mathrm{Aut}}(G)/O({\mathrm{Aut}}(G))\hookrightarrow A_{2}.$
The result follows. ∎
## Chapter 7 Applications to Division Rings
The classification of Sylow-cyclic fields can be used to prove Wedderburn’s
theorem. Along the way we will see an argument that every finite subgroup of a
field is cyclic. This section is independent of Example 3 where we used (1)
Wedderburn’s theorem and (2) the fact that $F^{\times}$ is cyclic for any
finite field. We start with the following.
###### Lemma 87.
Let $D$ be a division ring and let $F$ be its prime subfield. Then every
finite subgroup $G$ of $D^{\times}$ is freely representable over $F$.
###### Proof.
We let $G$ act on $V=D$ by left multiplication. Note that $V$ is an $F$-vector
space. This action is a free linear action since $D$ has no zero divisors. ∎
###### Remark.
In particular, if $F$ has characteristic zero then $G$ is freely representable
(and so is Sylow-cycloidal). This result is the starting point for Amitsur’s
classification of finite subgroups of $D^{\times}$ where $D$ is a division
ring (1955 [2]). Amitsur used class field theory to complete the
classification.
###### Lemma 88.
Let $G$ be a group of order $p^{2}$ where $p$ is a prime. Suppose $F$ is a
field of characteristic not equal to $p$. If $G$ is freely representable over
$F$ then $G$ is cyclic.
###### Proof.
Suppose $G$ is freely representable but not cyclic. This means that every
nonidentity element of $G$ is in a unique cyclic group of order $p$. Observe
that there are $k=(p^{2}-1)/(p-1)=p+1$ such cyclic groups. Let $\mathcal{C}$
be the collection of cyclic subgroups of $G$ of order $p$. Then
$\sum_{C\in\mathcal{C}}{\mathbf{N}}C=(k-1)\mathbf{1}+{\mathbf{N}}G=p\mathbf{1}+{\mathbf{N}}G.$
Since $p$ is nonzero in $F$, this contradicts Theorem 11. ∎
Suppose $D$ is a division ring and that $G$ is a finite subgroup of
$D^{\times}$. If $F$ is the prime subfield of $D$, then let $F(G)$ be the
$F$-span of $G$ in $D$. (Warning: $F(G)$ is analogous to the group ring
$F[G]$, but they are not the same since $G$ might not be linearly
independent.)
###### Lemma 89.
Let $D,G,F,F(G)$ be as above, and suppose $F$ is ${\mathbb{F}}_{p}$ for some
prime $p$. Then $F(G)$ is a finite division ring of order a power of $p$.
###### Proof.
Let $k$ be the size of a basis of $F(G)$ for scalar field $F$, and observe
that $F(G)$ has finite size $p^{k}$. Observe that $F(G)$ is closed under
multiplication, so we conclude that $F(G)$ is a subring of $D$. Next suppose
$a\in F(G)$ is nonzero. Then the map $x\mapsto ax$ is an injective map
$F(G)\to F(G)$ since $F(G)$ is contained in a division ring. Since $F(G)$ is
finite, this map is surjective, and $ab=1$ for some $b\in F(G)$. This implies
that $F(G)$ is a division ring. ∎
###### Lemma 90.
Let $D$ be a division ring whose prime field $F$ has prime characteristic $p$.
Then $D^{\times}$ has no elements of order $p$.
###### Proof.
Suppose $g\in D^{\times}$ has order $p$, and let $G$ be the group generated by
$g$. Observe that $G$ is a subgroup of $F(G)^{\times}$, and $F(G)^{\times}$
has order $p^{k}-1$ for some $k\geq 1$. So the order of $g\in F(G)^{\times}$
fails to divide the order of $F(G)^{\times}$, a contradiction. ∎
###### Corollary 91.
Let $G$ be a subgroup of $D^{\times}$ where $D$ is a division ring. Then every
subgroup of $G$ of order $p^{2}$, where $p$ is a prime, is cyclic.
###### Proof.
Let $H$ be a subgroup of $G$ of order $p^{2}$ where $p$ is a prime. By the
above lemma, we can assume that $p$ is not the characteristic of the prime
field $F$ of $D$. By Lemma 87, $H$ is freely representable over $F$. So $H$ is
cyclic by Lemma 88. ∎
###### Corollary 92.
Suppose $G$ is a finite subgroup of $F^{\times}$ where $F$ is a field. Then
$G$ is cyclic.
###### Proof.
By the previous corollary, every subgroup of $G$ of order $p^{2}$ is cyclic,
for any prime $p$. Since $G$ is Abelian, $G$ is cyclic by the finite structure
theorem of Abelian groups (or you can use the more elementary argument given
in the remark after Corollary 19). ∎
###### Proposition 93.
Let $D$ be a division algebra and let $G$ be a finite subgroup of
$D^{\times}$. Then $G$ is a Sylow-Cycloidal group
###### Proof.
If $p$ is odd, then any $q$-Sylow subgroup of $G$ is cyclic by Theorem 25. If
$p=2$ then any $q$-Sylow subgroup is either cyclic or a generalized quaternion
group by Corollary 37.
Now suppose $F$ has prime characteristic $p$. In this case $G$ is a subgroup
of $F(G)^{\times}$, and $F(G)^{\times}$ has order $p^{k}-1$ for some $k\geq
1$. So $p$ cannot divide the order of $G$, so there are no $p$-Sylow subgroups
of $G$ that we need to worry about.
We conclude that all Sylow-subgroups of $G$ have the desired form, and that
$G$ is a Sylow-cycloidal group. ∎
###### Lemma 94.
Let $D$ be a division algebra of prime characteristic $p$, and let $G$ be a
finite subgroup of $D^{\times}$. Then $G$ is a Sylow-cyclic group
###### Proof.
Let $q$ be any odd prime dividing the order of $G$. By Corollary 91 and
Theorem 25, the $q$-Sylow subgroups of $G$ are cyclic.
Suppose the 2-Sylow subgroup of $G$ are not also cyclic. By Corollary 91 and
Proposition 37 the $2$-Sylow subgroups of $G$ must be generalized quaternion
groups. By Proposition 41, $G$ must then contain a subgroup $Q$ that we can
identify with the quaternion group of size 8. By Lemma 89 we have the division
algebra ${\mathbb{F}}_{p}(Q)$. The idea of the remainder of the proof is to
argue that there cannot be a “quaternion ring” over ${\mathbb{F}}_{p}$.
To proceed we solve $1+x^{2}+y^{2}=0$ over ${\mathbb{F}}_{p}$. If $p=2$ then
$x=1,y=0$ is a solution. Otherwise, observe that there are $(p+1)/2$ squares
in ${\mathbb{F}}_{p}$. So as $x$ varies in ${\mathbb{F}}_{p}$, the expression
$-1-x^{2}$ takes on $(p+1)/2$ distinct values. At least one of these values
must be a square since there are only $(p-1)/2$ nonsquares in
${\mathbb{F}}_{p}$. So we choose $x$ so that $-1-x^{2}$ is a square, and we
choose $y$ so that $y^{2}$ is $-1-x^{2}$. We can exchange $x$ and $y$ if
necessary, and assume $y\neq 0$. Then in ${\mathbb{F}}_{p}(Q)$ we have
$(1+x\mathbf{i}+y\mathbf{j})(1-x\mathbf{i}-y\mathbf{j})=1+x^{2}+y^{2}=0.$
Since ${\mathbb{F}}_{p}(Q)$ has no zero divisors, we have
$1+x\mathbf{i}+y\mathbf{j}=0$ or $1-x\mathbf{i}-y\mathbf{j}=0$. Since $y\neq
0$ this means that $\mathbf{j}\in{\mathbb{F}}_{p}(\left<\mathbf{i}\right>)$.
But ${\mathbb{F}}_{p}(\left<\mathbf{i}\right>)$ is a field, and so
$\mathbf{i}$ and $\mathbf{j}$ commute, a contradiction. ∎
The following is a result of Herstein. He proved it as a corollary of
Wedderburn’s theorem, but we will prove Wedderburn’s theorem as a corollary of
this result.
###### Theorem 95 (Herstein).
Let $D$ be a division ring of prime characteristic $p$, and let $G$ be a
finite subgroup of $D^{\times}$. Then $G$ is cyclic.
###### Proof.
First we consider the case where $D$ is finite and $G=D^{\times}$. By the
above lemma $G$ is a Sylow-cyclic group. Let $C$ be the maximum cyclic
characteristic (MCC) subgroup of $G$. Such a subgroup $C$ exists by Corollary
63 and, by Corollary 70, $C$ is maximal among cyclic subgroups of $G$. Observe
that ${\mathbb{F}}_{p}(C)$ is a field, so ${\mathbb{F}}_{p}(C)^{\times}$ is
cyclic. By the maximality of $C$, this means that
$C={\mathbb{F}}_{p}(C)^{\times}$.
Let $q$ be the number of elements of ${\mathbb{F}}_{p}(C)$, and let $q^{k}$ be
the number of elements of $D$. (Here we use the fact that $D$ is a vector
space over any subfield). If $k>1$ then
$|G/C|=\frac{q^{k}-1}{q-1}=q^{k-1}+\ldots+q+1\geq q+1>q-1=|C|$
which contradicts Corollary 72. Thus $k=1$ and so $G=C$, and $G$ is cyclic.
In general, we consider ${\mathbb{F}}_{p}(G)$. Since
${\mathbb{F}}_{p}(G)^{\times}$ is cyclic, as we have just shown, and since $G$
is a subgroup of ${\mathbb{F}}_{p}(G)^{\times}$, we conclude that $G$ is
cyclic as well. ∎
###### Corollary 96 (Wedderburn).
Every finite division ring $D$ is a field.
###### Proof.
By the above theorem $D^{\times}$ is cyclic. So $D$ must be a commutative
ring. ∎
## Chapter 8 Sylow-Cycloidal Groups: The Solvable Case
Suppose $G$ is a solvable Sylow-cycloidal group and $O(G)$ is the maximal
normal subgroup of $G$ of odd order. Then our first important result will be
to describe the possible quotients $G/O(G)$. We will show that $G/O(G)$ is
isomorphic to either a cyclic $2$-group, a generalized quaternion group, the
binary tetrahedral group $2T$ or the binary octahedral group $2O$. In
particular, $G/O(G)$ is isomorphic to a solvable subgroup of
${\mathbb{H}}^{\times}$. This result divides Sylow-Cycloidal groups into four
mutually exclusive types. We then focus on each type individually.111I learned
the technique of classifying freely representable groups $G$ by their
quotients $G/O(G)$ from a recent paper by Daniel Allcock [1]. My approach
generalizes the scope of Allcock a bit. Although my proof is different and
more elementary that that in [17], I also lean on Wolf [17], and thus
indirectly on Zassenhaus (1936), for guidance. Zassenhaus adopts a more
general scope than mine by only restricting the $2$-Sylow subgroup (see Lemma
6.1.9 of Wolf [17] attributed to Zassenhaus).
### 1 The Quotient $G/O(G)$
We start more generally than with Sylow-cycloidal groups. We say a finite
group $G$ _satisfies the $(2,3)$ condition_ if every $2$-Sylow subgroup of $G$
is cyclic or is the quotient of a generalized quaternion group222This means
that the $2$-Sylow subgroups of $G$ are cyclic, dihedral, or generalized
quaternion, but we will not need this fact. and every $3$-Sylow subgroup is
cyclic.
###### Lemma 97.
Let $G$ be a group that satisfies the $(2,3)$ condition. Then every subgroup
and quotient of $G$ satisifies the $(2,3)$ condition.
###### Proof.
Observe that every $p$-Sylow subgroup of a quotient $G/N$ is a quotient of a
$p$-Sylow subgroup of $G$. Also every $p$-Sylow subgroup of a subgroup $H$ of
$G$ is a $p$-group and so is a subgroup of a $p$-Sylow subgroup of $G$.
The class of cyclic $p$-groups is closed under the processes of quotient and
subgroup. Since the class of cyclic and generalized quaternion $2$-groups is
closed under subgroup, the class of quotients of such groups is closed under
quotient and subgroups. ∎
###### Definition 7.
Let $G$ be a finite solvable group and let
$G,G^{\prime},G^{\prime\prime},\ldots,G^{(k)}=\\{1\\}$ be the derived series
of commutator subgroups. Then the _characteristic Abelian subgroup of $G$_,
which we denote as $\mathcal{A}(G)$, is defined to be the first Abelian term
of the series. Observe that $\mathcal{A}(G)$ is Abelian and characteristic,
and if $G$ is nontrivial then $\mathcal{A}(G)$ is also nontrivial.
###### Lemma 98.
Let $G$ be a solvable group satisfying the $(2,3)$ condition. Then either $G$
has order of the form $2^{m}3^{n}$ or there is a prime $p\geq 5$ such that
there is a $p$-subgroup $K$ of $G$ that is a characteristic subgroup of $G$.
###### Proof.
Let $q$ be a prime dividing $\mathcal{A}(G)$ (if no such $q$ exists then $G$
has order $2^{0}3^{0}$ and we are done). If $q\geq 5$ then we can choose $p=q$
and choose $K$ to be the $p$-Sylow subgroup of $\mathcal{A}(G)$, and we are
done. Otherwise we define an elementary characteristic subgroup $N$ as the
solutions of $x^{q}=1$ in $\mathcal{A}(G)$.
Since $G$ satisfies the $(2,3)$ condition, the same is true of $N$ and $G/N$.
If $q=3$ this means that $N$ is cyclic of order $3$. If $q=2$ then $N$ is
generated by one or two elements, so is either cyclic of order $2$ or is the
Klein four group. So any automorphism of $N$ has order $1,2,$ or $3$.
If $G/N$ has order of the form $2^{m}3^{n}$ we are done, so we will now assume
that $G/N$ is not of that form. By induction we can assume that $G/N$ has a
$p$-subgroup $L/N$ where $p\geq 5$ is prime and where $L$ is a subgroup of $G$
containing $N$ such that $L/N$ is a characteristic subgroup of $G/N$. Since
$N$ is a characteristic subgroup, this implies that $L$ must also be a
characteristic subgroup of $G$. Let $K$ be a $p$-Sylow subgroup of $L$ and
observe that $K$ is a complement of $N$ in $L$, so $L\cong N\rtimes K$. Note
that $K$ acts trivially on $N$ since all automorphism of $N$ have order prime
to $p$. This means that $L$ is isomorphic to $N\times K$, and so $K$ is the
unique $p$-Sylow subgroup of $L$. Since $L$ is a characteristic subgroup of
$G$, its unique $p$-Sylow subgroup $K$ is also characteristic subgroup of $G$.
∎
###### Corollary 99.
Let $G$ be a solvable group satisfying the $(2,3)$ condition. Let $O_{6}(G)$
be the maximal normal subgroup of $G$ of order relatively prime to $6$. Then
$G/O_{6}(G)$ has order of the form $2^{m}3^{n}$.
###### Corollary 100.
Let $G$ be a solvable group satisfying the $(2,3)$ condition. Let $O(G)$ be
the maximal normal subgroup of $G$ of odd order. Then $G/O(G)$ has order of
the form $2^{m}3^{n}$, and every nontrivial normal subgroup of $G/O(G)$ has
even order.
Motivated by the above corollary, we focus on Sylow-Cycloidal groups whose
order is of the form $2^{m}3^{n}$. Of course we know what such group are when
$m=0$ or when $n=0$, so we focus on the case where $m$ and $n$ are positive.
###### Lemma 101.
Let $G$ be a solvable Sylow-cycloidal group of order $2^{m}3^{n}$ with $m$ and
$n$ positive. Assume also that every normal subgroup of $G$ is of even order.
Then the following hold
* •
$G$ contains a normal subgroup $N$ isomorphic to the quaternion group $Q_{8}$.
* •
Every element of $G$ of order $3$ acts nontrivially on $N$.
* •
$G$ contains four subgroups of order $3$, and the action of $G$ on these
subgroups gives a homomorphism $G\to S_{4}$ with kernel $Z(G)$.
* •
The center $Z(G)$ has two elements.
* •
$G/Z(G)$ is isomorphic to either $A_{4}$ or $S_{4}$, and so $G$ has order 24
or 48.
###### Proof.
Let $N$ be a maximal normal $2$-subgroup (i.e., the intersection of the
$2$-Sylow subgroups of $G$). Observe that characteristic Abelian subgroup
$\mathcal{A}(G/N)$ of $G/N$ must be a nontrivial $3$-group since if it had
order divisible by $2$ then we could violate the maximality of $N$ via the
$2$-Sylow subgroup of $\mathcal{A}(G/N)$. Let $H/N$ be the unique cyclic group
of order $3$ in $\mathcal{A}(G/N)$. Here $H$ is a subgroup of $G$ containing
$N$, and since $H/N$ is characteristic in $\mathcal{A}(G/N)$, we see that
$H/N$ is characteristic in $G/N$. Since $N$ is characteristic in $G$ we then
see that $H$ is characteristic in $G$. Let $C_{3}$ be a cyclic subgroup of $H$
of order $3$. Then we have that $H=NC_{3}$ is a semidirect product $N\rtimes
C_{3}$. If $C_{3}$ acts trivially on $N$ then $H$ would be isomorphic to
$N\times C_{3}$ and would contain a characteristic subgroup of order 3. This
violates our assumption on normal subgroups of $G$. So $C_{3}$ acts
nontrivially on $N$.
Up to isomorphism the only $2$-Cycloidal group with an automorphism of order 3
is the quaternion group $Q_{8}$, so $N$ is isomorphic to $Q_{8}$. Thus $H$ has
$24$ elements. Denote by $-1$ the unique element of $N$ of order $2$, which
must be the unique element of $H$ of order 2. Note that $-1$ is fixed by the
action of $C_{3}$, and so must be in the center of $H=NC_{3}$. In fact, the
action of $C_{3}$ on the quaternions only fixes $\pm 1$ so the center $Z(H)$
is $\\{\pm 1\\}$.
Let $L$ be the normalizer of $C_{3}$ in $H$. Observe that $L\cap N$ and
$C_{3}$ are normal in $L$, and so $L$ is isomorphic $(L\cap N)\times C_{3}$.
This means $C_{3}$ is characteristic in $L$. So $L$ cannot be normal in $H$,
otherwise $C_{3}$ would be normal in $H$, and hence in $G$, a contradiction.
So $L$ must have index at least $4$ in $H$. However, $L$ contains $C_{3}$ and
$Z(H)$. Thus $L$ has order $6$. Since $L$ is isomorphic to $(L\cap N)\times
C_{3}=Z(H)\times C_{3}$, it is cyclic of order $6$. By the orbit-stabilizer
theorem (and the fact that $p$-Sylow groups are conjugate) we get that $H$ has
exactly four $3$-Sylow subgroups. This action gives a homomorphism $H\to
S_{4}$. The kernel of this action is the intersection of the normalizers. Our
description of $L$ applies to all the normalizers, and the intersection is
seen to be $Z(H)$. So $Z(H)$ is the kernel of the action $H\to S_{4}$. Observe
that the image of $L$ under $H\to S_{4}$ is isomorphic to $L/Z(H)$, so is
cyclic of order $3$, and it fixes the element corresponding to $C_{3}$. This
observation applies not just to $C_{3}$ but to all 3-Sylow subgroups of $H$.
So the image of $H\to S_{4}$ contains all subgroup of order $3$. This means
that the image contains all of $A_{4}$, and must in fact be $A_{4}$ since
$H/\\{\pm 1\\}$ is a group of size $12$.
If $G=H$, then we are done. So for the remainder of the proof we assume $H$ is
not all of $G$. We claim that $G$ has has order $2^{m}3$ where $m>3$.
Otherwise every element $h$ of order $3$ in $H$ is of the form $g^{3}$ for
some element of $g$ of order $9$. Since such an $g$ acts on $N$ as an
automorphism of order 1 or 3, this forces $h$ to act trivially on $N$, a
contradiction to a previous conclusion.
The $2$-Sylow subgroups of $G$ cannot be normal in $G$ by maximality of $N$.
So there are three $2$-Sylow subgroups $S_{1},S_{2},S_{3}$ since the number of
such groups must divide $|G|/2^{m}$. Since $G$ acts transitively on
$\\{S_{1},S_{2},S_{3}\\}$ by conjugation, we get a homomorphism $G\to S_{3}$
whose image is $S_{3}$ or $A_{3}$. The kernel is a normal $2$-subgroup of $G$
of index 3 or 6 in $G$. By maximality of $N$, this kernel is contained in $N$.
So the index of $N$ in $G$ is a divisor of $6$, and since it has size at least
$6$ it is equal to $6$. Hence (1) $G$ has size $48$, (2) each $S_{i}$ is a
generalized quaternion group of size 16, (3) $H$ is the kernel of $G\to
S_{3}$, (4) the homomorphism $G\to S_{3}$ is surjective with each $S_{i}$
mapping to a different subgroup of order $2$ in $S_{3}$, and (5) $H$ has index
2 in $G$ and its image under $G\to S_{3}$ is $A_{3}$.
Observe that every $3$-Sylow subgroup of $G$ is contained in $H$ since
$[G:H]=2$. Thus the set of $3$-Sylow subgroups of $G$ has size 4, giving a
homomorphism $G\to S_{4}$ extending the earlier surjection $H\to A_{4}$. This
means that the kernel $K$ of $G\to S_{4}$ has order $2$ or $4$. The image of
$K$ under $G\to S_{3}$ is a normal $2$-subgroup of $S_{3}$, so $K$ has trivial
image in $S_{3}$. This means that $K\subseteq H$. But we already know that the
kernel of $H\to S_{4}$ is $Z(H)$. Hence $K=Z(H)$. This implies that $G/Z(H)$
is isomorphic to $S_{4}$.
We conclude by showing that $Z(H)=Z(G)$. First observe that under the map
$G\to S_{4}$ the center $Z(G)$ maps to the center of $S_{4}$ which is trivial.
So $Z(G)$ is contained in the kernel $Z(H)$. Conversely, since $S_{1}\cap
S_{2}\cap S_{3}$ is the maximal normal 2-subgroup of $G$, this intersection is
$H$. Thus $Z(H)\subseteq S_{i}$, and so $Z(H)$ is the unique subgroup of
$S_{i}$ of size 2. So $Z(H)$ is contained in $Z(S_{i})$ for each $i$. Since
$S_{1},S_{2},S_{3}$ generates $G$ we have that $Z(H)\subseteq Z(G)$. ∎
###### Remark.
By a theorem of Burnside, all groups of order $p^{m}q^{n}$ are solvable for
distinct primes $p$ and $q$. So accepting this result allows us to drop the
solvability assumption for groups of order $2^{m}3^{n}$. I will keep this
assumption just to make the proofs a bit more accessible.
There are two cases in the above lemma. We now link these cases to specific
subgroups of ${\mathbb{H}}^{\times}$.
###### Proposition 102.
If $G$ is a solvable Sylow-cycloidal group of order 24 with no normal subgroup
of order 3, then $G$ is isomorphic to the binary tetrahedral group $2T$.
###### Proof.
By the above lemma, we can identify $Q_{8}$ with a normal subgroup of $G$.
Also, $G$ contains four subgroups of order $3$, each which acts nontrivially
on $Q_{8}$, and the center of $G$ has two elements. Note that we have an
embedding of $G/Z(G)$ into ${\mathrm{Aut}}(Q_{8})$. This implies that if
$\alpha$ and $\beta$ are distinct elements of order 3, their actions on
$Q_{8}$ are distinct (otherwise $\alpha=\epsilon\beta$ where $\epsilon\in
Z(G)$, and $\epsilon$ must be $1$ since $\alpha$ has order $3$).
Note that any automorphism $\phi$ of order $3$ of $Q_{8}$ must permute its
three subgroups of order 4, and cannot fix any such subgroup (otherwise it
would have to fix all three, and then $\phi^{2}$ would fix all of $Q_{8}$). So
there are only 4 possible values of $\phi(\mathbf{i})$, and they are contained
in $\\{\pm\mathbf{j},\pm\mathbf{k}\\}$. Once $\phi(\mathbf{i})$ is known,
there are only two possibilities for $\phi(\mathbf{j})$. Since $\mathbf{i}$
and $\mathbf{j}$ generate $Q_{8}$, this gives $8$ possibilities. All of these
possibilities must occur as the automorphism associated with elements of order
3, since there are 8 such elements.
Thus we can find an element $g\in G$ of order 3 such that it acts on $Q_{8}$
by sending $\mathbf{i}$ to $\mathbf{j}$ and sending $\mathbf{j}$ to
$\mathbf{k}$. Let $C_{3}$ be the subgroup generated by $g$. Then
$G\cong Q_{8}\rtimes C_{3}$
where $C_{3}$ acts on $G$ as specified. Thus $G$ is unique up to isomorphism.
Since $2T$ is freely representable, it is a Sylow-cycloidal group. Also
$2T/\\{\pm 1\\}$ is isomorphic to $A_{4}$. Since $A_{4}$ is solvable and has
no normal subgroups of order 3, the same is true of $2T$. So $2T$ satisfies
our assumptions for $G$, and so must be isomorphic to any such $G$. ∎
Here is a variant of the above:
###### Corollary 103.
Let $G$ be a finite group of order 24 that has a unique element of order $2$
but does not have a unique subgroup of order $3$. Then $G$ is isomorphic to
the binary tetrahedral group $2T$.
###### Proof.
Let $P_{2}$ be a $2$-Sylow subgroup of $G$. Then $P_{2}$ has a unique element
of order $2$, so must be isomorphic to $Q_{8}$ or a cyclic group of order $8$.
In particular, $G$ is a Sylow-cycloidal group. Note that the subgroups of
order $3$ of $G$ are conjugate since they are Sylow subgroups, so they cannot
be normal since there is more than one. The result now follows from
Proposition 98 if we grant that $G$ is solvable.
It is well-known that all subgroups of order 24 are solvable, but to see this
directly for $G$ here let $C_{2}$ be the unique subgroup of order $2$ and let
$C_{3}$ be any subgroup of order $3$. Then the subgroup
$C_{2}C_{3}=C_{2}\rtimes C_{3}$ must be cyclic since $C_{2}$ has no
automorphism. So the normalizer of $C_{3}$ has at least 6 elements. This means
that there are at most $4$ subgroups of $G$ of order $3$. Since the number of
such Sylow subgroups is congruent to 1 modulo $3$, we see there are exactly
$4$ subgroups of order $3$ and $C_{2}C_{3}$ is the normalizer of $C_{3}$. The
action of $G$ on the set of subgroups of order $4$ gives a homomorphism $G\to
S_{4}$. The kernel of the action is the intersection of the normalizers for
subgroups of order 3. Our description of the normalizer shows this
intersection is $C_{2}$. Since $S_{4}$ is solvable, this implies that $G$ is
solvable. ∎
###### Proposition 104.
Let $G$ be a solvable Sylow-cycloidal group of order 48 such that $G/Z(G)$ is
isomorphic to $S_{4}$. Then $G$ is isomorphic to the binary octahedral group
$2O$. Furthermore $G$ has $8$ elements of order 3, and $4$ subgroups of order
3, and these elements generate the unique subgroup of $G$ of index $2$, and
this index 2 subgroup is a binary tetrahedral group.
###### Proof.
We start by checking that $G=2O$ satisfies the hypothesis. Recall that $2O$
has $48$ elements. We note that $2O$ has center containing the elements $\pm
1$, and since $2O/\\{\pm 1\\}\cong O\cong S_{4}$ the center is exactly $\\{\pm
1\\}$ (since $S_{4}$ has trivial center). So $2O/Z(2O)$ is isomorphic to
$S_{4}$.
Now let $G$ be any subgroup of size 48 such that $G/Z(G)\cong S_{4}$. Observe
that $Z(G)$ has order $2$ since $G$ has order 48 and $S_{4}$ has order $24$.
So the generator of $Z(G)$ is the unique element of $G$ of order 2 (Lemma 49).
For now, we fix a particular isomorphism $G/Z(G)\to S_{4}.$ Since $Z(G)$ has
order $2$, the order of the image of $g\in G$ in $S_{4}$ is either the same as
the order of $g$ or has half the order of $g$. For example if $g$ maps to an
element of order 2 then $g$ must have order 2 or 4. But in this case $g$ is
not the unique element of order 2 since that maps to the identity element, so
$g$ has order $4$ and $g^{2}$ is the unique element of order $2$.
In contrast if $g$ has order 3, then $g$ must map to an element of order $3$
since $3$ is odd. In particular if $g$ has order 3 then $g$ maps to a 3 cycle
in $S_{4}$. So if $\mathcal{S}_{3}$ is the set of elements of order $3$ in $G$
then we have a map $\mathcal{S}_{3}\to\mathcal{T}_{3}$ where $\mathcal{T}_{3}$
is the set of three cycles. Now let $t\in\mathcal{T}_{3}$ be given. The
subgroup $\left<t\right>$ of $S_{4}$ corresponds to a subgroup $H_{t}$ of $G$
of order $6$ containing $Z(G)$ as a normal subgroup. In fact,
$H_{t}=Z(G)C=Z(G)\rtimes C$ where $C$ is a 3-Sylow subgroup of $H_{t}$. Since
$Z(G)$ has no nontrivial automorphisms, $H_{t}$ is just $Z(G)\times C$, so $C$
is the unique subgroup of $G$ order 3 whose image in $S_{3}$ is
$\left<t\right>$. Thus there is exactly one element of $G$ that maps to $t$.
This means that $\mathcal{S}_{3}\to\mathcal{T}_{3}$ is a bijection. In
particular, there are $8$ elements of order 3 in $G$, and $4$ subgroups of
order 3 in $G$.
Let $H$ be the subgroup of $G$ generated by the set $\mathcal{S}_{3}$ of
elements of order 3. The image in $S_{4}$ is $A_{4}$ (since $\mathcal{T}_{3}$
generates $A_{4}$). This implies that $H$ is also of even order and so must
contain the unique subgroup $Z(G)$ of $G$ of order $2$. Thus $H/Z(G)\cong
A_{4}$, and so $H$ has order $24$. Also if $H$ has a normal subgroup of size
3, then its image in $A_{4}$ would also have a normal subgroup of size 3. But
this is not the case. So we have that $H$ is isomorphic to the binary
tetrahedral group $2T$ (see previous proposition). We also note that $H$ is
the only subgroup of $G$ of order 24. So see this note that any other such
group $H^{\prime}$ would have to contain the unique element of $G$ order $2$
and so would contain $Z(G)$. Thus its image in $S_{4}$ would have size 12. But
$A_{4}$ is the only subgroup of $S_{4}$ of index two333A subgroup $N$ of
$S_{4}$ of index 2 is normal, and contains all three cycles since $G/N$ has
order 2. Since three cycles generate $A_{4}$ any such $N$ would have to be
$A_{4}$. and so $H^{\prime}$ and $H$ would have the same image in $S_{4}$ and
so would be equal.
This gives us enough information to describe $G$ in terms of relations. The
final specification of $G$ will not depend on a particular isomorphism
$G/Z(G)\cong S_{4}$ but only on the fact that such an isomorphism exists. Note
that the subgroup $H$ (the unique subgroup of $G$ of index 2) and the subset
$\mathcal{S}_{3}$ do not depend on the map. Choose an element $g_{1}\in G$ of
order $3$. It can be any such element. Next choose $g_{2}$ to be any element
of order $3$ such that the product $g_{1}g_{2}$ has order 4. To show this can
be done it is useful to choose a map $G/Z(G)\cong S_{4}$ (by permuting the
number if necessary from a given map) so that $g_{1}$ corresponds to
$(1\,2\,3)$. Then the element $g_{2}$ corresponding to $(1\,2\,4)$ will work
since $g_{1}g_{2}$ maps to $(1\,3)(2\,4)$ of order $2$ in $S_{4}$ and so
$g_{1}g_{2}$ has order $4$ in $G$. In fact, once we have chosen a suitable
$g_{1}$ and $g_{2}$, we can permute the numbering of the four elements
permuted by $S_{4}$ so that $g_{1}$ corresponds to $(1\,2\,3)$ and $g_{2}$
corresponds to $(1\,2\,4)$. Note that $(34)(1\,2\,3)(34)=(1\,2\,4)$ so if we
choose $\tau\in G-H$ mapping to $(34)$ then $\tau g_{1}\tau^{-1}$ is an
element of order 3 corresponding to $(1\,2\,4)$. Since
$\mathcal{S}_{3}\to\mathcal{T}_{3}$ is a bijection, this means that
$\tau g_{1}\tau^{-1}=g_{2}.$
We also have
$\tau g_{1}^{-1}\tau^{-1}=g_{2}^{-1},\quad\tau g_{2}\tau^{-1}=g_{1},\quad\tau
g_{1}^{-1}\tau^{-1}=g^{-1}_{1}.$
Here we made use of $\tau^{-1}g\tau=\tau g\tau^{-1}$ for all $g\in G$ since
$\tau^{2}$ has order 2 in $Z(G)$. By conjugating other 3-cycles in $S_{4}$ by
$(3\,4)$ we can see that
$\tau g\tau^{-1}=g^{-1}$
for the other four elements $g\in\mathcal{S}_{3}$. This gives us eight
relations, one for each element of $\mathcal{S}_{3}$. We verified they held by
using a particular map $G/Z(G)\cong S_{4}$, but the actual relations
themselves do not depend on the map. In addition we have a ninth relation
$\tau^{2}=-1$ where $-1$ is the unique element of order $2$ in $H$.
Now consider the free product $H*C_{4}$ where $C_{4}$ is an abstract cyclic
group of order $4$ with generator called $\tau$. Let $K$ be the normal
subgroup generated by the 9 relations discussed above. Note that every element
of $H*C_{4}/K$ can be written as $a$ or $a\overline{\tau}$ where $a$ is in the
image of $H$ and $\overline{\tau}$ is the image of $\tau$. This follows from
the fact that $\mathcal{S}_{3}$ generates $H$. Thus the group $H*C_{4}/K$ has
at most $48$ elements. However, $G$ satisfies these relations so there is a
homomorphism $H*C_{4}/K\to G$, and this is a surjection since $G$ is generated
by $\tau$ and $H$. Thus $G$ is isomorphic to $H*C_{4}/K$.
So if $G_{1}$ and $G_{2}$ are solvable Sylow-cycloidal groups of order 48 that
contain a common subgroup $H$ of size 24, and if $G_{1}/Z(G_{1})$ and
$G_{2}/Z(G_{2})$ are both isomorphic to $S_{4}$, then $G_{1}$ must be
isomorphic to $G_{2}$ since both are isomorphic to $H*C_{4}/K$ (and $K$ does
not depend on $G_{i}$ but only on $H$). More generally, if $G_{1}$ and $G_{2}$
are are solvable Sylow-cycloidal groups of order 48 such that $G_{1}/Z(G_{1})$
and $G_{2}/Z(G_{2})$ are isomorphic to $S_{4}$, then as observed above each
$G_{i}$ has a subgroup isomorphic to $2T$. By identifying these subgroups we
reduce to the situation where $G_{1}$ and $G_{2}$ share a subgroup of order
$24$. We conclude that $G_{1}$ and $G_{2}$ are isomorphic under these
conditions. ∎
Here are some useful observations linking the $2$-Sylow subgroup of $G$ to the
quotient $G/O(G)$.
###### Proposition 105.
Let $G$ be finite group and let $O(G)$ be its maximal normal subgroup of odd
order. Then $G$ and $G/O(G)$ have isomorphic $2$-Sylow subgroups.
###### Corollary 106.
Let $G$ be a solvable Sylow-cycloidal group whose $2$-Sylow subgroup $S$ is
not quaternionic of order 8 or 16. Then $G/O(G)$ is isomorphic to $S$.
###### Corollary 107.
Let $G$ be a Sylow-cycloidal group. Then $G$ is a Sylow-cyclic group if and
only if $G/O(G)$ is a cyclic $2$-group.
So we divide the solvable Sylow-cycloidal groups $G$ into four mutually
exclusive types:
1. 1.
Sylow-cyclic groups. These are the Sylow-cycloidal groups where $G/O(G)$ is
cyclic.
2. 2.
Quaternion type. These are defined to encompass the Sylow-cycloidal groups
where $G/O(G)$ is a generalized quaternion group.
3. 3.
Binary tetrahedral type. These are defined to encompass the Sylow-cycloidal
groups where $G/O(G)\cong 2T$.
4. 4.
Binary octahedral type. These are defined to encompass the Sylow-cycloidal
groups where $G/O(G)\cong 2O$.
In addition, there are non-solvable Sylow-cycloidal groups that we will
consider later. These are Sylow-cycloidal groups that contain a perfect Sylow-
cycloidal subgroup.
### 2 Type 1: Sylow-cyclic groups
As noted out above, a Sylow-cycloidal group $G$ is of this type if and only if
$G/O(G)$ is a cyclic $2$-group. Such groups were treated in Section 6. In
particular, this type of group $G$ is freely representable if and only if it
has a unique subgroup of order $p$ for each prime $p$ dividing the order of
$G$.
In order to compare with later results it is convenient to break out the odd
part from the even part:
###### Proposition 108.
Let $G$ be a Sylow-cyclic group of order $2^{k}n$ where $n$ is odd and where
$k\geq 1$. Then $G$ has a normal Sylow-cyclic subgroup $M$ of order $n$. For
such $M$, the group $G$ is freely representable if and only if (1) $M$ is
freely representable and (2) $G$ has a unique element of order $2$.
###### Proof.
Let $M=O(G)$, so $G/M$ is a cyclic $2$-group (Corollary 107), and $O(G)$ has
odd order by definition, so $M$ has order $n$.
If $G$ is freely representable then (1) and (2) hold by earlier results. So
assume (1) and (2). To show $G$ is freely representable, it is enough to show
that $G$ has a unique subgroup of order $p$ for each $p$ dividing $2^{k}n$.
For $p=2$ we are covered by assumption (2). For an odd prime $p$ we note that
every subgroup of $G$ of order $p$ is a subgroup of $M$ since $G/M$ is a
$2$-group. By (1) there is a unique subgroup of $M$ (and hence of $G$) of
order $p$. ∎
### 3 Case 2: Quaternion Type
As noted in Proposition 105, these groups have $2$-Sylow subgroups that are
generalized quaternion groups, and if conversely if $G$ is a solvable Sylow-
cycloidal group with $2$-Sylow subgroups that are generalized quaternion
groups of order $32$ or more then $G$ must be of this type. (If $G$ is a
solvable Sylow-cycloidal group with $2$-Sylow subgroups isomorphic to the
quaternion group $Q_{8}$ of order $8$, it can either be of this type or of
binary tetrahedral type. If $G$ is a solvable Sylow-cycloidal group with
$2$-Sylow subgroups isomorphic to the generalized quaternion group $Q_{16}$ of
order $16$, it can either be of this type or of binary octahedral type.)
Observe that for Sylow-cycloidal groups $G$ of quaternion type, any $2$-Sylow
subgroup $Q$ of $G$ functions as a complement for the normal subgroup $O(G)$
of $G$. Thus
$G=O(G)Q=O(G)\rtimes Q.$
In particular, up to isomorphism $G$ is determined by $O(G)$ and the action of
$Q$ on $O(G)$.
###### Proposition 109.
Let $G$ be a Sylow-cycloidal group of quaternion type. Then $G$ has a unique
element of order 2. This element is in the center of $G$.
###### Proof.
Let $Q$ be a $2$-Sylow subgroup of $G$, which is a generalized quaternion
group (Proposition 105). We start by considering the action of $Q$ on $G$. By
Proposition 86, we have that $A={\mathrm{Aut}}(O(G))/O({\mathrm{Aut}}(O(G)))$
is an Abelian $2$-group. The action of $Q$ gives a map into $A$:
$Q\to{\mathrm{Aut}}(O(G))\to{\mathrm{Aut}}(O(G))/O({\mathrm{Aut}}(O(G)))=A.$
Since $A$ is Abelian, this map has a nontrivial kernel. Thus if $C_{2}$ is the
unique subgroup of $Q$ of order 2, then $C_{2}$ is in this Kernel. In
particular, the image of $C_{2}$ in ${\mathrm{Aut}}(O(G))$ must land in
$O({\mathrm{Aut}}(O(G)))$. But $O({\mathrm{Aut}}(O(G)))$ has odd order, so the
image of $C_{2}$ in ${\mathrm{Aut}}(O(G))$ is trivial. Thus $C_{2}$ acts
trivially on $O(G)$. Since $G=O(G)Q$ we have that $C_{2}$ is in the center of
$G$.
Observe that the subgroup $O(G)C_{2}$ of $G$ corresponds to the unique
subgroup of $G/O(G)$ of order two. If $g\in G$ has order 2, then its image in
$G/O(G)$ is the unique element of order 2, and so $g\in O(G)C_{2}$. Since
$C_{2}$ is in the center of $G$, we have $O(G)C_{2}=O(G)\times C_{2}$, and so
every element of $O(G)C_{2}$ of order 2 must be in $C_{2}$. We conclude that
every element of order $2$ is in $C_{2}$. In other words, there is a unique
element of order $2$ in $G$. ∎
Let $G$ be a Sylow-cycloidal group of quaternion type and let $C_{2}$ be its
unique subgroup of order 2. Let $M=O(G)C_{2}$. Observe that every element of
odd prime order must be in $O(G)$, so every element of $G$ of prime order is
in $M$. By Corollary 80, $G$ is freely representable if and only if $M$ is
freely representable. Since $M\cong O(G)\times C_{2}$ we have the $M$ is
freely representable if and only if $O(G)$ and $C_{2}$ are freely
representable (Corollary 21), but of course $C_{2}$ is freely representable.
Thus we get the following:
###### Theorem 110.
Let $G$ be a Sylow-cycloidal group of quaternion type. Then the following are
equivalent:
1. 1.
$G$ is freely representable.
2. 2.
$O(G)$ is freely representable.
3. 3.
For each prime $p$ dividing the order of $G$, there is exactly one subgroup of
$G$ of order $p$.
###### Proof.
The equivalence $(1)\iff(2)$ was addressed in the discussion proceeding the
statement of the theorem. The implication $(3)\implies(1)$ follows from
Proposition 84.
So we just need to verify that $(2)$ implies $(3)$. We know that $(3)$ holds
for $p=2$ by the previous proposition (independent of whether $(2)$ is true or
not). So suppose that $(2)$ holds and that $p$ is an odd prime dividing the
order of $G$. Observe that every subgroup of order $p$ of $G$ must actually be
a subgroup of $O(G)$ since $G/O(G)$ has even order. Finally, $(2)$ implies
$O(G)$ has exactly one subgroup of order $p$ by Theorem 81. ∎
Now we consider other characterizations of this type of group.
###### Proposition 111.
Let $G$ be a finite group. Then $G$ is a Sylow-cycloidal group of quaternion
type if and only if it is a semidirect product $M\rtimes Q$ where $M$ is a
Sylow-cyclic group of odd order and $Q$ is a generalized quaternion group.
If $G$ is a semidirect product $M\rtimes Q$ where $M$ is a Sylow-cyclic group
of odd order and $Q$ is a generalized quaternion group, then $G$ is freely
representable if and only if $M$ is freely representable.
###### Proof.
We mentioned earlier that if $G$ is a Sylow-cycloidal group of quaternion type
then $G=O(G)\rtimes Q$ where $Q$ is any $2$-Sylow subgroup of $G$ and where
$Q$ is a generalized quaternion group. Conversely suppose $G$ is of the form
$M\rtimes Q$. Then $M$ is isomorphic to a normal subgroup of $G$ of odd order,
and the $2$-group $Q$ is isomorphic to the corresponding quotient of $G$. Thus
$O(G)$ must be isomorphic to $M$, and $G/O(G)$ is isomorphic to $Q$. So by the
above theorem, $G$ is freely representable if and only if $M$ is freely
representable. ∎
###### Proposition 112.
Let $G$ be a finite group. Then $G$ is a Sylow-cycloidal group of quaternion
type if and only _(1)_ the $2$-Sylow subgroups of $G$ are generalized
quaternion groups, and _(2)_ $G$ has a Sylow-cyclic subgroup $M$ of index $2$.
In this case $G$ is freely representable if and only if $M$ is freely
representable.
###### Proof.
Suppose is a Sylow-cycloidal group of quaternion type, so $G/O(G)$ is a
generalized quaternion group. This quotient contains a cyclic subgroup of
index 2 which we can write as $M/O(G)$ where $M$ is a subgroup of $G$
containing $O(G)$. Note that $M$ has index $2$ in $G$. Since $O(G)$ has odd
order, the $2$-Sylow subgroups of $M$ are isomorphic to the cyclic group
$M/O(G)$. Thus $M$ is a Sylow-cyclic subgroup of $G$. We also note that
$G/O(G)$ is isomorphic to the 2-Sylow subgroups of $G$, so these $2$-Sylow
subgroups of $G$ are generalized quaternion groups.
Conversely, suppose (1) and (2) hold. Since $M$ is a Sylow-cyclic group, the
quotient $M/O(M)$ is a cyclic $2$-group. Since $G/M$ has order 2, the subgroup
$O(G)$ must be a subgroup of $M$, so $O(G)\subseteq O(M)$ by the maximality of
$O(M)$. Since $M$ is normal in $G$ and since $O(M)$ is characteristic in $M$
it follows that $O(M)$ is normal in $G$. Thus $O(G)=O(M)$. Since
$M/O(M)=M/O(G)$ is a $2$-group and since $G/M$ is a $2$-group, it follows that
$G/O(G)$ is a $2$-group. Any Sylow $2$-group of $G$ is thus isomorphic to
$G/O(G)$, and so $G/O(G)$ is a generalized quaternion group. Also note that
since $G/M$ has order 2, all odd order Sylow-subgroups of $G$ are contained in
$M$, and so are cyclic. Thus $G$ is a Sylow-cycloidal group of quaternion
type.
If $G$ is freely representable, then so is the subgroup $M$. Conversely, if
$M$ is freely representable then so is $O(G)$ since, as noted above, $O(G)$ is
a subgroup of $M$. Thus $G$ is freely representable by Theorem 110. ∎
###### Proposition 113.
Let $G$ be a finite group and let $S$ be a $2$-Sylow subgroup of $G$. Then $G$
is a Sylow-cyclic group if and only if $S$ is cyclic and $G$ has a normal
Sylow-cyclic subgroup $M$ of index $|S|$. Similarly, $G$ is a Sylow-cycloidal
group of quaternion type if and only if $S$ is a generalized quaternion group
and $G$ has a normal Sylow-cyclic subgroup $M$ of index $|S|$.
###### Proof.
If $G$ is Sylow-cyclic group or a Sylow-cycloidal group of quaternion type,
then let $M=O(G)$ which is normal. Conversely, suppose there is a normal
Sylow-cyclic subgroup $M$ of index $|S|$. This group $M$ has odd order, and is
maximal with this property, so $M=O(G)$. It follows that $G/O(G)\cong S$. Note
also that every odd ordered Sylow subgroup of $G$ is actually in $M$ since
$G/M$ has even order. Thus every odd ordered Sylow subgroup of $G$ is cyclic.
∎
### 4 Case 3: Binary Tetrahedral Type
Let $G$ be a Sylow-cycloidal group of binary tetrahedral type and let $O(G)$
be its maximal normal subgroup of odd order. Then $G/O(G)$ is isomorphic to
the binary tetrahedral group $2T$, so the $2$-Sylow subgroups of $G$ are all
isomorphic to the quaternion group of order 8 (Proposition 105). What is
interesting about this case is that the $2$-Sylow subgroup of $G$ is unique,
and so is characteristic:
###### Proposition 114.
Let $G$ be a Sylow-cycloidal group of binary tetrahedral type. Then $G$ has a
unique $2$-Sylow subgroup of order $8$, and this $2$-Sylow subgroup is
isomorphic to the quaternion group with 8 elements. Furthermore, the $2$-Sylow
subgroup of $G$ centralizes $O(G)$.
###### Proof.
We start with the fact that $2T$ has a cyclic quotient $C_{3}$ of order 3. Let
$H$ be the kernel of the composition $G\to G/O(G)\cong 2T\to C_{3}$. Note that
every $2$-Sylow subgroup of $G$ must be in the kernel $H$. So we just need to
show that $H$ has a unique $2$-Sylow subgroup. Let $Q$ be a $2$-Sylow subgroup
of $G$, and observe that $Q$ is a complement for $O(G)$ in $H$. So
$H=O(G)Q=O(G)\rtimes Q.$
To prove the uniqueness result for $H$, and hence for $G$, it is enough to
show that $Q$ acts trivially on $O(G)$.
By Proposition 86, $A={\mathrm{Aut}}(O(G))/O({\mathrm{Aut}}(O(G)))$ is an
Abelian $2$-group. Let $G$ act on $O(G)$ by conjugation. Then we have the
composition
$G\to{\mathrm{Aut}}(O(G))\to{\mathrm{Aut}}(O(G))/O({\mathrm{Aut}}(O(G)))=A.$
Since the codomain is a $2$-group, we have $O(G)$ is in the Kernel. So we get
a map
$2T\cong G/O(G)\to A.$
The kernel must contain the commutator subgroup of $2T$. The commutator
subgroup of $2T$ contains the commutator subgroup of $Q_{8}$, so contains the
unique subgroup $C_{2}$ of $2T$ of order $2$. But $2T/C_{2}$ is isomorphic to
$T=A_{4}$, whose commutator subgroup is the normal subgroup of order $4$. Thus
the commutator subgroup $(2T)^{\prime}$ of $2T$ has index $3$ in $2T$. Thus we
get a homomorphism
$(2T)/(2T)^{\prime}\to A.$
Since $(2T)/(2T)^{\prime}$ has order 3, and $A$ is a $2$-group, we get that
the image of $G$ in $A$ is trivial. In other words, the image of $G$ in
${\mathrm{Aut}}(O(G))$ has odd order. In particular, the image of $Q$ in
${\mathrm{Aut}}(O(G))$ must be trivial. So $Q$ acts trivially on $O(G)$. ∎
###### Corollary 115.
Let $G$ be a Sylow-cycloidal group of binary tetrahedral type. Then $G$ has a
unique element of order $2$.
Let $M/O(G)$ be the subgroup of $G/O(G)$ of order $8$. Here $M$ is a subgroup
of $G$ containing $O(G)$, and since $M/O(G)$ is normal in $G/O(G)$ we have
that $M$ is a normal subgroup of $G$. Let $Q_{8}$ be the $2$-Sylow subgroup of
$G$. Observe that $Q_{8}$ is in $M$ since $G/M$ has order 3. In fact $Q_{8}$
is a complement for $O(G)$ in $M$, so
$M=O(G)Q_{8}=O(G)\times Q_{8}$
since $Q_{8}$ acts trivially on $O(G)$. We have that $Q_{8}$ is freely
representable, so $M$ is freely representable if and only if $O(G)$ is freely
representable (Corollary 21). The further analysis of freely representable
depends on whether or not $9$ divides the order of $G$.
###### Theorem 116.
Let $G$ be a Sylow-cycloidal group of binary tetrahedral type. If $9$ divides
the order of $G$ then the following are equivalent.
1. 1.
$G$ is freely representable.
2. 2.
$O(G)$ is freely representable.
3. 3.
For each prime $p$ dividing the order of $G$, there is exactly one subgroup of
$G$ of order $p$.
###### Proof.
The property of being freely representable is inherited by subgroups so
$(1)\implies(2)$.
Now suppose that $(2)$ holds. Note that $(3)$ holds for $p=2$ (independent of
whether $(2)$ is true or not) by the above Corollary. Suppose that $C$ is a
subgroup of $G$ of order $p$ where $p$ is an odd prime. If $p\neq 3$ then $C$
is in $O(G)$ since $G/O(G)$ has order prime to $p$. If $p=3$, then $C$ is
contained in a $3$-Sylow subgroup $P$ of order at least $9$. The map $P$ to
$G/O(G)$ has image of size $3$, so it has a nontrivial kernel. Since $P$ is
cyclic, all nontrivial subgroups of $P$ contain $C$. So $C$ is in the kernel
of $P\to G/O(G)$. In other words, $C$ is in $O(G)$. Since all subgroups of odd
prime order of $G$ are in $O(G)$, we have uniqueness for each prime order by
Theorem 81. So $(3)$ holds.
Finally the implication $(3)\implies(1)$ follows from Proposition 84. ∎
###### Theorem 117.
Let $G$ be a Sylow-cycloidal group of binary tetrahedral type. If $9$ does not
divide the order of $G$ then $G$ has a subgroup $H$ isomorphic to $2T$ and
$G=O(G)H\cong O(G)\rtimes 2T.$
Furthermore, the following are equivalent.
1. 1.
$G$ is freely representable.
2. 2.
$O(G)$ is freely representable and $G\cong O(G)\times 2T$.
3. 3.
For each prime $p\neq 3$ dividing the order of $G$ there is exactly one
subgroup of $G$ of order $p$, and there are $4$ subgroups of order $3$ in $G$.
###### Proof.
Let $C_{3}$ be a $3$-Sylow subgroup of $G$, and let $Q_{8}$ be the unique
$2$-Sylow subgroup of $G$. Then $Q_{8}$ is normal in $G$, so $H=Q_{8}C_{3}$ is
a subgroup of $G$ of order $24$. Note that $O(G)$ has order prime to $24$, so
$H$ maps isomorphically onto the quotient $G/O(G)$. Hence $H$ is isomorphic to
$2T$. Also observe that $H$ is a complement to $O(G)$ so
$G=O(G)H\cong O(G)\rtimes 2T.$
Next observe that $O(G)C_{3}=O(G)\rtimes C_{3}$ is a Sylow-cyclic subgroup of
$G$. So if $G$ is freely representable, then same is true of $L=O(G)C_{3}$. If
$L=O(G)C_{3}$ is freely representable then, by Theorem 81, $C_{3}$ is the only
subgroup of $L$ of order 3. Thus $C_{3}$ is normal in $L$, and so
$L=O(G)\times C_{3}$. In particular $C_{3}$ acts trivially on $O(G)$. Since
$H=Q_{8}C_{3}$ and since $Q_{8}$ acts trivially on $O(G)$ then $H$ acts
trivially on $O(G)$. So $(1)\implies(2)$ holds.
Note that $(2)\implies(1)$ by Corollary 21.
Next observe that if $G\cong O(G)\times 2T$ then every subgroup of prime order
$p\neq 2,3$ of $G$ must be in $O(G)$ and the subgroups of prime order $p=2$ or
$p=3$ correspond to the subgroup of $2T$ of prime order. If $O(G)$ is freely
representable then there is one subgroup of order $p$ for each $p\neq 2,3$
dividing the order of $G$ (and hence the order of $O(G)$) (Theorem 81).
Observe that $2T$ has one subgroup of order 2 and four subgroups of order 3.
So $(2)\implies(3)$.
Finally suppose $(3)$ holds. We can conclude that $O(G)$ is freely
representable by Proposition 84. Since $H$ itself has four subgroups of order
3, we have at least four subgroups of order $3$. Note the four subgroups of
$H$ of order $3$ map to distinct subgroups of $G/O(G)$. As before, let
$L=O(G)C_{3}$ where $C_{3}$ is a cyclic subgroup of $H$. Any subgroup of order
3 in $L$ maps to the same subgroup of $G/O(G)$ as $C_{3}$. Since there are
only 4 subgroups of order 3 in $G$ we conclude that $C_{3}$ is the unique
subgroup of order $3$ in $L$. So $C_{3}$ is normal in $L$, and $L=O(G)\times
C_{3}$. Thus $C_{3}$ acts trivially on $O(G)$. Since $H=Q_{8}C_{3}$ and since
$Q_{8}$ acts trivially on $O(G)$, we conclude that $G=O(G)\times H$. So $(3)$
implies $(2)$. ∎
For convenience we have divided into cases depending on whether or not $9$
divides the order of $G$. Now we will see another more unified approach. As
before let $Q_{8}$ be the unique $2$-Sylow subgroup. Let $S_{3}$ be a
$3$-Sylow subgroup of $G$. We note that $Q_{8}S_{3}$ maps onto $G/O(G)$, which
implies that $S_{3}$ is not in the centralizer of $Q_{8}$. So $S_{3}$ must act
on $Q_{8}$ by sending a generator to an automorphism of $Q_{8}$ of order 3.
Recall that $O(G)$ acts trivially on $Q_{8}$. Consider
$M=O(G)S_{3}=O(G)\rtimes S_{3}$. Note that the image of $M$ in
${\mathrm{Aut}}(Q_{8})$ has order 3. Also note that
$G=Q_{8}M=Q_{8}\rtimes M.$
###### Proposition 118.
Let $G$ be a Sylow-cycloidal group of binary tetrahedral type, and let $Q$ be
its unique $2$-Sylow subgroup of of $G$, which is isomorphic to the quaternion
group with 8 elements. Then there is a complement $M$ to $Q$ in $G$ which is a
Sylow-cycloidal group of odd order:
$G=QM=Q\rtimes M.$
Here $M\to{\mathrm{Aut}}Q$ has image of order $3$. Moreover, $G$ is freely
representable if and only if $M$ is freely representable.
###### Proof.
The only thing left to show is that if $M$ is freely representable, then $G$
is freely representable. Since $M$ is freely representable, then the same is
true of $O(G)$ since it is isomorphic to a subgroup of $M$. If $9$ divides the
order of $G$ then $G$ must be freely representable by Theorem 116. So from now
on we assume that $9$ does not divide the order of $G$.
Since $M$ is freely representable, it has a unique subgroup $C_{3}$ of order
$3$ (Theorem 81). Since $O(G)$ and $C_{3}$ are normal in $M$ we have
$M=O(G)\times C_{3}$ and $C_{3}$ acts trivially on $O(G)$. Note that $Q$ acts
trivially on $O(G)$, so $H=QC_{3}$ acts trivially on $O(G)$. This means that
$G=O(G)\times H$. However, $H$ is isomorphic to $G/O(G)$. Thus $G\cong
O(G)\times 2T$. So $G$ is freely representable by Theorem 117. ∎
We can strengthen the above:
###### Proposition 119.
Let $G$ be a finite group and let $Q$ be a $2$-Sylow subgroup of $G$. Then $G$
is a Sylow-cycloidal group of binary tetrahedral type if and only if $Q$ is a
normal quaternionic subgroup of $G$ and there exists a non-normal Sylow-cyclic
subgroup $M$ of index $|S|$ in $G$. In this case $G\cong Q\rtimes M$ where $Q$
is a quaternion group with 8 elements and where the action map
$M\to{\mathrm{Aut}}Q$ has image of size 3. Additionally, in this case $G$ is
freely representable if and only if $M$ is freely representable.
###### Proof.
In light of the previous proposition we just need to show that $G$ is a Sylow-
cycloidal group of binary tetrahedral type under the assumption that $Q$ is a
normal quaternionic group in $G$ and there exists a non-normal Sylow-cyclic
subgroup $M$ of index $|S|$ in $G$.
Under these assumption, every Sylow-subgroup of $M$ of odd order is actually a
Sylow-subgroup of $G$ since $[G:M]$ is even, and every Sylow-subgroup of $G$
is conjugate to a Sylow-subgroup of $M$ by the Sylow theorems. Thus every
Sylow-subgroup of $G$ of odd order is cyclic. Note also that $M$ is isomorphic
to $G/Q$ since $M$ is of odd order. Thus $G/Q$ and $Q$ are solvable, and so
$G$ is solvable.
Thus $G$ is a solvable Sylow-cycloidal group. In addition $O(G)$ is a proper
subgroup of $M$ since $O(G)$ is normal. So $G/O(G)$ is not a $2$-group. Thus
$G$ is not Sylow-cyclic, and is not of quaternion type. So $G$ is either of
binary tetrahedral or binary octahedral type. Note that the image of $Q$ in
$G/O(G)$ is a $2$-Sylow subgroup of $G/O(G)$ that is normal. This rules out
the binary octahedral type (since the existence of such a normal $2$-Sylow
subgroup in $2O$ gives a unique $2$-Sylow subgroup in its quotient $S_{4}$,
contradicting the fact that two-cycles of $S_{4}$ generates $S_{4}$). Thus $G$
is a Sylow-cycloidal group of binary tetrahedral type. ∎
### 5 Case 3: Binary Octahedral Type
We start with some basic observations about key subgroups of this type of
Sylow-cycloidal group:
###### Proposition 120.
Let $G$ be a Sylow-cycloidal group of binary octahedral type. Then every
$2$-Sylow subgroup of $G$ is a generalized quaternion group of order 16, and
these $2$-Sylow subgroups are not normal in $G$. In addition $G$ contains a
unique subgroup $H$ of index $2$, and this subgroup is of binary tetrahedral
type. Moreover, $G$ contains a unique quaternion subgroup $Q$ of order 8, and
this group $Q$ is contained in $H$. Finally, $G$ contains exactly four
subgroups of index $16$; these subgroups are conjugate subgroups of $H$ hence
are conjugate in $G$; these subgroups are not normal in $H$ hence are not
normal in $G$; these subgroups each contain $O(G)$ as a subgroup of index 3;
these subgroups are Sylow-cyclic groups of odd order; and these subgroups are
maximal among subgroups of odd order.
###### Proof.
Every $2$-Sylow subgroup $S$ of $G$ is isomorphic to a $2$-Sylow subgroup of
$G/O(G)\cong 2O$, so is a generalized quaternion group of order 16
(Proposition 105). The image $\overline{S}$ in $G/O(G)$ of a $2$-Sylow
subgroup $S$ contains the unique element of $G/O(G)$ of order 2, so these
Sylow subgroups $\overline{S}$ correspond to Sylow subgroups of the quotient
$O\cong S_{4}$. But Sylow subgroups of $S_{4}$ are not normal (a normal
$2$-Sylow subgroup $N$ in $S_{4}$ would have to contain all two cycles since
$S_{4}/N$ has order 3, but the collection of two cycles generate $S_{4}$).
Thus such an $\overline{S}$ is not normal in $G/O(G)$, and so $S$ cannot be
normal in $G$.
Let $H$ be the subgroup of $G$ containing $O(G)$ such that $H/O(G)$
corresponds to the binary tetrahedral subgroup of $G/O(G)$. Clearly
$O(G)\subseteq O(H)$, but since $H$ is normal in $G$ and since $O(H)$ is
characteristic in $H$, it follows that $O(H)$ is normal in $G$, and so
$O(G)=O(H)$. Hence $H/O(H)$ is a binary tetrahedral group.
Suppose $L$ is any subgroup of $G$ of index 2. Thus $L$ is normal in $G$.
Since the quotient $G/L$ has two elements and $O(G)$ has odd order, $O(G)$ is
contained in $L$. Note that $L/O(G)$ has index 2 in $G/O(G)$, so
$L/O(G)=H/O(G)$ (Proposition 104). This gives us $L=H$, and so $H$ is the
unique subgroup of index 2 in $G$.
By Proposition 114, $H$ has a unique subgroup $Q_{H}$ of order $8$ and this
group is a quaternion group. This group is characteristic in $H$ and so is
normal in $G$. By the Sylow theorems, $Q_{H}$ is contained in some $2$-Sylow
subgroup of $G$, and hence in all since $Q_{H}$ is normal (and all $2$-Sylow
subgroups are conjugate). Every quaternion subgroup $Q$ of $G$ of order $8$ is
contained in some $2$-Sylow subgroup $S$ of $G$ by the Sylow theorems, so both
$Q_{H}$ and $Q$ are subgroups of $S$. Thus $Q=Q_{H}$ since every general
quaternion group contains a unique subgroup isomorphic to the quaternion group
with 8 elements. So $Q_{H}$ is the unique subgroup $Q_{H}$ isomorphic to the
quaternion group with 8 elements.
Each subgroup of order $3$ in $G/O(G)$ is uniquely of the form $M/O(G)$ where
$M$ is a subgroup of $G$ containing $O(G)$. Each such $M$ is of index $16$ in
$G$ and contains $O(G)$ as a subgroup of index 3. Since each such $M$ is of
odd order, $M$ is a Sylow-cyclic group. Having index $16$ in $G$, each such
$M$ must be maximal among subgroups of odd order in $M$. Since $M\neq O(G)$
this means $M$ cannot be normal in $G$ since $O(G)$ is the maximal subgroup of
odd order.
Next we argue that each subgroup of index 16 in $G$ arises in this way.
Suppose $M$ has index $16$ in $G$. Then $O(G)M/O(G)$ is isomorphic to
$M/(M\cap O(G))$ which has odd order. Thus $O(G)M$ has odd order. But since
$M$ has index 16 in $G$, there is no strictly larger subgroup of odd order. So
$M=O(G)M$, and so $O(G)$ is contained in $M$. In particular, such a group
corresponds to a subgroup $M/O(G)$ of $G/O(G)$ of index 16 and order 3.
By Proposition 104, there are $4$ subgroups of $G/O(G)$ of order 3, and they
are all contained in $H/O(G)$. They constitute the $3$-Sylow subgroups of
$H/O(G)$ hence are conjugate in $H/O(G)$ by the Sylow theorems, and thus
cannot be normal in $H/O(G)$. This means that there are 4 subgroups of $G$ of
index 16, they are all contained in $H$, they are conjugate in $H$, and cannot
be normal in $H$. ∎
In particular, there is a unique subgroup of order $2$ in any group of this
type:
###### Proposition 121.
Let $G$ be a Sylow-cycloidal group of binary octahedral type. Then $G$ has a
unique element of order $2$.
###### Proof.
By the above proposition, we have a normal subgroup of order 8 in $G$, and
this group has a unique subgroup of order 2. Thus we have a normal subgroup of
order 2 in G. The result follows from Lemma 49. ∎
The main theorem about freely representable groups of this type is as follows:
###### Theorem 122.
Let $G$ be a Sylow-cycloidal group of binary octahedral type. Let $H$ be the
unique subgroup of $G$ of index $2$, which is of binary tetrahedral type. Let
$M$ be any of the four subgroups of $G$ of index 16, which is Sylow-cyclic
subgroup of $H$ of odd order. Then the following are equivalent
1. 1.
$G$ is freely representable.
2. 2.
$H$ is freely representable.
3. 3.
$M$ is freely representable.
Furthermore, if $9$ divides the order of $G$ then if $O(G)$ is freely
representable, then so is $G$.
###### Proof.
The implication $(1)\implies(2)\implies(3)$ is clear since a subgroup of a
freely representable group is freely representable (Proposition 13). So we
just need to show that $(3)\implies(1)$. In fact we will proceed by showing
$(3)\implies(2)\implies(1)$.
Suppose that $M$ is freely representable. By Proposition 118, $H$ is also
freely representable. Since $G/H$ has order 2, any subgroup of $G$ odd prime
order $p$ is a subgroup of $H$. Also, by the previous proposition, $G$ has a
unique subgroup of order $2$ and this is a subgroup of $H$ since $H$ has even
order. By Corollary 80 we conclude that $G$ is freely representable.
Now suppose $9$ divides the order of $G$ and that $O(G)$ is freely
representable. Note that $O(H)$ is characteristic is $H$, and $H$ is normal in
$G$, so $O(H)$ is normal in $G$. This implies that $O(H)\subseteq O(G)$ so
that $O(H)$ is freely representable. Thus $H$ is freely representable by
Theorem 116, which as we have seen implies that $G$ is freely representable. ∎
### 6 Final Observations for the Solvable Case
We make some observations about solvable Sylow-cycloidal groups in general.
###### Proposition 123.
Let $G$ be a solvable Sylow-cycloidal group. If $G$ is not a Sylow-cyclic
group, then $G$ always has a unique element of order 2. If $G$ is a Sylow-
cyclic group then $G$ has a unique element of order 2 if and only if $G$ has a
normal subgroup of order 2.
###### Proof.
This was proved for each type individually. See Lemma 49, Proposition 109,
Corollary 115, and Proposition 121. ∎
###### Proposition 124.
Let $G$ be a solvable Sylow-cycloidal group of order $2^{k}n$ where $n$ is odd
and where $k\geq 1$. Then $G$ has a Sylow-cyclic subgroup $M$ of order $n$
with the following property: $G$ is freely representable if and only if (1)
$M$ is freely representable and (2) $G$ has a unique element of order $2$. In
particular, if $G$ is not itself Sylow-cyclic then $G$ is freely representable
if and only if $M$ is freely representable.
###### Proof.
This was proved for each type individually. See Proposition 108, Proposition
111, Proposition 118, and Theorem 122. ∎
## Chapter 9 The Non-Solvable Case: the Group
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$.
A very important family of Sylow-cycloidal groups is
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$. The first goal here is to show that
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ is Sylow-cycloidal for all odd primes
$p$. In some sense these, together with the Sylow-cycloidal groups we have
considered up to now, are all that are needed to form the most general Sylow-
cycloidal groups. Along the way we see some very striking results about the
cyclic (or equivalently the Abelian) subgroups of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$. These results leads naturally to the
classification of normal subgroups of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$
and nonsolvability results. We will also consider an amusing necessary
condition for these groups to be freely-representable: $p$ is a Fermat prime.
It turns out that $p=3$ or $5$ is a necessary and sufficient condition, but we
will not prove this here (at least not in this version of the document).
Interestingly, $p=3$ and $p=5$ are the two cases that occur as subgroups of
${\mathbb{H}}^{\times}$.
A few of the initial results can be proved for any finite field $F$, but we
will need to specialize to $F$ of odd prime order if we want Sylow-cycloidal
groups.
###### Proposition 125.
Let $F$ be a finite field of order $q$. Then $\mathrm{SL}_{2}(F)$ is a group
of order $(q-1)q(q+1)$. If $q$ is odd then there is a unique element of order
$2$ in $\mathrm{SL}_{2}(F)$.
###### Proof.
The first row of an invertible $2$-by-$2$ matrix is nonzero, so we can limit
our attention to $q^{2}-1$ candidates for the top row. For each of these
candidates, we form an invertible matrix if and only if we choose the second
row not to be in the span of the first row. This gives $q^{2}-q$ choices for
the second row for each given top row. So there are
$(q^{2}-1)(q^{2}-q)=(q-1)^{2}q(q+1)$
elements of $\mathrm{GL}_{2}(F)$. The determinant homomorphism
$\mathrm{GL}_{2}(F)\to F^{\times}$ is surjective since even the invertible
diagonal matrices map onto $F^{\times}$. So the kernel of this map, which is
$\mathrm{SL}_{2}(F)$, has order $(q-1)q(q+1)$.
Now we assume $q$ is odd. Every element $\alpha$ of order 2 in
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ has has an eigenvalue $1$ or $-1$ since
$(\alpha-I)(\alpha+I)=0$. Any element of $\mathrm{SL}_{2}({\mathbb{F}}_{p})$
with eigenvalue $\pm 1$ has the following form (with respect to some basis):
$\alpha=\pm\begin{pmatrix}1&a\\\
0&1\end{pmatrix},\qquad\alpha^{2}=\begin{pmatrix}1&2a\\\ 0&1\end{pmatrix}.$
We conclude that the only element of order 2 in
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ is $-I$. ∎
Next we classify elements of order greater than 2 by the number of eigenvalues
in $F$, starting with two and working down to zero eigenvalues.
###### Lemma 126.
Let $F$ be a finite field of order $q$ and let $\alpha\in\mathrm{SL}_{2}(F)$
be an element with two distinct eigenvalues in $F$. Then $\alpha$ is contained
in a cyclic subgroup $C$ of $\mathrm{SL}_{2}(F)$ of order $q-1$ where $C$ has
the property that that there is a basis such that every element of $C$ is a
diagonal matrix. Furthermore, the existence of such an $\alpha$ implies that
$q>3$.
###### Proof.
Fix a basis of eigenvectors for $A$. Then $A$ is contained in the subgroup $C$
consisting of matrices of the form
$\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}$
with respect to the chosen basis, where $a\in F^{\times}$. Since $F^{\times}$
is cyclic, this group is a cyclic group of order $q-1$.
Since the product of the eigenvalues of $\alpha$ is $1$ and since they are
distinct, we must have $q>3$. ∎
###### Lemma 127.
Let $F$ be a finite field of order $q$ and characteristic $p$. Let $v\in
F^{2}$ be a nonzero vector. Then the set of elements $D_{v}$ of
$\mathrm{SL}_{2}(F)$ with exactly one eigenvalue and with eigenvector $v$ is a
subgroup of $\mathrm{SL}_{2}(F)$ isomorphic to $\\{\pm 1\\}\times F$ where $F$
is the additive group of $F$ (of size $q$). In particular $D_{v}$ is Abelian
and each element of $F$ has order divisible by $2p$ if $q$ is odd, and
divisible by $p=2$ if $q$ is even.
Let $\alpha\in\mathrm{SL}_{2}(F)$ be an element with exactly one eigenvalue in
$F$. If $q$ is odd then $\alpha$ is a contained in a cyclic subgroup $C$ of
$\mathrm{SL}_{2}(F)$ of order $2p$. If $q$ is even then $\alpha$ is contained
in a cyclic subgroup $C$ of $\mathrm{SL}_{2}(F)$ of order $p=2$.
###### Proof.
Choose a basis whose first element is $v$. Then $D_{v}$ consists of the
matrices whose representation with respect to this basis is of the form
$\pm\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}$
for some $a\in F$. Observe that
$\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}\begin{pmatrix}1&b\\\
0&1\end{pmatrix}=\begin{pmatrix}1&a+b\\\ 0&1\end{pmatrix},$
and in fact that $D_{v}$ is isomorphic to $\\{\pm 1\\}\times F$. In
particular, every element of $D_{v}$ is contained in a cyclic subgroup of
order $2p$ if $q$ is odd, and a subgroup of order $p=2$ if $q$ is even. ∎
###### Lemma 128.
Let $F$ be a finite field of order $q$ and let $\alpha\in\mathrm{SL}_{2}(F)$
be an element with no eigenvalues in $F$. Then $\alpha$ is contained in a
cyclic subgroup of $\mathrm{SL}_{2}(F)$ of order $q+1$. More specifically, let
$M_{2}(F)$ be the ring of $2$-by-$2$ matrices with entries in $F$, where we
view $F$ as a subring via the diagonal embedding. Then the subring
$E=F[\alpha]$ of $M_{2}(F)$ generated by $F$ and $\alpha$ is a field. This
field $E$ has the following properties:
* •
$E$ has size $q^{2}$.
* •
The group $K={\mathrm{SL}}_{2}(F)\cap E^{\times}$ is a cyclic group of order
$q+1$ that contains $\alpha$.
* •
The determinant homomorphism $E^{\times}\to F^{\times}$ is the map $x\mapsto
x^{q+1}$, and the kernel of this map is $K={\mathrm{SL}}_{2}(F)\cap
E^{\times}$.
* •
The Galois group of $E$ over $F$ has two elements. Its nontrivial element
$\sigma$ is the automorphism $x\mapsto x^{q}$. For all $x\in
K={\mathrm{SL}}_{2}(F)\cap E^{\times}$ we have $\sigma x=x^{-1}$.
###### Proof.
Using the ring homomorphism $F[X]\to M_{2}(F)$ sending $X$ to $\alpha$, we see
that $F[\alpha]$ is isomorphic to $F[X]/\left<f\right>$ where $F[X]$ is the
polynomial ring in one-variable and $f$ is the minimal polynomial of $\alpha$
in $F[X]$. By the Cayley-Hamilton theorem, $f$ divides the characteristic
polynomial of $\alpha$, so in this case $f$ must be an irreducible quadratic
polynomial since the characteristic polynomial of $\alpha$ has no roots in
$F$. This implies that $F[X]/\left<f\right>$ is a quadratic field extension of
$F$. In particular, $E=F[\alpha]$ is a field of size $q^{2}$. Note that
$E^{\times}$ is cyclic of order $q^{2}-1$ (Corollary 92).
Because $F^{\times}$ is the subgroup of $E^{\times}$ of size $q-1$, we have
that that $\beta\in E^{\times}$ is in $F^{\times}$ if and only if
$\beta^{q-1}=1$. This implies that $\beta\in E$ is in $F$ if and only if
$\beta^{q}=\beta$.
Now let $\beta$ be a generator of the cyclic group $E^{\times}$. In $E[X]$ we
have the polynomial
$(X-\beta)(X-\beta^{q})=X^{2}-(\beta+\beta^{q})X+\beta^{q+1}$
Observe that
$(\beta+\beta^{q})^{q}=\beta^{q}+\beta^{q^{2}}=\beta^{q}+\beta$
and that
$(\beta^{q+1})^{q}=\beta^{q^{2}}\beta^{q}=\beta^{1+q}.$
These follow since $\beta^{q^{2}-1}=1$, so $\beta^{q^{2}}=\beta$. Also $q$ is
a power of the characteristic $p$ of $E$ and in fields of characteristic $p$
we have the identity $(a+b)^{p}=a^{p}+b^{p}$, so $(a+b)^{q}=a^{q}+b^{q}$ for
all $a,b\in E$. We conclude that $X^{2}-(\beta+\beta^{q})X+\beta^{q+1}$ lies
in $F[X]$ and so must be the minimal polynomial of $\beta$ in $F[X]$ (since
$\beta$ is not in $F$). By the Cauchy-Hamilton theorem, it is the
characteristic polynomial of $\beta$. In particular the determinant of $\beta$
is $\beta^{q+1}$. Note also that we have established that $x\mapsto x^{q}$ is
an automorphism $\sigma$ of the field $E$, and that it fixes $F$.
Consider the determinant homomorphism $E^{\times}\to F^{\times}$. This sends
the generator $\beta$ to $\beta^{q+1}$, so sends any element in $E^{\times}$
to its $q+1$ power. In particular the kernel $K$ must be the cyclic subgroup
of $E^{\times}$ of size $q+1$ since $q+1$ divides $q^{2}-1$. Note also that
$K=E^{\times}\cap{\mathrm{SL}}_{2}(F)$ since it is in the kernel of the
determinant map. So $\alpha\in K$.
Also note that any automorphism of $E$ fixing $F$ is determined by its action
on the generator $\beta$ of $E^{\times}$, and that $\beta$ must map to a root
of $(X-\beta)(X-\beta^{q})$. Thus there are only two elements of the Galois
group of $E$ over $F$ (i.e., the automorphisms of $E$ fixing $F$): the
identity $x\mapsto x$ and $x\mapsto x^{q}$. Let $\sigma$ be the map $x\mapsto
x^{q}$. Note that if $x\in K$ then $x^{q+1}=1$ so $x^{q}=x^{-1}$. Thus the
restriction of $\sigma$ to $K$ is the map $x\mapsto x^{-1}$. ∎
We can combine these three lemmas:
###### Lemma 129.
Let $F$ be a finite field of order $q$ and characteristic $p$. Let
$\alpha\in\mathrm{SL}_{2}(F)$ be an element not equal to $1$ or $-1$. Then the
following hold:
* •
The element $\alpha$ has two distinct eigenvalues in $F$ if and only if
$\alpha$ has order dividing $q-1$.
* •
The element $\alpha$ has exactly one eigenvalue in $F$ if and only if $\alpha$
has order dividing $2p$.
* •
The element $\alpha$ has no eigenvalues in $F$ if and only if $\alpha$ has
order dividing $q+1$.
Furthermore $\alpha$ can only have two distinct eigenvalues if $q>3$.
###### Proof.
One direction of each implication follows directly from Lemmas 126, 127, and
128. To see the converse, observe that the GCD of any two distinct elements of
$\\{q-1,2p,q+1\\}$ is $2$ is $q$ is odd, and is $1$ is $q$ is even. Now if $q$
is odd then $-1$ is the only element of order $2$ (Proposition 125). Since
$\alpha$ is not $1$ or $-1$, we conclude that the order of $\alpha$ cannot
divide two of $\\{q-1,2p,q+1\\}$. From this the converse follows. ∎
A _maximal_ cyclic subgroup of a finite group $G$ is defined to be a cyclic
subgroup of $G$ that is not contained in a cyclic group of $G$ of larger
order. Of course every cyclic subgroup is contained in an maximal cyclic group
(since $G$ is finite), but a given cyclic subgroup might be contained in
several maximal cyclic groups in a general group $G$. For
$G={\mathrm{SL}}_{2}(F)$ we can establish uniqueness.
###### Corollary 130.
Let $F$ be a finite field of order $q$ and characteristic $p$ and let $C$ be a
cyclic subgroup of ${\mathrm{SL}}_{2}(F)$.
* •
If $q>3$ is odd then $C$ is a maximal cyclic subgroup if and only if it has
order $q-1$, $2p$, or $q+1$.
* •
If $q=3$ then $C$ is a maximal cyclic subgroup if and only if it has order
$2p=6$ or $q+1=4$.
* •
If $q>2$ is even then $C$ is a maximal cyclic subgroup if and only if it has
order $q-1$, $2$, or $q+1$.
* •
If $q=2$ then $C$ is a maximal cyclic subgroup if and only if it has order $2$
or $3$.
###### Proof.
Let $\alpha$ be a generator of $C$. Since $C$ is a maximal cyclic group, it
has the listed orders by Lemmas 126, 127, and 128 depending on the number of
eigenvalues of $\alpha$ in $F$.
Conversely, suppose $q>3$ is odd and suppose $C$ has order exactly $q-1$,
$2p$, or $q+1$. Let $D$ be a maximal cyclic subgroup containing $C$. Then $D$
also has order $q-1$, $2p$, or $q+1$ by the above. Note that (1) the order of
$C$ divides the order of $D$, (2) the order of $C$ is at least $3$ (since
$q>3$), and (3) the GCD of any two of $\\{q-1,2p,q+1\\}$ is $2$. So it is
impossible for $C$ and $D$ to have different orders. So $C=D$. A similar but
modified argument works for $q=3$, or $q>2$ even, or $q=2$. ∎
We are now ready for the first main theorem.
###### Theorem 131.
Let $p$ be an odd prime. Then ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ is a
Sylow-cycloidal group whose $2$-Sylow subgroups are not cyclic.
###### Proof.
Let $q$ be an odd prime dividing the order $(p-1)p(p+1)$ of the group
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ and let $q^{k}$ be the maximal power that
$q$ divides $(p-1)p(p+1)$. By Cauchy’s theorem, there is an element $\alpha$
of order $q$. Let $C$ be a maximal cyclic subgroup
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ containing $\alpha$. By the above
corollary, $C$ has order $p-1$ or $2p$ or $p+1$. Since $q\geq 3$, we have that
$q$ and hence $q^{k}$ divides exactly one of $p-1$ or $p$ or $p+1$. Since $q$
divides $|C|$ this means that if $|C|=p-1$ or $p+1$ then $q^{k}$ divides
$|C|$. If $|C|=2p$ then $q=p$ (since $q$ is odd) and so $q^{k}$ divides $|C|$
as well (and $k=1$ in this case). In any case, $C$ has a cyclic subgroup of
order $q^{k}$. Since all $q$-Sylow subgroups are conjugate, we have
established that all $q$-Sylow subgroups are cyclic.
Since ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ has a unique element of order $2$
(Proposition 125), each $2$-Sylow subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ has a unique element of order $2$ (by
Cauchy’s theorem). Thus each $2$-Sylow subgroup $S$ of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ is either cyclic or quaternionic
(Corollary 37). So ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ is a Sylow-cycloidal
group.
Suppose $S$ is a cyclic $2$-subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$,
and let $C$ be a maximal cyclic subgroup containing $S$. By the above
corollary, $S$ has order dividing $|C|$ which is either $p-1$ or $2p$ or
$p-1$. Since $2$ divides both $p-1$ and $p+1$, the largest power of $2$ diving
$|C|$ is less than the largest power of $2$ dividing the order $(p-1)p(p+1)$
of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$. Thus $S$ is not a 2-Sylow subgroup.
∎
Here is a partial converse.
###### Proposition 132.
Suppose that $F$ is a finite field. If ${\mathrm{SL}}_{2}(F)$ is a Sylow-
cycloidal group then $F={\mathbb{F}}_{p}$ for some prime $p$.
###### Proof.
By Lemma 127 there is a subgroup $D$ of ${\mathrm{SL}}_{2}(F)$ isomorphic to
the additive group $F$. If ${\mathrm{SL}}_{2}(F)$ is a Sylow-cycloidal group
then the Abelian group $D\cong F$ must be cyclic. This can only happen if
$F={\mathbb{F}}_{p}$ where $p$ is the characteristic of $F$. ∎
###### Remark.
The case ${\mathrm{SL}}_{2}({\mathbb{F}}_{2})$ is special. It has order $6$,
and so is Sylow-cyclic since it is of prime free order. As we have seen, all
its cyclic subgroups are of order $2$ or $3$, so it is not cyclic and so must
be dihedral. Note that ${\mathrm{SL}}_{2}({\mathbb{F}}_{2})$ fails to have a
unique element of order $2$.
The next major result is that all cyclic subgroups of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of the same order are conjugate when $p$
is a prime. It is a bit easier to show that such groups are conjugate in
$\mathrm{GL}_{p}({\mathbb{F}}_{p})$, but the following two lemmas will give us
tools to achieve conjugacy in ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ instead of
$\mathrm{GL}_{p}({\mathbb{F}}_{p})$.
###### Lemma 133.
Let $\alpha\in\mathrm{GL}_{2}(F)$ where $F$ is a finite field of order $q$ and
let $d\in F^{\times}$. If $\alpha$ has two distinct eigenvalues in $F$ or no
eigenvalues in $F$ then there is an element $\beta\in\mathrm{GL}_{2}(F)$ such
that $\beta\alpha\beta^{-1}=\alpha$ and $\det\beta=d$.
###### Proof.
Suppose $\alpha$ has two eigenvalues in $F$ and let $v_{1},v_{2}\in F^{2}$ be
a basis of eigenvectors. Then let $\beta$ be such that its associated linear
tranformation maps $v_{1}\mapsto dv_{1}$ and $v_{2}\mapsto v_{2}$. Then
$\beta$ clearly works.
Suppose that $\alpha$ has no eigenvalues in $F$. By Lemma 128, $\alpha\in
E^{\times}$ where $E^{\times}$ is a cyclic subgroup of $\mathrm{GL}_{2}(F)$ of
order $q^{2}-1$ (in fact $E^{\times}$ is the multiplicative group of a field);
furthermore, the determinant map on $E^{\times}\to F^{\times}$ is given by
$x\mapsto x^{q+1}$ and so is surjective since $F^{\times}$ has order $q-1$.
Now just let $\beta\in E^{\times}$ be an element of determinant $d$. ∎
###### Lemma 134.
Let $p$ be a prime and let $C$ be a cyclic subgroup of
${\mathrm{SL}}({\mathbb{F}}_{p})$. Let $d\in{\mathbb{F}}_{p}^{\times}$. Then
there is an there is an element $\beta\in\mathrm{GL}_{2}(F)$ such that $\beta
C\beta^{-1}=C$ and $\det\beta=d$.
###### Proof.
Let $\alpha$ be a generator of $C$. If $\alpha$ has two distinct eigenvalues
or no eigenvalues then the result follows from the previous lemma. So suppose
that $\alpha$ has exactly one eigenvalue in $F$ and let $v_{1}$ be an
eigenvector. Let $v_{2}$ be such that $v_{1},v_{2}$ form a basis for $F^{2}$.
The representation for $\alpha$ in this basis is of the form
$\begin{pmatrix}e&a\\\ 0&e\end{pmatrix}$
where $e$ is $1$ or $-1$. So let $\beta$ be an element with the following
matrix representation (for this same basis $v_{1},v_{2}$):
$\begin{pmatrix}d&0\\\ 0&1\end{pmatrix}$
Note that
$\begin{pmatrix}d&0\\\ 0&1\end{pmatrix}\begin{pmatrix}e&a\\\
0&e\end{pmatrix}\begin{pmatrix}d^{-1}&0\\\
0&1\end{pmatrix}=\begin{pmatrix}e&da\\\ 0&e\end{pmatrix}=\begin{pmatrix}e&a\\\
0&e\end{pmatrix}^{m}$
where $m$ is an odd positive integer such that $m\equiv d\pmod{p}$. ∎
As mentioned above, our next goal is to establish that cyclic subgroups
(equivalently, Abelian subgroups) of the same order of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ are conjugate. We start with cyclic
groups of order $p-1$, followed by order $p$, then order $p+1$, ending with
general order.
###### Lemma 135.
Let $F$ be a finite field of order $q$. The vector space $F^{2}$ has $q+1$
distinct one-dimensional subspaces. Let $L_{1}$ and $L_{2}$ be two distinct
one-dimensional subspaces of $F^{2}$. Then the set elements of
${\mathrm{SL}}_{2}(F)$ with a basis of eigenvectors in $L_{1}\cup L_{2}$ forms
a cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of order $q-1$.
Now assume $q>3$. Then all cyclic subgroups of ${\mathrm{SL}}_{2}(F)$ of order
$q-1$ arise in this way. There are $\frac{1}{2}q(q+1)$ cyclic subgroups of
order $q-1$ and they are conjugate. Two distinct cyclic subgroups of order
$q-1$ have intersection $\\{\pm 1\\}$.
###### Proof.
Counting the number of one-dimensional subspaces is straightforward.
Let $v_{1},v_{2}$ be a basis for $F^{2}$ such that $v_{1}\in L_{1}$ and
$v_{2}\in L_{2}$. Then $\alpha\in{\mathrm{SL}}_{2}(F)$ has a basis of
eigenvectors in $L_{1}\cup L_{2}$ if and only if it has form
$\alpha=\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}$
with respect to this basis, where $a\in F^{\times}$. So the set of such
matrices is a cyclic subgroup of order $q-1$.
Assume $q>3$ for the remainder of the proof.
Let $C$ be a cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of order $q-1$, and let
$\alpha$ be a generator of $C$. So $\alpha$ is not $1$ or $-1$ since $q-1>2$.
By Lemma 129, $\alpha$ has two distinct eigenvalues, and its eigenvectors
determine two distinct one-dimensional subspaces $L_{1}$ and $L_{2}$ of
$F^{2}$. So $\alpha$ is in a cyclic group of order $q-1$ of the form described
above. Hence $C$ has the desired form.
Suppose $C$ and $C^{\prime}$ are cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of
order $q-1$. Suppose $C$ is defined using $L_{1},L_{2}$ and $C^{\prime}$ is
define using $L_{1}^{\prime},L_{2}^{\prime}$. Suppose $g\in C\cap C^{\prime}$
is not $\pm 1$. Then $g$ has distinct eigenvalues (Lemma 129). Let $v_{1}$ be
an eigenvector of $g$. Then $v_{1}\in L_{i}$ and $v_{1}\in L^{\prime}_{j}$ for
some $i,j$, so $L_{i}=L^{\prime}_{j}$ since these are one-dimensional. After
renumbering we can assume $L_{1}=L^{\prime}_{1}$. Let $v_{2}$ be an
eigenvector of $g$ not in the span of $v_{1}$. Then $v_{2}$ is in $L_{2}$ and
$L_{2}^{\prime}$ so $L_{2}=L^{\prime}_{2}$. Thus $C=C^{\prime}$. In other
words, distinct cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of order $q-1$
intersect in $\\{\pm 1\\}$. (Note that $-1\in C$ and $-1\in C^{\prime}$ by
Proposition 125 if $q$ is odd, and trivially if $q$ is even).
So different choices of $\\{L_{1},L_{2}\\}$ will produce difference cyclic
subgroups of order $q-1$ since $q-1>2$. So there are $\frac{1}{2}q(q+1)$ such
cyclic subgroups.
Suppose $C$ and $C^{\prime}$ are cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of
order $q-1$. Suppose $C$ is defined using $L_{1},L_{2}$ and $C^{\prime}$ is
define using $L_{1}^{\prime},L_{2}^{\prime}$. Let $\alpha$ be a generator for
$C$. Let $\beta_{1}\in\mathrm{GL}_{2}(F)$ give a linear transformation mapping
$L_{1}$ to $L_{1}^{\prime}$ and $L_{2}$ to $L_{2}^{\prime}$. Let
$\beta_{2}\in\mathrm{GL}_{2}(F)$ be such that
$\beta_{2}\alpha\beta^{-1}_{2}=\alpha$ and $\det(\beta_{1}\beta_{2})=1$ (see
Lemma 133). Let $\beta=\beta_{1}\beta_{2}$. Observe that $C^{\prime}=\beta
C\beta^{-1}$. So $C$ and $C^{\prime}$ are conjugate. ∎
###### Lemma 136.
Let $p$ be an odd prime. The vector space ${\mathbb{F}}_{p}^{2}$ has $p+1$
distinct one-dimensional subspaces. Let $L$ be a one-dimensional subspaces of
${\mathbb{F}}_{p}^{2}$. Then the set elements of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ with exactly one eigenvalue, and with an
eigenvector in $L$, forms a cyclic subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $2p$.
All cyclic subgroups of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $2p$
arise in this way. There are $p+1$ cyclic subgroups of order $2p$ and they are
conjugate. Two distinct cyclic subgroups of order $2p$ have intersection
$\\{\pm I\\}$.
###### Proof.
Counting the number of one-dimensional subspaces is straightforward.
Let $v_{1},v_{2}$ be a basis for ${\mathbb{F}}_{p}^{2}$ such that $v_{1}\in
L$. Then $\alpha\in{\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ has exactly one
eigenvalue and has an eigenvector in $L$ if and only if can be written as
$\alpha=\pm\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}$
with respect to this basis, where $a\in{\mathbb{F}}_{p}$. Observe that the set
of such matrices forms a cyclic group isomorphic to $\\{\pm
1\\}\times{\mathbb{F}}_{p}$ where ${\mathbb{F}}_{p}$ here is the additive
group.
Let $C$ be a cyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order
$2p$, and let $\alpha$ be a generator of $C$. Then $\alpha$ has exactly one
eigenvalue by Lemma 129. If $L$ is an eigenspace for $\alpha$, then as above
$\alpha$ is contained in a group of order $2p$ of the given form. So $C$ is a
group of the desired form.
Suppose $C$ and $C^{\prime}$ are cyclic subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $2p$ where $C$ is defined using
$L$ and $C^{\prime}$ is define using $L^{\prime}$. Suppose $g\in C\cap
C^{\prime}$ is not $\pm 1$. Then $g$ has a unique eigenvalue, and the
eigenspace associated to that eigenvalue has dimension one (since $g\neq\pm
1$). Thus $L=L^{\prime}$ and so $C=C^{\prime}$. In other words, distinct
cyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $2p$
intersect in $\\{\pm 1\\}$.
So different choices of $L$ will produce difference cyclic subgroups of order
$2p$. So there are $p+1$ such cyclic subgroups.
Suppose $C$ and $C^{\prime}$ are cyclic subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $2p$. Suppose $C$ is defined
using $L$ and $C^{\prime}$ is define using $L^{\prime}$. Let
$\beta_{1}\in\mathrm{GL}_{2}({\mathbb{F}}_{p})$ represent a linear
transformation sending $L$ to $L^{\prime}$. Let
$\beta_{2}\in\mathrm{GL}_{2}(F)$ be such that $\beta_{2}C\beta^{-1}_{2}=C$ and
so that $\det(\beta_{1}\beta_{2})=1$ (see Lemma 134). Let
$\beta=\beta_{1}\beta_{2}$. Observe that $C^{\prime}=\beta C\beta^{-1}$. So
$C$ and $C^{\prime}$ are conjugate. ∎
###### Lemma 137.
Let $F$ be a finite field of order $q$ and let $M_{2}(F)$ be the ring of
$2$-by-$2$ matrices with entries in $F$. Then the following hold:
* •
If $C$ is a cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of order $q+1$ then
there is a unique quadratic field extension $E$ of $F$ in $M_{2}(F)$
containing $C$, and $C$ is the unique subgroup of $E^{\times}$ of order $q+1$.
* •
Two distinct cyclic subgroups of ${\mathrm{SL}}_{2}(F)$ of order $q+1$ are
conjugate in ${\mathrm{SL}}(F)$ and have intersection $\\{\pm 1\\}$.
* •
If $C$ is a cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of order $q+1$ then
there is a $\gamma\in{\mathrm{SL}}_{2}(F)$ such that $x\mapsto\gamma
x\gamma^{-1}$ is an automorphism of the group $C$ sending any $x\in C$ to
$x^{-1}$.
###### Proof.
Let $\alpha$ be a generator of $C$. By Lemma 129, $\alpha$ cannot have
eigenvalues in $F$. So by Lemma 128, the subring $E=F[\alpha]$ of of
$M_{2}(F)$ is a field of order $q^{2}$, and $\alpha$ is contained in
$K={\mathrm{SL}}_{2}(F)\cap E^{\times}$, which is the unique subgroup of
$E^{\times}$ of order $q+1$. So in fact $C=K$. Since $E=F[\alpha]$ any
quadratic field extension $E$ of $F$ in $M_{2}(F)$ containing $C$ would
contain $\alpha$ and so $F[\alpha]$, and hence be equal to $E$. So $E$ is the
unique such field.
Suppose $C$ and $C^{\prime}$ are two subgroups of ${\mathrm{SL}}_{2}(F)$ size
$q+1$ and that $E$ and $E^{\prime}$ are the respective quadratic extensions of
$F$ containing $C$ and $C^{\prime}$. Suppose $\beta\in C\cap C^{\prime}$ is
not $\pm 1$. Then $\beta$ cannot be of the form $c1$ for some $c\in F$ since
the determinant of $\beta$ is $1$. Thus $\beta$ generates a proper extension
of $F$ contained in the intersection $E\cap E^{\prime}$. So
$E=E^{\prime}=F[\beta]$ since $[E:F]=[E:F]=2$ and $[E^{\prime}:F]\geq 2$. We
conclude that $C=C^{\prime}$. In other words, if $C$ and $C^{\prime}$ are
distinct subgroups of ${\mathrm{SL}}_{2}(F)$ of size $q+1$ then their
intersection is $\\{\pm 1\\}$.
Next fix a generator $g$ of $F^{\times}$. In other words, $g$ has order $q-1$.
Suppose $C$ is a cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of order $q+1$ and
let $E$ be the field extension of $F$ inside $M_{2}(F)$ of size $q^{2}$
containing $C$. Since $E^{\times}$ has order $q^{2}-1$, and since $q+1$ is
even, there is a subgroup of $E^{\times}$ of order $2(q-1)$. This implies that
$E^{\times}$ has an element $\beta$ such that $\beta^{2}=g$. Fix a nonzero
vector $v_{1}\in F^{2}$ and let $v_{2}=\beta v$ (viewing $v$ as a column
vector). By the Cayley-Hamilton theorem, the characteristic polynomial of
$\beta$ is $X^{2}-g$, so $\beta$ has no eigenvalues in $F$. Thus $v_{1},v_{2}$
must be a basis of $F^{2}$. Note also that $\beta v_{2}=gv_{1}$.
Suppose $C^{\prime}$ is another cyclic subgroup of ${\mathrm{SL}}_{2}(F)$ of
order $q+1$ and let $E^{\prime}$ be the field extension of $F$ inside
$M_{2}(F)$ of size $q^{2}$ containing $C^{\prime}$. Let
$\beta^{\prime}\in(E^{\prime})^{\times}$ be such that $v_{1}^{\prime}=v_{1}$
and $v^{\prime}_{2}=\beta^{\prime}v^{\prime}_{1}$ forms a basis with
$\beta^{\prime}v_{2}^{\prime}=gv_{1}^{\prime}$.
Let $\gamma_{1}\in\mathrm{GL}_{2}(F)$ be chosen so that $\gamma_{1}$ maps
$v_{1}=v_{1}^{\prime}$ to itself, and maps $v_{2}$ to $v_{2}^{\prime}$. Note
that $\gamma_{1}\beta\gamma_{1}^{-1}=\beta^{\prime}$ since both sides of this
equation map $v^{\prime}_{1}\mapsto v^{\prime}_{2}$ and $v^{\prime}_{2}\mapsto
gv^{\prime}_{1}$. Since $F[\beta]=E$ and $F[\beta^{\prime}]=E^{\prime}$ this
means that the map
$x\mapsto\gamma_{1}x\gamma_{1}^{-1}$
is an isomorphism $E\to E^{\prime}$ between fields. Hence it sends the unique
subgroup $C$ of $E^{\times}$ of order $q+1$ to the unique subgroup
$C^{\prime}$ of $(E^{\prime})^{\times}$ of order $q+1$.
In other words, $\gamma_{1}C\gamma_{1}^{-1}=C^{\prime}$. Let
$\gamma_{2}\in\mathrm{GL}_{2}(F)$ be chosen so that $\gamma_{1}\gamma_{2}$ has
determinant 1 and so that $\gamma_{2}C\gamma_{2}^{-1}=C$ (Lemma 133 applied to
a generator $\alpha$ of $C$ and to $d=\det\gamma_{1}^{-1}$). If
$\gamma=\gamma_{1}\gamma_{2}$ then $\gamma C\gamma^{-1}=C^{\prime}$ and
$\gamma\in{\mathrm{SL}}_{2}(F)$. So $C$ and $C^{\prime}$ are conjugate in
${\mathrm{SL}}_{2}(F)$.
A special case of this construction is where $C=C^{\prime}$ and where
$\beta^{\prime}=-\beta$ so that $\gamma_{1}\beta\gamma_{1}^{-1}=-\beta$.
Observe that $x\mapsto\gamma_{1}x\gamma_{1}^{-1}$ must then be a field
automorphism fixing $F$ (thought of as diagonal matrices). Since $F^{\times}$
has $C$ for its unique subgroup of order $q+1$, this automorphism acts on $C$.
Lemma 128 implies that it sends $x\in C$ to $x^{-1}$. So if
$\gamma=\gamma_{1}\gamma_{2}$ then $\gamma x\gamma^{-1}=x^{-1}$ for all $x\in
C$. ∎
These lemmas combine to yield the following:
###### Lemma 138.
Let $p$ be an odd prime, and let $C$ and $C^{\prime}$ be two distinct maximal
cyclic subgroups of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$. Then $C\cap
C^{\prime}=\\{\pm 1\\}$. Furthermore, if $C$ and $C^{\prime}$ have the same
order then $C$ and $C^{\prime}$ are conjugate subgroups of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$.
###### Proof.
By Corollary 130, the orders of $C$ and $C^{\prime}$ are in the set
$\\{p-1,2p,p+1\\}$. If $C$ and $C^{\prime}$ happen to have different orders
then $C\cap C^{\prime}$ has order dividing $2$ since the GCD of any two of
$\\{p-1,2p,p+1\\}$ is $2$. Since $C$ and $C^{\prime}$ are of even order they
both contained $-1$ (since $-1$ is the unique element of order 2). So $C\cap
C^{\prime}=\\{\pm 1\\}$.
Now we consider the case where $C$ and $C^{\prime}$ have the same order in
$\\{p-1,2p,p+1\\}$. Then $C$ and $C^{\prime}$ are conjugate and $C\cap
C^{\prime}=\\{\pm 1\\}$ by Lemma 135, Lemma 136, or Lemma 136 (depending on
the order of $C$). ∎
The above lemma yields the conjugacy results we want but only for _maximal_
cyclic subgroups of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$. The following lemma
and proposition make it possible to establish the result for cyclic subgroups
more generally.
###### Lemma 139.
Let $C$ be a cyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ where
$p$ is an odd prime. If the order of $C$ is at least $3$ then there is a
unique maximal cyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$
containing $C$.
###### Proof.
Suppose $D_{1}$ and $D_{2}$ are distinct maximal cyclic subgroups of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ containing $C$. By the above lemma
$D_{1}\cap D_{2}=\\{\pm 1\\}$, contradicting the assumption on the size of
$C$. ∎
###### Proposition 140.
Let $C$ be a cyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ where
$p$ is a odd prime. If $C$ has order at least $3$ then the centralizer $Z(C)$
is the unique maximal cyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$
containing $C$. So if $C$ is a maximal cyclic subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ then $C=Z(C)$.
###### Proof.
Let $D$ be the unique maximal cyclic subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ containing $C$ (see previous lemma).
Suppose that $h\in Z(C)$ and let $H$ be the subgroup generated by $h$. Observe
that $HC$ is an Abelian subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$.
Since ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ is Sylow-cycloidal (Theorem 131)
the group $HC$ must be cyclic and so is contained in a maximal cyclic subgroup
$D^{\prime}$. By the previous lemma $D=D^{\prime}$ since both contain $C$.
Thus $h\in D$. We have established that $Z(C)\subseteq D$. The other inclusion
is clear. ∎
We are ready for the second main theorem.
###### Theorem 141.
Let $p$ be an odd prime, and let $C$ and $C^{\prime}$ be two cyclic subgroups
of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of the same order. Then $C$ and
$C^{\prime}$ are conjugate subgroups of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$.
If, in addition, $C\neq C^{\prime}$ then $C\cap C^{\prime}\subseteq\\{\pm
1\\}$.
###### Proof.
If $C$ and $C^{\prime}$ have order $1$ or $2$ then the result is clear since
$-1$ is the unique element of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order
$2$. So from now on assume that $C$ and $C^{\prime}$ have equal order at least
$3$. By the above proposition, $Z(C)$ and $Z(C^{\prime})$ are maximal cyclic
subgroups. By Corollary 130, the orders of $Z(C)$ and $Z(C^{\prime})$ are
restricted to the set $\\{p-1,2p,p+1\\}$. So the order of $C$ and $C^{\prime}$
must divide an element of the set $\\{p-1,2p,p+1\\}$, and in fact divides a
unique element of $\\{p-1,2p,p+1\\}$ since the GCD of any two distinct
elements is $2$. This means that $Z(C)$ and $Z(C^{\prime})$ have the same
order, namely the unique element of $\\{p-1,2p,p+1\\}$ that is a multiple of
the order of $C$.
Thus $Z(C)$ and $Z(C^{\prime})$ are conjugate subgroups by Lemma 138. Since
$C$ is the unique subgroup of $Z(C)$ of its order, and the same is true of
$C^{\prime}$ in $Z(C^{\prime})$, we conclude that $C$ and $C^{\prime}$ are
conjugate as well.
If, in addition, $C\neq C^{\prime}$ then $Z(C)$ cannot equal $Z(C^{\prime})$
since $C$ is the unique subgroup of $Z(C)$ of its order, and the same is true
of $C^{\prime}$ in $Z(C^{\prime})$. So the intersection $Z(C)\cap
Z(C^{\prime})$ is $\\{\pm 1\\}$ by Lemma 138. Thus $C\cap
C^{\prime}\subseteq\\{\pm 1\\}$. ∎
Next we wish to count the number of cyclic subgroups of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of each order. We have already counted
such groups when the order is $p-1$ or $2p$, so we start with the order $p+1$
case.
###### Lemma 142.
Let $p$ be an odd prime. The number of cyclic subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $p+1$ is equal to
$\frac{1}{2}(p-1)p$.
###### Proof.
We consider two approaches. One is to note that each element $g$ of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ outside of $\\{\pm 1\\}$ is contained in
a unique maximal cyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$,
namely the centralizer $Z(g)$ (see Lemma 140). Also a cyclic subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ is maximal if and only if its order is
$p-1$ (when $p\neq 3$), $2p$, or $p+1$. So we can partition the elements of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ outside of $\\{\pm 1\\}$: if $c_{n}$ is
the number of cyclic subgroups of order $n$ we have
$|{\mathrm{SL}}_{2}({\mathbb{F}}_{p})|-2=c_{p-1}((p-1)-2)+c_{2p}(2p-2)+c_{p+1}((p+1)-2)$
(this works even for $p=3$ since $(p-1)-2=0$). Based on what we know so far
$(p-1)p(p+1)-2=\frac{1}{2}p(p+1)(p-3)+(p+1)(2p-2)+c_{p+1}(p-1).$
Now we solve for $c_{p+1}$.
This first approach is legitmate, but there is also a more group theoretical /
Galois theoretical approach. Let $C$ be any cyclic group of order $p+1$, and
let $E$ be the field $F[C]$ inside of $M_{2}({\mathbb{F}}_{p})$ generated by
the elements of $C$. Then the normalizer $N(C)$ acts via conjugation not just
on $C$ but on all of $E=F[C]$. In fact we get a homomorphism from $N(C)$ to
the Galois group $G$ of $E$ over $F$, and kernel of this homomorphism is just
$Z(C)$. Since $E$ is a quadratic extension of ${\mathbb{F}}_{p}$, the Galois
group $G$ has order 2 (see Lemma 128). So $Z(C)$ has index $1$ or $2$ in
$N(C)$. Lemma 137 shows that $N(C)$ is not $Z(C)$ so the index is actually 2,
and Proposition 140 show that $Z(C)=C$. Thus $[N(C):C]=2$ and so $N(C)$ has
order $2(p+1)$. By the orbit-stabilizer theorem we have that the number of
conjugates of $C$ is
$\frac{(p-1)p(p+1)}{2(p+1)}=\frac{1}{2}(p-1)p$
which counts the number of cyclic subgroups of order $p+1$ since all such
groups are conjugate (Theorem 141). ∎
Our calculations culminate with a full census of the cyclic subgroups of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$. Since $\mathrm{SL}_{2}({\mathbb{F}}_{p})$
is Sylow-cyloidal, this gives a full census of Abelian subgroups as well.
###### Proposition 143.
Let $p$ be an odd prime and let $c_{m}$ be the number of cyclic subgroups of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ of order $m$. Then
$c_{1}=c_{2}=1.$
If $m>2$ divides $p-1$ then
$c_{m}=\frac{1}{2}p(p+1).$
If $m>2$ divides $2q$ then
$c_{m}=p+1.$
If $m>2$ divides $p+1$ then
$c_{m}=\frac{1}{2}p(p-1).$
Otherwise $c_{m}=0$.
###### Proof.
Since there is a unique element of order 2 (Proposition 125) and order 1, we
have $c_{1}=c_{2}=2$. So from now on we assume $m>2$.
Suppose that $m$ divides $p-1$ (so $p>3$ since $m>2$). Each cyclic subgroup
$C$ of order $p-1$ has $Z(C)$ for the maximum cyclic subgroup containing $C$
(Proposition 140). By Corollary 130, $Z(C)$ has order in the set
$\\{p-1,2p,p+1\\}$, and $m$ divides the order of $Z(C)$ since $C$ is a
subgroup of $Z(C)$. Since $m$ divides $p-1$, it cannot divide $2p$ or $p+1$ as
well (since the GCD of $p-1$ with each is $2$ and $m>2$). Thus $Z(C)$ is a
cyclic group of order $p-1$. Conversely each cyclic group of order $p-1$ has a
unique subgroup of order $m$. Thus there is a one-to-one correspondences
between cyclic subgroups of order $m$ and subgroups of order $p-1$. By Lemma
135 there are $\frac{1}{2}p(p+1)$ cyclic subgroups of order $p-1$. Thus
$c_{m}=\frac{1}{2}p(p+1)$.
The case where $m$ divides $2p$ is similar: there is a one-to-one
correspondence $C\mapsto Z(C)$ between cyclic subgroups of order $m$ and
cyclic subgroups of order $2p$. We can use Lemma 136 to conclude that there
are $c_{m}=p+1$ cyclic subgroups of order $m$.
The case where $m$ divides $p+1$ is also similar: there is a one-to-one
correspondence $C\mapsto Z(C)$ between cyclic subgroups of order $m$ and
cyclic subgroups of order $p+1$. We can use Lemma 142 to conclude that there
are $c_{m}=\frac{1}{2}(p-1)p$ cyclic subgroups of order $m$. ∎
###### Corollary 144.
Suppose that $p$ is an odd prime. The only normal cyclic subgroups of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ are $\\{1\\}$ and $\\{\pm 1\\}$.
###### Proof.
Suppose that $C$ is a normal subgroup of order $m$. By Theorem 141 all cyclic
subgroups of order $m$ are conjugate. Since $C$ is normal, this means that
there is only one subgroup of order $C$. According to the previous
proposition, this can only happen if $m=1,2$. The result follows now from the
fact that $-1$ is the only element of order two (Proposition 125). ∎
###### Corollary 145.
Suppose that $p$ is an odd prime. Then the only normal Sylow-cyclic subgroups
of $\mathrm{SL}_{2}({\mathbb{F}}_{p})$ are $\\{1\\}$ and $\\{\pm 1\\}$.
###### Proof.
Suppose that $N$ is a normal Sylow-cyclic subgroup of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$. By Theorem 59 the commutator subgroup
$N^{\prime}$ of $N$ is cyclic of odd order. Since $N^{\prime}$ is
characteristic in $N$ it must be normal in
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$. By the above corollary, $N^{\prime}$ is
the trivial group. So $N$ is Abelian, hence cyclic since $N$ is Sylow-cyclic.
So by the above corollary $N$ is $\\{1\\}$ or $\\{\pm 1\\}$. ∎
###### Lemma 146.
Suppose that $p$ is an odd prime and that $N$ is a normal subgroup of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ not equal to $\\{1\\}$ or $\\{\pm 1\\}$.
Then $N$ has odd index in $\mathrm{SL}_{2}({\mathbb{F}}_{p})$.
###### Proof.
Let $N_{2}$ be a $2$-Sylow subgroup of $N$. By the above corollary, $N_{2}$
cannot be cyclic, so $N_{2}$ must be a general quaternion group. In
particular, $N$ contains a cyclic subgroup of order $4$. All cyclic subgroups
of order $4$ are conjugate in $\mathrm{SL}_{2}({\mathbb{F}}_{p})$ by Theorem
141. Since $N$ is normal, it must contain all cyclic subgroups of order $4$.
Let $P_{2}$ be a $2$-Sylow subgroup of $\mathrm{SL}_{2}({\mathbb{F}}_{p})$.
Note that $P_{2}$ is generated by its cyclic subgroups of order $4$. So
$P_{2}$ must be contained in $N$. The result follows. ∎
###### Lemma 147.
Let $p$ be an odd prime. Suppose $\mathrm{SL}_{2}({\mathbb{F}}_{p})$ has a
normal subgroup not equal to $\\{1\\}$ or $\\{\pm 1\\}$ or
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$. Then $\mathrm{SL}_{2}({\mathbb{F}}_{p})$
has a normal subgroup of odd prime index.
###### Proof.
Let $N$ be a normal subgroup not equal to $\\{1\\}$ or $\\{\pm 1\\}$ or
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$. Then $N$ has odd index in
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ by the previous lemma. Observe that
$G=\mathrm{SL}_{2}({\mathbb{F}}_{p})/N$ is a Sylow-cyclic group of odd order.
By Corollary 56, $G$ has a nontrivial cyclic quotient. Hence $G$ has a
quotient of odd prime order. This means that
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ has a quotient of odd prime order as well,
and the result follows. ∎
Now we are ready for the third main result:
###### Theorem 148.
Let $p\geq 5$ be an odd prime. Then the normal subgroups of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ are $\\{1\\}$, $\\{\pm 1\\}$, and
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ itself.
###### Proof.
Suppose there are normal subgroups that differ from $\\{1\\}$, $\\{\pm 1\\}$,
and $\mathrm{SL}_{2}({\mathbb{F}}_{p})$ itself. By the previous lemma, there
is a normal subgroup $N$ such that $N$ has prime index $q$ in
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$. Since the order of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ is $(p-1)p(p+1)$ we have that $q$ divides
an element of the set $\\{p-1,p,p+1\\}$. It cannot divide two elements of this
set since $q\geq 5$. So there are three cases based on which element is
divisible by $q$.
The next step is to partition $\mathrm{SL}_{2}({\mathbb{F}}_{p})$ as follows:
* •
Let $\Gamma_{0}=\\{\pm 1\\}$.
* •
Let $\Gamma_{1}$ be all elements $g$ of order $m\geq 3$ such that the
centralizer $Z(\left<g\right>)$ of $\left<g\right>$ has order $p-1$.
* •
Let $\Gamma_{2}$ be all elements $g$ of order $m\geq 3$ such that the
centralizer $Z(\left<g\right>)$ of $\left<g\right>$ has order $2p$.
* •
Let $\Gamma_{3}$ be all elements $g$ of order $m\geq 3$ such that the
centralizer $Z(\left<g\right>)$ of $\left<g\right>$ has order $p+1$.
These sets are clearly disjoint. They form partition of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ since every element outside $\Gamma_{0}$
has order at least $3$ (Proposition 125), and $Z(\left<g\right>)$ has order in
the set $\\{p-1,2p,p+1\\}$ (see Proposition 140 and Corollary 130).
Next we further partition each $\Gamma_{i}$. We start with $\Gamma_{1}$. Every
element $g\in\Gamma_{1}$ is in exactly one cyclic subgroup of order $p-1$,
namely $Z(\left<g\right>)$ (it cannot be in a second cyclic group of order
$p-1$ by Theorem 141). So $\Gamma_{1}$ can be partitioned into sets of the
form $C\cap\Gamma_{1}$ where $C$ is a cyclic subgroup of order $p-1$. What is
the size of $C\cap\Gamma_{1}$? By Corollary 130, if $C$ is a cyclic subgroup
of order $p-1$ then it is a maximal cyclic group, and so $C$ equal to
$Z(\left<g\right>)$ for all $g\in C$ of order $m\geq 3$ (Proposition 140). The
only elements of $C$ not in $C\cap\Gamma_{1}$ are the elements of order $1$ or
$2$, namely the elements $\pm 1$. So each partition $C\cap\Gamma_{1}$ has size
$(p-1)-2=p-3$. From Proposition 143 we know that the number of partitions is
$c_{p-1}=\frac{1}{2}p(p+1)$. So $\Gamma_{1}$ has
$c_{p-1}(p-3)=\frac{1}{2}p(p+1)(p-3)$ elements. A similar calculation can be
made for $\Gamma_{2}$ and $\Gamma_{3}$.
In fact, we can list the proportion of elements of
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ in $\Gamma_{1},\Gamma_{2},\Gamma_{3}$ as
follows:
* •
The proportion of elements of $\mathrm{SL}_{2}({\mathbb{F}}_{p})$ in
$\Gamma_{1}$ is
$\frac{\frac{1}{2}p(p+1)(p-1-2)}{(p-1)p(p+1)}=\frac{1}{2}\frac{p-3}{p-1}<\frac{1}{2}.$
* •
The proportion of elements of $\mathrm{SL}_{2}({\mathbb{F}}_{p})$ in
$\Gamma_{2}$ is
$\frac{(p+1)(2p-2)}{(p-1)p(p+1)}=\frac{2}{p}.$
* •
The proportion of elements of $\mathrm{SL}_{2}({\mathbb{F}}_{p})$ in
$\Gamma_{3}$ is
$\frac{\frac{1}{2}(p-1)p(p+1-2)}{(p-1)p(p+1)}=\frac{1}{2}\frac{p-1}{p+1}<\frac{1}{2}.$
Claim: _the complement of $N$ is contained in $\Gamma_{i}$ for some
$i\in\\{1,2,3\\}$._ For instance, suppose $q$ divides $p-1$. Then clearly
$\Gamma_{0}\subseteq N$. Note that $q$ cannot divide the order of any
$g\in\Gamma_{2}$ since $q$ does not divide $2p$. So each $g\in\Gamma_{2}$ must
have trivial image under
$\mathrm{SL}_{2}({\mathbb{F}}_{p})\to\mathrm{SL}_{2}({\mathbb{F}}_{p})/N$. In
other words, $\Gamma_{2}\subseteq N$. Similarly $q$ cannot divide the order of
any $g\in\Gamma_{3}$ since $q$ does not divide $p+1$. So each $g\in\Gamma_{3}$
must have trivial image under
$\mathrm{SL}_{2}({\mathbb{F}}_{p})\to\mathrm{SL}_{2}({\mathbb{F}}_{p})/N$. In
other words, $\Gamma_{3}\subseteq N$. So the complement of $N$ in
$\mathrm{SL}_{2}({\mathbb{F}}_{p})$ must be contained in $\Gamma_{1}$. The
other cases are similar. In fact, if $N^{c}$ is the complement
$\mathrm{SL}_{2}({\mathbb{F}}_{p})-N$ then
* •
If $q$ divides $p-1$ then $N^{c}\subseteq\Gamma_{1}$.
* •
If $q$ divides $p$ then $N^{c}\subseteq\Gamma_{2}$.
* •
If $q$ divides $p+1$ then $N^{c}\subseteq\Gamma_{3}$.
This leads to a contradiction when $p\geq 5$. Since $N$ is a proper subgroup
of $\mathrm{SL}_{2}({\mathbb{F}}_{p})$, the complement $N^{c}$ must contain at
least one-half of the elements of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$. But
the formulas for proportion of the elements of
$\Gamma_{1},\Gamma_{2},\Gamma_{3}$ in ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$
shows that these proportions are each strictly smaller than one-half. So no
such normal subgroup $N$ of index $q$ exists. ∎
###### Corollary 149.
Let $p\geq 5$ be an odd prime. Then ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ is a
non-solvable Sylow-cycloidal group, and the quotient
$\mathrm{PSL}_{2}({\mathbb{F}}_{p})\;\stackrel{{\scriptstyle\mathrm{def}}}{{\,=\,}}\;{\mathrm{SL}}_{2}({\mathbb{F}}_{p})/\\{\pm
1\\}$
is a simple group.
###### Remark.
The simple groups $\mathrm{PSL}_{2}({\mathbb{F}}_{p})$ were studied even by
Galois, and were the earliest known non-Abelian simple groups outside of the
alternating groups.
The above takes care of the case $p\geq 5$. The case $p=3$ is of interest as
well:
###### Proposition 150.
The group ${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ is a solvable Sylow-cycloidal
group isomorphic to the binary tetrahedral group $2T$.
###### Proof.
The order of ${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ is $2\cdot 3\cdot 4=24$
(Proposition 125). By Theorem 131, ${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ is a
Sylow-cycloidal group whose $2$-Sylow subgroups are not cyclic. Thus the
$2$-Sylow subgroups of ${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ must be
isomorphic to the quaternion group $Q_{8}$ with $8$ elements.
According to Proposition 143, there are $3$ cyclic subgroups of
${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ of order $4$ which is exactly the number
of such cyclic subgroups of $Q_{8}$. Thus there is a unique $2$-Sylow subgroup
of ${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$, and so it is normal. This explains
why ${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ is solvable. Proposition 143 also
asserts that there are $4$ cyclic subgroups of
${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ of order $3$, and these subgroups are
conjugate (Theorem 141). Thus ${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ has no
normal subgroup of order 3. By Proposition 102,
${\mathrm{SL}}_{2}({\mathbb{F}}_{3})$ is isomorphic to the binary tetrahedral
group $2T$. ∎
We have considered cyclic subgroups and normal subgroups of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$. Now we will consider subgroups of odd
order. We will do so using the following result on normalizers of cyclic
subgroups.
###### Lemma 151.
Let $C$ be a cyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ where
$p$ is an odd prime. Then the normalizer $N(C)$ of $C$ is the normalizer
$N(Z(C))$ of the maximal cyclic subgroup $Z(C)$ containing $C$. Furthermore,
* •
If $Z(C)$ has order $p-1$ then $N(Z(C))$ is a non-Abelian subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $2(p-1)$.
* •
If $Z(C)$ has order $2p$ then $N(Z(C))$ is a non-Abelian subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $(p-1)p$.
* •
If $Z(C)$ has order $p+1$ then $N(Z(C))$ is a non-Abelian subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $2(p+1))$.
###### Proof.
Since $Z(C)$ is cyclic, $C$ is a characteristic subgroup of $Z(C)$. Thus $C$
is normal in $N(Z(C))$ since $Z(C)$ is normal in $N(Z(C))$. Hence
$N(Z(C))\subseteq N(C).$
We show equality by showing that both subgroups in this inclusion have the
same order.
We focus on the case where $Z(C)$ has order $p-1$. The other cases are
similar. By Theorem 141 the group ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ acts
transitively on the collection of cyclic subgroups of order $|C|$. By
Proposition 143 the orbit of $C$ under this action has size
$\frac{1}{2}p(p+1)$. So by the orbit-stabilizer theorem, the stabilizer $N(C)$
under conjugation has the following size:
$|N(C)|=\frac{(p-1)p(p+1)}{\frac{1}{2}p(p+1)}=2(p-1).$
This calculation is valid for $N(Z(C))$ as well since $Z(C)$ is cyclic of size
$p+1$. So $N(Z(C))$ has order $2(p-1)$ as well. Equality follows. ∎
###### Corollary 152.
Suppose $H$ is a noncyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$
of odd order, where $p$ is an odd prime. Then $H$ has order dividing $(p-1)p$.
Furthermore $H$ contains a normal subgroup $C$ of order $p$, and $H$ is
subgroup of the normalizer $N(C)$ of $C$ in
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$.
###### Proof.
Since ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ is a Sylow-cycloidal group, the
subgroup $H$ of odd order is a Sylow-cyclic group. By Theorem 59, $H$ has a
nontrivial normal cyclic subgroup $C$, and so $H\subseteq N(C)$. If $C$ has
order dividing $p-1$ or $p-2$ then $N(C)/Z(C)$ is even, and the image of $H$
in $N(C)/Z(C)$ is trivial. Thus $H\subseteq Z(C)$ and so $H$ is cyclic. So we
are left with the case where $C$ has order dividing $2p$. In other words, $C$
has order $p$. By the above proposition $N(C)$ has order $(p-1)p$. ∎
Given the above corollary, the normalizer $N(C)$ of a cyclic subgroup $C$ of
order $p$ warrants our further attention.
###### Lemma 153.
Let $C_{p}$ be a subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order
$p$, where $p$ is an odd prime. Then $C_{p}$ is conjugate to the subgroup
$\left\\{\begin{pmatrix}1&b\\\
0&1\end{pmatrix}\;\middle|\;b\in{\mathbb{F}}_{p}\right\\}\cong{\mathbb{F}}_{p}.$
The normalizer $N(C_{p})$ is conjugate to the subgroup
$\left\\{\begin{pmatrix}a&b\\\
0&a^{-1}\end{pmatrix}\;\middle|\;a\in{\mathbb{F}}^{\times}_{p},\;b\in{\mathbb{F}}_{p}\right\\}.$
Furthermore, $N(C_{p})$ is a semidirect product $C_{p}\rtimes C_{p-1}$ where
$C_{p-1}$ is the cyclic group
$\left\\{\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}\;\middle|\;a\in{\mathbb{F}}^{\times}_{p}\right\\}\cong{\mathbb{F}}_{p}^{\times}.$
Viewing $C_{p-1}$ as ${\mathbb{F}}_{p}^{\times}$ and $C_{p}$ as
${\mathbb{F}}_{p}$, the action of $a\in C_{p-1}$ on $C_{p}$ is given by
$b\mapsto a^{2}b.$
###### Proof.
Let $\alpha$ be a generator of $C_{p}$. By Lemma 129, $\alpha$ has exactly one
eigenvalue in ${\mathbb{F}}_{p}$. If we work in a basis $v_{1},v_{2}$ where
$v_{1}$ is an eigenvector of $\alpha$, then $\alpha$ has the form
$\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}$
for some nonzero $b\in{\mathbb{F}}_{p}$. It follows that $C_{p}$ has the
desired form (up to conjugation). From now on we assume $C_{p}$ is this
particular subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$.
Next consider the subgroup
$H\;\stackrel{{\scriptstyle\mathrm{def}}}{{\,=\,}}\;\left\\{\begin{pmatrix}a&b\\\
0&a^{-1}\end{pmatrix}\;\middle|\;a\in{\mathbb{F}}^{\times}_{p},\;b\in{\mathbb{F}}_{p}\right\\}.$
Observe that
$\begin{pmatrix}a&b\\\ 0&a^{-1}\end{pmatrix}\mapsto a$
is a surjective homomorphism $H\to{\mathbb{F}}_{p}^{\times}$ with kernel
$C_{p}$. So $C_{p}$ is a normal subgroup of $H$, and so $H\subseteq N(C_{p})$.
By the above lemma $N(C_{p})$ has size $(p-1)p$, which is also the size of
$H$. So $H=N(C_{p})$. Note that $C_{p-1}$ (as defined in the statement of the
lemma) is a complement to $C_{p}$ in $H=N(C_{p})$ and so $N(C_{p})$ is a
semidirect product $C_{p}\rtimes C_{p-1}$. The last assertion follows from the
calculation:
$\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\begin{pmatrix}1&b\\\
0&1\end{pmatrix}\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}^{-1}=\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}\begin{pmatrix}1&b\\\
0&1\end{pmatrix}\begin{pmatrix}a^{-1}&0\\\
0&a\end{pmatrix}=\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}$
∎
Here is the next major result:
###### Theorem 154.
Let $H$ be a noncyclic subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of
odd order, where $p$ is an odd prime. Then $H$ contains a cyclic subgroup of
order $p$ and $H$ is contained in the normalizer $N(C)$ of $C$ in
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$. Each such $H$ is isomorphic to
${\mathbb{F}}_{p}\rtimes T$ where $T$ is a nontrivial subgroup of
${\mathbb{F}}_{p}^{\times}$ of odd order. Here the action of $a\in T$ is given
by $b\mapsto a^{2}b$.
Furthermore, given such a semidirect product, ${\mathbb{F}}_{p}\rtimes T$
there is a non-Abelian subgroup of ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$
isomorphic to ${\mathbb{F}}_{p}\rtimes T$.
###### Proof.
The existence of $C$ follows from Corollary 152. By the above lemma, any such
$H$ is isomorphic to a subgroup of
${\mathbb{F}}_{p}\rtimes{\mathbb{F}}_{p}^{\times}$ where ${\mathbb{F}}_{p}$
corresponds to $C$. Observe that $H$ is actually isomorphism to
${\mathbb{F}}_{p}\rtimes T$ where $T$ is the image of $H$ in
${\mathbb{F}}_{p}^{\times}$. Note that $T$ is nontrivial since $H$ is
noncyclic.
Conversely, given a nontrivial odd order subgroup $T$ of
${\mathbb{F}}_{p}^{\times}$, we have a subgroup of $N(C)$ isomorphic
${\mathbb{F}}_{p}\rtimes T$ where here $C$ is any subgroup of
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ of order $p$. (This follows from the
above lemma). ∎
This leads to some important corollaries:
###### Corollary 155.
Suppose $p$ is an odd prime. Then ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$
contains a noncyclic subgroup of order the product of two primes if and only
if $p$ is not a Fermat prime.
###### Proof.
First suppose ${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ has a noncyclic subgroup
$H$ of order $|H|=q_{1}q_{2}$ where $q_{1}\leq q_{2}$ are primes. Since
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ is a Sylow-cycloidal group, any subgroup
of prime squared order is cyclic. Thus $q_{1}\neq q_{2}$.
First suppose that $q_{1}=2$. By Theorem 59 we have that $H$ is isomorphic to
a semidirect product $C_{q}\rtimes C_{2}$ where $C_{q}$ and $C_{2}$ are cyclic
subgroups of order $q=q_{2}$ and $2$ respectively. The only nontrivial action
of $C_{2}$ is the one where the nontrivial element of $C_{2}$ acts on $C_{q}$
by $x\mapsto x^{-1}$. Thus $H$ is a dihedral group of order $2q$, and so
contains $q$ elements of order 2. However,
${\mathrm{SL}}_{2}({\mathbb{F}}_{p})$ has a unique element of order $2$. Thus |
# Construction of smooth rhythms through a monotone invariant measure
Fumio HAZAMA
Tokyo Denki University
Hatoyama, Hiki-Gun, Saitama JAPAN
e-mail<EMAIL_ADDRESS>
###### Abstract
The present article introduces the notion of quasi-smoothness of marked
rhythms through a certain transformation $Ref$, called reformation map. A
marked rhythm consists of a rhythm together with a marker, and the map $Ref$
modifies the marked onset of the rhythm. It is shown that an iteration of the
map $Ref$ transforms an arbitrary marked rhythm into a quasi-smooth one. A
numerical criterion for a marked rhythm to be quasi-smooth is given in terms
of the difference of its rhythm part. Through this criterion, the rhythm part
of any quasi-smooth marked rhythm is shown to be smooth.
$\mathbf{Keywords}$: marked rhythm, reformation map, deformation map, quasi-
smoothness, finite dynamical system, monotone invariant measure.
## 0 Introduction
The main purpose of this paper is to propose a new method of construction of
smooth rhythms. The notion of smoothness of a rhythm is introduced in (Hazama
2022), and is shown to give us a larger category of rhythms than that of
maximally even ones, which have been widely studied (Clough et a. 2000;
Demaine 2009; Toussaint 2013). In (Hazama 2022), a self-map $Rav$ on the space
$\mathbf{R}_{N}^{n}$ of rhythms of $N$ beats with $n$ onsets is introduced,
and a rhythm is defined to be smooth if it is periodic under $Rav$. One of the
main results of (Hazama 2022) gives us a numerical criterion for a rhythm to
be smooth in terms of its width. The proof of the validity of the criterion,
given there, is quite involved however, and the author of the paper expresses
his hope about a possible simplification of its original proof. Our main
result realizes his hope by introducing a self-map $Ref$, called reformation
map, on the space $m\mathbf{R}_{N}^{n}=\mathbf{Z}_{n}\times\mathbf{R}_{N}^{n}$
of marked rhythms. The first coordinate $k$ of a marked rhythm
$(k,\mbox{\boldmath$a$})$ indicates which entry of the rhythm $a$ is modified
by $Ref$. Since $Ref$ does not change any other entries, the dynamical
behavior of $Ref$ will be seen to be much simpler than that of $Rav$, which
changes all the entries at the same time. Our main result shows that, if a
marked rhythm is periodic under $Ref$, then its rhythm part is smooth in the
sense of (Hazama 2022).
The plan of this paper is as follows. Section one introduces our main tool Ref
on $m\mathbf{R}_{N}^{n}$ for our study. The map leads us to a finite dynamical
system $(m\mathbf{R}_{N}^{n},Ref)$, which we denote by $\mathbf{Ref}_{N}^{n}$.
The notion of quasi-smoothness of a marked rhythm is defined by the property
that it is periodic under $Ref$. We investigate the dynamics of $Ref$ through
its descent to a quotient dynamical system
$\mathbf{Def}_{N}^{n}=(m\mathbf{D}_{N}^{n},Def)$ by an automorphism of
$\mathbf{Ref}_{N}^{n}$. The compatibility of $Ref$ and $Def$ is proved at the
end of this section. Section two axiomatizes a crucial point which emerges in
our treatment of the dynamical system $\mathbf{Ref}_{N}^{n}$ together with its
quotient system $\mathbf{Def}_{N}^{n}$. We introduce the notion of “monotone
invariant measure” on an arbitrary finite dynamical system $\mathbf{F}=(X,F)$
with an automorphism $\varphi$. It is shown to descend to the quotient
dynamical system $\overline{\mathbf{F}}=(\overline{X},\overline{F})$ by the
action of $\varphi$. The measure enables one to relate the periodic behavior
of the self-map $F$ on $X$ with that of the self-map $\overline{F}$ on
$\overline{X}$. Section three constructs a monotone invariant measure
$\mu_{m\mathbf{R}}$ on $\mathbf{Ref}_{N}^{n}$. This is modeled after one of
the measures proposed and investigated in (Rota 2001). A general result in the
previous section allows $\mu_{m}{\mathbf{R}}$ to descend to a measure
$\mu_{m\mathbf{D}}$ on $\mathbf{Def}_{N}^{n}$, and enables us to relate the
periodicity of $Def$ and that of $Ref$. Section four introduces the notion of
$\mu_{m\mathbf{D}}$-stability, which is shown to be a necessary condition for
the periodicity. The final step toward our goal is provided by the fact that
$\mu_{m\mathbf{D}}$-stability of $\mbox{\boldmath$D$}\in m\mathbf{D}_{N}^{n}$
implies that the width of $D$ is less than or equal to 1. Thus we arrive at
our main theorem which shows the equivalence of the quasi-smoothness and the
smoothness.
## 1 Reformation map for marked rhythms
In this section we fix some notation and terminology, and introduce our main
subject, the reformation map.
### 1.1 Basic setup
For any integer $m\geq 3$, let $\mathbf{Z}_{m}=\\{0,1,\cdots,m-1\\}$, and let
$R_{m}:\mathbf{Z}\rightarrow\mathbf{Z}_{m}$ denote the function which maps an
integer to its least nonnegative remainder modulo $m$. The usual addition and
subtraction operations on $\mathbf{Z}/m\mathbf{Z}$ are translated to the ones
on the set $\mathbf{Z}_{m}$, which we denote by “$+_{m}$” and “$-_{m}$”
respectively. Namely we define
$\displaystyle a+_{m}b$ $\displaystyle=$ $\displaystyle R_{m}(a+b),$
$\displaystyle a-_{m}b$ $\displaystyle=$ $\displaystyle R_{m}(a-b),$
for any pair $a,b\in\mathbf{Z}_{m}$. For any pair $a,b$ of distinct elements
in $\mathbf{Z}_{m}$, we define the close interval by
$[a,b]_{m}=\\{a,a+_{m}1,\cdots,b\\}\subset\mathbf{Z}_{m}$.
Let $N$ be an integer $\geq 3$, and let $n$ be an integer with $3\leq n\leq
N$. Throughout the paper the integers $N,n$ are assumed to satisfy these
conditions. An $n$-tuple
$\mbox{\boldmath$a$}=(a_{0},a_{1},\cdots,a_{n-1})\in\mathbf{Z}_{N}^{n}$ of
mutually distinct elements of $\mathbf{Z}_{N}$ is a rhythm of length $N$ with
$n$ onsets, if it satisfies the condition
$\displaystyle\sum_{k=0}^{n-1}(a_{k}-_{N}a_{k-_{n}1})=N.$ (1.1)
Here the summation on the left hand side means the usual addition on
$\mathbf{Z}$. We denote by $\mathbf{R}_{N}^{n}$ the set of rhythms of length
$N$ with $n$ onsets. A marked rhythm is the pair $(k,\mbox{\boldmath$a$})$ of
an element of $k\in\mathbf{Z}_{n}$, called the marker, and a rhythm
$\mbox{\boldmath$a$}\in\mathbf{R}_{N}^{n}$, called the rhythm part. The $k$-th
coordinate $a_{k}$ is called the marked entry of the marked rhythm
$(k,\mbox{\boldmath$a$})$. The set of marked rhythm is denoted by
$m\mathbf{R}_{N}^{n}$:
$\displaystyle m\mathbf{R}_{N}^{n}=\mathbf{Z}_{n}\times\mathbf{R}_{N}^{n}.$
Our main ingredient $Ref$ of this papr will be defined as a self-map on
$m\mathbf{R}_{N}^{n}$.
For our investigation of rhythms and marked rhythms, it is indispensable to
consider their differences, whose effectiveness is also fully illustrated in
the comprehensive book (Toussaint 2013). For any rhythm $a$, its difference
$d(\mbox{\boldmath$a$})$ is defined by
$\displaystyle
d(\mbox{\boldmath$a$})=(a_{0}-_{N}a_{n-1},a_{1}-_{N}a_{0},\cdots,a_{n-1}-_{N}a_{n-2})$
Since a rhythm consists of mutually distinct elements of $\mathbf{Z}_{N}$, it
follows from (1.1) that the difference belongs to the set
$\displaystyle\mathbf{D}_{N}^{n}=\\{(d_{0},d_{1},\cdots,d_{n-1})\in\mathbf{Z}_{N}^{*}|\sum_{k=0}^{n-1}d_{k}=N\\},$
where $\mathbf{Z}_{N}^{*}=\mathbf{Z}_{N}\setminus\\{0\\}$. For any marked
rhythm $(k,\mbox{\boldmath$a$})\in m\mathbf{R}_{N}^{n}$, the difference
$\Delta(k,\mbox{\boldmath$a$})$ is defined by
$\displaystyle\Delta(k,\mbox{\boldmath$a$})=(k,d(\mbox{\boldmath$a$})).$
Hence it belongs to
$\displaystyle m\mathbf{D}_{N}^{n}=\mathbf{Z}_{n}\times\mathbf{D}_{N}^{n}.$
Its elements are called marked differences. The first component is called the
marker and the second is the difference part.
Let $tr$ denote the bijective self-map on $\mathbf{R}_{N}^{n}$ defined by
$\displaystyle tr(\mbox{\boldmath$a$})=(a_{0}+_{N}1,\cdots,a_{n-1}+_{N}1)$
(1.2)
for any $\mbox{\boldmath$a$}=(a_{0},\cdots,a_{n-1})\in\mathbf{R}_{N}^{n}$, and
let $\sim_{tr}$ denote the equivalence relation on $\mathbf{R}_{N}^{n}$
generated by the bijection $tr$. It is not hard to see that the space
$\mathbf{D}_{N}^{n}$ is identified with the quotient space
$\mathbf{R}_{N}^{n}/\sim_{tr}$ of $\mathbf{R}_{N}^{n}$ by the equivalence
relation $\sim_{tr}$. Similarly, if we define the bijective self-map $mtr$ on
$m\mathbf{R}_{N}^{n}$ by
$\displaystyle mtr(k,\mbox{\boldmath$a$})=(k,tr(\mbox{\boldmath$a$})),$ (1.3)
and denote by $\sim_{mtr}$ the equivalence relation generated by the bijection
$mtr$, then we see that the space $m\mathbf{D}_{N}^{n}$ is identified with the
quotient space $m\mathbf{R}_{N}^{n}/\sim_{mtr}$ of $m\mathbf{R}_{N}^{n}$ by
the equivalence relation $\sim_{mtr}$.
### 1.2 Definition of the reformation map $Ref$
In this section, we define a self map “Ref”, called reformation map, on
$m\mathbf{R}_{N}^{n}$. It is based on the discrete average
$rav:\mathbf{Z}_{N}\times\mathbf{Z}_{N}\rightarrow\mathbf{Z}_{N}$ defined as
follows.
###### Definition 1.1
For any pair $(a,b)\in\mathbf{Z}_{N}^{2}$, its discrete average
$rav_{N}(a,b)\in\mathbf{Z}_{N}$ is defined by
$\displaystyle rav_{N}(a,b)=a+_{N}\left\lfloor\frac{b-_{N}a}{2}\right\rfloor.$
Note that the discrete average is compatible with the addition by one modulo
$N$, namely we have
$\displaystyle rav_{N}(a+_{N}1,b+_{N}1)=rav_{N}(a,b)+_{N}1.$ (1.4)
Although this appears to be trivial, the whole our study in this paper is
built on this simple equality. We define a self map
$ref:\mathbf{R}_{N}^{3}\rightarrow\mathbf{R}_{N}^{3}$ as follows.
###### Definition 1.2
For any $(a,b,c)\in\mathbf{R}_{N}^{3}$, we put
$\displaystyle ref(a,b,c)=(a,rav_{N}(a,c),c).$ (1.5)
Note that the map $ref$ transforms only the middle entry into the discrete
average of the first and the third entries. It follows from (1.4) that
$\displaystyle ref\circ tr=tr\circ ref.$ (1.6)
Based on this map, we define the reformation map, denoted by $Ref$, on the
space $m\mathbf{R}_{N}^{n}$ as follows.
###### Definition 1.3
For any $(k,\mbox{\boldmath$a$})\in m\mathbf{R}_{N}^{n}$ with
$\mbox{\boldmath$a$}=(a_{0},a_{1},\cdots,a_{n-1})$, we define
$Ref(k,\mbox{\boldmath$a$})$ to be $(k+_{n}1,\mbox{\boldmath$b$})$, where
$\mbox{\boldmath$b$}=(b_{0},b_{1},\cdots,b_{n-1})$ is given in the following
way: Let $(c_{k-_{n}1},c_{k},c_{k+_{n}1})=ref(a_{k-_{n}1},a_{k},a_{k+_{n}1})$,
and define
$\displaystyle b_{i}=\begin{cases}c_{k-_{n}1},&\mbox{ if }i=k-_{n}1,\\\
c_{k},&\mbox{ if }i=k,\\\ c_{k+_{n}1},&\mbox{ if }i=k+_{n}1,\\\ a_{i},&\mbox{
if }i\notin\\{k-_{n}1,k,k+_{n}1\\}\\\ \end{cases}$
We see below through a few examples how the reformation map acts on marked
rhythms. In order to visualize marked rhythms, we employ their circle graphs,
whose construction follows. When
$\mbox{\boldmath$A$}=(k,\mbox{\boldmath$a$})=(k,(a_{0},\cdots,a_{n-1}))$ is an
element of $m\mathbf{R}_{N}^{n}$, we draw a small disk at
$\chi_{N}(a_{j})\in\mathbf{C}$ for any $j\in\mathbf{Z}_{n}$, where
$\chi_{N}:\mathbf{Z}_{N}\rightarrow\mathbf{C}$ is defined by
$\chi_{N}(a)=e^{2\pi ia/N}$, and connect the $n$ disks by a polygonal line.
The marked entry is enclosed in a circle. The numbers $m=0,1,\cdots,N-1$,
surrounding the unit circle, indicate the spots where the nodes $\chi_{N}(m)$
locate.
Example 1 For the marked rhythm $\mbox{\boldmath$A$}=(0,(0,1,2))\in
m\mathbf{R}_{8}^{3}$, the marked entry “$0$” is moved by $Ref$ to the discrete
average “$5$” of “$2$” and “$1$”, and the marker “$0$” goes to the next “$1$”:
Fig 1: $\mbox{\boldmath$A$}=(0,(0,1,2))$ and
$Ref(\mbox{\boldmath$A$})=\mbox{\boldmath$A$}^{(1)}=(1,(5,1,2))$
By iterating the application of $Ref$, the marked rhythm
$\mbox{\boldmath$A$}^{(1)}=(1,(5,1,2))$ on the right of Figure 1 is mapped to
$\mbox{\boldmath$A$}^{(2)}=(2,(5,7,2))$:
Fig 2: $\mbox{\boldmath$A$}^{(1)}=(1,(5,1,2))$ and
$Ref(\mbox{\boldmath$A$}^{(1)})=\mbox{\boldmath$A$}^{(2)}=(2,(5,7,2))$
We see that the rhythm part of the marked rhythm $\mbox{\boldmath$A$}^{(2)}$
is smooth, since its width is equal to one. The main purpose of this paper is
to show that this is a general phenomenon. Namely, we can make the rhythm part
of an arbitrary marked rhythm smooth by iterated applications of $Ref$.
The equality (1.6) implies that the compatibility relation
$\displaystyle Ref\circ mtr=mtr\circ Ref$ (1.7)
holds. Hence the self-map $Ref$ on $m\mathbf{R}_{N}^{n}$ descends to
$m\mathbf{D}_{N}^{n}=m\mathbf{R}_{N}^{n}/\sim_{mtr}$. A concrete expression of
the descended map is given in the next subsection.
### 1.3 Descent of $Ref$ to $m\mathbf{D}_{N}^{n}$
We introduce a self map $Def$, called deformation map, on
$m\mathbf{D}_{N}^{n}$, and show that it is compatible with the reformation map
$Ref$ through the difference map $\Delta$.
###### Definition 1.4
For any $(k,\mbox{\boldmath$d$})\in m\mathbf{D}_{N}^{n}$ with
$\mbox{\boldmath$d$}=(d_{0},d_{1},\cdots,d_{n-1})$, we define
$Def(k,\mbox{\boldmath$d$})$ to be $(k+_{n}1,\mbox{\boldmath$e$})$, where
$\mbox{\boldmath$e$}=(e_{0},e_{1},\cdots,e_{n-1})$ is given in the following
way: Let
$\displaystyle(f_{k},f_{k+_{n}1})=(\left\lfloor\frac{d_{k}+_{N}d_{k+_{n}1}}{2}\right\rfloor,\left\lceil\frac{d_{k}+_{N}d_{k+_{n}1}}{2}\right\rceil),$
and define
$\displaystyle e_{i}=\begin{cases}f_{k},&\mbox{ if }i=k,\\\
f_{k+_{n}1},&\mbox{ if }i=k+_{n}1,\\\ d_{i},&\mbox{ if
}i\notin\\{k,k+_{n}1\\}\\\ \end{cases}$
###### Proposition 1.1
We have the following commutative diagram:
$\displaystyle\begin{CD}m\mathbf{R}_{N}^{n}@>{\Delta}>{}>m\mathbf{D}_{N}^{n}\\\
@V{Ref}V{}V@V{Def}V{}V\\\
m\mathbf{R}_{N}^{n}@>{\Delta}>{}>m\mathbf{D}_{N}^{n}\\\ \end{CD}$ (1.8)
###### Proof.
Let $(k,\mbox{\boldmath$a$})=(k,(a_{0},\cdots,a_{n-1}))\in
m\mathbf{R}_{N}^{n}$ be an arbitrary marked rhythm, and let
$Ref(k,\mbox{\boldmath$a$})=(k+_{n}1,\mbox{\boldmath$b$})$. It follows from
Definition 1.3 that $b_{i}=a_{i}$ for any
$i\in\mathbf{Z}_{n}\setminus\\{k\\}$. Furthermore, if we put
$\Delta(k,\mbox{\boldmath$a$})=(k,\mbox{\boldmath$d$})$ and
$Def(k,\mbox{\boldmath$d$})=(k+_{n}1,\mbox{\boldmath$e$})$, then it follows
from Definition 1.4 that $e_{i}=d_{i}$ for any
$i\in\mathbf{Z}_{n}\setminus\\{k,k+_{n}1\\}$.Therefore, for the proof of the
commutativity, we have only to focus on the behavior of the subtriple
$(a_{k-_{n}1},a_{k},a_{k+_{n}1})$ of the rhythm part $a$. It follows that we
are reduced to proving the assertion for the case when $n=3$. Let
$(1,(a_{0},a_{1},a_{2}))\in m\mathbf{R}_{N}^{3}$ be an arbitrary marked rhythm
with marker equal to 1. It is mapped by $\Delta\circ Ref$ to
$\displaystyle\Delta(Ref(1,(a_{0},a_{1},a_{2})))$
$\displaystyle=\Delta(2,(a_{0},rav_{N}(a_{0},a_{2}),a_{2}))$
$\displaystyle=(2,(a_{0}-_{N}a_{2},rav_{N}(a_{0},a_{2})-_{N}a_{0},a_{2}-_{N}rav_{N}(a_{0},a_{2})))$
$\displaystyle=(2,(a_{0}-_{N}a_{2},\left\lfloor\frac{a_{2}-_{N}a_{0}}{2}\right\rfloor,(a_{2}-_{N}a_{0})-_{N}\left\lfloor\frac{a_{2}-_{N}a_{0}}{2}\right\rfloor))$
$\displaystyle=(2,(a_{0}-_{N}a_{2},\left\lfloor\frac{a_{2}-_{N}a_{0}}{2}\right\rfloor,\left\lceil\frac{a_{2}-_{N}a_{0}}{2}\right\rceil)),$
(1.9)
the last equality being a consequence of the equality
$x=\left\lfloor\frac{x}{2}\right\rfloor+_{N}\left\lceil\frac{x}{2}\right\rceil$,
which holds for any $x\in\mathbf{Z}_{N}$. On the other hand, the value of
$Def\circ\Delta$ at $(1,(a_{0},a_{1},a_{2}))$ is equal to
$\displaystyle Def(\Delta(1,(a_{0},a_{1},a_{2})))$
$\displaystyle=Def(1,(a_{0}-_{N}a_{2},a_{1}-_{N}a_{0},a_{2}-_{N}a_{1}))$
$\displaystyle=(1,(a_{0}-_{N}a_{2},\left\lfloor\frac{(a_{1}-_{N}a_{0})+_{N}(a_{2}-_{N}a_{1})}{2}\right\rfloor,$
$\displaystyle\hskip
142.26378pt\left\lceil\frac{(a_{1}-_{N}a_{0})+_{N}(a_{2}-_{N}a_{1})}{2}\right\rceil)$
$\displaystyle=(1,(a_{0}-_{N}a_{2},\left\lfloor\frac{a_{2}-_{N}a_{0}}{2}\right\rfloor,\left\lceil\frac{a_{2}-_{N}a_{0}}{2}\right\rceil)).$
(1.10)
Since the rightmost sides of (1.9) and (1.10) coincide, we see that the two
maps $Def\circ\Delta$ and $\Delta\circ Def$ give one and the same value at
$(1,(a_{0},a_{1},a_{2}))$. Since even if the marker is not equal to 1, a
similar proof can be given, this completes the proof. ∎
### 1.4 Finite dynamical systems $\mathbf{Ref}_{N}^{n}$ and
$\mathbf{Def}_{N}^{n}$
We introduce two finite dynamical systems employing the maps $Ref$ and $Def$.
###### Definition 1.5
(1) Let $\mathbf{Ref}_{N}^{n}$ denote the dynamical system
$(m\mathbf{R}_{N}^{n},Ref)$.
(2) Let $\mathbf{Def}_{N}^{n}$ denote the dynamical system
$(m\mathbf{D}_{N}^{n},Def)$.
The notion of quasi-smoothness is introduced through the periodicity of the
self map $Ref$. For any finite dynamical system $\mathbf{F}=(X,F)$, let
$Per(\mathbf{F})$ denote the set of periodic points:
$\displaystyle Per(\mathbf{F})=\\{x\in X|\mbox{ There exists a positive
integer }m\mbox{ such that }F^{m}(x)=x\\}.$
###### Definition 1.6
A marked rhythm $\mbox{\boldmath$A$}\in m\mathbf{R}_{N}^{n}$ is quasi-smooth
if $\mbox{\boldmath$A$}\in Per(\mathbf{Ref}_{N}^{n})$. A rhythm
$\mbox{\boldmath$a$}\in\mathbf{R}_{N}^{n}$ is quasi-smooth if there exists a
$k\in\mathbf{Z}_{n}$ such that $\iota_{k}(\mbox{\boldmath$a$})$ is quasi-
smooth as a marked rhythm, where $\iota_{k}:\mathbf{R}_{N}^{n}\rightarrow
m\mathbf{R}_{N}^{n}$ denotes an inclusion defined by
$\iota_{k}(\mbox{\boldmath$a$})=(k,\mbox{\boldmath$a$})\in
m\mathbf{R}_{N}^{n}$.
The main theorem in this article is summarized as follows:
A rhythm is smooth if and only if it is quasi-smooth.
Thus we obtain another iterative method of construction for smooth rhythms,
which is much simpler than the one employed in (Hazama 2022). Furthermore, as
the reader will see in Section four, the long-term behavior of the self map
$Ref$ is far more easier for us to understand than that of $Rav$.
## 2 Monotone invariant measure
We axiomatize the method of construction of quasi-smooth rhythms. This will
enable us to prove the validity not only of our iterative method, but also of
other possible methods which produce certain target objects with a desired
property in a given dynamical system.
### 2.1 Automorphism, invariance, and monotonicity
###### Definition 2.1
Let $\mathbf{F}=(X,F)$ be a finite dynamical system.
(1) A bijection $\varphi:X\rightarrow X$ is called an automorphism of
$\mathbf{F}$ if it commutes with $F$, namely the equality
$\displaystyle F\circ\varphi=\varphi\circ F$ (2.1)
holds. We denote by “$\sim_{\varphi}$” the equivalence relation on $X$
generated by $\varphi$, and put $\overline{X}=X/\sim_{\varphi}$, the quotient
space of $X$ by $\sim_{\varphi}$.
(2) A real valued function $\mu$ on $X$ is called an invariant measure on
$\mathbf{F}$ with respect to an automorphism $\varphi$ of $\mathbf{F}$, if
$\displaystyle\mu(\varphi(x))=\mu(x)$ (2.2)
holds for any $x\in X$.
(3) A real valued function $\mu$ on $X$ is called a monotone measure on
$\mathbf{F}$, if
$\displaystyle\mu(F(x))\geq\mu(x)$ (2.3)
holds for any $x\in X$.
Under the conditions (2.1) and (2.2), the self-map $F$ as well as the function
$\mu$ on $X$ descends to the quotient space $\overline{X}$. We record this
fact for later use.
###### Proposition 2.1
(1) There exists a unique self map $\overline{F}$ on the quotient
$\overline{X}$ such that the following diagram commutes:
$\displaystyle\begin{CD}X@>{F}>{}>X\\\ @V{\pi}V{}V@V{{\pi}}V{}V\\\
\overline{X}@>{\overline{F}}>{}>\overline{X}\\\ \end{CD}$ (2.4)
where $\pi:X\rightarrow\overline{X}$ denotes the quotient map.
(2) The function $\mu$ descends to a unique function $\overline{\mu}$ on the
quotient $\overline{X}$ such that the following diagram commutes:
$\displaystyle\begin{CD}X@>{\mu}>{}>\mathbf{R}\\\
@V{\pi}V{}V@V{{id_{\mathbf{R}}}}V{}V\\\
\overline{X}@>{\overline{\mu}}>{}>\mathbf{R}\\\ \end{CD}$ (2.5)
The following proposition relates the monotonicity of $\mu$ with that of
$\overline{\mu}$.
###### Proposition 2.2
Suppose that $\mu$ is an invariant measure on $\mathbf{F}$ with respect to an
automorphism $\varphi$ of $\mathbf{F}$. Let $\overline{F}$ and
$\overline{\mu}$ denote the induced maps defined as above. Then $\mu$ is a
monotone measure on $\mathbf{F}$ if and only if $\overline{\mu}$ is a monotone
measure on $\overline{\mathbf{F}}$.
###### Proof.
For any $\overline{x}\in\overline{X}$, take an arbitrary
$x\in\pi^{-1}(\\{\overline{x}\\})$. Then we have
$\displaystyle\overline{\mu}(\overline{F}(\overline{x}))$ $\displaystyle=$
$\displaystyle\overline{\mu}(\overline{F}(\pi(x)))$ $\displaystyle=$
$\displaystyle\overline{\mu}(\pi(F(x)))\hskip 42.67912pt(\mbox{by (2.4)})$
$\displaystyle=$ $\displaystyle\mu(F(x))\hskip 56.9055pt(\mbox{by (2.5)})$
Since $\overline{\mu}(\overline{x})=\overline{\mu}(\pi(x))=\mu(x)$, the above
computation implies that the inequality
$\overline{\mu}(\overline{F}(\overline{x}))\geq\overline{\mu}(\overline{x})$
holds if and only if $\mu(F(x))\geq\mu(x)$. This completes the proof. ∎
For any dynamical system $(X,F)$, let $Per(X,F)$ denote the set of periodic
points, namely
$\displaystyle Per(X,F)=\\{x\in X|\mbox{ there exists a positive integer $m$
such that $F^{m}(x)=x$}\\}.$
A monotone measure provides us with a useful necessary condition for an
element $x\in X$ to be periodic.
###### Proposition 2.3
Let $\mu$ be a monotone measure on $(X,F)$. If $x\in Per(X,F)$, then we have
$\displaystyle\mu(F(x))=\mu(x).$ (2.6)
Furthermore, for any positive integer $k$, we have
$\displaystyle\mu(F^{k+1}(x))=\mu(F^{k}(x)).$ (2.7)
###### Proof.
Contrary to (2.6), suppose that we have
$\displaystyle\mu(F(x))>\mu(x).$
Let $m$ be a period of $x$ with respect to $F$. By the monotonicity (2.3) we
have the following chain of inequalities.
$\displaystyle\mu(x)<\mu(F(x))\leq\mu(F^{2}(x))\leq\mu(F^{m}(x))=\mu(x).$
This contradiction shows that we must have $\mu(F(x))=\mu(x)$. The equality
(2.7) follows from (2.6), since $F^{k}(x)$ is also a periodic point. This
completes the proof. ∎
In view of the importance of the condition (2.6) and (2.7) for the study of
the periodic points of a dynamical system, we introduce the following.
###### Definition 2.2
${\rm(1)}$ A point $x\in X$ is said to be $(\mu,F)$-invariant if the equality
$\mu(F(x))=\mu(x)$ holds.
${\rm(2)}$ A point $x\in X$ is said to be $\mu$-stable if it is
$(\mu,F^{\ell})$-invariant for any positive integer $\ell$. The set of
$\mu$-stable points is denoted by $St(\mu)$.
Through this terminology, Proposition 2.3 can be rephrased as follows:
###### Corollary 2.1
Let $\mu$ be a monotone measure on $(X,F)$. Then we have
$\displaystyle Per(X,F)\subset St(\mu).$
In other words, the $\mu$-stability is a necessary condition for the
periodicity.
Under the existence of an automorphism of a dynamical system $(X,F)$, we can
relate the periodic points of $(X,F)$ with the periodic points of the quotient
system $(\overline{X},\overline{F})$.
###### Proposition 2.4
Let $\varphi$ be an automorphism of a finite dynamical system $(X,F)$. For an
element $x\in X$ to belong $Per(X,F)$, it is necessary and sufficient that
$\pi(x)\in Per(\overline{X},\overline{F})$.
###### Proof.
(Necessity) Let $x\in Per(X,F)$ so that there exists a positive integer $m$
such that
$\displaystyle F^{m}(x)=x.$ (2.8)
The left hand side is mapped through $\pi$ to
$\displaystyle\pi(F^{m}(x))=\overline{F}^{m}(\pi(x))$ (2.9)
by the very definition of $\overline{F}$, namely by the equality $\pi\circ
F=\overline{F}\circ\pi$. Since $\pi$ maps the right hand side to $\pi(x)$,
this shows the equality $\overline{F}^{m}(\pi(x))=\pi(x)$. Hence $\pi(x)\in
Per(\overline{X},\overline{F})$.
(Sufficiency). Let $\pi(x)\in Per(\overline{X},\overline{F})$ so that there
exists a positive integer $m$ such that
$\displaystyle\overline{F}^{m}(\pi(x))=\pi(x).$ (2.10)
The left hand side is equal to $\overline{F}^{m}(\pi(x))=\pi(F^{m}(x))$ by
(2.8). Hence, together with (2.9), it implies that
$\displaystyle F^{m}(x)\sim_{\varphi}x.$
It follows from the definition of the relation $\sim_{\varphi}$ that there
exists a positive integer $k$ such that
$\displaystyle F^{m}(x)=\varphi^{k}(x).$ (2.11)
Since $\varphi$ commutes with $F$, applying $F^{m}$ on both sides, we see that
$\displaystyle
F^{2m}(x)=F^{m}(\varphi^{k}(x))=\varphi^{k}(F^{m}(x))=\varphi^{2k}(x),$
the last equality being a consequence of (2.10). Repeating the process, we see
that, for any positive integer $p$, we have
$\displaystyle F^{pm}(x)=\varphi^{pk}(x).$
Therefore, when $p$ is the order of the automorphism $\varphi$, we have
$\displaystyle F^{pm}(x)=\varphi^{pk}(x)=x,$
which shows that $x\in Per(X,F)$. This completes the proof. ∎
## 3 Monotone invariant measure $\mu_{m\mathbf{R}}$ on
$m\mathbf{Ref}_{N}^{n}$
We introduce a $\mathbf{Z}$-valued function $\mu_{m\mathbf{R}}$ on the
dynamical system $\mathbf{Ref}_{N}^{n}=(m\mathbf{R}_{N}^{n},Ref)$. Recall that
the self-map $mtr$ on $m\mathbf{R}_{N}^{n}$ is an automorphism of
$\mathbf{Ref}_{N}^{n}$ in the sense of Definition 2.1 (1), since we have the
equality (1.7). We will show that $\mu_{m\mathbf{R}}$ is a monotone invariant
measure with respect to the automorphism $mtr$.
###### Definition 3.1
For any $\mbox{\boldmath$a$}=(a_{0},\cdots,a_{n-1})\in\mathbf{R}_{N}^{n}$, let
$\mu_{\mathbf{R}}:\mathbf{R}_{N}^{n}\rightarrow\mathbf{Z}$ be defined by
$\displaystyle\mu_{\mathbf{R}}(\mbox{\boldmath$a$})=\prod_{i=0}^{n-1}(a_{i+_{n}1}-_{N}a_{i}),$
(3.1)
where the product on the right hand side is that of integers. For any marked
rhythm $(k,\mbox{\boldmath$a$})\in m\mathbf{R}_{N}^{n}$, we put
$\displaystyle\mu_{m\mathbf{R}}(k,\mbox{\boldmath$a$})=\mu_{\mathbf{R}}(\mbox{\boldmath$a$}).$
(3.2)
By the very definition, the function $\mu_{\mathbf{R}}$ satisfies the
invariance property
$\displaystyle\mu_{\mathbf{R}}\circ tr=\mu_{\mathbf{R}},$ (3.3)
since each factor on the right hand side of (3.1) depends only on the
difference of the neighboring entries. Furthermore, it follows from (3.2) and
(3.3) that
$\displaystyle\mu_{m\mathbf{R}}\circ mtr=\mu_{m\mathbf{R}}.$ (3.4)
Hence we obtain the following.
###### Proposition 3.1
The function $\mu_{m\mathbf{R}}$ on $m\mathbf{R}_{N}^{n}$ defines an invariant
measure on the dynamical system $\mathbf{Ref}_{N}^{n}$ with respect to the
automorphism $mtr$ of $\mathbf{Ref}_{N}^{n}$.
By the invariance (3.3), $\mu_{\mathbf{R}}$ descends to the quotient
$\mathbf{D}_{N}^{n}$. To be more specific, let
$\mu_{\mathbf{D}}:\mathbf{D}_{N}^{n}\rightarrow\mathbf{Z}$ be defined by
$\displaystyle\mu_{\mathbf{D}}(\mbox{\boldmath$d$})=\prod_{i=0}^{n-1}d_{i}$
for any
$\mbox{\boldmath$d$}=(d_{0},d_{1},\cdots,d_{n-1})\in\mathbf{D}_{N}^{n}$. Then
we have the commutative diagram:
$\displaystyle\begin{CD}\mathbf{R}_{N}^{n}@>{\mu_{\mathbf{R}}}>{}>\mathbf{Z}\\\
@V{d}V{}V@V{id_{\mathbf{Z}}}V{}V\\\
\mathbf{D}_{N}^{n}@>{\mu_{\mathbf{D}}}>{}>\mathbf{Z}\\\ \end{CD}$ (3.5)
We extend the domain $\mathbf{D}_{N}^{n}$ of $\mu_{\mathbf{D}}$ to
$m\mathbf{D}_{N}^{n}$ in the same way as above, and denote the extended
measure by $\mu_{m\mathbf{D}}$. Namely, we put
$\displaystyle\mu_{m\mathbf{D}}(k,\mbox{\boldmath$d$})=\mu_{\mathbf{D}}(\mbox{\boldmath$d$}).$
It follows that we have the commutative diagram.
$\displaystyle\begin{CD}m\mathbf{R}_{N}^{n}@>{\mu_{m\mathbf{R}}}>{}>\mathbf{Z}\\\
@V{\Delta}V{}V@V{id_{\mathbf{Z}}}V{}V\\\
m\mathbf{D}_{N}^{n}@>{\mu_{m\mathbf{D}}}>{}>\mathbf{Z}\\\ \end{CD}$ (3.6)
Thus $\mu_{m\mathbf{D}}$ is identified with $\overline{\mu_{m\mathbf{R}}}$,
the map on the quotient $m\mathbf{D}_{N}^{n}=m\mathbf{R}_{N}^{n}/\sim_{mtr}$
induce from $\mu_{m\mathbf{R}}$. Therefore the general result Proposition 2.2
gives us the following specific result.
###### Proposition 3.2
For any $\mbox{\boldmath$A$}\in m\mathbf{R}_{N}^{n}$, it belongs to
$Per(\mathbf{Ref})$ if and only if $\Delta(\mbox{\boldmath$A$})\in
Per(\mathbf{Def})$.
Furthermore we can show the monotonicity of the measures $\mu_{m\mathbf{R}}$
and $\mu_{m\mathbf{D}}$.
###### Proposition 3.3
${\rm(1)}$ For any $(k,\mbox{\boldmath$a$})\in m\mathbf{R}_{N}^{n}$, we have
$\displaystyle\mu_{m\mathbf{R}}(Ref(k,\mbox{\boldmath$a$}))\geq\mu_{m\mathbf{R}}(k,\mbox{\boldmath$a$}).$
${\rm(2)}$ For any $(k,\mbox{\boldmath$d$})\in m\mathbf{D}_{N}^{n}$, we have
$\displaystyle\mu_{m\mathbf{D}}(Def(k,\mbox{\boldmath$d$}))\geq\mu_{m\mathbf{D}}(k,\mbox{\boldmath$d$}).$
(3.7)
${\rm(3)}$ When $\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$, the equality in
$(3.7)$ holds if and only if $|d_{k}-d_{k+_{n}1}|\leq 1$.
###### Proof.
By Proposition 2.2, the item (1) follows from the item (2). Hence we have only
to prove the items (2) and (3). Furthermore the deformation map changes only
an adjacent pair of the entries, we are reduced to showing the following:
###### Lemma 3.1
For any pair $(a,b)$ of positive integers, we have
$\displaystyle a\cdot
b\leq\left\lfloor\frac{a+b}{2}\right\rfloor\cdot\left\lceil\frac{a+b}{2}\right\rceil.$
(3.8)
The equality holds if and only if $|a-b|\leq 1$.
We may assume that $a\leq b$. When $a=b$, both sides of (3.8) are equal to
$a^{2}$, and hence the equality holds. When $b=a+1$, the right hand side of
(3.8) is equal to $a\cdot(a+1)$, and hence the equality holds too. Suppose
that $b>a+1$. In case $a\equiv b\pmod{2}$, the right hand side of (3.8) is
equal to $\displaystyle{\left(\frac{a+b}{2}\right)^{2}}$, and hence the
difference (RHS)-(LHS) becomes $((a+b)^{2}-4ab)/4=(a-b)^{2}/4\geq 0$, which
implies the validity of (3.8). In case $a\not\equiv b\pmod{2}$, let $a+b=2k+1$
for an integer $k$. Then the right hand side is equal to $k(k+1)$. On the
other hand, the left hand side becomes $a((2k+1)-a)$. Note that the quadratic
function $x((2k+1)-x)$ on $\mathbf{Z}$ takes the maximum value at $x=k,k+1$.
Since we are in the case that $b>a+1$, $a$ cannot equal to $k$ and hence the
product $ab=a((2k+1)-a)$ is smaller than $k(k+1)$. This completes the proof. ∎
Combining Proposition 3.1 and 3.3, we arrive at the following objective of
this section.
###### Proposition 3.4
(1) $\mu_{m\mathbf{R}}$ is a monotone invariant measure on the dynamical
system $\mathbf{Ref}_{N}^{n}$ with respect to the automorphism $mtr$ on
$\mathbf{Ref}_{N}^{n}$.
(2) $\mu_{m\mathbf{D}}$ is a monotone measure on the dynamical system
$\mathbf{Def}_{N}^{n}$.
## 4 Main theorem and its proof
In this section we prove our main theorem which characterizes quasi-smooth
rhythms in $m\mathbf{R}_{N}^{n}$. For our proof it is indispensable to focus
on the structure of the set $St(\mu_{m\mathbf{D}})$ of
$\mu_{m\mathbf{D}}$-stable points in the dynamical system
$\mathbf{Def}_{N}^{n}=(m\mathbf{D}_{N}^{n},Def)$. The following property is
fundamental.
###### Proposition 4.1
Assume that $\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})$ is
$(\mu_{m\mathbf{D}},Def)$-invariant, where
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$. Then we have
$(1)$ $|d_{k}-d_{k+_{n}1}|\leq 1$.
$(2.1)$ When $d_{k+_{n}1}=d_{k}+1$, we have
$\displaystyle Def(k,\mbox{\boldmath$d$})=(k+_{n}1,\mbox{\boldmath$d$}).$
(4.1)
$(2.2)$ When $d_{k+_{n}1}=d_{k}-1$ or $d_{k+_{n}1}=d_{k}$, we have
$\displaystyle
Def(k,\mbox{\boldmath$d$})=(k+_{n}1,ad_{k}(\mbox{\boldmath$d$})),$ (4.2)
where $ad_{k}:\mathbf{D}_{N}^{n}\rightarrow\mathbf{D}_{N}^{n}$ denotes the map
which transposes the $k$-th and the $(k+_{n}1)$-th coordinates.
###### Proof.
Let $Def(k,\mbox{\boldmath$d$})=(k+_{n}1,\mbox{\boldmath$e$})$ with
$\mbox{\boldmath$e$}=(e_{0},\cdots,e_{n-1})$. The item (1) is already proved
in Proposition 3.3 (3). When $d_{k+_{n}1}=d_{k}+1$, we have
$(e_{k},e_{k+_{n}1})=(d_{k},d_{k+_{n}1})$ by the definition of the map $Def$,
and hence (4.1) follows. When $d_{k+_{n}1}=d_{k}-1$, we have
$e_{k}=\lfloor\frac{2d_{k}-1}{2}\rfloor=d_{k}-1=d_{k+_{n}1}$, and
$e_{k+_{n}1}=\lceil\frac{2d_{k}-1}{2}\rceil=d_{k}$, and hence the equality
(4.2) holds. Furthermore when $d_{k+_{n}1}=d_{k}$, the equality (4.2) holds
trivially. This completes the proof. ∎
For ease of description, we introduce the following notation.
###### Definition 4.1
Let $\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})\in\mathbf{D}_{N}^{n}$ with
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$.
(1) For any positive integer $\ell$, we express
$Def^{\ell}(\mbox{\boldmath$D$})$ as
$\displaystyle\mbox{\boldmath$D$}^{(\ell)}=(k+_{n}\ell,\mbox{\boldmath$d$}^{(\ell)})=(k+_{n}\ell,(d_{0}^{(\ell)},\cdots,d_{n-1}^{(\ell)})).$
(2) The multiset of all the entries of $d$ is called the content of $D$, and
is denoted by $Cont(\mbox{\boldmath$D$})$.
As a direct consequence of Proposition 4.1, we obtain the following.
###### Corollary 4.1
If $\mbox{\boldmath$D$}\in St(\mu_{m\mathbf{D}})$, then for any positive
integer $\ell$ we have
$\displaystyle Cont(\mbox{\boldmath$D$}^{(\ell)})=Cont(\mbox{\boldmath$D$}).$
Now we can prove the following result which plays a crucial role for our
characterization of the quasi-smoothness. Recall that the difference of the
maximum and the minimum of the entries of
$\mbox{\boldmath$d$}\in\mathbf{D}_{N}^{n}$ is called the width of $d$, and is
denoted by $w(\mbox{\boldmath$d$})$. For any
$\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})\in m\mathbf{D}_{N}^{n}$ we set
$w(\mbox{\boldmath$D$})=w(\mbox{\boldmath$d$})$, and call it the width of $D$
too.
###### Proposition 4.2
If $\mbox{\boldmath$D$}\in St(\mu_{m\mathbf{D}})$, then we have
$\displaystyle w(\mbox{\boldmath$D$})\leq 1.$
###### Proof.
We may assume that the marker of $D$ is equal to 0, and hence we set
$\mbox{\boldmath$D$}=(0,\mbox{\boldmath$d$})$ with
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$. Let
$M=\max\\{d_{i}|i\in\mathbf{Z}_{n}\\}$ and let $i_{m}=\min\\{i|d_{i}=M\\}$,
the first index of the entry which attains the maximum $M$. Then it follows
from Proposition 4.1 (2.2) that, for any $\ell\geq i_{m}$, the marked entry of
$\mbox{\boldmath$D$}^{(\ell)}$ is equal to $M$, and the subsequent entry
belongs to $\\{M,M-1\\}$. Applying $Def$ repeatedly, the same argument shows
that $Cont(\mbox{\boldmath$D$}^{(i_{m}+n-1)})\subset\\{M,M-1\\}$. Therefore it
follows from Corollary 4.1 that $Cont(\mbox{\boldmath$D$})\subset\\{M,M-1\\}$,
and hence we have $w(\mbox{\boldmath$D$})\leq 1$. This completes the proof. ∎
In order to characterize the set $Per(\mathbf{Def}_{N}^{n})$ in
$m\mathbf{D}_{N}^{n}$, we need to specify the value which the marked entry
takes.
###### Definition 4.2
For any $\mbox{\boldmath$D$}\in m\mathbf{D}_{N}^{n}$, we denote the maximum of
its difference part by $\max(\mbox{\boldmath$D$})$. We say that $D$ is max-
marked, if the value of the marked entry is equal to
$\max(\mbox{\boldmath$D$})$.
The following lemma enables us to characterize the subset
$Per(\mathbf{Def}_{N}^{n})$ in $m\mathbf{D}_{N}^{n}$.
###### Lemma 4.1
Suppose that $\mbox{\boldmath$D$}\in m\mathbf{D}_{N}^{n}$ has width $\leq 1$.
$(1)$ If it is max-marked, then for any positive integer $\ell$, the $\ell$-th
deformation $\mbox{\boldmath$D$}^{(\ell)}$ is max-marked.
$(2)$ If it is not max-marked, it is not periodic under the deformation map
$Def$.
###### Proof.
(1) By the assumption that $D$ is max-marked, the alternative (2.2) in
Proposition 4.1 holds, and as a result $\mbox{\boldmath$D$}^{(1)}$ is max-
marked. Repeating this argument we see that every
$\mbox{\boldmath$D$}^{(\ell)}$ is max-marked for any positive integer $\ell$.
(2) Let $\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})$ with
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$, and
$M=max(\mbox{\boldmath$D$})$. It follows from the assumption that $d_{k}=M-1$.
Hence there exists an index $\ell\in\mathbf{Z}_{n}$ such that $d_{\ell}=M$ and
that $d_{m}=M-1$ for every $m\in[k,\ell-_{n}1]$. Then it follows from
Proposition 4.1 that $\mbox{\boldmath$D$}^{(\ell-_{n}k)}$ is max-marked. This
implies by (1) that $\mbox{\boldmath$D$}^{m)}$ is max-marked for any
$m\geq\ell-_{n}k$, and hence the original $D$, which is not max-marked, cannot
belong to a cycle. This completes the proof. ∎
###### Theorem 4.1
${\rm(A)}$ A marked difference $\mbox{\boldmath$D$}\in m\mathbf{D}_{N}^{n}$
belongs to $Per(\mathbf{Def}_{N}^{n})$ if and only if it satisfies the
following two conditions.
$(A.1)$ $w(\mbox{\boldmath$D$})\leq 1$.
$(A.2)$ It is max-marked.
${\rm(B)}$ A marked rhythm $\mbox{\boldmath$A$}\in m\mathbf{R}_{N}^{n}$ is
quasi-smooth if and only if its difference $\Delta(\mbox{\boldmath$A$})$
satisfies the above two conditions $(A.1)$ and $(A.2)$.
###### Proof.
(A) Only-if-part: Suppose that $\mbox{\boldmath$D$}\in
Per(\mathbf{Def}_{N}^{n})$. Since $\mu_{m\mathbf{D}}$ is a monotone measure by
Proposition 3.3 (2), it follows from Corollary 2.1 that $D$ is
$\mu_{m\mathbf{D}}$-stable, which implies by Proposition 4.2 that
$w(\mbox{\boldmath$D$})\leq 1$. Hence the icondition (A.1) holds. The validity
of the condition (A.2) is a direct consequence of Lemma 4.1 (2).
If-part: Let $\mbox{\boldmath$D$}=(k,\mbox{\boldmath$d$})$ with
$\mbox{\boldmath$d$}=(d_{0},\cdots,d_{n-1})$. Under the conditions (A.1) and
(A.2), we have
$\displaystyle\mbox{\boldmath$D$}^{(1)}=(k+_{n}1,ad_{k}(\mbox{\boldmath$d$}))$
by Proposition 4.1 (2.2). Since $\mbox{\boldmath$D$}^{(1)}$ is max-marked
itself, we have
$\displaystyle\mbox{\boldmath$D$}^{(2)}=(k+_{n}2,ad_{k+_{n}1}ad_{k}(\mbox{\boldmath$d$})).$
By repeating these processes, we see that
$\displaystyle\mbox{\boldmath$D$}^{(n-1)}=(k+_{n}(n-1),ad_{k+_{n}(n-2)}\cdots
ad_{k+_{n}1}ad_{k}(\mbox{\boldmath$d$})).$
Notice here that, as the product of adjacent transpositions on
$\mathbf{Z}_{n}$, we have the equality
$\displaystyle(n-2\hskip 5.69054ptn-1)\cdots(1\hskip 5.69054pt2)(0\hskip
5.69054pt1)=(n-1\hskip 5.69054ptn-2\hskip 5.69054pt\cdots\hskip
5.69054pt2\hskip 5.69054pt1\hskip 5.69054pt0),$
and a slightly general one
$\displaystyle(k+_{n}(n-2)\hskip 5.69054ptk+_{n}(n-1))\cdots(k+_{n}1\hskip
5.69054ptk+_{n}2)(k\hskip 5.69054ptk+_{n}1)$ $\displaystyle=(k+_{n}(n-1)\hskip
5.69054ptk+_{n}(n-2)\hskip 5.69054pt\cdots\hskip 5.69054ptk+_{n}2\hskip
5.69054ptk+_{n}1\hskip 5.69054ptk)$ $\displaystyle=(n-1\hskip
5.69054ptn-2\hskip 5.69054pt\cdots\hskip 5.69054pt2\hskip 5.69054pt1\hskip
5.69054pt0),$
the last equality coming from the fact that both of the last two cyclic
permutations map $m$ to $m-_{n}1$ for any $m\in\mathbf{Z}_{n}$. Since the
cycle $(n-1\hskip 5.69054ptn-2\hskip 5.69054pt\cdots\hskip 5.69054pt2\hskip
5.69054pt1\hskip 5.69054pt0)$ is of order $n$, we see that
$\displaystyle\mbox{\boldmath$D$}^{(n(n-1))}=\mbox{\boldmath$D$},$
and hence $\mbox{\boldmath$D$}\in Per(\mathbf{Def}_{N}^{n})$.
(B) The assertion is a direct consequence of Proposition 3.2 which establishes
the equivalence of the periodicity of $A$ and that of
$\Delta(\mbox{\boldmath$A$})$. This completes the proof. ∎
Thus we arrive at the main theorem of this paper.
###### Theorem 4.2
A rhythm is smooth if and only if it is quasi-smooth. In particular, for any
rhythm $\mbox{\boldmath$a$}\in\mathbf{R}_{N}^{n}$, there exists a nonnegative
integer $\ell$ such that $p_{2}(Ref^{\ell}(\iota_{0}(\mbox{\boldmath$a$})))$
is smooth, where $p_{2}:m\mathbf{R}_{N}^{n}\rightarrow\mathbf{R}_{N}^{n}$
denotes the projection onto the second factor.
###### Proof.
Recall that, by the main theorem of (Hazama 2022), a rhythm $a$ is smooth if
and only if
$\displaystyle w(d(\mbox{\boldmath$a$}))\leq 1.$ (4.3)
Therefore the if-part follows from Theorem 4.1 (B). As for the only-if part,
assume that $a$ is smooth. Let
$\mbox{\boldmath$d$}=d(\mbox{\boldmath$a$})=(d_{0},\cdots,d_{n-1})$, and
choose an index $k\in\mathbf{Z}_{n}$ such that
$d_{k}=\max(\mbox{\boldmath$d$})$. Then
$\mbox{\boldmath$D$}=\iota_{k}(\mbox{\boldmath$d$})\in m\mathbf{D}_{N}^{n}$ is
max-marked and satisfies the condition $w(\mbox{\boldmath$D$})\leq 1$ by
(4.3). Therefore Theorem 4.1 (B) implies that $\iota_{k}(\mbox{\boldmath$a$})$
is quasi-smooth, and hence $a$ is quasi-smooth.
∎
References
Clough J, Douthett J, Krantz R (2000), Maximally Even Sets: A Discovery in
Mathematical Music Theory is Found to Apply in Physics. In: Reza S(ed)
Bridges: Mathematical Connections in Art, Music, and Science, Conference
Proceedings 2000. Central Plain Book Manufacturing, Kansas, pp 193-200.
Demaine ED, Gomez-Martin F, Meijer H, Rappaport D, Taslakian P, Toussaint GT,
Winograd T, Wood DR (2009), The distance geometry of music. Computational
Geometry 42: 429-454.
Hazama F (2022) Iterative method of construction for smooth rhythms. Journal
of Mathematics and Music 16:216-235.
Rota GC (2001) The adventures of measure theory. In Crapo H, Senato D(eds)
Algebraic Combinatorics and Computer Science. Springer-Verlag Italia, pp
27-39.
Toussaint. G (2013) The Geometry of Musical Rhythm: What Makes a “Good” Rhythm
Good? CRC Press, New York.
|
# Differential Subordinations for functions with positive real part using
Admissibility conditions
Meghna Sharma Department of Mathematics, University of Delhi, Delhi–110 007,
India<EMAIL_ADDRESS>, Sushil Kumar Bharati Vidyapeeth’s college
of Engineering, Delhi-110063, India<EMAIL_ADDRESS>and Naveen
Kumar Jain Department of Mathematics, Aryabhatta College, Delhi-110021,India
<EMAIL_ADDRESS>
###### Abstract.
Some sufficient conditions on certain constants which are involved in some
first, second and third order differential subordinations associated with
certain functions with positive real part like modified Sigmoid function,
exponential function and Janowski function are obtained so that the analytic
function $p$ normalized by the condition $p(0)=1$, is subordinate to Janowski
function. The admissibility conditions for Janowski function are used as a
tool in the proof of the results. As application, several sufficient
conditions are also computed for Janowski starlikeness.
###### Key words and phrases:
Differential subordination; admissibilty condition; modified sigmoid function;
Janowski function; Exponential function
###### 2010 Mathematics Subject Classification:
30C45, 30C50
The first author is supported by Junior Research Fellowship from Council of
Scientific and Industrial Research, New Delhi, Ref. No.:1753/(CSIR-UGC NET
JUNE, 2018).
## 1\. Introduction
Let $\mathcal{A}$ denote the class of all analytic functions $f$ on the open
unit disc $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ normalized by the
conditions $f(0)=0$ and $f^{\prime}(0)=1$. An analytic function defined on
$\mathbb{D}$ is univalent if $f$ is one-to-one in $\mathcal{A}$. Let
$\mathcal{S}\subset\mathcal{A}$ be the class of all univalent functions.
Denote the class of all analytic functions $f$ having Taylor series expansion
$f(z)=a+a_{n}z^{n}+a_{n+1}z^{n+1}+...$, for some $a\in\mathbb{C}$ and fixed
integer $n$ by $\mathcal{H}[a,n]$. Let $f$ and $g$ be analytic in
$\mathbb{D}$. The function $f$ is subordinate to $g$, and write $f\prec g$, if
there exists an analytic function $w:\mathbb{D}\rightarrow\mathbb{D}$ with
$|w(z)|\leq|z|$ such that $f(z)=g(w(z))$ for all $z\in\mathbb{D}$. In
particular, if $g\in\mathscr{U}$ then, $f\prec g$ if and only if $f(0)=g(0)$
and $f(\mathbb{D})\subseteq g(\mathbb{D})$.
Let $\mathcal{P}$ be the class of functions with positive real part of the
form $p(z)=1+c_{1}z+c_{2}z+\cdots$ over $\mathbb{D}$. Let $A$ and $B$ be
arbitrary fixed numbers which are satisfying the inequality $-1\leq B<A\leq
1$, then the analytic function $p\in\mathcal{P}$ is known as the Janowski
functions associated with right half plane if it satisfies the subordination
relation $p(z)\prec(1+Az)/(1+Bz)$ for all $z\in\mathbb{D}$. The class of such
functions is denoted by $\mathcal{P}[A,B]$. Let $\mathcal{S}^{*}[A,\,B]$ be
the class of Janowski starlike functions $f\in\mathcal{A}$ such that
${zf^{\prime}(z)}/{f(z)}\in\mathcal{P}[A,\,B]$ for $z\in\mathbb{D},$
introduced by Janowski [17]. Let $\mathcal{S}^{*}[A,B]$ be the class of the
functions $f\in\mathcal{A}$ such that the quantity $zf^{\prime}(z)/f(z)$ lies
in the region $\Delta=\\{w\in\mathbb{C}:|(w-1)/(A-Bw)|<1\\}$. As a special
case, we note that $S^{*}[1-2\alpha,-1]=\mathcal{S}^{*}(\alpha)$ that contains
starlike functions of order $\alpha$ [16, 31]. In 2015, authors [24]
introduced the class $\mathcal{S}^{*}_{e}$ which contains the functions
$f\in\mathcal{A}$ satisfying the subordination relation
${zf^{\prime}(z)}/{f(z)}\prec e^{z}$ for all $z\in\mathbb{D}$. In addition, if
$f\in\mathcal{S}^{*}_{e}$, then the quantity $zf^{\prime}(z)/f(z)$ lies in the
domain $\\{w\in\mathbb{C}\colon|\log w|<1\\}$. Recently, Goel and Kumar [14]
introduced and studied the class $\mathcal{S}^{*}_{SG}$ which contains
starlike functions associated with modified sigmoid function
$\phi_{SG}(z):=2/(1+e^{-z})$ and satisfy the subordination relation
$zf^{\prime}(z)/f(z)\prec\phi_{SG}$ for all $z\in\mathbb{D}$. In similar way,
if the function $f\in\mathcal{S}^{*}_{SG}$, then the quantity
$zf^{\prime}(z)/f(z)$ lies in the domain $\\{w\in\mathbb{C}\colon|\log
w/(2-w)|<1\\}$. For details, see[11, 2, 3].
Goluzin [15] studied initially the first order differential subordination
$zp^{\prime}(z)\prec zq^{\prime}(z)$, whenever $zq^{\prime}(z)$ is convex, the
subordination $p\prec q$ holds and the function $q$ is the best dominant.
After this basic result, many authors established several generalizations of
differential subordination implications. In 1981, an article titled
”Differential subordination and univalent functions” by Miller and Mocanu [25]
commenced the study of differential subordination as a generalized version of
differential inequalities. For more details, see [8, 22, 29, 26]. In 1989,
Nunokawa _et al._ [28] studied the first order differential subordination and
proved that $1+zp^{\prime}(z)\prec 1+z$ implies $p(z)\prec 1+z$. They used
this result to provide a criterion so that a normalized analytic function is
univalent in $\mathbb{D}$. Then, Ali _et al._ [4] generalized this result and
proved that $p(z)$ is subordinate to the Janowski function whenever $1+\beta
zp^{\prime}(z)/p^{j}(z)\prec(1+Dz)/(1+Ez)$ for $j=0,1,2$. Here,
$A,B,D,E\in[-1,1]$. Further, Ali _et al._ [5] determined the estimate on
$\beta$ so that the subordination $1+\beta zp^{\prime}(z)/p^{j}(z)$ is
subordinate to the function $\sqrt{1+z}$, $(j=0,1,2)$ which implies that
$p(z)$ is subordinate to $\sqrt{1+z}$. Later, Kumar _et al._ [21] computed a
bound on $\beta$ so that $p(z)\prec\sqrt{1+z}$, whenever $1+\beta
zp^{\prime}(z)/p^{j}(z)\prec(1+Dz)/(1+Ez)$, $(j=0,1,2)$ with $|D|\leq 1$ and
$-1<E<1$. Some of these results were not sharp. Also, it was difficult to
establish analogous results for certain functions with positive real parts
such as $\phi_{0}(z):=1+(z/k)((k+z)/(k-z))$, $(k=1+\sqrt{2})$,
$\phi_{\sin}(z):=1+\sin z$, $\mathcal{Q}(z):=e^{e^{z}-1}$ by the approach used
in above discussed research work. Later in 2018, Kumar and Ravichandran [19]
used some different approach and were able to established best possible bounds
on $\beta$ so that $1+\beta zp^{\prime}(z)/p^{j}(z)$ is subordinate to
$\sqrt{1+z},(1+Az)/(1+Bz)$ which implies that $p(z)\prec e^{z},(1+Az)/(1+Bz)$.
In 2018, Ahuja _et al._ [1] also obtained sharp subordination implications
results for the functions associated with lemniscate of Bernoulli. For recent
work related to first order differential subordinations, reader may refer [9,
10, 12, 13, 32, 33].
In 2018, Madaan _et al._ [23] established first and second order differential
subordinations associated with the lemniscate of Bernoulli using admissibility
technique. Further, Anand _et al._ [7] also studied the generalized first
order differential subordination for the Janowski functions. In 2019, Dorina
Răducanu [30] established second order differential subordination implications
associated with generalized Mittag-Leffler function. For related work, readers
may see [20, 27, 18].
Motivated by the aforesaid work, using admissibility conditions for Janowski
functions, we determine certain conditions on $\beta$, $\gamma$, $A$ and $B$
where $-1\leq B<A\leq 1$ so that $p$ belongs to the class $\mathcal{P}[A,B]$
whenever $1+\beta{zp^{\prime}(z)}/{p^{k}(z)}$,
$1+\beta{(zp^{\prime}(z))^{2}}/{p^{k}(z)}$, $(1-\alpha)p(z)+\alpha
p^{2}(z)+\beta z{p^{\prime}(z)}/{p^{k}(z)}$ $(\alpha\in[0,\,1])$,
$\left({1}/{p(z)}\right)-\beta z{p^{\prime}(z)}/{p^{k}(z)}$,
$p(z)+{zp^{\prime}(z)}/{(\beta p(z)+\gamma)^{k}}$ $(\gamma>0)$, $1+\gamma
zp^{\prime}(z)+\beta z^{2}p^{\prime\prime}(z)$ and $p(z)+\gamma
zp^{\prime}(z)+\beta z^{2}p^{\prime\prime}(z)$ are subordinate to some
functions with positive real part like $e^{z}$, $2/(1+e^{-z})$ and
$(1+Az)/(1+Bz)$, where $k$ is a positive integer. Certain implications of
these results are also discussed which gives sufficient conditions for an
analytic function $f$ to be in the class $\mathcal{S}^{*}[A,B]$.
## 2\. The Admissibility Condition
This section provides some basic facts related to admissibility conditions
associated with Janowski function that will be needed in the proving our main
results.
###### Definition 2.1.
Let $\psi(r,s,t;z):\mathbb{C}^{3}\times\mathbb{D}\to\mathbb{D}$ be analytic
and $h\in\mathcal{S}$. Then a function $p\in\mathcal{A}$, satisfying following
differential subordination relation
$\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\prec h(z)$
is called its _solution_.
Let $\mathfrak{Q}$ denote the class of all functions $q\in\mathcal{S}$ defined
on $\overline{\mathbb{D}}\setminus\mathbf{E}(q)$, where
$\mathbf{E}=\\{\zeta\in\partial\mathbb{D}:\lim\limits_{z\rightarrow\zeta}q(z)=\infty\\}$
such that $q^{\prime}(\zeta)\neq 0$ for
$\zeta\in\partial\mathbb{D}\setminus\mathbf{E}(q)$.
###### Definition 2.2.
Let $\Omega\subset\mathbb{C},q\in\mathfrak{Q}$ and $n\geq 1$. Consider the
class of admissible functions $\Psi_{n}[\Omega,q]$, consists of those
functions $\psi:\mathbb{C}^{3}\times\mathbb{D}\to\mathbb{C}$ which satisfy the
admissibility condition:
$\psi(r,s,t;z)\notin\Omega$
whenever
$r=q(\zeta),s=m\zeta
q^{\prime}(\zeta)\,\,\text{and}\,\,\operatorname{Re}\left(\frac{t}{s}+1\right)\geq
m\operatorname{Re}\left(\frac{\zeta
q^{\prime\prime}(\zeta)}{q^{\prime}(\zeta)}+1\right)$
for $z\in\mathbb{D}$, $\zeta\in\partial\mathbb{D}\setminus\mathbf{E}(q)$ and
$m\geq n\geq 1$.
We write the class $\Psi_{1}[\Omega,q]$ as $\Psi[\Omega,q]$.
###### Theorem 2.3.
[25, Theorem 2.3b, p.28] Let the function $\psi\in\Psi_{n}[\Omega,q]$ and
$q(0)=a$. If $p\in\mathcal{H}[a,n]$, then
$\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\in\Omega\implies
p(z)\prec q(z).$ (2.1)
Let $\Omega$ be a simply connected domain which is not the entire complex
plane, then there exists a conformal mapping $h$ from $\mathbb{D}$ onto
$\Omega$ with $h(0)=\psi(a,0,0;0)$. Therefore, if $p\in\mathcal{H}[a,n]$,
equation (2.1) can be written as
$\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\prec h(z)\implies
p(z)\prec q(z).$ (2.2)
The univalent function $q$ is called _dominant of the solutions_ of the
differential subordination (2.2). The function $\tilde{q}$ is called the _best
dominant_ of (2.2) if $\tilde{q}\prec q$ for all dominants of (2.2).
Consider the function $q(z)=(1+Az)/(1+Bz)$ for $-1\leq A<B\leq 1$. Denote the
class $\Psi_{n}[\Omega,(1+Az)/(1+Bz)]$ by $\Psi_{n}[\Omega;A,B]$. Therefore,
the admissibility conditions for the function $q$ are given as follows:
###### Theorem 2.4.
[7] Let the function $p\in\mathcal{H}[1,n]$ such that $p(z)\not\equiv 1$ and
$n\geq 1$ and $\Omega$ be a subset of $\mathbb{C}$. The class
$\Psi_{n}[\Omega;A,B]$ is defined as the class of all those functions
$\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ such that
$\psi(r,s,t;z)\not\in\Omega\quad\text{whenever}\quad(r,s,t;z)\in\operatorname{Dom}\psi\quad\text{and}$
$r=q(\zeta)=\frac{1+Ae^{i\theta}}{1+Be^{i\theta}},s=m\zeta
q^{\prime}(\zeta)=\frac{m(A-B)e^{i\theta}}{(1+Be^{i\theta})^{2}}\text{ and
}\operatorname{Re}\left(\frac{t}{s}+1\right)\geq\frac{m(1-B^{2})}{1+B^{2}+2B\cos\theta}$
for $z\in\mathbb{D},\theta\in(0,2\pi)$ and $m\geq 1$.
On taking $\psi\in\Psi_{n}[\Omega;A,B]$ in Theorem (2.4), we have
###### Corollary 2.5.
If $(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\in\operatorname{Dom}\psi$
and $\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)\in\Omega$ for
$z\in\mathbb{D}$, then $p$ belongs to $\mathcal{P}[A,B]$.
For $\zeta=e^{i\theta}$, where $\theta\in[0,2\pi)$, let us consider
$|q(\zeta)|=\sqrt{\frac{1+A^{2}+2A\cos\theta}{1+B^{2}+2B\cos\theta}}:=k(\theta)$
(2.3)
whose minimum value is $\frac{1-A}{1-B}$, attained at $\theta=\pi$. Also,
observe that
$|q^{\prime}(\zeta)|=\frac{A-B}{1+B^{2}+2B\cos\theta}:=d(\theta)$ (2.4)
and the minimum value of $d(\theta)$ is $d(0)=\frac{A-B}{1+B^{2}+2B}$ for
$B>0$ and $d(\pi)=\frac{A-B}{1+B^{2}-2B}$ for $B<0$. Note that
$\operatorname{Re}\left(\frac{\zeta
q^{\prime\prime}(\zeta)}{q^{\prime}(\zeta)}\right)=\frac{-2B(B+\cos\theta)}{1+B^{2}+2B\cos\theta}:=g(\theta)$
(2.5)
and the minimum value of $g(\theta)$ is $g(0)=\frac{-2B(B+1)}{1+B^{2}+2B}$ for
$B>0$ and $g(\pi)=\frac{-2B(B-1)}{1+B^{2}-2B}$ for $B<0$.
Using above values, we get the admissibility condition for third order
differential subordination as follows:
In order to the prove our main results, we will use the following lemmas
extensively.
###### Lemma 2.6.
[20] Let $z$ be a complex number. Then
$|\log(1+z)|\geq 1\quad\text{if and only if}\quad|z|\geq e-1.$
###### Lemma 2.7.
Consider the disc $\Delta_{\beta}=\\{w\in\mathbb{C}:|w|<\beta,0<\beta\leq
1\\}$. Then the inequality
$\left|\log\left(\frac{w}{2-w}\right)\right|\geq 1$
holds if and only if $|w|\geq\beta_{0}\approx 0.473519$, where $\beta_{0}$ is
the positive real root of the equation
$(e^{2}-1)\beta^{4}-2(e^{2}-4)\beta^{3}+4(e^{2}-6)\beta^{2}+32\beta-16=0$.
###### Proof.
For $\theta\in[0,2\pi]$, let $w=\beta e^{i\theta}$ be a boundary point of the
disc $\Delta_{\beta}$. Consider
$\displaystyle\left|\log\left(\frac{\beta e^{i\theta}}{2-\beta
e^{i\theta}}\right)\right|^{2}$
$\displaystyle=\left|\frac{1}{2}\log\left(\frac{4\beta^{2}+\beta^{4}-4\beta^{3}\cos\theta}{(4+\beta^{2}-4\beta\cos\theta)^{2}}\right)+i\arctan\left(\frac{2\beta\sin{\theta}}{2\beta\cos{\theta}-\beta^{2}}\right)\right|^{2}$
$\displaystyle=\left(\log\left(\frac{\sqrt{4\beta^{2}+\beta^{4}-2\beta^{3}\cos\theta}}{4+\beta^{2}-4\beta\cos\theta}\right)\right)^{2}+\left(\arctan\left(\frac{2\sin{\theta}}{2\cos{\theta}-\beta}\right)\right)^{2}$
$\displaystyle:=f(\beta,\theta)$
In the interval $[0,2\pi]$, the function $f(\beta,\theta)$ attains its
absolute minimum at $\theta=0$ and therefore, $f(\beta,\theta)\geq f(\beta,0)$
for all $\theta\in[0,2\pi]$. Thus, the inequality
$\left|\log\left(\frac{\beta e^{i\theta}}{2-\beta
e^{i\theta}}\right)\right|\geq 1$
holds if and only if
$f(\beta,0)\geq 1$
or equivalently,
$(e^{2}-1)\beta^{4}-2(e^{2}-4)\beta^{3}+4(e^{2}-6)\beta^{2}+32\beta-16\geq 0.$
(2.6)
Therefore, by Intermediate Value Theorem, the inequality(2.6) holds for
$\beta\geq\beta_{0}\approx 0.473519$ which is the positive real root of the
equation in (2.6). ∎
## 3\. First Order Differential Subordination
In order to prove first order differential subordination relations, we need
the following result due to Swati _et al._ [19], which is a specific case of
Theorem (2.4).
###### Theorem 3.1.
Let $p\in\mathcal{H}[1,n]$ such that $p(z)\not\equiv 1$ and $n\geq 1$. Let
$\Omega$ be a set in $\mathbb{C}$. The class $\Psi_{n}[\Omega;A,B]$ is defined
as the class of all those functions
$\psi:\mathbb{C}^{2}\times\mathbb{D}\to\mathbb{C}$ such that
$\psi(r,s;z)\not\in\Omega\quad\text{whenever}\quad(r,s;z)\in\operatorname{Dom}\psi\quad\text{and}$
$r=\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}\quad\text{ and }\quad
s=\frac{m(A-B)e^{i\theta}}{(1+Be^{i\theta})^{2}}$ (3.1)
for $z\in\mathbb{D},\theta\in(0,2\pi)$ and $m\geq 1$.
Consequently, when $\psi\in\Psi_{n}[\Omega;A,B]$, the above theorem becomes:
Let $z\in\mathbb{D}$. If $(p(z),zp^{\prime}(z);z)\in\operatorname{Dom}\psi$
and $\psi(p(z),zp^{\prime}(z);z)\in\Omega$, then $p\in\mathcal{P}[A,B]$.
Using above theorem, we determine the conditions on $A$, $B$ and $\beta$ so
that the function $\psi(p(z),zp^{\prime}(z);z)$ is subordinate to Modified
Sigmoid function $\phi_{SG}$ and exponential function $e^{z}$ implies $p(z)$
is subordinate to $(1+Az)/(1+Bz)$.
###### Theorem 3.2.
Let $-1\leq B<A\leq 1$ and $k$ be a non-negative integer. Let $p$ be an
analytic function defined on $\mathbb{D}$, which satisfies $p(0)=1$ and
$\beta_{0}\approx 0.475319$. Then the following are sufficient for
$p\in\mathcal{P}[A,B]$.
1. (a)
$1+\beta\frac{zp^{\prime}(z)}{p^{k}(z)}\prec\phi_{SG}$, where
$|\beta|\geq\begin{cases}\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{2-k}}{(A-B)},&\text{
when }0\leq k\leq 2\\\ \frac{\beta_{0}(1+|A|)^{k}}{(A-B)(1-|B|)^{k-2}},&\text{
when }k>2\end{cases}$.
2. (b)
$1+\beta\frac{zp^{\prime}(z)}{p^{k}(z)}\prec e^{z}$, where
$|\beta|\geq\begin{cases}\frac{(e-1)(1+|A|)^{k}(1+|B|)^{2-k}}{(A-B)},&\text{
when }0\leq k\leq 2\\\ \frac{(e-1)(1+|A|)^{k}}{(A-B)(1-|B|)^{k-2}},&\text{
when }k>2\end{cases}$.
###### Proof.
1. (a)
Let
$\Omega=\phi_{SG}(\mathbb{D})=\\{w\in\mathbb{C}:\left|\log\left(\frac{w}{2-w}\right)\right|<1\\}$.
Consider the analytic function
$\psi:\mathbb{C}\setminus\\{0\\}\times\mathbb{C}\times\mathbb{D}\rightarrow\mathbb{C}$
defined as
$\psi(r,s;z)=1+\beta\frac{s}{r^{k}}.$
In accordance with Theorem (3.1), $\psi\in\Psi[\Omega;A,B]$, if
$\psi(r,s;z)\notin\Omega$ where $r$ and $s$ are given in the equation (3.1).
Therefore, it is enough to show that the required subordination holds if
$\left|\log\left(\frac{\psi(r,s;z)}{2-\psi(r,s;z)}\right)\right|\geq 1.$ (3.2)
1. (i)
When $0\leq k\leq 2$, let us consider
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|1+\beta\frac{m(A-B)e^{i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2-k}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2-k}+\beta
m(A-B)e^{i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2-k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)-|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{2-k}|}{|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{2-k}|}$
$\displaystyle\geq\frac{|\beta|m(A-B)-(1+|A|)^{k}(1+|B|)^{2-k}}{(1+|A|)^{k}(1+|B|)^{2-k}}$
$\displaystyle=:\phi(m).$
By First Derivative Test, $\phi(m)$ is an increasing function for $m\geq 1$.
This implies $\phi(m)\geq\phi(1)$ for all $m\geq 1$. Hence, the last
inequality reduces to
$|\psi(r,s;z)|\geq\phi(1)$
where
$|\phi(1)|=\frac{|\beta|(A-B)-(1+|A|)^{k}(1+|B|)^{2-k}}{(1+|A|)^{k}(1+|B|)^{2-k}}.$
Using Lemma (2.7), the inequality (3.2) is true if
$|\phi(1)|\geq\beta_{0}$
or
$\frac{|\beta|(A-B)-(1+|A|)^{k}(1+|B|)^{2-k}}{(1+|A|)^{k}(1+|B|)^{2-k}}\geq\beta_{0}.$
or
$|\beta|(A-B)-(1+|A|)^{k}(1+|B|)^{2-k}\geq\beta_{0}(1+|A|)^{k}(1+|B|)^{2-k},$
which shows that $\psi(r,s;z)\notin\Omega$ for
$|\beta|\geq\beta_{0}\frac{(1+|A|)^{k}(1+|B|)^{2-k}}{(A-B)}.$
2. (ii)
When $k>2$, observe that
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|1+\beta\frac{m(A-B)(1+Be^{i\theta})^{k-2}e^{i\theta}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}+\beta
m(A-B)(1+Be^{i\theta})^{k-2}e^{i\theta}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)|(1+Be^{i\theta})^{k-2}|-|(1+Ae^{i\theta})^{k}|}{|(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|B|)^{k-2}-(1+|A|)^{k}}{(1+|A|)^{k}}$
$\displaystyle=:\phi(m)$
As in the earlier case, note that $\phi(m)$ is an increasing function of $m$.
Hence, $\phi(m)\geq\phi(1)$ for all $m\geq 1$. By Lemma (2.7), (3.2) holds if
$|\psi(r,s;z)|\geq|\phi(1)|\geq\beta_{0}$
where
$\phi(1)=\frac{|\beta|(A-B)(1-|B|)^{k-2}-(1+|A|)^{k}}{(1+|A|)^{k}}.$
Therefore, for $|\beta|\geq\beta_{0}\frac{(1+|A|)^{k}}{(A-B)(1-|B|)^{k-2}}$,
$\psi\in\Psi[\Omega;A,B]$ and hence, $p\in\mathcal{P}[A,B]$.
2. (b)
Consider the domain $\Omega=\\{w\in\mathbb{C}:|\log w|<1\\}$. Let
$\psi:\mathbb{C}\backslash\\{0\\}\times\mathbb{C}\times\mathbb{D}\rightarrow\mathbb{C}$
be defined by
$\psi(r,s;z)=1+\beta\frac{s}{r^{k}}.$
By Theorem (3.1), $\psi$ belongs to $\Psi[\Omega;A,B]$ if
$\psi(r,s;z)\notin\Omega$ for $z\in\mathbb{D}$. This implication holds if
$|\log(\psi(r,s;z))|\geq 1$ (3.3)
Since
$\displaystyle|\log(\psi(r,s;z))|$
$\displaystyle=\left|\log\left(1+\beta\frac{s}{r^{k}}\right)\right|$
$\displaystyle=\left|\log\left(1+\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Be^{i\theta})^{2}(1+Ae^{i\theta})^{k}}\right)\right|,$
by Lemma(2.6), inequality (3.3) holds if and only if
$\left|\frac{\beta
m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Be^{i\theta})^{2}(1+Ae^{i\theta})^{k}}\right|\geq
e-1.$ (3.4)
1. (i)
When $0\leq k\leq 2$ , consider
$\displaystyle\left|\frac{\beta
m(A-B)e^{i\theta}}{(1+Be^{i\theta})^{2-k}(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)}{|(1+Be^{i\theta})^{2-k}||(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|(A-B)}{(1+|B|)^{2-k}(1+|A|)^{k}}\quad(\because
m\geq 1)$ $\displaystyle\geq e-1$
if
$|\beta|\geq\frac{(e-1)(1+|A|)^{k}(1+|B|)^{2-k}}{(A-B)}.$
which shows that $\psi(r,s;z)\notin\Omega$ and hence the required result
holds.
2. (ii)
When $k>2$, observe that
$\displaystyle\left|\frac{\beta
m(A-B)e^{i\theta}(1+Be^{i\theta})^{k-2}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)|(1+Be^{i\theta})^{k-2}|}{|(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|(A-B)(1-|B|)^{k-2}}{(1+|A|)^{k}}\quad(\because
m\geq 1)$
Therefore, (3.4) holds if
$\frac{|\beta|(A-B)(1-|B|)^{k-2}}{(1+|A|)^{k}}\geq e-1$
or equivalently,
$|\beta|\geq\frac{(e-1)(1+|A|)^{k}}{(A-B)(1-|B|)^{k-2}}.$
which proves that $\psi(r,s;z)\notin\Omega$ and hence, $p\in\mathcal{P}[A,B]$.
∎
###### Corollary 3.3.
Let $f\in\mathcal{A}$ and $\beta_{0}\approx 0.473519$. Set
$\mathcal{G}(z):=\frac{f^{\prime}(z)}{f(z)}-z\left(\frac{f^{\prime}(z)}{f(z)}\right)^{2}+\frac{zf^{\prime\prime}(z)}{f(z)}$.
If one of the following subordination holds, then $f\in S^{*}[A,B].$
1. (a)
$1+\beta z\mathcal{G}(z)\prec\phi_{SG}$ for
$|\beta|(A-B)\geq\beta_{0}(1+|B|)^{2}$,
2. (b)
$1+\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)\mathcal{G}(z)\prec\phi_{SG}$
for $|\beta|(A-B)\geq\beta_{0}(1+|A|)(1+|B|)$,
3. (c)
$1+\frac{\beta}{z}\mathcal{G}(z)\prec\phi_{SG}$ for
$|\beta|(A-B)\geq\beta_{0}(1+|A|)^{2}$,
4. (d)
$1+\beta z\mathcal{G}(z)\prec e^{z}$ for $|\beta|(A-B)\geq(e-1)(1+|B|)^{2}$,
5. (e)
$1+\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)\mathcal{G}(z)\prec e^{z}$ for
$|\beta|(A-B)\geq(e-1)(1+|A|)(1+|B|)$,
6. (f)
$1+\frac{\beta}{z}\mathcal{G}(z)\prec e^{z}$ for
$|\beta|(A-B)\geq(e-1)(1+|A|)^{2}$.
###### Theorem 3.4.
Let $-1\leq B<A\leq 1$, $\beta_{0}\approx 0.473519$, $k$ be a non-negative
integer and $p\in\mathcal{A}$ such that $p(0)=1$. If any of the following
subordinations holds true, then $p(z)\in\mathcal{P}[A,B]$.
1. (a)
$1+\beta\frac{(zp^{\prime}(z))^{2}}{p^{k}(z)}\prec\phi_{SG}$, where
$|\beta|\geq\begin{cases}\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{4-k}}{(A-B)^{2}},&\text{
when }0\leq k\leq 4\\\
\frac{\beta_{0}(1+|A|)^{k}}{(A-B)^{2}(1-|B|)^{k-4}},&\text{ when
}k>4\end{cases}$
2. (b)
$1+\beta\frac{(zp^{\prime}(z))^{2}}{p^{k}(z)}\prec e^{z}$, where
$|\beta|\geq\begin{cases}\frac{(e-1)(1+|A|)^{k}(1+|B|)^{4-k}}{(A-B)^{2}},&\text{
when }0\leq k\leq 4\\\ \frac{(e-1)(1+|A|)^{k}}{(A-B)^{2}(1-|B|)^{k-4}},&\text{
when }k>4\end{cases}$
###### Proof.
1. (a)
Consider $\Omega$ as in Theorem (3.2)(a). Define the analytic function
$\psi:\mathbb{C}\setminus\\{0\\}\times\mathbb{C}\times\mathbb{D}\to\mathbb{C}$
as
$\psi(r,s;z)=1+\beta\frac{s^{2}}{r^{k}}.$
Therefore, we have
$\psi(r,s;z)=1+\beta\frac{m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4}}.$
Proceeding in the similar manner as in Theorem (3.2), we have the following
two cases.
1. (i)
When $0\leq k\leq 4$, let us consider
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|1+\beta\frac{m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4-k}+\beta
m^{2}(A-B)^{2}e^{2i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4-k}}\right|$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}-|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{4-k}|}{|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{4-k}|}$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}-(1+|A|)^{k}(1+|B|)^{4-k}}{(1+|A|)^{k}(1+|B|)^{4-k}}$
$\displaystyle=:\phi(m)$
Simple observation shows that $\phi(m)$ in an increasing function for $m\geq
1$. The required subordination result holds if $\psi(r,s;z)\notin\Omega$. So,
using Lemma (2.7), it is concluded that $p\in\mathcal{P}[A,B]$ if
$|\beta|\geq\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{4-k}}{(A-B)^{2}}$.
2. (ii)
When $k>4$, observe that
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|1+\beta\frac{m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}+\beta
m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k-4}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}|(1+Be^{i\theta})^{k-4}|-|(1+Ae^{i\theta})^{k}|}{|(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}(1-|B|)^{k-4}-(1+|A|)^{k}}{(1+|A|)^{k}}$
$\displaystyle=:\phi(m)$
Noting that $\phi^{\prime}(m)>0$ for $m\geq 1$ and proceeding as in the part
(i), we get the desired subordination result.
2. (b)
Let $\Omega=\\{w\in\mathbb{C}:|\log w|<1\\}$ be the domain. Let
$\psi:\mathbb{C}\backslash\\{0\\}\times\mathbb{C}\times\mathbb{D}\rightarrow\mathbb{C}$
be defined as
$\psi(r,s;z)=1+\beta\frac{s^{2}}{r^{k}}.$
On the similar lines of the proof of Theorem (3.2) and using Lemma (2.6), we
get the desired result if
$\left|\frac{\beta
m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4}}\right|\geq
e-1.$ (3.5)
1. (i)
When $0\leq k\leq 4$ , consider
$\displaystyle\left|\frac{\beta
m^{2}(A-B)^{2}e^{2i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{4-k}}\right|$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}}{|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{4-k}|}$
$\displaystyle\geq\frac{|\beta|(A-B)^{2}}{(1+|A|)^{k}(1+|B|)^{4-k}}\quad(\because
m\geq 1)$
Now
$\frac{|\beta|(A-B)^{2}}{(1+|A|)^{k}(1+|B|)^{4-k}}\geq e-1$
if and only if
$|\beta|\geq\frac{(e-1)(1+|A|)^{k}(1+|B|)^{4-k}}{(A-B)^{2}}.$
2. (ii)
When $k>4$, observe that
$\displaystyle\left|\frac{\beta
m^{2}(A-B)^{2}e^{2i\theta}(1+Be^{i\theta})^{k-4}}{(1+Ae^{i\theta})^{k}}\right|$
$\displaystyle\geq\frac{|\beta|m^{2}(A-B)^{2}|(1-Be^{i\theta})^{k-4}|}{|(1+Ae^{i\theta})^{k}|}$
$\displaystyle\geq\frac{|\beta|(A-B)^{2}(1-|B|)^{k-4}}{(1+|A|)^{k}}\quad(\because
m\geq 1)$ $\displaystyle\geq e-1.$
which yields the desired estimate on $\beta$.∎
###### Corollary 3.5.
Let $f\in\mathcal{A}$ and $\mathcal{G}(z)$ be same as defined in Corollary
(3.3). Then each of the following subordination imply $f\in S^{*}[A,B]$.
1. (a)
$1+\beta(z\mathcal{G}(z))^{2}\prec\phi_{SG}$ for
$|\beta|(A-B)^{2}\geq\beta_{0}(1+|B|)^{4}$,
2. (b)
$1+\beta
z\left(\frac{f(z)}{f^{\prime}(z)}\right)(\mathcal{G}(z))^{2}\prec\phi_{SG}$
for $|\beta|(A-B)^{2}\geq\beta_{0}(1+|A|)(1+|B|)^{3}$,
3. (c)
$1+\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)^{2}(\mathcal{G}(z))^{2}\prec\phi_{SG}$
for $|\beta|(A-B)^{2}\geq\beta_{0}(1+|A|)^{2}(1+|B|)^{2}$,
4. (d)
$1+\beta(z\mathcal{G}(z))^{2}\prec e^{z}$ for
$|\beta|(A-B)^{2}\geq(e-1)(1+|B|)^{4}$,
5. (e)
$1+\beta z\left(\frac{f(z)}{f^{\prime}(z)}\right)(\mathcal{G}(z))^{2}\prec
e^{z}$ for $|\beta|(A-B)^{2}\geq(e-1)(1+|A|)(1+|B|)^{3}$,
6. (f)
$1+\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)^{2}(\mathcal{G}(z))^{2}\prec
e^{z}$ for $|\beta|(A-B)^{2}\geq(e-1)(1+|A|)^{2}(1+|B|)^{2}$.
###### Theorem 3.6.
Let $-1\leq B<A\leq 1$, $\beta_{0}\approx 0.475319$ and $\alpha\in[0,1]$. If
$p\in\mathcal{P}$ satisfy the differential subordination
$(1-\alpha)p(z)+\alpha p^{2}(z)+\beta
z\frac{p^{\prime}(z)}{p^{k}(z)}\prec\phi_{SG}(z),\text{ where }$
$|\beta|\geq\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{2}+(1-\alpha){(1+|A|)^{k+1}(1+|B|)+\alpha(1+|A|)^{k+2}}}{(A-B)(1-|B|)^{k}}.$
then $p(z)\prec(1+Az)/(1+Bz)$.
###### Proof.
Let $\Omega$ be same as in Theorem (3.2)(a). Let the function $\psi$ be
defined as
$\psi(r,s;z)=(1-\alpha)r+\alpha r^{2}+\beta\frac{s}{r^{k}}.$
Substituting the values of $r$ and $s$ from equation(3.1), we get
$\psi(r,s;z)=(1-\alpha)\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\alpha\frac{(1+Ae^{i\theta})^{2}}{(1+Be^{i\theta})^{2}}+\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}$
Then
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|(1-\alpha)\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\alpha\frac{(1+Ae^{i\theta})^{2}}{(1+Be^{i\theta})^{2}}+\beta\frac{m(A-B)e^{i\theta}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2-k}}\right|$
$\displaystyle=\left|\frac{(1-\alpha)(1+Ae^{i\theta})^{k+1}(1+Be^{i\theta})+\alpha(1+Ae^{i\theta})^{k+2}+\beta
m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|B|)^{k}-(1-\alpha)|(1+Ae^{i\theta})^{k+1}||(1+Be^{i\theta})|-\alpha|(1+Ae^{i\theta})^{k+2}|}{|(1+Ae^{i\theta})^{k}||(1+Be^{i\theta})^{2}|}$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|B|)^{k}-(1-\alpha)(1+|A|)^{k+1}(1+|B|)-\alpha(1+|A|)^{k+2}}{(1+|A|)^{k}(1+|B|)^{2}}$
$\displaystyle=:\phi(m)$
Verify that the function $\phi(m)$ is increasing $\forall$ $m\geq 1$ and
hence, attains its minimum value at $m=1$. Since
$\frac{|\beta|(A-B)(1-|B|)^{k}-(1-\alpha)(1+|A|)^{k+1}(1+|B|)-\alpha(1+|A|)^{k+2}}{(1+|A|)^{k}(1+|B|)^{2}}\geq\beta_{0}$
by Theorem (3.1) and Lemma (2.7), we get the desired result.
∎
###### Remark 3.7.
For $\alpha=0$, the above theorem reduces to the following result.
###### Corollary 3.8.
Let $p$ be an analytic function satisfying $p(0)=1$ and
$\beta>\beta_{0}\approx 0.475319$. Then each of the following subordinations
is sufficient to imply $p\in\mathcal{P}[A,B]$.
$p(z)+\beta z\frac{p^{\prime}(z)}{p^{k}(z)}\prec\phi_{SG},\text{ where
}|\beta|\geq\frac{\beta_{0}(1+|A|)^{k}(1+|B|)^{2}+{(1+|A|)^{k+1}(1+|B|)}}{(A-B)(1-|B|)^{k}}.$
###### Theorem 3.9.
Let $\beta_{0}\approx 0.475319$ and $k$ be a non-negative integer. If
$p\in\mathcal{P}$ and satisfies the differential subordination
$\left(\frac{1}{p(z)}\right)-\beta
z\frac{p^{\prime}(z)}{p^{k}(z)}\prec\phi_{SG}(z),\text{ where
}|\beta|\geq\begin{cases}\frac{\beta_{0}(1+|A|)^{k+1}(1+|B|)^{2-k}+(1+|A|)^{k}(1+|B|)^{3-k}}{(A-B)(1-|A|)},&\text{
when }0\leq k\leq 2\\\
\frac{\beta_{0}(1+|A|)^{k}+(1+|A|)^{k}(1+|B|)}{(A-B)(1-|A|)(1-|B|)^{k-2}},&\text{
when }k>2\end{cases},$
then $p(z)\prec(1+Az)/(1+Bz)$.
###### Proof.
Let $\Omega$ be same as in Theorem(3.2)(a). Consider the analytic function
$\psi$ defined as
$\psi(r,s;z)=\frac{1}{r}-\beta\frac{s}{r^{k}}.$
Substituting the values of $r$ and $s$ as given in equation (3.1), we get
$\psi(r,s;z)=\frac{1+Be^{i\theta}}{1+Ae^{i\theta}}-\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}$
Proceeding as in Theorem (3.2)(a), the following two cases arises.
1. (i)
When $0\leq k\leq 2$, consider
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|\frac{1+Be^{i\theta}}{1+Ae^{i\theta}}-\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{3-k}-\beta
m(A-B)e^{i\theta}(1+Ae^{i\theta})}{(1+Ae^{i\theta})^{k+1}(1+Be^{i\theta})^{2-k}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|A|)-(1+|A|)^{k}(1+|B|)^{3-k}}{(1+|A|)^{k+1}(1+|B|)^{2-k}}$
$\displaystyle=:\phi(m)$
Observe that $\phi^{\prime}(m)>0$ for $m\geq 1$. In view of above and Lemma
(2.7), simple computations gives the desired bound on $\beta$ in terms of $A$
and $B$.
2. (ii)
When $k>2$, consider
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|\frac{1+Be^{i\theta}}{1+Ae^{i\theta}}-\beta\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k}}{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})^{2}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})^{k}(1+Be^{i\theta})-\beta
m(A-B)e^{i\theta}(1+Ae^{i\theta})(1+Be^{i\theta})^{k-2}}{(1+Ae^{i\theta})^{k+1}}\right|$
$\displaystyle\geq\frac{|\beta|m(A-B)(1-|A|)(1-|B|)^{k-2}-(1+|A|)^{k}(1+|B|)}{(1+|A|)^{k+1}}$
$\displaystyle=:\phi(m)$
Since, the function $\phi(m)$ is increasing for $m\geq 1$, similar
computations as done in case (i) gives the required result.
∎
###### Corollary 3.10.
Let $\beta_{0}\approx 0.475319$, $f$ be an analytic function and
$\mathcal{G}(z)$ be same as in Corollary (3.3). Then each of the following
subordinations imply that $f\in\mathcal{S}^{*}[A,B]$.
1. (a)
$\frac{f(z)}{zf^{\prime}(z)}-\beta z\mathcal{G}(z)\prec\phi_{SG}$ for
$|\beta|(A-B)(1-|A|)\geq\beta_{0}(1+|A|)(1+|B|)^{2}+(1+|B|)^{3}$,
2. (b)
$\frac{f(z)}{zf^{\prime}(z)}-\beta\frac{f(z)}{f^{\prime}(z)}\mathcal{G}(z)\prec\phi_{SG}$
for $|\beta|(A-B)(1-|A|)\geq\beta_{0}(1+|A|)^{2}(1+|B|)+(1+|A|)(1+|B|)^{2}$,
3. (c)
$\frac{f(z)}{zf^{\prime}(z)}-\beta\left(\frac{f(z)}{f^{\prime}(z)}\right)^{2}\mathcal{G}(z)\prec\phi_{SG}$
for $|\beta|(A-B)(1-|A|)\geq\beta_{0}(1+|A|)^{3}+(1+|A|)^{2}(1+|B|)$.
###### Theorem 3.11.
Suppose $-1\leq B<A\leq 1$, $\gamma>0$, $\beta_{0}\approx 0.475319$ and $k$ be
a non-negative integer. Let $p$ be an analytic function satisfying the
differential subordination
$p(z)+\frac{zp^{\prime}(z)}{(\beta p(z)+\gamma)^{k}}\prec\phi_{SG}(z),\text{
where }$
$(A-B)(1-|B|)\geq\beta_{0}(1+|B|)^{2-k}(\beta(1+|A|)+\gamma(1+|B|))^{k}(2+|A|+|B|),\text{
when }0\leq k\leq 2,$
$(A-B)(1-|B|)^{k-1}\geq\beta_{0}(2+|A|+|B|)(\beta(1+|A|)+\gamma(1+|B|))^{k},\text{
when }k>2.$
Then $p\in\mathcal{P}[A,B]$.
###### Proof.
Let $\Omega$ be the domain as defined in Theorem(3.2)(a). Define the function
$\psi(r,s;z):\mathbb{C}\backslash\\{0\\}\times\mathbb{C}\times\mathbb{D}\rightarrow\mathbb{C}$
as
$\psi(r,s;z)=r+\frac{s}{(\beta r+\gamma)^{k}}.$
Then using equation(3.1), the function $\psi$ becomes
$\psi(r,s;z)=\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\frac{m(A-B)(1+Be^{i\theta})^{k}e^{i\theta}}{(1+Be^{i\theta})^{2}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}$
In view of Theorem (3.1), the desired subordination $p\prec(1+Az)/(1+Bz)$ will
follow if we show that $\psi\in\Psi[\Omega;A,B]$. For this, it suffices to
show that
$\left|\log\left(\frac{\psi(r,s;z)}{2-\psi(r,s;z)}\right)\right|\geq 1.$
1. (i)
When $0\leq k\leq 2$, observe that
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\frac{m(A-B)e^{i\theta}}{(1+Be^{i\theta})^{2-k}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}\right|$
$\displaystyle=\left|\frac{\begin{multlined}(1+Ae^{i\theta})(1+Be^{i\theta})^{2-k}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}\\\
+m(A-B)e^{i\theta}(1+Be^{i\theta})\end{multlined}(1+Ae^{i\theta})(1+Be^{i\theta})^{2-k}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}\\\
+m(A-B)e^{i\theta}(1+Be^{i\theta})}{(1+Be^{i\theta})^{3-k}(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}\right|$
$\displaystyle\geq\frac{\begin{multlined}m(A-B)|(1+Be^{i\theta})|-|(1+Ae^{i\theta})||(1+Be^{i\theta})^{2-k}|\\\
|(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|\end{multlined}m(A-B)|(1+Be^{i\theta})|-|(1+Ae^{i\theta})||(1+Be^{i\theta})^{2-k}|\\\
|(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}{|(1+Be^{i\theta})^{3-k}||(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}$
$\displaystyle\geq\frac{m(A-B)(1-|B|)-(1+|A|)(1+|B|)^{2-k}|(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}{(1+|B|)^{3-k}(\beta(1+|A|)+\gamma(1+|B|))^{k}}$
Similar analysis as done in Theorem (3.2)(a) gives that
$\psi(r,s;z)\notin\Omega$ for
$(A-B)(1-|B|)\geq\beta_{0}(1+|B|)^{2-k}(\beta(1+|A|)+\gamma(1+|B|))^{k}(2+|A|+|B|).$
2. (ii)
When $k>2$, consider
$\displaystyle|\psi(r,s;z)|$
$\displaystyle=\left|\frac{1+Ae^{i\theta}}{1+Be^{i\theta}}+\frac{m(A-B)e^{i\theta}(1+Be^{i\theta})^{k-2}}{(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}\right|$
$\displaystyle=\left|\frac{(1+Ae^{i\theta})(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}+m(A-B)e^{i\theta}(1+Be^{i\theta})^{k-1}}{(1+Be^{i\theta})(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}}\right|$
$\displaystyle\geq\frac{m(A-B)|(1+Be^{i\theta})^{k-1}|-|(1+Ae^{i\theta})||(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}{|(1+Be^{i\theta})||(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}$
$\displaystyle\geq\frac{m(A-B)(1-|B|)^{k-1}-(1+|A|)|(\beta(1+Ae^{i\theta})+\gamma(1+Be^{i\theta}))^{k}|}{(1+|B|)|(\beta(1+|A|)+\gamma(1+|B|))^{k}|}$
On the similar lines as in proof of part (i), we get the desired result.
∎
## 4\. Second order differential subordination
In this section, sufficient conditions are obtained so that the subordination
implication
$p(z)\prec\frac{1+Az}{1+Bz}$
holds whenever $\psi(p(z),zp^{\prime}(z),z^{2}p^{\prime\prime}(z);z)$ is
subordinate to Modified Sigmoid function, exponential function and Janowski
function.
###### Theorem 4.1.
Let $-1<B<A<1$, $\gamma>0$, $\beta>0$ and $\beta_{0}\approx 0.475319$. Let $p$
be an analytic function satisfying $p(0)=1$. Then, each of the following is
sufficient for $p\in\mathcal{P}[A,B]$.
$\displaystyle(a)$ $\displaystyle 1+\gamma zp^{\prime}(z)+\beta
z^{2}p^{\prime\prime}(z)\prec\phi_{SG}(z),\text{where}$
$\displaystyle(A-B)[\gamma(1+B^{2}+2B)-2B\beta(B+1)]\geq(\beta_{0}+1)(1+B^{2}+2B)^{2}\text{
for }B>0\text{ and }$
$\displaystyle(A-B)[\gamma(1+B^{2}-2B)-2B\beta(B-1)]\geq(\beta_{0}+1)(1+B^{2}-2B)^{2}\text{
for }B<0.$ $\displaystyle(b)$ $\displaystyle 1+\gamma zp^{\prime}(z)+\beta
z^{2}p^{\prime\prime}(z)\prec e^{z},\text{where}$
$\displaystyle(A-B)[\gamma(1+B^{2}+2B)-2B\beta(B+1)]\geq(e-1)(1+B^{2}+2B)^{2}\text{
for }B>0\text{ and }$
$\displaystyle(A-B)[\gamma(1+B^{2}-2B)-2B\beta(B-1)]\geq(e-1)(1+B^{2}-2B)^{2}\text{
for }B<0.$ $\displaystyle(c)$ $\displaystyle 1+\gamma zp^{\prime}(z)+\beta
z^{2}p^{\prime\prime}(z)\prec(1+Az)/(1+Bz),\text{where}$
$\displaystyle(A-B)(1-B^{2})[\gamma(1+B^{2}+2B)-2B\beta(B+1)]-|B|(A-B)(1+B^{2}+2B)^{2}$
$\displaystyle\geq(A-B)(1+B^{2}+2B)^{2},\text{ for }B>0\text{ and }$
$\displaystyle(A-B)(1-B^{2})[\gamma(1+B^{2}-2B)-2B\beta(B-1)]-|B|(A-B)(1+B^{2}-2B)^{2}$
$\displaystyle\geq(A-B)(1+B^{2}-2B)^{2},\text{ for }B<0.$
###### Proof.
1. (a)
Let
$\Omega=\phi_{SG}(\mathbb{D})=\\{w\in\mathbb{C}:\left|\log\left({w}/{(2-w)}\right)\right|<1\\}$.
Consider the analytic function
$\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ defined as
$\psi(r,s,t;z)=1+\gamma s+\beta t$
For $\psi\in\Psi[\Omega;A,B]$, we must have $\psi(r,s,t;z)\notin\Omega$. By
Theorem (2.4), this implication is true if
$\left|\log\left(\frac{\psi(r,s,t;z)}{2-\psi(r,s,t;z)}\right)\right|\geq 1.$
(4.1)
By Lemma (2.7), the inequality (4.1) holds if and only if
$|\psi(r,s,t;z)|\geq\beta_{0}$. A calculation shows that
$\displaystyle|\psi(r,s,t;z)|$ $\displaystyle=|1+\gamma s+\beta t|$
$\displaystyle\geq
1+\gamma|s|\left|1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right|-1$
$\displaystyle\geq\gamma|s|\operatorname{Re}\left(1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right)-1$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)-1$
$\displaystyle\geq\begin{cases}\frac{m(A-B)}{1+B^{2}+2B}\left(\gamma+\frac{-2B(B+1)\beta
m}{1+B^{2}+2B}\right)-1,B>0\\\
\frac{m(A-B)}{1+B^{2}-2B}\left(\gamma+\frac{-2B(B-1)\beta
m}{1+B^{2}-2B}\right)-1,B<0\end{cases}$ $\displaystyle:=\phi(m),$
where $d(\theta)$ and $g(\theta)$ are given by (2.4) and (2.5) respectively.
Observe that $\phi(m)$ is increasing function for $m\geq 1$. Therefore, we
have $|\psi(r,s,t;z)|\geq\phi(1)\geq\beta_{0}$ and hence,
$\psi\in\Psi[\Omega;A,B]$. By Theorem 2.4, $p(z)\prec(1+Az)/(1+Bz)$.
2. (b)
Consider the domain $\Omega=\\{w\in\mathbb{C}:|\log w|<1\\}$. Let the function
$\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ be defined as
$\psi(r,s,t;z)=1+\gamma s+\beta t.$
For $\psi\in\Psi[\Omega;A,B]$, we must have $\psi(r,s,t;z)\notin\Omega$. In
order to satisfy this relation, it is sufficient to show that
$|\log(\psi(r,s,t;z))|\geq 1.$
Since
$\displaystyle\left|\gamma s\left(1+\frac{\beta t}{\gamma s}\right)\right|$
$\displaystyle\geq\gamma|s|\left|1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right|$
$\displaystyle\geq\gamma|s|\operatorname{Re}\left(1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right)$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)$
$\displaystyle\geq\begin{cases}\frac{m(A-B)}{1+B^{2}+2B}\left(\gamma-\frac{2B(B+1)\beta
m}{1+B^{2}+2B}\right),B>0\\\
\frac{m(A-B)}{1+B^{2}-2B}\left(\gamma-\frac{2B(B-1)\beta
m}{1+B^{2}-2B}\right),B<0\end{cases}$ $\displaystyle:=\phi(m)$
and $\phi(m)$ is increasing function of $\phi$, by Lemma 2.6, $\left|\gamma
s\left(1+\frac{\beta t}{\gamma s}\right)\right|\geq e-1$. Thus, by Theorem
2.4, $p(z)\prec(1+Az)/(1+Bz)$.
3. (c)
Consider the domain
$\Omega=\left\\{w\in\mathbb{C}:\left|w-\frac{1-AB}{1-B^{2}}\right|<\frac{A-B}{1-B^{2}}\right\\}.$
Let $\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ be defined as
$\psi(r,s,t;z)=1+\gamma s+\beta t$. Now, $\psi\in\Psi[\Omega;A,B]$, if
$\psi(r,s,t;z)\notin\Omega$. On the similar lines on the proof of part(a),
$\displaystyle\left|\psi(r,s,t;z)-\frac{1-AB}{1-B^{2}}\right|$
$\displaystyle=\left|1+\gamma s+\beta t-\frac{1-AB}{1-B^{2}}\right|$
$\displaystyle\geq\gamma|s|\operatorname{Re}\left(1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right)-\frac{|B|(A-B)}{1-B^{2}}$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)-\frac{|B|(A-B)}{1-B^{2}}$
$\displaystyle:=\phi(m).$
Using the values of $d(\theta)$ and $g(\theta)$ as given in the equations
(2.4) and (2.5), and first derivative test for function $\phi$ we have,
for $B>0$,
$\displaystyle\left|\psi(r,s,t;z)-\frac{1-AB}{1-B^{2}}\right|$
$\displaystyle\geq\frac{(A-B)}{1+B^{2}+2B}\left(\gamma-\frac{2B(B+1)\beta}{1+B^{2}+2B}\right)-\frac{|B|(A-B)}{1-B^{2}}$
$\displaystyle\geq\frac{A-B}{1-B^{2}},$
and for $B<0$,
$\displaystyle\left|\psi(r,s,t;z)-\frac{1-AB}{1-B^{2}}\right|$
$\displaystyle\geq\frac{(A-B)}{1+B^{2}-2B}\left(\gamma-\frac{2B(B-1)\beta}{1+B^{2}-2B}\right)-\frac{|B|(A-B)}{1-B^{2}}$
$\displaystyle\geq\frac{A-B}{1-B^{2}}.$
Therefore, $\psi\in\Psi[\Omega;A,B]$ and hence, by Theorem 2.4,
$p(z)\prec(1+Az)/(1+Bz)$.∎
###### Corollary 4.2.
Let $\gamma$ and $\beta$ be positive integers and $f$ be an analytic function.
Set
$\displaystyle\mathcal{H}(z)=$ $\displaystyle
1+\gamma\Bigg{(}\frac{z^{2}f^{\prime\prime}(z)}{f(z)}-\Bigg{(}\frac{zf^{\prime}(z)}{f(z)}\Bigg{)}^{2}+\frac{zf^{\prime}(z)}{f(z)}\Bigg{)}+\beta\Bigg{(}\frac{z^{3}f^{\prime\prime\prime}(z)}{f(z)}$
$\displaystyle+\frac{2z^{2}f^{\prime\prime}(z)}{f(z)}+2\Bigg{(}\frac{zf^{\prime}(z)}{f(z)}\Bigg{)}^{3}-2\Bigg{(}\frac{zf^{\prime}(z)}{f(z)}\Bigg{)}^{2}-\frac{3z^{3}f^{\prime}(z)f^{\prime\prime}(z)}{f(z)^{2}}\Bigg{)}.$
Then, $f\in\mathcal{S}^{*}[A,B]$ if any one of the following condition hold.
$\displaystyle(a)$ $\displaystyle\mathcal{H}(z)\prec\phi_{SG}(z),\text{where}$
$\displaystyle(A-B)[\gamma(1+B^{2}+2B)-2B\beta(B+1)]\geq(\beta_{0}+1)(1+B^{2}+2B)^{2}\text{
for }B>0\text{ and }$
$\displaystyle(A-B)[\gamma(1+B^{2}-2B)-2B\beta(B-1)]\geq(\beta_{0}+1)(1+B^{2}-2B)^{2}\text{
for }B<0.$ $\displaystyle(b)$ $\displaystyle\mathcal{H}(z)\prec
e^{z},\text{where}$
$\displaystyle(A-B)[\gamma(1+B^{2}+2B)-2B\beta(B+1)]\geq(e-1)(1+B^{2}+2B)^{2}\text{
for }B>0\text{ and }$
$\displaystyle(A-B)[\gamma(1+B^{2}-2B)-2B\beta(B-1)]\geq(e-1)(1+B^{2}-2B)^{2}\text{
for }B<0.$ $\displaystyle(c)$
$\displaystyle\mathcal{H}(z)\prec(1+Az)/(1+Bz),\text{where}$
$\displaystyle(A-B)(1-B^{2})[\gamma(1+B^{2}+2B)-2B\beta(B+1)]-|B|(A-B)(1+B^{2}+2B)^{2}$
$\displaystyle\geq(A-B)(1+B^{2}+2B)^{2},\text{ for }B>0\text{ and }$
$\displaystyle(A-B)(1-B^{2})[\gamma(1+B^{2}-2B)-2B\beta(B-1)]-|B|(A-B)(1+B^{2}-2B)^{2}$
$\displaystyle\geq(A-B)(1+B^{2}-2B)^{2},\text{ for }B<0.$
###### Theorem 4.3.
Suppose $-1<B<A<1$, $\beta_{0}\approx 0.475319$, $\beta>0$ and $\gamma>0$. Let
$p$ be an analytic function which satisfies the condition $p(0)=1$ and the
following inequalities holds:
$(A-B)(1+B)[\gamma(1+B^{2}+2B)-2B(B+1)\beta]-(1+A)(1+B^{2}+2B)^{2}\geq\beta_{0}(1+B)(1+B^{2}+2B)^{2}$
for $B>0$, and
$(A-B)(1+B)[\gamma(1+B^{2}-2B)-2B(B-1)\beta]-(1+A)(1+B^{2}-2B)^{2}\geq\beta_{0}(1+B)(1+B^{2}-2B)^{2}$
for $B<0$. Then,
$p(z)+\gamma zp^{\prime}(z)+\beta z^{2}p^{\prime\prime}(z)\prec\phi_{SG}(z)$
implies
$p\prec\frac{1+Az}{1+Bz}.$
###### Proof.
Let $\Omega$ be the domain defined in Theorem (3.2)(a). Consider the function
$\psi:\mathbb{C}^{3}\times\mathbb{D}\rightarrow\mathbb{C}$ defined as
$\psi(r,s,t;z)=r+\gamma s+\beta t.$
For $\psi$ to be in $\Psi[\Omega;A,B]$, we must have
$\psi(r,s,t;z)\notin\Omega$. By Theorem (2.4), this result is true if
$\left|\log\left(\frac{\psi(r,s,t;z)}{2-\psi(r,s,t;z)}\right)\right|\geq 1$
Using Lemma (2.7), this inequality holds if and only if
$|\psi(r,s,t;z)|\geq\beta_{0}$ (4.2)
where $\beta_{0}$ is the positive real root of the equation in (2.6). If
$k(\theta)$, $g(\theta)$ and $d(\theta)$ are given by the equations (2.3),
(2.5) and (2.4) respectively, then
$\displaystyle|\psi(r,s,t;z)|$ $\displaystyle=|r+\gamma s+\beta t|$
$\displaystyle\geq\gamma|s|\left|1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right|-|r|$
$\displaystyle\geq\gamma|s|\operatorname{Re}\left(1+\left(\frac{\beta}{\gamma}\right)\frac{t}{s}\right)-|r|$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)-|k(\theta)|$
$\displaystyle\geq m\gamma
d(\theta)\left(1+\left(\frac{\beta}{\gamma}\right)(mg(\theta)+m-1)\right)-\frac{1+A}{1+B}$
$\displaystyle:=\phi(m).$
Since $\phi(m)$ is increasing function, we have, for $B>0$,
$|\psi(r,s,t;z)|\geq\frac{(A-B)}{1+B^{2}+2B}\left(\gamma-\frac{2B(B+1)\beta}{1+B^{2}+2B}\right)-\frac{1+A}{1+B}\text{
and }$
for $B<0$,
$|\psi(r,s,t;z)|\geq\frac{(A-B)}{1+B^{2}-2B}\left(\gamma-\frac{2B(B-1)\beta}{1+B^{2}-2B}\right)-\frac{1+A}{1+B}.$
Therefore, inequality (4.2) is satisfied and hence, we get the required
result. ∎
## References
* [1] O. P. Ahuja, S. Kumar and V. Ravichandran, Applications of first order differential subordination for functions with positive real part, Stud. Univ. Babeş-Bolyai Math. 63 (2018), no. 3, 303–311.
* [2] Rosihan M. Ali, Naveen kumar Jain and V. Ravichandran,Convolutions of certain analytic functions, J. Analysis, Vol. 18 (2010), pp. 1-8.
* [3] R. M. Ali, N. K. Jain and V. Ravichandran, Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane, Appl. Math. Comput. 218 (2012), no. 11, 6557–6565.
* [4] R. M. Ali, V. Ravichandran and N. Seenivasagan, Sufficient conditions for Janowski starlikeness, Int. J. Math. Math. Sci. 2007, Art. ID 62925, 7 pp.
* [5] R. M. Ali, N. E. Cho, V. Ravichandran, and S. S. Kumar, Differential subordination for functions associated with the lemniscate of Bernoulli, Taiwanese J. Math. 16 (2012), no. 3, 1017–1026.
* [6] O. Altıntaş, Certain applications of subordination associated with neighborhoods, Hacet. J. Math. Stat. 39 (2010), no. 4, 527–534.
* [7] S. Anand, S. Kumar and V. Ravichandran, Differential subordination for Janowski functions with positive real part, Stud. Univ. Babeş-Bolyai Math. to appear (2020).
* [8] J. A. Antonino and S. Romaguera, Strong differential subordination to Briot-Bouquet differential equations, J. Differential Equations 114 (1994), no. 1, 101–105.
* [9] N. Bohra, S. Kumar and V. Ravichandran, Some special differential subordinations, Hacet. J. Math. Stat. 48 (2019), no. 4, 1017–1034.
* [10] N. E. Cho, S. Kumar, V. Kumar and V. Ravichandran, Differential subordination and radius estimates for starlike functions associated with the Booth lemniscate, Turkish J. Math., 42 (2018), 1380–1399.
* [11] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York, 1983.
* [12] A. Ebadian, N. H. Mohammed, E. A. Adegani , and T. Bulboacă, New results for some generalizations of starlike and convex functions, J. Funct. Spaces 2020, Art. ID 7428648, 12 pp.
* [13] S. Gandhi, S. Kumar and V. Ravichandran, First order differential subordinations for Carathéodory functions, Kyungpook Math. J. 58 (2018), no. 2, 257–270.
* [14] P. Goel and S. Sivaprasad Kumar, Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 1, 957–991.
* [15] G. M. Goluzin, On the majorization principle in function theory, Dokl. Akad. Nauk. SSSR, 42 (1935), pp. 647–650.
* [16] A. W. Goodman, Univalent functions. Vol. II, Mariner Publishing Co., Inc., Tampa, FL, 1983.
* [17] W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970/71), 159–177.
* [18] S. Kanas, Differential subordination related to conic sections, J. Math. Anal. Appl. 317 (2006), no. 2, 650–658.
* [19] S. Kumar and V. Ravichandran, Subordinations for functions with positive real part, Complex Anal. Oper. Theory 12 (2018), no. 5, 1179–1191.
* [20] S. S. Kumar and P. Goel, Starlike functions and higher order differential subordinations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2020), no. 4, 192.
* [21] S. S. Kumar, V. Kumar, V. Ravichandran and N. E. Cho, Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli, J. Inequal. Appl. 2013, 2013:176, 13 pp.
* [22] M. Liu, On certain class of analytic functions defined by differential subordination, Acta Math. Sci. Ser. B (Engl. Ed.) 22 (2002), no. 3, 388–392.
* [23] V. Madaan, A. Kumar and V. Ravichandran, Starlikeness associated with lemniscate of Bernoulli, Filomat 33 (2019), no. 7, 1937–1955.
* [24] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 1, 365–386.
* [25] S. S. Miller and P. T. Mocanu, Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker, Inc., New York, 2000.
* [26] S. S. Miller and P. T. Mocanu, Subordinants of differential superordinations, Complex Var. Theory Appl. 48 (2003), no. 10, 815–826.
* [27] A. Naz, S. Nagpal and V. Ravichandran, Star-likeness associated with the exponential function, Turkish J. Math. 43 (2019), no. 3, 1353–1371.
* [28] M. Nunokawa, M. Obradović and S. Owa, One criterion for univalency, Proc. Amer. Math. Soc. 106 (1989), no. 4, 1035–1037.
* [29] Özkan and O. Altıntaş, Applications of differential subordination, Appl. Math. Lett. 19 (2006), no. 8, 728–734.
* [30] D. Răducanu, Differential subordination associated with generalized Mittag-Leffler function, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), no. 2, 435–452.
* [31] M. S. Robertson, Certain classes of starlike functions, Michigan Math. J. 32 (1985), no. 2, 135–140.
* [32] K. Sharma, N. E. Cho, and V. Ravichandran, Sufficient conditions for strong starlikeness. Bull. Iran. Math. Soc. appeared online (2020).
* [33] L. A. Wani and A. Swaminathan, Starlike and convex functions associated with a Nephroid domain. Bull. Malays. Math. Sci. Soc. to appear (2020).
|
# Self-Consistent Determination of Long-Range Electrostatics in Neural Network
Potentials
Ang Gao<EMAIL_ADDRESS>Department of Physics, Beijing University of Posts
and Telecommunications, Beijing, China 100876 Richard C. Remsing
<EMAIL_ADDRESS>Department of Chemistry and Chemical Biology, Rutgers
University, Piscataway, NJ, USA 08854
###### Abstract
Machine learning has the potential to revolutionize the field of molecular
simulation through the development of efficient and accurate models of
interatomic interactions. In particular, neural network models can describe
interactions at the level of accuracy of quantum mechanics-based calculations,
but with a fraction of the cost, enabling the simulation of large systems over
long timescales with ab initio accuracy. However, implicit in the construction
of neural network potentials is an assumption of locality, wherein atomic
arrangements on the scale of about a nanometer are used to learn interatomic
interactions. Because of this assumption, the resulting neural network models
cannot describe long-range interactions that play critical roles in dielectric
screening and chemical reactivity. To address this issue, we introduce the
self-consistent field neural network (SCFNN) model — a general approach for
learning the long-range response of molecular systems in neural network
potentials. The SCFNN model relies on a physically meaningful separation of
the interatomic interactions into short- and long-range components, with a
separate network to handle each component. We demonstrate the success of the
SCFNN approach in modeling the dielectric properties of bulk liquid water, and
show that the SCFNN model accurately predicts long-range polarization
correlations and the response of water to applied electrostatic fields.
Importantly, because of the separation of interactions inherent in our
approach, the SCFNN model can be combined with many existing approaches for
building neural network potentials. Therefore, we expect the SCFNN model to
facilitate the proper description of long-range interactions in a wide-variety
of machine learning-based force fields.
Computer simulations have transformed our understanding of molecular systems
by providing atomic-level insights phenomena of wide importance. The earliest
models used efficient empirical descriptions of interatomic interactions, and
similar force field-based simulations form the foundation of molecular
simulations today Allen and Tildesley (2017). However, it is difficult to
describe processes like chemical reactions that involve bond breakage and
formation, as well as electronic polarization effects within empirical force
fields. The development of quantum mechanics-based ab initio simulations
enabled the description of these complex processes, leading to profound
insights across scientific disciplines Tuckerman _et al._ (1996); Car and
Parrinello (1985); Chen _et al._ (2018); Geissler _et al._ (2001); Lee _et
al._ (2008); Walker, Crowley, and Case (2008); Senn and Thiel (2006); Dal
Peraro _et al._ (2007). The vast majority of these first principles
approaches rely on density functional theory (DFT), and the development of
increasingly accurate density functionals has greatly improved the reliability
of ab initio predictions Sun, Ruzsinszky, and Perdew (2015); Sun _et al._
(2016); Chen _et al._ (2017); Furness _et al._ (2020); Zhang _et al._
(2021a); Adamo and Barone (1999). But, performing electronic structure
calculations are expensive, and first principles simulations are limited to
small system sizes and short time scales.
The prohibitive expense of ab initio simulations can be overcome through
machine learning. Armed with a set of ab initio data, machine learning can be
used to train neural network (NN) potentials that describe interatomic
interactions at the same level of accuracy as the ab initio methods, but with
a fraction of the cost. Consequently, NN potentials enable ab initio quality
simulations to reach the large system sizes and long time scales needed to
model complex phenomena, such as phase diagrams Gartner _et al._ (2020);
Zhang _et al._ (2021b); Deringer _et al._ (2018); Niu _et al._ (2020);
Deringer _et al._ (2021) and nucleation Khaliullin _et al._ (2011); Bonati
and Parrinello (2018).
Despite the significant advances made in this area, there are still practical
and conceptual difficulties with NN potential development, especially with
regard to long-range electrostatics. To make NN potential construction
computationally feasible, most approaches learn only local arrangements of
atoms around a central particle, where the meaning of “local” is defined by a
distance cutoff usually less than 1 nm. Because of this locality, the
resulting NN potentials are inherently short-ranged. The lack of long-range
interactions in NN potentials can lead to both quantitative and qualitative
errors, especially when describing polar and charged species Yue _et al._
(2021); Grisafi and Ceriotti (2019); Niblett, Galib, and Limmer (2021).
The need for incorporating long-range electrostatics into NN potentials has
led to the development of several new approaches Xie, Persson, and Small
(2020); Yao _et al._ (2018); Grisafi and Ceriotti (2019); Yue _et al._
(2021); Ko _et al._ (2021). Many of these approaches exclude all or some of
the electrostatic interactions from training and then assign effective partial
charges to each atomic nucleus that are used to calculate long-range
electrostatic interactions using traditional methods Xie, Persson, and Small
(2020); Yao _et al._ (2018); Yue _et al._ (2021); Ko _et al._ (2021);
Niblett, Galib, and Limmer (2021). The values of these effective charges can
be determined using machine learning methods. For example, 4G-HDNNP Ko _et
al._ (2021) employs deep neural networks to predict the electronegativities of
each nucleus, which are subsequently used within a charge equilibration
process to determine the effective charges. These approaches can predict
binding energies and charge transfer between molecules, but they also
introduce quantities that are not direct physical observables, such as the
effective charges and electronegativities. Another approach introduced feature
functions to explicitly incorporate nonlocal geometric information into the
construction of NN potentials Grisafi and Ceriotti (2019). However, these
feature functions depend on the system size. The resulting NN models cannot be
used to accurately model systems that are larger than the original training
set. This size restriction severely limits applicability by making this
approach unable to model extended system sizes.
The difficulties that current approaches to NN potentials have when treating
long-range interactions can be resolved by a purely ab initio strategy that
uses no effective quantities. Such a strategy can be informed by our
understanding of the roles of short- and long-range interactions in condensed
phases Widom (1967); Weeks, Chandler, and Andersen (1971); Chandler, Weeks,
and Andersen (1983); Rodgers and Weeks (2008a). In uniform liquids,
appropriately chosen uniformly slowly-varying components of the long-range
forces — van der Waals attractions and long-range Coulomb interactions —
cancel to a good approximation in every relevant configuration. As a result,
the local structure is determined almost entirely by short-range interactions.
In water, these short-range interactions correspond to hydrogen-bonding and
packing Rodgers and Weeks (2009); Remsing, Rodgers, and Weeks (2011); Rodgers
and Weeks (2008b); Rodgers, Hu, and Weeks (2011). Therefore, short-range
models, including current NN potentials, can describe the structure of uniform
systems. This idea, that short-range forces determine the structure of uniform
systems, forms the foundation for the modern theory of bulk liquids Widom
(1967); Weeks, Chandler, and Andersen (1971); Chandler, Weeks, and Andersen
(1983), in which the averaged effects of long-range interactions can be
treated as a small correction to the purely short-range system.
In contrast, the effects of long-range interactions are more subtle and play a
role in collective effects that are important for dielectric screening.
Moreover, long-range forces do not cancel at extended interfaces and instead
play a key role in interfacial physics. As a result, short-range systems
cannot describe interfacial structure and thermodynamics, as they do in the
bulk, and standard NN models fail to describe even the simplest liquid-vapor
interfaces Niblett, Galib, and Limmer (2021). The local molecular field (LMF)
theory of Weeks and coworkers provides a framework for capturing the average
effects of long-range interactions at interfaces through an effective external
field Rodgers and Weeks (2008a); Remsing, Liu, and Weeks (2016); Gao _et al._
(2018); Gao, Remsing, and Weeks (2020); Remsing (2019). LMF theory also
provides physically intuitive insights into the roles of short- and long-range
forces at interfaces that can be leveraged to model nonuniform systems.
Here, we exploit the physical picture provided by liquid-state theory to
develop a general approach for learning long-range interactions in NN
potentials from ab initio calculations. We separate the atomic interactions
into appropriate short-range and long-range components and construct a
separate network to handle each part. Importantly, the short-range model is
isolated from the long-range interactions, such that each component is treated
independently. This separation also isolates the long-range response of the
system, enabling it to be learned. Short-range interactions can be learned
using established approaches. The short- and long-range components of the
potential are then connected through a rapidly-converging self-consistent
loop. The resulting self-consistent field neural network (SCFNN) model is able
to describe the effects of long-range interactions without the use of
effective charges or similar artificial quantities. We illustrate this point
through the development of a SCFNN model of liquid water. In addition to
capturing the local structure of liquid water, as evidenced by the radial
distribution function, the SCFNN model accurately describes long-range
structural correlations connected to dielectric screening, as well as the
response of liquid water to electrostatic fields.
## I Results
### I.1 Workflow of the Self-Consistent Field Neural Network Model
The SCFNN model consists of two modules that each target a specific response
of the system (Fig. 1). Module 1 predicts the electronic response via the
position of the maximally localized Wannier function centers (MLWFCs). Module
2 predicts the forces on the nuclear sites. In turn, each module consists of
two networks: one to describe the short-range interactions and one to describe
perturbations to the short-range system from long-range electric fields.
Together, these two modules (four networks) enable the model to predict the
total electrostatic properties of the system.
In the short-range system, the $v(r)=1/r$ portion of the Coulomb potential is
replaced by the short-range potential $v_{0}(r)=\mathop{\rm
erfc}(r/\sigma)/r$. Physically, $v_{0}(r)$ corresponds to screening the charge
distributions in the system through the addition of neutralizing Gaussian
charge distributions of opposite sign - the interactions are truncated by
Gaussians. Therefore, we refer to this system as the Gaussian-truncated (GT)
system Rodgers and Weeks (2008b); Remsing, Rodgers, and Weeks (2011); Rodgers
and Weeks (2009); Rodgers, Hu, and Weeks (2011); Rodgers and Weeks (2008a). By
making a physically meaningful choice for $\sigma$, the GT system can describe
the structure of bulk liquids with high accuracy but with a fraction of the
computational cost. Moreover, the GT system has served as a useful short-range
component system when modeling the effects of long-range fields Rodgers and
Weeks (2008b); Remsing, Liu, and Weeks (2016); Gao, Remsing, and Weeks (2020);
Baker III, Rodgers, and Weeks (2020); Cox (2020). Here, we choose $\sigma$ to
be 4.2 Å (8 Bohr), which is large enough for the GT system to accurately
describe hydrogen bonding and the local structure of liquid water Rodgers and
Weeks (2008b); Remsing, Rodgers, and Weeks (2011); Rodgers and Weeks (2009);
Rodgers, Hu, and Weeks (2011); Rodgers and Weeks (2008a).
Figure 1: Schematic of the self-consistent field neural network (SCFNN). The
SCFNN consists of two modules, each with two networks. One networks learns the
short-range interactions (S) and the other learns the effects of long-range
interactions (L). Module 1 learns the positions of maximally localized Wannier
function centers, $\mathbf{r}_{w}$, and Module 2 learns the forces,
$\mathbf{F}$, on the atomic nuclei, the positions of which are indicated by
$\mathbf{R}$.
The remaining part of the Coulomb interaction,
$v_{1}(r)=v(r)-v_{0}(r)=\mathop{\rm erf}(r/\sigma)/r$, is long ranged, but
varies slowly over the scale of $\sigma$. Because $v_{1}(r)$ is uniformly
slowly-varying, the effective field produced by $v_{1}(r)$ usually induces a
linear response in the GT system. The linear nature of the response makes the
effects of $v_{1}(r)$ able to captured by linear models. In the context of
neural networks, we demonstrate below that a linear network is sufficient to
learn the linear response induced by long-range interactions.
#### I.1.1 Module 1
The separation of interactions into short- and long-range components is
crucial to the SCFNN model. In particular, the two networks of each module are
used to handle this separation. Network 1S of Module 1 predicts the positions
of the MLWFCs in the short-range GT system, while Network 1L predicts the
perturbations to the MLWFC positions induced by the effective long-range
field. Networks 1S and 1L leverage Kohn’s theory on the nearsightedness of
electronic matter (NEM) Kohn (1996); Prodans and Kohn (2005). The NEM states
that Prodans and Kohn (2005) “local electronic properties, such as the density
$n(r)$, depend significantly on the effective external potential only at
nearby points.” Here, the effective external potential refers to the Kohn-Sham
effective potential, which includes the external potential and the self-
consistently determined long-range electric fields. Therefore, the NEM
suggests that the electronic density, and consequently the positions of the
MLWFCs, are ‘nearsighted’ with respect to the effective potential, but not to
the atomic coordinates, contrary to what has been assumed in previous work
that also uses local geometric information of atoms as input to neural
networks Krishnamoorthy _et al._ (2021); Zhang _et al._ (2020). An atom
located at $\mathbf{r}^{\prime}$ will affect the effective potential at
$\mathbf{r}$, even if $\mathbf{r}^{\prime}$ is far from $\mathbf{r}$, through
long-range electrostatic interactions. Consequently, current approaches to
generating NN models can only predict the position of MLWFCs for a purely
short-range system without long-range electrostatics, such as the GT system
Krishnamoorthy _et al._ (2021); Zhang _et al._ (2020). We exploit this fact
and use established NNs to predict the locations of the MLWFCs in the GT
system Krishnamoorthy _et al._ (2021). To do so, we create a local reference
frame around each water molecule (Fig. 2) and use the coordinates of the
surrounding atoms as inputs to the neural network. The local reference system
preserves the rotational and translational symmetry of the system. The network
outputs the positions of the four MLWFCs around the central water, which are
then transformed to the laboratory frame of reference.
Network 1L predicts the response of the MLWFC positions to the effective field
$\mathbf{E}(\mathbf{r})$, defined as the sum of the external field,
$\mathbf{E}_{\rm ext}(\mathbf{r})$, and the long-range field from $v_{1}(r)$:
$\mathbf{E}(\mathbf{r})=\mathbf{E}_{\rm ext}(\mathbf{r})+\int
d\mathbf{r}\mkern 2.0mu\vphantom{r}^{\prime}\rho(\mathbf{r}\mkern
2.0mu\vphantom{r}^{\prime})\nabla v_{1}(|\mathbf{r}-\mathbf{r}\mkern
2.0mu\vphantom{r}^{\prime}|)\,,$ (1)
where $\rho(\mathbf{r}\mkern 2.0mu\vphantom{r}^{\prime})$ is the
_instantaneous_ charge density of the system, including nuclear and electronic
charges. Network 1L also introduces a local reference frame for each water
molecule. However, Network 1L takes as input both the local coordinates _and_
local effective electric fields. The NEM suggests that this local information
is sufficient to determine the perturbation in the MLWFC positions. Network 1L
outputs this change in the positions of the water molecule’s four MLWFCs, and
this perturbation is added to the MLWFC position determined in the GT system
to obtain the MLWFCs in the full system. We note that $\mathbf{E}(\mathbf{r})$
is a slowly-varying long-range field, such that the MLWFCs respond linearly to
this field. Therefore, Network 1L is constructed to be linear in
$\mathbf{E}(\mathbf{r})$. Table 1 demonstrates that the linear response
embodied by Network 1L predicts the perturbation of the MLWFCs with reasonable
accuracy.
Figure 2: Local frame around the central water. The $y$-axis is along the OH
bond. The $z$-axis is perpendicular to the plane of the molecule. The $x$-axis
is perpendicular to the $x$ and $z$ axis.
We now need to determine the effective field $\mathbf{E}(\mathbf{r})$. This
effective field depends on the electron density distribution, but evaluating
and including the full three-dimensional electron density for every
configuration in a training set requires a prohibitively large amount of
storage space. Instead, we approximate the electron density by the charge
density of the MLWFCs, assuming each MLWFC is a point charge of magnitude
$-2e_{0}$. This approximation is often used when computing molecular
multipoles, as needed to predict vibrational spectra, for example Zhang _et
al._ (2020, 2021a). Here, it is important to note that the MLWFs of water are
highly localized, so that the center gives a reasonable representation of the
location of the MLWF. Moreover, the electron density is essentially smeared
over the scale of $\sigma$ through a convolution with $v_{1}(r)$, which makes
the resulting fields relatively insensitive to small-wavelength variations in
the charge density. As a result, the electron density can be accurately
approximated by the MLWFC charge density within our approach.
The effective field is a functional of the set of MLWFC positions,
$\mathbf{E}[\left\\{\mathbf{r}_{w}\right\\}]$, and the positions of the MLWFCs
themselves depend on the field, $\mathbf{r}_{w}[\mathbf{E}]$. Therefore, we
determine $\mathbf{E}$ and $\left\\{\mathbf{r}_{w}\right\\}$ through self-
consistent iteration. Our initial guess for $\mathbf{E}$ is obtained from the
positions of the MLWFCs in the GT system. We then iterate this self-consistent
loop until the MLWFC positions no longer change, within a tolerance of
$2.6\times 10^{-4}$ Å. In practice, we find that self-consistency is achieved
quickly.
#### I.1.2 Module 2
After Module 1 predicts the positions of the MLWFCs, Module 2 predicts the
forces on the atomic sites. As with the first module, Module 2 consists of two
networks: one that predicts the forces of the GT system and another that
predicts the forces produced by $\mathbf{E}(\mathbf{r})$. To predict the
forces in the GT system, we adopt the network used by Behler and coworkers
Morawietz _et al._ (2016). This network, Network 2S, takes local geometric
information of the atoms as inputs and, consequently, cannot capture long-
range interactions. To describe long-range interactions, we introduce a second
network (Network 2L in Fig. 1). This additional network predicts the forces on
atomic sites due to the effective field $\mathbf{E}(\mathbf{r})$, which
properly accounts for long-range interactions in the system. In practice, we
again introduce a local reference frame for each water molecule and use local
atomic coordinates and local electric fields as inputs. In this case, we also
find that a network that is linear in $\mathbf{E}(\mathbf{r})$ accurately
predicts the resulting long-range forces, consistent with the linear response
of the system to a slowly-varying field.
In practice, separating the data obtained from standard DFT calculations into
the GT system and the long-range effective field is not straightforward. To
solve this problem, we apply homogeneous electric fields of varying strength
while keeping the atomic coordinates fixed. The fields only perturb the
positions of the MLWFCs and the forces on the atoms — these perturbations are
not related to the GT system. The changes induced by these electric fields are
directly obtained from DFT calculations and are used to train Networks 1L and
2L, which learn the response to long-range effective fields. The remaining
part of the DFT data is used to train Networks 1S and 2S, which learn the
response of the short-ranged GT system. See the Methods section for a more
detailed discussion of the networks and the training procedure.
We emphasize that our approach to partitioning the system into a short-range
GT piece and a long-range perturbation piece is different from other machine
learning approaches for handling long-range electrostatics. The standard
approaches usually partition the total energy into two parts, a short-ranged
energy and an Ewald energy that is used to evaluate the long-range
interactions. However, this partitioning results in a coupling between the
short- and long-range interactions. For example, the short-range part of the
energy in the 4G-HDNNP model depends on the effective charges that are
assigned to the atoms, but these effective charges depend on long-range
electrostatic interactions through the global charge equilibration process
used to determine their values Ko _et al._ (2021). In contrast, the approach
we propose here isolates the short-range and long-range physics. The GT system
does not depend on long-range electrostatics even implicitly; it is completely
uncoupled from the long-range interactions. The effects of long-range
electrostatic interactions are completely isolated within the second network
of each module, Network 1L and Network 2L in Fig. 1. This separation of short-
and long-ranged effects is similar in spirit to the principles underlying LMF
theory Rodgers and Weeks (2008a); Remsing, Liu, and Weeks (2016); Gao,
Remsing, and Weeks (2020) and related theories of uniform liquids Weeks,
Chandler, and Andersen (1971); Chandler, Weeks, and Andersen (1983); Rodgers,
Hu, and Weeks (2011); Rodgers and Weeks (2009); Remsing, Rodgers, and Weeks
(2011).
| $0.1$ V/Å | $0.2$ V/Å
---|---|---
MLWFC | $F_{\rm O}$ | $F_{\rm H}$ | MLWFC | $F_{\rm O}$ | $F_{\rm H}$
MAE ($\times 100$) | $0.028$ | $1.4$ | $0.98$ | $0.056$ | $2.8$ | $2.0$
Table 1: When homogeneous external fields are applied, the location of the
maximally localized Wanner function center (MLWFC) and the forces on the
oxygen and hydrogen nuclei, $F_{\rm O}$ and $F_{\rm H}$, respectively, are
changed. This table shows the Mean Absolute Error (MAE) of Network 1L and 2L
in predicting the changes in the MLWFC positions (Å) and the forces (eV/Å)
along the $z$-direction when fields of strength 0.1 V/Å and 0.2 V/Å are
applied along $z$-direction. The predictions are made for the test sets and
the error is computed with respect to the DFT results. Figure 3: Comparison of
the radial distribution functions for (a) O-O, (b) O-H and (c) H-H
correlations in liquid water, as predicted by molecular dynamics simulations
of the self-consistent field neural network (SCFNN) and Behler-Parrinello (BP)
models.
### I.2 Water’s Local Structure is Insensitive to Long-Range Interactions
We demonstrate the success of the SCFNN approach by modeling liquid water.
Water is the most important liquid on Earth. Yet, the importance of both
short- and long-range interactions makes it difficult to model. Short-range
interactions are responsible for water’s hydrogen bond network that is
essential to its structure and unusual but important thermodynamic properties
Ball (2008); Remsing, Rodgers, and Weeks (2011). Long-range interactions play
key roles in water’s dielectric response, interfacial structure, and can even
influence water-mediated interactions Prelesnik _et al._ (2021). Because of
this broad importance, liquid water has served as a prototypical test system
for many machine learning-based models Morawietz _et al._ (2016); Cheng _et
al._ (2019); Zhang _et al._ (2020, 2021b); Grisafi and Ceriotti (2019) Here,
we test our SCFNN model on a system of bulk liquid water by performing
molecular dynamics simulations of 1000 molecules in the canonical ensemble
under periodic boundary conditions.
One conventional test on the validity of a NN potential is to compare the
radial distribution function, $g(r)$, between atomic sites for the different
models. The $g(r)$ predicted by the SCFNN model is the same as that predicted
by the Behler-Parrinello model Morawietz _et al._ (2016) for all three site-
site correlations in water (Fig. 3). This level of agreement may be expected,
based on previous work examining the structure of bulk water Rodgers and Weeks
(2008b); Rodgers, Hu, and Weeks (2011); Remsing, Rodgers, and Weeks (2011);
Gao _et al._ (2018); Gao, Remsing, and Weeks (2020). The radial distribution
functions of water are determined mainly by short-range, nearest-neighbor
interactions, which arise from packing and hydrogen bonding; long-range
interactions have little effect on the main features of $g(r)$. Consequently,
purely short-range models, like the GT system, can quantitatively reproduce
the $g(r)$ of water Rodgers and Weeks (2008b); Rodgers, Hu, and Weeks (2011);
Remsing, Rodgers, and Weeks (2011); Gao _et al._ (2018); Gao, Remsing, and
Weeks (2020). Similarly, the short-range Behler-Parrinello model accurately
describes the radial distribution functions, as does the SCFNN model, which
includes long-range interactions.
### I.3 Long-Range Electrostatics and Dielectric Response
Though the short-range structure exemplified by the radial distribution
function is insensitive to long-range interactions, long-range correlations
are not. For example, the longitudinal component of the dipole density or
polarization correlation function evaluated in reciprocal space,
$\chi^{0}_{zz}(\mathbf{k})$, was recently shown to be sensitive to long-range
interactions Cox (2020). This correlation function is defined according to
$\chi^{0}_{zz}(\mathbf{k})=\frac{1}{V}\sum_{l,j}\frac{(\mathbf{k}\cdot\mathbf{p}_{l})\,(\mathbf{k}\cdot\mathbf{p}_{j})}{k^{2}}\,e^{-i\mathbf{k}\cdot\left(\mathbf{r}_{l}-\mathbf{r}_{j}\right)}\,,\,\,\mathrm{with}\,\,\mathbf{k}=k\hat{\mathbf{z}}\,.$
(2)
Here $\mathbf{p}_{j}$ is the dipole moment of water molecule $j$ and
$\mathbf{r}_{j}$ is the position of the oxygen atom of water molecule $j$.
The longitudinal polarization correlation function predicted by our SCFNN
model and the Behler-Parrinello agree everywhere except at small $k$,
indicating that long-range correlations are different in the two models (Fig.
4a). The long-wavelength behavior of the polarization correlation function is
related to the dielectric constant via
$\lim_{k\rightarrow 0}\chi^{0}_{zz}(\mathbf{k})=\epsilon_{0}k_{\rm
B}T\frac{\epsilon-1}{\epsilon}\,,$ (3)
where $\epsilon=78.4$ is the dielectric constant of water. The
$\chi^{0}_{zz}(\mathbf{k})$ predicted by our SCFNN model is consistent with
the expected behavior at small $k$. In contrast, short-range models, like the
GT system Cox (2020) and the Behler-Parrinello model, significantly deviate
from the expected asymptotic value. Consequently, these short-range models are
expected to have difficulties describing the dielectric screening that is
important in nonuniform systems Rodgers and Weeks (2008b); Remsing, Liu, and
Weeks (2016); Gao, Remsing, and Weeks (2020); Cox (2020); Niblett, Galib, and
Limmer (2021), for example.
To further examine the dielectric properties of the NN models, we can apply
homogeneous fields of varying strength to the system and examine its response.
To do so, we performed finite-field simulations at constant displacement
field, $\mathbf{D}$. These finite-$\mathbf{D}$ simulations Zhang and Sprik
(2016) can be naturally combined with our SCFNN model, unlike many other
neural network models. Following previous work Cox (2020), we use
$\mathbf{D}=D\hat{\mathbf{z}}$, vary the magnitude of the displacement field
from $D=0$ V/Å to $D=0.4$ V/Å, and examine the polarization, $P$, induced in
water. As shown in Fig. 4b, the polarization response of water to the external
field is accurately predicted by dielectric continuum theory, as expected,
further suggesting that the SCFNN model properly describes the dielectric
response of water. To the best of our knowledge, this is the first NN model
that can accurately describe the response of a system to external fields. We
emphasize that this response is achieved by learning the long-range response
via Networks 1L and 2L.
Figure 4: (a) The longitudinal polarization correlation function in reciprocal
space, $\chi^{0}_{zz}(\mathbf{k})$, shows differences between the self-
consistent field neural network (SCFNN) and Behler-Parrinello (BP) models at
low $k$. In particular, the SCFNN model plateaus as $k\rightarrow 0$ in a
manner consistent with the theoretical prediction (green line), while the BP
(short-range) model does not. (b) The polarization, $P$, induced by a
homogeneous displacement field of magnitude $D$ along the $z$-axis is
accurately predicted by the SCFNN model, evidenced by the agreement with
dielectric continuum theory (DCT) predictions.
## II Discussion
In this work, we have presented a general strategy to construct NN potentials
that can properly account for the long-range response of molecular systems
that is responsible for dielectric screening and related phenomena. We
demonstrated that this model produces the correct long-range polarization
correlations in liquid water, as well as the correct response of liquid water
to external electrostatic fields. Both of these quantities are related to the
dielectric constant and require a proper description of long-range
interactions. In contrast, current derivations of NN potentials result in
short-range models that cannot capture these effects.
We anticipate that this approach will be of broad use to the molecular machine
learning and simulation community for modeling electrostatic and dielectric
properties of molecular systems. In contrast to short-range interactions that
must be properly learned to describe the different local environments
encountered at extended interfaces and at solute surfaces, the response of the
system to long-range, slowly-varying fields is quite general. Learning the
long-range response (through Networks 1L and 2L) is analogous to learning a
linear response, and we expect the resulting model to be relatively
transferable. As such, our resulting SCFNN model can make _predictions_ about
conditions on which it was not trained. For example, we trained the model for
_electric fields_ of magnitude 0, 0.1, and 0.2 V/Å, and then used this model
to successfully predict the response of the system to _displacement fields_
with magnitudes between 0 and 0.4 V/Å. This suggests that our approach can be
used to train NN models in more complex environments, like water at extended
interfaces, and then accurately predict the response of water to long-range
fields in those environments.
Finally, we note that our SCFNN approach is complementary to many established
methods for creating NN potentials. Learning the short-range, GT system
interactions can be accomplished with any method that uses local geometric
information. In this case, the precise form of Networks 1S and 2S can be
replaced with an alternative NN. Then, Networks 1L and 2L can be used as
defined here, within the general SCFNN workflow, resulting in a variant of the
desired NN potential that can describe the effects of long-range interactions.
Because of this, we expect our SCFNN approach to be transferable and readily
interfaced with current and future machine learning methods for modeling
short-range molecular interactions.
## III Methods
### III.1 Neural Network Potentials.
Our training and test set consists of 1571 configurations of 64 water
molecules. Homogeneous electric fields were applied to the system, as
described further in the next section. We used two thirds of the
configurations for training and one third to test the training of the network.
To train the networks we need to separate the DFT data into the GT system and
the long-range effective field. However, that separation is not
straightforward in practice. To achieve this, we use the differences in the
MLWFC locations and forces induced by different fields to fit Networks 1L and
2L. We now describe this procedure in detail for fitting Network 1L, and
Network 2L was fit following a similar approach.
To learn the effects of long-range interactions, we consider perturbations to
the positions of the MLWFCs induced by external electric fields of different
magnitudes. Consider applying two fields of strength $\left|\mathbf{E}\right|$
and $\left|\mathbf{E^{\prime}}\right|$. These fields will alter the MLWFC
positions by $\Delta\mathbf{r}_{w}[\mathbf{R},\mathbf{E}]$ and
$\Delta\mathbf{r}\mkern
2.0mu\vphantom{r}^{\prime}_{w}[\mathbf{R},\mathbf{E^{\prime}}]$, respectively.
However, both $\Delta\mathbf{r}_{w}$ and $\Delta\mathbf{r}\mkern
2.0mu\vphantom{r}^{\prime}_{w}$ are not directly obtainable from a single DFT
calculation. Instead, we can readily compute the difference in perturbations,
$\Delta\mathbf{r}_{w}-\Delta\mathbf{r}\mkern 2.0mu\vphantom{r}^{\prime}_{w}$,
directly from the DFT data, because
$\Delta\mathbf{r}_{w}-\Delta\mathbf{r}\mkern
2.0mu\vphantom{r}^{\prime}_{w}=\mathbf{r}_{w}-\mathbf{r}\mkern
2.0mu\vphantom{r}^{\prime}_{w}\,.$ (4)
Here, $\mathbf{r}_{w}$ and $\mathbf{r}\mkern 2.0mu\vphantom{r}^{\prime}_{w}$
are the locations of the MLWFCs in the full system in the presence of the
field $\mathbf{E}$ and $\mathbf{E^{\prime}}$, respectively, and these
positions can be readily computed in the simulations. These differences in the
MLWFC positions are used to fit Network 1L. In addition, we also exploit the
fact that $\Delta\mathbf{r}_{w}=0$ when $\mathbf{E}=0$. This allows us to fix
the zero point of Network 1L.
After fitting Networks 1L and 2L, we use them to predict the contribution of
the effective field to the MLWFC locations and forces. We then subtract that
part from the DFT data. What remains corresponds to the short-range GT system,
and this is used to train Networks 1S and 2S.
We now describe the detailed structure of the four networks used here.
Network 1S. In the local frame of water molecule $i$, we construct two types
of symmetry functions as inputs to Network 1S. The first type is the type 2
Behler-Parrinello symmetry function Behler (2011),
$G_{i}^{2}=\sum_{j\neq
i}\exp\left(-\eta(r_{ij}-r_{s})^{2}\right)f_{c}(r_{ij}).$ (5)
Here, $\eta$ and $r_{s}$ are parameters that adjust the width and center of
the Gaussian, and $f_{c}$ is a cutoff function whose value and slope goes to
zero at the radial cutoff $r_{c}$. We adopted the same cutoff function as
previous work Morawietz _et al._ (2016), and the cutoff $r_{c}$ is set equal
to 12 Bohr.
The second type of symmetry function is similar to the type 4 Behler-
Parrinello symmetry functionBehler (2011). This symmetry function depends on
the angle between $\mathbf{r}_{ij}$ and the axis of the local frame,
$\mathbf{G}_{i}^{4}=\sum_{j\neq
i}2^{1-\zeta}(1+\lambda\frac{\mathbf{r}_{ij}}{r_{ij}})^{\zeta}\exp(-\eta
r_{ij}^{2})f_{c}(r_{ij}).$ (6)
Here, $\zeta$ and $\lambda$ are parameters that adjust the dependence of the
angular term.
We use 36 symmetry functions as input to Network 1S. Network 1S itself
consists of two hidden layers that contain 24 and 16 nodes. The output layer
consists of 12 nodes, corresponding to the three-dimensional coordinates of
the four MLWFCs of a central water molecule. Network 1S is a fully connected
feed-forward network, and we use $\tanh(x)$ as its activation function.
Network 1L. In the local frame of water molecule $i$, we construct one type of
symmetry function as input to Network 1L,
$\mathbf{EG}_{i}^{2}=\sum_{j}\mathbf{E}_{j}\exp(-\eta(r_{ij}-r_{s})^{2})f_{c}(r_{ij}).$
(7)
Here, $\mathbf{E}_{j}$ is the effective field exerted on atom $j$. We use 36
symmetry functions as inputs to Network 1L. Network 1L has no hidden layers.
The output layer consists of 12 nodes, corresponding to the three-dimensional
coordinates of the _perturbations_ of a water molecule’s four MLWFCs induced
by the external field.
Network 2S. Network 2S is exactly the same as the Behler-Parrinello Network
employed in previous work Morawietz _et al._ (2016). In brief, the network
contains 2 hidden layers, each containing 25 nodes. Type 2 and 4 Behler-
Parrinello symmetry functions are used as inputs to the network. The network
for oxygen takes 30 symmetry functions as inputs, while the network for
hydrogen takes 27 symmetry functions as inputs. A hyperbolic tangent is used
as the activation function.
Network 2L. Network 2L uses the same type of symmetry function as Network 1L.
The network for the force on the oxygen and for the force on hydrogen are
trained independently. To predict the force on the oxygen, we center the local
frame on the oxygen atom. When the force on a hydrogen atom is the target, we
center the local frame on a hydrogen atom. We use 36 symmetry functions as
inputs to Network 2L. Network 2L has no hidden layers. The inputs map linearly
onto the forces on the atoms.
### III.2 DFT Calculations.
The DFT calculations followed previous work Cheng _et al._ (2019); Marsalek
and Markland (2017) and used published configurations of water as the training
set Cheng _et al._ (2019). In short, all calculations were performed with
CP2K (version 7) Kühne _et al._ (2020); VandeVondele _et al._ (2005), using
the revPBE0 hybrid functional with 25% exact exchange Zhang and Yang (1998);
Adamo and Barone (1999); Goerigk and Grimme (2011), the D3 dispersion
correction of Grimme Grimme _et al._ (2010), Goedecker-Tetter-Hutter
pseudopotentials Goedecker, Teter, and Hutter (1996), and TZV2P basis sets
VandeVondele and Hutter (2007), with a plane wave cutoff of 400 Ry. Maximally
localized Wannier function centers Marzari _et al._ (2012) were evaluated
with CP2K, using the LOCALIZE option. The maximally localized Wannier function
spreads were minimized according to previous work Berghold _et al._ (2000). A
homogeneous, external electric field was applied to the system using the Berry
phase approach, with the PERIODIC_EFIELD option in CP2K Souza, Íniguez, and
Vanderbilt (2002); Umari and Pasquarello (2002); Zhang, Hutter, and Sprik
(2016). Electric fields of magnitude 0, 0.1, and 0.2 V/Å were applied along
the $z$-direction of the simulation cell. Sample input files are given at
Zenodo (https://doi.org/10.5281/zenodo.5521328).
### III.3 Molecular Dynamics Simulations
MD simulations are performed in the canonical (NVT) ensemble, with a constant
temperature of 300 K maintained using a Berendsen thermostat. The system
consisted of 1000 water molecules in a cubic box 31.2 Å in length. The
equations of motion were integrated with a timestep of 0.5 fs. Radial
distribution functions and longitudinal polarization correlations functions
were computed from 100 independent trajectories that were each 50 ps in
length. Finite-$\mathbf{D}$ simulations were performed under the same
simulation conditions, and each trajectory was 50 ps long at each magnitude of
$D$.
### III.4 Data Availability.
Datasets used to train and test the NNP can be found at Zenodo
(https://doi.org/10.5281/zenodo.5521328).
### III.5 Code Availability.
All DFT calculations were performed with CP2K version 7. In-house code was
used to construct the NN potentials and perform the MD simulations. These
codes are available at Github (https://github.com/andy90/SCFNN).
###### Acknowledgements.
We acknowledge the Office of Advanced Research Computing (OARC) at Rutgers,
The State University of New Jersey for providing access to the Amarel cluster
and associated research computing resources that have contributed to the
results reported here. We thank John Weeks for helpful comments on the
manuscript.
## References
* Allen and Tildesley (2017) M. P. Allen and D. J. Tildesley, _Computer simulation of liquids_ (Oxford university press, 2017).
* Tuckerman _et al._ (1996) M. E. Tuckerman, P. J. Ungar, T. Von Rosenvinge, and M. L. Klein, The Journal of Physical Chemistry 100, 12878 (1996).
* Car and Parrinello (1985) R. Car and M. Parrinello, Physical review letters 55, 2471 (1985).
* Chen _et al._ (2018) M. Chen, L. Zheng, B. Santra, H.-Y. Ko, R. A. DiStasio Jr, M. L. Klein, R. Car, and X. Wu, Nature chemistry 10, 413 (2018).
* Geissler _et al._ (2001) P. L. Geissler, C. Dellago, D. Chandler, J. Hutter, and M. Parrinello, Science 291, 2121 (2001).
* Lee _et al._ (2008) T.-S. Lee, C. S. López, G. M. Giambaşu, M. Martick, W. G. Scott, and D. M. York, Journal of the American Chemical Society 130, 3053 (2008).
* Walker, Crowley, and Case (2008) R. C. Walker, M. F. Crowley, and D. A. Case, Journal of computational chemistry 29, 1019 (2008).
* Senn and Thiel (2006) H. M. Senn and W. Thiel, Atomistic approaches in modern biology , 173 (2006).
* Dal Peraro _et al._ (2007) M. Dal Peraro, P. Ruggerone, S. Raugei, F. L. Gervasio, and P. Carloni, Current opinion in structural biology 17, 149 (2007).
* Sun, Ruzsinszky, and Perdew (2015) J. Sun, A. Ruzsinszky, and J. P. Perdew, Physical review letters 115, 036402 (2015).
* Sun _et al._ (2016) J. Sun, R. C. Remsing, Y. Zhang, Z. Sun, A. Ruzsinszky, H. Peng, Z. Yang, A. Paul, U. Waghmare, X. Wu, M. L. Klein, and J. P. Perdew, Nat. Chem. 8, 831 (2016).
* Chen _et al._ (2017) M. Chen, H.-H. Ko, R. C. Remsing, M. F. C. Andrade, B. Santra, Z. Sun, A. Selloni, R. Car, M. L. Klein, J. P. Perdew, and X. Wu, Proc. Natl. Acad. Sci. U.S.A. 114, 10846 (2017).
* Furness _et al._ (2020) J. W. Furness, A. D. Kaplan, J. Ning, J. P. Perdew, and J. Sun, The journal of physical chemistry letters 11, 8208 (2020).
* Zhang _et al._ (2021a) C. Zhang, F. Tang, M. Chen, J. Xu, L. Zhang, D. Y. Qiu, J. P. Perdew, M. L. Klein, and X. Wu, The Journal of Physical Chemistry B (2021a).
* Adamo and Barone (1999) C. Adamo and V. Barone, The Journal of chemical physics 110, 6158 (1999).
* Gartner _et al._ (2020) T. E. Gartner, L. Zhang, P. M. Piaggi, R. Car, A. Z. Panagiotopoulos, and P. G. Debenedetti, Proceedings of the National Academy of Sciences 117, 26040 (2020).
* Zhang _et al._ (2021b) L. Zhang, H. Wang, R. Car, and E. Weinan, Physical Review Letters 126, 236001 (2021b).
* Deringer _et al._ (2018) V. L. Deringer, N. Bernstein, A. P. Bartók, M. J. Cliffe, R. N. Kerber, L. E. Marbella, C. P. Grey, S. R. Elliott, and G. Csányi, The journal of physical chemistry letters 9, 2879 (2018).
* Niu _et al._ (2020) H. Niu, L. Bonati, P. M. Piaggi, and M. Parrinello, Nature communications 11, 1 (2020).
* Deringer _et al._ (2021) V. L. Deringer, N. Bernstein, G. Csányi, C. Ben Mahmoud, M. Ceriotti, M. Wilson, D. A. Drabold, and S. R. Elliott, Nature 589, 59 (2021).
* Khaliullin _et al._ (2011) R. Z. Khaliullin, H. Eshet, T. D. Kühne, J. Behler, and M. Parrinello, Nature materials 10, 693 (2011).
* Bonati and Parrinello (2018) L. Bonati and M. Parrinello, Physical review letters 121, 265701 (2018).
* Yue _et al._ (2021) S. Yue, M. C. Muniz, M. F. Andrade, L. Zhang, R. Car, and A. Z. Panagiotopoulos, Journal of Chemical Physics 154 (2021), 10.1063/5.0031215.
* Grisafi and Ceriotti (2019) A. Grisafi and M. Ceriotti, Journal of Chemical Physics 151 (2019), 10.1063/1.5128375, 1909.04512 .
* Niblett, Galib, and Limmer (2021) S. P. Niblett, M. Galib, and D. T. Limmer, arXiv preprint arXiv:2107.06208 (2021).
* Xie, Persson, and Small (2020) X. Xie, K. A. Persson, and D. W. Small, Journal of Chemical Theory and Computation 16, 4256 (2020).
* Yao _et al._ (2018) K. Yao, J. E. Herr, D. W. Toth, R. McKintyre, and J. Parkhill, Chemical Science 9, 2261 (2018).
* Ko _et al._ (2021) T. W. Ko, J. A. Finkler, S. Goedecker, and J. Behler, Nature Communications 12, 1 (2021).
* Widom (1967) B. Widom, Science 157, 375 (1967).
* Weeks, Chandler, and Andersen (1971) J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys. 54, 5237 (1971).
* Chandler, Weeks, and Andersen (1983) D. Chandler, J. D. Weeks, and H. C. Andersen, Science 220, 787 (1983).
* Rodgers and Weeks (2008a) J. M. Rodgers and J. D. Weeks, Journal of Physics: Condensed Matter 20, 494206 (2008a).
* Rodgers and Weeks (2009) J. M. Rodgers and J. D. Weeks, The Journal of chemical physics 131, 244108 (2009).
* Remsing, Rodgers, and Weeks (2011) R. C. Remsing, J. M. Rodgers, and J. D. Weeks, J. Stat. Phys. 145, 313 (2011).
* Rodgers and Weeks (2008b) J. M. Rodgers and J. D. Weeks, Proceedings of the National Academy of Sciences 105, 19136 (2008b), https://www.pnas.org/content/105/49/19136.full.pdf .
* Rodgers, Hu, and Weeks (2011) J. M. Rodgers, Z. Hu, and J. D. Weeks, Molecular Physics 109, 1195 (2011).
* Remsing, Liu, and Weeks (2016) R. C. Remsing, S. Liu, and J. D. Weeks, Proc. Natl. Acad. Sci. USA 113, 2819 (2016).
* Gao _et al._ (2018) A. Gao, L. Tan, M. I. Chaudhari, D. Asthagiri, L. R. Pratt, S. B. Rempe, and J. D. Weeks, The Journal of Physical Chemistry B 122, 6272 (2018).
* Gao, Remsing, and Weeks (2020) A. Gao, R. C. Remsing, and J. D. Weeks, Proc. Natl. Acad. Sci. U.S.A. 117, 1293 (2020).
* Remsing (2019) R. C. Remsing, Proceedings of the National Academy of Sciences 116, 23874 (2019), https://www.pnas.org/content/116/48/23874.full.pdf .
* Baker III, Rodgers, and Weeks (2020) E. B. Baker III, J. M. Rodgers, and J. D. Weeks, The Journal of Physical Chemistry B 124, 5676 (2020).
* Cox (2020) S. J. Cox, Proceedings of the National Academy of Sciences of the United States of America 117, 19746 (2020), 2008.01682 .
* Kohn (1996) W. Kohn, Physical Review Letters 76, 3168 (1996).
* Prodans and Kohn (2005) E. Prodans and W. Kohn, Proceedings of the National Academy of Sciences of the United States of America 102, 11635 (2005).
* Krishnamoorthy _et al._ (2021) A. Krishnamoorthy, K. I. Nomura, N. Baradwaj, K. Shimamura, P. Rajak, A. Mishra, S. Fukushima, F. Shimojo, R. Kalia, A. Nakano, and P. Vashishta, Physical Review Letters 126, 216403 (2021).
* Zhang _et al._ (2020) L. Zhang, M. Chen, X. Wu, H. Wang, E. Weinan, and R. Car, Physical Review B 102, 1 (2020), 1906.11434 .
* Morawietz _et al._ (2016) T. Morawietz, A. Singraber, C. Dellago, and J. Behler, Proceedings of the National Academy of Sciences of the United States of America 113, 8368 (2016).
* Ball (2008) P. Ball, _Chemical Reviews_ , Chemical Reviews 108, 74 (2008).
* Prelesnik _et al._ (2021) J. L. Prelesnik, R. G. Alberstein, S. Zhang, H. Pyles, D. Baker, J. Pfaendtner, J. J. De Yoreo, F. A. Tezcan, R. C. Remsing, and C. J. Mundy, Proceedings of the National Academy of Sciences 118 (2021).
* Cheng _et al._ (2019) B. Cheng, E. A. Engel, J. Behler, C. Dellago, and M. Ceriotti, Proceedings of the National Academy of Sciences 116, 1110 (2019).
* Zhang and Sprik (2016) C. Zhang and M. Sprik, Physical Review B 93, 1 (2016), 1510.00643 .
* Behler (2011) J. Behler, Journal of Chemical Physics 134 (2011), 10.1063/1.3553717.
* Marsalek and Markland (2017) O. Marsalek and T. E. Markland, The journal of physical chemistry letters 8, 1545 (2017).
* Kühne _et al._ (2020) T. D. Kühne, M. Iannuzzi, M. Del Ben, V. V. Rybkin, P. Seewald, F. Stein, T. Laino, R. Z. Khaliullin, O. Schütt, F. Schiffmann, _et al._ , The Journal of Chemical Physics 152, 194103 (2020).
* VandeVondele _et al._ (2005) J. VandeVondele, M. Krack, F. Mohamed, M. Parrinello, T. Chassaing, and J. Hutter, Comput. Phys. Commun. 167, 103 (2005).
* Zhang and Yang (1998) Y. Zhang and W. Yang, Physical Review Letters 80, 890 (1998).
* Goerigk and Grimme (2011) L. Goerigk and S. Grimme, Physical Chemistry Chemical Physics 13, 6670 (2011).
* Grimme _et al._ (2010) S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem. Phys. 132, 154104 (2010).
* Goedecker, Teter, and Hutter (1996) S. Goedecker, M. Teter, and J. Hutter, Phys. Rev. B 54, 1703 (1996).
* VandeVondele and Hutter (2007) J. VandeVondele and J. Hutter, J. Chem. Phys. 127, 114105 (2007).
* Marzari _et al._ (2012) N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012).
* Berghold _et al._ (2000) G. Berghold, C. J. Mundy, A. H. Romero, J. Hutter, and M. Parrinello, Phys. Rev. B 61, 10040 (2000).
* Souza, Íniguez, and Vanderbilt (2002) I. Souza, J. Íniguez, and D. Vanderbilt, Physical review letters 89, 117602 (2002).
* Umari and Pasquarello (2002) P. Umari and A. Pasquarello, Physical Review Letters 89, 157602 (2002).
* Zhang, Hutter, and Sprik (2016) C. Zhang, J. Hutter, and M. Sprik, The journal of physical chemistry letters 7, 2696 (2016).
|
# Absence of anomalous dissipation for weak solutions of the Maxwell–Stefan
system
Luigi C. Berselli, Stefanos Georgiadis and Athanasios E. Tzavaras Luigi C.
Berselli, Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti
1/c, I56127, Pisa, Italy<EMAIL_ADDRESS>Stefanos Georgiadis,
Computer, Electrical and Mathematical Science and Engineering Division, King
Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900,
Saudi Arabia and Institute for Analysis and Scientific Computing, Vienna
University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
<EMAIL_ADDRESS>Athanasios E. Tzavaras, Computer, Electrical
and Mathematical Science and Engineering Division, King Abdullah University of
Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
<EMAIL_ADDRESS>
###### Abstract.
In this paper we give a short and self-contained proof of the fact that weak
solutions to the Maxwell-Stefan system automatically satisfy an entropy
equality, establishing the absence of anomalous dissipation.
###### Key words and phrases:
Gas mixture, Maxwell–Stefan equations, isothermal model, nonequilibrium
thermodynamics, anomalous dissipation.
###### 2000 Mathematics Subject Classification:
35D30, 35Q35, 76N15, 76R50, 76T30.
## 1\. Introduction
Energy balance is an important tool in the understanding of fluid systems.
Already from the work of Kolmogorov in the 1940s, it became evident that the
evolution of energy in fluids plays a crucial role in the mechanism of
creation of turbulence. To this extent, establishing an energy equality
implies that the flow is not chaotic, as there is no turbulence (with some
caveat due to the variety of phenomena related with fluid flows). It was
formalized in the famous Onsager conjecture, stating that Hölder continuous
solutions of the incompessible Euler equations flows with exponent
$\alpha>1/3$ conserve the kinetic energy. The positive part of the conjecture
was proved in Constantin, E, and Titi [5], according to which weak solutions
to the Euler equations with some additional regularity conserve their energy.
The negative part of the conjecture was settled in an important work by Isett
[10] and Buckmaster et al. [3] building on several previous works (see
references therein). Following the work [5], several studies investigated
regularity criteria under which an energy conservation (or dissipation)
identity can be attained, including the incompressible [7] or compressible
[13] Navier-Stokes equations, the system of magnetohydrodynamics [4], the
Euler-Korteweg system [6] and general conservation laws [9]. See [1] for a
review of results on incompressible flows under various regularity assumptions
that induce conservation of energy.
The study of multicomponent systems is an active research field of
mathematical fluid mechanics for compressible flows. The Maxwell-Stefan
system, see (1.1) below, is one of the simplest models describing the
diffusive transport of the components of a gaseous mixture with zero mean
flow. It consists of a system of continuity equations, one for the mass
fraction of each component, together with an algebraic system describing
frictional balance among the components, and coupling the molar fluxes with
the mass fractions. The system is equipped with an entropy dissipation
structure obtained from thermodynamic considerations which plays an important
role in its analysis. It induces a gradient flow structure [11] which renders
it formally a parabolic system. Local existence of smooth solutions is shown
in [2] while global existence of weak solutions is shown in [12]. At present
it is not known if smooth solutions break down and in what norm.
The solutions constructed in [12] inherit from construction an energy
dissipation inequality. The objective of the present work is to show that weak
solutions to the Maxwell-Stefan system satisfy an energy dissipation identity,
making a first qualitative connection with similar results about standard
turbulence theory. The main result we prove is the absence of anomalous
dissipation for weak solutions to the Maxwell-Stefan system in the same class
where we have global existence, see Theorem 2. This property is obtained as a
consequence of the regularity of the weak solutions, and of the structure of
the equations, without additional assumptions.
### 1.1. Setting
We consider the system of Maxwell-Stefan equations which describes
multicomponent diffusive phenomena in a gas mixture. The system consists of
the partial differential equations
(1.1) $\partial_{t}c_{i}+\operatorname{div}J_{i}=0,\quad\textnormal{in
}\Omega\times(0,T),$
describing the evolution of the mass fractions $c_{i}$ for each component
$i=1,\dots,n$, where $J_{i}$ is the molar flux of the $i$-th component and the
evolution takes place in a bounded domain $\Omega\subset{\mathbb{R}}^{3}$ with
smooth boundary $\partial\Omega$.
Initially, the mass fractions are given by
(1.2) $c_{i}(x,0)=c_{i}^{0}(x)\quad\textnormal{ in }\Omega,$
where $c_{i}^{0}(x)\geq 0$ and $\sum_{i=1}^{n}c_{i}^{0}(x)=1$ and we assume
that no mass can enter or leave the domain, i.e., that
(1.3) $J_{i}\cdot\nu=0,\quad\textnormal{ on }\partial\Omega\times(0,T).$
where $\nu$ is the exterior normal unit vector to $\partial\Omega$.
The molar fluxes $J_{i}$ are solutions of the linear system
(1.4) $-\sum_{j\not=i,j=1}^{n}\frac{c_{j}J_{i}-c_{i}J_{j}}{D_{ij}}=\nabla
c_{i}\quad\textnormal{in }\Omega\times(0,T),$
where $D_{ij}$, for $i\not=j$, are positive and symmetric coefficients
describing binary interactions between the $i$-th and $j$-th components, and
the molar fluxes are subject to the constraint
(1.5) $\sum_{i=1}^{n}J_{i}=0.$
In the case of nonvanishing concentrations, the molar fluxes $J_{i}$ are
defined as the product of the mass fractions $c_{i}$ and the diffusional
velocities $u_{i}$, i.e. $J_{i}:=c_{i}u_{i}$. In [11] it was shown that system
(1.4)-(1.5) can be rewritten in terms of the quantities
$m_{i}:=\sqrt{c_{i}}u_{i},$
which make sense even for $c_{i}=0$, so that $J_{i}$ should be understood as
$J_{i}:=\sqrt{c_{i}}m_{i}$ and $m_{i}$ can be found by inverting the algebraic
linear system
(1.6) $-\sum_{j=1}^{n}A_{ij}m_{j}=2\nabla\sqrt{c_{i}},$
with
$A_{ij}=\begin{cases}-\frac{\sqrt{c_{i}c_{j}}}{D_{ij}},&i\not=j\\\
\sum_{k\not=i}\frac{c_{k}}{D_{ik}},&i=j\end{cases},$
subject to the constraint
(1.7) $\sum_{i=1}^{n}\sqrt{c}_{i}m_{i}=0.$
The main difficulty of the Maxwell-Stefan system is that system (1.4) (and
hence also when reformulated as (1.6)) is singular and thus yields no unique
solution. Nevertheless, under the additional constraint (1.5) (equivalently
(1.7)), which ensures conservation of total mass, the linear system can be
inverted on its image by means of the Bott-Duffin inverse (for more details we
refer to [11]).
To properly define the class of solution which we will handle, we define a
weak solution as follows:
###### Definition 1.
We call the vector field $\bm{c}=(c_{1},\dots,c_{n})$, such that $c_{i}\in
L^{2}(0,T;H^{1}(\Omega))$ and $\partial_{t}c_{i}\in
L^{2}(0,T;(H^{1}(\Omega))^{*})$, a weak solution of (1.1)-(1.4) if:
* (i)
it satisfies (1.1) in the weak sense, i.e.
(1.8)
$\int_{0}^{T}\int_{\Omega}c_{i}\partial_{t}\varphi_{i}\,\textnormal{d}x\textnormal{d}t+\int_{0}^{T}\int_{\Omega}\sqrt{c_{i}}m_{i}\cdot\nabla\varphi_{i}\,\textnormal{d}x\textnormal{d}t=0,$
for all $i=1,\dots,n$ and for all $\varphi_{i}\in
C^{1}_{c}(\overline{\Omega}\times(0,T))$, where $m_{i}=\sqrt{c_{i}}u_{i}$ is
the solution of (1.6)-(1.7);
* (ii)
$c_{i}\in L^{\infty}(0,T;L^{\infty}(\Omega))$, for all $i=1,\dots,n$,;
* (iii)
$c_{i}\geq 0$ with $\sum_{i=1}^{n}c_{i}=1$, a.e. in $\Omega\times[0,T]$ for
all $i=1,\dots,n$,;
* (iv)
$\sqrt{c_{i}}\in L^{2}(0,T;H^{1}(\Omega))$ and $c_{i}\in
C([0,T];L^{2}(\Omega))$ for all $i=1,\dots,n$, so that the initial conditions
are satisfied in the strong $L^{2}(\Omega)$ sense.
###### Remark 1.
Note that from the weak formulation (1.8) one can obtain the one considered in
[11], namely
$\int_{\Omega}c_{i}(T)\psi_{i}(T)\,\textnormal{d}x-\int_{\Omega}c_{i}^{0}\psi_{i}(0)\,\textnormal{d}x-\int_{0}^{T}\int_{\Omega}c_{i}\partial_{t}\psi_{i}\,\textnormal{d}x\textnormal{d}t-\int_{0}^{T}\int_{\Omega}\sqrt{c_{i}}m_{i}\cdot\nabla\psi_{i}\,\textnormal{d}x\textnormal{d}t=0,$
for all $\psi_{i}\in C^{1}([0,T];C^{1}(\overline{\Omega}))$, by choosing in
(1.8) $\varphi_{i}(x,t)=\eta_{\sigma}(t)\psi_{i}(x,t)$, where $\eta_{\sigma}$
is the cut-off in time defined by
(1.9) $\eta_{\sigma}(t)=\begin{cases}\frac{t-\sigma}{\sigma},&\sigma\leq t\leq
2\sigma\\\ 1,&2\sigma\leq t\leq T-2\sigma\\\
\frac{T-t-\sigma}{\sigma},&T-2\sigma\leq t\leq T-\sigma\\\
0,&t\leq\sigma,~{}t\geq T-\sigma,\end{cases}$
for $0<\sigma<\frac{T}{4}$ and $\psi_{i}\in
C^{1}([0,T];C^{1}(\overline{\Omega}))$ and then pass to the limit $\sigma\to
0$.
We note, here, that since weak solutions satisfy $c_{i}\in
L^{\infty}(0,T;L^{\infty}(\Omega))$ and $\nabla\sqrt{c_{i}}\in
L^{2}(0,T;L^{2}(\Omega))$, the inversion of system (1.6) yields $m_{i}\in
L^{2}(0,T;L^{2}(\Omega))$.
The existence of such a weak solution, satisfying all the above conditions was
proven in [12]. In addition to the properties of Definition 1, weak solutions
satisfy an entropy inequality. Let us define the quantity
(1.10)
$H(\bm{c}):=\sum_{i=1}^{n}\int_{\Omega}c_{i}(\ln{c_{i}}-1)\textnormal{d}x,$
which is the entropy functional associated to the system. Notice that
$H(\bm{c})\leq 0$, since $0\leq c_{i}\leq 1$ almost everywhere. Moreover, the
entropy is well-defined even for zero concentrations, since the limit of the
function $s\ln s$, as $s\to 0^{+}$, is equal to zero. Previous works,
especially [12, 11], have shown that weak solutions, as in Definition 1,
satisfy the following entropy inequality:
(1.11)
$H(\bm{c}(T))+\frac{1}{2}\sum_{i,j=1}^{n}\int_{0}^{T}\int_{\Omega}\frac{c_{i}c_{j}}{D_{ij}}|u_{i}-u_{j}|^{2}\,\textnormal{d}x\textnormal{d}t\leq
H(\bm{c}^{0}),$
where the dissipation term can be rewritten in terms of $\sqrt{c_{i}}$ and
$m_{i}$ as follows:
$\frac{1}{2}\sum_{i,j=1}^{n}\int_{0}^{T}\int_{\Omega}\frac{1}{D_{ij}}|\sqrt{c_{j}}m_{i}-\sqrt{c_{i}}m_{j}|^{2}\,\textnormal{d}x\textnormal{d}t,$
and thus is well-defined, since $c_{i}\in L^{\infty}(0,T;L^{\infty}(\Omega))$
and $m_{i}\in L^{2}(0,T;L^{2}(\Omega))$. In this work, we will show that, in
fact, weak solutions to the Maxwell-Stefan system as in Definition 1 satisfy
(1.11) as an identity. It should be noted that, according to the results from
[8], weak solutions are in fact renormalized solutions, meaning that they
further satisfy the identity
(1.12)
$\partial_{t}\beta(c_{i})+\operatorname{div}(\beta^{\prime}(c_{i})c_{i}u_{i})=c_{i}u_{i}\cdot\nabla\beta^{\prime}(c_{i}),$
in the weak sense, for all functions $\beta\in C^{2}([0,+\infty))$. Even
though a formal computation suggests that the entropy identity could be
obtained by choosing in (1.12) $\beta(s)=s\ln{s}-s$, such a $\beta$ is not
admissible, due to the fact that $s\ln{s}-s$ is not a $C^{1}$ function up to
$s=0$. To this degree, the aim of this paper is to make this formal
computation rigorous.
### 1.2. Main result
The fact that weak solutions satisfy instead of (1.11) the entropy equality
(1.13), is known in the literature as “absence of anomalous dissipation” and
it is generally a basic, but additional, assumptions used in the proofs of
weak-strong uniqueness results (see Theorem 1 in [11]). In particular, this
assumption is at the basis of the relative entropy method, which assumes that
the strong solution satisfies an entropy identity, allowing for the derivation
of the relative entropy inequality.
In this work we prove that all weak solutions satisfy the entropy equality,
implying that there is no anomalous dissipation. This can be interpreted in
analogy with incompressible fluids as the fact that the flow is not turbulent.
Since the entropy equality is “formally” obtained by testing the equations by
$\ln{c_{i}}$ and integrating by parts over the domain, the essential steps
will be the “rigorous” justification of these computations. As we will see,
this is not just a matter of regularity of the solution: the fact that the
solution may not be bounded away from zero (it is not enough simply to assume
that $c_{i}>0$ since estimates involving the logarithm are not uniform near
zero and one should ask the unrealistic $c_{i}\geq\overline{c}>0$) is a
peculiar issue and makes the justification non trivial.
The main result of this paper is, then, the following:
###### Theorem 2.
Let $\bm{c}$ be a weak solution to the Maxwell-Stefan system (1.1)-(1.4) and
let the initial data be such that $c_{i}^{0}(x)\geq 0$ and
$\sum_{i=1}^{n}c_{i}^{0}(x)=1$ a.e. in $\Omega$. Then $\bm{c}$ satisfies the
entropy identity
(1.13)
$H(\bm{c}(t))+\frac{1}{2}\sum_{i,j=1}^{n}\int_{0}^{t}\int_{\Omega}\frac{c_{i}c_{j}}{D_{ij}}|u_{i}-u_{j}|^{2}\,\textnormal{d}x\textnormal{d}t=H(\bm{c}^{0})\qquad\forall\,t\in(0,T].$
## 2\. Proof of Theorem 2
We first notice that even if we use the weak formulation in terms of
$\sqrt{c_{i}}$ and $m_{i}$ as in [11], the mass fractions $c_{i}$ of a weak
solution satisfy the regularity stated in Definition 1 and especially the
strong $L^{2}$-continuity.
###### Lemma 3.
Let $\bm{c}$ be a weak solution such that $c_{i}\in
L^{\infty}(0,T;L^{\infty}(\Omega))$ and $m_{i}\in L^{2}(0,T;L^{2}(\Omega))$.
Then, it follows that
(2.1) $\partial_{t}c_{i}\in L^{2}(0,T;H^{1}(\Omega)^{*}),$
and thus
(2.2) $c_{i}\in C([0,T];L^{2}(\Omega)).$
###### Proof.
Let $\langle\cdot,\cdot\rangle$ denote the duality pairing between the spaces
$L^{2}(0,T;H^{1}(\Omega)^{*})$ and $L^{2}(0,T;H^{1}(\Omega))$. Then, the
computation
$\begin{split}\int_{0}^{T}|\langle\partial_{t}c_{i},\varphi\rangle|\,\textnormal{d}t&=\int_{0}^{T}|\langle-\operatorname{div}(c_{i}u_{i}),\varphi\rangle|\,\textnormal{d}t\\\
&=\int_{0}^{T}|\langle c_{i}u_{i},\nabla\varphi\rangle|\,\textnormal{d}t\\\
&\leq\|\sqrt{c_{i}}\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\|m_{i}\|_{L^{2}(0,T;L^{2}(\Omega))}\|\nabla\varphi\|_{L^{2}(0,T;L^{2}(\Omega))},\end{split}$
shows that due to $m_{i}$ being in $L^{2}(0,T;L^{2}(\Omega))$,
$\partial_{t}c_{i}\in L^{2}(0,T;(H^{1}(\Omega)^{*}).$
Moreover, since
$c_{i}\in L^{2}(0,T;H^{1}(\Omega))\quad\textnormal{ and
}\quad\partial_{t}c_{i}\in L^{2}(0,T;H^{1}(\Omega)^{*}),$
and since
$H^{1}(\Omega)\hookrightarrow L^{2}(\Omega)\hookrightarrow H^{1}(\Omega)^{*}$
is a Gelfand evolution triple (by continuous and dense embedding), (2.2)
follows by Lions-Magenes interpolation theory. ∎
We can now give the proof of the main result of this paper.
###### Proof of Theorem 2.
In the weak formulation (1.8) choose
$\varphi_{i}(x,t)=(\eta_{\sigma}(t)\ln(c_{i}^{\epsilon}(x,t)+\delta))^{\epsilon}$,
where $\eta_{\sigma}$ is the cut-off in time defined in (1.9) for
$0<\sigma<\frac{T}{4}$ and $0<\delta<1$, a constant. Moreover, for
$0<\epsilon<\sigma$ we define $v^{\epsilon}$ to be the mollification of $v$ in
time, i.e.
$v^{\epsilon}(x,t):=\int_{0}^{T}\rho_{\epsilon}(t-\tau)v(x,\tau)\,\textnormal{d}\tau,$
where $\rho$ is a standard symmetric mollifier. Defining mollification this
way, we assume that the function $v$ is extended by zero outside $[0,T]$. This
gives the usual convergence $v^{\epsilon}\to v$ in the interior of $[0,T]$ and
eventually problems on the boundary; this is why we work with the cut-off in
time. We note that admissible test functions belong to the space
$C^{1}_{c}(\overline{\Omega}\times(0,T))$, but by a density argument it is
enough to consider them in $H^{1}(0,T;H^{1}(\Omega))$.
We first notice that
(2.3)
$\int_{0}^{T}\int_{\Omega}v\,\partial_{t}\phi^{\epsilon}\,\textnormal{d}x\textnormal{d}t=\int_{0}^{T}\int_{\Omega}v^{\epsilon}\,\partial_{t}\phi\,\textnormal{d}x\textnormal{d}t,$
for $\phi$ such that $\phi(x,0)=\phi(x,T)=0$.
Indeed, we have:
$\displaystyle\int_{0}^{T}\int_{\Omega}v(x,t)\partial_{t}\phi^{\epsilon}(x,t)\,\textnormal{d}x\textnormal{d}t$
$\displaystyle=\int_{0}^{T}\int_{\Omega}v(x,t)\left(\int_{0}^{T}\partial_{t}\rho_{\epsilon}(t-\tau)\phi(x,\tau)\,\textnormal{d}\tau\right)\,\textnormal{d}x\textnormal{d}t$
$\displaystyle=-\int_{0}^{T}\int_{\Omega}v(x,t)\left(\int_{0}^{T}\partial_{\tau}\rho_{\epsilon}(t-\tau)\phi(x,\tau)\,\textnormal{d}\tau\right)\,\textnormal{d}x\textnormal{d}t$
$\displaystyle=\int_{0}^{T}\int_{\Omega}v(x,t)\left(\int_{0}^{T}\rho_{\epsilon}(t-\tau)\partial_{\tau}\phi(x,\tau)\,\textnormal{d}\tau\right)\,\textnormal{d}x\textnormal{d}t$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\partial_{\tau}\phi(x,\tau)\left(\int_{0}^{T}\rho_{\epsilon}(\tau-t)v(x,t)\,\textnormal{d}t\right)\,\textnormal{d}x\textnormal{d}\tau$
$\displaystyle=\int_{0}^{T}\int_{\Omega}v^{\epsilon}(x,t)\partial_{t}\phi(x,t)\,\textnormal{d}x\textnormal{d}t,$
where in the second equality we used
$\partial_{t}\rho_{\epsilon}(t-\tau)=-\partial_{\tau}\rho_{\epsilon}(t-\tau)$,
in the third one an integration by parts combined with $\phi(x,0)=\phi(x,T)=0$
and in the fourth equality we used Fubini-Tonelli and the fact that $\rho$ is
a symmetric mollifier.
Then, because of (2.3) and since
$\eta_{\sigma}(T)\ln(c_{i}^{\epsilon}(T)+\delta)=\eta_{\sigma}(0)\ln(c_{i}^{\epsilon}(0)+\delta)=0$,
the first term of (1.8) yields:
$\begin{split}\int_{0}^{T}\int_{\Omega}c_{i}\partial_{t}\varphi_{i}\,\textnormal{d}x\textnormal{d}t&=\int_{0}^{T}\int_{\Omega}c_{i}^{\epsilon}\partial_{t}\left[\eta_{\sigma}(t)\ln(c_{i}^{\epsilon}+\delta)\right]\,\textnormal{d}x\textnormal{d}t\\\
&=\int_{0}^{T}\int_{\Omega}c_{i}^{\epsilon}\eta_{\sigma}^{\prime}(t)\ln(c_{i}^{\epsilon}+\delta)\,\textnormal{d}x\textnormal{d}t+\int_{0}^{T}\int_{\Omega}c_{i}^{\epsilon}\eta_{\sigma}(t)\partial_{t}\ln(c_{i}^{\epsilon}+\delta)\,\textnormal{d}x\textnormal{d}t,\end{split}$
where
$\begin{split}\int_{0}^{T}\int_{\Omega}c_{i}^{\epsilon}\eta_{\sigma}^{\prime}(t)\ln(c_{i}^{\epsilon}+\delta)\,\textnormal{d}x\textnormal{d}t&=\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}c_{i}^{\epsilon}\ln{(c_{i}^{\epsilon}+\delta)}\,\textnormal{d}x\textnormal{d}t-\frac{1}{\sigma}\int_{T-2\sigma}^{T-\sigma}\int_{\Omega}c_{i}^{\epsilon}\ln{(c_{i}^{\epsilon}+\delta)}\,\textnormal{d}x\textnormal{d}t,\end{split}$
and
$\begin{split}\int_{0}^{T}\int_{\Omega}c_{i}^{\epsilon}\eta_{\sigma}(t)\partial_{t}\ln(c_{i}^{\epsilon}+\delta)\,\textnormal{d}x\textnormal{d}t&=\int_{0}^{T}\int_{\Omega}c_{i}^{\epsilon}\eta_{\sigma}(t)\frac{\partial_{t}c_{i}^{\epsilon}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t\\\
&=\int_{0}^{T}\int_{\Omega}\eta_{\sigma}(t)\partial_{t}c_{i}^{\epsilon}\,\textnormal{d}x\textnormal{d}t-\delta\int_{0}^{T}\int_{\Omega}\eta_{\sigma}(t)\partial_{t}\ln(c_{i}^{\epsilon}+\delta)\,\textnormal{d}x\textnormal{d}t\\\
&=-\int_{0}^{T}\int_{\Omega}\eta_{\sigma}^{\prime}(t)c_{i}^{\epsilon}\,\textnormal{d}x\textnormal{d}t+\delta\int_{0}^{T}\int_{\Omega}\eta_{\sigma}^{\prime}(t)\ln(c_{i}^{\epsilon}+\delta)\,\textnormal{d}x\textnormal{d}t\\\
&=-\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}(c_{i}^{\epsilon}-\delta\ln{(c_{i}^{\epsilon}+\delta)})\,\textnormal{d}x\textnormal{d}t\\\
&\phantom{xx}+\frac{1}{\sigma}\int_{T-2\sigma}^{T-\sigma}\int_{\Omega}(c_{i}^{\epsilon}-\delta\ln{(c_{i}^{\epsilon}+\delta)})\,\textnormal{d}x\textnormal{d}t.\end{split}$
The second term of (1.8) reads
$\begin{split}\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\cdot\nabla\varphi_{i}\,\textnormal{d}x\textnormal{d}t&=\int_{0}^{T}\int_{\Omega}(c_{i}u_{i})^{\epsilon}\eta_{\sigma}(t)\cdot\nabla\ln{(c_{i}^{\epsilon}+\delta)}\,\textnormal{d}x\textnormal{d}t\\\
&=\int_{0}^{T}\int_{\Omega}(c_{i}u_{i})^{\epsilon}\eta_{\sigma}(t)\cdot\frac{\nabla
c_{i}^{\epsilon}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t,\end{split}$
and putting everything together we arrive at
(2.4)
$\begin{split}&\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}c_{i}^{\epsilon}\ln{(c_{i}^{\epsilon}+\delta)}\,\textnormal{d}x\textnormal{d}t-\frac{1}{\sigma}\int_{T-2\sigma}^{T-\sigma}\int_{\Omega}c_{i}^{\epsilon}\ln{(c_{i}^{\epsilon}+\delta)}\,\textnormal{d}x\textnormal{d}t\\\
&\phantom{x}-\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}(c_{i}^{\epsilon}-\delta\ln{(c_{i}^{\epsilon}+\delta)})\,\textnormal{d}x\textnormal{d}t+\frac{1}{\sigma}\int_{T-2\sigma}^{T-\sigma}\int_{\Omega}(c_{i}^{\epsilon}-\delta\ln{(c_{i}^{\epsilon}+\delta)})\,\textnormal{d}x\textnormal{d}t\\\
&\phantom{xx}+\int_{0}^{T}\int_{\Omega}(c_{i}u_{i})^{\epsilon}\eta_{\sigma}(t)\cdot\frac{\nabla
c_{i}^{\epsilon}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t=0.\end{split}$
The plan is to first let $\epsilon\to 0^{+}$ keeping $\sigma>0$ and $\delta>0$
fixed, then keeping $\delta>0$ fixed take the limit $\sigma\to 0^{+}$ and
finally let $\delta\to 0^{+}$.
Step 1: Keeping $\sigma>0$ and $\delta>0$ fixed, we pass to the limit
$\epsilon\to 0^{+}$. For the first two terms, due to mollification properties,
$0\leq c_{i}\leq 1$ and $\Omega$ being bounded, we have
$\begin{split}&c_{i}^{\epsilon}\ln{(c_{i}^{\epsilon}+\delta)}\to
c_{i}\ln{(c_{i}+\delta)}~{}\textnormal{almost everywhere in
}\Omega\times[0,T]\\\
&|c_{i}^{\epsilon}\ln{(c_{i}^{\epsilon}+\delta)}|\leq\max\\{\ln{(1+\delta)},|\ln{\delta}|\\}\in
L^{1}(0,T;L^{1}(\Omega)),\end{split}$
so that by the Dominated Convergence Theorem we can pass to the limit. The
next two terms are treated in a similar way.
For the last term, we have
$\begin{split}&\int_{0}^{T}\int_{\Omega}(c_{i}u_{i})^{\epsilon}\eta_{\sigma}\cdot\frac{\nabla
c_{i}^{\epsilon}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t-\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\eta_{\sigma}\cdot\frac{\nabla
c_{i}}{c_{i}+\delta}\,\textnormal{d}x\textnormal{d}t\\\
&\phantom{xx}=\int_{0}^{T}\int_{\Omega}[(c_{i}u_{i})^{\epsilon}-c_{i}u_{i}]\eta_{\sigma}\cdot\frac{\nabla
c_{i}^{\epsilon}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t+\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\eta_{\sigma}\cdot\frac{\nabla
c_{i}^{\epsilon}-\nabla
c_{i}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t\\\
&\phantom{xxxx}+\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\eta_{\sigma}\cdot\nabla
c_{i}\left(\frac{1}{c_{i}^{\epsilon}+\delta}-\frac{1}{c_{i}+\delta}\right)\,\textnormal{d}x\textnormal{d}t,\end{split}$
and notice that $c_{i}u_{i}$ and $\nabla c_{i}$ are both in
$L^{2}(0,T;L^{2}(\Omega))$, due to $\sqrt{c_{i}}u_{i}$ and
$\nabla\sqrt{c_{i}}$ being in $L^{2}(0,T;L^{2}(\Omega))$ and $c_{i}\in
L^{\infty}(0,T;L^{\infty}(\Omega))$. This implies that
$(c_{i}u_{i})^{\epsilon}\to c_{i}u_{i}$ and $\nabla c_{i}^{\epsilon}=(\nabla
c_{i})^{\epsilon}\to\nabla c_{i}$ as $\epsilon\to 0$, both in
$L^{2}(0,T;L^{2}(\Omega))$. Then, due to the bounds $|\eta_{\sigma}(t)|\leq 1$
and $\frac{1}{c_{i}^{\epsilon}+\delta}\leq\frac{1}{\delta}$, we have
$\begin{split}\left|\int_{0}^{T}\int_{\Omega}[(c_{i}u_{i})^{\epsilon}-c_{i}u_{i}]\eta_{\sigma}\cdot\frac{\nabla
c_{i}^{\epsilon}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t\right|&\leq\frac{1}{\delta}\int_{0}^{T}\int_{\Omega}|(c_{i}u_{i})^{\epsilon}-c_{i}u_{i}||\nabla
c_{i}^{\epsilon}|\,\textnormal{d}x\textnormal{d}t\\\
&\leq\frac{1}{\delta}\|(c_{i}u_{i})^{\epsilon}-c_{i}u_{i}\|_{L^{2}(0,T;L^{2}(\Omega))}\|\nabla
c_{i}\|_{L^{2}(0,T;L^{2}(\Omega))}\\\ &\to 0,\textnormal{ as $\epsilon\to
0$,}\end{split}$
and similarly
$\begin{split}\left|\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\eta_{\sigma}\cdot\frac{\nabla
c_{i}^{\epsilon}-\nabla
c_{i}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t\right|&\leq\frac{1}{\delta}\int_{0}^{T}\int_{\Omega}|c_{i}u_{i}||\nabla
c_{i}^{\epsilon}-\nabla c_{i}|\,\textnormal{d}x\textnormal{d}t\\\
&\leq\frac{1}{\delta}\|c_{i}u_{i}\|_{L^{2}(0,T;L^{2}(\Omega))}\|(\nabla
c_{i})^{\epsilon}-\nabla c_{i}\|_{L^{2}(0,T;L^{2}(\Omega))}\\\ &\to
0,\textnormal{ as $\epsilon\to 0$}.\end{split}$
For the third term, we apply the Dominated Convergence Theorem, since
$c_{i}u_{i}\eta_{\sigma}\cdot\nabla
c_{i}\left(\frac{1}{c_{i}^{\epsilon}+\delta}-\frac{1}{c_{i}+\delta}\right)\to
0\textnormal{ a.e. in $\Omega\times(0,T)$, as $\epsilon\to 0$,}$
and
$\begin{split}\left|c_{i}u_{i}\eta_{\sigma}\cdot\nabla
c_{i}\left(\frac{1}{c_{i}^{\epsilon}+\delta}-\frac{1}{c_{i}+\delta}\right)\right|&=|c_{i}u_{i}||\eta_{\sigma}||\nabla
c_{i}|\frac{|c_{i}-c_{i}^{\epsilon}|}{|(c_{i}+\delta)(c_{i}^{\epsilon}+\delta)|}\\\
&\leq\frac{2}{\delta^{2}}|c_{i}u_{i}||\nabla c_{i}|\in
L^{1}(0,T;L^{1}(\Omega)),\end{split}$
thus establishing
$\int_{0}^{T}\int_{\Omega}(c_{i}u_{i})^{\epsilon}\eta_{\sigma}\cdot\frac{\nabla
c_{i}^{\epsilon}}{c_{i}^{\epsilon}+\delta}\,\textnormal{d}x\textnormal{d}t\to\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\eta_{\sigma}\cdot\frac{\nabla
c_{i}}{c_{i}+\delta}\,\textnormal{d}x\textnormal{d}t.$
Therefore, letting $\epsilon\to 0^{+}$ in (2.4), we obtain
(2.5)
$\begin{split}&\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}(c_{i}+\delta)\ln{(c_{i}+\delta)}\,\textnormal{d}x\textnormal{d}t-\frac{1}{\sigma}\int_{T-2\sigma}^{T-\sigma}\int_{\Omega}(c_{i}+\delta)\ln{(c_{i}+\delta)}\,\textnormal{d}x\textnormal{d}t\\\
&\phantom{x}-\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}c_{i}\,\textnormal{d}x\textnormal{d}t+\frac{1}{\sigma}\int_{T-2\sigma}^{T-\sigma}\int_{\Omega}c_{i}\,\textnormal{d}x\textnormal{d}t+\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\eta_{\sigma}(t)\cdot\frac{\nabla
c_{i}}{c_{i}+\delta}\,\textnormal{d}x\textnormal{d}t=0.\end{split}$
Step 2: Now keeping $\delta>0$ fixed, let $\sigma\to 0^{+}$. We start with the
third term and see that
$\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}c_{i}\,\textnormal{d}x\textnormal{d}t\to\int_{\Omega}c_{i}^{0}\,\textnormal{d}x.$
The reason is that
$\displaystyle\left|\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}c_{i}\,\textnormal{d}x\textnormal{d}t-\int_{\Omega}c_{i}^{0}\,\textnormal{d}x\right|$
$\displaystyle=\left|\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}(c_{i}-c_{i}^{0})\,\textnormal{d}x\textnormal{d}t\right|$
$\displaystyle\leq\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}|c_{i}-c_{i}^{0}|\,\textnormal{d}x\textnormal{d}t$
$\displaystyle=\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\|c_{i}-c_{i}^{0}\|_{L^{1}(\Omega)}\,\textnormal{d}t,$
which converges to zero as $\sigma\to 0$ since the integrand is continuous due
to $c_{i}\in C([0,T];L^{2}(\Omega))$. The fourth term is treated the exact
same way.
For the first term of (2.5), we use the Mean Value Theorem to write:
$\displaystyle\left|\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}(c_{i}+\delta)\ln{(c_{i}+\delta)}\,\textnormal{d}x\textnormal{d}t-\int_{\Omega}(c_{i}^{0}+\delta)\ln{(c_{i}^{0}+\delta)}\,\textnormal{d}x\right|$
$\displaystyle\phantom{xxxxxx}\leq\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}\left|(c_{i}+\delta)\ln{(c_{i}+\delta)}-(c_{i}^{0}+\delta)\ln{(c_{i}^{0}+\delta)}\right|\,\textnormal{d}x\textnormal{d}t$
$\displaystyle\phantom{xxxxxx}=\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}|\ln{(\xi+\delta)}+1||c_{i}-c_{i}^{0}|\,\textnormal{d}x\textnormal{d}t$
$\displaystyle\phantom{xxxxxx}\leq
M(\delta)\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\|c_{i}-c_{i}^{0}\|_{L^{1}(\Omega)}\,\textnormal{d}x\textnormal{d}t,$
where $\xi$ lies between $c_{i}$ and $c_{i}^{0}$ and thus $0<\xi<1$ and
$M(\delta)$ is a positive constant which only depends on $\delta$. Then,
taking the limit $\sigma\to 0^{+}$ we conclude that
$\frac{1}{\sigma}\int_{\sigma}^{2\sigma}\int_{\Omega}(c_{i}+\delta)\ln{(c_{i}+\delta)}\,\textnormal{d}x\textnormal{d}t\to\int_{\Omega}(c_{i}^{0}+\delta)\ln{(c_{i}^{0}+\delta)}\,\textnormal{d}x,$
and similarly for the second term of (2.5).
For the last term of (2.5), since $\eta_{\sigma}(t)\to 1$ almost everywhere in
$(0,T)$ we see that
$c_{i}u_{i}\eta_{\sigma}(t)\cdot\frac{\nabla c_{i}}{c_{i}+\delta}\to
c_{i}u_{i}\cdot\frac{\nabla c_{i}}{c_{i}+\delta}~{}\textnormal{almost
everywhere in }\Omega\times(0,T),$
and again due to the bounds $|\eta_{\sigma}(t)|\leq 1$ and
$\frac{1}{c_{i}^{\epsilon}+\delta}\leq\frac{1}{\delta}$ and the fact that
$c_{i}u_{i}$ and $\nabla c_{i}$ are in $L^{2}(0,T;L^{2}(\Omega))$ we obtain
$\left|c_{i}u_{i}\eta_{\sigma}(t)\cdot\frac{\nabla
c_{i}}{c_{i}+\delta}\right|\leq\frac{1}{\delta}|c_{i}u_{i}||\nabla c_{i}|\in
L^{1}(0,T;L^{1}(\Omega)).$
and Lebesgue’s Dominated Convergence Theorem can be applied.
Then, passing to the limit $\sigma\to 0^{+}$ in (2.5) gives
(2.6)
$\begin{split}&\int_{\Omega}(c_{i}^{0}+\delta)\ln{(c_{i}^{0}+\delta)}\,\textnormal{d}x-\int_{\Omega}(c_{i}(T)+\delta)\ln{(c_{i}(T)+\delta)}\,\textnormal{d}x\\\
&\phantom{xx}-\int_{\Omega}c_{i}^{0}\,\textnormal{d}x+\int_{\Omega}c_{i}(T)\,\textnormal{d}x+\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\cdot\frac{\nabla
c_{i}}{c_{i}+\delta}\,\textnormal{d}x\textnormal{d}t=0.\end{split}$
Step 3: We remove $\delta$, needed to regularize the logarithm, by letting
$\delta\to 0^{+}$. The first term gives
$\int_{\Omega}(c_{i}^{0}+\delta)\ln(c_{i}^{0}+\delta)\,\textnormal{d}x\to\int_{\Omega}c_{i}^{0}\ln
c_{i}^{0}\,\textnormal{d}x,$
by the Dominated Convergence Theorem, since
$(c_{i}^{0}+\delta)\ln(c_{i}^{0}+\delta)\to c_{i}^{0}\ln c_{i}^{0}$ almost
everywhere as $\delta\to 0^{+}$ and
$|(c_{i}^{0}+\delta)\ln(c_{i}^{0}+\delta)|\leq C$ uniformly in $\delta$ for
some constant $C>0$ independent of $\delta$, since $0\leq c_{i}^{0}\leq 1$ and
$0<\delta<1$ and constants are integrable in $\Omega$. The second term is
treated similarly.
Finally, for the last term we obtain
$\int_{0}^{T}\int_{\Omega}c_{i}u_{i}\cdot\frac{\nabla
c_{i}}{c_{i}+\delta}\,\textnormal{d}x\textnormal{d}t\to
2\int_{0}^{T}\int_{\Omega}m_{i}\cdot\nabla\sqrt{c_{i}}\,\textnormal{d}x\textnormal{d}t,$
by the Dominated Convergence Theorem, since for $m_{i}=\sqrt{c_{i}}u_{i}$
$c_{i}u_{i}\cdot\frac{\nabla c_{i}}{c_{i}+\delta}\to
2m_{i}\cdot\nabla\sqrt{c_{i}}\quad\textnormal{ almost everywhere as $\delta\to
0^{+}$},$
and
$\begin{split}\left|c_{i}u_{i}\cdot\frac{\nabla
c_{i}}{c_{i}+\delta}\right|&=|m_{i}||2\nabla\sqrt{c_{i}}|\frac{c_{i}}{c_{i}+\delta}\\\
&\leq|m_{i}||2\nabla\sqrt{c_{i}}|\in L^{1}(0,T;L^{1}(\Omega)),\end{split}$
since by the regularity of weak solutions $m_{i}$ and $\nabla\sqrt{c_{i}}$ are
in $L^{2}(0,T;L^{2}(\Omega))$ and the bound is uniform in $\delta$.
Therefore, letting $\delta\to 0^{+}$ we obtain the entropy equality
(2.7) $\int_{\Omega}c_{i}(T)(\ln
c_{i}(T)-1)\,\textnormal{d}x-2\int_{0}^{T}\int_{\Omega}m_{i}\cdot\nabla\sqrt{c_{i}}\,\textnormal{d}x\textnormal{d}t=\int_{\Omega}c_{i}^{0}(\ln
c_{i}^{0}-1)\,\textnormal{d}x.$
We sum (2.7) over all $i\in\\{1,\dots,n\\}$ and use the definition of the
entropy (1.10), to arrive at
$H(\bm{c}(T))-2\sum_{i=1}^{n}\int_{0}^{T}\int_{\Omega}m_{i}\cdot\nabla\sqrt{c_{i}}\,\textnormal{d}x\textnormal{d}t=H(\bm{c}^{0}).$
Performing the following computation, which derives from (1.6) and the
symmetry of $D_{ij}$
$\begin{split}-\sum_{i=1}^{n}2m_{i}\cdot\nabla\sqrt{c_{i}}&=\sum_{i=1}^{n}\sum_{j=1}^{n}A_{ij}m_{j}m_{i}\\\
&=-\sum_{i=1}^{n}\sum_{j\neq i}\frac{1}{D_{ij}}c_{i}u_{i}\cdot
c_{j}u_{j}+\sum_{i=1}^{n}\sum_{j\neq i}\frac{1}{D_{ij}}c_{j}m_{i}^{2}\\\
&=\frac{1}{2}\sum_{i=1}^{n}\sum_{j\neq
i}\frac{c_{i}c_{j}}{D_{ij}}|u_{i}-u_{j}|^{2},\end{split}$
we obtain
(2.8)
$H(\bm{c}(T))+\frac{1}{2}\int_{0}^{T}\int_{\Omega}\sum_{i=1}^{n}\sum_{j\neq
i}\frac{c_{i}c_{j}}{D_{ij}}|u_{i}-u_{j}|^{2}\,\textnormal{d}x\textnormal{d}t=H(\bm{c}^{0}).$
The fact that $T>0$ is arbitrary, implies that (2.8) holds for all times
$t\in(0,T]$ and the proof is complete. ∎
## Acknowledgments
LCB acknowledges support by INdAM GNAMPA and also by MIUR, within project
PRIN20204NT8W4-004: Nonlinear evolution PDEs, fluid dynamics and transport
equations: theoretical foundations and applications. LCB also thanks King
Abdullah University of Science and Technology (KAUST) for the support and
hospitality during the preparation of the paper. The research of SG and AET
was partially supported by King Abdullah University of Science and Technology
(KAUST) baseline funds. SG also acknowledges partial support from the Austrian
Science Fund (FWF), grants P33010 and F65. This work has received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme, ERC Advanced Grant no. 101018153.
## References
* [1] L.C. Berselli, _Three-dimensional Navier–Stokes equations for turbulence_ , Mathematics in Science and Engineering, Academic Press, London, [2021].
* [2] D. Bothe. On the Maxwell–Stefan equations to multicomponent diffusion. In: J. Escher et al. (eds). Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications, pp. 81–93. Springer, Basel, 2011.
* [3] T. Buckmaster, C. De Lellis, L. Székelyhidi Jr., and V. Vicol, Onsager’s conjecture for admissible weak solutions. Comm. Pure Appl. Math. 72 (2019), 229–274.
* [4] R. Caflisch, I. Klapper, G. Steele. Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD. Commun. Math. Phys. 184 (1997), 443–455.
* [5] P. Constantin, W. E, E. Titi. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165 (1994), 207–209.
* [6] T. Debiec, P. Gwiazda, A. Swierczewska-Gwiazda, A.E. Tzavaras. Conservation of energy for the Euler–Korteweg equations. Calc. Var. 57 (2018), 160.
* [7] J. Duchon, R. Robert. Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13 (2000), 249-255.
* [8] S. Georgiadis, H. Kim, A.E. Tzavaras. Uniqueness of renormalized solutions for the Maxwell–Stefan system. Under review, arXiv: 2311.10465 (2023).
* [9] P. Gwiazda, M. Michalek, A. Swierczewska-Gwiazda. A Note on Weak Solutions of Conservation Laws and Energy/Entropy Conservation. Arch. Rational Mech. Anal. 229 (2018), 1223–1238.
* [10] P. Isett. A proof of Onsager’s conjecture. Ann. of Math. (2) 188 (2018), 871–963.
* [11] X. Huo, A. Jüngel, A.E. Tzavaras, Weak-Strong Uniqueness for Maxwell–Stefan Systems, SIAM J. Math. Anal. 54(3) (2022), 3215-–3252.
* [12] A. Jüngel, I.V. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures, SIAM J. Math. Anal. 45 (2013), 2421–2440.
* [13] C. Yu. Energy Conservation for the Weak Solutions of the Compressible Navier–Stokes Equations. Arch. Rational Mech. Anal. 225 (2017), 1073–1087.
|
# Artificial intelligence for online characterization of ultrashort X-ray
free-electron laser pulses
Kristina Dingel Intelligent Embedded Systems, University of Kassel,
Wilhelmshöher Allee 73, 34121 Kassel, Germany Artificial Intelligence Methods
for Experiment Design (AIM-ED), Joint Lab Helmholtzzentrum für Materialien und
Energie, Berlin (HZB) and University of Kassel these authors contributed
equally to this work<EMAIL_ADDRESS>Thorsten Otto
Intelligent Embedded Systems, University of Kassel, Wilhelmshöher Allee 73,
34121 Kassel, Germany these authors contributed equally to this work Lutz
Marder Institute for Physics and CINSaT, University of Kassel, Heinrich-
Plett-Straße 40, 34132 Kassel, Germany Artificial Intelligence Methods for
Experiment Design (AIM-ED), Joint Lab Helmholtzzentrum für Materialien und
Energie, Berlin (HZB) and University of Kassel Lars Funke Technische
Universität Dortmund, Fakultät Physik, Maria-Goeppert-Mayer-Straße 2, 44227
Dortmund, Germany. Arne Held Technische Universität Dortmund, Fakultät
Physik, Maria-Goeppert-Mayer-Straße 2, 44227 Dortmund, Germany. Sara Savio
Technische Universität Dortmund, Fakultät Physik, Maria-Goeppert-Mayer-Straße
2, 44227 Dortmund, Germany. Andreas Hans Institute for Physics and CINSaT,
University of Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany
Artificial Intelligence Methods for Experiment Design (AIM-ED), Joint Lab
Helmholtzzentrum für Materialien und Energie, Berlin (HZB) and University of
Kassel Gregor Hartmann Helmholtz-Zentrum Berlin für Materialien und Energie,
Hahn-Meitner-Platz 1, 14109 Berlin, Germany Artificial Intelligence Methods
for Experiment Design (AIM-ED), Joint Lab Helmholtzzentrum für Materialien und
Energie, Berlin (HZB) and University of Kassel David Meier Intelligent
Embedded Systems, University of Kassel, Wilhelmshöher Allee 73, 34121 Kassel,
Germany Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-
Platz 1, 14109 Berlin, Germany Artificial Intelligence Methods for Experiment
Design (AIM-ED), Joint Lab Helmholtzzentrum für Materialien und Energie,
Berlin (HZB) and University of Kassel Jens Viefhaus Helmholtz-Zentrum Berlin
für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
Artificial Intelligence Methods for Experiment Design (AIM-ED), Joint Lab
Helmholtzzentrum für Materialien und Energie, Berlin (HZB) and University of
Kassel Bernhard Sick Intelligent Embedded Systems, University of Kassel,
Wilhelmshöher Allee 73, 34121 Kassel, Germany Artificial Intelligence Methods
for Experiment Design (AIM-ED), Joint Lab Helmholtzzentrum für Materialien und
Energie, Berlin (HZB) and University of Kassel Arno Ehresmann Institute for
Physics and CINSaT, University of Kassel, Heinrich-Plett-Straße 40, 34132
Kassel, Germany Artificial Intelligence Methods for Experiment Design (AIM-
ED), Joint Lab Helmholtzzentrum für Materialien und Energie, Berlin (HZB) and
University of Kassel Markus Ilchen Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany European XFEL GmbH, Holzkoppel 4, 22869
Schenefeld, Germany Institute for Physics and CINSaT, University of Kassel,
Heinrich-Plett-Straße 40, 34132 Kassel, Germany Wolfram Helml Technische
Universität Dortmund, Fakultät Physik, Maria-Goeppert-Mayer-Straße 2, 44227
Dortmund, Germany.
###### Abstract
X-ray free-electron lasers (XFELs) as the world’s brightest light sources
provide ultrashort X-ray pulses with a duration typically in the order of
femtoseconds. Recently, they have approached and entered the attosecond
regime, which holds new promises for single-molecule imaging and studying
nonlinear and ultrafast phenomena such as localized electron dynamics. The
technological evolution of XFELs toward well-controllable light sources for
precise metrology of ultrafast processes has been, however, hampered by the
diagnostic capabilities for characterizing X-ray pulses at the attosecond
frontier. In this regard, the spectroscopic technique of photoelectron angular
streaking has successfully proven how to non-destructively retrieve the exact
time–energy structure of XFEL pulses on a single-shot basis. By using
artificial intelligence techniques, in particular convolutional neural
networks, we here show how this technique can be leveraged from its proof-of-
principle stage toward routine diagnostics even at high-repetition-rate XFELs,
thus enhancing and refining their scientific accessibility in all related
disciplines.
###### keywords:
Angular streaking, Detector image analysis, Free-electron laser, Pulse
characterization, Machine learning, Artificial intelligence, Simulation,
Convolutional neural network
## Introduction
X-ray free-electron lasers (XFELs) are the world’s fastest X-ray cameras,
providing ultrashort exposure times in combination with a spatial resolution
limit down to the sub-nanometer range, which allows for time-resolved
experiments ’freezing’ the motion of atoms and molecules. In fact, XFELs have
revolutionized several fields of science enabling us to observe the role of
transient structures and resonances in atoms [1] as well as single-molecule or
cluster imaging [2], investigations of ultrafast processes at element-specific
observer sites [3], and the study of nonlinear light–matter interaction in the
X-ray regime [4].
Over the past decade, further development of the underlying machine operation
techniques has enabled increasingly sophisticated control over the photon
pulse parameters. One of the most recent major upgrades is the increased
repetition rate of XFELs that is anticipated to initiate a leap from proof-of-
principle experiments to advanced applications of interdisciplinary
importance, thus representing a cornerstone of modern XFEL science [5].
Most of the FELs and in fact all XFELs worldwide are currently based on the
principle of self-amplification of spontaneous emission (SASE) [6]. More
precisely, their pulses are formed stochastically through the interplay
between the relativistically accelerated electron bunches themselves and the
spontaneously emitted synchrotron radiation, caused by their sinusoidal
trajectories inside magnetic structures with periodically changing polarity,
so-called undulators. This feedback interaction leads to a subsequent density
modulation of the undulating electrons, resulting in bursts of ultrashort
X-ray pulses with a peak brightness up to and exceeding $10^{32}$
$\rm\frac{photons}{sec\cdot mrad^{2}\cdot mm^{2}\cdot 0.1\%BW}$ (with
bandwidth $BW$). The amplification process generates a non-predictable
time–energy structure for every single pulse, constituting one of the biggest
limitations for XFEL science so far. There are currently no control mechanisms
and no routinely available diagnostics in place to directly measure the
temporal properties of these X-ray pulses. This is due to the stochastic
nature of each individual XFEL pulse, making their single-shot
characterization necessary and precluding standard integrated methods
developed for attosecond pulses from lab-based sources [7, 8, 9]. Hence, the
bulk of dynamics on attosecond to femtosecond time scales occurring during the
exposure to the X-rays is unfortunately only inferred via indirect pulse
measurements such as spectral analysis [10] or electron beam diagnostics [11].
Recently, we have demonstrated a new technique termed angular streaking that
is capable of retrieving the time–energy structure of all incoming SASE X-ray
pulses non-destructively with attosecond resolution [12]. Besides the major
diagnostic breakthrough, this generally paves the way for time-resolved and
nonlinear attoscience in the X-ray regime. In fact, the onset of all
structural dynamics in matter can now be studied in detail even from specific
observer sites through strongly localized electrons. Fast and reliable
feedback of this novel diagnostic regarding the experiment and the machine
itself is of utmost importance for upcoming scientific applications at XFELs
[13, 14]. For high-repetition-rate XFELs such as the European XFEL near
Hamburg, Germany, conventional analysis approaches fail to accommodate the
enormous amount and complexity of angular-streaking data in full depth.
Especially for online analysis and, ultimately, (re)active control and pulse
shaping during beam times, conventional data processing methods are not
suited. Therefore, several core challenges of XFELs are anticipated to be
tackled by artificial intelligence (AI), in particular machine-learning (ML)
techniques.
Figure 1: Illustration of this work’s topical orientation. A series of
previously unknown XFEL pulses releases electrons (indicated in yellow) from a
gas target with characteristic kinetic energies. Their kinetic energy shifts
caused by a circularly polarized infrared streaking field are recorded in an
angle-resolving spectrometer (indicated by the cones and eyes). In the
artificial brain, the thus obtained information about the electrons is
processed and reveals the underlying attosecond substructure of the X-ray
pulses. The precise knowledge of the time–energy substructure can then be used
to shape the X-ray pulses via a feedback loop to the XFEL machine settings or
to enable access to ultrafast electron dynamics at the attosecond frontier, as
here indicated by a molecular movie of an interatomic Auger decay in water
with subsequent dissociation.
In this article, we present a machine-learning-based proof-of-concept on
retrieving the full and detailed XFEL pulse temporal profile, including the
pulse duration and its intensity substructure. In addition, we show that it is
possible to extract temporal information on the electronic processes after
photoionization initiated by the X-rays, that is a subsequent Auger decay, via
the method of angular streaking paired with analysis through neural networks
(NNs). Moreover, by using simulated streaking data with various degrees of
instrument noise and different electron emission signatures, we demonstrate
the flexibility of the NN-based online diagnostic tool for XFELs. It is thus
robust against detector noise and machine fluctuations, and covering the vast
majority of current and future operation modes.
(a) Exemplary shot showing a detector image measured during an angular
streaking experiment. The image was taken for X-ray pulses at a photon energy
of 1180 eV, ionizing photoelectrons from the 1s shell in neon, and streaked by
a circularly polarized infrared laser with a wavelength of 10.6 $\mu$m (as
published in [12]).
(b) Example for a simulated detector image after adding $\pm 30\ \%$ noise.
The simulation parameters are chosen similar to those used in the experiment
shown in panel (a).
(c) Simulated spectrogram of an X-ray pulse with a simple Gaussian intensity
distribution.
(d) Simulated spectrogram after adding a spiky structure to mimick a SASE
pulse.
(e) Simulated spectrogram of Auger electrons after X-ray 1s ionization in neon
for decay time of 7.6 fs.
(f) Simulated angularly streaked detector image generated from (c).
(g) Simulated angularly streaked detector image generated from (d).
Supplementary Fig. 1 highlights the difference between this detector image and
(f).
(h) Simulated angularly streaked detector image generated from (e).
Figure 2: Experimental (a) and simulated (b)–(h) data for an angular streaking
measurement. (a) shows real data from Hartmann et al. [12] in neon. (b)
displays a simulated detector image under similar conditions with the addition
of $\pm 30\leavevmode\nobreak\ \%$ noise. In [(c) and (f)] a simulated
spectrogram and the corresponding detector image, respectively, are shown
before and after [(d) and (g)] adding a pulse structure to a Gaussian
intensity distribution. Note the very slight differences between (f) and (g),
which are caused by the additional SASE pulse structure and give rise to an
accordingly spiky pulse reconstruction. (e) and (h) display a simulated Auger
electron spectrogram and the respective angularly streaked detector image,
after ionization of the Ne 1s shell with an arbitrarily chosen decay time of
7.6 fs. All displayed figures are normalized to the interval $[0,1]$.
## Application Case: Angular Streaking
A long-standing goal in laser and X-ray research is to enable measurements
providing both temporal and spatial real-time information about electronic and
consequential structural changes on a molecular level with element-site
specificity—a so-called molecular movie. For this, a suitable ultrashort X-ray
pulse duration is one of the key parameters, which is both hard to facilitate
and difficult to measure. Yet, a reliable time-resolving experimental method
is essential for determining parameters such as the detailed intensity profile
of the SASE FEL pulse [15], corresponding damaging thresholds for materials
under investigation [16], the nanoscale interpretation of ultrafast single-
shot diffraction imaging [17], and the probabilities for multi-photon
processes [18, 19], to name a few. Applying the angular-streaking technique
[20] to the field of XFELs leverages a versatile approach for the temporal and
spectral characterization of individual (X)FEL pulses [12].
The applied scientific instruments for this new method are angle-resolving
electron spectrometers [12, 13]. In case of the first demonstration of angular
streaking at XFELs and also in the present case, 16 individually working time-
of-flight (TOF) spectrometers are arranged in a ring-like structure around the
target region, perpendicular to the propagation direction of the incoming
X-rays [12]. Together with a co-propagating circularly polarized infrared
laser, spatially and temporally overlapped with the XFEL at the target region,
this setup enables angular streaking. Atoms from a target gas are ionized with
the XFEL pulse and the emitted electrons are swept, i.e., streaked, in energy
and angle by the concomitant rotating electric field vector of the infrared
laser. In a simplified picture, the streaking field vector can be understood
as the hand of a clock that encodes the parameter time via the angles at which
electrons are detected with accordingly shifted energies (see illustration in
Fig. 1). Given sufficiently many electrons to "report" on their ionization
time within the SASE pulse, the measured electron emission patterns contain
the information of the full time–energy structure of the ionizing XFEL pulse
with attosecond resolution. The method can be adapted for pulses with
different photon energy by selecting target gases of suitable electron binding
energies and photoionization cross sections. The mechanism and experimental
setup for the angular-streaking technique as applied to SASE X-ray pulses is
described by Hartmann et al.[12] and the general principle can be found here
[21, 22, 23].
In the experiment under consideration, a time trace is measured for
photoelectrons emitted by each X-ray shot and in each TOF spectrometer, hence,
generating 16 traces at a rate set by the repetition frequency of the XFEL and
of the overlapped streaking laser. For single-shot spectroscopy, a trace
represents the number of electrons arriving after specific flight times. These
time-domain traces can be converted to the energy domain (spectra) by taking
into account the length of the flight path and the actions of additional
electric fields along their paths, which are routinely used for enhancing the
achievable energy resolution and the electron collection efficiency. The
combined representation of a full angle-resolved streaking measurement forms
an image with $16$ columns, representing the respective detector angles, and
several rows corresponding to the range of electron energies detected in the
specific measurement (detector image, cf. Fig. 2a). Time-dependent electron
spectra (spectrograms) are then generated by converting the emission angles to
times using the known rotation period of the electric field vector for the
circularly polarized streaking laser. (cf. Figs. 2c & 2d). In the present ML
case study, we have simulated the spectrograms and the according detector
images based on the equations for streaking previously derived [22], following
the procedure established and described in detail in the SI of Hartmann et
al.[12]. Thus, we can at will generate both a huge "data" set for training the
ML algorithms, as well as an additional set of completely known target shots
for testing the developed NN predictions.
For an X-ray pulse with no temporal and spatial overlap in the interaction
region with an external streaking field (unstreaked shot), the spectra in all
detectors are showing the characteristic electron energy distributions
(spectral lines) for the target under investigation. Typically each line also
shows an angular dependence in signal intensity. In Figs. 2f & 2g one can see
this variation in the low intensity regions around $0^{\circ}$ and
$180^{\circ}$ in contrast to the high-intensity parts at $90^{\circ}$ and
$270^{\circ}$, with intermediate intensities in the columns at angles in
between.
If a circularly polarized streaking laser is present, the detector image is
modulated according to the instantaneous streaking laser vector potential,
leading to a sinusoidal variation of the spectral lines along the angle axis.
Figure 3: There are two ways to extract important pulse characteristics from
the detector image. The dotted line refers to a full reconstruction
(spectrogram) and the extraction of relevant parameters out of this
reconstruction. The other way is to skip the full reconstruction and to
concentrate on typically most relevant parameters only. In this article, we
choose the second approach.
The goal regarding SASE FEL X-ray pulse characterization and their potential
control is to reconstruct the spectrogram from a measured detector image,
which gives the full information about the X-ray time–energy structure (cf.
Fig. 3, dotted line). In many experimental situations, however, it is
sufficient to restrict our analysis to some of the most relevant SASE X-ray
parameters (cf. Fig. 3, red line). In the subsequent discussion, we have,
therefore, focused on NN predictions of the temporal aspects of ultrashort FEL
pulses. Details about the NNs’ framework conditions, the chosen architecture,
and hyperparameter optimization can be found in the Methods below.
We picked the following pulse characteristics for a comparison of their
reconstruction by the NNs (prediction) with the originally simulated data
(target):
##### Kick
The kick is the maximum streaking shift in electron kinetic energies for each
X-ray shot, and thus for a given temporal delay and phase relation between the
X-ray pulse and the infrared streaking laser. There are two main reasons for a
change in the kick from shot to shot. The first is the relative timing jitter
between the X-ray and the streaking pulse [24, 25, 26], which is unavoidable
due to the stochastic generation process of the SASE mechanism and additional
fluctuations in arrival time caused by air fluctuations, thermal expansion in
optomechanical components, and general synchronization errors between the two
separate laser pulses. The second reason for variations of the kick is the
random change of the carrier–envelope phase of the streaking laser from shot
to shot. One can solve this by stabilizing the carrier–envelope phase [27],
which is a rather difficult technical requirement, or by using the technique
of angular streaking [12], which is the basis for the simulations studied in
this article.
Since we translate a temporal distribution (X-ray intensity structure) into an
energy distribution (kinetic energy of the streaked photoelectrons), a lower
kick value means a more shallow gradient of the streaking ramp, which is given
by the kick over the cycle period of the electric field corresponding to the
streaking laser wavelength. Thus, the resolution of the measurement is
directly degrading with decreasing kick. In this sense, the determination of
the kick strength is not so much interesting in itself, but is a measure for
the quality of the reconstruction and can be used as a filtering handle of the
data. It is also a good consistency check for the functioning of the applied
NNs, since the kick is a parameter that can also be readily assessed with
other analysis methods.
Figure 4: Three examples of different SASE XFEL pulse intensity structures
showing varying total duration and complexity generated with OCELOT [28]. The
pulse durations given as FWHM/RMS, respectively, for the exemplary shots are
1.5 fs/640 as (blue), 4.5 fs/1.9 fs (orange) und 10.4 fs/4.4 fs (green). The
corresponding Gaussian pulse envelopes are shown as dashed grey curves,
including the FWHM durations, and their projections onto the time axes as
black lines.
##### Pulse Duration
The pulse duration is the most important parameter for many ultrafast free-
electron laser experiments, e.g., a variety of pump/probe measurements of
electronic state changes or investigations of nonlinear excitation dynamics
[19, 29, 30, 31], albeit it is one of the most difficult to measure directly.
Especially for XFEL SASE pulses, each pulse has a different duration and
erratic intensity structure that even complicates the definition of the term
pulse duration. In this article, we use the root-mean-square (RMS) duration,
i.e., the square root of the time variance of the temporal intensity profile
[32],
$t_{\rm p,RMS}=\sqrt{\langle t^{2}\rangle-\langle t\rangle^{2}},$ (1)
where
$\langle
t^{n}\rangle=\frac{1}{N}\int_{-\infty}^{\infty}t^{n}I(t)\;dt\quad\text{and}\quad
N=\int_{-\infty}^{\infty}I(t)\;dt$ (2)
are the $n$-th moment and the normalization constant, respectively, as the
basic definition of the pulse duration.
A common choice for more well-behaved Gaussian-like laser pulses from table-
top systems is the full width at half-maximum (FWHM). Given a Gaussian
distribution with standard deviation $\sigma$, corresponding to the RMS
duration in this case, the FWHM is calculated as follows:
$FWHM=2\sqrt{2\ln 2}\sigma\approx 2.35\cdot\sigma.$ (3)
As SASE pulses are generally spiky and irregular (cf. Fig. 4), this metric is
not fully applicable. In our simulation case, however, this quantity is
nevertheless of interest due to the fact that we use the FWHM to generate (and
in fact define) the Gaussian distribution envelopes in Fig. 4 and because the
FWHM better relates to the intuitive concept of a full-length pulse duration.
The RMS duration, however, gives a more complete measure of the temporal
distribution of the pulse energy including possible pulse wings [32] or
substructure (see also the next paragraph).
##### Pulse Structure
Due to the microbunching in the FEL each SASE pulse has an individual
intensity profile, made up of several shorter ‘spikes’ with random intensities
(cf. Fig. 4). The average number of spikes per pulse is determined by the
specific operation parameters of the XFEL. It can be expressed in a
statistical treatment as the number of individual energy modes contributing to
the XFEL pulse [33]. The ensuing pulse shape can be arbitrarily complex. The
shorter the overall pulse duration in relation to the single-spike length,
i.e., the fewer spikes per complete pulse, the more important individual
spikes are becoming (Fig. 4). Especially for estimating the damage thresholds
of investigated probes as well as for experiments sensitive to the
instantaneous X-ray intensity, or for ultrafast pump/probe measurements, the
XFEL pulse structure needs to be known exactly to interpret the observed data
on a shot-to-shot basis unambiguously.
##### Auger Decay Time
Many of the scientifically interesting processes of non-equilibrium physics
and structure-changing chemistry are not directly triggered by the exciting
X-ray pulse but are the result of subsequent complex relaxation dynamics.
These dynamics are determined by the time-dependent, i.e., transient
electronic structure of the system under study. One of the most fundamental
electronic processes after inner-shell ionization of matter by X-rays is the
Auger decay, whereby a second electron from an outer shell fills the generated
core hole and transfers the excess energy to a third electron (Auger
electron), which is then emitted from the ion. This process is specific to the
contributing discrete electronic states of an atomic or molecular system and
has a characteristic time constant for the emission of the third electron
(Auger decay time). In our simulations, we assume that one Auger decay channel
dominates for neon (Ne) after 1s ionization. The corresponding Auger decay
time on the order of 2 fs to 3 fs [34] can serve as a fundamental benchmark
for demonstrating the capability of the method to retrieve ultrafast timing
information from recorded data.
## Results
(a) Difference between the target and the predicted kick label, demonstrated
on a test set with $4\cdot 10^{4}$ samples. 96 % of all predictions have a
smaller deviation than 10 % of the respective target value.
(b) Difference between the target and the predicted FWHM pulse duration label,
demonstrated on a test set with $4\cdot 10^{4}$ samples. In 94 % of all cases
where the kick was greater than $5$ eV, the deviation between prediction and
target value was less than 1 fs.
(c) Difference between the target and the predicted Auger decay time,
demonstrated on a test set with $4\cdot 10^{4}$ samples. In 92 % of all cases
where the kick was greater than 3 eV, the deviation between prediction and
target value was less than 0.5 fs.
(d) Deviation of the computed RMS pulse duration from the predicted pulse
structure to the RMS pulse duration of the target pulse structure.
(e) Comparison of the FWHM pulse duration prediction as a function of energy
kick. The accuracy of the pulse length estimate depends on the respective kick
value of the shot. The higher the kick, the more accurate the prediction of
the FWHM pulse duration gets.
(f) Comparison of the decay prediction as a function of energy kick. The
accuracy of the Auger decay time estimate depends on the respective kick value
of the shot. The higher the kick, the more accurate the prediction of the
Auger decay time gets.
Figure 5: Prediction accuracies and dependencies of the labels kick, RMS pulse
duration, FWHM pulse duration, and decay using data sets containing all levels
of noise [$\pm 0\leavevmode\nobreak\ \%$, $\pm 10\leavevmode\nobreak\ \%$,
$\pm 20\leavevmode\nobreak\ \%$, $\pm 30\leavevmode\nobreak\ \%$]. The slight
skew of the distributions in Figs. (b), (c) & (d) stems from the inherent
tendency of NNs to predict values closer to the mean of the learned parameter
space for unclear inputs, i.e. from detector images with too low kicks.
All of the above described SASE XFEL pulse characteristics can be predicted
with varying degrees of accuracy by utilizing convolutional NNs. For each
pulse characteristic, we will examine the results of the trained models in
more detail.
### Kick
Of all the characteristics studied, the kick turned out to be the easiest to
predict. Fig. 5a shows that most of the predictions only slightly deviate from
the respective targets. It turns out that $96\leavevmode\nobreak\ \%$ of all
predictions have a smaller deviation than $10\leavevmode\nobreak\ \%$ of the
respective target value. Though the kick can easily be derived from the
detector images, an accurate estimate of this parameter is necessary for
better judging the reliability of the reconstruction for the FWHM pulse
duration, pulse structure and Auger decay time. This will become apparent in
the following paragraphs.
### FWHM Pulse Duration
The comparison of predicted and target FWHM pulse durations is shown in Fig.
5b. As for the kick estimates, the majority of the values are well
reconstructed. However, it is evident that a few of the predictions deviate
strongly from the target values. One hypothesis explaining this behavior is
that for smaller kicks the resolution of the measurement degrades and
predicting a pulse duration can become arbitrarily difficult. That is why we
have investigated the accuracy of the FWHM pulse duration estimate against the
true kick value.
Fig. 5e confirms the previously stated hypothesis. Above a (true) kick value
of approximately $5$ eV, estimating the FWHM pulse duration becomes feasible.
In fact, in $94\leavevmode\nobreak\ \%$ of all cases where the kick was
greater than $5$ eV, the deviation between prediction and target value was
less than $1$ fs. This is due to the fact that small kick values on the order
of the nominal SASE bandwidth correlate to unsuccessful angular streaking
shots that need to be discarded anyway. Supplementary Figs. 2a and 2b display
exemplary shots with a large and a small kick, respectively.
### Auger Decay Time
The auger decay time can also be well approximated by the respective NN (cf.
Fig. 5c). Most of the estimates hardly deviate from the zero line of the
difference between the target value and the prediction. However, as for the
FWHM pulse duration estimates, there are some outliers strongly deviating from
the true Auger decay time value. Using the same reasoning as for the FWHM
pulse duration estimates, we have compared the prediction of the Auger decay
time value with the true kick value. Here, the same behavior can be observed
as for the FWHM pulse duration (cf. Fig. 5f). It is evident that a reasonable
determination of the Auger decay time is only possible for a kick value of $3$
eV or higher. In fact, in $92\leavevmode\nobreak\ \%$ of all cases where the
kick was greater than $3$ eV, the deviation between prediction and target
value was less than $0.5$ fs. It follows that shots with small kick values
should be discarded in advance to appropriately approximate the true Auger
decay time. As before, Supplementary Figs. 2c and 2d display exemplary shots
with a small and a large kick for the decay reconstruction, respectively.
### Pulse Structure and RMS Pulse Duration
The full temporal pulse structure of the SASE pulse is probably the most
difficult property to predict in our study. This is not surprising, as it is
also the most complex one, being represented by a vector instead of a single
value in case of the kick or the pulse duration, which holds the information
of the intensity distribution over time. Altogether, the trained network works
relatively well in its objective to predict the trend, i.e., peak positions
and their relative intensities of the pulse structure. However, as has been
expected, these predictions get less accurate for more complex pulse
structures. This behavior can be seen for two different exemplary simulated
SASE pulses in Fig. 6, one relatively simple 6a and one more complex 6b, in
which not all of the finer structures could be reliably reproduced.
Nevertheless, the main features including the larger peaks can always be
predicted.
(a) Simple SASE pulse structure.
(b) Complex SASE pulse structure.
Figure 6: Examples for simple (a) and more complex (b) simulated and
reconstructed SASE pulse structures with RMS pulse durations of $6.2$ fs and
$11.7$ fs, respectively.
Remarkably, the quality of the predicted pulse structure does not show a
significant dependency on the value of the kick; except for a kick very close
to or equalling zero, which is not surprising, as this again corresponds to an
unsuccessful event where basically no streaking occurred. There is also no
significant dependency on the duration of the pulse, as one might have
expected. An absolute value for the mean squared error (MSE) does indeed
increase with the pulse duration; normalized to it, however, the average ‘MSE
per time step’ is more or less constant (cf. Supplementary Fig. 3). Now that
our model is capable of extracting the pulse structure, it is possible to
compute the RMS pulse duration by using Eq. 1. Fig. 5d shows the deviation of
this computed RMS pulse duration to the RMS pulse duration of the target pulse
structure. The average deviation lies below $1$ fs. Only for very long pulse
durations there is a trend towards a slight underestimation, although the
error remains just around 10 % in most cases. We note that an additional NN
for the direct prediction of the RMS pulse duration could serve as a
comparative measure for the RMS value calculated from the pulse structure,
allowing a coarse estimation of the quality of the reconstructions.
### Influence of Noise on the NN Performance
In order to investigate the influence of noise on the predictions, we
generated a test set comprising of detector images with several different
noise levels ($p=[0.0,0.1,0.2,0.3]$) as shown in Eq. 4 in the Methods section
and fed it into the NNs.
As expected, additional noise influences the result of the NN’s prediction
(cf. Tab. 1). An example for a noisy simulated detector image is given in Fig.
2b, more expressive examples for different simulation settings are shown in
Supplementary Fig. 4. The prediction for non-noisy data is nearly perfect,
whereas the prediction for noisy data slightly differs from the target.
Despite the decreased prediction accuracies, it is evident that the NNs can
handle noise robustly.
Noise | Kick | FWHM | RMS | Decay
---|---|---|---|---
$\pm 0\%$ | $0.19$ eV | $0.52$ fs | $0.67$ fs | $0.08$ fs
$\pm 10\%$ | $0.24$ eV | $0.88$ fs | $0.65$ fs | $0.17$ fs
$\pm 20\%$ | $0.34$ eV | $1.05$ fs | $0.65$ fs | $0.26$ fs
$\pm 30\%$ | $0.43$ eV | $1.2$ fs | $0.73$ fs | $0.34$ fs
Table 1: Standard deviations of the predicted labels with regard to the
respective targets kick, pulse duration, and Auger decay computed on $1000$
samples, respectively. Settings for the kick, FWHM/RMS pulse duration, Auger
decay sample data: kick = $22.5$ eV, pulse duration = $4.85$ fs, Auger decay =
$7.0$ fs.
## Discussion & Outlook: Online SASE-Pulse Characterization and Shaping
So far, we have shown that several characteristics of XFEL pulses are
predictable with varying degrees of accuracy. To investigate how close the
current status comes to real-time pulse characterization during experimental
campaigns, we need to address a number of different issues, which can each be
tackled exploiting the specific analytical strength made accessible by the
methods of NNs:
##### Output Speed
For evaluating the input images at full speed of the XFEL repetition rate in
the kHz to MHz regime, an efficient analysis is inevitable. NNs are known for
delivering outputs quickly. We have used several batch sizes, starting from
one image up to $4096$ as input. Investigating this is important as such a
comparison determines whether a batch-wise and therefore highly parallelized
analysis is performing better than analyzing image-wise. Batch-wise evaluation
is specifically suited for the European XFEL facility, since a train with a
very fast succession of pulses (maximum 2700 per 600 $\mathrm{\mu}$s) is
followed by a pause of several milliseconds that can be used for analysis
purposes [5, 35]. We have tested how fast the NN output is generated on a
GeForce RTX 2070 GPU using single precision floating point format (Tab. 2).
BS | $1$ | $64$ | $128$ | $256$ | $512$ | $1024$ | $2048$ | $4096$
---|---|---|---|---|---|---|---|---
Dur [ms] | $1.38$ | $1.44$ | $1.48$ | $1.54$ | $1.62$ | $1.54$ | $1.38$ | $1.44$
Table 2: Time measurements for predictions of the trained model on a GeForce
RTX 2070 card with different batch sizes (BS). We ran hundred experiments and
averaged the results.
The model is able to reach quick predictions mostly independent of the batch
size as the computation on a GPU runs all tasks, i.e., computes a prediction
for each image within the batch, in parallel. In general, the number of input
images in one batch is only limited by the RAM of the used GPU. Thus, it is
apparent that it is advantageous to analyze a larger batch of data than
individual images. With a batch size of $4096$ our current model is already
able to keep up with the European XFEL in high-repetition mode for online
predictions.
##### Reliability Estimation
Next to fast evaluation, a degree of certainty in the NN predictions must be
ensured. As shown in the results, the NN prediction may deviate quite
substantially from the target. Some of the difficulties can be directly
circumvented. By determining the kick, for example, we can already filter
whether a prediction regarding the labels Auger decay time or pulse duration
is reasonable. But this still does not give us a direct statement specifying
how certain the NN’s prediction is. Optimally, we would want to have a
reliable measure of how good the predictions of the trained models are, even
for unknown shots without a target.
There are several ways to determine the prediction uncertainties of NNs. The
epistemic uncertainty determines the uncertainty due to insufficient
knowledge. This can be implemented, for example, via Monte Carlo dropout [36]
or Monte Carlo batch normalization [37]. The aleatoric uncertainty determines
the uncertainty due to the complexity of the problem, and can be investigated
by creating a fitted cost function [38]. This topic is currently a very active
area of ML research. We are in the process of developing our own approach to
benchmark the specific task of stochastic X-ray pulse reconstruction, ideally
combining both uncertainty determinations in one procedure as previously
shown[38].
##### Gap between Simulation and Reality
So far, our NNs are suited only for data that look exactly like the input
shown in Fig. 2. Whether the NNs are suitable for predictions on experimental
data (cf. Fig. 2a) is not easy to validate, especially since we do not have
true labels for the experimental data. In addition, we need to identify to
what extent our modeled noise replicates real noise of the spectrometer, e.g.,
electronic ringing of the detector readout or background signals from
undesired processes. There are two ways to tackle the gap between simulation
and experiment. Either the real data must be denoised before the analysis
(e.g., Denoising Autoencoders [39]) or the simulation data must be provided
with additional, appropriately modeled noise. Simultaneous approaches in both
directions should give a more complete understanding for mitigating this issue
in future efforts.
##### Responding to Changes
We have shown that our developed NNs work on data with several levels of
noise. However, when utilizing real-life TOF spectrometers, it may occur that
TOF sensors fail or produce unrealistic results. In such cases NN re-training
or knowledge extension is inevitable. Here, online learning [40] is a helpful
tool. In this case, the model is trained continuously on newly generated data.
Thus, the training can be quickly adapted to new environments. To circumvent
the catastrophic forgetting [41] of NNs, continual learning [42, 43] may be
utilized.
##### Pulse Shaping
The term pulse shaping can be interpreted in two technically different ways.
With the online analysis methods already demonstrated in the current
manuscript, we can establish an X-ray sorting scheme based on the full
characterization of every single pulse and the possibility to filter for
desired pulse shapes and durations. Especially with high-repetition rate
XFELs, this can be a vital form of “passive shaping”. The second, and more
exciting, route refers to an actual pulse shaping in terms of intelligent
experimentation schemes and dynamical interaction of the machine with
simultaneous photon-based measurements. Thus, the live updates on X-ray pulse
changes may be used for a more detailed control of the parameters and for
actual SASE pulse shaping. First steps for this interaction have been
identified to entail a feedback loop to the accelerator that provides an
online data stream of, e.g., X-ray pulse duration and spectrum. The machine
operators can choose to engage this loop into their electron bunch compression
algorithms with a preset goal of optimization to be pursued. Further steps
towards intelligent and active experimentation are in development and will be
presented in future studies.
##### Conclusion
In this article, we demonstrated a path toward online characterization of
free-electron laser pulses by applying NNs on detector images captured with
angular streaking. In addition to several predictable characteristics, we have
been able to identify and confirm dependencies between the respective
characteristics that can be used to control the machine settings during
experimental campaigns. This way, the angular streaking technique has the
potential to be leveraged from the proof-of-principle stage to a robust and
highly advanced diagnostic tool for all free-electron laser facilities,
including high-repetition rate operation. In addition, these novel ML
reconstruction procedures may also be used for better online X-ray pulse
control and future FEL pulse shaping on demand. Further steps to a successful
implementation of these advanced methods involve closing the gap between
simulation and experimental data through an instrument-specific treatment of
the measurement noise and a reliable concept for error and reliability
estimation, which we will investigate in future work.
## Methods: Machine Learning Procedure Design
In real-world XFEL experiments, the spectrograms or pulse characteristics of
individual SASE pulses have to be reconstructed from the detector image. There
are first approaches for deriving single-shot characteristics of rapid pulse
sequences from high-repetition rate XFELs [44]. Unfortunately, they are only
suitable to a limited degree for providing detailed insights via real-time
processing during experimental campaigns.
Figure 7: We use a simulated spectrogram to construct according detector
images. These images are used for training the NNs and to extract several
different pulse characteristics afterward.
Here, we apply specifically developed NNs on the angular streaking approach to
demonstrate the possibility of a fast online pulse characterization, as NNs,
particularly convolutional NNs, have proven to be suitable for similar
challenges [45, 46].
### General Machine Learning Problem Formulation
For each pulse characteristic, we need to train a NN that takes detector
images as inputs (cf. Fig. 7). The outputs for each of the NNs vary and are
listed below:
##### Kick
The kick is the amplitude of the wave-like intensity distribution within the
detector image (cf. Fig. 2). When changing the kick, the spectrogram stays as
is as the kick only affects the streaking signal captured in the detector
image. That is the reason why the kick is easily extractable from the detector
images. The NN has to solve a regression task, where the output is one number
in the unit eV.
##### FWHM Pulse Duration
The FWHM pulse duration is well extractable from the spectrogram, as it can be
seen as $2.35\cdot\sigma$ (cf. Eq. 3) in the direction of $x$ (time scale),
with $\sigma$ being the standard deviation of the 2D gaussian distribution in
x direction. The longer the FWHM pulse duration, the longer the distribution
stretches in x direction. Within the detector image, a change of the pulse
duration mostly affects the width and the peculiarity of the wave form. Here
again, the NN has to solve a regression task, where the output is one number
in the unit fs.
##### Auger Decay Time
The Auger decay is visible in the spectrograms as well as the detector images.
Within the spectrogram, the length of the tail after the 2D Gaussian
distribution indicates the decay time. The longer the tail, the larger the
decay. Within the detector image, a larger decay affects the distortion of the
wave. Once more, the NN has to solve a regression task, where the output is
one number in the unit fs.
##### Pulse Structure
The pulse structure is the most challenging feature to extract, as the output
itself consists of several values indicating the intensities of multiple
spikes within the SASE pulses. By looking at the spectrogram in Fig. 2 (c),
one can see that the pulse structure can be derived by summing up the
intensities at each point in time along the vertical axis. The pulse
characteristic will be determined here as the intensity as a function of
arrival time, where the intensity is integrated over all photon energies
within the 6 eV spectral bandwidth. This leads to an output similar to Fig. 4.
In this case, the NN has to solve a regression task, where the output consists
of several time steps in arbitrary intensity units.
A note regarding the RMS Pulse Duration: As the RMS pulse duration can be
directly derived from the pulse structure, there is no need to train an
independent NN for this pulse characteristic.
After the general examination of the ML problem, the next sections will look
at how the ML pipeline looks in detail and how to successively address the
individual ML problems above.
### Framework Conditions
In order to train NNs in a supervised manner, we require training data
$\mathcal{D}_{K}$ of size $K\in\mathbb{N}$, which comprises $K$ simulated
detector images $\mathcal{X}=\\{\mathbf{X}_{i}\in\mathbf{M}^{m\times
n}(\mathbb{R}),\ i=1,\dots,K\\}$ and $K$ corresponding pulse characteristics
$\mathcal{L}=\\{L_{i}\in\mathbb{R}^{j}_{+},\ i=1,\dots,K\\}$. Here, $m$ is the
number of TOF detectors used within the complete spectrometer setup and $n$
displays the electron kinetic energy in intervals. The size of $j$ changes
according to the pulse characteristic that has to be predicted. In the
following, we will refer to pulse characteristics as labels. To verify the
performance of a NN, we split $\mathcal{D}_{K}$ into two distinct sets,
$\mathcal{D}_{\rm train}$ and $\mathcal{D}_{\rm test}$, such that
$\mathcal{D}_{K}=\mathcal{D}_{\rm train}\cup\mathcal{D}_{\rm test}$. The
accuracy of the NN is determined by $\mathcal{D}_{\rm test}$. We train the NNs
with $n$ distinct batches of detector images
$\mathcal{B}_{n}\in\mathcal{D}_{\rm train}$. Similarly, we test the
performance of the NNs with $m$ batches of detector images
$\mathcal{B}_{m}\in\mathcal{D}_{\rm test}$. To avoid overfitting of the NN, we
utilize cross-validation [47].
Although we work in a simulation environment, it is reasonable to choose
values that would correspond to real experimental data (cf. Fig. 2a).
Therefore, we take previously acquired data from earlier experimental
campaigns [12] as an example. In particular this means:
* •
For each of the two use cases, Ne 1s and KLL Auger electrons data, we generate
a size of $K=4.4\cdot 10^{6}$ samples to be predicted. Of these, $4\cdot
10^{6}$ are used for training and $4\cdot 10^{5}$ for testing.
* •
Our angle-resolved spectrometer consists of $m=16$ TOF detectors.
* •
We fix the intervals in the TOF detectors to $n=200$, with a varying energy
bin size.
More particularly, this means that the following NN architecture depends on
the chosen parameters, though it can be adapted easily if, e.g., more TOF
detectors are added.
### Preparing the Simulation Data
We derive artificial detector images for Ne 1s and KLL Auger electrons from
the simulation environment as introduced by Hartmann et al. [12]. The kinetic
energy of the 1s photoelectrons depends on the ionizing X-ray photon energy,
which, in this case, is set to 1180 eV. We include a spectral bandwidth of 6
eV for the X-ray pulses, but omit the effect of a potential chirp in these
simulations, which would only have a marginal effect on the parameters
reconstructed in this study. A photon energy of 1180 eV results in Ne 1s
photoelectron kinetic energies centered at $\sim$310 eV. The Auger electron
kinetic energies are independent of the X-ray photon energy and bandwidth,
with the main peak lying at $\sim$804 eV and a standard deviation determined
by the detector resolution. The angular streaking maps the TOF measurements of
16 detectors distributed over 360 degree to a window of $35.3$ fs, as this is
the duration of one optical cycle for the chosen streaking wavelength
$\lambda=10.6\,\mathrm{\mu}$m of the circularly polarized laser in Hartmann et
al.[12]
We chose the range and the precision based on the experimental implementation
expected for real streaking measurements at XFELs. We want our models to
estimate kicks in the range of $0-30$ eV, FWHM pulse durations in the range of
$0.4-13.4$ fs, and decays in the range of $0-10$ fs. The temporal resolution
of the pulse structure was equally chosen in accordance with the expected
duration of the shortest features in the X-ray pulse intensity structures (the
SASE spikes), leading to a grid size along the time axis for the pulse
structure reconstruction of $441$ as.
Figs. 2c and 2f are simulated without artifacts. In 2c, there is only one
randomly placed Gaussian distribution present in the spectrogram. The
underlying pulse structure is neglected so far. To get closer to the real data
(cf. Fig. 2a), we implement three steps. We add a pulse structure to the
spectrogram and noise to the simulated detector images, and prepare the data
for NN training by utilizing data normalization.
##### Step 1: Adding a Pulse Structure to the Spectrogram
To achieve a SASE-like temporal structure in the spectrogram, we modulate the
original Gaussian time distribution with a spiky intensity profile (cf. Fig.
4). We obtain the latter by generating a comb of Gaussian spikes with
randomized amplitudes and spike durations as predicted by theory for a typical
setting of an XFEL in ultrashort-pulse mode [33].
##### Step 2: Adding Noise to the Detector Image
Additional noise is added to $\mathbf{X}\in\mathcal{X}$, representing the
intensity values for each pixel in the detector image, during training and
testing as shown in Eq. 4. A given percentage $p$ of the maximum intensity
value $x_{\rm max}$ of $\mathbf{X}\in\mathcal{X}$ is used as an upper and
lower bound of an equal distribution $\mathcal{G}$ to draw $w$ from:
$\mathbf{X}_{\rm noisy}=\left(x_{i,j}+(x_{\rm max}\cdot
w)\right)_{i=1,\dots,m,j=1,\dots,n},w\sim\mathcal{G}(-p,p),$ (4)
Figure 2b displays a detector image with added noise ($p=0.3$).
##### Step 3: Normalizing the Data
It is evident, that the range of the intensity values differs from case to
case. To counteract this, we perform a min-max-normalization for each
$\mathbf{X}\in\mathcal{X}$. Therefore, the minimum ($x_{\rm min}$) and maximum
($x_{\rm max}$) intensity value of $\mathbf{X}$ are used to perform the
transformation for each pixel value $x_{k,l}$:
$\mathbf{X}_{\rm norm}=\left(\frac{x_{k,l}-x_{\rm min}}{x_{\rm max}-x_{\rm
min}}\right)_{k=1,\dots,m,l=1,\dots,n},$ (5)
After normalization, all values in $\mathbf{X}_{\rm norm}$ lie within the
interval $[0,1]$.
### Designing the Machine Learning Models
As we want to extract information from images, the most intuitive solution is
to use a convolutional NN [48, 49], which uses convolutions and pooling to
extract low- and high-level features such as edges and predicts an estimate of
those features using fully-connected layers. In our case, the estimates
ideally should correspond to the target pulse characteristics.
#### Architecture
One key problem regarding the choice of a proper NN architecture is the
dimensionality of the detector images, which are not equal in size and
therefore not symmetrical. This fact needed to be taken into account in the
design of the NN. Furthermore, we wanted our NN architecture to be as dense as
possible to ensure generalization and performance for reliable online
operation. After testing several architecture configurations with different
numbers of layers and neurons, the most suitable network architecture for our
problem is an NN with three convolutional blocks (cf. Fig. 8). Each block
contains a convolutional layer, followed by an activation function and a max-
pooling layer. The convolutional layers use a $3x3$ kernel, stride of $1x1$
and $1x1$ zero-padding. The pooling layers use a $3x3$ kernel, stride of $2x2$
and $1x1$ zero-padding. Architectures with less than three convolutional
blocks could not grasp all required features necessary to derive the
underlying mapping from detector image to the desired label. The NN is
specifically designed to cut both dimensions, i.e., width and height of the
image, in half after each block. A filter size of $[16,32,64]$ for the
respective convolutional layers has proven to be sufficient. For the fully-
connected stage (except the last layer), we use three layers with
$[3200,1600,800]$ neurons, respectively. Architectures with less than three
layers within the fully-connected part did not transmit enough information to
be able to solve the problem decently. The size of the last layer depends on
the label to predict, i.e., the size of $j$ in $\mathcal{L}$.
Figure 8: The convolutional neural network architecture used in the present
approach. The dimensions of the first layers (orange) are displayed in
[channel, height, width]. The dimensions of the fully-connected layers
(violett) show the number of used neurons.
When predicting the kick, FWHM pulse duration, or decay time, $j=1$. When
predicting the pulse structure, $j$ corresponds to the dimension of the
spectrogram’s x-axis. We utilize the mean squared error loss function to train
and optimize the network as the prediction of the pulse characteristics is a
regression task in all cases.
#### Hyperparameter Optimization
The NN architecture is not the only choice to be considered. Especially when
training the NNs, appropriately chosen hyperparameters are important to
achieve efficient and goal-oriented training. Important parameters in this
context are the batch size, type of activation function, optimizer, and
learning rate. To find the best suitable combination of hyperparameters, we
performed a grid-search on distinct data sets using the approach from before
with the following values:
* •
Batch size: [64, 128, 256, 512, 1024]
* •
Activation function: [ReLU, Sigmoid]
* •
Optimizer: [Adam, SGD (with Momentum)]
* •
Learning rate: [0.01, 0.001, 0.0001, 0.00001]
After NN training, we evaluated the respective parameter combinations
according to the following criteria:
* •
Criterion 1: The test loss (after inputting $\mathcal{D}_{\rm test}$ into the
trained NN) should be minimal.
* •
Criterion 2: The standard deviation of the test loss curve should be minimal
to penalize slow convergence and overfitting.
In general, it should be noted that there is not only one combination of
hyperparameters that achieves good results during training. Nevertheless,
there has been an evident leader. The best hyperparameter configuration for
all labels is a batch size of $64$, a ReLU activation function, a learning
rate of $0.0001$, and Adam as optimizer.
## References
* [1] Mazza, T. _et al._ Mapping Resonance Structures in Transient Core-Ionized Atoms. _Physical Review X_ 10, 041056 (2020).
* [2] Ho, P. J. _et al._ The role of transient resonances for ultra-fast imaging of single sucrose nanoclusters. _Nature communications_ 11, 1–9 (2020).
* [3] Young, L. _et al._ Roadmap of ultrafast x-ray atomic and molecular physics. _Journal of Physics B: Atomic, Molecular and Optical Physics_ 51, 032003 (2018).
* [4] Eichmann, U. _et al._ Photon-recoil imaging: Expanding the view of nonlinear x-ray physics. _Science_ 369, 1630–1633 (2020).
* [5] Decking, W. _et al._ A MHz-repetition-rate hard X-ray free-electron laser driven by a superconducting linear accelerator. _Nature Photonics_ 14, 391–397 (2020).
* [6] Milton, S. _et al._ Exponential gain and saturation of a self-amplified spontaneous emission free-electron laser. _Science_ 292, 2037–2041 (2001).
* [7] Tzallas, P., Charalambidis, D., Papadogiannis, N. A., Witte, K. & Tsakiris, G. D. Direct observation of attosecond light bunching. _Nature_ 426, 267–271 (2003).
* [8] Paul, P. M. _et al._ Observation of a train of attosecond pulses from high harmonic generation. _Science_ 292, 1689–92 (2001).
* [9] Kienberger, R. _et al._ Atomic transient recorder. _Nature_ 427, 817–821 (2004).
* [10] Serkez, S. _et al._ Opportunities for two-color experiments in the soft X-ray regime at the european XFEL. _Applied Sciences_ 10, 2728 (2020).
* [11] Behrens, C. _et al._ Few-femtosecond time-resolved measurements of X-ray free-electron lasers. _Nature communications_ 5, 1–7 (2014).
* [12] Hartmann, N. _et al._ Attosecond time–energy structure of X-ray free-electron laser pulses. _Nature Photonics_ 12, 215 (2018).
* [13] Duris, J. _et al._ Tunable isolated attosecond X-ray pulses with gigawatt peak power from a free-electron laser. _Nature Photonics_ 14, 30–36 (2020).
* [14] Driver, T. _et al._ Attosecond transient absorption spooktroscopy: a ghost imaging approach to ultrafast absorption spectroscopy. _Physical Chemistry Chemical Physics_ 22, 2704–2712 (2020).
* [15] Bonifacio, R., De Salvo, L., Pierini, P., Piovella, N. & Pellegrini, C. Spectrum, temporal structure, and fluctuations in a high-gain free-electron laser starting from noise. _Phys. Rev. Lett._ 73, 70–73 (1994).
* [16] Neutze, R., Wouts, R., van der Spoel, D., Weckert, E. & Hajdu, J. Potential for biomolecular imaging with femtosecond X-ray pulses. _Nature_ 406, 752–7 (2000).
* [17] Barty, A. _et al._ Ultrafast single-shot diffraction imaging of nanoscale dynamics. _Nature Photonics_ 2, 415–419 (2008).
* [18] Young, L. _et al._ Femtosecond electronic response of atoms to ultra-intense X-rays. _Nature_ 466, 56–61 (2010).
* [19] Rudenko, A. _et al._ Femtosecond response of polyatomic molecules to ultra-intense hard X-rays. _Nature_ 546, 129–132 (2017).
* [20] Eckle, P. _et al._ Attosecond angular streaking. _Nature Physics_ 4, 565–570 (2008).
* [21] Constant, E., Taranukhin, V., Stolow, A. & Corkum, P. Methods for the measurement of the duration of high-harmonic pulses. _Physical Review A_ 56, 3870–3878 (1997).
* [22] Itatani, J. _et al._ Attosecond Streak Camera. _Physical Review Letters_ 88, 173903 (2002).
* [23] Kazansky, A. K., Bozhevolnov, A. V., Sazhina, I. P. & Kabachnik, N. M. Interference effects in angular streaking with a rotating terahertz field. _Physical Review A_ 93, 013407 (2016).
* [24] Bionta, M. R. _et al._ Spectral encoding of x-ray/optical relative delay. _Optics Express_ 19, 21855–65 (2011).
* [25] Hartmann, N. _et al._ Sub-femtosecond precision measurement of relative X-ray arrival time for free-electron lasers. _Nature Photonics_ 8, 706–709 (2014).
* [26] Diez, M. _et al._ A self-referenced in-situ arrival time monitor for X-ray free-electron lasers. _Scientific Reports_ 11, 3562 (2021).
* [27] Fuji, T. _et al._ Monolithic carrier-envelope phase-stabilization scheme. _Optics Letters_ 30, 332 (2005).
* [28] Agapov, I., Geloni, G., Tomin, S. & Zagorodnov, I. OCELOT: A software framework for synchrotron light source and FEL studies. _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 768, 151–156 (2014).
* [29] Rudek, B. _et al._ Ultra-efficient ionization of heavy atoms by intense X-ray free-electron laser pulses. _Nature photonics_ 6, 858–865 (2012).
* [30] Picón, A. _et al._ Hetero-site-specific X-ray pump-probe spectroscopy for femtosecond intramolecular dynamics. _Nature Communications_ 7, 11652 (2016).
* [31] Li, X. _et al._ Electron-ion coincidence measurements of molecular dynamics with intense X-ray pulses. _Scientific reports_ 11, 1–12 (2021).
* [32] Sorokin, E., Tempea, G. & Brabec, T. Measurement of the root-mean-square width and the root-mean-square chirp in ultrafast optics. _JOSA B_ 17, 146–150 (2000).
* [33] Krinsky, S. & Gluckstern, R. Analysis of statistical correlations and intensity spiking in the self-amplified spontaneous-emission free-electron laser. _Physical Review Special Topics - Accelerators and Beams_ 6, 050701 (2003).
* [34] Haynes, D. C. _et al._ Clocking Auger electrons. _Nature Physics_ 17, 512–518 (2021).
* [35] Pergament, M. _et al._ Versatile optical laser system for experiments at the European X-ray free-electron laser facility. _Opt. Express_ 24, 29349–29359 (2016).
* [36] Gal, Y. & Ghahramani, Z. Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning. In Balcan, M. F. & Weinberger, K. Q. (eds.) _Proceedings of The 33rd International Conference on Machine Learning_ , vol. 48 of _Proceedings of Machine Learning Research_ , 1050–1059 (PMLR, New York, New York, USA, 2016).
* [37] Teye, M., Azizpour, H. & Smith, K. Bayesian uncertainty estimation for batch normalized deep networks. In Dy, J. & Krause, A. (eds.) _Proceedings of the 35th International Conference on Machine Learning_ , vol. 80 of _Proceedings of Machine Learning Research_ , 4907–4916 (PMLR, 2018).
* [38] Kendall, A. & Gal, Y. What uncertainties do we need in Bayesian deep learning for computer vision? In _Proceedings of the 31st International Conference on Neural Information Processing Systems_ , 5580–5590 (2017).
* [39] Vincent, P., Larochelle, H., Lajoie, I., Bengio, Y. & Manzagol, P.-A. Stacked Denoising Autoencoders: Learning Useful Representations in a Deep Network with a Local Denoising Criterion. _Journal of Machine Learning Research_ 11, 3371–3408 (2010).
* [40] Fontenla-Romero, Ó., Martinez-Rego, D., Guijarro-Berdiñas, B., Pérez-Sánchez, B. & Peteiro-Barral, D. Online machine learning. In _Efficiency and Scalability Methods for Computational Intellect_ , 27–54 (IGI Global, 2013).
* [41] Kirkpatrick, J. _et al._ Overcoming catastrophic forgetting in neural networks. _Proceedings of the National Academy of Sciences_ 114, 3521–3526 (2017).
* [42] Parisi, G. I., Kemker, R., Part, J. L., Kanan, C. & Wermter, S. Continual lifelong learning with neural networks: A review. _Neural Networks_ 113, 54–71 (2019).
* [43] He, Y. & Sick, B. CLeaR: An adaptive continual learning framework for regression tasks. _AI Perspectives 2021 3:1_ 3, 1–16 (2021).
* [44] Heider, R. _et al._ Megahertz-compatible angular streaking with few-femtosecond resolution at x-ray free-electron lasers. _Phys. Rev. A_ 100, 053420 (2019).
* [45] Ren, S., He, K., Girshick, R. & Sun, J. Faster r-cnn: Towards real-time object detection with region proposal networks. In Cortes, C., Lawrence, N., Lee, D., Sugiyama, M. & Garnett, R. (eds.) _Advances in Neural Information Processing Systems_ , vol. 28 (Curran Associates, Inc., 2015).
* [46] Kyrkou, C., Plastiras, G., Theocharides, T., Venieris, S. I. & Bouganis, C.-S. DroNet: Efficient convolutional neural network detector for real-time UAV applications. In _2018 Design, Automation Test in Europe Conference Exhibition_ , 967–972 (2018).
* [47] Berrar, D. Cross-Validation. In Ranganathan, S., Gribskov, M., Nakai, K. & Schönbach, C. (eds.) _Encyclopedia of Bioinformatics and Computational Biology_ , 542–545 (Academic Press, Oxford, 2019).
* [48] Krizhevsky, A., Sutskever, I. & Hinton, G. E. Imagenet classification with deep convolutional neural networks. _Advances in neural information processing systems_ 25, 1097–1105 (2012).
* [49] Gu, J. _et al._ Recent advances in convolutional neural networks. _Pattern Recognition_ 77, 354–377 (2018).
## Acknowledgements
This research was supported by the project SpeAR_XFEL (05K2019) funded by the
BMBF (German Federal Ministry of Education and Research). We gratefully
acknowledge the assistance and support of the Joint Laboratory Artificial
Intelligence Methods for Experiment Design (AIM-ED) between Helmholtzzentrum
für Materialien und Energie, Berlin, and the University of Kassel. MI
acknowledges funding by the VW foundation for a Peter-Paul-Ewald Fellowship.
This version of the article has been accepted for publication, after peer
review but is not the Version of Record and does not reflect post-acceptance
improvements, or any corrections. The Version of Record is available online
at:
https://doi.org/10.1038/s41598-022-21646-x
## Author contributions statement
Kristina Dingel: Conceptualization, Methodology, Software, Investigation,
Writing - Original Draft, Visualization.
Thorsten Otto: Conceptualization, Methodology, Software, Investigation,
Writing - Original Draft, Visualization.
Lutz Marder: Methodology, Software, Investigation, Writing - Original Draft,
Visualization.
Lars Funke: Writing - Review & Editing, Visualization.
Arne Held: Writing - Review & Editing.
Sara Savio: Writing - Review & Editing.
Andreas Hans: Writing - Review & Editing.
Gregor Hartmann: Software, Writing - Review & Editing.
David Meier: Writing - Review & Editing.
Jens Viefhaus: Writing - Review & Editing.
Bernhard Sick: Writing - Review & Editing, Supervision, Funding acquisition.
Arno Ehresmann: Writing - Review & Editing.
Markus Ilchen: Conceptualization, Writing - Original Draft, Supervision.
Wolfram Helml: Conceptualization, Writing - Original Draft, Supervision,
Project administration, Funding acquisition.
## Additional information
Data and Code Availability A repository containing the detector image analysis
software and according simulation data will be provided on request;
Competing interests The authors declare no competing interests.
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